vii
TABLE OF CONTENTS
List of contributors
ix
1. Introduction Jaroslav Peregrin
1
PART I: FOUNDATIONS 2. Structural Properties of Dynamic Reasoning Johan van Benthem 3. Construction by Description in Discourse Representation Noor van Leusen and Reinhard Muskens 4. On the Dynamic Turn in the Study of Meaning and Interpretation Richard Breheny 5. Real Dynamics Wolfram Hinzen
15 33 69 91
PART II: SYNTAX, SEMANTICS, DISCOURSE 6. Growth of Logical Form: the Dynamics of Syntax Ruth Kempson, Wilfried Meyer-Viol and Masayuki Otsuka 7. The Double Dynamics of Definite Descriptions Klaus von Heusinger 8. Dynamics in the Meaning of the Sentence and of Discourse Petr Sgall 9. On the Meaning of Prescriptions Timothy Childers and Vladimír Svoboda 10. Imperative Negation and Dynamic Semantics Berislav Žarnić
121 149 169 185 201
PART III: SEMANTIC GAMES 11. Dynamic Game Semantics Tapio Janasik and Gabriel Sandu 12. About Games and Substitution Manuel Rebuschi 13. In Defense of (Some) Verificationism: Verificationism and Game-Theoretical Semantics Louise Vigeant
215 241 259
1
INTRODUCTION Jaroslav Peregrin, Academy of Sciences of the Czech Republic, Prague*
1 CONTEXTS IN FORMAL SEMANTICS It is, of course, not obligatory, to take ‘folk semantics’, according to which to be a (‘meaningful’) expression is to have a ghostly pendant called meaning, at face value. There are many linguists and philosophers who argue that it is futile to see the meaningfulness of an expression as a matter of a thing it possesses. However, there is also a strong tradition which does take meanings at face value, and tries to find out what they are, or at least in which way they should be reconstructed or explicated. The modern history of this tradition starts especially with Gottlob Frege, who proposed to exempt meanings from the legislation of philosophy and to subordinate them rather to logic. The logico-philosophical approach to meaning, elaborated by Wittgenstein, Russell and Carnap and resulting in the Montagovian and post-Montagovian formal semantics, wanted to abstract from the way language is actually put to use and account for the semantics of language as an abstract system1. The pioneers of this approach to semantics also concentrated on language as a means of encoding knowledge; and hence first and foremost as a means of expressing contextindependent propositions, which thus appeared as the primary subject matter of semantics. Context-dependence was seen mostly as a marginal and unimportant phenomenon: everything, so the story went, worth being said can be articulated by means of a context-independent sentence or sentences – and using context-dependent means is only an oblique (though sometimes perhaps handy) way of doing the same. Such view of language was imputed also into the foundations of formal semantics and governed it for some time. * 1
I am grateful to Klaus von Heusinger for helpful comments on a previous version of this text.
Elsewhere (Peregrin, 2001) I pointed out that that ambition may be seen as establishing an interesting kinship between this program and Ferdinand de Saussure’s program of concentrating solely on language as langue (as contrasted to parole) and that, consequently, formal semantics can be seen as a way of carrying out his structuralistic program.
2
Meaning: the Dynamic Turn
However, even within this framework there were, from the beginning, serious attempts to incorporate context-dependence into the agenda2. In his path-breaking paper on formal semantics, David Lewis (1970) made some proposals in this direction; but the first systematic accounts of context dependence within this tradition were provided by David Kaplan and Robert Stalnaker. Kaplan (1970) pointed out that ‘circumstances’ intervene into the process of determining an utterance’s truth value twice, rather than only once, as possible-worlds semantics appeared to suggest. He urged that the utterance must be first confronted with its context to yield an intension which can be only then confronted with a possible world to yield the truth value of the utterance w.r.t. the world. Thus, deciphering an utterance such as I am hungry we must first exploit the context to unfold the indexical I, whereby we reach a proposition which then can be true w.r.t. some possible worlds (namely those in which the current utterer is hungry) and false w.r.t. others. Whereas Kaplan concentrated on contexts as inputs to utterances, Stalnaker (1970; 1978) was the first to study contexts also as outputs from utterances, thus introducing a truly dynamic perspective. According to him, an utterance ‘consumes’ the actual context to yield a proposition (which may than be evaluated w.r.t. various possible worlds), and parallelly produces a new context which can be consumed by subsequent utterances. In the simplest case, where both contexts and propositions are seen as sets of possible worlds, the new context may be seen as simply the intersection of the context with the proposition. Thus when I say I am hungry, two things happen: the content of the sentence uttered gets amended by the context to yield a proposition which becomes the utterance meaning; and the context gets in turn amended by this proposition to yield a new context.
2 SYSTEMS OF DYNAMIC SEMANTICS Despite such pioneering work, up to almost the beginning of the 1980s, the mainstream of formal semantics still approached language as a static system. Then, especially after pathbreaking papers of Lewis (1979), Kamp (1981) and others, the situation started to change: it started to be ever more clear that the anaphoric structure of natural language, which is being put to work in almost any kind of discourse, cannot be accounted for without some kind of ‘going dynamic’. Moreover, Kamp has indicated how to carry out a dynamic turn without abandoning the framework of formal semantics in the wide sense (although perhaps abandoning one in the narrow, i.e. strictly model-theoretic sense). And later Groenendijk & Stokhof (1991), van Benthem (1997) and others showed that the turn can be accomplished even within the narrow bounds of logic and model theory: they presented logical systems which were dynamic just in the way needed for the semantic analysis of the anaphoric structure of language3.
2
Prior to the establishment of formal semantics, context-dependence was explicitely addressed for example by Bar-Hillel (1954).
3
Earlier versions of dynamic logic originated especially within the context of computer science.
Introduction
3
The most general idea behind the dynamic turn in semantics appears to be the acceptance of the Stalnakerian insight that our pronouncements must be seen both as ‘context-consumers’ and ‘context-producers’. A sentence such as He wears a hat is not truly intelligible (and is neither true, nor false) without there being a context for it to ‘consume’ and to furnish its he with a referent. Such a context might be constituted by the circumstances of utterance of the sentence (which may include pointing at a person), but often it is produced by some previous utterances: e.g. by the previous utterance of a sentence like Look, there is a man passing by! Turning the idea of sentences being context-producers and context-consumers into the formally-semantic cash leads to thinking of a sentence as something which has a contextchange potential: i.e. is able to bring about a change of the actual context. This, in turn, often leads to viewing the denotation of the sentence as something which involves (or simply is) a function mapping contexts on contexts (which is taken to be the formal reconstruction of the potential). Contexts are often seen also as information states: they are constituted by all the information amassed by the discourse so far. (And, importantly, they contain what can be called an individuary4 – a collection of individuals having been made salient by the discourse – which acts as the casting agency for the needs of the pronouns and other anaphoric expressions of language.) Hence the resulting semantic model consists of (i) a set S of information states and (ii) a family of functions from S to S. Thus, while on the Kaplanian picture a proposition came to interact with the context to produce the content of the actual utterance (‘what is said’), now it is recognized to fulfill also one more important task: to create a new context. And if the Kaplanian picture led to the explication of the proposition as a function from contexts to contents (which were explicated as intensions), now they, prima facie, come to be explicable as a pair of functions: one mapping contexts on contents (the character, in Kaplan’s term) and the other one mapping contexts on contexts (the context-change potential). But in fact these two functions need not be independent: one of the possible ways in which an utterance may produce the new context is to take its content and somehow add it to the old context. In this way, the production of the content may appear as a by-product (or an intermediate product) of the production of the new context; and the contextchange potential might be looked at as fulfilling both the tasks. This appears to be what has led some of the semanticists to the identification of propositions with context-change potentials5. A variety of dynamic semantic models of language emerge from various kinds of explications of the notion of context or information state. Take, as an example of one kind of such approach, Hans Kamp’s discourse representation theory (DRT)6: there the role of the information states is played by discourse representation structures (DRSs), each of which consists of a set of individuals and a set of conditions supplied for them by the discourse. Thus a sentence like 4
See Peregrin (2000a). Note, however, that as a sentence must interact with the context to yield not only a new context, but also the content of the actual utterance (‘what is said’), identifying its meaning with the context change potential is subsuming the latter task under the former, which may not be unproblematic (cf. Stalnaker, 1999). 6 Kamp (1981); Kamp and Reyle (1993); van Eijck and Kamp (1997). 5
4
Meaning: the Dynamic Turn A man owns a donkey
introduces two individuals together with the conditions that the first of them is a man, the second one is a donkey and the first owns the second. This gives rise to the following DRS:
x
y
man(x) donkey(y) owns(x,y)
Hence sentences produce DRSs; but they can, and some of them need to, consume DRSs. Thus a sentence such as He beats it needs a context delivering the reference for its pronouns. If it is uttered in the context represented by the above DRS, it enriches it with the condition that the first of the individuals of the DRS beats the second; hence produces, in effect, the following DRS:
x
y
man(x) donkey(y) owns(x,y) beats(x,y)
All in all, a sentence is to be seen as something which in general upgrades a given DRS to another, more elaborated one. Quite a different explication of the notion of context is provided by such systems as Groenendijk & Stokhof’s (1991) DPL7. Here the role of information states is played by sets of assignments of objects to the so called discourse markers. The intuitive idea behind this could be explained as follows: uttering a man introduces an individual which becomes the potential referent for he; hence the context produced by the utterance can be reconstructed as assigning this object to he. In general, the context is an assignment of all potential referents to all possible anaphoric phrases of language, which are established by the previous utterances. And discourse markers are what assume the role of the phrases within the language of DPL.
7
See also van Benthem (1997) and Muskens et al. (1997).
Introduction
5
Thus, (1) is analyzed as ∃d1d2(man(d1)∧donkey(d2)∧owns(d1,d2)) and produces the context consisting of all functions which map d1 and d2 on some elements of the universe such that the first of them is a man, the second one is a donkey and the first owns the second. The subsequent utterance of (2) which is analyzed as beats(d1,d2) then reduces this set to the set of only those functions which map d1 onto an individual which beats d2. (Thus, within DPL, ∃d1d2(man(d1)∧donkey(d2)∧owns(d1,d2))∧beats(d1,d2) is – surprisingly – equivalent to ∃d1d2(man(d1)∧donkey(d2)∧owns(d1,d2)∧beats(d1,d2)). The reason is that the operators ∃ and ∧ are not the well-known operators of predicate logic: the former in fact establishes an assignment of an ‘arbitrary’ value to a discourse marker, whereas the latter represents a concatenation.) A dynamic approach also based on logic but in a rather different way (and in fact older than the previous ones) is represented by Hintikka’s proposals to account for the semantics of sentences in terms of games8. It is well known that the usual semantics of standard logic (first-order predicate calculus) can be alternatively formulated as based on games: each formula is taken to represent a game of two players (Me, attempting to show that the formula is true, and Nature, attempting the opposite), such that one of them (Me) has a winning strategy just in case the sentence is true. Thus the formula ∃x1x2(man(x1)∧donkey(x2)∧owns(x1,x2)) of the standard predicate logic is thought of as representing a game in which I am first to pick up two individuals of the universe, a1 and a2, so that the game is to continue with the formula man(a1)∧donkey(a2)∧owns(a1,a2). (What I aim at is picking up some such individuals that the latter formula will be true.) In the next turn, Nature chooses one of the three conjuncts with respect to which the game is to continue then. (What it aims at is choosing that which is most unlikely to be true.) And then the game comes to an end: if the selected atomic formula is true, I win, if not, the victory belongs to Nature. Hintikka proposed various kinds of broadening of the range of games representable by sentences, going behind the boundaries of those naturally associated with the formulas of firstorder logic. And we can also think of modifying the rules in such a way that they produce
8
Hintikka (1973); Hintikka and Sandu (1997); and also the contributions in the last part of this volume.
6
Meaning: the Dynamic Turn
‘contexts’ with ‘individuaries’. (Keeping track of the moves of the game produces, besides other things, a list of individuals picked up by Me in the course of the game9.)
3 EXPRESSION MEANING VS. UTTERANCE MEANING Whereas formal semantics in its original form saw context-dependence as a form of imperfection, and the relationship between ‘true’ semantics and discourse analysis or pragmatics as akin to that between pure and applied mathematics, the above sketched dynamic turn has necessarily implied a blurring of this boundary. Formal semanticists have recognized that we can hardly account for meanings of a nontrivial part of natural language without taking into account how discourse functions, how what is said can depend on context, and how it can in turn change the context. In this way formal semantics has come into contact with a different semantic tradition, the tradition which has, from the beginning, concentrated on the actual processes of communication rather than on language as an abstract system and thus has given pride of place to utterance meaning over expression meaning – for it is, after all, the utterance meaning which does the job of communication. Within philosophy, this tradition has its roots especially in the Oxford ordinary language philosophy of J. L. Austin, P. Strawson and G. Ryle; but it has been later reinforced by many linguists and cognitive scientists who aimed at an account of how people actually communicate. The most influential conceptual framework for studying utterance meaning (and its relationship to expression meaning) is due to the Oxford philosopher H. Paul Grice (1957; 1989). His idea was that to understand an utterance is to grasp what the utterer intended to convey, i.e. to grasp the ‘speaker meaning’, which is only indirectly related to the meaning of the sentence he uttered. (Thus, by uttering It is late one may want to say that he should go home, or that his neighbor should not make such a noise etc.) Grice argued that the meaning of the sentence uttered is used only as an indirect indicator of the speaker’s meaning, which it yields only in interaction with many general assumptions about the speaker’s goals and intentions (such that the speaker intends to convey something, that she intends to convey something which is relevant to her audience etc.). The idea that to understand somebody’s utterance is to find out what he intends to say (using the meaning of the expression he uttered as a mere clue) resulted, within the context of cognitive science, into conceptions of linguistic communication as based on something like a ‘mind-reading’ (see, e.g., Sperber and Wilson, 2002; and also Breheny, this volume). In a slightly different way, Noam Chomsky and his followers have also contributed to this tradition in that they have turned the attention of linguists to the processes of generating utterances, i.e. to the processes which result in the utterer’s making the particular utterance she makes (see Hinzen, this volume).
9
See Hintikka and Kulas (1985); and also Janasik and Sandu (this volume).
Introduction
7
4 BUT ... WHAT IS IT ALL ABOUT? After the dynamic turn, the conceptual foundations of formal semantics became a little bit ambiguous. The static semantic theories of possible worlds or situations were replaced by theories centering around the concepts of context or information state. However, while possible worlds were readily explainable as our means of accounting for “what (empirical) truth is relative to” (Stalnaker, 1986) and hence possible-worlds semantics could be seen as a relatively straightforward account for truth conditions, the foundations of dynamic semantics are still much less clear. To be sure, there seems to exist an almost universal agreement that what dynamic theories are about is something like an ‘information flow’ or ‘kinematics of discourse’. Thus, Chierchia (1994, 141), claims that what we are now after is the question “how [meaning] affects the information available to illocutionary agents”; and van Benthem (1997, p. ix) urges that even logic is now to concentrate on “the logical structure of cognitive actions, underlying human reasoning or natural language understanding”. However, what exactly does this mean? Prima facie there might seem to be no great fuzziness here. There are information states, so the story goes, and there is communication which changes them; and dynamic semantics is to account for this. However, in what sense do information states exist and how do they actually get changed by means of communication? There seems to be at least two quite different answers available, answers which can be called individualistic and collectivistic, respectively. According to the first, the contexts we describe are a matter of (i.e. ‘are in the head of’) an individual speaker, according to the other they have a status of some socially-constructed reality. Both many cognitive scientists and some linguists with the Chomskyan background appear to be more sympathetic to the individualistic interpretation. Many of them saw a great deal of previous formal semantics as simply chimerical in that it addressed something which did not really exist, and now hail dynamic semantics as a means of turning attention to what is really there: to the manipulation of some mental representations or to workings of some language faculty. After the dynamic turn, these theoreticians have been joined by a host of those which were inspired by DRT interpreted in a mentalist way. Many logicians and some philosophers, on the other hand, are more or less reluctant to make meaning into a matter of mind or cognition. After all, they feel, it was the most crucial lesson Frege taught us that to subordinate logic, and consequently also semantics, to psychology is a way to hell; and some of them also accept Wittgenstein’s or Quine’s arguments to the effect that language cannot but be social through and through. Hence they often prefer the collectivistic understanding of information states. According to this, information states are not within the heads of individual speakers, they are rather our means of hypostasing certain states of groups of speakers: states of sharing the acceptance of some beliefs or pieces of knowledge. Now notice that whereas on the individualistic reading the dynamic turn involves a substantial change of the subject matter of semantics (from studying meanings to studying processes
8
Meaning: the Dynamic Turn
behind the production or comprehension of utterances), on the collectivistic reading this is not necessarily so. True, we can still see semantics as turning its attention to a description of certain processes (though now not cognitive, but rather ‘socio-logical’ ones); but we can also go still further on the way of rendering information states ‘virtual’ and claim that they are not something which we describe, but rather something which we only use as a tool for describing the semantic aspect of language and of communication. This ‘instrumental’ approach to dynamic semantics should be better recognized as a reading of its own: as it usually assumes that the ultimate data semantic theory is to account for with the help of meanings are our linguistic inferences and inferential rules, let us call it inferentialistic. The situation is similar to the one which obtained during the heyday of possible worlds. Though there almost did not exist the possibility of a mentalist interpretation (more precisely those who wanted such an interpretation found possible world semantics straightforwardly inadequate10), there still remained two possible ways of making sense of possible worlds, a ‘metaphysical’ one and an ‘instrumental’ one. According to the first, possible worlds existed in some metaphysical sense (and prior to language) and to do possible world semantics was to reveal their relationship to natural language expressions. According to the other, possible worlds were only in the eye of the beholder (semanticist), who used them as a tool for explicating facts about meaning (which were usually seen as a matter of the rules of language, especially the inferential ones). Thus, a possible world semanticist of the latter kind might say that what he is after is simply and straightforwardly meaning, but as he wants to explicate also meanings of modal and counterfactual locutions, he needs a richer semantics than offered by standard first-order logic, and arguably the most handy one, due to Kripke, employs entities which are suitably called possible worlds. And similarly, an information-state semanticist might say that as what he is after is explication also of anaphoric locutions, he arguably needs a semantics of ‘updates’ and entities suitably called contexts or information states11. It might seem that these three interpretations of dynamic turn differ only in some not so important ideology. But this is not so; their differentiation has profound consequences for the way we are to assess the success of the individual theories. Suppose that two theoreticians disagree about the meaning of an expression and want to find a criterion for resolving their disagreement. If they are both subjectivists, then they are bound to search for the criterion within the minds of the speakers, for then their theories are meant to depict something which is to be found there. If, on the other hand, they are both collectivists, they will have to look at the praxis of communication. And if they are both inferentialists, then they will have to check the inferences governing the usage of the expression. Of course the worst situation obtains if each of them interprets the dynamic turn in a different way – then they can hardly not talk past each other. And, as a matter of fact, this is quite a frequent situation.
10
As, e.g., Partee (1980).
11
See Peregrin (2000b).
Introduction
9
5 IS MEANING DYNAMIC? Different interpretations of the dynamic turn also lead to quite different answers to the question what the turn tells us about the nature of meaning. Some exponents of the turn would see it as abandoning the (‘futile’) search for expression meanings and concentrating on utterance meanings or on the very process of communication. Others would see it as disclosing the real nature of meanings – telling us that what meaning really is is not a class of possible worlds or a situation type or something of this kind, but rather an update, a function from information states to information states (where the information state may be explicated as a class of possible worlds, as a DRS, as a set of mappings of discourse markers on individuals etc.). And still others would see it as opening a new way of explicating meaning, which can coexist alongside the older ones, but which is particularly suitable for the explication of the anaphoric features of language. It was with these questions in mind that we organized the workshop “Is Meaning Dynamic?”, which was held in Prague in September 2001 and at which most of the papers included in this volume were presented. Its topic was suggested by the fact that the problems concerning the interplay between the dynamics of language and the nature of meaning have been affecting different areas of logic, linguistics and philosophy of language in such a way that they appear to foster a true paradigm shift: indeed, this question itself appears to be not only a ‘contextconsumer’ (i.e. is the product of the recent status of the discussions of the nature of meaning), but also a ‘context-producer’ since it actually changes the character of the discussions. And though the papers do not provide ready-made answers to this question or to questions related to it, the book can hopefully facilitate a kind of brainstorming which might lead us either to some answers, or to reconsidering the questions. The book consists of three parts. The papers in the first deal with foundational questions: why should we think that meaning is dynamic, what exactly does it mean that it is dynamic and what does this imply for the methodologies of semantics. The fact that the dynamic view has also already achieved a certain tradition in forming theories is reflected by the papers of the second part. These papers show different applications of dynamic theories / views / approaches in different areas of the theory of language, such as the compositional process of interpretation, the lexical meaning of the articles, the interaction with a discourse structure or the semantics of imperatives. Papers of the last part then address the game-theoretical variety of dynamic semantics. Johan van Benthem realizes that if we take the shift from the static perspective on language to the dynamic one, we will have to reconsider our very concept of inference: for the traditional view of inference as reflexive, transitive, obeying cut etc. turns out to be obsolete. Therefore he considers quite different kinds of inference and indicates that it is this very kind which is of a piece with the usual kinds of dynamic semantics. The paper by Noor van Leusen and Reinhard Muskens considers the problem of ‘procedurality’ versus ‘declarativity’ of dynamic theories. Whereas the common wisdom appears to be that whereas DRT is decidedly procedural, DPL and related theories are declarative, the authors
10
Meaning: the Dynamic Turn
argue that a synthesis between the two perspectives can be reached if logic is used as a metalanguage for the description of natural languages. Procedural aspects, they claim, are then associated with the proof theory of this logic, declarative aspects with its model theory. A thorough criticism of the foundations of dynamic semantics is presented by Richard Breheny. He concentrates on the question of what exactly it is that is accounted for by dynamic theories of formal semantics; and he argues that some of these theories are based on an unwarranted ‘semantization’ of what are essentially pragmatic effects of utterances. Wolfram Hinzen confronts the kind of dynamics which appears to fuel current dynamic theories of the kind of DPL with the different kind of dynamics implicit in the Chomskyan view of language. His point is that “given that the philosophical theory of meaning has long since moved on to the philosophy of mind, it remains surprising that the Chomskyan enterprise continues to be as marginal to it” and that “it seems desirable to have more discussions on how these partings of ways could be fruitfully overcome.” Ruth Kempson, Wilfried Meyer-Viol and Masayuki Otsuka argue that there is a sense in which we should see the dynamics of language as grounded already within syntax; and moreover, that syntax may be the only place of the system of language where we need to ‘go dynamic’. What they propose is to concentrate on the process of interpretation of strings, which they account for as a process of building syntactico-semantic trees; and argue that “the old question of whether we need structures in representing interpretation in semantics is transformed into a new question: Do we need any mode of representation other than growth of logical forms in order to express generalisations about natural-language syntax?” The nature of the dynamic turn can be clearly seen in the analyses of the English articles. While the static versions of formal semantics often accepted the obviously inadequate Russellian approach (according to which a amounted to existence and the to unique existence), the dynamic turn has opened the door for much more adequate analyses replacing existence with referential availability and uniqueness with top salience. Klaus von Heusinger’s paper discusses this very development and attempts to elaborate on it. While the focus on the dynamic aspect of language is a relative novum for formal semantics, it has been long discussed within some linguistico-semanticist circles, notably within the so-called Prague School. And this is the background of the paper of Petr Sgall: he discusses the interplay of semantics and the workings of discourse, trying to accommodate the insights of the Praguian theory (with its concepts of communicative dynamism, topic-focus articulation etc.) within the formally semantic framework. Dynamic semantics can be, in fact, seen as switching from seeing our assertoric statements as representing or expressing something (classes of possible worlds, DRSs etc.) to seeing them as effecting something – namely the change of the actual information state or context. This not only maneuvers semantics of natural language into the vicinity of that of programming languages (Groenendijk and Stokhof, for example, explicitly admitted theories of semantics of programming languages as a source of inspiration for their DPL), but it also reduces the semantic gap between the assertoric mode of speech and the imperative one. And the semantics
Introduction
11
of imperatives is the subject matter of two papers closing the second part: while Berislav Žarnić discusses more the technical side of the semantic theories of imperative discourse, Timothy Childers and Vladimír Svoboda deal more with the foundational problems. The last part of the book is devoted to the game-theoretical approaches to semantics. Tapio Janasik and Gabriel Sandu discuss the way in which such approaches can deal with anaphoric reference. Manuel Rebuschi argues that game-theoretical semantics should be based on substitutional, rather than on the objectual, quantification. And Louise Vigeant scrutinizes the step from the original version of game-theoretical semantics to the independence-friendly logic and concludes that the step contains an undesirable ‘intensionalization’, which takes it away from laudable verificationism. What all the discussions appear to indicate is that the ‘dynamics’ which is in play is more than a mere hypothesis – that it rather aspires to become an integral part of our general view of meaning. The dynamic turn is thus responsible not only for some new theories, but also for a change in our overall notion of semantics. Hence, however we want to answer the question whether meaning is dynamic, it is simply no longer possible for us to ignore it.
REFERENCES Bar-Hillel, Y. (1954). Indexical Expressions. Mind, 63, 359-379. van Benthem, J. (1997). Exploring Logical Dynamics. CSLI, Stanford. van Benthem, J. and A. ter Meulen, eds. (1997). Handbook of Logic and Language. Elsevier / MIT Press, Oxford / Cambridge (Mass.). Chierchia, G. (1994). Intensionality and Context Change. Journal of Logic, Language and Information, 3, 141-168. van Eijck, J. and H. Kamp (1997). Representing Discourse in Context. In van Benthem and ter Meulen (1997), pp. 179-238. Grice, H. P. (1957). Meaning. Philosophical Review, 66; reprinted in Grice (1989). Grice, H. P. (1989). Studies in the Way of Words. Harvard University Press, Cambridge (Mass.). Groenendijk, J. and M. Stokhof (1991). Dynamic Predicate Logic. Linguistics and Philosophy, 14, 39-101. Hintikka, J. (1973). Logic, Language-Games and Information. Clarendon Press, Oxford. Hintikka, J. and J. Kulas (1985). Anaphora and Definite Descriptions. Reidel, Dordrecht. Hintikka, J. and G. Sandu (1997). Game-theoretical semantics. In van Benthem and ter Meulen (1997), pp. 277-322. Kamp, H. (1981). A Theory of Truth and Semantic Representation. In: Formal Methods in the Study of Language (J. Groenendijk, T. Janssen and M. Stokhof, eds.), pp. 277-322. Mathematical Centre, Amsterdam. Kamp, H. and U. Reyle (1993). From Discourse to Logic. Kluwer, Dordrecht. Kaplan, D. (1970). On the Logic of Demonstratives. Journal of Philosophical Logic, 8, 81-98. Muskens, R., J. van Benthem and A. Visser (1997). Dynamics. In van Benthem and ter Meulen (1997), pp. 587-648.
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Lewis, D. K. (1970). General Semantics. Synthèse, 22, 18-67. Lewis, D. K. (1979). Scorekeeping in a Language-Game. Journal of Philosophical Logic, 8, 339-59. Partee, B. (1980). Montague Grammar, Mental Representation and Reality. In: Philosophy and Grammar (S. Kanger and S. Oehman, eds.), Reidel, Dordrecht. Peregrin, J. (2000a). Reference and Inference. In: Reference and Anaphoric Relations (K. von Heusinger and U. Egli, eds.), pp. 269-286. Kluwer, Dordrecht. Peregrin, J. (2000b). The Logic of Anaphora. In: The Logica Yearbook 1999 (T. Childers, ed.), pp. 191-205. Filosofia, Prague. Peregrin, J. (2001). Meaning and Structure. Ashgate, Aldershot. Sperber, D. and D. Wilson (2002). Pragmatics, Modularity and Mind-reading. Mind & Language, 17, 3-23. Stalnaker, R. (1970). Pragmatics. Synthèse, 22, 272-89. Stalnaker, R. (1978). Assertion. In: Syntax and Semantics 9: Pragmatics (P. Cole, ed.), pp. 315-22. Academic Press, New York. Stalnaker, R. (1986). Possible Worlds and Situations. Journal of Philosophical Logic, 15, 109123. Stalnaker, R. (1999): ‘Introduction’, in Stalnaker: Context and Content, pp. 1-27. Oxford University Press, Oxford.
THE ‘CAUSAL STORY’ AND THE ‘JUSTIFICATORY STORY’ (reflections on a McDowellian theme) Jaroslav Peregrin* www.cuni.cz/~peregrin
Suppose for a moment, that J.R.R. Tolkien, the famous author of the cult fantasy saga Lord of the Rings, did not publish anything of his writings during his lifetime; suppose that after his death the manuscripts of all his writings are lying on his table. Where, then, is the Middlearth, the glorious land of hobbits, dwarfs, elfs and human heroes, situated? We might be tempted to say that it is within our world, namely inside the pile of the papers on the writer’s table for it exists solely through the letters written on these papers. However, to say this would be wrong (or at least strongly misleading) - surely we do not expect that should the heroes of the book walk in a straight line long enough, they would cross the boundaries of the book and appear in Mr. Tolkien’s room. Middlearth is, of course, not within our world - despite existing solely due to certain things which are within it. Now the situation is not substantially different actually, when Middlearth does not exist solely through a single pile of papers, but rather through millions of printed copies of Tolkien’s books and through the minds of millions of their readers. Again, the land exists exclusively through the existence of entities which are parts of our world (albeit that they are now scattered throughout the whole Earth), but this does not mean that the land itself is a part of our world. The point of this anecdotal excursion is now that this relationship between our world and Middlearth is, in a sense, similar to the relationship between our physical space of things and „the space of reasons“ (Sellars, 1956, §36); or between „the causal story“ and „the justificatory story“ (Rorty, 1991, 148). Like Middlearth, the space of reasons exists exclusively due to us, humans, and our minds (and perhaps also of some of our artifacts), and in this sense we might be tempted to situate it in our world, to see it as a certain, perhaps scattered, compartment of the world of things within which we live; but just as in the case of Middlearth, this might be dangerously misguiding. The rationale of talking about something as the space of reasons comes from Sellars’ argument, recognized as sound by his followers, that we have to distinguish carefully between thing-like entities, particulars, which enter into causal (in a broad sense) relationships, and proposition-like entities, facts (and potential facts, which we may call simply propositions), which enter into justificatory relationships. These are two essentially different kinds of entities, living essentially different kinds of ‘lives’ within their different realms. Particulars typically inhabit our spatiotemporal world and are denoted by names; whereas propositions inhabit the space of reasons and are expressed by sentences. And as Brandom (1984, 6) stresses, it is the grasp of propositional contents that in an important sense distinguishes rational or sapient beings.
*
This work was supported by the Research Support Scheme of the OSI/HESP, grant No. 280/1997.
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The necessity of separating these two kinds of entities was what underlay Sellars’ rejection of the traditional empiricism with its ‘sense data’ - for the ‘sense data’ are nothing else than entities that are supposed to belong to both these categories of entities at once. The sense-data-theorist assumes that the sense datum is a point in which the causal chain going from the outside world to the subject’s mind changes into a justificatory chain, he „insists both that sensing is a knowing and that it is particulars which are sensed.“ (Sellars, 1956, §3) Thus what is sensed is assumed to be knowledge, a true belief, but knowledge which is immediately ‘given’ to the mind, for it is directly delivered into it by the world itself (and is thus infallible). This is what Sellars famously called the Myth of the Given. However, if a particular cannot be a reason for a belief, we inevitably have to conclude, as Davidson (1986, 310) did, that „nothing can serve as reason for a belief save another belief“. But if this is true - if the world has no way of penetrating the space of beliefs -, beliefs appear to be turned loose from the world, to be condemned to blindly and aimlessly revolve within the mind. John McDowell (1994, p. 7) writes: „The idea of the Given is the idea that the space of reasons, the space of justifications or warrants, extends more widely than the conceptual sphere. The extra extent of the space of reasons is supposed to allow it to incorporate non-conceptual impacts from outside the realm of thought.“ However, he continues (p.8), „it can seem that if we reject the Given, we merely reopen ourselves to the threat to which the idea of the Given is a response, the threat that our picture does not accommodate any external constraint on our activity in empirical thought and judgment.“ This is what McDowell does not like, and why he seeks a third path, a path that would lead us safely between the Scylla of the Myth of the Given, and the Charybda of Davidsonian coherentism. What we would like to indicate here is that both the Myth of the Given and the threat of leaving our thought externally unconstrained, broken loose from the outside world, presupposes the picture on which the space of reasons is somehow inside the space of things, so that causal chains from the outside world can penetrate into the inner one (thereby changing their nature to justificatory chains). I am going to argue that although this picture might appear to be extremely natural or even unavoidable, it is one more picture which „holds us captive“ (Wittgenstein, 1953, §115) - and that what is really needed is to abandon it. Before we turn to the discussion of the relationship of the space of reasons to the realm of things of our everyday life, and thereby of the ‘justificatory story’ to the ‘causal story’, let me point out that the distinctions between the two realms and the two stories are related to another interesting distinction, the distinction between two ways we can approach a mind (and, I think, also a language1). We can look at a mind ‘from without’: to look at it as one of the objects which feature within our the causal story (and, indeed, also within our justificatory story). We could hardly have failed to notice that among the objects which surround us there are some quite specific ones, which we have come to classify as mind-havers, thereby positing minds, specific objects the having of which distinguishes mind-havers. (It is, of course, not necessary to treat
1
Cf. Peregrin (1995, esp. §11.6).
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minds as genuine objects, rather than only as ‘properties’ or ‘aspects’ of mind-havers, but this is not a question we will consider now.) But we can also try to look at a mind ‘from within’: we may notice that the justificatory story urgently points to somebody who has invented it and who ‘harbors’ it - who is in the business of justification. (This contrasts this story with the causal story, which, in a sense can be imagined to be ‘told’ - i.e. performed - by the inanimate world itself.) Thus, the justificatory story points out to a mind (or, indeed a ‘community of minds’), which is ‘behind’ it, to a ‘transcendental ego’. So telling this story we are in a sense assuming the standpoint of a mind, we approach it ‘from inside’. Now I think that the advise of keeping apart the causal and the justificatory story should be understood as also entailing the advice not to try to be simultaneously inside and outside a mind. And if we do follow this advice, the relationship between the mind and the world is no mystery: If we look at the mind from without, then there is nothing mysterious about its relationship to the rest of the world: mind-havers, and thereby minds, enter in all kinds of causal interactions with their surroundings. And if we approach the mind from within, then asking about its relationship to the outside world makes no sense at all: then the mind, the thinking subject, is not part of the world (but rather its boundary, as Wittgenstein, 1922, §5.632, duly points out2) and hence there simply is no outside for it to have. This vantage point may also help us distinguish the question we are considering, the question of how to cope with the „threat that our picture [of the relationship of the mind and the world] does not accommodate any external constraint on our activity in empirical thought and judgment“, from some other, related questions, with which it sometimes appears to be intermingled in McDowell’s book. First, there is a question which arises from looking at mind from outside, the question about the nature of mind and about the specificity of its role within the causal story. We have seen that from this perspective minds cannot be anything else than kinds of objects (or properties of objects) causally interacting with other objects. However, one can legitimately wonder whether the causal story really gives us resources to account for the peculiarity of minds in the first place. Do we need a specific kind of vocabulary to account for them, say a normative vocabulary?3 Second, there is a question that arises from looking at the world from within a mind, the question of whether we do see the world through the prism of the mind adequately. We
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Cf. Kripke’s (1982, p.123ff.) discussion of this Wittgenstein’s passage.
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This seems to be a question which divides Quine and Rorty (whose answer to the question seems to be no) from Davidson and Brandom (who seem to accept that need for some kind of a specific vocabulary). However, we could also see the question as directly challenging the very constitution of the causal story: Since the dawn of modern science, from Descartes, Leibniz and Newton, we have come to see the causal world as made exclusively of passive materia; but in view of the existence of minds, is this really right? Do we not need also an active ‘pateria’ (to use the terminology of the Czech mathematician and philosopher Petr Vopìnka) to describe the world containing minds? Do we not need to assume that there may exist entities which are not only subject to causal law, but are also able to insert new causes into the causal chains?
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may wonder whether our conceptualizations which underlie our justificatory story do not corrupt the world, whether the story we thus tell presents the world ‘as it really is’. This is the question about the nature of the unconceptualized world, about the outlook of bare facts stripped of our values4. Both these questions, which we are not going to address here, are to be distinguished from our question of the relationship of the thoughts inside mind to the things outside it. How do elements of the causal world of things manage to restrain the elements of our inner space of reasons, to make our minds work somehow dependently on what is going on outside them? Note that this question arises only if we attempt to account for minds within the world by making the causal story continuous with the justificatory story in such a way that the justificatory story would account for minds and the causal story for the rest of the world. This leads to the picture of minds as spaces of their own within the physical space, as certain islands governed by the justificatory relations within the vast sea governed by the causal ones. To understand the real nature of this question, we now turn our attention back to the concept of the space of reasons. What is the space of reasons and where is it situated? What is the nature of propositions which constitute it? On my construal, the concept of the space of reasons and the concept of proposition are two sides of the same coin. Intuitively, it is very hard to say what a proposition is, to get any kind of a firm grip on them. However, there are facts about propositions which seem to be obvious: we would, for example hardly call something a proposition unless it has a negation. Similarly, it seems to be constitutive of the concept of proposition that propositions can be conjoined, that a proposition can imply something etc. In short, propositions necessarily exist within a network, or a space, of logical relationships. And it is these logical relationships which constitute the most general shape of the space of reasons. Now as a matter of fact, some propositions happen to be true, or, in other words, are facts. It was Wittgenstein (1922), who famously insisted that it is facts, and not things, of which our world consists. Why does Wittgenstein find it so important to deny that the building blocks of the worlds are things, despite the fact that probably any normal, philosophically uncontaminated person would say that world does consist of things (perhaps things which stand in certain relationships)? Well, one answer might be that as he wants to put forward his correspondence theory of language, he needs the world cut into pieces corresponding to the pieces of language, and thus he invokes facts, the „sentence-shaped items“ (Strawson5), or „ghostly doubles of the grammarian’s sentence“ (Collingwood6). A little bit more sympathetic answer would be that this „linguistic“ structuring of the world is not only something Wittgenstein needs to accomplish his project, but in fact something that is
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This is a question many philosophers have warned us is illusory: the idea that there is a story which would be told by the world itself - as contrasted by the stories told by the mind-havers -, they say, is an idea not worth being taken seriously.
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As quoted by Rorty (1988, p. 35).
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Quoted by Putnam (1994, p. 301).
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in a sense how the world sometimes really looks to us, language-users. Although we perceive our worlds as the world of things, having language we sometimes reflect it, and reflecting it we see the world as the world not of things, but of facts. Thus, our „language instinct“ (as Pinker, 1994, dubbed our ability to use language) makes us see our worlds in terms of facts and propositions. „Language is“, as Davidson (1997, 22) put it recently, „the means of propositional perception“. Thus, the causal story (featuring things in their causal interaction) and the justificatory story (featuring propositions in their inferential dependencies) are, in an important sense, two different stories with the same subject, namely our world (which is what makes this case different from the Middlearth one). We may say that the sun sends its rays thus causing the air to become warm; and we may also say that the fact that the sun shines implies (via the „observational categorical“ saying that if the sun shines, the air becomes warm) that it is warm. Seen from this perspective, the space of reasons is not embedded within the realm of things, it is merely the very same realm differently conceived. However, propositions do not merely reside within the abstract space of reasons; some of them come to be entertained or endorsed by rational individuals, thereby becoming the individuals’ thoughts or beliefs7. It is, for example, me, who believes that the sun is shining and that (therefore) it is warm outside. Are then not my beliefs, the propositions that I endorse, situated inside the physical world, namely inside my head? And is it not necessary to secure that they do properly reflect the world outside the mind? It is this picture which makes the Myth of the Given so attractive - it seems that if we do not want believers’ minds to be completely independent of the world, there must be a path from the outside space of things into the inside realm of beliefs. There must be a boundary of the space of beliefs at which the causal chain gets transformed into the evidential and justificatory chain, there must be a spot on the boundary between mind and world at which a particular becomes a proposition which thus constitutes direct, given knowledge. The answer to this temptation is, again, the rejection of the conflation of the causal story with the justificatory story and to situate the space of beliefs inside the physical space. Beliefs are better not imagined as being within one’s head. We have the causal story: the world, e.g. the sun sending its rays, impinges on my (or whoever’s) sensory receptors, the receptors send signals to the brain, there some kind of causal interaction between the neurons takes place, and then the brain perhaps sends a signal to some motoric nerves which do something, e.g. make the hands put off the coat and hang it into the wardrobe. What is important is that this story is causal through and through, the causal chain nowhere changes into anything non-causal. Now we could perhaps improve on this causal story by assuming what Dennett (1987) calls the intentional stance: instead of addressing the proceedings of one’s neural machinery (which we can hardly really know), we can adopt a much more rough and a much more useful 7
Of course the very existence of propositions and of the space of reasons is parasitic upon the rational (predominantly linguistic) practices of us humans, the space being in fact nothing more that a hypostasis vividly envisaging the structure of our linguistic practices. However, once we accept this hypostasis, it is clear that propositions may exist without being anybody’s beliefs.
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way of speaking, and characterize the person as, e.g., believing that it is warm. There is still nothing non-causal about this: believes that it is warm is our rough way to specify the physical state of the person in question. And believes here should not be construed as is in the possession of a thing called belief let alone has a belief floating somewhere inside his head. And it is important to realize that the situation does not change even when we grant Davidson and Brandom that assuming the intentional stance means a more substantial change than simply starting to discern more global patterns, namely that it is the place where some kind of normativity creeps in. Ascribing beliefs and even thoughts to somebody is still, as Davidson stresses, not situating propositions into an inner space of that person, it is using propositions as a classificatory scale in the same way in which we use the number scale for the purpose of classifying weights of things: „In thinking and talking of the weights of physical objects we do not need to suppose there are such things as weights for objects to have. Similarly in thinking and talking about the beliefs of people we needn’t suppose there are such entities as beliefs. ... The entities we mention to help specify a state of mind do not have to play any psychological or epistemological role at all, just as numbers play no physical role.“ (Davidson, 1989, 11) Now it is important to keep in mind that from this vantage point, we have to distinguish between the properties which a propositions has simply in itself, and those which it may have in virtue of being endorsed by a believer. The proposition that it is warm may be, for example, true (i.e. be a fact), which is, of course, independent of whether anybody believes it. On the other hand, the same proposition, happening to be my belief, might be, e.g., caused by the sun rays coming into my eye - which is obviously only the property of my belief, not of the proposition as such. Now to ask what is the reason of something is to ask about a property of the first kind, whereas to ask why did somebody come to believe something is to ask about a property of the second kind. To say that the reason it is warm is that the sun shines (and, possibly, that whenever the sun shines, it is warm) is to say something that does not depend on anybody’s in fact believing that it is warm. It is something essentially different from saying why X believes that it is warm. Now if I say „The reason why it is warm is that the sun shines“, I give a reason, I tell the justificatory story; whereas when I say „I believe that it is warm, because I believe that the sun shines“, I do not give reasons, I tell the causal story (or some its enhancement tailored to account for agents8) about myself. This means that the term belief is systematically ambiguous: it may mean a potential belief, a proposition from the space of reasons that may (or may not) become somebody’s belief, and it may also mean an actual belief of a concrete person. There is a belief as such, i.e. a proposition, and there is a belief of somebody similarly as there is a pint as such (the unit of measure) and there is a pint of something. If we ask whether beliefs are broken loose from the (rest of the) world, we must first clarify which sense of belief do we mean: if it is the first one, then the question does not make much sense, for there is nothing for abstract propositions to be broken loose from (similarly as pints and meters are not broken loose from anything); and if we mean the second sense, then beliefs are
8
See footnote 3.
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trivially not broken loose from the world: they are part of the causal world and as such they causally interact with their environment in various ways (similarly as pints of beer do, e.g. by being drunk by people). What about, then, McDowell’s worry that the rejection of the Myth of the Given „threatens to make what was meant to be empirical thinking degenerate, in our picture, into a frictionless spinning in a void“ (p. 66)? Well the upshot of our considerations so far is that we should see thinking either as a causal matter, in which case it „spins“ unproblematically within its causal surroundings, or as a matter of the justificatory relationships, in which case it does not „spin“ in anything. I am convinced that this should be the right response to the „in the void“ threat. However, there is also the „frictionless“ threat. The causal story and the justificatory story differ in that although both must come to an end, there is no end to causes, whereas there has to be an end to reasons. Everything has, as we believe, its cause, and any causal chain can be traced back indefinitely; but there are reasons which do not require further justification, which are, so to say, justified in themselves. (This is not to say that such reasons could be distinguished once and for all - which reasons do not require further justification depends on the context of the justification, but in each context there are such reasons.) And does this not mean that such reasons are „unwarranted“, that accepting them our mind draws on willful on arbitrary foundations? That thinking is „frictionless“? Of course not: once we see that there is no outside from where such „unwarranted“, „border“ reasons could (fail to) be sustained, we should be bound to see that they are not representations of something outside there in the world, but rather parts of the world itself. Beliefs are propositions purported to be true, and if some of them are obviously true, their purport thus being veridical, then they are simply what true propositions are, viz facts. As Brandom (1994, p. 333) puts it: „Thus a demolition of semantic categories of correspondence relative to those of expression does not involve ‘loss of the world’ in the sense that our discursive practice is then conceived as unconstrained by how things actually are. ... What is lost is only the bifurcation that makes knowledge seem to require the bridging of a gap that opens up between sayable and thinkable contents - thought of as existing self-contained on their side of the epistemic crevasse - and the wordly facts, existing on their side“. If I look from the window and claim that the sun is shining, and somebody standing besides me asks „why?“, my reaction is probably not going to be to give a reason, but rather to cease to see him as a serious partner within the ‘practice of giving and asking for reasons’ (maybe only for that moment - maybe what he says is only a kind of joke, or his way of doing poetry). The fact that a claim does not need further justification does not mean that it is somehow broken loose from the world and thereby basically dubious - on the contrary, it means that it is the most indubitable9.
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We must not be confused by the fact that we sometimes appear to voice further justifications by switching from the justificatory to the causal story. If somebody asks why X believes that it is warm, I can answer „because he believes that the sun shines and he infers that it is warm from it“, or „because he feels it“ or whatever: in short, I can investigate and describe the causes of his adopting the belief. Also if I claim „It is warm“ and somebody asks „why do you think so?“, I can sometimes
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This gets us to the Kantian story about spontaneity: it tells us that while the inanimate world is the realm of law, the mind constitutes the realm of freedom. The mind does not simply behave according to rules, but it rather acts according to conceptions of rules - as it is thus possible for it to disobey a rule (see Brandom, 1994, §4.1). Thus, the mind is free in a way other objects are not. And this leads to a further problem: if mind is free, how is it that the world forces upon it, in perception? Does it mean that perception takes place somewhere still behind the bulwarks of mind, or does it mean that mind is not as free it seems to be? This is an important theme for McDowell; and his answer is, in effect, that the space of one’s beliefs does not coincide with the realm of his freedom. This squares with the fact which we urged above: namely that justification must come to an end, that every justificatory claim must end with a reason for which no justifications appears to be required (in the corresponding context). If there were any freedom with respect of the acceptance of such a reason, there would be necessarily a further „why?“. Thus I think that if McDowell speaks about the „threat of empirical thinking degenerating into a frictionless spinning in a void“ we should see this rather as two different kinds of challenges: we have to explain why our thinking is not „frictionless“, and we have to show why it is not „in the void“. To show that it is not „frictionless“ we need to show that the realm of our beliefs does not coincide with the realm of our freedom - and making this obvious is one of the achievements of McDowell’s book. On the other hand, to show that it is not „in the void“ requires, I am convinced, to show that the whole picture in which our thinking is „in something“ is basically misleading - which appears to be something McDowell is not willing to settle for. „Thus the fate of all ‘philosophical problems’ is this: Some of them will disappear by being shown to be mistakes and misunderstandings of our language and the others will be found to be ordinary scientific problems in disguise“, wrote Moritz Schlick in 1932 thus expressing the opinion of a great majority of analytic philosophers of his age that philosophical problems could be dispensed with by means of a careful analysis, or indeed an adjustment, of the semantics of language. This was, no doubt, an exaggeration; but the conviction surely did have a certain rational core. Some of the problems we try to solve in philosophy can be dissolved by means of changing the way we see certain things and the way we speak about them. I think that the relationship between mind and the world is one of them: it is the postCartesian picture of mind as an ‘inner space’ which has given rise to most of the questions we ask now. And it is, I think, the consequential abandonment of this picture which may help us deal with them. People like Rorty, Davidson and Brandom have done very much to bring out the misleadingness of the „representational model“ of thought. McDowell seems to think that in some respects they might have been to hasty: that in the course of cleaning away pseudoproblems they swept under the table also some genuine problems, such as the problem construe the question as „What has caused you to have the belief?“ and give similar kinds of explanations; otherwise the only thing I can do is to voice a reason for it being warm - if there is one.
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of „empiricism“. Although his book is surely a deep discussion of many issues concerning human mind, I cannot help feeling that the author, by resurrecting the problem of empiricism, restores also a picture which we should be glad to have gotten rid of.
References: Davidson, D. (1986): ‘A Coherence Theory of Truth and Knowledge’, in Truth and Interpretation: Perspectives on the Philosophy of Donald Davidson (ed. LePore, E.), Blackwell, Oxford, 307-319; reprinted (with added 'Afterthoughts') in Reading Rorty (ed. R.Malachowski), Blackwell, Oxford, 1990, pp. 120-138. Davidson, D. (1989): ‘What is Present to the Mind?’, in The Mind of Donald Davidson (eds. J. Brandl, W.L. Gombocz), Rodopi, Amsterdam, 3-18. Davidson, D. (1997): ‘Seeing Through Language’, in Thought and Language (ed. Preston, J.), Cambridge University Press, Cambridge, 15-27. Dennett, D. (1987): The Intentional Stance, MIT Press, Cambridge (Mass.). Kripke, S. (1982): Wittgenstein on Rules and Private Language, Harvard University Press, Cambridge (Mass.). McDowell, J. (1994): Mind and World, Harvard University Press, Cambridge (Mass.). Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Pinker, S. (1994): The Language Instinct, W. Morrow, New York. Putnam, H. (1994): Words and Life, Harvard University Press, Cambridge, Mass.. Rorty, R. (1991): ‘Pragmatism, Davidson and Truth, in Objectivity, Relativism and Truth (Philosophical Papers, vol. 1), Cambridge University Press, Cambridge. Rorty, R. (1998): Truth and Progress (Philosophical Papers, vol. 3), Cambridge University Press, Cambridge (Mass.). Schlick, M. (1932): ‘The Future of Philosophy’, College of the Pacific Publications in Philosophy 1, 45-62; reprinted in and quoted from R. Rorty, ed.: The Linguistic Turn, University of Chicago Press, Chicago, 1967. Sellars, W. (1956): ‘The Myth of the Given: Three Lectures on Empiricism and the Philosophy of Mind’, in The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minnesota Studies in the Philosophy of Science 1; eds. H. Feigl & M. Scriven), University of Minnesota Press, Minneapolis; reprinted as and quoted from Sellars: Empiricism and the Philosophy of Mind, Harvard University Press, Cambridge (Mass.), 1997. Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London; English translation Routledge, London, 1961.
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POSSIBLE WORLDS: A CRITICAL ANALYSIS Jaroslav Peregrin1 www.cuni.cz/~peregrin Prague Bulletin of Mathematical Linguistics 59-60, 1993, pp.9-21] [appeared in
1 The Emergence of Possible Worlds Frege has proposed to consider names as denoting objects, predicates as standing for concepts and sentences as denoting truth values. He was, however, aware that such denotation does not exhaust all what is to be said about meaning. Therefore he has urged that in addition to such denotation (Bedeutung) an expression has sense (Sinn). The sense is the "way of presentation" of denotation; hence the expressions Morning Star and Evening Star have identical denotations, but different senses. Carnap has proposed to replace Frege's distinction between Sinn and Bedeutung by the distinction between intension and extension. Extension is, according to Carnap, what is shared by every two expressions which can be truthfully declared equivalent; where equivalence means a bit different things for different grammatical categories. In the formalism of the classical logic, the equivalence of two terms T1 and T2 is expressed by the formula T1=T2, that of two sentences S1 and S2 is by the formula S1↔S2, and that of two unary predicates P1 and P2 by the formula ∀xP1(x)↔P2(x). Hence the extension of a term can be identified with the individual the term stands for, that of a statement with its truth value, and the one of a unary predicate with the class of individuals which fall under the predicate. This means that Carnap's intension is something quite close to Frege's sense. Carnap's intension is, on the other hand, not a mere reincarnation of Frege's sense. Frege has been never quite explicit with respect to the nature of sense, but his reluctance to speak about it quite explicitly was conscious and purposeful: Frege was convinced that the only entities that can be really talked about are objects, and sense was not an object for him. To speak about a sense we would have to name it, and hence make the sense into a denotation of an expression. Carnap's intension is, on the other hand, not an entity of a category essentially different from that of extension, it is only something that casts a finer net over the space of expressions. Carnap has defined both extension and intension as derivatives of an equivalence, as that what so-and-so related expressions share. However, while in the case of extension there has been suitable things that could be considered as reifications of such equivalence (namely individuals, truth values and classes of individuals), in the case of intensions no such candidates were at hand. 1
The first version of my views of the matters discussed in this paper were presented at the '87 LOGICA symposium at Bechynì and published in the proceedings under the title The Notion of Possible World in Logic and in Logical Analysis of Natural Language. The present paper is based on my contribution read at the '88 LOGIC AND LANGUAGE conference at Hajdúszoboszló, Hungary, but (from reasons unknown to me) not included in the proceedings. The present version draws heavily on detailed comments of Petr Sgall and especially of Barbara Partee.
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Help came from formal logic. In 1963 Kripke has developed the model theory for modal propositional calculi; he has based it on the concept of possible world. This concept, and the whole Kripkean semantics has proven itself to offer a fruitful explication of the notion of intension. The era of possible-worlds-semantics begun. Kripke's idea has come to be amalgamated with an idea of another logician, with the ideas underlying Church's formulation of the simple theory of types, and this has given rise to the grandiose systems of intensional logic as we know them from the writings of Montague, Cresswell, Tichý and others. The notion of a possible world, reminiscent of the philosophical views of Leibniz, thus has moved to the centre of discussion of theoreticians of language. However, although the results reached by means of the intensional logic (and its possible-worlds semantics) have been considered great, the nature of the concept of the possible world has been subject to various discussions. What is such a possible world? Are possible worlds "real", are they elements of our "mental world", or are they objects of a Platonic "third realm"? Is it at all reasonable to transfer this more or less obscure notion into a rigorous conceptual framework of formal semantics? These are questions which bothered people dealing with possible-worlds semantics and which in fact have not stopped bothering us yet.2 The first step to a proper solution of a philosophical problem is to achieve its proper formulation. In the present context this means that we have to consider the above mentioned questions and inspect them to see if we are able to make a proper sense of them. The first thing to see is that it is not right to speak about a transfer with respect to the concept of possible worlds: the terms of a formal system necessarily have a rigorous formal definition (explicit or implicit) and by giving them one or another name we do not "give" them anything belonging to the intuitive sense of the name. Such a term can only more or less faithfully render the character of the corresponding intuitive notion; i.e. it can be more or less considered as its explication3. By calling an element of our formal system a possible world we in no way guarantee that it has anything in common (besides the name) with possible worlds in the intuitive sense; that this is so is something to be shown.
2 The model theory for modal calculi For the Kripkean introduction of the concept of possible world as well as for the Montagovian logistic formalization of natural language the methods of the model theory are essential. Let us thus now investigate into the very nature of the notion of model and into the place of the concept of possible world within the theory. A model of a theory, i.e. of a class of statements (propositional formulas) of a language, is a special kind of interpretation of the language. An interpretation is considered as a function assigning extralinguistic entities to expressions of the language; a model of a theory is then such an interpretation which to all and only statements belonging to the theory assigns a distinguished element (typically the value T, "the truth"). However, not every mapping of the class of expressions can act as an interpretation. For specific logical systems restricted classes of such mappings are defined as admissible 2
See e.g. Cresswell (1973), Putnam (1975) or Partee (1980).
3
In the sense of Carnap (1950) and Quine (1960).
10
interpretations. An interpretation of the traditional propositional calculus is, for example, a mapping which assigns (i) one of the two truth values T or F to any atomic propositional formula; (ii) the usual truth tables (i.e. certain functions from {T,F} or {T,F}x{T,F} to {T,F}) to propositional operators; and (iii) the relevant function value of the relevant truth table to any non-atomic propositional formula. The model theory for the classical first-order predicate calculus employs, in addition, a universe of discourse (an arbitrary set), such that individual terms are interpreted by elements of the set and predicates by relations over the set. In general we can say that we have defined a model theory for a given logical calculus iff we have specified a class of interpretations of the language underlying the calculus such that any propositional formula is derivable in the calculus if and only if it is assigned the distinguished element by any of the class of interpretations. We then also say that the calculus is sound and complete with respect to this class of interpretations4. Let us also notice what kinds of functions interpretations in general are. First, it seems that a substantial characteristic of any interpretation is that it is compositional, i.e. that the interpretation of a complex expression is a function of the interpretations of its parts. Moreover, the interpretation of one of the parts of a complex expression is usually a function and the interpretation of the complex is the function value of this function applied to the interpretations of the other parts. Second, the interpretation of expressions which are considered as "logical" is fixed once for all while that of the other, "non-logical" expressions is left open5. Let us try to consider what kind of modifications the model theory of the non-modal logic would require to be applicable to the modal calculus. It can be shown easily that if L is a logical calculus and I an admissible interpretation of it, then two propositional formulas which have the same interpretation must be "L-intersubstitutive", i.e. that if we substitute one for the other in a complex propositional formula, then the resulting formula belongs to L if and only if the original did. It is clear that if L is the classical propositional calculus, then two propositional formulas are L-intersubstitutive if and only if they have the same truth value; this enables us to interpret propositional formulas directly by their truth values. However, the situation is different with respect to the modal propositional calculus: there are propositional formulas which coincide in truth values, although they are not intersubstitutive. From this it follows that propositional formulas of the modal calculus cannot be interpreted by means of their truth values. So let us assume that propositional formulas will be interpreted by elements of a set P; members of P will be called propositions. Then, however, to retain compositionality, the logical operators have to be interpreted by functions from P or PxP into P. This seems to be implausible: the classical logical operators have been explicated in terms of the non-modal calculus in an exhaustive way, and now we seem to have to abandon their truth-functional interpretation. It seems that it would be plausible to define their interpretation within the modal 4
Note that such a class of interpretations exists not only for first-order logic. Leblanc (1976), for example, shows that in spite of the well-known results of Gödel there exists a class of interpretations with respect to which the second-order predicate calculus is sound and complete. In fact, by means of the well-known method of Henkin (1950) the model theory in the present sense can be defined for a predicate calculus of any order.
5
For a detailed discussion see Peregrin (1992b).
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calculus in such a way that it would be reducible to their classical interpretation. The solution of this problem is, of course, simple; it is based on the identification of the members of P with subsets of a set L, or, which is clearly the same, with functions from L to {T,F}. If we denote the class of all functions from a set A into a set B as [A→B], the propositions become members of [L→{T,F}] and the interpretations of the logical operators become functions from [L→{T,F}], or [L→{T,F}]x[L→{T,F}], into [L→{T,F}]; they then can be induced in a natural way by functions from {T,F}, or {T,F}x{T,F}, into {T,F}.6 Besides this it is possible to interpret also the operator of necessity in a plausible way within this framework; the relevant completeness proofs have been presented by Kripke (1963)7. Kripke has called members of such a set L possible worlds. So returning to the questions of the nature of possible worlds we can conclude that a possible world is something on the basis of which a proposition yields a definite truth-value. This means that (Kripkean) possible worlds are entities relative to which the truth values of propositions obtain. Let us add some explanations. It is clear that the definition of the concept of possible world within Kripke's approach is not explicit; we cannot put the concept on one side of an equality sign. However, an implicit definition is also a definition; the concept of possible world is defined in the same way as the concept of number within Peano arithmetic or the concept of a set within the axiomatic set theory. In contrast to the explicit definition, the implicit one, however, does not allow us to reduce the concept to simpler ones, we cannot explicate it. Roughly speaking, anything could be a possible world, providing it has the capacity of sorting propositions into true and false.8 Let us now try to look for an explicit definition of the concept.
3 Which worlds are 'really' possible? The Kripkean theory thus assumes that there are possible worlds (i.e. that there is something relative to which statements have truth values), but it makes no attempt to inquire into the 'real nature' of the space of possible worlds. On the contrary, the theory purposefully aims at truths valid independently of what such a space might in reality look like, just as the predicate calculus aims at truths independent of what the domain of individuals might be. However, there has to be one 'real' space of possible worlds, just as there seems to be one 'real' domain of individuals. To expose the problem from this side, let us return to the point of departure of the model theory for modal logic from that of the non-modal one. We have concluded that we have to interpret propositional formulas by means of elements of a set P which has to be "richer" than 6
The "modal" interpretation of a classical binary logical operator is then that function which to any two propositions (i.e. functions from L to {T,F}) P and P' assigns such a proposition P'' that, for any element l of L, P''(l) equals the "extensional" interpretation of the operator applied to the truth values P(l) and P'(l).
7
Kripke introduces a binary relation over L, called the relation of accessibility, and he interprets modal operators as functions assigning to a subclass L' of L the subclass of all elements of L accessible from L'.
8
Stalnaker (1986) writes: "What is a possible world? It is not a particular kind of thing or place; it is what truth is relative to, what it is the point of rational activities such as deliberation, communication and inquiry to distinguish between."
12
the set of the two truth values. The reason is that some formulas coinciding in truth values are not intersubstitutive within the modal calculus and thus have to be interpreted by different items. Formulas which surely are intersubstitutive even in this calculus are those the equivalence of which is derivable, i.e. which have necessarily the same truth value; such necessarily equivalent formulas thus need not be interpreted differently. This indicates that it would be possible to identify propositions with the classes of necessarily equivalent propositional formulas. Explicating propositions in this way we can see that the class P constitutes a Boolean algebra. Indeed, if we consider logical entailment as inclusion, then all axioms of Boolean algebra are fulfilled (see Rieger, 1967); the supremum of two propositions is their disjunction9 and their infimum is their conjunction. The well known theorem of Stone states that any Boolean algebra is isomorphic with the Boolean algebra of subsets of a set, in the typical case of the set of the atoms of the original algebra. An atom is in our case a proposition which is not inconsistent and from which any other proposition or its negation follows. In this way a natural transfer from propositions to subclasses of a set L can be achieved; just as required by the Kripkean strategy. The elements of L, i.e. of the Boolean algebra of propositions, then do really have the capacity of sorting propositions into true and false: we have observed that truth or falsity of any proposition is, indeed, implied by any possible world. Atoms are in fact infima of some "maximal" classes of propositions, and as infima are conjunctions, possible worlds are conjunctions of certain maximal lists of propositions. It seems to be reasonable to assume that for every maximal consistent class of propositions there is a corresponding possible world and vice versa. Here we have come close to the way in which the model theory for the modal calculi has been approached by Hintikka (1969). Let us call the resulting notion of a possible world Hintikkian. Now we can give an alternative answer to the question of the nature of possible worlds: (Hintikkian) possible worlds are maximal consistent classes of propositions.10 In contrast to the Kripkean definition, this definition is explicit and it allows us to reduce the concept of a possible world to the concept of proposition, which in turn can be reduced to the quite transparent concept of an equivalence class of propositional formulas. This seems to deprive the concept of a possible world of whatever mysteries might be connected with it.
4 Are possible worlds POSSIBLE WORLDS? So far we have discussed the definition(s) of the concept of possible world in terms of formal logic. If we now want to conclude that possible worlds do belong to logic, we have to show something more: that what has been defined within logic under the name of possible world really corresponds to what we normally understand under this name. Let us write POSSIBLE WORLDS for possible worlds in the intuitive sense of the expression, 9
If [F] and [F'] are two equivalence classes of propositional formulas, then [F&F'] is their conjunction. It is easy to prove that this is a correct definition; similarly for the other operators. This means that we can speak about conjunction, disjunction, negation etc. of propositions.
10
The case of truth value gaps is left out of considerations here; however, under suitable restrictions this case need not be incompatible with the present view.
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not for the possible worlds of formal logic. Let us investigate into the correspondence between POSSIBLE WORLDS and classes of statements of our natural language. Let us associate with any POSSIBLE WORLD the class of all and only statements which hold in this world. What can be said about such a correspondence? First, let us recall that for me to posit two distinct POSSIBLE WORLDS makes sense only when I am able to tell one from the other. This leads to the following principle THE PRINCIPLE OF DISCERNIBILITY For any two distinct POSSIBLE WORLDS there exists a statement which is true in one of them and false in the other. From this principle it follows that the mapping of possible worlds onto the classes of statements true in them is injective. From the other side, whatever is possible, holds in some POSSIBLE WORLD. This gives THE PRINCIPLE OF ACTUALIZATION OF POSSIBILITIES Any consistent class of statements (i.e. any class of statements which is imaginable as true simultaneously) is true in some POSSIBLE WORLD. Further we surely cannot refute THE PRINCIPLE OF CONSISTENCY The class of all statements which are true in a POSSIBLE WORLD is consistent. Finally, a POSSIBLE WORLD is a world, it is maximal, it cannot be a proper part of another world.11 Thus we can formulate THE PRINCIPLE OF MAXIMALITY If a statement is consistent with what holds in a POSSIBLE WORLD, then the statement holds in the POSSIBLE WORLD. The four principles clearly guarantee that there obtains a bijective correspondence between POSSIBLE WORLDS and maximal consistent classes of statements of our language. The notions of consistency and maximality now, of course, cannot be defined formally; however, they possess a clear sense: a class of statements is consistent if all the statements can be true simultaneously, and it is maximal if adding any statement to it makes it inconsistent. From these four principles which seem to be indisputable it now follows that a possible world in the intuitive sense can be explicated as a maximal consistent class of statements. This shows that (Hintikkian) possible worlds may serve as good explications for POSSIBLE WORLDS.
11
It is, of course, possible to work with "partial possible worlds", but we prefer to call these situations or something like that.
14
5 Possible Worlds vs. Classes of Statements The first outcome of our identification of possible worlds12 with maximal consistent classes of statements is that some possible mysteries may be swept away from the concept in this way. This is not to say that all problems regarding possible worlds vanish; however, they are no more mysterious. We cannot say for sure if there is a possible world in which there is a round square; however, now we know that this problem can be reduced to the problem if the sentence there is a round rectangle may be true and that our indecision regarding it arises due to an inner obscurity of the statement and not due to an obscurity of our notion of a possible world. We are not sure if there is a POSSIBLE WORLD in which the fifth Euclid's postulate is false; however, we now know that this is the problem of our uncertainty of whether the axiom might or might not be false. The possible worlds actualize possibilities; and they do it in a most straightforward and transparent way. If we were able to decide perfectly which statements are consistent and which are not, possible worlds would be specified without any vagueness. However, as we are vague with respect to consistency, also possible worlds reflect this vagueness. Pythagoras' theorem states a quite precise relation between the sides of a triangle; however, if we measure the two sides with some vagueness, we cannot expect that the third one will be computed precisely. Another outcome is that we can reveal possible worlds even where they are not declared as such. For this it is essential to realize that what follows from our analysis is not that possible worlds are reducible to classes of propositions, but rather that they are such classes, and any class meeting the conditions stated above is a possible world13. In this way we have gained a substantial criterion for marking approaches to semantics as possible-worlds-based, a criterion which is independent of what proponents of these approaches declare.14 Let us take as an example the logical system presented Bealer and Mönnich (1988). It is a system the interpretation of which is based on model structures which the authors specify as follows: "Omitting certain details for heuristic purposes, we may characterize an algebraic model structure as a structure containing (i) a domain D comprised of (items playing the role of) individuals, propositions, properties, and relations, (ii) a set K of functions which tell us the actual and possible extensions of the items in D, and (iii) various fundamental logical operations on the items in D" (p.215). Then they describe the basic difference between this kind of model structures and the more traditional ones: "In a possible-worlds model structure, ..., (i) is typically replaced by a domain consisting of actual individuals and 'nonactual individuals'; then PRP-surrogates are constructed from these items by means of set-theoretical operations; and (ii) and (iii) are omitted" (p.216). Now I agree that there is a substantial difference between the two kinds of model structures, and I also agree that the employment of the algebraic model structures 12
As we have concluded that POSSIBLE WORLDS coincide with possible worlds, there is no more need of capitalization.
13
This is what Wittgenstein (1921) says: "The world is a collection of facts."
14
Such an independence seems to be quite important: I think that if one presents a theory equivalent to the set theory, then he does use sets even if he happens to call the members of the universe of his theory, say, lattices. Or, to offer a more metaphorical but a more lively example, an absolutist society remains absolutist no matter how intensely it may declare itself democratic.
15
is really superior to the more traditional approaches; however, what I do not agree with is that this difference or superiority has something to do with an absence of possible worlds. In fact, they are not absent within Bealer's and Mönnich's approach, as (for all that has been concluded so far) their set K is nothing other than a set of possible worlds. Thus, I think, we can conclude that this approach does not abandon possible worlds.15
6 Possible Worlds as First-order Structures Various explications of the notion of possible world have been historically presented; from Leibnitz' considerations16 to Cresswell's complicated formal metaphysics17. Our results need not be in contradiction with such explications18. However, if what has been said so far is correct, then any acceptable explication of the notion of possible world should be compatible with our notion of such a world as a class of propositions. Let us use an example to show how a concretization of the notion corresponds to positing certain restrictions with respect to the space of propositions. One of the most straightforward approaches to possible worlds is to view them as classes of all distributions of some set of basic properties among a given universe of individuals. This would mean that the space of possible worlds can be considered as the class of all structures of a relational language with a given carrier, i.e. as the class of all models of a first-order theory without special axioms with a fixed universe19. Such a space would be plausible to deal with; therefore this approach is tempting20. We shall not consider the question of adequacy of such a view here; we only want to show what its adoption would mean for our framework. Without going into technical details, it is clear that as under the view in question the class of possible worlds coincides with the class of all structures of a relational language, any possible world is to contain the diagram21 of one and only one structure and the diagram of any structure is to be contained in one and only one world. This, roughly speaking, means that there exists a class P of "protocol statements" such that22 15
What it does abandon is the reduction of meanings to possible-worlds based functions, and this is what the ingenuity of the approach consists in.
16
See Mates (1968).
17
Cresswell (1973).
18
As Stalnaker (1986) claims, "it is not part of any widely shared conception of possible worlds semantics that possible worlds are indefinable or unanalyzable."
19
Thus logical truth, i.e. truth in all possible worlds is closely connected in such a framework with first-order logical validity.
20
The approach is not only promising from the point of view of knowledge representation; also, e.g., a thorough ontological analysis of Tichý (1988) leads to a similar view.
21
I.e. the class of all first-order propositional formulas without quantifiers holding in the model structure.
22
The situation is quite simple if any individual is supposed to have a name; in the opposite case everything is much more complicated; however, these are technical complications, by the treating of which we would not like to obscure the main idea.
16
(i) any statement of P is of an atomic structure, i.e. it is a statement of a certain relation holding among certain individuals; (ii) the truth values of all statements are fixed by fixing the truth values of the statements of P; (iii) the statements of P are independent, i.e. each of them can be true or false independently of the truth values of other statements; and (iv) if a statement belongs to P, then a statement which arises from the substitution of a subject term for another subject term in P also belongs to P. I do not say that these four conditions are not fulfilled in reality, but I insist that we cannot simply assume that they are fulfilled without a thorough discussion. In fact the discussion has been started by Carnap, Schlick and the other logical positivists, and it does not seem to have yielded a definite conclusion.
7 Possible Worlds and Beliefs Probably the most tricky problem the possible world semantics has to face is the interpretation of belief sentences. The problem reappears in writings of nearly anyone who inquires into the semantics of natural language.23 The problem is that while the sentences One plus one equals two. Every first-order theory which has a model has an at most denumerable model.
(1) (2)
are both true under any imaginable circumstances, the statements John believes that one plus one equals two. (3) John believes that any first-order theory which has a model has an at most denumerable model. (4) may well differ in truth value. As it seems to be clear, two statements which are not intersubstitutive salva veritate cannot coincide in meaning24, (1) and (2) differ in meaning, i.e. express different propositions. However, if we then apply the conclusions of the previous paragraphs, we come to the conclusion that there has to exist a possible world in which (1) and (2) differ in truth value (there has to exist an atom of the corresponding Boolean algebra of propositions which belongs to one of them and does not belong to the other). So do there exist possible worlds in which some truths of mathematics do not hold? The answer to this question is not at all simple. A positive answer to the question seems to be in contradiction with our Principle of Consistency. However, should to be consistent be understood as to be imaginable as true, then either the negation of (2) would have to be consistent, or (4) could not be false. The point is that if belief sentences are accepted into the 23
For a survey see Bäuerle and Cresswell (1989). Cf. also Peregrin (1987).
24
Elsewhere (Peregrin, 1992b) I have called this fact the Principle of Verifoundation; Bäurle and Cresswell (1989) call it the Most Certain Principle.
17
range of our investigation, then in fact any subject we are able to speak about is granted the power to establish a distinction between the meanings of two statements. This leads to the conclusion that the possible worlds which arise in this way are something of the kind of Hintikka's (1978) epistemically possible worlds.25 However, it is important to realize that the existence of such epistemically possible worlds is not a conclusion of metaphysical considerations; simply if we insist on calling the atoms of the Boolean algebra of propositions possible worlds, then the possibility mentioned cannot be logical possibility. Elsewhere (Peregrin, 1988) we have argued for a "gradual" view of meaning in correspondence to what kinds of contexts are included in semantic analysis. Four kinds of contexts were distinguished: (i) extensional contexts; (ii) intensional contexts; (iii) contexts involving propositional attitudes; and (iv) quotational contexts. To this inclusive chain of four kinds of contexts there correspond four semantic correlates of sentences ("levels of meaning") in a natural way: (i) extension (sentences denote truth values, predicates denote classes of individuals); (ii) intension (sentences denote classes of possible worlds, predicates denote functions from possible worlds to classes of individuals); (iii) "hyperintensions" (disambiguated correlates of sentences such as Lewis', 1972, structured meanings, Barwise and Perry's, 1983, situations-based meanings or tectogrammatical representations of Sgall et al., 1986); and (iv) surface shapes. I think that here we have another fact corroborating this view: if we consider the algebras of propositions corresponding to these contexts and their atoms we can see that we have: (i) a single element (the "actual world") for extensional contexts26; (ii) logically possible worlds in intensional contexts; (iii) epistemically possible worlds in contexts involving propositional attitudes. As for quotational contexts, for them also the outer form of statements comes into consideration; this means that corresponding atoms become something like "inventories of sentential expressions".
8 Conclusion Possible worlds are, as far as semantics is concerned, primarily mere indices relative to which propositions have truth values. However, if we accept the principles of Section 4 (which I consider indispensable), then we see that there is an intrinsic interconnection between possible worlds and propositions, namely that a possible world can be identified with a maximal consistent class of propositions. Any asserted structure of possible worlds and/or their space means certain structure of propositions and/or their consistent classes. For example, to say that possible worlds consist of individuals and relations is to say that atomic propositions have the predicate-terms structure; and to say that there is a one-to-one correspondence between possible worlds and first-order structures is to say that atomic propositions are independent. It is, of course, more usual to go the other way round and to identify a proposition with a class of possible worlds. Such identification is surely useful for various purposes, but is not useful for the explication of either the concept of possible world (which it takes as primitive), or 25
Cf. also an interesting essay of Kroy (1976), in which the author argues for an identification of possible worlds and "products of imagination".
26
The Boolean algebra of propositions contains in this case two elements ("truth" and "falsity"), and hence is isomorphic to the Boolean algebra of the power set of a set with a single element.
18
the concept of proposition (which is reduced to the problematical concept of possible world). The viewpoint proposed here hints, on the other hand, at a real explication: the concept of possible world is reduced to the concepts of proposition and consistency, whereas that of proposition is in turn reduced to the concepts of sentence and logical equivalence. As the concepts of consistency and logical equivalence are definable in terms of (logical) consequence, the terms we have to take as basic and unanalyzed are sentence and consequence; and this seems to be really the concepts which are given to us quite directly27. Someone might object that my conclusion implies that possible worlds are languagedependent. However, I am convinced that this is indeed the case: there is no absolute, unbiased reality structured independently of any language, there is, as Goodman (1960) puts it, no "innocent eye"28. Semantics is no metaphysics; but I do not believe that there is a metaphysics beyond semantics.
27
This indicates that, contrary to the usual opinion, I believe that model-theory should be reduced to proof theory, not vice versa. For the general grounds for such a belief see Peregrin (1992a,b).
28
See Peregrin (1992b).
19
References Barwise, J., Perry, J.(1983): Situations and Attitudes, MIT Press, Cambridge (Massachusetts). Bäuerle, and Cresswell, M.J.(1989): Propositional Attitudes, in: Handbook of Philosophical Logic, ed. by D.Gabbay and F.Guenthner, Reidel, Dordrecht. Bealer, G. and Mönnich, U.(1989): Property Theories, in: Handbook of Philosophical Logic, ed. by D.Gabbay and F.Guenthner, Reidel, Dordrecht. Carnap, R.(1950): The Logical Foundations of Probability, Univ. of Chicago Press. Church, A.(1940): A Formulation of the Simple Theory of Types, Jour. of Symb.Logic 5, 56-68. Cresswell, M.J.(1973): Logic and Languages, Meuthen, London. Goodman, N.(1960): The Way the World is, The Review of Metaphysics 14. Henkin, L.(1950): Completeness in the Theory of Types, Jour. of Symb. Logic 15, 81-91. Hintikka, J.(1969): Models for Modalities, Reidel, Dordrecht. Hintikka, J.(1978): Impossible Possible Worlds Vindicated, in: Game-Theoretical Semantics, ed. by E.Saarinen, Reidel, Dordrecht. Kemeny, J.G.(1948): Models of Logical Systems, Jour. of Symb. Logic 13. Kripke, S.(1963): Semantical Considerations on Modal Logic, Acta Philosophica Fennica XVI, 83-94. Kroy, M.(1976): Mentalism and Modal Logic, Athenaion, Wiesbaden, 1976. Leblanc, H.(1976): Truth-value Semantics, North Holland, Amsterdam. Mates, B.(1968): Leibnitz on Possible worlds, in: Logic, Methodology and Philosophy of Science III, North Holland, Amsterdam. Lewis, D.K.(1970): General Semantics, Synthese 22, 18-67; reprinted in: Semantics of Natural Language (ed. by D.Davidson and G.Harman), Dordrecht, 1972). Materna, P.(1983): On Understanding and Believing, Prague Studies in Math.Linguistics 8. Montague, R.(1974): Formal Philosophy: Selected Papers of R.M. (ed. by R.Thomason), Yale Univ.Press. Partee, B.H.(1980): Montague Grammar, Mental Representations, and Reality, in: Philosophy and Grammar (ed. by S.Kanger and S.Oehman), Reidel, Dordrecht. Partee, B.(1989): Possible Worlds in Humanities, Arts and Sciences, Proceedings of Nobel Symposium 65 (ed. by S.Allen), de Gruyter, Berlin/New York, pp. 93-123. Peregrin, J.(1987): A Contribution to the Theory of Propositional Attitudes, Prague Bulletin of Mathematical Linguistics 48, 13-36. Peregrin, J.(1988): Intersubstitutivity Scales and Intension, in: Proceedings of the '87 Debrecen Symposium on Logic and Language (ed. by I.Ruzsa and A.Szabolczi), Budapest. Peregrin, J.(1992a): Meaning, Truth and Models, From the Logical Point of View 2/92, 67-75. Peregrin, J.(1992b): Words and Worlds, Studies in Theoretical and Computational Linguistics, Charles University, Prague. Putnam, H.(1975): The Meaning of "Meaning", in: Language, Mind and Knowledge (ed. by K.Gunderson), Univ. of Minnesota Press, Mineapolis. Quine, W.V.O.(1960): Word and Object, M.I.T.Press. Rieger, L.(1967): Algebraic Methods of Mathematical Logic, Academia, Prague. Rieger, L.(1967): Algebraic Methods of Mathematical Logic, Academia, Prague. Sgall, P., Hajièová, E., Panevová, J.(1986): The Meaning of the Sentence in its Semantic and Pragmatic Aspects, Academia, Prague.
20
Stalnaker, R.(1986): Possible Worlds and Situations, Journal of Philosophical Logic 15, 109-123. Tichý, P.(1980): Logic of Temporal Discourse, Linguistics and Philosophy 3, 313-369. Tichý, P.(1988): Foundations of Fregean Logic, Springer. Wittgenstein, L.(1921): Logisch-Philosophische Abhandlung. Tractatus logico-philosophicus, Annalen der Naturphilosophie 14; printed in Werkausgabe, Band 1, Suhrkamp, Frankfurt am Main, 1969.
21
JAROSLAV PEREGRIN
INTERPRETING FORMAL LOGIC* ABSTRACT. The concept of semantic interpretation is a source of chronic confusion: the introduction of a notion of interpretation can be the result of several quite different kinds of considerations. Interpretation can be understood in at least three ways: as a process of "dis-abstraction" of formulas, as a technical tool for the sake of characterizing truth, or as a reconstruction of meaning-assignment. However essentially different these motifs are and however properly they must be kept apart, they can all be brought to one and the same notion of interpretation: to the notion of a compositional evaluation of expressions inducing a "possible" distribution of truth values among statements.
1 WHAT IS A SEMANTIC INTERPRETATION? The concept of semantic interpretation might seem quite unproblematic. Expressions of our natural language stand for a kind of objects; therefore, if we make a logical formalization of language, we should make the expressions of the formal language also stand for something. Semantic interpretation is then what establishes the link between the formulas and what they stand for. According to this view, semantic interpretation is a formal imitation of the real denotandum/denotatum relation. However, such a view, although accepted by many theoreticians of language, is utterly naive; it rests on the identification of language with a kind of nomenclature. We assume that expressions are sort of labels which we attach to pre-existing, real-world objects to make the objects capable of being referred to. Accordingly, our world is a great museum, the exhibits of which are waiting to be classified and named by us; therefore Quine (1969) calls this view the museum myth. However, there are also other meanings in which we use the term interpretation. Even if we disregard the sense which underlies the enterprise of hermeneutics, there remain at least two other meanings, both of which are essential for formal logic and for the logical analysis of language. In one of these meanings interpretation is an assignment of concrete instances to items of an abstract formal system, typically an assignment of concrete language expressions to abstract formulas. Then there is the technical sense of interpretation used in textbooks of mathematical logic, where interpretation is regarded as a technical means 1 of characterizing truth. Erkenntnis 40: 5-20, 1994.
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However, are these three meanings really essentially different? Some authors appear to feel free to pass from one to another (Tarski's, 1936, introduction of the concept of model is an example of a fluent passage from the second meaning to the third; while Cresswell's, 1973, easygoing switch from semantics to "metaphysics" does not seem to make any distinction between the third and the first meaning, nor do many other contemporary "metaphysical" considerations based on model theory). In order to be able to understand the proper role of the concept of interpretation within the enterprise of formal logic and logical semantics, we must first clarify the basic tenets of logical formalization.
2 TWO MODES OF DOING FORMALIZATION Formal logic is based on the utilization of symbols. Symbols function as substitutes for natural language expressions; it is the utilizations of symbols that help us to ignore the irrelevant aspects of natural language expressions and to point out patterns relevant for consequence. Thus, the symbolization is that which helps us, as Frege (1879, p. IV) put it, "die Bündigkeit einer Schlu_kette auf die sicherste Weise zu prüfen und jede Voraussetzung, die sich unbemerkt einschleichen will, anzuzeigen, damit letztere auf ihren Ursprung untersucht werden könne." However, symbols may be utilized in different ways. It is especially important to distinguish between two quite disparate modes of their employment, between the regimentative mode and the abstractive mode. Employing symbols in the regimentative mode means no more than to disregard irrelevant peculiarities of grammar of natural language. To express the fact that John loves Mary we may use various natural language statements, e.g. John loves Mary, It is John who loves Mary, Mary is loved by John; however, on the level of logical schematization all these ways may boil down to canonical loves(John,Mary), or, if we employ P to represent loves, T1 to represent John and T2 to represent 2 Mary, to P(T1,T2) . Hence regimentation means only the reduction of redundancies in the lexicon and/or grammar of natural language. This means that regimentation is simply a kind of sifting of natural language through the sieve of relevance; what is relevant is unambiguously retained in the resulting formal representation, that which is not, vanishes. Symbols and their concatenations utilized for the purposes of regimentation may be truly called constant: each of these stands constantly for a definite natural language expression or at least for a definite "pattern" common to several synonymous natural language
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expressions. The other mode of employment of symbols - abstraction - is of a different nature. While doing regimentation we do not abandon the level of concrete expressions, abstraction leads to articulating types of expressions. We may use the symbol P to represent an arbitrary binary predicate and the symbols T1 and T2 to represent arbitrary terms; P(T1,T2) then represents every statement which shares the form with John loves Mary. Symbols employed in the abstractive mode may be looked at in varying ways; T1 used as a means of abstraction can be considered once as John, once as Mary, etc. The symbols are thus not constants in the proper sense of the word, they are rather a kind of parameters. In contrast to constant formal expressions, formal expressions containing such parameters shall 3 be called parametric formal expressions ; those parametric formal expressions which represent statements also will be called schemata. Hence a schema does not stand for a concrete natural language statement, it is a mere matrix. If we now look at standard logic, we can distinguish two kinds of symbols. There are symbols that are used unequivocally in the regimentative mode. The examples of these symbols are logical connectives, quantifiers, or the equality sign. Such a symbol as & is surely not meant to be considered once as and and once as or, it is meant to represent a definite way of conjoining statements, a way which is, in the prototypical case, expressed by and. Symbols of this kind are usually called logical constants. The other symbols, called extralogical constants, are ambiguous between constants proper and parameters. Their examples are nonspecific terms or predicates. The term constant T1, for example, can be understood as being a constant proper (representing, e.g., the name John), or it can be understood to be a mere schematic representation of an arbitrary term. This ambiguity then extends to all statements containing extralogical constants: P(T1,T2) may be understood to represent a concrete statement, such as John loves Mary, or it can be understood as an abstract schema amounting to all statements of the relevant form. The reason why we can treat extralogical, in contrast to logical, constants in this way is that properties of statements which are interesting from the point of view of
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logic (especially their behaviour from the viewpoint of consequence) are invariant under the replacement of an extralogical constant by another one. Thus, we can replace T1 by any other term in an instance of consequence without disturbing its "consequencehood" (while we cannot so replace, e.g., & by another logical connective). The fact that it is not always fully clear in which sense some of the symbols of the languages of formal logic are employed is the reason for a profound ambiguity. We may view the predicate calculus (and indeed any other formal calculus) in two ways: the first view, the regimentative view, is that every statement of the calculus is constant, that it stands for a definite natural statement (although when it contains extralogical constants, we need not and do not say for which natural statement it stands); the second view, the abstractive view, is that each statement containing extralogical constants is a mere schema, that it amounts to all those natural language statements that conform to it.
3 INTERPRETATION AS "DIS-ABSTRACTION" The first notion of interpretation we are going to address is based on the abstractive view. If we understand a formula as an abstract schema, then the formula, and any argument or proof including it, covers a multiplicity of individual instances. If we say that, e.g., P(T1,T2) entails ∃x.P(x,T2), then what we say is that John loves Mary entails Someone loves Mary, that Peter hates Jane entails Someone hates Jane, etc. If reasoning about John's loving Mary, I can use the abstract schema understanding P as loves, T1 as John and T2 as Mary, if what I have in mind is Peter's hating Jane, then I may use it understanding P as hates, T1 as Peter and T2 as Jane. In other words, I may interpret P as loves, T1 as John and T2 as Mary, or P as hates, T1 as Peter and T2 as Jane. This leads to the first sense of interpretation, which can be called interpretation as "disabstraction"; dis-abstraction consists in pinning down parameters to constants, it is the assignment of constants to parameters (or constants proper to extralogical constants). Any assignment of constants to parameters induces an assignment of a constant expression to every parametric one, especially an assignment of a constant statement to every schema. Every schema is assigned one of its instances. Such an interpretation represents one of many possible "temporary" systematic identifications of abstract entities (parametric expressions) with their concrete instances (constant expressions).
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This notion of dis-abstraction is thus quite straightforward; however, only up to the point when we try to do justice to our intuitive view that the ultimate instances of abstract formulas are concrete things rather than concrete expressions; that instances of P(T1,T2) are not the statements loves(John, Mary), hates(Peter, Jane), etc., but rather the facts of John's loving Mary, of Peter's hating Jane, etc. But to embrace this intuition seems to be the only way to gain the notion of dis-abstraction which is not trivially dependent on the resources of a particular language: there may clearly be an "obejctual" instance without there being a corresponding substitutional instance; we may imagine that Peter loves someone for whom we have no name. From this point of view it seems to be appropriate to consider interpretation as not an assignment of constants, but rather of things (some of which are denotations of constants and others possibly not). However, this way we face the problem of identifying the definite realworld objects which expressions stand or may stand for; the problem that has been recognized as essentially tricky. How should we investigate the world in order to find out the "objectual" instances of P(T1,T2)? Is John's yesterday's loving Mary an instance other than his loving her today? Is an instance's instantiating the formula itself an instance? To answer these questions by thinking about the world means to do speculative metaphysics; and was it not just this kind of speculative metaphysics that had to be overcome by means of logical analysis of language?
4 INTERPRETATION AS CHARACTERIZATION OF TRUTH An interpretation, as just revealed, maps every schema on a constant statement; and as every constant statement has a truth value, an interpretation induces a mapping of schemata on truth values. If a schema is mapped on truth in this manner, we shall say that it is verified by the interpretation; otherwise we shall say that it is falsified. Now if we switch from the abstractive mode to the regimentative mode, we cannot consider interpretation as a matter of schemata, but rather as a matter of constant statements; interpretation is then something that maps constants on constants, hence constant statements on other constant statements, and consequently constant statements on truth values.
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Let us distinguish between true statements which are true contingently and those which are true necessarily. A statement is true contingently if it can be false; it is true necessarily if its falsity is impossible, if such a 4 falsity is beyond the scope of our imagination . Now, and this is of crucial importance, every necessarily true statement comes out verified by every interpretation. This follows from the way extralogical constants are chosen - they are the expressions which are freely interchangeable without disturbing consequence. Therefore, as consequence and 5 necessary truth are merely two sides of the same coin , they are also freely interchangeable without disturbing necessary truth. This means that no permutation of extralogical constants can turn a necessary truth into a falsity, and therefore no interpretation can falsify a necessary truth. Hence verification by every interpretation is a necessary condition of necessary truth. The idea behind the proposal of using interpretations for the purposes of characterization of necessary truth is to consider verification by every interpretation as not only a necessary, but also a sufficient condition of necessary truth. Such a proposal was clearly articulated by Tarski (1936); 6 so let us call it Tarski's thesis . If we call a statement verified by every interpretation logically true, then Tarski's thesis amounts to the identification of necessary and logical truth, or, better, to the explication of the concept of necessary truth (which is a natural, empirical concept) by means of the concept of logical truth (which is a technical, defined concept). Let us stress that Tarski's proposal can be considered as a purely technical matter, as a general version of the matrix method as proposed (for the propositional calculus) e.g. by Lukasiewicz and Tarski (1930). The basis of this method is to consider certain collection of assignments of objects (prototypically of the two truth values TRUE and FALSE) to statements; some of the assigned objects are in a way distinguished (in the prototypical case it is the value TRUE) and a statement is declared logically true if it is always assigned a distinguished element. The matrix method, as Lukasiewicz and Tarski clearly saw, is simply an alternative to the axiomatic method of defining a formal calculus. However, Tarski immediately realized that if we use the "substitutional" notion of interpretation considered so far, then there is no guarantee that logical truth would always come out identical with necessary truth; there is no reason to be sure that every statement verified by every substitutional interpretation will in fact be a statement intuitively understood as necessarily true. Let us consider loves, John and Mary as
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constants, and let us suppose that these three words are the only constants of our language; in that case every substitutional instance of P(T1,T2) may be true (in the imaginable case when John loves Mary and himself and Mary loves John and herself), and hence loves(John,Mary) may be logically true, although John's love for Mary is surely not a necessary matter. This means that if we accept Tarski's thesis, we are left with a kind of modification of the notion of interpretation: if there are some logical truths which are not necessary, then we have to postulate more interpretations and thus have to reduce the number of logical truths. The fact that loves(John,Mary) is not necessarily true implies (if we take Tarski's thesis for granted) that there must be an interpretation of P(T1,T2) which falsifies it; if there is no such substitutional interpretation, we have to postulate interpretations beyond the substitutional ones. If there is to be an interpretation of P(T1,T2) beyond the four interpretations making it into loves(John, Mary), loves(Mary, John), loves(John, John), loves(Mary, Mary), then there must be either an instance of T1 or T2 other than John and Mary, or an instance of P other than loves. Anyway, there must be instances of parameters beyond expressions; there must be something other than constant expressions on which parameters can be mapped by interpretations. The case of P(T1,T2) can be solved by assuming that besides John and Mary there is a third instance for parametric terms, an instance that leads to the needed falsifying interpretation; for example that there is some X such that if T2 is considered as this X then loves(John,T2) comes out false. What is the nature of this X? It is not a constant term; for we have fixed our language to have John and Mary as the only two terms. We may think of it as of a 'potential' term and assume that to consider it means to consider a potential extension of our language. However, the more straightforward way seems to be to give up the whole idea that term parameters are interpreted by constants, and to consider them as interpreted by some more abstract entities. The entities may be called 7 individuals . In our case we have to assume that we have three individuals: John, Mary and X. The situation is similar in the general case. Whenever a statement is not necessarily true while it is verified by every interpretation, we need more
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interpretations, concretely we need an interpretation which would falsify the statement in question. But we need not make do with adding instances of parametric terms: e.g. in the above example the case is that every interpretation verifies ∃x.∃y.P(x,y) although such a statement as ∃x.∃y.loves(x,y) is not necessarily true, and this can be rectified only by postulating an instance of P other than loves. Concluding we cannot make do with constant predicates as instances, we must introduce new kinds of instances of parametric predicates and call them e.g. relations (properties in the unary case). In this way we reach a wholly new notion of interpretation that is no longer substitutional; it is denotational. However, the passage from substitutional to denotational interpretation is not entirely unproblematic. The point is that every substitutional interpretation induces a mapping of schemata on truth values which makes it possible to talk about verification and falsification in connection with an interpretation and thus to make sense of Tarski's thesis; but if we pass from terms to individuals, from predicates to properties and relations, and from expressions to "abstract" objects in general, then the induction does not work any longer. To make it work we need the interpretations of parts of a whole to add up into an interpretation of the whole; furthermore we need the interpretation of statement somehow to yield a truth value (or directly to be a truth value). For an atomic statement (consisting of a predicate and terms), we need the relation interpreting the predicate together with the individuals interpreting the terms to add up into the interpretation of the statement and this interpretation either to have, or to be a truth value. The most straightforward way to achieve this is, of course, to identify n-ary relations with functions from n-tuples of individuals to truth values and to interpret an atomic statement by the value of the application of the interpretation of its predicate to the interpretations of its terms. The concept of substitutional interpretation was based on the assignment of expressions to expressions; only extralogical constants had to be interpreted. Denotational interpretation is in this respect different: interpretation now means mapping of expressions on extralinguistic objects; hence all constants (logical as well as extralogical) have to be interpreted. Now the distinction between logical and extralogical constants may be considered to consist in the fact that a logical constant is always interpreted by one and the same object, whereas an extralogical constant may be interpreted by whatever object of the corresponding domain. However, it appears that this kind of change is necessary independently
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of the passage from substitution to denotation. The point is that the sharp boundary drawn between logical and extralogical constants is in fact untenable. As Etchemendy (1988) points out, there are statements which consist purely of logical constants and which are, nevertheless, intuitively 8 not necessary . The picture saying that there are expressions whose interpretation is completely fixed, and that there are, on the other hand, expressions the interpretation of which is completely free (within the bounds of the corresponding domain), is oversimplified; there are in fact 9 also expressions whose interpretation is partially fixed and partially free . This fact can be naturally accounted for just by considering an interpretation an assignment of values to all expressions, some expressions (the purely logical ones) always being interpreted in the same way, others (the purely extralogical ones) quite freely, and the remaining ones in some partly limited way.
5 PROVISIONAL SUMMARY There are assignments of truth values to statements which are "possible" and others which are "impossible". An assigment of truth values to statements is in this sense "impossible" iff it violates consequence (consequence can, in fact, be considered as delimitation of the space of 10 "possible" truth valuations) . Thus to say that something is necessarily true is to say that it is verified by every "possible" distribution of truth values. This implies that, as we need all and only necessary truths to come out as verified by every interpretation, that the class of distributions of truth values induced by interpretations should coincide with the class of "possible" distributions, that is, that an interpretation should be defined as inducing one of the "possible" distributions. As we no longer require that values of interpretation be expressions, what we have reached is the following "abstract" notion of interpretation: an interpretation is a compositional assignment of objects to parameters leading to a "possible" distribution of truth values among statements. A kind of a by-product of the new, denotational notion of interpretation is the explication of the obejctual notion of instance encountered in the end of Section 3. We have concluded that the substitutional notion of an instance need not be in accordance with our intuition; that there may be
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objectual instances for which no corresponding substitutional instances obtain. Tarski's thesis has now led us to the needed criterion for the demarcation of the space of instances: there are just enough instances to make logical truth coincide with necessary truth. Tarski's proposal, of course, was based on the inverse perspective: Tarski assumed that it is intuitively clear as to what the instances of a given schema are and that the concept of logical truth can thus be considered as a more or less empirical one. Hence the perspective is the one questioned by Wittgenstein (1984, I.8): "Die Logik ist eine Art von Ultra-Physik, die Beschreibung des 'logischen Baus' der Welt, den wir durch eine Art von Ultra-Erfahrung wahrnehmen (mit dem Verstand etwa)." But we deny (together with Wittgenstein, Quine and others) the possibility of inquiring into the world by-passing language, and hence we consider Tarski's perspective doomed. To know what the instances of a schema are is the same thing as to know what is possible and, therefore, what is necessary; and to know what is necessary is the same thing as to know what is necessarily true. Hence knowing instances does not precede knowing necessary truth; and it is thus more appropriate to consider necessary truth constitutive to the space of instances (and interpretations), not vice versa. This means that the notion of interpretation that we have just reached does justice both to the notion of interpretation as dis-abstraction, and to the notion of interpretation as characterization of truth. The general principles governing this notion of interpretation are as follows: such an interpretation is an assignment of objects (the nature of which is irrelevant) to expressions such that it fulfils three principles. First, the value of the interpretation of a whole is to be "computable" from those of its parts; hence the interpretation is to be compositional. (Note that compositionality is a purely technical matter here - it guarantees that interpretation will be something we shall be able reasonably to work with.) Moreover, the simpler the "computation" of the value of 11 interpretation of a whole from those of its parts, the better . Second, among the values assigned to statements there are some distinguished 12 values . Third, the assignment should be as economic as possible, i.e. the 13 fewer interpreting entities, the better .
6 INTERPRETATION AS MEANING ASSIGNMENT Now we can return to the considerations we have started from, to the notion of interpretation as meaning assignment. We have stated that
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meanings are not to be found by a direct empirical investigation of the world, but we have not rejected the plausibility of considering meaning as 14 an object . The way in which meaning must be approached has been formulated in the frequently quoted sentence due to Lewis (1972, p.173): "In order to say what a meaning is, we may first ask what a meaning does, and then find something that does that." This is not to say that we give up the effort of revealing the right nature of meaning; this is the recognition of the fact that there is really nothing to meaning beyond what meaning does. However, if we try to analyze what we are able definitely to say about meaning, we come to the conclusion that the assignment of meanings to expressions must be something very close to interpretation in the sense of the previous section. Thus, meaning, as we handle it intuitively, also 15 seems to be a compositional matter . Besides this, meaning is what can be called verifounded: difference in truth value clearly implies difference 16 in meaning . Third, there seems to be something like Occam's razor, 17 something that pushes the number of meanings as far down as possible . Meaning assignment can thus be identified with that interpretation (in the above sense) of our language which leads to that distribution of truth values among its statements which really obtains. Thus, it seems that the notion of interpretation reached above can do justice even to the notion of interpretation as meaning assignment. It can be proved that a meaning assignment fulfils the above characterization (compositionality, verifoundation plus Occam's razor) if and only if sameness of meaning coincides with intersubstitutivity salva 18 veritate . This result should not be too surprising: the idea that the meaning of an expression is the contribution of the expression to the truth value of the statements in which it occurs is nothing new, it can be traced back to Frege and Wittgenstein (not to speak of Leibniz). What we have arrived at is especially close to the idea of semantics put forward by Davidson (1984): "I suggest that a theory of truth for a language does, in a minimal but important respect, do what we want, that is, give the meanings of all independently meaningful expressions on the basis of an analysis of their structure." This is precisely what semantic interpretation in the sense spoken of here does: it distinguishes between truth and falsity and it propagates this
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distinction "on the basis of an analysis of structure" of complex expressions (i.e. by doing justice to the principle of compositionality plus the principle of Occam's razor) up to their ultimate parts. The meaning of an expression is thus the simplest thing which allows for a compositional characterization of truth. However, should this mean that meaning is a matter of the actual distribution of truth values among statements and that it thus depends on contingent facts? Not at all; at least not for natural language. Every change in semantic interpretation is conditioned by a change of the truth value of a statement; this, however, in no way means that every change of the truth value of a statement would really bring about a change of semantic interpretation. It may well be the case that the only change of truth value that leads to a change of semantic interpretation is a change of the truth value of a necessary statement; and this is indeed the case of natural language. Natural language is inherently intensional; whereas some artificial languages (for example the traditional predicate calculus), for which semantic interpretation does depend on contingent truth, are extensional. This throws quite a peculiar light on extensional languages, but the reason is simple: with respect to these languages it is simply not possible to speak about anything like meaning worth its name. Intensionality is an essential property of natural language; and extensional languages are from this point of view rather "pseudolanguages": they share some essential features with real languages, but not enough to be on a par with them in respect to meaning. Extensional languages can, however, help us in clarifying another problematic semantic concept, namely that of reference. These languages, such as the classical predicate calculus, are results of the formalization of a certain restricted part of our language, of a part which is in a sense distinguishable from the rest of language. It is a kind of language within 19 language , and it is a plausible hypothesis that what we call reference is just semantic interpretation of this extensional core of our language.
7 CONCLUSION We have tried to show that the diverse senses in which the concept of semantic interpretation is used can be brought to one and the same notion. It is the notion of interpretation as an assignment of entities to expressions compositionally characterizing truth. Thus, a semantic interpretation can be seen as a certain kind of account for truth; as an
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account for which Tarski's idea of identification of necessary truth with verification by every interpretation is crucial. It is, however, necessary truth, i.e. the fact that we understand some statements as not capable of being false, that is basic: hence not "a statement is necessarily true because it is verified by every interpretation", but rather "a statement is necessarily true and therefore there cannot be an interpretation which falsifies it". Formal interpretation can be considered a matter of semantics because it accounts for necessary truth, not because it imitates a real denotandum/denotatum relation. Every theory aims at an explication of some facts which are prior to it. To understand such a theory properly, it is essential to distinguish carefully between that which was here from the beginning and that which we built on top of it in our effort to "make sense" of it. The former can only be described (or in a way "systematized"), the latter can be explained (away) by pointing out its role in the pursuit of the description of the former. It is a basic error to try to explicate the former in the same way as the latter. We may explain the fact that expressions have different meanings by pointing out that they make different contributions to truth values of statements in which they occur, but we can hardly explicate the fact that there are statements which are true, or, using the material mode of speaking, that there is anything that is.
NOTES *
Work on this paper has been supported by the Alexander von Humboldt-Foundation and by the Research Support Scheme of the Central European University. The author thanks P. Sgall for stimulating discussions and P.Stekeler-Weithofer for valuable critical comments to earlier versions of the manuscript. 1. E.g. Robinson (1965). 2. Slight semantic differences - e.g. different felicity conditions - are likely to be found even between John loves Mary and It is John who loves Mary. Hence whether they really do boil down, as they usually do, to a single formula, depends on the threshold of difference we decide to take as significant. 3. It would be more accurate to call them variable formal expressions, but the term variable is traditionally used in a different sense within logic. Note the essential difference between variables and parameters: variables are tools of quantificational theory which are in fact unessential (it is possible - although a bit cumbersome - to do logic without them; see Quine, 1960 and also Peregrin, 1992a).
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4. The boundary between these two kinds of truth is surely fuzzy (witness Quine's Everything green is extended). Moreover, as Wittgenstein and Quine stressed, it is in fact not an absolute matter, it is rather a matter relative to an adopted conceptual framework. However, despite both of these facts the boundary surely is something quite meaningful. 5. S is a necessary truth iff it is entailed by the empty set; and S is entailed by S1,...,Sn iff (S1->(...(Sn->S)...)) is a necessary truth. 6. The idea, however, can be traced back to Bolzano. See Etchemendy (1990). 7. Note however, that such a notion of an individual is not a product of metaphysical speculation, but rather the outcome of our insistence on Tarski's thesis. Let us avoid the usual misguided idea that by giving an object a name we transform it into that which the name usually denotes. That we call what we have invented individuals is to keep within the boundaries of the usual logical terminology, it is not to attribute hands and feet to these entities! 8. A statement consisting of only logical constants is clearly inevitably accounted for as necessary: there are no alternative ways of interpreting it. However, this is appropriate at most in the case of the pure predicate calculus; it is enough to add = to it in the customary manner, and the picture gets distorted, as there emerge statements which consist purely of logical constants, which are, nevertheless, intuitively contingent. This is the case of ∃x∃y¬(x=y) as discussed by Etchemendy. 9. Etchemendy's example implies that this is in fact the case with quantifiers. A simpler example is that of adding the axiom ∀x(P(x)->Q(x)) to the predicate calculus (for some definite predicate constants P and Q). P and Q are then no longer extralogicals: they are not replaceable by any other predicates. On the other hand, they are also not definite in the sense of &: they can be replaced, but only by some pairs of predicates and not by others. 10. To say that S1,...,Sn entail S is to say that it is impossible that S is false and S1,...,Sn are at the same time all true. See Peregrin (1992b). 11. Thus within the classical predicate calculus the interpretation of an atomic statement is not only uniquely determined by the interpretation of the predicate and the interpretation(s) of the term(s), but it is computable simply as the functional application of the former to the latter. The value assigned to a complex statement can again be computed as the application of the interpretation of the connective (which is the usual truth function) to the interpretations of the substatements. 12. In classical logic there is simply one distinguished and one undistinguished value, the case of modal logic, however, shows that a more general approach is appropriate. 13. If it were not for this third requirement, an identical mapping could be considered as an interpretation; and this is clearly absurd. 14. There are outstanding philosophers that would question this assumption - e.g. Austin or Quine. 15. Compositionality is also what underlies the most influential theories of meaning since Frege. In fact if we accept Frege's conviction that what is primarily meaningful are sentences, then we need the principle of compositionality in order to be at all able to individuate meanings of parts of sentences. It is, however, important to realize that understood this way compositionality is not a thesis to be verified or falsified, that it is rather a postulate which is constitutive to semantics. 16. Cresswell (1982) considers this the most certain principle of semantics. 17. "Senses are not to be multiplied beyond necessity", as Grice (1989, 47) puts it. 18. The problem can be formulated in algebraic terms: if we view language as a many-
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sorted algebra (in the spirit of Janssen, 1983), then a meaning assignment fulfils the three conditions iff it is a homomorphism with its kernel equal to the maximal congruence for which it holds that no true statement is congruent with a false one. It is easy to see that it is just this congruence which coincides with the relation of intersubstitutivity salva veritate. 19. Its exceptional status manifests itself in the fact that many of the greatest philosophers of language, such as Frege, Wittgenstein, or Church, tried hard to restrict themselves to this very part. We can hardly ascribe to them an inability to see that there are intensional contexts in our language; rather they clearly considered these contexts as is in some sense secondary to the extensional core.
REFERENCES Barwise, J. and Perry, J.: 1983, Situations and Attitudes, MIT Press, Cambridge (Mass.). Cresswell, M.J.: 1973, Logic and Languages, Methuen, London. Cresswell, M.J.: 1982, 'The Autonomy of Semantics', Processes, Beliefs and Questions (ed. by S.Peters and E.Saarinen), Reidel, Dordrecht. Davidson, D.: 1984, Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Etchemendy, J.: 1988, 'Tarski on Truth and Logical Consequence', Journal of Symbolic Logic 53, pp. 51-79. Etchemendy, J.: 1990, The Concept of Logical Consequence, Harvard University Press, Cambridge/London. Frege, G.: 1879, Begriffsschrift, Nebert, Halle. Goodman, N.: 1960, 'The Way the World is', The Review of Metaphysics 14, pp. 48-56. Grice, P.: 1989, Studies in the Ways of Words, Harvard University Press, Cambridge (Mass.). Janssen, T.M.V.: 1983, Foundations and Applications of Montague Grammar, dissertation, Mathematisch Centrum, Amsterdam. Lewis, D.: 1972, 'General Semantics', Semantics of Natural Language (ed. by D.Davidson and G.Harman), Reidel, Dordrecht. Lukasiewicz, J. and Tarski, A.: 1930, 'Untersuchungen Über den Aussagenkalkül', Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie 23, pp. 30-50. Peregrin, J.: 1992a, 'The Role of Variables in Logical Formalization of Natural Language', Proceedings of the Eight Amsterdam Colloquium, University of Amsterdam, pp. 463-481. Peregrin, J.: 1992b, 'Meaning, Truth and Models', From the Logical Point of View 2/92, pp. 67-75.
20
JAROSLAV PEREGRIN
Quine, W.V.O.: 1960, Variables explained away, Proceedings of the American Philosophical Society 104, pp. 343-347; reprinted in Quine: Selected Logic Papers, Random House, New York, 1966. Quine, W.V.O.: 1963, From the Logical Point of View, Harper and Row, New York. Quine, W.V.O.: 1969, Ontological Relativity and Other Essays, Columbia University Press, New York. Robinson, A.: 1965, Introduction to Model Theory and to the Metamathematics of Algebra, North-Holland, Amsterdam. Tarski, A.: 1936, 'Über den Begriff der logischen Folgerung', Actes du Congrés International de Philosophie Scientifique 7, pp. 1-11. Wittgenstein, L.: 1984, Bemerkungen über die Grundlagen der Mathematik (Werkausgabge Band 6), ed. by G.E.M.Anscombe, R.Rhees and G.H.von Wright, Suhrkamp, Frankfurt am Main.
TOPIC-FOCUS ARTICULATION AS GENERALIZED QUANTIFICATION Jaroslav Peregrin
[In: P. Bosch and R. van der Sandt (eds): Proceedings of “Focus and natural language processing”, 1995, 49-57. Heidelberg: IBM Deutschland; reprinted in: E. Hajičová et al. (eds):Travaux du Cercle Linguistique de Prague, vol 4, Benjamins, Amsterdam, 2002, 263-273]
Abstract Recent results of Partee, Rooth, Krifka and other formal semanticians confirm that topic-focus articulation (TFA) of sentence is relevant for its semantics. The essential import of TFA, which is more apparent in case of a language with relatively free word order such as Czech than in case of English, has been traditionally intensively studied by Czech linguists. In this paper we would like to indicate the possibility of the account for TFA in terms of the theory of generalized quantifiers, drawing on the results of both these groups of theoreticians. The basic intuition which we accept as our point of departure is the intuition of topic as the “semantic subject” and focus as the “semantic predicate”; we point out that the role of topic is to specify the entity the sentence is “about” (thereby triggering a presupposition), while that of the focus is to reveal a characterization of this entity, and usually a characterization that is in some sense exhaustive. Then we show that it may be plausible to consider topic and focus as arguments to an implicit generalized quantifier, which may get overridden by an explicit focalizer.
0. A preliminary remark: symbols in semantics There are two principal ways to employ formal or symbolic means to semantically analyze a sentence (more generally an expression or an utterance). The first of the ways consists in using symbols to picture some aspects of the sentence to emphasize or to point out those aspects of its syntactic structure which we consider relevant loves for its semantics. We may, for example, depict the statement John loves Mary as JOHN MARY to emphasize that John is the agent of the action expressed by loves and Mary its objective; or we may indicate the same by writing [John]Ag loves [Mary]Ob. It is the graphical shape of such a representation that is important: it is just the graphics that does the job of explication. We are free to employ symbols of any kind, the only point is to emphasise the relevant, and to suppress the irrelevant, features of the sentence analyzed. 1
The other way to use formal means is to do logical analysis, and its nature is essentially different. Its primary aim is not to depict features of the sentence’s structure, but rather to point out the proposition which is to explicate the meaning of the sentence (and to point out this proposition means – in effect – to point out the truth conditions of the sentence, or the contribution the sentence brings to truth conditions of suprasentential wholes, i.e. the “context-change potential” of the sentence). We again use symbolic means: we may, for example use the formula LOVES(JOHN,MARY) of the first-order predicate calculus to schematize John loves Mary. However, now it is not the graphical shape of the symbols which is relevant – we know that the formula we have used is equivalent to many other formulas and it would have been the same if we have used any of them – the proposition pointed out would be the same. The basic difference between the two approaches is that the latter works with closed systems of symbols with strictly defined equivalence – in their case, it is not the formula itself that does the explication, but rather the proposition pointed out by it. And it is also only this approach that facilitates semantic analysis in the strict sense (cf. Lewis, 1972).
1. Topic and Focus within the Prague Tradition Contemporary semantic theories are inclined to view focus as something extraordinary, as something which a sentence may, or may not, have, and which becomes semantically relevant only when it gets combined with a focus-sensitive operator. The Praguian approach, on the other hand, considers TFA as the universal basis of semantic structuring of sentence. Let us briefly recapitulate its basic tenets – for a more detailed exposition see Sgall, Hajičová and Panevová (1986) and also Sgall (1995) and Hajičová (1995). The syntactic characterization of sentence underlying this approach works with the concept of dependency. The default order of items depending on a verb (i.e. the order of thematic roles and adverbials) is considered to be fixed for a given language; it is called the systemic ordering. However, this order together with the order of other items not depending directly on the main verb is modified in a concrete utterance, so that the resulting “deep” order is that of the communicative dynamism (CD). The CD order of contextually bound items dependent on the same head is determined by the speaker’s discourse strategy rather than by grammar; on the other hand, the CD of the unbound items dependent on the same head is in accordance with the systemic ordering. An item is less dynamic than its head iff the dependent item is bound. The least dynamic element of the sentence constitutes the topic proper. All the contextually bound items depending on the main verb together with all that depends on them and together with the main verb if this is contextually bound, constitute the topic of the sentence; the rest of the 2
sentence constitutes the focus. Hence the topic/focus classification is exhaustive: every element of the sentence belongs either to the topic or to the focus.
2. Topic & Focus as “Subject” & “Predicate” The basic idea is that the topic-focus articulation has to do with the dynamism of discourse. A linguistic utterance says – in the prototypical case – something new about something old. There is a subject which the sentence is about, and there is a predicate which is ascribed to the subject. The utterer first points out an element of the common ground shared by the participants of the discourse and then reveals something which he proposed to be added to the common ground. If his utterance is successful, then the new item is added and it can become the starting point of a following utterance. In the prototypical case, this semantic notion of subject and predicate coincides with the syntactic notion – the grammatical subject refers to what the sentence is about, and the grammatical predicate expresses what is said about it. However, in general almost any syntactic part of sentence can act as the semantic subject, leaving the rest of the sentence for the role of semantic predicate. Thus, while under normal condition the sentence (1) expresses the ascription of the property of walking to the individual John, its variant (2) (with the stress on John) seems rather to express something like the ascription of the property “being instantiated by John” to the “entity” walking. John walks
(1)
JOHN walks
(2)
An instructive way to see topic and focus is to see them just as “semantic subject” and “semantic predicate”. Topic is that which the sentence is about, by which it gets anchored to the common ground, and focus brings about the very information the sentence was assembled to convey. What does this mean for logical analysis? The straightforward way to analyze (1) is (1’); and it seems that to account for the fact that in (2) the roles of John and Walks get exchanged it is enough to employ the mechanism of lambda-abstraction and to use the well-known Montagovian trick of type-raising. (2) thus seems to be analyzable as (2’). Walk(John)
(1’)
λf.f(John)(Walks)
(2’)
3
However, this idea is misguided, because it disregards the very sense of logical analysis. The point is that (1’) and (2’) are equivalent formulas ((2’) is straightforwardly lambda-reducible to (1’)), and hence (2’) fosters precisely the same analysis as (1’). To analyze something as a “semantic subject” it is not enough to provide a formula in which it corresponds to the subject – because we can always provide another, equivalent formula in which it would correspond to the predicate. This means that if we want to speak about “semantic subject” and “semantic predicate”, we must understand these terms as expressing two different ways of contributing to truth conditions. We cannot simply say that the contribution of the predicate differs from that of the subject in that the truth conditions of the whole sentence result from applying the former to the latter; because we saw it is easy to provide an alternative analysisŘeknu-li někomu například ". The real semantic difference can be seen as stemming from the fact that the subject purports to point out something known. If it fails to do so, the sentence is not false, but rather infelicitous, because it makes the listener wonder what the speaker is talking about. On the other hand, if the subject succeeds in pointing out a known anchor and the predicate fails to be true, the sentence is simply false. In other words, the subject is the matter of presupposing, whereas the predicate is the matter of stating. What kind of logic do we need to account for this?. First, we need a logic that is partial: its formulas can posses not only the values T and F, but can also have no truth value at all (i.e. they can be not only true or false, but also “infelicitous”). Then we can associate define what can be called the “propositional content” of an expression (a propositional content of a statement is the statement itself, that of a term is the proposition claiming the existence of the corresponding individual and that of a predicate is the proposition claiming the non-emptiness of the corresponding class); and we can define “presuppositional” predication as predication posing the presupposition associated with the subject. Let us do it – for the sake of simplicity – on the level of extension (everything can be straightforwardly relativized to possible worlds); thus, ║X║ will denote the extension of X (i.e a truth value if X is a sentence). Definition 1. (“propositional content” of X) │X│ = ║X║ if X is a sentence = ║∃y.y=X║ if X is a term = ║∃y.X(y)║ if X is an unary predicate = T otherwise Definition 2. (“presuppositional” predication) ║P{S}║ = T iff (│S│=T & ║P(S)║=T) = F iff (│S│=T & ║P(S)║=F) = 0 iff │S│=F 4
Only now we can give the analyses of (1) and (2) in spirit of the above considerations – viz (1”) and (2”). Walk{John}
(1”)
λf.f(John){Walk}
(2”)
These two formulas now express different propositions: although there are no circumstances under which one of them would be true and the other false, there are circumstances under which one is false and the other lacks a truth value altogether (if there is no John, then (2”) yields F while (1”) yields 0; if nobody walks then it is the other way round). If we take the sentence John loves Mary, there are formally nine straightforward ways to draw the boundary between topic and focus; they lead to the analyzes (3a) through (3h) (where x, y are variables for individuals, p for properties of individuals, r for binary relations among individuals, f 1 for properties of propositions and g for properties of properties of propositions) . John loves Mary
(3)
λy.love(y,Mary){John}
(3a)
λx.love(John,x){Mary}
(3b)
λp.p(Mary){λy.love(John,y)}
(3c)
λp.p(John){λx.love(x,Mary)}
(3d)
(λf.f){love(John,Mary)}
(3e)
λg.g(love(John,Mary)){λf.f}
(3f)
λr.r(John,Mary){love}
(3g)
λx,y.love(x,y){John,Mary}
(3h)
Out of these, (3e), in which the whole sentence is contained in the topic, appears to be ruled out – a sentence seems to have to bring in something new. All the other ways of TFA are acceptable. Moreover, there are other possibilities to draw the boundary, non-aligned to the boundary of words. Thus, to respond to the statement John loved Mary by stating John LOVES Mary means to make 2 only the present tense grammateme of the verb into the focus . What is important is that in case of sentences with a generic noun phrase the subject-predicate structuring that is yielded by the TFA determines the scope and thus can mean differences not only regarding felicity conditions, but also regarding truth conditions proper, like in case of (4). 5
Every man loves one woman
(4)
λP.P(λx.∃!y.(woman(y)&love(x,y))){λp.∀x.(man(x)→p(x))}
(4a)
λP.P(λy.∀x.(man(x)→love(x,y))){λp.∃!y.(woman(y)&p(y))}
(4b)
λp.∃!y.(woman(y)&p(y)){λy.∀x.(man(x)→love(x,y))}
(4c)
λp.∀x.(man(x)→p(x)){λx.∃!y.(woman(y)&love(x,y))}
(4d)
Besides this, there is another semantically relevant feature of TFA. The focus not only predicates something of the topic; it appears to tend to exhaust all what there is to be predicated about the topic. Uttering (2), we not only state that walking is instantiated by John, but also that John is the only (or at least “the principal”) walker. (2) would be inappropriate in the situation where John would be only one of a plenty of those who walk around. To be able to account for the phenomenon of exhaustiveness, we have to introduce the concept 3 of alternative. We assume that for every entity there is a class of alternatives (of the same type), which count as its alternatives. We can introduce a functor ALT such that if e is an entity of the type t, then ALT(e) is a set of entities of type t (containing e). The trivial version of ALT would be such that ALT(e) would be the set of all the entities of the type t. In fact, it is clear that the nature of ALT is pragmatic – what counts as an alternative to a given item and what not is given by the context; and from this point of view it seems to be reasonable to see ALT as something which gets changed by ongoing utterances. This means that a logic which incorporates an adequate version of ALT would have to be dynamic – changing ALT in the way dynamic logic of Groenendijk and Stokhof (1991) changes valuation of discourse markers. Anyway, once we have ALT, we can define the “exhaustive” predication as predication such that the predicate has no alternative which would also hold of the subject, i.e. as follows (where ║X║I means the extension of X under the interpretation I and I[y→Y] denotes the interpretation which is just like I save for the single exception that I(Y)=y. Definition 3. (“exhaustive” predication) ║P!(S)║I =T iff ║P(S)║I =T & ∀p.[p∈ALT(║P║I)&║P(S)║I[p→P]=T→║P║I =p] Combining definitions 2 and 3 we arrive at the concept of “exhaustive presuppositional” predication.
6
Definition 4. (“exhaustive presuppositional” predication) ║P!{S}║I = T iff │S│I = T & ║P(S)║I = T & ∀p.[(p∈ALT(║P║I)&║P(S)║I[p→P] = T)→║P║I = p] = F iff │S│I = T & (║P(S)║I = F v ∃p.[p∈ALT(║P║I)&║P(S)║I[p→P] = T&║P║I ≠ p])) = 0 iff │S│I = F Now we can analyze (3) as (3a’) through (3h’). λy.love(y,Mary)!{John}
(3a’)
λx.love(John,x)!{Mary}
(3b’)
λp.p(Mary)!{λy.love(John,y)}
(3c’)
λp.p(John)!{λx.love(x,Mary)}
(3d’)
(λf.f)!{love(John,Mary)}
(3e’)
λg.g(love(John,Mary))!{λf.f}
(3f’)
λr.r(John,Mary)!{love}
(3g’)
λx,y.love(x,y)!{John,Mary}
(3h’)
(3a’) through (3h’) differ not only in their felicity conditions, but they may also differ in truth conditions (this depends on how we interpret the operator ALT). If both John and Tom love Mary and if Tom counts as an alternative to John, then (3d’) is false, whereas (3c’) may well be true (in the case when Mary is the only person loved by John). However, it seems not to be adequate to take TFA to always amount to this exhaustive version of predication. The extent to which the focus of a given utterance really purports to exhaust all what there is to be said about the topic (relative to context) seems to vary (the heavy stress on focus usually strongly facilitating exhaustiveness), and thus the TFA should be considered to be somewhere in between simple presuppositional and exhaustive presuppositional predication.
3. Introducing the generalized quantifier PRED The linguistic significance of the theory of generalized quantifiers, as elaborated by van Benthem and others, bears on the idea that the truth value of a sentence can be seen as the matter of a pair of 4 sets being in certain relation. This theory seems to be in accordance with certain intuitions regarding topic and focus, namely with the intuition that “topic specifies a class and focus lists the 7
elements of the class”. This is in fact an idea underlying the first attempts of semantic capturing of the Prague notion of topic and focus within the framework of intensional logic (Materna and Sgall, 1980, Materna, Hajičová and Sgall, 1987). This approach is also closely allied to that based on the concept of tripartite structure as proposed by Partee (1995). There is a straightforward way from the analysis by means of strong presuppositional predication given above to the analysis by means of a generalized quantifier. Let us introduce the (type-polymorphic) generalized quantifier PRED in the following way. Let us assume that we have a sentence analyzed as P!{S}; we shall show how to turn this analysis into the new one. We shall distinguish two cases. If P is P = λp.p(T) for some T (i.e. if P is a “type-raised T”), like in case (3c’), then we turn P!{S} into PRED(S,λx.x=T); thus we turn (3c’) into (3c”). PRED(λy.love(John,y),λx.x=Mary)
(3c”)
In the other case (when P is not a λp.p(T), e.g. (3a’)), we turn P!{S} into PRED(λp.p(S),P); thus we turn (3a’) into (3a”). PRED(λp.p(John),λy.love(y,Mary))
(3a”)
However, in this new way we can account also for cases which are not so easily analyzable in the old one. Thus, we can straightforwardly analyze (5) as (5’) and (6) as (6’). John loves MARY, MINIE and HILLARY
(5)
PRED(λx.loves(John,x),λx.(x=Mary∨x=Minie∨x=Hillary) John loves those who BILL loves
(5’) (6)
PRED(λx.loves(John,x),λx.loves(Bill,x))
(6’)
What is the character of the generalized quantifier PRED introduced in this way? It can be characterized by three points. (1) PRED(A,B) implies that all Bs are As, hence B⊆A; which accounts for the intuition that “focus lists only members of the topic-class”. (2) PRED(A,B) presupposes nonempty A (if A is empty, then PRED(A,B) is ‘undefined’); which is the formal counterpart of the intuition that “topic assumes knowledge shared by communicants”. Then there is the most controversial point: (3) PRED(A,B) holds only if B exhausts, or ‘almost exhausts’, A (“focus lists all, or the most important, members of the topic-class”). If we took (3) as stating that PRED(A,B) implies that B⊇A, then PRED would turn out to be very close to ONLY. But it would 8
be probably necessary to distinguish at least two versions of PRED: the basic version defined only by means of (1)+(2), and the exhaustive version, PREDexhaust, defined by (1)+(2)+(3). The idea is that PRED is the default, implicit generalized quantifier which can be overridden by 5 an overt focalizer. Thus, (7) is analyzed as (7’), whereas (8) as (8’). John loves only Mary
(7)
ONLY(λx.loves(John,x),λx.x=Mary)
(7’)
John loves only those who BILL loves
(8)
ONLY(λx.loves(John,x),λx.loves(Bill,x))
(8’)
However, this does not mean that a word like only would automatically assume the place of PRED – in cases with the so called “free focus” the quantifier expressed by the focalizer stays embedded within the topic – viz (9’). JOHN loves only Mary
(9)
PRED(λx.ONLY(λx.loves(y,x),λx.x=Mary),λx.x=John)
(9’)
The problem of focalization can thus be seen as the problem of overriding PRED. In some cases, like (7) and (8), it is overriden, in other cases it is not. Partee (1995) points out, that “adverbial” quantifiers are usually focalizers, whereas “determiner quantifiers” are usually not. Negation can be considered as a principal focalizer – as the primary aim of negation seems to be to deny that “the focus holds of the topic” (see Hajičová, 1995), it can be seen as changing PRED into another, complementary quantifier. And possibly also the relationship between PRED and PREDexhaust can be understood in this way: we might see PREDexhaust as a non-default quantifier which may in some cases (especially in cases with the heavy stress on focus) override the default PRED. However, let us once more stress that if what we are after is semantics proper, i.e. truth conditions, then we must be careful about claims to the effect of presence of quantifiers or other similar items within that which the sentence expresses, means or denotes. Quantifiers are constituents of formulas, and logical formulas are, as we have stressed, only ways to point out propositions. The same propositions can be pointed out in various ways, some of them employing a quantifier, others not. Let us consider the sentence (10). Every man walks
(10)
9
In fact, it is sentences of this kind, which oil the wheels of linguist’s embracing the theory of generalized quantifiers. We have seen that the intuition behind generalized quantifiers in linguistics stems from the fact that the truth of the sentence of the shape Det N V can be computed by comparing the extensions of N and V in the way prescribed by Det. Thus, (10) is true if and only if the extension of man is included in the extension of walk. The TFA-based approach proposed in this paper now might seem to detach the tools of the theory of generalized quantifiers from this powerful intuition. However, this is not exactly true, because the analysis we proposed need not contradict the traditional one. Let us, for the sake of illustration, assume the standard TFA of (10), with every man being in the topic and walks in the focus. If we disregard presupposition and the requirement of exhaustiveness, then our analysis becomes equivalent to the standard one. To see this, let us consider the analysis (10’) of (10) yielded by our approach. It is easy to see that (10’) is – under the given conditions – equivalent to (11), which can be straightforwardly reduced to (12). And if we define the generalized quantifier EVERY in the usual way (i.e. EVERY(A,B) iff ║A║⊆║B║), then (12) is further equivalent to (13). PRED(λp.∀x.man(x)→p(x),λp.p=walk)
(10’)
(p=walk)→(∀x.man(x)→p(x))
(11)
∀x.man(x)→walk(x)
(12)
EVERY(man,walk)
(13)
Thus it does not always make sense to ask whether the semantic analysis of a sentence contains this or that quantifier. What does make sense is to ask whether if we decide to employ a quantifier, its operands coincide with topic and focus (i.e. whether the quantifier can be seen as overriding PRED).
4. Conclusion The aim of this paper was to survey the “logic” of topic and focus and to indicate how this “logic” can be done justice to by means of the currently popular formal approaches to semantics. We have claimed that it is the theory of generalized quantifiers, which could provide for a simple and adequate framework to capture TFA. Contemporary semantic theories are inclined to view TFA as something that is relevant only for the scope of focus-sensitive operators. The Praguian approach, on the other hand, considers TFA as the universal basis of semantic structuring of sentence. The approach proposed here can be – I believe – seen as compatible with the approach of Rooth and Krifka – the cases of focus-sensitive 10
operators are considered only special cases of the general pattern. Moreover, the approach can be seen as a step towards an elaboration of Partee’s insight regarding the tripartite structuring of 6 sentence.
References Groenendijk, J. and Stokhof, M.1991. “Dynamic Predicate Logic”. Linguistics and Philosophy 14: 39-101. Hajičová, E 1984. “On Presupposition and Allegation”. In Contributions to Functional Syntax, Semantics and Language Comprehension, P. Sgall (ed.), 99-122. Amsterdam: Benjamins and Praha: Academia. Hajičová, E. 1995. “Topic, focus and negation”. In Proceedings of Focus and natural language processing, P. Bosch and R. van der Sandt (eds.), Heidelberg: IBM Deutschland. Jakobs, J. 1991. “Focus Ambiguities”. In Journal of Semantics 8: 1-6. Krifka, M. 1991. “A Compositional Semantics for Multiple Focus Constructions”. In Proceedings of SALT I, Cornell Working Papers 11. Krifka, M. (ms). “Focus, Quantification and Dynamic Interpretation”. Manuscript. Lewis, D. 1972. “General Semantics”. In Semantics of Natural Language, D. Davidson and G. Harman (eds.). Dordrecht: Reidel. Materna, P. and Sgall, P. 1980. “Functional Sentence perspective, the Question Test, and Intensional Semantics”. In SMIL 1-2: 141-160. Materna, P., Hajičová, E. and Sgall, P. 1987. “Redundant Answers and Topic-Focus Articulation”. In Linguistics and Philosophy 10: 101-113. Partee, B. 1991. “Topic, Focus and Quantification”. In Proceedings of SALT I. Cornell Working Papers 10. Partee, B. 1995. “Focus, quantification and semantics-pragmatics issues”. In Proceedings of Focus and natural language processing, P. Bosch and R. van der Sandt (eds.). Heidelberg: IBM Deutschland. Peregrin, J. and Sgall, P. 1986. ”An Attempt at a Framework for Semantic Interpretation of Natural Language”. In Theoretical Linguisitcs 13:37-73. Peregrin, J. 1996. “Topic and Focus in a Formal Framework.” In Discourse and Meaning, B. Partee and P. Sgall (eds.), 235-254. Amsterdam: Benjamins. Rooth, M. 1992. “A Theory of Focus Interpretation”. In Natural Language Semantics 1: 75-116.
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Sgall, P., Hajičová, E. and Panevová, J. 1986. The Meaning of the Sentence in Its Semantic and Pragmatic Aspects. J. L. Mey (ed.). Dordrecht: Reidel and Prague: Academia. Sgall, P. 1995. “Focus and focalizers”. In Proceedings of Focus and natural language processing, P. Bosch and R. van der Sandt (eds.). Heidelberg: IBM Deutschland.
Notes 1
[x F] If S is a sentence with the focus F, then the general way to analyze it would be as λx.S → {F} (where stands for S in which x is substituted for F; x being the variable of the appropriate type). S [x→F]
2
This shows that we need semantic representation fine-grained enough to allow for capturing all possible TFAs; everything that can by itself constitute the focus must obtain its own counterpart within semantic representation. We see that the tense grammateme constitutes such an item. This might seem an obstacle, but is it not rather a test of what are the ultimate constituents of semantic representation? 3
Employed for the first time – at least to my knowledge – by Rooth.
4
And it can thus be seen as a sort of vindication of the Aristotelian, syllogistic approach to logic.
5
We can – in a sense – say hat also negation behaves as a focalizer, because in the typical case it can be seen as replacing PRED by the complementary generalized quantifier NPRED. This is to say that in the typical case it is the TFA that determines the scope of negation – the scope being the focus. See Hajičová (1995) and also Peregrin (1996).
12
Structural Linguistics And Formal Semantics Jaroslav Peregrin [Travaux du Cercle Linguistique de Prague I, Benjamins, Amsterdam, 1995; original pagination]
Introduction The beginning of this century hailed a new paradigm in linguistics, the paradigm brought about by de Saussure's Cours de Linguistique Genérále and subsequently elaborated by Jakobson, Hjelmslev and other linguists. It seemed that the linguistics of this century was destined to be structuralistic. However, half of the century later a brand new paradigm was introduced by Chomsky's Syntactic Structures followed by Montague's formalization of semantics. This new turn has brought linguistics surprisingly close to mathematics and logic, and has facilitated a direct practical exploitation of linguistic theory by computer science. One of the claims of this paper is that the post-Saussurian structuralism, both in linguistics and in philosophy, is partly based on ideas quite alien to de Saussure. The main aim then is to explain the ideas driving the formalistic turn of linguistics and to investigate the problem of the extent to which they can be accommodated within the framework of the Saussurian paradigm. The main thesis advocated is that the point of using formalisms in linguistics is more methodological than substantial and that it can be well accommodated within the conceptual framework posited by de Saussure.
1 De Saussure vs. Structuralism Before beginning to discuss structuralism, let us stress the distinction between the genuine views of Ferdinand de Saussure and the teachings of his various avowed followers, be they linguists or philosophers. In fact, de Saussure's theory, as presented in his Course, is an austere and utterly rational scientific theory articulated with a rigour commonly associated with linguistic theories of the 'post-Chomskian' period, though differing from them by the absence of
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formalisms. Many of the de Saussure's followers tried to turn his approach into something quite different: into a tool of questioning scientific rationalism overemphasizing the "literary" aspect of language. This is true particularly of French philosophers who used the structural insight to fight the analytic approach of their Anglo-American colleagues. It is beyond doubt that French structuralism constitutes one of the most significant philosophical movements of this century; however, its affiliation to de Saussure is an intricate matter. These philosophers have eagerly reassumed the view of language as a self-contained phenomenon to be explained by an appeal to its intrinsic properties; however, they have almost completely ignored other aspects of de Saussure's approach to language, notably his calm scientific rigour.1 Linguists such as Jakobson and Hjelmslev, of course, remained far more faithful to the teaching of their predecessor, but they failed to match his rigour. Thus Hjelmslev's theory, although guided by the promising goal of finding "the system beyond the process" and "the constancy beyond the variability",2 is overloaded with more or less mysterious concepts which he is not willing to make sufficiently precise; and Jakobson, although on the one hand ready for such exquisitely "Saussurian" claims as "if topology is defined as the study of those qualitative properties which are invariant under isomorphic transformations, this is exactly what we did in structural linguistics"3, on the other hand considers theory of language to be akin to literary criticism and claims that "only as a poetry is language essential"4. 2 De Saussure de-mythicized In what sense then was de Saussure himself a structuralist? Structuralism, as developed by de Saussure, consists in viewing abstract linguistic objects (especially meanings, but everything that he calls linguistic reality) as values of elements of the system of the expressions that make up language. Let us explain this in detail5. First, let us notice that to speak about a structure is possible only there where it is possible to speak about parts and wholes. Indeed: structure is the way of organizing parts into a whole. So to base one's theory of language on the concept of structure presupposes viewing language as a part-whole system. Let us stress that the notion of a part-whole structure of language may be far from trivial. Expressions are indeed strings of words and as such they consist of substrings (thus John loves Mary consists of John loves and Mary, or of John and loves Mary, or of John and loves and Mary), but this trivial part-whole structuring is not what linguistics is about. Besides it there is another, nontrivial part-whole structure which can be imposed on the class of expressions of
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language and which stems from centuries of investigations by grammarians. According to this notion John loves Mary consists of John and to love Mary, or of John, to love and Mary, where loves is considered to be only a kind of "form" (or a "manifestation") of to love. Let us further notice that to speak about a structure is necessary only there where two different wholes may consist of the same parts. Indeed, structure then is what makes the difference. Otherwise there is no reason for not considering all wholes as having the same structure. We saw that the sentences John loves Mary and Mary loves John can be considered as consisting of the same parts. But these two sentences are different, and hence there must be something which makes them so; and it is this something that is addressed as their structure. The part-whole view of language implies the perceiving of expressions as building-blocks, as constituents of more complex expressions, the ultimate wholes being sentences. (Sentences themselves can thus be viewed both as complete wholes and as blocks used to build more complex wholes.) Any block is suitable for some ways of building some wholes, and not suitable for other ways and other wholes; and the situation may arise in which the usability of two blocks coincides. This is the case when using one of the blocks instead of the other leads always to the result which we consider equivalent to the original one. (If we build houses and equate all houses of the same shape, i.e., differing only in colour, then we thereby equate also all bricks differing only in colour.) This is to say that considering some wholes equivalent engenders our also taking some blocks to have equal values. Hence every equivalence on the class of expressions of language induces an assignment of values to expressions. The concept of equivalence, or, in de Saussure's term, identity, is thus interdependent with the concept of value. This is de Saussure's (1931, p.110) claim that "the notion of identity blends with that of a value and vice versa." Now, roughly speaking, the main claim of de Saussure's is that all the abstract entities associated with expressions can be considered as values and hence as certain "spin-offs" (using the term as used by Quine) of certain equivalences (or oppositions, which are complements of equivalences). 3 Chomsky, Montague and Formal Semantics Chomsky's path-breaking theory occasioned the reconstruction of language as a formal algebraic structure. Chomsky proposed to account for a language via a set
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of formal generative rules, the recursive application of which to a given initial symbol generates all and only syntactically well-formed sentences of the language. The notion of natural language as a bundle of rules is clearly nothing new. In fact, the very idea of grammar is based on this view: to write a grammar of a given language means to articulate rules accounting for well-formedness of that language. Chomsky's novum was that he proposed organizing the rules into a hierarchical system allowing for systematical generation, and basing all this upon setting up of the grammar as a real mathematical structure6. Such a mathematization entailed an exceptional increase of rigour and perspicuity and, moreover, it led to the development of a metatheory, investigating into the formal properties of grammars (e.g. their relative strengths). Chomsky's approach proved to be extremely fruitful in the realm of syntax, and linguists immediately tried to extend it to semantics. They attempted to generate meanings in the same way as Chomsky's theory generated surface structures. However, these attempts, be they presented as semantic markers of Katz and Postal (1964), or as generative semantics due to Lakoff (1971), in general failed to be satisfactory. The reason for this failure was diagnosed by Lewis (1972): it was the failure to account for truth conditions, which is a conditio sine qua non of semantics7. Montague, Lewis and others thefore offered a new way to account formally for semantics based on the results of formal logic. The basic idea was to treat meanings as set-theoretical objects on which expressions are mapped. The first approximation, going back to Gottlob Frege, was to reify the two truth values and to consider the meaning of a sentence to be directly its truth value. However, this approach had the unpleasant consequence that any and every pair of sentences that are either both true, or both false, are synonymous; which proves such an approach to be essentially untenable. The lesson to be learned seemed to be that the meaning of the sentence does not amount to its truth value, but rather to its truth conditions. This obstacle was resolved by introducing the concept of possible world into semantics and this is where Montague enters the scene. (However, it is fair to stress that possible-world semantics was not discovered by Montague; he was neither the first one to use possible worlds as a tool of logical theory - the first to use them systematically were Stig Kanger and Saul Kripke - nor the only one to employ possible-worlds-based logic in an effort to formulate a systematic semantics of natural language; concurrently other theoreticians presented similar theories - at least Tichý's (1971) transparent intensional logic is surely worth mentioning. But Montague is the one who has become the legend.)8 Possible worlds were considered as the entities to which truth is relative; hence to say that the meaning of sentence was its truth conditions became to say that it was a certain function assigning truth values to possible worlds. This turned truth
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conditions into entities accommodable within the framework of set theory. The step from truth values to truth values relativized to possible worlds (and in general from extensions to intensions) was a good one, but not good enough. It soon became clear that even to consider every pair of sentences being true in the same possible worlds as synonymous is inadequate. Every truth of mathematics is true in every possible world; but it is surely inadequate to consider all truths of mathematics as synonymous. The solution accepted by the majority of semanticians was to consider meaning of sentence as something structured. According to an old proposal of Carnap (1957), two expressions were taken as really synonymous if they not only shared intensions, but were intensionally isomorphic, i.e. if they consisted of the same number of constituents and if their respective constituents shared intensions; and this idea, revived by Lewis, has served to ground the "hyperintensional" semantics that became prevalent in the eighties. Lewis has proposed to consider meaning as a Chomskian tree whose leaves are occupied by intensions. His proposal has been further elaborated especially by Cresswell (1985). Other proposals to the effect of considering meaning as a structure, were articulated within the framework of situation semantics of Barwise and Perry (1983) and within that of discourse representation theory due to Kamp (1981)9. Tichý (1988) has reached the conclusion that the intension of a complex expression is constructed from the intensions of its components and has proposed to consider not the result of the construction, but rather the construction itself as meaning. The shift from functions to structures hailed a rapproachement between the theories of logically-minded semanticists operating within set theory and those of the more traditionally-minded ones using formalisms more loosely. If we free ourselves from the 'ideologies' of individual schools, we can see that the gap between Kamp's discourse representation structure or Tichý's construction, on the one side, and Chomsky's deep structure or the tectogrammatical representation of Sgall et al. (1986), on the other, need not be crucial10. 4 Language as an Algebra Now what we are claiming is that formal linguistics does not in general bring insights essentially incompatible with the structuralist paradigm; rather, it carries out the "mathematization" of language thereby creating a framework in which the Saussurian point can appear quite intelligible.
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We have stated that de Saussure's approach presupposes the view of language as a part-whole structure, i.e. as a class of items (expressions) some of them consisting of others. Thus we are viewing language as a class of expressions plus a collection of operations which enable more complex expressions to be made out of simpler ones. So, for example, the class of expressions contains the expressions John, to love, Mary and John loves Mary; and among the operations there is one that makes John loves Mary out of John, to love, Mary. Now in order to state this in mathematical terms, let us present a tiny fragment of algebra. We shall restrict ourselves to three definitions, and we are not going to press for exactness; the aim of the whole enterprise is merely to illustrate the role which mathematics can play within a theory of language.
Definition 1. An algebra is an ordered pair A=
, where C is a set, called the carrier of A; and F=j∈J is a family of functions, called the operations of A, each of which maps the Cartesian power of C on C (i.e. each of which is an n-ary function on C). Definition 2. Let A=j∈J> and A'=j∈J> be algebras with the same number of operations. Let G be a function from the carrier of A to that of A'. We shall say that G is a homomorphism from A to A' if G(Fj(x1,...,xn)) = Fj'(G(x1),...,G(xn)) for every x1,...,xn from the domain of Fj and for every j∈J. Definition 3. Let A=j∈J> be an algebra and E an equivalence (i.e. a transitive, symmetric and reflexive binary relation) on the carrier of C. Let us call a subclass of C an E-subclass if each its two elements are equivalent according to E. Let us call an E-subclass of C maximal if it is not included in another E-subclass of C. Let C' be the class of all maximal E-subclasses of C; and for every j∈J let Fj' be the function on C' such that Fj'(y1,...,yn)=y if and only if there exist elements x1,...,xn,x of C such that x1∈y1, ... ,xn∈yn, x∈y and Fj(x1,...,xn)=x. The algebra j∈J> is called the factor-algebra of A according to E and it is denoted by A/E.
Using Definition 1 we can restate our previous consideration more mathematically. The consideration effectively views language as an algebra: the carrier of the algebra contains the expressions John, to love, Mary and John loves Mary; and among the operations of the algebra there is a ternary one, say Fk, such that Fk(John, to love, Mary) = John loves Mary. Chomsky's contribution was not to abolish this view, but rather to
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explicate it and to articulate the rules of the algebra of syntax in the rigorous mathematical way. The contribution of the semanticians can then be seen in articulating the semantic aspect of language into another, semantic algebra connected with the first one by the function of meaning assignment. The point is that if Chomsky's theory can be seen as the reconstruction of language as the class of lexical items plus the class of grammatical rules constructing more complex expressions out of simpler ones, then Montague's contribution can be seen in mapping lexical items on certain basic set-theoretical objects and in paralleling the syntactic rules operating on expressions by rules operating on their set-theoretical meanings. Every expression E is thus supposed to be furnished with a set-theoretical meaning ÂEÂ; the simple expressions being assigned their meanings directly, the complex ones via semantic rules. If we - for the sake of simplicity - temporally shelve possible worlds, then we can say that ÂJohn and ÂMary are supposed to be elements of a basic set understood as the universe of discourse, Âto love is considered as a function assigning truth values to pairs of elements of the universe (the sentence holds true for some pairs and is false for other ones), and ÂJohn loves Mary as a truth value (true or false)11. The meanings of simple expressions such as John, to love, and Mary are supposed to be given in a direct way, whereas those of complex ones like John loves Mary are considered to be "computable" out of meanings of its parts. Thus, the value ÂJohn loves Mary is considered to be computable out of ÂJohnÂ, Âto love and ÂMaryÂ; namely as Âto loveÂ(ÂJohnÂ,ÂMaryÂ). In general, the meaning of a complex expression is a function of meanings of its parts. This approach thus sanctions the so called principle of compositionality, which has been considered basic for the theory of meaning since Frege. The set-theoretical meanings of lexical items plus the rules to compute the meanings of complex expressions out of their parts thus yield an algebra with the same number of operations as the algebra of expressions. Meaning assignment then comes out as a mapping M of the algebra of expressions on the algebra of meanings such that to every operation F of the algebra of expressions there corresponds an operation F' of the algebra of meanings such that for every e1,...,en from the domain of F it holds that M(F(e1,...,en))=F'(M(e1),...,M(en))). Hence, referring to the Definition 2, we can say that the meaning assignment is a homomorphism from the algebra of expressions to the algebra of meanings.
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5 Saussure Mathematized With the help of this framework, we find that many points previously difficult to articulate, become surprisingly simple. An example is the way we have just expressed the principle of compositionality: this principle, which has been constantly subject to misunderstandings, now becomes the simple and unequivocal claim of the homomorphic character of meaning- assignment. Everybody who is familiar with the basics of algebra easily understands; a misunderstanding is hardly possible12. We have stressed that de Saussure's claim is that the meaning of an expression is its value resulting from oppositions present in the system of language. We have stressed also that the value is a reification of the way the expression functions as a building-block for building wholes suitable for various purposes, notably true sentences. Translated into our algebraic framework, the algebra of semantics owes its being to certain oppositions present within the system of language, notably to the opposition between truth and falsity, or, which is the same, the equivalence of sameness of truth value. Algebraic theory allows us to clarify how an algebra plus an equivalence between elements of its carrier yields a new algebra: our Definition 3 articulates this in explicating the term factor algebra; it amounts to the "coalescing" of the equivalent elements of the original algebra and to the corresponding adjustment of its operations. This suggests the idea of considering the algebra of meanings as the factor algebra of the algebra of expressions factored according to the equivalence of sameness of truth value. The obvious objection to embracing this conclusion is that it leads to identifying meanings with classes of expressions, which seems to be highly implausible. However, saying that the algebra of meanings can be considered as an algebra of classes of expressions is not to say that meaning be a class of expressions - the point of the structural view is that meaning is not this or that kind of thing, that what there is to meaning is rather only the structure of the algebra of meaning. This is to say, in algebraic terms, that the algebra of meanings is definite only up to isomorphism; the factor algebra of the algebra of expressions must be seen as a mere representative of the whole class of isomorphic algebras, each of which can be considered to represent the algebra of meaning, and none of which can be directly identified with it. In fact, formal semantics can be seen as placing additional, pragmatic requirements on the algebra which is to be considered as the algebra of meanings; it endeavours to select that of the isomorphic algebras which would be the easiest to work with. In particular, it is usual to require that the operations of the algebra of semantics be as simple as possible. Frege proposed that the meaning of a
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sentence should be considered as the result of application of the meaning of its predicate to those of its terms. This idea was subsequently generalized to yield the general requirement that the operators of the algebra of meaning should be operators taking as one of its arguments a function and yielding what this function yields when it is applied to the remaining arguments. This means that if F is an nary operator of the algebra of expressions, then there exists an i such that for every n-tuple e1,...,en of expressions from the domain of F it holds that ÂF(e1,...,en)Â=ÂeiÂ(Âe1Â,...,Âei-1Â,Âei+1Â,...,ÂenÂ). The relation of an algebra supplemented by an equivalence to the corresponding factor algebra is thus the prototype of the relationship between the system of language expressions and a system of values which the expressions acquire with respect to some opposition or equivalence. Although the algebraic model, if taken literally, might lead to an oversimplified view of language, it evidently dramatically improves the intelligibility and comprehensibility of the Saussurian point of the value-like character of meanings and other linguistic abstracta. And, this is, in general, the true role of mathematics within a theory of language: it is neither to improve language, nor to give its precise and exhaustive description, but rather to facilitate comprehensibility and intelligibility of language via confronting it with models13. 6 Structuralism Rejoined? De Saussure's structuralistic insights invited generalization: the idea of structuralism is far more sweeping than to restrict itself to linguistic reality. The French structuralists took one direction of generalization: they snesed that natural sciences were threatening to swallow up humanities and they have ended up with philosophy as a kind of literary genre (viz Derrida). But there were other people more fascinated by, than fearful of, the sciences and they instead merged their own structuralist insight with the rigorous scientific thinking. Analytic philosophers, from Russell, Carnap and Wittgenstein to Quine and Davidson, were, of course, not influenced directly by de Saussure, but their teachings seem to be in certain aspects more congenial to the spirit of Cours de linguistique générale than the teaching of those who are usually considered as de Saussure's direct followers. And it was analytic philosophy whose advancement is inseparably interlocked with the advancement of formal semantics.
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I do not want to claim that analytic philosophy and formal semantics are necessarily intrinsically structuralist. The views of many analytic philosophers, and of even more formal semanticians, amount to a kind of nomenclatural view of language, which is the direct opposite of structuralism. Many analytic philosophers did not manage to resist the temptation to embrace a form of naive scientism and ended up with a kind of systematic metaphysics (now approached via language and done with the help of mathematics and set theory) which the structuralist insight vanquishes. But a significant number of these thinkers, people such as Wittgenstein, Quine or Davidson, avoided such traps and their approach can be truly called structuralistic. Quine's (1992) recent conclusions about the concept of structure are even more radical than those of his predecessors: instead of urging the reduction of abstract objects to relations, he questions the very idea of an object: "The very notion of object, or of one and many, is indeed as parochially human as the parts of speech; to ask what reality is really like, however, apart from human categories, is self-stultifying. It is like asking how long the Nile really is, apart from parochial matters of miles and meters." (p.9) Hence aphoristically: "Save the structure and you save all." (p.8) The views close to Quine's and especially relevant for what we have been pursuing here are due to Donald Davidson - it is him who has made it clear that formal semantics need not be understood as a naively-metaphysical, nomenclatural matter and who has shown the plausibility of the theory of meaning as derivative to the theory of truth. And it is these views which we try to show to be congenial to the basic insights of de Saussure14. We have stated that the ways of analytic philosophy and structuralism essentially parted. However, their recent offspring - post-analytic philosophy and poststructuralism - are no longer antagonistic and indeed are sometimes surprisingly close. I think that the inseparability of the question about the nature of reality from the question about the nature of the language we use to cope with the reality, as urged in the above quotation by Quine, is in fact the same problem as that which irritates people like Foucault and Derrida. And I think, on the other hand, that the "careful mathematization of language" which is urged by Derrida (1972) is nothing else than the non-nomenclatural, non-metaphysically founded usage of formal logic as pursued by Quine and Davidson15.
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7 Conclusion De Saussure's structuralistic view of language is quite compatible with the formal trend of linguistics appearing during the recent decades. In fact formalization and mathematization help to make the structuralist point intelligible. Many theoreticians believe that formal semantics and analytic philosophy is connected with a nomenclatural view of language and hence is incompatible with the structural insight. But this is wrong - formal semantics is in itself neutral, and it is capable of being explicated both in the naive nomenclatural way and in the way congenial to de Saussure's structuralism; and among analytic philosophers we can find outstanding representatives not only of the former, but also of the latter view.
NOTES 1
There are philosophers who evaluate even more harshly the way in which French structuralists handled the heritage of de Saussure. Thus Pavel (1989, p. vii) characterizes their efforts as follows: "They mistook the results of a specialized science for a collection of speculative generalities. They believed that breathtaking metaphysical pronouncements could be inferred from simple-minded descriptive statements." 2
Hjelmslev (1943, p.11).
3
Jakobson (1971, pp. 223-4).
4
See Holenstein (1987, p. 25).
5
For a more detailed expositions of the issues presented in this section see Peregrin (1994b).
6
Chomsky himself, of course, would consider his approach not a mere improvement of methodology, but as an empirical discovery concerning human's innate inner workings; we leave this conviction of his aside, because it is peculiar to his own line of thought and it is not essential to the formalistic turn as such. 7
Lewis claimed that linguistic theories of meaning are mere translations of natural language into another, formal language, namely 'markerese'. However, I think that this caveat, as it stands, is misguided: every explicit semantic theory is clearly a translation of natural language into another language, be it 'markerese', the language of set theory, or whatever. The only way to do explicit semantics is to make statements 's' means m, where m is an expression of a language. 8
For general information about the concept of possible world see Partee (1989); for
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conceptual analysis see Peregrin (1993a). 9
Kamp's framework aims, besides this, at capturing what can be called dynamics of language, especially its anaphoric capacities; and it slowly becomes a paradigm of the semantic theory for the nineties. 10
This point was made quite clear by Davidson (1984, p.30): "Philosophers of a logical bent have tended to start where the theory was and work out towards the complications of natural language. Contemporary linguists, with an aim that cannot easily be seen to be different, start with the ordinary and work toward a general theory. If either party is successful, there must be a meeting." 11
Taking the intensional aspect of language at face value, we have to relativize all of this to possible worlds: the denotations of ÂJohn and ÂMary (if we do not treat them as rigid, i.e. possible-worlds-independent, designators) will be functions from possible worlds to the universe, Âto love a function from possible worlds to pairs of elements of the universe, and ÂJohn loves Mary will be a function from possible worlds to truth values: in some worlds (situations, time-spans etc. the sentence holds true, in other worlds it does not. 12
The objection that such an explication is simple only due to the backlog of the complicated theory of algebra, is not sound - algebra is nothing ad hoc, it is a well established theory whose meaningfullness is independent of whether we do or do not use it within a theory of language. 13
See Peregrin (1993b).
14
For Davidson's way of understanding semantic theory see Davidson (1984); see also Peregrin (1994a). 15
The recent philosophical development of Richard Rorty documents that these two seemingly disparate approaches to philosophy could lead to a unified stance.
REFERENCES Barwise, J. and Perry, J. (1983): Situations and Attitudes, MIT Press, Cambridge (Massachusetts). Carnap, R. (1928): Der Logische Aufbau der Welt, Berlin. Carnap, R. (1957): Meaning and Necessity, University of Chicago Press, Chicago. Chomsky, N. (1957): Syntactic Structures, Mouton, The Hague. Cresswell, M. (1985): Structured Meanings, MIT Press, Cambridge (Massachusetts). Davidson, D. (1984): Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Derrida, J. (1972): Sémiologie and Grammatologie, Entretien Avec Julia Kristeva; in: Positions, Minuit, Paris. Hjelmslev, L. (1943): Omkring sprogteoriens grundlaeggelse, Coppenhagen. Holenstein, E. (1987): Jakobson's Philosophical Background, in: Language, Poetry and Poetics (ed. by K.Pomorska), de Gruyter, Berlin. Jakobson, R. (1971): Selected Writings II, Mouton, The Hague. Janssen, T.M.V. (1983): Foundations and Applications of Montague Grammar, dissertation, Mathematisch Centrum, Amsterdam.
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Kamp, H. (1981): A Theory of Truth and Semantic Representation, Truth, Interpretation and Information (ed. by J.Groenendijk, M.Stokhof and T.M.V.Janssen), Foris, Dordrecht, pp. 1-41. Katz, J. and Postal, P. (1964): An Integrated Theory of Linguistics Descriptions, MIT Press, Cambridge (Masachusetts). Kripke, S. (1963): Semantical Considerations on Modal Logic, Acta Philosophica Fennica 16, pp. 83-94. Lakoff, G. (1971): On Generative Semantics, Semantics: An Interdisciplinary Reader in Philosophy, Linguistics and Psychology (ed. by D.D.Steinberg and L.A.Jakobovits), Cambridge. Lewis, D. (1972): General Semantics, Semantics of Natural Language (ed. by D.Davidson and G.Harman), Reidel, Dordrecht. Montague, R. (1974): Formal Philosophy: selected papers of R.Montague (ed. by R.Thomason), Yale University Press, Yale. Partee, B. (1989): Possible Worlds in Humanities, Arts and Sciences, Proceedings of Nobel Symposium 65 (ed. by S.Allen), de Gruyter, Berlin/New York, pp. 93-123. Pavel, T.G. (1989): The Feud of Language, Blackwell, Cambridge (Massachusetts). Peregrin, J. (1993a): Possible Worlds: A Critical Analysis, Prague Bulletin of Mathematical Linguistics 59-60, 9-21. Peregrin, J. (1993b): Language and Formalization, Linguistica Pragensia 2/1993, 74-80. Peregrin, J. (1994a): Interpreting Formal Logic, Erkenntnis 40, 1994, 5-20. Peregrin, J. (1994b): Structuralism: Slogan or Concept, Lingua e Stile (to appear). Quine, W.V. (1992): Structure and Nature, The Journal of Philosophy 89, 5-9. Russell, B. (1924): Logical Atomism, in: Contemporary British Philosophy: Personal Statements, First Serise, London. Saussure, F. de (1931): Cours de linguistique generale, Payot, Paris. Sgall, P., Hajicová, E., Panevová, J. (1986): The Meaning of the Sentence in its Semantic and Pragmatic Aspects, Academia, Praha. Tichý, P. (1971): An Approach to Intensional Analysis, Nous 5, pp. 273-297. Tichý, P. (1988): The Foundations of Frege's Logic, de Gruyter, Berlin/New York.
TOPIC, FOCUS AND THE LOGIC OF LANGUAGE 1
Jaroslav Peregrin
[appeared in Sprachtheoretische Grundlagen für die Computerlinguistik (Proceedings of the Göttingen Focus Workshop, 17. DGfS, IBM Deuschland, Heidelberg, 1995]
1 Topic as the "point of departure" of an utterance The terms topic and focus are used by many theoreticians, but they often mean different things. In the most usual informal sense, topic is what an utterance is about (as contrasted to comment), and focus is what is emphasized in the utterance (as contrasted to background). Let us consider how this intuition can be sharpened. Let us start with the notion of topic, of "that which an utterance (or a sentence) is about". Consider the sentence John loves Mary. What is it about? Clearly we can say that it is (says something) about John. However, it is hard to deny that it is - also - about Mary. And we can equally well say that it is about John loving Mary, or about John's love for Mary. "What a sentence is about" is thus an extremely vague notion; and to use it unregimented to define the concept of topic is hardly meaningful. However, we can consider another, narrower sense of "being about" - a prototypical utterance of English (or indeed any other language) can be seen as consisting in picking out an item and saying something about it; thus the sentence John walks, in a "neutral" context and with a "neutral" articulation, picks out the person John and states that this person walks. In this sense, the sentence is about John, and not about, say, John's walking. Hence, in this sense, aboutness has to do with the subject-predicate structure; a sentence is "normally" taken to be about its subject. However, it is important to realize that the subject-predicate structuring which is relevant from this semantic - or "informatoric" - viewpoint does not necessarily coincide with the syntactic structuring - what functions as a subject from the semantic viewpoint need not be identical with the syntactic subject. When one says John walks (answering the question What does John do?), he is addressing John and claiming that he is walking; whereas when he says JOHN walks (answering, e.g., the question Who walks?), he is rather addressing walking and is claiming that this activity is carried out by John. We can imagine that the separation of the syntactic from the semantic subject-predicate structure may represent a certain step in the genesis of language; a step by which language has gained another degree of flexibility - but this is of course nothing more than speculation. 1
The paper has been essentially improved thanks to the helpful criticism of Petr Sgall and Barbara Partee.
1
However, if we want to claim meaningfully that semantic subject and syntactic subject need not be one and the same thing, we must be more exact about what a semantic subject is (syntactic subject seems to be defined clearly enough). We have said that semantic subject is that part of the utterance which the utterance is about; but to take this as the desired definition would be begging the question, for we have explicated the concept of aboutness (in the narrower sense) by means of the concept of the semantic subject. To prevent this circle from being vicious we must find a way of defining semantic subject independently of aboutness. A distinctive feature of the semantic subject is displayed by the fact that the failure of an utterance, seen as a communicative act, has different consequences when concerning the subject than when concerning the predicate. The former case of failure means the failure to point out something common to the speaker and the hearer, some shared basis which would anchor the following core of the utterance within the framework of knowledge and awareness shared by the speaker and the hearer; and by consequence generally making the whole utterance not quite intelligible. Thus this kind of failure leads to the utterance being either unintelligible or at least infelicitous. On the other hand, a later failure concerning that part of the utterance which aims at claiming something about the subject results in the utterance being simply false. If I say (as in the well-known example used by Strawson) The exhibition was visited by the king in the context where there is no exhibition to be referred to, my audience will simply fail to grasp what I am talking about and is likely to see my utterance as inappropriate; whereas if there is an exhibition which is understood to be referred to and if there is no king to visit it, the utterance is simply false. Thus what we have called semantic subject is actually a basis, an information-anchoring point of departure for an utterance; whereas that which we have called semantic predicate is the information-conveying core of the utterance. Indeed it is precisely these two concepts - the point of departure and the core of an utterance - which were established as the basic means of analyzing a sentence by Mathesius (see esp. 1939) and his colleagues of the Prague Linguistics Circle (whose precursors were in France and Germany). In the more recent literature (see esp. Sgall et al., 1986), it has been proposed to identify this pair of concepts with the concepts of topic and focus; we have now indicated which considerations can substantiate such an identification in the case of topic.
2 ... and focus as its "core" Having presented some evidence that topic, as the term is usually used today, is comparable to what the fathers of the Prague Linguistic School called the point of departure of an utterance, we would now like to indicate that it is the Prague core of the utterance that can serve as a
2
plausible generalization of the notion of focus. To be sure, focus is often conceived today as something occurring only within some utterances, and which becomes especially relevant in combination with focalizers (i.e. particles like only, even, also etc.). Focus is an emphasized part of the utterance commonly distinguished by way of intonation and stress; it is, as Krifka (ms.) put it, the "intonationally highlighted" part of the utterance. However, the notion stemming from the Prague tradition is different: focus, taken as the core of the utterance, is not some accidental surplus of certain utterances, but an essential constituent of every meaningful and "pointful" utterance; and intonational highlighting can be seen as a mere way of marking focus in cases where the topic-focus articulation cannot be read off the syntax (or sometimes possibly as a means also of marking a strengthened exhaustiveness or contrastiveness claim of the utterance). The plausibility of such a view grows if we do not restrict our attention to English but consider also languages with relatively free word-order like Czech. Such languages allow for organizing the utterance according to the 'functional sentence perspective' - going from the topic, the point of departure, to the focus, the core of the utterance. The topic picks up an entity (in a very broad sense) familiar to the participants of the communicative act (thus presupposing its existence, or, more precisely, its referential availability) and thereby anchors the utterance within the informational pool shared by the communicants (thereby making the utterance intelligible). Focus then says something about the entity thus specified by the topic (usually without presupposing anything) and develops a new theme (extending the common pool). It might seem that topic (what the utterance is about) is something to a large extent independent from focus (conceived as that which the utterance stresses). However, the view advocated here is based on the conviction that all utterances have a "logic" which prevents these two things from being totally independent. The point of an utterance is to get from something known and agreed to something new and informative, and it is the new information that is primarily stressed; so that the topic and the focus emerge as two aspects of a single articulational pattern. This is to say that any "intonational highlighting", characteristic of the focus, makes eo ipso that which is highlighted into that which is the "point" of the utterance, and hence into that which is the core of the utterance.
3 Three perspectives The intuitions discussed so far allow for various forms of explication of the topic-focus articulation; these in turn lend themselves to various kinds of formal articulation. Let us present at least three "metaphors". The three viewpoints lead to three ways of formalization; but these are not intended to represent three different ways of topic-focus structuring - they are simply
3
three ways of presentation of a single ("triune") pattern. (1) Topic is a subject (picking up a piece of information "as an object", thereby triggering an existential presupposition) and focus is a predicate (presenting some further specification of the object). (2) Topic and focus are arguments of an implicit generalized quantifier, or they are - in terms of Partee (1991) - the restrictor and the nuclear scope of a "tripartite structure". In certain cases, the implicit topic-focus-binding quantifier can be overridden by an explicit focalizer, such as always or only, but also by negation (cf. Haji_ová, 1984; 1994). (3) Topic and focus are two phases of an information-conveying act (and they can be pictured as two segments of a dynamically viewed proposition). Topic corresponds to the phase where the information gets anchored to the existing "informational structures", and focus to that where the genuine new information is being added. Therefore, the failure of the act during the topic-phase (i.e. the falsity of presupposition) means the failure of the whole act (which may precipitate a - possibly temporary - breakdown of communication), whereas that during the focus-phase (i.e. the falsity of assertion) engenders the failure to add new information.
4 General Questions of Formalization The three perspectives outlined in the preceding section lead to three different ways in which we can develop a logical formalism to account for the topic and focus. (1) The elaboration of the idea of topic and focus being semantic subject and semantic predicate, respectively, calls for a formalization allowing for the notion of presupposition, i.e. for one based on partial or three-valued logic. (To say that A is a presupposition of B is to say that if B is not true, A cannot be but truth-valueless, which makes a nontrivial sense only if we allow for a nontrivial truth-valuelessness.) The basic subject-predicate nexus has to be analyzed in such a way that subject, in contrast to predicate, triggers a kind of existential presupposition. The subject-predicate pattern of the formal language we use to analyze its natural counterpart thus loses its role as a reflection of the overt, syntactic structure and is intended instead to reflect the topic-focus articulation. This means that we analyze an utterance consisting of the topic T and focus F as a subject-predicate statement F(T) which has a truth value only if any presupposition associated with T is true. The sentence John walks (with the "neutral" intonation) may be thus analyzed as walk{John} (the curly braces indicating non-classical, "presuppositional" predication) with the presupposition ∃x.x=John; whereas the sentence JOHN walks as SS -RKQ ^ZDON` presupposing something like ∃x.walk(x). Moreover, the fact that the focus is usually in a sense exhaustive (JOHN walks is usually understood as not only claiming that walking is carried out by John, but also as indicating that John is the only, or at least the
4
most significant, walker) can be accounted for by a further modification of the apparatus of predication yielding the formula SS -RKQ ^ZDON` - where P!{T}, roughly speaking, presupposes the existence of T, claims that P applies to T and that there is no alternative P' to P that would apply to T. For details see Peregrin (1994 and 1995). (2) From the second viewpoint we see an utterance as essentially consisting of three parts (not all of which have to be overt). Each utterance is seen to consist of a topic (which may be void), a focus, and an operator binding them together (which may be implicit). In the prototypical case the operator can be seen as nothing more than a "higher-level" realization of predication, so that PRED(John,walk) yields walks!{John} and PRED(walk,John) gives SS -RKQ ^ZDON`. (For details see Peregrin, 1994). The operator PRED is in fact nothing else than Jacobs' (1984) operator ASSERT; the place of this operator can be assumed by overt focalizers such as ONLY, but also by negation. (3) If we stick to a dynamic view of language, as articulated by various dynamic semantic theories, we have to see an utterance as something that alters the context, or the information state, in which it is produced. The utterance uses the input information state and works towards an articulation of a message yielding a new, output information state. (This perspective has been suitably formalized by Groenendijk and Stokhof's, 1989a and 1989b, dynamic logic; see also Peregrin & von Heusinger, 1995). Topic and focus now present themselves as two different phases in this process; thus, an utterance consisting of a topic T and a focus F has to be analyzed as T }&! F, where }&! is a new kind of concatenation operator signalling the switch between two modes of evaluation (see Peregrin, 1995). In contrast to the Groenendijko-Stokhofian logic, it is vital here that the logic needed to accommodate such a proposal can distinguish between two kinds of failure of an utterance: between falsity and infelicity: the formula A }&! B is then (1) true w.r.t. an input evaluation f iff there exists an HYDOXDWLRQI VXFKWKDW II ! $DQGDQHYDOXDWLRQJVXFKWKDW I J! % LWLV false w.r.t. f iff WKHUH H[LVWV D I VXFK WKDW II ! $ EXW WKHUH H[LVWV QR I
VXFK WKDW I I
! % DQG LW LV infelicitousZUWILIIWKHUHH[LVWVQRI VXFKWKDW II ! $`
5 A Case Study: Bill Meets the Dallas Clan To illustrate the foreceding of this, let us take the sentence (1) borrowed from Krifka (1991). John only introduced Bill to Sue
(1)
This sentence, devoid of topic-focus articulation, can be schematized as (1') (for simplicity's sake we remain on the level of extensions)
5
(1')
introduce(J,B,S)
Notice that the same sentence (still without any topic-focus articulation) could be equivalently analyzed in many other ways; for (1) is equivalent to many other formulas, and the particular one we employ is a purely technical matter. Thus, instead of (1) we could alternatively use any of, say, (1'a) through (1'c) (where x is a variable ranging over individuals, p a variable ranging over properties of individuals, r a variable ranging over binary relations among individuals, and q a variable ranging over properties of binary relations among individuals): xintroduce(J,x,S)(B) p.p(B)( x.introduce(J,x,S)) q.q( xy.introduce(x,B\ ^ UU J,S)}
(1'a) (1'b) (1'c)
All these formulas are provably equivalent (by lambda-conversion) to (1'), and so they furnish the very same semantic analysis of (1'). Now let us take (1) with Bill stressed, i.e. (2) John introduced BILL to Sue
(2)
Speaking informally, this sentence expresses the claim that the person introduced by John to Sue is Bill, hence it is about the property analyzable as x.introduce(J,x,S) and it claims that this property is instantiated by Bill (and moreover that Bill is in some sense the only significant - in the simplest case the unique - instantiant of the property). To account for this intuition, we cannot simply take (1'b) (for this would in effect be to license no semantic difference between (1) and (2)); we have to make use of the modified predication p.p(B)!{ x.introduce(J,x,S)}
(2')
(2'), in contrast to (1'), has a truth value only if there is someone whom we introduced to Sue, and it is true only if Bill is - in the present context - the only "relevant" person we introduced to her. In fact, (2) is compatible also with another topic-focus articulation, with not only Bill, but introduced Bill constituting the focus, which yields the following analysis: q.q( xy.introduce(x,B,y)){ r.r(J,S)}
(2'')
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An alternative way to express (2') and (2'') would be PRED( x.introduce(J,x,S),{B}) PRED( r.r(J,S),{ xy.introduce(x,B,y)})
(2'a) (2''a)
These analyses amount to comparing two classes and claiming that the latter exhausts the "significant" part of the former; in case of (2'a) the two classes are the class of persons introduced by John to Sue and the class consisting of Bill, in case of (2''a) the class of all that John does to Sue and the class consisting of introducing Bill. (That these new analyses are equivalent to the old ones is ensured because PRED is defined so that PRED(A,{B}) = SS % ^$` This perspective is plausible if we consider vocalizers, as in the sentence (3) John only introduced BILL to Sue
(3)
In this case we may assume the focalizer to simply assume the place of the implicit general quantifier PRED; the two resulting analyses are ONLY( x.introduce(J,x,S),{B}) ONLY( r.r(J,S),{ xy.introduce(x,B,y)})
(3') (3'')
claiming now that the two compared classes coincide. This perspective can be also used to account for what Krifka (ms.) calls "second occurrence focus", like in (4), which can be analyzed as (4') or (4''): John also only introduced Bill to Pamela ALSO( y.ONLY( x.introduce(J,x,y),{B}),{P}) ALSO( y.ONLY( r.r(J,y),{ xy.introduce(x,B,y)}),{P})
(4) (4') (4'')
Back to (2) - another way to analyze it, by means of a suitably modified version of dynamic logic, would lead to the analyses introduce(J,d,S) }!& d=B r(J,S ` U [\introduce(x,B,y)
(2'a) (2''a)
7
6 Conclusion In this paper we have tried to bring forward some arguments for the following theses: 1. Topic and focus are two aspects of a single articulational pattern which is basic for every sentence. 2. It is just this pattern that was pointed out by the linguists of the Prague School under such names as the 'topical structuring' ('aktuální _len_ní') or 'functional sentence perspective'. 3. This pattern is relevant semantically, namely in that it triggers an existential presupposition connected with the topic, and that it gives the focus a certain claim of exhaustivity of the significant. 4. The pattern can be viewed from three different perspectives which lead to formalizations in the spirit of three different formal semantic theories (predicate logic, theory of generalized quantifiers, dynamic logic).
References. Groenendijk, J., M.Stokhof. 1989a. "Dynamic Predicate Logic". ITLI Prepublication Series LP89-02, University of Amsterdam. Printed in Linguistics and Philosophy 14.39-101. 1991. -. 1989b. "Dynamic Montague Grammar, a first sketch". ITLI Prepublication Series X-89-04. University of Amsterdam. Haji_ová, E. 1984. "On Presupposition and Allegation". Sgall 1984. 99-122. -. 1994. "Topic, Focus and Negation." Focus & Natural Language Processing (ed. P.Bosch & R.van der Sandt), IBM Deutschland, Heidelberg. Jacobs, J. 1984. "Funktionale Satzperspektive und Illokutionssemantik". Linguistische Berichte 91.25-28. Krifka, M. 1991. "A Compositional Semantics for Multiple Focus Constructions". Proceedings of SALT I. Cornell Working Papers 11. -. Ms.: Focus and/or Context: A Second Look at Second Occurence Expressions. To appear. Mathesius, V. 1939: "O takzvaném aktuálním _len_ní v_tném". Slovo a Slovesnost 5.171-174. Translated as "On Information-bearing Structure of the Sentence". Harvard Studies in Syntax and Semantics ed. by S.Kuno. Harvard: Harvard University 1975.467-480. Partee, B. 1991. "Topic, Focus and Quantification". Proceedings of SALT I. Cornell Working Papers 10. -. 1994. "Focus, Quantification, and Semantic-Pragmatic Issues." Focus & Natural Language Processing (ed. P.Bosch & R.van der Sandt), IBM Deutschland, Heidelberg.
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Peregrin, J. 1994. "Topic-focus articulation as Generalized Quantification." Focus & Natural Language Processing (ed. P.Bosch & R.van der Sandt), IBM Deutschland, Heidelberg. -. 1995. "Topic and Focus in a formal Framework". To appear in Discourse and Meaning (ed. by B.Partee & P.Sgall). Peregrin, J. and von Heusinger, K. 1995. "Dynamic Semantics with Choice Functions". In: U. Egli & K. von Heusinger (eds.). Choice Functions in Natural Language Semantics, Konstanz: Universität Konstanz, 43-67. Sgall, P. ed. 1984. Contributions to Functional Syntax, Semantics and Language Comprehension. Amsterdam: Bejnamins - Praha: Academia. Sgall, P., E. Haji_ová & J.Panevová. 1986. The Meaning of the Sentence in its Semantic and Pragmatic Aspects. Praha: Academia.
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TOPIC AND FOCUS IN A FORMAL FRAMEWORK Jaroslav Peregrin1 [from B.H. Partee and P. Sgall (eds.): Discourse and Meaning: Papers in Honor of Eva HajiþRYi%HQMDPLQV$PVWHUGDP]
1. Introduction The concepts of topic and focus have first begun to figure in linguistics in the middle of the nineteenth century. They have been recognized as interesting and they have received the casual attention of linguists of various proveniences, but they have never moved to the centre of interest. One of the linguistic schools in which the problems of topic and focus have not been considered marginal was the school of Prague structuralists. It was especially Mathesius (1929; 1939) who pointed out the import of this range of problems. Firbas (1957; 1971) then continued the study of the phenomena under the heading of 'functional sentence perspective'; Daneš (1974) studied intonation and word order as a means of articulation of topic-focus structuring. The elaboration of the problem of topic-focus articulation within a formal framework of linguistic description has been carried out by Sgall and his collaborators (especially Sgall et al., 1973, and Sgall et al., 1986). Besides Czech linguists there have also been also various other scholars who have displayed interest in this kind of phenomena (e.g. Kuno, 1972, Dahl, 1974); but the mainstream of the Chomskian movement, which has dominated the linguistic world since the sixties, has left them almost unnoticed. Now the situation seems to be changing: results such as those of Rochemont (1986), von Stechow (1989), Jacobs (1991), Krifka (1991), Partee (1991) or Rooth (1992) indicate that topic-focus articulation (hereafter TFA) is being increasingly recognized as a real challenge. From this viewpoint the results of the long tradition of Czech linguistics might hold a renewed interest. To facilitate discussion about the various approaches to TFA and to related phenomena it may be worthwhile to summarize the possibilities of formal accommodation of TFA as developed in Prague. This is the aim of the present paper - not to offer
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ready-made solutions, but rather to overview the problems and possible leads towards solving them.
2. Basic Concepts of the Prague Approach The framework of the Prague research group of formal linguistics has been presented in detail by Sgall et al. (1986). Let us summarize the main points. The aim of the summarization is to facilitate the understanding of the main concepts independently of the particular framework in terms of which they might be articulated - hence the framework is to some extent oversimplified. 1. The framework, designed to capture grammar, is primarily based on dependency, not on constituency. The meaning of a sentence is considered in the form of a tree, called tectogrammatical representation, which contains no non-terminals and captures the dependential structuring. The items that are considered to depend on the main verb are classified according to their thematic roles (which are, however, more closely related to grammar than the Θ-roles common in American linguistics). 2. Each of the elements of the tectogrammatical structure (corresponding to autosemantic lexical elements of the sentence being represented) is either contextually bound or contextually nonbound. A prototypical example of a contextually bound item is one corresponding to an expression which appears also within the immediately preceding part of the discourse. However, contextually bound elements are not only those which are explicitly used before, they are also elements which are in an indirect way implied by the context, where context means not only verbal co-text, but also the situation of the discourse, including the common cultural background shared by the speaker and the hearer. 3. The default order of items depending on a verb (i.e. the order of thematic roles and adverbials) is considered to be fixed for a given language; it is called the systemic ordering. However, this order together with the order of other items not depending directly on the main verb is modified in a concrete utterance, so that the resulting order of the items of the tectogrammatical structure is that of the communicative dynamism (CD). The CD order of contextually bound items dependent on the same head is determined by the speaker's discourse strategy rather than by grammar; on the other hand, the CD of the unbound items dependent on the same head is in accordance with the systemic ordering. An item is less dynamic than its head iff the
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dependent item is bound. 4. The least dynamic element of the sentence constitutes the topic proper. 5. All the contextually bound items depending on the main verb together with all that depends on them and together with the main verb if this is contextually bound, constitute the topic of the sentence; the rest of the sentence constitutes the focus. Hence the topic/focus classification is exhaustive: every element of the sentence belongs either to the topic or to the focus.
3. Formal means There have been several attempts to account for the Prague notion of TFA in formal frameworks. One group of such attempts has been carried out within the framework of an intensional logic, namely of Tichý's transparent intensional logic (see Tichý, 1980). The basic issues of such kind of formalization have been discussed by Materna and Sgall (1980) and Materna et al. (1987); Vlk (1988) has outlined a procedure for the translation of surface forms into the logical representation. The attempts gave rise to an account of TFA in which the topic is taken roughly to be the specification of a class and the focus is taken as giving a kind of exhaustive listing of the elements of that class. There has also been an attempt to account for TFA in a framework similar to DRT; this attempt is due to Peregrin and Sgall (1986). In this framework, each sentence is associated with a situation-like structure (the "content" of the sentence); the "meaning" of a sentence is then understood as the class of all the embeddings of its "content" into the model. A sentence articulated into a topic and a focus is considered as true if every embedding of the "content" of its topic is meaningfully extensible to an embedding of the "content" of the whole sentence. Meanwhile, other approaches to semantic analysis which appear to be worthwhile from the point of view of capturing TFA have appeared as well. There are three impulses which we consider to be particularly promissing in this context: Rooth's alternative semantics, Groenendijk's and Stokhof's dynamization of logic, and Partee's elaboration of the notion of a tripartite structure.
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4. Topic = Subject & Focus = Predicate? The subject-predicate pattern has been considered as central to language since Antiquity. On the syntactic level this means that a typical sentence consists of a subject (nominal phrase) and a predicate (verbal phrase). On the semantic level it means that the content of a typical sentence can be considered as an assignment of a property to an object. In the typical case the syntactic (grammatical) subject coincides with the semantic (logical) subject, and the syntactic predicate with the semantic one. Most linguists have restricted their attention to the syntactic side of the pattern; philosophers and logicians, who are intrinsically interested in the semantic pattern, have, on the other hand, usually tacitly considered it to coincide with the syntactic one. However, the identification of the syntactic subject-predicate pattern with the semantic one is unwarranted; and those who have really understood the nature of language have avoided it. Thus Frege, who sees the semantic subject-predicate pattern as constitutive of the object-concept opposition, remarks, that it need not be the grammatical subject which acts as the semantic or logical one: "Die Sprache hat Mittel, bald diesen, bald jenen Teil des Gedankens als Subjekt erscheinen zu lassen." (Frege, 1892, p.74). TFA can be considered as just this kind of means. Let us consider a simple sentence (1) and its first-order formalization (1') John walks Walk(John)
(1) (1')
The syntactic subject-predicate pattern of (1) is unequivocal: John is the subject and walks is the predicate. Sgall et al. (1986) suggested that due to the impact of TFA the pattern comes to be modified: if we say (2), then what we express seems to be not the property of walking assigned to the individual John, but rather the property of being John assigned to an anonymous walker. JOHN walks
(2)
One might here evoke the idea that the power of TFA is reminiscent of lambda abstraction: what we do when focusing John resembles what we do when making a predicate, λf.f(John), out of John and then applying it to
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Walks. Hence (2) might seem to be appropriately formalizable by (2'). λf.f(John)(Walks)
(2')
However, this idea, although based on a sound intuition, is not without its drawbacks. The point is that (2') lambda-converges to (1') and that it is hence equivalent to (1'). Thus if we consider a logical formula a mere means of presenting a proposition, then it makes no difference whether we analyze (1) as (1') or as (2'): in both cases we render (1) as expressing the same proposition. This might seem to lead to the conclusion that what has been considered as the articulation of the subject-predicate pattern brought about by TFA is not a semantically relevant matter; and this would mean that it is a matter relevant in no way, since it is surely not relevant syntactically. The syntactic patterning is left unchanged in (2). However, what really makes a sentence into a predication is the fact that one of its parts is "about" the other part. The (semantic) subject is what the sentence is about, predicate is what it says about the subject. What does this "about" mean? Well, it, first and foremost, means that the subject is taken for granted for the whole sentence, its existence is not being disputed. This is to say that the subject is connected with a presupposition. If I say about John that he walks, then the fact that there is no John (i.e. nobody known to the interlocutors under this name) makes the statement meaningless rather than false; in contrast to this, if there is a John, but he does not walk, then the statement is simply false. (Classical Russellian examples with the king of France are probably more perspicuous, but proper names are also subject to presuppositions). Let us write ÂXÂ for the extension of an expression 'X' (hence ÂXÂ will be a truth value if 'X' is a sentence, an individual if 'X' is a term, and a class of individuals if 'X' is an unary predicate)2. Let us with every expression X associate a proposition whose extension we denote by ·X· (to be understood as a presupposition associated with X) in the following way: ·X· = ÂXÂ if X is a sentence = Â∃y.y=XÂ if X is a term = Â∃y.X(y)Â if X is an unary predicate
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Now we can define the semantics of a formula P{S} (to be understood as the "real" predication of P of S) as follows: ÂP{S}Â = T iff ·S·= T & ÂP(S)Â = T = F iff ·S·= T & ÂP(S)Â = F = 0 iff ·S·=F Appropriate formalizations of (1) and (2) now are (1'') and (2''), respectively. Walk{John} λf.f(John){Walk}
(1'') (2'')
If both (1'') and (2'') do have truth values, then these truth values coincide; however, it might be the case that (1'') is false, while (2'') lacks a truth value (in the case when ∃x.Walk(x) is false), as well as vice versa (in the case when ∃x(x=John) is false). Let us consider a complex sentence, (3). If we accept the subjectpredicate pattern articulated by means of {} as a standard pattern of a natural language sentence, then there seem to be the formalizations (3a)(3d), and possibly also the two degenerated cases (3e) and (3f). John loves Mary λy.love(y,Mary){John} λx.love(John,x){Mary} λf.f(Mary){λy.love(John,y)} λf.f(John){λx.love(x,Mary)} λf.f(love(John,Mary)){} {λf.f(love(John,Mary))}
(3) (3a) (3b) (3c) (3d) (3e) (3f)
From these, (3f) would seem to be ruled out by the fact that a sentence has to contain some nonpresupposed information; the 'thetic' reading (3e), however, would seem to be, possible3. And it is this 'thetic' reading, as well as the reading (3a) in which the semantic subject-predicate pattern coincides with the syntactic one, which may be considered the preferred reading of (3). (3c) is then the preferred reading of (3') and (3d) of (3''); but (3') may also be read as (3a). The reading of (3'') as (3b) might also ge feasible, although not quite regular.
TOPIC AND FOCUS IN A FORMAL FRAMEWORK John loves MARY JOHN loves Mary
241 (3') (3'')
5. Quantifier Scope In the case of (1), or of (3), the difference between various TFA's is the difference in presuppositions, and hence can be considered as a matter of felicity conditions rather than of truth conditions in the strict sense. Not so, however, if we take into account sentences with two quantifiers. Let us consider (4): if we adhere to the Montagovian treatment of quantified noun phrases, then the analogues of (3a)-(3d) would be (4a)-(4d). Every man loves a woman λM.M(λx.∃y.(woman(y)&love(x,y))){λQ.∀x.(man(x)→Q(x))} λM.M(λy.∀x.(man(x)→love(x,y))){λQ.∃y.(woman(y)&Q(y))} λQ.∃y.(woman(y)&Q(y)){λy.∀x.(man(x)→love(x,y))} λQ.∀x.(man(x)→Q(x)){λx.∃y.(woman(y)&love(x,y))}
(4) (4a) (4b) (4c) (4d)
If we disregard felicity conditions, i.e if we replace {} by simple parentheses, then (4a) would reduce to (4a') and (4b) to (4b'), while (4c) in turn to (4c') and (4d) to (4d'). (4a') then further reduces to (4c') and (4b') to (4d'); but (4c') and (4d') are substantially different, hence different TFA's of (4) lead not only to different felicity conditions, but to quite different propositions. λQ.∃y.(woman(y)&Q(y))(λy.∀x.(man(x)→love(x,y))) λQ.∀x.(man(x)→Q(x))(λx.∃y.(woman(y)&love(x,y))) ∃y.(woman(y)&∀x.(man(x)→love(x,y))) ∀x.(man(x)→∃ ∃y.(woman(y)&love(x,y)))
(4a') (4b') (4c') (4d')
This implies that in this case TFA is not just a matter of felicity conditions, it is something that results in different orders (and hence different scopes) of quantifiers. Hence (4a) and (4b) (or (4c) and (4d)) can have different truth values; and they will have different values in the case when every man will have a loved woman of his own, but there will be no single woman that would be loved by every man.
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6. Focus = Exhaustive Listing? Besides the subject-predicate character of the topic-focus pattern, there is another characteristic feature of the pattern, namely that focus in some sense has the character of exhaustive listing. If we utter (2), then what we say is not only that there is a man who walks, but also that the walking man is the only entity that walks. The only is, of course, not quite determinate: the range of entities with respect to which it is meant may be in various senses determined by the context. If we, following Rooth (1985) and Krifka (ms.), assume that for any expression X there is a class ALT(X) of its alternatives, then we can define the sentence P!(T) (to be understood as the "unique" predication of P of T) by the following prescription: ÂP!(S)Â=T iff ÂP(S)Â=T & ∀P'[P'∈ALT(P)&ÂP'(S)Â=T→ÂPÂ=ÂP'Â] However, it would seem to be more plausible to consider ALT as operating on the level of semantics rather than on that of syntax. If we write ÂXÂI for the extension of X under the interpretation I, and if I[P/p] denotes the interpretation which is like I with the only possible exception that it assigns p to P, then we can write4 ÂP!(S)ÂI=T iff ÂP(S)ÂI=T & ∀p.[p∈ALT(ÂPÂI)&ÂP(S)ÂI[P/p]=T→ÂPÂI=p] In this case ALT is a function mapping elements of the model structure on classes of such elements. It is reasonable to assume that if p is an element of a domain D, then ALT(p)⊆D. In the simplest case we may let ALT(p)=D, i.e. we may let the set of alternatives coincide with the whole domain. In such a case P!(S) says the same as λf.f(S)={P} or ιf.f(S)=P; i.e. it says that P is the only property instantiated by S. For nontrivial choices of ALT, P!(S) says that P is the only one of some restricted classes of properties that are instantiated by S. Let us now forget (for the sake of simplicity) about presuppositions and let us consider (3) from the point of view of exhaustiveness of focus. The four basic readings are
TOPIC AND FOCUS IN A FORMAL FRAMEWORK λy.love(Mary,y)!(John) λx.love(John,x)!(Mary) λf.f(Mary)!(λy.love(John,y)) λf.f(John)!(λx.love(x,Mary))
243 (3a') (3b') (3c') (3d')
Let us first turn our attention to (3c') and (3d'). (3c') says that the class of all Mary's properties is instantiated by the property of being loved by John, and that none of its alternative is; (3c') says that the class of all John's properties is instantiated by the property of loving Mary, and that none of its alternative is. As any property surely belongs to more than one class of properties, we clearly need a nontrivial notion of an alternative. However, it seems to be clear what should count as such an alternative: any class of all the properties of an individual. If we conceive alternatives in this way, then (3c') says that Mary is the only individual loved by John and (3d') says that John is the only individual that loves Mary. In the case of (3a') and (3b') the need of a nontrivial notion of an alternative is also quite evident (to love Mary is surely in no case the only property of John), but in this case no plausible notion seems to be at hand. The problem is that under the standard treatment of properties, any class including an individual (or any function from possible worlds to classes of individuals such that its value in the actual world includes the individual) is considered a property of the individual; so any individual is sure to instantiate a vast amount of properties. It seems that the trouble is grounded partly in the very nature of properties and partly in the way in which properties are approached within modern logic. (If we, contrary to the usual way, treated properties as primitives and individuals as classes of properties, then the problem might be with (3c') and (3d') rather than with (3a') and (3b').) Hence it seems to be in general more appropriate to consider ALT nontrivial, to add it as a new element of the model structure that can be changed by the ongoing utterances. From such a notion there leads a direct path to both the concept of stock of shared knowledge as discussed by Sgall et al. (1986), and, on the other hand, to the dynamic notion of semantics to be discussed below. 7. Falsity vs. Inappropriateness Combining {} and ! we come to the following definition
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ÂP!{S}ÂI = T iff ·S·I = T & ÂP(S)ÂI=T & ∀p.[p∈ALT(ÂPÂI)&ÂP(S)ÂI[P/p]=T→ÂPÂI=p] = F iff ·S·I=T & (ÂP(S)ÂI=F ∨ ∃p.[p∈ALT(ÂPÂI)&ÂP(S)ÂI[P/p]=T&ÂPÂI≠d]) = 0 iff ·S·I=F This seems to be an adequate expression of the way in which a sentence can be considered as articulated out of the topic and focus: it takes into account both the predicative character of the articulation, and the exhaustive character of the focus. We may distinguish three cases of situations in which P!{S} comes to be false: 1. P!(S) is false, whereas P(S) is not. In this case we may speak about a failure of exhaustiveness. An example of a sentence which would not be true purely due to the failure of exhaustiveness is German is spoken in AUSTRIA. Disregarding TFA the sentence is surely true; however, with the indicated stress and the consequent TFA it is a sentence the utterance of which may cause a serious misguidance (suggesting that Austria is the country, or at least the most representative country in which German is spoken). 2. P{S} is not true, whereas P(S) is. This is the case of failure of presupposition. A case of presupposition failure is the sentence François MITTERAND is the present king of France. Disregarding TFA the sentence would not be true (notice, however, that in that case no presupposition would fail). With the indicated TFA (i.e. with the present king of France in the topic) it is not simply false, it makes the hearer wonder what the speaker is talking about. 3. P(S) is false. In this case we can speak simply about failure of the subject matter. Hence we have three levels of falsity (or, better put in a weaker way, of a breakdown in communication). The extreme, straightforward level is the failure of the subject matter; with respect to the other two levels it is dubious whether it is appropriate to speak about falsity at all. In the case of the falsity of presupposition, the sentence is usually regarded not as false, but rather as lacking a truth value. In the case of the failure of exhaustiveness the falsity is even more subtle.
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8. Negation A positive sentence is usually considered as a means of asserting that its predicate holds about its subject; a negative sentence as that of asserting that this is not the case. If we use the classical predicate calculus, then a positive sentence is understood as P(S), whereas its negative as ¬P(S), as being true just in case P(S) is false. However, we have seen that the semantically relevant subject-predicate patterning of a sentence is a matter far from being this simple. We have seen that TFA makes it possible to make almost any part of a sentence into a semantic subject, and that it is this part which is usually connected with a presupposition. There is a corresponding reading of its negative counterpart for every reading of a positive sentence, hence in the case of (5), the negation of (3), we have the following possibilities5 John does not love Mary ¬ λy.love(y,Mary)!{John} ¬ λx.love(John,x)!{Mary} ¬ λf.f(Mary)!{λy.love(John,y)} ¬ λf.f(John)!{λx.love(x,Mary)} ¬ λf.f(love(John,Mary))!{}
(5) (5a) (5b) (5c) (5d) (5e)
However, these do not exhaust the readings of the negative sentence: there are, moreover, readings in which the negation is "internal", and which should thus be considered as cases of positive predication of a negative predicate. These additional readings are (5f) and (5g). λf.f(Mary)!{λy.¬love(John,y)} λf.f(John)!{λx.¬love(x,Mary)}
(5f) (5g)
Note that the presupposition of (5f) is that there is someone whom John does not love, and that of (5g) is that there is someone who does not love Mary; hence these two readings are really different from all the previous ones. If no stress is on John, then (5b), (5d) and (5g) are ruled out, and thus, leaving the thetic reading (5e) aside, we have three basic readings of (5), namely (5a), (5c) and (5f) (cf. Hajièová, 1984).
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What is beyond discussion is that the negation of a reading of a sentence is true if the reading of the sentence is false due to the fallacy of the subject matter. The cases of the other two fallacies are open to discussion: it seems to be commonly accepted that in the case of presupposition failure the relevant sentence lacks a truth value; hence its negation also lacks a truth value. 9. Dynamic Semantics There is a background, an environment, of an utterance which is determined by the context of the utterance, especially by the utterances immediately preceding it. Any utterance changes this background. If we see the background as a kind of a stack (a stock of shared knowledge, as Sgall et al., 1986, put it), then we can say that an utterance may add new items and in so doing it may concurrently force the less salient items out of the stack. We may see these changes in the environment as a mere side-effect of discourse; it is, however, ever more clear that they should be rather seen as something quite essential. One way to account for this dynamic aspect of language has been formulated within the framework of the dynamic logic due to Groenendijk and Stokhof (1989a; 1989b). This theory identifies the phenomenon of the environment's influencing of individual utterances (and then itself being changed by them) with an assignment of values (discourse referents) to special kind of terms (discourse markers). This engenders an essential perspectival change upon the meaning of a sentence: the meaning is now no longer considered as a truth value or a class of possible worlds, but as a medium of changing environment, hence a function from states of the environment into states of the environment. In this way the semantic account of natural language more nearly approaches that proposed earlier for programming languages - natural language becomes to be envisaged as an implicit command language instead of as a declarative language. Each formula of the dynamic logic is associated with a set of ordered pairs of assignments (of objects to discourse markers); we shall denote the set of pairs of assignments associated by F as [F]. If ∈[F] for a formula F, then this means that the evaluation of F with g as the "input environment" succeeds, and yields g' as the "output environment". If ∈[man(a)] (where a is a discourse marker), then g is an assignment which does not associate a non-man with a (i.e. that g(a) either is a man or it is undefined), and g' is like g with the single possible difference that it
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associates a man with a. ÂFÂ, the truth value of F, can be now considered as T if and only if [F] is nonempty. The rule for dynamic conjunction is as follows: ∈[F&G] iff there is an assignment h such that ∈[F] and ∈[G]. This means that & no longer plays the role of the usual classical connective within dynamic logic, but has become instead an operator of concatenation. The evaluation of a formula F changes the environment E of its evaluation to a new environment E'; the meaning of F is the way in which it changes the environment. If the evaluation of F changes E to E' and that of G changes E' to E'', then the evaluation of F&G changes E to E''. This indicates that F&G need not be the same as G&F: if e.g. F changes E1 to E2 and fails in E3 and if G changes E2 to E4 and E1 to E3, then F&G changes E1 to E3, while G&F fails in E1. This brings about a new kind of sensitivity which may be utilized for capturing the linear order of items in an utterance. However, and here is where TFA may enter the scene, it is not the surface word order that is really relevant, it is rather the "deep" one, the one corresponding to the scale of communicative dynamism (with topic being always less dynamic than focus). The dynamic framework seems to be well suited for the treatment of TFA; it seems that topic and focus can be treated as two ongoing utterances. If we interpret a sentence and fail during the evaluation of the topic, the sentence is meaningless or at least inappropriate; whereas when we succeed in evaluating the topic, but fail during the evaluation of focus, the sentence is simply false6. However, we have so far treated topic and focus as the subject and the predicate of a single sentence; to accommodate our treatment within the dynamic framework we would have to treat them as two sentences uttered subsequently. But this can in principle be done: it is enough to realize that to say that P(S) is true is to say that there exists an assignment of a value to x that satisfies P(x) & x=S. The two new modes of predication we have denoted by {} and !() can now be turned into two new modes of conjunction (or, better, concatenation): the idea is that F}&G (corresponding to G{F}) has a truth value iff F is true, and it is true iff F&G is; F&!G is true iff F&G is true and F&G' is true for no nontrivial alternative G' of G. However, the embodiment of these ideas into the dynamic framework is far from being trivial. The embodiment of }& meets the complication that dynamic logic is essentially two-valued. We may
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introduce }& by the rule below; however, this rule does not establish a real semantics for }&, since it does not specify the value [F}&G] and hence is not applicable recursively. ÂF}&GÂ = T iff [F&G] ≠ ∅ = F iff [F] ≠ ∅ and [F&G] = ∅ = 0 iff [F] = ∅ Far deeper problems concern the embodiment of &!. If we articulate the above mentioned idea, we have ÂF&!GÂ = T iff ÂF&GÂ = T & ∀G'[G'∈ALT(G)&ÂF&G'Â=T→ÂGÂ=ÂG'Â] But if ÂGÂ is, as so far, the truth value of G, then ÂGÂ=ÂG'Â is a condition far too weak to be satisfactory. We may try to substitute [G] for ÂGÂ and so we can write ∈[F&!G] iff ∃k ∈[F&G] & ∀G'[G'∈ALT(G)&∈[F&G']→[G]=[G']] but neither is this what we need; since for any two sentences G, G' containing no discourse markers [G]=[G'] trivially. Moreover, as Krifka(ms.) duly points out, the notion of alternative should depend on the environment of evaluation, so that the rule should look somewhat as follows: ∈[F&!G] iff ∃k ∈[F] & ∈[G] & ∀G'∀h[G'∈ALTk(G)&∈[G]→"G≈G'"] Leaving technical problems aside, we can indicate the way in which (1) should be rendered in the dynamic framework by (1''), while (2) would be rendered by (2''). (a=John) }&! walk(a) walk(a) }&! (a=John)
(1'') (2'')
The first formula is to be true if there is someone who is John, if he walks, and if everyone who is John walks. Hence it is to be true just when walk!(John) is true. The second formula is to be true if there is someone
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who walks, if it is John, and if everyone who walks is John; this means that it is to be true just when λf.f(John)!(walk) is true.
10. The Tripartite Structure Partee (1991) has stressed the role of the notion of a tripartite structure as a universal pattern usable also for the analysis of TFA. A sentence of the shape Det NP VP can be considered as a tripartite structure consisting of an operator (determiner), a restrictor (noun phrase) and a nuclear scope (verb phrase). Such a sentence is true (w.r.t. a possible world) if the extension of the restrictor and that of the nuclear scope are in a relation determined by the operator. Partee claims that the tripartite pattern is of greater generality than simply to cover the case of the basic grammatical structure of simple sentences; and she claims that the pattern is also useful for the analysis of the sentence considered not as a grammatical structure, but rather as a structure consisting of a topic and a focus. This is in accordance with what has been claimed above: that topic and focus are "in fact" subject and predicate. Moreover, what has been stated with respect to the exhaustive listing character of the focus means that the determiner that is to be implicitly present is of the kind of only7. The concept of a tripartite structure is closely connected with the theory of generalized quantifiers. If both the restrictor and the nuclear scope can be considered to stand for a class, then the determiner can be considered as a relation between sets or as a function assigning a set of sets to a set. The determiner a can be, for example, considered as a function that to every set s assigns the class S of sets such that s'∈S holds just in case s and s' are not disjoint; A man walks is then true if this relation holds between the extension of man and that of walks. However, the picture that a sentence of the structure Det NP VP is true iff the extension of its NP and that of its VP stand in the relation determined by its Det is plausible provided that we disregard TFA. Let us consider the sentence A MAN walks. Its grammatical structure leads to its analysis as the generalized quantifier a applied to man and walk, which renders the sentence true iff the set of men is not disjoint with the set of walkers. However, the TF-tripartite structure of the sentence leads rather to
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its analysis as an implicit determiner only applied to walks and a man, rendering the sentence true iff the set of walkers is included in a set consisting of a single man. But to treat the combination of topic and focus as a tripartite structure based on the implicit generalized quantifier only we would have to interpret a man as a set, which seems to be implausible. The general picture is that the operator of the "grammatical" tripartite structure somehow moves into the restrictor of the "topic-focus" tripartite structure making place for only. The problem is that while originally we have a determiner (e.g. a) of the type (in an extensionalized Montagovian notation) <<e,t>,<<e,t>,t> applied to two predicates (e.g. man and walk) of the type <e,t>, then we have the determiner only applied to a quantifier of the type <<e,t>,t> (a man) and a predicate (walk); and no plausible type-theoretical treatment seems to be at hand. We would need a man to be of the type <e,t>; but this would mean that we would have to consider a as of the type <<e,t>,<e,t>>. One way to substantiate considering a man as an expression of the type <e,t> is to consider it as having "distributed reference", referring to a class of individuals, not, however, to one definite class, but rather to any element of a class of classes of individuals, namely to any one-element class of men. Such a view engenders a substantial change to the whole semantic framework: expressions come to be considered as having alternative references and a sentence is considered true if there exists at least one reference that renders it true. However, this change in the framework is not totally alien to the spirit of contemporary semantics: it is in fact quite close to the basic idea of DRT. If we consider man as capable of referring to any class of men, then a can be considered as a filter retaining some of these interpretations (the one-element ones) for a man and filtering out the others. That this treatment is universally available for every monotone increasing quantifier follows from the following consideration. If we denote the standard interpretation of Det, a function mapping sets on classes of sets, as ÂDetÂ, then the truth value of the sentence will be ÂDetÂ(ÂNPÂ)(ÂVPÂ). ÂeveryÂ(X) will be {Y |X⊆Y} for every set X; ÂaÂ(X) will be {Y|X∩Y≠∅} for every set X. The quantifier ÂDetÂ(X) is supposed to "live on" X for every Det and S; this means that Y∈ÂDetÂ(X) if and only if Y∩X∈ÂDetÂ(X)8. If we denote the class {Y|∃Z.(Z∈ÂDetÂ(X) & Y=X∩Z)} as ÂDetÂ*(X), then clearly Y∈ÂDetÂ(X) iff Y∩X = ÂDetÂ*(X), i.e. iff ∃Z.(Z∈ ÂDetÂ*(X) & Z=Y∩X). Moreover, if ÂDetÂ(X) is monotone increasing,
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then Y∈ÂDetÂ(X) iff ∃Z.(Z∈ÂDetÂ*(X) & Z⊆Y∩X); hence, as Z∈ÂDetÂ*(X) implies Z⊆X, iff ∃Z.(Z∈ÂDetÂ*(X) & Z⊆Y)9. Let us consider the sentence Every man walks. Let us consider Âman the class of all men and Âwalk the class of all walkers. ÂeveryÂ(X) is the class of all supersets of X; hence ÂeveryÂ*(X) = {Y|∃Z.(Z∈ÂeveryÂ(X) & Y=X∩Z)} = {Y|Y=X} = {X}. This means that the sentence is true iff there is an X∈ÂeveryÂ*(ÂmanÂ) such that X is included in ÂwalkÂ. Hence, as the only member of ÂeveryÂ*(ÂmanÂ) is ÂmanÂ, i.e. the set of all men, the sentence is true if and only if the set of all men is included in the set of all walkers. Similarly for A man walks: ÂaÂ(X) is the class of all the sets not disjoint with X; hence ÂaÂ*(X) = {Y|∃Z.(Z∈ÂaÂ(X) & Y=X∩Z)} = {Y|Y⊆X & Y≠∅}. Thus, the sentence is true iff there is an X∈ÂaÂ*(ÂmanÂ) such that X is included in the set of all walkers; hence, iff there is a nonempty subset of the set of all men that is also a subset of the set of all of walkers. These ideas have been in a somewhat more cumbersome way outlined by Peregrin and Sgall (1986) and also by Peregrin (1987). It has been shown that they can be useful for the purpose of accounting for TFA in a dynamic framework based on the DRT-like ideas (although DRT did not serve as its explicit foundation). Here we can see that they can illuminate the interplay between the "grammatical" tripartite pattern and that which results from TFA considerations.
11. Conclusion The aim of the present paper has not been to develop a definite formal theory of TFA, its aim was rather to overview the different possibilities of the formal description of topic and focus holding the Prague notion of TFA particularly in regard. From the logico-semantic point of view there are three aspects of TFA which are relevant. First, differences in TFA mean differences in the "deep word order" which may result in differences in the scopes of quantifiers. Second, topic is usually connected with a presupposition; hence different TFA's may lead to different felicity conditions (to different classes of possible worlds in which the sentence has a truth value). Third, at least
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under some conditions, TFA purports the exhaustive listing readings with respect to focus, it causes a sentence to behave ina way as if it contained an overt only. We have seen that some of these aspects can be accommodated as fairly straightforward extensions of the predicate calculus; but we have also indicated that dynamic logic might be a more suitable framework. We have, moreover, shown that picturing the TF-articulated sentence as a tripartite structure is fruitful, and we have floated the possibility of merging this view with that of dynamic semantics.
Notes 1
2 3
4
5 6
7 8 9
The author thanks Petr Sgall, Eva Hajièová, Barbara Partee and Mannfred Krifka for their willingness to share their views of the matters addressed here with him. Moreover, the exposition of the Praguian approach to topic-focus articulation presented here would hardly be adequate without the thorough help of Petr Sgall. For the sake of simplicity we shall work with extensions where possible. However, the relativization to possible worlds is obvious. The possibility of the thetic reading of (3) may be disputable due to the proper name in the subject position; in the case of a sentence such as A man loves a woman it would be quite regular. Note that the formal difference between the "syntactic" variant of ALT and the "semantic" one is parallel to the difference between substitutional and objectual quantification. If there are no "nameless" entities, then these notions give the same results. As was found with the thetic reading (3e) itself, its negation (5e) is also not quite regular. This is indeed a simplification: the situation is more complicated due to the existence of presuppositions triggered by (parts of) the focus, e.g. by the object clauses of factive verbs. The situation can be, indeed, more complicated when explicit focalizers other than only are involved. See Barwise and Cooper(1981). ÂDetÂ*(S) is in fact the set of all the sets which Barwise and Cooper(1981) call witness sets for ÂDetÂ(S).
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Materna, P., P.Sgall. 1980. "Functional Sentence perspective, the Question Test, and Intensional Semantics". SMIL 1-2.141-160. Materna, P., E. Hajièová & P. Sgall. 1987. "Redundant Answers and TopicFocus Articulation". Linguistics and Philosophy 10.101-113. Partee, B. 1991. "Topic, Focus and Quantification". Proceedings of SALT I. Cornell Working Papers 10. Peregrin, J., P. Sgall. 1986. "An Attempt at a Framework for Semantic Interpretation of Natural Language". Theoretical Linguisitcs 13.37-73. Peregrin, J. 1987. "Quantification and Reference in Natural Language." Studies in Logic, Grammar and Rhetorics (Bialystok) VI.159-185. Rochemont, M.S. 1986. Focus in Generative Grammar. Benjamins: Amsterdam. Rooth, M. 1985. Association with Focus. Ph.D.-dissertation. University of Massachusets: Amherst. Rooth, M. 1992. "A Theory of Focus Interpretation." Natural Language Semantics 1.75-116. von Stechow, A. 1989. Focusing and Backgrounding Operators. Universität Konstanz. Sgall, P., E. Hajicová & E. Benešová. 1973. Topic, Focus and Generative Semantics. Kronberg: Taunus. Sgall, P. ed. 1984. Contributions to Functional Syntax, Semantics and Language Comprehension. Amsterdam: Bejnamins - Praha: Academia. Sgall, P., E. Hajicová & J.Panevová. 1986. The Meaning of the Sentence in its Semantic and Pragmatic Aspects. Praha: Academia. Tichý, P. 1980. "Logic of Temporal Discourse". Linguistics and Philosophy 3.345-369. Vlk, T. 1988. "Towards a Transduction of Underlying Structures into Intensional Logic". Prague Bulletin of Mathematical Linguistics 50.35-70.
STRUCTURE AND MEANING Jaroslav Peregrin [Semiotica 113-1/2 (1997); roughly original pagination]
Two Approaches to Language It seems that the theories of language of the present century can be classified into two basic groups. The approaches of the first group perceive language as a mathematical structure and understand any theory of language as a kind of application of mathematics or logic. Their ideological background is furnished by logical positivism and analytical philosophy (esp. by Russell, Carnap, Wittgenstein and their followers); and their practical output is Chomskian formal syntax and subsequent formal semantics. The approaches of the other group do not approve of formalization and consider a theory of language closer to psychology than to mathematics. The specific position within this group is occupied by the so-called structuralists (de Saussure, Hjelmslev, Derrida). However, this kind of classification is rather superficial; there is another classification that is really significant, and this classification cuts across standard philosophical schools. What makes the really fundamental difference is the way one views the link of an expression to its meaning. From this vantage point we can distinguish between approaches we shall 1 call semiotic and those we are going to call structural. The approaches we classify as semiotic rest on the assumption that the significance or meaning of an expression does not depend on whether the expression is a part of language or of some other system of signs. Thus, a semiotic approach is characterized by considering language a means of codification of a pre-existing, language-independent world. The scholars who subscribe to this approach see the world as a definite system of objects; and they see expressions as labels which we must stick on these objects if we want to talk about them. This is a naive picture (Quine justly calls it the museum myth); nevertheless a picture that underlies (explicitly or implicitly) many theories of language of the present century. The basic principle of the structural approach to language, on the other hand, consists in the conviction that any significance, or any meaning of an 71
expression comes to it from its being part of the system of language. In contrast to the semiotic notion, this notion of language is based on the reluctance to see language as a kind of nomenclature. The meaning of an expression is, according to it, not a language-independent object casually linked to the expression, rather it is the value of the expression, its position within the system of language or within the language game to be played. However, let us stress that the distinction between the semiotic and the structural view of language is far from coinciding with the distinction between logical positivism and structuralism, or indeed between any two philosophical schools. De Saussure, the father of structuralism, and Wittgenstein, the great dissenter of logical positivism, are personalities representing the structural approach to language most consistently and consequentially (and it is quite surprising how close the opinions of these 2 two otherwise so different scholars in this point are ). The majority of other scholars, including both the avowed followers of de Saussure and those of Wittgenstein, on the other hand, have not succeeded in maintaining this view quite so systematically and consequentially. The aim of the present paper is not only to survey the claims of what we have characterized as the structural approach to language; we would also like to document that such an approach is, contrary to common opinion, that of the most outstanding representatives of analytical philosophy. The main purpose of the paper then is to show how meanings can be considered to materialize out of the oppositions of the system of language.
The Untenability of the Semiotic View Let us begin with neutral, commonly acceptable facts. Language expressions are subject to two kinds of relations: the 'horizontal' relations, relating them to other expressions, and the 'vertical' relations, linking them to their meanings. The terms horizontal and vertical amount to the schematic picture (due, in effect, to de Saussure) of Figure 1. The horizontal and vertical relations are clearly mutually interdependent, and the majority of philosophers would probably agree that they rise and fall, so to say, hand in hand. The main difference
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Figure 1: Horizontal and vertical relations 72
between the semiotic and structural approaches to language is the matter of which kind of these relations is granted primacy. The semiotic view is based on the assumption that it is the vertical relations which are primary. (Let us stress that it is just this primacy that makes incompatible the semiotic with the structural view - that expressions are signs in the sense that they simply are significant - that they 'have meanings' - is a truism that would hardly cause a quarrel.) One of the clearest articulations of the semiotic approach was given by Charles Morris: 'The properties of being a sign, a designatum, an interpreter, or an interpretant are relational properties which things take on by participating in the functional process of semiosis. Semiotics, then, is not concerned with the study of a particular kind of object, but with ordinary objects in so far (and only in so far) as they participate in semiosis.' (Morris 1966: 4) If language has any structure, then it owes it to the fact that its expressions are signs (ibid: 17). The structural view, on the other hand, grants primacy to the horizontal relations; it does not grant meanings any existence independent of the system of language. De Saussure pointed out that language cannot be considered as a nomenclature of real-world objects (de Saussure 1931: 65). Whatever it is that is signified, the signification relation arises from the interdependence of expressions within the system of language. It is thus the horizontal relations between expressions which are primary; and the vertical relations associating expressions with their meanings are derivative. De Saussure writes: '...the language has neither ideas, nor sounds, that existed before the linguistic system, but only conceptual and phonic differences that have issued from the system.' (de Saussure 1931: 65). There is little doubt that it is the semiotic view that accords with common sense; moreover, it seems that this view is also (consciously or unconsciously) adopted by many scholars. However, it is hard to imagine that the view could be correct. De Saussure and some of the other structuralists criticised it explicitely; however, and this is often not noticed, very strong arguments against it are offered also by the results of the work of some of the most outstanding analytical philosophers. In fact, since the inception of analytical philosophy by Frege the semiotic view of language has had to suffer one blow after another. Frege (1892) recognized that if we identify the meaning of an expression with the real-world object which it is felt the expression normally stands for (the meaning of the expression Morning star with the planet Venus, for example), then we shall not be always able to recover the meaning of a complex expression from the meanings of its parts. Frege and Wittgenstein also realized that the 73
elements of language that are primarily significant are not words, but 3 rather sentences, and that the discernment of the meanings of parts of a 4 sentence is 'parasitic upon its structure' . Quine (1960) documented that if there were a relation of reference connecting expressions with objects of the world, then there would not be a possibility of determining it; and Davidson (1977) used this result to show that in such a case the whole concept of reference lacks sense. All these results suggest that it is not possible to hold the semiotic view consistently.
Meaning vs. Synonymy Semantics is by definition the matter of the vertical relations linking expressions to their extralinguistic meanings. However, there are also horizontal relations, relations between expressions, which are semantic in nature. These relations amount to similarity and dissimilarity of meaning, their most clear-cut representative being the relation of synonymy. Within the semiotic approach such relations are considered to be mere auxiliaries: saying that two expressions are synonymous is considered to be a mere shortcut for saying that they are linked to the same object. Not so within the structural view. It is sameness-of-meaning and difference-in-meaning that are primary according to this view; meanings are mere reifications of these relations. In the same way as we posit colours by saying that two objects perceived as in a certain sense similar 'share a colour', we posit meanings by saying that two expressions understood as in a certain sense similar 'share a meaning'. Thus, the existence of meaning is considered as grounded in certain equivalences of expressions. However, what is synonymy, if it is not the linkage to the same object? The answer, crucial for the structural approach, is that it is the sameness of position within the system of language, or the sameness of 'usability' within the language game to be played. It is the language itself and the way we use it that is primary; meanings (as well as other linguistic abstracta) are only our means of representing these primary facts, of 'making sense' of them. Wittgenstein (1953: §654): 'Unser Fehler ist dort nach einer Erklärung zu suchen, wo wir die Tatsachen als "Urphänomene" sehen sollten. D.h. wo wir sagen sollten: dieses Sprachspiel wird gespielt.' And further (ibid: §656): 'Sieh auf das Sprachspiel als das Primäre! Und auf die Gefühle, etc. als auf eine Betrachtungsweise, eine Deutung, des Sprachspiels!' When we use language for the purpose of communication, we come to 74
perceive any expression as a tool more or less suitable for our purposes; we come to see it as possessing a certain value. (The task of an expression may - in a particular case - be seen as representing an object, as being a name; in such a case its value may possibly be identified with the object. But this would be quite a special case.) Expressions which are usable to the same effect have equal values, they are equivalent; and synonymy is primarily just this kind of equivalence. It is illuminating to consider the similarities between language and a game such as chess (and it is characteristic that both de Saussure and Wittgenstein, not to speak of Frege, Husserl, Jakobson and others, made use of this comparison quite frequently). What turns a piece of wood into a chess knight, what makes it meaningful within the chess game, is not primarily some link to a material entity ('knighthood'), but rather the rules according to which it is treated. If there is a 'knighthood', then it owes its being to the rules of the chess game. Similarly, what makes an expression meaningful within the system of language is not its link to a pre-existing meaning, but rather the rules of language; and meanings owe their existence to these rules. The semiotic view is based on the assumption that what is primary are the vertical relations, and that the horizontal relations of synonymy or of likeness and difference in meaning are mere auxiliaries which help us speak about those that are vertical. However, the structural approach recognizes that this is not an assumption adequate to the real functioning of language: what is ultimately primary ('Urphänomene') are the similarities and differences between the usability of expressions, and the concept of meaning is our hypostatic means which facilitates speaking about these similarities and differences and making efficient theories from them. The Indeterminacy of Meaning One of the basic characteristics of the structural approach, as contrasted with the semiotic one, is the fact that it leads to the notion of meaning that is in a sense indeterminate or relativistic. If it is the process of semiosis that establishes the link between an expression and its meaning, then there is no room for any kind of indeterminacy: there must be a determinate, pre-existing object that becomes meaning through semiosis. If it is, on the other hand, the position of the expression within the system of language hat establishes its semantics, then any entity the expression may be seen to stand for can be determined only differentially, as opposed to the entities other expressions stand for. 75
Let us imagine that speaker S1 uses the name 'N' for an object A, while his fellow speaker S2 misuses the same name for another object B. How can they find out that they use the name in different ways? If A and B can be unequivocally pointed at, then there is, of course, no problem; but this is usually not the case. However, even if there is no direct pointing at, the resolution seems to be quite simple. S1 can say: 'You call B "N", but "N" is in fact the name of A!' This possibility is nevertheless limited to the case when both speakers are in possession not only of the name 'N', but also of the names 'A' and 'B'. What about the case when they have no alternative names? In such a case S1 cannot, of course, say 'What you call "N" is not N!'; because for S2 what he calls 'N' surely is N. However, if there is something that is true for A, but not for B, then S1 can make use of this fact. If, e.g., A is green, whereas B is blue, then S1 can put forward the statement 'N is green'; S2 will disagree, and S1 and S2 then have to conclude that they use 'N' in different ways (presupposing that neither of them is colour-blind). Thus S2's misuse of 'N' may be discovered as soon as the speakers encounter a sentence containing 'N' and such that it would be true in case 'N' were A and false in case 'N' were B. If there is no such sentence (and if A and B cannot be distinguished by direct ostension), S1 can never find out that S2 is talking about another object. This is the case of Quine's (1960) illustrious example of a rabbit and its undetached part. Another prominent example is that of the meaning of numerals: all of Peano arithmetic does not allow us to distinguish whether a number is the set of equipotent sets (as Frege, in effect, maintained), or the set of all the numbers smaller than itself (as von Neumann proposed), or a primitive object. This means that if A differs from B in a property expressible in the language in question, the confusion of their names can be discovered. On the other hand, if no such difference exists, then there is no possibility of recognizing the confusion, and it is thus altogether problematic to speak about a confusion. To say that 'N' is used to name two different objects makes sense only with the background of a language which has two different names for the two objects. If we can distinguish the two objects by means of what can be said about them, then the different names can possibly be formed as definite descriptions, otherwise not. This is Quine's inscrutability of reference. This precisely is the contrast between the the structural and the semiotic views. If we accepted the semiotic view, then we could not see any problem in determining whether 'N' refers to A or B even if A and B had precisely the same properties; 'N' would simply unequivocally refer to that object which took part in the relevant process of semiosis. Not so within 76
the structural approach: we may differentiate horizontal links only in terms of vertical ones, and where there is no difference on the horizontal level, there cannot be a difference on the vertical one. Any difference between meanings of two expressions must be based on a difference between the functioning of the expressions within language. However, it would be essentially wrong to interpret this structural indeterminacy as implying that we cannot know what the meanings of our expressions are. The right interpretation is rather that meanings are not definite in the sense presupposed by the semiotic view. There is little doubt that natural numbers do exist; at least to the extent to which objects which cannot be touched and tasted can exist. And we consider them unique existing objects although we know that for all we care they can be identified with objects of various kinds. A number is not one of the things it can be reduced to, it is rather what all of these things have in common, it 5 is a number . And what can be said about numbers, can, more generally, be said about meanings: the meaning of an expression is not one or another definite thing conforming to our usage of the expressions within language; at most it may be said to be what all such things have in common. As de Saussure puts it: 'Language is a form and not a substance' (de Saussure 1931: 122). What is basically relevant for meaning is the structure of language.
Structure and Parts & Wholes Structure is the way in which a whole is composed out of its parts; therefore there is no structure (and also no structuralism) where there are no parts and wholes. This indicates that the notion of structure is meaningful only within the context of a part-whole system. Let us, therefore, survey several important facts concerning part-whole relations and part-whole systems. To make the summarization rigorous, we shall need some mathematics, but to make it comprehensible, we shall not go into any mathematical details. A part-whole system can be considered as a set U of items plus a set of certain 'operators' each of which maps an n-tuple of items of U (parts) on an item of U (whole). Hence a part-whole system can be considered as the ordered pair ∈I>where U is a set and each Oi is a (in general n partial) function from the Cartesian power U into U; in other words a part-whole system is a certain (partial) algebra. If e=Oi(e1,...,en) for some e,e1,...,en∈8DQGIRUDQi∈IWKHQZHVD\WKDWHLVFRPSRVHGRXWRIH1,...,en (in the way Oi). L
77
What kind of algebra is a part-whole system? Can any partial algebra be meaningfully considered as a part-whole system? Certainly not - if the operators of an algebra are to be considered as operators of compositions, then they must fulfil certain restrictions. If A= ∈I> is an algebra, let us define the relation PA in such a way that e PAH LIIWKHUHLVDQ i∈I VXFK that e'=Oi(...,e,...). In case A is a part-whole system, PA is the relation of an immediate (proper) part - the relation of a general (proper) part can then be defined as its transitive closure. This means that A can be a part-whole system only if PA fulfils certain conditions - if it is acyclic (its transitive closure is antisymmetric) and if it fulfils some other more intricate 6 conditions, which we are not going to discuss here . Hence we may say that a part-whole system is an algebra A, for which the relation PA has the properties appropriate for the relation of an immediate (proper) part. One basically important thing concerning part-whole systems is that the partwhole structure offers the basis for induction: for a property to be instantiated by all the elements of a part-whole system, it is enough to be instantiated by all the simple elements (those which have no parts) and to be inherited from parts to wholes. Several important concepts are now intrinsically connected with the notion of part-whole system, especially the concepts of substitution, interchangeability and compositionality. Let us outline general definitions 7 of these notions . (1) A substitution of an item ea for an item eb within an item e, denoted by e[ea/eb], is the item that is composed in the same way as e with the single exception that eb is used everywhere instead of ea. The precise inductive definition could run as follows: (basis) if e is simple, then either e is ea and then e[ea/eb]=eb, or e is not ea and then e[ea/eb]=e; and (induction step) if e=Oi(e1,...,en), then either e is ea and then e[ea/eb]=eb, or e is not ea and then e[ea/eb]= Oi(e1[ea/eb],...,en[ea/eb]). The restrictions imposed on algebra to be a part-whole system should guarantee that this definition is correct: that Oi(e1[ea/eb],...,en[ea/eb]) is the same object as Oj(e1'[ea/eb],...,em'[ea/eb]) whenever Oi(e1,...,en) is the same object as Oj(e1',...,em'). (2) If R is a binary relation on U, then ea and eb are said to be interchangeable w.r.t. R iff e[ea/eb] R e for every item e. If R is a binary relation on U, then the minimal relation R' containing R and such that any two elements equivalent according to R' are interchangeable w.r.t. R' is called the interchangeability closure of R. The interchangeability closure of a binary relation is always existent and unique; an interchangeability closure of an equivalence is again an equivalence. (3) A function F on a part-whole system is called compositional iff L
78
there is a general rule how to recover the value of F for the whole from its YDOXHV IRU WKH SDUWV LH LI WKHUH LV IRU HYHU\ i∈I, D IXQFWLRQ )i such that F(Oi(e1,...,en)) = Fi(F(e1),...,F(en)). (This, as well as all the following equations, is to be understood in the extended sense, such as stating that the equated values are either equal or are both undefined.) It can be proven that the function F is compositional if and only if its kernel (i.e. the relation KERF such that e KERF e' iff F(e)=F(e')) contains its own interchangeability closure; therefore the following definition seems to be plausible: a relation is called compositional iff it contains its own interchangeability closure. Thus, a function is compositional if and only if its kernel is. If R is a compositional relation and e1 R e2, then e[e1/e2] R e for every e; hence if F is a compositional function and F(e1)=F(e2) then F(e[e1/e2])= F(e) for every e. Compositional functions and relations occupy a somewhat outstanding position among other functions on a part-whole systems. (If we see partwhole systems simply as algebras, then compositional functions coincide with homomorphisms and compositional relations with congruences.) We shall see later that there may be good reasons for approving only compositional relations and functions; and in such a case we must consider any non-compositional relation or function a mere approximative or incomplete representation of a compositional one. This poses the problem of 'compositionalization', of finding a relation (function) that could be considered a reasonable approximation of a given relation (function). This problem is easily solvable for relations, and not so easily for functions. Every relation can be extended to a compositional relation, the smallest of such relations always being its interchangeability closure. R is clearly compositional if and only if it contains (and consequently is identical with) its own interchangeability closure. A compositionalization of a function is a more problematic matter. In contrast to the case of relation, it is surely not the case that every function could be extended to a compositional function. However, a function can be refined to a compositional function. This means that if F is a function from a part-whole system into a range S, then there exists a function F' with the range S' and a contraction c of S' to S such that c(F'(x)) = F(x). More precisely, if F is a function from a part-whole system P into a set S, then there is a part-whole system P', a compositional function F1 from P to P' and a compositional function F2 from P' into S such that F is the composition of F1 and F2. Thus, a noncompositional function can be decomposed into two compositional functions: but this may require an intermediary system of objects (P'). This indicates that turning an noncompositional into a compositional function may necessitate new kinds 79
of objects.
An Example Let us imagine we have the objects a, b, c, d, e, f such that c is composed out of a and b and d is in the same way composed out of b and a; and that f is composed out of c and e, and g is in the same way composed out of d and e. We can imagine the objects as the geometrical shapes of Figure 2. Hence we have the part-whole system >, where U={a,b,c,d,e,f,g} and O1 and O2 are partial binary function on U such that O1={<,c>,<,d>} and O2={<,f>,<,g>}. d is an immediate part of g, and a is an immediate part of d; hence a is a part of g, but it is not its immediate part. The substitution f[c/d] of d for c in f yields g; the substitution f[d/c] of c for d in f yields f (because d is not a part of f); the substitution f[c/a] of a for c in f yields nothing at all (because O2(a,e) is undefined). Let R be the equivalence on U that yields the two equivalence classes {a,b,c,d,e} and {f,g}. a and b are interchangeable w.r.t. R and so are c and d; but a and c are not interchangeable w.r.t. R, since, e.g., f[c/a] is undefined. This means that a and c are not interchangeable w.r.t. R, although a R c; hence R is not compositional. The interchangeability closure of R is the compositional equivalence R' that yields four equivalence classes: {a,b}, {c,d}, {e} and {f,g}. /HWXVLPDJLQHZHKDYHWKHIXQFWLRQ)WKDWDVVLJQWKHYDOXH.WRa, b, c, d and e and the value ß to f and g. Then F is not compositional. To see
Figure 2: Example of a part-whole system 80
this, let us assume that there is a function O2' such that F(O2(x,y))=O2'(F(x),F(y)) for every x and y of U; then ß = F(f) = F(O2(c,e)) = O2'(F(c),F(e)) = O2 .. 22'(F(a),F(b)) = F(O2(a,b)), but O2(a,b) is undefined. (The noncompositionality of F follows directly also from the noncompositionality of KERF - and this noncompositionality was proved DERYH 7RPDNH)FRPSRVLWLRQDOZHKDYHWRUHILQHLWWRVSOLW.LQWRWKUHH values: a and b will be assigned the first of them, c and d the second, and e WKHWKLUG+HQFHZHPD\KDYHWKHFRPSRVLWLRQDOIXQFWLRQ) WKDWDVVLJQV.1 to a and b.2 to c and d.3 to e and ß to f and g. In terms of F we can distinguish two kinds of objects: if we say x is a(n) y instead of F(x)=y and if we say non-house LQVWHDGRI.DQGhouse instead of ß, then F allows us to articulate statements to the effect that an object of U either is or is not a house. But to make this distinction compositional we need a finer classification of non-houses: we must distinguish among an oblong, a square and a triangle. The abstract objects ('categories') oblong .1), square .2) and triangle .3), which have been employed to 'refine' the object non-house . FDQWKXVEHORRNHGXSRQDVEURXJKWLQWREHLQJE\ the compositionalization of the opposition between houses and nonhouses.
Language as a Part-whole System It is clear that language can be seen as a part-whole system; and that it is just this view that underlies the very idea of grammar. To compile a grammar means to explicitly reconstruct the factual phenomenon of language as a part-whole system. A meaning-assignment is then a compositional mapping of the part-whole system of language on a system of denotations. What the structural approach to language claims is that meanings are kind of outgrowths of the oppositions present within the system of language. In algebraic words, that the meaning-assignment is somehow induced by some kind of binary relations on the part-whole system of language. We may say that meanings owe their existence to synonymy, that they are 'reifications' of synonymy; in the sense of Quine's (1992) 'there is no existence without equivalence'. But what we are going to argue for here is a much less trivial thesis: that meanings are 'reifications' of relations much more elementary than that of synonymy, in the paradigmatic case of the opposition between truth and falsity. Sameness of truth values is an equivalence; and it directly reifies into the two truth values. However, if we try to distinguish between truth and falsity compositionally, we may 81
need many more values, and what we argue for is that meanings can be plausibly considered as just these values. To say that sameness of truth values is a compositional relation is to say that language is extensional. The point is that compositionality of the sameness of truth values implies that for every sentences S,S1,S2 the sameness of truth values of S[S1/S2] and S follows from that of S1 and S2; and this is just what can be considered as the general definition of extensionality. However, the profusion of modal and intensional contexts of our language documents that language is in general far from being extensional: to take the most traditional example, Necessary S1 may well differ in truth value from Necessary S2 (i.e. from Necessary S1 [S1/S2]) even in the case when S1 and S2 do not differ. Hence the relation of sameness of truth value of sentences of our language is in general surely not compositional. We have seen what we have to do when we want to make a function compositional: we have to refine its range. This means that if we want to distinguish between truth and falsity compositionally, we do not make do with the two truth values themselves. In order to be able to classify sentences into true and false in a compositional way, we need a much more fine-grained ('auxiliary') classification. If the truth value of Necessary S1 differs from that of Necessary S2, then S1 and S2 must have different values even if these have the same truth value. From the viewpoint of truth, S1 and S2 do not differ when taken as wholes per se, but they differ when taken as parts, as building-blocks. This means that as soon as we take compositionality as a principle, then the value issuing from the differentiation between truth and falsity is a value quite different form the truth value. The sameness of the former implies the sameness of the latter value; but not vice versa. Hence as soon as we take compositionality as a principle (and if we realize the nonextensionality of our language) we cannot account for truth without an elaborate system of values, without meanings.
Infinity and Compositionality We have shown that if we want to treat truth compositionally, we need meanings. What we have not shown so far is why compositionality is crucial. To show this, we must consider the very role of abstract entities for our comprehending our world. There are 'real' objects we perceive, and there are abstract objects we posit. We may posit objects explicitly with the knowledge that they will 82
help us formulate efficient theories of our world, or we may posit them as 8 tools implicit to our very act of comprehending the world . The simplest situation of positing an abstract object is that of grasping a similarity between 'real' objects as something they have in common. However, there are also less trivial ways of positing objects. One of those situations is connected with the way we treat infinity; and it is just this situation that is crucial for our understanding of the nature of meaning. Our view of our world is so infiltrated by contemporary mathematics based on Cantor's set theory that we count infinite sets among the uttermost realities. However, on second thought it is quite clear that there is no infinite set we could really encounter within our 'real' world. Everyone of us can be confronted (at once, but also during the whole span of his life) with at most a finite number of objects. The only aspect of reality that may be felt as amounting to infinity is unlimitedness, the possibility to continue various processes over and over without any limit. In other words, there is no actual infinity, there is at most potential infinity. There is no 'real' infinite set of objects, but there are 'real' devices which make it possible to 'generate' objects without any limitation ('ad infinitum'). We can never encounter the set of all natural numbers; we can at most encounter a rule of the kind of '0 is a number x is a number, then also x' is a number'. The rule in itself is in no way infinite (it is simply a fourteen-word phrase), it is only in a sense felt to insinuate infinity. This nature of infinity has been repeatedly pointed out - probably for the first time by Aristotle, and more recently by several of the most outstanding 9 mathematicians and philosophers of this century - but many theoreticians simply ignore it. However, if there are no real infinite sets beyond those grounded in a finite number of generating rules, then there are also no functions with infinite domains beyond functions defined via finite rules. There is no way of defining a function on an infinite domain, save by giving its values for a finite number of basic elements and then giving a finite number of rules to compute values for new elements from those already computed. We may define function f with its domain equal to the set of all natural numbers by defining f(0) and giving the recipe how to compute f(x') out of f(x), or we may base the definition on a more complicated system of rules generating natural numbers, but it is impossible to define such a function directly, bypassing generating rules altogether. If we understand generating rules as rules of composition of wholes out of parts (which can be in some cases taken literally and in others as an illuminating metaphor), then we can say that there is no 'real' infinite set without a part-whole structure; and that there is no 'real' function on such a 83
set that would not follow the part-whole structure, i.e. which would not be compositional. Thus, reality is restricted to part-whole systems and to 10 functions compositional on part-whole systems. Due to the Cantorian indoctrination we comprehend the world in terms of functions that are not necessarily compositional. This leads to the conviction that any opposition expressed in terms of a classificatory function is a function with merely two values. The opposition between truth and falsity means simply a function that assigns truth to the true statements, and falsity to the false ones. However, we have seen that such an opposition may call for a much wider range if it is to be accomplished compositionally. We have seen, in our oversimplified example, that to distinguish between houses and non-houses compositionally, we cannot make do with two values, that we need a finer system of values. Similarly to distinguish between truth and falsity compositionally we need a much finer system of values than the pair of the two truth values themselves: we need meanings. We might be tempted to say that meanings result from compositionalization of truth. However this statement may be misleading: it seems to suggest that there is a direct, noncompositional characterization of truth which we would compositionalize positing meanings. But there is in fact no such direct characterization. Maybe there is a God who is able to grasp infinity directly and who thus perceives our positing meanings in this way; but for us the compositional way is the only way. Any more direct differentiation is our mythical creature that is licensed by set theory which embraces actual infinity and which can only be talked about, but never really demonstrated. We may well talk about the infinite class of all the expressions of our language, but such talk makes real sense only because we can understand it as a 'shortcut talk' about our ability to form new expressions 'forever', or about the grammar of our language. And we can also well talk about direct, noncompositional assignment of truth values to sentences, but we can make real sense of this talk only in so far as we can understand it as a 'shortcut talk' about (compositional) meaningassignment. There is no possibility of characterizing true statements without specifying the meaning of some basic expressions and specifying how meanings of wholes depend on meanings of their parts. Two Approaches to Language Revisited The concept of meaning is trivially interrelated with that of synonymy: if we have meaning we eo ipso have synonymy ('sameness of meaning'), and if we have synonymy we eo ipso have meaning ('value shared by 84
synonymous expressions'). The semiotic approach tries to offer an independent explication of the concept of meaning: meaning of an expression is a real-world object casually connected with the expression; two expressions are hence synonymous if they are so linked to the same objects. The structural view proceeds the other way round: it tries to reduce the concept of meaning to the concept of synonymy. Synonymy is, according to this approach, sameness of role within the system of language or within a relevant language game, and this in turn is the role of the building-block within the account for certain oppositions, especially for the opposition between truth and falsity. Thus, the meaning of an expression is a value, it is the value that determines the kind of brick the expression is, how it can be utilized within the opposition-based building of language. We have concentrated on meaning; but also other linguistic abstracta can be accounted for in similar terms. Grammatical categories, for example, can be conceived analogously, as a means of differentiating well-formed expressions from mere nonsensical sequences of words. Evidently the differentiation alone would amount to no more than two categories (expressions and non-expressions); if it is, however, accomplished compositionally (i.e. if the category of a complex expression is uniquely determined by the categories of its parts), then what is brought into being are grammatical categories in the usual sense. Thus, compositional differentiation is responsible not only for meanings, but for the whole of what de Saussure calls linguistic reality. That meanings and other linguistic abstracta issue from oppositions does not contradict the fact that the oppositions do not manifest themselves in language as perfectly sharp and clear-cut, that the boundary between, e.g., truth and falsity is notoriously fuzzy. The concept of synonymy, and hence also that of meaning, is based on a intentional simplification: we consider two expressions synonymous if their likeness exceeds such or another boundary. This may lead us to split meaning into 11 various levels corresponding to some natural boundaries , and eventually it may lead us to deny the reasonableness of the treatment of meanings as 12 objects altogether . However, such a simplification necessarily occurs with every abstraction: to speak about expressions being synonymous does not seem to be more far-fetched than to speak about, say, objects having the same colour. Conclusion There is meaning only in so far as there are synonymous expressions. And 85
expressions are synonymous only if they contribute in the same way to truth, i.e. only if they are intersubstitutive salva veritate (or, more generally, if they are intersubstitutive with respect to the usability of language statements in various language games). Hence there is meaning only in so far as there is truth. On the other hand, there is no truth without meaning. In some simple artificial languages there can be truth without meaning; in a universal natural language comprising an infinite number of truths this is, however, not possible. Truth as a property of statements is a compositional property (for there is, strictly speaking, no 'real' noncompositional property), and as a compositional property it rests on meaning. Meaning is the value of an expression with respect to the opposition between truth and falsity; meaning is the product (or, if one prefers, a by-product) of the 13 compositionality of truth . Meaning is in this sense implicit to truth; and we can make it explicit ('explicate it') in that we make a theoretical reconstruction of language as a part-whole system and if we evaluate its 14 expression to make a compositional (recursive) characterization of truth . Philosophers of diverse provenance claim that the hierarchical structure of abstract entities which are the means of our comprehending 15 our world arise from some kind of associations . What has been said here implies that associations, or at least some of them, need not be accounted for as primary phenomena, rather that they may be considered necessary implicit means of our making sense of oppositions, of our understanding oppositions, since understanding is finitistic and hence compositional. We associate synonymous expressions, but we do so because we want to distinguish between true and false sentences and since synonymy represents the only way to accomplish this compositionally. Notes 1. A less ambiguous way to call this approach would be to use the term nomenclatural, employed by Peregrin (forthcoming b), but here we want to stress the roots of this approach, which are connected with understanding language as arising out of a process of semiosis. 2. Cf. Harris (1988). See also Peregrin (forthcoming a). 3. "Nur im Zusammenhang eines Satzes bedeuten die Wörter etwas." (Frege 1884: 73). "Nur der Satz hat Sinn; nur im Zusammenhange des Satzes hat ein Name Bedeutung" (Wittgenstein 1922: §3.3. Following the usual praxis we quote Qittgenstein by paragraph numbers rather than page numbers.) 4. In Dummett's (1988) term. 5. As Quine puts it: "there is no saying absolutely what the numbers are; there is only arithmetic" (Quine 1969: 45). 6. See, e.g., Simmons (1987).
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7. It is not usual to address these notions on such a general level; the only algebraic theory that does go into this and that is known to the author is that of Aczel and Lunnon (1991). 8. For detailed analysis of the situation we refer the reader to the writings of W.V.O.Quine, esp. Quine (1992). 9. E.g. Hilbert (1925), Wittgenstein (1964), Lorenzen (1957) or Vopenka (1979). Vopenka also provides an insightful analysis of how we are "held captive" by the Cantorian framework. Cf. also Hailperin (1992). 10. Thus, if we insist on direct reconstruction of reality in terms of infinite sets, then infinite sets in the sense of Brouwer (1920) are much more appropriate than those in the traditional sense. 11. The Fregean split of meaning into Sinn and Bedeutung is well known, but we can go even further. Cf. Peregrin (1992). 12. And this is indeed the conclusion reached by such scholars as Austin, Wittgenstein and Quine. 13. This way of viewing meaning is indeed quite close to that put forward by some analytical philosophers, especially Davidson. Thus. Davidson writes: 'I suggest that a theory of truth for a language does, in a minimal but important respect, do what we want, that is, give the meanings of all independently meaningful expressions on the basis of an analysis of their structure' (Davidson 1984: 55). See also Peregrin (1994). 14. See Peregrin (1992; forthcoming b). 15. Statements to this effect can be found in the writings of Aristotle and Kant as well as in those of the modern philosophers from Husserl (1900/1) and Carnap (1928) on. The paradigmatic picture is the Aristotelian one, as characterized by Cassirer: 'Nothing is presupposed save the existence of things in their inexhaustible multiplicity, and the power of the mind to select from this wealth of particular existences those features that are common to several of them.' (Cassirer 1910: 4)
References Aczel, Peter and Lunnon, Rachel (1991): Universes and Parameters. In Situation Theory and its Applications, vol.2, J.Barwise, J.M.Gawron, G.Plotkin and S.Tutiya (eds.), 3-24. Stanford: Standford University. Brouwer, L .E. J. (1920): Intuitionistische Mengenlehre. Jahresbericht der Deutschen Mathematiker-Vereinigung 28, 203-208. Carnap, Rudolf (1928): Der Logische Aufbau der Welt. Berlin: Weltkreis-Verlag. Translated as The Logical Structure of the World, Berkeley: University of California Press, 1967. Cassirer, Ernst (1910): Substanzbegriff und Functionsbegriff, Berlin. Translated as and quotted from Substance and Function, New York: Dover, 1953. Davidson, Donald (1977): Reality Without Reference. Dialectica 31. Reprinted in Davidson (1984). - (1984): Inquiries into Truth and Interpretation. Oxford: Clarendon Press. Dummett, Michael (1988): The Origins of Analytical Philosophy I. Lingua e Stile 23, 3-49. Frege, Gottlob (1892): Über Sinn und Bedeutung. Zeitschrift für Philosophie und
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philosophische Kritik 100, 25-50. Hailperin, Theodore (1992): Herbrand Semantics, the Potential Infinite, and Ontology-Free Logic. History and Philosophy of Logic 13, 69-90. Harris, Roy (1988): Language, Saussure and Wittgenstein, London: Routledge. Hilbert, David (1925): Über das Unendliche. Mathematische Annalen 95. Translated as On the Infinite, in From Frege to Gödel: A Source Book from Mathematical Logic, Jan van Heijenoort (ed.), Harvard: Harvard University Press, 1971. Husserl, Edmund (1900/1), Logische Untersuchungen, Halle: Niemeyer. Lorenzen, Paul (1957): Das Aktual-Unendliche in der Mathematik. Philosophia Naturalis 4, 2-11. Reprinted in Lorenzen: Methodisches denken, Frankfurt: Suhrkamp, 1988. Morris, Charles W. (1966): Foundations of the Theory of Signs. (=International Encyclopedia of Unified Science 1), Chicago: University of Chicago Press. Peregrin, Jaroslav (1992): Sprache und ihre Formalisierung. Deutsche Zeitschrift für Philosophie 40, 237-244. - (1994): Interpreting Formal Logic. Erkenntnis 40, 5-20. - (forthcoming a): Structuralism: Slogan or Concept? Lingua e Stile, to appear. - (forthcoming b): How To Do Worlds with Words, Dordrecht: Kluwer, to appear. Quine, Willard Van Orman (1960): World and Object. Cambridge (mass.): MIT Press. - (1969): Ontological Relativity and Other Essays. New York: Columbia University Press. - (1992): Pursuit of Truth. Revised edition. Cambridge (Mass.): Harvard University Press. Saussure, Ferdidand de (1931): Cours de linguistique générale. Paris: Payot. Translated as and quoted from Course in General Linguistics, New York: Philosophical Library, 1959. Simons, Peter (1987): Parts. A Study in Ontology. Oxford: Clarendon Press. Vopenka, Petr (1979): Mathematics in the Alternative Set Theory, Leipzig: Teubner. Wittgenstein, Ludwig (1922): Tractatus Logico-Philosophicus, London: Routledge. - (1964): Philosophische Bemerkungen. Frankfurt: Suhrkamp. - (1953): Philosophische Untersuchungen. Oxford: Blackwell.
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Jaroslav Peregrin LANGUAGE AND ITS MODELS: IS MODEL THEORY A THEORY OF SEMANTICS?
1. Introduction Tarskian model theory is almost universally understood as a formal counterpart of the preformal notion of semantics, of the “linkage between words and things”. The wide-spread opinion is that to account for the semantics of natural language is to furnish its settheoretic interpretation in a suitable model structure; as exemplified by Montague 1974. The thesis advocated in this paper is that model theory cannot be considered as semantics in this straightforward sense. We try to show that model theory is more adequately understood as shining light on considerations concerning the relation of consequence than on those concerning the relation of expressions to extralinguistic objects; and that it makes little sense to use model theory for the purposes of answering such questions as what is meaning? or when is a sentence true?. The organization of the paper is the following: We start by considering various formal reconstructions of natural languages utilizing standard logic. Section 2 points out that the usual way of explicating the semantics of natural language, namely the way of Tarskian modeltheoretic interpretation, is problematic. In Section 3 we propose a more adequate way: to explicate semantics via the delimitation of the space of possible truth-valuations of sentences, especially in terms of the specification of the relation of consequence (indicating that logic, as an account for consequence, is eo ipso a theory of semantics). Section 4 demonstrates that a plausible way to account for the space of possible truth-valuations is to locate a “basis”, i.e. a class of mutually independent sentences each truth-valuation of which uniquely extends to a truth-valuation of the whole language; and it is noted that some of Nordic Journal of Philosophical Logic, Vol. 2, No. 1, pp. 1–23. c 1997 Scandinavian University Press.
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the most usual traditional logical calculi, notably the first-order predicate calculus, may lack such a basis. In Section 5 we consider another example of a baseless language, namely the modal propositional calculus, and we indicate the general pattern which appears to invoke the idea of a model. Section 6 puts forward the thesis that the idea behind model theory is that of making such baseless languages based. In Section 7 we analyze the concept of quantifier, which appears to be crucial for the fact that predicate calculus is in general baseless; we point out that the seemingly indirect way in which a quantifier alludes to the infinite cannot be replaced by any direct way. In Section 8 we draw the general conclusion regarding the nature of model theory: we conclude that model theory can be understood as concerned with extending baseless languages to based ones; however, that doing model theory usually does not mean presenting such an extension, but rather only to stipulate its existence. Section 9 concludes that model theory should not be considered as a direct presentation of semantics, not only because it connects words with words (rather than words with things), but ultimately because it cannot succeed in explicitly connecting semantically non-transparent (baseless) languages with semantically transparent (based) ones. In Section 10 we point out that the common “ideology” behind model theory is closely connected with the doctrine of logical atomism as pursued at the beginning of this century by Russell, Carnap and Wittgenstein. Section 11 then lays out our conclusion that insofar as model theory can be considered a theory of semantics, it is not in force of its being an imitation of the “real” denotandum/denotatum relation, but rather in force of its being an account for consequence. 2. Language and the Ways of its Reconstruction What is language? It is now commonplace to consider language as a bundle of generative and/or transformational rules, as some kind of algebra, as a set-theoretically interpreted logical calculus, etc. Such views need no substantiation as long as we stay within the realm of mathematics: there we are relatively free to define the concepts we use. But, outside of mathematics we must not forget, however banal this may sound, that language is first and foremost something to be found in our world, that it is something factual. This is what is stressed by Wittgenstein (1953, §494): “Der Apparat unserer gew¨ ohnlichen Sprache, unserer Wortsprache, ist vor allem das, was wir ‘Sprache’ nennen; und
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dann anderes nach seiner Analogie oder Vergleichbarkeit mit ihr”.1 It is obviously legitimate, and indeed helpful, to try to reconstruct factual language in terms of a mathematical theory. The attempt may be guided by diverse purposes—we may want to throw light on the workings of some restricted range of expressions (such as logical connectives), to regiment a part of language in a form manageable by computers, etc. What concerns us here is the purpose of explication, i.e. that usage of formal means which aims at fostering our understanding of the phenomenon of language. If some kind of a mathematical structure is to be considered a reconstruction of language relevant in this respect, then it has to share some substantial amount of the characteristic features of the factual language. Which features are considered relevant, and which not, is again partly a matter of viewpoint; but once we are clear about what functions of language we consider relevant we may, in terms of Quine’s (1960, p. 259) classic definition of explication, “devise a substitute, clear and couched in terms of our liking, that fills the functions.” However, there seem to be also certain general principles governing adequate explication, principles that can be articulated independently of a viewpoint. One such principle can be called the principle of persistence of individuation (hereafter PPI ): it states that different explicanda should obtain different explicata and vice versa, that the correspondence between the domain of the pre-theoretic entities that are to be explicated and that of those which are used to explicate them should be one-to-one. If we try to explicate the intuitive concept of ordered pair, then we cannot furnish an explication such that two intuitively different ordered pairs would be explicated by one and the same object, or that one ordered pair would be explicated by two different things. If we try to explicate the concept of language, then we should maintain the one-to-one correspondence between languages in the intuitive sense and the theoretic constructs we employ as their explications.2 One possible way of reconstructing natural language is to reconstruct it as a grammatical system, as a set of basic strings of letters (lexicon) plus a set of rules to combine strings of letters into strings 1 “It is primarily the apparatus of our ordinary language, of our word-language, that we call language; and then other things by analogy or comparability with this.” 2 The factual boundaries between languages are, of course, both fuzzy and ambiguous (in one sense, we can regard, e.g., British and American English as two subspecies of a single language; in another sense we can regard them as two different languages). However, this is not in itself problematic—it is in the nature of explication that it replaces fuzzy lines with crisp ones and sticks to a single sense where ambiguity occurs.
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of strings of letters (grammar). This is quite a useful reconstruction employed by syntacticians; however, it is based on the (purposeful) disregard of the semantic aspect, it clearly violates PPI (for two different languages may conceivably share one and the same syntax) and it is thus essentially incapable of being taken as a genuine explication of the concept of language. What makes language into more than a grammatical system is that its expressions are in some sense significant, that they have meaning—the reason two different languages with the same syntax may still be different is that they can differ in meaning, in what their expressions ‘stand for’. So it may seem that to account for the whole of language we need a grammatical system plus an assignment of some kind of entities to expressions. This intuition appears to be done justice to by the notion of language established by Carnap (with support from Morris, Tarski and others), which is nowadays taken almost for granted. According to this commonly accepted doctrine a language consists of (i) a list of basic expressions (words; the lexicon ); (ii) a list of rules which produce more complex expressions out of simpler ones (extending the lexicon into an algebra generated by its words; the syntax ); (iii) an assignment of some basic set-theoretic objects to basic expressions; (iv) an assignment of operations over the set-theoreticals to rules (extending the interpretation of words into the homomorphism of the whole algebra). As Montague claims, this notion of language is so encompassing that it bans any need to make a difference between natural language and formal ones.3 However, it is necessary to realize that to take this kind of reconstruction directly as an explication would still violate PPI ; in this case because there would be more than one explicatum for a single explicandum, for a single language. First, it is quite clear that the grammar of a language can be articulated in various ways. If we take English, then, e.g., John, to love and Mary can be composed into John loves Mary, but we can account for it either by postulating a rule which combines a transitive verb phrase with a noun phrase into an intransitive verb phrase and a rule combining an intransitive verb phrase and a noun phrase into a sentence, or we can have a single rule combining a transitive verb phrase with two noun phrases into a sentence (not to speak about other possibilities). Second, also the semantics of a language can be accounted for by means of various set-theoretic interpretations. We know, for instance, that it makes no substantial difference whether we interpret a predicate by a subclass of the universe, or by the charac3
Montague 1974, pp. 188 and 222.
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teristic function of the subclass; however, a class and its characteristic function are two different things and corresponding interpretations thus make up different objects. Similarly, we know that as far as the usual principles of arithmetic are concerned we may consistently consider a number to be a primitive object, a set of sets of objects (as Frege in effect proposed), or the set of all the smaller numbers (as proposed by von Neumann). This indicates that what makes up a language in the intuitive sense should be considered not as a definite grammatical system plus a definite set-theoretic interpretation, but something ‘more abstract’, something which is shared both by different but equivalent grammatical systems, and different but equivalent set-theoretical interpretations. This is to say that to furnish a real explication of the concept of language we should have to be able to decide when two different grammatical systems are mere alternative ways of capturing the syntax of the same language, and when they concern different languages; and similarly when two set-theoretical interpretations are mere alternative accounts of its semantics and when they differ substantially. In other words, we should have to specify identity criteria, a criterion of identity for grammatical systems and a criterion of identity for interpretations. The situation is relatively straightforward in the case of syntax: two grammatical systems seem to be two different accounts for the same language if and only if they generate the same class of sentences. The class of sentences (or, more basically, the speakers’ ability to tell a well-formed sentence from a mere haphazard string of letters) is what can be considered as genuinely empirically capturable, the grammar is our way of accounting for this class.4 What two grammatical systems relating to one and the same language have in common is the class of sentences; hence we can identify the explication of the syntactical part of language with this set.5 However, the situation is more complicated in respect to interpretation; here no such straightforward criterion is in sight. To volunteer a reconstruction of semantics doing justice to PPI, we need to differentiate interpretations only up to some kind of “semantic equivalence” or 4
Within linguistics a grammar is sometimes considered adequate if it generates the right class of sentences and if it associates the sentences with some kind of “right” inner structures. However, it is problematic on which empirical grounds it could be decided which inner structures are right and which not; it rather seems that the inner structure is a mere by-product of the reconstruction of the class of sentences and that hence the structure is right if and only if it is the product of an adequate reconstruction. This debate, though, is beyond the scope of the present paper (an interested reader is referred to Peregrin 1996b). 5 Cf. Quine 1969, pp. 48–49.
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“isomorphism” (or, to use a term coined by Quine, up to some kind of proxy function ). Arithmetic is more than a bare class of sentences; but it is in the same time less than the class plus an assignment of definite objects. Language is a class of sentences plus that part of its interpretation which it shares with all interpretations which are “semantically equivalent” (elementarily equivalent, in terms of current model theory). If we managed to reify the structure which any two so “semantically equivalent” interpretations share, than we have that which must be added to syntax to make up a good explication of language. 3. Meaning and Truth So what must two set-theoretic interpretations share in order to be understood as two representations of the same semantics, rather than as representations of two different semantics? Let us consider arithmetic. It is not relevant whether we consider numbers in von Neumann’s sense and < an inclusion, or numbers in Frege’s sense and < the relation λXY.∀x ∈ X.∃y ∈ Y.x ⊂ y; all that counts is that the interpretation makes the right sentences true: that 1 < 2 comes out true, while 2 < 1 false. This is what Quine (1953, p. 45) seems to have in mind stating: “there is no saying absolutely what the numbers are, there is only arithmetic.” In the same sense it is inessential whether a predicate P is interpreted by a set or by the characteristic function of the set; all that counts is the truth values which result from the combination of P with terms and quantifiers. This suggests that what might be added to grammar to make up a fully-fledged language is something like a specification of truth. The situation is relatively straightforward in the case of arithmetic or any other mathematical language. For such languages truth coincides with eternal, necessary truth; what is actually true is always true. In such a case we can speak about the class of true sentences, or about the truth-valuation (assignment of truth values to sentences). However, in the general case there may be truths which are not necessary, truths which may become false at any moment. There is no definite set of truths, no definite truth-valuation. A sentence such as ‘It is raining’ may be true in the morning, but false in the afternoon. In the general case it is thus not possible to speak about the class of true sentences, or the truth-valuation; we can at most speak either about the class of actually true sentences, (the actual truth-valuation), or about the class of all the possible classes of true sentences (truthvaluations). The enterprise of the account for truth for a general language thus may be thought as split into two distinct enterprises: first,
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the delimitation of the space of all the possible truth-valuations and, second, the specification of the actual truth-valuation. We can call the first of these enterprises, borrowing from Schlick (1932), the pursuit of meaning, while calling the latter the pursuit of truth. The former is a matter for philosophy and logic, the latter one for the sciences.6 A logical account of truth for a general language thus means the delimitation of the space of all the possible truth-valuations of the sentences of the language. For English, such a space would contain truth-valuations which would verify (assign the truth to) ‘Aristotle is human’ and it would also contain those which would verify ‘Aristotle is not human’, but it would contain no truth-valuation which would verify both of these sentences, and it would probably also not contain one which would falsify both of them. However, the delimitation of the space of possible truth-valuations seems to be far from enough to really account for semantics. It seems to be insufficient to know that ‘Aristotle is human’ cannot be true simultaneously with ‘Aristotle is not human’. We have to know when ‘Aristotle is human’ is true; we must know its truth conditions. However, are we really in a position to articulate the truth conditions in a nontrivial way? We can surely say that ‘Aristotle is human’ is true just in case that Aristotle is human. There seem to be many logicians and philosophers who, since Tarski, consider this kind of ‘correspondencetheoretic’ explication of truth conditions satisfactory; but there are also more skeptical theoreticians of language, such as the late Wittgenstein or Quine, who consider this to say nothing nontrivial.7 The statement ‘“Aristotle is human” is true if and only if Aristotle is human’ can never bring us nontrivial information, because it begs the question: to be able to understand it we have to know what it means for Aristotle to be human, and to know this is nothing other than to know when the sentence ‘Aristotle is human’ is true.8 6
Of course it is just this division which was challenged by Quine’s famous attack on the analytic/synthetic distinction. However, as we have stressed before (see footnote 2), a theoretical explication consists, besides others, in positing sharp boundaries where there are in fact none. Thus, although Quine is certainly right that we cannot turn the distinction between the pursuit of truth and the pursuit of meaning into philosophical cash in the way Schlick proposed, we do need such a boundary when building a formal model of language. 7 As Wittgenstein (1977, p. 27) puts it: “Die Grenze der Sprache zeigt sich in der Unm¨ oglichkeit, die Tatsache zu beschreiben, die einem Satz entspricht (seine ¨ Ubersetzung ist), ohne eben den Satz zu wiederholen.” [“The limit of language is shown by it being impossible to describe the fact which corresponds to (is the translation of) a sentence, without simply repeating the sentence.”] 8 Caveat (due to P.Tich´ y): The statement ‘“Aristotle is human” means that Aristotle is human’ (or ‘“Aristotle is human” is true if and only if Aristotle is
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This indicates that the only way theoretically to account for the semantics of a language over and above the demarcation of the space of its possible truth-valuations is to translate it into another language taking the semantics of the other language for granted. This claim may immediately raise suspicion: would it in such a case ever be possible to learn a first language? However, to learn a language and to articulate its semantics explicitly are two essentially different matters: learning a language is an enterprise that is essentially practical and the wordthing (or, better, sentence-circumstances) links which are established during it cannot be directly spelled out within a theory—we may teach someone to use the term ‘dog’ to refer to dogs, but by putting this into words we get the trivial ‘“dog” refers to dogs’. A theory can pair words with words; it is only practice that can literally “pair words with things”. However, do not modern formal theories of semantics aim higher than that: do they not explain the semantics of language in an absolute way (not only relatively to another language)? Do they not provide a direct access to meanings by way of their set-theoretic articulation? Of course not: such claims as that ‘dog’ stands for the class of dogs, or that the sentence ‘Aristotle is mortal’ is true in those possible worlds in which Aristotle is mortal are subject to the same objection as above; namely that to understand them we would have to know what they try to communicate to us in advance. Only if we know what ‘dog’ means can we understand what the class of dogs is; and only if we know when ‘Aristotle is mortal’ is true we can understand which are the worlds in which Aristotle is mortal. This is why Quine (1953, p. 49) suggests that the problem of meaning boils down to the problem of synonymy.9 However, both Quine’s and, especially, Davidson’s (1984) considerations also suggest that meaning can be somehow approached via truth: and it is this idea on which we elaborate here. The decoding of an unknown language involves, as Quine’s and Davidson’s thought experiments with radical translation or interpretation make plausible, finding out which human’) is informative because it is a literal English translation of the German statement ‘“Aristotle is human” bedeutet daß Aristoteles menschlich sei’, which is evidently informative. However, if this were true, then the English translation of the Englisch–Deutsch W¨ orterbuch, containing statements to the effect that ‘dog’ means dog, etc., would have to be as informative as the original. Nevertheless, while the Englisch–Deutsch W¨ orterbuch is surely a bestseller, one could hardly believe that a single copy of the English translation could be sold. 9 Elsewhere Quine puts this point even more vividly (1953, p. 50): “What makes sense is to say not what the objects of a theory are, absolutely speaking, but how one theory of objects is interpretable or reinterpretable in another.”
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sentences are true when, and also how the truth of one sentence depends on that of other sentences. We suggest that if we allow ourselves of positing a sharp boundary between possible and impossible truthvaluations (which is, as we indicated, admittedly an oversimplification in respect to natural language, but a helpful one in respect to its theoretical explication), we can see semantic explication of a language as the delimitation of the space of its possible truth-valuations. What we tried to indicate above was that a set-theoretic interpretation can do no better than the delimitation of this space. In other words, our suggestion is that if we want to furnish a formal explication of a language, then a suitable one, not violating PPI, would be (the specification of) a class of sentences plus (the specification of) a space of possible truthvaluations of these sentences. Hence the first thesis of the present paper: A language is most adequately considered as a class of sentences plus a space of their truthvaluations; hence to account for the semantics of a language means to delimit the space of its truth-valuations. 4. Entailment and Bases The space of possible truth-valuations can be considered one side of the coin the other side of which is consequence. Indeed, consequence can be understood as a specification of the boundaries within which it is reasonable to consider alternative distributions of truth values among sentences: to say that the sentences S1, . . . , Sn entail the sentence S is to say that it is impossible for S to be false if S1 , . . . , Sn are true; hence it is to say that a truth-valuation which would assign T to all of S1, . . . , Sn and F to S is unacceptable. Hence any account for consequence is eo ipso an account of the space of possible truth-valuations, and vice versa. This establishes the important link between logic and semantics: to do logic means to work toward an account of consequence, and hence, in view of the conclusion reached in the previous section, towards an account of semantics. Now the most straightforward way to account for consequence and hence for the space of the possible truth-valuations is clearly the axiomatic method. (The axiomatic method aims simply at a list of all the instances of consequence—or, which is, for a “reasonable” language, the same, of necessary truths; it is only because such instances are infinite in number that it is forced to provide, in proxy, merely a potential generator of the list.) We have seen that an axiom can be understood as a statement to the effect of the exclusion of some truth-valuations; hence an exhaustive system of axioms can be understood as a (negative)
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demarcation of the class of those truth-valuations which are possible. However, the axiomatic method is not the only possible approach to such a demarcation. There is also another approach, an approach based on the idea of finding a “comprehensible core” of language the truth-valuations of which uniquely determine the truth-valuations of the whole language. To clarify the nature of this alternative approach let us first introduce some auxiliary concepts. We shall say that a sentence S is partially determined by a class C of sentences, if there is a distribution of truth values among the sentences in C which forces S to have a definite truth value. (If C is {S1 , . . . , Sn}, then to say that C partially determines S is to say that there is a necessarily true sentence S1?, . . . , Sn? → S ?, where each X ? is either X or ¬X). We shall say that a sentence S is (totally) determined by a class C of sentences, if every distribution of truth values among the sentences in C forces a definite truth value of S. We call a class of sentences of a language L a quasibasis of L if it totally determines every sentence of L. The definition of the concept of quasibasis says that every truthvaluation of a quasibasis extends uniquely to a truth-valuation of the class of all sentences; hence that the truth-valuation of the whole class is uniquely determined by that of the quasibasis. This means that the study of truth-valuations of the entire language can be reduced to the study of their restrictions to the quasibasis. Every language obviously has some quasibases; this follows from the fact that the whole language itself is its own quasibasis. However, only a nontrivial quasibasis, a quasibasis whose complexity is in some sense remarkably smaller than that of the whole language, can be of interest from the viewpoint of the demarcation of the class of possible truth-valuations. We shall call a sentence S independent of a class C of sentences if S is not partially determined by C. A set of sentences is called independent, if every one of its elements is independent of the other sentences of the set. This means that a set of sentences is independent if all the distributions of truth-values among its elements are possible. We call a quasibasis a basis if it is independent. Although every language has a quasibasis, there is evidently no guarantee that it has a basis. Let us call a language based if it does have a basis; and let us call it baseless if it has none. Notice that a set which is independent can be considered as comprehensible from the viewpoint of truth-valuations: every distribution of truth values among elements is possible. The problem of demarcating the class of its possible truth-valuations is hence in fact—in a sense—solved by finding a basis: there is evidently a one-to-one corre-
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spondence between possible truth-valuations and subsets of the basis; the space of possible truth-valuations can thus be identified with the power set of the basis. So to find a basis is one of the possible ways of solving the problem of the demarcation of the class of possible truth-valuations, and hence of the problem of the pursuit of meaning. Hence: To specify a basis is to specify the space of the possible truth-valuations and hence to account for semantics. It is an alternative way to handle the task which is otherwise handled by the axiomatic method. 5. Model Theory as the Completion of Basis Now it seems that we somehow assume that a language must (or at least should) have a basis. If it has none, then we have the impression that the basis must be only somehow “hidden” and we feel urged to “fix up” the language to make the basis visible. The main claim of the present paper is that it is plausible to see precisely this kind of urge for a fix up as that which is constitutive of model theory. But before we discuss the thesis explicitly, let us consider the existence of bases within the most traditional logical systems. There is evidently a nontrivial quasibasis within the classical propositional calculus: it is constituted by the atomic sentences. The truth value of every sentence is uniquely determined by the truth values of some atomic sentences (it is determined by its atomic subsentences; and this is the reason why the calculus is decidable). Moreover, the truth value of an atomic sentence is independent of those of any other atomic sentences, hence the quasibasis is a basis. The classical propositional calculus thus fulfills our expectations in respect of the existence of basis. The troubles begin when we pass to predicate calculus, i.e. when we introduce the apparatus of quantification. The atomic sentences of the first-order predicate calculus do not in general constitute a basis: the truth value of a sentence such as ∀xP (x) is not totally determined by those of all the atomic sentences. If all the atomic sentences (especially all the sentences P (t) for any term t) are true, then ∀xP (x) may be true as well as false. However, it is not possible to add ∀xP (x) itself to the basis: it is not independent of the class of atomic sentences, because it cannot be true if some of the P (t)’s are false. The point is hence that ∀xP (x) is partially determined by the class of atomic sentences (in particular by those of the form P (t)), but it is not totally determined by it. Now having found out that there is no visible basis within a firstorder theory, one feels somehow urged to say that there is a “latent”
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one: to say that the falsity of ∀xP (x) “means” that there exists a “nameless” individual i which does not have the property P . If we embrace this intuition, then in fact we compensate the lack of a name for i by considering the “metaname” ‘i’, and the lack of the (false) sentence “P (i)” by postulating the (false) “metasentence” ‘i has the property P ’. To see what is going on in such a case, imagine the situation where ∀xP (x) is false, but every instance of P (t) is true. The conclusion will be that the universe contains a nameless individual which is not P . This conclusion sounds like an empirical discovery which explains the falsity of ∀xP (x): we have found a certain individual with a certain property which makes the statement false! However, in fact the existence of the individual results from mere reshaping of the information implicit in the falsity of ∀xP (x) and the truth of all the P (t)’s. The conclusion that the universe contains an individual which is not P requires nothing more that the trivial transformation of ¬∀xP (x) into ∃x¬P (x); whereas the conclusion that the individual is nameless is the direct consequence of the fact that no P (t) is true. Thus, it is misguided to say that the existence of such an individual in the universe explains the truth values of the formulas—for it is only another way of stating the truth values (see further Peregrin 1995, Chapter 5). Now if we accomplish such a compensation (of the lack of names by a supply of metanames) systematically in frames of set theory, then what we reach is model theory. If a ∀xP (x) is false although P (t) is true for every term t, then the universe of the model is to contain such an i that i ∈ / ||P || (where ||P || is the denotation of P , a subset of the universe). This nature of model theory is especially manifest if we consider the famous completeness proofs presented by Henkin (1949, 1950). What Henkin in fact showed is that every (“reasonable”) language (or theory) can be conservatively extended to a language with respect to which “there are no nameless objects” (i.e. to a theory which has the property that whenever a ∀xP (x) is false, then there is a t such that P (t) is false).10 What is important is that to understand Henkin’s construction we need not consider interpretations and models at all; it is enough to consider embeddings of language into another 10
Henkin’s procedure incorporates two steps. First, we add names to guarantee that whenever there is a true existentially quantified formula ∃x.F (x), then there is a name t such that F (t) is true. Second, we equate names displaying no difference in behaviour with respect to consequence. (If our system allows for definite descriptions, then we can drop the first step; if it does not include the identity predicate, then we must drop the second one). Then it is possible to assume the one-to-one correspondence between equivalence classes of names and objects and so the relationship between the language and the model may be seen as transparent.
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language. For what Henkin’s proof really says is that every theory can be conservatively embedded into a theory the atomic sentences of which form a basis. If it were guaranteed that every language does have a basis, then Henkin’s proof would lose its import. This indicates that considering a model of a language may be seen as considering an embedding of the language into another, based language. (Notice that only the case discussed above makes the need for a model theory really urgent: if there were a false P (t) for every false ∀xP (x), then there would always be a trivial model constituted by the language itself.) 6. Modal Logic The predicate calculus with its overt quantification is not the only case of a well-established calculus which lacks a basis. Modal propositional logic is another instance. The status of the formula 2S of S5 is similar to that of ∀xP (x) of the classical predicate calculus: it is neither totally determined by, nor quite independent of the class of all the atomic sentences. If S is false, then 2S is forced to be false too, but if S is true, 2S may be true as well as false. Whatever differences one may find between ∀xP (x) of the predicate calculus and 2S of the modal propositional calculus, the modeltheoretic solution is surprisingly similar. If S is true (in the “actual world”) and 2S is not, then this is understood as “meaning” that there is a possible world w, different from the actual world, with respect to which S is false. 2 is thus understood as latently quantifying over possible worlds in the same way in which ∀ quantifies over individuals. While within the classical predicate calculus the basis is restored on the meta-level as the class of all sentences of the form ‘the individual i has the property P ’, within S5 it is restored as the class of all the sentences of the form ‘S is true w.r.t. the world w’. ‘Ideological’ differences aside, S5 is quite analogous to the monadic predicate calculus.11 This indicates that the pattern underlying the ‘pull towards models’ is more general than that of the overt quantification. The relevant pattern is that of the ‘semi-independent quasibasis’: a quasibasis Q with an independent subset B such that B determines the elements of Q\B partially, but not totally. Model theory for such a language then amounts to the embedding of the language into a language in which 11 If we consider other modal calculi, then the situation is technically slightly more involved (the truth value of 2S depends on those of the elements of the basis in a complicated way—requiring a relation of “accessibility”), the principle is, however, the same.
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B gets extended to B0 such that B0 is still independent, and totally determines all the elements of Q\B. The new items constituting B0 \B are then naturally felt as being in a way implicit to those of Q\B. 7. Model Theory as Relating Words to Things This is, of course, an unusual way of conceiving model theory. The usual idea is that model theory depicts the relationships between words and things. Expressions of natural languages, the story goes, are expressions because they are somehow conventionally linked to extralinguistic objects, and model theory does the job of convention for formal languages—it guarantees that expressions of these languages are also linked to something extralinguistic, namely to certain sets. Model theory done for a (fragment of) natural language thus appears to be a direct account of its semantics. However, it is naive to take this idea at face value. It would be tantamount to what Quine (1969) calls the museum myth—to the idea that expressions are stuck on objects as labels in a great museum; and this is far from how language really works. As we said above, it is clear that no theory could succeed in directly relating words to things; it can do nothing over and above relating words to other words. It can give meanings of expressions of a language; but only given a “meta”language which is taken as the unquestioned background. (You cannot put a real dog into your theory to show what ‘dog’ means—you inevitably have to employ such or another representation.) And our natural language is the ultimate metalanguage, we cannot step out of it. There is no hope of doing model theory for our natural language, and hence of modeltheoretic answers to philosophical questions about language.12 This is why we stress that model theory should be primarily understood as amounting to extending languages, to making baseless languages based. On the other hand, philosophy is only enlightened common sense (as Popper once remarked), and therefore it would be unwise simply to ignore the intuition which equates model theory with semantics in the pre-theoretic sense: rather, we should seek the rational core of this intuition. Perhaps there might be a way of reconciling the traditional view of model theory (as the words-things question) with the one proposed here (as the words-words question). There are languages, we might say, which are “semantically transparent”, whose relation to the world is straightforward, and model theory can be considered as relating lan12 I discuss this predicament in detail in Peregrin (1995, Chapter 11 and 1996a) but see Hintikka 1988 for a defense of the contrary claim.
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guage to the world because it relates semantically non-transparent languages with semantically transparent ones. Moreover, it seems to be plausible to conclude that for a language to be semantically transparent means to have a basis. Henkin’s proof shows that establishing a basis means having the job of model theory essentially done. So, after all, there is a sense in which relating a baseless language to a based one, embedding the former into the latter, can be understood as establishing the link between words and things, i.e. as doing semantics in the direct sense. Is not model theory, therefore, the “direct” theory of semantics after all? The problem is that what is understood as furnishing model theory for a given language is in fact usually not furnishing a based language into which the language could be conservatively embedded. On the contrary: in the nontrivial cases it is impossible really to furnish a based extension, so model theory is left with merely postulating the existence of such an extension, and it is ultimately this point which precludes taking model theory as a direct theory of semantics. However, to articulate this intelligibly, we must examine more closely the point at which the nature of model theory comes into the open. 8. The Nature of the Quantifier From what has been said it follows that the lack of basis in the predicate calculus is somehow connected with quantification; let us thus analyze the notion of the quantifier. An excellent study of the nature of quantification in formal languages and of its early development has been presented by Goldfarb (1979). Commenting on the views of Bertrand Russell the author writes (p. 354): . . . the incorporation of quantifiers enables us to have principles that deal with infinity . . . . The meaning of our signs is only finitely complex ( . . . ). By the use of signs with such an intelligible meaning, we finite human beings reason about the infinite. But the nature of only finitely complex meaning of the quantifier cannot be given by some explanation from without. No such explanation can have any force. Rather, the meaning of quantification is shown by how we employ the signs, that is, by the logical rules of inference.
Hence for Russell, as for Frege before him, quantification was a means of reasoning about the infinite. It was an indirect means of such reasoning; however, this is not to be understood as claiming that Russell would have entertained the possibility of a reasoning about the infinite that would be direct. On the contrary, the conviction was, as Goldfarb aptly points out, that “the meaning of a quantifier cannot be given from without”.
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It was Tarski who took the essential point of departure in this respect. It is not by chance that for the argument Tarski has used to document the insufficiency of the Frego-Hilbertian proof-theoretic approach and to substantiate the need of an alternative, model-theoretic approach, quantification was crucial. In the paper which is usually considered as the point of departure of model theory Tarski (1936) argues that the proof-theoretic approach is not able to account for the fact that the sentences ‘n has the property E’ for every natural n together entail the sentence ‘All natural numbers have the property E’. Let us reproduce the relevant part of Tarski’s paper: Schon vor mehreren Jahren habe ich ein Beispiel, u ¨brigens ein ganz elementares, einer derartigen Theorie gegeben, die folgende Eigent¨ umlichkeit aufweist: unter den Lehrs¨atzen dieser Theorie kommen solche S¨atze vor wie: A0 . 0 besitzt die gegebene Eigenschaft E, A1 . 1 besitzt die gegebene Eigenschaft E, u.s.w., im allgemeinen alle speziellen S¨atze der Form: An . n besitzt die gegebene Eigenschaft E, wobei ‘n’ ein beliebiges Symbol vertritt, das eine nat¨ urliche Zahl in einem bestimmten (z.B. dekadischen) Zahlensystem bezeichnet; dagegen l¨aßt sich der allgemeine Satz A. Jede nat¨ urliche Zahl besitzt die gegebene Eigenschaft E, auf Grund der betrachteten Theorie mit Hilfe der normalen Schlußregeln nicht beweisen. Diese Tatsache spricht, wie mir scheint, f¨ ur sich selbst: sie zeigt, daß der formalisierte Folgerungsbegriff, so wie er von den mathematischen Logikern allgemein verwendet wurde, sich mit dem u ¨ blichen keineswegs deckt. Inhaltlich scheint es doch sicher zu sein, daß der allgemeine Satz A aus der gesammtheit aller speziellen S¨atze A0 , A1 , . . . , An , . . . im u ¨blichen Sinne folgt: falls nur alle diese S¨atze wahr sind, so muß auch der Satz A wahr sind.13
Tarski’s proposal is to improve this via the introduction of his, model-theoretic, notion of consequence. His proposal runs as follows: 13 “Some years ago I gave a quite elementary example of a theory which shows the following peculiarity: among its theorems there occur such sentences as: A0 . 0 possesses the given property P , A1 . 1 possesses the given property P , and, in general, all particular sentences of the form An . n possesses the given property P , where ‘n’ represents any symbol which denotes a natural number in a given (e.g. decimal) number system. On the other hand the universal sentence: A. Every natural number possesses the given property P , cannot be proved on the basis of the theory in question by means of the normal rules of inference. This fact seems to me to speak for itself. It shows that the formalized concept of consequence as it is generally used by mathematical logicians, by no means coincides with the common concept. Yet intuitively it seems certain that the universal sentence A follows in the usual sense from the totality of particular sentences A0 , A1 , . . . , An , . . . . Provided all these sentences are true, the sentence A must also be true.”
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Die Aussage X folgt logisch aus den Aussagen der Klasse K dann und nur dann, wenn jedes Modell der Klasse K zugleich ein Modell der Aussage X ist14 .
Let us analyze Tarski’s argument in detail. What he is urging is in fact the reduction of the truth of the sentence ‘All natural numbers have the property E’ to the truth of the infinite set of sentences containing the sentence ‘n has the property E’ for every natural n. However, what does the truth of such an infinite set really amount to? To say that a definite sentence is true may be considered a kind of paraphrastic way of asserting the sentence itself. To state that the sentence ‘1 has the property E’ is true is to say that 1 has the property E. However, to say that the sentence ‘n has the property E’ is true for every n is no paraphrase: nobody can succeed in asserting an infinite number of sentences directly. Let us suppose we are to put down an instance of consequence. If the instance amounts to the fact that something follows from the truth of the sentence ‘1 has the property E’ (i.e. from the fact that 1 has the property E), we may consider it to follow either from the (“meta”)sentence ‘“1 has the property E” is true’ or from the sentence ‘1 has the property E’. However, if we say that something follows from the fact that the sentence ‘n has the property E’ is true for every n, then there is no alternative: we have no choice but ‘“n has the property E” is true for every n’. We cannot list all the sentences of an infinite set, so the articulation of the truth of all its members inevitably boils down to the truth of such or another “meta”sentence. What Tarski himself in fact says is that it is impossible to account for the fact that the “meta”sentence ‘ “n has the property E” is true for every natural n’ entails the sentence ‘All natural numbers have the property E’. And the employment of the “meta”sentence is no shortcut: there is no way of avoiding it.15 However, what is the real difference between ‘ “n has the property E” is true for every natural n’ and ‘All natural numbers have the property E’ ? To state ‘P (x)’ is true for every x is nothing other than to state P (x) for every x and hence, if we stick to the usual predicate-calculus formalism, to state ∀xP (x); the only substantial difference is that (from the viewpoint of the calculus) the first sentence is a metasentence, whereas the last one is a sentence proper. What makes the difference between ‘P (x)’ is true for every x and ∀xP (x) 14
“The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X.” 15 The whole problem is in fact only a special case of the general problem of the infinite; see Peregrin 1995, Chapter 9.
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is that moving from the former to the latter we cross the boundary between that which we take as metalanguage and that which we take as language. However, to say that between two sentences, such as ‘P (x)’ is true for every x and ∀xP (x), there lies a boundary is not an empirical discovery, it is rather something we impose on the language to treat it in a way we want to; the concept of metalanguage is in fact a methodological, not a directly empirical one. It is an outcome of Tarski’s conceptual framework, and we can hardly accept that two things should be different only because we have decided to classify them into different compartments. 9. The Nature of Model Theory So let us return to our considerations concerning the nature of model theory. The point made at the end of the previous section can now be articulated as follows: model theory is (usually) not an embedding of a based language into a baseless “meta”language; it is only about such an embedding. And in fact, where the explicit presentation of the extended language actually is a possibility, i.e. where it suffices to introduce a finite number of new (“meta”)constants, there the situation is trivial. Let us consider the language of arithmetic. Furnishing model theory for this language amounts to introducing the infinity of numbers. But obviously we cannot introduce all the numbers explicitly (by listing them); we can at most stipulate the existence of the domain of all the numbers. Within the original language, there might be a valid sentence ∀xP (x). Model theory is incapable of explicitly furnishing the infinite number of “meta”statements 1 has the property P , 2 has the property P , etc.; it can only cover all these “meta”statements employing the “meta-meta”statement every n of the domain has the property P. Model theory for the predicate calculus can be seen as motivated by a desire to reduce quantificational truth to truth simpliciter of infinitely many instances. However, this cannot be accomplished by model theory; at least not in nontrivial cases. Model theory in these cases is not the embedding of a baseless language into a based “meta”language; but rather the translation of the baseless language into another baseless language, which is purported to be the “meta”language of a based “meta”language of the original language. Thus, far from actually reducing quantificational truth to truth simpliciter it is only stipulating such a reduction. Does all of this mean that model theory is nonsensical or vacuous? Not at all. Model theory amounts to relating two theories—the object
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language theory and a particular theory of sets. Such a relation can be interesting and can give rise to a respectable mathematical discipline; however, it should not be considered as relating expressions of the object language with things. The ultimate reason is not only that it amounts to a relation between two languages and not between a language and reality—we know that no theory could do better. The decisive reason is that it is not capable of relating a baseless language, such as that of a first-order theory, with a language that would be based and hence “semantically transparent”. This is to say that the idea of model theory, however interesting a mathematical discipline it might give birth to, is misguided as soon as it is considered to be capable of directly accounting for semantics of natural language and of explaining such pre-theoretic notions as truth, consequence or quantification. The underlying error consists in the failure to see the real nature of the infinite. A sentence is assumed to allude to infinity if it entails an unlimited number of certain consequences, and this is quite sound; what is not sound is to assume that this allusion to infinity can be made explicit. In that we manipulate our finite signs in a certain way, we do in a sense attain to the infinite; however, such an indirect way is in fact the only way in which we can really attain to it. 10. Logical Atomism Our considerations have been based on the assumption that it somehow belongs to the nature of language, or at least of a “semantically transparent” language, that it has a basis. Where does this feeling come from? The reason seems to be that we take for granted that the world has something as a basis, and that hence the language that faithfully accounts for it must have a basis, too. We usually consider the world as a kind of structure made of some basic building blocks, of what Putnam (1984) aptly calls the Furniture of the Universe. This idea found its most straightforward expression within several influential philosophical doctrines of the present century, namely within Russell’s logical atomism, Carnap’s logical empiricism and the philosophical doctrine of Wittgenstein’s Tractatus. All these approaches share the conviction that in the foundation of our language there is an underlying basis of “atomic”, “protocol”, or “elementary” sentences; and they postulate the principal coincidence of the order of language with that of the world and that of our knowledge, so that they see the existence of the basis of atomic statements from which all other statements can be deduced as tantamount to the existence of a basis of atomic facts from which all other facts can be constituted and to the existence of a basis
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of atomic pieces of our knowledge (“observations”) from which all our knowledge can be inferred. Logical empiricists even discussed the nature of concrete bases;16 but the fact is that no definite basis has been established for natural language. Nevertheless, if we accept that our world actually is as Russell, Carnap and the early Wittgenstein saw it, then the conclusion seems to be inevitable, that our natural language really is based—for otherwise we would be left with the conclusion that there are some parts of the world we are principally not able to speak about. Being a language—in the genuine sense of the word—thus seems synonymous with being a based language. However, there is no reason to believe that every part of language is necessarily itself a language; similarly we do not believe that every part of, say, an algebra must itself be an algebra. It seems to be the case that certain closure properties are essential for the concept of language, and that besides syntactic closure properties certain semantic or truthvaluational closure properties are also relevant. If this is right, then what looks like a language may sometimes be a mere pseudolanguage: it may be a fragment of a language whose boundaries are not chosen properly. A formal language can be considered as metaphorically “cut out” from its metalanguage; all languages being ultimately “cut out” from natural language, from the “metalanguage of all metalanguages”. Our conviction that there should be a basis within every language indicates that a language in which there is none may be in some sense be faulty, that it has been erroneously gerrymandered. Model theory can then be helpfully considered as an attempt to restore the ‘right’ boundaries, to cut out the language anew without the error, to round off the original language. Hence model theory, viewed from this perspective, can be considered as an attempt at a readjustment of a language cut out in a wrong way, an attempt to realign the language. If this is acceptable, then model-theoretic semantics of natural language may be considered as an attempt to realign natural language; however, out of what could natural language be considered to be improperly cut out? Logical atomism may suggest the idea that it is not our natural language, but the world itself that should be taken as the ultimate metalanguage—but it is hard to see how this idea could be substantiated. Indeed it is highly debatable that the atomistic view of our world is adequate. The linguistic turn taught us to see the structure of the world in terms of the structure of the language we use to cope with 16
From Carnap 1928 on.
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the world, rather than the other way round. Logic shows us that there are many sentences that can be seen as following from other, simpler sentences; and this may be understood to mean that there are many facts that can be seen as constituted out of other, simpler facts. But logic does not guarantee that there is a basis of “atomic” statements from which all other statements would follow—this is a philosophical doctrine; and it is worth considering whether it is really so plausible as it usually seems. Giving up atomism means embracing holism and consequently pluralism (if there is no definite basis, then the decision what to take as basic and what as inferred is ours), and this may seem to be incompatible with the rigor of logic; but scholars like Quine have made it seem plausible that logic may be of a piece with holism.17 11. Conclusion We have tried to indicate that it is preferable to see semantics as primarily a matter of the consequence relation, rather than as a matter of Tarskian assignment of set-theoretic objects. This supports a perspective on logical calculi the reverse of that which is widespread nowadays: the perspective of viewing proof-theory as more basic than model-theory. From this vantage point it is more appropriate to see, for example, incompleteness not as a failure on the part of axiomatics to capture a model-theoretically delimited class of sentences, but rather as a failure on the part of model-theory. That second-order predicate calculus is incomplete does not mean that its axiomatics is unsatisfactory, but rather that its classical model theory is inappropriate, that it is Henkin’s models that are the “right” ones. An orthodox model-theoretician may consider the majority of such considerations irrelevant; what she is after is a mathematical theory and model theory has proven itself to be an interesting mathematical theory. Completeness and incompleteness results are mathematical results which need not invoke any considerations of what is primary or what is appropriate. This is, indeed, true; however, there are not many model-theoreticians consequently orthodox in this sense. The relation between an expression and its model-theoretic interpretation is often understood not as a relation between two abstract entities of a mathematical theory, but rather directly as a relation between an expression and its meaning. Model-theory is increasingly understood as semantics in the intuitive, pre-theoretic sense of the word. While Tarski was— despite some problematical claims—principally aware that it is tricky to see model theory otherwise than in terms of extending languages (or 17
See also Peregrin (In preparation).
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mapping languages on their metalanguages), many of his followers, be they mathematicians or theoreticians of language, lack this awareness and mistake model theory for metaphysics. Contrary to Montague’s popular claim that there is no substantial difference between natural and formal languages we are convinced that there is quite a fundamental difference which manifests itself also in connection with model theory: it makes sense to make a model theory of a fragment of natural language or of a formal counterpart of such a fragment (and so to possibly repair its “false alignment”), there is, however, no clear sense in applying model theory to an entire natural language. The idea that model theory is semantics because it deals with assignments of objects to expressions and because semantics is also a matter of objects connected with expressions, is misguided. Model theory can indeed be considered a theory of semantics, but not in force of being an imitation of the ‘real’ denotandum/denotatum relation, but rather in force of being an account for consequence.18 References Carnap, R. 1928. Der logische Aufbau der Welt. Berlin. Davidson, D. 1984. Inquiries into Truth and Interpretation. Clarendon Press, Oxford. Goldfarb, Warren D. 1979. Logic in the twenties: The nature of the quantifier. Journal of Symbolic Logic, 44, pp. 351–368. Henkin, L. 1949. The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14, pp. 159–166. Henkin, L. 1950. Completeness in the theory of types. Journal of Symbolic Logic, 15, pp. 81–91. Hintikka, J. 1988. On the development of the model-theoretic viewpoint in logical theory. Synth`ese, 77, pp. 1–36. Reprinted in J. Hintikka, Lingua Universalis vs. Calculus Ratiocinator (Selected Papers 2), Kluwer, Dordrecht, 1997, pp. 104–139. Montague, R. 1974. Formal Philosophy: Selected Papers of R. Montague, edited by R. Thomason. Yale University Press, Yale. Peregrin, J. 1994. Interpreting formal logic. Erkenntnis, 40, pp. 5–20. Peregrin, J. 1995. Doing Worlds with Words. Kluwer, Dordrecht. 18 See also Peregrin 1994, where a similar conclusion was reached starting from considerations of another kind, namely from considerations of different meanings of the term semantic interpretation.
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Peregrin, J. 1996a. Logic and the pursuit of meaning. Logic and Linguistic in the Pursuit of Meaning (Studies in Logic and Philosophy—Department of Logic preprints). Available via the author’s homepage: http://www.cuni.cz/∼peregrin. Peregrin, J. 1996b. Linguistics and philosophy. Logic and Linguistic in the Pursuit of Meaning (Studies in Logic and Philosophy – Department of Logic preprints). Available via the author’s homepage: http://www.cuni.cz/∼peregrin. Peregrin, J. In preparation. When meaning goes by the board, what about philosophy? Proceedings of ANALYOMEN 2. Putnam, H. 1984. After Ayer, after empiricism. Partisan Review, 51. Reprinted as ”After empiricism” in Post-analytic Philosophy, edited by J. Rajchmann and C. West, Columbia University Press, New York, 1985, pp. 20–30. Quine, W.V.O. 1953. From a Logical Point of View. Harvard University Press, Cambridge, Massachusetts. Quine, W.V.O. 1960. Word and Object. MIT Press, Cambridge, Massachusetts. Quine, W.V.O. 1969. Ontological Relativity and Other Essays. Columbia University Press, New York. Schlick, M. 1932. The future of philosophy. College of the Pacific Publications in Philosophy, 1, pp. 45–62. ¨ Tarski, A. 1936. Uber den Begriff der logischen Folgerung. Actes du Congr´es International de Philosophique Scientifique, 7, pp. 1–11. Translated as ”On the concept of logical consequence”, in Tarski 1956, pp. 409-420. Tarski, A. 1956. Logic, Semantics, Metamathematics. Clarendon Press, Oxford. Wittgenstein, L. 1953. Philosophische Untersuchungen. Blackwell, Oxford. Translated as Philosophical Investigations, Blackwell, Oxford, 1953. Wittgenstein, L. 1977. Vermischte Bemerkungen, edited by G.H. von Wright. Suhrkamp, Frankfurt. Translated as Culture and Value, Blackwell, Oxford, 1980.
Department of Logic, Institute of Philosophy Academy of Sciences, Czech Republic
When meaning goes by the board, what about philosophy? Jaroslav Peregrin [from G.Meggle amd J. Nida-Rümelin (ed.): Analyomen 2: Proceedings of the 2nd Conference „Perspectives in Analytical Philosophy“, Vol II: Philosophy of Language; Metaphysics, de Gruyter, Berlin, 1997; original pagination]
1. Philosophy as the Pursuit of Meaning Philosophy is usually considered to be searching out the most general, and hence also the most necessary and the most eternal, truth; its central part, ontology, is often assumed to be fastening upon whatever might be "the form of the world". And because our world is the world as formed by the way we comprehend it and by the way we cope with it by means of our language, it is often assumed that its form must be brought out by the analysis of the interrelations between the meanings of our words and our statements. This is why many philosophers, and analytic philosophers in particular, say that philosophy consists in the analysis of meaning. However, to maintain that philosophy is in this way interlocked with necessary truth and with meaning is no longer a simple matter. Both these concepts are being constantly challenged. Is there, after all, something like necessary truth (pace Quine) to be captured by philosophy? Can we still maintain that there is a meaning (pace not only Quine, but also Austin, Wittgenstein and other sceptics), which can constitute the subject to philosophical analyses? And if these concepts should become endangered species, then what about philosophy?
2. Philosophical Theory? To answer these questions, let us reconsider the question as to what it is that philosophy is searching out. The roots of the way in which contemporary analytical philosophers would generally answer this question can be traced back to the pioneers of the analytic movement, in particular to the Vienesse
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logical empiricists. In a programmatic article, Moritz Schlick proposed to distinguish between the pursuit of truth (yielding contingent theses "about the world") and the pursuit of meaning (yielding necessary interconnections between concepts we use); he identified science with the former and philosophy with the latter (Schlick 1932). In this way, the philosophers of the linguistic turn claimed to have found a new definition of philosophy establishing it as a scientifically respectable enterprise and blocking attempts to decry it as what they considered unscientific metaphysical rubbish; and it is in fact this way of thinking which continues to provide the underlay for the intuitions of the majority of analytical philosophers. However, proposals such as Schlick's are tricky. Does the pursuit of meaning mean the same as the pursuit of truth about meaning? It seems that in whichever way we answer this question, we are in trouble. If we do identify the pursuit of meaning with the pursuit of truth about meaning, then philosophy turns out to be identical with a branch of science, namely empirical semantics; and if we reject the identification, then we are left with the conclusion that philosophy is not to yield truths, which seems to be absurd. In the former case we would in fact have no philosophy at all, while in the latter philosophy would be something mysterious and there could surely be no philosophical theory. Logical positivists do not seem to have realized the full significance of this dilemma; they wished to consider philosophy as a theoretical discipline par excellance and so embraced the first answer. Thus Carnap considers meaningful philosophical statements simply as "second-order" contingent truths, as truths about the language in which we formulate the "first-order" truths about the world; philosophy, according to him, is "the logical syntax of the language of science" (Carnap 1934). It would seem to follow from this that philosophy is simply a science, a particular brand of linguistics; but Carnap insists that philosophy, at the very same time, nevertheless constitutes the foundation to science. Carnap's position is thus untenable: we cannot have philosophy both as a theory and, simultaneously, as something fundamental to every theory. Either we must relinguish the theoretical character of philosophy, or we must relinguish its foundational character. This predicament became the central problem for those philosophers of the analytical tradition, most significantly Wittgenstein and Quine, who realized the failure of positivism. These two philosophers chose different ways: Wittgenstein believed that philosophy is in a sense foundational and he realized that in such circumstances it can be neither theoretic, nor systematic; thus he concluded that philosophy must be understood as something as a collection of "therapeutic" hints. Quine, on the
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other hand, wanted to uphold the theoretical status of philosophy and saw that consequently he must relinguish the idea of philosophy as fundamental to science. This led him to his rejecting the very idea of a fundament, of a Cartesian prima philosophia, and to his considering philosophical theory as a bundle of scientific results marked not by being firmer or more fundamental than the rest of science, but by being relevant for questions which are traditionally considered as philosophical. However, there might seem to be a third way, a way which allows us to defend Carnap's stance and to have philosophy both as a theoretical, and a foundational discipline. We may hold that philosophy does yield truths, but truths which are firmer than those yielded by science, because they are analytic. Under this view, philosophy appears to be a special kind of theory which - in force of the firmness of its truths - is fundamental to every other theory. Seen from this viewpoint, philosophy also appears to coincide with logic, which can be seen as a theoretical account of necessary truth. To illuminate the real nature of this proposal, we must consider the nature of necessary truth.
3 The Nature of Necessary Truth It seems to be obvious that some true statements merely happen to be true, whereas others are true necessarily; no matter whether one speaks about truths of fact and truths of reason, real and verbal truths, matters of fact and relations of ideas, analytic and synthetic truths or contingent and necessary truths. Statements true in the former way are usually considered to express some kind of junctions of things and they are thought of as made true by things really being joined in the manner they declare them to be; statements true in the latter way are taken to amount to limitations of the joinability of things (where the limitations can again be seen as certain junctions of objects - in this case as everlasting junctions of some "higher-order" objects in a "third realm"). It is this picture which was challenged by Quine in his celebrated Two Dogmas of Empiricism (Quine 1951). Quine's point is that the boundary between necessary and contingent truth is not an absolute one, that it is rather a more or less pragmatic matter; that we can never verify a single isolated statement, because it is only whole theories, or at least their nontrivial parts, that ever can be really verified. To be necessary thus, according to him, means to be so interwoven with the "web of our beliefs" that we cannot imagine giving it up without doing the whole web a serious harm. A similar picture emerges from the enigmatic writings of the late Wittgenstein: necessity is the matter of the rules of the language game being played, of the theory we employ. Such rules are unsurmountable as long as we
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are inside of the game, but they appear deliberate as soon as we move outside of it. Thus, Quine's claim, "any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system" (Quine 1963, 43), can be paralleled by Wittgenstein's (1953, p.69) "it is in a sense arbitrary what is called possible and what is not called possible" (Wittgenstein 1953, 69). Rules are in this clear sense arbitrary; and so are analytic truths as articulations of rules. Hence the arbitrariness of what is necessary, and consequently of what is possible. However, we must be careful not to take this Wittgensteino-Quinean stance in an oversimplified way. To accept a language means to approve some of its sentences as necessarily true. There is no principal need to adopt a particular language (although our native language is in a sense forced upon us and it is just this language which furnished us with our ultimate means of theoretical coping with the world); however, doing theory (in the broadest sense of the word) and engaging oneself in rational argumentation presupposes adoption of a language as an unquestionable basis. Language is a Janus-faced being: its rules, its necessary truths, when seen "from outside", are contingent and deliberate, but when seen "from inside", are necessary and obligatory. To say that something is necessarily true is not to say anything factual, rather it amounts to (as Ayer, 1936, pointed out) declaring one's conformity with a certain language game, one's willingness to accept a certain language. If I say that All humans are mortal is necessarily true, then what I say is that if I encounter an immortal being I am not going to consider it a man. If someone disagrees, then our disagreement cannot be settled by investigation of the world; his disagreement simply means that he is playing a language game different from the one that I am. I can persuade him (for example by showing him some books from which it would follow that my game is the one played by the majority of speakers of English), but I cannot show that he is false in the sense in which I could if, for example, he denied that Bill Clinton is mortal. Thus it is, indeed, in a sense arbitrary which statements are necessarily true - but only if we view language from outside. Accepting the language as our means of communication - and we need a language all the time - we surrender the possibility of querying its necessary truths, on pain of blurring the boundary between consistency and inconsistency and so loosing the firm ground beneath our feet, downgrading language to a bundle of expressive shrieks. However, it is to some extent possible to tamper with necessary truth without breaking down rational communication; and that language is in fact nothing other than an equilibrium between the stability guaranteeing ongoing understanding and the variability making language into something more than a mere set of symbols.
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Our language can be seen as the stage we set up for the world to make its appearance; necessary truth is our setting up the stage, contingent truths are then the way the world appears. Changing language we change the appearance of the world. However, we must beware of taking this scheme-content way of viewing the language absolutely; the decomposition into a scheme and a content should rather be seen as our way of viewing how language works, as our making sense of the working. Hence the necessary/contingent boundary can be seen as the matter of the outside observer's way of viewing the game; and one and the same game may be viewed in different ways drawing the boundary at various places. This is the point of the holistic insight of Wittgenstein and Quine. However, there is more to necessary truth than this. Some necessary truths are constitutive to the actual language game; and the explicit adherence to them makes it possible for the speakers to retain the common ground. To play the game is to take these truths as necessary. The necessary truth of such statements is not the mere matter of the outside observer's conclusion with respect to the way they are handled by the speakers; it is the matter of the speaker's credo, of their making it explicit that they are willing to accept the language game which is being played.
4 Logic and Philosophy after the Fall of the Dogma The recognition of the real character of the necessity/contingency opposition as initiated by Wittgenstein and Quine, constitutes a real challenge to logic logic is to be the pursuit of necessary truth, but does not necessary truth turn out to be something like a mere chimera? However, logic does continue as the summarization of necessary truths - a finite grasp on the infinity of instances of consequences. The difference is only that as necessity turns out to amount not to an ultimate structure of reality, but rather "merely" to the rules of the way we spell out reality, logic turns out to be not the "true canon of the Universe", but rather only the code of our way of theoretizing and our argumentation. Logical truths are necessary not because they reach beyond language into a realm of unchanging and ever-lasting propositions, but because they summarize certain basic relevant patterns of our language on which our overall interaction with the world rests. What in particular is put into doubt is the atomistic view of the world, which has underpinned our understanding of logic for almost the whole century. Logic keeps us yielding cases for the reduction of truth of some statements to the truth of other statements; and these we can continue to view as amounting to the reduction of more complex facts to simpler ones, or to the reduction of some more advanced pieces of our knowledge to other, more
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primitive ones. However, it seems no longer feasible simply to assume that there is an absolute, ultimate basis of elementary statements (or of elementary facts, or of elementary pieces of knowledge) to which all other statements (facts, pieces of knowledge) must be reducible. What counts as elementary from one visual angle can count as complex from another; and there is no absolute viewpoint, no God's eye view of the Universe as one closed system (Putnam, 1984, p.27). As Hacking puts it, "logic, depth grammar, structuralism, and the like should postulate points of convergence or condensation, not atoms." (Hacking 1979, 315) Philosophy can still, in a sense, be understood as the pursuit of meaning; but we must not take such a definition as implying that meaning is something absolutely fixed and that the task of philosophy is simply to point it out. We can no longer see necessary truth as something given once and forever by the way our words hook onto the world, and we can no longer claim that the task of philosophy is to discover the true structure of the world beyond all languages and the relationship of words to nodes of this structure. Philosophy is a matter of the critique of the usefulness of the languages we use, it does not result in theories in our common language, but rather at practical hints that are to make us see, to use Austin's popular turn of phrase, how we do things with words, and how else we could do them. We can imagine that viewed "from Nowhere", "by God's eye", the world can be considered as sheer contingency displaying regularity of only the causal kind; linguistic utterances appearing to be merely peculiar kinds of events obeying the all-encompassing web of causes and effects. But insofar as we are no Gods, each of us has to dwell "within a language", to observe the world through its prism and to perceive some of its God's-eye-view-contingencies as necessities. We have to admit that Heidegger (taken, since Carnap, as the prototypical enemy of an analytical philosopher) was right in stressing that language is not a tool, but rather the way of our existence; that it is the House of our Being (Heidegger 1959). Logic and philosophy can be considered as searching out the description of this house established by our language; helping us to figure out its architecture and so to help us feel at home in it.
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5. References Ayer, Alfred J.: Language, Truth and Logic, London 1936. Schlick, M.: The Future of Philosophy. College of the Pacific Publications in Philosophy 1 (1932), 45-62. Carnap, R.: On the Character of Philosophic Problems. Philosophy of Science 1 (1934), 5-19. Hacking, Ian: What is Logic? Journal of Philosophy 76 (1979), 285-319. Heidegger, Martin: Unterwegs zur Sprache, Pfullingen 1959. Putnam, Hilary: After Empiricism. In Post-Analytic Philosophy, ed. by J.Rajchmann and C.West, New York 1984, 21-30. Quine, Willard Van Orman: Two Dogmas of Empiricism. Philosophical Review 60 (1951), 20-43. Reprinted in Quine (1963). Quine, Willard Van Orman: From the Logical Point of View, New York 1963. Wittgenstein, Ludwig: Philosophische Untersuchungen, Oxford 1953.
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DYNAMIC SEMANTICS WITH CHOICE FUNCTIONS Jaroslav Peregrin and Klaus von Heusinger1
1 Introduction Over the last two decades, semantic theory has been marked by a continuing shift from a static view of meaning to a dynamic one. The increasing interest in extending semantic analysis from isolated sentences to larger units of discourse has fostered the intensive study of anaphora and coreference, and this has engendered a shift from viewing meaning as truth conditions to viewing it as the potential to change the "informational context". One of the central problems of discourse analysis is the treatment of anaphorical expressions, of discourse pronouns and definite NPs. The traditional solution is to take pronouns as bound variables and to analyze definite NPs by means of the Russellian iota inversum operator; and this solution is usually taken over also by those semantic theories which take the dynamics of language at face value. However, as we try to show, theories falling in with this approach necessarily stop half way in the dynamic enterprise and, therefore, cannot achieve a satisfactory semantical analysis of anaphora. In the present paper we propose to apply the dynamic approach uniformly to all expressions. We give a dynamic treatment not only to meanings of sentences and supersentential units of discourse, but also to those of pronouns and NPs. This is possible by exploring the intuitive idea of salience and by its formalization by means of choice functions. A definite NP the P is taken to refer to the most salient P (to that entity which is yielded by the actual choice function for the set of all P's); discourse pronouns are then considered as equivalent to certain definite NPs. Thus, the mechanism of choice manages to supplant both the apparatus of binding of variables (or discourse markers), on the one hand, and the Russellian analysis of definite NPs, on the other. This enables us to represent all anaphorical expressions uniformly. Salience, however is not a global and static property of a discourse, but rather a local and dynamic one - an indefinite NP can change a given salience by raising a new entity to the most salient one.2 Therefore, a linguistic expression is semantically characterized neither by its truth conditions, nor by its potential to change assignments of individuals to some artificial syntactic objects such as variables or discourse markers, but rather by its salience change potential, which we formalize as its potential to change the actual choice function. In an utterance such as a man walks, the indefinite NP a man is taken to refer to an arbitrarily chosen element of the class of men, and this individual is in the same time taken to update the actual choice function in such a way that it becomes the value of the function for the class of men (i.e. it becomes the most salient man).3 A subsequent anaphorical expression like the man in the man whistles is then interpreted according to this updated choice function - and, therefore, comes to refer to the same individual that was introduced (arbitrarily chosen) by the indefinite NP in the preceding utterance. The pronoun he in he whistles is interpreted in the same way; it refers to the referent that was
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introduced by the indefinite NP a man. The present theory is worked out as a modification of Groenendijk & Stokhof's (1991) dynamic logic and is hence based on the idea of treating the meaning of a sentence as the relation between its "input" and "output" referential links; it differs from Groenendijk and Stokhof's classical version of dynamic logic in formalizing referential links not by means of assignments of objects to discourse markers, but rather by means of choice functions. This makes it possible to analyze NPs not via quantifiers, but instead simply as terms (already Hintikka 1974); and consequently to dispose of any kind of variables or discourse markers, thus making the structure of semantic representation straightforwardly correspond to that of the analyzed discourse4. Furthermore, atomic formulas are taken to be both internally and externally dynamic. The theory assumes, together with discourse representation theories (cf. Kamp 1981 and Heim 1982), that anaphora and definiteness are two sides of the same familiarity principle; we take definiteness to presuppose not simply the unique existence, but rather the unique referential availability. However, we do not assume that this forces us to represent anaphora by means of binding. We agree with Etype analyses (Evans 1977, Neale 1990, Heim 1990, van Eijck, 1993) that discourse pronouns are definite descriptions, but we do not agree that this kind of definiteness could be explicated in the Russellian way. We prefer to employ an Hilbertian treatment of definite expressions with the epsilon operator, which substitutes the Russellian uniqueness condition by the principle of choice. Since Hilbert's epsilon operator, which we thus engage instead of Russell's iota inversum, expresses a global and static choice function, we have to make this function context dependent (cf. Egli 1991, and Egli & von Heusinger 1995), which we realize by embedding them into the Groenendijko-Stokhofian framework5. The paper is organized in the following way. In section 2 we discuss the motivation for dynamic semantics as rooted in the shift from sentence semantics to discourse semantics. Connected with this shift we can perceive at least three kinds of important, and mutually interrelated problems: the basic problem of the compositional analysis of an anaphora-laden discourse, the problem of how to account for crossreference - where possible - by linguistic means, and the problem of a uniform representation of anaphorical expressions. We show that dynamic logic, although it solves - besides other - the first of these problems in a very elegant way, does not really tackle the other two. Anaphorical NPs continue to be analyzed via Russellian definite descriptions governed by the problematic uniqueness condition, and the right resolution of anaphorical links is simply presupposed. In section 3 we propose our modification of dynamic logic that would solve the latter two of the mentioned problems. We give the definition of the apparatus and we show how to apply it. Then we hint at various ways of extending the apparatus with the aim to cover further linguistic data; and, of course, we cannot avoid giving our own account of donkey sentences. Finally, we discuss further extensions of our theory to other quantifiers and events. In section 4 we give the conclusion.
2 The dynamic view and its intricacies 2.1 The classical problem The new problems brought about by extending analysis from isolated sentences to larger units of discourse and texts were extensively discussed by Geach (1962). He used classical predicate logic; examining (1) he arrived at the straightforward analysis (1'). A man walks. He whistles ∃x (man(x) & walk(x) & whistle(x))
(1) (1')
It is easy to see that (1') is an adequate analysis of (1); and we should expect the two sentences constituting (1) to be analyzable in such a way that the two analyses would add up to (1'). However,
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although there is no problem with analyzing (2) as (2'), there is no straightforward way to analyze (3): A man walks He whistles ∃x (man(x) & walk(x)) ???
(2) (3) (2') (3')
This is the most general problem; we can call it the problem of compositionality. Another, affiliated and more specific, problem, concerns the link between the anaphorical pronoun he and its antecedent a man in (1). We cannot analyze (2) and (3) independently, because the argument of man in (2) and that of whistle in (3) are to be bound by the same quantifier. We shall call this the problem of correference. A third problem was mentioned by Evans (1977), who criticized the Geachian way of representing pronouns as bound variables: he showed that this representation is too strong and yields inadequate truth conditions. Evans himself proposed to represent pronouns as definite descriptions which obtain their descriptive material from the antecedent sentence (plus context). However, this analysis raises new problems: the uniqueness condition implied by the Russellian analysis of definite descriptions is still too strong. We will call this problem the problem of definite description. The development of a new family of semantic theories, all of which are based on a dynamic notion of meaning, was stimulated especially by a desire to solve the general problem of compositionality. Some of the most influential among them (esp. Groenendijk & Stokhof, 1991) adhere to the Geachian tradition in representing anaphorical pronouns as bound variables. An alternative view stems from Evans and represents discourse pronouns as definite descriptions (Evans 1977, Slater 1988, Neale 1990, Heim 1990, Chierchia 1992, van der Does 1993). However, such theories fail, or have difficulties, in solving the problem of definite descriptions.
2.2 The way of dynamic logic Groenendijk & Stokhof (1991) offered an elegant solution to the problem of compositionality they introduced a "dynamic version" of the existential quantifier, a version which binds variables even outside what would traditionally be considered as its scope. (The side effect of this move is that variables loose something of their essential character and that they become what Groenendijk & Stokhof call discourse markers.)6 If we denote this version of the existential quantifier as ∃dyn, we can analyze (2) as (2'') and (3) as (3''); the concatenation of these two formulas is then taken to yield (1'') (by way of a suitable dynamic redefinition of &), which can be seen to be - after "dedynamization" - equivalent to (1'). ∃dynd1 (man(d1) &dyn walk(d1)) whistle(d1)
(2'') (3'')
∃dynd1 (man(d1) &dyn walk(d1)) &dyn whistle(d1) (1'') The key to the dynamization of ∃ is that instead of taking the semantic value of a sentence to be a truth value it is taken to be a relation between assignments of values to discourse markers. However, Groenendijk & Stokhof's solution is less satisfactory on closer inspection. Our success in analyzing (2) and (3) was conditioned by the fact that we used the same discourse marker in both formulas - if we used d1 in (2) and d2 in (3), then we would have not been successful. However, if the problem is to give independent analyses of (2) and (3) in such a way that they could be straightforwardly concatenated into the adequate analysis of (1), then Groenendijk and Stokhof's
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method evidently falls short of solving it - if we analyze (3) independently of (2), then the composition could work only by pure chance. The point is that analyzing (3) we can see no reason for employing d1 rather than, say, d2 - we can see no reason to favour one discourse marker over another. The problem seems to be rooted in the nature of discourse markers - like variables they are not anchored to any overt items of natural language, but unlike variables they are not equivalent whistle(d1) is not a formula equivalent with whistle(d2). In fact, this is only a new reincarnation of the traditional problem of how to get the antecedent of an anaphorical term. The traditional way to handle this is to interlock corefering expressions by means of coindexing - but this clearly does not solve the problem, but only moves it outside of the theory. It is assumed that our formal model starts where coindexing, and hence anaphora resolution, is already done. Groenendijk & Stokhof accepted this common policy; they assume that the choice of right discourse markers is something which is either taken for granted, or at least a job for someone else.
2.3 Definite descriptions Discourse representation theories (Kamp 1981, Heim 1982) reduce definiteness to the principle of familiarity according to which a definite expression refers to a familiar or already introduced discourse referent. In this way, definite and indefinite NPs can both be represented as terms (or as variables). However, such theories usually do not distinguish between different familiar discourse referents, i.e. they do not solve the problem of coreference. Dynamic logic, by contrast, usually keeps - in effect - to the classical representation of indefinite NPs as existential quantifiers and definite NPs as Russellian iota terms. Even the dynamic interpretation of indefinite NPs with nondeterministic programs, as proposed by van Eijck, although in some aspects similar to our employment of choice functions, still cannot escape this Russellian predicament. Van Eijck (1993, p. 240) writes: "[...] in a dynamic setup the interpretation of an indefinite description can be viewed as an act of picking an arbitrary individual, i.e. as an indeterministic action". Therefore, instead of employing the (dynamic) existential quantifier, he uses a dynamic eta term for representing indefinite NPs. The eta operator is not interpreted as a term-creating operator, but rather as dynamic quantifier: "Note that η and ι are program building operators (in fact, dynamic quantifiers) rather than term building operators, as in the logic of Hilbert & Bernays" (van Eijck 1993, p. 245). Furthermore, definite noun phrases are treated according to the Russellian uniqueness condition interpreting definite noun phrases: "It is not difficult to see that this [i.e. the interpretation conditions for ι] results in the Russell treatment for definite descriptions." (ibid.). This indicates that even this modification of dynamic logic does not suceed in applying the dynamic approach in a way such as to dispose of the classical framework which would seem to be responsible for posing the classical problems.
3 Dynamic logic with choice functions 3.1 A sketch of the solution Here we would like to attack the classical problem via a new approach; we propose a modification of dynamic logic which makes do without discourse makers and hence without quantifiers. The idea, taking its cue from the considerations of von Heusinger (1992), is simply to replace assignment of values to discourse markers by Hilbertian choice functions. Roughly speaking, we can say that if we encounter an indefinite noun phrase, then we do not introduce anything like a discourse marker that would be assigned a referent; instead we let the referent be associated with the noun phrase itself via
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the extension of its predicative part. Formally, we introduce epsilon functions as functions from the power-set of the universe into the universe such that if e is an epsilon function, then e(s) ∈ s for every subset s of the universe.7 An epsilon function is thus a choice function: it chooses "representatives" of subsets of the universe. Semantic values of sentences are then taken to be relations between epsilon functions - this means that we can look at a sentence as something that gets an input epsilon function and yields (indeterministically) an output epsilon function. The basic idea is that an occurrence of an indefinite noun phrase a(n) N obliges the actual epsilon function to choose a representative of the extension of N, and that a subsequent occurrence of the definite noun phrase the N can be taken to refer to this very element. Let us consider (1); i.e. (2) followed by (3). Let us assume that there is an epsilon function e which is input to (2). The evaluation of (2) results in an output epsilon function e' which is just like the input function e save for the single possible difference that there is a representative of the class of men (who walks), i.e. that there is a d ∈ ||man|| such that e'(||man||) = d and d ∈ ||walk||. This function is passed to (3) which can then identify d as the referent for the man. According to this intuitive motivation we make the preliminary assumption that the definite article is formalized as an "epsilon test" whereas the indefinite article is represented as an epsilon update that changes the actual epsilon function.
3.2 The theory Let us assume that we have the non-empty universe U of individuals. An epsilon function (or a choice function) f is a partial function from the power-set of U into U such that f(s)∈s for every s⊆U for which f is defined. This means that the class EPSU of all epsilon functions based on U is defined as follows (where D(f) and R(f) denote the domain and the range of f, respectively)8: DEF1. EPSU = {f | D(f)⊆Pow(U) and R(f)⊆U and f(s)∈s for every s∈D(f)} We further define update functions for epsilon functions, or epsilon updates in short. An epsilon update is a function of three arguments: if it is fed an epsilon-function, an element of the universe and a subset of the universe, it yields a new epsilon function. So formally we define DEF2. UPD = {f | D(f)=EPS×U×Pow(U) and R(f)⊆EPS} The basic epsilon-update, which we shall denote as upd1, is such that if it is applied to an epsilon function e, an individual d and a set s, then it outputs the epsilon function e' which is just like e with the single possible exception that if d∈s, then e'(s)=d. DEF3. upd1 is the element of UPD defined as follows upd1(e,d,s)(s') = d if s'=s and d∈s = e(s') otherwise We shall write e'=es as the shorthand for ∃d.e'=upd1(e,d,s)9. If e2=e1s and e3=e2s', then we shall also write e3=e1s,s'. upd1 can be seen as the first approximation to the explication of an indefinite NP's salience change potential: an indefinite NP a man can be thought of as selecting an arbitrary man and changing the actual epsilon function so that the arbitrarily chosen man becomes the current
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representative for the class of men. Now we can define a fragmentary language to illustrate how epsilon functions can be used to build dynamic semantics. We do it in three steps, describing the lexicon, the syntax and the semantics. DEF4a. (lexicon) There are the following categories of expressions; the constants of each category (if any) are listed in brackets: 1. sentences 2. terms (constants he, she, it) 3. n-ary predicates for n>0 (constants man, walk, whistles, farmer, boring, woman, thing for n=1; own, beat for n=2) 4. determiners (constants a, the) 5. n-ary logical operators for n=1,2 (the constant ¬ for n=1; &, ∨ for n=2) 6. quantifiers (constants some, every) Note that there are neither variables, nor discourse markers within our language. This is made possible by the fact that we represent indefinite NPs as terms expressing epsilon updates, and that we do not treat anaphorical relations in any way evocative of the binding of predicate logic. DEF4b. (syntax) 1. If P is a unary predicate and D a determiner, then D(P) is a term. 2. if T1,...,Tn are terms and R an n-ary predicate, then R(T1,...,Tn) is a sentence. 3. If S is a sentence and o a unary logical operator, then oS is a sentence. 4. If S1 and S2 are sentences and o a binary logical operator, then S1oS2 is a sentence. 5. If Q is a quantifier and S1, S2 sentences, then Q(S1,S2) is a sentence. DEF4c. (semantics) A model is a pair , where U is a set and || || is a function such that (i) if T is a term, then ||T||∈U; (ii) if R is an n-ary predicate, then ||R||∈Un. If T is a term and e∈EPSU, then we define the value ||T||e in the following way: ||T|| if T is a constant term ||T||e = e(||P||) if T is D(P) for a determiner D and a predicate P We extend the function || || to the categories of terms and sentences so that if E is a term or a sentence, then ||E||⊆EPSxEPS: 1a. ||a(P)|| = {<e,e'>| e'=e||P||} 1b. ||the(P)|| = {<e,e'>| e'=e and e'(||P||) is defined} 1c. ||he|| = ||the(man)|| 1d. ||she|| = ||the(woman)|| 1e. ||it|| = ||the(thing)|| 2. ||P(T1,...,Tn)|| = {<e,e'>| there exist e0,...,en so that e=e0 and e'=en and <e0,e1>∈||T1|| and ... and <en1,en>∈||Tn|| and <||T1||e1,...,||Tn||en>∈||P||} 3. ||¬S|| = {<e,e'>| e=e' and there is no e'' such that <e,e''>∈||S||} 4a. ||S1;S2|| = {<e,e'>| there is an e'' such that <e,e''>∈||S1|| and <e'',e'>∈||S2||} 4b. ||S1&S2|| = {<e,e'>| there is an e'' such that <e,e''>∈||S1|| and <e'',e'>∈||S2||} (= ||S1;S2||)
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4c. ||S1∨S2|| = {<e,e'>| e=e' and there is an e'' such that <e,e''>∈||S1|| or <e,e''>∈||S2||} 5a. ||every(S1,S2)|| = {<e,e'>| e=e' and for every e1, if <e,e1>∈||S1||, then there is an e2 such that <e1,e2>∈||S2||} 5b. ||some(S1,S2)|| = {<e,e'>| e=e' and there exist e1 and e2 such that <e,e1>∈||S1||, and <e1,e2>∈||S2||} Let us add some explanations to bring out the insights behind the apparatus. An indefinite NP a P is taken to express an epsilon-update, i.e. its function is taken to be the updating of the actual epsilon function e to a new epsilon function e'. e' then differs from e at most in the representative of the set of P; the NP is taken to refer to this representative. We write e||P|| for an e' resulting from the evaluation of a P with the input e. A definite NP the P is taken to refer to the representative of the set of P's according to the current epsilon function; it is taken to express the trivial epsilon-update.10 Further, it is required that there is at least one P - this expresses the existential presupposition of definite NPs. There is no uniqueness condition - it is replaced by the condition that there exists the representative of the set of P's. A pronoun is defined to be semantically equivalent to the impoverished definite NP expressing merely the corresponding gender.11 The atomic sentence is semantically characterized via its potential to change the current epsilon function e to the updated function e' by way of the subsequent application of the updates expressed by its terms. Thus, e and e' must be connected by a sequence of epsilon functions such that the adjacent pairs of the sequence fall into the respective updates expressed by the terms; and the referents of the terms must fall into the extension of the predicate. Here we differ essentially from usual dynamic logic in that we consider atomic sentences as internally and externally dynamic. The logical operators ¬ and ∨ are static (they act as tests); they are in fact the classical operators only formally dynamized. & is the dynamic conjunction suitable for conjoining subsequent sentences. The actual choice of operators is not important (it would, of course, be possible to employ others); it is not important for our present purpose. Note too that our quantifiers are simply other logical operators. The semantic interpretation for the quantifier every consists in the condition that every epsilon function which is an output of its first argument sentence can serve as an input function for the second argument sentence; thus, our every is in fact Groenendijk & Stokhof's →. every(S1,S2) can
be thought about as corresponding to the classical formula ∀...(S1→S2), where ∀... indicates ∀ unselectively binding all what there is to bind in S1. The condition for some is that there is at least one epsilon function which is an output of its first argument acceptable as an input for the second argument; hence it can be imagined as corresponding to ∃...(S1→S2). (some corresponds to the cardinality reading of English some; that reading of the English word which licenses anaphorical relationships is considered as representable by the operator a). Both the quantifiers are externally static, they act as tests; but they are internally dynamic. This accounts for the fact that a term in the first argument sentence can be the antecedent of an anaphorical expression occurring in the second argument.
3.3 Examples of discourse anaphora In this section we discuss some simple examples illustrating the mechanism of establishing
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anaphorical links by linguistic means. As already mentioned, we assume that the semantic contribution to finding the adequate antecedent can be described in terms of the salience change potential of expressions.12 This salience change can be thought about as a first formal approximation to the intuitive notion of familiarity that is used in traditional grammars and that was introduced into dynamic frameworks by Heim (1982). We start our analysis with a simple atomic sentence with an indefinite NP. A man walks (4) Walk(a(man)) (4') ||Walk(a(man))|| = {<e,e'>| <e,e'>∈||a(man)|| and ||a(man)||e'∈||walk||} = {<e,e'>| e'=e||man|| and e'(||man||)∈||walk||} Sentence (4) is assigned the formula (4'); this formula is then interpreted according to the definitions given above. As we have noted, a pair of epsilons <e,e'> falls into the update expressed by an atomic sentence iff e and e' are connected by a sequence of epsilons such that the adjacent pairs of the sequence fall into the respective updates expressed by the terms and the referents of the terms fall into the extension of the predicate. Since we have only one term in (4), this reduces to the condition that <e,e'> falls into the update expressed by a(man) and that the referent of a(man) falls into the extension of walk. This yields e'=e||man|| and e'(||man||)∈||walk||. The resulting set of pairs is clearly non-empty just in case ∃d.d∈||man||&d∈||walk|| (i.e. if the intersection of ||man|| and ||walk|| is nonempty); and our formula (4') is thus in this sense equivalent to the classical formula ∃x(man(x) &walk(x)). The man whistles (5) Whistle(the(man)) (5') ||Whistle(the(man))|| = {<e,e'>| <e,e'>∈||the(man)|| & ||the(man)||e'∈||whistle||} = {<e,e'>| e=e' & e'(||man||)∈||whistle||} Sentence (5) with the definite NP the man is represented and interpreted similarly to (4). The only difference is the condition on the epsilon function - the interpretation of the definite NP is static. The only condition is that the referent of the NP, determined by the actual epsilon function, falls within the extension of the predicate. The difference between the definite and the indefinite NP thus lies in the different behaviour with respect to the epsilon function - the indefinite NP updates it, whereas the definite NP acts merely as a test. In both cases, the referent of the NP is yielded by the actual epsilon function. The analysis of the concatenation of (4) and (5) now shows how the referent of the anaphorical NP the man gets identified with that of its antecedent a man. A man walks. The man whistles (6) Walk(a(man)); Whistle(the(man)) (6') ||Walk(a(man)); Whistle(the(man))|| = {<e,e'>| there is an e'' such that <e,e''>∈||Walk(a(man))|| and <e'',e'>∈||Whistle(the(man))||} = {<e,e'>| there is an e'' such that <e,e''>∈{<e,e'>| e'=e||man|| and e'(||man||)∈||walk||} and <e'',e'>∈ {<e,e'>| e=e' and e'(||man||)∈||whistle||}} = {<e,e'>| there is an e'' such that e''=e||man|| and e''(||man||)∈||walk|| and e''=e' and e'(||man||)∈||whistle||} = {<e,e'>| e'=e||man|| and e'(||man||)∈||walk|| and e'(||man||)∈||whistle||}
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<e,e'> falls into the update expressed by (6') if and only if there is an epsilon function e'' such that <e,e''> falls into the update expressed by (4') and <e'',e'> falls into the update expressed by (5'). Using the results of the above analyses and eliminating redundancies, we reach the result that <e,e'> falls into the update expressed by (6') iff e' differs from e at most in the representative of the class of men and this representative is a walker and a whistler. This derivation shows the basic mechanism of relating anaphorical expressions to their antecedents; it illustrates how the semantic principle of familiarity can be explicated in terms of the salience change potentials of the expressions involved. The information that a new object is raised to salience can be thought about as passed on by the updated epsilon functions; and since the subsequent definite term is interpreted according to these updated epsilon functions, it comes to refer to the very same object. Let us now turn our attention to sentences about farmers and donkeys. A farmer owns a donkey (7) Own(a(farmer),a(donkey)) (7') ||Own(a(farmer),a(donkey))|| = {<e,e'>| there is an e'' such that <e,e''>∈||a(farmer)|| and <e'',e'>∈||a(donkey)|| and <||a(farmer)||e'',||a (donkey)||e'>∈||own||} = {<e,e'>| there is an e'' such that e''=e||farmer|| and e'=e''||donkey|| and <e''(||farmer||),e'(||donkey||) >∈||own||} = {<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||)>∈||own||} The farmer beats the donkey (8) Beats(the(farmer),the(donkey)) (8') ||Beat(the(farmer),the(donkey))|| = {<e,e'>| there is an e'' such that <e,e''>∈||the(farmer)|| and <e'',e'>∈||the(donkey)|| and <||the(farmer) ||e'',||the(donkey)||e'>∈||beat||} = {<e,e'>| there is an e'' such that e''=e and e'=e'' and <e''(||farmer||),e'(||donkey||)>∈||beat||} = {<e,e'>| e'=e and <e'(||farmer||),e'(||donkey||)>∈||beat||} The atomic sentence (7) contains two indefinite NPs. Since both terms lead to modifications of the actual epsilon function, the eventual output function is modified in respect to its values for both the set of farmer and the set of donkeys. For each of the two sets, a new representative is chosen; the two representatives must be such as to stand in the relation of owning. The sentence (8) does not lead to any updating, since both its terms are definite and act as mere tests. Therefore, its output epsilon function is identical with the input one. Again, the concatenation of the two sentences shows how the epsilon function updated by the first conjunct creates a new context according to which the definite expressions in the second conjunct are interpreted. A farmer owns a donkey. The farmer beats the donkey. (9) Own(a(farmer),a(donkey)); Beat(the(farmer),the(donkey)) (9') ||Own(a(farmer),a(donkey)); Beat(the(farmer),the(donkey))|| = {<e,e'>| there is an e'' such that <e,e''>∈||Own(a(farmer),a(donkey))|| and <e'',e'>∈||Beat(the (farmer),the(donkey))||} = {<e,e'>| there is an e'' such that <e,e''>∈{<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||) >∈||own||} and <e'',e'>∈{<e,e'>| e'=e and <e'(||farmer||),e'(||donkey||)>∈||beat||}} = {<e,e'>| there is an e'' such that e''=e||farmer||,||donkey|| and <e''(||farmer||),e''(||donkey||)>∈||own|| and e''=e' and <e'(||farmer||),e'(||donkey||)>∈||beat||} =
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{<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||)>∈||own|| and <e'(||farmer||),e'(||donkey||) >∈||beat||} These simple examples show the general principles of linking anaphorical expressions to their antecedents. The indefinite expression changes the informational context so that the subsequent definite expression refers to the object which it has selected. The last example has demonstrated that the epsilon function can carry information about more than one object.
3.4 Improving updates We have seen that classical examples are adequately analyzable by our apparatus introduced so far; however, there are still simple cases which remain outside its scope. We are now going to show that such cases are analyzable by way of various simple enhancements of the apparatus. We saw that we were able to adequately analyze (6); and in force of DEF4c (1c) the same holds for (10) - the point is that the semantics of he is - by definition - the same as that of the man. A man walks. He whistles (10) Walk(a(man)); Whistle(he) (10') ||Walk(a(man));Whistle(he)|| = ||Walk(a(man));Whistle(the man)|| This does not work in case of (11) or (12) - he is not defined to be semantically equivalent to the farmer. A farmer walks. He whistles A farmer owns a donkey. He beats it
(11) (12)
However, this can be improved by replacing our epsilon update upd1 in DEF4c (1a) by a slightly more sophisticated version. We distinguish certain mutually disjoint subsets of U, which we shall call categories, namely M=||man||, W=||woman|| and T=||thing||, and we assume that the occurrence of a P fixes not only the value chosen from ||P||, but also that chosen from the category containing ||P|| (if any). This means that the occurrence of a farmer triggers not only the choice of a representative of the class of farmers, but that it also forces the same individual as the representative of the class of men and thereby as the potential referent for he13. DEF5. Let CAT={M,W,T} be a subset of Pow(U) such that M=||man||, W=||woman|| and T=||thing||. Let Ct be the function from Pow(U) into CAT such that if there is a c∈CAT such that s⊆c, then Ct(s) =c and Ct(s) is undefined otherwise. upd2 is now the element of UPD defined as follows upd2(e,d,s)(s') = d if (s'=s or s'=Ct(s)) and d∈s = e(s') otherwise If we now modify 1a of DEF4c by substituting upd2 for upd1, we can readily handle both (11) and (12). The point is that now not only ||he|| = ||the(man)||, but if e'=e||farmer||, then also e'(||man||) = e'(||farmer||); similarly if e'=e||donkey||, then also e'(||thing||) = e'(||donkey||). Walk(a(farmer)); Whistle(he) ||Walk(a(farmer)); Whistle(he)|| =
(11')
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{<e,e'>| there is an e'' such that <e,e''>∈||Walk(a(farmer))|| and <e'',e'>∈||Whistle(he))||}= {<e,e'>| there is an e'' such that <e,e''>∈{<e,e'>| e'=e||farmer|| and e'(||farmer||)∈||walk||} and <e'',e'>∈ {<e,e'>| e=e' and e'(||man||)∈||whistle||}} = {<e,e'>| there is an e'' such that e''=e||farmer|| and e''(||farmer||)∈||walk|| and e''=e' and e'(||farmer||) ∈||whistle||} = {<e,e'>| e'=e||farmer|| and e'(||farmer||)∈||walk|| and e'(||farmer||)∈||whistle||} Own(a(farmer),a(donkey)); Beat(he,it) (12') ||Own(a(farmer),a(donkey)); Beat(he,it)|| = {<e,e'>| there is an e'' such that <e,e''>∈||Own(a(farmer),a(donkey))|| and <e'',e'>∈||Beat(he,it)||} = {<e,e'>| there is an e'' such that <e,e''>∈{<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||) >∈||own||} and <e'',e'>∈{<e,e'>| e'=e and <e'(||man||),e'(||thing||)>∈||beat||}}= {<e,e'>| there is an e'' such that <e,e''>∈{<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||) >∈||own||} and <e'',e'>∈{<e,e'>| e'=e and <e'(||farmer||),e'(||donkey||)>∈||beat||}} = {<e,e'>| there is an e'' such that e''=e||farmer||,||donkey|| and <e''(||farmer||),e''(||donkey||)>∈||own|| and e''=e' and <e'(||farmer||),e'(||donkey||)>∈||beat||} = {<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||)>∈||own|| and <e'(||farmer||),e'(||donkey||) >∈||beat||} We can, of course, go further and let an indefinite noun phrase a P update not only the value chosen from ||P|| and that chosen from the corresponding category, but the values chosen from all supersets of ||P|| whatsoever. This would lead to the following simple epsilon-update. DEF6. upd3 is the element of UPD defined as follows upd3(e,d,s)(s') = d if s⊆s' and d∈s = e(s') otherwise Plugging upd3 instead of upd1 into 1a of Def4c would result into an apparatus which would allow for the right resolution in cases like (12); on the other hand, it might lead to successful resolution also in cases when it is doubtful that such a resolution factually takes place, like in (13). A boxer walks. The sportsman whistles. (12) A woodchuck from Hradec Králové walks. The East-Bohemian groundhog whistles. (13) The right solution seems to be somewhere in between upd2 and upd3 - it seems that in order to crosslink a P and the P' it is not enough that ||P||⊆||P'||, it is in addition required that it is in some sense obvious that ||P||⊆||P'||. However, on the other hand, if the hearer perceives that there is nothing else to be cross-linked with the P' save a P, then (s)he may take it as an information about the fact that ||P||⊆||P'||14. Another kind of problem is posed by proper names. We are as yet unable to account for (14). However, this is easy to repair. Let us add John as a new constant to our category of terms, let us write e'=ed as the shorthand for e'=upd3(e,d,{d}); and let us enrich DEf4c by the new clause 1f: 1f. ||T|| = {<e,e'>| e'=e||T|| if T is John (or any other proper name) This allows us to give the analysis of (14) (note that if e'=e||John||, then e'(||John||) = e'(||man||)): John walks. He whistles
(14)
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Walk(John); Whistle(he) (14') ||Walk(John); Whistle(he)|| = {<e,e'>| there is an e'' such that <e,e''>∈||Walk(John)|| and <e'',e'>∈||Whistle(he))||} = {<e,e'>| there is an e'' such that <e,e''>∈{<e,e'>| e'=e||John|| and e'(||John||)∈||walk||} and <e'',e'>∈ {<e,e'>| e=e' and e'(||man||)∈||whistle||}} = {<e,e'>| there is an e'' such that e''=e||John|| and e''(||John||)∈||walk|| and e''=e' and e'(||man||) ∈||whistle||}= {<e,e'>| e'=e||John|| and e'(||John||)∈||walk|| and e'(||man||)∈||whistle||}= {<e,e'>| e'=e||John|| and e'(||John||)∈||walk|| and e'(||John||)∈||whistle||}
3.5 Donkey sentences Donkey sentences serve as a touchstone for every semantic theory since they combine several intricate problems. In what follows, we try to show how we can cope with this kind of sentence within the proposed framework. The main problem is how an indefinite NP embedded in the restrictive clause can furnish the antecedent for a subsequent expression. This is the case of the indefinite NP a donkey in (15). Every farmer who owns a donkey beats it
(15)
Before we analyze this sentence, let us inspect a simpler one to see how the definition of the universal quantifier works.15 Every man who is a farmer is boring (16) every(farmer(a(man)),boring(the(man))) (16') ||every(farmer(a(man)),boring(the(man)))|| = {<e,e'>| e=e' and for every e1, if <e,e1>∈||farmer(a(man))||,then there is an e2 such that <e1,e2>∈||boring(the(man))||} = {<e,e'>| e=e' and for every e1, if <e,e1>∈{<e,e'>| e'=e||man|| and e'(||man||)∈||farmer||}, then there is an e2 such that <e1,e2>∈{<e,e'>| e'=e and e'(||man||)∈||boring||}} = {<e,e'>| e=e' and for every e1, if e1=e||man|| and e1(||man||)∈||farmer||}, then there is an e2 such that e2=e1 and e2(||man||)∈||boring||} = {<e,e'>| e=e' and for every e1, if e1=e||man|| and e1(||man||)∈||farmer||}, then e1(||man||)∈||boring||}= {<e,e'>| e=e' and ||man||∩||farmer||⊆||boring||} The analysis of the sentence yields a test that succeeds if every output epsilon function of the first clause can serve as an input to the second clause. One branch of the test can be imagined as, first, picking up a new representative of the set of men such that he is a farmer (the first clause), and, second, testing whether this representative is boring (the second clause); the whole test is successful if all such possible branches are. The testing is independent of an input epsilon function, i.e. either every epsilon function passes it, or none does. The crucial problem of donkey sentences is solved by quantifying over epsilon functions instead of over individuals, which is the result of our disposal of variables and binding. The problem concerns the anaphorical link between the embedded indefinite NP a donkey and the pronoun it - this link cannot be accounted for by the classical means of variable binding, since the pronoun has to be located outside the scope of the quantifier governing the indefinite NP. Since we do not establish
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anaphorical links in terms of binding, but rather via context information, this problem fades away. The analysis of a sentence such as (15) proceeds straightforwardly. every(Own(a(farmer),a(donkey)),Beat(he,it)) (15') ||every(Own(a(farmer),a(donkey)),Beat(he,it))|| = {<e,e'>| e=e' and for every e1, if <e,e1>∈||Own(a(farmer),a(donkey))||, then there is an e2 such that <e1,e2>∈||Beat(he,it)||} = {<e,e'>| e=e' and for every e1, if <e,e1>∈{<e,e'>| e'=e||farmer||,||donkey|| and <e'(||farmer||),e'(||donkey||) >∈||own||}, then there is an e2 such that <e1,e2>∈{<e,e'>| e'=e and <e'(||farmer||),e'(||donkey||) >∈||beat||}} = {<e,e'>| e=e' and for every e1, if e1=e||farmer||,||donkey|| and <e1(||farmer||),e1(||donkey||)>∈||own||}, then there is an e2 such that e2=e1 and <e2(||farmer||),e2(||donkey||)>∈||beat||} = {<e,e'>| e=e' and for every e1, if e1=e||farmer||,||donkey|| and <e1(||farmer||),e1(||donkey||)>∈||own||}, then <e1(||farmer||),e1(||donkey||)>∈||beat||} Again, the whole expression acts as a test that succeeds if every epsilon function resulting from interpreting the first clause can successfully act as input for the second clause. Since there are two indefinite NPs within the first clause, they both modify the input epsilon function, and so we have to apply the test connected with the second clause to all such functions differing from the basic output of the whole sentence in the representatives of the class of farmers and donkeys which pass the test connected with the first clause. Thus the anaphorical relation is encoded in the context information and in this way it can cross the scope of the indefinite NP.16
3.6 Events Another way to enhance our apparatus is by the introduction of events and event-quantification. A way to do this is to introduce the new category E coinciding with the extension of the new predicate event, and to add the new constants terms then, once, whose semantics is defined as follows: ||then|| = ||the(event)|| ||once|| = ||a(event)|| Then we can analyze (17) as follows: If a farmer owns a donkey, he beats it. (17) every(Own(once,a(farmer),a(donkey)),Beat(then,he,it)). (17') ||every(Own(once,a(farmer),a(donkey)),Beat(then,he,it))||= {<e,e'>| e=e' and for every e1, if <e,e1>∈||Own(once,a(farmer),a(donkey))||, then there is an e2 such that <e1,e2>∈||Beat(then,he,it)||} = {<e,e'>| e=e' and for every e1, if <e,e1>∈{<e,e'>| e'=e||event||,||farmer||,||donkey|| and <e'(||event||),e'(||farmer||),e'(||donkey||)>∈||own||}, then there is an e2 such that <e1,e2>∈{<e,e'>| e'=e and <e'(||event||),e'(||farmer||),e'(||donkey||)>∈||beat||}} = {<e,e'>| e=e' and for every e1, if e1=e||event||,||farmer||,||donkey|| and <e1(||event||),e1(||farmer||),e1 (||donkey||)>∈||own||}, then there is an e2 such that e2=e1 and <e2(||event||),e2(||farmer||),e2(||donkey||) >∈||beat||} =
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{<e,e'>| e=e' and for every e1, if e1=e||event||,||farmer||,||donkey|| and <e1(||event||),e1(||farmer||),e1 (||donkey||)>∈||own||}, then <e1(||event||),e1(||farmer||),e1(||donkey||)>∈||beat||} The relationship between (15') and (17') now depends on how we treat events, namely which kind of relationship between sets like {<x,y>| farmer(x)&donkey(y)&own(x,y)} and {<x,y,e>| farmer(x) &donkey(y)&event(e)&own(e,x,y)} we stipulate. It seems to be clear that the latter set can be projected on the former and hence that (17') implies (15'); the other direction of inclusion and implication is, however, doubtful - but we do not elaborate on this here.
3.7 Further ways of elaboration Even after the enhancements outsketched in this chapter, the framework remains in many respects too simple to allow for a satisfactory analysis of some statements and pieces of discourse. However, this was - at least partly - intentional: we wanted first and foremost to bring out the basic principles of the way we think the idea of the exploitation of salience can be formalized and put to work. Let us now mention - briefly - directions in which the possibility of further elaboration of the framework seems to be obvious. 1. Adding more structure to choice functions. The way we have exploited the concept of salience so far makes it possible for every class to acquire a representative. The real content of the concept of salience is, of course, much richer. This could be accounted for by means of replacing of our simple choice functions by some more complicated ones: we could, for example employ functions which would allow to pick up not merely a single representative, but both a representative and a "vicerepresentative". Or we may utilize functions which for each class would yield a partial ordering of elements of the class. The fine-tuning of the optimal version of such a function would have to draw on the empirical results concerning salience, activation, and the stock of shared knowledge. 2. Going beyond extensions. In some cases, we cannot make do working with extensions, we need something more fine-grained. Thus for example in case when discourse participants do not see coextensionality of two predicates, they may well let their respective predicates license different references. This would call for going beyond extensions - we could allow the representatives to be assigned to intensions or to the predicates themselves. However, before accepting the latter solution, we should have to realize that we would then have to painfully stipulate the relations between predicates which we get for free on the level of extension (such as the relation between farmers and men - we saw that stating the rule that the choice of a new representant of a set forces the representant to represent also some/all supersets of the sets automatically leads to the right resolution of many basic cases). 3. Reintroducing variables. In claiming that we can account for anaphora without making use of the traditional apparatus of variables and binding, we did not claim that this apparatus is of no use whatsoever. There is nothing in our framework which would prevent it from being combinable with the more classical apparatus - we may well enrich it by introducing variables for the sake of coping with, say, relative clauses. This may call for making predicates dynamic, but this need not pose a major problem. 4. Accounting for the topic/focus articulation. Anaphora and crossreference are not satisfactorily analyzable without taking into account their interplay with the topic/focus structure. As argued elsewhere (Peregrin 1994; to appear), the logical formalization of this structure requires that we see linguistic utterances as consisting of two subsequent parts, one differing from the other in that the failure of the interpretation of the first of them (the topic, determining what the utterance is about) results in the whole utterance being meaningless (or at least infelicitous), whereas that of the second
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(the focus, spelling out something new) results in it being simply false. This interacts with the anaphorical links (the consequences of the failure of referential availability in connection with the definite description differ according to whether the failure concerns the topic or the focus - viz Hajieová's (1993) distinction between presupposition and allegation). Therefore, we would have to somehow render this two-stage process making the difference between an utterance which is infelicitous and that which is false. Here no obvious ways of modification of our framework are in sight; but it is clear how to seek them.
4 Conclusion In this paper we have proposed a modification of Groenendijk and Stokhof's dynamic logic, a modification based on the employment of choice functions. Dynamic logic was developed to give a compositional semantic account of discourse, i.e. primarily to cope with the problem of coreference extending sentence boundaries. However, it did not tackle two important problems - the problem of resolving linguistically detectable anaphorical links and the problem of a uniform representation of anaphoric expressions, i.e. of definite NPs and pronouns. We have tried to show that modifying it in the proposed way can yield a feasible account of both these problems. Both definite and indefinite NPs are represented as proper terms; they are taken to express choice-function-updates and to refer to actually chosen representatives. The definiteness lies in the fact that a definite NP refers to the already established representative, whereas an indefinite NP picks up a new element, which then becomes the new representative. In this way, the anaphorical linkage is encoded in the contextual information, which is passed on by epsilon functions. No preceding coindexing is necessary. Furthermore, the formalism yields a uniform representation of all definite expressions, i.e. of definite NPs and pronouns.
References. Egli, Urs 1991. (In)definite Nominalphrase und Typentheorie. In: U. Egli & K. von Heusinger (eds.). Zwei Aufsätze zur definiten Kennzeichnung. Arbeitspapier 27. Fachgruppe Sprachwissenschaft Universität Konstanz. Egli, Urs & von Heusinger, Klaus 1995. The EpsilonOperator and EType Pronouns. In: U.Egli et al. (eds.). Lexical Knowledge in the Organization of Language. Amsterdam: Benjamins, 121141. (Current Issues in Linguistic Theory 114) Evans, Gareth [1977] 1980. Pronouns, Quantifiers and Relative Clauses (I). Canadian Journal of Philosophy 7, 467536. Reprinted in: M. Platts (ed.). Reference, Truth, and Reality. London: Routledge and Kegan Paul, 255317. Geach, Peter [1962] 1968. Reference and Generality. An Examination of Some Medieval and Modern Theories. Emended Edition. Ithaca/N.Y.: Cornell Univ. Pr. Groenendijk, Jeroen & Stokhof, Martin 1991. Dynamic Predicate Logic. Linguistics and Philosophy 14, 39100. Hajieová, Eva 1993. Issues of Sentence Structure and Discourse Patterns, Faculty of Philosophy, Charles University, Prague. Heim, Irene 1982. The Semantics of Definite and Indefinite Noun Phrases. Ph.D. Dissertation. University of Massachusetts, Amherst. Ann Arbor: University Microfilms.
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Heim, Irene 1990. EType Pronouns and Donkey Anaphora. Linguistics and Philosophy 13, 137177. Hilbert, David & Bernays, Paul [1939] 1970. Grundlagen der Mathematik. Vol. II. 2. ed. Berlin; Heidelberg; New York: Springer. Hintikka, Jaakko 1974. Quantifiers vs. Quantification Theory. Linguistic Inquiry 5, 15377. Kamp, Hans [1981] 1984. A Theory of Truth and Semantic Interpretation. In: J. Groenendijk & T. M. V. Janssen & M. Stokhof (eds.). Truth, Interpretation and Information. Dordrecht: Foris, 141. Neale, Stephen 1990. Descriptions. Cambridge/Mass.: MIT Press. (Bradford Book) Peregrin, Jaroslav 1992a. Logical Approach to Discourse Dynamics. Prague Bulletin of Mathematical Linguistics 58, 5-10. Peregrin, Jaroslav 1992b. The Role of Variables in the Formalization of Natural Language, Proceedings of the Eight Amsterdam Colloquium. University of Amsterdam, 463-481. Peregrin, Jaroslav 1994. Topic-focus articulation as generalized quantification. In: P.Bosch & R.van der Sandt (eds.). Focus & Natural Language Processing. Heidelberg: IBM Deutschland, 379-387. Peregrin, Jaroslav to appear. Topic and Focus in a Formal Framework. In: B.Partee and P.Sgall (eds.). Meaning and Discourse, Amsterdam: Benjamins. Quine, Willard Van Orman 1960. Word and Object. Cambridge/Mass.: MIT Press. Quine, Willard Van Orman 1986. Autobiography, in: The Philosophy of W.V.Quine (ed. by E.Hahn & P.A.Schilpp), Open Court, La Salle. Sgall, Petr, Hajieová, Eva & Panevová, Jarmila 1986: The Meaning of the Sentence in its Semantic and Pragmatic Aspects. Amsteradm: Benjamins and Academia: Praha. Slater, B. H. 1986. Etype Pronouns and EpsilonTerms. Canadian Journal of Philosophy 16, 2738. van der Does, Jaap 1993. Dynamics of Sophisticated Laziness. In: J. Groenendijk (ed.). Plurals and Anaphora. Dyana2 Deliv. R.2.2.A Part I. August 1993, 152. van Eijck, Jan 1993. The Dynamics of Description. Journal of Semantics 10, 239267. von Heusinger, Klaus 1992. EpsilonAusdrücke als Semanteme für definite und indefinite Nominalphrasen und anaphorische Pronomen. Dissertation. Konstanz
Footnotes. 1. This paper has resulted from (1) the research project "Semantics of natural language" developed by Peregrin under the support of the research grant 70/94 of the Charles University; (2) the research of choice functions and natural language that were developed by von Heusinger together with U. Egli in the project 645/94 "Interaktion von Wort und Semantik" supported by the German Science Foundation. Further, we should like to express our thanks to the Deutscher Akademische
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Austauschdienst, who supported us in the exchange program between the Charles University in Prague and the University of Konstanz. (Back to text) 2. See Sgall et al. (1986), esp. §1.35. Choice functions, as we employ them, can plausibly be seen as a way of explicating "the hierarchy of activation of the elements in the stock of knowledge that the speaker assumes to be shared by the hearer" (ibid., p.55). See also Peregrin (1992a).(Back to text) 3. Let us clear away a possible misunderstanding, which often occurs with respect to dynamic semantics. To facilitate readability, we sometimes speak about the meaning of a dynamically interpreted statement as if about a process - but it must be kept in mind that this is nothing more than a metaphor. The sole aim of logical analysis and logical formalization, which is what we pursue in this paper, is the articulation of truth conditions, it is not a description of any processes which might go in speakers' heads or wherever. The shift from truth conditions to a context change potential must not be literally seen as a shift to the description of some mental events underlying discourse utterances - we do not see semantics as a part of psychology. The specificum of dynamic logic is that it realizes that truth conditions of certain larger units of discourse (A man walks. He whistles) cannot be constructed by simply adding up truth conditions of its parts - some of the parts (He whistles) might simply lack their own truth conditions. So in order to arrive at truth conditions of some larger wholes we need more than truth conditions of the basic units, i.e. of sentences, we need what has come to be called context-change potential. However, what we do in specifying this context-change potential cannot be literally understood as describing the raising of an object to salience - we merely say that a sentence containing the NP may be true only if there exists an object which may be thought of as associated with the NP - whichever this object may be. If there are more such objects, then there is no question of the choice of a particular one.(Back to text) 4. This accomplishment is what Quine (1986, p.15) considers "a major step in conceptual analysis". Cf. also Peregrin 1992b.(Back to text) 5. The idea of employing Hilbert's epsilon operator for the purpose of analyzing indefinite NPs in a way analogous to the employment of Russell's iota inversum for the purpose of analyzing definite NPs by means of his iota inversum is of course far from really new. However, this idea is not easy to exploit, as the nature of the epsilon operator differs from that of the iota inversum one (the former clearly does not render the indefinite article as the name of a definite function in a way corresponding to the latter's rendering the definite article as the name of the function mapping singletons on their single elements).(Back to text) 6. In contrast to the theories of Kamp and Heim, Groenendijk and Stokhof tried to retain as much traditional logic as possible; and they managed to show that the dynamics of discourse does not necessarily call for a framework which would eschew the basic principles of traditional logical analysis. See also Peregrin (1992a).(Back to text) 7. Epsilon functions have been extensively studied by Hilbert and Bernays (1939); however, let us keep in mind that Hilbert's motivations, and consequently also his theory of epsilon functions, was quite different from the present one.(Back to text) 8. Some further restrictions might be added; we might for example require that for every epsilon function e, e(A) is undefined, i.e. that the empty class is never assigned a representative. Or we might stipulate that for every epsilon function e and for every element d of the universe, e({d}) = d (i.e. that the singleton of d is always automatically represented by d); this would make our analysis work also in cases where the classical Russellian mechanism for interpreting definite descriptions were required. We do not explore this here -since this paper aims only to bring out the basic principles, not to elaborate details.(Back to text) 9. e' = es thus expresses a relation between two epsilon functions, it says that e' differs from e at most
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in respect to the representative of s.(Back to text) 10. This semantics provides a unified treatment of definite and indefinite NPs. Both determine their referents by using an epsilon function. This similarity of definite and indefinite NPs was pointed out by Egli (1991).(Back to text) 11. This view meets the Etype analysis of discourse pronouns; it could be seen as stemming from Quine (1960, 102f.): "Often the object is so patently intended that even the general term can be omitted. Then, since 'the' (unlike 'this' and 'that') is never substantival, a pro forma substantive is supplied: thus 'the man', 'the woman', 'the king'. These minimum descriptions are abbreviated as 'he', 'she', 'it'. Such a pronoun may be seen thus as a short singular description, while its grammatical antecedent is another singular term referring to the same object (if any) at a time when more particulars are needed for its identification."(Back to text) 12. Obviously there are several other nonlinguistic factors that influence and rule the relation between the antecedent and the corresponding anaphorical expression, but they are not subject to the present considerations.(Back to text) 13. In the English case - in contrast to languages like German or Czech - this is not wholly correct, for a NP like a farmer does not force a subsequent pronoun to be masculine, it may equally well be followed by she. This indicates that the extension of farmer must be taken to comprise both male and female farmers, and consequently it does not fall into any single category; thus no representative of the class of men is established, and the use of he is not licensed. This means that the success of anaphora resolution in this case is conditioned by the accommodation of the presupposition The farmer is male triggered by the second sentence. Again, we do not elaborate on this here but keep with the simple transparent version of the apparatus.(Back to text) 14. It might seem that we need an update changing not only supersets of the class referred to by the predicative part of the indefinite NP, but also its subsets. We may follow the utterance A sportsman walks by The boxer whistles and get the referent of the boxer to be identified by that of a sportsman, although the class of boxers is a subclass of that of sportsmen rather than its superclass. But this anaphora resolution is not "automated" -it is conditioned by the accommodation of a substantial presupposition triggered by the second utterance, namely The sportsman is a boxer; therefore we do not recommend that it should be accounted for in our present terms.(Back to text) 15. Note that as our semantics is such that sentences are assigned the same type of semantic values as terms (namely sets of pairs of epsilon functions), it would be only a trivial matter to extend our quantifiers so as to accept as arguments not only sentences, but also terms. Then (16') could be alternatively stated as every(a(farmer),boring(the farmer)), which corresponds straightforwardly to the natural language Every farmer is boring.(Back to text) 16. This treatment of donkey sentences uses a kind of unselective binding and, therefore, it cannot represent asymmetric readings. As in other theories, additional constraints must be imposed to license asymmetric readings.(Back to text)
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LINGUISTICS AND PHILOSOPHY Jaroslav Peregrin* [Theoretical Linguistics 25, 1998, 245-264] www.cuni.cz/~peregrin
1. How philosophers became linguists Alle Philosophie ist "Sprachkritik". Wittgenstein (1922, §4.0031) During the first half of the present century a number of outstanding philosophers realized that language theory could profitably be viewed as far more than merely a means of studying one among the many human faculties, or merely sharpening the tool we use to philosophize - they realized that there is a sense in which philosophy of language comprises (almost) the whole of philosophy. This was the famous linguistic turn: philosophers came to accept that everything that is is in a sense through language, and that to study what there is is to study what our words mean.1 The enigma of the language-world relationship was brought to the centre of philosophical discussion early in this century by Frege, Russell, Wittgenstein, Austin and others. Their original point was that we cannot take the representing capacities of language at face value, that in order to treat of things - which cannot be done save with the help of words - we must first treat of words and make sure which of them are really capable of treating of things. Thus the philosophers undergoing the linguistic turn slowly gave up asking what is consciousness (matter, evil etc.)? in favour of asking what is the meaning of 'consciousness' ('matter', 'evil' etc.)? This simple turn seemed to have tremendous consequences for philosophy. By replacing the question what is consciousness? by the question what is the meaning of 'consciousness'? we seem to lose nothing (any meaningful answer to the former question seems to be recoverable from an answer to the latter), and yet it seems to take us from the weird realms of mind to the commonplace domain of language, from the troublesome immersing into people's heads to straightforward observing how they use words. It also seems to guard against the "bewitchment of our reason by language" (Wittgenstein) caused by words which are only seemingly meaningful: such questions as what does the word 'ether' stand for? can be answered simply by nothing, whereas the question what is ether? presupposes that there is something as ether (as *I would like to thank people whose comments on earlier versions of the paper have helped me to improve it in an essential way: Hans Kamp, Pavel Materna, Barbara Partee and Petr Sgall. 1 This concerns especially those philosophers who later came to be called analytic (see Rorty, 1967); but not only them - Heidegger, e.g., has accomplished a turn of a very similar kind. 1
that about the nature of which we are asking) and hence that there is something for which the word stands; thus the latter question, in contrast to the former, forces a certain view of the world on us, simply by our acceptance of it as a question. Ayer (1936, p.35), for one, concludes that The propositions of philosophy are not factual, but linguistic in character - that is, they do not describe the behaviour of physical, or even mental, objects; they express definitions, or the formal consequences of definitions. Therefore, the proponents of the linguistic turn argue, philosophy can be nothing more and nothing else than a certain kind of analysis of language, "the pursuit of meaning", as Schlick (1932) puts it. Metaphysics is thus aufgehoben - it is exposed as a worthless enterprise stemming from the failure to understand the true role of language; it boils down to expressing one's "life feeling" (Carnap, 1931). Thus, philosophers became linguists.
2. How linguists became philosophers I've puzzled for a long time about what the difference is between certain kinds of philosophy and certain kinds of linguistics and finally decided that the main difference lies in whether you're embarrassed about not knowing about a paper in 'Linguistic Inquiry' or the 'Journal of Philosophy' Bach (1985, p.593) Linguists, of course, have been in pursuit of meaning - in their own way - since the very time linguistics came into being; some of them, the semanticians, being even the specialists. And it was only several decades after the linguistic turn of philosophy that something which could be called the model-theoretic turn of semantics occured: many of the linguists who tried to get hold of meaning in an explicit way have come to appreciate the usefulness of Tarskian logical semantics and model theory. This way of approaching the problem of meaning appeared to be particularly promising to the purposes of philosophy; and, in fact, this turn was to a large extent inspired by the heirs of the linguistic turn (especially by Carnap, 1957). It was this approach which seemed to provide the needed framework for making meanings explicit, by reconstructing them as set-theoretical objects. It apparently augured the reconcilation of the intuition of the platonistic character of meanings with the modern mistrust of any 'ghostly entities' like ideas: we only have to presuppose the existence of the ordinary things and the possibility to group entities together - set theory has taught us that this alone is enough to yield us a platonistic heaven. However, the traditional logic with its extensional semantics was quickly deemed to be insufficient - the range of natural language phenomena which could be directly captured by its means was only had to be found scanty. It was necessary either to develop a more sophisticated logical system, or to find ways how to capture the interesting aspects of natural language in an indirect fashion. The first such new way is inseparably connected with the name of Richard Montague (1974), who was the first to show (or at least the first to persuade a broad audience) 2
that if we accept intensional logic with possible-world semantics, we can account for many nontrivial problems of natural language which are beyond the scope of extensional logic. The key concept was that of possible world - a concept introduced implicitly by Rudolf Carnap (esp. 1957; under the name of state of affairs) and explicitly by Saul Kripke (1963).2 Some philosophers, like Quine and Davidson, rejected intensional logic in favour of the good, old, austere classical first-order logic. Davidson (1967), e.g., tried to show how it is possible to analyze certain nontrivial natural language locutions if we let the first-order quantifiers range over what he called events - thus he rejected logic which would implicitly necessitate possible worlds (as objects in the universe of its metatheory) in favour of logic which would explicitly necessitate events (as objects inside the universe of the logic itself). There were others who rejected the concept of possible world on the grounds of the incomprehensible immensity of such an entity - they proposed to replace it by something smaller and more comprehensible, such as situation (thus Barwise and Perry, 1983). Others urged the necessity of having something like situations, which, however, would be the subject of dynamic development (Kamp's, 1981, discourse representational structures, or the informational states of dynamic predicate logic as expounded by Groenendijk & Stokhof, 1991). And others felt the necessity to work with still other entities of diverse natures, like, e.g., Hintikka's (1978) impossible possible worlds, Heim's (1982) files, Tichý's (1988) constructions, etc. All in all, the activities of linguists-semanticians have come increasingly to resemble those of philosophers-metaphysicians; and some of the semanticians have explicitly spoken about doing metaphysics (see, e.g., Cresswell, 1973), or at least 'natural language metaphysics' (Bach, 1986 but see the motto of this section). Thus, the old monsters of metaphysics, once thrown out through the front door, now strike back through the window; and hence linguists became philosophers. My point in this paper is that this alliance of linguistics and philosophy, or, more precisely, of semantics and metaphysics, is, despite all its apparent fruitfulness, rather tricky; and I would like to indicate some of its dangers. On the general level we can say that it is tricky in that it fosters dangerous vicious circularities: linguists explicate some phenomena by relying on certain philosophical entities or doctrines, whose explanation, however, has in turn come to rest on the linguistic phenomena being explicated. A simple example: linguists sometimes like to explain words like necessary simply by referring to possible worlds, whose real nature they take as something they need not bother very much about, because it is explained by philosophers. However, the (post-linguistic-turn) philosophers would reduce explaining possible worlds to explaining the talk about possible worlds, which is nothing but the linguistic (or logicolinguistic) talk about words like necessary.
3. Two senses of 'semantics' I think that it is of crucial importance to point out immediately that the term semantics is used to cover what are in fact two different enterprises, only one of which is directly relevant for linguistics and philosophy. The term covers themes pertaining to two essentially distinct realms: the realm of language and the realm of the links between language and things in the world. Let 2
See also Peregrin (1993). 3
us call that part of semantics which addresses the issues of the former kind semanticsL, while calling the part addressing those of the latter kind semanticsW. The central theme of semanticsL is meaning, and consequently also analytic truth (for analytic truth is "truth in virtue of meaning"). The central theme of semanticsW is (contingent) truth, and consequently reference (for reference is what is needed to compositionally yield truth). The crucial difference is that semanticsL addresses things which one knows in virtue of knowing language: to know the meaning of, say, the king of France it is enough to know English3, there is no need to know anything about the present state of the world. SemanticsW, on the other hand, addresses things which one knows when she knows language and something about the present state of the world: to know what the phrase the king of France refers to one has to know its meaning plus certain facts about France.4 Roots of many puzzles and problems of modern semantic theory can be traced back to confusions between semanticsL and semanticsW. These confusions begin with the unhappy way in which Frege used the term meaning (Bedeutung) for what we now call reference; this usage had the consequence that the knowledge of language seemed to presuppose and to imply knowledge of many extralinguistic facts (for a detailed analysis see Tichý, 1992). This move instantiated an undesirable ambiguity of the term meaning - we should now rather speak about meaningL, which is, in accordance with common sense, a matter of language alone, and about meaningW which is, in accordance with Frege, a matter of relating words to things5. MeaningW of an expression amounts to some causal or "intentional" link between the expression and an extralinguistic thing (a real thing, a 'content of consciousness' or something like that). MeaningL, on the other hand, is the matter of relations between expressions; hence the meaningL of an expression is best seen as something like materialisation of the place of the expression within the system of language, or of its role within the actual language game.6 Approaches to language may then be classified according to which of these notions of meaning they grant primacy: the "nomenclatural", or representational, ones take the relations between 3
This is not literally true because of the proper name; but this is clearly peculiar to the Russellian example. 4The opposition between meaningL and meaningW is sometimes, especially in the context of Saussurian linguistics, reflected by such distinctions as ‘meaning’ vs. ‘content’, ‘Bedeutung’ vs. ‘Bezeichnung.’ or ‘form of content’ vs. ‘substance of content’. Cf. Sgall et. al. (1986, p. 13). 5
It follows from the considerations of Dummett (1974), that even if we consider that of the Fregean terms which is really closer to the intuitive concept of meaning, namely his Sinn (sense), we are likely to encounter a parallel ambiguity, for Fregean senses have come to be taken to play two incompatible roles: to explicate what a linguistic agent grasps when she grasps words, and to determine the corresponding Bedeutung, i.e. extension. In this way, it appears caught on the horns of the dilemma popularized by Putnam (1975). 6
This is not to say that we must accept an absolute boundary between meaningL and meaningW. As Quine showed, the holistic character of language makes it impossible to distribute the relatively clearcut boundary between semanticsL and semanticsW to individual statements and expressions in any unique way; and this makes the boundary between meaningL and meaningW of an individual linguistic item rather illusory. However, an expression surely can be seen as fulfilling two distinct, however inextricably linked, functions: to cope with the world and to collaborate with its fellow expressions. 4
expressions to be parasitic upon the way words are linked to things; whereas the structural, or inferential, approaches claim that the relations between words and things are, the other way around, grounded in the interrelations of words.7 Anyway, it seems to be quite clear that what is in the province of a linguist or a philosopher of language is meaningL, not meaningW: the project of discovering who is the present king of France, required in order to determine the meaningW of the expression the king of France and hence belonging to the project of semanticsW, is clearly not a part of the semantic theory of English. (As Dummett, 1991, p.151 puts it, "in so far as a knowledge of the semantic value of an expression goes beyond what is required for an understanding of it ... its semantic value is not an ingredient in its meaning, and the specification of it no part of a meaning theory.") Meaning, in the ordinary sense of the word, is a matter of semanticsL - knowing meaning is a part of knowing language, not of knowing facts about the extralinguistic world. This implies that the meaning of an expression is not a thing to be discovered within the extralinguistic world, but rather something as the value of the expression, the materialisation of the role of the expression within the system of language and within the language games that we play. Wittgenstein (1984, §64) writes: "Compare the meaning of a word with the 'function' of a clerk. And 'different meanings' with 'different functions'."
4. What is a 'semantic analysis'? This seems to indicate that it is misguided to see the semantic analysis of language as a matter of pairing words and things; that it is more appropriate to see it as a matter of 'finding the position of the expression within the structure of language'. Let us look how things are in practise; let us inspect what the semanticians do when they analyze language. Doing semantic analysis of an expression usually results in providing a formula, a diagram or another expression. Let consider some examples, chosen more or less at random, of various kinds of formulas and diagrams which one can find in books about semantics - (1) is Montague’s (1974, p. 238) logical analysis of one of the reading of the sentence John seeks a unicorn; (2) is Chomsky’s (1986, s. 76) description of the logical form of the sentence I wonder who gave the book to whom; (3) is Kamp’s (15) discourse representation structure (DRS) corresponding to the sentence Every farmer who owns a donkey beats it; and (4) is the ‘tectogrammatical representation’ of one of the articulations of the sentence The professor of chemistry will come tomorrow as given by Sgall et al. (1986). ! seek’(∧j, P ∨u[unicorn’∗(u)∧ P{∧u}])
(1)
I wonder [whomj , whoi [ei gave the book to ej ]]
(2)
7
See Peregrin (1995a, Chapter 8; and 1997). For a detailed analysis of the representational/inferential dichotomy see Brandom (1994). 5
┌──────────────────────────────────────────────────┐ │ ┌───────────────────┐ ┌───────────────────┐ │ │ │ x v │ │ x v │ │ │ │ . . │ │ . . │ │ │ │ farmer(x) │ │ farmer(x) │ │ │ │ x owns a donkey │ => │ x owns a donkey │ │ │ │ donkey(v) │ │ donkey(v) │ │ │ │ x owns v │ │ x owns v │ │ │ │ │ │ x beats it │ │ │ │ │ │ x beats v │ │ │ └───────────────────┘ └───────────────────┘ │ └──────────────────────────────────────────────────┘
comePosterior Dir here
Act
(3)
(4)
Time-when
professor
tomorrow
Obj chemistry
What are we doing in furnishing some such formula or some such diagram? In what sense do we explain the analyzed sentence? In general, providing a diagram may encapsule one of two essentially different enterprises: providing a translation, or providing a description. Providing a translation of the analyzed expression into a language which is taken as understood, or which is in some sense more "semantically transparent", surely means explicating meaning - but, equally of course, only relatively to the uncritical acceptance of the language into which we translate. Providing a description elucidates the meaning to the extent to which it is the description of the meaning, or of that to which we hold the meaning to be reducible, e.g. the use of the expression, or a 'cognitive content' for which the expression is supposed to stand. Restricting ourselves to the two most prominent reducienda of the meaning of an expression, namely the use of the expression and the mental entity ('cognitive content') 'behind' the expression, the following main possibilities seem to emerge as to what a diagram associated with a sentence, or, more generally, with an expression, can amount to: (i) a description of the meaning of the expression (ii) a description of the way the expression is used 6
(iii) a description of a mental entity associated with the expression (iv) a translation of the sentence into another language The first alternative seems to offer the most promising route: what could be a more direct realisation of the task of semantics than displaying expressions alongside with their meanings?8 However, this proposal is rather tricky; for what could count as a description of meaning, which, as we have concluded in the preceding section, is best seen not as a 'real' object, but rather as a value? The most secure way to describe the meaning of an expression is to use the expression itself - to describe the meaning of, say every farmer owns a donkey we best use the description the meaning of 'every farmer owns a donkey' or, possibly, that every farmer owns a donkey. However, using these would lead to trivialities like The meaning of 'Every farmer' is the meaning of 'every farmer' 'Every farmer owns a donkey' means that every farmer owns a donkey We may, of course, also say something less trivial, e.g. 'Every farmer owns a donkey' means that for every x, if x is a farmer, then x owns a donkey; however, what is nontrivial with this is the purported synonymy of 'Every farmer owns a donkey' and 'For every x, if x is a farmer, then x owns a donkey'; i.e. the fact that the latter is - in a certain sense - a faithful translation of the former. Thus it seems that there is no interesting direct describing of the meaning of an expression which would not rest on finding an interesting translation of the expression into another language (or an interesting paraphrase of the expression in the same language); and providing (i) seems to be in this sense parasitic upon providing (iv)9. As Quine (1969, p. 53) puts it, "A question of the form 'What is an F?' can be answered only by recourse to a further term: 'An F is a G.'" It is important to realize that the same applies, mutatis mutandis, to (ii). The most straightforward way to characterize the use of 'Every farmer owns a donkey' is to say something like The sentence 'Every farmer owns a donkey' is (correctly) produced if and only if every farmer owns a donkey (this is, of course, a severe oversimplification, but not one that affects the point made here). And if we use another sentence on the right hand side of the biconditional, then it is the purported synonymy of this sentence with the characterized sentence which is nontrivial. Moreover, despite appearances, the situation does not differ substantially even in the case of (iii). It might seem that in this case we may be able to pick up some relevant 'content of consciousness' independently of any linguistic articulation; however, it is hard to see how we could identify contentful mental entities save by way of language; we cannot describe the mental entity 'beyond' the sentence 'Every farmer owns a donkey' save by saying that it is the thought (or idea, or whatever) that every farmer owns a donkey, or the thought that for every x, if x is a farmer, then x owns a donkey etc. What is worse, even if we could give an independent 8
See Chomsky (1967).
9
For details see Peregrin (1993; 1995a, Chapter 11). 7
characterization of such a mental entity (e.g. by means of some 'mentographic coordinates'), this would not really further our attempts to grip meaning: being told that an expression is associated with such or another lump of a mental stuff can never by itself reveal us what the expression means, for knowing what it means involves knowing how it behaves relatively to other expressions, what follows from it etc.10 There is also no help in recourse to talking of 'neural events' or the like: it is true that these, unlike mental entities, are specifiable independently of the sentences whose usage they may accompany (at least in principle); however they are quite like thoughts in that if they are specified in this way, they cannot really provide us with meanings.11 So it seems that diagrams offered by semantic analysts cannot be taken as descriptions of meanings in a direct sense (in the sense in which a photo is the description of the bearer of a name). This indicates that the only real sense which can be made of formulas and diagrams as exemplified above is in terms of translating the analyzed language into another language. However, does this not suggest that this kind of semantic analysis is circular and consequently futile?
5. The Myth of the Structure One of the common way to avoid this ‘intractability of meaning’ is to move the concept of meaning to the periphery of one’s teaching and to concentrate on the word struture. The enterprise of semantic analysis, it is then claimed, consists in revealing the "semantic structure" of an expression (or of the mental content of an expression). Thus, for many theoreticians of language, meaning has come to coincide with something like the semantic structure; and semantic analysis with pinpointing this structure. This might be understood as accepting the structural approach to language urged above - but usually it is not. The point is whereas what we have urged is an approach which sees meaning of an expression as the possitionof the expression within the network of language, the common way of engaging the concept of structure is based on the picture that an expression is like, say, a mineral: that it can be analyzed and examined with tools akin to microscopes up to the point where we see its structure. This picture essentially obscures the fact that an expression does not have any inherent structure in the sense in which a mineral has - at least no interesting inherent structure. (An expression does have an inherent structure in that it consists of words and letters but this is not the structure held in mind by those who use the term structure to make sense of semantics.) The fact that the structures which linguistic theories ascribe to an expression are not really to be found on the expression itself has forced many linguists to acquire the conviction that what they are studying are - ultimately - not expressions, but rather mental objects which the 10
This is, of course, only an anecdotic hint at the case made against mentalism by Frege, Wittgenstein and others. It is, of course, also only another expression of the fact spelled out earlier in the paper: namely that meaning is not a thing, but rather a value. 11
Hans Kamp has suggested to me that one of the ways to express this is the following: „A theory of the implementation of memory presupposes a theory of understanding of meaning“. 8
expressions stand for. The structure of the expression, the story goes, is the structure of a mental entity behind the expression - be it called an idea, an intention, a cognitive content, or whatever. Thus the situation has arisen where many linguists begin calling themselves 'cognitive scientists'. This semantic mentalism is often complemented with a kind of 'reduction axiom': everything mental is physical, every event in the mind is (in fact) an event in the underlying brain etc. This seems to guard against the suspicion that what is going on is the old mentalism which has been seriously challenged by so many philosophers - the structures which are studied are ultimately tangible structures of the human brain. However, this is illusiory - the structures postulated by linguists are clearly not results of studying the brain - the books which present them do not map neural synapses nor anything of their kind (and, in fact, as pointed out in the previous section, if they did so, they would not be about semantics). The structures are obviously the results of studying language - which is, however, understood as studying mind, which is in turn postulated to be studying brain. The thesis advocated here is that the structure of an expression is essentially a quite different kind - it is a theoretical construct which locates the expression within the system of the language to which it belongs. We first reconstruct language as a rule-based system; and this reconstruction causes expressions to fall into certain categories. If the rules which we consider are the rules of syntax (i.e. if they provide for the criterial reconstruction of well-formedness), then the resulting categories are known as syntactic categories (they express the expressions' behaviour from the viewpoint of constituting well-formed expressions and statements); if they are the rules of semantics (i.e. if they amount to truth, assertibility, or use in general), then the categories are meanings (they express the expressions' behaviour from the viewpoint of truth, or, more generally, from the viewpoint of their employability within language games). Anyway, given such a reconstruction we come to observe every expression as a construct built according to certain rules from parts of certain categories12. And this is a holistic matter - the expression only has this kind of structure when considered as belonging to the system of language. In fact, this applies to all abstract entities and their structures - in contrast to concrete entities like minerals. A mineral does have its structure independently of (the existence of) any other minerals (at least independently of those which are not its spatial parts) - it is enough to use a microscope which would enable us to identify it. The structure of an abstract entity, on the other hand, is always the matter of the entity's position within the web of other abstract entities of the same category - there is no "mental microscope" to examine it in isolation and penetrate inside it. This has become especially clear with the development of the mathematical theory of categories (see, e.g., Herrlich and Strecker, 1973), whereby any kind of formal structure is defined solely by means of morphisms between objects displaying this kind of structure (thus, e.g., to be a set is to be a member of a family of objects interrelated by a certain web of relationships). Let's, for the sake of illustration take a diagram of the kind of the Kampian DRS (3). What does it depict? As far as my experience goes, the majority of people practising DRT would answer to the effect that it depicts something like the (structure of the) mental content which is expressed by the expression analyzed, or that it somehow records what is going on with speakers' and/or hearers' mental representations. However, this is nothing but a cheap readymade universal answer - (3) is not the result of an introspection or of an extrospective 12
For details see Peregrin (1995b; 1997). 9
psychological analysis, it is the result of examining the linguistic properties of the analyzed sentence, namely its relations to other sentences, especially to those which imply it and those which are implied by it. Another story, however, can be told: a story which construes the switch from the more traditional, "static" semantic theories to the more recent, "dynamic" ones, like DRT, in terms of acknowledging certain inferential properties of certain sentences (prototypically those involving anaphora) - properties which are hard to account for with recourse only to traditional tools. Evincing Kamp's own example (personal communication), if we analyze the sentences One of the three candidates is over forty and Two of the three candidates are under forty by traditional means, we are unable to account for the important difference between them, namely that the former can, while the latter cannot, be followed by We eliminate him. This vantage point lets us see DRT, and semantic theory in general, as an explicit reconstruction of structural, inferential patterns governing our use of language carried out via explicating the roles of individual expressions within these patterns.
6. Semantic analysis as envisaging inferential structure This line of thought leads to a picture of semantic analysis quite different from the one envisaged by the usual uncritical construal. What we do in explicating semantics of words and sentences via formulas and diagrams is not picturing extralinguistic things or concepts or structures purported to be the meanings of the expressions; we rather envisage the roles of the words and sentences within the structure (esp. inferential structure) of language13. We achieve this by developing languages (or quasilanguages) whose expressions wear their inferential roles more or less on their sleeves. To assess the adequacy and reasonability of a diagram used to pursue semantic analysis we thus should not try to probe the speaker's and hearer's minds to find out whether we glimpse something which could be pictured by the diagram, but we should rather consider the following two points: (A) Is the analysandum adequate to the analysatum, does the inferential role of the former within the analyzing language 'reasonably approximate' that of the latter within the analyzed one?; and (B) is the inferential role of the analysatum, as a part of the analyzing formal language, in some sense explicit? Let us return to (3) once more. Does it provide us with a useful semantic analysis of Every farmer who owns a donkey beats it? To answer this question, it is not enough to consider (3) in isolation: if it is isolated from the body of DRT, it obviously provides us with no semantic analysis at all, for any formula or diagram can successfully play the role of semantic analysatum only as a node within a large structure expounding relevant relations. (Note that this would not be the case if (3) were the picture of the meaning of the analyzed sentence.) To ask whether (3) is a reasonable semantic analysis is to ask whether DRS's can be put into correspondence with English sentences in such a way that (A) there is a 'reasonable' extent to which DRS's defined to imply (to be implied by) other DRS's correspond to sentences intuitively implying (being 13
Thus providing what Sellars (1974) calls their functional classification. 10
implied by) sentences corresponding to the other DRS's; (B) the inferential properties of DRS's are in some sense more explicit than those of English sentences (the properties can be somehow read off from the DRS's themselves); and (C) (3) corresponds to Every farmer who owns a donkey beats it. This yields an understanding of the nature of the praxis of semantic analysis which may differ dramatically from the commonsense view. It may not really tackle the praxis itself; for this praxis largely consists in collecting and cataloguing facts about language, and this is something that is largely independent of an 'ideologic' background. However, it has tremendous consequences for grasping the possibilities and limitations of drawing philosophical consequences from such a semantic analysis; and by corollary also for understanding the nature of semantic analysis itself.
7. Realism? Some of the arguments of the last two sections can rightly be seen as decrying mentalism in semantics. Does this mean that I am siding with 'realism' as against 'conceptualism' in the sense of Katz and Postal (1991)? Not quite - for the best way to see this paper is as fighting on two fronts: against the construal of semantics as parasitic upon psychology, and against its construal as based on a realistic metaphysics. One reason for my reluctance to be seen as engaging myself in the struggle for realism is that there is a straightforward sense of 'realism' for which no such struggle would warrant itself - for every minimally plausible semantic theory trivially has to be 'realistic' in this sense. I am convinced that nobody, not even the most diehard mentalists and conceptualists, would claim that semantics is the matter of describing some mental (neural) particulars within the head of an individual speaker - for this would be no theory of English (nor of any other language), but rather the theory of some features of a particuar person. Even if we accept the assumption that semantics is a matter of particulars of such a kind, we simply have to assume that these particulars can be somehow equated over speakers; that they have some properties which make them treatable as different tokens of same types14. So the semanticist must talk about some nonparticulars - be they construed as intersubjective identities of particulars, or some abstract entities borne by these identities. In any case, talk about meaning is in the clear sense talk about types, not about tokens; and semantics is - in this sense - inevitably realistic. On the other hand, even the most diehard realist has to assume that there are some contingent facts that elicit which meaning an individual expression has. We do not discover meaning by an 'intellectual trip' into a realm of abstracta where we would see them attached to expressions; but rather by observing and recording certain concreta. It is the occurrence of certain particular events or entities (the occurrence of certain contents within the heads of speakers, or the occurrence of certain utterances of speakers) which establishes the meaning of an expression15. 14
A detailed argument against a particularistic construal of mind in general has been presented in the famous paper of Sellars (1956). 15
It is precisely this fact which Quine (1960) took seriously to gain his well-known robust 'behavioristic' constraints of the theory of meaning, which then led to the indeterminacy theses and subsequent dismantling of the atomistic view of language. 11
Therefore, both the conceptualist and the realist apparently must agree that meanings are abstracta (universals) which are in a certain sense determined by (parasitic upon) certain concreta (particulars). So, if the only thing that realism claimed were that semantics is a matter of abstracta rather than of concreta, of types rather than of tokens, then realism would seem to be unobjectionable. And if the only thing which conceptualism asserted were that abstracta make no sense unless they are in the sense outlined ‘parasitic’ upon concreta, then it too would be unobjectionable. Hence, this modest conceptualism and modest realism coincide - for our knowledge (in general) arises out of apprehending particular occurrences as displaying universal structures. The only clash is then a terminological one: whether this situation justifies us in saying that linguistics is about the particular occurrences, or about the universal structures. This is a legitimate subject of a quarrel, but not of one which would go very deep.16 The trouble is that both of them seem to claim something more. Conceptualism seems to claim that, first, the particulars which are relevant in lingustics are mental entities (or contents of consciousness, or the internal wirings of our ‘language faculty’), and, second, that the theoretician of language has no use of abstract entities whatsoever. I have indicated why I think this conception of a theory of language is futile: I have indicated why the mentalistic conception of meaning is problematic (only hinting at all the complexities discussed at length by Wittgenstein and his direct and indirect followers - in the American context especially by Sellars, Quine and Davidson); and I have also indicated that any theory worth its name must concern itself with public universals rather than with private particulars, and must envisage an intersubjectively understandable "form" or "structure". Realism (in the spirit of Katz and Postal), on the other hand, seems to claim not only that linguistic data, to be construable as such, must display some regularities and appear as instances of a realistic "form"; they seem to claim also that these realistic entities are accessible in a direct way. Katz and Postal write about "sentential structure" which can be examined to see if it is "at some grammatical level logically significant" (ibid., 519). This invokes the picture of our descending into the depths of the sentence in question, and inspecting a certain floor in its underground to see whether it displays a certain feature; the picture criticised in Section 5. This is why I prefer adjudicating between that which I argue to be an adequate theory of language and that which I claim to be inadequate not in terms of the realist versus conceptualist distinction, but in terms of the difference between the structuralistic, or inferentialistic, and the nomenclaturistic, or representationalistic, theory.
16
This is to say that there is one sense of "about" in which linguistics is about concreta, and another sense of "about" in which linguistics is about abstracta. It is, of course, a severe error to construe linguistics to be about abstract entities in the former sense of "about" (i.e., roughly speaking, in the sense of having abstract entities as the ultimate source of evidence). If this is what Chomsky criticises, then he is surely right. 12
8. Conclusion We must not try to resolve the metaphysical questions first, and then construct a meaning-theory in the light of the answers. We should investigate how our language actually functions, and how we can construct a workable systematic description of how it functions; the answers to those questions will then determine the answers to the metaphysical ones. Dummett (1991, p.338) Philosophy, at least in its analytic variety, has in a certain sense come to rest on the analysis of language; any notion of metaphysics over and above 'natural language metaphysics' has proven itself to be rather precarious. Therefore it is hardly possible to base natural language semantics on a metaphysics. It is futile to see the enterprise of semantics as secondary to that of some (real or would-be) metaphysics; to think that we must first clarify and formally depict the structure of the word and only then to pair expressions with the elements of the word thus depicted. At the same time it is futile to see semantics as parasitic upon a psychology of language use. Semantics is primarily neither a matter of relating words with things, or of words with thoughts, it is a matter of displaying a certain kind of structure of language. Thus, semantic analysis is always ultimately a matter of translating the language that is to be analyzed into another language - it makes sense if the latter is in some relevant sense more perspicuous than the former. There is no absolute measure of what is or is not more perspicuous - it all depends on the purpose and on the visual angle. Montague grammar, e.g., can be extremely perspicuous for some people (those educated in logic and model theory and familiar with the symbolism), while extremely obscure for others. Lewis (1972) correctly points out that trading expressions for other expressions is not in itself a semantic analysis, but this should not be understood as saying that the touchstone of a true semantic analysis is that it pairs expressions with things (for no theory can do better than to pair expressions with expressions); the touchstone is rather that it pairs expressions with expressions of a specific kind, namely with expressions of a (quasi)formal language with its (inferential) structure explicitly articulated. The paradigmatic cases of such 'inferentially explicit' languages are, of course, the languages of logic. One of the important consequences of this view of semantics is that there is nothing as the structure of language. Every structure we ascribe to language and to individual expressions is the result of our theoretical reconstruction, and every theory is guided by a purpose. Therefore, there is not much sense in striving for something as "the right and absolutely adequate semantic theory". A theory is like a scheme someone draws up to help us see the principles of operation of a complicated machine, or to help us find our way through a town: it makes us see something which is otherwise obscured - and this may be accomplished at the cost of purposefully neglecting something else. The analysis of language is indeed crucial for many (if not all) traditional philosophical problems. Unfortunately not all the philosophers who have undergone the linguistic turn have really bothered to penetrate into the depths of the true semantic structure of language; and not all of those linguists who have succeeded in discerning the real nature and perplexities of various 13
parts of language have avoided seeing language uncritically as a kind of nomenclature of some 'cognitive contents'. True, it is not the business of philosophers to study details of our grammar; and it is not the business of linguists to answer the philosophical questions about the nature of our language. However, the sagest abstract philosophical conception of language is empty if it does not reflect the facts of how language really works; and the most detailed atlas of the landscape of language is impotent if it is not clear which questions it purports to answer. I think that the recent results of semantics are overwhelming. Take for example the large body of studies concerning the nature of definite and indefinite descriptions, which have persuasively shown that to see these locutions directly in terms of classical, Fregean quantification is inadequate and may be severely misguiding. Or take the interesting results of the systematic investigations of the linguistic evidence for the count/mass, event/process or individual/stage distinctions. Or take the rich results of the inquiry into the vast gallery of kinds and workings of presuppositions. All these results have greatly advanced us in our understanding of the nature and structures of our language; however, I think that to become really operative, they must be placed within the framework of a more sophisticated theory of language; a theory which would not rest on some naive picture of expressions as signs which we use to label exhibits of the world-museum, or to externalize our thoughts.
References Ayer, A.J. (1936): Language, Truth and Logic, Gollancz, London. Bach, E. (1986): 'Natural Language Metaphysics', in Logic, Methodology and Philosophy of Science 7 (ed. by R.B.Marcus, G.J.W.Dorn and P.Weingartner), North-Holland, Amsterdam, pp. 573-595. Barwise J. and Perry, J. (1983): Situations and Attitudes, MIT Press, Cambridge (Mass.). Brandom, R. (1994): Making it Explicit, Harvard University Press, Cambridge (Mass.). Carnap, R. (1931): 'Überwindung der Metaphysik durch logische Analyze der Sprache', Erkenntnis 2, pp. 219-241. Carnap, R. (1957): Meaning and Necessity, University of Chicago Press, Chicago. Chomsky, N. (1967): ‘The Formal Nature of Language,’ Appendix A to E.H. Lenniberg: Biological Foundations of Language, Wiley, New York. Chomsky, N. (1986): Knowledge of Language: its Nature, Origin, and Use, Praeger, Westport. Cresswell, M.J. (1973): Logic and Languages, Meuthen, London. Davidson, D. (1967): 'The Logical Form of Action Sentences', in The Logic of Decision and Action (ed. N.Rescher), The University of Pittsburgh, Pittsburgh; reprinted in Davidson (1980), pp. 105-122. Davidson, D. (1980): Essays on Actions and Events, Clarendon Press, Oxford. Dummett, M. (1974): '"Postscript"', Synthèse 27, 523-534. Dummett, M. (1991): The Logical Basis of Metaphysics, Duckworth, London. Groenendijk, J. and Stokhof, M. (1991): 'Dynamic Predicate Logic', Linguistics and Philosophy 14, 39-101 Heim, I. (1982): The Semantics of Definite and Indefinite Noun Phrases, University of Massachusetts, Amherst. 14
Herrlich, H. and Strecker, G.E. (1973): Category Theory, Allyn and Bacon, Boston. Hintikka, J. (1978): 'Impossible Possible Worlds Vindicated', in Game-Theoretical Semantics (ed. by E.Saarinen), Reidel, Dordrecht, pp. 367-379. Kamp, H. (1981): 'A Theory of Truth and Semantic Representation', in Formal Methods in the Study of Language (ed. by J.Groenendijk, T.Janssen and M.Stokhof), Mathematical Centre, Amsterdam; reprinted in Truth, Interpretation and Information (ed. by J.Groenendijk, M.Stokhof and T.Janssen), Foris, Dordrecht, 1984, pp. 1-41. Katz, J.J. & Postal, P.M. (1991): 'Realism vs. Conceptualism in Linguistics', Linguistics and Philosophy 14, 515-554. Kripke, S. (1963): 'Semantical Considerations on Modal Logic', Acta Philosophica Fennica 16, pp. 83-94. Lewis, D. (1972): 'General Semantics', Semantics of Natural Language (ed. by D.Davidson and G.Harman), Reidel, Dordrecht. Montague, R. (1974): Formal Philosophy: selected papers of R.Montague (ed. by R.Thomason), Yale University Press, New Haven. Peregrin, J. (1993): 'Possible Worlds: a Critical Analysis', Prague Bulletin of Mathematical Linguistics No. 59-60, pp. 9-21. Peregrin, J. (1995a): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (1995b): 'Structuralism: Slogan or Concept?', Lingua e Stile 30, 445-464. Peregrin, J. (1997): 'Structure and Meaning', to apear in Semiotica. Putnam, H. (1975): 'The Meaning of "Meaning"', Language, Mind and Knowledge (Studies in the Philosophy of Science VII, ed. by K.Gunderson), University of Minnesota Press, Minneapolis; reprinted in Putnam: Mind, Language and Reality (Philosophical Papers 2), Cambridge University Press, Cambridge, pp. 215-271. Rorty, R., ed. (1967): The Linguistic Turn, The University of Chicago Press, Chicago. Sellars, W. (1956): 'The Myth of the Given: Three Lectures on Empiricism and the Philosophy of Mind', in The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minnesota Studies in the Philosophy of Science I, ed. by H.Feigl and M.Scriven), University of Minnesota Press, Minneapolis. Sellars, W. (1974): ‘Meaning as Functional Classification’, Synthèse 27, 417-437. Schlick, M. (1932): 'The Future of Philosophy', College of the Pacific Publications in Philosophy 1, pp. 45-62. Tichý, P. (1988): The Foundations of Frege's Logic, de Gruyter, Berlin. Tichý, P. (1992), 'Sinn and Bedeutung reconsidered', From the Logical Point of View, N.2, 1-10. Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London; translated as Tractatus Logico-Philosophicus, Routledge, London, 1961. Wittgenstein, L. (1984): Über Gewissheit, Suhrkamp, Frankfurt.
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Mark Norris Lance and John O’Leary-Hawthorne: THE GRAMMAR OF MEANING Cambridge University Press, Cambridge, 1997 xiii + 450 pp. The linguistic turn associated with the ‘classical’ period of analytic philosophy fostered the conception of philosophy as a kind of pursuit of meaning (Schlick). Philosophers desired, first and foremost, to get their grip on meaning, be it simply by meticulous studying and mapping of the intricacies of ordinary language, or by improving of this language via logical or mathematical ‘engineering’. Meaning was usually seen as something hidden (within a Platonic heaven or within our minds) which philosophers were to disclose, analyse (and, perhaps, ‘fix’). Some of the (post)analytic philosophers of the second half of the twentieth century, notably the late Wittgenstein, challenged this approach by pointing out that the picture it suggests, namely the picture of language as a set of labels stuck on some ready-made things, can be severely misguiding. Among American philosophers, the most significant critics were Quine and Sellars (as Rorty, 1980, duly pointed out): Quine indicated that it is problematic to see meaning as a thing because it is usually too vaguely and indeterminately delimited (and concluded that it would be better to try to entirely relinguish the concept) while Sellars urged that meaning is not a thing because it is an essentially normative matter. Sellars’ criticism has not become so popular as Quine’s (probably because Sellars’ writings are very hard to read), but recently a number of philosophers seem to be realising that it is perhaps even deeper and more far-reaching. The view of language urged by Sellars has been recently also remarkably elucidated and developed by the book by Sellars’ disciple Robert Brandom (1994): language is, according to Sellars and Brandom, primarily a tool of our, human ‘game of giving and asking for reasons’, and as such it is essentially a matter of implicit proprieties and norms (which can be made explicit - thus becoming accessible to critical assessment and, as the case may be, alteration - in semantic and logical discourse.) Thus, according to this line of thought, when we speak about meaning, we spell out the norms implicit to our linguistic practices. The authors of the book under the present review accept this Sellarsian insistence on the essentiality of the normative and the consequent non-naturalistic account of meaning, but what they suggest is that even this kind of appreciation of normativity is still not radical enough. When speaking about meaning we, according to them, are not describing or spelling out the actual norms governing our (nor anybody else’s) linguistic behaviour (the rules of the relevant language game), what we are doing is making essentially normative (‘ought to’) statements (aiming at modifying the rules). Thus, if we say that ‘gavagai’ means rabbit, or that bachelor is an unmarried man, we are not stating what is the case (we neitner describe relationships in a platonic realm of concepts, nor the way words are used by our or by somebody else’s community), we are, rather, suggesting what ought to be done: how words ought to be used or how they ought to be translated, and that we ought to ‘censure’ the people failing to use or translate them in this way. This is a surprising, and for somebody maybe even preposterous, thesis, so let us examine how the authors arrive at it and what they have to say in its support. This is the content of the first part of their book. The first chapter discusses the problem of translation (so popular within analytic philosophy since Quine’s seminal thought experiments with radical translation) and interlinguistic semantic discourse in general. After a critical summary of the Quinean views, the authors call the attention to an aspect of translational claims which is not reflected by Quine, namely to the fact that these claims „are speech acts whose point is to influence a structure of social practices, to impose a (possibly new) socially recognised constraint upon behaviour“ (p. 61). This is to say that when we establish a translational manual, we do not simply record regularities of the natives’ linguistic behaviour, we propose a way to build a bridge, between the native and our community, to „form one large community where there previously were two“ (p. 64). This is to say that adopting a translational manual is „not a process of describing a prior set of standards, either implicit or explicit. Rather, it is a matter of agreeing to a normatively binding document, a set of constraints on further behaviour“ (p. 63). Consequently, „meaning claims license certain inferences and license censure to those who do
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not acquiesce in such inference, censure that does not take the form of mere disagreement with the person censured, nor even of the attribution of irrationality, but which instead treats them as at least partially exempt from the ‘language game’“ (p. 64). Thus, the principal effect of the claim ‘gavagai’ means rabbit is, according to the authors’ conception, the license to treat the natives’ talk about ‘gavagai’s as a talk about rabbits, and the license to banish those who do not treat them so from our newly established natives-us linguistic alliance. In the second chapter the authors turn their attention to intralinguistic semantic claims, with the goal of showing that also claims of this kind are best seen in terms of licensing and censuring, as purporting to establish normative proprieties. They start their argument with a reappraisal Quine’s challenge to the analytic/synthetic distinction: is there a kind of analyticity, they ask, which would be compatible with the Quinean picture of language (pace Quine himself)? And their answer is positive: a sentence is analytic in this workable sense if „failure to assent to it is (or would be) taken as excellent evidence that the person has failed to understand one word or other (and thus, relatedly, as good grounds for moving from the realm of substantive argument to that of stipulation, paraphrase, or pedagogy)“ (96). This means, the authors suggest, that although there does exist a feasible notion of analyticity, it can only be understood in normative terms, viz in terms of a censure to those who do not accept statements analytic in this sense. From this finding they move towards a general conclusion about the normativity of meaning claims: „To claim, for example, that ‘F’ means ‘G’ on our account is to license a certain sort of inference [namely the inference from ‘F’ to ‘G’] and to license a certain sort of censure [namely of those who refuse to endorse the inference]“ (127). Thus, „meaning talk is primarily used to provide normative guidance for inferential behaviour“ (138). Chapter three then addresses the heart of the matter - here the authors develop their own conception of the normative and show how it can lend further support to their conclusions about the normativity of semantic discourse. Normative assertions are, according to them, neither declaratives, nor imperatives: they are „of a grammatical category which, while having the same sort of criteria of application as descriptive assertions, have in certain crucial respects the same sort of consequences of applications as imperatives“ (198). Normative utterances are, according to them, to be understood in terms of an effort to change the form of a practice, to change the rules of a game we are playing. „Normative assertions,“ the authors say, „are to be seen, on the side of their constitutive consequences, as efforts to bring into explicit question the future development of a particular practice“ (209). This means that we have to reject both the transcendental conception of norms (which sees norms as absolute and independent of any factual practice, while normative discourse makes sense only within the framework of a practice), and the attributive conception (which sees norms as simply rules of a game, and thus leaves no room for questioning them - for to question a rule of a game is to question whether we should play this very game, not to question whether the game should have this very rule). The authors claim that the common failure of both the conceptions consists in the fact that they postulate some level of unchangeability (the absolute norms in case of the transcendental conception, and game-constitutive rules in case of the attributive one), while genuine normative discourse is marked by the absence of such an unchangeability, its constitutive point being, we could perhaps say, the entertainment of human freedom (or spontaneity, to use the Kantian term recently resurrected by McDowell, 1994). Thus, the verdict is the following: „The goal of asserting a normative propriety ... is to attempt to constrain the future proprieties of play within a game, the existing practice of which is provisionally assumed to be generally in order and which thereby forms the context within which the normative proposal has its sense“ (213). I think that this part of the book possesses everything that warrants an excellent book: it opens an entirely new vista on traditional problems while being extremely intelligible and well argued. And it challenges some of the most central pillars of the standardly held views with such vehemence that if one accepts the arguments of the authors, one is indeed likely to experience a real „shock of recognition“. Also, I think that the authors do point out something of basic importance: that the assumption that semantics amounts to some or other kind of description (which appears to be an „unspoken dogma“ common to the majority of contemporary analytic philosophers, from Quine and Lewis to Dennett and Searle) may obliterate something vital. Thus, while Sellars and Brandom urged that to understand the point of semantics it is not enough to cease seeing it as disclosing some ‘things-of-
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the-mind’, but that it is also necessary to cease viewing it as reporting regularities of linguistic usage and to start viewing it as spelling out norms and proprieties, the authors of the present book indicate that even this may still not suffice. And they give good arguments in support of their thesis. Moreover, their original conception of norms steers ingenuously between the traditional absolutistic Scylla and the relativistic Charybda: it is enough to accept that we humans somehow have the ability to assess and modify the rules we live by (perhaps it is the very ability we call, since Kant, our Vernunft?), and we can render the normative as something which is neither absolute (for it does not make sense outside of the context of an existing set of rules), nor simple relative (for it does make sense to criticise the existing rules and such criticism can be right or wrong). If this is right, then it is the space for the exercising of human freedom, lacking from those rather rigid pictures traditionally offered to us by the majority of analytic philosophers, which is needed to account for the real nature of semantics. This is not to say that the conception of the authors does not raise doubts - it does provoke various kinds of doubts (and it would be a wonder if a truly novel conception did not do so). Let us mention at least two. First, it is hard to accept the authors’ ‘normative radicalism’ which appears to suggest that semantic discourse is normative through and through. What their arguments do make plausible is that some meaning claims are utterly normative, and perhaps that many of them are partly normative, but it is hard to conclude that all of them are utterly normative. This is to say that although one may well feel persuaded that the point of some utterances of ‘gavagai’ means rabbit is simply to propose or establish a rule for a newly created linguistic alliance, it is hard to believe that there are not cases in which it simply reports, if not the natives use the term ‘gavagai’ as we use ‘rabbit’, then what the natives take to be the correct usage of the term ‘gavagai’ is what we take to be the correct usage of ‘rabbit’. Such utterances appear to be straightforwardly descriptive, and not normative in the authors’ sense. (There is, of course, a sense in which every pronouncement whatsoever is normative: if I say This is a horse, then I license a censure to accept the entity pointed at as a candidate for the UN chancellorship - but this is clearly not very interesting.) Second, the authors claim that normative claims lie somewhere between indicatives and imperatives, but what they say about their role within our linguistic practice seems only to elucidate their imperative aspect: to propose an alteration of an accepted practice, which is the alleged point of semantic claims, is to urge we ought to use these linguistic items thus and so. (Let us note in passing that if we assimilated such claims to fully-fledged imperatives of the kind of let us use these linguistic items thus and so, we would be echoing proposal of Ayer, 1936, and others to see analytic truths as suggestions to use words in certain ways). We are left in virtual darkness about the indicative aspect of the pronouncements. There is the fact, to be sure, that they can enter into inferences, but is this all? Are we to see normative claims as, besides proposing something, also reporting something? This last worry is perhaps partially addressed by the second part of the book, but unfortunatelly in a way which is likely to raise more new doubts than it resolves. Here the authors turn to problems of what they call the metaphysics of meaning: „Are there facts about meaning? If so, what sort of entities are they? What sorts of facts, if any, do facts about meaning supervene upon? How are claims about meaning to be analysed? By virtue of what do words and sentences have the particular meanings they do? What is the relationship between semantic facts and non-semantic facts?“ (p. 242) I think the reader may rightly expect that the reaction to such questions, implicit in the conclusions of the first part of the book, will be that meaning talk is not the kind of talk for which a ‘metaphysics’ would make much sense, and consequently that all the above questions are either beside the point, or capable of being answered in some trivial, uninteresting way. After all, if semantic claims are „attempts to constrain the future proprieties of play within a game“, what sense, over and above a trivial one, could it make to see it as expressing facts? And in fact, in the next three chapters, chapters four to six, the authors seem to fulfil this expectation: they try to subvert various attempts to render ‘meaning facts’ as consisting in, or supervening on, ‘naturalistic facts’; i.e. attempts to naturalise the meaning talk. However, this part of the book is rather less comprehensible and less persuasive than the first one: the trouble is that the authors seem to be not content with arguing against the naturalistic theories of meaning from their
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own position (if they are right that the meaning talk is irreducibly normative, then any kind of a theory which tries to translate the talk into naturalistic terms is eo ipso simply wrong) and to wish, additionally, to defeat their opponents using the opponents’ own weapons. Thus, they engage in lengthy discussions of issues not directly relating to the central argument of their book. The real surprise comes in the last chapter of the book, where the authors, despite their rejection of the possibility of naturalising the meaning talk, argue for the necessity of a ‘metaphysics of meaning’ and indicate their own way of approaching it. They claim that although they have so far provided „some illumination concerning why we need meaning discourse ... a number of important metaphysical questions remain“ (375). However, as I indicated earlier, this seems to be far from clear - and I think that the reader may question, why, if we accept that „to make a claim about meaning is not to attempt to describe, but to attempt to legislate, rigidify, amend, or codify“ (374), we need, in addition, any kind of „metaphysics“. When we consider, e.g., the question are there facts about meaning?: does the authors’ conception not imply the (uninteresting) answer that there are no facts in the strict sense in which facts are correlates of the descriptive talk, and that there are facts in the loose sense in which facts are correlates of any kind of talk which admits of rightness or wrongness? Does it add something to our understanding of the meaning talk if we see it as a factexpressing enterprise? The authors try to reject all kinds of reasons which undermine seeing semantic claims as truthvalueless or not fact-expressing - however, they somehow fail to consider the one which is perhaps the most obvious, namely that the claims are ‘ought-to’ statements. This might square with their earlier insistence that the normative, ‘ought-to’ statements have not only an imperative, but also an indicative aspect - but the reader would probably expect an explanation of this fact, whereas this is only its reassertion. Thus, if the authors say that „if ... semantic decisions are intertwined with one’s theoretical decisions, then metaphysics cannot but be regarded as an exercise in fallible theoretizing“ (376), the objection which comes to mind is that this is not true, for metaphysics can also simply be spurned as an enterprise inappropriate to the nature of the matter. Despite all such reservations, I think the book belongs to the very best of recent publications in the field of philosophy of language: it dismantles many superstitions, opens new and surprising horizons, and suggests many novel answers to traditional questions. And even if you do not accept all the answers, the hard work you are going to have with substantiating your disagreement is likely to do much good to your personal understanding of the nature of language. Briefly, a bright book which is likely to provoke deep and interesting discussions.
References: Ayer, A.J. (1936): Language, Truth and Logic, Gollancz, London. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). McDowell, J. (1994): Mind and World, Harvard University Press, Cambridge (Mass.). Rorty, R. (1980): Philosophy and the Mirror of Nature, Princeton University Press, Princeton. Jaroslav Peregrin
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Johan van Benthem and Alice ter Meulen, eds. Handbook of Logic and Language Elsevier Science B.V. / The MIT Press 1997, Oxford/Shannon/Tokio / Cambridge (Massachusetts). xxiii + 1247 pp. Reviewed by Jaroslav Peregrin The relationships between logic and natural language are multiverse. On the one hand, logic is a theory of argumentation, proving and giving reasons, and such activities are primarily carried out in natural language. This means that logic is, in a certain loose sense, about natural language. On the other hand, logic has found it useful to develop its own linguistic means which sometimes in a sense compete with those of natural language. This has led to the situation where the systems of logic can be taken as interesting "models" of various aspects of natural language. The alliance of logic and linguistics has flowered especially from the beginning of the seventies, when scholars like Montague, Lewis, Cresswell, Partee and others showed how semantics of natural language can be explicated with the help certain suitable logical calculi and the corresponding model theory. (Montague went so far as to claim that in view of this, there is no principal difference between natural and formal languages - but this is, as far as I can see, rather misguiding.) Since that time, the interdisciplinary movement of formal semantics (associating not only linguists and logicians, but also philosophers, computer scientists, cognitive psychologists and others) has yielded a rich repertoire of formal theories of natural language, some of them (like Hintikka's game-theoretical semantics or the dynamic logic of Groenendijk and Stokhof) being based directly on logic, others (like the situation semantics of Barwise and Perry or DRT of Kamp) exploiting different formal strategies. Moreover, although the enterprise of formal semantics (i.e. of modeling natural language semantics by means of certain formal structures) seems to be the principal point of contact between linguistics and logic, there are also other cooperative enterprises. One of the most fruitful ones seems to be the logical analysis of syntax, which has resulted from elaboration of what was originally called categorial grammar. (However, even this enterprise can be seen as importantly stimulated by Montague.) All in all, the region in which logic and theoretical linguistics overlap has grown both in size and fertility. And so the task set out by the book under review, namely a comprehensive survey of this field, is far from simple; and the authors of the volume had to produce not less than some twelve hundreds pages of text to carry it out. Fortunately, the book has such a distinguished cast that the guide through the realms of logicolinguistics which the reader gets is genuinely firsthand. In view of this, the book aspires to become a real Principia Semantica of our age. The book is divided into three parts: Frameworks, General Topics and Descriptive Topics, and into twenty chapters. The opening chapter of the book is devoted (what else?) to Montague Grammar; it is written jointly by Barbara Partee and Herman Hendriks. The ninety pages of this contribution offer an excellent and deep introduction into both the motives behind Montague's deed ("Montagovian revolution", as the authors term it), and the resulting theory; and add a comprehensive bibliography of Montagovian books and papers. The twenty five years which have elapsed since the publication of Montague's crucial articles enable the authors to assess the real import and impact of this revolution. The next chapter, Michael Moortgart's Categorial Type Logics, focuses on the relatively recent, but intensively flourishing, enterprise of applying the methods of logic to syntactic structure. The basic idea of this enterprise is that of treating syntactic combination as inference: we read the fact that expressions of the respective categories A1,...,An combine into an expression of the category A as, in effect, A1,...,An implies A. Given suitable ways of categorial indexings, what we gain are various kinds of interesting logical systems. Moortgart's exposition of the state of the art is detailed
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and exhaustive. The third chapter, written by Jan van Eijck and Hans Kamp, deals with the broad theme of Representing discourse in context; it is in fact largely devoted to the popular framework of Kamp's discourse representation theory. The underlying idea of this framework is that "each new sentence of a discourse is interpreted in the context provided by the sentences preceding it" (p. 181). This yields the theory which is well known for its illustrious box-based formalism (an aspect which is perhaps more attractive, for a working linguist, than it is usually assumed), but which has also solid formal backing - and this is what the authors demonstrate in this chapter. The next chapter is devoted to Situation Theory - the metaphysical, or model-theoretical part of the once famous, but now perhaps not so much popular, Situation Semantics. As Jerry Seligman and Lawrence S. Moss, the authors of this chapter, put it, Situation Theory investigates "foundational questions about the emerging [from Barwise and Perry's work] ontology, with the hope of providing a unified mathematical framework in which the linguists' work could be interpreted" and it "was intended to stand to Situation Semantics as Type Theory stands to Montague Grammar" (241). The task of the authors was a difficult one, for after the short heyday of Barwise's and Perry's proposal the enterprise splintered into a couple of different enterprises which had to be reconstituted into a single framework. But I think they succeeded - they offered a rich gallery of metaphysical denizens borne by situation theory, and a rich mathematics to handle them. The penultimate chapter of the first part of the book presents an introduction to GB theory, the recent outgrowth of the Chomskian approach to language. The author of this chapter is James Higginbotam. This is a slightly different kind of semantic theory: neither a model-theoretic, nor a set-theoretic reconstruction of meanings or representations behind linguistic items, but rather a theory of semantics as one of the layers of human language faculty - as argued for by Chomsky and his followers. And it also does not rest on logic (with the exception of the usage of the term 'logical form', which was chosen by Chomsky, I think unhappily, for the semantic layer). However, although the Chomskian approach differs considerably from those inspired by logic, the latter would hardly be possible without the former's prior 'mathematization' of linguistics. And besides this, the great majority of people involved with the theory of natural language are in this or another way significantly influenced by the Chomskian picture; so the elements of the picture often loom even in theories which are otherwise developed from different perspectives. The closing chapter of the foundational part of the book is devoted to the framework of Gametheoretical semantics, a dynamic semantic framework stalwartly championed and elaborated by Jaako Hintikka since the beginning of the seventies. The present contribution is written by Hintikka together with Gabriel Sandu. Hintikka originally considered the fact that standard logic allows for a game-theoretical interpretation (its formulas being construed as encodings of certain games between 'Me' and 'Nature') and concluded that there is no decisive reason to restrict oneself to precisely those kinds of games which happened to be catered for by this very logic. Hintikka's and Sandu's current result is what they call independence-friendly logic: a logico-semantical system superficially close to standard predicate calculus, but possessing a number of remarkable logical properties (the most shocking of them being the ability to express its own truth predicate). I think that the six chapters of the first part of the Handbook, which are to cover the basic logicolinguistic frameworks, have been chosen with real ingenuity - they encompass the most substantial and the most self-contained systems of 'formal' and 'logical' approaches to natural language and especially its semantics. First, there are three paradigmatic frameworks which can be seen as representing the three most important stages of development which have occurred since the original marriage of linguistics and logic within Montague's writings: the first of them, which can be called intensional and which culminated in the seventies, is represented by Montague Grammar, the second, hyperintensional, one, whose heyday was roughly the following decade, can be seen as represented by Situation Semantics, whereas the third, dynamic, paradigm, whose culmination we are currently witnessing, is represented by DRT. Besides them, there is the current stage of the not to be neglected Chomskian approach to language, which has opened the door for formal methods in linguistics and in various ways deeply influenced many of the more logical approaches. And I also agree that the game-theoretical semantics is a framework worth being included: this approach, though perhaps less popular than the other, exploited the dynamic view of language long before it became common.
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The second part of the book, General Topics, is opened by the chapter on Compositionality, written by Theo M. V. Janssen. Janssen explains what compositionality is (discussing various kinds of formulations of the principle of compositionality), gives examples and then outlines mathematical theories of compositionality. He concludes is that semantics can be either compositional or noncompositional, and that there are decisive reasons to prefer compositional theories - so that compositionality is a substantial desideratum of a good semantic theory. My impression is that despite the exhaustive discussions of many aspects of compositionality which Janssen presents, he fuses together, unhappily, two different problems: the philosophical problem of the status of compositionality (is compositionality constitutive to the very concept of meaning, or is compositionality only a contingent property of some meanings?) and a mathematical question about properties of compositional mappings (which may be construed, relatively noncontroversially, as homomorphisms of certain kinds of algebras). Janssen gives a thorough answer to the latter question, but his claim that compositionality is not essential is ambiguous and in both senses problematical: if it is to be interpreted as saying that there are not only homomorphic mappings, then it is a platitude not worth stating, and if it is to be interpreted as claiming that compositionality is not constitutive of meaning, then it would have to be supported by a discussion of what meaning is - a discussion which Janssen does not offer. Chapter VIII, written by William C. Rounds, describes the framework of Feature Logic. This logic was devised to deal with formal structures emerging from the formal analysis of natural language which can be seen as sets of attribute-value pairs. Rounds' contribution is a detailed overview of the corresponding formal apparatus. Raymond Turner, the author of Chapter IX called Types, addresses the general logical concept of type, as a semantic counterpart of the syntactic concept of category. (The concept was introduced into modern logic by Russell, whose name, curiously enough, is not mentioned in Turner's paper.) He introduces the basic logical formalism which can be taken as framing general theory of types, namely Church's typed lambda-calculus, and discusses its relationship to other logical formalisms, like categorial logic or higher order predicate calculus. Then he discusses a number of problems connected with the idea of a type: various kinds of type-polymorphism, constructivity etc. In Chapter X, Dynamics, Reinhard Muskens, Johan van Benthem and Albert Visser analyze the foundations of the current widespread tendencies toward "dynamization" of semantics. The common idea behind them is recognized to be that of context change: like a command of a computer program, a natural language utterance comes to be seen not as statically expressing a content, but as dynamically changing the current context or information state. The authors then discuss the ways in which this general idea is captured within various kinds of semantic theories: the most common way seems to be to see the change of context as the change of assignment of values to some variable elements of a representation (this is the way of Kamp's discourse representation theory, Heim's filechange semantics and Groenendijk and Stokhof's dynamic predicate logic), but it is also possible, as the authors point out, to construe it in terms of change of the attentional state, the change of assumptions, or the change of belief. (The last perspective then yields the intensively studied logical theory of belief-revision.) The formal aspects of dynamic logical frameworks are then discussed in detail. Chapter XI, written by Jens Erik Fenstad, analyzes the concept of Partiality; on a very general level, perhaps too general. The range of phenomena which the author sees as instances of partiality is vast: truth-value gaps, elliptical utterances, presuppositions, incomplete knowledge, partial algorithms and many more. Unfortunately, this makes the concept of partiality so general that it becomes almost empty of content. The most interesting part of the chapter seems to be that which concentrates on the clearly graspable topic of truth-value gaps and partial logic. Chapter XII bears the title of Mathematical Linguistics and Proof Theory and is written by Wojciech Buszkowski; it partly overlaps with Mortgaart's exposition of categorial type logic from Part I of the book. The author sees mathematical linguistics, originally mostly the theory of syntax of formal languages based on generative grammars and automata, as now married with proof theory (in the way analyzed by Moortgart). Chapter XIII slightly departs from the other topics of this part, it does not provide a theory of an aspect of language, but rather of language learning. The authors are Daniel Osherson, Dick de Jongh, Eric Martin and Scott Weinstein; and the title of the chapter is Formal Learning Theory. What is in
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focus are various models of knowledge acquisition and especially language learning, often built with the help of logical and model-theoretical tools. Richmond H. Thomason, the author of the last, fourteenth, chapter of this part, addresses the topic of Nonmonotonicity in Linguistics. His contribution consists of the discussions of various settings which may lead to nonmonotonic inference, of formal apparatuses which may be employed to account for it (especially feature structures), and of the case studies of nonmonotonicity within phonology, morphology, syntax and semantics. On the whole, I think that Part II is slightly less well-balanced than Part I. First, there seem to exist "categorical differences" between the topics of the individual contributions. The core of the contributions concentrate on particular general concepts (compositionality, types, dynamics, partiality, nonmonotonicity), while others rather discuss specific fields (mathematical linguistics, learning theory) or even specific frameworks (feature logics). Besides this, some place their emphasis on the analysis of the very concept they address, whereas others concentrate on the corresponding mathematics. However, this is not meant as a real criticism - it is only a matter of the ingenious layout of Part I having aroused expectations of the same ingeniousness of organization within subsequent parts. The third part of the book, Descriptive topics, contains contributions addressing various specific aspects of language and of those formal theories which have been developed to account for these aspects. The first of this part's chapters is devoted to Generalized Quantifiers; the authors are Edward L. Keenan and Dag Westert?hl. The contribution gives the motivation for developing a theory of generalized quantifiers and summarizes the basics of the corresponding mathematics. However, some of the aspects of the authors' exposition should perhaps be more clearly explained: I doubt that I am alone in failing to see, e.g., why, if we abbreviate [P(E)->2] as <1>, we should abbreviate [P(E)->[P(E)->2]] as <1,1>; or what is the rationale behind seeing the sentence Most critics reviewed just three films as a single quantifier most ... just three applied to critics, films and reviewed. Mark Steedman's chapter Temporality accounts for its subject in an extraordinarily illuminating way. It divides the theme of temporality into three basic compartments: (i) temporal ontology, which addresses all kinds of entities which are being employed to account for temporality (such as activities, accomplishments, achievements); (ii) temporal relations, which accounts for the types of relationships which may be considered to obtain between entities of these kinds; and (iii) temporal reference, which deals with the reference points of utterances and with temporal anaphora. Also the next contribution, devoted to Presuppositions and written by David Ian Beaver, belongs to the best chapters of the book: it not only analyzes all relevant aspects of the phenomenon in question and all important ways to account for it, but presents the analysis in an extremely systematic and comprehensible way. Beaver first distinguishes between semantic presuppositions (A is presupposed by B if B cannot have a truth value unless A is true) and pragmatic presuppositions (A is a pragmatic presupposition of a speaker S if S simply takes A for granted), then overviews locutions which trigger presuppositions and then goes on to discuss, in detail, the formal means used to account for presuppositions. These means are basically of two kinds: static (multivalence and partiality) and dynamic (based on contexts and context-change potentials). The next, shorter chapter, Plurals and collectivity by Jan Tore L?nning, discusses semantic analysis of plural nouns. Within the framework of standard logic, the author discerns two basic kinds of approaches: either to take the denotations of plurals as some suitable higher-order objects, or to take them simply as members of the universe, in which case the universe is to be given a 'mereological' structure. Ways which bypass standard logic altogether are also mentioned. The penultimate chapter of the book concentrates on Questions and is written by Jeroen Groenendijk and Martin Stokhof. The authors consider two possible approaches to questions: the pragmatic approach (which amounts to accounting for questions on the level of speech acts, not on the level of semantic content) and the semantic one (which has it that questions have semantic content in the same way as assertions do and can be analyzed by analyzing the specific nature of this content). Groenendijk and Stokhof discuss various logical, computational and linguistic theories based on either of the two approaches. The closing chapter, by Francis Jeffry Pelletier and Nicholas Asher deals with the phenomena of Generics and Defaults. Its authors expose the peculiar status of generics (which are in a sense
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'law-like' despite allowing for exceptions) and survey the usual ways to account for them (such as relevant quantification, approaches based on prototypes or stereotypes etc.). Of these, attend primarily to default logic. At the end of the paper they develop their own formal theory of generics. The handbook is an extremely successful attempt at a state of the art summary of the interdisciplinary field which centers around the intersection of logic and linguistics (and partly also philosophy of language, computer science etc.) All the contributions are written by competent authors; many of them probably by the most competent ones. Moreover, they are almost all written in the disciplined way which validates them as true encapsulations of the state of the art of the problematic they address, rather than as vehicles of popularization for their authors. This makes the book into an indispensable compendium for anyone working with language. However, it seems to me that it is precisely such a representative survey of the whole field in question, and the possibility of seeing the whole field from a 'bird's eye view' it offers, which could (and perhaps should) also provoke some very general, foundational (and maybe sometimes heretical) questions about the nature of the whole enterprise. I myself am plagued by the obsessive thought that now, faced with such a vast amount of answers, it is time to pay more attention to understanding (and in some cases plainly finding out) the questions to which they belong.
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PRAGMATIZATION OF SEMANTICS Jaroslav Peregrin* (The Semantics/Pragmatics Interface from Different Points of View, ed. K.Turner, Elsevier, Amsterdam, 1999, 419-442)
1. Introduction During the middle of the present century, Charles Morris and Rudolf Carnap established their triadic division of the theory of language, which has since become one of the determinants of our theoretical understanding of language1: syntax was to deal with the relations between expressions; semantics was to address the relations between expressions and what they stand for; and pragmatics was to examine the relations between expressions and those who use it. The aim of this paper is to summarize some recent considerations of the nature of language and linguistic theory which seem to challenge the usefulness and adequacy of such a division and to indicate that these considerations may provide for a new paradigm. I attempt to show that these considerations indicate that the Carnapian boundary between syntax and semantics is, in the case of natural languages, misconceived; while that between semantic and pragmatics is more stipulated than discovered. The Carnapian paradigm has been challenged, during recent decades, in two ways; we may call these the internal and the external way. By the internal challenge I am referring to the mutation of the Carnapian model evoked ‘from inside’, namely a development of linguistics and logic which extends Carnapian semantics far beyond its original boundaries to swallow up much of what was originally counted to pragmatics; while by the external challenge I mean the questioning of the whole model, namely a development within the philosophy of language which casts doubt on the entire Carnapian way of viewing language. These two movements are largely independent of each other, but they may be seen as manifesting the common drift to what can be called the pragmatization of semantics.
2. The ‘Internal’ Challenge to the Carnapian Paradigm 2.1 The Problem of Indexical and Anaphorical Expressions It was especially Carnap’s book Meaning and Necessity (1956) that pointed out the direction for those who wanted to account for the semantics of natural language via modelling it with the *
I am grateful to Petr Sgall and Ken Turner for valuable critical commnets to the earlier version of this paper. 1
See, e.g. Carnap (1939).
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help of formal languages of logic. Before Carnap indicated how logic can surpass extensions, the models logic had been able to offer were, from the point of view of natural language, hopelessly oversimplified. Carnap initiated the process which culminated in Montague’s (1974) intensional logic, thereby establishing a firm foundation for what we now call formal semantics of natural language2. The intensional model finally convinced many linguists and philosophers of language that to see natural language ‘as a logic’ and to develop logical languages for the purpose of modelling natural language may be enlightening. Thus Carnapian semantics came to fruition3. However, it soon became clear that to reach an exhaustive semantic analysis of natural language we unavoidably trespass on the boundary which Carnap drew between semantics and pragmatics. The meaning of an expression in the sense of Carnapian semantics was supposedly something ‘context-independent’, i.e. something which had nothing to do with the context or circumstances under which the expressions happen to be uttered (for these were matters of pragmatics which Carnap obviously considered not addressable with the rigour he requested for semantics). However, the more practising semanticists extended the range of natural language phenomena under semantic analysis, the less they found they could eliminate ‘contextdependence’. The first kind of terms which resisted an ‘context-independent’ analysis were indexicals: words like I, you, here, there, now, before etc. It was apparently impossible to say what these expressions meant without speaking about the circumstances of their utterance. Their denotations are obviously dependent on such or other aspects of the context; and in fact it is natural to see their meanings as some kinds of functions which yield a denotation when applied to the context. Thus, I may be seen as denoting an individual in a similar way as Jaroslav Peregrin, but the individual is determined only by the application of the meaning of I to the actual context (for what the application does is extract the utterer out of it). Hence indexicals appeared to be what I have elsewhere called context consumers (see Peregrin, to appear): to yield semantically relevant values, they had to be fed by the context. The kind of expressions whose analysis then attracted the attention of most natural language semanticists towards the concept of context, some two decades ago, were pronouns: again, it is hard to say what a pronoun means without talking about context: a pronoun’s function seems to be, just like that of I, to denote an individual somehow picked up from the context of its utterance. However, there is an important distinction between I and he: whereas I 2 3
For more about this establishing see Partee and Hendriks (1997).
In contrast to Montague, Carnap did not see the investigation of natural languages as the very same kind of enterprise as the investigation of formal languages of logic. In fact, in the beginning of his paper ‘Meaning and Synonymy in Natural Languages’ he identifies „the empirical investigation of historically given natural languages“ with pragmatics and introduces the term „semantics“ only within the context of „constructed language systems“ (see Carnap, 1956, p. 233). However, he clearly did see his „constructed language systems“ as models in terms of which we were to grasp natural languages (as witnessed, for example, by the fact that only two paragraphs later in the same paper he claims that the description of a language, such as German „may well begin with the theory of intension“, i.e. with the theory of its semantics).
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utilizes context in the sense of the ‘non-linguistic’ circumstances of utterance, he may utilize ‘linguistic’ context resulting from a preceding discourse. Thus, to understand the working of pronouns, we have not only to understand that some expressions may be context-consumers, but also to realize that some other expressions may be context-producers: that they can provide such contexts on which pronouns (and other anaphoric expressions) then live. Another kind of expressions the analysis of which has proved to require the concept of context are articles. Their classical, Russellian, analysis (see Russell, 1905; and Peregrin, to appear, for a recapitulation) resulted in understanding a as expressing existence and the as expressing unique existence, but this has now been recognized as generally inadequate; for a great deal of the functioning of articles has likewise proved specifiable only in a ‘contextdependent’ way4. The indefinite article, as it turned out, is generally best seen as a means of introducing a new item into the ‘context’ (and attaching a label to it); whereas the definite one is best seen as a means of pointing at a specifically labelled individual present within the ‘context’. Thus, we may see a man as storing an item with the label man, and we can see the man as searching out an individual with precisely the same label. This allows for the intricate anaphorical structure of discourse (see also Peregrin, in prep.).
2.2 Topic and Focus Such challenges to the traditional ways of construing the ‘logic of language’ have led also to the reassessment of the basic semantically relevant parsing of our pronouncements. Even the traditional concepts of subject and predicate, which were usually seen as expressing the most basic backbone of our sentences, demand a ‘dynamic’ approach: it seems that the concepts of subject and predicate, if they are to have the semantic import they are usually credited with, cannot be seen as delimited by the traditional rigid grammatical criteria. That part of the sentence which is most reasonably seen as its semantical subject need not always coincide with the grammatical subject, and similarly for the predicate. This is what was urged by Frege (1892b, p.74): „Die Sprache hat Mittel, bald diesen, bald jenen Teil des Gedankens als Subjekt erscheinen zu lassen.“ My suggestion is that what we should see as underlying the semantically relevant subject-predicate patterning of sentence is not its grammatical counterpart, but rather that which the linguists of the Prague Linguistic Circle once called ‘aktuální členění větné’, now usually translated as ‘topic-focus articulation’ (see Peregrin, 1996; for topic-focus articulation as an ingredient of the sentence structure see Sgall et al., 1986, and Hajičová et al., in print)5. The semantical subject coincides with the topic of the utterance (‘východisko výpovědi’), whereas the semantical
4 5
In fact, this finding goes back to Strawson (1950).
The stress which the exponents of the Circle put on this articulation is to be seen in the context of their functionalism, which took language to be first and foremost „a functioning system, adapted to its
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predicate coincides with its focus (‘jádro výpovědi’)6. This intuition can be accommodated within the framework of formal semantics in various ways. Perhaps the most straightforward of these is the one just sketched (and elaborated in Peregrin, 1996): to treat the topic as the semantic subject (picking up a piece of information ‘as an object’, thereby triggering an ‘existential’ presupposition) and focus as the semantic predicate (presenting some further specification of the object). There are, however, other, perhaps less perspicuous ways, which nevertheless may better fit with current techniques of formal semantics. One is to base the account for topic and focus on the theory of generalized quantifiers and see them as arguments of an implicit generalized quantifier, or as - in terms of Partee (1991) - the restrictor and the nuclear scope of a tripartite structure (see Peregrin, 1994). In certain cases, the implicit quantifier can be overridden by an explicit focalizer, such as always or only, but also by negation (cf. Hajičová, 1994). However, if we adopt a consequentially dynamic stance, it is best to see topic and focus as two phases of an information-conveying act (and they can be pictured as two segments of a dynamically viewed proposition). Topic corresponds to the phase where the information gets anchored to the existing ‘informational structures’, and focus to that where the genuine new information is being added. Therefore, the failure of the act during the topic-phase (i.e. the falsity of the relevant presupposition) means the failure of the whole act (which may precipitate a - possibly temporary - breakdown of communication), whereas that during the focus-phase (i.e. the falsity of the assertion) engenders merely the failure to add new information.
2.3 Meaning as ‘Context-Change Potential’ These and similar conclusions have led many semanticists to see meanings of natural language sentences as context-change potentials. (The term is, as far as I know, due to Irene Heim, who belonged, together with Hans Kamp, to the main initiators of the ‘dynamic turn’ of semantic theory of natural language; see Kamp, 1981, Heim,19827.) It has also led to the development of a new kind of logic which reflects this change of perspective and which gives logical analysis of natural language a surprising proximity to the theory of programming languages (see Groenendijk & Stokhof, 1991, van Benthem, 1997)8. In fact, we can see the development of formal semantics as the struggle for dominating increasing ranges of natural language locutions. We may see intensional logic as resulting from communicative role“ (Sgall, 1987, p.169). Such a position naturally leads to general tendencies to ground semantics in pragmatics. 6
It is necessary to keep in mind that the terms topic and focus are sometimes used also in different senses.
7
Similar ideas have been presented by Hintikka & Kulas (1985), Seuren (1985), Sgall et al. (1986), Chierchia (1992), as well as by others.
8
The principles of this whole turn are summarized by Muskens et al. (1997).
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the effort to master the various modal aspects of language; we may see ‘hyperintensional’ semantic theories (like Cresswell’s, 1985, theory of structured meanings, Barwise’s and Perry’s, 1983, situation semantics, or Tichý’s, 1988, theory of constructions) as the result of turning attention to those aspects of language which concentrate within propositional attitude reports; and we can now see dynamic semantics as resulting from the effort to account for anaphoric aspects of language. And the same holds for the semantic entities brought about by these theories: for possible worlds and intensions of intensional semantics, for the various kinds of structures, situations or constructions of the ‘hyperintensional’ semantic theories, and for the contexts or information states and context-change potentials of dynamic semantics. Note that the identification of the meaning of a sentence with its context-change potential does not entail that the difference between semantics and pragmatics vanishes. Take I: it remains a matter of pragmatics that when I now utter the sentence I am hungry, it will refer to me, while if you do so, it will refer to you. However, it is the matter of semantics that it always refers to the speaker (whoever she or he might be). Similarly it is a matter of pragmatics that he in He is hungry refers to me if the utterance of the sentence is accompanied by pointing at me, or if it follows the utterance of Here comes Peregrin; but it is a matter of semantics that he gets its referent, in a certain way, from the context. Anyway, it no longer seems feasible to do formal semantics of natural language disregarding the concepts of context and context-dependence. The range of semantic phenomena which cannot be adequately explained without their help is vast, and the very working of language is essentially oversimplified if meanings are explicated in a way which does not account for how utterances interact with each other via contexts.
3. The ‘External’ Challenge 3.1 Language as a Toolbox Along with this ‘internal’ challenge, the Carnapian paradigm has been challenged also in a quite different way; namely by the development of a wholly alternative view on language which has been claimed, by its partisans, to be philosophically more adequate and more fruitful. This view started to lurk in the writings of several analytic philosophers during the second half of this century. It is an approach based on viewing language not as a set of labels stuck on things, but rather as a kind of a toolbox, the significance of its elements thus lying in the way they function rather than in their attachment to things. The first prominent propagator of such a view was the ‘later’ Wittgenstein, who showed that to try to separate the question what does a word mean? from the question how is the word used? is futile; and concluded that thus what a word means consists in how it is used. „The meaning of a word is its use within language,“ as he puts it in his Philosophical Investigations (1953, §43).
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A similar visual angle has been adopted by Willard Van Orman Quine and subsequently by Donald Davidson, who tried to approach the question what is meaning? via considering the question how do we find out about meaning?. In Quine’s hands, this gave rise to the instructive contemplations of the problem of radical translation (see esp. Quine, 1960); Davidson speaks about radical interpretation (Davidson, 1984). This stance has served the philosophers to identify what it is that we learn when we learn an expression; with the conclusion, akin to Wittgenstein’s, that it is the way the expression is used. From this angle, it seems that it must be pragmatics, as the theory of how people use linguistic signs, rather than semantics, which should be the heart of a theory of language. The impression that semantics becomes, from this viewpoint, in a sense parasitic upon pragmatics (rather than the other way around), has been underlined by the considerations of another outstanding American philosopher of the second half of this century, Wilfrid Sellars. Sellars pointed out that what appears as semantics is often rather ‘pragmatics in disguise’: that what we really do when we seemingly state the semantic relationship between a word and a thing is to specify the function of the word in question by means of invoking that of some other, familiar word. Thus, ‘Kanninchen means rabbit’, according to Sellars, does not state a relation between two entities, the (German) word Kanninchen and, say, rabbithood, it rather describes the function of Kanninchen within German as a function analogous to the one the English word rabbit has in English (see Sellars, 1963)9. Perhaps the most symptomatic, and probably also the most popular, picture of language reached in this way can be found in the writings of Davidson, who has most consequentially assumed the stance from which language appears to be essentially a tool or an aspect of human action, inextricable from the network of other actions. Therefore I shall call this kind of approach to language, which I shall see as in competition with the Carnapian one, the Davidsonian approach (without thereby claiming that all its details are ascribable to Davidson).
3.2 The Roots of the Carnapian Paradigm We can hardly deny that the new, Davidsonian paradigm is prima facie much less plausible than the old, Carnapian one. To explain what makes us nevertheless recommend it, we have to inspect the source of the apparent plausibility of its rival; we shall try to indicate that this plausibility is dubious. Carnap’s triadic classification is based on a picture which is very natural: on a picture of language as a ‘nomenclature’ (as I called it elsewhere - see, e.g. Peregrin, 1995, Chap. 8), i.e. as a set of labels conventionally attached to certain extralinguistic entities. What makes up language in the first place is, then, the linkage between its signs and some things which the signs stand for; and it is the theory of this linkage which is the subject matter of Carnapian semantics. 9
In fact, this position is close to that of Carnap before he embraced what we call the Carnapian paradigm. See Carnap (1934).
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This theory is then supplemented, on the one hand, by a theory of the idiosyncratic nature of the signs themselves; and, on the other hand, by the theory of how the signs are employed by human agents. (It is unnecessary to add a specific theory of the entities which the signs stand for, because these are supposed to be ordinary, non-linguistic things falling under general theories of concrete or abstract things.) The two other theories, syntax resp. pragmatics, may be considered as secondary to semantics: in so far as something qualifies as a language simply by its elements standing for their meanings, to analyze this ‘standing for’ relation is to analyze what is really essential. Pragmatics is then, and this was as Carnap indeed seemed to see it, a not very interesting matter of the idiosyncratic ways speakers employ words and sentences when they use language (what they feel and imagine when using its expressions etc.). And syntax, although surely interesting in its own right, is inessential in the sense that language could fulfil the same function even if the idiosyncratic syntactic features of its expressions were quite different provided the ‘standing for’ relation were retained. I can see two main reasons leading Carnap to his triad (if we disregard the support it gains from its accordance with common sense): firstly that it is straightforward for artificial, formal languages, and secondly that it accords with the doctrine of logical atomism which tacitly underlaid the philosophical views of the majority of logicians and analytical philosophers of the first half of the present century. Formal languages of logic have been usually defined by defining their syntax and semantics: syntax established how their formulas were to be formed and ‘reformed’ (‘umgeformt’, in Carnap’s, 1934, German term), whereas semantics established what the formulas and their parts were to stand for. (In his 1934 book, Carnap’s intention was to make do with syntax only, but later, especially under the influence of Tarski’s development of ‘scientific semantics’, viz Tarski, 1936, he fully embraced also the set-theoretically constructed semantics.) These two compartments of the theory of language had to be complemented, Carnap obviously thought, by a third one which would comprise everything that could not be directly subjected to logic - i.e. matters concerning the idiosyncratic ways people actually employ language. That applying this view also to natural language well accorded with the doctrine of logical atomism is not hard to see. The doctrine, explicitly entertained by Russell (1914; 1918/19; 1924), but implicitly endorsed also by Wittgenstein (1922) and Carnap himself (1928) is based on seeing the language and the world as edifices erected upon certain atomistic bases; and seeing the link between language and the world in terms of an isomorphism of the edifices: what a complex statement (a logically complex, that is, for what really counts is the logical structure, which can be covert, not the superficial, overt structure) stands for is a certain conglomerate of that which is stood for by its parts (see Peregrin, 1995, §5.6). However, I think that none of the reasons for the Carnapian paradigm is to be embraced; albeit the grounds for rejecting them are quite different. The reason stemming from the analogy between natural and formal languages is to be rejected because the analogy does not really exist; while the reason of atomism is to be rejected because the atomistic doctrine failed. Let us first turn our attention to the failure of atomism.
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3.3 ‘The Dismantling of Atomism’ That the atomistic picture is far too naive to underlie an adequate account of the language-world relationship soon became clear. It is interesting that Wittgenstein, whose Tractatus offered probably the most philosophically fruitful elaboration of the atomistic picture, himself early recognized its weak points; and what took place in his thinking since 1929 is aptly called, by Kenny (1972), ‘the dismantling of atomism’. It is instructive to consider the reasons for this dismantling. The basic point was that Wittgenstein realized that the assumption of a basis of atomic statements underlying our language (and hence of a basis of atomic facts laying the foundations of our world) has no true support in reality. The constitutive feature of such a basis is the independence of its elements, i.e. the fact that each of them can be true or false independently of the truth or falsity of any other. However, when Wittgenstein examined more closely the most basic sentences of natural language, especially the ascriptions of colours which appeared to be exemplary cases of elementary statements, he realized that they are far from conforming to this picture. Such sentences, although constituting the most ‘primitive’ level of our language, are clearly not independent of each other: x is blue, for example cannot be true if x is red is (assuming, of course, that x is not a kind of object which can be both blue and red; viz the well known Sophists’ argument criticized by Aristotle). Thus we have either to conclude that such statements are still not atomic, that they are compounds of some other, more primitive statements, whose nature is then, however, unavoidably mysterious (they are surely not statements of the overt language), or we have to give up the whole picture of an atomic basis. Some of the passages of the Tractatus indicate that Wittgenstein was at least dimly aware of this predicament already while writing this book and tried to resolve it in the former way, by indicating that his theory addresses some hidden structure underlying natural language rather than the language itself, which in fact immunizes his theory against any findings about real language. What he realized later was that a theory of language of this kind is nothing else than another kind of infallible metaphysics (just because of its immunity from any findings about the real language) which he himself always struggled to eradicate. However, considerations of the ascription of colours and of the nature of atoms of our language were neither the sole, nor the most decisive, reason for Wittgenstein’s later change of mind. The crucial factor appeared to be the realization of the fact that to see meanings as things and their relation to their expressions as correspondence is both unwarranted and futile. Meanings, as Wittgenstein clearly saw in the later period of his life, are best seen as ways of usage, not as things named. Quine, Davidson, Sellars and other American (post)analytic philosophers later came with related critique; although sometimes put in rather different ways. Quine pointed out that atomism breaks down once we appreciate the essentially holistic character of our language, which becomes evident when we consider, e.g., the process of translating an unknown language or the process of verifying scientific hypotheses. (This led him also to his famous claim that we cannot keep positing an insurmountable hedge between truths which are analytic and those which are contingent.) Sellars’ criticism is similar, although originally based predominantly on considerations of the nature of our knowledge and resulting in the rejection of the boundary
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between ‘the given’ and ‘the inferred’, implicit in the atomistic picture as the boundary between the knowledge of atomic facts (which is direct, we simply ‘accept such facts into our heads’) and of facts that are complex (which we then fabricate out of the direct intake). Given all this, it becomes less easy to see language as a collection of expressions each of which mirrors its own particular bit of the world independently of the others10. Moreover, it almost inevitably leads to the view of meaning as a tool employed by an interpreter to comprehend language via classifying its elements11. However, if this is the case, then the Carnapian triadic division of linguistic matters becomes dubious. From this visual angle, it may seem as if pragmatics swallows up everything else. Everything that we learn when we decipher a language (and hence everything that there is to know in order to know the language) is how the speakers of the language use it. If language is no nomenclature, if meaning is only a classificatory tool of an interpreter, then there is no sharp boundary between those aspects of linguistic behaviour which are to be viewed by the prism of meaning and those which are not. We posit meaning where we see it helpful; and we do not posit it where we think we can make do without it.
3.4 Natural and Formal Languages It is clear that there is no difficulty in separating semantics from syntax and pragmatics in a formal language. In fact, it is usual for an exposition of such a language to be given in three sharply separated chapters: syntax proper (delimiting well-formedness, i.e. the class of the expressions of the language), proof theory or ‘logical syntax’ (delimiting provability, i.e. the class of the theorems of the language) and model theory or semantics (delimiting validity, i.e. the class of tautologies or analytic truths of the system). Each chapter constitutes its own selfcontained field of study. The first two may be combined, following Carnap, under the common heading of syntax, while the third is left under that of semantics. Pragmatics then may be considered as not a matter of the system itself, but rather of the way in which the system is handled by those who employ it. It is beyond doubt that the development of languages of modern logic advanced our understanding of natural language. It is also true that these languages can often be, beneficially, conceived of as models of natural language. Nevertheless, we should be careful in concluding that therefore the nature of formal and natural languages is the same: after all, although any model must be in some sense similar to what it models, the very fact that it is capable of serving as a model means that it is also, in some other sense, quite dissimilar to it. If we want a model of an atom, then we cannot simply take another atom, we have to make a metal (or plastic, or 10 11
This has been discussed in detail by Rorty (1980).
This is stressed by Sellars (1974), who speaks directly about „meaning as functional classification“. Elsewhere (see Peregrin, 1995; 1997a) I have pointed out that this means that any theory of meaning has to be ‘structuralistic’.
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whatever) construction, which is in some important aspect (‘structure’) similar to the atom while in some other aspect (scale) utterly dissimilar. It is the neglect of this important point that leads to pronouncements such as Montague’s denial of any important difference between natural and formal languages, and it was the very same neglect that underlaid Carnap’s approach. (To avoid misunderstanding: of course there is a sense in which we have to neglect the differences between the model and that which it models. Such neglect underlies the very employment of the model as a means of reasoning about the modelled. What we want to say is that if our aim is to account for the very relation of modelling and its terms, then we have to reflect their essential asymmetry.) The idea was that we could get syntax by studying the system of signs itself, semantics by studying on which things its individual signs hook, and pragmatics by studying how the signs get employed by their users. However, as Quine and Davidson demonstrated with their thought experiments of radical translation resp. interpretation, to observe which expressions speakers employ and how they employ them is all there is to observe and all there is to understand; there is no observing of how words hook on things over and above this. However, this looks like an evaporation of semantics: there are matters of which expressions constitute language, i.e. which expressions are well-formed, which undoubtedly fall under the heading of syntax; and there are matters of how their users employ them, which appear to fall under that of pragmatics. There appears to be no room for an intervening semantics. Does this mean that natural languages have in fact no semantics (but only pragmatics) and that formal languages, in force of having one, are inadequate to them? Of course not: it only means that the Carnapian division may not be applicable to natural language so straightforwardly as many seem to think; that it may not be a matter of the phenomenon studied (natural language), but rather of our way, or our means, of studying it (the formal language model). We should not be blind to the fact that natural language itself does not come in the three chapters in which formal languages do; we make it look so only when we devise a suitable formal language as a model, as a prism through which we look at it. (And note that doing this is not cheating, it is simply imposing an organizatory scheme which promotes our understanding.) In applying the model, we do our best to make it fit, to make all junctures of the latter be precisely superimposed on the corresponding junctures of the former. However, there are no natural junctures to be superimposed by the formal boundaries of Carnapian semantics, and so we have a certain latitude over our placing of it (we can move it here and there to the extent to which it does not pull other junctures which do have natural counterparts to superimpose). Hence, it is crucial not to confuse the (natural) language which is the object of our investigation with the (formal) language which is a means of the investigation. As Putnam (1962) puts it, discussing the problem of the analytic/synthetic distinction, „we have a model of natural language according to which a natural language has ‘rules’, and a model with some explanatory and predictive value, but what we badly need to know are the respects in which the model is exact, and the respects in which the model is misleading. ... The dispositions of speakers of a natural language are not rules of a formal language, the latter are only used to represent them in a certain technique of representation; and the difficulty lies in being sure that other elements of the model, e.g. the sharp analytic-synthetic distinction, correspond to anything
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at all in reality.“ The same holds for the boundaries of Carnapian semantics: they are boundaries essential (and straightforward) for formal languages, but rather chimerical for natural languages. The problem is that we are so accustomed to viewing natural language through the prism of its formal-language model that we often mistake the latter for the former.
3.5 Meaning as ‘interpretational construct’ In order to fully understand the Davidsonian paradigm, we must break radically from viewing language in the Carnapian way. We must forget about ‘standing for things’; we must stop construing expressions like labels and instead see them as tools. To learn the meaning of a word is no longer to discover the thing (or the chunk of reality, or the thought) which the expression labels, but rather to learn the ways in which the expression is used within its ‘language game’. If it is not a word of our first language, then this usually involves finding a word or phrase of ours which is useful in the same way as the foreign word in question; to find the component of our toolbox with which we achieve (more or less) the same as the foreigners achieve with that element of their toolbox. Thus, from this perspective, translating a foreign language does not consist in finding the things which are the common nominata of the foreign and our words, it rather consists in something like comparing toolboxes. However, if we took this idea at face value, would it not lead to absurd consequences? If we took the meaning of an expression to consist in the totality of cases in which the expression is really put to use, would it not mean that meaning is something which we can never really learn (for we surely cannot witness all the cases), and, moreover, something too idiosyncratic to an individual speaker? Would it not mean that we could almost never find an expression of the foreign language which would mean the same as an expression of ours? (It would be enough if my foreign informant once mistook a cat for a rabbit, while I did see it was a cat, and his gavagai would be bound to mean something else than my rabbit - no matter how many other times we would use the two words concurrently.). The answer is that, of course, we cannot see meaning as consisting in all the cases of utterance. It is clear that we have to allow for marginal discrepancies in usage and not to construe meaning as the entire way an expression is actually put to use, but by something as ‘the most substantial part of this way’12. However, having given up the notion of language as a nomenclature, we cannot say that this ‘most substantial part’ is simply that part which is the matter of the expression’s ‘standing for’. Is there another feasible way of delimiting it? 12
Besides this, we must realize that the fact that our conjectures about the ways the foreigners use their words are bound to be based on restricted evidence (for we can realistically witness only a severely limited subset of the cases of their utterances) is not an argument against knowing them. Drawing general conjectures from restricted evidence is the general way we put together our theories of our world, and we know that they do work despite this (although we must grant Hume and Popper that we can never be sure that our general theories are really true). So in this respect, our learning of foreign language is no more problematic than finding out about anything else within the world.
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We may think of identifying the boundary of the ‘most substantial’, ‘meaningdetermining’ part of an expression’s functioning with the boundary between the analytic part of language and the synthetic part. Some statements, it is usually assumed, are analytic and meaning-constituting (hence they are, in effect, explicit or implicit definitions), others are synthetic and fact-expressing (they are reports of how things are). However, this way is in general precluded to us too - for as Quine (1951) has famously pointed out, even this boundary goes by the board with the notion of language as a nomenclature13. This brings us to the central point of Quine and Davidson: linguistic holism. Language is a co-operative enterprise, and its working cannot be construed as a resultant of self-standing workings of mutually independent items. Let us imagine a clock: it shows time, and it does so by consisting of a number of co-operating parts. However, it would be futile to see its parts as carrying out each its independent subtask of time-showing, and to claim that the working of the whole clock is the resultant of such individual subtasks. Of course there are ways to specify the role of a part of the clock, but we can usually do so only relatively to the roles of other parts; and we can do it in more than one way. Is the role of the clockface simply to show the layout of the hours of the day; or to underlie the hands, or perhaps to give meaning to the positions of the hands? And is the role of the predicate to apply to its subject; or rather to let the subject be applied to itself; or perhaps something else? This indicates that meaning, viewed from this perspective, becomes something as an ‘interpretational construct’14. Assigning meaning is specifying a role, or possible roles, within a co-operative enterprise; it is to state how an expression could be useful for the purposes for which we use language. Thus, assigning a meaning to a word is not like discovering a thing effecting the word, but rather like the determination of a value which the word has from the viewpoint of a certain enterprise. When a speaker S utters a statement s, then our way of perceiving it is that S has a belief b and that this belief is the meaning of s. However the belief is not something that could be found by opening S’s head, and similarly the meaning is not something that could be found by inspecting s’s linkage to a piece of the world; both are something we stipulate to ‘make sense’ of S’s utterances. We start from the observed facts about the speakers’ linguistic behaviour and ‘decompose’ the body of these facts into a theory of what the speakers believe and a theory of what their words mean. Thus we are to understand the pronouncement ‘the meaning of s is 13
In fact, once we assume the Davidsonian stance, the same conclusion is bound to be forthcoming. Observing the natives, how could we tell an analytic pronouncement from a synthetic one? How could we tell difference in meaning from difference in claims? Suppose that the foreigner whose language we think we have translated into our language satisfactorily utters a sentence which we translate as ‘Pigs have wings’. How could we decide whether he really believes that pigs have wings, or whether we have only misinterpreted some of the words? Of course by asking him questions about pigs and wings (and indeed about having) to find out to which extent what he says differs from what we think - and if the differences are deep, we would be inclined to vote for an error in our translation manual, while if they are not, we could vote for differences in opinions. However, there is no criterion to draw a sharp boundary.
14
Cf. Abel (1994).
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such-and-such’ more or less as only a metaphoric expression of ‘the way s gets employed within a certain language game is substantially such-and-such’ or ‘the position of s within a certain language is substantially such-and-such’; just as we understand the pronouncement ‘the price of x is such-and-such’ as being a shorthand for ‘the position of x within the selling-and-buying relations among people is such-and-such’.
3.6 ‘Semantics must answer to pragmatics’ While Davidson’s own writings do not pay much attention to the concepts of pragmatics and semantics, an explicit reflection of these concepts and their relationship is fostered by the writings of Wilfrid Sellars and carried out in detail by Sellars’ disciple Robert Brandom. Brandom (1994, p. 83) diagnoses the situation in the following way: „Semantics must answer to pragmatics. The theoretical point of attributing semantic content to intentional states, attitudes, and performances is to determine the pragmatic significance of their occurrence in various contexts.“ According to him, the talk about the meaning of a statement is a disguised (sometimes quite misguidingly disguised) talk about what the statement is good for, and the talk about the meaning of a part of a statement spells out the contribution which this expression brings to the usefulness of those statements in which it may occur. „Semantic contents corresponding to subsentential expressions,“ as Brandom (ibid.) puts it, „are significant only insofar as they contribute to the determination of the sorts of semantic contents expressed by full sentences.“ Following Sellars, Brandom moreover stresses the essentially normative nature of pragmatics: pragmatics, as he understands it, is essentially a matter of norms: it is a matter of rules which institute what is right and what is wrong within the realm of usage of language. According to him, the meaning of a statement, i.e. what the statement is good for, consists first and foremost in the commitments and entitlements which the assertion of the statement brings about, and these commitments and entitlements are in turn reflected by the inferences in which the statement participates. Thus, the meaning of a statement is, according to Brandom, its inferential role15. In this way any ‘semantic interpretation’ is merely a spelling out of the ‘pragmatic significance’: „It is possible,“ writes Brandom (ibid., p. 84), „to associate all sorts of abstract objects with strings of symbols in formalized languages, from sets of models to Gödel numbers. Such an association amounts to specifically semantic interpretation just insofar as it serves to determine how those strings are correctly used. For example Tarski’s mapping of well-formed formulas of the first-order predicate calculus onto topological domains qualifies as a semantic interpretation of them only because he can derive from it a notion of valid inference, a way of telling what follows from what - that is, a notion of their correct use.“
15
Cf. Also Peregrin (to appear).
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Thus, I think that Brandom’s book, appropriately called Making it explicit, offers the clearest exposition of the pragmatization of semantics implicit to what we have called the Davidsonian approach: it makes explicit that once we give up the notion of language as a nomenclature, there is no way to extricate semantics from pragmatics.
4. New Boundaries? 4.1 Semantics and Syntax The considerations of the previous sections have posed serious challenges to the Carnapian paradigm and to those boundaries between syntax, semantics and pragmatics on which this paradigm is based. However, their upshot should not be that there are no such boundaries, but rather that the Carnapian way of drawing them is inadequate. Let us first look at what separates semantics from syntax; we shall denote the Carnapian notions as syntaxC and semanticsC. Thus, syntaxC is supposed to be about relations between expressions, whereas semanticsC is supposed to be about those between expressions and extralinguistic objects. It is clear that this definition makes real sense if there is a matter of fact underlying the distinction between properties of expressions and relations linking expressions to extralinguistic objects, i.e. if those relations between words and things which be the subject matter of semantics are in some sense ‘real’. Given the atomistic character of the language-world relationship and assuming a ‘real’, ‘matter-of-factual’ link between a thing and that expression which denotes this thing, the boundary between syntax and semantics becomes an empirical matter. It is like a boundary between relations among trees in a forest (e.g. ‘a tree T1 is bigger than a tree T2’) on the one hand and relations between trees and men taking care of the forest (e.g. ‘a tree T has been planted by a man M’) on the other. To ascertain which properties of a tree are of the former kind (i.e. which are ‘natural’) and which are of the latter kind (i.e. which are caused by an extraneous agent) is the matter of an empirical inquiry: it is in principle possible (though, as the case may be, in practice difficult) to find out whether, say, a scratch in the trunk of a tree is ‘natural’ or whether it was caused by an action on the part of a forester. And it would seem likewise possible to ascertain which aspects of an expression are ‘natural’ and which were caused by an external agent - its meaning. However, if we relinquish the notion of language as a nomenclature and embrace the ensuing holism, this boundary - just like the boundary between the analytic and the synthetic ceases to be directly grounded in the nature of things and becomes - in this sense - rather visionary. Once we deny the possibility of empirically discovering the factual link between an individual expression and its meaning, then we must accept that there is no real boundary at all. Once meaning becomes an ‘interpretational construct’, there ceases to be an absolute boundary delimiting the cases of its employment. Whenever we have an expression E having a property P (a syntacticC matter), we can treat P as an object associated with E (i.e. as a semanticC matter). On the other hand, whenever there is an object O associated with an expression E (a semanticC
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matter), we can speak about E’s having the property ‘being-associated-with-O’ (i.e. view it as a syntacticC matter). Let us take an example. What does it mean for a statement to be true? To be true is surely a property; but we can just as well articulate it, as Frege did, as the relation between the statement and a posited extralinguistic object (‘the truth’). If we do nothing more than reify it in this way, then we surely do not do anything substantial; however, we stop speaking about a property of expressions (i.e. about a unary relation and hence about a syntacticC matter) and start instead to speak about a relation of the expression to an extralinguistic object (a semanticC matter). Another example: Let us take the statement A man walks and he whistles. We say that its part a man and he refer to the same object - which amounts to speaking about a relation between words and a thing. However, the situation can be equally well accounted for by saying that the statement in question is synonymous with the statement A man walks and the (same) man whistles - and we are talking about a relation between two statements to the same effect16. This leads to the conclusion that the nature of the syntax/semantic boundary is quite similar to the analytic/contingent boundary - that both rise and fall with the atomistic view of language. This is, indeed, true - however only if we keep with the Carnapian definition drawing the boundary as that between relations linking words to words and relations linking words to things. It is not true if we construe the syntax/semantics boundary along different lines. The basic intuition underlying our employment of the pre-theoretical concepts of syntax and semantics is that there is a distinction between questions concerning words per se and questions concerning words-as-meaning-what-they-mean. However, this intuitive distinction is misconstrued by the Carnapian definition: ‘to be about’ is not a naturalistic matter of ‘hard facts’, it is rather the matter of interpretation and hence cannot mark a distinction which would be absolute. The right way to construe the intuitive syntax/semantic distinction is another, and if we adopt the Davidsonian stance, it is rather straightforward. It takes the distinction as marking the boundary between what we use to communicate and how we use it. Thus, we suggest to consider syntax as the theory of which expressions people use to communicate (i.e. a theory of well-formedness), while semantics17 as the theory of how they use them. The boundary between syntax and semantics conceived in this way is surely an existing and observable one: to delimit the range of expressions encountered by the natives is one thing, and to describe when and how they employ them is another (though the former may be thought of as a prerequisite for the latter)18. However, it is a boundary substantially different from the 16
The situation is reminiscent of the old philosophical question about the reality of univerasalia: it is clear that if the question is simply whether to be red is to have a property, or rather be connected to an object (‘redness’), then it is a pseudoquestion. 17
Here we, of course, use the term semantics broadly, in the sense of semantics-cum-pragmatics. How to separate these two ingredients will be the subject of the next section.
18
The former task amounts to formulating some recursive rules characterizing the class of well-formed expressions, whereas the latter amounts to articulating relevant likenesses and differences between expressions (between the foreign expressions as well as between them and our expressions). Quine (1953, p. 49) puts it in the following way: „What had been the problem of meaning boils down now to a pair of problems in which meaning is best not mentioned; one is the problem of making sense of
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Carnapian one: what Carnap called logical syntax now falls on the side of semantics, not on the side of syntax - for inference is clearly a matter of how statements are used. It is nevertheless precisely this which it takes to gain a solid and real boundary19.
4.2 Semantics and Pragmatics In this way we have managed, after all, to draw a clear and distinct dividing line between syntax and semantics (or we should rather say between syntax and the rest, i.e. semantics-cumpragmatics); albeit differently than Carnap. However, what about the opposite boundary of semantics, that which delimits it from pragmatics? We have seen that the considerations of the ‘external challengers’ indicate that we could or should see meanings as ‘ways of usage’, ‘inferential roles’ or some other usage-based creatures; and that those of the ‘internal challengers’ suggest that we can no longer find an adequate theory of meaning without employing the concept of context (or some equivalent). Is there still room for distinguishing between meanings and ways of putting to use? I think there is; but again, to find it we should better forget about the Carnapian paradigm, for seeking a boundary between ‘hooking on things’ and ‘being employed by speakers’ can give us no cue. What I suggest is that we should attend to the concept of invariance: Meaning of an expression is, roughly speaking, that which is invariant among various cases of employment of this expression. (Interestingly, this is what was urged by Roman Jakobson as the core of linguistic structuralism: „If topology is defined as the study of those qualitative properties which are invariant under isomorphic transformations, this is exactly what the notion of significant sentence, and the other is the problem of making sense of the problem of synonymy.“ 19
To indicate the consequences of the shift, let us consider the concept of provability. Carnap would count it to (logical) syntax, for to prove something is a matter of applying some ‘reformation’ rules transforming statements into other statements; and so would undoubtedly also many of the contemporary logicians. But it is important to see that this usage institutes an ambiguity of the word ‘syntactic’: to be provable is not syntactic in the sense in which, say, to begin with the letter ‘a’ is - it is not a property of statements per se, i.e. statements as sequences of marks on a paper, but rather of statements as meaning something, as used within a certain language. To be provable primarily means to be demonstrably true, or to be the result of recursive application of some obviously truth-preserving inferential rules to some obviously true statements (only secondarily is it sometimes used also within the context of formal languages in the sense of to be the result of recursive application of whatever counts as inferential rules to whatever counts as axioms). It clearly makes no sense to say whether a sentence is provable unless it means something (whereas we can say whether it begins with ‘a’ even so). (Of course we can be given some instructions with the help of which we can identify some provable statements even without understanding them, but this is equally true for any property of expressions, however purely semantic it may be.) The boundary proposed here does away with this ambiguity: proving is clearly a matter of what statements mean, i.e. how they are used, so provability now falls on the side of semantics.
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we did in structural linguistics“20). This is obviously a very abstract and perhaps ambiguous specification, for there can be various kinds of invariances; but what has been concluded above indicates that we should not see the boundary between semantics and pragmatics as altogether unambiguous. (Thus the status of this boundary is essentially different from that of the boundary between semantics and syntax - we found, in the end, that syntax can be delimited quite unambiguously.) Let us hint at what kind of invariances we may have in mind: On one level, semantics may be held to be that which is invariant to individual utterances of the same expression, pragmatics as that which is not. Thus, when I say I am hungry and you say I am hungry too, the fact that the first I refers to me, whereas the second to you, is a pragmatic matter. What is invariant is that it always refers to the speaker. On another level, we can see semantics as that which is invariant to the idiosyncratic ways individual speakers employ words, and pragmatics as that which is not. The fact that it is true to say I am hungry when one is hungry is a matter of semantics; while the fact that I never utter the sentence when I am in somebody else’s place, whereas someone else does unscrupulously announce her hunger even in such situations is a matter of pragmatics. Similarly, the fact that I always imagine a roasted boar when I say that I am hungry, whereas my companion imagines the emptiness of her belly, and somebody else imagines nothing at all is the matter of pragmatics: what is invariant is the very utterance and the norms governing it. Of course all such boundaries are fuzzy: but this is of a piece with the conclusion that meaning should not be seen as a thing, but rather as an interpretational construct. In Section 3.5 we concluded that assuming the Davidsonian stance implies taking the meaning of an expression as the ‘most substantial part’ of the way the expression is being put to use; and we stressed the important fact that the boundary is thus not ‘in the nature of things’, but is, in an important sense, in the hands of the interpreter. Now the idea of invariance indicates that the interpreter at least has some cues what to count to the substantial part and what not - the interpretation is, in effect, nothing else than a case of the old pursuit of the e pluribus unum which seems to be the basic pattern of our grasping the world.
4.3 Objections The proposals made in the previous sections will, no doubt, be felt by many people as counterintuitive. Here I shall try to indicate that this feeling may be the result of, to paraphrase Wittgenstein’s (1953, §115) bonmot, ‘being held captive by the Carnapian picture’, rather than the insufficiency of the proposal. Let us consider some of the possible objections to it. Perhaps the most obvious objection is that the new picture fails at doing justice to the most basic ‘semantic intuition’, namely that words are a matter of labelling things. Is, say, the
20
Jakobson (1971, pp. 223-4).
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word cat not a label used to label cats? Does it not fly in the face of reason to deny that, say, the name the cat which chases Jerry is the label of Tom? I think that this ‘semantic intuition’ is in fact a malign (i.e. misguiding) mixture of various heterogeneous, more or less benign (i.e. just), intuitions. First, it is obviously true that some expressions are intimately connected with some kind of objects (viz cat with cats) in the sense that the way we use them would be hardly specifiable without mentioning (or indeed pointing at) objects of this kind. However this hardly establishes a name-like relation. For a general term like cat it holds that if we want to treat it as a name of an entity, then this entity would, of course, have to be not a cat, but rather something as a ‘cathood’. However, as millennia of disputations of philosophers have made plausible, ‘cathood’ seems to be nothing else than an entity brought to life precisely only by our desire to have something for cat to name - and hence can be hardly taken as something which prompted us for being labelled by the word. If we, on the other hand, consider a singular term like the cat which chases Jerry, then it is true that the term can be seen as naming an object, but it is notoriously well-known that the object has little to do with the meaning of the phrase: you can surely know the meaning of the phrase without knowing Tom (or even without knowing whether there really is any such cat). This fact has been clearly analyzed, for the first time in the modern context, in Frege’s (1982) famous Über Sinn und Bedeutung. Thus it seems that whereas using language may involve various kinds of relations which may be seen in terms of labelling, it does not follow that the word-meaning link is one of them. Another objection might be that if we give up the old good notion of semantics as semanticsC, we will not be able to explain truth. If we do not see words as essentially denoting things, then we cannot say that a statement is true if things are the way the statement declares them to be. Language, the argument may continue, is the matter of a words-things relation, so an explanatory account of language must be semanticC. This objection is, of course, question begging - it simply presupposes the view of language as a nomenclature which we have eschewed as inadequate. Our Davidsonian stance implies that language is primarily not a collection of word-thing relations, but rather a collection of human actions and dispositions to such actions; and also that truth neither needs, nor admits, an explanation in terms of correspondence. (How truth is to be explained in such a case is another question; but proposals are numerous. See, e.g., Peregrin, in print.) Then there are objections which are likely to come from logicians. One such may be that if we give up semanticsC, then we shall not be able to give truth conditions for quantifiers - for the substitutional view of quantification is usually considered as unsatisfactory. We cannot give the truth conditions for ∀x.P(x), this argument says, otherwise than via amounting to those things which P can be considered to apply to, namely as P applies to every individual. If ∀x.P(x) is understood as an expression of a formal calculus (e.g. of the first-order predicate calculus), then there is no problem: formal languages do have their explicit model theories and nothing that has been said here can prevent them from keeping them. On the other hand, if ∀x.P(x) is taken to be not an expression of a formal calculus, but rather a mere regimentation of a natural language phrase, then there is - anyway - no nontrivial way of stating its truth conditions over and above repeating or rephrasing the sentence itself - for an utterance like
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∀x.P(x) iff P applies to every individual de-schematizes to the trivial Everything is P iff P applies to everything. (For details see Peregrin, 1997b). However, the objection which most logicians would be likely to take as the most crucial is that, as Gödel proved, we need semanticsC in order to make any sense whatsoever of some theories. Some theories, it is claimed, such as arithmetic, are not axiomatizable, we can specify them only model-theoretically; and this is taken to mean that the very ‘grasp’ or ‘apprehension’ of these theories (on our part) must be based on semanticsC. A thorough discussion of this problem would exceed the bounds of the present paper; so let us only note that the key is again in carefully distinguishing between formal and natural languages. We have very little to say about formal theories in formal languages; that the class of theorems of formal Peano arithmetic is not complete is a mathematical fact (the fact that a certain formally constructed set has a certain property), and nothing which has been said here is meant to interfere with this kind of truth of mathematics. On the other hand, if Gödel’s results are interpreted as stating that there are truths which are accessible only semanticallyC, then this is hardly true: there does exist a proof of Gödel’s undecidable sentence (its truth is easily demonstrable to any adept of mathematical logic by means of a gapless chain of reasoning), albeit it is a proof which cannot be, curiously enough, squeezed into a single formal system21.
5. Conclusion The ‘external challengers’ have disputed the view that expressions are basically names which stand for their nominata; they have done so by looking at language ‘in action’. They have concluded that what makes an expression ‘meaningful’ is not a thing for which it would stand, but rather the fact that it can serve as a tool of communication - that there is a way we can use it for certain communicative purposes22. Thus, meaning is better not seen as an object, but rather as something as a role or a value, a reification of the way in which its expression is useful. The ‘internal challengers’ keep explicating meaning as a (set-theoretical) object, but they have made it plausible that if we want to account for the riches of natural languages, then this object is bound to become something which is no longer reasonably seen as a real ‘thing’, it is again more a reification of the way the corresponding expression ‘works’, i.e. in which it alters the context into which it is uttered. In this way, the two challenges seem to be almost complementary expressions of what I called the pragmatization of semantics. 21
The reason is that the proof requires switching from reasoning on the ‘object level’ to reasoning on the ‘metalevel’. See also Dummett (1963) who duly points out that the nature of Gödel’s discovery is rather obscured by the talk about models.
22
The fact that we have concentrated on the Davidsonian approach, which we consider as the most penetrating, should not conceal the fact that the repudiation of the ‘language as nomenclature’ notion is a much broader matter. It is for example surprising to see how close, in this respect, Chomsky (1993) appears to be to the Davidsonian view - despite all the grave differences between his and Quine’s or Davidson’s view of the nature of language.
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The Carnapian trichotomy which has underpinned, explicitly or implicitly, our grasping of language for much of this century, should be reassessed: it is a product of a particular, now rather outworn, philosophical doctrine, and of the unscrupulous assimilating of natural language to formal ones. This paradigm played a positive and a stimulating role for some time, but now it seems to be more misguiding than fruitful. The new paradigm which grows out of the writings of various recent philosophers, linguists and logicians is the paradigm of language as not a nomenclature, but rather a toolbox. Whereas Carnapian theories saw a theory of language as consisting of syntax (the theory of relations between expressions and expressions; further divisible into syntax proper and logical syntax, i.e. proof theory), semantics (the theory of relations between expressions and things) and pragmatics (the theory of relations between expressions and speakers), this new, Davidsonian theory of language, the usefulness of which has been urged here, envisages a theory of language partitioned instead into syntax (proper) amounting to which expressions come into the language, semantics, amounting to the ‘principal’, ‘core’ or ‘invariant’ part of the way the expressions are employed, and pragmatics amounting to the remaining, ‘peripheral’ aspects of the way they are employed.
References Abel, G. (1994): ‘Indeterminacy and Interpretation’, Synthèse 37, 403-19. Barwise, J. and J. Perry (1983): Situations and Attitudes, MIT Press, Cambridge (Mass.). Bosch, P. and R. van der Sandt (1994): Proceedings of ‘Focus and Natural Language Processing’, IBM Deutschland, Heidelberg. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Carnap, R. (1928): Der Logische Aufbau der Welt, Berlin. Carnap, R. (1934): Logische Syntax der Sprache, Vienna. Carnap, R.(1939): Foundation of Logic and Mathematics (International Encyclopedia of Unified Sciences 1), Chicago. Carnap, R. (1942): Introduction to Semantics, Cambridge (Mass.). Carnap, R. (1956): Meaning and Necessity, enlarged edition, University of Chicago Press, Chicago. Cresswell, M.J. (1985): Structured meanings: The Semantics of Propositional Attitudes, MIT Press, Cambridge (Mass.) Chierchia, G. (1992): ‘Anaphora and Dynamic Binding’, Linguistics and Philosophy 15, 111183. Chomsky, N. (1993): Language and Thought, Moyer Bell, Wakefield. Davidson, D. (1984): Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Dummett, M. (1963): ‘The Philosophical Significance of Gödel’s Theorem’, Ratio V, 140-155 (reprinted in Dummett: Truth and other Enigmas, Duckworth, London, 1978).
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Frege, G. (1892a): ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und philosophische Kritik 100, 25-50; reprinted in Frege: Funktion, Begriff, Bedeutung, Fandenhoeck & Ruprecht, Göttingen. Frege, G. (1892b): ‘Über Begriff und Gegenstand’, Vierteljahrschrift für wissentschaftliche Philosophie 16, 192-205; reprinted in and quoted from Frege: Funktion, Begriff, Bedeutung, Fandenhoeck & Ruprecht, Göttingen. Groenendijk, J. and M. Stokhof (1991): ‘Dynamic Predicate Logic’, Linguistics and Philosophy 14, 39-101. Hajičová, E. (1994): ‘Topic, Focus and Negation’, in Bosch and van der Sandt (1994). Hajičová, E., B. Partee and P. Sgall (in print): Topic-focus articulation, tripartite structures, and semantic content, Kluwer, Dordrecht. Heim, I. (1982): The Semantics of Definite and Indefinite Noun Phrases, dissertation, University of Massachusetts, Amherst. Hintikka J. and J. Kulas (1985): Anaphora and Definite Descriptions, Reidel, Dordrecht. Jakobson, R. (1971): Selected Writings II, Mouton, The Hague. Kamp, H. (1981): ‘A Theory of Truth and Semantic Representation ‘, Formal Methods in the Study of Language (ed. J. Groenendijk et al.), Mathematical Centre, Amsterdam. Kenny, A. (1972): Wittgenstein, Penguin, London. Montague, R. (1974): Formal Philosophy: Selected Papers of R.Montague (ed. R. Thomason), Yale University Press, New Haven. Muskens, R., J. van Benthem and A. Visser (1997): ‘Dynamics’, in van Benthem and ter Meulen (1997), 587-648. Partee, B. (1994): ‘Focus, Quantification, and Semantic-Pragmatic Issues’, in Bosch and van der Sandt (1994). Partee, B. and H. Hendriks (1997): ‘Montague Grammar’, in van Benthem and ter Meulen (1997), 5-91. Peregrin, J. (1994): ‘Topic-focus articulation as generalized quantification’, in Bosch and van der Sandt (1994), 379-387. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (1996): ‘Topic and Focus in a formal Framework’, Discourse and Meaning (ed. B. Partee and P. Sgall), Benjamins, Amsterdam, 235-254. Peregrin, J. (1997a): ‘Structure and Meaning’, Semiotica 113, 71-88. Peregrin, J. (1997b): ‘Language and its Models’, Nordic Journal of Philosophical Logic 2, 123. Peregrin, J. (to appear): ‘Reference and Inference: the Case of Anaphora’, Reference and Anaphorical Relations (ed. K. von Heusinger and U. Egli), Kluwer, Dordrecht. Peregrin, J., ed. (in print): Truth and Its Nature (if Any), Kluwer, Dordrecht. Peregrin, J. (in prep.): ‘The Logic of Anaphora’, in preparation. Putnam, H. (1962): ‘The Analytic and the Synthetic’, Minnesota Studies in the Philosophy of Science III (ed. by H.Feigl and G.Maxwell), University of Minnesota Press, Maples. Quine, W.V.O. (1952): ‘Two Dogmas of Empiricism’, Philosophical Review 60, 20-43 (reprinted in Quine: From a Logical Point of View, 1953). Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.).
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Quine, W.V.O. (1953): From the Logical Point of View, Harper and Row, New York. Rorty, R. (1980): Philosophy and the Mirror of Nature, Princeton University Press, Princeton. Russell, B. (1905): ‘On denoting’, Mind 14, 479-493. Russell, B. (1914): Our Knowledge of the External World, Allen & Unwin, London. Russell, B. (1918/19): ‘The Philosophy of Logical Atomism’, Monist 28, 495-527; Monist 29, 32-53, 190-222, 345-380. Russell, B. (1924): ‘Logical Atomism’, Contemporary British Philosophy: Personal Statements, First Series, Allen & Unwin, London, 356-383. Sellars, W. (1963): Science, Perception and Reality, Routledge, New York. Sellars, W. (1974): ‘Meaning as Functional Classification’, Synthèse 27, 417-437. Seuren, P.M. (1985): Discourse Semantics, Blackwell, Oxford. Sgall, P. (1987): ‘Prague Functionalism and Topic vs. Focus’, Functionalism in Linguistics (ed. R. Dirven and V. Fried), Benjamins, Amsterdam. Sgall, P., E. Hajičová and J. Panevová (1986): The Meaning of the Sentence in its Semantic and Pragmatic Aspects, Academia, Prague. Strawson, P.F. (1950): ‘On Referring’, Mind 59, 320-344. Tarski, A. (1936): ‘Grundlegung der wissenschaftlichen Semantik’, Actes du Congrès International de Philosophie Scientifique 3, 1-8; translated as ‘The Establishment of Scientific Semantics’, in Tarski: Logic, Semantics, Metamathematics, Oxford University Press, Oxford, 1956. Tichý, P. (1988): The Foundations of Frege’s Logic, de Gruyter, Berlin. van Benthem, J. (1997): Exploring Logical Dynamics, CSLI, Stanford. van Benthem, J. and A. ter Meulen (1997): Handbook of Logic and Language, Elsevier / MIT Press, Oxford / Cambridge (Mass.). Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford.
JAROSLAV PEREGRIN
THE ‘NATURAL’ AND THE ‘FORMAL’ Received in revised version on 30 March 1999 ABSTRACT. The paper presents an argument against a “metaphysical” conception of logic according to which logic spells out a specific kind of mathematical structure that is somehow inherently related to our factual reasoning. In contrast, it is argued that it is always an empirical question as to whether a given mathematical structure really does captures a principle of reasoning. (More generally, it is argued that it is not meaningful to replace an empirical investigation of a thing by an investigation of its a priori analyzable structure without paying due attention to the question of whether it really is the structure of the thing in question.) It is proposed to elucidate the situation by distinguishing two essentially different realms with which our reason must deal: “the realm of the natural”, constituted by the things of our empirical world, and “the realm of the formal”, constituted by the structures that we use as “prisms” to view, to make sense of, and to reconstruct the world. It is suggested that this vantage point may throw light on many foundational problems of logic. KEY WORDS: philosophy of logic, logical form, logical truth, structuralism, mathematical models
1. W HAT ARE S TATEMENTS ?
In the course of contemplating the nature of necessary truth, Hilary Putnam (for one) considers the feasibility of giving up a claims like (∗)
For all statements p, p(p&¬p)q is true.1
I think that a good way to illuminate many crucial problems of this centuryþs logic and analytic philosophy is to inquire into the nature of objects quantified over in claims of this kind; i.e. to examine the nature of statements (or thoughts, when (∗) is disguised as something like ‘it is not possible to think a thought together with its negation’) that are dealt with by logicians and analytic philosophers. Two basic responses are clear: (1) statements can be taken to be some factual objects that exist and can be identified independently of logic, typically sentences of some real language; or (2) the realm of statements can be taken to be constituted by logic. For simplicity’s sake, let us take a claim simpler than (∗): (∗∗)
For all true statements p, p¬pq is false.
Journal of Philosophical Logic 29: 75–101, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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Let us concentrate on an individual instance of (∗∗): let us suppose we know which statement is represented by the sign ‘p’, and let us consider the sign p¬pq. Which statement does this sign represent? Various answers are possible: If we subscribe to (1), then we must have a procedure to determine which statement p¬pq represents (given p is determined) without any recourse to logic. If we see statements as (uniquely determined by) sentences of a factual language, then this procedure could be identified with the application of a specific grammatical construction (say a construction which typically results in a modification of the main verb of p by the particle ‘not’). In this case, (∗∗) is clearly a meaningful claim with nontrivial empirical content. On the other hand, if we subscribe to (2), then we have to understand the statement represented by p¬pq as determined via the stipulation that ¬p is the statement which is true just in case p is false. In this case, statements are understood as objects constituted by logical laws; and in this case we shall speak, as usual, about propositions. Understood in this way, (∗∗) is evidently trivial – it is no more than the direct consequence of our way of understanding the sign p¬pq. In this sense, (∗∗) obviously has no relevance for our factual reasoning. However, neither of these answers is satisfactory: neither the claim that the truths of logic are empirical, nor that they have no relevance for our factual reasoning is acceptable. This seems to force a standpoint somewhere in between (1) and (2), resulting in the claim that statements do exist independently of logic and of logical laws (and hence that claims like (∗∗) are nontrivial, that they are more than consequences of our definitions), but that their compliance to the laws of logic is nevertheless not an empirical matter – that these laws express some necessary and eternal relationships between them. This yields a metaphysical (or ultra-physical, as Wittgenstein, 1956, §I.8, would put it) conception of logic: according to this, logic reports facts about a realm of non-empirical things. My point in this paper is that accepting this view can easily engender the trivialization of the most important problems with which logic and analytic philosophy are devised to cope, and leads to an intrinsically corrupted view of the way our language and our knowledge functions. This does not mean that we are not to avoid both the conclusion that the truths of logic are empirical, and that they are mere consequences of our definitions; the point, however, is that the metaphysical conception of logic is of no real help. Thus, what I am going to say should not be taken to imply that we have no alternative other than to grasp logic as a chapter either of descriptive linguistics or of algebra;
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I do think logic indeed is situated somewhere ‘in between’, but not in the way suggested by the metaphysical conception. 2. D IGRESSION I: S UNSPOTS AND HEADACHES
Let me illustrate this point by a little story. Let us imagine a person, call him X, who complains that he often has headaches and claims that his headaches are caused by the occurrence of sunspots. However, what he claims is not simply that he has headaches whenever there are sunspots; he claims that what the sunspots do to him depends on a further factor, namely on the influence mode of the sun on him. His thesis is that if this influence mode is positive, then his head aches if and only if there are sunspots, while if it is negative, then his head aches if and only if there are no sunspots. In such a case, the contentfulness of X’s claim would clearly depend on how he explains his notion of influence mode. If he specifies it as a matter of, say, his blood pressure – so that the influence would be positive, e.g., if and only if his blood pressure were high – then the claim would clearly be a perfectly meaningful, empirical thesis that could be tested and verified or falsified in a straightforward way. The claims that somebody’s head aches if and only if either there are sunspots and his blood pressure is high, or there are no sunspots and his blood pressure is low is something that may, or may not, be true – depending on how things really are. However, imagine that the only thing X is willing to say about the influence mode is something like “it is that which determines whether sunspots are causing me headaches or rather the other way around”. It is clear that in this case the contentfulness of his claim becomes highly suspect. The point is that now his thesis is defensible come what may, and consequently is empty of any real, empirical content. It could never, for instance, happen that there would be sunspots, that the influence of the sun on X would be positive, and yet that he would nevertheless have no headache – for in such a case he would always say that the influence is not positive and thereby save his thesis. Thus he could always brand the influence mode so that the thesis would keep holding – independently of the real pattern of co-occurrence of sunspots and his headaches. In this situation X’s thesis is merely something like an (implicit) definition of his concept of influence mode (a concept which would be not only probably useless, but possibly even harmful: for its very employment would tend to deceive us into believing that there is a real, empirically significant correlation between the sunspots and the headaches). If X insisted that the thesis is contentful, we would probably suspect him of either
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being hopelessly slow-witted, or trying to cheat – for his insistence would look like a foolish, or a treacherous, attempt to conjure up content where there is none. Now my point is that the metaphysical conception of logic (and, indeed, any stipulation of metaphysical reality of this kind) does, in a sense, an analogous thing, namely tries to produce claims which are simultaneously both infallible and contentful. There is, no doubt, a clear sense in which we, when doing logic and giving philosophical accounts of language, deal with propositions rather than with sentences. When we speak, for instance, about one sentence implying another, we cannot see the sentences as mere syntactic objects. We are obliged to see them as meaning what they normally mean in a certain language (such as English). And when we say: ‘meaning what they normally mean in English’, we can equally well say ‘expressing the propositions they normally express in English’. If we say that Prague is in Europe implies Something is in Europe, we obviously mean the two sentences as English sentences, or, we can say, as sentences expressing the propositions which they normally express in English. This is straightforward and indubitable. The trouble is that if we take the picture of expressing propositions (or thoughts) at face value, propositions may easily come to acquire a role similar to that of the influence mode of the previous anecdote: they may come to make logic and philosophy into an enterprise which is (seemingly) both contentful (in the sense that it can tell us something about our factual reasoning) and infallible. The starting point of the previous story was X’s claim about sunspots causing him headaches. However, his headaches were really not co-occurrent with sunspots; and what X did was to posit a ‘ghostly entity’, the influence mode, which immunized his claim from falsification. Now the starting point of logic is the presumption that logical laws apply to our factual reasoning and hence to our factual language, which is the basic medium of our reasoning. But if we saw logical laws as being directly about natural language, then their validity would be a dubious empirical matter: if we saw ‘¬’ as representing an English grammatical construction, then we could never be sure that (∗∗) is universally valid. And hence we might come up with a ‘remedy’ akin to X’s move, namely to posit ‘ghostly entities’, in this case propositions, that would immunize logical laws from being falsified: whenever we found a sentence apparently violating (∗∗), we would conclude that this merely shows that the sentence does not express the proposition which it seems to express after all, and therefore is not an instance of (∗∗).
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Let us imagine sunspots and headaches as acquiring the values 1 and 0 (in the sense ‘occurring’ resp. ‘non-occurring’). Then what X did could be depicted as replacing the invalid equation sunspots = headaches, by sunspots + influence mode = headaches, whose validity is secured by taking influence mode = headaches − sunspots. 2 Thus, if there are sunspots, but no headaches, the influence mode can always be blamed. The general structure of this trick is this: we would like A to ‘yield’ B (A = B), but the fact is that A does not ‘yield’ it (A 6= B). Hence we say that A does not ‘yield’ B ‘by itself’, but rather ‘via’ C (A + C = B), where C is precisely what is needed to neutralize the disparity of A and B (C = B − A). Doing this, though, is turning the original claim into an infallible triviality. However, we can not only substitute sunspots for A, headaches for B and influence mode for C to gain the story of the brave Mr. X; we can also substitute logical laws for A, their real-language instances for B, and the propositions expressed by these instances for C — and we have the metaphysical conception of logic I warn against. The conclusion, therefore, is that if X’s employment of the influence mode was misleading because it pretended to establish a regularity regardless of whether there really was one, then the introduction of propositions may be similarly misleading – if we take them to establish ‘logical’ regularities independently of whether these really do obtain within our factual language and our factual enterprise of ‘giving and asking for reasons’.3 The point is not that if we pay insufficient attention to our factual language and our factual reasoning, we might reach logical laws which would be invalid (after all, the ‘law’ sunspots + influence mode = headaches is also not invalid), it is rather that such laws may be simply without purpose.
3. T HE TWO REALMS
In his Timaeus, Plato claims: “Now first of all we must, in my judgment, make the following distinction. What is that which is Existent always and has no Becoming? And what is that which is Becoming always and never is Existent? Now the one of these is apprehensible by thought with the aid
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of reasoning, since it is ever uniformly existent; whereas the other is an object of opinion with the aid of unreasoning sensation, since it becomes and perishes and is never really existent.” In this way, Plato became the first of many philosophers to notice that we, humans, somehow come to deal with two essentially different kinds of entities: that besides the ordinary things like dogs, trees, secret service agents or Rolls-Royces, whose properties continually change and which we never know for absolutely certain, we also operate with other kinds of entities, such as categories, geometrical shapes and structures, which are strangely rigid and somehow completely seized by our reason. The soundness of such a distinction itself is hard to doubt; irrespectively of if and how we cash it out philosophically. It is not my purpose here to address the general philosophical questions regarding the nature and status of the two kinds of entities and of their mutual relationship (in particular I do not aim to answer the traditional philosophical question about the extent to which our categories are ‘within the things themselves’ and that to which they are ‘in the eye of the beholder’). Nevertheless, it is my conviction that the nature of such problems as the one mentioned at the beginning of this paper may be helpfully elucidated by realizing that our understanding is often the result of an interplay of entities of such two radically different kinds: that we often understand the ‘empirical world’ with the help of formal ‘prisms’. To bring Plato’s high-flown cogitation down to earth, imagine a more mundane situation: imagine that you move to a city wholly unknown to you, and that a friend of yours who has lived there for a long time draws a simple plan of the city for you. The plan contains marks representing some basic orientation points (the City Hall, the theatre, a famous Chinese restaurant, the railway station etc.) and the main streets connecting them. Despite the fact that such a plan is a drastically simplified and compressed picture of the city, it can obviously help you to acquaint yourself with your new environment. You begin to see the streets and buildings via the prism of the plan and thus you start to see them occupying their places within a general layout. And note that what is crucial is the extent of simplification and condensation of the plan: if it were too simplified, it would cease to be a plan of the city and hence would be of no use; but at the same time it would also be of no use if it were as complicated and as large as the city itself. The point I want to make is that many things we employ within our enterprise of coping with our environment play roles analogous to such a map of an unknown city. Mathematical representations are a typical case: when we say that something is a triangle, or that something can be captured
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by a set of differential equations, what we usually mean is that it can be viewed as a triangle or as something governed by the dependencies spelled out by the equations on a certain level of abstraction – i.e. disregarding some amount of discrepancy. After all, nothing we can find around us is a precise, geometrical triangle, nothing displays a pure mathematical structure. Although I have no wish to embrace Plato’s general philosophical standpoint, I think it may be helpful to accept Plato’s illuminating terminology. Hence I propose to consider the situation in terms of two essentially distinct ‘realms’ related to our reason. First, there is what we can call ‘The Realm of the Natural’ (RN; the realm of Plato’s Becoming) – the realm of the ‘things’ with which we live our lives. In this realm, things and matters can be found and described, but they are essentially vague andfuzzy (in the sense that nothing has a pure mathematical structure). Nothing regarding this realm can be proven in the mathematical sense. It can be seen as inhabited with things (in the prototypical sense of the word) and events, and prototypically it is the subject of natural science. Contrasting with this, there is what we will call ‘The Realm of the Formal’ (RF; the realm of Plato’s Being). Here everything is precisely defined and sharply delimited; things are stipulated and facts about this world can be unambiguously proved. The inhabitants of this realm can, with a certain amount of oversimplification, be called structures; they are addressed most directly by mathematics. The metaphysical conception of logic criticized above can now be seen as resting on the assumption that what logic addresses is something belonging both to RF and RN – something which, on the one hand, is rigid and directly susceptible to mathematical treatment, and yet, on the other hand, is a matter of our factual language and our factual reasoning. Truths of logic are taken to be true come what may, i.e. independently of what may or may not happen within the real world, but simultaneously they are assumed to be somehow inherently related to the language we happen to use and to the way we happen to think. Our point, then, is that the inherentness of this relationship is a pernicious illusion; we claim that whether a real thing can be reasonably seen as having this or another structure, or whether a structure can be helpfully ‘projected’ on this or that thing, is always an empirical matter. Thus, proving something about a structure from RF can be taken as proving something about a thing from RN only if it is taken for granted that the thing has this structure – which is itself something that is beyond a formal proof. From this viewpoint the metaphysical conception of logic results from the misconstrual of the relationship between RN and RF, namely from
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neglecting the fact that the roles of these two realms within our coping with the world are quite distinct: RN is the very world with which we are destined to cope, whereas RF is the realm of prisms we employ (maybe have to employ, constituted as we are) to ‘tame’ it and indeed to ‘make sense’ of it, to understand it. Thus, structures from RF serve as prisms through which we see and understand the world, and which we may employ to explicitly reconstruct its regularities and to point out the ‘forms’ or ‘structures’ of things or events.4
4. D IGRESSION II: G EOMETRY
It is of essential importance to recognize the difference between dealing with RN via RF and dealing with RF itself, i.e. between addressing reality via the prism of a structure and addressing the structure itself. To provide a vivid illustration of what kind of confusions may arise if this difference is not properly acknowledged, let us return, for a moment, to the beginning of this century, to the time when modern, formal mathematics, as a powerful way of addressing RF, was establishing itself. Some mathematicians, by that time, had begun to see mathematics no longer as the study of some parts or features of reality carried out by analyzing their mathematical structures, but rather as the study of the structures themselves; and they consequently begun to urge that it is only this conception of mathematics that can guarantee that mathematics is truly rigorous. It was in geometry where this process took place most spectacularly – for geometry was the mathematical discipline traditionally most strongly tethered to a specific aspect of reality. Hence, while geometry traditionally was conceived of as a way of accounting for certain aspect of reality (of RN) by means of analyzing its structure, some mathematicians with the new vision now shifted their attention to the very structure itself (i.e. RF) seeing in it the real subject matter of geometry. Thus, “geometry gradually moved from the study of absolute or perceived space – matter and extension – to the study of freestanding structures” (Shapiro, 1996, p. 149).5 The new formalistic conception of geometry was most systematically presented in Hilbert’s (1899) Grundlagen der Geometrie: Hilbert’s idea was that all basic concepts of geometry, like point, line, plane, is situated, parallel etc. are delimited by nothing more than by their mutual relationships, which are spelled out by the axioms of geometry. (As Poincaré, 1900, p. 78, put it, “if one wants to isolate a term and abstract from its relations to other terms, what remains is nothing”.)
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But obviously those mathematicians who did not make this ‘formalistic turn’ and continued to treat mathematical structures as mere tools of an account for reality (rather than the very subject matter of mathematics) were puzzled – for them such proposals were tantamount to making zoological concepts like dog, elephant, mammal etc. wholly independent of any real animals and taking them to be constituted simply by their relations to each other. Gottlob Frege famously protested that what Hilbert calls “axioms” (and what thus, according to Frege, should have been the most indubitable truths) are not truths at all, but just mere definitions, which point out certain structures, but do not say to what these structures are ascribed. However, this, of course, was precisely what Hilbert meant – so no wonder he rejected Frege’s objections as misguided: Sie schreiben: “. . . Aus der Wahrheit der Axiome folgt, dass sie einander nicht widersprechen”. Es hat mich sehr interessirt, gerade diesen Satz bei Ihnen zu lesen, da ich nämlich, solange ich über solche Dinge denke, schreibe und vortrage, immer gerade umgekehrt sage: Wenn sich die willkürlich gesetzten Axiome nicht einander widersprechen mit sämtlichen Folgen, so sind sie wahr, so existieren die durch die Axiome definirten dinge. . . . Ja, es ist doch selbstverständlich eine jede Theorie nur ein Fachwerk oder Schema von Begriffen nebst ihren nothwendigen Beziehungen zu einander, und die Grundelemente können in beliebiger Weise gedacht werden (printed in Frege, 1976, pp. 66, 67)6
For Frege, this was clearly simply preposterous: what he desperately missed was the projection of the ‘free-standing structure’ delimited by Hilbert’s axioms onto reality, a projection which would made it into a prism through which to see real things. Without such a projection, for him no axioms could make sense: Ich weiss nicht, wie ich mit Ihren Definitionen die Frage entscheiden soll, ob meine Taschenuhr ein Punkt sei. Gleich das erste Axiom handelt von zwei Punkten; wenn ich also wissen wollte, ob es von meiner Uhr gälte, müsste ich zunächst von einem anderen Gegenstande wissen, dass er ein Punkt wäre. Aber selbst wenn ich das z.B. von meinem Federhalter wüsste, so könnte ich noch immer nicht entscheiden, ob meine Uhr und mein Federhalter eine Gerade bestimmten, weil ich nicht wüsste, was eine Gerade wäre (ibid., p. 73)7
Hilbert’s reply was again rather laconic: Meine Meinung ist Eben die, dass ein Begriff nur durch seine Beziehungen zu anderen Begriffen logisch festgelegt werden kann. Diese Beziehungen, in bestimmten Aussagen formulirt, nenne ich Axiome und komme so dazu, dass die Axiome . . . die Definitionen der Begriffe sind (ibid., 79)8
In this way, the dispute soon foundered in deadlock. Now it is hard to get rid of the impression that both parties of this quarrel are at least partly right, each in its own way. And indeed what we claim is that if we distinguish properly between RF itself and RN viewed via RF, we can see that the differences between the standpoints of Frege
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and Hilbert may consist more in an overaccentuation of different aspects of a common picture than in proposing incompatible pictures. What Frege (and Russell and other ‘realists’) demanded was that there be a way to use geometrical axioms and the structure they spell out to address real things like watches, pen-cases etc., that the structure be somehow projected on the real world. This is indeed a reasonable demand, but we should add, on behalf of Hilbert, that if geometry is to be exercised with mathematical exactitude, then the projection cannot play a real role within the system as such. However, it is hard to believe that Frege, the depth of whose contributions to the development of modern, exact logic and mathematics is indubitable, would not have seen this. It seems unlikely that the author of the Begriffsschrift, which has set the standard of logical regimentation of our judging, would have underestimated exactitude and theoretical precision. He stressed rather that exactitude and precision have real value only when addressing something ‘real’, something that is not a mere conclusion of our definitions and stipulations. On the other hand, what Hilbert (and Poincaré and other ‘formalists’) insisted was that if we want to do with geometry what Peano did with arithmetic, if we want to leave nothing unproved, then we must treat it as an abstract, ‘ideal’ system whose terms are significant only as its nodes. Again, this is surely true, but on behalf of Frege we must add that we call something ‘geometry’ only if it is capable of serving a certain specific purpose, namely to help us cope with certain spatial aspects of the things which surround us. And again, it is hard to believe that Hilbert would be blind to this; it would be more than difficult to believe that he would accept that any system of axioms, say that of Peano arithmetic, would constitute as good a geometry as that which is constituted by Hilbert’s own axioms of geometry. I think that the truth is rather that he believed this to be too self-evident to dwell on; and he simply wanted to stress that the matters concerning the projectibility of the geometrical structure on reality are not capable of being included into mathematics itself. The emerging conclusion is that geometry originated as a matter of addressing, and thereby explicating, certain aspects of RN with the help of a certain prism from the RF. It is consequently a relatively uninteresting, terminological problem whether we should use the term ‘geometry’ for the abstract prism alone, or for the prism together with the projection. In the first case, we would have to keep in mind that the prism is called so only in virtue of its relevant projectibility; in the second we would have to realize that it is only the prism, not the projection, that can be subjected to ‘mathematical’ treatment.
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From this point of view, the Frege–Hilbert controversy may appear more a misunderstanding than a real disagreement. This, of course, is not to say that there are no substantial differences between the views of the two theoreticians; it is to suggest that some of the differences may be less deep than generally supposed.9
5. M ATHEMATICAL MODELS AND REALITY
The fact that we have to distinguish between an abstract mathematical structure and reality captured via that mathematical structure – between ‘pure’ and ‘applied’ mathematics, somewhat oversimplifying – is, of course, no breath-taking discovery. Probably everybody dealing with mathematics recognizes it on a general level; and those who reflect upon the workings of modern, formal mathematics have sometimes even articulated it with remarkable clarity. Thus Reichenbach (1920, pp. 32–35): Der mathematische Gegenstand ist durch die Axiome und die Definitionen der Mathematik vollständig definiert. ... Für den physikalischen Gegenstand aber ist eine derartige Definition unmöglich. Denn er ist ein Ding der Wirklichkeit, nicht jener konstruierten Welt der Mathematik. . . . Es ist Methode der Physik geworden, eine Größe durch andere zu definieren, indem man sie zu immer weiter zurückliegenden Größen in Beziehung setzt und schließlich ein System von Axiomen, Grundgleichungen der Physik, an die Spitze stellt. Aber was wir auf diese Weise erreichen ist immer nur ein System von verflochtenen mathematischen Sätzen, und es fehlt innerhalb dieses Systems gerade diejenige Behauptung, daß dies System von Gleichungen Geltung für die Wirklichkeit hat. Das ist eine ganz andere Beziehung als die immanente Wahrheitsrelation der Mathematik. Wir können sie als eine Zuordnung auffassen: die wirklichen Dinge werden Gleichungen zugeordnet. . . . Nennen wir die Erde eine Kugel, so ist das eine Zuordnung der mathematischen Figur “Kugel” zu gewissen Wahrnemungen unserer Augen und unseres Tastsinns, die wir, bereits eine primitive Stufe der Zuordnung vollziehend, als “Wahrnemungsbilder der Erde” bezeichnen.10
Thus, a mathematical term can have two kinds of ‘meanings’: the ‘internal’ meaning which it acquires by denoting a node in a mathematical structure, and possibly also an ‘external’ meaning which it acquires as the consequence of the fact that the structure is somehow projected on reality. The term “sphere” or the numeral “five” represent certain nodes within certain ‘free-standing’ mathematical structures (which have been brought out by the axioms of the corresponding theories, geometry resp. arithmetic), but they can also be seen as representing, or accounting for, something from the non-mathematical world: objects of certain shapes, resp. groups of objects of certain cardinality. The difference is, in our words, whether what we have in mind is merely an element of RF, or rather those elements of RN onto which this formal element is taken to be adequately ‘projectible’.
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Unfortunately, although the distinction between a mathematical object and a real object captured by it – i.e. between mathematical model and reality – may be clear on the general level, the fact that in many circumstances we can simply neglect it makes us sometimes disregard it even in cases when its acknowledgment is crucial. Imagine that you inspect a room, see that it is empty and then you see three, and later four people enter it. You claim: “The room now contains seven people”; and if this claim is challenged, you offer the proof of the fact that three plus four equals seven within Peano arithmetic. Is this really the proof of the claim? This is clearly so only provided your ‘mathematization’ of the problem has been adequate. If you later opened the room and discovered there not seven, but only six people, you would surely not blame Peano arithmetic. You would certainly say that, apparently, one person has left the room without your noticing; so that the correct thing to prove was not 3 + 4 = 7, but rather 3 + 4 − 1 = 6. But perhaps the slack between the ‘mathematical model’ and reality can be dispensed with, perhaps the only thing needed is to make the model really adequate? Perhaps we only have to pay attention to achieve the perfect fit? Perhaps, in our case, if we took pains to achieve the perfect fit, the proof of the mathematical theorem would be, eo ipso, a proof of the number of people within the room? However, what would such a perfect fit amount to? We have seen that you should have guarded against people entering or leaving the room unnoticed. But this is in no way the only thing that could spoil the fit. Some person in the room might have killed and eaten another person; or some person might have borne a child. Or somebody may be so positioned that it is unclear whether she is still in the room or not. Or it may turn out that the room you were observing does not, in fact, exist, for what you took to be the walls of the room was only an optical illusion. It seems clear that it is never possible to really exclude or even spell out all of these potentialities to secure the ‘perfect fit’. We rather assume such fit, but carry somewhere in the back of our minds that there is this assumption, which might turn out to be false. And troubles begin if we forget about it. This is, of course, not to say that ‘mathematical’ or ‘logical’ models cannot capture reality and help us solve real problems. It would, of course, be absurd to claim that we do not build bridges, planes or cyclotrons with the help of mathematics, or that we cannot, say, build systems of knowledge representation with the help of logic. It is to say that any such ‘mathematization’ necessarily has an empirical ingredient brought about by the fact that the fit of a structure from RF to a problem from RN is bound to be an empirical matter. In practice, this is not particularly problematic,
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for the fit we can achieve is usually quite sufficient. However, the situation may alter when we turn our attention to foundational questions about the nature of logic, philosophy or mathematics: disregarding the matter here may result into grave misconceptions. This prompts us to give the following answer to the question about the nature of logic: the truths of logic are necessary and ‘mathematically treatable’ because (and only insofar as) they constitute a system within RF, while they are ‘factual’ and ‘about our reasoning’ because (and only insofar as) this system is projected onto RN and used as a prism to capture the relevant aspects of our reasoning. In other words, they are necessary, for they spell out a constant structure, and they are about our reasoning, for this reasoning can be seen to display this very structure. Their ‘necessity’ and their ‘factualness’ are thus properties of different levels, and it is crucial to hold them in a certain equilibrium: concentrating exclusively on the former puts us in danger of losing the connection with the real world (thus falling into scholastic speculations, or, in the better case, into pure mathematics); while concentrating exclusively on the latter puts us in danger that we shall not be able to really understand (for understanding requires an appreciation of regularities, the application of a ‘mathematical’ prism). 6. L OGIC VS . METAPHYSICS
Laying the foundations of modern logic, Frege realized that in order to get a grip on the ‘content’ of sentences (i.e. on the propositions they express), he had to strip them of everything not relevant from the viewpoint of the consequence relation.11 At this point, finding out which formula was to regiment a given sentence and thereby which proposition the sentence expressed, seemed to be a matter of a certain simplification of syntax. Frege himself, and especially his followers (notably Russell), subsequently concluded that sometimes it is necessary to accept that finding a correct logical regimentation of a sentence (hence locating the proposition expressed by the sentence) might itself be a nontrivial task (this was exemplified, e.g. by the Russellian analysis of sentences containing definite descriptions12 ). A logician or a philosopher who admits the possibility of such a nontrivial gap between a sentence and the proposition it expresses must recognize the possibility of investigating the world of propositions, bypassing sentences expressing the propositions. However, there seems to be a bifurcation of ways of understanding this enterprise. The ‘metaphysical’ standpoint amounts to the conclusion that it is this investigation that is the ultimate task of logic. After all, propositions are
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what are really substantial; and the question about which propositions are expressed by which sentences is the business of empirical linguistics. “That my logic does not apply to natural language,” such a theoretician is likely to say, “is not my business – the worse for the language. What I am investigating are propositions; and I do not care which propositions happen to be expressed by sentences of a factual language.” But he would also reject that what he is doing is simply mathematics (an investigation of a certain abstract structure). He would insist that the world of propositions he is addressing is in some intimate way connected with our reasoning. This is a standpoint that Norman Malcolm (1940, p. 197) characterized as follows: “Philosophers and logicians have the idea that when a question as to whether one statement entails another arises, verbal considerations enter only because of ambiguity, and that the real question is not a verbal one, but one to be settled by the intellect’s fixing its gaze upon the proposition, after the ambiguity has been cleared up”. The trouble with the metaphysical view of logic is not that it accepts propositions, but rather that it accepts the notion that propositions can be investigated by “the intellect’s fixing its gaze upon them.” The alternative standpoint we advocate amounts to seeing the investigation into the realm of propositions as an enterprise internal to logic – as the analysis of the formal structures logic uses to account for our reasoning. Such an enterprise then has its substantiation not simply in itself; it is substantiated only insofar as the tool it analyzes is a useful tool. Thus, while the metaphysical conception of logic simply assumes that the ‘mathematics of propositions’ equals logic because the propositions are somehow inherently related to our reasoning, we urge that it is logic only because, and only insofar as, we are able to use the propositions as the nodes of the prism that helps us understand reasoning. If this is correct, then the sense of the system of propositions which is susceptible to mathematical treatment should be seen in its capacity to account for the regularities of the way we use language and of the rules implicit to this usage. It does its job only if it can be projected on our factual usage of language and our factual reasoning in such a way that it explicates its substantial regularities and rules. It is the possibility of such a projection which generally substantiates the usage of items from RF outside of mathematics; and as the projection is a matter of the relationship between RF and RN, its existence can never be proved or subjected to formal criteria (for these make sense only inside RF). Thus, to see logic or philosophy as the study of a world of propositions makes real sense only insofar as the world can be seen as parasitic upon our factual games of ‘giving and asking for reasons’.
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7. DAVIDSON ON PROPOSITIONS
The warning against promoting the abstract world of propositions (or thoughts in the Fregean sense) to an independently accessible reality is a point which plays, I am convinced, an important role in the writings of several key figures of this century’s philosophy of logic and analytic philosophy. I think that this was precisely what Wittgenstein had in mind when he insisted that the real subject matter of the philosophy of language is constituted by the factual language games we play, and that the mental entities that we tend to see as making our expressions meaningful are better seen as our way of accounting for the games. (“Sieh auf das Sprachspiel als das Primäre! Und auf die Gefühle, etc. als auf eine Betrachtungsweise, eine Deutung, des Sprachspiels!”13 – 1953, §656.) I also think that this view is central to many of the founding fathers of American analytic philosophy, namely Quine, Sellars, Davidson etc., in their effort to revise the picture of the relationship between language and the world provided by their European predecessors.14 Quine is so vehemently against seeing sentences as expressing propositions (which, according to him, may so easily lead to what he calls the museum myth) that he insists on rejecting the very notion of proposition and of meaning in general. Sellars’ way of rejecting the metaphysics of propositions concentrates in his claim that meanings (and especially propositions) are inherently functional entities; that they – in an important sense – do not exist apart from their embodiment.15 However, the most vivid elaboration of the anti-metaphysical standpoint urged here is offered by Donald Davidson. Davidson’s claim is that propositions should be construed as the units of measurement we use to characterize certain aspects of the world surrounding us (namely rational beings) in the same sense in which we use meters and kilograms to characterize other aspects of it. Thus, saying that a speaker believes a proposition (or that a sentence expresses the proposition) is like saying that something is five meters long, or that something can be captured by a certain mathematical equation. Davidson (1989, p. 11) claims: Just as in measuring weight we need a collection of entities which have a structure in which we can reflect the relations between weighty objects, so in attributing states of belief (and other propositional attitudes) we need a collection of entities related in ways that will allow us to keep track of the relevant properties of the various psychological states. In thinking and talking of the weights we need not suppose there are such things as weights for objects to have. Similarly in thinking and talking about the beliefs of people we needn’t suppose there are such entities as beliefs.
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What Davidson denies is that there are such things as propositions, which would be, on the one hand, associated with sentences (thus becoming their meanings), and which would, on the other hand, come to inhabit people’s heads (thus becoming their beliefs). However, saying that propositions are not things in this sense does not amount to saying that they are nothing at all – it amounts to saying, as we would put it, that they are not to be sought within the RN. Thus, according to Davidson, propositions do exist in the same way as meters and kilograms do: as nodes within a structure that we cast over a certain part of our world to make it intelligible in the way our reason seeks it. In the same way as it is helpful for us to see, say, a stone as assuming a place on the scales of meters, kilograms etc., it is also helpful to see a sentence as assuming a place within the network of propositions, and to see a rational agent as assuming a place within the network of theories, i.e. sets of such propositions.16 8. F OUNDATIONAL PROBLEMS OF LOGIC
It is my conviction that the view urged above can not only provide a clearer insight into the conceptual framework of modern logic and into the nature of the entities this framework employs, but can also throw some new light on some of the most frequently discussed foundational problems of logic. Let us briefly review some of the cases where carefully distinguishing RF and RN may, I believe, be enlightening. Before turning to genuine problems, let me mention an instance of the rare case where the relationship between RF and RN comes to the open to such an extent that it usually does not cause any serious confusion. Church’s thesis states explicitly that a formal concept, namely recursivness (or, equivalently, Turing-computability, representability in lambdacalculus etc.), matches a natural one, namely computability in the intuitive sense.17 It is clear that Church’s thesis cannot be proved, for we can prove the equivalence only of two formal concepts (like recursivness and Turingcomputability); we cannot prove that a formal concept does capture an informal one. On the other hand, the thesis provides an excellent example of how we can obtain an informal, but compelling justification for a thesis of this kind: if we find out, as we have, that all, or almost all, independently developed formal concepts purporting to capture a given informal one come to the same, there are good grounds for concluding that they do capture it successfully. (Note however, that such a justification, unlike a formal proof, does not guarantee that any further disagreement is bound to be a matter of misunderstanding – in fact there continue to be people
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who undoubtedly understand the issue very well and who nevertheless do challenge Church’s thesis.18 ) A case where the relationship between the RF and RN plays a less perspicuous role is the problem of the significance of Gödel’s incompleteness proof. This result is often interpreted as a stunning discovery stating ‘the limits of human reason’, the ‘inscrutability of mathematical truth’ or, in Roger Penrose’s words, the “unalgorithmicity of the mind”.19 However, the perspective urged here leads to the conclusion that such interpretations should only be accepted with caution. Was Gödel’s proof a formal proof (of the kind of that of the equivalence of recursivness and Turing-computability), or was it a finding of something factual (about something like computability in the intuitive sense, i.e. about what we humans, as a matter of fact, can do)? It seems that the first is the case: Gödel’s proof appears to be a mathematical matter which establishes the theorem proved with the certainty, which guarantees that everybody who does not believe simply does not understand it. However, if this is the case and if our above conclusions are right, then Gödel’s result must concern merely an abstract mathematical structure and be thus confined to RF.20 It can tell us nothing whatsoever about anything factual, such as how our human reason, as a matter of fact, works, or how our mathematical practices may or may not proceed. On the other hand, if we took Gödel’s result as saying something about RN, then we would have to give up the idea that it is a proof in the mathematical sense, clearing away any possibility of doubt. We have seen that the competence of a mathematical proof within RN is always only conditional: the applicability of the proof to factual matters always depends crucially on the adequacy of the relevant projection, which is itself beyond any proof. In other words, we can interpret Gödel’s result as being about something from our real world only to the extent to which the thing in question can be adequately ascribed the structure which Gödel’s proof concerns directly. This is to say that it can be taken as being about what we humans do when we count and when we do what we call arithmetic only if we take this activity of ours to display the very structure which is envisaged by formal Peano arithmetic. Of course, we have various kinds of convincing reasons to think that it does display it; never, though, can we have a formal proof. The problem is that the shining of Gödel’s result derives, at least partly, precisely from the fact that it is taken to be a formal proof of something factual, namely of how we humans do or can think – and this, if our above conclusions are right, is simply not possible. To be formally provable (to be ‘mathematically certain’) and to say something about our real, human world (to ‘refer to reality’) are mutually exclusive properties: a formal
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proof, as we have seen, is the matter of RF, it can directly concern neither RN, nor a projection of RF onto RN. (As Einstein, 1983, p. 28, put it: “So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.”) We may either take Gödel’s proof to be a proof in the strong sense, but then we cannot take it as directly addressing ‘human arithmetic capacities’ (let alone human reason as a whole), or insist that it is about ‘real’ arithmetic, but then we cannot see it as a proof in the strict mathematical sense. Another foundational problem I will mention is the theory of truth as established by Alfred Tarski21 : in this case, too, the failure to distinguish between ‘the natural’ and ‘the formal’ is likely to cause much confusion. It is often claimed that the status of Tarskian T-sentences, i.e. sentences like (T)
‘Snow is white’ is trueT if and only if snow is white,
is essentially problematic, for if we take trueT to be the predicate introduced by Tarski’s theory, then there is a sense in which we can say that (T) is a truth of logic – for it follows from nothing else but the principles of logic (including, possibly, set theory) and definitions. This means that (T) must be uninformative in the way logical truths are, it has to be “true in every possible world” (see Putnam, 1985, p. 63). However, this seems to be in contradiction with Tarski’s declared intention to address the preformal concept of truth. Some theoreticians (e.g. Putnam) conclude that Tarski managed to develop a logical theory but failed to tell us anything about truth; others (e.g. Etchemendy, 1988) suggest that what is missing from Tarski’s theory is the stipulation that a sentence is trueT if and only if it is true (what Davidson, 1990, calls the “truth axiom”). I think the best way to describe what is going on here is to say that what Tarski was after was to point out a formal structure capable of serving as a reconstruction of our language with its truth-predicate. Tarski’s theory is a formal theory in that it constitutes a system within which some statements can be proven to be ‘logical truths’ (notably those of the shape of (T)); but it is also a theory of truth in that it can be projected onto our natural language and thus helps us understand the functioning of the predicate true and the concept expressed by it. (As Davidson, 1990, p. 314, stresses, the Tarskian theory of the “formal properties” of the concept of truth must be supplemented by an indication of “how a theory of truth can be applied to particular speakers or groups of speakers”.) If we see Tarski’s theory as pointing out an item within RF, then his trueT is a formal predicate governed by its formal definition; but if we see it as using this item for the purpose of capturing an item from RN, then we may see it as an explication of our informal concept of truth. However, formulating a “truth axiom” stipulating that Tarski’s formal system is adequate to that which
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it is devised to capture is no more meaningful than stipulating Church’s thesis – adding axioms can make items from RF into larger items of RF only, it can never pin them down to items of RN. The point is that an axiom, being by its nature a matter of RF, can never guarantee that a formal theory is adequate to which it is supposed to be a theory of. This is always necessarily a matter of practical assessment and evaluation by human subjects, who may, or may not, find it useful for their enterprises of coping with the RN. Nothing can belong both to RF and to RN (be both a stipulation and a ‘phenomenon’); nevertheless, an item from RF may turn out to be a helpful prism to observe an item from RN.22 Another cluster of problems which might be clarified by adopting the vantage point urged here centers on the concept of semantic interpretation. An interpretation is a mapping of a system of items (‘expressions’) onto another system of items (‘denotations’, ‘meanings’). Within logic, semantic interpretations of this kind are studied by model theory. It was Montague (1974, p. 188) who voiced the claim that from the point of view of semantics, there is no real difference between natural language and formal languages; he was also one of the first to demonstrate how to apply model-theoretic notions to natural languages in an interesting way, and thus he laid the foundations of what is nowadays called formal semantics. There are two essentially different ways of interpreting a formal language (i.e. a certain system of items of RF). We can map it either (i) on another system of items within RF, or (ii) on a system of items within RN. In the case (i) we do not leave the RF, i.e. the province of mathematics; so even the mapping itself is a formal object which can be studied in a mathematical way. This is exactly what model theory does: it addresses mappings of certain kinds of formal structures (‘formal languages’) onto another kind of formal structures (‘model structures’). In case (ii), interpretation amounts instead to what we have called projection: it is the means of perceiving a part, or a feature, of RN through the prism of the structure from RF (typically perceiving a factual language as a formal structure); and in this way it also licenses us to see the structure from RF as being ‘about’ something factual, as being a structure ‘of’ something factual (e.g. our natural language). It is important to fully appreciate the depth of the difference between these two kinds of mappings; and to realize that calling them both interpretations may even be misguiding23 . If we say that a formal language, a mere system of strings of items, has to be interpreted in order to become a language worth its name, what this should typically mean is that an element of RF has to be projected onto RN in order to become useful for the purposes of explicating factual matters.24 However, it is often assumed
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that the same can be achieved by mapping the formal language onto a formal system of denotations, by furnishing the language with a formal, set-theoretic interpretation. But a formal interpretation makes a formal language into merely a more complex formal system; it can never make it really ‘meaningful’.25 What is possible, and sometimes indeed useful, is to take a formal mapping of a formal language (an item of RF) onto a formal system of denotations (another item of RF) as a ‘picture’ of a ‘natural’ mapping of a natural language (an item of the RN) onto the system of meanings of the expressions of the language (another item of RN). Personally, I think that this is the only viable sense in which we can take ‘formal semantics’ as a theory of natural language (see Peregrin, 1997). However, this can be also deeply misleading and hence dangerous: it may suggest that natural language is a set of labels stuck on pre-existing things; and this is a view that is essentially problematical (in fact, to see language thus involves falling for what Quine calls the museum myth – see Peregrin, 1995; 1998). 9. D IGRESSION III: T HE NATURE OF LINGUISTIC THEORY
The last issue discussed in the previous section is connected to problems concerning the very nature of a theory of language. As this is a deep and interesting problem and as it can, I believe, throw some further light on the conceptual framework introduced here, let us make a short excursion into linguistics (and its philosophy) and say a few words about it. (Elsewhere I have discussed it at length – see Peregrin, 1998.) The question to be answered is this: is linguistic theory about individual speakers (be it about their minds, language faculties, behavior or whatever), or about a realm of abstracta? As a starting point, let us take Katz and Postal’s (1991) paper, where the authors urge the replacement of Chomskyan “conceptualism” with “realism”. The “conceptualist” view which Katz and Postal challenge (and which they ascribe to Chomsky) is that “grammars and grammatical theory describe a psychological reality” (p. 517); their own “realistic” view takes “natural languages to be abstract objects rather than concrete psychological or acoustic ones” (p. 515). Although I think Katz’s and Postal’s criticism is basically sound, I am persuaded that the two standpoints do not exclude each other to the extent to which they would seem to do so – that if we look at the dispute from the proper angle, we may see it at least partly as a terminological matter. On the one hand, there is a straightforward sense of ‘realism’ in which every minimally plausible semantic theory trivially has to be realistic. It is hard to believe that anybody, even the most diehard mentalists and con-
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ceptualists, would claim that semantics is a matter of describing some mental (neural) particulars within the head of an individual speaker – for this would be no theory of English (nor of any other language), but rather the theory of some features of a particular person. Even if we accept the assumption that semantics is a matter of particulars of such a kind, we simply have to assume that these particulars can somehow be equated across speakers; that they have some properties which make them treatable as tokens of recurrent types. So the linguist must talk about some non-particulars – be they construed as cross-subjective type-identities of particulars, or some abstract entities borne by these identities. In any case, talk about meaning is in a clear sense a talk about types, not about tokens; and semantics is – in this sense – inevitably (and trivially) realistic. On the other hand, even the most diehard realist has to assume that there are some contingent facts that elicit which meaning an individual expression has. We do not discover meanings by an ‘intellectual trip’ into a realm of abstracta where we could see them attached to expressions; but rather by observing and recording certain concreta. It is the occurrence of certain particular events or entities (be it the occurrence of certain utterances of speakers, or the occurrence of certain contents within the heads of speakers) which establishes the meaning of an expression. Therefore, both the conceptualist and the realist apparently must agree that meanings are abstracta (universals) which are in a certain sense determined by (parasitic upon) certain concreta (particulars). So, if the only thing that realism claimed were that semantics is a matter of abstracta rather than of concreta, of types rather than of tokens, then realism would seem to be unobjectionable. And if the only thing which conceptualism asserted were that abstracta make no sense unless they are in the sense outlined ‘parasitic’ upon concreta, then it too would be hardly objectionable. Hence, such modest conceptualism and modest realism might even coincide – if we accept that our knowledge (in general) arises out of apprehending some particular arrays of occurrences as displaying universal structures, out of seeing items of RN via items from RF. The only clash is then, again, a terminological one: whether this situation justifies us in saying that linguistics is about the particular occurrences, or about the universal structures. This is a legitimate subject for a quarrel, but not for one with great significance. Troubles begin when conceptualism or realism ar taken as claiming something more. Conceptualism sometimes seems to claim that the theoretician of language has no use of abstract entities whatsoever, whereas realism sometimes appears to claim that that these abstract entities are accessible in a direct way, wholly bypassing their concrete embodiments.
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Extreme conceptualism thus disregards the fact that to understand is to discern a pattern, a structure, to see tokens as tokens of types; whereas extreme realism forgets that the abstract structures we discern are interesting only insofar as they are the structures of that which we have set out to study. Thus, I think that this kind of quarrel can be again largely clarified by pointing out that in making a theory such as the theory of our linguistic performance we usually address something from RN by means of something from RF. We address the potential infinity of concrete utterances of speakers by means of a certain structure; we ‘capture’ the former by the latter. This is to say that if we ask what it is that linguistic theory is about, then there are, just like in the case of geometry discussed above, two different kinds of answers available, corresponding to two different senses of “about”. In the first sense, linguistics is about the part of RN which it addresses (i.e. about certain ‘concreta’); in the second sense, it is about the part of RF which is capable of providing an adequate reconstruction the part of RN in question (i.e. about certain abstracta). The quarrel between ‘realists’ and ‘conceptualists’ thus may again turn out to be more a misunderstanding than a real discrepancy. 10. C ONCLUSION
The main thesis of this paper, the usefulness of distinguishing between the ‘natural’ and the ‘formal’ should not be read as a metaphysical pronouncement. The talk about the two ‘realms’ should be read not as a report of a (re)discovery, but rather as a vivid way of making the point that something may be susceptible to a more geometrico treatment only if it is a thing we can somehow completely seize by our reason, not one of the things we encounter within the world of our everyday experience. This point was duly made long ago by Brouwer (1907, p. 76), who stressed that people err when they think that they “could reason logically about other subjects than mathematical structures built by themselves”. We can prove things only about entities we ourselves stipulate, not about entities we encounter. However, this is not to say that the ‘formal’ entities we stipulate cannot help us comprehend and grasp the ‘natural’. On the contrary, they are essentially important, for the imposing of structures (which is what they basically are) is our fundamental means of theoretically coping with our environment (or indeed of handling it, predicting its behavior etc.). We can, nevertheless, never restrict ourselves to the structures alone: we have to constantly check and assess whether they are fully adequate to that to which they have been ascribed, whether they are sufficiently helpful as
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prisms through which to look. We can never eradicate a ‘pragmatic factor’: any mathematical theory of anything outside of mathematics, in order to be helpful, has not only to be in an internal order (be consistent), but also has to fit adequately to that of which it purports to be a theory. And the second requirement always involves an assessment from the viewpoint of some of our specifically human interests.26 In particular, when we return to logic, it is often helpful to reconstruct our reasoning, our language and our thought as various formal or mathematical structures; however, this helps us only insofar as we remember that these structures are nothing more (and, indeed, also nothing less) than our way of getting a grip on that which we thus reconstruct.
ACKNOWLEDGEMENT This work was supported by the Research Support Scheme of the OSI/ HESP, grant No. 280/1997. I want to express my thanks to Timothy Childers and the anonymous reviewer of this journal for insightful comments which helped me to bring the paper to this final form.
N OTES 1 See Putnam (1994, p. 250; I have replaced quotes by the more appropriate Quinean quasiquotes). The author’s claim is that mathematical and logical necessity is a matter of our not being able to give up, and indeed not being able make any sense of the falsity of, such claims. 2 Where the value 0 of the influence mode is to be understood as representing positive influence, while the other two possible values as representing the negative one. 3 A phrase due to Brandom (1994). 4 Cf. Stekeler-Weithofer (1994). 5 Shapiro’s paper also contains a more detailed discussion of the Frege–Hilbert controversy outlined below and of its historical context. 6 “You write: ‘ . . . From the truth of the axioms it follows that they do not contradict one another’. I was very interested to read this particular sentence of yours, because for my part, ever since I have been thinking, writing and lecturing about such matters, I have been accustomed to say just the reverse: if the arbitrarily posited axioms are not in mutual contradiction with the totality of their consequences, then they are true – the things defined by the axioms exist. . . . . Yes, it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes.” 7 “I do not know how to use your definitions to decide the question whether my pocket watch is a point. Already the first axiom treats of two points; thus if I wanted to know whether it is valid for my watch, I would first have to know about some other object that it is a point. But even if I knew this, e.g., about my pen-case, I could still not decide whether
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my watch and my pen-case determine a straight line, for I would not know what a straight line is.” 8 “It is my opinion just that a concept can be fixed logically only by its relations to other concepts. These relations, formulated in certain statements, I call axioms, thus arriving at the new view that axioms . . . are the definitions of the concepts.” 9 This distinction is also closely connected to that between the two ways of understanding mathematics as the “science of structure” discussed by Shapiro (1996), namely between the “ante rem” structuralism and the “in re” or “eliminative” structuralism. The former consists in seeing mathematics as directly addressing structures from RF, while the latter sees it as addressing items from RN as instances of these structures and sees the structures as not independent, but rather only as parasitic on their instances. 10 “The mathematical object is fully defined by the axioms and definitions of mathematics. . . . Such a definition is, however, not possible for the physicalistic object, for it is a thing of the real world, not of the constructed world of mathematics. . . . It has become the method of physics to define one magnitude through others, in that it is related to magnitudes lying ever more in the background, finally putting the system of axioms, of the basic equations of physics, on the top. However, what we reach in this way is still only a system of entangled mathematical sentences, and this system does not contain the assertion that the system of equations is valid for reality. This is a quite different relation than the immanent truth-relation of mathematics. We can grasp it as an assignment: real things are assigned to the equations. . . . If we call the Earth a sphere, then it is the assignment of the mathematical figure ‘sphere’ to certain perceptions of our eyes and our taste, which we denote, thereby accomplishing a primitive level of the assignment, as ‘perception of the Earth’.” (My translation.) 11 See Frege (1879, p. IV). 12 See Russell (1905). 13 “See the language game as the primary! And the feelings etc. as a way of dealing with, or of accounting for, the language game!” 14 Cf. Peregrin (1999a). 15 Cf. Brandom (1994). 16 Cf. Hofman (1995). 17 See, e.g., Boolos and Jeffrey (1974, p. 20). 18 See, e.g., Hintikka and Mutanen (1998). 19 See Penrose (1990). 20 To highlight the formal character of the proof, we can characterize Gödel’s incompleteness proof, e.g., as follows. Let A be an alphabet (a finite set of objects) and let us call L the set of all strings over A. If a, b ∈ L, let a ∩ b denote the concatenation of a and b (i.e. the string which arises out of appending b to a) and let ⊕ denote some binary operation defined on the basis of concatenation (a ⊕ b might be, for example, substituting b for a given symbol within a). Let M be a subset of the powerset of L, i.e. a set of subsets of L. We define the relation =M among the elements of L in such a way that a =M b if and only if every m ∈ M contains either both a and b, or neither a nor b. If x ∈ A, then by the x-variant of an a ∈ L we shall call the string x ∩ a, i.e. such a string which arises out of a by the prefixation of x. A subset of L will be called x-open, if it contains no string together with its x-variant; and we shall call it x-saturated, if it contains x-variants of all such strings from L which it does not contain. Now it clearly holds that if, for some x ∈ A, some a ∈ L and every b ∈ L, a ⊕ b =M x ∩ b ⊕ b, then no x-open set from M contains a ⊕ a (and hence x ∩ a ⊕ a); thus no x-open set from M is x-saturated. Now we can see Gödel’s proof as the
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proof of the fact that a particular structure fulfills the premises of this general theorem: that if we take L to be the language (the set of wffs) of Peano arithmetic, M the set of consistent theories in this language containing the axioms of Peano arithmetic, x the negation-sign, and ⊕ the appropriate kind of substitution of the numeral expressing the Gödel’s number, then there will indeed be an a ∈ L so that for every b ∈ L, a ⊕ b =M x ∩ b ⊕ b; hence no x-open (= consistent) theory is x-saturated (= complete). 21 See Tarski (1932; 1944). See also Peregrin (1999b). 22 For a similar argumentation see García-Carpintero (1999), who argues, in effect, that it is precisely this what is constitutive of an explication (in the Carnapian and Quinean sense). 23 One of the bad habits of contemporary ‘formal semantics’ is to confuse the two senses of interpretation. See Stekeler-Weithofer (1986) for a discussion. 24 As Brandom (1994, p. 144) puts it, “it is only in so far as it is appealed to in explaining the circumstances under which judgments and inferences are properly made and the proper consequences of doing so that something associated by the theorist with interpreted states or expressions qualifies as a semantic interpretant, or deserves to be called a theoretical concept of content”. Brandom’s way of reflecting the distinction we stress here is distinguishing between what he calls formal and philosophical semantics (where the former is an enterprise internal to what we call RF, while the latter’s concern is, in our terms, to explicate a relevant portion of RN, perhaps with the help of some tools from RF). 25 This ambiguity of the concept of interpretation also engenders the ambiguity of concepts which are based on it, especially of the concepts of soundness and completeness. Formal soundness and completeness amounts to capturing all and only sentences universally valid w.r.t. a given class of model structures; natural soundness and completeness means an exhaustive capturing of a pre-formal range of truths. The former, not the latter, can be subject to mathematical proof – and the proof of the former is not a proof of the latter (pace Kreisel, 1967). See also Peregrin (1995, §4.9). 26 It may be illuminating to invoke the good old Kantian dualism of Verstand and Vernunft here: the adequacy assessment, we can say, is always a matter not of the calculating Verstand, but rather of the understanding Vernunft inseparable from the ability to perceive things from the distinctively human visual angle. Cf. Stekeler-Weithofer’s (1992) attempt to explain Hegel’s criticism of Kant in these terms.
R EFERENCES Boolos, G. S. and Jeffrey, R. C. (1974): Computability and Logic, Cambridge University Press, Cambridge. Brandom, R. (1994): Making it Explicit, Harvard University Press, Cambridge, MA. Brouwer, L. E. J. (1907): Over de grondslagen der wiskunde, Thesis, Amsterdam; reprinted in A. Heyting (ed.), Brouwer: Collected Works I: Philosophy and Foundations of Mathematics, North-Holland, Amsterdam, 1975. Davidson, D. (1989): What is present to the mind?, in J. Brandl and W. L. Gombocz (eds), The Mind of Donald Davidson, Rodopi, Amsterdam, pp. 3–18. Davidson, D. (1990): The structure and contents of truth, J. Philos. 87: 279–328. Einstein, A. (1983): Geometry and Experience, Sidelights on Relativity, Dover, New York (translation of the German original ‘Geometrie und Erfahrung’ from 1921).
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Etchemendy, J. (1988): Tarski on truth and logical consequence, J. Symbolic Logic 53: 51–79. Frege, G. (1879): Begriffsschrift, Nebert, Halle; English translation in van Heijenoort (1971), pp. 1–82. Frege, G. (1976): Wissenschaftlicher Briefwechsel (ed. by G. Gabriel et al.), Meiner, Hamburg; English translations in Frege: Philosophical and Mathematical Correspondence, Blackwell, Oxford, 1980. García-Carpintero, M. (1999): The explanatory value of truth theories embodying the semantic conception, in Peregrin (1999b), pp. 185–212. Gödel, K. (1931): Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38: 173–198. van Heijenoort, J. (ed.) (1971): From Frege to Gödel: A Source Book in Mathematical Logic, Harvard University Press, Cambridge, MA. Hilbert, D. (1899): Grundlagen der Geometrie, Teubner, Leipzig. Hintikka, J. and Mutanen, A. (1998): An alternative concept of computability, in J. Hintikka (ed.), Language, Truth and Logic in Mathematics (Selected Papers 3), Kluwer Acad. Publ., Dordrecht. Hofman, A. (1995): On the nature of meaning and its indeterminacy: Davidson’s view in perspective, Erkenntnis 42: 15–40. Katz, J. J. and Postal, P. M. (1991): Realism vs. conceptualism in linguistics’, Linguistics and Philosophy 14: 515–554. Kreisel, G. (1967): Informal rigour and completeness proofs, in I. Lakatos (ed.), Problems in the Philosophy of Mathematics, North-Holland, Amsterdam; reprinted in J. Hintikka (ed.), The Philosophy of Mathematics, Oxford University Press, Oxford, 1969, pp. 78– 94. Malcolm, N. (1940): Are necessary propositions really verbal?, Mind 49: 189–203. Montague, R. (1974): Formal Philosophy: Selected Papers of R. Montague (ed. R. Thomason), Yale University Press, New Haven. Penrose, R. (1990): The Emperor’s New Mind, Vintage, London. Peregrin, J. (1995): Doing Worlds with Words, Kluwer Acad. Publ., Dordrecht. Peregrin, J. (1997): Language and its models, Nordic J. Philos. Logic 2: 1–23. Peregrin, J. (1998): Linguistics and philosophy, Theoretical Linguistics 24: 245–264. Peregrin, J. (1999a): The pragmatization of semantics, in K. Turner (ed.), The Semantics/Pragmatics Interface from Different Points of View, Elsevier, Amsterdam, pp. 419–442. Peregrin, J. (ed.) (1999b): Truth and its Nature (if Any), Kluwer Acad. Publ., Dordrecht. Poincaré, H. (1908): Science and method (English translation), in Foundations of Science, Science Press, New York, 1921, pp. 359–546. Putnam, H. (1985): A comparison of something with something else, New Literary History 17: 61–79. Putnam, H. (1994): Rethinking mathematical necessity, in J. Conant (ed.), Words and Life, Harvard University Press, Cambridge, MA. Quine, W. V. O. (1960): Word and Object, MIT Press, Cambridge, MA. Reichenbach, H. (1920): Relativitätstheorie und Erkenntnis Apriori, Springer, Berlin. Russell, B. (1905): On denoting, Mind 14: 479–493. Sellars, W. (1963): Science, Perception and Reality, Routledge, New York. Shapiro, S. (1996): Space, number and structure: A tale of two debates, Philos. Math. 4: 148–173. Stekeler-Weithofer, P. (1986): Grundprobleme der Logik, de Gruyter, Berlin.
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Stekeler-Weithofer, P. (1992): Hegels Analytische Philosophie, Schöningh, Paderborn. Stekeler-Weithofer, P. (1994): Ideation und Projektion, Deutsche Zeitschrift für Philosophie 42: 783–798. Tarski, A. (1932): Der Wahrheitsbegriff in den Sprachen der deduktiven Disziplinen, in Akademischer Anzeiger der Akademie der Wissenschaften in Wien 69; English translation ‘The Concept of Truth in Formalized Languages’ in Tarski (1956), pp. 152–278. Tarski, A. (1944): The semantic conception of truth, Philosophy and Phenomenological Research 4: 341–375. Tarski, A. (1956): Logic, Semantics, Metamathematics, Clarendon Press, Oxford. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford; English translation Philosophical Investigations, Blackwell, Oxford, 1953. Wittgenstein, L. (1956): Bemerkungen über die Grundlagen der Mathematik, Blackwell, Oxford; English translation Remarks on the Foundations of Mathematics, Blackwell, Oxford, 1956.
Institute of Philosophy, Academy of Sciences, Jilská, 11000 Praha, Czech Republic
‘Fregean’ Logic and ‘Russellian’ Logic Jaroslav Peregrin* www.cuni.cz/~peregrin Australasian Journal of Philosophy 78, 2000, 557-575
Summary. I do not think there is one true answer to the question What is logic?. There are, clearly, good and less good answers, and there are answers which are plainly wrong; but the term ‘logic’ has been employed, throughout the history of the subject matter, in such diverse ways that no single one of the uses can be said to be the correct one. However, even among the answers which are acceptable on historico-semantical grounds there are still, without doubt, good and less good ones, in the sense of more and less useful. In this paper, I will argue for a certain, rather narrow conception of logic; and I am going to argue that it is not only an acceptable answer, but also one which is more useful and fruitful than its alternatives. I will argue that when setting the agenda for logic we must keep ourselves grounded; for, as I will try to indicate, it was precisely a down-to-earth conception of logic which underlay the jump start into the era of modern symbolic logic that occurred in the late nineteenth century, notably within the work of Gottlob Frege. I will compare his notion of logic with some rival ones and aim to show that the alternatives are either wrong or unmanageable.
1. Frege’s Begriffsschrift Philosophers have always dreamt of a language which would be more suitable for the purpose of solving their problems (or maybe even all problems) than the ordinary language with which God or Nature equipped us. Many of them suspected that philosophical problems arise partly or wholly due to the fact that our ordinary language does not allow us to express our ideas and thoughts precisely enough – and that all would improve if we had a language which would be in perfect accordance with our thinking and/or our world. According to this view, the basic requirement of a philosopher (or also of a scientist) is a kind of ‘alphabet of human ideas’ which would enable them ‘through the connection of its letters and the analysis of the words, which consist of them, to discover and to assess everything else’ (Leibniz). However, the creation of such an alphabet presupposes a small ‘detail’: it is necessary to collect and classify all our thoughts and ideas, to find out which of them are not adequately expressed in our language and which are expressed somehow distortedly, and to clear away these shortcomings. The problem is, of course, that nobody had any idea how to achieve this, viz how to get hold of ‘ideas’ by-passing words expressing them. Thus, Leibniz’s project of a calculus ratiocinator, analogously with similar proposals, has remained a mere utopian ideal. Nevertheless, today we do have something at least partly resembling Leibniz’s ‘calculus of rationality’: we have the symbolic languages developed by logicians. True, they are far cry from a tool facilitating the immediate solution or dissolution of philosophical problems; it is, however, undeniable that in the case of many philosophical (and scientific) problems they have helped us if not solve them, then at least to make them more perspicuous or to gain new, helpful *
Work on this paper has been supported by a research grant of the Grant Agency of the Czech Republic No. 401/99/0619. I am grateful to Timothy Childers for helpful critical comments.
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insights into them1. What has lead us, philosophers of the twentieth century, to this partial success – when so many of our predecessors failed? I think that it is now generally accepted that the man to whom we are most indebted is Gottlob Frege, and I am going to argue that what is especially important in this respect is what I would like to call Frege’s down-to-earthness. The point is that when Frege launched his investigations, which resulted in his Begriffsschrift and later in his subsequent writings, he had in mind no such magnificent and all-embracing aim as setting up the ‘alphabet of human thought’: his primary goal was relatively humble and modest, namely to contribute to the possibility of articulating mathematical proofs with such precision and clarity that no doubts about their validity could arise. However, as it turned out, less may sometimes be more: it was precisely the modesty of his goal which enabled him to draw the project of making human language rigorous down from heavenly heights to the realistic earth. What is a mathematical proof? Basically, it is a means of demonstrating the validity of a mathematical theorem; where a mathematical theorem is a claim that whenever some premises are fulfilled, then a conclusion is also bound to be fulfilled. A mathematical theorem thus claims that a conclusion is a consequence of some premises (where the set of premises may, of course, be empty); and a proof is a demonstration of the fact that a statement follows from a list of statements. How can we demonstrate that something follows from something else? How can we demonstrate that the statement ‘Prague is in Europe’ follows from the statement ‘Prague is in Europe and Tokyo is in Asia’? In fact in no way at all: should somebody doubt this, we would probably conclude either that he simply does not know enough English, or that he is insane; in any case that we are not communicating with him. However, what about the fact that the statement ‘z is a number divisible by six’ follows from ‘x is a number divisible by two’, ‘y is a number divisible by three’ and ‘z is the product of x and y’ (i.e. that the product of a number divisible by two with a number divisible by three is divisible by six)? If somebody did not know, or doubted, this, we would not necessarily take him to be a lost case; we might feel that we could demonstrate this to him. What we would say to him in such a case would be, roughly, the following: ‘That x is divisible by two means that there is a number n such that x = n×2; that y is divisible by three means that there is an m such that x = m×3. But as z = x×y, it is the case that z = (n×2)×(m×3). And this further means that z = (n×m)×(2×3), i.e. that z = (m×n)×6. However, this is nothing else than that z is divisible by 6.’ What did our demonstration of the validity of the instance of consequence in question consist in? Its point clearly was in decomposing the instance into simpler steps: to show that S is the consequence of S1, S2 and S3 we presented a sequence S1, S2, S3, ... S such that every Si (for i = 4, ...n) is a consequence, and an obvious one2, of S1, ..., Si-1; and that Sn = S. No reasonable being can fail to see both the validity of the individual steps and the fact that they add up to the original instance of consequence to be proved. A typical proof thus consists in the decomposition of an instance of consequence into a chain of obvious instances of consequence – what makes it possible is the fact that some instances of consequence are more obvious than others. 1
There were (and indeed there still are) philosophers who believed that formal logic can actuallysolve (or dissolve) all philosophical problems (perhaps Rudolf Carnap in his Vienna Circle period). However, I am afraid that this attitude has done the interaction between logic and philosophy more harm than good.
2
Note that this requirement makes the definition different from the usual formal definition of a proof.
2
Frege wanted to develop a method enabling us immediately and unambiguously to recognize the validity of a given proof; he wanted to assemble a manual of elementary logical transitions which are capable of constituting the basic steps of proofs – so that a proof could be checked simply by comparing its steps with this manual. (As I have noted elsewhere3, what he was after was an inventory of acceptable elementary logical transitions analogous to the inventory of reprehensible deeds assembled in the laws.) Doing this he realized that the task could be effectively accomplished only if he made language more precise, in the sense of ridding it of everything which is not substantial from the viewpoint of consequence. In natural language, the same thing can be expressed in many different ways and our talk can be augmented by many things which do not influence the ‘judgeable content’ – and it is thus necessary to distinguish between matters of mere stylistic variation and the real differences in content. Notice that in this phase, Frege’s efforts are no longer so distant from what Leibniz talked about: we are heading for a language containing only those linguistic means which really do express content; those which have other functions and could thus mislead us are dispensed with. However, Frege, unlike Leibniz, now has a method to tell these two kinds of means apart – the criterion is the nontriviality of ‘inferential role’. Frege states that ‘die Inhalte von zwei Urtheilen in doppelter Weise verschieden sein können: erstens so, dass die Folgerungen, die aus dem einen in Verbindung mit bestimmten andern gezogen werden können, immer auch aus dem zweiten in Verbindung mit denselben andern Urtheilen folgen; zweitens so, dass dies nicht der Fall ist’4 ([7], p. 2-3). For two judgments S and S’, to differ in the former way means that for all judgments S1, ..., Si-1, Si+1, ..., Sn, Sn+1 it holds that Sn+1 follows from S1, ..., Si-1, S, Si+1, ..., Sn, if and only if it follows from S1, ..., Si-1, S’, Si+1, ..., Sn; whereas the judgments S and S’ differ in the latter way iff there exist some S1, ..., Si-1, Si+1, ..., Sn, Sn+1 such that Sn+1 follows from S1, ..., Si-1, S, Si+1, ..., Sn, but not from S1, ..., Si-1, S’, Si+1, ..., Sn. The ‘judgeable content’ is thus, for him, precisely that part of content which is shared by any two judgments which differ at most in the first way; and it is merely this part which is to be expressed by the ‘concept script’. Thus ‘alles was für eine Richtige Schlussfolge nöthig ist, wird voll ausgedrückt; was aber nicht nöthig ist, wird meistens auch nicht angedeutet; nichts wird dem Errathen überlassen.’5 (ibid, p. 3) And it is his concentration on inferential behavior that enables Frege to dispense with everything which is not substantial from the viewpoint of consequence and to let the ‘substantial backbone’ shine – which then leads not only to an effective rendering of a criterion of consequence, but also to the materialization of the unity of sense within the multiplicity of surface forms. Thus, whereas before Frege it seemed that devising a perfect language, assembling a Leibnizian ‘alphabet of human thoughts’, would presuppose gripping raw ideas or thoughts, comparing them with possible expressive means and choosing the most adequate ones, from Frege’s considerations there emerges a quite different methodology6. We have to erase all 3
See [24], §2.1.
4
‘The contents of two judgments may differ in two ways: either the consequences derivable from the first, when it is combined with certain other judgments, always follow also from the second, when it is combined with these same judgments, or this is not the case.’
5
‘Everything necessary for a correct inference is expressed in full, but what is not necessary is generally not indicated; nothing is left to guesswork.’
6
Compare this situation with the situation of someone who wants to check the truth of some statements about the world. What seems to be needed is a comparison of the statements with the raw
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differences between expressions which do not influence the expressions’ ‘inferential properties’. Needless to say that in contrast to the previous one, this methodology is manageable; and thus it constitutes a real breakthrough towards assembling a ‘perfect’ language. Hence the brand new period in the development of logic (with its profound influence on philosophy) which Frege’s approach initiated is the result of precisely the reasonable ‘down-toearthness’ of his original aim and of the ensuing relatively narrow construal of the concept of logic, a construal of logic as basically a matter of a canonization of consequence. I think that this indicates that it is this narrow delimitation of logic which we should stick to.
2. Morals of the Fregean approach What is distinctive of this conception of logic? Besides the issues discussed in the previous sections, I would like to stress two crucial points: first, according to the Fregean approach, the primary subject matter of logic are not objects and their properties as constituting the world, but rather propositions as constituting the ‘logical space’ of inferential relationships; second, propositions are to be approached via the sentences which express them. The substantiality of the move from considering relations among objects (particulars) to those among propositions may not be immediately apparent; but it is, as far as I can see, the very crux of Frege’s approach to logic. Frege himself explicitly urges the point in an essay called ‘Booles rechnende Logik und die Begriffsschrift’ ([8]), in which he compares the merits of his Begriffsschrift with those of Boole’s Algebra of Logic. Frege points out that what is distinctive of his approach (as contrasted to that of Boole, which he considers as a direct continuation of the project of Leibniz), is that he does not proceed from concepts to judgments, but rather the other way around. ‘Im Gegensatz zu Boole,’ writes Frege (p. 17), ‘gehe ich von den Urteilen und deren Inhalten statt von den Begriffen aus. ... Das Bilden der Begriffe lasse ich erst aus den Urteilen hervorgehen.’7 And he continues (p. 18): ‘Statt also das Urteil aus einem Einzeldinge als Subjecte mit einem schon vorher gebildeten Begriffe als Praedicate zusammen zu fügen, lassen wir umgekehrt den beurteilbaren Inhalt zerfallen und gewinnen so den Begriff.’8,9. Why did Frege choose this approach? The crucial point seems to be that if we want to explicate concepts independently of judgments, we have no feasible criteria of individuation, and hence no feasible way to grasp concepts as real, demarcated entities. We would be left with conceiving concepts as something mental and to seek criteria of their individuation in reality which they claim to capture and to find out whether the two things are in accordance. However, what could ‘accordance’ between the string of letters ‘There are Elephants in Africa’ and the African elephants consist in? Surely not in a similarity akin to that which obtains between, say, a thing and its image in a mirror! In this way, the seemingly straightforward correspondence theory of truth gives way to various other accounts of truth which do not assume that we could get hold of a raw reality to be checked for being correctly mirrored by our statements. 7
‘Unlike Boole, I start out from judgements and their contents, and not from concepts. ... I let the building of concepts proceed from the judgements.’
8
‘Instead of putting a judgement together out of an individual as subject and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of a possible judgment.’
9
Cf. [1], §2.II.2.
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psychology – which would cause us to erase the boundary between concepts (Begriffe) and mental presentations (Vorstellungen), and this would, in turn, lead us to erasing the boundary between what is really the case and what is thought or imagined to be the case. In contrast to this, we do have a relatively clear criterion of individuation for propositions: propositions are ‘judgeable contents’ (‘beurteilbare Inhalte’), and as such they are ‘that which is shared by all sentences not differing by their inferential properties’. Thus, Frege’s strategy appears to be the following: First, we must study and systematize correct inferences. Then we can divide differences between sentences into those which are, and those which are not, inferentially significant. Thereby we establish an equivalence relation between statements (‘not to differ significantly from the viewpoint of inference’) and we can proceed, via abstraction, to (judgeable) contents of statements – such judgeable contents are what Frege called thoughts and what we call propositions. And then we can decompose thoughts into unsaturated functions and their saturated arguments, objects (by means of ‘subtracting’ the objects), thus gaining concepts understood as functions mapping objects onto truth values. The bite of Frege’s conception can be appreciated if we consider, e.g., his quarrel with Husserl about the nature of number concepts10. What Husserl proposed was a conception of numbers which was of a piece with his overall phenomenological inclination: a number is, according to him, to be abstracted from our presentations of ‘sets, multiplicities and definite objects’ ([19], p. 9) and is thus based on ‘elementary psychic data’ (ibid., p. 131). Frege’s disagreement with such an approach stemmed not only from his general disagreement with any attempts to base logic and mathematics on facts of psychology (and in fact here Husserl’s defense could have been that what he had in mind was transcendental psychology), but especially from the fact that Husserl’s attitude does not provide for the vital distinction between what really is the case and what somebody takes to be the case, i.e. between ‘being true’ and ‘being taken as true’. For Frege any logical consideration importantly rests on the concept of objective truth; and hence it presupposes entities to which the concept applies, viz judgments or thoughts. As he puts it in an overview of his approach to logic: ‘Das Eigenartige meiner Auffassung der Logik wird zunächst dadurch kenntlich, dass ich den Inhalt der Wortes ‘Wahr’ an die Spitze Stelle, und dann dadurch, dass ich den Gedanken sogleich folgen lasse als dasjenige, bei dem Wahrsein überhaupt in Frage kommen kann.’11 ([10], p. 273) And as he adds in another posthumously published paper: ‘Eine Erkenntnis kommt dadurch zustande, dass ein Gedanke als wahr anerkannt wird. ... Als Erkenntnisquelle sehe ich das an, wodurch die Anerkennung der Wahrheit, das Urteil, gerechtfertigt its.’ ([11], p. 287)12. Thus, what, according to Frege, establishes the objectivity of a concept (e.g., a number concept, like three) is the objectivity of the truth values of sentences or judgments containing the corresponding word (‘three’). We all agree that a sentence like ‘Venus has three moons’ has a definite truth value independently of whether anybody thinks it is true or false. (Our agreement is a matter of the fact that our language is governed by certain rules, to which all of 10
For a detailed discussion of this controversy see [6].
11
‘What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word ‘true’, and then immediately go on to introduce a thought as that to which the question ‘Is it true?’ is in principle applicable.’
12
‘When someone comes to know something it is by his recognizing a thought to be true. ... What I regard as a source of knowledge is what justifies the recognition of truth, the judgement.’ For a thorough discussion of this point see [34].
5
us, its speakers, subject ourselves to be able to communicate.) The concept three is what remains when we subtract (in modern terminology: ‘lambda-abstract’) from a thought expressed by such a sentence everything save the part corresponding to ‘three’13. Note that this strategy is based not only on the acknowledgment of the primacy of the propositional, but also on the assumption that propositions are inseparably connected with sentences which express them – that a thought is always the sense of a sentence. The reason appears to be that propositions are individuated in terms of inferences, and inferences are relations between (meaningful) sentences. Hence we can get a grip on propositions via gripping sentences in their inferential relationships; and there is no other way. Thus, Frege sowed the seed of the linguistic turn with its conviction that ‘a philosophical account of thought can be attained through a philosophical account of language, and ... that a comprehensive account can only be so attained.’ ([4], p. 5). It may be helpful to elucidate the idiosyncrasy of propositions from a different angle, by briefly discussing Wilfrid Sellars’s attack on the doctrine of the traditional empiricism (see [31]). What Sellars pointed out was that the doctrine’s apparent acceptability depended upon its proponents ignoring the distinction between particular objects and propositions. The empiricist picture criticized by Sellars is based on the idea that the outer world impinges on a subject producing various sorts of ‘sense data’, which then constitute the most basic and infallible layer of the subject’s knowledge, upon which she may build further, non-immediate layers. Thus, rays of light reflected from a green tree make the subject see that there is a green tree before her, and from this ‘perceptual knowledge’ she can further infer, e.g., that there is something which is green before her. This picture, Sellars argued, presupposes that there is a causal chain leading from an external source to a sense organ of the subject in question and which is then continued, ‘within’ the subject, by a justificatory chain leading to the ‘less immediate’ varieties of knowledge. What Sellars pointed out was, in effect, that causal chains and justificatory chains inhabit different spaces (the first of them belonging to the spatiotemporal world, the other to the ‘logical space of reasons’) and hence cannot be made continuous with each other. In other words, if we perceive a green tree, then the endpoint of the relevant causal chain is a perception of the green tree (a kind of constellation of something within our eye and/or brain), whereas the starting point of the relevant justificatory chain is the belief that there is a green tree. And whereas we can have a perception of a green tree without possessing the concepts green and tree (thus in this sense not knowing that what we perceive is a tree and is green), we cannot have the belief. Considerations of this kind made Brandom ([1]) count Sellars as an ally of Frege, and both of them as followers of Kant, in regarding propositions as a crucial sui generis which is more basic than that of concepts. Moreover, Brandom argues that it is precisely propositional knowledge which is crucially characteristic for rationality of the distinctively human kind. To be rational in this sense is to have beliefs, desires etc. and to act according to them; and beliefs are propositional in nature. As Brandom would put it, rational agents are first and foremost those who are able to give and ask for reasons, and reasons are what can figure within inferences, i.e propositions. ‘Behavior is made intelligible,’ Brandom writes (p. 83), ‘by exhibiting it as rational, given various beliefs and pro-attitudes, and to do that is to exhibit a piece of practical reasoning that is taken somehow to stand behind or be implicit in the behavior. The imputed reasoning shows why an organism with the states and attitudes that 13
Frege then argues that in this case what we have to substract is a concept, perhaps ‘the moon of Venus’, so that the entity expressed by ‘three’ is a second-order concept.
6
provide the premises ought, rationally, to behave in the way specified by the conclusion. But what can serve as a premise in reasoning must have a propositional content.’ Objects are, in the typical case, entities which we can see, touch, or kick (in short causally interact with); propositions are those which we can know, deny or infer from each other. Thus whereas the modus vivendi of (spatiotemporal) objects is causal interaction, the modus vivendi of propositions are logical relationships – nothing can be a proposition unless it can be negated, conjoined with other propositions, etc14. Propositions are creatures and vehicles of reasoning and hence should be in the primary focus of logic. And the way to get a grip on them is to investigate the overt tools of reasoning, statements – propositions are what emerges when we rid statements of everything irrelevant from the viewpoint of inference.
3. The Principal Alternative: The ‘Russellian’ Notion of Logic The most common alternative to this understanding of logic is the conception according to which logic captures the most general traits of the world, especially the boundaries of what is possible within the world. According to this conception, the truths of logic are about the world and about the objects to be encountered within the world in the same way in which the truths of natural science are – the only substantial difference being a matter of their generality. This means that unless we want to see the truths of logic as contingent, we have to assume that truths about the world which are spelled out by logic are, in contrast to those spelled out by natural science, somehow so general that they are no longer contingent, but become necessary. This is the conception of logic put forward, within the post-Fregean context, most illustriously by Bertrand Russell: ‘[L]ogic is concerned with the real world just as truly as zoology, though with its more abstract and general features’ ([30], 169-70). What can be said about the object/proposition dichotomy from this viewpoint? Well, for Russell, there are really no propositions in our sense (he uses the term ‘proposition’ for statements, the linguistic entities). The only entity which is semantically relevant for a statement is a certain fact: the statement is true if the fact is present, whereas it is false if it is absent. Facts are kinds of conglomerates of objects which, ‘just as much as particular chairs and tables, are part of the real world’ ([29], p. 42). Thus, the fact that a tree is green appears to be a specific kind of complex object, somehow consisting of the tree and greenness. Russell (ibid., p. 80) claims: ‘... I should always wish to be engaged in the study of some actual fact or set of facts, and it seems to me that that is so in logic just as much as in zoology’. The idea behind such a construal of logic is that whereas a statement like ‘The king of France is bald’ may be true or false depending on the actual status of a relevant part of the world (i.e. on the presence, respectively absence, of the fact made up of the king of France and baldness), when we move to more general statements we ultimately reach those which are so general that they no longer concern only a specific part of the world, but somehow the world as a whole, and thereby they lose their contingency. The example of such a statement is ‘If one class is part of another, the term which is the member of the one is also a member of the other’. Russell (ibid., p. 43) describes the situation as follows: ‘There are facts concerning particular things or particular qualities or relations, and, apart from them, the completely general facts of the sort that you have in logic, where there is no mention of any constituent
14
See [27].
7
whatever of the actual world, no mention of any particular thing or particular quality or particular relation, indeed strictly you may say no mention of anything’. Since the establishment of model theory in the modern sense, this approach to logic has tended to be slightly mutated (reflecting the shift from what van Heijenoort calls ‘logic as language’ to what he calls ‘logic as calculus’15). According to this modified version of the conception, logic reports to us what holds in every member of a certain class of formal structures, which represent all and only possible states of our world (see [5]). This seems to render Russell’s ‘more abstract and general’ as ‘universally valid’. That this is a notion of logic taken as virtually self-evident during recent decades (especially among mathematicallyminded logicians) is documented by the fact that Kreisel’s replacement of the problem of the relationship between logical validity in the intuitive sense and ‘model-theoretic validity’ with the problem of the relationship between the truth w.r.t. all possible model structures and the truth w.r.t. the restricted class model structures worked with by model theory16 has been almost universally accepted as an unproblematic move. I think that this way of understanding logic, although it might seem prima facie plausible, is in fact plainly untenable, as it stands, for the simple reason popularised long ago by David Hume. The reason is that we can report what is the case, but not what must be the case, nor what cannot fail to be the case. These are simply not the kinds of things which can be reported – however many times we see something happen or be the case, we cannot be sure that it is bound to happen or is necessarily the case. So the idea that proceeding from the usual, observable facts towards ever greater abstractness and generality will ultimately lead us to some kind of facts whose superior abstractness and generality secure for them necessity and inevitability is not really plausible. Transposed into the modern, ‘model-theoretical’ setting, the Humean line of thought suggests that we can never find out, by observing the world, whether a proposed class of model structures really represents all its possible states. If Sher ([33], p. 139) notes that ‘Tarski has never shown that the set-theoretic structures that make up models constitute adequate representations of all (formally) possible states of affairs’, then our point is that the only way to show this would be to show that the class of structures does justice to what is logically true – and consequently that explaining logical truth as that which holds in all of them would be circular17. To make this point obvious, suppose somebody asks how we know that a statement, say, ¬(P(a)&¬P(a)), is logically true. Surely our answer cannot be ‘It holds in all (the model structures capturing) the possible states of the world – I have gone through them all and have not encountered a single one in which it would not hold’; it would have to be something like ‘a thing simply cannot be P and simultaneously not-P’, or perhaps ‘[“]to be P and not-P[”] makes no intelligible sense’. This indicates that not ‘a logical truth is true because it is valid in all possible structures’, but rather ‘because something is a logical truth, there cannot be a structure in which it does not hold’. (It is true that we can sometimes discover that something holds in all structures of a certain class – but unless the class is finite, we can hardly do so by
15
See [15].
16
See [22]. The difference, in Kreisel’s view, lies in the fact that among the model structures with which model theory normally works there is, for example, no one with the whole set-theoretical universe as its carrier.
17
Cf. [24], Chapter 4.
8
going through all the structures; we have to somehow deduce it from some properties constitutive of the very class18.) This indicates that the ‘Russellian’ notion of logic, if it is to be minimally feasible, has to be modified in the sense that the truths of logic are somehow a matter of what is ‘within us’, of what is somehow ‘imposed’ on the world by us. This leads to the well-known Kantian response to the Humean challenge: We can know what is necessary within the world because the necessities somehow stem from us, are somehow a matter of ‘the structure of our epistemological apparatus’. This permits us to save a part of the original intuition constitutive of the ‘Russellian’ conception of logic; however, it also demands its significant modification. The modified conception claims that logic reports the most general traits of the world as we think it. However, this clearly cannot be taken to imply that logic is simply a matter of studying our actual thinking (or cognition) – the case against all kinds of psychological construals of logic was made so vehemently by Frege and by many others since, that there is, I hope, no reason to repeat it here (cf. the previous section). Hence again, to make this construal of logic feasible, we must accept that what it investigates somehow ‘transcends’ our actual thinking, that it is a matter of the boundaries of what we can and what we cannot think. However, even this conception of logic may still be subject to serious objections. The principal one is, I think, that which can be found sketched in Wittgenstein’s Tractatus: it is the objection that we cannot think about (let alone form a theory of) what we cannot think19. How can we see something as a boundary without being able to conceive of the outside of the boundary? It seems that with this conception of logic we would need to see the boundaries of thinking as something which we can, but at the same time cannot surpass20. I think that it is the concept of rule which helps us overcome this difficulty (and this is also the reason why the concept plays such a crucial role within Wittgenstein’s later philosophy). A rule is something which can be, but at the same time ought not to be, violated. A rule draws a boundary, but not one which is utterly unsurpassable: the boundary drawn by a rule can be surpassed – even though at some cost. We can violate the rules of chess – at the cost of ceasing to play chess. However, the straightforward embedding of the concept of rule into the modified ‘Russellian’ conception of logic still yields a picture which is questionable: namely to the notion of logic as the theory of the rules of correct thinking, i.e. of how we ought to think. This notion takes logic to be a kind of specification of what is to be happening within our heads if we are to ‘think correctly’. However, I think that the Fregean arguments against the psychological construal of logic extend even to this case: logic is not about what is going on 18
Notice that this is not an argument against the ‘platonistic’ construal of mathematics. Even if we granted that facts about a mathematical structure of the kind of those employed by model theory can be reported using an ‘inner eye’ in the same way in which we report facts about the empirical world using the real eye, it would still not follow that it would be possible to report what holds in every member of an infinite class of such structures.
19
See Wittgenstein ([35], §5.61): ‘Was wir nicht denken können, das können wir nicht denken; wir können also auch nicht sagen, was wir nicht denken können.’ [‘What we cannot think, that we cannot think: we cannot therefore say what we cannot think.’]
20
Somebody could try to overcome this difficulty by appealing to some kind of ‘metalevel thinking’ from which we somehow can cross the boundaries not crossable within the ‘object-level thinking’. (Cf. Kleene’s, [21], pp. 2-3, proposal that ‘we simply put the logic which we are studying into one compartment, and the logic we are using to study it in another’.) But this is obviously futile: if we are able to cross them in any sense at all, they are clearly not the real boundaries of our thinking.
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in anybody’s head, not even in this normative sense, i.e. as being a recipe. The reason is, again, that logic is concerned with what is true (it is, of course, directly concerned only with the noncontingent side of what is true, i.e. with necessary truth and with inference), which does not directly depend on any goings-on within a head21. That ‘Someone is bald’ is true if ‘The king of France is bald’ is, is an objective fact independent of the fact of how a real person moves from the knowledge of the truth of the former to that of the truth of the latter. However, at this point the dismantling of the ‘Russellian’ conception appears to be completed – there seems to be nothing more left of it. If what we have claimed so far is right, then logic is best seen, just like Frege urged, as the study of objective inferential relationships between propositions, which result from the rules which govern our way of handling sentences which express them. Viewed thus, logic is the systematic study and ‘canonization’ of inferential rules which are constitutive of the core of our language.
4. Logic as a Matter of Inference Rules However, is the upshot of the previous section that logic is a specific part of linguistics? Does it mean that logical studies are in fact peculiar grammatical studies? Surely not: there is, of course, an important sense in which logic is non-empirical and normative. The first thing to realize is that, as we have stated, what logic canonizes are rules. This means that it does not merely spell out regularities of the way people use language; what it spells out are, as Brandom ([1]) puts it, proprieties22. And logic does not put them forward as mere ‘linguistic’ reports of what is held to be correct (‘The speakers of English take it to be correct to infer “Someone is bald” from “The king of France is bald”’), it puts them forward as claims with genuine normative import (‘It is correct – for us speakers of English – to infer “Someone is bald” from “The king of France is bald”’.) Moreover, logic deals with the core of the inferential structure of language, which is its target, as with a rigid, unchanging structure. This is what makes it possible to apply mathematics to logical investigations, to have a ‘mathematical logic’. Thus we can say that logic addresses a ‘mathematical’ structure which is – which happens to be, we can say – embodied – imperfectly, as the case may be – by the inferential structure of the core of our language. (This is, in itself, no specialty of logic – consider, e.g., geometers studying
21
It is, of course, indirectly dependent on ‘what is going on in heads’ in the sense that if nothing went on in any head, there would be no thoughts and no sentences to be true, and hence there would be, in this sense, no truth. Therefore Frege ([9], §26) says: ‘So verstehe ich unter Objektivität eine Unabhängigkeit von unserem Empfinden, Anschauen und Vorstellen, von dem Entwerfen innerer Bilder aus den Erinnerungen früherer Empfindungen, aber nicht eine Unabhängigkeit von der Vernunft.’ [‘It is in this way that I understand objective to mean what is independent of our sensations, intuition and imagination, and of all construction of mental pictures out of memories or earlier sensations, but not what is independent of the reason.’]
22
Of course this presupposes a theory which renders the rules of human linguistic conduct as more than acknowledged regularities – but such theory indeed can be found in Brandom’s book. We should also note that the proprieties in questions are again ultimately based on some regularities, viz on regularities of what Brandom calls normative attitudes, i.e. of the ‘takings as (in)correct’ of the linguistic behaviour of one’s fellow speakers and herself.
10
geometrical forms ‘imperfectly embodied’ by things of our world) 23. In this way it produces claims about this structure which are mathematically certain – but these claims are to be understood not directly about the language, but rather about a formal prism which is taken to be a ‘reasonable reconstruction’ of the language and which is indeed used as an ‘ideal norm’24. What is important is that the kind of structure which is studied by logic appears to be so essential for our language that it would not make sense to use the term ‘language’ for anything which lacks it. (In particular, the structure appears to have to be embodied by anything capable of serving as a means of communication and ‘information exchange’.) Note that this is nothing else than what characterizes other words of our vocabulary: we would not call a ‘car’ or a ‘crocodile’ anything which does not display the most essential features characteristic of those things for which the names were introduced. This observation also yields an answer to the question of whether on this construal of logic we are perhaps bound to have merely a logic of English, a logic of German etc., and no logic simpliciter. Our logic expresses the normative structure constitutive of our language, but thereby the one which is bound to be embodied by all languages worth the name. We simply use the term ‘language’ for certain kind of entities; and one of the criteria of calling something ‘language’ is that it shares the basic structure we know from our language – similarly as the criterion of calling an alien entity a ‘car’ or a ‘crocodile’ is that it is close enough to our cars or our crocodiles. This means that the fact that all languages must share a basic logical structure is not a fact of metaphysics, but rather a fact of semantics, concerning the meaning of the word ‘language’25. To elucidate the consequences of our proposal, let us discuss the claim of Nagel ([23], p. 38-39) to the effect that logic cannot be extracted from the grammar of our language: ‘To the extent that linguistic practices display principles of reasoning or show us, for example, something about the nature of arithmetical propositions, it is not because logic is grammar but because grammar obeys logic. No “language” in which modus ponens was not a valid inference or identity was not transitive could be used to express thoughts at all.’ Does our construal of logic result into the claim that ‘logic is grammar’? To enable us to answer this question, let us first consider Nagel’s latter claim. What does it mean that ‘modus ponens is a valid inference in a language’? Is modus ponens a valid inference in English? To answer this question, we have to specify what is to be understood as implication in English, for modus ponens tells us that an implication together with the antecedent of the implication entail the consequent of the implication. Could we simply identify implications with sentences of the shape if A, then B? It seems that if we did so, then we would have to admit that modus ponens in fact is not valid in English – for there are 23
This account of the role of mathematics in logic coheres, in my view, with the overall characterization of the role of mathematics within our coping with the world as put forward by Kitcher ([20]).
24
See [26]. In this paper I also urge that it is futile to conceive of logic as being about a non-empirical reality – as being the inquiry into a world of eternal propositions independent of the fact whether these propositions happen to be expressed by sentences of a language. I argue that the value of studying any ‘pure’ mathematical structure is in that it is a structure of a thing with which we have to do within our world; and especially that studying inferential relations within a structure of abstract propositions makes (extra-mathematical) sense only if this structure can be used as a ‘prism’ to look at our real language and our real argumentation.
25
It is a consideration of this kind that appears to have led Davidson ([2]) to the conclusion that the very idea of a language untranslatable into ours is incoherent.
11
surely many cases in which a sentence B is not entailed by if A, then B. (Consider. e.g., the cases where the consequent of the if ... then sentence is not a self-contained sentence, like If a farmer owns a donkey, he beats it.26) Moreover, it is unclear what ‘a language in which modus ponens is not a valid inference’ would amount to. Would it be a language containing implication not governed by modus ponens? But then why would we call the connective in question ‘implication’ in the first place? Suppose somebody argued that modus ponens is not valid in English and tried to justify the claim by pointing out that the sentences ‘Paris is in France’, ‘Paris is in France or Paris is in China’ are true, but the sentence ‘Paris is in China’ is false. We would surely protest that ‘or’ is not an implication. However, how else could we justify our protest save by pointing out that the inferential behavior of ‘or’ is different from that of implication – viz that ‘or’ does not obey modus ponens (and other inferential rules constitutive of implication)? So the concept of ‘an implication not obeying modus ponens’ is problematic in itself. Moreover, it follows from our considerations that it may be, more generally, problematic to think about a language not obeying our basic logical principles (such as modus ponens). The point is, as we have seen, not that ‘such “language” could not be used to express thoughts at all’, but that it would be not clear whether we should call such an entity ‘language’. (And this is not a metaphysical pronouncement about the essence of language, but merely a semantic gloss on how we (happen to) use the term ‘language’.) Now if this is right, then Nagel’s crucial verdict, namely that ‘linguistic practices display principles of reasoning ... not because logic is grammar but because grammar obeys logic’, is really not intelligible. From our viewpoint, the question Is logic grammar or does rather grammar obey logic? is simply a bad question – bad in a way analogous to the badness of the question does our world obey geometry or is geometry the (idealized) structure of our world?. The principles we recognize as logical are, as a matter of fact, embodied in our language(s) (although not quite directly, but in the sense that the language(s) can be seen as their imperfect embodiment(s)). It is also true that any language must so embody these principles – for otherwise we would not call it ‘language’ and we would not call its rules ‘grammar’. In this sense ‘grammar obeys logic’. However, the rules of logic are idealized versions of grammatical rules (they regulate what follows from what), and so in this sense, ‘logic is grammar’.
5. The ‘Formality’ of Logic There may seem to be one more source of the idea that logic is ‘transcendent’ to the rules of our language which we have not yet tackled, namely what is usually called the formality of logic. Logic is not concerned with the fact that it is correct to infer ‘X is an animal’ from ‘X is a dog’ or ‘X is a number’ from ‘X is a prime’. Such material inferences, so the story goes, are a matter of the content of our language (in this particular case of the content of the words ‘dog’, ‘animal’, ‘prime’, ‘number’); and hence in this sense are a matter of the ‘grammar’ (if we construe the term so as to comprise semantics) of our language. However, the laws of
26
It might seem that this problem could be dispensed with by some easy gerrymandering: that all that need be stipulated is that If A, then B is an implication (proper) only if both A and B express selfcontained statements. However, what exactly is self-contained? Is the sentence ‘Clinton is the president of the USA’ self-contained? There surely exists more than one Clinton!
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logic are purely formal, they have nothing to do with content and hence nothing to do with ‘grammar’. I think this argument is utterly misguided. What is the form of an expression? Given the normal meaning of the term ‘form’, the form of an expression is that which remains if we abstract from all particular expressions (and indeed, we use ‘form’ in this sense when we speak, e.g. about the ‘subject-predicate form’ of a statement). However, it is clear that in fact all inferences are valid (also) in virtue of the meanings of some expressions involved – none of them is valid purely in virtue of the form (no statement may be a logical truth on the basis of, e.g., the fact that it has a subject-predicate form, or that it consists of two sentences linked together by a connective). Logical inferences are valid in virtue of the meanings of the expressions like ‘and’, ‘or’, ‘something’ etc., others in virtue of the meaning of such expressions as ‘animal’, ‘dog’ etc. On our construal then, logic is formal for it deals with only those norms of our language which cut across all varieties of our (rational) discourse, for they concern the expressions which can be called ‘topic-neutral’. (We can indeed study inferences which are a matter of expressions pertaining to various specific domains, like the inference of ‘X is a animal’ from ‘X is a dog’; but the term ‘logic’ has been simply reserved for the study of the most general ones.) Now the fact is that these norms are presupposed by any kind of discourse which is about something – thus logic, as Quine ([28], p. 52) puts it, ‘has no objects of its own’ (in the sense in which arithmetic has natural numbers and zoology has animals) and in this sense it expresses a form, not a content27. This way of understanding the formality of logic underscores the contrast between the construal of logic proposed here and the ‘Russellian’ construals according to which the laws of logic report some very general facts about the world. Let us consider the argumentation Hintikka ([18]) uses to forward his logical system, the so-called ‘independence-friendly logic’. What Hintikka claims is that quantified formulas of predicate logic spells out, in effect, the ways we can choose elements from the universe: thus e.g. the formula ∀x∃yR(x,y) states that for every x we can choose a y which stands in the relation R to it. Hintikka concludes that any formula is in fact a codification of a game, the moves of which consist in choosing of individuals from the universe. Given this, he asks: is there a sound reason to restrict the games codified to only those games which are expressible by the formulas of the standard predicate calculus? Why accept only games with full information (i.e. those in which all the information about previous moves is available), why exclude other kinds of games; hence why accept only linearly ordered quantifiers and not branching ones28? And as he does not see any such sound reason, he sets up his ‘independence-friendly logic’ which he views as releasing real logic from the unwarranted chains of linearity. 27
Russell ([30], p. 201) famously claimed that ‘there are words which express form’. I think that without further ado this is a bizarre thing to say – I think that words which express form in the normal sense of the word would be not ‘and’ or ‘every’, but rather ‘mould’, or indeed, ‘form’. And I think that the only possible way to give such a claim an intelligible sense is to say that ‘to express form’ is to be understood as ‘to belong to that stratum of our language which is necessiatated by any talk about any objects’.
28
An example of a formula with branching quantifiers would be the formula ∀x∃y R(x,y,u,v), ∀u∃v which is to be interpreted as claiming that for every x there is a y, and independently of it for every u there is a v so that R(x,y,u,v). This is provably expressible by no standard first-order formula.
13
Hintikka’s argumentation is indeed persuasive, however, only provided we agree that the standard logic is about choosing elements from the universe. In this case it seems indeed peculiar that logic is not capable of expressing such things as two mutually independent choices – and the laws of traditional logic would appear as perhaps a haphazard selection of principles characteristic of choosing. However, if we opt instead for the conception of logic which we urge here and which we ascribe to Frege, the situation changes: for those basic laws of logic which were articulated by Frege and which have survived, without substantial changes, to the present, do apear to be the most primitive and elementary instances of inference (and hence possible moves in argumentations and proofs).
6. A Case Study: Frege vs. Dipert The fact that different people understand different things by ‘logic’ of course causes misunderstandings. In particular, those who subscribe to the ‘Russellian’ notion of logic and who thus expect it to yield a ‘super-theory of the universe’ are bound to be disappointed by the results logic in fact achieves. The point is not only that the tasks they expect logic to solve are not acknowledged by those logicians who think about logic in the ‘Fregean’ way, but, more fundamentally, that these tasks are such that logic simply cannot live up to them. As an example of a criticism of the current status of logic based on such a misapprehension, let us discuss the recent paper of Randall R. Dipert ([3]), which diagnoses an overall failure of logic and as a remedy proposes the replacement of logic with graph theory. The crux of Dipert’s argumentation is that logic simply cannot underpin an adequate general theory of the world – for no less than six decisive reasons. However it is not diffecult to see that measured by the Fregean conception of logic his criticism turns out to be misplaced: 1. ‘Aural and visual structures ... do not seem to be in a logical form, even if they can be wrestled into it.’ (ibid., p. 333) Indeed – for logical form is a form of propositions (Fregean thoughts), not of things, like perceptions. Logic, of course, can be used as a framework for constructing theories of structures of things (by introducing new, extralogical constants and new, extralogical axioms); but it would still remain essential to distinguish between the structure of a proposition of such a theory and the structure of a thing described by the proposition. 2. ‘Logical structure is historically associated with highly conceptualized thought and in fact with thoughts that are easily expressed in natural language.’ (ibid., p. 334). Again, it is Frege’s achievement to differentiate the sense of ‘thought’ which is interesting for logic from possible broader senses, in which ‘thoughts’ are a matter for psychology. As Frege tried to show, to trespass into the realm of psychology is fatal for logic. 3. ‘Current logical notations, as linear sequences of symbols or ‘strings’, seem to be irredeemably awkward, or even inadequate, at representing certain quantificational phenomena.’ (ibid., p. 334) This clearly echoes the argument of Hintikka discussed in the previous section; but as we have concluded, this argument is inconclusive. The point is that it is not clear that the inadequacy of logic to express the quantificational phenomena in question must be seen as a deep shortcoming – for, as we have argued above, it is at least dubious whether the phenomena do belong to the ‘logical backbone’ of our language and hence whether it is a basic duty of logic to provide for their direct representation. (This is of course not to say that the choice of the particular inferential patterns which Frege elevated to the cornerstones of our logic would be immune to criticism; it is even not to say that the 14
Hintikkian line of cricicism would be utterly misplaced. It is only to say that it does not in itself provide for a refutation of the Fregean approach to logic.) 4. ‘The notion of a logical individual ... has been enormously problematic.’ (ibid., p. 335) Again, I am afraid that from the Fregean angle there is no such notion: logic simply assumes that there are individuals (in force of there being grammatical subjects), but is (or should be) absolutely neutral to any considerations about the nature of such individuals. (And again, we should not mistake logic for individual substantial theories constructed on top of logic, or within the framework it establishes.) As we have said, logic is characterized by the fact that there are no objects which it could take as its own. 5. ‘Predicate logic does not answer, or even frame the question, for example, of which one-place properties are basic, and which reducible, or which (if any) two-place relations are basic, and so on.’ (ibid., p. 335-336). Indeed; and once we drop the idea that logic should spell out properties or structures of things, we can see no reason why it should answer or frame such questions. (We must beware mistaking logic for the Russellian, essentially philosophical, doctrine of logical atomism, which was, in effect, a pursuit of the ultimate atoms of the world. As Hacking, [14], p. 315, puts it, ‘logic ... should postulate points of convergence or condensation, not atoms.’ See also [25].) 6. ‘Logic gives nonperspicuous accounts of large and important structural features in the world – the organization of the planetary system or of a Ludwig Beethoven symphony.’ Once more, logic in itself does not (or should not) do anything like giving accounts of this kind; but here the author probably does not mean logic itself, but any theory couched in the formal means offered by logic. However, again we have to make the crucial distinctions between things (which can be named and depicted), and propositions (which can be expressed by a statement). That a detailed propositional description of a Beethoven symphony would consist of a large number of atomic propositions and hence would probably be a long conjunction does not seem embarrassing – unless we confuse the description for a picture or a diagram. In summary then, logic, in the Fregean sense of the word which we urge to be the most promising, is not meant to solve the tasks Dipert claims it has failed to solve. However is there anything nontrivial now left for our logic to solve? Have we not reduced it to a mere triviality which really has to do with nothing? Surely not: as pointed out above, the reason why logic appears to be ‘about nothing’ is that it is prior to any ‘about something’, that it articulates those basic structures of our reasoning which enable it to be ‘about something’ in the first place. And I think this articulation is something to achieve. Besides this, it is essential to distinguish between two ways of seeing the world, in a sense engendered by the object/proposition dichotomy urged above. Viewed from one angle, the world is the world of objects in various ways (causally) connected one to another (by means of the various relations which are the subject matter of science). However, seen from another angle, it consists of facts, and as facts are nothing other than true propositions, they are interconnected by logical relations, especially by the relations of entailing, of being a reason for. Thus, on this construal, facts are not simply complex objects belonging to the same world as objects proper: the ‘world of objects’ and the ‘world of facts’ are alternative ways to grasp the same world. Now logic has very little to do with the world seen in the first of these ways – it only prepares (universal) framework for (‘scientific’) theories which do deal with it. It has far more to do with the world seen in the second way; for it spells out its structure. Thus, if we want to see logic as a theory of the world, then we have to see it as a theory of the world in this sense: it can be reasonably seen as revealing the structure only of entities of a very specific sort, viz propositions. To compare logical formulas with paintings, 15
graphs or schemes is like criticizing a hammer on the score that it is not good for extracting aching teeth29.
6. Conclusion We claim that it is reasonable to construe the term logic relatively narrowly, as a theory of (correct) inference (which primarily follows the aim of ‘inventarization’ and ‘canonization’ of the most general and the most elementary steps in inferences). Inference, then, is at best considered as a relationship between sentences (and only via this as a relation between ‘thoughts’ or propositions) – as a relationship which is a matter of the most general norms and rules constitutive of our language (and which are necessarily shared by everything which we would be willing to call ‘language’). We have indicated that this construal of logic is congenial with the approach pioneered by Gottlob Frege. It can be objected that the difference between Frege and Russell has been exaggerated – indeed, Russell, in many respects, is himself surely a smooth continuator of Frege. This is true; and I am not claiming that there is a grave difference between the logical praxis of the two scholars. However, when it comes to the way they understand the praxis, the difference, as I have tried to show, is no longer insignificant; and what is important, amplified through the works of their followers it may become a true source of misunderstandings and misapprehensions.
References [1] Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). [2] Davidson, D. (1974): ‘On the Very Idea of a Conceptual Scheme’, Proceedings and Addresses of the American Philosophical Association 47; reprinted in Davidson: Inquiries into Truth and Interpretation, Clarendon Press, Oxford, 1984, 183-198. [3] Dipert, R. R. (1997): ‘The Mathematical Structure of the World: the World as Graph’ Journal of Philosophy XCIV, 329-358. [4] Dummett, M. (1988): ‘The Origins of Analytical Philosophy I-II’, Lingua e Stile 23, 3-49 a 171-210. [5] Etchemendy, J. (1990): The Concept of Logical Consequence, Harvard University Press, Cambridge (Mass.). [6] Føllesdal, D. (1958): Husserl and Frege: A Contribution to Elucidating the Origins of Phenomenological Philosophy, Norwegian Academy of Science and Letters, Oslo; reprinted in Haaparanta (1994), pp. 3-47.
29
All of this is not to say that the graph-theoretical framework presented by Dipert is not interesting. On the contrary, I think that such mathematical frameworks indeed can help us throw new and interesting light on some old philosophical problems. One example: using the framework, we can translate Quine’s thesis of the inscrutability of translation into the interesting thesis that human languages, seen as networks of expressions interconnected by semantic relations, are graphs which are auto-isomorphic; in the particular case of English there being an auto-isomorphism mapping ‘rabbit’ on ‘undetached rabbit part’ and vice versa.
16
[7] Frege, G. (1879): Begriffsschrift, Nebert, Halle; English translation in van Heijenoort (1971). [8] Frege, G. (1880/81): ‘Booles rechnende Logik und die Begriffsschrift’, printed in Frege (1983), pp. 9-52. [9] Frege, G. (1884): Grundlagen der Arithmetik, Koebner, Breslau; English translation Foundations of Arithmetics, Blackwell, Oxford, 1953. [10] Frege, G. (1919): ‘Aufzeichnungen für Ludwig Darmstaedter’, printed in Frege (1983), pp. 273-277. [11] Frege, G. (1924/25): ‘Erkenntnisquellen der Mathematik und der matematischen Naturwissenschaften’, printed in Frege (1983), pp. 286-294. [12] Frege, G. (1983): Nachgelassene Schriften (ed. H. Hermes, F. Kambartel and F. Kaulbach), Meiner, Hamburg; English translation Posthumous writings, University of Chicago Press, Chicago, 1979. [13] Haaparanta, L., ed. (1994): Mind, Meaning and Mathematics (Essays on the Philosophical Views of Husserl and Frege), Kluwer, Dordrecht. [14] Hacking, I. (1979): ‘What is logic?’, Journal of Philosophy 76, 285-319. [15] Heijenoort, J. van (1967): ‘Logic as Calculus and Logic and Language’, Synthèse 17, 324-330. [16] Heijenoort, J. van, ed. (1971): From Frege to Gödel: A Source Book in Mathematical Logic, Harvard University Press, Cambridge (Mass.). [17] Hintikka, J., ed. (1969): The Philosophy of Mathematics, Oxford University Press, Oxford. [18] Hintikka, J. (1997): Lingua Universalis vs. Calculus Ratiocinator (Selected Paters, vol. 2), Kluwer, Dordrecht. [19] Husserl, E. (1891): Philosophie der Arithmetik, Pfeffer, Halle. [20] Kitcher, P. (1984): The Nature of Mathematical Knowledge, Oxford University Press, New York. [21] Kleene, S.C. (1967): Mathematical Logic, John Wiley, New York. [22] Kreisel, G. (1967): ‘Informal Rigour and Completeness Proofs’, in Problems in the Philosophy of Mathematics (ed. I.Lakatos), North-Holland, Amsterdam; reprinted in Hintikka (1969), pp. 78-94. [23] Nagel, T. (1997): The Last Word, Oxford University Press, New York. [24] Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. [25] Peregrin, J. (1997): 'Language and its Models', Nordic Journal of Philosophical Logic 2, 1-23. [26] Peregrin, J. (2000): The ‘Natural’ and the ‘Formal’, Journal of Philosophical Logic 29, 75-101. [27] Peregrin, J. (to appear): ‘The “Causal Story“ and the “Justificatory Story“’, Proceedings of the workshop on John McDowell’s Mind and World, Pécs, Hungary. [28] Quine, W.V.O. (1995): From Stimulus to Science, Harvard University Press, Cambridge (Mass.). [29] Russell, B. (1918/9): ‘The Philosophy of Logical Atomism’, Monist 28/29, 495-527/3253, 190-222, 345-380; reprinted in and quoted from Russell: The Philosophy of Logical Atomism, Open Court, Chicago, 1985. [30] Russell, B. (1919): Introduction to Mathematical Philosophy, Allen & Unwin, London. [31] Sellars, W. (1956): ‘The Myth of the Given: Three Lectures on Empiricism and the Philosophy of Mind’, in The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minnesota Studies in the Philosophy of Science 1) (ed. H. Feigl 17
and M. Scriven), University of Minnesota Press, Minneapolis; reprinted in Sellars (1963) . [32] Sellars, W. (1963): Science, Perception and Reality, Routledge, New York. [33] Sher, G. (1991): The Bounds of Logic, MIT Press, Cambridge (Mass.). [34] Tieszen, R. (1994), ‘The Philosophy of Arithmetic: Frege and Husserl’, in Haaparanta (1994), pp. 85-112. [35] Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London.
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From: K. von Heusinger & U. Egli (eds.): Reference and Anaphoric Relations. Dordrecht: Kluwer (Studies in Linguistics and Philosophy), 2000.
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This paper discusses the relationship between the concept of reference and that of inference; the point is to indicate that contrary to the usual view it may be good to see the former as “parasitic” on the latter, not the other way around. The paper is divided into two parts. In part one, I give an (unsystematic) overview of the development of logical tools which have been employed in the course of the analysis of referring expressions, i.e. definite and (specific) indefinite singular terms, of natural language. I present Russell’s celebrated theory of definite descriptions which I see as an attempt to explain definite reference in terms of unique existence (and reference in general in terms of existence simpliciter); and I present Hilbert’s ε-calculus as an attempt to explain existence in terms of choice. Then I turn to contemporary, dynamic approaches to the analysis of singular terms and point out that only within a dynamic framework can the Russellian and Hilbertian ideas yield a truly satisfactory analysis of singular terms, and consequently of reference and coreference. I call attention to the fact that current results of formal semantics demonstrate the advantages of viewing singular terms as denoting updates, i.e. as a means of changing the context (information state), and especially that part of the context which I call the individuary. In part two I turn to the discussion of the nature of such explications; especially to the question whether it forces the acceptance of a representational view of language. I answer the question negatively; I deny that we should see discourse representations, information states, or individuaries, which play the central roles within contemporary semantic theories, as descriptions of a mental reality; I try to show that these entities can, and indeed should, be seen as tools internal to our accounts for singular terms’ inferential capacities. Therefore I conclude that we should not take the obscure concept of reference at face value, but rather as parasitic upon the clear concept of inference. 1
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1.1 Russell Some expressions of our language are seen as doing their linguistic jobs by referring to definite things of our world. How do they manage to do this? The classical analysis of definite descriptions (and of the English definite article, which is their linguistic hallmark), as presented by Russell (1905), consists in explicating definiteness in terms of unique existence. To say that the king of France is bald * I am grateful to Klaus von Heusinger and Vladimír Svoboda for valuable comments and criticism. This work was supported by the Grant Agency of the Czech Republic, grant No. 401/99/0619.
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is to say that there is one and only one entity which is the king of France, and that this entity is bald. Thus, the sentence (1) The king is bald. should, according to Russell, be construed as (1’)
9x
(K(x)
8
& B(x) & y (K(y)
!
(y = x)))
Thus, Russell’s claim is that definite singular terms which are not proper names (‘the king’) are not in fact referring expressions at all but that they, when properly logically analyzed, give rise to a certain quantificational structure: a sentence consisting of a definite description and a predicate, according to him, says that there is one and only one object satisfying the description and is such that it has the property expressed by the predicate. This claim means reconstructing the definite article as a “syncategorematic term”, i.e. as something which is not itself a fully-fledged constituent of the sentence. However, if we employ stronger formal means than those entertained by Russell (which were in fact those of the first-order predicate calculus), we can reconstruct ‘the’ “categorematically” (in the sense of granting it its own denotation) without violating the spirit of the Russellian analysis. Thus, helping ourselves to the machinery of lambda-abstraction, we can rewrite (1’) as
9
8
λ f x (K(x) & f (x) & y(K(y) and this further as
9
8
!y
λg λ f x (g(x) & f (x) & y(g(y)
( = x)))(B )
!y
( = x)))(K)(B)
This yields a formula consisting of three parts which may be put into natural correspondence with the three components of the analyzed sentence; the definite article (y = x))), i.e. as a functhus gets formalized as λgλ f x(g(x) & f (x) & y(g(y) tion which takes sets into sets of sets; or, probably more perspicuously, as a relation between sets. The relation holds between two sets iff the first is a singleton and has a non-empty intersection with the second. This is the analysis which has become standard after Montague (1974); and which has given rise to the so called theory of generalized quantifiers (viz Barwise & Cooper 1981). An alternative elaboration of (1’) can issue from the following consideration. First, assume that x(K(x) & y(K(y) (y = x))) is true, i.e. that the extension of K is a singleton. Under such an assumption, if we denote the single element of the extension of K as c, the whole formula becomes equivalent to B(c). Next, assume that x(K(x) & y(K(y) (y = x))) is not true, i.e. that the extension of K is either empty or contains more than one element. Then the whole formula (1’) is patently false, i.e. equivalent to . This means that if we were able to define a (second-order) function F which maps singletons on their single elements, and all other sets on something of which B is inevitably false, we could rewrite (1’) as B(F(K)); or, writing x K(x) instead of F(K), as B( x K(x)).
9
8
8
! ?
!
ι
9
9
8
!
ι
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The most straightforward way to devise such a function would be to stipulate an object of which everything is necessarily false and to let this object be the value of the function for all non-singletons; or, which is the same, to let the function be defined only for singletons, and to stipulate that any predicate applied to a term which lacks denotation yields a false sentence. In the latter way, we would reach an analysis which would be clearly effectively equivalent to the previous one, which treated the definite article as a generalized quantifier. However, a modification suggests itself: we can also stipulate that a predicate applied to a denotationless term yields a sentence which is not false, but truthvalueless – in this way we can clearly accommodate the idea of an existential presupposition associated with a definite description (as urged by Strawson 1950). Anyway, we have seen that Russell himself claimed that a sentence consisting of a definite description and a predicate says that there is one and only one object satisfying the description and such that it has the property expressed by the predicate (viz (1’)). Similarly he claimed that a sentence consisting of an indefinite description (‘a king’) and a predicate says that there is an object satisfying the description and having the property expressed by the predicate; thus, the adequate analysis of (2) is, according to him, (2’). (2) A king is bald. (2’)
9x K x (
( )
& B(x))
In this way, the functioning of a definite singular term gets reduced to the unique existence of the corresponding referent; and that of an indefinite one to existence simpliciter. We have seen that by allowing ourselves more powerful logical means than Russell himself, we can extrapolate Russell’s analysis to explicate definite singular terms via taking the denotation of the definite article to be a function mapping singletons onto their unique elements (and nonsingletons onto some kind of “nothing”).1 However, no analogous straightforward explication is available for the meaning of the indefinite article, and hence for reference in general. 1.2 Hilbert Russell’s treatment of definite descriptions illustrates the intuitive intimate tie between definiteness and choosability: definiteness, i.e. unique existence, turns out to be a matter of unique choosability. This seems to invite generalization: why not see existence in general as choosability in general? The idea is that something exists if and only if it can be chosen (“picked out”); not necessarily by a particular human subject (whose capacities to actually carry out the choice could be limited), but “in principle”, or “by God”. That there is an F means that an F can be chosen. If we render the possibility of choosing as the existence of the corresponding choice function, we can say that the existence of an item is tantamount to the existence of the appropriate choice function: to say that there is an F is to say that there is a choice function which chooses an F. Two kinds of objections can be raised against the identification of the existence of an object with the existence of a choice function choosing the object. First, there is the
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“constructivistic” objection claiming that to be choosable is always more than merely to exist – that claims about the former necessarily violate bivalence, while those of the latter do (ex definitio) not. Then there are the scruples of set-theoreticians cash out the intuition directly by embracing the axiom of choice. The first objection can be turned into a purely terminological matter: to say that something exists is to acknowledge that it exists and in this sense to choose the thing from among other things. Thus, in this sense, if it makes sense to speak about existence of a thing, it makes the same sense to speak about that thing being chosen – although we have to keep in mind the broad (“bivalent”) sense in which the term choice is being used. The second objection invokes the well-known set-theoretical perplexities swarming around the axiom of choice, which we are not going to discuss in this paper.2 One of the possible ways of developing this idea is to stipulate the reduction of the axioms of existence to the axiom of choice: this development was carried out by Hilbert. To see the semantic point of the enterprise, let us consider functions mapping nonempty sets onto their elements: a function f is called a choice function on the set U iff the domain of f is included in the power set of U and f (s) s whenever s is not empty. If M is a model, then the set of all total choice functions on the universe of M will be denoted as CHFM. It follows that if c CHFM and if s is a subset of the universe, then c(s) s iff s is nonempty, i.e. iff there is an element of s. If s is the denotation of a unary predicate F ([[F]]M = s), then c(s) s iff there is an F, i.e. iff x Fx:
9
2
2
9
[[ x Fx]] = 1 iff c([[F]])
2
2
2 [[F]] for an arbitrary c 2 CHF 3
M
If we now understand ε as denoting an arbitrary function from CHFM in such a way that εx Fx denotes the value of ε for the set, and if t is an arbitrary term, then the following formulas will clearly be valid:
! 9 $ 8 $
F(εx Fx) Ft x Fx F(εx Fx) x Fx F(εx Fx)
:
Hilbert (1925) showed that if we accept the first of these as an axiom characterizing the ε-operator, we will be able to prove the other two and hence justifiably reduce quantification (and especially existence) to choice.4 In this way the ε-operator might seem to be capable of explicating existence simpliciter, and consequently the semantics of the indefinite article, in a way parallel to that in which the -operator explicates unique existence and hence the definite article. In one sense, this is indeed the case;5 however, we should notice an essential difference between the two operators: while can be seen as a logical constant denoting a definite function (namely the function which maps singletons on their unique elements and is undefined for non-singletons) and thus furnishing a definite model-theoretical explication of the meaning of the definite article, ε is in fact more like an extralogical constant, ranging over the whole set of choice functions and thus not furnishing any definite explication of the meaning of the indefinite article. Moreover, one of the unmistakable features of the articles is their interplay: if the F follows an F, then the two ι
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noun phrases are in the typical case coreferential; and this is something which is not directly reflected by the Russellian and Hilbertian treatments. Therefore the direct exploitability of the ideas of Russell and Hilbert seems to be limited; and my opinion is that to progress we must “go dynamic” – we cannot have an explication of the meaning of the indefinite article (nor indeed a satisfactory explication of that of the definite one) until we start seeing meanings as “contextchange potentials”. 1.3 Dynamic semantics To do this, let us first return to the Russellian analysis: it is clear that as it stands, it is not adequate to explicate our everyday use of the definite article. Saying the F does not usually involve claiming that there is one and only one F, but rather that there is one and only one “salient” F. With respect to this point two principal candidate ways to amend the analysis seem to emerge. The first possibility is to retain the Russellian analysis as such and to retract the assumption that all evaluations take place with respect to the general, all-embracing universe. This is to assume that the evaluation of a particular sentence may be based on a local, restricted universe, which is the result of the (“pragmatic”) circumstances, in particular of the preceding discourse. Thus, we keep assuming that the sentence ‘The king is bald’ implies that there is one and only one king – not, however, in the general universe, but rather in a local universe determined by the context in which the sentence is being uttered. Hence it implies not that one and only one king exists, but rather that one and only one king is salient. The second possibility then is to assume that the Russellian analysis itself must be amended, that the choice function represented by the definite article is defined not only for singletons, but instead that it is capable of using strategies to successfully pick out an element even from some other nonempty sets. In both cases the analysis rests on one or another formalization of the notion of context. In a context, some objects of the universe are salient, while others are not. The first approximation of the formalization of the concept of context could be thought of as simply a set of objects, a subset of the universe. This kind of context is utilizable by both of the above mentioned strategies: in the first case, we apply the Russellian analysis not to the universe, but rather to the context-delimited subset; in the second case we let the Russellian operator choose not the only element of the extension in question, but the only element which is in the context-delimited subset. Taking the concept of context seriously entails subscribing to some version of the dynamic view of semantics, as proposed by a number of semanticists.6 It leads to reconstructing meaning as resting on (or at least involving) a kind of “context-change potential”,7 of mapping of contexts on contexts. Contexts may be, and indeed are, captured in a variety of ways – for the problems discussed here it is nevertheless vital that a context somehow “contains” a class of (salient) individuals. In this way we can reconstruct definiteness not as presupposing unique existence, but rather as presupposing unique “referential availability”. To sensitize the denotation of a sentence to context, the denotation must become a “context-consumer”, it must become functionally dependent on the context (formal-
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ized in one or another way). For the semantics to become really dynamic, we must turn denotations not only into “context-consumers”, but also into “context-producers” – so that an utterance might consume a context produced by a preceding one. From the point of view of anaphora it is again vital that the context somehow contains individuals – so that one utterance can introduce (raise to salience) an individual, to which other utterances may then refer back. Our current interests lie only in that part of the context which functions as such an individuals-container (which other parts a context can or should have is not a theme for us now) – to have a name for it, let us introduce the term individuary (devised in analogy to the well- established term bestiary). In the simplest case, an utterance introduces a single object by using an indefinite description, and a subsequent utterance uses a definite description, or a pronoun, to pick it up. This is the case of (3) A man walks. He (the man) whistles. For this example, it suffices to construe the individuary as a single slot which can be occupied by an object, or be empty. The phrase ‘a man’ fills the slot with a (“fixed but arbitrary”) individual, and the phrase ‘he’ is then interpreted as referring to this very individual. (There are essentially two ways of formalizing this “fixed but arbitrary”: we can either represent the assignment of such an “arbitrary” object by means of a whole class of assignments of “real” objects, or we can introduce some kind of genuine “arbitrary objects”, pegs, which are capable of being identified with real objects. The former is the way of Groenendijk & Stokhof 1991, the latter is that of Kamp 1981, and also of Groenendijk & Stokhof & Veltman 1996). The situation is more complicated once there is more than one object which is passed from one utterance to another. Take (4) A man meets a woman. He greets her. In this case, when it comes to the anaphoric reference, it is necessary to choose the right referent from among more salient objects. This is not to say that there is always the right choice, but in many cases, such as in this one, there clearly is. This possibility of the right choice implies that the items in the individuary, the salient individuals, have to be in some way characterized, for it is only on the basis of some characteristic specification that we can distinguish between them and carry out the choice. 1.4 The structure of the individuary Usually, approaches to dynamic semantics take the resolution of anaphora to be a matter of “coindexing”; they in fact assume that the “real” semantic analysis begins only after coreference has been settled. Thus, they assume that getting the right semantic analysis of (4) is a matter of being given (4a), and not, say, (4b), as the input of the semantic analysis. (4) a. A man1 meets a woman2 . He1 greets her2 . b. A man1 meets a woman2 . He2 greets her1 .
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Under such an approach, the salient item stored into the individuary is characterized in such a way that it is associated with an index, or with some other formal item working to the same effect (Groenendijk & Stokhof’s discourse marker) – the individuary can thus be constructed as a unique assignment of objects to indices. However, in this way semanticians get rid of a part of the burden which is evidently their own. It seems obvious that the semantic analysis resulting from (4a) is the right one, while that resulting from (4b) is a wrong one – and this comes from understanding (4) – hence, it is the matter of the semantics of (4) and it has to be brought out by the semantic analysis of (4). Thus, the semanticians should not wait for someone to give them a coindexing, they should aim at yielding the right analysis directly. And this requires a more substantial characterization of the items in the individuary. The idea that comes to mind is to store the individual with the “attribute” which is employed to introduce it: to store the item that is raised to salience by means of the phrase ‘a man’ with the attribute man, and, more generally, that raised to salience by means of the phrase a(n)N with the attribute N. This would turn the individuary into an assembly of the individual, attribute pairs – a phrase the N would then look for the individual which is paired with the attribute N. Thus, the first sentence of (4) would fill the individuary with the pairs I1 , man and I2 , woman ; and for the resolution to succeed it would be enough to secure that the pronoun ‘he’ seeks an individual with the attribute man (i.e. it is in effect equivalent to ‘the man’) and the pronoun ‘she’ seeks one with the attribute woman (it is equivalent to ‘the woman’).8 The question now is how to construe the word “attribute”. We may take attributes simply as words – and thus form the individuary as a collection of individual-term pairs. However, this would block the resolution in intuitively clear cases, where one uses a slightly different word, like
h
h
i
i h
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A man walks. The guy whistles. This may lead us to abandon taking attributes as terms in favor of taking them more as meanings of terms – where meanings can be, in turn, taken to be extensions, intensions, or something else. (Then, however, we may have to face the opposite problem: the problem of “overgeneration” of anaphora resolutions. In general, the optimality of the account for anaphora is a matter of fine-tuning the fine-grainedness of the attributes – and due to the heterogeneity of language there is little hope that we can find one universally optimal solution.9) Anyway, within the dynamic approach, both the meaning of an indefinite noun phrase and that of a definite one get explicated as updates, as means of innovating the current context and especially its individuary. An indefinite noun phrase changes the individuary by introducing a new inhabitant characterized in a certain way; a definite noun phrase does not change the individuary (in this sense it is a trivial update, a “test”), but searches it for the existence of an individual with a certain specification, thereby triggering a presupposition that such an individual is indeed present there, i.e. that it is “referentially available”.10 This opens the possibility of adequately explicating the meanings of the indefinite and definite articles: both get explicated as functions which map properties onto corresponding updates. Thus, the denotation of
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h
i
‘a’ is the function which maps the denotation [[N]] of the common noun N onto the update which stores I, [[N]] into the current individuary and makes the whole singular term refer to the I thus introduced; and the denotation of ‘the’ is the function which maps [[N]] on the update which searches the current individuary for an I, [[N]] and, if successful, makes the whole singular term refer to the I thus found. One possibility is to take the attributes as sets of potential referents (this is straightforward if we stay on the level of extensions; but attributes can be constructed as sets of individuals even when we embrace intensions – in this case they are sets of not only actual, but rather also possible, individuals11). In that case, individuary is explicated as a choice function taking sets of individuals into their members – this is the framework introduced by Peregrin & von Heusinger (1995). In this setting, choosing the right referent is reconstructed as bringing the choice function to bear on the relevant subpart of the class of potential referents. 2
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2.1 The nature of the individuary In this second part of the essay we now descend to a more foundational level and turn to a different question concerning the dynamic semantic framework: what are the individuaries, and more generally the “discourse representation structures” of all sorts, supposed to be? Few people practicing dynamic semantics seem bothered by this problem: they apparently assume it to be straightforwardly answerable in terms of “mental representations” or “cognitive states”. I do not think they are right: I am convinced that to explain the linguistic by means of the mental is to explain the clearer by the more obscure; and, moreover, it is to block the requisite possibility of going the other way around, namely to use the linguistic to account for the mental. Therefore, I want to propose an alternative answer: the answer that the individuary, and indeed the relation of reference connecting words with the inhabitants of the individuary, is our (i.e. of us, theoreticians) way of accounting for the inferential properties of anaphoric expressions. This question is also essentially relevant for the proper understanding of the analysis of definiteness in terms of choice outlined above: if we see the apparatus of choice functions as descriptive of mind or cognition, we will be likely to see choice functions as reports of actual “mental actions” carried out by speakers and hearers and we will be inclined to pose questions such as how they carry out the relevant choices or why they choose as they do and not otherwise; whereas if we see it as our way of accounting for certain valid inferences, such questions do not really make sense. In the former case, ultimate answers will appear to be buried within people’s heads and we will have to set upon the slippery path of introspection; whereas in the latter case we shall be able to rest on the relatively solid notion of inference as based in the norms of verbal behavior. To illustrate what I mean, let us compare the “semantics of anaphora” with a more ordinary semantics of logical connectives, such as & (the regimentation of natural language ‘and’). It is well known that & can be characterized either by saying that
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it denotes a certain binary truth function (namely the familiar function assigning T to two T’s and F to any other pair of truth values), or by stating that it is governed by certain inferential rules, namely that X&Y can be inferred from X and Y; and that X; as well as Y can be inferred from X&Y: The inferential characterization amounts to the following “introduction” and “elimination” rules:12 (&I) XX&YY (&E1 ) X&Y X (&E2 ) X&Y Y It is clear that the denotational and the inferential ways are equivalent from the formal point of view; but it should be also clear that from the point of view of analyzing natural language the second is superior. The point is that if somebody asks why we should regiment the natural language ‘and’ as &, two answers, corresponding to the two characterizations of &, are possible: it is possible to say either that ‘and’ indeed denotes the truth function stood for by &, or that it is indeed governed by the inferential rules governing &. However, the first answer can hardly be made sense of otherwise than as resting upon the second: if someone goes to ask how do we know that ‘and’ denotes such function, we can hardly do anything else than refer him to the inferences which sentences with ‘and’ license. (We can hardly show him - inside the speakers’ heads or wherever – ‘and’ to be associated with the function.) On the other hand, the second answer does make a direct sense: we indeed can find out what are the inferential patterns governing the proper use of ‘and’ – it is enough to study the (publicly accessible) ways English speakers use their language. Thus, from the point of view of the analysis of natural language, propositional logic is best seen as the theory of the inferential behavior of our basic “logical” voetc. cabulary: of those particles and connectives which get regimented as , &, , (Further logical calculi then can be seen as theories of the inferential behavior of more advanced “logical” words and aspects of our language; thus, e.g. modal logic as the theory of the inferential behavior of adverbs like ‘necessarily’ and ‘possibly’, yielding the operators 2 and 3). And what I want to claim here is that dynamic logic, and consequently dynamic semantics of the kind envisaged in the first part of the paper, should be seen as the theory of the inferential behavior of further items of our “logical” vocabulary, namely of pronouns and articles – rather than a theory of how certain words refer to things via our mental repositories. (In the case of pronouns this is about the first systematic attempt at such an account; in the case of articles, especially the definite one, it is an improvement on previous attempts, like the Russellian one.) If this view is to be tenable, then we have to be able to specify the basic inferential patterns governing pronouns and articles. I propose that we think about those like the following ones (all of them are to be seen as bidirectional inferences, i.e. so that not only the consequent is inferable from the antecedent, but also vice versa; M stands for masculine terms, F for feminine ones, and P; Q; R; S for predicates):
: _!
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M Ps. He Qs (John walks. He whistles) M Ps and Qs (John walks and whistles) Somebody Ps. He Qs (Somebody walks. He whistles) Somebody Ps and Qs (Somebody walks and whistles) M Ps and F Qs. He Rs and she Ts M Ps and Rs. F Qs and Ts (John walks and Mary sits. He whistles and she sings) (John walks and whistles. Mary sits and sings) An R Ps. The R Qs (A man walks. The man whistles) An R Ps and Qs (A man walks and whistles)
2.2 The emergence of a context-change potential Let us now show how contemplating inferential patterns like the above may lead us directly to thinking about terms as denoting context-change potentials and thus to positing an individuary on which the potentials could rest. Let us restrict ourselves, for simplicity’s sake, to the first of the above patterns (ignoring, moreover, the genderdependence of the pronoun) and let us see how we could accommodate it within a logical calculus.13 To do so, we need a constant corresponding to ‘he’; let us use the sign . To say that the constant should correspond to ‘he’ as governed by the first inferential pattern above (stripped of the gender-dependence) is to say that P(A) & Q( ) should be equivalent to P(A) & Q(A) for every singular term A and all predicates P and Q, i.e. to characterize by the following natural deduction rules:14 ( I)
P(A) & Q(A) P(A) & Q( ) P(A) & Q( )
( E) P(A) & Q(A)
Now let us think about a semantics (or about a model theory) for the resulting calculus; for the sake of simplicity let us assume that the calculus contains, besides , nothing more than what is needed to spell out ( I) and ( E), i.e. individual constants, unary predicate constants and conjunction. We soon discover that we cannot make do with the simple semantics of elementary logic, namely with treating terms as names of objects of the universe. (It is not difficult to see that this would, given ( I) and ( E), imply that all terms denote a single object.) Similarly we could eliminate other candidate semantics; and after some trial-and-error searching we are likely to realize that since the inferential role of is such that it “behaves like a if it follows a clause containing a, it behaves like b if it follows a clause containing b etc.”, we could make denote something like the identity function from individuals to do with letting individuals. This idea can be put to work by making the semantic values of statements into functions from individuals to individuals, and letting & work as a concatenation in the following way.15 First, we change the semantics of our fragment of the predicate calculus by letting statements denote mappings of a class C onto itself: we let true statements denote some function defined for every element of C, and false statements
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denote the function which is defined for no element of C. (This is clearly an entirely trivial move: it is clear that we can have any two distinct objects playing the role of the two truth values. However, note that in this setting, & denotes functional composition.) Then, we let different true statements denote different functions defined everywhere on C, and treat any such function as the truth value the truth. We can, for example, identify C with the universe and define [[P(a)]] to be such function that for every x C, if [[a]] [[P]], then [[P(a)]](x) = [[a]], and if [[a]] [[P]], then [[P(a)]](x) is undefined. [[S&S ]](x) can then be defined as [[S ]]([[S]](x)) (where this is meant to be undefined where [[S]](x) is undefined). All of this is still trivial in the important sense of not tampering with the logical properties of the calculus. Now, however, we can easily provide an adequate semantics for statements containing : we can define [[P( )]] to be such function that [[P( )]](x) = x if x [[P]], and is undefined otherwise. And if we do this, the inferences ( I) and ( E) hold. The final step is then to project the new semantics onto terms. We can define the semantics e.g. in the following way (we can take S to be true if [[S]](x) is defined for every element x of the universe; or else if it is defined for at least one element x of the universe – the difference affects only formulas with a “free occurrence” of , i.e. formulas in which does not follow an individual constant):
2
0
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0
2
[[a]] is a constant function defined everywhere on U (where a is an individual constant) [[
]] is the identity function defined everywhere on U
2
[[P(t )]] is a function such that [[P(t )]](x) = [[t]](x) if [[t]](x) [[P]], and is undefined if [[t]](x) [[P]] (where t is an individual constant or )
62
[[S&S ]] is the composition of [[S ]] and [[S]], i.e. [[S&S ]](x) = [[S ]]([[S]](x)) 0
0
0
0
In this way we have turned denotations of sentences and terms of our fragment into context-change potentials; and we have created an individuary (a particularly tiny one, which can contain at most one individual). What is important is that this creation has resulted from our attempt to account semantically for certain inferences – not to depict some mental or real machinery. In other words, the semantics based on this individuary has been employed as a tool of our account; not as a picture.16 The introduction of the whole machinery of definite and indefinite singular terms, which leads to individuaries of more complex kinds, is now only a more complicated version of the same process. We have more inferences to account for: inferences like that of an R Ps and Qs (‘A man walks and whistles’) from an R Ps and the R Qs (‘A man walks and the man whistles’); hence we need more “slots” to store individuals, and we need labels to tell different slots apart. However, the individuary is again no more than a creature of our theory of drawing inferences. The crux of this kind of semantic treatment is that the link between a singular noun phrase and a following pronoun (or between an indefinite noun phrase and a following definite noun phrase) is established by linking both phrases to the same inhabitant of
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the individuary. Now insofar as reference is taken as the link between a word and an inhabitant of an individuary, and coreference consequently as the property of being linked to the same item, saying that two noun phrases are coreferential thus becomes a short way of stating that their inferential roles are in a certain way interconnected (to say that within ‘A man walks and he whistles’, ‘a man’ is coreferential with ‘he’ is to say that the sentence entails, and is entailed by, ‘A man walks and whistles’), and talking about reference in turn becomes only a particularly illustrative way of rendering coreference. If we recognize individuaries and their inhabitants as mere tools to account for inferences, then the talk about reference becomes essentially parasitic upon the talk about inference – a referent is nothing more than an illustrious clamp holding certain inferentially related expressions together. 2.3 Capturing inference as reference We have given an example, admittedly oversimplified, of how talk about an individuary, and about expressions’ referring to elements of the individuary, can be rendered as talk about inferential patterns. Now if we want to make a more general claim concerning the reducibility of “referential talk” to “inferential talk”, such an example is surely not enough. We have to indicate that rendering talk about reference as talk about inferential patterns is possible in general, for all “referring expressions”, and besides this, we have to indicate that we do not need the concept of reference to underpin language in the first place. The former task will be the topic for the present section, the latter will be left for the next one. Let us first introduce some terminology in order to be able to talk about the inferential roles of expressions. In general, we can see two expressions as inferentially connected if, informally stated, the inferences licensed by one of them are licensed also by the other. There are two levels of inferential relationships between expressions, the first level concerning (material) implication (licensing inferences “here and now”), and the second concerning entailment (licensing inferences “everywhere and always”). We say that a statement S implies a statement S if S is not false unless S S ; and we say that S entails S if S cannot be false if S is not; i.e. if is, i.e. if S =S S . Let us then call an expression e weakly (inferentially) subordinated to an expression e iff for every atomic statement S, S implies S[e =e] (where S[e =e] is the statement which arises from S by replacing e by e ); and let us call e strongly (inferentially) subordinated to e iff for every atomic statement S, S entails S[e =e]. Finally, let us call e and e weakly (inferentially) equivalent iff e is weakly subordinated to e and e is at the same time weakly subordinated to e (i.e. iff S S[e =e] for every statement S); and analogously for strong (inferential) equivalence. With the help of this terminology, we can characterize the inferential behavior of expressions which are traditionally considered as “referring”, i.e. of names (in a broad sense). First, names are characterized by the fact that no name is subordinated to another name without being equivalent to it. This means that the inferential structure of the domain of names is more or less trivial (in contrast to that of predicates which constitutes a Boolean algebra).17 As the consequence, we inferentially characterize
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a name simply by saying with which other names it is weakly equivalent. If we use the term coreferential as a synonym of weakly equivalent, then we can say that to inferentially characterize a name is to state with which other names it is coreferential; and as the relation of coreferentiality is clearly an equivalence relation which thereby decomposes the class of names into the corresponding equivalence classes, this is to specify the coreferentiality class to which the name belongs. And if we further use the term what it refers to as a shortcut for which coreferentiality class it belongs to, we can say that by inferentially characterizing a name we pinpoint what it refers to – thus explicating “referent” as “that which is shared by all coreferential expressions”. We can also make the “inferential sense” of distinctions which are usually drawn from the referential perspective: such distinctions as those between proper names (“rigid designators”), descriptions (“contingent designators”) and pronouns (“contextdependent designators”). First, we can single out the class of proper names as the maximal subclass of the class of names which has the property that any two of its members are weakly equivalent (coreferential) if and only if they are strongly equivalent. Coreferentiality is thus a standing property for proper names, and the inferential behavior of a proper name is thus exhaustively characterized by specifying its (standing) coreferentiality class, i.e. by its (standing) referent. Then we can single out the subspecies of descriptions analogously by stating that two descriptions can be coreferential without being strongly equivalent, and no description can ever be strongly equivalent to a proper name (although it can be coreferential with it). The relation of coreferentiality among descriptions is a fluctuating, contingent matter. Thus, the exhaustive inferential characterization of a description cannot consist simply in pointing out its (momentary) coreferentiality class; we must somehow say to which coreferentiality class it belongs when. This could be done by specifying its coreferentiality class relative to the truth-valuation of sentences, i.e. to a possible world. Thus, a description can be inferentially characterized by being assigned a function from possible worlds to referents; as it is the case within Montagovian frameworks. Finally we can characterize the third basic kind of names, pronouns, by stating that no pronoun is coreferential with a proper name nor with a description (which may suggest that pronouns are “in fact” not names and have no referents), that, however, pronouns are what can be called “strongly locally” equivalent with names – sentences containing pronouns are often strongly equivalent with sentences containing proper names in their place (viz ( I) and ( E)). One way to accommodate this is to make coreferentiality, and hence reference, somehow relative to the context – to associate pronouns with functions from contexts to referents. It then becomes vital to articulate the notion of context appropriately (as discussed in section (1.4) and to embody this into a tractable compositional semantics. 2.4 The essentiality of inference Thus, we can, at least in principle, render the concept of reference, and referential links between expressions and occupants of “slots” of individuaries, as a means of
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characterizing inferential behavior. However, it may be objected that reference is primarily a relation between expressions and real-world things which is only mediated by inhabitants of mental individuaries; and that as such it is something which underpins language in the first place. Is thus not explaining reference in terms of inference putting the cart before the horse? Is reference not the thing via which language “hooks on the world” and without which no language could exist; and is thus reference not the key to everything else in language, including inference? Without being able to go into detail here, let me point out that the picture of language centered around the concept of reference (justified by the claim that reference is what gives language its hook on the world) can be counterposed to a picture centered around the concept of inference; justified by the fact that what distinguishes a language is its capability of serving as the medium of the human “game of giving and asking for reasons” (Brandom 1994). The latter perspective denies that language would be, as the referential perspective seems to have it, only a rich and complexly interrelated system of names; it sees the referential view as a misguided “museum myth” (Quine 1969). According to this inferential picture, drawn most vividly by Wilfrid Sellars and Robert Brandom (but in fact going back to Kant), the distinctive feature of language is that it is capable of conveying propositional content – and hence that the constitutive characteristics of words is not that they name (refer to, represent, stand for) things, but rather that they can add up to propositionally contentful utterances. If we subscribe to this story, then we have to conclude that it is the relation of inference, not that of reference, which is the backbone of language on which everything else should be seen as supervening – for it is only inferential articulation which can confer propositional contents on sentences. Even the relation of reference is then seen as parasitic on the relation of inference – which inevitably leads to a deflationary view of reference (in the sense of Horwich 1990, Sec. 39).18 The obvious objection is that this leads to an absurdly idealistic picture of language, to a picture in which language is completely cut loose from the world. But this is a misunderstanding. Of course if we want to see language generally as a matter of inferences, then we have to construe the term inference broadly enough to comprise what Sellars (1974) calls language entry transitions (roughly, inferences from situations to statements) and language exit transitions (inferences from statements to actions). (However, equally of course, only inferences in the narrow sense, inferences from statements to statements are capable of being the subject of semantic theory. The other ones do not yield a nontrivial theory over and above trivialities like We (correctly) say ‘A rabbit runs’ iff a rabbit runs.) So according to this view, language, of course, is connected to the world, but not via denotative word-thing links, but rather via normative statement-occasion links. According to this view, the unequivocal aim of semantics is to account for the relation of inferability among statements – and any semantic value which semantic theory associates with an expression should be seen not as a depiction of a real thing on which the expression is claimed to (causally or otherwise) hook, but rather as a kind of hypostasis of the way the expression functions within inferences.19 And it is important to see that urging this view is not merely an
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exercise in speculative philosophy, that it has most important consequences for the practice of semantic analysis: 1. It implies that the way of semantics is essentially parallel to the way of logic – for logic is precisely the account for inferability. The main difference is that semantics endeavors to see the inferential behavior of an expression materialized in its meaning, i.e. in an object associated with the expression. (In contrast to this, the view which makes reference basic is bound to see semantics allied to cognitive psychology investigating the ways in which we attach names to things.) 2. It suggests that semantic formalisms should not be seen as describing platonistic entities or mental representations “behind” expressions, but rather as explicitly articulating expressions’ inferential properties. That is to say, a formula or a schema associated with a sentence should be seen as explicating the “inferential potential” of the sentence, i.e. as encapsulating what is implied by the sentence and what implies it; and the formula associated with a subsentential expression as that which adds up to the inferential potentials of the sentences in which it occurs. 3. It entails that the criteria of adequacy of a formal semantic theory are a matter of success in capturing inferences. Such success then can be checked by finding out about the publicly accessible rules of correct usage of the language in question. (In contrast, the mentalistic approach is bound to see such criteria as a matter of faithfully depicting the hopelessly private mind or cognition). What, then, about the common temptation to see semantic representations, individuaries and the like as depicting some structures of the minds or brains of the speakers who draw the inferences? In a certain weak sense, this need not be incompatible with the vantage point advocated here: if we accept the inseparability of language and thought, we have to see any account of one’s usage of language as eo ipso an account of her thinking. However, what I cannot find any substantiation of is seeing the referential apparatus as a depiction of structures and processes going on within speakers’ minds/brains (and thus holding that to assess semantic theory we should observe what is going on within our heads). Chomsky (1986, 45) writes: “One can speak of ‘reference’ and ‘coreference’ with some intelligibility if one postulates a domain of mental objects associated with formal entities of language by the relation with many of the properties of language, but all of this is internal to the theory of mental representations; it is a form of syntax.” If we use the term syntax in such a way that we see inference as a syntactic matter (which is usual, although perhaps misguiding), then the standpoint advocated here can be seen as consisting precisely in taking the theory of reference as “a form of syntax”. However, Chomsky’s pronouncement is problematic, it seems to me, because he speaks about “mental objects” and “mental representations” – which I find simply unwarrantable. I suggest that we replace the term “mental” in claims of this kind simply with the neutral term “semantic”: perhaps some of the people speaking about the mental in this context really do not mean anything over and above the semantic.
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N OTES ι
1
Russell himself introduced the -operator as a mere notational short-cut governed by a contextual definition, not as a fully-fledged term (see Russell & Whitehead 1913). 2 However, notice that, as Lavine (1994, 104) points out, the axiom of choice is problematic only under a certain specific notion of set, namely under the “logical” notion, claiming that items can be collected into a set only if they can be delimited by a criterion.
The definition of CHFM clearly guaranties that this holds for at least one c 2 CHFM if and only if it holds for every c 2 CHFM . 3
Hilbert’s treatment of the ε-operator was purely axiomatic; he did not consider any kind of model theory. Nevertheless, choice functions clearly represent the straightforward way to put his ideas into the semantic cash. (Notice, however, that Hilbert’s axiom allows for “intensional” choice – the value of εx Fx may differ from that of εx F0 x even if F and F’ are coextensional. See, e.g. Meyer Viol 1995). 4
5
In fact, it seems to be this idea which lays the foundations of Egli’s and von Heusinger’s exploitation of the Hilbertian ideas for semantic analysis (see their contributions in Egli & von Heusinger 1995). 6
The pioneers of the kind of dynamic semantics which is relevant here are especially Kamp (1981) and Heim (1982); but other kinds of dynamic semantic theories were proposed earlier (e.g. by Hintikka 1973). 7
As far as I know, this term is due to Irene Heim.
8
The most straightforward way of exploiting this idea is perhaps Heim’s (1982) File Change Semantics. 9
This is analogous to the case of objects of propositional attitudes: they also seem to be sometimes like propositions, whereas sometimes rather like sentences. 10
An individual can, of course, enter the individuary also in a “non-linguistic” way, e.g. via ostension.
11
It is not without interest to note that this is in fact the way in which the concept of intension was approached by Rudolf Carnap (see esp. Carnap 1955), who is mostly responsible for its current dissemination.
12
For the general notion of introduction and elimination rules see, e.g. Prawitz (1965).
13
For a more detailed elaboration see Peregrin (1998).
14
From the purely logical viewpoint, these rules, and consequently , are clearly not of great interest, for what they introduce is in fact nothing but certain notational variants of certain conjunctive statements.
15
From the formal point of view, the enterprise is, of course, only a minor variation on the basic theme of dynamization of semantics as presented by Groenendijk & Stokhof (1991).
16 17
For the discussion of this dichotomy see Peregrin (1999).
Making a similar point, Brandom (1994, 372) says: “Singular terms are grouped into equivalence classes by the good substitution inferences in which they are materially involved, while predicates are grouped into reflexive, transitive, asymmetric structures or families. That is to say that some predicates are simply inferentially weaker than others, in the sense that everything that follows from the applicability of the weaker one follows also from the applicability of the
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stronger one, but not vice versa. ( : : : ) Singular terms, by contrast, are not materially involved in substitution inferences whose conclusions are inferentially weaker than their premises.” 18
In some of my recent writings (see esp. Peregrin 1995) I have urged a distinction between two views of language, which I have called the nomenclatural and the structural view, respectively (Brandom 1994 speaks about the representational and the inferential view to the same effect). The former view is based on a view of language as a nomenclature of some kind of things; the latter view sees language rather as a kind of toolbox. What I tried to indicate, and what I am trying to indicate also here, is that it is the latter which can help us gain real insight into the workings of language.
19
As Brandom (1994, 84) puts it, “an association [of abstract objects with strings] amounts to specifically semantic interpretation just insofar as it serves to determine how those strings are correctly used”. This also implies that within logic, proof theory is in an important sense primary to model theory – see Peregrin (1998).
R EFERENCES Barwise, J. & Cooper, R. 1981. Generalized Quantifiers and Natural Language. Linguistics and Philosophy 4, 159-219. Brandom, R. 1994. Making It Explicit. Cambridge/MA: Harvard University Press. Carnap, R. 1955. Meaning and Synonymy in Natural Languages. Philosophical Studies 7, 33-47. Chomsky, N. 1986. Knowledge of Language. Westport: Praeger. Egli, U. & von Heusinger, K. 1995. The Epsilon Operator and E-Type Pronouns. In: U. Egli et al. (eds.). Lexical Knowledge in the Organization of Language. Amsterdam: Benjamins, 121-141. Groenendijk, J. & Stokhof, M. 1991. Dynamic Predicate Logic. Linguistics and Philosophy 14, 39-100. Groenendijk, J. & Stokhof, M. & Veltman, F. 1996. Coreference and Modality. In: S. Lappin (ed.). The Handbook of Contemporary Semantic Theory. Oxford: Blackwell, 179-213. Heim, I. 1982. The Semantics of Definite and Indefinite Noun Phrases. Ph.D. dissertation. University of Massachusetts, Amherst. Distributed by Ann Arbor: University Microfilms. Hilbert, D. 1925. Über das Unendliche. Mathematische Annalen 95, 161-190. [Translated as “On the Infinite” in: J. van Heijenoort (ed.) 1971. From Frege to Gödel: A Source Book from Mathematical Logic. Cambridge/MA: Harvard University Press, 367-392.] Hintikka, J. 1973. Logic, Language-Games and Information. Oxford: Clarendon Press. Horwich, P. 1990. Truth. Oxford: Blackwell. Kamp, H. 1981. A Theory of Truth and Semantic Representation. In: J. Groenendijk & T. Janssen & M. Stokhof (eds.). Formal Methods in the Study of Language. Amsterdam: Mathematical Centre, 277-322. [Reprinted in: J. Groenendijk & T. Janssen & M. Stokhof (eds.) 1984. Truth, Interpretation and Information. Dordrecht: Foris, 1-41.] Lavine, S. 1994. Understanding the Infinite. Cambridge/MA: Harvard University Press. Meyer Viol, W. 1995. Instantial Logic. Ph.D. dissertation. University of Amsterdam. Distributed by Institute for Logic, Language and Computation (ILLC), University of Amsterdam.
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Montague, R. 1974. Formal Philosophy. Selected Papers of Richard Montague. Ed. and with an introd. by R. Thomason. New Haven: Yale University Press. Peregrin, J. 1995. Doing Worlds with Words. Dordrecht: Kluwer. Peregrin, J. 1998. The Logic of Anaphora. Ms. Prague University. Peregrin, J. 1999. Linguistics and Philosophy. Theoretical Linguistics 24. Peregrin, J. & von Heusinger, K. 1995. Dynamic Semantics with Choice Functions. In: U. Egli & K. von Heusinger (eds.). Choice Functions in Natural Language Semantics. Arbeitspapier 71. Fachgruppe Sprachwissenschaft, Universität Konstanz. [Also in: H. Kamp & B. Partee (eds.) 1997. Proceedings of the Workshop “Context Dependence in the Analysis of Linguistic Meaning”. Vol. I. Institut für maschinelle Sprachverarbeitung, Universität Stuttgart, 329-353.] Prawitz, D. 1965. Natural Deduction. Stockholm: Almqvist & Wiksell. Quine, W. 1969. Ontological Relativity and Other Essays. New York: Columbia University Press. Russell, B. 1905. On Denoting. Mind 14, 479-493. Russell, B. & Whitehead, A. 1913. Principia Mathematica. Vol. I. 2nd ed. Cambridge: Cambridge University Press. Sellars, W. 1974. Meaning as Functional Classification. Synthese 27, 417-437. Strawson, P. 1950. On Referring. Mind 59, 320-344.
VARIABLES IN NATURAL LANGUAGE: WHERE DO THEY COME FROM? Jaroslav Peregrin* www.cuni.cz/~peregrin [Variable-Free Semantics (ed. M.Boettner and W.Thümmel), Secolo, Osnabrück, 2000, 46-65]
0. Introduction What is a variable and in which sense can we say that natural language contains variables? Inspecting the Oxford Advanced Learner’s Dictionary we learn that a variable is a ‘variable thing or quantity’. The American Heritage Dictionary of the English Language tells us that a variable is (1) ‘something that varies or is prone to variation’; (2) within astronomy, ‘a variable star’; and (3) within mathematics, ‘a quantity capable of assuming any of a set of values’ or ‘a symbol representing such a quantity’. Penguin’s Dictionary of Mathematics states that a variable is (1) ‘a mathematical entity that can stand for any of the members of a set’ and (2) ‘an expression in logic that can stand for any element of a set (called the domain) over which it is said to range’. The Oxford Dictionary of Philosophy gives no explicit definition, but explicates a variable informally as something which can replace a word in a sentence and which can then be seen as ‘pointing’ at different members of a domain. From this mini-recherché two senses of ‘variable’ seem to emerge a narrower sense, in which a variable is something which inhabits the realms of logics and mathematics, and a wider sense in which a variable is simply anything that varies. It seems also clear that this latter concept of variable is wide to the point of not being a concept at all (for surely any thing can be seen as somehow varying, i.e. changing, evolving, or displaying varying aspects etc.), and therefore the only concept we really have is the narrower one. This means that we should see variables as primarily a matter of logical and mathematical calculi. Does this mean that it makes no real sense to speak about variables in connection with natural language? Of course not: natural language is in various respects similar to formal calculi; and formal calculi are employed, in various ways, to regiment, analyze or explicate it. It might therefore be both possible and reasonable to transfer the concept from the latter to the former. To find out whether and in which sense this is justified, we must investigate (i) the role of variables within logical calculi, and (ii) the ways in which we can, or should, see natural language through the prism of such calculi.
1. Variables within logic 1.1 The birth of variable The concept of variable, as nowadays employed within logic and mathematics, is inseparably connected with the concept of quantifier developed by Gottlob Frege and his followers. Frege, when introducing quantifiers, says roughly this: Imagine a sentence decomposed into two *
The author is grateful to Vladimír Svoboda for helpful criticism.
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parts, and imagine one of the parts „abstracted away“, thus turning the sentence into an „unsaturated“, gappy torso. Then think of the gap in this matrix as being filled with various things and consider the truth values of individual cases - the truth value of a corresponding quantificational sentence then can be computed from these values1. A variable is then a symbol that is employed to mark the gap(s). Thus, in his Begriffsschrift (Frege, 1879, p.19), he writes: In dem Ausdrucke eines Urtheils kann man die rechts von ├─ stehende Verbindung von Zeichen immer als Funktion eines der darin vorkommenden Zeichen ansehen. Setzt man an die Stelle dieses Argumentes einen deutschen Buchstaben, und giebt man dem Inhaltsstriche eine Höhlung, in der dieser selbe Buchstabe steht, wie in a ├─∪─ X(a) so bedeutet dies das Urtheil, daß jene Function eine Thatsache sei, was man auch als ihr argument ansehen möge.2 Later, Frege came to see the ‘de-saturation’ which underlies quantification as only a special case of ‘functionalization’, i.e. of the process of making a linguistic gap to define a function. Thus in Funktion und Begriff (Frege, 1891, p.8), he writes: In dem Ausdruck erkennen wir die Funktion dadurch, daß wir ihn zerlegt denken; und eine solche mögliche Zerlegung wird durch seine Bildung nahe gelegt. ... Wenn ich nun z.B. sage „die Funktion 2•x3+x“, so ist x nicht als zur Funktion gohörig zu betrachten, sondern dieser Buchstabe dient nur dazu, die Art der Ergänzungsbedürftigkeit anzudeuten, indem er die Stelle kenntlich macht, wo das Zeichen des Arguments einzutreten hat3. 1
This consideration has later come to be seen as ambiguous between the ‘substitutional’ and the ‘objectual’ version. According to the ‘substitutional’ version of the story, we fill the gap literally: we replace the variable by suitable expressions, thus saturating the matrix ‘syntactically’. According to the ‘objectual’ version, on the other hand, we make the variable stand for suitable object, thus saturating the matrix ‘semantically’. If we consider the matrix ‘x conquered Gaul’ (which might have arisen, e.g., out of the sentence ‘Caesar conquered Gaul’ via taking away ‘Caesar’), then the ‘substitutional’ ‘re-saturation’ would consist in replacing x by various names (‘Caesar’, ‘Aristotle’, ‘Clinton’, ...) and thus turning the the matrix into various sentences (‘Caesar conquered Gaul’, ‘Aristotle conquered Gaul’, ‘Clinton conquered Gaul’, ...), whereas the ‘objectual’ one would consist in making x stand for various individuals (Caesar, Aristotle, Clinton, ...) and thus making the matrix express various propositions (that Caesar conquered Gaul, that Aristotle conquered Gaul, that Clinton conquered Gaul, ...). 2 „In the expression for a judgement, the complex symbol to the right of ├─ may always be regarded as a function of one of the symbols that occur in it. Let us replace this argument with a Gothic letter, and insert a concavity in the content-stroke, and make this same Gothic letter stand over the concavity, e.g.:
a
├─∪─ X(a) This signifies the judgement that the function is a fact whatever we take its argument to be.“ 3 „We recognize the function in the expression by imagining the latter as split up, and the possibility of thus splitting it up is suggested by its structure. ... For instance, if I say ‘the function 2•x3+x’, x must not be considered as belonging to the function; this letter only serves to indicate the kind of
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This indicates that for Frege variables played a merely auxiliary role of ‘gap-markers’: they helped turn expressions into unsaturated torsos which can be seen as indicating functions (not denoting functions, for unsaturated expressions are no names and thus do not denote anything)4 and which can yield saturated expressions (names, especially sentences) not only by their gaps being filled with names, but also by the gaps (i.e. variables) being ‘bound’ by quantifiers (or ‘abstracted away’ by Frege’s operator ’, which was the counterpart of the modern λ).
1.2 Bound variables Let us now restrict our attention to bound variables, i.e. to variables which are within the scope of a quantifier. The important thing to notice is that they are not essential: we can do logic (predicate calculus) wholly without them. What does this mean? It is well known that we can (as Polish logicians demonstrated) do predicate calculus wholly without parentheses; and this fact is usually taken to show that parentheses are idiosyncratic to one specific way of articulating the syntax of the calculus, and they are not essential for the calculus as such. It is less well known that precisely the same holds for (bound) variables. A particularly instructive way of reformulating the syntax of the first-order predicate calculus enabling us to rid ourselves of variables was presented by Quine (1960); and this way can be generalized to higher-order calculi as well (for second-order predicate calculus, this has actually been carried out by Došen, 1988). Indeed, that predicate calculus of any order can be articulated without variables follows from the proof of the equivalence of lambda calculus and combinatory logic (see Curry and Feys, 1957). Why are bound variables not essential? Informally speaking, the reason is that their role is merely auxiliary, it is a role which can be played also by other things. The analogy with parentheses may be helpful: parentheses are dispensable for their task is to specify the order in which logical operators apply, and they are no longer needed if we use prefix (instead of the infix) notation - for then the order is unique. Likewise, variables are inessential for their role is simply that of a clamp (they connect quantifiers with the appropriate places within the matrix), and this can be provably accomplished using other means (even utilizing things evocative of real clamps, such as the arrows in Bourbaki, 1958). To illustrate how it is possible to rid ourselves of variables, first think about quantifiers attached only to atomic formulas with unary predicate constants. Then we obviously need no variables at all: we can treat quantifiers syntactically on par with ordinary individual constants and write, e.g., P(∃) instead of ∃xP(x) (if we want to make explicit the essential semantic difference between quantifiers and individual constants, we can make predicates the arguments of quantifiers, rather than vice versa, and thus write ∃(P)). If we now allow for binary predicate supplementation that is needed; it enables one to recognize the places where the sign for the argument must go in.“ 4 Frege distinguished between a function and the course of values of the function. His function is something which is almost linguistic, something which we now would probably call rule. His course of values is the function in the modern sense - in effect a certain set of ordered pairs of objects. Whereas what he calls a function is no object, what he calls the course of values of a function is, and hence it can be named.
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constants, the situation changes: we cannot write R(∃,∀), for this would be ambiguous between ∃x∀yR(x,y) and ∀y∃xR(x,y). Another problem arises when quantifiers are attached to nonatomic formulas (even when these contain only unary predicate constants): P(∃)∧Q(∃) would be ambiguous between ∃x(P(x)∧Q(x)) and (∃xP(x))∧(∃xQ(x)). However, the needed disambiguation can be carried in ways which avoid variables: we can, e.g., state the linear order of the quantifiers using superscripts, so that ∃x∀yR(x,y) would be R(∃1,∀2), whereas ∀y∃x R(x,y) would be R(∃2,∀1); and similarly ∃x(P(x)∧Q(x)) would be P(∃1)∧Q(∃1), whereas (∃xP(x))∧(∃xQ(x)) would be P(∃1)∧Q(∃2). An obvious objection is that such superscript notation would not lead to a syntax which could support a reasonable compositional semantics. This may be right, but what Quine and others have demonstrated is precisely that there does exist a syntax which underlies compositional semantics and which manages without variables. Let us sketch the idea of these proposals. First, for the sake of simplicity, we shall consider monadic first-order predicate calculus (i.e. first-order predicate calculus involving no predicate constants of arity greater than one). The syntax of this calculus is based on the following rules: (i) A (unary) predicate plus a term (where a term is an individual constant or a variable; we omit functors for simplicity) give a formula. (ii) ¬ plus a formula give a formula. (iii) ∧ (∨, →) plus two formulas give a formula. (iv) ∃ (∀) plus a variable plus a formula give a formula. The basic idea for dispensing with variables is to treat a quantifier as something which can yield a formula together with a predicate. This is fine, we have seen, with formulas like ∃xP(x), which can be seen directly as the combination of a quantifier with a predicate constant (∃ with P). Problems, though, arise with formulas like ∃x(P(x)∧Q(x)) - for such formulas need to be seen as the combination of a quantifier with a complex predicate (∃ with the conjunction of P and Q), and we lack the way to form a conjunction of predicates. The remedy, however, is to add such a way to our syntax; one of the possibilities to do this is to replace (iv) by (iv’) and to add (v) and (vi): (iv’) ∃ (∀) plus a predicate give a formula. (v) ¬ plus a predicate give a predicate. (vi) ∧ (∨, →) plus two predicates give a predicate. Then, writing predicates as arguments of quantifiers5, we can clearly reflect the difference between ∃x(P(x)∧Q(x)) and (∃xP(x))∧(∃xQ(x)) as that between ∃(P&Q) and ∃(P)∧∃(Q). The situation complicates itself, of course, when we consider full (i.e. nonmonadic) first-order predicate calculus; for the presence of predicate constants of arities greater than one necessitate further rules for predicates. (Thus in this case we have, e.g., formulas such as ∃xR(x,x), which must be treated as ∃ applied to something as the ‘reflexivization’ of the binary predicate R; so we need a rule which takes R and yields a unary predicate which applies to an individual just in case the individual stands in the relation of R to itself.). However, that all of this can be handled in an analogous vein is precisely what Quine has demonstrated. All of this justifies our claim that (bound) variables are not essential elements of the predicate calculus; they are better seen as syncategorematic symbols on par with parentheses. 5
This kind of syntax does justice to the fact that if we treat quantifiers as categorematic terms (like within lambda calculus - see Church, 1940) we usually treat them as second-order predicates (unary predicates predicable of unary predicates), hence as denoting classes of classes of individuals.
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1.3 Free variables Now what about free variables and open formulas? For them, the most straightforward approach appears to be to take them as mere steping stones on the way to closed formulas. Seen thus, a variable x and a formula F containing x free are only intermediaries to formulas such as ∃xF and ∀xF (or, more generally, λxF6); and they can be ‘kicked away’ when the destination is achieved. Their usefulness consists in the fact that as the steping stones they are extremely simple and elegant; and they provide for the usual seminally simple grammar of the predicate calculus. Seen thus, free variables are simply to-be-bound variables. Nevertheless what we have stated in the previous section means that bound variables are dispensable, and hence that they are better seen not as real constituents of the calculus; so if free variables are only their pre-bound stages, they are surely dispensable too. However, it may be insisted that open formulas are more than this, that they are needed for something more than for arriving at closed formulas. (We can perhaps distinguish two versions of this claim: one is that open formulas are needed as schemes, i.e. as means of envisaging types of closed formulas; the other is simply that there is no reason for denigrating open formulas as less fully-fledged constituents of logical calculi as closed ones.) As what interests us here is natural language, it is crucial to see how this view fares from the point of view of the logical analysis of natural language; we should ask whether open formulas are in some sense necessary for the analysis of language. Hence the question facing us now is the following: are there natural language sentences which can be reasonably seen as counterparts of (corresponding to, adequately analyzable by means of, having as their logical forms) open logical formulas? To answer this, we must first investigate precisely in which sense a natural language sentence can be seen as a counterpart to a formula of a logical calculus, in which the former’s ‘logical form’ can be expressed by the latter.
2. Logic and Natural Language 2.1 Origins of Logical Analysis At the beginning of this paper we said that logical calculi can be used to regiment natural language; but more is true: logical calculi resulted from regimentation of natural language. Going back to Frege once more, we see that his concept script, a predecessor of our predicate calculus, emerged directly from his desire to bring natural language to the form where we are rid of everything which is not important from the point of view of consequence.7 It is clear that λ-abstraction can be seen as the only variable binding operation - once we see quantifiers as second-order predicates, quantification gets decomposed into λ-abstraction and simple application: ‘∃xF’ turns into a shorthand for ‘∃(λxF)’. This is also, I think, what made Frege see quantification as based on a special case of ‘functionalization’. 7 Thus, Frege (1879, p. IV) writes: „[Die vorliegende Begriffsschrift] soll also zunächst dazu dienen, die Bündigkeit einer Schlußkette auf die sicherste Weise zu prüfen und jede Voraussetzung, die sich unbemerkt einschleichen will, anzuzeigen, damit letztere auf ihren Ursprung untersucht werden 6
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When we look at the syntax of predicate calculus, we can see that it is based on three kinds of rules: (i) rules of predication, connecting predicates with the appropriate number of terms; (ii) rules of logical operators, constructing more complex statements out of simpler statements with the help of logical operators (negation, conjunction, disjunction, implication); and (iii) rules of quantification, prefixing quantifiers to statements. It is clear that the rules of the first two kinds straightforwardly reflect basic syntactic operations of natural language: these are, respectively, that we can associate a verb phrase with noun phrases to form a sentence, and that we can negate a sentence or put two sentences together by means of a connective, respectively. The rules of the third kind, however, are different, they do not reflect a basic syntactic structure of natural language; in fact, they stem, as we have seen, from certain metalinguistic considerations. We have already seen how Frege employed variables to allow him to say such things as ‘when we saturate a gappy expression in certain way(s), we get certain value(s)’. Thus, the structure of a quantificational formula (i.e. of a formula of the form quantifier + variable + formula) does not reflect the structure of a common natural language sentence, it is rather a shortcut for saying ‘if we fill that gap, which in the formula is marked by the variable, with various things, we get a true sentence (at least) so many times as claimed by the quantifier’. Hence to say that, e.g., the formula ∃xPx is true is to say that if we keep filling the gap x in Px with various things, we gain a statement which is true at least once. The adoption of this kind of quantificational rules has had far-reaching consequences. On one hand, it has fostered the impressive development of modern logic and mathematics which we have been witnessing; on the other hand it has caused much confusion concerning the logical analysis of natural language. Frege’s choosing of this way to regiment our way of talking has caused many people to feel that quantifiers and variables are entities somehow covertly contained within natural language. But we should keep in mind that, as we have seen, it is perfectly possible (although perhaps less effective) to do Fregean logic without variables and variable-binding quantifiers.
2.2 Logical Form? We have just seen that Frege’s original view of the ‘logical form’ of an expression was something like ‘that which remains of the expression if we strip off everything which is irrelevant from the viewpoint of consequence’. However, soon Frege himself (and then especially his followers, notably Russell) concluded that the logical form is something which may differ wildly from the surface form and that logical analysis is thus a far more intricate enterprise than simple ‘stripping off’. (Russell’s famous paper ‘On Denoting’ is particularly instructive from this viewpoint: there the author tries to demonstrate that the logical form of könne. Deshalb ist auf den Ausdruck alles dessen verzichtet worden, was für die Schlußfolge ohne Bedeutung ist." ("That is why I decided to forego expressing anything that is without significance for the inferential sequence. Its [of the present ideography] first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed, so that its origin can be investigated.")
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sentences containing descriptions differs radically from what the surface of the sentences would suggest.8) This has lead to the view that logical form is something buried deep inside each expression and waiting for a ‘logical analysis’ to dig it out and bring it to light. Accepting this picture seems to offer a suitable basis for saying that ‘natural language contains variables’: if the logical form of an expression contains variables, then (as the form seems to be something really present ‘inside’ the expression) the expression itself can be said to contain variables. However, we are going to indicate that this whole picture of logical form as a definite thing extractable from inside an expression is dubious. (Moreover, it follows from the above considerations that even if we accepted this picture, or some weakened version of it, expressions could be seen as containing variables only if they had open formulas as their logical forms.) To see the dubiousness of the whole picture we must consider the general question of what counts as a criterion for being a logical form of a given sentence. What justifies us in saying that the logical form of a sentence S is given by a formula F? A necessary condition seems to be that F must capture the truth conditions of S. But if this were also a sufficient condition, it would mean that an expression has infinitely many logical forms: for if F captures the truth conditions of S, then so do all formulas logically equivalent to it, notably F∧F, F∧F∧F etc. This means that if we want to have logical form as a unique thing extractable from the expression, we must add some other necessary condition. An obvious candidate seems to be the condition that the logical form of S is rendered by the ‘simplest’ or ‘least redundant’ of those formulas which capture the truth conditions of S. However, even if these vague notions could be made precise enough within a given logical calculus, it is hard to see how they could make sense across logical calculi: is the Russellian analysis of A man walks, namely ∃x(man(x)∧ ∧walks(x)), ‘simpler’ than that which is offered by, say, the theory of generalized quantifiers, namely (a(man))(walk)9? An improvement may be (and I think indeed is) to speak about ‘closeness to surface form’ instead of about ‘simplicity’ or ‘nonredundancy’: to say that the logical form of S is that of the formulas capturing the truth conditions of S whose structure is the closest to the (surface) structure of S. However, it seems to be clear that if we employ a logical calculus with a sufficiently rich syntax, we can make the analysis as close to the surface as we desire, and this would mean that there is no nontrivial concept of logical form (as opposed to surface form) after all. (And as it would render every two sentences differing on the surface as differing in logical form, it would apparently contradict the basic intuition behind the concept of logical form.) The moral of these considerations seems to be the following: we may regiment A man walks as ∃x(man(x)∧ ∧walks(x)) (which has the advantage of utilizing only the simple and perspicuous apparatus of first-order logic), or we may regiment it as (a(man))(walk) (which is closer to the surface, but logically more involved), or we may regiment it, say, within dynamic logic as ∃Dx(man(x)∧ ∧Dwalks(x))10 (which captures some semantic aspects of the analyzed sentence, which are ignored by the ‘static’ analysis, which are nevertheless not obviously a matter of truth conditions), or as (walk)(aD(man))11 (which does the same as the 8
See Russell (1905). See Barwise & Cooper (1981). 10 See Groenendijk & Stokhof (1991); the indices indicate that the indexed operators are not the ones of standard logic, but rather their dynamic counterparts. 11 See Peregrin & von Heusinger (1995). 9
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previous one while being again closer to the surface). Each of these analyses may be useful for some purposes, but none of them is the analysis (there also seems to be a certain trade-off between exhaustiveness of analysis and simplicity and prosperousness of the means employed). Hence there seems to be nothing as the logical form, but rather only various ways to envisage truth conditions (or perhaps some more general semantic features)12. However, what if we accept that logical form may be nothing over and above surface form, and claim that this structure sometimes corresponds to an open formula? What if we argue that there are some expressions, e.g. pronouns, which behave expressly like free variables? Two kinds of objections to such a view can again be raised. First and foremost, it is not entirely clear what is meant, in this context, by ‘behave like a variable’. We have seen that the original role of a variable was to mark a place within a gappy expression - and this is a metalinguistic role, a role within a certain enterprise of decomposing expressions and reasoning about them (especially examining what happens when we fill their gaps in various ways). And this does not seem to be a role meaningfully ascribable to an expression of natural language. So what about the more general characterization of variable, like: that which can stand for any element of a domain? Does natural language not comprise expressions which behave in this way? Does natural language not contain expressions which are said to have variable or distributive reference? Here the trouble is that to classify a natural language expression as something ‘which can stand for any element of a domain’, we would require the fixed relation of standing for or reference - and elsewhere I have argued at length that the relation of reference is too tricky to be taken as an ‘unexplained explainer’ (see Peregrin, 1995, Chapter 8; and especially Peregrin, to appear). The second trouble is that even if we grant that some elements of natural language, e.g. pronouns, stand variably for different things, it is hard to deny that in other respects they are often very much unlike variables. Take the sentence He walks. In which respects is it like the open formula walk(x) (or taking into account the gender of he, walk(x)∧ ∧man(x))? I am afraid that only in a single one: in that its subject is not a name (an individual constant). For when one says he walks, he surely does not invite the hearers to substitute every (or any) conceivable individual for he. He walks is always synonymous with a sentence containing a name (or at least a description) in place of he (if I say he walks pointing at Salman Rushdie, it is synonymous with Salman Rushdie walks), albeit it is synonymous with different sentences on different occasions. In contrast, walk(x) is clearly equivalent with no formula of the shape walk(c) with constant c; and the context in which it occurs play no role with respect to its being equivalent to other formulas. A different way to salvage the absolutist notion of logical form may appeal to the analyses of the human ‘language faculty’ as provided by Noam Chomsky. Chomsky’s theories seem to imply that logical form is something ‘tangible’, which in some sense really exists and can be depicted. However, a little reflection reveals that the Chomskian notion of logical form has little to do with logic and hence with the realm where we have concluded the nature of the concept of variable should be sought. Chomsky’s logical form is one of the ‘levels of representation’ which constitute his reconstruction of ‘human language faculty’, a level which constitutes, as he puts is, the interface between language and other cognitive systems (see, e.g., Chomsky, 1986). Chomsky claims that his logical form „does in fact have many of the notational properties of familiar logical form, including the use of quantifier12
As Quine (1972, p.453) puts it: „Logical analysis ... may go one way or another depending on one’s specific logical purpose.“
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variable notation“, that this is, however, a contingent fact. (ibid., 156)13. Moreover, Chomsky defines variable as a certain species of what he calls the empty category, namely an „ A bound r-expression that must have Case by the visibility condition“ (ibid., 164). I think it is quite clear that these concepts of logical form and of variable have nothing to do with the Fregean, and hence with the principal logical, ones.14
3. Variables as Means of ‘Functional Classification’ 3.1 Variables and Rules of Composition The moral of the previous chapters is that from the viewpoint of analysis of natural language, variables are, in principle, dispensable. However, if we look at the way logical analysis is usually being carried out, what we see is that they are, in fact, far from dispensed with; and this indicates that despite of being dispensable, variables must be remarkably useful. In this chapter we are going to examine what this usefulness consists in, and we are going to try to indicate how the role of variables can be easily misunderstood. Elsewhere I claimed that variables are best seen as „auxiliaries helping us to codify complicated rules“ (see Peregrin, 1995, p. 103). Here I want to elaborate on this point: variables, I am going to show, are useful tools to characterize complex rules of composition (especially grammatical rules of linguistic systems), and also to specify the behavior (functioning) of some elements of these systems with respect to the rules. I claim that this is what creates the illusion of variables being ‘contained’ within natural language expressions: we use variables to articulate the linguistic, and especially semantic, functions of expressions, and mistake the articulations for expositions of the expressions’ ‘insides’. However, variables are no more inside expressions than holes for screws are inside screwdrivers. To justify this thesis, we must first articulate a general framework to talk about objects and their components. So let us imagine a realm of objects some of which are parts of others. The realm can be clearly seen as a set, call it R, and the part-whole relationships can be articulated in terms of a family i∈I of operators over R which represent the ‘ways of composing’ elements of R into other elements of R. This means that e=Oi(e1,...,en) renders the fact that e is composed of e1,...,en (in the way Oi). Seen thus, the part-whole system can be considered as the ordered pair i∈I> and hence as the (partial) algebra whose carrier is R and whose operators are Oi. Alternatively we can imagine the elements of R as categorized into sorts j∈J according to their behavior w.r.t. the ways of composition, and also imagine the ways of composition i∈I refined into i∈I’ so that each Oi’ is from the Cartesian product of some sorts into a sort; and we can thus see the system as the many-sorted algebra <j∈J, i∈I’> (where for each Oi’ there exist j1,...,jm,jm+1 so that the domain D(Oi’) of Oi’ equals the Cartesian product Rj1×...×Rjm and the range R(Oi’) of Oi’ is included 13
Personally, I find it hard to see the positive contents of this claim: for what does it mean, in this context, to have ‘notational properties’? 14 Unfortunatelly many people do not realize that the logical form of logicians and that of Chomsky are two essentially different things, and that if we use the same word to refer to both of them, it is nothing more than a pure homonymy. I think that not seeing this and merging logical analysis with Chomskian linguistics is tantamount to aiming at theories akin to such as would result from merging metallurgy and biology into a unified theory of ‘nails’.
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in Rjm+1)15. If O is an operator of a many-sorted algebra A and R is a sort of A, then we shall say that O takes (elements of) R (as its ith arguments) iff D(O) = R1×...×Ri-1×R×Ri+1×...×Rn for some sorts R1,..., Ri-1,Ri+1,...,Rn of A; and if O takes elements of R as its ith arguments and does not take elements of R as its jth arguments for any j≠i, then we shall say that O takes (elements of) R uniquely. Language is one of the important things which can be seen as part-whole systems and hence as partial or many-sorted algebras: the carrier of the algebra is constituted by the expressions, and the operators are the grammatical ways of putting expressions together to yield more complex expressions (in the case of language, we shall thus sometimes speak about rules instead of about operators). For the sake of illustration, let us consider a very simple language L0 containing names, unary predicates, and sentences each of which is the concatenation of a name and a predicate. If N is the set of names, P the set of predicates, S the set of sentences, and PRED the operation of concatenating a name with a predicate into a sentence (thus PRED(n,p)= n∩p, where n∩p symbolizes the concatenation of n and p16), then we can see L0 as the partial algebra >; or, better, as the many-sorted algebra <,>, where PRED is from N×P into S. According to our definitions, PRED takes names as first arguments, and it takes predicates as second arguments; it takes both names and predicates uniquely. Let us also consider the extension L1 of L0: L1 contains names, predicates and sentences like L0, but in addition to L0 it contains binary sentential connectives and such sentences which consist of two sentences linked by a connective. This means that L1 can be seen as the many-sorted algebra <,>, where CON is from S×C×S into S and CON(s1,c,s2)= s1∩c∩s2. CON takes sentences as its first arguments and also as its third arguments; so it does not take sentences uniquely. Let us further assume that CON contains the expression ‘and’. Now given the rules of L1, we can take a name n and two predicates p1 and p2 and form a sentence consisting of two subsentences, n∩p1 and n∩p2, connected by ‘and’. We can do this by first applying PRED to p1 and n, then PRED to p2 and n, and then CON to ‘and’ and the results of the previous two operations. Thus, there is a (‘complex’) operator which takes a name n and two predicates p1 and p2 into the sentence n∩p1∩‘and’∩n∩p2. The existence of this complex rule is in a sense implicit to the existence of PRED and CON (and ‘and’). Now to characterize this complex operator, we have to say something like: first, PRED is applied to the first predicate and the name, then PRED is applied to the second predicate and the name, and then CON is applied to ‘and’ and the results of the previous two operations; or better to say, as we did, first PRED is applied to p1 and n, then PRED is applied to p2 and n, and then CON is applied to ‘and’ and the results of the previous two operations. When we 15
For the concept of many-sorted algebra and for its applications to natural language see Janssen (1983). Of course not every algebra could be reasonably seen as amounting to a part-whole system: the relation of being a proper part is essentially acyclic (i.e. its transitive closure is antisymmetric); and hence if an algebra is to be seen as (amounting to) a part-whole system, it has to satisfy this restriction. Thus, the operators of, e.g., an algebra in which, for some elements a, b, c, d and some operators O and O’, a=O(b,c) and b=O’(a,d) cannot be reasonably seen as ‘rules of composition’ - for this would mean that we would have to see b as a proper part of a and in the same time a as the proper part of b. 16 As we want L0 to be English-like, we in fact assume that PRED is more than mere concatenation, that it modifies the predicate in the appropriate way (in the simplest case by appending the suffix ‘s’). However, for the sake of simplicity we shall speak simply about concatenation.
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choose the latter way of articulation, we need some symbols to regiment the expressions ’the first predicate’, ‘the second predicate’ and ‘the name’ of the former articulation. If we employ such variables, we can directly designate the complex operator by some self-explicating schema like CON(PRED(n,p1),’and’,PRED(n,p2)). This means that in this context we employ variables to derive designators of complex operators from those of basic operators (plus, as the case may be, those of elements). Complex operators which are in this way implicit to the operators of an algebra A are sometimes called polynomials over A. Polynomials arise out of composing and iterating the operators of A (and some trivial operators, namely projections, i.e. functions mapping n-tuples on their ith constituents). The operator described in the previous paragraph, CON(PRED(n,p1),’and’,PRED(n,p2)), is a polynomial over L1. To say that operators of an algebra are closed under forming polynomials is to say that the operators include projections and that they can be composed and iterated (and it seems that the operations of forming wholes from parts should be closed in this sense). Now if we look at a book where polynomials are defined (Grätzer, 1979, Janssen, 1983), what we see is that the employment of some kind of variables is quite essential to the whole enterprise. However, in this context, variables are simply tools employed to form selfexplicating designators for complex operators or rules. In general, if we accept composibility of operators (and the existence of projections) as a general principle, we accept that the existence of any family of operators involves the existence of all polynomials based on the family; and variables are indispensable tools of articulating canonical names for the polynomials. Notwithstanding this though, we must realize that here we are using variables on the metalevel, we use them to talk about language, more precisely about rules of language. There is thus no question of these variables being ‘inside’ expressions.
3.2 Variables and Expressions’ Functioning Let us now examine a part-whole system from the viewpoint of the ‘behavior’ of its elements. The behavior clearly consists in the ways in which the elements, together with other elements, constitute compounds. To specify the behavior of an object it is necessary to summarize all cases of composition into which it can enter. Let us articulate this idea formally: let A be a manysorted algebra, a an element of a sort R of A, and O an operator of A which takes R as its ith argument, i.e. such that there are some sorts R1,...,Ri-1,Ri+1,...,Rn,Rn+1 of A such that D(O) = R1×...×Ri-1×R×Ri+1×...×Rn and R(O) = Rn+1. The (O,i)-trace of a will be the function fO,i,a from R1×...×Ri-1×Ri+1×...×Rn into Rn+1 such that if ∈D(O) for some a1,...,ai-1,ai+1,...,an, then a1,...,ai-1,ai+1,...,an∈D(fO,i,a) and fO,i,a(a1,...,ai-1,ai+1,...,an) = O(a1,...,ai-1,a,ai+1,...,an). If O takes a uniquely, then we shall also speak about the O-trace of a instead of about its (O,i)-trace. The O-trace of a thus in a sense characterizes ‘the behavior of a with respect to O’; and all the traces of a w.r.t. all operators which take a characterize the behavior of a w.r.t. the whole system. Now the only aspect of an item which is relevant from the viewpoint of a system is clearly its behavior w.r.t. the system; thus, from this viewpoint, if we are able to capture the behavior as an object (function), we could well deal directly with the behavior instead of with the element itself. Let us assume that an element is uniquely determined by its behavior, i.e. that there are no two elements with exactly the same behavior. A particularly instructive case 11
obtains when items of a sort are taken by a single rule and are taken by it uniquely; or if, more generally, the behavior of the items w.r.t. all the rules which take them is uniquely determined by their behavior w.r.t. the single rule. This means: Let O be an operator and R a sort taken uniquely by O and such that any two elements of R sharing the same O-trace are identical (and hence share the same O’-trace for every O’ which takes them). In this case, an element of R is uniquely determined by its O-trace; and then we can justifiably identify it with its Otrace. Now suppose that we do this for all elements of R; that is, for every a∈R, we identify a with fO,a. Then obviously for every ∈D(O), O(a1,...,an) = a(a1,...,ai-1,ai+1,...,an). Hence we are, in a sense, ‘delegating’ the working of O to the objects of the sort R; the work of the new O is merely to bring the capacities of these objects to bear on their fellowarguments. Such ‘functionalization’ of items of a sort thus deprives the rule involved of all its substantial content rendering it a purely formal ‘applicator’ - all content is localized in the objectual form. Note also, that the function fO,a coincides with (the course of values of) the polynomial O(x1,...,xi-1,a,xi+1,...,xn): the domain of both of them is R1×...×Ri-1×Ri+1×...×Rn, their common range is Rn+1, and for every a1,...,ai-1,ai+1,...,an from the common domain, the value of both is O(a1,...,ai-1,a,ai+1,...,an). So by identifying a with its behavior, we are identifying it with a certain polynomial operator; and since we have found variables indispensable to articulate polynomials, we now see that they may be useful to articulate objects’ behavior. Let us return to our example part-whole system L0. The behavior of the name n is characterized by its PRED-trace, i.e. by the function fPRED,n such that for every predicate p, fPRED,n(p) = PRED(n,p) (= n∩p). Similarly, the behavior of a predicate p is characterized by the function fPRED,p such that for every name n, fPRED,p(n) = PRED(n,p). Suppose now that we identify names with their ‘behaviors’: i.e. that we identify each n with its trace fPRED,n. The operator PRED then becomes the application of a name to a predicate - since for every such new name n and every predicate p, PRED(n,p) = n(p). Alternatively, we could identify predicates (instead of names) with their behaviors, i.e. each p with its trace fPRED,p, which would result in PRED becoming the application of a predicate to a name. Now imagine that we replaced both names and predicates with their behaviors: then, clearly, PRED(n,p) would be neither n(p), nor p(n) (for then names would be functions from P into S, predicates would be functions from N into S, leaving none of them within the domain of the other). This indicates that optimally we might ‘functionalize’ only one of the sorts. The reason for this, informally speaking, is that the working of PRED can be delegated either to names, or to predicates17, but to delegate it to both at once is to duplicate it to a harmful effect. Thus, if our aim is to shift as much of functioning as possible from rules to elements, our best course is to find for each rule its own sort to whose elements the working of the rule is delegated. (Whether we can thereby manage to ‘empty’ all rules depends on the structure of the system18).
17
In fact, traditional logic has it in the latter way, while, e.g. Montague Grammar exploits the former one. 18 The necessary condition is that there exists an injective mapping M of operators on sorts such that for every operator O, O takes M(O) and takes it uniquely. However, this will not suffice: in order to avoid circularity in the definition of functions, there has to exist a well-ordering of sorts such that for every operator O, the sort M(O) is strictly greater than all other sorts taken by O. The existence of such an ordering is tantamount to the existence of a ‘categorial indexing’, i.e. to the possibility of
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Now if an element of the system is uniquely determined by its behavior, we can say that it is the element which behaves thus and so. We can, e.g., say that a predicate, say walk, is the element of L0 which takes ‘Peter’ into ‘Peter walks’, ‘Mary’ into ‘Mary walks’ etc.; i.e. that it behaves in the way fPRED,’walk’. Moreover, as every predicate’s behavior coincides with the workings of a certain polynomial rule, we can specify its behavior by pointing out the rule. Thus, if we want to specify the behavior of ‘walk’, we can point out the polynomial PRED(‘walk’,x) (or, expanding the definition of PRED, x∩’walk’). To indicate that what is now referred to is an object, rather than a rule, we usually make use of something like the lambda notation: we say that ‘walk’ is the item λx.PRED(‘walk’,x), or that it is λx.x∩’walk’19. However, to say this is not to say that the word ‘walk’ contains a variable, it is rather to envisage its behavior. Hence, the conclusion is that as we can profitably employ variables to codify complex rules (polynomials), and as the behavior of some expressions is identifiable with the working of some such rules, variables are helpful tools for specifying expressions’ behavior, tools of the ‘functional classification’ of expressions20.
3.3 Semantic Interpretation Language is not simply a system of expressions composible into more complex expressions; its crucial feature is that its expressions have meaning. And indeed, the relevant functional classification of expressions, which invokes variables, is usually a matter of semantic functioning; i.e. not only a matter of the ways in which expressions take part in constituting more complex expressions, but rather a matter of the ways in which they contribute to the meaning of the wholes in which they occur. Within our algebraic setting, this means that we should not reconstruct language as simply an algebra of expressions, but that we should see each element of the algebra associated with a meaning (whatever it may be21). And as meanings are assumed to be compositional, we can see mappings of expressions on their meanings, semantic indexing sorts in such a way that each operator comes to combine an element of a sort A/B1...Bn with elements of the respective sorts B1,...,Bn into an element of the sort A. See also Peregrin (1992). 19 We have been speaking rather loosely abour ‘identifying an element with a function’, about ‘an operator becoming an application’, etc. Each such identification involves the replacement of the original algebra by another algebra, which is nevertheless isomorphic with the original one. What makes it possible to pass freely from the one algebra to the other is that we are investigating structural properties. To algebraically reconstruct a factual range of items (e.g. the range of expressions of a factual language) is to define an algebra embodying the relevant structure of the range - what is important is the structure alone, the factual nature of the elements of the algebra is no more important than the nature of the marks chosen to site towns on a map. So we can also employ such items which, so to say, wear their functioning on their sleeves - and it is precisely this kind of reconstruction which often appears most useful. 20 The term functional classification is purposefully chosen to allude to the same term employed by Wilfrid Sellars (1974). 21 I have repeatedly stressed (see esp. Peregrin, 1995) that language cannot be seen as a nomenclature, as a set of labels stuck on pre-existing real-world objects. However, once we drop the assumption that meanings are ‘real’, ‘pre-linguistic’ objects, then the picture of meanings as associated with expressions becomes harmless and is often useful.
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interpretations, as homomorphisms from the algebra of expressions into an algebra of denotations22. Now such a semantic interpretation can be seen as having two parts, corresponding to the two different ways of defining the denotations, and hence explicating the semantic functions, of individual sorts of expressions. (i) The semantic functioning of expressions of some sorts, of the ‘basic’ ones, are ‘explicated’ only in an utterly trivial way - it is simply stipulated that each of these expressions has some denotation. This means that for the basic sorts of expressions it is simply assumed that there exist corresponding domains of denotations and that the elements of the former are mapped on those of the latter. Nothing nontrivial is said about the nature of the domains and their elements. (ii) The semantic functioning of expressions of the other, ‘nonbasic’ sorts is then explicated relatively to the functioning of those of the ‘basic’ ones. This is to say, the denotations of expressions of the nonbasic sorts are identified with their behviors, i.e. with the ways in which they constitute, together with the denotations of expressions composible with their ones, denotations of the resulting compounds. To see this working, let us consider L0 once more. The usual way to define semantics for a language of this kind is to assume a domain U of individuals and a domain B of propositions (which might be, in the simplest case, the two truth values), to map each name on an element of U, each predicate on a function from U to B (we can call such functions properties), and each sentence consisting of a name n and a predicate p on that element of B which arises out of the application of the function denoted by p on the denotation of n. This is to say that the semantic interpretation of L0 is a homomorphism from L0 into the algebra D0=<,<APPL>>, where [U→B] is the set of functions from U to B and APPL takes an element from U and an element from [U→B] into the value of the application of the latter to the former (APPL(x,y)=y(x)). Thus, N and S are treated as semantically ‘basic’ categories of L0: their expressions are simply taken to denote some elements of the corresponding domains U and B. In contrast to them, the semantic function of the elements of P is really explicated - albeit only relatively to the semantic functions of the elements of the basic categories. Passing from L0 to L1, we add another non-basic category: its elements are semantically interpreted by being mapped on functions from B×B to B (call such functions propositional junctors) so that the denotation of a sentence consisting of two sentences connected by the connective results from the application of the function denoted by the connective to the denotations of the two contained sentences. The denotation-algebra is now D1=<,<APPL1,APPL2>>, where APPL1 is from U×[U→B] into B such that APPL1(x,y)=y(x), and APPL2 is from B×[B×B→B]×B into B such that APPL2(x,y,z)=y(x,z). Now let us consider the operation taking an individual i and two properties r1 and r2 into a proposition that the individual has both the properties. This operation consists in first applying APPL1 to i and r1, then applying APPL1 to i and r2, and finally applying APPL2 to the conjunction function and the result of the previous two operations. If we denote the conjunction function by &, the operator is the polynomial APPL2(APPL1(i,r1),&,APPL1(i,r2)). Allowing the explicit names of the application operators to give way to the usual bracket notation gives us &(r1(i), r2(i)); and by writing & in the usual infix way, we have r1(i) & r2(i). Now if we consider the semantic interpretation of L1 in D1 and assume that ‘and’ denotes &, we see that this polynomial operator over D1 is the semantic 22
See Janssen (1983, Chapter I).
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counterpart of the polynomial CON(PRED(n,p1),’and’,PRED(n, p2)) over L1; we can say that the former embodies the semantic function of the latter (or that the latter expresses, or represents, or stands for, the former). The same applies to the elements of the algebras. That kind of function of an element of L1 which was discussed in the previous section can be called its syntactic function, whereas its semantic function is actually the function of its denotation within the algebra D1; and since the denotation has been devised precisely so as to directly coincide with its function, it is this denotation itself. We have seen that the syntactic function of the predicate ‘walk’ is the function λx.PRED(x,’walk’) (= λx.x∩’walk’). The denotation of ‘walk’ is now the function which takes the individual denoted by the name ‘Peter’ into the proposition denoted by the sentence ‘Peter walks’, the individual denoted by the name ‘Mary’ into the proposition denoted by the sentence ‘Mary walks’, etc. If we designate that element of D1 on which an expression of L1 is mapped by the italicized form of the expression (so that the individual denoted by the name ‘Peter’ is Peter, the proposition denoted by the sentence ‘Peter walks’ is Peter walks etc.), we can say that the denotation of ‘walk’ is the item walk such that walk(Peter) = Peter walks, walk(Mary) = Mary walks, etc.; hence, in other words (or rather signs), the denotation of ‘walk’ is λx.walk(x). Thus, while the syntactic function of ‘walk’ (= its behavior w.r.t. the algebra L1) is λx.x∩’walk’, its semantic function (= the behavior of its denotation w.r.t. the algebra D1) is λx.walk(x). Saying that the semantic function of ‘walk’ is λx.walk(x), however, has only little informative value: λx.walk(x) is nothing else than walk, and saying that the denotation of ‘walk’ is walk is nothing more than saying that the denotation of ‘walk’ is that element of the denotation algebra which is denoted by ‘walk’. To see how a semantic function can be pointed out in a less trivial way, let us consider the extension L2 of L1. L2 has the same carrier as L1, but in addition to the operators of L1 it contains the operator PCON taking a predicate, a connective, and a predicate into a predicate: thus if p1 and p2 are predicates and c a connective, then PCON(p1,c,p2) is a predicate; and thus if n is a name, then PRED(n,PCON(p1,c,p2)) is a sentence. Hence, PCON takes, e.g., the expressions ‘walk’, ‘and’ and ‘whistle’ into the complex predicate ‘walk and whistle’, which can then be combined with ‘Peter’ into ‘Peter walks and whistles’. In order to extend a semantic interpretation of L1 in D1 to the interpretation of L2, we must extend D1 with an operator which would be the semantic counterpart of PCON - let us call it PCON*. It seems to be natural to require that ‘Peter walks and whistles’ denote the same proposition as ‘Peter walks and Peter whistles’23; that is, to require that for every n, p1, c and p2, the denotation of PRED(n,PCON(p1,c,p2)) coincides with that of CON(PRED(n,p1),c,PRED(n,p2)). If this is so, then for every individual i, every properties r1 and r2, and every propositional junctor o, PCON*(r1,o,r2)(i) = r1(i) o r2(i). That is, the predicate PCON*(r1,o,r2) takes an individual i into the proposition r1(i) o r2(i); hence PCON*(r1,o,r2) = λi. r1(i) o r2(i). This means that, e.g., PCON*(walk,&,whistle) (= the denotation of PCON(‘walk’,’and’,‘whistle’), i.e. the denotation of ‘walk and whistle’) is λi. walk(i) & whistle(i)24. In this way we describe certain elements of the denotation algebra, and thereby the semantic functions of certain elements of the expression algebra, by means of variables. We take for granted the denotations of the
23
It certainly is natural if we assume that ‘denote the same proposition’ is to explicate ‘have the same truth conditions’. 24 Remember that ‘&’ was introduced earlier as a name for what we now see as the denotation of ‘and’, i.e. for and.
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basic predicates and connectives (‘walk’, ‘whistle’, ‘and’), and we explicate the denotations of, e.g., complex predicates (‘walk and whistle’) in their terms. Therefore we are using variables to designate the functioning of the elements of the denotation algebra, and as we also take the elements of the expression algebra to be endowed with meanings via being mapped on the elements of the denotation algebra, we thus use variables to explicate their meaning. However, by saying that ‘walk and whistles’ means (or denotes, or stands for, or expresses, or whatever) λi. walk(i) & whistle(i) we are not saying that ‘walk and whistles’ contains the variable i: what we say is that the semantic behavior (= meaning) of the expression is describable in terms of the result of its application to individuals.
3.4 Composing and Decomposing Constructions Now let us take an even more abstract vantage point: let us think about constructing entities in general. We construct complex entities by putting simpler entities ‘together’: within the physical world (as when we construct a chair out of logs and nails), or within the world of abstract entities (as when we construct the sum out of two numbers). The ways of ‘putting together’ are ‘domain-specific’: when we construct a chair, then they consist in certain fastenings of objects one to another with the help of other objects; when we construct a number, they consist in adding, multiplying and the like. However, there seem to be some quite general principles governing all kinds of constructings. It seems that it is in the nature of the concept of constructing that constructions can be composed and iterated, and that the trivial constructions, like choice, are generally available. This is reflected by the assumption that by explicitly accepting a set of basic constructions we implicitly accept all the polynomials based on them. The idea is that if we are able to construct x out of y and y out of z, then we are surely able to construct x out of z; more generally, if we are able to construct x out of x1,...,xn and if we are able to construct xi out of xi1,..., ximi (for every i), then we are able to construct x out of x11,...,x1m1,..., xn1,...,xnmn. In short, it seems to be in the nature of constructing that constructions are composible (and hence closed under forming polynomials). Thus, addition of polynomials only makes explicit what is already implicit. In algebraic terms, this means that an algebra A+ which is the polynomial extension of a given algebra A (i.e. which differs from A in that its family of operators contains some operators which, while not operators of A, are nevertheless polynomial over A) amounts to the same constructional system as A. Janssen (1983) has proven the following theorem: if A and B are many-sorted algebras, h a homomorphism from A to B, and A+ a polynomial extension of A, then there is a polynomial extension B+ of B, and a unique extension h+ to a homomorphism from A+ to B+. If we assume that semantic interpretation is by its nature a homomorphism, then the theorem implies that addition of polynomial symbols is trivial also in the sense of not being able to tamper with semantic interpretation: every semantic interpretation of a language can be extended to a semantic interpretation of a polynomial extension of the language. Now besides the fact that the space of constructions is closed under composition, we may consider its being closed under the inverse operation, decomposition. Whereas the principle of compositibility of constructions says that we can always merge two subsequent constructional steps into one, the inverse principle of decomposition says that we can always divide a nontrivial constructional step into two subsequent simpler steps. The idea is that if 16
we can use x, y and v to construct z, then it is also possible to, first, use x and y to construct an intermediary entity w, and then use w and v to construct z. In general, the principle of decomposition says that if we can construct x out of x1,...,xn and if {i1,...,im} and {j1,...jk} are disjoint sets whose union is {1,...,n}, then there is a y which can be constructed out of xi1,...,xim and such that we can construct x out of y,xj1,...,xjk. It is, of course, dubious, whether we can assume such a principle to hold generally: when we are constructing physical things, then, as experience teaches us, we cannot decompose each step as we please: if we, e.g., fasten two logs together by a string, we cannot combine one of the logs with the string into a ‘half-way construct’ ready for subsequent combination with the other log to yield the ultimate construct (the two logs tied together). However, the situation is different within the realm of abstract entities: there seems to be no reason not to accept decomposibility as a general principle. (Indeed, this fact might perhaps be one of the characterizing differences between the two realms.) Now assume that decomposition not only always exists, but also is unique. This means that if C is a unary construction and {i1,...,im} and {j1,...jk} are disjoint sets whose union is {1,...,n}, then there is a uniquely determined pair of constructions C1 and C2 so that C(x1,...,xn) = C2(C1(xi1,...,xim),xj1,...,xjk); and hence for any concrete objects a1,...,an from the domain of C there is an intermediary, ‘half-way’ construct C1(ai1,...,aim) which is constructed out of ai1,...,aim and is capable of yielding the ultimate construct C(a1,...,an) together with the rest of the arguments aj1,...,ajk. The object C1(ai1,...,aim) can be characterized by its disposition to take its part in constructing C(a1,...,an) - and the plausible way to designate it seems to be something like the lambda notation. When we consider composing operators, we need to designate complex, composed operators (polynomials), and we employ variables to help us build complex designators for these operators from the simple names of basic operators. When we consider decomposing operators, certain objects become known as the ‘half-way constructs’ and they become illuminatingly designatable as such. Thus an object may become viewable as, e.g., a ‘half-way construct’ on the way from the objects x1,...,xn to the object C(x1,...,xn): as that ‘half-way construct’ which arises out of exploiting xi1,...,xim, but still not exploiting xj1,...,xjk. Such an object is then plausibly designated by means of an expression such as ‘λxj1...xjk.C(x1,...,xn)’. As polynomial symbols characterize the range polynomial operators, the new symbols I proposed earlier (see Peregrin, 1992) to call abstractive, characterize a more general range of operators, abstractive operators. What I then claimed was that the addition of such abstractive operators to an algebra is still trivial in a sense analogous to that in which it is trivial to add polynomials. To see this, we must realize that semantic interpretation, as the term is usually understood, is not simply a homomorphism, but typically a homomorphism into an algebra of a specific kind. In the terminology of the present paper, it is a homomorphism into an algebra the operators of which are formal, i.e. devoid of content (which means, as explained in the previous chapter, that they only bring the capacities of some elements of the algebra to bear on some other elements). In Peregrin (1992) I called such algebras applicative and I have proven the following theorem: if A is an algebra, B an applicative algebra, h a homomorhpism from A to B, and A+ an abstractive extension of A, then there is an extension B+ of B, and a unique extension h+ of h such that h+ is a homomorphism from A+ to B+. This means that by employing not only variables, but also the lambda operator, we still only make explicit the implicit semantic capacities. However, while polynomial symbols are bound to stand for operators, abstractive symbols (lambda-terms) may, in the limit case when all variables are ‘lambda-abstracted away’, stand for objects. In this way variables 17
appear to be revealing us something about things (in particular about expressions); but what they really do is only revealing (making explicit, explicating) certain implicit capacities of the things’ behavior.
4. Conclusion Regimenting a natural language expression by a formula containing variables should be seen as revealing neither the variables’ covert presence ‘within’ the expression, nor their presence ‘within the expression’s meaning’, but rather as a functional, usually semantic, characterization of the expression. Variables have been introduced, and are still best seen as certain metalinguistic tools, as means of designating functions.
References Barwise, J., Cooper, R.(1981): ‘Generalized Quantifiers and Natural Language’, Linguistics and Philosophy 4, 159-219. Bourbaki, N. (1958): Élements de Mathématique I: Théorie des ensembles, Hermann, Paris. Chomsky, N.(1986): Knowledge of Language, Praeger, New York. Church, A. (1940): 'A Formulation of the Simple Theory of Types', Journal of Symbolic Logic 5, 56-68. Došen, K. (1988): ‘Second-order logic without variables’, in Categorial Grammar (ed. W.Buszkowski, W.Marcizsewski & J. van Benthem), Benjamins, Amsterdam. Frege, G. (1879): Begriffsschrift, Nebert, Halle; translated as Begriffsschrift, in van Heijenoort (1971), 1-82. Frege, G.(1891): ‘Function und Begriff’, ein Vortrag, gehlaten in der Sitzung vom 9.1.1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft, Jena; translated as Function and Concept in Geach and Black (1952), 21-41. Geach P. and M. Black (1952): Translations from the Philosophical Writings of Gottlob Frege, Blackwell, Oxford. Grätzer, G.(1979): Universal Algebra, Springer, New York. Groenendijk, J., M.Stokhof (1991): ‘Dynamic Predicate Logic’, Linguistics and Philosophy 14, 39-101. Janssen, T.M.V. (1983): Foundations and Applications of Montague Grammar, dissertation, Mathematical Centre, Amsterdam. Peregrin, J. (1992): 'The Role of Variables in the Formalization of Natural Language', in Proceedings of Eight Amsterdam Colloquium, Amsterdam, 463-481. Peregrin, J. and von Heusinger, K. (1995): ‘Dynamic Semantics with Choice Functions’, in Choice Functions in Natural Language Semantics (ed. U. Egli & K. von Heusinger), Universität Konstanz, 43-67. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (to appear): ‘Reference and Inference’, Proceedings of the Workshop „Reference and Anaphorical Relations“, to be published by Universität Konstanz Quine, W.V. (1960): 'Variables explained away', Proceedings of the American Philosophical Society 104, 343-347. 18
Quine, W.V. (1972): 'Methodological Reflections on Current Linguistic Theory', in Semantics of Natural Language (ed. D.Davidson and G.Harman), Reidel, Dordrecht, 442-454. Russell, B. (1905): 'On denoting', Mind 14, 479-493. Sellars, W. (1974): ‘Meaning as Functional Classification’, Synthèse 27, 417-437. van Heijenoort, J., ed. (1971): From Frege to Gödel: A Source Book from Mathematical Logic, Harvard University Press, Cambridge (Mass.)
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THE LOGIC OF ANAPHORA Jaroslav Peregrin* www.cuni.cz/~peregrin [The Logica Yearbook 1999 (ed. T. Childers), FILOSOFIA, Prague, 2000, 191-205]
Abstract. The paper addresses foundational questions concerning the dynamic semantics of natural language based on dynamic logic of the Groenendijko-Stokhofian kind. Discussing a series of model calculi of increasing complexity, it shows in detail how the usual semantics of dynamic logic can be seen as emerging from the account for certain inferential patterns of natural language, namely those governing anaphora. In this way, the current ‘dynamic turn’ of logic is argued to be reasonably seen not as the product of changing the focus of logic from the relation of entailment to „a structure of human cognitive action“ (van Benthem), but rather as merely another step in our long-term effort to master more and more inferential patterns.
1. The dynamic turn of logic Classical logic may be seen as having given a faithful account of a certain core part of language. Some of the logics which go beyond its boundaries have then helped to account for other, more intricate parts. One of those elaborations which have proven to be especially interesting has been modal and intensional logic. The idea behind it is that we should let statements denote subsets of a set of indices rather than simply truth values. (Kripke called such indices possible worlds, but this term might be misleading, because it seemingly and unwarrantedly transposes Kripke’s originally purely logical achievement into the realm of metaphysics.) The fact is that the step from extensional to intensional logic made by Kripke and other semanticists of modal and intensional logic (notably Montague) can be seen as necessitated by the desire to master the modal aspect of language, primarily especially the particles necessary and possibly, usually regimented as and ◊, respectively. (It has turned out that other grammatical constructions of natural language, like tenses, are of an essentially similar character too.) And the basic trick of the step, to say it once more, was the passage from seeing statements as names of truth values to seeing them as names of sets. At present we are facing the effort to master another important aspect of language, which escapes even the modal and intensional conception of logic, the anaphoric aspect1. The target particles of this dynamic turn of logic appear to be pronouns and articles. It is now also quite clear what this turn amounts to in general2: we should stop seeing statements as names *
I would like to thank Barbara Partee for valuable critical comments. A previous version of the text was presented at the 3rd conference in Information-Theoretic Approaches to Logic, Language and Computation (Hsitou, Taivan, June 1998). 1
A formal account of anaphora has become an important problem for semanticists of natural language especially due to the pioneering work of Hans Kamp resulting in his DRT - see Kamp, 1981, and Kamp & Reyle, 1993. Other approaches to anaphora are due to, e.g., Hintikka and Kulas, 1985, or Heim, 1990; purely logical tools of such an account were presented by Greonendijk & Stokhof, 1991.
2
See Muskens et al. (1987) or van Benthem (1997).
1
of sets and start to see them as names of certain transformations of ‘contexts’ or ‘information states’. Thus intensions are being replaced by context-change potentials; statements are no longer characterized by the possible worlds in which they are true, but rather by the way in which they alter the information states which constitute the contexts of their utterances. What I think may still not be quite clear enough is the purely logical rationale of this step. The engagement of information states within logic is usually explained and justified by epistemological reasons: we need them, so the story goes, to do justice to how human subjects acquire their knowledge or use their language. (Van Benthem, 1997, p. ix, speaks about „the logical structure of cognitive actions, underlying human reasoning or natural language understanding“.) This seems to indicate that logic after the ‘dynamic turn’ is no longer logic in the good old sense of the word: that it is no longer about the inferential patterns governing our language, but now rather about the ways we exercise the patterns; no longer about truth but rather about the way we find out about truth. I think this can be treacherous (one of the basic points of Frege’s jump-start into the modern era of logic was the sharp distinction of matters of truth and consequence, which are independent of how anybody may come to know them, from matters of human subjects’ coming to recognize that something is true or that something is the consequence of something else); and therefore I think that we should seek a more conservative substantiation of the turn. The situation is similar within semantics: there too the switch from ‘static’ notions of meaning to the notion of meaning as a context-change potential is normally explained in terms of the need to account for „how [meaning] affects the information available to illocutionary agents" (Chierchia, 1994, 141). This again indicates that the ‘dynamic turn’ alters the subject matter of semantics: while earlier semantics addressed what expressions meant (perhaps their truth conditions), now it is to address (also) how they change the context. The turn thus might seem to amount to simply a deliberate shift of the boundaries of semantics, which made it include also some parts of what has been previously considered a matter of, say, the theory of communication. Again, I think this may be misleading, for the engagement of context-change potentials need not be the result of moving the boundaries of semantics, but merely of paying due attention to some of those expressions and locutions of natural language which were ignored before (pronouns, articles, anaphoric locutions). Thus also here a more conservative substantiation might be in place. What I mean by ‘a more conservative substantiation’ can be elucidated by volunteering a parallel between the dynamic turn with its information states and the intensional turn with its possible worlds. We can, of course, give various epistemological (or even psychological) reasons for the employment of intensional logic and possible worlds; but we can justify it also purely logically. The fact is that we need the operator as governed by certain axioms for the purpose of capturing certain inferential patterns which play an important role within our language and thereby within our reasoning. And another fact is that arguably the simplest and most perspicuous model theory for the relevant calculus treats statements as names of elements of the powerset of a fixed set and as a designation of a mapping of the power-set on itself. From this vantage point, possible-worlds-semantics is the natural outcome of the effort to account for certain inferential patterns of our language. And what I want to do in this paper is to render the information-states-semantics (or the semantics of context-change potentials) as the natural outcome of the effort to account for other kind of inferential patterns; namely those involving anaphora. This is to say that what I want to do is to point out the basic inferential patterns characterizing ‘the logic of anaphora’ and show how they can be seen to yield the basic framework of dynamic semantics. 2
2. Anaphora from the Viewpoint of Inference As the paradigmatic cases of inferences involving anaphora we shall consider (1)-(4): John walks. He whistles. ⇒ John walks and whistles. Somebody walks. He whistles. ⇒ Somebody walks and whistles. A man walks and a woman runs. He whistles and she smiles. ⇒ A man walks and whistles. A woman runs and smiles. A man walks. The man whistles ⇒ A man walks and whistles
(1) (2) (3) (4)
When we consider the possibility of capturing these instances within the framework of a predicate-calculus-like logic, we probably come to the conclusion that at least the first three of them could be accommodated quite easily if we managed to extend the calculus by means of pronoun-like terms. Then they may acquire the form of (1’)-(3’). Wa(J) & Wh(he) ⇒ Wa(J) & Wh(J) ∃xWa(x) & Wh(he) ⇒ ∃x(Wa(x) & Wh(x)) ∃x(Ma(x) & Wa(x)) & ∃x(Wo(x) & Ru(x)) & Wh(he) & Sm(she) ⇒ ∃x(Ma(x) & Wa(x) & Wh(x)) & ∃x(Wo(x) & Ru(x) & Sm(x))
(1’) (2’) (3’)
The question now is how we could add terms of this kind to the predicate calculus.
3. The Semantics of ‘Backwards-Looking’ Let us start from (1’). As the only types of expressions it contains aside of the problematic he are individual constants, unary predicate constants and conjunction, let us, for simplicity’s sake, start from a very simple language whose vocabulary is restricted just to these expressions. Hence the following language, which we shall call L. Expressions of L fall into three categories: individual constants, unary predicate constants and connectives (the last category consisting of the single constant &). If p is a predicate constant and i is an individual constant, p(i) is a statement; and if s1 and s2 are statements, s1&s2 is a statement. The standard semantics of L is then as one would expect: If U is a set, then ║║ is an interpretation of L in U iff ║i║∈U for every individual constant i, ║p║⊆U for every predicate constant p. The interpretation then extends to an assignment of truth values to statements in the natural way: for every individual constant i and every predicate constant p, ║p(i)║ = T iff ║i║∈║p║; and for every statements s1 and s2, ║s1&s2║ = T iff ║s1║ = T and ║s2║ = T. (The exact specifications of all the calculi discussed within the text can be found in the appendix.) In order to be able to enrich L with a constant behaving like he in (1’), we shall consider an alternative semantics for L, which is as follows. Individual constants are taken to denote functions from U to U, each of them being assigned a constant function defined everywhere on U. The denotations of predicate constants are as before. This interpretation induces an assignment to statements of functions from U to U in the following way: If i is an individual constant and p is a predicate constant, then ║p(i)║ is a (partial) function from U to U such that for every x∈U, ║p(i)║(x) is defined iff ║i║(x)∈║p║, and in this case ║p(i)║(x) 3
=║i║(x). (Thus, ║p(i)║ = {<x,y> | <x,y>∈║i║ and y∈║p║}; and as ║i║ is a constant function, ║p(i)║ is either identical with ║i║, or is empty.) If s1 and s2 are statements, then ║s1&s2║ is such function that for every x∈U, ║s1&s2║(x) = ║s2║(║s1║(x)); i.e. ║s1&s2║ = {<x,y> | there is a z such that <x,z>∈║s1║ and ∈║s2║}. A statement is true, by definition, if its denotation (which is a function) is defined everywhere on U; it is false if it is defined nowhere on U. It is easy to see that under any such alternative interpretation, every statement is either true or false. It is also easy to see that for each standard interpretation of L there exists an equivalent alternative interpretation and vice versa. (If ║║S is a standard interpretation, then the equivalent alternative interpretation ║║A is defined in the following way: ║i║A = {<x,y> | x∈U and y = ║i║S} for every individual constant i and ║p║A = ║p║S for every predicate constant p. If, conversely, ║║A is an alternative interpretation, then the corresponding ║║S is defined in such a way that ║i║S is the constant value of ║i║A.) Now let us enrich L with a ‘backward-looking’ term ← having the property that for ←) is equivalent every individual constant i and every predicate constant p1 and p2, p1(i)&p2(← to p1(i)&p2(i). We can do this easily if we start from the just defined alternative semantics for L - for then the desired equivalence is secured by taking ║← ←║ to be the identity function defined everywhere on U. To see that this works, let us consider the value of ║p1(i)&p2(← ←)║ applied to an element x of the universe. If we denote the single value of the constant function ║i║ as xi, then ║p1(i)&p2(← ←)║(x) = ║p2(← ←)║(║p1(i)║(x)), where ║p1(i)║ is either a constant function (if xi∈║p1║) or a function defined nowhere (otherwise), and ║p2(← ←)║ is an identical function defined for those and only those x for which x∈║p2║. Thus, ║p2(← ←)║(║p1(i)║(x)) is defined (and yields xi) iff xi∈║p1║ and xi∈║p2║, thus it is defined if and only if ║p2(i)║(║p1(i)║(x)) is. Hence, ║p1(i)&p2(← ←)║ is true (= defined for every x) if and only if ║p1(i)&p2(i)║ is true (defined for every x). Let us call this language L←. L← differs from L (considered with the alternative semantics) in that it contains formulas denoting functions which are defined for some elements of the universe and undefined for others, hence formulas which are, according to our definition of truth, neither true, nor false. This is the case of, for instance, any formula p(← ←) with ║p║ being a nonempty, proper subset of U. If we realize that ← should play the role of a pronoun, then this should not surprise us: if whistle(← ←) is to render he whistles, then no wonder than it is, when uttered out of the blue, neither true, nor false.
4. ‘Backward-Looking’ Terms and Indeterminacy In this way, we have developed a language which does justice to (1) (our ← being the regimentation of (1)’s he); so let us proceed to (2). To be able to accommodate it, we need existential quantification; however, we shall not introduce it in the usual way. Let us return back to L and let us now consider another kind of alternative semantics for it. Individual constants will now be taken to denote subsets of the universe, each individual constant being assigned a singleton; and if i is an individual constant and p a predicate constant, then ║p(i)║ = T iff ║i║∩║p║ ≠ ∅. Everything else is as in the standard case. That this alternative semantics is equivalent to the standard one is obvious. Starting from this semantics, we can define the language Lε by adding to L the new term ε which is taken to denote the whole universe, ║εε║ = U. Then it is easy to see that p(εε) is 4
true if and only if there is something which is p, i.e. that what the statement claims is, in traditional notation, ∃x.p(x). This form of existential quantification is now suitable to interact with the backwardlooking term ←. What we need to do is to combine Lε with L←; so what we first need is to merge the two alternative semantics of L which we have employed to produce L← and Lε, respectively. This can be done in a straightforward way: denotations of terms (where terms are now individual constants, ε and ←) will now be binary relations over U such that for every ordinary individual constant i, ║i║ = {<x,xi> | x∈U} for some xi∈U, ║εε║ = {<x,y> | x∈U and y∈U} and ║← ←║ = {<x,x> | x∈U}. The denotation of predicates remains unchanged (i.e. they keep denoting subsets of U). Also statements now denote binary relations over U, their denotations being defined in the following way: ║p(t)║ = {<x,y> | <x,y>∈║t║ and y∈║p║}; ║s1&s2║={<x,y> | there is a z such that <x,z>∈║s1║ and ∈║s2║}. A statement s is true iff for every x∈U there exists a y∈U such that <x,y>∈║s║; it is false iff for no x∈U is there a y∈U such that <x,y>∈║s║ (i.e. iff ║s║ = ∅). It is now easy to see that within the resulting language Lε,← not only is p1(i)&p2(← ←) equivalent to p1(i)&p2(i) for every individual constant i and every predicate constant p1 and p2, but p1(εε)&p2(← ←) is true if and only if there is an x which is both p1 and p2. The proof of the former claim is straightforwardly analogous to the proof of the same claim within L←; so let us prove the latter one only. If s is a sentence of Lε,←, then we shall write x║s║y instead of <x,y>∈║s║. Then we can see that x║p1(εε)&p2(← ←)║y iff there is a z such that x║p1(εε)║z and z║p2(← ←)║y. Moreover, x║p1(εε)║z iff x║εε║z and z∈║p1║, i.e. iff z∈║p1║ (for x║εε║z holds for ←)║y iff z = y and y∈║p2║. Hence x║p1(εε)&p2(← ←)║y iff y∈║p1║ every x and z); and z║p2(← and y∈║p2║. This means that ║p1(εε)&p2(← ←)║ is true iff there is a y such that y∈║p1║ and ←)║y for every x∈U). y∈║p2║ (for then x║p1(εε)&p2(← The fact that statements of Lε,← denote relations between individuals of the universe invites a ‘dynamic’ reading: what a statement denotes can be seen as an (indeterministic) transition from an individual to an individual, as something that ‘consumes’ an individual (which is yielded by a previous statement - if any) and ‘produces’ a (possibly different) individual (which is then consumed by a subsequent statement - if any). We may see it also in terms of a ‘saliency box’ the content of which may be supplied by one statement and subsequently utilized by another one: the statement’s input is what is in the box when the statement is uttered (the box may be empty or filled by an individual), and its output is the content of the box as established by the utterance (the box may be unchanged, or (re)filled by a new individual). In such terms we can describe the semantics of Lε,← in an illuminating way: an ordinary constant always (re)fills the saliency box by a fixed individual; ← leaves the contents of the box unchanged, and ε fills the box ‘indeterministically’ by an arbitrary individual. A subject-predicate statement then ‘works’ (and thereby is true) in a given context if the contents of the saliency box as produced by the subject of the statement (in that context) belongs to the subset of the universe which is denoted by the predicate of the statement. However, it is good to notice that this story is meant neither as a depiction of something going on within speakers’ heads, nor as a description of a structure of ‘cognitive actions’; it is a metaphorical way of envisaging a model theory for a language which we have found to do justice to inferences like (1) and (2).
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5. Multiple ‘Backward-Looking’ Terms Now let us turn our attention to (3). What we need in this case is a plurality of different ‘pronouns’ (‘backward-looking’ terms). To provide for it, let us modify Lε,← in the following way: let us divide the terms of the language into n disjoint sorts, so that every individual constant falls into one of the sorts, and ε and ← are replaced by εk and ←k for every sort k. ║║ is an interpretation of this language iff ║t║∈UnxUn for every term t and ║p║⊆U for every predicate constant p. Let us write [x]n instead of <x1,...,xn> and omit the subscript n where no confusion is likely to arise; and let us write [x]i for the i-th constituent of the sequence [x]. For every k, ║tk║ is some set of pairs of n-tuples <[x],[y]> such that for j≠k, [y]j=[x]j; if the tk is εk then it is the set of all such pairs, if it is ←k, then it is the set of all such pairs for which also [y]k =[x]k, and for an ordinary individual constant it is the set of all such pairs for which [y]k is a fixed element of the universe. In symbols, let ║εεk║ = {<[x],[y]> | [y]j=[x]j for j≠k}, ║← ←k║ = {<[x],[y]> | [y]=[x]}, and for every individual constant ik of the sort k let there exist an individual xik so that ║ik║ = {<[x],[y]> | [y]k= xik; [y]j=[x]j for j≠k}. Such an interpretation now induces an assignment of elements of UnxUn to statements in the following way: ║p(tk)║ = {<[x],[y]> | <[x],[y]>∈║tk║ and [y]k ∈║p║} if tk is a term of the sort k, and ║s1&s2║ = {<[x],[y]> | there is a [z] so that <[x],[z]>∈║s1║ and <[z],[y]>∈║s2║}. A sentence s is true iff for every [x]∈Un, there is an [y]∈Un so that <[x],[y]>∈║s║; it is false if for no [x]∈Un there is an [y]∈Un so that <[x],[y]>∈║s║ (i.e. iff ║s║=∅). Now let us show that p1(ik)&p2(← ←k) is equivalent to p1(ik)&p2(ik) for every individual constant ik of the sort k (and every predicate p1 and p2). By definition, [x]║p1(ik)&p2(← ←k)║[y] iff there is a [z] so that [x]║p1(ik)║[z] and [z]║p2(← ←k)║[y]. Furthermore, [x]║p1(ik)║[z] iff [x]║ik║[z] and [z]k∈║p1║; and [z]║p2(← ←k)║[y] iff [z]║← ←k║[y], and [y]k∈║p2║. Hence [x]║p1(ik)&p2(← ←k)║[y] iff [x]║ik║[z] and [z]k∈║p1║ and [z]║← ←k║[y], and [y]k∈║p2║. But as [z]║← ←k║[y] iff [z] = [y], this reduces to [y]k = xik and [y]j=[x]j for j≠k and xik∈║p1║ and xik∈║p2║, which is clearly also a necessary and sufficient condition for [x]║p1(ik)&p2(ik)║[y]. Thus, [x]║p1(ik)&p2(← ←k)║[y] if and only if [x]║p1(ik)&p2(ik)║ [y]. ←r) is not equivalent to p1(ik)&p2(ik) for In contrast to this, let us show that p1(ik)&p2(← k≠r. By reasoning analogous to that of the previous paragraph, we see that [x]║p1(ik)&p2(← ←r)║[y] iff [x]║ik║[y] and [y]k∈║p1║ and [y]r∈║p2║; and as [x]║ik║[y] implies r r [x] = [y] , this further reduces to [x]║ik║[y] and [y]k∈║p1║, and [x]r∈║p2║. Now let ║║ be such an interpretation that the constant value of ║ik║, xik, is a member of both ║p1║ and ║p2║, but ║p2║ ≠ U. Let us take an n-tuple of objects [x] so that [x]r∉║p2║. Then, as we have just ←r)║[y]. However, at the same time it is obvious seen, there is no [y] so that [x]║p1(ik)&p2(← that there is an [y] so that [x]║p1(ik)&p2(ik)║[y], namely the [y] defined as follows: [y]k = xik and [y]j = [x]j for j≠k. This means that p1(ik)&p2(← ←r) and p1(ik)&p2(ik) are not equivalent. ←k) is true iff there is something which is Similarly we could show that p1(εεk)&p2(← both p1 and p2, whereas p1(εεk)&p2(← ←r) may be true even if there is no such x (and may be false even if there is). It would be also possible to introduce constants which are of more sorts than one: if ║i║ = {<[x],[y]> | [y]k=[y]r=c; [y]j=[x]j for k≠j≠l}, then both p1(i)&p2(← ←k) and p1(i)&p2(← ←r) p p not p are equivalent to 1(i)&p2(i); but 1(i)&p2(← ←s) is equivalent to 1(i)&p2(i) for k≠s≠r.
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Returning back to our inferential patterns, we can see that to do justice to (3) we need L with the three sorts rendering natural language grammatical gender (masculina, feminina, neutra). The operators ←1, ←2, ←3 would then correspond to ‘he’, ‘she’, ‘it’, respectively; and the operators ε1, ε2, ε3 would provide a more refined resource for which English only has the two particles ‘somebody’ and ‘something’. In general, we can see the expressions of Lε,←,n in the following way. We have ‘introducers’, which are either definite (ordinary individual constants) or indefinite (epsilons), and ‘pickers’ (backarrows). Introducers interact with pickers to yield the anaphoric structure of the discourse, but not just any introducer with just anypicker; to interact with an introducer, the picker has to be ‘tuned to the same frequency’ as the introducer. The ‘frequency’ to which a picker or an introducer is ‘tuned’ is its sort. We have a finite number of ‘frequencies’. It is even more illuminating to see the situation again in terms of a saliency box, this time with n slots instead of just one, the slots corresponding to that which has been just envisaged as ‘frequencies’. The introducers of the sort k fill the k-th slot of the box; the pickers of the sort l pick up the contents of the l-th slot. ε,←,3
6. Complex ‘Backward-Looking’ Terms So far, the only ‘introducers’ and ‘pickers’ were words (primitive constants). Thus we have developed a framework to account for the inferential patterns of the kind of (3); but what about those of the kind of (4)? Here we have to depart from the structure of the predicate calculus more substantially; but our previous investigations can give us a lead. It seems that besides the simple ones, in natural language we have also complex ‘introducers’ and ‘pickers’: a man and the man underlying the inferential pattern (4) seem to be an example. Introducers like a man, a logician, or a killer, despite being all tuned to the general ‘masculine frequency’ (and thus being capable of interacting with he), seem to provide also a finer frequency key. They do not interact with the corresponding pickers, the man, the logician, or the killer, indiscriminately: A logician walks and the killer whistles does not in general say that there is somebody who walks and whistles3. We have to imagine an infinite number of frequencies and a mechanism which turns a general name into either an ‘introducer’ or a ‘picker’. And indeed, this seems to be what we have in English and what underlies the inferential pattern (4): articles. Thus instead of employing the multiple ε’s and ←’s, we can use two constants, say a and the (to make them wear their function on their sleeves), constituting a new category of expressions applicable to predicates to form terms. To see what semantics this new calculus, call it Lthe,a, should have, let us return to the image of the saliency box once more. So far, the box consisted of n slots, each of which was suited to contain the current salient item of the sort n. Thus, the slots of the box may be seen as labeled by integers up to n. Now what we obviously need is an infinity of slots; more precisely we need an individual for each semantically distinct predicate. This is to say that the slots of the box are now to be labeled by denotations of predicates, i.e. subsets of the universe; which means that statements of Lthe,a are to be seen as denoting relations between functions which assign elements of the universe to subsets of the universe. Realizing that it is plausible to work only with such functions which map a set always on an element of itself 3
Although this might be felt as a kind of a ‘conversational implicature’.
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(‘the most salient p’ should, of course, be a p), we reach the framework of Peregrin & von Heusinger (1997). To be more rigorous: the terms of Lthe,a are individual constants plus complex terms formed by applying a and the to predicates. Let U be the universe and let CHFU denote the set of choice functions over U, i.e. the set of all functions C such that (i) the domain of C is included in Pow(U), (ii) the range of C is included in U, and (iii) C(s)∈s for every s from the domain of C. The semantics of Lthe,a is as follows: each term and each statement is assigned a subset of CHFUxCHFU; each predicate is assigned a subset of U. The ‘value referred to by tk in the context c’, |tk|c, where the context is identified with a choice function, is defined in the following way: if tk is a(p) or the(p), then |tk|C = C(║p║), whereas if tk is an individual constant ik, then |tk|c = xik for some fixed xik∈U (independently of C). If p is a predicate and t a term, then ║p(t)║={ | ∈║t║ and |t|C’∈║p║}. If i is an individual constant, then ║i║ = { | C∈CHFU}4; if p is a predicate, then ║a(p)║ = { | C(s)= C’(s) for s ≠ ║p║ and C’(║p║) ∈║p║}; and ║the(p)║={ | C(║p║)∈║p║}. If we again stipulate that a statement s is true (false) iff for every (no) C∈CHFU there exists a C’∈CHFU so that ∈║s║, then we can easily prove that p1(a(p))&p2(the(p)) is true if and only if there is an item which is p, p1 and p2; and hence the inferential pattern (4) is validated. To prove it, let us compute the denotation of p1(a(p))&p2(the(p)). By definition, C║p1(a(p))&p2(the(p))║C’ iff there is a C’’ such that C║p1(a(p))║C’’ and C’’║p2(the(p))║C’. But as C║p1(a(p))║C’’ iff C║a(p)║C’’ and C’’(║a(p)║)∈║p1║, while C’’║p2(the(p))║C’ iff C’’║the(p)║C’ and C’(║the(p)║)∈║p2║, it is obvious that C║p1(a(p))&p2(the(p))║C’ iff C║a(p)║C’’ and C’’(║a(p)║)∈║p1║ and C’’║the(p)║C’ and C’(║the(p)║)∈║p2║. And as C║a(p)║C’’ iff C(s)= C’(s) for s ≠ ║p║ and C’(║p║) ∈║p║, while C’’║the(p)║C’ iff C’’=C’ and C’’(║p║)∈║p║, this further reduces to C(s)= C’(s) for s ≠ ║p║ and C’(║p║) ∈║p║ and C’(║a(p)║)∈║p1║ and C’║p║∈║p║ and C’(║the(p)║)∈║p2║, which means that C’ differs from C at most in the value it assigns to ║p║ and this value is an element of ║p║, ║p1║ and ║p2║. Hence, given C, the existence of C’ for which C║p1(a(p))&p2(the(p))║C’ is tantamount to the existence of something which is p, p1 and p2. Q.e.d.
7. De-simplification The formal languages we have discussed so far are, of course, so simple that they cannot be taken too seriously. The reason for employing such impoverished languages was to make the exposition of the basic logical backbone of the apparatus quite transparent; and now we are going to indicate how we could put flesh back on the bones. We shall talk about Lε,←, the cases of the more complex languages are straightforwardly analogous. First, we have restricted ourselves to unary predicates. However, there is no problem of principle in accepting predicates of greater arities. In fact, there are at least two distinct ways to handle them. Informally speaking, these two ways differ in the order in which we compute the referents of terms of a two- or more-place predicate (which are to be tested 4
This means that individual constants are taken to trigger no anaphoric effects. This is, of course, a substantial oversimplification. For a more elaborate treatment see Peregrin & von Heusinger (1997).
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against the denotation of the predicate): we may compute a term’s referent either immediately after the very term exercises its context-change potential, or else after every term does so. For Lε,←, we have the following two alternative rules: <x,y>∈║p(t1,...,tn)║ iff there are x0,...,xn so that x = x0, y = xn, <xj-1,xj>∈║tj║ for j = 1,...,n, and <x1,...,xn>∈║p║ <x,y>∈║p(t1,...,tn)║ iff there are x1,...,xn so that y = xn, <x,xj>∈║tj║ for j = 1,...,n, and <x1,...,xn>∈║p║ Which of these options is more adequate from the viewpoint of the language’s capacity to adequately reconstruct real inferential patterns of natural language is a matter of careful linguistic analysis which will not be discussed here. Second: negation, implication and other usual logical operators. The accommodation of these operators within a dynamic logical framework cause problems discussed in detail in the literature. They are again mostly problems of having to choose between various alternatives. The most straightforward way to accommodate the operators within Lε,← seems to be (in analogy with what has been proposed by Greonendijk and Stokhof, 1991): ║¬s║ = {<x,x> | there is no y so that <x,y>∈║s║} ║s1 → s2║ = {<x,x> | for every <x,y>∈║s1║ there is a z so that ∈║s2║} However, the most controversial feature of the approach proposed here is that the languages presented avoided ordinary quantification in favor of indefinite terms with existential import. The reason for this is that this is, I am convinced, the right way to make the structure of the languages close enough to that of natural language to be able to account for the inferential patterns involving anaphora. However, if we want to make the present model into more than a toy, we undoubtedly have to indicate how it could be enriched with something approaching the power of quantifiers of ordinary logic. Lε,← obviously does not enable us even to spell out that there is something which is p1 and p2. The formula p1(εε)&p2(εε) says that there is something which is p1 and something which is p2, not that there is necessarily something with both the properties. How could we do away with this restriction? Remember that we invented ε because this seemed to be more congenial to the way in which indeterminacy is articulated in natural language; so we should go on taking lessons from natural language. And if we do this, we can see that the fact that there is something which is both p1 and p2 should be articulated as (p1&p2)(εε). This would, of course, require us to enrich Lε,← with the possibility of forming complex predicates, which might seem to take us far beyond the boundaries of first-order predicate calculus. However, this is not true: as Quine (1996) points out, to do predicate logic in terms of what he calls predicate functors is a natural (albeit unusual) possibility5. Now what about universal quantification? There are again several ways to introduce it ε,← into L . The first is to make it simply dual to the existential quantification, i.e. to introduce a universal term π functioning in such a way that for every predicate p it holds that (where ∼ is the predicate-functor of negation):
5
By this reformulation of the predicate calculus we effectively get rid of variables, which is a good way to make logic closer to natural language - as I argued elsewhere (see Peregrin, in press).
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p(π π) = ¬(∼p)(εε) (Note that one of the consequences of introducing predicate functors is that every statement can be articulated into a subject-predicate form, so the definition may be seen as entirely general.) This definition, however, of course depends on the way we decide to define negation. If we define ¬ in the most straightforward way shown above (and if ∼ is taken to denote simply the set-theoretical complement), it is easy to see that ║¬(∼p)(εε)║ = {<x,x> | there is no y so that <x,y>∈║(∼p)(εε)║} = {<x,x> | there is no y so that <x,y>∈║εε║ and y∈║∼p║ } = {<x,x> | there is no y so that y∈║∼p║ } = {<x,x> | for every y, y∈║p║ } Another possibility would be to accept the ordinary general quantification of predicate logic. This would amount to the (re)introduction of the whole machinery of variables and binding of predicate logic, and it would make our logic into ordinary logic with the dynamic machinery besides the ordinary, static one. (Of course we then could define classical existential quantification in terms of the universal one, and dynamic universal quantification in terms of the dynamic existential one; so we would have both full sets of quantifiers.) A third possibility would be to have no explicit universal quantification at all and to analyze corresponding locutions of natural language in terms of dynamic implication. (In contrast to negation, implication as defined above is not merely a straightforward transposition of the material implication of static logic; and this is what makes it possible to use it for the purposes of analyzing universal statements). Thus we could analyze Every human is mortal as human(εε) → mortal(← ←) which may be read roughly as If something is human, then it is mortal. It is again easy to see that ║human(εε) → mortal(← ←)║ = {<x,x> | for every <x,y>∈║human(εε)║ there is a z so that ∈║ mortal(← ←)║} = {<x,x> | for every <x,y> such that <x,y>∈║εε║ and y∈║human║ there is a z so that ∈║← ←║ and z∈║mortal║} = {<x,x> | for every <x,y> such that y∈║human║ there is a z so that y = z and z∈║mortal║} = {<x,x> | for every <x,y> such that y∈║human║ it holds that y∈║mortal║} = {<x,x> | for every y such that y∈║human║ it holds that y∈║mortal║} = {<x,x> | ║human║ ⊆║mortal║} This restriction to ‘unselective’ universal quantification is known, e.g., from DRT.
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8. Concluding Remarks The basic trick which allows for the logical accommodation of anaphoric effects is taking sentences as denoting not sets of indices, but rather relations between sets of indices. The more fine-grained structure we then give to the indices, the richer repertoire of anaphoric effects we can have. Within Lε,←, we took the indices to be, in effect, simply individual elements of the universe, and thus we were able to have only one, ‘unselective’ ‘backwardlooking term’. Within Lε,←,n we improved on the situation by taking the indices to be n-tuples of elements of the universe: this allowed us to have n distinct, selective ‘backward-looking terms’. Within Lthe,a, the structure of the indices was still richer: they were taken to be choice functions; and as the consequence, we gained an infinite number of distinct backward-looking terms6. One can imagine also various further generalizations. Now like in the case of introducing ‘modal indices’ (possible worlds), this introduction of ‘dynamic indices’ (underlying information states) can (and indeed, as I am convinced, should) be interpreted neither as a metaphysical, nor as a psychological, but rather as a purely logical achievement. We did not model any ‘storage facilities’ which we would find within human mind/brain; what we have done was to identify certain expressions with certain interesting inferential behavior within our language (he, she, a man, the woman, etc.), we have devised (rudimentary) logical systems in which it is possible to reflect the workings of such expressions (i.e. the inferential patterns governing their usage), and we have devised some simple model theories for these logics.
References Chierchia, G. (1994): ‘Intensionality and Context Change’, Journal of Logic, Language and Information 3, 141-168. Groenendijk, J. and M. Stokhof (1991): ‘Dynamic Predicate Logic’, Linguistics and Philosophy 14, 39-101. Heim, I. (1990): ‘E-Type Pronouns and Donkey Anaphora ’, Linguistics and Philosophy 13, 137-177. Hintikka J. and J. Kulas (1985): Anaphora and Definite Descriptions, Reidel, Dordrecht. Kamp, H. (1981): ‘A Theory of Truth and Semantic Representation ’, in Formal Methods in the Study of Language (ed J. Groenendijk et al.), Mathematical Centre, Amsterdam. Kamp, H. and U. Reyle (1993): From Discourse to Logic, Kluwer, Dordrecht. Muskens, R., J. van Benthem and A. Visser (1997): ‘Dynamics’, Handbook of Logic and Language (ed. J. van Benthem and A. ter Meulen), Elsevier / MIT Press, Oxford / Cambridge (Mass.). Peregrin, J. and K. von Heusinger (1997): ‘Dynamic Semantics with Choice Functions’, in Context-Dependence in the Analysis of Linguistic Meaning I (ed. H. Kamp and B.Partee), Universität Stuttgart, Stuttgart, 329-354. Peregrin, J. (in press): ‘Variables in semantics - where do they come from?’, to appear in Variable-free Semantics (ed. M. Böttner and W. Thümmel), Secolo, Osnabrück. 6
A different possibility is to structure the indices by means of some ad hoc linguistic items like the discourse markers of Groenenedijk’s & Stokhof’s DPL. See Peregrin and von Heusinger, 1997, for an indication of why this might be considered as unsatisfactory.
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Peregrin, J. (2000): ‘Reference and Inference: the Case of Anaphora’, Reference and Anaphoric Relations (ed. K. von Heusinger and U. Egli), Kluwer, Dordrecht, 269-286. Quine, W.V.O. (1996): From Stimulus to Science, Harvard University Press, Cambridge (Mass.). van Benthem, J. (1997): Exploring Logical Dynamics, CSLI, Stanford.
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Appendix: the Calculi The language L Vocabulary: unary predicate constants (pc’s), individual constants (ic’s), & Grammar: if p is a pc and i an ic then p(i) is a statement (st) if s1 and s2 are st’s, then s1 & s2 is a st. Standard semantics: if i is an ic, then ║i║∈U (where U is the universe) if p is a pc, then ║p║ ⊆U if s is a st, then ║s║ is a truth value; s is true iff ║s║ = T, it is false iff ║s║ = F ║p(i)║ = T iff ║i║∈║p║ ║s1 & s2║ = T iff ║s1║= T and ║s2║ = T Alternative semantics 1: if i is an ic, then ║i║ is a total constant function from U into U if p is a pc, then ║p║⊆U if s is a st, then ║s║ is a function from U into U; s is true iff ║s║ is total, it is false iff ║s║ is empty ║p(i)║ = {<x,y> | <x,y>∈║i║ and y∈║p║} (thus: if the constant value of ║i║ is an element of ║p║, then ║p(i)║ = ║i║, else ║p(i)║ = ∅) ║s1 & s2║ = ║s2║(║s1║) Alternative semantics 2: if i is an ic, then ║i║⊆U and ║i║ is a singleton if p is a pc, then ║p║⊆U if s is a st, then ║s║ is a truth value; s is true iff ║s║ = T, it is false iff ║s║ = F ║p(i)║ = T iff ║p║∩║i║ ≠ ∅ ║s1 & s2║ = T iff ║s1║ = T and ║s2║ = T Alternative semantics 3: if i is an ic, then ║i║⊆UxU, such that ║i║ = {<x,xi> | x∈U} for some xi∈U if p is a pc, then ║p║⊆U if s is a st, then ║s║⊆UxU; s is true iff for every x∈U there exists an y such that <x,y>∈║s║, it is false iff for no x∈U there exists an y such that <x,y>∈║s║ ║p(i)║ = {<x,y> | <x,y>∈║i║ and y∈║p║} ║s1 & s2║ = {<x,y> | there is a z such that <x,z>∈║s1║ and ∈║s2║}
13
← The language Lε,←
Vocabulary: pc’s; terms (t’s); & t’s are ic’s, ←, ε Grammar: if p is a pc and t a t, then p(t) is a st if s1 and s2 are st’s, then s1 & s2 is a st Semantics: if t is a t, then ║t║⊆UxU; ║εε║ = UxU, ║← ←║ = {<x,x> | x∈U}; for every ic i there exists a xi∈U such that ║i║ = {<x,xi> | x∈U} if p is a pc, then ║p║⊆U if s is a st, then ║s║⊆UxU; s is true iff for every x∈U there exists an y such that <x,y>∈║s║, it is false iff for no x∈U there exists an y such that <x,y>∈║s║ ║p(i)║ = {<x,y> | <x,y>∈║i║ and y∈║p║} ║s1 & s2║ = {<x,y> | there is a z such that <x,z>∈║s1║ and ∈║s2║}
14
←,n The language Lε,←
Vocabulary: pc’s; terms of category k (tk’s) for k=1,...,n; & tk’s are individual constants of category k (ick’s), ←k, εk Grammar: if p is a pc and t a t, then p(t) is a st if s1 and s2 are st’s, then s1 & s2 is a st Semantics: if t is a tk, then ║t║⊆UnxUn; ║εεk║ = {<[x]n,[y]n,> | [y]nj=[x]nj for j≠k}, ║← ←k║ = {<[x]n,[y]n,> | [y]n=[x]n}; for every ick i there exists an xi∈U such that ║i║={<[x]n,[y]n,> | [y]nk=xi and [y]nj=[x]nj for j≠k} if p is a pc, then ║p║⊆U if s is a st, then ║s║⊆UnxUn; s is true iff for every [x]n∈Un there exists an [y]n such that <[x]n,[y]n>∈║s║, it is false iff for no [x]n∈Un there exists an y such that <[x]n,[y]n>∈║s║ ║p(tk)║ = {<[x]n,[y]n > | <[x]n,[y]n >∈║tk║ and [y]nk ∈║p║} ║s1 & s2║ ={<[x]n,[y]n > | there is a [z]n such that < [x]n,[z]n >∈║s1║ and < [z]n,[y]n > ∈║s2║}
15
The language Lthe, a Vocabulary: pc’s; ic’s; a, the; & Grammar: ic is a t if p is a pc, then a(p) and the(p) are t’s if p is a pc and t a t, then p(t) is a st if s1 and s2 are st’s, then s1 & s2 is a st Semantics: Let CHFU = {C | C is a partial function from Pow(U) to U such that if s belongs to the domain of C, then C(s)∈s}. if t is a t, then, ║t║⊆CHFUxCHFU; for every ic i, ║i║= { | C∈CHFU} if t is a t and C∈CHFU, then |t|C∈U; for every ic i there exists a xi∈U such that |i|C = xi for every C∈CHFU if p is a pc, then ║p║⊆U if s is a st, then ║s║⊆CHFUxCHFU; s is true iff for every C there exists a C’ such that ∈║s║, it is false iff for no C there exists a C’ such that ∈║s║ ║a(p)║={ | C(s) = C’(s) for every s ≠ ║p║ and C’(║p║) ∈║p║} ║the(p)║={ | C(║p║) ∈║p║ } |a(p)|C =|the(p)|C = C(║p║) ║p(t)║= { | ∈║t║ and |t|C’∈║p║ } ║s1 & s2║ = { | there is a C’’ such that ∈║s1║ and ∈║s2║}
16
Constructions and Concepts Jaroslav Peregrin www.cuni.cz/~peregrin Between Words and Worlds (A Festschrift for Pavel Materna), ed. T. Childers & J. Palomäki, Filosofia, Prague, 2000, 34-48
Tichý’s hyperintensional semantics Some twenty years ago, semanticists of natural language came to be overwhelmed by the problem of semantic analysis of belief sentences (and sentences reporting other kinds of propositional attitudes): the trouble was that sentences of the shapes X believes that A and X believes that B appeared to be able to have different truth values even in cases when A and B shared the same intension, i.e. were, from the viewpoint of intensional semantics, synonymous1. Thus, taking intensional semantics for granted, belief sentences appeared to violate the principle of intersubstitutivity of synonyms. The verdict of the gurus of intensional semantics was that hence intensional semantics is inadequate, or at least insufficient for the purposes of analysis of propositional attitudes; and that we need a kind of a ‘hyperintensional semantics’. The locus classicus of considerations of this kind is Lewis (1972): the author concludes that to be able to give a fair account of meaning of belief sentences we need an augmentation of the intensional semantics in the spirit of Carnap’s (1957) notion of intensional isomorphism. Carnap’s proposal amounts, in effect, to the notion of meaning according to which the meaning of a complex expression would be an ordered n-tuple of the intensions of its parts; so that if we denote the intension of E as ║E║, the meaning of (1) would be (1′): John walks <║John║,║walk║>
(1) (1′)
Lewis improved on this suggestion by requiring that the meaning should somehow directly reflect the syntactic structure of the corresponding expression, so that the meaning of (1) would be not (1′), but rather something like (1′′): S
1
(1′)
NP
VP
N
V
║John║
║walk║
See, e.g., Bigelow (1978); and also Materna (1983).
1
This is a proposal that clearly solves the problem at hand; nevertheless the solution appears to be a bit too ad hoc. Pavel Tichý, the author of transparent intensional logic (TIL) came with a similar proposal, which, however, was backed by a more intuitive story. The story goes as follows. According to extensional semantics, the meaning of (1) results from the application of the meaning of walk (a function mapping individuals on truth values) to that of John (an individual). According to intensional semantics, the situation is somewhat more complicated for all of this gets relativized to possible worlds; but again, the meaning of (1) results from a certain kind of combination of the meaning of walk (which now is a function mapping possible worlds on functions from individuals to truth values) with that of John (which may be a function mapping possible worlds on individuals; or, as Tichý had it, directly the individual named). Because this combination will play an important role for us later on, let us coin a name for it: intensional application. Thus, intensional application of an object to another object or other objects is like ordinary application save for an additional ‘pre- and post-processing’: all of the involved objects that are intensions (i.e. functions from possible worlds) get, prior to the application, applied to the variable w; and the result of the application then gets abstracted upon the very variable. (Later on, when Tichý started to relativize extensions not only to worlds, but also to time moments, the application and abstraction came to involve the variables w and t instead of merely w.) Now Tichý’s suggestion is that although what a complex expression like (1) amounts to is this kind of intensional application, what should be seen as the meaning of the expression is not the result of such an application, i.e. an intension, but rather the very application itself - the construction, as he puts it.
Constructions What is clearly crucially presupposed by Tichý’s proposal is getting a firm grip on the concept of construction. What, then, is a construction in Tichý’s sense? Does Tichý manage to explicate the concept in an acceptable and sufficient way? The trouble is that for many decades the paradigm of a reasonable explication of a pre-theoretical entity (and at the same time the proof that the entity ‘indeed exists’) has been its reconstruction within set theory. Take, say, the concept of lattice: its canonical explication has come to be identified with a certain kind of algebra, i.e. a certain kind of ordered pair consisting of a set (the ‘carrier’ of the algebra) and a set of sets of ordered (n+1)-tuples of the elements of the carrier (the ‘operations’ of the algebra). Or take the concept of intension: it has come to be identified with a function taking possible worlds (and, as the case may be, time moments) to extensions (which are again construed as set-theoretic objects). Also when Lewis proposed his structured meanings mentioned above, he hastened to add that rigorously explained, they of course are certain set-theoretical entities. Tichý’s ambition, on the other hand, was to somehow forego set theory: he assumed that his constructions were not to be accomodable within set theory, for they were to be more fundamental then anything like sets. However, it is clear that not even every kind of abstract entity accepted by settheoretically oriented mathematicians can be reduced to a set-theoretical construct. There is obviously at least one for which this is not possible, namely set itself. How, then, is the concept of set usually explicated? The standard way is the formulation of an axiomatic set theory, usually within the framework of first-order logic (although, e.g., second-order set theory is, of course, also conceivable). So would this not be also a way of explicating the 2
concept of construction? Could we not try to articulate an axiomatic theory of constructions, analogous to set theory? Tichý, to be sure, would not think so: for him, forming axiomatic theories of the kind of set theory was a useless formalistic game. (His standpoint was, in this point, similar to that of Frege in his well-known quarrel with Hilbert over the nature of geometry2.) Hence, he purposefully tried to explicate the concept of construction on a less formal level. My opinion is, nevertheless, that indicating how an axiomatic theory of constructions might look like may be instructive, that it can help us throw some more light on the nature of Tichý’s approach. Hence before I proceed to the central theme of the paper, Materna’s explication of the concept of concept, I attempt to outline such a theory. I will try to ‘translate’ what Tichý says about the nature of constructions into the language of first-order logic and try to articulate it as an axiomatic theory. (However, it should be kept in mind that what I aspire to is nothing more than a sketch.) Probably the first thing to notice when we try to form a ‘construction theory’ in the spirit of set theory is that whereas set theory does not presuppose anything but general logic (the only non-logical term of the formal theory is the predicate of set membership), construction theory presupposes some nontrivial concepts, namely the concepts of function and functional application. Explaining the nature of constructions, Tichý takes these concepts simply for granted. Thus, we have to assume that the language in which we formulate the theory contains not only logical primitives, but rather also, for every natural number n, the unary predicate Fncn and the (n+1)-ary predicate Appln together with some axioms fixing the intended meanings of Fncn(x) to „x is an n-ary function“ and Appln(y,x1,... xn,z) to „z is the result of the application of y to x1,..., xn“3. (We leave it open precisely what kind of theory this is supposed to be. Clearly one option would be to let the construction theory be underlain by set theory with its explication of the concepts of function and application; but this would obviously frustrate Tichý’s effort to get beyond set theory.) So given we have a theory of functions, a way of building a theory of constructions atop of it might be the following. Tichý speaks about seven types of constructions, so we should have five unary predicates true of all and only constructions of the respective types. Let the predicates be Var („is a variable“), Triv („is a trivialization“), Exe („is an execution“), DExe („is a double execution“), Comp („is a composition“) and Clos („is a closure“). We can define the predicate Cons („to be a construction“) as Cons(x) ≡Def Var(x) ∨ Triv(x) ∨ Exe(x) ∨ DExe(x) ∨ Comp(x) ∨ Clos(x) Moreover, as we assume that nothing can instantiate two different types of constructions, we should have axioms of the kind of ∀x (Var(x) → ¬(Triv(x) ∨ Exe(x) ∨ DExe(x) ∨ Comp(x) ∨ Clos(x)) ∀x (Triv(x) → ¬(Var(x) ∨ Exe(x) ∨ DExe(x) ∨ Comp(x) ∨ Clos(x)) ...
2
See Frege (1976). See also Peregrin (to appear a; §4).
3
We could also make do with taking only Appln as primitive and define Fncn(y) as ∃x1...xnz Appln(y,x1,... xn,z); but this is not important now.
3
How are individual constructions to be characterized? Constructions construct, so we should have a relation connecting them to what they construct. However, some of the constructions (variables and open constructions) construct something only relative to a valuation of variables. It follows that in order to be able to address constructing, we first have to deal with v-constructing, i.e. constructing relative to a given valuation. And hence it follows that before we are able to address constructing in general, we have to address one specific kind of constructions, namely variables. 1. Variables. A variable is an entity characterized simply by the fact that it constructs (i.e., expressed more traditionally, has) a value, relative to a given ‘valuation’. Hence to characterize variables within our axiomatic system, we have to assume that our universe contains, in addition to variables themselves, also entities called valuations, which somehow ‘make’ variables acquire objects of the universe as their values. (Don’t be puzzled by the fact that we do not treat variables as linguistic items and valuations as ‘metalinguistic’ ones; the ‘objectual’ treatment of variables is one of the key points of Tichý’s approach. And don’t confuse the variables of TIL, which thus come to be objects of the universe of our theory, with the variables which are part of the first-order language by means of which we articulate the theory.) One of the ways of axiomatically reflecting this is introducing the unary predicate Val („is a valuation“) and a ternary relation Value („the value which the valuation ... assigns to the variable ... is ...“) governed by at least the following axioms: ∀vxy (Value(v,x,y) → (Val(v)∧Var(x)) ∀vx((Val(v)∧Var(x)) → ∃yValue(v,x,y)) ∀vxyy′((Value(v,x, y)∧Value(v,x,y′)) → (y=y′)) ∀x1...xny1...yn((Var(x1)∧...∧Var(xn)) → ∃v(Val(v)∧(Value(v,x1,y1)∧ ...∧(Value(v,xn,yn))) ∀vv′((Val(v)∧Val(v′)) →(∀xyy′((Value(v,x, y)∧Value(v′,x,y′)) → (y=y′)) → (v = v′))) The first of the axioms states that what assigns values are valuations whereas what is assigned values are variables4. The next two axioms state that for every variable and every valuation there is one and only one object assigned to the variable by the valuation. The fourth axiom states that for every n-tuple of variables and every n-tuple of objects there exists a valuation assigning the objects to the variables. The last axiom then states that two valuations which assign the same objects to the same variables are identical. Given this, we can introduce the ternary relation VConstr („with respect to the valuation ..., ... constructs ...“) such that it always holds between a valuation, a construction and an object, and it can be seen as inducing a (partial) function assigning its last argument to its first two: ∀vxy (VConstr(v,x,y) → (Val(v) ∧ Cons(x))). ∀vxyy′ ((VConstr(v,x,y) ∧VConstr(v,x,y′)) → (y = y′)). The relation is to be further characterized specifically for individual types of constructions, to which we can now turn our attention.
4
The axiom could incorporate also some restriction with respect to the kind of objects assigned to variables by valuations: it might be, e.g. excluded that an object assigned to a variable is a valuation.
4
2. Trivialization. As trivialization is always ‘of something’, we need a functor to take us from this something to the respective trivialization. So let Triv* be an unary functor such that5 ∀x Triv(Triv*(x)). A further axiom should probably guarantee that Triv* is injective: ∀xy ((Triv*(x) = Triv*(y)) → (x = y)) The most essential axiom concerning trivialization now would be one spelling out that a trivialization of an entity constructs the very entity: ∀vxy (VConstr(v,Triv(x),y) ↔ (y=x)) 3. Execution. An execution is, like a trivialization, of something; so we introduce the predicate Exe* and stipulate ∀x Exe(Exe*(x)). ∀xy ((Exe*(x) = Exe*(y)) → (x = y)) The crucial axiom here is one stating that an execution of a construction constructs what is constructed by the very construction (if anything): ∀vxy VConstr(v,Exe*(x),y) ↔ VConstr(v,x,y) 4. Double Execution. The situation is again analogous: ∀x DExe(DExe*(x)). ∀xy ((DExe*(x) = DExe*(y)) → (x = y)) Here the crucial axiom spells out that a double execution of a construction constructs what is constructed by what is constructed by the construction (if anything): ∀vxy (VConstr(v,DExe*(x),y) ↔ ∃z (VConstr(v,x,z) ∧ VConstr(v,z,y))) 5. Composition. Here the situation is more complicated, for composition is of more than one object; we can have a composition of n objects for any natural n>1. So what we need is an (n+1)-ary functor Compn for every natural n, governed by the axioms of the following kind: ∀yx1...xn Comp(Compn(y,x1,...,xn)) Of course that we could define Triv in terms of Triv*, viz Triv(x) ≡Def ∃y(x = Triv*(y)); but again we do not dwell on such details. 5
5
∀vyx1...xnzv′y′x1′...xn′ (VConstr(v,Compn(y,x1,...,xn),z) ↔ (VConstr(v,y,y′) ∧ VConstr(v,x1,x1′) ∧ ... ∧ VConstr(v,xn,xn′) ∧ Appln(y′,x1′,...,xn′,z))) The last axiom states that a composition constructs the result of application of what is constructed by its first argument to what is constructed by the rest of the arguments. 6. Closure. We again need, for every natural n, an (n+1)-ary functor Closn: ∀yx1...xn ((Var(x1) ∧ ... ∧ Var(xn)) → Clos(Closn(x1,...,xn,y))) ∀vyx1...xnz (VConstr(v,Closn(x1,... xn,y),z) ↔ (Var(x1) ∧ ... ∧ Var(xn) ∧ ∀v′a1...an (Val(v′) ∧ (Value(v′,x1,a1) ∧ ... ∧ (Value(v′,xn,an) ∧ ∀x(((x≠x1) ∧ ... ∧ (x≠xn)) → ∀w(Value(v′,x,w)↔Value(v,x,w)))) → ∃u (Appln(z,a1,...,an,u) ∧ VConstr(v′,y,u))) Here the last axiom spells out the fact that a closure is a way of turning a construction of an object into that of a function: Closn(x1,...,xn,y) constructs the function which applied to the ntuple a1,...,an, yields what would be constructed by the construction y if the variables x1,...,xn constructed a1,...,an, respectively. This completes our sketch of an axiomatic theory of constructions. I think that despite the fact we have refrained from dwelling on details (which would, of course, become important were somebody to take the task of constructing this kind of theory seriously), our exercise has indicated that it does not seem to be impossible to explicate the concept of construction in this way, i.e. in the usual axiomatic manner. However, the resulting theory appears to be quite different from set theory: it is not so ground level as set theory is (it presupposes the concepts of function and functional application), it is much less ‘elegant’, and it also does not seem to give rise to nontrivial mathematical problems in the way set theory does (at least not directly).
Systems of constructions Tichý’s system of constructions (TSC, for short), as we have just seen, is based on six types of constructions. An interesting question now is: Why should we accept that there are just the constructions he claims there are? Suppose I would insist that there is a peculiar kind of construction called addition which turns a pair of constructions of natural numbers into a construction of the sum of the numbers. More precisely, if C1 and C2 are constructions such that C1 constructs the natural number n1 and C2 constructs the natural number n2, then [C1 C2]+ constructs the number n+m (and it constructs nothing if either C1 or C2 does not construct a natural number). Tichý would probably respond that what I see as a peculiar kind of construction is in fact a specific case of his composition, that what I see as [C1 C2]+ is ‘in fact’ his [0+ C1 C2], i.e. the application of the trivialization of + to C1 and C2. However, how can one justify a claim that one construction is in fact another construction? We can claim, to be sure, that one system of constructions can be reduced to another system of constructions - but why should we say that the constructions of the former are therefore ‘in fact’ those of the latter? 6
An argument against my proposal might, of course, be based on the fact that if we had a separate construction for addition, we would need one for subtraction, multiplication etc., and that this would lead to a proliferation of constructions. Thus, whereas TSC makes do with altogether six types of constructions, what I have proposed would yield a many times more numerous denizens of constructions. Thus, if what we are after is a minimalist stock of basic primitives, TSC is undoubtedly better6. However, what if we proposed a system of constructions which is different from TSC, but equally simple? Let us consider two such proposals: 1. It is, it seems to me, slightly counterintuitive to say that the expression „1+2“ expresses a construction of the application of the trivialization of + to the trivializations of 1 and 2: it would seem to be more natural to say that it is simply the construction of application of the object + to the objects 1 and 2. My opinion is that the source of this counterintuitivness is the fact that Tichý’s construction of composition meshes together two intuitively different constructions: the construction of application proper (just exemplified) and the construction of composition of constructions. An example of the latter would be the application of + to the results of some sub-operations, like in the construction expressed by „(1+2)+(3+4)“. In view of this fact we might attempt to improve on TSC by replacing Tichý’s composition by the following two constructions: Application: If f is an n-ary function and x1,...,xn are objects, then (f x1 ... xn) is a construction constructing the value of f for the arguments x1,...,xn. Composition: If C is a construction of the object x from the objects x1,...,xn and C′ is the constructions of xi from x1i,..., xmi, then the result of replacing xi in C by C′ is a construction of x from x1, ..., xi-1, x1i,..., xmi, xi+1,...,xn. Now the construction expressed by „1+2“ would be simply (+ 1 2), while that expressed by „(1+2)+(3+4)“ would be (+ (+ 1 2)(+ 3 4)). So here we have an example of a rival system of constructions: it has one more basic construction than TSC, but the gain is that it might be, at least in some respects, more conceptually perspicuous. (Moreover, the new system could in fact not need more types of constructions that TSC, for I suspect that once we have the above construction of application, we no longer need trivialization. But this is not a theme for us now.) 2. The second example I am going to present is more important for our subsequent considerations. There is obviously a close parallel between the structure of TSC and that of the two-sorted variant of the language of typed lambda-calculus of Church (1940) - in fact, it seems as if the two-sorted lambda-calculus were just the canonical language expressing the constructions of TSC. However, Church’s typed lambda calculus is known to be translatable into combinatory logic7 - so why not consider the system of constructions for which combinatory logic would be the canonical language in the very sense in which two-sorted type theory is for TSC? If we call this system of constructions the combinatory system of constructions, CSC, we may ask: what makes TSC the ‘right’ system and CSC a ‘false’ one? And as far as I can see, the only feasible answer is that TSC appears to be, in some sense, 6
Let me note in passing that it seems that Platonistically-minded logicians like Tichý and Materna do not appear to be after something like „the smallest store of materials with which a given logical or semantic edifice can be constructed“ (Russell, 1914, p.51) - what they do seem to be after is rather the discovery of the matter-of-factual system of constructions underlying what we say.
7
See Curry and Feys (1957). 7
more handy. However, in what follows we will see that ridding ourselves of variables, which is what the replacement of TSC by CSC would effect, might be a useful thing. However, to see this, we must consider further aspects of TIL.
Constructions and natural language expressions Another important question related to TSC concerns the criterion for deciding which construction is expressed by a given natural language statement. Let us restrict ourselves to sentences and let us ask: how should we tell which construction is expressed by a given sentence? Notice that if we believed, as the (pre-hyper-)intensional logicians did, that what is expressed by a sentence is (capturable as) a class of possible worlds8, the criterion would be clear: a given sentence would express the class of those and only those possible worlds in which it is true. It is obvious that the criterion pins down a single unique entity (disregarding, of course, the vagueness of natural language sentences): there is one and only one class of those possible worlds in which a sentence is true. This criterion is still usable as a constraint even if we now claim that what sentences express are constructions: a given sentence is bound to express a construction constructing the class of those and only those possible worlds in which the sentence is true. However, this constraint now no longer pins down a unique entity: for every construction, there clearly exists an infinite number of different constructions equivalent to it in the sense of constructing the same class of possible worlds. Which one, then, is the construction expressed by the sentence? Let us take an example; let us consider the sentence Venus is a planet.
(2)
According to Materna (1998, p. 43), this sentence expresses the construction λwt.[0planetwt 0Venus],
(2′)
where planet is an object of the type (οι)τω and Venus is an object of the type ι. Now there is a number of constructions equivalent to (2′), among others λwt.[ 0& [0planetwt 0Venus] [0= [0+ 01 01] 02]]],
(2*)
where & is the usual conjunction (the object of the type (οοο)); or 0
Venus-is-a-planet,
(2**)
where Venus-is-a-planet is a suitable object of the type οτω; or [0is-a 0planet 0Venus]
8
(2***)
Or, better, a (generally partial) function from possible worlds to truth values.
8
where is is a suitable of object of the type (οτω)((οι)τω)ι. Why should we say that it is (2′), and not (2*), (2**) or (2***), that is expressed by (2)? The only way of answering this question I can see is that the appropriateness of (2′) somehow rests on the (syntactic) structure of (2). That is to say that (2′) is a candidate for the construction expressed by (2) superior to, e.g., (2*), because (2*) contains numbers which are not mentioned at all in (2). A similar argument can be used if we want to justify the superiority of (2′) over (2**): (2**) consists of a single component, whereas (2) is a compound, it contains more than one term. However, then it would seem that a wholly analogous argument could be used to show that (2***) is superior to (2′): for (2***) contains an object corresponding to the part ‘is’ of (2) which is lacking in (2′). (And of course it would be possible to further refine (2***) into a construction containing separate parts for ‘is’ and ‘a’.) Imagine that we accept a system of constructions, different from TSC, which contains the following type of construction: Basic intensional application. If P is an (οι)τω-object and T a ι-object, then {T is a P} constructs a οτω-object which takes w and t into truth just in case T belongs to Pwt. (The string ‘is a’ is to be considered as a syncategorematic sign akin to the brackets ‘{‘ and ‘}’.)9 Given this definition, we could analyze (2) by means of {Venus is a planet}, the structure of which is in the straightforward correspondence with the syntactic structure of (2). So it seems that if we accept that a construction is the better expression of what a given sentence says the closer it is to the syntactical structure of the sentence, our last proposal would fare better than the standard TSC-based analysis. Moreover, it would be generally preferable to switch from TSC to CSC - natural language expressions do not contain anything like variables and so the constructions of TSC, unlike those of CSC, are bound to be structurally deviant from natural language expressions.
Materna’s way from constructions to concepts Materna, who has been interested in the concept of concept for a long time, realized that constructions can well serve for the purposes of explicating this concept. In fact, closed constructions are something very close to what concepts, as he uses the term, are10. The only problem which Materna saw was that constructions are a bit more fine-grained than concepts are supposed to be. This can be seen from the fact that, e.g., (3) and (3′) are different
9
Note that this construction can be easily interpreted within TSC: from the viewpoint of Tichý’s system, {T is a P} would ‘in fact’ be λwt.[Pwt T]
10
The term „concept“ is sometimes interpreted as almost generally synonymous with „meaning“, while sometimes it is rather interpreted more narrowly, as something like „meaning of a predicative expression“. Whereas Frege, as is well known, endorsed the latter interpretation, Materna sticks to the former one, which, as he argues, was endorsed also, e.g., by Bolzano.
9
constructions, but intuitively they amount to one and the same concept, namely the concept of addition: λxy [0+ x y] λyx [0+ y x]
(3) (3′)
The remedy Materna proposes is to explicate concepts not directly as constructions, but rather as certain equivalence classes of constructions. For this purpose he defines two equivalence relations among constructions, namely the α-equivalence and the β-equivalence. Two constructions are α-equivalent iff one can be obtained from the other by systematic renaming of its lambda-bound variables; while they are β-equivalent iff one can be obtained from the other by erasing ‘idle lambda-abstractions’, rewriting λxP(x) as P. The relation of quasiidentity is then the transitive closure of the union of the relations of α- and βequivalence. Eventually, a concept is the equivalence class of closed constructions according to the quasiidentity.11 I think the trouble with this move is that it largely spoils the intuitiveness of Tichý’s picture and introduces an ad hoc element of the kind of that which impairs Lewis’ theory. It seems to me that the basic attractiveness of Tichý’s proposal consists in the fact that it is plausible to assume that the complex expressions of our language express ways in which meanings or concepts associated with their parts add up and construct something - and once we start to say that what the expressions express are in fact not constructions, but rather classes, the illuminating picture is gone. What I think would be a more promising route is to develop an alternative notion of construction which would not be ‘over-fine-grained’, i.e. which would enable us to identify concepts directly with constructions. A moment reflection reveals that what causes the troubles Materna’s definition of concept is devised to overcome are variables. No two different variable-free constructions are either α- or β-reducible. So if we restricted ourselves to variable-free constructions, we could identify concepts directly with constructions. And we have indicated above that there might be a way to accomplish this. Let us return to our sentence (2): Venus is a planet
(2)
We have seen that there does not appear to be a decisive reason not to see it as expressing the construction [0is-a 0planet 0Venus],
(2***)
or, allowing ourselves of the above construction of basic intensional application, as {Venus is a planet}. Now this can be generalized. Let us introduce the following type of construction:
11
For rigorous definitions, see Materna (op. cit., §5.3).
10
(General) intensional application. Let I be a subset of the set {0,...,n}. Let C0, C1,...,Cn be such constructions that C0 constructs either an object of the type ((βα1...αn)τω) or an object of the type (βα1...αn), depending on whether 0∈I or 0∉I; and similarly every Ci constructs either an object of the type (αiτω) or an object of the type αi, depending on whether i∈I or not. Then [C0 C1 ... Cn]I constructs an object of the type (βτω) such that its value for a world W and a time moment T is the result of the application of the function O0 to the objects O1,...,On, where Oi is the object constructed by Ci if i∈I and it is the value of the object constructed by Ci for W and T otherwise. (The equivalent of [C0 C1 ... Cn]I in TSC is λwt.[C0* C1* ... Cn*] where X* is Xwt for i∈I and is X otherwise.) Let us illustrate the definition by examples of instances of intensional applications with their equivalents within TSC: [planet Venus]{0} λwt.[0planetwt 0Venus] [finds John a-unicorn]{0,2} λwt.[0findswt 0John 0a-unicornwt] [seeks John a-unicorn]{0} λwt.[0seekswt 0John 0a-unicorn] [& [planet Venus]{0} [finds John a-unicorn]{0,2}]{1,2} λwt.[0& [0planetwt 0Venus] [0findswt 0John 0a-unicornwt]] Let us return to the step which led from extensional semantics to the intensional one. Within extensional semantics, we have (besides others) subject terms meaning individuals, predicates meaning functions from individuals to truth values and sentences meaning truth values. So the meaning of a predicate and that of a term fit nicely together, in the sense that the former gets simply applied to the latter and what results is the meaning of their combination. Now the intensionalization of this model, which has turned out to be necessary to make the model at least minimally semantically plausible, threatened to spoil this nice fit. Within the intensional model, meanings get relativized to possible worlds (and, as the case may be, time moments), so that the meaning of a term now is a function from worlds to individuals, that of a predicate a function from worlds to the functions from individuals to truth values and that of a sentence a function from worlds to truth values. Hence the meaning of a predicate can no longer be simply applied to the meaning of a term to produce the meaning of the corresponding sentence. What Tichý employed to save the situation were variables for possible worlds: the mechanism of lambda-abstraction, which was part of the apparatus anyway, then did all the rescue work, via the ‘pre- and post-processing’ mentioned above. (Montague originally attempted to do it differently, but his followers soon found out that to assimilate intensional logic to the two-sorted lambda-calculus, as Tichý, in effect, did from the beginning, is the most efficient way12). However, if we accept a new kind of construction as the intensional application defined above, we solve the problem bypassing the engagement of variables. And this may be, as we saw, convenient.
12
Cf. Gallin (1975).
11
Logic without variables? The indicated way of banishing the variables w and t from the usual constructions is only a specific case of a much more general mechanism (which, in effect, underlies the transformation of lambda-calculus into combinatory logic). Within categorial grammar, which underlies logics like TIL, you can only combine an expression of the type C/C1 ... Cn with expressions of the types C1, ..., Cn. Now imagine you want to combine an expression of the type C1/C2 with an expression of the type C2/C3 into an expression of the type C1/C3. (Imagine, e.g., that you want to combine an (οο)-expression, like negation, with an (οι)expression.) With the help of lambda-abstraction, you can accomplish this by, first, applying the latter expression to a variable of the category C3, then applying the former expression to the result, and then abstracting the result upon the variable used in the first step. The question, though, is why not allow for a direct combination, especially when the combination makes a clear semantic sense: it amounts to composition of functions. Hence although the traditional categorial grammar allows for the direct expression of only functional application, it would be straightforward to extend it in such a way that it would allow for expressing also all kinds of functional compositions. (Our intensional application would then be one of the possible grammatical rules of such an extended language.) In fact, ideas of this kind were expressed as early as in the fifties by Lambek (1958), and have come to flavor especially during the last decade13. Does, then, what I propose amount to banishing variables from logic altogether? Yes and no. The fact is that we can do logic without variables. It can be shown that any ‘reasonable’ logical calculus can be reformulated without the employment of variables. Quine (1960) has envisaged this for first-order predicate calculus; and this kind of treatment can be generalized (see Peregrin, to appear b, for some details). And, as I have tried to show, such a reformulation can be more than a formalistic game - it may be a way of gaining conceptual clarity. However, I do not suggest that we should simply forget about variables - variables are hardly disapensable technical means of doing logic. What we, I think, should do, is to „speak with the vulgar, but think with the learned“, i.e. to continue using variables, only not counting them to the essential, ‘categorematic’ inventory of our language, but seeing them rather as dispensable items on par with, say, brackets. We know, due to the Polish logicians, that brackets are dispensable, but we keep using them, for it is convinient - but we are not tempted to claim that they belong to our language in the same sense as predicates or logical connectives do. And we should, I suggest, see variables in an analogous way: employ them as an efficient technical tool, but disregard them when we ponder foundational questions such as the nature of concepts. All of this is, of course, connected to the very way one sees the enterprise of logical analysis. What I am convinced is that we should see it not as a ‘(quasi)metaphysical’ enterprise of discovering and reporting entities ‘making up’ the meanings of expressions, but rather as an explicative enterprise of envisaging the semantic, especially inferential, properties of expressions by means of building formal models (see Peregrin, 1998). I am convinced that the metaphysical stance entraps us into the net of really unanswerable (pseudo)questions like Are variables really out there, within the meanings?. I think that in contrast to this, the explicative stance allows us to replace such questions by much clearer and more contentful 13
See, e.g., Morrill (1994).
12
ones, like Is a model with variables in some sense more useful (e.g. more comprehensible, or more easy to handle) than one without them?
References Bigelow, J.C. (1978): ‘Believing in Semantics’, Linguistics and Philosophy 2, 101-144. Carnap, R. (1957): Meaning and Necessity, University of Chicago Press, Chicago. Curry, H.B. and Feys, R. (1957): Combinatory Logic, North Holland, Amsterdam. Frege, G. (1976): Wissenschaftlicher Briefwechsel (ed. by G. Gabriel et al.), Meiner, Hamburg. Gallin, D. (1975): Intensional and Higher-order Modal Logic, North-Holland, Amsterdam. Church, A. (1940): ‘A Formulation of the Simple Theory of Types’, Journal of Symbolic Logic 5, 56-68. Lambek, J. (1958): ‘The Mathematics of Sentence Structure’, Amer. Math. Monthly 65, 154170. Lewis, D. (1972): ‘General Semantics’, in Semantics of Natural Language (ed. D. Davidson & G. Harman), Reidel, Dordrecht. Materna, P. (1983): ‘On Understanding and Believing’, Prague Studies in Mathematical Linguistics 8, 211-218. Materna, P. (1998): Concepts and Objects (Acta Philosophica Fennica 63), Societas Philosophica Fennica, Helsinki. Morrill, G.V. (1994): Type Logical Grammar (Categorial Logic of Signs), Kluwer, Dordrecht. Peregrin (1998): ‘Linguistics and Philosophy’, Theoretical Linguistics 24, 245-264. Peregrin (to appear a): ‘The „Natural“ and the „Formal“’, Journal of Philosophical Logic. Peregrin (to appear b): ‘Variables in natural language: where do they come from?’, Variablefree semantics (ed. M. Boettner & W. Thümmel), Secolo, Osnabrück. Quine, W.V.O. (1960): ‘Variables explained away’, Proceedings of the American Philosophical Society 104, 343-347. Russell, B. (1914): Our Knowledge of the External World, Allen and Unwin, London.
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Paul Horwich: MEANING, Clarendon Press, Oxford, 1998. ix + 241 pp.
Berkeley once compared philosophical problems to the situation when "we have first rais'd a dust, and then complain, we cannot see"; and this quote could be the motto underwriting Horwich's attempt to dismantle the enigma of meaning in the way he attacked that of truth earlier (viz Horwich, 1990). Horwich's strategy is to show that meaning is in fact a simple and perspicuous concept and that it appears perplexing for philosophers only because they have piled it with unreasonable onuses. If the task of a theory of meaning is, to borrow David Lewis' (1972, p. 173) often quoted dictum, "to find out what meaning does and then to find something that does this", then Horwich's suggestion is that once we discard distorted views of what meaning does, the task of finding something that does that becomes rather easy. In particular, Horwich claims we will then safely be able to say, together with Wittgenstein, that meaning is use1. Horwich claims that each meaningful expression has a property, meaning property, in virtue of which it means what it does. He introduces a 'disquotational-capitalizing' notation: he designates the meaning property currently possessed by the English word "dog" by the predicate "meaning DOG", that possessed by "and" by "meaning AND" etc.; and he says that the meaning property of an expression consists in the way the expression is used. In particular, the meaning property of any word is constituted by the fact that certain sentences containing the word are being accepted. Thus, the word "and" means what it does in virtue of the fact that we tend to accept "p and q" if and only if we accept "p" and "q"; whereas the meaning of "red" consists in our having "the disposition to apply 'red' to an observed surface when and only when it is clearly red" (p. 45; let us note in passing that if meaning is to consist in an acceptance of sentences, then "to apply 'red' to an observed surface" has to be construed as something like "to accept '(This is) red' when observing a surface"). The pervading idea is that the entire usage of a word is reducible to a basic regularity, which is thus constitutive of the meaning of the word. The core of the book is devoted to showing that this simple theory fulfills all reasonable demands which can be made on a theory of meaning and to defending this conclusion against all possible kinds of objections. Horwich counts with seven basic constraints which, according to him, a theory of meaning is generally expected; and he tries to show that if they are not misinterpreted, then the use theory of meaning fulfills them almost trivially. The discussion of the seven demands constitutes Chapter 2 of the book; some of them, however, are discussed in greater detail in subsequent chapters. The constraints, in Horwich's view, are the following (p. 13): (1) The Understanding Constraint. The theory must explain how facts about meaning can be known; for understanding a language involves knowing what its words mean. (2) The Relationality Constraint. The theory must specify the general nature of the relation between terms and their meanings; that is, it must provide some analysis of the notion, 'x means y'. (3) The Representation Constraint. The theory must explain how language can represent reality - how, in virtue of their meanings, singular terms are used to refer to objects in the world, predicates pick out sets of things, and sentences express thoughts that are true or false. (4) The Aprioricity Constraint. The theory must accommodate the arbitrary, conventional, aprioristic character of the meanings of words. For the choice of which language to use to express one's empirical beliefs is not itself an empirical matter. (5) The Compositionality Constraint. It must explain how meanings are compositional how it happens that the meanings of complex expressions are determined by the meanings of their parts.
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(6) The Normativity Constraint. It must explain how meanings can have normative consequences - how it is, for example, that given what the word "dog" means, one ought to apply it to this thing but not to that. (7) The Use Constraint. It must account for the relationship between the meaning of an expression and its use - the possibility of explaining the way words are used on the basis of how they are understood. Horwich claims that if these constraints are understood properly, none of them poses a serious problem for a simple use theory of meaning. The problem with Constraint (1), according to him, is that it is often construed as amounting to explicit knowledge of meaning, whereas it should properly be understood as speaking merely about an implicit one. Thus, understanding is "a skill or a practical ability with no propositional content" (p. 18) - to understand an expression is to be able to use the expression in question in an appropriate way, not to have an explicit knowledge of an object. In the case of (2), Horwich claims that although it is possible to construe "'dog' means DOG" as "the utterance of 'dog' indicates (i.e. justifies belief in) the presence (within some mental state of the speaker) of the concept, DOG", this possibility concerns only the "superficial composition of meaning properties", not their underlying nature, which is non-relational. Thus, we can say, meaning-talk is relational as such, whereas it becomes non-relational when translated into the language of physics. Regarding (3) Horwich states that this constraint is again not controversial in itself; that it becomes problematic only if construed in an unwarrantedly strong way, namely as saying "that the meaning properties of terms must reduce to relations between those terms and the aspects of reality to which they apply" (p.27). That "dog" refers to dogs or is true of dogs is a platitude explainable in a disquotation theory; it does not mean that there must be a real (e.g. causal) link between the word and something within the world. Likewise (4) presents Horwich with no difficulty - unless it is construed to imply that what thus comes to be known a priori are "substantive postulates". Horwich claims that "the commitment to a substantive theory of fness, '#f', is the product of two independent decisions: one of them is to hold true the existential thesis 'x(#x)' (...), the other is to hold true the conditional 'x(#x) #f'" (p. 31-32). It is the latter conditional which constitutes the real implicit meaning of "f" - and this conditional is not, according to Horwich, "substantive". To satisfy (5), Horwich claims, it is enough "to find accounts of word meanings consistent with the fact that the meanings of words engender the meanings of sentences" - and the use conception of meaning, according to him, is precisely suited for this. Also (6) is readily satisfied by the use theory once we realize that "the normative import of meaning does not preclude a reduction of meaning properties in non-semantic, non-normative (...) terms", for it is clear that "situations characterized in non-normative terms" may have "normative import" (p. 38-39). And (7) is obviously satisfied trivially. In the following chapter, Horwich sketches the basic features of his use theory. His crucial theses are the following: (i) Meanings are concepts, "abstract entities from which beliefs, desires and other states of mind are composed"; (ii) the overall use of each word stems from its possessing a basic "acceptance property"; and (iii) two words express the same concept in virtue of having the same basic acceptance property (p. 44-46). Hence, in Horwich's view, according to Horwich, the meaning of any word is engendered, as noted above, by the fact that the competent users of the word accept certain sentences which contain the word. The remainder of the chapter is devoted to countering any possible kinds of criticisms of this conception of meaning. The ensuing five chapters elaborate on some of the points sketched in Chapter 2. In particular, Chapters 4 and 5 summarize Horwich's disquotational theory of truth and show how it engenders a
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deflationary theory of reference. Chapter 6 gives more consideration to the problem of implicit definitions and a priori knowledge, while Chapter 7 focuses on the problem of compositionality and Chapter 8 on that of normativity of meaning. The final two chapters of the book, Chapters 9 and 10, then present a discussion of controversial theses of Quine and Kripke, respectively. It would seem that there are two basic ways to write a book expounding a question as general as that about the nature of meaning. One is a 'scientific' way, which consists in discussing all the answers proposed by earlier inquirers, weighting their respective pros and cons, and then working towards an answer accommodating those aspects of the previous views which are taken to be warranted and correcting those which are not. The other is the way chosen, e.g. by Wittgenstein for his Tractatus: the way consisting in paying less attention to what has been published about the theme so far, and more to the very exposition, in retelling the whole story in one's own words. Horwich chooses the latter, 'Tractarian' way, and thus his approach resembles not only the late Wittgenstein in holding the conviction that solutions of philosophical problems are "open to view" when delusive prejudices and preconceptions are removed, but also the early Wittgenstein in trying to present a self-contained, 'crystal-clear' exposition. I think this choice is understandable; and it has enabled Horwich to write an extremely readable and duly provocative book. His edifice is self-contained, perspicuous and 'seamless' to such an extent that it is difficult for an opponent to find a fissure to wedge it. (Which is not to predict that proponents of more 'substantial' theories of meaning will feel overwhelmed - I suspect that most of them will take the 'seamlessness' of Horwich's exposition to be engendered by the fact that the author did not appreciate 'the real depth' of the problem.) As far as I can see, the most heterogeneous brick within the walls of Horwich's edifice is his argument concerning the constraint (2). There his otherwise anti-psychologistic semantic theory adopts what to me seems to be a discordantly psychologistical appendage. "Occurrences of the word 'dog'," says Horwich (p. 4), "provide reason to believe in the presence (within the thought of the speaker) of the concept DOG." However, what if the word "dog" occurs in a newspaper article - in whose thought should we believe the concept to be present then? The author of the article in question? But what if the article is a joke put together by a computer program? And even if we dismiss doubts of this kind, it would seem that Horwich's proposal rests on an assumption which, I think, surely should not go without saying: that to have meaning is not only to be used in a certain way, but also to be uniformly accompanied, when uttered, by a specific mental state or activity, 'presence of a concept within the mind'. As far as I can see this assumption is not only problematic (cf. Dummett, 1988, p. 185), but also incongruous with the rest of Horwich's story. (After all, the author claims that understanding is a skill, not a possession of something.) But the truth is that if we do not buy this mentalistic account, we will have troubles with understanding why there should exist the two levels of the meaning talk which play such an important role in Horwich's account: the 'surface' level on which meanings appear to be things associated with expressions, and the underlying level on which there are only expressions employed in certain ways. Besides this, Horwich sometimes seems too hasty in dismissing opposing ways to view meanings. Thus on p. 16 he writes: "It is not easy to identify the fact regarding, say, the word 'dog', the knowledge of which constitutes our understanding of that word. The obvious candidate is the fact that it means what it does, i.e. the fact that 'dog' means DOG. But there appears to be a decisive objection to this suggestion - as clearly correct as it may initially seem. For in light of how the meaning designator, 'DOG', has been introduced (...), it is a trivial matter to work out that 'dog' means DOG. ... So it cannot be that the understanding the word 'dog' consists in knowing that dog means DOG. But this is puzzling; for what other fact could possibly be relevant?" Now suppose that somebody is of the opinion that the meaning of "dog" is, say, a kind of a mental content and that understanding the word consists in possessing the very content. The reply of such a person to Horwich's rhetorical question would them probably be: "The fact that the word is associated with the
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relevant content, of course! - When you say that 'dog' means DOG, then your 'DOG' is bound to refer to the very content!" The point is that Horwich's argumentation appears to be on a par with the following: "For every country C, let 'CPC' denote the capital of C. Now the knowledge that the capital of C is a certain town (namely CPC), cannot be substantial, for to work out that the capital of C is CPC (i.e. that 'the capital of C is CPC' is true) is a trivial matter."
The 'Tractarian' form of Horwich's exposition also engenders a certain lack of clarity in respect to the relationship of his views to those of other similarly-minded philosophers. Horwich pays some explicit attention to the relationship between his doctrine and that of Donald Davidson; however, he appears to almost overlook another prominent approach which would seem even closer than Davidson's to his own stance, namely the approach introduced by Wilfrid Sellars (see, e.g., Sellars, 1963) and recently further developed by Robert Brandom (1994). Sellars' basic claim bears much similarity to Horwich's, namely that to have meaning is to be used in a certain way, especially to play a role within certain 'transitions', such as those from seeing a red surface to 'accepting' "This is red", or from 'accepting' "This is red" to accepting "This is not green". However, the crucial issue between the Sellarsian approach and that proposed by Horwich is the possibility of 'naturalizing' the meaning talk, of reducing it to behaviorist talk (couched in physicalist terms). Horwich claims, as we saw, that the meaning of a word consists in our accepting certain sentences which contain it. Now what does it mean, according to him, to accept a sentence? Horwich gives a tentative behaviorist theory of 'accepting' (p. 96) based on postulates of the kind of For each observable fact O there is a sentence type "o" such that: O is instantiated in view of S S accepts "o" The crucial question is whether such a theory is actually feasible at all - even if we invested it with more sophistication. The trouble is that it does not seem to be capable of allowing for the notion of an error - and any notion of accepting which does not provide for the possibility of misaccepting does not seem to fit to underlie meaning talk. At least so Sellars would insist. In fact, as far as I can see, the situation is very similar in respect to the relationship between the approach of Horwich and that of Davidson. Davidson's claim that we have to base semantics on the irreducible concept of truth (and hence construe meanings not as uses, but rather as truth conditions) stems precisely from his conviction that the 'acceptance' relation holding between speakers and sentences, if it is to underlie semantics, cannot be reduced to behaviorist terms. Thus, both Davidson and Sellars argue at length that to take an 'accepting' crucial for semantics as reducible is to commit a naturalistic fallacy (for Davidson such an 'accepting' would consist in the irreducibly idiosyncratic 'holding for true', for Sellars in the irreducibly normative 'holding for correctly assertible') - and I am not sure whether Horwich's gesture towards a behaviorist reduction can be taken as a substantial counterargument. Another point which seems to me somewhat unclear and which is crucial for assessing the depth of the disagreement between Horwich and Sellars concerns the question of holism. Horwich claims that his approach is only moderately holistic (for it embraces only those "mild" varieties of holism which are engendered by the fact that the sentences whose acceptance establishes the meaning of a word may contain also other words). However, it is not clear whether this should be understood as saying that a word means what it does exclusively thanks to its acceptance property and independently of the presence of any words which are not needed to articulate its property. Imagine a language containing a single sentence, namely "Red(!)", governed by the same acceptance property as the homophonic English sentence (namely "the disposition to apply 'red' to an observed surface when and only when it is clearly red"). Would it follow that "red" in this language means the same as in English? Or
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would this be somehow dependent on whether the language in question contains also some other words? From the Sellarsian viewpoint, it is crucial that "red" could not express our usual concept of redness, nor indeed any concept at all, if it were not able to figure in judgments - and nothing can be a judgment unless it can be negated, conjoined with other judgments etc. Horwich seems to reject this - however, if this is the case, does he not have to grant a sensor reacting to red color the concept of red? So much for the viewpoint of a basically sympathetic reader. As for unsympathetic ones, i.e. those who take for granted that meanings are some real entities labeled by corresponding expressions, as I said earlier, I doubt that they would be swayed by Horwich's book. However, I am convinced that even the unsympathetic reader - one who hastens to dismiss the 'crystal-clarity' of the author's exposition as simply a form of superficiality - should ask herself: Is it really the superficiality of the exposition which is at fault, might it not be that the illusion of depth which apparently escapes the author is itself the product of a certain picture which "holds us captive"? This is not to say that some criticism of this kind cannot be substantiated, but it is to urge that Horwich's arguments should not be judged by a first impression. The founding fathers of analytic philosophy based their revolt against bad philosophy on the conviction that many (if not all) philosophical problems are solvable by 'conceptual analysis'. Their conviction that this would lead to the ultimate solution or a complete dissolution of traditional philosophical problems subsequently proved itself too naive, and some of the current successors of analytic philosophy appear to take this to show that the whole analytic movement was a philosophical blind alley. I think that Horwich's book is one of those which indicate that 'conceptual analysis' is still something to pursue, something which can strongly aid the understanding and solving of philosophical problems.
References. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Dummett, M. (1988): 'The Origins of Analytical Philosophy I-II', Lingua e Stile 23, 3-49, 171-210. Horwich, P. (1990): Truth, Blackwell, Oxford. Lewis, D. (1972): 'General Semantics', in Semantics of Natural Language (ed. D. Davidson and G. Harman), Reidel, Dordrecht. Sellars, W. (1963): Science, Perception and Reality, Routledge, New York. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford.
Notes: 1. See
Wittgenstein (1953, §43): "Die Bedeutung eines Wortes ist sein Gebrauch in der Sprache."
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Absolute and relative concepts in logic Jaroslav Peregrin*
www.cuni.cz/~peregrin [Logica Yearbook 2000 (ed. O. Majer), Filosofia, Prague, 2001, 71-77]
Incompleteness of arithmetic It is a common wisdom that whereas consequence or entailment is a semantic concept, provability is a syntactic concept. However, what exactly does this mean? What is provability? In the traditional, intuitive sense, to prove something is to demonstrate its truth, and indeed the Latin word for proof is demonstratio. Hence in this sense, we cannot prove something unless it is true. Now in the course of his well known proof of the incompleteness of arithmetic, Gödel showed that provability within the axiomatic system of Peano arithmetic (PA) can be reconstructed as a number-theoretic predicate Pr and he showed that within the system there exists a statement G so that (*) G ⇔ ¬Pr(G). Now if we assume that for every statement X of PA it is the case that X is true whenever Pr(X), we have the following argument: Pr(G) ⇒ G is true ⇒ G ⇒ ¬Pr(G). This means that, in pain of contradiction, it cannot be the case that Pr(G). Hence Pr(G) is bound to be false, and hence ¬Pr(G) is bound to be true. However, this means that, by (*), G is bound to be true. Hence G is true (although it is not provable). Now is what we have just displayed not a (sketch of a) proof of G? The answer of Gödel and his followers is No; it is not a proof in the strict (sic!) sense of the word. (For if G were provable, it would follow that it is unprovable, and the whole system of PA would be in ruins – whereas we take for granted that PA is consistent, for what it spells out are our most basic intuitions regarding the manipulatability of natural numbers; intuitions which are axiomatic in the primordial sense of the word1.) Nevertheless, it is hard to deny that it is a demonstration of the truth of G (everybody who reads it – and has basic skills in mathematics *
Work on this paper was supported by the research grant No. A0009704 of the Grant Agency of the Academy of Sciences of the Czech Republic. 1
Hence G is valid not merely in force of the validity of the axioms of PA, but also of the consistency of PA, which may be seen as warranted by what Gaifman (2000, p. 467) calls the „recognizable validity“ of PA. And, as Gaifman points out, „a recognizably valid T cannot capture our full mathematical reasoning; because we shall reason that T is consistent, and this reasoning is something T cannot deliver.“ Thus, according to Gaifman, it is not the case that there are statements which are true but unprovable, but rather that there are statements which are provable only within a theory we cannot ‚get outside of‘. Hence the moral we should draw is not the one drawn by Penrose (1990) and others, namely that we are not machines, but rather that „we (qua mathematicians) are machines that are unable to recognize the fact that they are machines“. I think that Gaifman is fully right (see also Peregrin, 1998).
1
– sees that G has to be true), hence that it is a proof in the intuitive sense – so why do we introduce an unnatural, ‘strict’ sense of “proof”? Well, the reason appears to be that we cannot make a mathematically rigorous sense of the intuitive concept, and so we have to make do with its relativist ‘circumscriptions‘. In particular, the concept of proof as introduced by Hilbert (amounting to derivability from axioms by means of rules) turns out to be incapable of accommodating intuitive proofs of the kind shown above. Hence we seem to be committed to accepting that not every statement of PA which is intuitively provable falls within the range of Pr; and we must see Pr as the direct regimentation not of the intuitive, absolute concept of provability, but rather of a relative concept of provability-in-PA (which is not the partialization of the absolute concept of provability to statements expressible in PA!). This leads us to the concept of provability relative to an axiomatic system, which now appears to be an imperfect, but the only mathematically tractable, counterpart of the intuitive concept of provability (simpliciter). However, note that if this is the case, then we must condone the fact that we can never conclude that something is ultimately unprovable in the absolutist sense. In particular, we cannot interpret Gödel’s result as saying that there are ultimately unprovable truths; and we cannot use Gödel’s result to argue against the coincidence of the concepts of truth and provability (in the intuitive sense). Viewed from this angle, Gödel’s result does not entail that there are unprovable truths, but rather that the concept of proof escapes rendering in Hilbertian terms. As Dummett (1963, p. 199) puts it: “Gödel’s discovery amounted to the demonstration that the class of [principles for recognizing as true a statement which involves quantification] cannot be specified exactly once and for all, but must be acknowledged to be an indefinitely extensible class.“2 In fact, the idea that we could recognize a mathematical statement as true without having proved it, is odd – how else could we come to know a truth of mathematics than by means of acknowledging its demonstrations? (There are, to be sure, philosophers and mathematicians who would insist that we can perceive mathematical reality in the way we perceive the outer world – but to claim we can establish the truth of G by observing natural numbers seems, at the very least, implausible.) Should we then maintain that truth and proof, within mathematics, remain inseparably connected? It is clear that if we do so, i.e. if we accept that Gödel’s incompleteness proof yields us a proof of G, then we have to see Peano arithmetic as contradictory. The point is that as G can be seen to claim its own unprovability, if it is in fact provable and hence true, we have a variant of the liar paradox. If we erase the gap between truth and proof, we no longer have any reason to see truth in PA as more than provability, and hence we are presented with G as simply a liar sentence. (This is how the situation was assessed by Finsler, 1926; 1944).
Is there a set of everything? Gödel’s reasoning suggests that the intuitive concept of proof should be replaced by a formal concept which is narrower. There is clearly a sense in which we have proved G above; however, as accepting this proof at face value would render PA inconsistent, we conclude that it is not a proof in the regimented sense. Is this a feasible solution? Consider a different, but parallel problem: the problem of the relation between the intuitive concept of set and its regimented variant as yielded by formal set theory. One of the arguments constitutive of the latter was put forward by Georg Cantor, and it aims at showing 2
See also Gaifman‘s paper quoted above in footnote 1; and cf. also Grim (1991).
2
that the cardinality of any set is necessarily smaller that that of its power set. We can reconstruct this well known argument as follows (see, e.g. Lavine, 1994): Let S be a set and f an injection from S to Pow(S). Let s* = {x∈S : x∉f(x)}. Then clearly f(y)=s* for no y∈S. (For if this were the case for a y, then y∈s* would entail y∉s* and vice versa). Hence there is no bijective mapping of S on Pow(S), and hence Pow(S) is always larger than S. Russell’s objection to this line of reasoning was that it cannot hold if S is the set of everything. If U is the set, then, by definition, U contains everything, including all its own subsets; hence Pow(U) ⊆U. This means that there is a bijective mapping of a part of U on Pow(U) (namely the identity mapping), and hence the cardinality of Pow(U) cannot exceed that of U. The response to this objection, which has now become so commonplace that it is not easy to see how it could be challenged, is that it only shows that everything does not constitute a set. However, consider the following anecdotic dispute between two persons called PseudoCantor and PseudoRussell: PseudoCantor: Every group of cars consists of black cars. PseudoRussell: But on the parking lot in front of my office there is a group of cars not all of which are black. PseudoCantor: But this only proves that what is in front of your office is not a group of cars. One of the basic intuitions behind the concept of set is that anything can be grouped together into a set3. My pencil, my left shoe and the Empire State Building constitute a set, and so do all chipmunks born in America or all natural numbers. And similarly everything whatsoever would appear to constitute a set. If someone devises a sense of “set” in which everything does not make up a set, then so be it – but this sense seems incapable of doing justice to the underlying intuition. The fact that there should be the set of everything is especially embarrassing from the viewpoint of the standard logical analysis of language. Such analysis usually counts with a universe of discourse, a set comprising all objects about which we can talk by means of the language in question; hence if there is no set of everything, it would seem there are things we cannot talk about – things which for some peculiar reason escape our linguistic snares. Obviously, though, any thing is an element of myriads of sets, and hence of potential universes of discourse, so we can talk about everything – but can do so only by somehow switching among different universes of discourses and thereby between languages. Thus, what is common to the Hilbertian concept of proof and the Cantorian concept of set is that they both apparently fall somewhat short of doing justice to our underlying intuitions. Both have been defending by claiming that our intuitions are inconsistent and that hence the regimented variants of the respective concepts necessarily cannot do justice to all these intuitions at once. Also, both of them lead to a kind of relativism. In the first case, we have no proofs simpliciter, but only various proofs-relative-to-systems; in the second we have no universe simpliciter but only various universes-relative-to-discourses.
3
More precisely, it is one of the basic intuitions underlying the Cantorian concept of set: for there is also an alternative conception (which might be called Fregean) and which takes sets as parasitic upon concepts. According to the latter, only such things which constitute the extension of a concept constitute a set.
3
Two approaches to logic The above considerations lead to the conclusion that we can distinguish two kinds of logical pursuit. The first is the pursuit of the explication of the absolute concepts of truth, proof, consequence etc.4; the second is the pursuit of their relativist counterparts. These two pursuits are, of course, not unrelated (the latter may be taken as a way of carrying out the former); however, many logicians appear to simply identify logic with one of them. And while it is feasible to identify logic with the first of them (possibly taking the second as a means of accomplishing it), it is not so feasible to identify it with the second. According to what we can call the absolutist conception of logic (Frege, Russell, early Carnap), logic provides the ultimate language for spelling out the facts of the world (improving on the inherently mutilating natural language). Hence we cannot get ‘outside’ of it, semantics is ineffable. In contrast to this, according to what can be called the relativist conception of logic (Hilbert, Tarski, late Carnap), logic provides multiple calculi (which can be studied either ‘in their own right’, or as applied to something). Hence to every language there exists a metalanguage in which we can speak about it and about its semantics In one of the most discussed papers published in the philosophy of logic, van Heijenoort (1967) referred to the two kinds of logical pursuits as logic as a language and logic as a calculus, respectively. He described how the absolutist pursuit of truth and consequence initiated by Frege and continued by Russell slowly gave way to the study of logical calculi and their model theories, started before Frege by Boole and Peirce and revived by Löwenheim, Skolem, Hilbert and Tarski5. My point is that it is important to beware mistaking the study of the mathematical properties of formally defined calculi for the study of the consequence relation in the (‘absolutist’) sense relevant to our factual reasoning (Peregrin, 2000). It may be that since the publication of van Heijenoort’s seminal analysis, the majority of logicians now realize the importance of the distinction, and perhaps even the inability of the relativist conception to stand on its own feet (save as a piece of pure mathematics); but I think it is still not fully appreciated that the majority of logical results constituting the commonly accepted core of logic are simply not intelligible outside of the relativist conception. To illustrate this, let us consider the completeness proofs which are nowadays usually considered to lie within the very hearth of logic6. Within the framework of the absolutist conception of logic, we cannot show that there exists a model of a statement of the (sic!) language save within the very language (for the 4
Of these three, consequence appears most fundamental. Truth is too wide for logic – the truth value of a sentence such as ‚The Earth is round‘ is surely not a matter of logic. (Logic can at most aspire to capture what other sentences must be true in order for this sentence to be true or vice versa – i.e. to capture the instances of consequence featuring this sentence.) Proof is usually taken to be too narrow, for it cannot cover all instances of consequence (but in the first section of this paper we indicated that this is a tricky matter).
5
The role of Hilbert in this is rather controversial. (See also Hintikka, 1988.) He is usually understood as the augur of formalism and hence of the relativist conception of logic; but the fact is that his divorce of the absolute and relative concepts was purely methodological. He realized that if we want to study consequence effectively, we may profit from rendering it in mathematical terms so that mathematical methods can be applied to it, disregarding that it is a formal reconstruction of something factual. It was only later that some logicians began to forget that the study of formal calculi is not a goal in itself.
6
For details see Coffa (1991, Chapter 15).
4
language has ‘no outside’7), hence every proof of the existence of a model of a statement is eo ipso a proof of the statement (perhaps a specific, ‘constructive’ case of one, as Carnap, 1928, has it). Hence there is no room for anything like model-theoretic completeness proofs. It is instructive to see how the phenomenon of completeness is handled by Carnap before he converted from the absolutist conception to the relativist one (whose champion he later became). Carnap (1928, p. 98-99) gives, in effect, the following ‘completeness proof’: 1. Let T have no model. 2. Then T ⇒ ┴ (for ┴ is true in every model in which T is, namely none) 3. But then T is inconsistent (for ‘T ⇒ ┴’ is Carnap’s definition of inconsistency of T). 4. Hence every consistent theory has a model. Obviously this is utterly trivial; and the reason is that within Carnap’s absolutist setting, there was no room for the distinction between ⇒ (consequence) and → (derivability). In contrast to this, if we stick to the relativist conception, showing that there exists a model of a statement of a language becomes a matter of a metalanguage of the language, and hence need not yield a proof of the statement (within the object language). We have the possibility of defining the concepts of consistency and inconsistency independently of the concept of consequence (‘T is inconsistent’ no longer abbreviates ‘T ⇒ ┴’, but rather ‘T → ┴’, where “→” now represents the relation of derivability by means of the deductive machinery provided by the specific formal system underlying T). The gap between the model-theoretic notions and the proof-theoretic ones opened by the relativist conception of logic has proved to be fruitful in the sense of creating space for plenty of interesting mathematics. However, it has also created problems, problems which many logicians apparently overlook. (To put it crudely: if we are studying, say arithmetic, we want to know which of its statements are provable and hence true – who cares which of them are derivable within a peculiar formal system?) I think that one of the most important task for logic now is to improve its self-consciousness: to achieve a more distinct understanding of what the results it has achieved, especially since the Gödel’s upheaval, really mean.
References Carnap, R. (1928): Untersuchungen zur allgemeinen Axiomatik (Carnap’s unpublished manuscript from 1928; ed. T. Bonk & J. Mosterin), Wissenschaftliche Buchgesellschaft, Darmstadt, 2000. Coffa, A. (1991): The Semantic Tradition from Kant to Carnap, Cambridge University Press, Cambridge. Dummett, M. (1963): ‘The Philosophical Significance of Gödel’s Theorem’, Ratio V, 140155; reprinted in and quoted from Dummett: Truth and other Enigmas, Duckworth, London, 1978, pp. 186-201. Finsler, P. (1926): ‘Formale Beweise und die Entscheidbarkeit’, Mathematische Zeitschrift 25, 676-682; English translation in van Heijenoort (1969), pp. 438-445. Finsler, P. (1944): ‘Gibt es unentscheidbare Sätze?’, Commentarii mathematici helvetici 16, 310-320. Gaifman, H. (2000): ‘What Gödel's Incompleteness Result does and does not Show’, Journal of Philosophy 97, 462-70. 7
See also Peregrin (1995,§11.5).
5
Grim, P. (1991): The Incomplete Universe, MIT Press, Cambridge (Mass.). Heijenoort, J. van (1967): ‘Logic as Calculus and Logic and Language’, Synthèse 17, 324330. Heijenoort, J. van, ed. (1971): From Frege to Gödel: A Source Book of Mathematical Logic, Harvard University Press, Cambridge (Mass.). Hintikka, J. (1988): ‘On the Development of the Model-Theoretic Viewpoint in Logical Theory’, Synthèse 77, 1-36. Lavine, S. (1994): Understanding the Infinite, Harvard University Press, Cambridge (Mass.). Penrose, R. (1990): The Emperor's New Mind, Vintage, London. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (1998): ‘Logic and Consciousness’ , The Philosophers Magazine, Spring Issue, 46-47. Peregrin, J. (2000): ‘The ‘Natural’ and the ‘Formal’’, Journal of Philosophical Logic 29, 75101.
6
Metaphysics as an attempt to have one’s cake and eat it Jaroslav Peregrin* www.cuni.cz/~peregrin [to appear in Metaphysics in the Post-Metaphysical Age, ed. P. Simons and U. Meixner, HölderPichler-Tempsky, Wien]
Why not metaphysics? Metaphysics is usually understood as the investigation of being qua being and of its ultimate categories. Given this characterization, it may be hard to grasp why anyone might wish to oppose metaphysics, why anyone might claim that metaphysics ”leads the philosopher into complete darkness” (Wittgenstein, 1958, p.18)? What could be so misleading about the investigation of the most abstract vestiges of being? One source of disparagement towards metaphysics, of course, stems from the relativist conviction that there is no absolute being, and hence nor are there any ultimate categories of being. Another ‘reason’ for some philosophers of this century (notably Carnap) to reject metaphysics appeared to consist in their reinterpretation of the word ”metaphysics”, in effect, simply as ”nonsensical philosophy”. However neither of these reasons seems to be the Wittgensteinian one. The way I propose to envisage this reason takes us back to Plato with his distinction between Being and Becoming (echoed, in one or another form, within the conceptions of so many subsequent philosophers). It is the distinction between the ‘higher’ realm of ideas with its crisp, eternal truths, and the ‘lower’ realm of appearances in which anything appears only as a fuzzy and transient reflection of something from the higher realm. The original message harbored in this Platonistic picture is clearly that the vast and hardly graspable flux of appearances shelters something firm and invariable, something potentially fully capturable by human reason which can then use it as a prism to comprehend and understand the everchanging phenomena. However, as I will claim, the picture of a ‘higher’ reality behind phenomena is dangerous in that it can delude a philosopher into feeling that he can solve empirical questions by a quasiempirical investigation of a non-empirical realm: a ‘metaphysical reality’.
The fortunes of a fortune-teller Let us, for a moment, forget about metaphysics and consider fortune-telling. A fortune-teller’s prediction is supposed to spell out the future, how reality is to unfold. So what the fortuneteller (implicitly) claims, we may say, is the following: predicted event = real event
(1)
However, any fortune-teller is likely to be at least minimally cautious in the sense that he would probably admit the possibility of ‘disturbances’ capable of making his prediction fail. The cautious fortune-teller’s claim then would be: *
This work was supported partly by the Research Support Scheme of the OSI/HESP, grant No. 280/1997 and partly by the Grant Agency of the Czech Republic, grant No. 401/99/0619.
1
predicted event + disturbances = real event
(2)
Now some fortune-tellers may exhibit more than caution - they may become so cunning as to try to make their predictions correct come what may. They may attempt to make the ‘disturbances’ in (2) into something which can justify any kind of discrepancy between their predictions and reality, i.e. to define, in effect, disturbances = real event - predicted event
(3)
Needless to say that given (3), (2) reduces to the trivial real event = real event
(4)
Suppose, e.g., that there is somebody who claims to be able to predict the sex of an unborn child (of course without using ultra-sound or other ‘invasive’ means)1. Suppose that he claims that although his predictions sometimes fail due to some unpredictable disturbances, as a matter of fact he can see what is to happen. What could convince us to take him seriously, why should we not think that he is merely guessing? Surely one decisive reason would be if his predictions were successful to such an extent that this would not be explainable by mere chance, in this particular case if his predictions were correct in a number of cases which would ‘statistically significantly’ exceed fifty percent. In such a case we would probably accept, at least tentatively, that he was doing something over and above mere guessing. However, were this not the case, we would most likely be unwilling to say that what he was doing was real predicting; we would feel that he really made the disturbancesfactor of his claim into a ‘universal compensator’ capable of neutralizing any kind of discrepancy between his predictions and real happenings - thus violating the Popperian maxim of falsifiability of contentful claims. It is improbable that it would be of any help for the fortune-teller to insist that the reason why he should be taken seriously is that he possesses ‘clairvoyance’, the ability to see real things-to-happen (before they might get distorted by the alleged disturbances). Perhaps some centuries ago this indeed might have been accepted as a reason (there may even have existed allegedly decisive ways to decide whether someone was clairvoyant or not); however, with the development of modern science this has become simply obsolete. Now it is generally not believed that clairvoyance is an ability to be had (by a human creature).
The fortunes of a metaphysicians The cunningness of the fortune-teller ridiculed above consisted in the fact that trying to present an observable phenomenon, O, as a sum of two factors, of something substantial, S, and some kind of disturbances, D: O = S+D, 1
In fact, predicting of this kind appears to have had quite a status in the first half of this century, at least in my country. Journals from that time are full of commercials like ”I predict the sex of your child; guaranteed - I return money in case of failure!”
2
he made the disturbances into a ‘universal compensator’ which could neutralize any kind of discrepancy between S and O. Thus, he defined, in effect, D = O-S, which, of course, when substituted into the original equation, reduces it to the trivial tautology O = O. Now my point is that a metaphysician is in a somewhat similar position. What he seeks are some firm, eternal essences within (or ‘behind’) evasive phenomena. A worldly thing is then understood as the result of the dressing of an essence into worldly clothing; as an imperfect embodiment (‘methexis’) of the essence within a worldly body. Hence ‘the metaphysician’s claim’ can be articulated as essence + worldly ingredients = worldly thing,
(5)
and we have an equation resembling the fortune-teller’s one. There is ‘something substantial’, in this case the essence, which gets modified by ‘disturbances’, in this case the worldly ingredients, resulting in what we can really observe. Moreover, in this case we obviously lack that possibility of justification which was available to the fortune-teller: we cannot take recourse to statistical significance, for we cannot record essences independently of the corresponding things to compute their covariances. Hence the only way for the metaphysician to argue for the nonvacuousness of his claim, to meet the challenge that the ‘worldly ingredients’ in his equation constitute a ‘universal compensator’ which makes any metaphysical claim trivially true, appears to be the one analogous to the fortune-teller’s claim to clairvoyance - in this case a claim to the ability to somehow ‘perceive essences’. And in fact this is what the metaphysicians indeed claimed: this ability is supposed to be our intellectus, which is able to ‘read inside’ things (‘intellectus’ = ‘intus legit’)2. True, the claim of being able to sort properties of things into ‘essential’ and ‘accidental’ clearly makes more sense than the claim of being able to see the future. However, what I suggest is that the empirical spirit brought about by the boom of natural sciences at the end of the nineteenth and the beginning of the twentieth century has eroded the intelligibility even of this claim. We perceive and record ‘worldly things’ and their properties - what should be a criterion of dividing the properties into those pertaining to the essence and those being merely accidental? It is precisely the skepticism with respect to this division that underlies the famous rejection of the analytic/synthetic distinction by Quine (1952). What we can observe is that something is this-and-so; but there is no observing, over and above this, that something is this-and-so necessarily or essentially, whereas something else is this-and-so only contingently
2
Aquinas (1271-2, VI lect. 5 n. 1179) writes: „Dicitur autem intellectus ex eo quod intus legit intuendo essentiam rei.“ [„The intellectus is so called bacause it reads inside when it perceives the essence of a thing.“]
3
or accidentally3. What Quine adds to this essentially Humean line of empiricist thought is that we cannot save the situation, as Carnap tried to do, by taking recourse to the rules of language - for the rules of a natural language are again ‘worldly things’ or aspects of ‘worldly things’, namely of the behavior of the speakers4. All of this is not to be interpreted as implying the denial of the existence of pragmatic criteria for classifying properties of a thing into more and less ‘essential’ in the sense of being more interesting for us or more important for our current purposes. (Thus in the majority of cases it will be probably more important for us that someone is rational than that he is twolegged; or that something is carnivorous than that it is stripped.) However, we must remember that such a distinction is one of degree and cannot directly establishing a crisp dichotomy; and, in addition, that it is context- and purpose-dependent - what we find important and interesting clearly depends on circumstances and on the aims pursued.
The fortunes of a logical analysts It might seem that metaphysics as envisaged above, is de facto something which has played no substantial role within this passing century’s philosophy; and hence that we (together with the late Wittgenstein) are merely challenging a poor little strawman. After all, who still talks about essences and who bothers to argue whether having lungs belongs to the essence of a man? However, this is not true: what has been said above does not concern only those who refer to essences in so many words - it is applicable to all those who take empirical things to be based on non-empirical, ‘mathematical’ entities. And if within what we said above we substitute, e.g., form or structure for essence, we obtain claims which are far from obsolete. Moreover, the claims of a metaphysician are not exhausted by statements claiming that the essence, or the structure, of something is this-and-so. It is felt that once we have gripped onto structures, we can also study their relationships; and thereby reveal the ‘true’ relationships between the things that have the structures. And as the world of structures is, unlike that of empirical phenomena, susceptible to a ‘more geometrico’ treatment, this appears to offer an inviting prospect. In fact, logical analysis, as practiced by the partisans of analytic philosophy (with Russell’s, 1905, celebrated analysis of definite descriptions as the paradigm) can readily be understood as this kind of a metaphysical enterprise. (Needless to say this is paradoxical, for many of the partisans employed it precisely with the expectancy that it would enable them to dispense with metaphysics5.) The point is that what logical analysis pursues is the genuine logical form hidden behind the surface form of an expression; and hence the logical analyst’s claim, namely 3
Quine (1960, p. 199) claims: ”Mathematicians can conceivably be said to be necessarily rational and not necessarily two-legged; and cyclists necessarily two-legged and not necessarily rational. But what of an individual who counts among his eccentricities both mathematics and cycling? Is this concrete individual necessarily rational and contingently two-legged or vice versa? Just insofar as we are talking referentially of the object, with no special bias towards a background grouping of mathematicians as against cyclists or vice versa, there is no semblance of sense in rating some of his attributes as important and others as contingent.”
4
See also Wheeler (1986).
5
See, e.g., Carnap (1931).
4
logical form + ‘imperfection’ = real expression
(6)
is again of the form of (5): we see an observable entity, in this case a real expression, as a sum of two separately unobservable factors. Moreover, the character of one of the factors makes it tempting to make it into a ‘universal compensator’: it is easy to acquiesce in the feeling that doing logical analysis means inspecting ‘abstract’ logical forms, and that any question concerning whether a factual expression happens to have a particular logical form is to be left to empirical linguistics. (And should the task of associating natural language expressions with the logical forms thus studied turn out to be problematic, then the worse for natural language!) Consider the picture of the language-world relationship as presented in Wittgenstein’s Tractatus. According to this, the ”substance” of the world is constituted by a collection of simple objects which are denoted by simple names constituting the foundation of language. However, nowhere in the whole book is there any hint as to how to find such simple objects within our factual world, nor how to reveal the simple names within our language. It seems that no real expression locatable within our language is really a simple name in the Wittgensteinian sense; yet Wittgenstein did not seem worried. Why should we care about the imperfections of our contingent language, when we aim higher, at non-contingent logical forms? More recent instances of this attitude are numerous: formally-logical or formallymetaphysical analyses aspiring to be analyses of our factual language or our factual world, yet expecting to be able to accomplish this without leaving the realm of the formal. Let us mention at least one, exemplary case: when Kreisel (1967), in his celebrated paper, replaced the problem of the relationship between logical validity in the intuitive sense and modeltheoretic validity with the problem of the relationship between two kinds of model-theretical validity (namely the truth w.r.t. all the usually conceived model structures and the truth w.r.t. all model structures whatsoever), this has been almost universally accepted as an unproblematic move6. Does this mean that logical analysis of language is an essentially misguided enterprise? Of course not - it is just a particular view of what logical analysis anounts to which is misguided, and which robs the analysis of its purported sense, namely to reveal the logical structure of the very language we are employing to make claims and argue for them. (Investigations of abstract forms are, to be sure, not necessarily altogether senseless - they can yield interesting mathematical results). What then is the alternative?
Two ways of understanding logical analysis Elsewhere (see Peregrin, 1995, §11.9) I have pointed out that there are two principal approaches to the enterprise of logical analysis. First, there is the approach entertained by Russell, early Wittgenstein and their followers, according to which logical forms of expressions seem to be something quite definite and the task of logical analysis is to dig them from beneath the irregular delusive surface of language where they lie hidden. Contrasting with this is the approach of the late Wittgenstein and also of philosophers like Quine or Davidson: they propose, in effect, to see logic and logical analysis not in terms of prospecting for the 6
See also Peregrin (in prep.)
5
structure within the depths of language, but rather, we can say, in terms of building watchtowers over language to help us comprehend language and see a structure. In Wittgenstein’s (1953, §132) words, ”Wir wollen in unserem Wissen vom Gebrauch der Sprache eine Ordnung herstellen: eine Ordnung zu einem bestimmten Zweck; eine von vielen möglichen Ordnungen; nicht die Ordnung”. In particular, as Davidson (1970, p. 140) puts it, ”to give a logical form of a sentence is to give its logical location in the totality of sentences, to describe it in the way that explicitly determines what sentences it entails and what sentences it is entailed by.” If we adopt the latter vantage point, the situation of a logical analyst becomes similar not to that of an archeologist trying to dig out the layout of a buried city, but rather to that of somebody who is drawing a simplified plan of an existing city - which is open to view (”offen daliegt”) - for the purposes of helping somebody to gain orientation within it, to familiarize himself with it (”sich auszukennen”)7. Elsewhere (Peregrin, to appear) I have claimed that what this reflects is a general trait of human understanding, i.e. the fact that we humans tend to see complicated phenomena through the prisms of certain ‘structures’. And here is where we can helpfully engage the distinction between Plato’s ”realm of Being” and his ”realm of Becoming”; or, as I prefer to say, between the realm of the formal (RF) and the realm of the natural (RN). How should we imagine the two realms? RN is the very realm of our ‘physical’ life, within which we can engage ourselves in finding and describing, but where everything appears to us as essentially vague and fuzzy (in the sense of not having a pure mathematical structure). Nothing regarding this realm can be proven in the mathematical sense. It can be seen as inhabited with things (in the prototypical sense of the word) and events, and prototypically it is the subject matter of natural science. Contrasting with this, within RF everything is precisely defined and sharply delimited; things are stipulated and facts about this world can be unambiguously proven. The inhabitants of this realm, with a certain amount of oversimplification, can be called structures; they are addressed most directly by mathematics8. Using this terminology, we can say that the metaphysical conception of logic and logical analysis has come to rest on the assumption that logic deals with a peculiar realm located as if somewhere in the intersection of RF and RN - a realm comprising entities which are rigid and susceptible to mathematical treatment, and yet, at the same time, wield control of our factual reasoning. According to this view, the truths of logic are true independently of what is going on within the contingent world and yet they are somehow inherently related to the language we happen to use and to the way we, contingent beings, happen to think. Out contention, then, is that the existence of such a hybrid realm is a pernicious illusion: there is no intersection of RN and RF, for the constitutive properties of the two realms are mutually exclusive. Something is rigid and susceptible to direct mathematical treatment only to the extent to which it is not part of what we normally called reality. It follows that the question whether a real thing can be reasonably seen as having this or another structure, or whether a structure can be helpfully ‘projected’ on this or that thing, is always an empirical one. Thus, proving something about a structure from RF can be taken as
7
Cf. Wittgenstein (1953, §§ 126, 123).
8
Needless to say that the talk about the two realms is itself not to be understood ‘metaphysically’, i.e. as a report of a discovery; but rather merely as an illustrious way of calling attention to the fact that our understanding crucially rests on an interplay of two essentially different levels.
6
proving something about a thing from RN only if it is taken for granted that the thing has this structure - which is something that, by its very nature, evades formal proof. From this viewpoint the metaphysical conception of logic errs by misconstruing the relationship between RN and RF, namely by overlooking the fact that the nature of the two realms is essentially distinct. RN is the very world with which we are confronted, whereas RF is the realm of prisms we employ to reflect it and to understand it. The structures from RF serve as prisms through which we see and grasp the world, and which we may employ to explicitly reconstruct its regularities and to point out the ‘forms’ or ‘structures’ of things or events9.
The Error of Metaphysics Our criticism of the metaphysical conception of logical analysis can now be generalized. It is obviously not capricious to seek rigid structures - or, if one wants, ‘essences’ - behind the elusive stream of phenomena which impinges on us; to try to, as the Ancients put it, ”save the phenomena”10. On the contrary, it is the conditio sine qua non of our theoretical (and in fact also practical) coping with the world. In particular, it is clearly useful to see our language as embodying various kinds of structures, especially a ‘logical’ one. The basic point is that it is useful to investigate a thing via investigating its structure, for then what we are dealing with is a rigid unchangeable entity which can be subjected to the methods of mathematics, the most reliable scientific methods we have. Where this enterprise goes astray is when we start to take the pertinence of an essence to a thing to be itself an ‘essential’, non-empirical matter, and consequently treat the results of an investigation of the structure or essence as eo ipso concerning the original thing. In opposition to this, we stress that this is the case only provided the structure really is the structure of the thing, and that whether it really is is inevitably an empirical question. Thus, the pitfall of metaphysics, of which the late Wittgenstein warns us, is that it can easily mislead us, if we are too eager to ‘look not at appearances, but rather to the (metaphysical) reality behind them’, into understanding the philosophical enterprise as a direct, pseudo-empirical investigation of RF (= mathematical or quasi-mathematical structures) wholly bypassing the RN (= our real, everyday world). In this way one comes to mistake studying a prism for studying the world captured by the prism, and to devote oneself to the study of abstract ideas, categories or structures whilst losing from sight whether and how they really are relevant for our real world and our real problems. It is our ”craving for generality”, our ”contemptuous attitude towards the particular case” (Wittgenstein, 1958, p. 17), that leads us astray - if it becomes excessive. The price which the entities of the RF must pay for their ‘crispness’ and their consequent susceptibility to ‘mathematical’ treatment is that they are literally merely ‘formal’ (in the sense of being forms which must be filled by a content), that they, by themselves, cannot enter the factual, everyday world. Thus, the kind of metaphysics which I think we should depreciate - together with the late Wittgenstein - is the product of the illusion that we philosophers can somehow have our cake and eat it - that we can get hold of entities which 9
Cf. Stekeler-Weithofer (1994). For a more detailed treatment of these issues see Peregrin (to appear).
10
Cf. Mittelstraß (1962).
7
are both tractable in the rigorous, ‘mathematical’ vein and factual - that philosophy can ‘calculate’ truths of the real world.
References Aquinas, Thomas (1271-2) Sententia libri Ethicorum, printed in in Opera omnia, Vatican Polyglot Press, Rome, 1882. Carnap, R. (1931): ‘Überwindung der Metaphysik durch logische Analyse der Sprache’, Erkenntnis 2, 219-241. Davidson, D. (1970): ‘Action and Reaction’, Inquiry 13; reprinted in and quoted from Davidson (1980). Davidson, D. (1980): Essays on Actions and Events, Clarendon Press, Oxford. Mittelstraß, J. (1962): Die Rettung der Phänomene, Berlin. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (to appear): ‘The ”Natural” and the ”Formal”’, Journal of Philosophical Logic. Peregrin, J. (in prep.): ‘The ”Fregean” Logic and the ”Russellian” Logic’, in preparation. Quine, W.V.O (1952): ‘Two Dogmas of Empiricism’, Philosophical Review 60, 20-43; reprinted in Quine (1953). Quine, W.V.O. (1953): From a Logical Point of View, Harper and Row, New York. Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.). Russell, B. (1905): ‘On denoting’, Mind 14, 479-493. Stekeler-Weithofer, P. (1994): ‘Ideation und Projektion’, Deutsche Zeitschrift für Philosophie 42, 757-960. Wheeler, Samuel C. III (1986): ‘The Extension of Deconstruction’, Monist 69, 3-21. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford. Wittgenstein, L. (1956): Bemerkungen über die Grundlagen der Mathematik, Blackwell, Oxford. Wittgenstein, L. (1958): The Blue and Brown Books, Blackwell, Oxford.
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/91 The Philosophers Review Wittgenstein II
No change Jaroslav Peregrin on a valuable but flawed account of Wittgensteins intellectual consistency
The New Wittgenstein, eds. Alice Crary and Rupert Read (Routledge) £17.99/$29.99
I
n the Tractatus Wittgenstein famously claims: The right method of philosophy would be this: To say nothing except what can be said, i.e. the propositions of natural science, i.e. something that has nothing to do with philosophy: and then always, when someone else wished to say something metaphysical, to demonstrate to him that he had given no meaning to certain signs in his propositions. Many of his readers have taken this as implying that he was, together with Russell, Carnap and others, in the business of logical analysis, of sorting out reasonable talk from metaphysical rubbish. On the other hand, during the later phases of his development we find Wittgenstein saying things which are scarcely compatible with such an understanding of his efforts. As he himself once put it, he comes to be in a sense making propaganda for one style of thinking as opposed to another (and needless to say that making propaganda and carrying out logical analyses are enterprises almost opposed to each other). Some philosophers interpreted the late Wittgenstein, despite such pronouncements, as still pursuing ends constitutive of Russellian analytic philosophy, only by rather nonstandard sometimes perhaps even extravagant means. This appears to be the position of Saul Kripke in his influential Wittgenstein on Rules and Private Language; but also of some of the most vigorous Kripkes critics, such as G P Baker and P M S Hacker. Other philosophers
have urged that the late Wittgenstein cannot be seen as merely using nonanalytic means, but also as pursuing non-analytic ends that his later writings cannot be really understood if we do not realise that Wittgensteins struggle for the delimitation of the bounds of sense is not animated by his will to secure a firm, nonsense-free ground for science or rational philosophy, but rather by his fascination by what is beyond the bounds. Stanley Cavells book The Claim of Reason is a good example of such non-analytic reading of the late Wittgenstein. Although reading Wittgensteins later writings in this non-analytic way is no longer considered extravagant or controversial, it has been almost universally accepted that it implies rendering him as completely changing his views between his early and his late stage. It was the realisation of the futility of his own earlier analytic zeal, so the story usually goes, that made him assume his later standpoint. What is new on the Wittgenstein as portrayed by the majority of the essays in the volume reviewed here is that he is taken to be assuming the non-analytic standpoint from the very beginning. The first part of the book concentrates on Wittgensteins later philosophy, and is a bit heterogenous. The core of the volume is constituted by the second part, which concentrates on the early Wittgenstein. There, philosophers led by Cora Diamond and James Conant try to overthrow the received wisdoms about the sense of Wittgensteins Tractatus. It is well known that in the end of that text Wittgenstein claims that his sentences are in fact nonsensical: but it has been usually as-
sumed that what was meant by this is that although these sentences cannot say what they seem to say, they do have a point in that they manage to somehow show it. Diamond, Conant and their followers argue that the purpose of the Tractatus is not to show something unspeakable, but rather to develop a certain (metaphysical) viewpoint only to let us see that it is utterly nonsensical. How does such a reading of Wittgenstein fare in comparison with the standard one? The writers of the essays, I think, suceed in showing that it is not too implausible, and they suceed in using it to throw new and unexpected light on many of the things Wittgenstein says. However, as Hacker, whose paper comprises the third part of the book, takes pain to show, it would be hard to make it compatible with the entire body of Wittgensteins Tractatus-related texts and pronouncements. (Wittgenstein, for example, kept referring to what he had said within the Tractatus as his (earlier) views not as something he put on paper only in the role of devils advocate). Hence Hackers analysis, as far as I can see, shows quite convincingly that the way to read Wittgenstein propagated within the New Wittgenstein cannot be the right way to read him. However, the question is whether there is any right way to do so; and if not, then the further question is whether even presenting a way to read him, if it is succeeds in opening a new and stimulating vista, is not something to achieve. I, for one, would vote for a negative answer to the former question and for the positive one to the T PM latter.
The Philosophers Magazine/Winter 2001
/57 The Philosophers Review Rorty and His Critics
No weirdo Jaroslav Peregrin on a Rorty to be reckoned with Rorty and his Critics, ed. Robert B Brandom (Blackwell) £19.99/ $29.95
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n the beginning of philosophy, there was a dialogue; a dialogue between an Athenian weirdo and a host of his compatriots. It is interesting that philosophy, having accomplished the detour via which it has produced an immense number of monologic treatises, now returns as if back to its roots. More and more philosophy books contain not presentations of theories, but rather dialogues between philosophers. This trend has been initiated by the famous Library of Living Philosophers published by Open Court; but during the recent decade, series of this kind have been proliferating. The present volume is the product of one such series, published by Blackwell: it contains the essays of twelve philosophers reflecting on the philosophical views of Richard Rorty together with Rortys responses. Its contributors are mostly people quite sympathetic with Rortys philosophy (so one should not take the critics in its title too seriously), but they are firstclass and original philosophical personalities (including Robert Brandom, the editor of the volume, Donald Davidson, Daniel Dennett, Hilary Putnam, John McDowell, and also Jürgen Habermas), and as a result, their anatomisation of Rortys views is truly penetrating. Rorty has become famous especially after the publication of his 1980 book Philosophy and the Mirror of Nature, in which he not only challenged almost every dogma of contemporary analytic
philosophy, but portrayed the whole Western philosophy as put on a wrong track by Plato. It is, Rorty claims, on the wrong track for it wants to achieve the impossible: to bring us some special kind of truths (maybe even ones which are firmer than are those of our fallible, empirical science). Contrary to this, we should, according to Rorty, free ourselves from the notion that philosophy can explain what science leaves unexplained and put up with the fact that philosophy can be at most useful kibitzing. By this attitude to philosophy, reinforced by his subsequent books, Rorty divided the philosophical public into two contrary camps: some philosophers have hailed him as a prophet who has managed to bring out that philosophy had degenerated into a nonsensical self-contained game; whereas others have seen him as an irresponsible sophist helping rhetoric prevail over reason. The present volume can well serve as the compendium of the most distinctive features of Rortys views and the most typical objections raised to them by philosophers who are not utterly unsympathetic with him. What is now apparently felt as most emblematic of Rortys philosophical attitude and what is challenged in many of the included papers (most explicitly in those of Habermas and Davidson) is his conviction that we do not need any concept which would transcend our communal practices, especially that we do not need the concept of truth stronger than the concept of justification. What we, according to Rorty, should be after is broadening the range of the audiences to which we are able to justify our convictions but to
struggle for a universal justifiability (hence for truth) is to waste time chasing a chimera. You can only work, as he puts it, for what you can recognise. While Rorty follows Dewey, who thought that the desire for universality, and necessity was undesirable, because it led one away from the practical problems of democratic politics into a never-never land of theory, his opponents, according to Rorty, think that this is a desirable desire, one which one shares only when one reaches the highest level of moral development. This rejection of the transcendental concept of truth is, of course, only the tip of the iceberg of Rortys controversial philosophical views, which are underlain by the conviction that there is no world-in-itself which would be independent of what we do and how we talk. There is no way, he claims in his response to McDowells effort to rehabilitate objectivity, to drive a wedge between convincing your peers and directing your meaning to the world; hence there is no answerability to the world, no objectivity, independent of getting in phase with ones fellow humans, with solidarity. This and many other strands of Rortys attitude are also discussed in the book. Reading Rortys responses to the challenges can serve as a good antidote against taking him as a superficial thinker. The way he faces the objections of his opponents shows that there is definitely much more to his controversial stance than ignorance you may take his view to be stubbornly extravagant (one more weirdo?), and you may take him to be wrong; but you cannot take him as somebody who need not be T taken seriously. PM
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Brandom: Articulating Reasons
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Robert B. Brandom: Articulating Reasons (An Introduction to Inferentialism), Harvard University Press, Cambridge (Mass.), 2000, 230 pp. When Brandom’s Articulating Reasons appeared, I heard a philosopher commenting sarcastically that ”it is odd to write a book so big that you have to write a smaller book to give guidelines how to read the big one”. But although Brandom’s new book can hardly fail to be taken as a belated prolegomena to his huge Making it Explicit (hereafter MIE), and although the author himself does present the book as an ”introduction to inferentialism”, it is in fact something else. Rather than being a systematic introduction, it is a not entirely homogenized collection of essays picking up the most difficult and most controversial points of Brandom’s doctrine as presented within his opus magnum and attempting to enlighten them from various new angles. However, although it would indeed be desirable to have a concise and comprehensible introduction to the Brandomian doctrines, what Brandom presents in the book is no less important. Nevertheless I do not think the book would serve well as a first point of contact with Brandom’s thinking - it is, I think, more helpful for those who are conversant with his approach than for those who would like to learn what it is all about. The introduction of the book presents a minimal explanation of the rationale behind Brandom’s inferentialism; and then it suggests into which ideological boxes it should be classed. According to the author, his stance ”is opposed to many (if not most) of the large theoretical, explanatory, and strategic commitments that have shaped and motivated Anglo-American philosophy in the twentieth century (p. 31): it rejects empiricism and naturalism (in favor of rationalism), representationalism (which is what is replaced by inferentialism), semantic atomism (in favor of holism), formalistic construal of logic (in favor of an expressivistic construal), and also the instrumentalistic construal of practical rationality. I find at least one item in this list of classificatory suggestions potentially misleading, namely Brandom’s avowal of rationalism and his rejection of naturalism. It is clear that Brandom is not an empiricist, but I do not think that this automatically renders him a rationalist; indeed I think that within the context of contemporary philosophy the traditional empiricism/rationalism classification is now simply obsolete. (I doubt that Brandom wants to be associated with the doctrines of ”innate ideas” or ”intuition of self-evident propositions” which are usually taken as constitutive of rationalism.) Also he is certainly not a naturalist in the sense of claiming that all knowledge is bound to be the product of natural science, but I think that this is not the sense of ”naturalism” currently prevailing (note that, e.g., Davidson is usually classified – with his own approval – as a naturalist). Consider Brandom’s account of normative discourse in MIE: he rejects the reducibility of the normative idiom to a non-normative one (and hence rejects Quinean physicalism and behaviorism), but at the same time he hastens to provide a story about how such form of discourse can evolve from non-normative forms. If any kind of naturalism were so alien to him, why would he find such a story important? Hence I think that there is a sense of ”naturalism” (and one not too far-fetched) in which Brandom is a naturalist. The first chapter of the book, Semantic Inferentialism and Logical Expressivism, discusses the question of how the inferential understanding of semantics yields an unusual, namely an ‘expressivist’, understanding of logic. The idea is that accepting the inferential standpoint, from which we see expressions as semantically determined by inferential patterns governing them (and hence see the meanings of expressions as their ‘inferential roles’), we can classify the vocabulary of natural language into two different compartments. While some words (like ”dog”, ”kill” or ”slowly”) take part within our reasoning about the world and are governed by inferences which can be called material (as contrasted to formal), other ones take part within our reasoning about reasoning, within our making explicit of the material inferential patterns. Thus to claim ”if Pluto is a dog, then Pluto is a mammal” is to claim the propriety of the inference from ”Pluto is a dog” to ”Pluto is a mammal” – and the fundamental role of expressions like ”if ... then” consists in turning that which is done (or maybe better a rule which implicitly governs what is done) into what can be claimed. This is the
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‘making explicit’ emblematic of Brandom’s understanding of human reason. What I find slightly problematic about Brandom’s discussion of logic is the fact that he does not sufficiently distinguish between ”logic” in the sense of ”the logical part of our vocabulary” and ”logic” in the sense of the theory accounting for it. As far as I can see, what he means when he talks about the ”expressive role of logic” (p. 57) is that the role of the logical part of our vocabulary is expressive, that its role is to enable us to express in the form of claims what is otherwise implicitly governing our language use – but I am afraid that it can be read as claiming that the task of logic in the sense of the theory done by logicians is to express something which is not expressed (or perhaps not even expressible) by our normal language. I suspect that this might invoke the impression that logic is, according to Brandom, in pursuit of something covert behind our overt language. The next chapter, Actions, Norms and Practical Rationality revokes one of the most difficult themes of MIE: Brandom’s account of practical reasoning and its normative aspect. Here what Brandom puts forward is the thesis that the role of the normative vocabulary (words like ”should”, ”ought to”, ”be correct” etc.) with respect to practical reasoning is analogous to that of the logical vocabulary with respect to theoretical reasoning – namely that of making explicit. Thus, just like the claim ”if Pluto is a dog, then Pluto is a mammal” explicitates the propriety of the inference from ”Pluto is a dog” to ”Pluto is a mammal”, the claim ”As dogs are not allowed into the shop, I should leave Pluto outside” explicitates the propriety of the inference from the acknowledgement of the prohibition to the act of leaving Pluto outside of the shop. Thus, Brandom claims, ”normative vocabulary (...) makes explicit the endorsement (attributed or acknowledged) of material proprieties of practical reasoning” (p. 89). Working with Wilfrid Sellars’ classification of the rules of language into the language entry, intralinguistic, and language exit rules, Brandom further suggests that voluntary actions resulting from practical reasoning are to the exit side of language what perception is to the entry side. While the former amount to the ”capacity to respond reliably to one’s acknowledgement of a commitment (of a norm binding on one) by differentially producing performances corresponding to the content of the commitment acknowledged”, the latter is a matter of ”a capacity to respond differentially to the presence of, say, red things, by acknowledging a commitment with a corresponding content” (p. 94). Thus, the commitments (and entitlements) which Brandom diagnoses as central to language can both cause and be caused by extralinguistic happenings. In the next chapter, Insights and Blindspots of Reliabilism, Brandom discusses the theories of knowledge based on the assumption that in the case of perceptually acquired knowledge, we should replace the understanding of knowledge as justified true belief by its understanding as a true belief acquired by a reliable mechanism. Brandom argues that this is misguided insofar as it aims at replacing the normative concept of justification by the behaviorist concept of reliability. Brandom, though, does not deny that reliability is a concept crucial within the context of explaining perceptually acquired beliefs; however, not as a surrogate for justification, but rather as a classifier of inferences. ”What one is doing in taking someone to be reliable,” he claims (120), ”is endorsing a distinctive kind of inference: an inference, namely, from the attribution to another of a propositionally contentful commitment acquired under certain circumstances to the endorsement or undertaking oneself of a commitment with the same content.” This is to say that if I know that somebody is a reliable reporter of some kind of events, then I am justified in using his reports as premises in my own arguments. Hence ”reliabilism,” as Brandom concludes (ibid.), ”points to the fundamental social or interpersonal articulation of the practices of reason giving and reason assessing within which questions of who has knowledge arise”. The following chapter undertakes an ambitious project of answering the question What are singular terms and why are there any? (in a way different from because there are individuals). Brandom’s argument leading to his answer can, in effect, be summarized as follows: As we need to make an unlimited number of claims, we need claims to be composed of recomposable parts. Certain parts of
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statements can be interchanged, they are substituted for’s, while the unchangeable rests of the statements are substitutional frames. As we need claims that entail other claims without being entailed by them (‘one-way’ inferences), it cannot be the case that both the substituted for’s and the substitutional frames are ‘inferentially symmetric’ (in the sense that a frame F with a substituted-for x, F[x], would entail F[y] just in case it would be entailed by it, and at the same time would entail F’[x] just in case it would be entailed by it). And as our language is bound to contain the essential logical machinery, especially the conditional, and hence to any substitutional frame F there is bound to exist a frame F’ complementary in the sense that whenever F[a] entails F[b], F’[b] is bound to entail F’[a] (F’[x] can be constructed as F[x] → S, for a statement S), the substituted for’s cannot but be ‘inferentially symmetric’. Hence the only possible combination is that of inferentially asymmetric substitutional frames and inferentially symmetric substituted for’s. Brandom thus concludes that ”syntactically, singular terms play the substitutional-structural role of being substituted for, while predicates play the substitutional-structural role of sentence frames”, and ”semantically, singular terms are distinguished by their symmetric substitution-inferential significance” (150). He then concludes that his answer to the question why are there singular terms? (which is now seen as one side of the coin whose other side is the question why are there objects?) can be aphoristically summarized as ”because it is so important to have something that means what conditional means!” (155). It seems to me that this chapter is more controversial than the previous ones. The very concept of ”substituted-for’s” appears to be problematic. Why should we see singular terms, but not other kinds of expressions as ”substituted-for’s”? What is the crucial difference between these terms and say, intransitive verbs (or, for that matter, sentential connectives)? Why should we see Brandom thinks as a substitutional variant of Sellars thinks, but not of Brandom writes; and why could we not see Brandom thinks and writes as a variant of Brandom thinks or writes? In other words, the delimitation of the concept of substituted-for’s (and thereby of the complementary concept of substitutional frame) appears to be a bit too arbitrary. The penultimate chapter of the book, A Social Route from Reasoning to Representing, discusses the emergence of the representational dimension of language from the basic, inferential one. According to Brandom, this is engendered by the ‘internalization’ of the switch between perspectives of various participants of linguistic practices: a speaker needs to present how things are not only from his own perspective, but also from the perspectives of other speakers; and this yields the (de dicto) propositional attitude reports. These reports then develop into claims pointing out what the other speaker’s beliefs are about (de re belief reports) – which, Brandom claims, are really specifying the objects of the reportee’s believes from the reporter’s perspective. Hence the about of the representational discourse has, in Brandom’s view, its constitutive locus within the claims of the forms X believes about O that ... . There are, in Brandom’s view, two commitments in play here: one attributed to the believer and one undertaken by the reporter by making the report; and the de re belief in a sense ‘exports’ something from the former into the latter. Thus, ”representational locutions make explicit the sorting of commitments into those attributed and those undertaken”, and hence ”the representational dimension of propositional contents reflects the social structure of their inferential articulation in the game of giving and asking for reasons” (183). The last chapter of the book then undertakes the most difficult task: it tries to answer the question of how this relativistic clash of perspectives can give rise to an objectivity independent of anybody’s peculiar perspective. Brandom claims that objectivity emerges out of the interplay of components of the ‘fine structure’ of inferential articulation – namely commitment and entitlement. In particular, Brandom claims that we must distinguish between inferences which are commitment-preserving and those which are entitlement-preserving, and he further urges that the interplay of these gives rise to inferences which are incompatibility-preserving (where two claims are incompatible in his sense if commitment to one of them precludes entitlement to the other one). This last kind of inference, Brandom insists, is then constitutive of claims with a truly objective content. His point is that there are claims which are in this sense not incompatible with claims to the effect of nonexistence of
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human beings etc. – hence claims which are answerable not to anyone’s perspective, but to how things ‘really’, ‘in themselves’ are. I find this most important chapter of the book also the most controversial and the least persuasive. First, the way in which Brandom introduces his ‘fine structure’ of inference is, as far as I can see, too cryptic. Thus, for example, if he claims – without any further ado – that every commitmentpreserving inference is entitlement-preserving but not vice versa, or that the concept of commitmentpreserving inference generalizes the notion of deductive inference, while that of the entitlementpreserving one that of inductive inference, I find it hard to follow. There is clearly a ‘structural’ difference between commitment and entitlement (whereas one can be committed to incompatible claims, one cannot be entitled to them) – but how does this generate a difference between commitment-preserving and entitlement-preserving; and why should the difference be seen as analogous to that between deductive and inductive inference? As a result, I find it hard to assess whether Brandom’s way of accounting for the birth of objectivity out of the interplay of individual perspectives really works. Brandom is one of those philosophers not content with working within an established paradigm – solving ‘philosophical puzzles’ – who hanker for a complete change. Moreover, he is one of the rare few who have managed to present a sweeping new picture of the relationship between man, his language and his world – his MIE is undoubtedly a philosophical milestone. (Doubtlessly many will find his picture controversial – but should we expect a path-breaking picture of this kind not to be?) The present book demonstrates that Brandom does not want to restrict himself to sketching magnificent pictures, but also tries to fill in their tedious details. It also indicates that there is still work to be done in this respect; but it is work which is likely to pay. Jaroslav Peregrin
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Topic-Focus Articulation, Tripartite Structures, and Semantic Content
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Eva Hajičová, Barbara H. Partee and Petr Sgall, Topic-Focus Articulation, Tripartite Structures, and Semantic Content. Studies in Linguistics and Philosophy 71. Dordrecht: Kluwer, 1998. ix + 216 pp. $86 (hb.). Reviewed by Jaroslav Peregrin, Department of Logic, Institute of Philosophy, Academy of Sciences of the Czech Republic, Jilská 1, 110 00 Praha 1, Czech Republic. It is uncommon to contrive a book on theoretical linguistics as a kind of a Platonic dialogue. However, the authors of Topic-Focus Articulation, Tripartite Structures, and Semantic Content have chosen to do precisely this: what they present is a dialogic confrontation of two different kinds of views of a certain aspect of language, resulting into a partial synthesis. I should say immediately that this unusual form was a happy choice; but to explain why we must return to the roots of formal semantics. Not so long ago, people working within the post-Montagovian tradition saw their task as addressing those aspects of expressions which are independent of how and in which contexts the expressions happen to get employed. Everything concerning concrete utterances and their contexts, so the story went, is to be relegated to pragmatics (considered as something necessarily less rigorous, and thereby somewhat less ‘noble’ than semantics). However, over the last few decades it has become increasingly clear that to draw any sharp boundary between semantics and pragmatics within natural language is inherently problematic; and that turning a blind eye on anything like contexts is bound to preclude us from being able to understand the semantics of many expressions and grammatical constructions. The English definite article provides a good example of this development. The traditional, Russellian ‘context-independent’ analysis presents the article as, in effect, the name of a function mapping sets of individuals on their members, namely singletons on their single elements and all other kinds of sets on nothing. Construed in this way, the X presupposes the existence of a unique F. However, it is clear that the great majority of natural language locutions containing the do not in this way amount to unique existence, but rather only to unique ‘referential availability’, to unique existence within the relevant context. Thus, since the works of Kamp and Heim, contexts and the dynamics of context change figure as one of the crucial themes of formal semantics (see, e.g., van Eijck and Kamp, 1997). This ‘pragmatization of semantics’ (cf. Peregrin, 1999) has led to semanticians’ considering and reconsidering new aspects of language, especially those previously consigned to the domain of pragmatics. A particular case is intonation and stress: many theoreticians realized that we can change even the truth conditions of a sentence by changing the intonational focus. This has initiated a boom of ‘focus-studies’, trying to develop formal models of the way a change of the focus may influence the proposition expressed by a sentence. However, it could hardly escape the notice of the exponents of this boom that the phenomena they are now so preoccupied to account for have been studied for a long time by a traditional linguistic school, namely the school originating between the two world wars with the Prague Linguistic Circle (Jakobson, Mathesius etc.) and which still has its successors in Prague (and elsewhere; cf. Luellsdorf et al., 1994). Of course, the conceptual framework of the representatives of this school differs substantially from the conceptual equipment of current formal semantics; however, that formal semantics would profit from accommodating their results appeared to be self-evident. The trouble was that the prima facie substantially different conceptual frameworks constituted a serious hindrance. And here is where we should seek the roots of the current book with its unusual form. Barbara Partee, one of the leading figures of the post-Montagovian formal semantics, confronts her views with those of Eva Hajičová and Petr Sgall, two outstanding representatives of the most formal wing
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of the current Praguian linguistics, and they work together towards a synthesis. And although they have not managed to go all the way to it, the report of the convergence of their doctrines makes very interesting and extremely instructive reading. Partee’s point of departure is the idea that we could account for the topic-focus articulation in terms of her tripartite structures, which amount to a kind of ‘generalization of generalized quantification’. From the viewpoint of the theory of generalized quantifiers, we see a sentence like Some boys smoke as stating that the extensions of its two terms (boys and smoke) stand in a certain relationship determine by its determiner (some). Partee’s idea now is that topic and focus could likewise be seen as denoting sets of some kinds of entities being in a relation which is either determined by an overt focalizer, or is a default one. Thus Boys SMOKE, BOYS smoke, Only BOYSsmoke would all come to be seen as stating that the class of boys and that of smokers stand in a specific relationship. (Partee presented the idea of a tripartite structure and discussed its application to topic and focus in a number of papers; see, e.g., Partee, 1991.) This, in effect, presupposes a categorial analysis of syntax in the Montagovian vein. On the other hand, Hajičová and Sgall work within the framework of dependency syntax - they see the meaning of a sentence in the form of a tree, called tectogrammatical representation, which contains no non-terminals and captures the dependential structuring. Every element of the tectogrammatical structure is either contextually bound or contextually nonbound. All the contextually bound items depending on the main verb, together with everything that depends on them, and together with the main verb - if this is contextually bound - then constitute the topic of the sentence; the rest of the sentence constitutes the focus. Hence the topic/focus classification is exhaustive: every element of the sentence belongs either to the topic or to the focus. To illustrate these notions, let us take an example from Sgall et. al. (1986, p. 190-1): (1) Tomorrow I’ll give a student some BOOKS. The tectogrammatical representation corresponding to this sentence is the following:
In a context in which (1) is given as an answer to the question What will you do tomorrow?, Tomorrow and I come to be contextually bound; whereas all the other items stay nonbound. Hence in this case, Tomorrow and I constitute the topic of the utterance, whereas the focus is constituted by give, student, some and book. Thus, the ‘Praguian’ concept of focus (and the complementary concept of topic) is not identical with the concept of focus as an ‘intonationally highlighted item’ (BOOKS in (1)). The Praguian notion is based on the assumption that ‘intonational highlighting’ is not the focus itself, but rather that it is only one of various possible ways of marking focus. From this viewpoint, ‘intonational highlighting’ is merely a means of indicating focus when it is in a marked position; and another way of indicating focus is word-order (referring of course to languages like Czech, which permit this).
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Now the book records a process whereby these two very different standpoints approach each other: from the initial exposition of the two conceptual frameworks in Chapter 2 (Towards an investigation of the relation between topic-focus articulation and tripartite structures) and the inventory of which assumptions the authors share and which divide them in Chapter 3 (Remarks on common background and shared assumptions), it proceeds to overcome Obstacles to joint work (Chapter 4) via a Dialogue, progressing towards a common basis for discussion (Chapter 5) to the proposal and examination of common hypotheses (Chapter 6: Some hypotheses proposed and examined) and further to outlining Future directions (Chapter 7). What are the hypotheses the authors reach regarding topic-focus articulation? They concern two principle issues: (i) The recursivity of topic-focus articulation (TFA). Whereas Partee’s original idea was that the articulation, being a case of the tripartite structure, must be unrestrictedly recursive in the way these structures are, the view of Hajičová and Sgall was that TFA is a matter of merely the sentence as a whole. The hypothesis to which the authors ultimately converged is closer to the latter view; they conclude that “embedded TFA happens (if and?) only if there is an embedded clause” (p. 164). In Chapter 6, the authors discuss many cases which may seem to constitute counterexamples and argue that all of them are reconcilable with the hypothesis. (ii) The very possibility of viewing TFA as a tripartite structure and the determination of the scope and focus of the focalizer (focus-sensitive operators, like only). The basic idea entertained in the beginning of the work by Partee was that “the counterpart of the focalizer is the Operator, that of its background is the Restrictor, and that of its focus is the Nuclear Scope” (p. 162); and this idea proved to be, in the course of the book, in principle tenable. This implies that the focalizer can be seen as denoting a relation over a ‘Boolean algebra’: thus, e.g., the sentence Only BOYS smoke is true iff the set of boys stands in a certain relation to the set of smokers. In this particular case, the relation is such that not only is the former set to be included in the latter one; it is, moreover, to ‘exhaust’ it (due to the effect of the focalizer only). This means that the set of boys is to conicide with that of smokers. A related problem is how a sentence is parsed into the topic, the focus and possibly the focalizer, i.e. into the parts corresponding to the three compartments of the tripartite structure. Here the authors reach the hypothesis which can be articulated in dependential terms as “the scope of a focalizer is the maximal projection of the head on which it depends” (thus, “the focus of a focalizer consists of the elements of the scope of the focalizer that are more dynamic than the focalizer itself”), or in terms of LF as “the scope of a focalizer is its c-command domain” (pp. 164, 165). In this way, Hajičová, Partee and Sgall reach a stage in which they are able to formulate substantial theses intelligible from within both the conceptual framework of formal semantics and that of the ‘Prague School’, thus overcoming their apparent ‘incommensurability’. In this way, they provide for a substantial interconnection of the two perspectives: they clarify how, in principle, it is possible to integrate the ‘Praguian’ insights into the framework of formal semantics; and, conversely, how methods developed within formal semantics can integrate with dependential syntax and with ‘tectogrammatical’ semantics. There remain, to be sure, issues in which the authors, as the representatives of the perspectives, have not managed to reach a consensus; and even the partial rapprochement they have achieved has yet to be fully systematized. However, I think we should keep in mind that there are terrains where securing the safe foundations for a house is no less difficult than building the house itself.
References
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Luelsdorff, Philip, Jarmila Panevová and Petr Sgall, eds., 1994. Praguiana 1945 - 1990. Amsterdam: Benjamins. Partee, Barbara, 1991. Topic, Focus and Quantification. In: S. Moore and A. Wyner, eds., Proceedings from SALT I, 257-280. Ithaca: Cornell University. Peregrin, Jaroslav, 1999. The Pragmatization of Semantics. In: Ken Turner, ed., The Semantics/Pragmatics Interface from Different Points of View, 419-442. Amsterdam: Elsevier. Sgall, Petr, Eva Hajičová, and Jarmila Panevová, 1986. The Meaning of the Sentence in its Semantic and Pragmatic Aspects. Prague: Academia. van Eijck, Jan, and Hans Kamp, 1997. Representing discourse in context. In: Johann van Benthem and Alice ter Meulen, eds., Handbook of Logic and Language, 179-238. Oxford: Elsevier and Cambridge (Mass.): MIT Press.
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Stephen Neale, Facing Facts, Oxford: Clarendon Press, 2001, xv + 254 pp. Reviewed by Jaroslav Peregrin, Department of Logic, Institute of Philosophy, Academy of Sciences of the Czech Republic, Jilská 1, 110 00 Praha 1, Czech Republic. E-mail: [email protected]. It is now often taken for granted that facts are entia non grata, for there exists a powerful argument (dubbed the slingshot), which is backed by such great names as Frege or Gödel or Davidson (and so could hardly be wrong), that discredits their existence. There indeed is such an argument, and it indeed is not wrong on the straightforward sense of wrong. However, in how far it knocks down any conception of facts is another story, a story which is anything but simple and perspicuous. In his book, Stephen Neale takes pains to excavate the origins of the argument and the presuppositions which it needs to be usable for the purpose of exorcising facts. In the introduction of the book, Neale expresses his conviction that his analysis of the slingshot will not only compromise its usability for the purpose of discrediting facts, but also save representationalist conceptions of language and mind from the attacks of the antirepresentationalist philosophers like Davidson and Rorty. „Representational philosophy,“ he claims, „survives the Davidson-Rorty onslaught because non-truth-functional logics and ontologies of facts, states of affairs, situations and propositions survive not only the actual arguments deployed against them, but also the most precise and powerful slingshot arguments that can be constructed.“ However, what he does take his analyses to show is that „the most precise and powerful slingshot arguments demonstrate conclusively that the logical and ontological theories originally targeted must satisfy non-trivial conditions if they are to avoid logical or ontological collapse.“ (P. 12) The book starts with the discussion of the philosophy of Donald Davidson, who appears to have brought the slingshot argument to the current prominence within philosophical discussions. Here we encounter the first variant of the slingshot: Consider two sentences φ and ψ and a proper name d. Consider the definite descriptions ‘the object x such that (x = d and φ)’ and ‘the object x such that (x = d and ψ)’. It seems to be clear that the former yields d if and only if it is the case that φ, whereas the latter yields d if and only if it is the case that ψ. Hence φ appears to be logically equivalent with ιx(x=d) = ιx(x=d and φ) and hence the two sentences appear to express the same fact; and similarly for ψ and ιx(x=d) = ιx(x=d and ψ). Now assume that φ and ψ are both true; then ιx(x=d and φ) and ιx(x=d and ψ) both denote the same object (namely d), hence ιx(x=d) = ιx(x=d and ψ) can be obtained from ιx(x=d) = ιx(x=d and φ) by mere replacement of a name of an object by another name of the same object, and hence they again appear to express the same fact. Hence if φ and ψ are true, each pair of subsequent sentences in the following sequence appears to be bound to express the same fact: φ ιx(x=d) = ιx(x=d and φ) ιx(x=d) = ιx(x=d and ψ) ψ However, as φ and ψ were randomly chosen true sentences, all true sentences are bound to express one and the same Great Fact. Neale offers a meticulous anatomization of the argument and concludes that „if a slingshot argument is to succeed in riding the landscape of facts (or at least ridding it of traditional ways of construing facts and making them philosophically useful), it will need to be
supplemented with a precise theory of descriptions“ (p. 57). This makes him turn his attention to definite descriptions. In the next two chapters, he therefore inspects the roots of modern semantics in Frege’s writings and then especially the conception of definite descriptions within the writings of Russell. He explains what kinds of slingshotian considerations prevented Frege from accepting facts as denotations of sentences and made him see sentences rather as names of truth values. And he then explains Russell’s theory of facts as complexes and the way Russell arrived at his theory of definite descriptions. Thereafter he turns to Gödel’s criticism of Russell and offers a recapitulation of his version of the slingshot. If we assume that Fa and Gb are true and that a and b are different objects, then we again we appear to have a path from Fa to Gb in which every pair of subsequent sentences appears to express the same fact: Fa a = ιx(x=a and Fx) a = ιx(x=a and (x≠b)) b = ιx(x=b and (x≠a)) b = ιx(x=b and Gx) Gb This leads Gödel to the conclusion which Neale himself outsketched earlier in connection with Davidson. „If a true sentence,“ summarizes Neale Gödel’s standpoint, „stands for a fact, then in order to avoid the collapse of all facts into one, the friend of facts must give up either (a) an intuitive and straightforward Fregean Principle of Composition or (b) the idea that definite descriptions are expressions that purport to stand for things.“ (Pp. 124-5) To clarify things further, Neale develops a conceptual framework which enables him to address the situation on a very general level, in terms of various kinds of ‘substitutivities’. Let me rehearse this in my own, rather than Neale’s terms. The situation can be seen in the following way: we want an equivalence between sentences, expressing the same fact or simply SF, such that logically equivalent sentences always express the same fact, i.e. (1) LE ⊆ SF, two sentences which differ from each other only in that a definite description in one of them is replaced by an equal definite description in the other also always express the same fact, i.e. (2) IDD ⊆ SF, but not every pair of materially equivalent sentences expresses the same fact, i.e. (3) ME ⊆/ SF. However, (1) and (2) yield LE∪IDD ⊆ SF, and the slingshot is devised to show that ME ⊆ LE∪IDD. Hence (1), (2) and (3) turn out to be inconsistent.
Now if we had some neutral names for facts, we could construe SF as the relation of intersubstitutivity salva veritate within the contexts of the shape ... expresses the fact F: we want such contexts to allow for interchange of logical equivalents and of equal definite descriptions, but not to be extensional, i.e. to allow for the interchange of material equivalents. In this form, the problem comes to appear as a subproblem of the problem of defining ‘non-collapsing’ non-extensional operators, and the classical slingshot appears to be generalizable to an argument against some non-extensional logics. In fact, as Neale points out, it has been put to this work by W.V. Quine, in the following shape (where C[...] represents a non-extensional context, in Quine’s particular case a context of a belief report): ϕ≡ψ C[ϕ] C[(a = ιx((x=a ∧ ϕ) ∨ (x=b ∧ ¬ϕ))] ιx((x=a ∧ ϕ) ∨ (x=b ∧ ¬ϕ) = ιx((x=a ∧ ψ) ∨ (x=b ∧ ¬ψ) C[(a = ιx((x=a ∧ ψ) ∨ (x=b ∧ ¬ψ))] C[ψ] This means that if we assume that logically equivalent sentences as well as equal definite descriptions are intersubstitutive within the context C[...] and if we assume that ϕ is materially equivalent with ψ, C[ϕ] must be materially equivalent with C[ψ] too. (Let me note that the allegedly fallacious version of this proof on page 172 of Neale‘s book is in fact identical with the correct one on the next page.) Of course that all of this depends crucially on the assumption that the investigated nonextensional context allows for the interchange of equal definite descriptions, the plausibility of which is, in case of modal logics, at least dubious. Why could it be seen as more plausible in case of facts? The reason, in case of facts, appears to be clear: expressions referring to things contribute to the facts expressed by sentences containing them by the things they refer to, so exchanging a name for another name of the same thing must not tamper with the fact expressed („rose by any other name would smell as sweet ...“). However, as Neale points out, as a reason for accepting the premises of the slingshot this line of thought possesses a substantial gap: definite descriptions, in their Russellian form, are no referring terms, hence intersubstitutivity of coreferential singular terms does not guarantee intersubstitutivity of equal definite descriptions (neither, for that matter, vice versa). All this indicates that however interesting the slingshot as an argument is, it usability as a fact-detonator is questionable. To summarize, we can say that Neale’s book provides a maximally thorough discussion of the point, the variations and the presuppositions of application, of the slingshot. What are the morals to be drawn of it? Surely that slingshot is an interesting piece of argumentation, which has bearing not only on theories of facts, but also on many other aspects of semantics. And especially that it is, without additional assumptions, not so lethal to facts as it is often taken to be. Has Neale also managed to thereby undermine Rortyan anti-representationalism as he claims in the introduction? As a matter of fact I do not think so. I think that his results affects Rorty’s anti-representationalism much less than Neale appears to think. In fact, already Davidson’s disrespect to facts stems from two very distinct and largely independent sources which Neale does not clearly disentangle, the slingshot-considerations being only one of them. The other is the conviction that “the notion of ... fitting the facts, or of being true to the facts, adds nothing intelligible to the simple concept of being true” Davidson (Inquiries into Truth and Interpretation, Clarendon Press, Oxford, pp.193-194). (Hence while the first source leads to the conviction that we cannot have a coherent theory of facts, the second one implies
that such a theory would not be a theory worth its name, for it would offer only an illusion of an explanation.) Rorty, then, draws almost exclusively on the second source: his conviction being that facts are useless entities which have been invented only to grant our sentences something which they could represent. Hence independently of whether we agree with Rorty on this point or not, it has very little to do with the slingshot. (It should be also noted that neither Davidson, nor Rorty would probably protest against the claim that our thinking involves creating and manipulating various kinds of representations – their claim being that representing should not be seen as the crucial thing done by mind and language.) Despite this, I think that Neale’s extremely coherent and lucidly written book should be recommended not only to everybody who wants to understand the real nature of the slingshot, but also to everybody who is interested in logical analysis of language, formal semantics, and also metaphysics.
Scott Soames, Beyond Rigidity, The Unfinished Semantic Agenda of Naming and Necessity, Oxford: Oxford University Press, 2002, ix + 379 pp. Reviewed by Jaroslav Peregrin, Department of Logic, Institute of Philosophy, Academy of Sciences of the Czech Republic, Jilská 1, 110 00 Praha 1, Czech Republic. E-mail: [email protected]. Saul Kripke ’s book Naming and Necessity (which first appeared in 1972 as a paper within a volume on natural language semantics1) is felt, by many linguists and philosophers, as a milestone of the semantic analysis of natural language. Prior to it, many semanticists took for granted that the meaning of any expression must be a two-level matter, consisting of something of the kind of what Frege called Sinn and Bedeutung or what Carnap christened as intension and extension. The first of the components is what the speaker knows when she understands the expression (and the knowledge of which is independent of any knowledge of facts external to the language in question), while the second amounts to some kind of chunk of the real world which gets denoted or referred to by the expression. Thus, the intension of the king of France is what you get to know as soon as you come to understand the phrase, and the extension is what (if anything) happens to be picked out by the intension in the actual world in the actual moment. This is to say that an expression gets to into the contact with its extension only via the intension (which we can imagine also as a kind of a criterion for picking up the thing). Intension, then, is what amounts to the meaning of the expression in the intuitive sense of the word2. Now Kripke’s considerations challenged this very two-level structure of meaning: he argued that especially proper names get their semantics via a direct, inmediated contact with the world – that their meaning does not mediate their contact with a thing, but directly is the thing. This may be not so surprising in case of proper names (after all, it is even disputable whether they can be counted to the language – for you cannot find them in dictionaries), but Kripke went on to argue that the same holds also for some other expressions, especially natural kind terms such as “cow” or “water”. Kripke’s argumentation is concise and flattening; and it has been provoking a lot of discussions. However, the relatively sparse Naming and Necessity did not address everything connected with proper names, natural kind terms and their rigidity with the desirable explicitness, and Kripke has never come back to it to elaborate on it. Scott Soames, in the present book, undertakes this very task, namely to investigate the matter in greater detail, to fill the gaps of Kripke’s original exposition, and possibly also to correct inconsistencies of Kripke’s approach. And we can say that his book, several times more voluminous than Kripke’s one, comes to be a successful follow-up to Kripke’s work. Soames concentrates especially on two themes which he does not take to be exhaustively treated by Kripke himself, one concerning proper names and the other natural kind terms. In case of proper names he feels that Kripke did not manage to explain quite satisfactorily how, if names rigidly denote their bearers, there can be informative claims putting the equality sing between two names. If morning star and evning star are (pace Frege) rigid designators, how can it be that Morning star is evening star can convey something nontrivial to us? 1 2
In Semantics of Natural Language (eds. Davidson, D. & Harman, G.), Reidel, Dordrecht, 253-355.
Let us disregard the fact that many intensional semanticists found intensions not enough fine-grained for the purpose of explicating meanings, and switched to kinds of ‘hyperintensional‘ theories.
Soames’ answer to this question invokes the distinction between semantics and pragmatics: though such sentences do not mean anything nontrivial, they can be used to say something nontrivial. In particular, he claims that (p. 242) on a proper conception of semantic content there is a principled distinction between the vast majority of ordinary, linguistically simple proper names, the semantic contents of which are simply their referents, and the special, but still quite extensive, category of partially descriptive names, which are often syntactically complex expressions, and which have semantic contents that include both their referents and certain descriptive information, conventionally associated with them by speakers. In the former case, substitution of one member of a pair of coreferential proper names for another does not change the proposition semantically expressed – ever though it may change the proposition asserted, believed or attributed to others as one of their assertions or beliefs. In the latter case – involving partially descriptive names – even the propositions semantically expressed may change. The other major problem of Kripke’s doctrine thematized by Soames concerns natural kind terms. Soames points out that while it is quite clear what does it take for a name to be rigid, this is far from so in case of natural kind terms. What does is take for a term such as “cow” to be rigid? To be associated with an individual in the very same sense as a proper name is associated with its bearer? But which kind of individual? The set of all cows existing at the moment of the initial baptism? But this would appear to imply that the word does not apply to our present set of cows, which is surely different from the original one. The property of being a cow? But as every term, natural kind or not, can be seen as associated with a property, this would seem to render every term rigid. Hence a mereological sum of bovine spatiotemporal regions, which somehow manages to keep its identity across shrinking and expanding? Or to some primordial cow and, thereby, to everything which is ‘just like it’? Soames proposes that it would be better to see the characteristic mark which natural kind terms have in common with proper names not in rigidity, but in non-descriptivity: the feature that “the referent ... at a world is not semantically determined by the satisfaction of any descriptive condition at that world” (264). This shift of focus allows us to concentrate on how do natural kind terms designate, rahter than what exact kind of entity they designate. The solution Soames then envisages consist in maintaining that (277) natural kinds should not be identified with properties after all – where properties are those sorts of things that are the meanings of arbitrary predicates. Instead of taking natural kinds to be properties, I propose that we identify them with intensions – that is functions from worlds to extensions. In this way, the analogy between proper names and natural kind terms comes to be explicated as consisting in the fact that just like a bearer of a name may be determined by a description without the description becoming the meaning of the name, a natural kind may be determined by a ‘property’ without the property becoming the meaning of the natural kind term. I think Soames is very right in pointing out that this is a highly nontrivial problem which Kripke and his followers mostly fail to reflect. However, it seems to me that this, second part of his book is slightly inferior to the first one, as if the author did not have enough patience to
develop to the perfection comparable to that of his discussion of proper names. In particular, it lacks a clear summarization of Soames’ answer to the main question, namely What does it take for a general term to be rigid? True, parts of the answer are scattered within Soames’ exposition, but the view he maintains is not always optimally clear. In particular, it seems that he somehow glosses over the most basic presupposition of Kripke’s extension of his theory of proper names to natural kind terms, namely the existence of natural kinds that are enough like individuals. Saying that natural kinds are intensions may help us cope with problems of the kind of the paradox of analysis, but it does not seem to be very helpful in the way of explaining what natural kinds are and why we should believe in them in the first place. Surely not every possible intension is a natural kind – hence which intensions should be seen as such, and especially why should we think that there are some such prominent intensions? Despite this, Soames’ careful examination of the big Kripkean themes suceeds in providing a more systematic and a more comprehensive exposition than the original Kripke’s one. It points out where Kripke’s theory may be seen as containing gaps, and provides a fruitful discussion of what it would take to fill them. Therefore it should not be missed by not only any Kripke scholar, but by any semantician who appreciates the significance of Kripke’s attack on the two-level semantics.
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Meaning and Inference Jaroslav Peregrin
Introduction Contemporary theories of meaning can be divided, with a certain amount of oversimplification, into those which see the meaning of an expression as principally a matter of what the expression denotes or stands for, and those which see it as a matter of how the expression is used. A prominent place among the latter ones is assumed by those which identify the semantically relevant aspect of the usage of an expression with an inferential pattern governing it. According to these theories, the meaning of an expression is, principally, its inferential role.1 In this paper we first propose an exact definition of the concept of inferential role, and then go on to examine the question whether subscribing to inferentialism necessitates throwing away existing theories of formal semantics, as we know them from logic, or whether these could be somehow accomodated within the inferentialist framework. The conclusion we reach is that it is possible to make an inferentialist sense of even those common semantic theories which are usually considered as incompatible with inferentialism, such as the standard semantics of second-order logic.
Primary inferential roles (of sentences) Let us use the sign > for correct inferrability (we will also use the term entailment). Thus, if X is a set of sentences and S a sentence, we will write X > S to express that X entails S. Besides entailment, we may consider one other important inferential property of sets of sentences (which, as we will see later, may or may not be seen as reducible to entailment), namely incompatibility. Let us write IC(X) to express that nobody can be entitled to all elements of X. Let us, moreover, write X ⊥ S as a shorthand for IC(X ∪ {S}).2 See especially Brandom (1994; 2000) for the ideological background and Lance (1996; 2001) and Kaldernon (2001) for some more technical aspects. See also Peregrin (2001; esp. Chapters 7 and 8) and Peregrin (to appear). 2 The notation is partly due to Lance (2001). 1
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We will assume that entailment has the following basic properties (we shorten {S1, …, Sn} > S to S1, …, Sn > S): [R] [T] [M]
S>S if X > S and {S} ∪ Y ⇒ S′, then X ∪ Y ⇒ S′ if X > S, then X ∪ {S′} > S
Moreover, there is the following relationship between entailment and incompatibility: [*]
if X > S and IC(Y ∪ {S}), then IC(Y ∪ X)
This means that something is incompatible with the consequent of an entailment, it is bound to be incompatible also with its antecedent. We may also consider reducing incompatibility to entailment by the well known ‘ex falso quodlibet’ axiom: [**]
X ⊥ S iff X ∪ {S}> S′
Now with this notational apparatus, what we will call the primary inferential role of a sentence S can be exhaustively characterized by the following four sets: conditions of S: S ← = {X | X > S} consequences of S: S → = {〈X,S′〉 | X ∪ {S} > S′} contradicta of S: S × = {X | X ⊥ S} Hence as a first approximation, we can set PIR(S) ≡Def 〈S ←, S →, S ×〉 We define the inclusion of PIR’s as the inclusion of their respective components PIR(S1) ⊆ PIR(S2) ≡Def S1← ⊆ S2← and S1→ ⊆ S2→ and S1× ⊆ S2× and we define the relation PIR of sameness of primary inferential roles: S1 PIR S2 ≡Def PIR(S1) = PIR(S2). We will now prove that two sentences have the same PIR iff they are logically equivalent. First a lemma (where <> is the relation of logical equivalence, i.e. the relation which holds between S1 and S2 just in case S1 > S2 and S2 > S1): Lemma: If S1 <> S2, then PIR(S1) ⊆ PIR(S2) Proof: Let S1 <> S2. Then 1. If X > S1, then X > S2 (from S1 > S2 by [T]).
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2. If X ∪ {S1} > S′, then X ∪ {S2} > S′ (from X ∪ {S2} > S1 by [T], which from S2 > S1 by [M]). 3. If X ⊥ S1, then X ⊥ S2 (from S2 > S1 by [*]). It obviously follows that if S1 and S2 are logically equivalent, they share the same PIR; for if S1 <> S2, then PIR(S1) ⊆ PIR(S2) and PIR(S2) ⊆ PIR(S1), i.e. S1 PIR S2. The next simple lemma shows that the converse implication also holds: Lemma: If S1 PIR S2, then S1 <> S2. Proof: S1 ∈ S1← (by [R]), hence S1 ∈ S2←, hence S1 > S2; and the same vice versa. Combining the two lemmas we get the anticipated result, namely that S1 PIR S2 iff S1 <> S2. Admitting some further assumptions about the inferential structure of language, we can further simplify the definition of PIR (for the lack of space, we omit proofs of these facts). Thus, accepting [**] we can reduce contradicta to consequences, S × = {X | X ∪ {S} > S′ for every S′} = {X | 〈X,S′〉 ∈ S → for every S′}, and PIR(S) becomes uniquely determined by S → and S ←. If we assume that for every statement S there exists a ‘minimal incompatible’ MI(S) such that MI(S) ⊥ S and if X ⊥ S, then X > MI(S), then it can be shown that we can reduce contradicta to conditions of the minimal incompatible: S × = {X | X > MI(S)} = MI(S)← Moreover, there is a sense in which we can reduce consequences to conditions (where X ← is the set of sets of sentences entailing all elements of X): S → = {〈X,S′〉 | S ← ⊆ X ← ∪ S′←}
Secondary inferential roles Primary inferential roles are a matter of sentences only. Hence if we want to apply inferentialism also to subsentential expressions, we have to find a concept of an inferential role which would be applicable to them too. Therefore we introduce the concept of secondary inferential role, which we will not define explicitly, but only via the relation of sameness. If E1 and E2 are expressions, then we will call two sentences S and S′ [E1/E2]-variants iff S′ differs
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from S only in that some the occurrences of E1 are replaced by E2. Then we define the sameness of secondary inferential roles as follows: E1 SIR E2 ≡Def S PIR S′ for every [E1/E2]-variants S and S′ Thus, SIR’s are ‘what SIR-equivalent expressions have in common’. In other words, an SIR of an expression is the contribution which the expression brings to the primary inferential roles of the sentences in which it occurs. Now sentences have two kinds of inferential roles, PIR’s and SIR’s. Are they different? Though in general they are,3 we will characterize the class of languages for which they may be seen as coinciding. Let us call a language at most intensional (i.e.: not ‘hyperintensional’) iff for every two sentences S1 and S2 and every two [S1/S2]-variants S and S′, S1 <> S2 implies S <> S′. Claim: A language is at most intensional iff for every two sentences S1 and S2, S1 SIR S2 iff S1 PIR S2. Proof: Let us first consider a language which is at most intensional. The ‘only if’ part is trivial. Let S1 PIR S2 and let S and S′ be [S1/S2]-variants. Then S1 <> S2, and as the language is at most intensional, S <> S′. But this means that S PIR S′, and hence S1 PIR S2. Now consider a language which is not at most intensional. Then there are some S1, S2 and some [S1/S2]-variants S and S′ such that S1 <> S2, whereas not S <> S ′. This means that S1 PIR S2, but not S1 SIR S2. This means that if we restrict ourselves to languages which are at most intensional, we can identify the SIR of a sentence with its PIR. The SIR of a subsentential expression can then be seen as the contribution the expression brings to the inferential roles of the sentences in which it occurs.
Inferential patterns The crucial claim of inferentialism is that the inferential role of every expression is determined by some finite inferential pattern. (The idea is that to grasp the meaning of the expression is just to master the inferential pattern, and so the pattern must be something humanly masterable, i.e. finite.) However, what is an inferential pattern? As a first approximation, we will identify such a pattern with a finite set of (possibly parametric) instances of inference, such as Note that the distinction between the PIR of a sentence and its SIR is the inferentialist embodiment of Dummett’s (1973) distinction between the “freestanding sense” and the “ingredient sense”.
3
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Cat(Tom), Mouse(Jerry) > Cat(Tom) ∧ Mouse(Jerry) or S1, S2 > S1 ∧ S2. Note that we do not claim that the inferential role of an expression must be specifiable independently of those of other expressions – an inferential pattern may well characterize the role of an expression only relatively to those of some other expressions (which then implies that the language cannot possess the former without possessing the latter).4 Note also that in no way do we maintain the claim that any inferential pattern is as good as any other in furnishing an expression with a viable meaning (i.e. the claim attacked by Prior, 1960/61).5 Let us take an example. We may characterize the inferential role of the sentence Cat(Tom) ∧ Mouse(Jerry) by the following pattern: Cat(Tom), Mouse(Jerry) > Cat(Tom) ∧ Mouse(Jerry) Cat(Tom) ∧ Mouse(Jerry) > Cat(Tom) Cat(Tom) ∧ Mouse(Jerry) > Mouse(Jerry) Expressed in terms of positive conditions, this yields Cat(Tom)← ∩ Mouse(Jerry)← ⊆ Cat(Tom) ∧ Mouse(Jerry)← Cat(Tom) ∧ Mouse(Jerry)← ⊆ Cat(Tom)← ∩ Mouse(Jerry)← and hence Cat(Tom) ∧ Mouse(Jerry)← = Cat(Tom)← ∩ Mouse(Jerry)← In fact, in this way we only reduce the inferential role of Cat(Tom) ∧ Mouse(Jerry) to those of its components, namely Cat(Tom) and Mouse(Jerry); so this definition works only if we already have the inferential roles of the two atomic sentences.6 As a further example, let us consider the inferential role of ∧, which obviously derives from that of S1 ∧ S2 with parametric S1 and S2. The pattern is, of course, Take arithmetic: the roles of all its constants are inextricably characterized by Peano’s axioms. See Peregrin (2001, §8.5) for more details. 6 What, then, would be the role of an atomic sentence, such as Cat(Tom)? Well, if we want Cat and Tom mean approximately what they do in English, then to understand them inferentialistically, we would have to somehow broaden the concept of inference to comprise not only the ‘language-language’ instances, but also some ‘world-language’ instances (see Peregrin, 2001, §7.6). Of course the inferentialist does not aspire to reduce meanings of empirical expressions to inferences in the usual, ‘language-language’ sense – such inferences, though vital for any kind of meaning, exhaust only meanings of non-empirical, especially logical, expressions. 4 5
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S1, S2 > S1 ∧ S2 S1 ∧ S2 > S1 S1 ∧ S2 > S2 This yields S1 ∧ S2← = S1← ∩ S2← Unlike in the previous case, this is not a reduction, but rather a full specification of the inferential role: ∧ is characterized as an expression which combines two sentences into a complex sentence with positive conditions equal to the intersection of the positive conditions of its two components. Now consider ∨. Part of the inferential pattern characterizing it is clear: S1 > S1 ∨ S2 S2 > S1 ∨ S2 But we want also the disjunction to be true only if at least one of its disjuncts is true. If we may make use of negation, then we can complete the pattern by ¬S1, ¬S2 > ¬(S1 ∧ S2) yielding the desired S1 ∨ S2← = S1← ∪ S2←, but this presupposes that we have already established the inferential role of negation. However, there is obviously no way of articulating the inferential role of negation, and consequently disjunction, by means of what we have so far called an inferential pattern.7 A reaction to this might be to say that hence the classical operators are not accessible for the inferentialist (the worse for them!, from the inferentialist standpoint), but we will indicate that a generalization of the concept of an inferential pattern, which is not too unnatural, will allow us to make inferential sense even of them. (For lack of space we will say nothing about the logical constants specific to predicate logic, i.e. quantifiers.) 7 It might seem that an inferential specification of the meanings of the logical operators is provided already by the standard axiomatics of the propositional calculus. Does it not follow from the soundness and completeness of the calculus that the axioms pin down the denotations of the operators to the usual truth functions? In fact, it does not. As a matter of fact, the axioms are compatible with some non-standard interpretations of the operators – with negations of some falsities being false and with disjunctions of some pairs of falsities being true. What is the case is that if the axioms hold and if the denotations of the operators are truth functions, then they are bound to be the standard truth functions. But the axioms are compatible with the indicated non-truthfunctional interpretation of the constants. This is a fact noted already by Carnap (1943), but very rarely reflected (see Koslow, 1992, Chapter 19, for a discussion).
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Truth tables and inferential patterns Our goal, then, will now be to provide a way of reading the rows of the truth table characterizing any of the usual logical operators as generally recapitulating inferential patterns governing the operator. This is straightforward in some cases, but rather more tricky in others. We will consider general types of rows in a possible truth table and try to provide their ‘translation’ into instances of inference (note, however, that from the inferentialist viewpoint this will not be a translation, but rather a case of archeology—retrieving the patterns lying beneath the tables).8
Method 1. True arguments → true value First, consider a row with only T’s both in the argument columns and in the value column. Here the inferentialist reading is, of course, quite straightforward: the row states that the complex sentence is entailed by all its arguments: S1 ... Sn O(S1,...,Sn) T ... T T .. ... ... ...
S1, …, Sn > O(S1, …, Sn)
Example S1 S2 S1∧S2 T T T .. ... ...
S1, S2 > S1 ∧ S2
Method 1 (generalized). True or arbitrary arguments → true value This can obviously be generalized to clusters of rows with T in the value columns, T’s in some fixed argument columns and all possible combinations of values in the other argument columns. Let us introduce the convention of putting as asterisk into the argument column whose content is irrelevant for the content of the value column:
8 Note also that what we are after is something quite different from the fact underlying the most usual proof of the completeness of the classical propositional calculus due to Kalmár (1936), namely that using negation and implication, we can express every row of every table as a theorem of the predicate calculus (see, e.g. Mendelson, 1964, §1.4). What we want is to find, for every table, an inferential pattern featuring the single operator whose semantics is given by the table.
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S1 v1 v1 ..
... ... ... ...
Si-1 vi-1 vi-1 ..
→
Si T F ...
Si+1 vi+1 vi+1 ...
... ... ... ...
Sn O(S1,...,Sn) vn v vn v ... ...
S1 ... Si-1 Si Si+1 ... Sn O(S1,...,Sn) v1 ... vi-1 * vi+1 ... vn v .. ... .. ... ... ... ... ...
Then clearly a row containing only T’s and *’s in the argument columns and T in the value column also yields an inferential pattern quite straightforwardly: S1 ... Sn O(S1,...,Sn) T/* ... T/* T .. ... ... ...
Si1, …, Sin > O(S1, …, Sn), where i1, …, in are the columns containing T (rather than *)
Example: S1 S2 S1∨S2 * T T .. ... ...
S2 > S1 ∨ S2
Method 2. One false argument and others true or arbitrary → false value An inferential pattern is also easily seen to be yielded by a row containing precisely one F in the argument columns and F in the value column: S1 ... Si ... Sn O(S1,...,Sn) T/* ... F ... T/* F .. ... ... ... ... ...
Si1, …, Sin, O(S1, …, Sn) > Si, where i1, …, in are the columns containing T
Example: S1 S2 S1→S2 T F F .. ... ...
S1, S1 → S2 > S2
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Method 3. True or arbitrary arguments → false value For rows containing no F’s in the argument columns, but containing F in the value column we need to invoke either the concept of incompatibility or the ‘ex falso quodlibet’ axiom:
Example: S1 ... Sn O(S1,...,Sn) T/* ... T/* F .. ... ... ...
S1, …, Sn ⊥ O(S1, …, Sn), or S1, …, Sn, O(S1, …, Sn) > S
Example: S ¬S T F .. ...
S ⊥ ¬S S,¬S > S′
Method 4. The types of rows for which we have provided ‘translations’ so far still do not cover all those occurring in the tables of the standard operators: viz. the F F F row within the disjunction truth table or the F T row within the negation one. How are we to make inferential sense of them? Let us look at an inferential pattern for an operator O as a means of enumerating all those assignments of truth values to S1, …, Sn for which O(S1, …, Sn) yields T (or, alternatively, F). Thus, the pattern S1 > S1 ∨ S2 S2 > S1 ∨ S2 specifies that S1 ∨ S2 is true if at least one of S1 and S2 is, i.e. for the truth-value pairs 〈T,F〉, 〈F,T〉 and 〈T,T〉. Similarly S1, S1 → S2 > S2 says that S1 → S2 is false if S1 is true and S2 false. Now it is not unnatural to assume that if one gives an enumeration of cases, it is meant to be exhaustive. If I say “My children are Tom and Jerry”, then what is normally taken for granted is that these are all my children. (See McCarthy’s, 1980, discussion of
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this feature of our enumerative claims).9 This may suggest that part and parcel of our ‘enumerative’ claims, and especially of our ‘enumerative’ specification of inferential patterns, is what can be called the exhaustivity assumption (EA): The inferential pattern’s specification of what entails a sentence (or what is entailed by it) is assumed to be exhaustive; i.e. it is assumed that the sentence is entailed by (entails) only that which is specified by the pattern. If this is the case, then it is enough to find a pattern behind all the rows of a table with T in the value column, or all those with F in this column – EA then takes care of the rest. Consider disjunction. The above inferential pattern can be seen as recapitulated by three of the four rows of its truth table. Now assuming exhaustivity of this implies that S1 ∨ S2 is true for no other assignment of truth values to S1 and S2, hence that for the truth-value pair 〈F,F〉 it yields F. In other words, the set of pairs of truth values for which disjunction yields T is the minimal set of pairs containing all pairs with at least one component being T: Example1 (minimality): S1 T T F F
S2 S1∨S2 T T F T T T F F
S1 > S1 ∨ S2 S2 > S1 ∨ S2 if S1 > S and S2 > S, then S1 ∨ S2 > S
Analogously for implication, where the set of pairs of truth values for which the implication yields T is the maximal set of pairs not containing 〈T,F〉: Example 2 (maximality): S1 T T F F
S2 S1∨S2 T T F T T T F F
S1, S1 → S2 > S2 if S1, S > S2, then S > S1 → S2
9 McCarty’s considerations resulted, in effect, into the conclusion that an intended model of what we say is always the minimal one, which led him to his concept of circumscription. See Hintikka (1988) for a further elaboration of this idea.
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The conception of logical operators arrived at in this way is in fact that developed by Koslow (1992): To any logical operator ϕ there corresponds a condition Φ such that the operator assigns to each pair (if the operator acts on pairs) the set of all items in the [inferential] structure that are the weakest members of all those that satisfy the condition. Note that both the truth-functional viewpoint, the reduction of incompatibility to entailing everything, and the exhaustivity assumption lead to ‘desired’ results only if the language considered is rich enough. Consider a language with an explicitly defined truth-functional semantics. In this case, the stipulation S,¬S > S′ guarantees that the negation of T is F only if the language contains a contradiction. Similarly the stipulation that the disjunction of S1 and S2 is the minimal S for which S1 > S and S2 > S makes the disjunction denote the usual truth function only if there is an expression denoting the function within the language at all; otherwise it yields its ‘closest approximation’.
From inferential roles to possible worlds Let W be the set of all maximal consistent sets of statements (i.e. all such sets which do not entail a contradiction and are not contained in another noncontradictory set). Let us define the intension of S, S I, in the following way: S I = {w | w ∈ W and there is an X ∈ S ← so that X ⊆ w} In this way the inferential roles can be seen as giving rise to possible world semantics. An objection to this construction might be that it yields us only a language-dependent notion of possible-world. However, as I argued elsewhere (Peregrin, 1995, §11.5), this is precisely what we should want. The point is that if we want to use possible worlds for the analysis of meanings, then we must exempt the language we analyze out of the possible worlds – the association of the expressions with their meanings must be kept fixed across the space of possible worlds. (Otherwise every statement would come out as true only contingently.) Hence the space of the possible worlds is to be limited by what is possible within the framework established by the language – by what the language ‘takes to be possible’. But there is a deeper objection which concerns incomplete languages such as the languages of second-order logic (with standard semantics). In such a language, there may exist S1, …, Sn, S such that S1, …, Sn entail S, but not
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S1, …, Sn > S, and hence not S1I ∩ … ∩ SnI ⊆ S I. In this case we may feel we have ‘too many worlds’. Peano arithmetic, for instance, yields us a world in which, in contrast to the standard model (which is the only model of the second-order Peano arithmetic), the Gödel sentence is not true. Thus, inferential roles lead us directly always only to Henkin semantics. If inferentialism were correct, how could standard semantics, and ‘second-order’ consequence come into being? It might seem that the inferentialist’s response to this must be that this only shows that second-order semantics is simply illusory. But this is not the case: there is a way of making sense even of this kind of semantics within the inferentialist framework; and it is even possible to see it, in cases like the Peano arithmetic, as the semantics (thus vindicating the intuition that the standard model is the model). We have admitted that inferential patterns may involve the exhaustivity assumption: and reading the Peano axioms as involving the EA is precisely what is needed to exclude the non-standard models. If we say that 0 is a number, the successor of every number is a number, and nothing else is a number, then obviously we have the unique specification of the standard natural numbers – it is the EA which is enough to take us from the Henkin to the standard semantics (cf. Hintikka, 1988; 1989).
Jaroslav Peregrin Dept. of Logic, Institute of Philosophy Academy of Sciences of the Czech Republic Jilská 1, 110 00 Praha 1, Czech Republic [email protected] www.cuni.cz/~peregrin
References Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Carnap, R. (1943): Formalization of Logic, Harvard University Press, Cambridge (Mass.).
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Hintikka, J. (1988): ‘Model Minimization—an Alternative to Circumscription’, Journal of Automated Reasoning 4, 1–13; reprinted in Hintikka (1998), 212–224. Hintikka, J. (1989): ‘Is there completeness in mathematics after Gödel?’, Philosophical Topics 17, 69–90; reprinted in Hintikka (1998), 62–83. Hintikka, J. (1998): Language, Truth and Logic in Mathematics (Selected Papers, vol. 3), Kluwer, Dordrecht. Kalderon, M.E. (2001): ‘Reasoning and Representing’, Philosophical Studies 105, 129–160. Kalmár, L. (1936): ‚Zurückführung des Entscheidungsproblems auf den Fall von Formeln mit einer einzigen binären Funktionsvariablen‘, Comp. math. 4, 137–144. Koslow, A. (1992): A Structuralist Theory of Logic, Cambridge University Press, Cambridge. Lance, M. (1996): ‘Quantification, Substitution and Conceptual Content’, No s 30, 481–507. Lance, M. (2001): ‘The Logical Structure of Linguistic Commitment III: Brandomian Scorekeeping and Incompatibility’, Journal of Philosophical Logic 30, 439–464. McCarthy, J. (1980): ‘Circumscription – a form of non-monotonic reasoning’, Artificial Intelligence 13, 27–39. Mendelson, E. (1964): Introduction to Mathematical Logic, Wadsworth & Brooks, Pacific Grove. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Peregrin, J. (to appear): ‘Pragmatism and Semantics’, to appear in German in a book on pragmatism edited by E. Olsson and A. Fuhrmann (available in English from my home page).
BOOK REVIEW
Robert B. Brandom, Tales of the Mighty Dead, Harvard University Press, Cambridge, MA, 2002, x + 416 pp., ISBN 067400 9 037, $39.95 (cloth). In his “Making it Explicit”,1 Robert Brandom set up a new philosophical paradigm, concentrating especially on the link between language and the world, but extendable (in the way familiar from the dawn of the linguistic turn) to the rest of philosophy. He views modern philosophy in terms of the tension between “representationalist” and “inferentialist” approaches to language (which, according to him, also underlies the much more commonly cited struggle between empiricism and rationalism); and elaborating on the ideas of his teacher Wilfrid Sellars he develops a distinctive kind of inferentialist framework. After “Making it Explicit”, Brandom published the much more slender “Articulating Reasons”,2 which appeared, in comparison to his opus magnum, to be mere miscellanea; and his latest book, “Tales of the Mighty Dead”, though not slender at all, is also miscellaneous in nature: it consists predominantly of Brandom’s earlier published papers, from 1977 to 2000. Each of the papers focuses on an aspect of a great philosopher (a “Mighty Dead”), the gallery of personalities – Spinoza, Leibniz, Hegel, Frege, Heidegger, and Sellars – cutting across any standard philosophical school. What Brandom wants to show is that all these philosophers share an engagement with what he sees as a specific philosophical tradition, a tradition primarily characterized by its inferentialism (as opposed to representationalism), i.e. the conviction that the representational capacities of language are secondary to the fact that language is ‘inferentially articulated’. Brandom has pointed out that inferentialism is also inseparably connected with holism – indeed, if it is inferences that confer meaning, then having certain meaning presupposes assuming a certain place within a certain inferential structure. Moreover, insofar as having the meaning is assuming the place (which presupposes that we construe the concept of inference broadly enough), then this holism results into a kind of functionalism. Furthermore and less obviously, the kind of inferentialism Brandom has in mind is closely connected to pragmatism (in the sense of the primacy of praxis over theory) – for inferences are something we do. And as it counters the Erkenntnis 59: 421–424, 2003.
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most central tenets of empiricism, Brandom sees it also as the current stage of rationalism.3 Let me refer to this cluster of interconnected stances as the IHFPR(inferentialist-holist-functionalist-pragmatist-rationalist)-tradition. In his essays, Brandom traces the origins of the IHFPR-stance back to the dawn of modern philosophy, namely to Spinoza and Leibniz; Spinoza for his epistemological holism and Leibniz also for his rudimentary inferentialism. But surely his most interesting – as well as most controversial – analyses of modern philosophy classics are those dealing with Hegel (who seems to be Brandom’s ‘fatal attraction’). What Brandom claims is that Hegel is an inferentialist and holist, because, according to him, “conceptual contents are identified and individuated solely by the relations of material incompatibility” (p. 49), for it is nothing else than material incompatibility to which Hegel refers by his “determinate negation”. But Brandom’s reading of Hegel is also pragmatist, for he sees him as committed to “understanding determinately contentful empirical conceptual norms as instituted by experience, the process of using those concepts by applying them in practice: making judgments and performing actions.” (56) Of the three more recent philosophers, Sellars’ presence is of course to be expected: it is Sellars’ legacy that Brandom has developed into his own distinctive sort of inferentialism. The choice of Frege is also not overly controversial - though some scholars do regard the inferentialist strands within his works as merely marginal, the inferentialist reading is nothing unprecedential. Hence here the most novel is Brandom’s reading of Heidegger. Of course, connecting Heidegger to pragmatism, which Brandom explains in terms of the priority of the Zuhanden over the Vorhanden, is also not new.4 But Brandom wants to picture Heidegger as a more integrated exponent of the IHFPR-tradition; in particular, he reads him as almost a companion of Sellars: “The practical distance from things that distinctively semantic relations afford – the capacity merely to take in how things are, the capacity merely to entertain thoughts about how they might be [i.e. the availability of things as vorhandenseiende – JP] – is available only through the institution of the equipment with the practical significance of assertings and inferrings.” (80) Brandom sees philosophy as essentially a matter of “talking with a tradition” and what he calls “bebop history”: improvisation on the “chord structures” of significant philosophical works of one’s predecessors. Thus, in the papers reprinted in the book he reinterprets the various philosophers as distinctive bearers, and promulgators, of the IHFPR-tradition.
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Richard Rorty has suggested5 that one of the key tasks of philosophy is “the colligation of hitherto unrelated texts”; and if we accept this, then we are likely to see Brandom’s exercise in “bebop history” as an extraordinary philosophical achievement. (Those who reject this notion of philosophy, those who claim that there is either doing the ‘true’ history of philosophy, or contributing to it, tertium non datur, would probably think otherwise.6 However, I am convinced that rethinking the tradition does in fact constitute a great deal of what philosophers have always done.) And I think that Brandom’s conceptual constitution of the IHFPR-view as such is indeed a true achievement. Before Brandom, there was already, on the one hand, the obvious connection between inferentialism, understood as a theoretical view of language intimately related to logic, and holism; and also, on the other hand, the connection between inferentialism, understood more as a view of the praxis of using language, and pragmatism. Brandom’s achievement, as I see it, is the interconnection of the two perspectives, his showing that the two kinds of inferentialism can be seen as two aspects of a single kind. And I think this is what has enabled him to throw new (and controversial, I admit) light on the views of Hegel, Heidegger and others – for example on the fact that Hegel can be read both as an (arch-)idealist and a pragmatist. Moreover, in one of the four introductory chapters of the collection he performs a kind of ‘second-order bebop history’: he reinterprets his own papers as being integral parts of his inferentialist project, which some of them originally were not – at least not explicitly (some of them were written long before he managed to give the project clear and explicit shape). Hence Brandom carries out a kind of ‘self-application’; and moreover, in another introductory chapter he applies his own theory to his own practices in a different way: he tries to explicate his enterprise of “bebop history” in terms of his inferentialist theory of language. What we do all the time, he suggests, is oscillate between the de re interpretation of others’ claims (their interpretation from the viewpoint of how things ‘really are’ (for us), i.e. in the context of our beliefs) and de dicto interpretations (from the perspective of how their speakers think things are, i.e. in the context of their beliefs) – and the “bebop history” is nothing else than doing this when confronted with a tradition. In this way Brandom’s book appears to instantiate the pattern made popular by Douglas Hofstadter7 (and before him by the snake of the mystic texts biting its own tail); and characteristic of many pieces which are often classified as postmodern. However, Brandom’s book indicates that being postmodern in this sense does not mean being superficial. It is original and
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interesting, as well as duly provocative; though still more of a by-product of his “Making it explicit” saga than its substantial continuance.
NOTES 1 Harvard University Press, Cambridge, MA, 1994. 2 Harvard University Press, Cambridge, MA, 2000. 3 I have expressed my reservations over this classification of Brandom’s stance in my
review of his previous book (see Erkenntnis 55, 2001, 121–127). 4 Viz. e.g. M. Okrent’ Heidegger’s Pragmatism (Cornell, Ithaca, 1988). 5 See Rorty: Objectivity, Relativism and Truth (Philosophical Papers vol. I), Cambridge
University Press, Cambridge, 1991, p. 94. 6 It is clear that the lack of piety with which Brandom treats Hegel and others might not be hailed by those scholars who put premium on historical accuracy. 7 Gödel, Escher, Bach (Harvester, Hassocks, 1979). Academy of Sciences of the Czech Republic Czechia Manuscript submitted 6 July 2003 Final version received 6 July 2003
JAROSLAV PEREGRIN
LOGIC AS "MAKING IT EXPLICIT" Jaroslav Peregrin
Logical operators in the inferentialist perspective In considering the very possibility of deviant logic, we face the following question: what makes us see an operator of one logical system as a deviant version of an operator of another system? Why not see it simply as a different operator? Why do we see, say, intuitionist implication as an operator 'competing' with classical implication? Is it only because both happen to be called implications?1 It is clear that if we want to make cross-systemic comparisons, we need an 'Archimedean point' external to the systems compared. Some logicians and philosophers, including Quine (1986), come close to saying that no such Archimedean point is available, and hence that there can be no deviant logics, for if two operators are governed by different axioms, then they are simply two different operators (or, if you prefer, operators with different meanings). From this viewpoint, intuitionist implication is no less different from classical implication than, say, intuitionist or classical conjunction. This conclusion is indeed plausible if we consider logical calculi simply as algebraic structures2; however, things are different if we see them as a means of accounting for something that is already 'there' before we establish the structures and is to be explicated by them. From such a perspective, two operators of different systems may be variants of the same operator in virtue of the fact that they are both means of capturing the same pretheoretical item. What might these items be? Sometimes it seems that logicians tacitly assume that there are some mythical archetypes of implication, conjunction etc., located somewhere in some Platonic heaven or somehow underlying the a priori structures of our mind, which logic tries to capture (for better or worse). However, when it comes to the comparison of the concrete outcomes of our logical efforts, say the classical and the intuitionist implications with the archetypal Implication, the latter can never be materialized so distinctly as to be of any help. A more constructive proposal is that the operators are related to elements or constructions of our language. For example, both intuitionist and classical implication can be seen as means of reconstructing the natural language connective "if ... then ...". Clearly, the properties of classical implication far from totally coincide with those of "if ... then ..." (as far as this is possible to judge at all, because of the notorious vagueness of the latter). Hence should we say that classical implication is a modification of "if ... then ...", or rather that it is something different which replaces it? It is also possible, of course, to say that operators are devised to capture some preconceived semantic items, such as truth-functions or Boolean operators over sets of possible worlds. However, this would turn logic into pure mathematics, which it is surely not -- logic is 1
See Haack (1996) for a thorough discussion of these questions.
2
It seems that even so there might be a way of accommodating the intuition that two operators of two different systems are two variations on a single theme -- if we had a way to judge relative similarity of structures and consequent relative similarity of places (or roles) within the structures. Intuitively, this should be possible, but I know of no feasible development of this idea.
1
something essentially related to the ways we actually argue, justify and prove, and is to aid us with classifying the arguments, proofs etc. as right or wrong. Here I would like to consider a possibility inspired by Bob Brandom's (1994; 2000) notion of logic as a means of "making it explicit". According to Brandom, what underlies both human language and logic are inferences. Human language is structured in such a way that commitments to some claims bring about commitments to other claims -- i.e. that the former ones entail the latter ones and hence the latter are correctly inferable from the former. In fact, Brandom claims that there is a still deeper layer constituted by the concept of incompatibility. Brandom also claims that what we see as logical vocabulary is first and foremost a means of making explicit the proprieties implicit in our using language. Before we have "if ... then", the inference from "This is a dog" to "This is a mammal" can only be implicitly endorsed (or, as the case may be, violated), but once this connective is at hand, this inference can be explicitly expressed in the form of a claim, viz. "If this is a dog, then this is a mammal" (and with a more advanced logical vocabulary perhaps further transformed into "Every dog is a mammal"), and hence discussed w.r.t. its 'appropriateness' (and perhaps in the end rejected). Thus logical vocabulary helps us make the rules that are intractably implicit in our practices explicit and discussable3. Hence in this paper I would like to look at the usual logical operators as a means of explicitating inferential structure. I will claim that the need of such an explicitation leads to a set of 'inferentially native' operators, which, within various logical systems, can mutate into various forms.
Historical remarks It is instructive to see how logical constants were understood in some of the classics of modern logic and analytic philosophy. Thus, introducing his conditional sign, Frege (1879, p.5) writes Wenn A und B beurtheilbare Inhalte bedeuten, so gibt es folgende vier Möglichkeiten: 1) A wird bejaht und B wird bejaht; 2) A wird bejaht und B wird verneint; 3) A wird verneint und B wird bejaht; 4) A wird verneint und B wird verneint. ▐─┬─ A └─ B
3
Of course that as Brandom's proposal is not utterly unprecedential -- seeing connectives as means of articulating facts about inferability is to a certain extent common within some proof-theoretic approaches to logic, especially within the Gernan constructivistic tradition (Lorenzen and Schwemmer, 1975).
2
bedeutet nun das Urtheil, dass die dritte dieser Möglichkeiten nicht stattfinde, sondern eine der drei andern.4 This indicates that the sign Frege defines, and the natural language "if ... then" which it is clearly supposed to 'regiment', has a metalinguistic content: it expresses a fact about its content clauses being asserted and/or denied. The situation is similar in respect to quantifiers. Frege (ibid., p.19) writes In dem Ausdrucke eines Urtheils kann man die rechts von ▐── stehende Verbindung von Zeichen immer als Funktion eines der darin vorkommenden Zeichen ansehen. Setzt man an die Stelle dieses Argumentes einen deutschen Buchstaben, und giebt man dem Inhaltsstriche eine Höhlung, in der dieser selbe Buchstabe steht, wie in a─Φ(a) ▐─∪ so bedeutet dies das Urtheil, daß jene Function eine Thatsache sei, was man auch als ihr argument ansehen möge.5 In this way, a quantified sentence is not a 'regimentation' of a sentence of the object language, but rather again a shortcut for a metalinguistic statement: a statement about the results of replacing a part of an object language sentence by various (suitable) expressions. Hence to say that some object-language expressions, such as "something" or "everything", can be regimented by such quantifiers, is to say that these expressions are means of 'internalizing' certain metalinguistic pronouncements. Russell (1905, p. 480) takes, in this respect, a very close train of thought: ... everything and nothing and something (...) are to be interpreted as follows: C(everything) means 'C(x) is always true'; C(nothing) means '"C(x) is false'' is always true'; C(something) means 'It is false that "C(x) is false'' is always true.'
4
"If A and B stand for contents that can become judgments, there are the following four possibilities: (1) A is affirmed and B is affirmed; (2) A is affirmed and B is denied; (3) A is denied and B is affirmed; (4) A is denied and B is denied. Now ▐─┬─ A └─ B stands for the judgment that the third of these possibilities does not take place, but one of the three others does." 5
"In the expression for a judgement, the complex symbol to the right of ├─ may always be regarded as a function of one of the symbols that occur in it. Let us replace this argument with a Gothic letter, and insert a concavity in the content-stroke, and make this same Gothic letter stand over the concavity, e.g.:
a─Φ(a) ▐─∪ This signifies the judgement that the function is a fact whatever we take its argument to be." (Elsewhere I have discussed the consequences of this Fregean notion of quantification for the concept of variable -- see Peregrin, 2000.)
3
Hence again, sentences with "everything", "nothing" etc. are taken to express metalinguistic pronouncements; and their presence in a language thus enables us to say in the language what holds about the language and what is otherwise only expressible within a metalanguage.
'Inferentially native' operators Suppose we have an inferentially structured language, i.e. a set S of sentences with a relation ├─ which is a relation between finite sequences of elements of S and elements of S. Hence A1,...,An ├─ A. means that the statement A is (correctly) inferable from the sequence A1,...,An of statements. We will employ the letters A, A1, ..., An, B, C ..., to stand for statements, whereas X, Y, Z, ... to stand for finite sequences thereof. What would it take to make an inference, such as A ├─ B, explicit in the structure? We would need a statement which says that B is inferable from A. But what does it take for a statement of such a structure to say this? Presumably to be true iff B is inferable from A. But the relation ├─ is unchanging and hence the explicitating claim would be true necessarily; and what is the closest approximation to necessary truth within the structure is clearly inferability from nothing. Let us assume that we have a function, call it d (for deductor), mapping pairs of statements on statements in such a way that (*) A ├─ B iff ├─ d(A,B). In this way, to claim d(A,B) (as a necessary truth) would be to claim that B is inferable from A. Splitting (*) into its two halves and using "/" as a shorthand for "only if" we have DED: A ├─ B / ├─ d(A,B) CODED: ├─ d(A,B) / A ├─ B However, d allows us to say that a statement is inferable from another statement, but not yet that it is inferable from a sequence of statements. To achieve this, we would need either a way to amalgamate a sequence of statements into a single statement, or, alternatively, a deductor which would be applicable recursively. The former way amounts to a function a (for amalgamator) such that X,A,B,Y ├─ C iff X,a(A,B),Y ├─ C, or split into the two halves AMLG: X,A,B,Y ├─ C / X,a(A,B),Y ├─ C 4
DEAMLG: X,a(A,B),Y ├─ C / X,A,B,Y├─ C. The latter way amounts to strengthening the condition (*) to (**) X,A ├─ B iff X├─ d(A,B), i.e. DED*: X,A ├─ B / X├─ d(A,B) CODED*: X├─ d(A,B) / X,A ├─ B. Hence from the viewpoint of explicitating the relation ├─ we may think of two 'native operators', d and a. However, according to Brandom, the relation of inference rests on a more primitive concept, namely that of incompatibility: to say that A is correctly inferable from X is to say that whatever is incompatible with A is incompatible with X. Hence if we denote the fact that the sequence X is incompatible as ⊥X, we can write X ├─ A iff for every Y and Z it is the case that if ⊥Y,A,Z then ⊥Y,X,Z. From the other side, the best approximation of incompatibility achievable once we work with inference only is in terms of the "ex falso quodlibet" rule, hence ⊥X iff for every A, X ├─ A. Anyway, what we might also need is to explicitate the relation of incompatibility. Let us consider a unary function c (for complementator) such that ⊥A iff ├─ c(A). However, this again allows us merely to say that a statement is incompatible, whereas we would want to be able to say, more generally, that a sequence of statements is. This could be done if we were able not only to express that a statement is incompatible, ⊥A, but also that a statement is incompatible with a sequence X, i.e. ⊥ X,A. So what we may want is to generalize the stipulation that A is incompatible iff c(A) is inferable (from nothing) to the stipulation that A is incompatible with X iff c(A) is inferable from X. Hence we have ⊥ X,A iff X ├─ c(A), i.e.
5
COMP: ⊥ X,A / X ├─ c(A), DECOMP: X ├─ c(A) / ⊥ X,A. If we emulate incompatibility in terms of inference, we have COMP*: X,A├─ B for every B / X ├─ c(A), DECOMP*: X ├─ c(A) / X,A├─ B for every B. Hence it seems that we can think of at least three kinds of 'inferentially native' operators: deductors, amalgamators and complementators. Our goal now is to see to what extent the usual operators of the common logical calculi can be seen as their embodiments.
Theories and provability Logic reconstructs what our knowledge tends to result into as theories. A theory T (such as the Peano arithmetic, PA) can be seen simply as a set of sentences of a language. But it seems that if a theory is to be producible, learnable and manageable by us, finite human beings, then it must somehow be 'given by finite means'. The standard way is to see it as generated from a finite set A of axioms. (Thus, PA is seen as based on the axioms of first-order logic and the well-known extralogic axioms.) Hence we assume T = Cn(A), where Cn is a way of getting T from A. What, more concretely, is Cn? It is a function mapping sets of statements on the sets of their consequences (intensively studied especially by Tarski, 1930 and elsewhere). It is usually seen as induced by a relation between finite sequences (or sets) of statements and statements. Hence, to be more precise we should write T = CnI(A), where I is a relation between finite sequences of statements and statements and CnI(A) = {s | <X,s>∈I for some finite sequence X of statements from A}. Now the trouble with I, if construed in this way, is that it suffers from the same shortage as the original A: namely it is an infinite set (of ordered pairs). Hence we may try to repeat the same move as before, namely to construe it as generated from a finite collection of 'axioms'. An ordered pair consisting of a finite set of statements and a statement will be called an inference; and an inference relation will be a set of inferences. Thus, our I is an inference relation. Now we assume that I = CnI'(R),
6
where R is a finite set of 'inferential rules' (for PA, for example, it consists of modus ponens and generalization). However, at this point we face the question of the nature of our I' (and we may begin to worry whether we have not started an infinite regress). The standard solution would run as follows: I is the relation which holds between a sequence of statements X and a statement S just in case we can prove S from X using the inferential rules from R. This is to say that I holds between X and A iff there is a finite sequence of statements ending with A and such that each of its elements is either an element of X or there is a rule from R which allows us to derive the statement from statements occurring earlier in the sequence. And hence CnI' is the functor which takes us from a given R to the relation of 'provability by means of R'. Can we make the nature of I' explicit? Yes, indeed. It is easy to see that a relation ├─ of provability by means of a given collection of rules satisfies the Gentzenian structural rules: (REF) A├─ A (EXT) X,Y├─ B / X,A,Y├─ B (CON) X,A,A,Y├─ B / X,A,Y├─ B (PERM) X,A,B,Y├─ C / X,B,A,Y├─ C (CUT) X├─ A; Y,A,Z├─ B / Y,X,Z├─ B More importantly, it is precisely the closure of R w.r.t. these rules that amounts to the relation of 'provability by means of R'6. For let A be provable from X by means of the rules of R. Then there is a sequence A1,...,An of statements such that An = A and every Ai is either an element of X or is inferable by a rule of R from statements which are among A1,...,Ai-1. If n=1, then there are two possibilities: either A∈X and then X├─ A in force of REF and EXT; or A is a consequent of a rule of R with a void antecedent, and then ├─ A and hence X├─ A in force of EXT. If n>1 and An is inferable from some Ai1,...,Aim (where i1
Inferential structures To be able to address the explicitation of inference more rigorously, we need to establish a more rigorous framework. Hence the following definitions: Let S be a set ('of statements'). An inference over S is an ordered pair whose first constituent is a finite sequence of elements of S and whose second is an element of S. A metainference over S is an ordered pair whose first constituent is a finite sequence of inferences over S and whose second is an inference over S. If P is a set ('of parameters'), then a (P-)inferential pattern over S is an inference over S in which some elements of S are replaced by elements of P. An instance of an inferential pattern over S is any inference over S 6
Note that if we start investigating substructural logics, thereby denying that I' is bound to be of this kind, we are on the verge of the infinite regress hinted at above. If I' be arbitrary, then we should at least require that it be 'finitely based'. i.e. that I' = CnI′'(R') for some I'' and R'. The natural thing would be to take I'' as constituted by the Gentzenian rules, but if we rejected this for I', it seems we might as well reject it for I'' ...
7
which can be obtained from the pattern by a systematic replacement of the elements of P by elements of S. A (P-)metainferential pattern over S is a metainference over S with some elements of S in its constituents replaced by those of P. An instance of a metainferential pattern over S is any metainference over S which can be obtained from the pattern by a systematic replacement of the elements of P by elements of S. An inferential structure is the ordered triple F = <S,I,M>, where S is a set, I is a finite set of inferential patterns over S, and M is a finite set of metainferential patterns over S. The notion of inference valid in F is defined in the following obvious recursive way: (i) an instance of an inferential pattern from I is valid; (ii) if <,I> is an instance of a metainferential pattern from M and all of I1,..., In are valid, then so is I; (iii) nothing else is an inference valid in F. If <X,A> is an inference valid in F, then we shall write X ├─F A. Hence in terms of the notation of the previous section, ├─F ≡Def. CnM(I). We will leave out the subscript accompanying ├─ where no confusion would be likely to arise. We will say that A is a theorem of F if <<>,A>, where <> is the empty sequence, is an inference valid in F. We will say that an inferential structure is standard if I contains the pattern (REF) A ├─ A and M consists of the metapatterns (EXT) X,Y ├─ A / X,B,Y ├─ A (CON) X,A,A,Y ├─ B / X,A,Y ├─ B (PERM) X,A,B,Y ├─ C / X,B,A,Y ├─ C (CUT) X,A,Y ├─ B; Z├─ A / X,Z,Y ├─ B We will say that it is substandard if M consists of a subset of these patterns. As we saw in the previous section, standard structures are important in that they explicate the intuitive notion of 'provability by means of a collection of rules'. Now an addition of an operator to a standard inferential structure is to consist in the addition to the structure of some inferential patterns characteristic of the operator. Hence our next task is to try to turn the metainferential patterns we used to characterize our native operators into inferential ones.
Deductor Let us start with the deductor. It is easily seen that CODED* is equivalent to the modus ponens inferential pattern
8
MP: d(A,B),A ├─ B. The proof is as follows (we underline the structural rules employed within the proofs to emphasize on which of them the individual results depend): CODED* ⇒ MP 1. d(A,B)├─ d(A,B) 2. d(A,B),A ├─ B
REF from 1 by CODED*
MP ⇒ CODED* 1. X├─ d(A,B) 2. d(A,B),A ├─ B 3. X,A ├─ B
assumption MP from 1 and 2 by CUT
However, things are not that easy with DED*: there does not appear to be an inferential pattern which would be equivalent to it. But there is an indirect way of securing its validity by establishing inferential patterns: The point is that a way of making DED* valid is to stipulate its validity for the instances of I and to secure that the validity is preserved by all the instances of M. If X is A1,...,An, then let us write d(X,B) as a shorthand for d(A1, d(A2, ..., d(An,B))); if X is the concatenation of the sequences Y and Z, then d(Y,Z,B) will mean d(X,B); and let us interpret d(X,B) as B if X is the empty sequence. Then what is necessary and sufficient for REF, EXT, CON, PERM and CUT to preserve the validity of DED* is obviously the following: (1)├─ d(A,A) (2)├─ d(X,Y,B) / ├─ d(X,A,Y,B) (3)├─ d(X,A,A,Y,B) / ├─ d(X,A,Y,B) (4)├─ d(X,A,B,Y,C) / ├─ d(X,B,A,Y,C) (5)├─ d(X,A); ├─ d(Y,A,Z,B) / ├─ d(Y,X,Z,B) Given (4), we can reduce (2), (3) and (5) to the simpler (2'), (3') and (5'): (2') if ├─ B, then ├─ d(A,B) (3') if ├─ d(A,A,B), then ├─ d(A,B) (5') if ├─ d(X,A) and ├─ d(A,B), then ├─ d(X,B) However, (2'), (3'), (5') as well as (4) are still metapatterns; so we are as yet no better than with DED* itself. Nevertheless in contrast to DED*, the validity of these metapatterns can be obviously established by stipulating the patterns (2'') B├─ d(A,B) (3'') d(A,A,B)├─ d(A,B) (4') d(X,A,B,Y,C)├─ d(X,B,A,Y,C) (5'') d(X,A), d(A,B)├─ d(X,B) and hence by 9
(2*) ├─ d(B,d(A,B)) (3*) ├─ d(d(A,A,B),d(A,B)) (4*) ├─ d(d(X,A,B,Y,C),d(X,B,A,Y,C)) (5*) ├─ d(d(X,A),d(d(A,B),d(X,B))) Now it can easily be shown that (4*) can be reduced to (4**), whereas (5*) to (5**): (4**) ├─ d(d(A,B,C),d(B,A,C)) (5**) ├─ d(d(C,A),d(d(A,B),d(C,B))) and that (1) and (4**) then follow from (2*), (3*) and (5**). In this way we reach the axiomatization of what Hilbert and Bernays (1939, Supplement III) called positive logic and which constitutes the implicative part of the intuitionist propositional calculus. It is this system which thus appears to constitute the 'native standard logic of implication'. (The purely implicative axioms of the classical propositional calculus are the same, but as the addition of the classical axiom(s) for negation are not conservative over them, the class of purely implicative theorems of the classical propositional calculus is more inclusive.) Now assume that we are interested not in standard, but in some substandard kinds of systems. Suppose we suspend one of the structural rules, say CON. As the result, we will have only ├─ d(A,A) ├─ d(B,d(A,B)) ├─ d(d(C,A),d(d(A,B),d(C,B))) In this way we reach the implication of what is sometimes called BCK logic7, and also what constitutes the purely implicative part of Wajsberg's (1931) axiomatization of Łukasiewicz's (1930) three-valued calculus. Suppose we leave only REF and CUT and suspend all the other structural rules. Then MP still replaces CODED*; and we only have to strike out those of (1)-(5) which correspond to the suspended rules. If we suspend all save REF and CUT, we have ├─ d(A,A) d(X,A), d(Y,A,Z,B),├─ d(Y,X,Z,B)
Amalgamator Now let us consider the amalgamator -- it is not difficult to see that within a standard inferential structure, AMLG becomes equivalent to ICN whereas DEAMLG to ECN1 plus ECN2: 7
See, e.g., Došen (1993).
10
ICN: A,B ├─ a(A,B) ECN1: a(A,B) ├─ A ECN2: a(A,B) ├─ B This can be proved as follows: DEAMLG ⇒ ICN 1. a(A,B)├─ a(A,B) 2. A,B├─ a(A,B)
REF from 1 by DEAMLG
ICN ⇒ DEAMLG 1. X,a(A,B),Y ├─ C 2. A,B├─ a(A,B) 3. X,A,B,Y ├─ C
assumption ICN from 2 and 1 by CUT
ECN ⇒ AMLG 1. X,A,B,Y ├─ C 2. a(A,B)├─ A 3. a(A,B)├─ B 4. X,a(A,B),a(A,B),Y ├─ C 5. X,a(A,B),Y ├─ C
assumption ECN1 ECN2 from 1, 2 and 3 by CUT from 4 by CON
AMLG ⇒ ECN1 1. A├─ A REF 2. A,B├─ A from 1 by EXT 3. a(A,B)├─ A from 2 by AMLG (the proof of AMLG ⇒ ECN2 is analogous) Now if we have also the deductor, we have a way of making all inferential patterns, with the exception of modus ponens, explicit, i.e. of reducing them to axioms. (And, in fact, we must do so in order to keep DED* valid.) As for the ones constitutive of the amalgamator, i.e. ICN, ECN1 and ECN2, they are transformed into ├─ d(A,d(B,a(A,B))) ├─ d(a(A,B),A) ├─ d(a(A,B),B) What about nonstandard structures? If we suspend any structural rule other than PERM, the equivalence between AMLG+DEAMLG and ICN+ECN1+ECN2 will fail. However, as long as REF and CUT hold, the concept of amalgamator will coincide with what has come to be called fussion within the theory of substructural logics (Restall, 2000): IF: X├─ A; Y├─ B / X,Y├─ a(A,B) EF: X├─ a(A,B); Y,A,B,Z ├─ C / Y,X,Z├─ C
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The proofs of the equivalence to the definition of amalgamator are as follows: DEAMLG ⇒ IF 1. a(A,B)├─ a(A,B) 2 A,B├─ a(A,B) 3. X├─ A 4. Y├─B 5. X,Y├─ a(A,B)
REF from 1 by DEAMLG assumption assumption from 3, 4, and 1 by CUT
AMLG ⇒ EF 1. X├─ a(A,B) 2. Y,A,B,Z ├─ C 3. Y,a(A,B),Z ├─ C 4. Y,X,Z ├─ C
assumption assumption from 2 by AMLG from 1 and 3 by CUT
EF ⇒ AMLG 1. X,A,B,Y ├─ C 2. a(A,B)├─ a(A,B) 3. X,a(A,B),Y ├─ C
assumption REF from 2 and 1 by EF
IF ⇒ DEAMLG 1. X,a(A,B),Y ├─ C 2. A├─ A 3. B├─ B 4. A,B ├─ a(A,B) 5. X,A,B,Y ├─ C
assumption REF REF from 2 and 3 by IF from 4 and 1 by CUT
Complementator The situation is more complicated with the complementator, whose definition given by COMP* and DECOMP* cannot be turned into axioms so straightforwardly. However, we can reduce DECOMP* to (DECOMP**) c(A), A ├─ B: The proof is as follows: DECOMP** ⇒ DECOMP* 1. X ├─ c(A) 2. X,A├─ B
assumption from 1 and DECOMP** by CUT
DECOMP* ⇒ DECOMP** 1. c(A) ├─ c(A) 2. c(A), A ├─ B
REF from 1 by DECOMP*
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Moreover, in the presence of a deductor, we can reduce COMP* to: (COMP**) d(A,B), d(A,c(B)) ├─ c(A) This can be proven in the following way: COMP** ⇒ COMP* 1. X,A├─ B 2. X├─ d(A,B) 3. X,A├─ c(B) 4. X├─ d(A,c(B)) 5. X,X ├─ c(A) 4. X ├─ c(A)
from the assumption from 1 by DED* from the assumption from 3 by DED* from 1, 3 and COMP** by CUT from 5 by PERM and CON
COMP* ⇒ COMP** 1. d(A,B),A ├─ B 2. d(A,c(B)),A ├─ c(B) 3. d(A,c(B)),A, d(A,B),A ├─ C 4. d(A,B), d(A,c(B)),A ├─ C 5. d(A,B), d(A,c(B)) ├─ c(A)
MP MP from 1, 2 and DECOMP** by CUT from 3 by PERM and CON form 4 by COMP*
Now with the help of the deductor, we recast DECOMP** and COMP** as ├─ d(c(A),d(A,B)) ├─ d(d(A,B),d(d(A,c(B)),c(A)))
Conclusion Let us summarize the inferences we have amassed in trying to make the (standard) inference relation explicit: A,d(A,B))├─ B ├─ d(B,d(A,B)) ├─ d(d(A,A,B),d(A,B)) ├─ d(d(C,A),d(d(A,B),d(C,B))) ├─ d(A,d(B,a(A,B))) ├─ d(a(A,B),A) ├─ d(a(A,B),B) ├─ d(c(A),d(A,B)) ├─ d(d(A,B),d(d(A,c(B)),c(A))) It is easy to see that they make up an axiomatization of the intuitionist propositional calculus. This indicates that w.r.t. standard inferential structures, it is intuitionist logic which is the logic of inference. 13
What about classical logic? As is well known, it can be derived from intuitionist logic by the addition of an axiom which, in our notation, reads ├─ d(c(c(A)),A) Is there an inferentialist substantiation of this axiom? I do not see one (which, of course, does not mean that none exists8). However, this axiom appears to provide for an 'alignment' of the intuitionist calculus, which leads to a considerable simplification. What about other logical systems? We have seen that for them, deductors, amalgamators and complementators may be embodied in operators of different kinds than are those of classical ones. (From our viewpoint, if the deduction theorem for a calculus fails, then the upshot is that the implication of the calculus is not its appropriate deductor -- for deduction is a constitutive feature of the deductor.)
References Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Došen, K. (1993): 'A historical introduction to substructural logics', in K. Došen and P. SchröderHeister, eds.: Substructural Logics, Clarendon Press, Oxford. Haack, S. (1996): Deviant Logic, Fuzzy Logic, University of Chicago Press, Chicago. Hilbert, D. and P. Bernays (1939): Grundlagen der Mathematik, Springer, Berlin. Lorenzen, P. & Schwemmer, O. (1975): Konstruktive Logik, Ethik und Wissenschaftstheorie, Bibliographisches Institut, Mannheim. Łukasiewicz, J. (1930): ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’, Compt. Rend. Soc. d. Sc. Let. Varsovie, Cl. III 23, No. 51-77. Milne, P. (2002): 'Harmony, Purity, Simplicity and a "Seemingly Magical Fact"', Monist 85, 498-534. Peregrin, J. (2000): 'Variables in Natural Language: Where Do They Come From?', in Variable-free Semantics (eds. ed. M. Böttner & W. Thümmel), Secolo, Osnabrück, 46-65. Prawitz, D. (1965): Natural Deduction, Almqvist & Wiksell, Stockholm. Quine, W.V.O. (1986): Philosophy of Logic, Harvard University Press, Cambridge (Mass.). Restall, G. (2000): Introduction to Substructural Logics, Routledge, London. Russell, B. (1905): 'On denoting', Mind 14, 479-493. Tarski, A. (1930): 'Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I', Monatshefte für Mathematik und Physik 37; English translation in Tarski (1956), 60-109. Tarski, A. (1956): Logic, Semantics, Metamathematics, Clarendon Press, Oxford. Wajsberg, M. (1931): 'Aksjomatyzacja trójwartościowego rachunku zdan', Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie, cl.iii 24, No. 126-145.
Department of Logic, Institute of Philosophy Academy of Sciences of the Czech Republic Jilská 1, 110 00 Praha 1, Czech Republic [email protected] www.cuni.cz/~peregrin 8
See, example the discussion given by Milne (2002).
14
Pragmatist in a Babylonian library Vladimír Svoboda & Jaroslav Peregrin
The naturalization of Philosophy In the middle of the twnetieth century, the centre of gravity of analytic philosophy gradually shifted from Europe to the USA. The reason was partly the emigration of some of the crucial figures to the nazi-free world (Carnap, Hempel, Tarski, Gödel etc.), and partly the incipient flourishing of the USA’s own variety of analytic philosophy, due especially to Willard Van Orman Quine1. If we were to characterize the specificity of this new version of the analytic doctrine, then I think two terms would offer themselves: naturalism and pragmatism. Quine’s variety of naturalism stemmed, first and foremost, from the disrespect to any knowledge which could not be made continuous with natural science, and as the conviction that „knowledge, mind and meaning are part of the same world that they have to do with and that they are to be studied in the same empirical spirit that animates natural sciences“ (Quine, 1969, p. 26)2. His pragmatism comes to the open most flagrantly in the context of the way he and his followers treat the concept of meaning. Quine wholly eschews the Carnapian notion of meaning as designatum (Carnap, 1942, pp. 8-10) in favor of the Deweyian notion of meaning as „a property of behavior” (Dewey, 1925, p. 179). The reason was that Quine duly noticed that meaning as conceived by the classics of analytic philosophy tends to became the Trojan horse by means of which philosophy is recaptured by the speculative metaphysics which analytic philosophy was originally devised to overcome. Thus, the logical empirists’ idea that the pursuit of truth is to be preceded by a more foundational enterprise, the pursuit of meaning, carried out by philosophers, appeared to him to lead us astray. Any theoretical pursuit, he has urged, be it carried out by a scientist or a philosopher, is bound to be part of the pursuit of truth, it is an effort of finding how some things really are.
1
From our perspective, Wilfrid Sellars was, of course, another pilot figure; but his direct influcence on American philosophy was much less significant than Quine’s.
2
Naturalism was not unprecedented within the analytic tradition: it was especially Russell, in certain stage of his development, who also entertained such a stance. However, Quine was much more consequential in drawing the consequences from this attitude and reached his kind of antifoundationalism, which was unprecedented.
1
From this viewpoint, the study of meanings cannot be but the study of a part or an aspect of the world, namely of a certain kind of human behaviour. If somebody needs such entities as meanings to articulate this theory, so be it; but we must keep in mind that what the theory is ultimately about are concrete, ‘tangible’ episodes of human behaviour. And in view of the fact that the engagement of meanings tends to deceive us into the feeling that it is them which is the ultimate subject matter of language theory, we would do better, Quine advices, if we forget about them altogether. The naturalist and pragmatist ‘pull towards tangibility’, underlying Quine’s disrespect for meanings-as-entities is, we think, largely warranted, and can lead us to a vantage point from which we can see language in a way immunizing us against many traditional philosophical (pseudo)problems. However, and this is what we would like to bring to light in the present paper, to think that we can simply banish ‘esoteric’ meanings in favor of ‘tangible’ expressions, thereby bringing semantics from a Platonic heaven down to the earth of empirical science, is, in our view, to adopt a dangerous simplification.
Sentences vs. Propositions Let us concentrate on propositions, meanings of sentences. A traditional argument says they are indispensable: if there were no propositions, what would intertrasnslatable sentences share, and what would people adopting propositional attitudes relate themself to? Quine’s answer is that speaking about ‘what is common to synonymous sentences’ or ‘what is believed’ is a kind of figurative speech which refers to nothing but eventually the behaviour. However, even Quine accepts that it is often necessary to treat of such things as propositional attitudes - only he counters that we need not take them as relating people to propositions, that we can well use ‘eternal’ sentences instead of propositions. He says (1992, p. 77-78):
Many sentences in the same or different languages are deemed to be alike in meaning, and distinctions among them are indifferent to truth; so one narrowed the field by ascribing truth rather to the meanings. This motive would be excellent if the notion of sentence meaning were not so elusive. But as matter stands, we fare better by treating directly of sentences. These we can get our teeth into. 2
What we would like to point out in this paper is that even if we disregard several rather obvious awkward consequences of this recasting of propositional attitude reports3, it would still be hardly possible to take the picture of ‘getting our teeth into sentences’ quite at face value. Even if we leave aside the technical questions of such a replacement of propositions by eternal sentences, we can understand the replacement as a move from something despicably ‘esoteric’ to something laudably ‘tangible’ only with many nontrivial provisos. Why so? First, let us point out the clear fact that sentences, unlike individual utterances, are no less abstract than propositions. Saying that someone uttered a sentence is saying that she emitted a sound which we classify as an instantiation of the sentence - just like saying that someone expressed a proposition is saying that he emitted a sound which we classify as an expression of the proposition. An objection might be that what makes the crucial difference is that the abstractness of sentences - in contrast to that of propositions - is quite unproblematic, for they are products of a ’natural’ abstraction, the capacity of which is somehow inborn to us. We simply recognize, constituted as we are, two sounds as ‘the same’ in the sense of being the same sentence whereas to recognize two sounds as ‘the same’ in the sense of having the same meaning takes complicated learning. This argument is, of course, partly sound - it is clear that sentences are in this sense much more ‘well-behaved’ abstract objects than propositions. However, the oposition is not categorical - in fact it is not true that to recognize two sounds as instantiating the same sentence is a ‘natural’ ability akin to the ability of recognizing two sounds as being of the same pitch. It is well known that what counts as cases of the same sentence in one language may well count as cases of different sentences in another - so the recognizing of this sameness must at least partly be acquired along with learning language. Quine is, of course, aware of the fact that a sentence is not a concrete thing, but rather „a universal: a repeatable sound pattern, or repeatedly approximable norm“ (1960, p. 191). On the other hand part of the attractiveness of his program (and of the program of those who follow him) appears to be induced by the fact that it really replaces ‘abstracta’ by ‘concreta’. Thus, if he says such things as „it seems perverse to bypass the visible or audible sentences 3
We are probably not the only ones who find it difficult to agree with Quine that it is plausible to portray a cat waiting in front of a hole as related to the sentence „There is a mouse in there“ (see Quine, 1992, p. 68). After all, does the fact that people can be represented by their passport numbers justify me in saying that I am in love with the number 8897939?
3
and to center upon sentence meanings as truth vehicles“ (Quine, 1992, p. 77), it is hard not to imagine the contrast between sentences and meanings as that between the concrete and tangible and the abstract and esoteric.
A Babylon in the Jungle? However, let us waive this; let us grant Quine that the way in which sentences are abstract is so trivial that it is not worth dwelling on. We want to point out that there are more serisous problems with his stance. To bring these to light, we will follow Quine’s perpetual companion, the field linguist carrying out his radical translation. Imagine that this time the lone wolf turns up in an unexplored land where he finds a native community that is civilized in a rather surprising way. The natives are obsessed with their language and have spent much energy on an attempt to create a special kind of library that we will call (inspired by Borges’ famous story) a Babylonian library. Their aim is to conserve their language by writing down in the volumes stored in the library all the (reasonable, in the sense of not excessively long) well formed meaningful sentences of their language (say in a lexicographical ordering). The natives believe that some of the sentences in the library can be ascribed truth-values independently of the state of the world. They call such sentences eternal sentences and write them in a special color; and moreover, they underscore those they take to be (eternally) true. The story is, of course, somewhat bizarre: but we tell it because it focuses attention on the fact that Quine’s exposition of language often seems to presuppose that for any given language such a library – a library with the class of all the (reasonable) sentences open to view. Thus Quine (1986, p. 16):
The grammarian’s question is, then, what strings of phonemes belong to the language? What strings, that is, ever get uttered or could get uttered in the community as normal speech? The grammarian’s job is to demarcate, formally, the class of all such strings of phonemes. ... His job is to devise his lexicon and his constructions in such a way as to demarcate the desired class: the class of all the strings of phonemes that could be uttered in normal speech.
4
Moreover, Quine appears to assume that not only the class of all well-formed sentences is demarcated, but also that it is as if annotated in a way which makes it possible to distinguish between the eternal sentences and the rest. Moreover, what he appears to be implicitly suggesting seems be that the field linguist can reconstruct the library from the evidence available to him. Hence it seems as if the data available to the field linguist, and consequently to the philosopher of language, were to a large extent tantamount to those available to the visitor of the native Babylonian library of the kind envisaged above.
Truth values Is the task of reconstructing the library within the powers of even the most gifted linguist? Clearly, the initial problem is showing how the whole of sentences of the library can be constructed out of the relatively meager set of actual utterances the linguist is able to genuinely witness. How does he come to be able to tell a well-formed native sentence from an ill-formed string? It is clear that the linguist has a chance of hearing only a tiny fragment of all the wellformed sentences of the native language. Of course that Quine is right in that this is simply how we generally acquire our knowledge of anything in the world: that everybody of us can tell (in principle) a car from a non-car, a chair from a non-chair and a horse from a non-horse - despite the fact that he could have seen at most a finite number of instances of each. However, to use induction to extract the general concept of „a well-formed junglese sentence“ from observed instances appears to be a task incomparably more difficult than to extract the general concept of „a horse“ from perceived horses. Moreover, as Chomsky has argued4, empirical evidence shows us that learning to talk one’s mother tongue is much more like learning to walk than like learning the properties of animals by observing them; and that hence induction is an altogether bad way to think about it. This is again not to say that it is not reasonable to expect that the translator will indeed eventually come to be able to tell a sentence from a non-sentence (be the mechanism behind learning as it may); it is only to point out that to see the boundary, and consequently sentences as such, as something into which we can “get our teeths” is far from staightforward.
4
See Chomsky(1975)
5
But let us waive even this, to be able to concentrate on problems we consider crucial. Even if we take the task of reconstructing the class of well-formed native sentences as principally unproblematic, the translator‘s task of delimiting eternal sentences is barely at the beginning. To be able to accomplish it sucessfully, he would have to draw a clear boundary between sentences which do have truth values and those which do not; and a boundary between those which are capable of changing its truth value and those which are not. That is to say he would have to make a clear sense of the concept of „a sentence having a truth value“ and that of „a sentence (not) changing its truth value“. Let us start with the former: which sentences can be supposed to have truth values and which not? It is commonly agreed that there are many kinds of grammatical sentences which cannot be reasonable taken to have a truth value, and many which cannnot be taken to have it save within a specific kind of context. [dále rozvést] So it seems that we have to restrict ourselves to sentences which do not contain indexicals (but only unambiguous names), which contain only explicit references to dates and times and are free of any vagueness and ambigiuty. However, are there such sentences in a natural language at all? Are there any absolutely unambiguous names at all? If I say ‘I admire Bill Gates’, the name might seem to be unambiguous; but clearly this is only unless I have a dog of the same name. If I write down the date 7.11.1918, then this might seem to be as unambiguous as something can be; but what if I talk about what happened that day in Russia do I mean their November, or our November (which was their October)?5 Worse still, if we are to agree with Quine that truth and falsity is primarily not ascribable to individual sentences, but rather only to theories (to clusters of sentences having, as he puts it, greater than critical ‘semantic mass’), does an absolute division of sentences into those which do and those which do not have truth values make a real sense at all? If it is never possible to falsify a single sentence, is it possible to find out that a single sentence is truthvalue-less?
5
This is to say: is not some kind of context dependence a pervasive property of the entire natural language rather than being only the property of a specific part of its vocabulary? Cf. Travis (1985).
6
Changing truth value However, the worst is still to come. Even if we granted the translator both the ability to delimitat the class of well-formed sentences and to tell those which do have a truth value from those which do not have any, it is hard to grant him that no problems are posed by drawing the boundary between sentences having a truth value fixed once for all, and those which can change the truth value depending on circumstances. What is the basis for drawing this boundary? How could a radical translator tell the former from the latter? It seems that what the translator can, relatively easily, do, is to distinguish sentences observed to change the truth value from those which do not do so. So he is likely to find out that the native equivalents of „It rains“, „There is a rabbit running over there“ or „I am hungry“ are not eternal - for he will simply see the natives changing their reactions to them from assent to dissent, and vice versa, frequently. However, how can he distinguish between sentences which do not change change truth value only during the time he observes the community (be it simply by chance, or for some deep reason) and those which change it never? Consider the native equivalent of the sentence „There is a stranger studying us“ - it is clear that it will not change its truth value before the field linguist‘s eyes - just like that of „1+1=2“. How would he be able to tell the former from the latter? It seems that while he would be easily able to sort sentences into quasistanding (= not observed to change truth value) and non-standing (= observed to do so), it would be much harder for him to sort the quasistanding ones into the standing proper and the rest. (It also seems that all sentences not actually heard by the translator, or even those heard only once, would have to fall into the category of quasistanding sentences. Or would there be a way to somehow project the nonstandingness of the heard sentences to some of the unheard ones via a kind of „analogic synthesis“?) It is crucially important to realize that what is in play here is more than underdetermination of theory by data. For it is hard to accept that within the Quinean framework, the concept of eternal sentence can be given any real content at all. What would it mean for a sentence to have a fixed truth value? Let us realize that according to Quine’s official holistic doctrine, no sentence at all is immune from revision - no sentence can be ever guaranteed the right to keep its truth value forever. So it seems that either the concept of an eternal sentence is empty, or Quine’s holism is undermined. 7
In fact, it is an absolute triviality that any sentence can change its truth value - it is enough, as those who have not dispensed with the concept of meaning, would put it, to alter the meanings of the words of which it consists. If we reinterpret ‘+’ as subtraction, then clearly „1+1=2“ will cease to be true; and if we make „bachelor“ refer to butchers, „No bachelor is married“ will clearly become false. In view of this fact, an eternal sentence should be defined not as one not capable of changing its truth value, but rather as one not capable of changing its truth value without changing its meaning. However, needless to say that such a definition is not acceptable for a Quinean - for it would make him come to rest on the concept of meaning which his approach is devised to exorcise. However, there is hardly any other way to make sense of the concept of a sentence not capable of changing its truth value: if we understand sentences as mere strings of letters or phonemes, then the concept will be trivially empty, whereas if we understand them as a strings-cum-meanings, then it will possibly make sense, but surely will cease to be capable of accomplishing the job for which it is engaged.
Down-to-earthness vs. foundationalism The Quinean ‘flight from intensions’ (see Quine 1960, Chapter VI) is an understandable attempt to be as ‘down-to-earth’ as possible, to avoid the ‘esoteric’ in favour of the ‘tangible’. It is a healthy reaction to the new kind of metaphysics, growing from the Carnapian formal semantics (originally devised to overcome sterile metaphysics). However, it would be an illusion to assume that it can lead us to eliminate everything ‘abstract’ in favour of things into which we can ‘get our teeth’. And in general one who is pursuing down-to-earthness in the pragmatistic spirit must beware sliping into the foundationalist pursuit reducing all knowledge to ‘acquaintance‘ or ‘givenness’. Compare Quine’s tendency to eliminate the abstract in favour of the concrete with David Hilbert’s attempt to reconstruct mathematics as grounded in basic truths concerning ‘middlesized dry goods’:
It is more or less a precondition of the applicability of logical inference and of the employment of logical operations is more or less something given already in perception: certain extralogical concrete objects, which are here as immediate experience before any 8
thinking. If logical inference is to be certain, the objects must be entirely and in all their parts comprehensible ... (Hilbert, 1925, p. 171)
Quine tends to assume that a theory of language should be analogously grounded in basic truths characteristic of human conduct. This tendency is sound, but it becomes misleading if it is construed as saying that the truths are not to overreach the actually witnessed, the ‘known by acquaintance’ – for any reasonable theory of a language must go beyond the factual to the counterfactual, in many dimensions. It has to treat of not merely strings actually uttered by the natives, but also about those correctly utterable; it has to treat not only of sentences which do really elicit assent or dissent, but also of those which would elicit assent or dissent in a hypothetical situation (or even in any conceivable situation) and for which would be correct to assent or dissent to. Talking in terms of rules: the natives cannot be portrayed as simply displaying regularities in their conduct - they have to be understood as following rules6. Only by acknowledging a rule can the field linguist conclude that something is to be the case in an counterfactual situation, or that something cannot fail to be the case. (This is why the concept of rule became so crucial for the late Wittgenstein7.) Only if he is able to discover the rules of the native language games is he able to make sense of the distinction between „1+1=2“ and „There is a stranger studying us“ - it consists in the fact that there is a rule never give up „1+1=2“, but there is no such rule for „There is a stranger studying us“. And also to be able to make a real sense of the notion of a sentence not changing its truth value, eternal sentence in Quine’s sense, he would have to mobilize resuources going beyond pure physicalistic descriptions of what he sees.
Conclusion The conclusion, then, appears to be that the problems the field linguist faces when he strives to reconstruct the native Babylonian library are fatal. However, if this is the case, does it make sense to assume that the natives could construct one in the first place? Hardly: if there 6
Cf. Brandom, 1994, §1.III.
7
A penetrating discussion of this aspect of Wittgenstein‘s philosophy is given by Baker and Hacker (1984).
9
is no concept of eternal sentence available to the radical interpreter, then there is no such concept altogether - for as Quine reminds us, all of us are speakers of our language solely in virtue of having been the radical interpreters of our elders. Hence, for a Quinean naturalist and pragmatist, to take language as something that can be grasped in the form of the Babylonian library of the kind envisaged above is equally mythic as Platonic heavens of meanings. And to freely employ concepts and boundaries available only to a visitor of such a library is to rob the linguistic pragmatism of its most basic point.
References
Baker, G. P. & Hacker, P. M. S. (1984): Scepticism, Rules and Language, Blackwell, Oxford. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Carnap, R. (1942): Introduction to Semantics, Harvard University Press, Cambridge (Mass.). Chomsky, N. (1975): ‘Quine's Empirical Assumptions’, in Words and Objections (ed. D. Davidson & J. Hintikka), Reidel, Dordrecht. Dewey, J. (1925): Experience and Nature, Open Court, La Salle (Ill.). Heijenoort, J. van, ed. (1971): From Frege to Gödel: A Source Book from Mathematical Logic, Harvard University Press, Cambridge (Mass.). Hilbert, D. (1925): ‘Über das Unendliche’, Mathematische Annalen 95, 161-190; English translation 'On the Infinite' in Heijenoort (1971) pp. 367-392. Peregrin, J. (1999): ‘Pragmatization of Semantics’, in: Semantics/Pragmatics Interface from Different Points of View (ed. by K. Turner), Elsevier, Amsterdam. Quine, W.V.O. (1969): Ontological Relativity and Other Essays, Columbia University Press, New York. Quine, W.V.O. (1986): Philosophy of Logic, Harvard University Press, Cambridge (Mass.). Travis, C. (1985): ‘On What Is Strictly Speaking True’, Canadian Journal of Philosophy 15, 187-229.
10
Brandom and Davidson: what do we need to account for thinking and agency? Jaroslav Peregrin, Academy of Sciences of the Czech Republic, Prague http://www.cuni.cz/~peregrin*
Mind-having and the 'ubs-erasure' Minds seem to be things peculiarly different from the middle-sized dry goods which surround us, and having a mind appears to be peculiarly different from having a beard or a Japanese car. This difference was famously accounted for by Descartes: beards and Japanese cars are instances of res extensa, things existing in spacetime where they interact with each other, whereas minds are instances of res cogitans - existing somewhere outside of spacetime but nevertheless able to somehow 'aim at' the spacetime things. Nowadays, philosophers are no longer willing to subscribe to this kind of dualism; but it would be hard to deny that mind-having is something else than beard-having. Some of them (famously Ryle) have argued that the talk about “minds” and “having minds” is only metaphoric; but even so there remains what appears to be a qualitative difference between entities with minds (be it in whatever metaphoric sense) and entities without them. There are philosophers who see the difference only in the complexity of the former entities; but there are also those who are convinced that the prima facie qualitative difference cannot be so reduced to a quantitative one and that it must be explained in some 'dualistic' mode. (After all, as Popper once put it, all philosophy is enlightened common sense.) However, the linguistic turn has opened not only the Rylean way of dealing with minds: it has also provided for the possibility of what can be called the ubs-erasure: recasting Descartes’ s-ubs-tances as stances (and vocabularies). The new version of the Cartesian story tells us that we need one kind of stance and one kind of vocabulary to account for the entities which we find mindless, and another kind of stance and another kind of vocabulary for those which we see as minded. We assume what Dennett (1987) calls the physical stance to the former, whereas we assume the intentional stance to the latter. And whereas for the former the language of physics suffices for the latter we require a different kind of language. If we agree on this agenda, then the next question we face is the following: What (linguistic or conceptual) resources (over and above those of the language of physics1) do we need to account for mind-havers & agents?
*
I want to thank Peter Pagin and Pirmin Stekeler-Weithofer for helpful critical comments to an earlier version of the draft. 1
It should be noted that the term "language of physics" is, of course, rather vague.
1
There are various kinds of answers offered to this question. Some of them are quite close to the original dualism of Descartes. Thus, Searle (1983) can be seen as arguing that what we need is an irreducible concept of intentionality that is to refer to a peculiar feature of the mind-stuff which makes it quite close to the Cartesian res cogitans. Others deny any need whatsoever: thus, Quine (1960) and other physicalists argue that everything should be ultimately describable within the language of physics2. However, some of the offered answers lie between these two polar extremes. Rorty (1980), for instance, seems to be almost agreeing with Quine that it is not only possible, but also desirable to describe everything by means of the language of physics. However, he hastens to add that this must not be understood as ruling out the possibility of alternative ways of describing the world or its parts, or implying that any such alternative description would be either uninteresting or translatable into the language of physics. Davidson is one of those who admit the need of a conceptual dualism: he has proposed that what we need over and above the concepts of physics concentrates into the concept of truth: “Without a grasp of the concept of truth, not only language, but thought itself, is impossible. Truth is important ... because without the idea of truth we would not be thinking creatures, nor would we understand what it is for someone else to be a thinking creature.” (1999a, p. 114; my emphasis). An alternative proposal was put forward by Wilfrid Sellars and further elaborated by Robert Brandom: what we need, according to them, is not a specific concept or concepts, but rather the normative mode of speech: Thus, Brandom speaks about "a shift from a broadly Cartesian dualism of the mental and the physical to a broadly Kantian dualism of the normative and the factual" and he claims: "Descartes’s opposition of two kinds of descriptive properties (corresponding to ontological kinds of substances) was contrasted with a deeper opposition between descriptive and prescriptive attitudes - between attributing properties and attributing proprieties”. (1994, p. 623) What I would like to discuss in this paper is to what extent the Davidson's and Brandom's proposals differ; and my thesis is that they are far much closer than they prima facie seem.
Truth as a form of correctness From the Brandomian side, the rapprochement does not seem to be excessively problematic for Brandom sees truth as a matter of a specific kind of correctness: “Assessments of truth, no less than assessments of rationality, are normative assessments. Truth and rationality are both forms of correctness.” (1994, 17). (Already Sellars, 1992, p. 101, urged that truth is nothing else than correct assertibility: “for a proposition to be true is for it to be correctly assertible”.)
2
The locus classicus of this view is, of course, Carnap (1932).
2
So, from this side, it might appear that the only difference is that Davidson reduces the whole normative realm to a single normative concept, that of truth. Why does Brandom appeal to normativity in the first place? He notes, as many other philosophers have, that the necessary condition for an expression’s having a specific meaning is that it be employed in a way which is in some sense regular. However, at the same time he notes that this cannot be a sufficient condition. What he would give as an official reason is that Wittgenstein has shown that every factual way of using an expression can be seen as regular in uncountable ways. Any finite sequence of utterances can be prolonged as to be made regular, and can be done so in an unlimited number of ways. Therefore to explicate meaningfulness as mere regularity would overgenerate: it would render everything as meaningful. To this it might be objected that meaningfulness presupposes a narrower kind of regularity: after all we do - as a matter of fact - classify already finite, and often relatively short, sequences of events as regular/irregular. (I shall disregard the fact that I do not know about anybody who would have succeeded in making this idea of a 'narrower regularity' reasonably precise.) But even if we conceded this, the identification of meaningfulness with regular employment would still erase the intuitive boundary between true meaningfulness and meaninglessnes; and consequently between mindedness and mindlessness. For does a sound come to mean 'It is 200C' by being regularly emitted by a thermometer when the temperature reaches 200C? Hence, Brandom concludes, meaningfulness cannot be a matter of regularity alone - what is needed is a form of rule-governedness. However, as we know again from Wittgenstein, we cannot say that meaningfulness is governedness by explicit rules. (As any explicit rule must be itself meaningful, this would lead to an infinite regress.) Therefore Brandom, as Wittgenstein before him, sees the task of semantics as that of steering between the Scylla of 'regularism' and the Charybda of 'regulism'. Now Davidson is well-known for his derogatory pronouncements about the role of rules in language and in semantics. Thus, e.g., in his discussion with Dummett he asks: “What magic ingredient does holding oneself responsible to the usual way of speaking add to the usual way of speaking?” (Davidson, 1994, p.8), a question which some of his followers interpret as a claim to the effect that once linguistic behaviour is regular, there is no need of its being, in addition, rule-governed (see, e.g., Glüer, 2000, 458 f. 11). But is this really Davidson’s view? In fact, Davidson has repeatedly urged that the concept of truth is one side of the coin, the other side of which is the concept of error – and as there is hardly an error where there is no correct/incorrect distinction, there is a sense in which even for him truth is inseparably connected with a correctness: “It is difficult to exaggerate the magnitude of the step from native or learned disposition to respond to stimuli of a certain sort, to employing a concept with the awareness of the chance of error. ... This is where the concept of truth enters, for there is no 3
sense in saying a disposition is in error – one cannot fail to ‘follow’ a disposition, but one can fail to follow a rule” (1999a, p. 112) This suggests that even for Davidson truth does have a lot to do with correctness, and hence with normativity - insofar as the concepts of correctness and normativity are inseparable. Is, then, his account really at odds with that of Brandom? I do not think so. I am convinced that what Davidson so vehemently rejects is a role of rules within semantics which Brandom himself does not endorse - it is the conviction that the rules which are constitutive of semantics are to be taken as telling us how to talk. This point can be elucidated with the help of a distinction articulated by Glüer (2000), who says (p. 460; my translation from German): "'Correctness' in the sense of only semantic correctness thus does not yet mean anything more than that utterances can be conceptually categorized., i.e. here that they can be sorted out into true and false. This talk of 'correctness' appears thereby deontically wholly innocent; to show that a use 'correct' in this sense is at the same time already a prescribed use, we must answer the question about the source of the prescriptive force, i. e. the question why we the should do the 'correct'." Although the distinction urged by Glüer is important; I find the way she articulates it misleading, as the question why we should do the correct is a bit odd - in fact self-answering. That we should do the correct appears to be a matter of the semantics of "should" and "correct" alone. Therefore, it seems to me that whoever talks about a correctness cannot but talk about a kind of a should. So, in contrast to Glüer, I think that the idea of a "deontically wholly innocent correctness" is ill-conceived. Nevertheless, I do think that the contrast between what we could call 'merely semantic correctness' and a 'more deontically laden correctness' is something to notice. I think that what Davidson is protesting against is that the rules of language are a matter of the latter, i.e. that they oblige us to talk in a certain way - perhaps, as Dummett suggested, to talk as others do. However, this is not what Brandom talks about when he speaks about the normativity of meaning. Brandom - as I understand him - takes it as a plain fact that we not only use language, but also treat others' and our own instances of using language as correct or incorrect. This is not to say that we would explicitly classify them with the help of rules, but rather that we assume certain normative attitudes to them. And what is important, we help force the rules, especially the rules of inference, on the speakers in the sense that we hold them for responsible not only for what they literarlky claim, but also for what follows from what they claim. It is not controversial to say that one is responsible for the consequences of one’s deeds. If I hurt somebody and he consequently dies, then I would be punished for the deed. Analogously, if I make a claim, I will be held responsible for whatever follows from my claim. (Imagine I claimed that person X stole my car; and at the court I am asked: "So you have accused Mr. X of theft?" It is clear that saying "No, your Honour, I did not accuse him of theft, I only said he stole my car" would hardly help me - by claiming that he stole I commit myself to the claim that he committed a theft, for the latter claim follows from the former one.) The difference is that whereas the former consequence is causal, the latter is 'logical' - and it is precisely the distinguishing mark of us, agents, that we live not only within the causal order, which predates 4
us and is totally independent of us, but also within the 'logical order', which we have implemented and which we are maintaining. (That the latter order is no less 'hard' than the former was commented on already in the Ecclesiasticus: "The stroke of the whip maketh marks in the flesh: but the stroke of the tongue breaketh the bones. Many have fallen by the edge of the sword: but not so many as have fallen by the tongue.") Hence Brandom's claim is not that the inferential rules, which he takes to be constitutive of semantics, tell us how we should speak, let alone that they might tell us to speak as others do. (They perhaps do tell us, in a sense, how we should speak in some very specific contexts, e.g. when we want to draw inferences or articulate mathematical proofs; but they surely constitute no manual for making oneself understandable.) Let us invoke the worn out comparison of language and chess: the rules of chess also do not tell us how to play; they only restrict the spectrum of our possibilities. (And though the term "restrict" indicates a purely negative achievement, it is important to realize that, on the positive side, such a restriction may lead to the constitution of a kind of 'inner space', which, in the case of language, may be seen as a space of meaningfulness3.) However, there is an important and rarely noticed distinction between language and chess4. What is directly comparable to playing a game as chess is speaking a particular language. When I violate the rules of English, I cease to talk English, but I need not cease to talk in a way which is intelligible to my audience. Speaking a particular language is an enterprise instrumental to the more general task of communicating; and we may do justice to the general task even by violating rules of particular languages. I think this is the fact pointed out by Davidson. However, it does not seem to contradict the Brandomian stance. The rules of languages constitute certain spaces of meaningfulness without which communication of the kind we are used to would be impossible. But these spaces can be utilized in various ways: we may dwell within a single one, or fluctuate between them, or perhaps lean on some of them as if from outside - i.e. not respect any existing rules, but make use of the fact that these rules obtain. (This is what appears to lay behind the famous Gricean condiderations: what we communicate need not be what the sentence we communicate means; but in communicating it we usually somehow make use of the fact that the sentence means what it does.5)
Correctness as a matter of attitudes 3
Cf. Peregrin (2001, esp. §§ 7.8, 10.7). See also Brandom (1979).
4
There are, to be sure, other important differences, such as the one that chess is 'self-contained' whereas language is 'opened to the world' (see Peregrin, ibid., § 1.3).
5
When somebody asks me where to get petrol (Grice, 1989, p. 32) and I answer "There is a garage round the corner", I communicate that he might be able to get the petrol round the corner, which is certainly not what the sentence I uttered means; but I would not communicate what I did if the sentence did not mean what it does.
5
To clear the way for the Brandom-Davidson truce, we have to notice that Davidson agrees with Brandom also in that language is essentially not only a matter of using expressions and thereby producing utterances, but also of assuming attitudes to the utterances. This (as we have seen) is what lays the foundations of Brandom's normativism: “For brutes or bits of the inanimate world to qualify as engaging in practices that implicitly acknowledge the applicability of norms, they would have to exhibit the behavior that counts as treating conduct (their own or that of others) as correct or incorrect.” (1994, p. 33). As Sellars put it, what is essential for language and thought is “the ability to language about languagins, to criticize languagins” (1974, p. 425) However, the same is essential to Davidson's idea of triangulation, which constitutes the heart of his explanation of semantic content: “[T]riangulation in its pure state: two or more animals equipped to correlate the responses of the others with the events and situations they jointly distinguish. ... Most of the time, one assumes, the reactions of the troop to a threat or a treat are simultaneous. The exceptions provide the entering wedge for correction and the dawning of sense of an independent reality and of the possibility of error.” (Davidson, 1999b, p. 731). It might seem, then, that Brandom and Davidson differ in that although they both agree that truth is intimately connected with error and hence with correctness, Brandom is convinced that we can reduce the concept of truth to the concept of correctness, and that thus we need not give the concept of truth any special pride of place. Davidson, on the other hand, repeatedly urges that the concept of truth is essentially irreducible (he even claims that to try to define it is a "folly"6). Hence, is there a non-circular way of singling out the form of correctness of our utterances which amounts to their truth? It seems to be clear that the truth of an assertion could at most amount to one kind of its correctness – clearly there are assertions which can be said to be correct, despite not being true (it may be, e.g., in a clear sense correct to lie to a dying patient). And, what is important, if Davidson is right in that it is truth alone which grounds meaning, the other forms of correctness are semantically irrelevant. The question which apear to emerge, then, is the following: what species of correctness is truth? And the problem is whether we can answer it in a non-circular way. I think that Davidson’s position can be characterized as yielded by the conviction that this is not possible. I subscribe to this. My question for Brandom is therefore: is there a way of delimiting the kind of correctness which amounts to truth and which is, consequently, constitutive of semantics, or should we agree with Davidson that the concept of truth is irreducible?
6
See Davidson (1996).
6
Brandom and Davidson on the nature of language To conclude, let me present some considerations concerning the difference between the accounts of the nature of language as given by the two philosophers. According to Davidson as far as I understand him - the development of language starts from an activity which is wholly graspable in physicalistic terms, an activity of triangulation, which crucially hinges upon the concept of deviation. In the final development, deviations are replaced with errors, and the whole enterprise of language is no longer construable in terms of physics - we need the concept of truth, which is not reducible to them. Davidson’s story about the emergence of language thus contains a substantial gap: he never tells us how it comes about that mere deviations mutate into errors, and how our prelinguistic activities, which are nothing more than a grist to the eternal mill of causes and effects, develop into the fully-fledged language, and with it the fully-fledged thought and agency, which fall under the legislation of the concept of truth. (The gap is there: it is not the case, Davidson, 1999c, 129, urges, "that the consensus defines the concept of truth, but that it creates the space for its application".) He does not tell us because he thinks there is nothing to be told here - the development takes place within the no-man’s land between the physicalistic realm of law and (what I, following Brandom, would like to see as) the normative realm of freedom, a land which is not describable in terms of either of its neighbours: "The difficulty in describing the emergence of mental phenomena is a conceptual problem: it is the difficulty of describing the early stages in the maturing of reason, the stages that precede the situation in which concepts like intension, belief and desire have clear application. ... What we lack is a satisfactory vocabulary for describing the intermediary steps. ... We have many vocabularies for describing nature when we regard it as mindless, and we have a mentalistic vocabulary for describing thought and intentional action; what we lack is a way of describing what is in between." (Davidson, 1999c, p. 127-8) Hence the gap appears to be principally unbridgeable. Some of those who would like to do away with the gap appear to conclude that if it resists being bridged, it must be denied. They stick to the opinion that error cannot be, on some level of analysis, anything more than a kind of deviation, and hence that truth cannot be anything more than a form of agreement. Understood in this way, the whole enterprise of language appears no longer to escape the reach of the language of physics. Such attempts result into what Blackburn (1984) terms “democratic harmony” theories (and I think that the deepest source of Davidson’s quarrel with Dummett is Davidson’s feeling that Dummett is aiming at something dangerously close to such a theory). Brandom is not willing to subscribe to anything like this. He agrees with Davidson that there is a qualitative gap between the prelinguistic, which can be accounted for in physicalistic terms, and the fully-fledged linguistic, for which we need, according to him, the normative mode of speech. In this sense, he agrees that the gap is essentially unbridgeable: there is no way to translate the talk of fully-fledged language - and thought and agency - into the language of physics. Nevertheless, he is convinced that the gap can be narrowed down in 7
the sense of giving a more explicit description of how language comes into being than Davidson would allow. Hence what I see as the crucial difference between the stances of Davidson and Brandom here is that Brandom wants to say a bit more about the hic sunt leones zone between the prelinguistic and the linguistic. This is what he achieves in terms of his concept of normative attitude: attitudes by means of which we treat what other people do as right or wrong. It is these attitudes, implicit to our behaviour, that provide for the emergence of the normative: utterances become right or wrong (and especially true or false) not because they would be uniform across speakers, but because they would be ‘accepted’ or ‘rejected’, ‘awarded’ or ‘penalized’, ‘taken as right’ or as ‘wrong’. Of course this does not bridge or close the gap entirely. The gap inevitably reappears as that between pre-normative attitudes describable physicalistically, like the tendency to beat someone with sticks if he does A, or give him money if he does B; and truly normative attitudes which can only be characterized with the help of concepts like right and wrong.) The bridge from the non-normative to the normative can never really be built: for if we try to do so using physicalistic terms, we cannot reach the normative shore; whereas using normative terms we are there before we start.
References Blackburn, S. (1984): Spreading the Word, Clarendon Press, Oxford. Brandom, R. (1979): ‘Freedom and Constraint by Norms’, American Philosophical Quarterly 16, No. 3, 187-196. Brandom, R. (1994): Making It Explicit, Harvard UP, Cambridge (Mass.). Carnap, R (1932): ‘Die physikalische Sprache als Universalsprache der Wissenschaft’, Erkenntnis 2, 432-65. Davidson, D. (1994): ‘The Social Aspect of Language’, in B. McGuinness & G. Olivieri, G. (eds.): The Philosophy of Michael Dummett, Kluwer, Dordrecht. Davidson, D. (1996): ‘The Folly of Trying to Define Truth’, Journal of Philosophy 93, 263278. Davidson, D. (1999a): ‘The Centrality of Truth’, in J. Peregrin (ed.): Truth and its Nature (if Any), Kluwer, Dordrecht. Davidson, D. (1999b): ‘Reply to Dagfinn Følesdal’, in L. E. Hahn (ed.): The Philosophy of Donald Davidson (Library of Living Philosophers), Open Court, La Salle (Ill.). Davidson, D. (1999c): 'The Emergence of Thought', Erkenntnis 51, 7-17; reprinted in and quoted from Davidson: Subjective, Intersubjective, Objective, Clarendon Press, Oxford, 2001, 123-134. Dennett, D. (1987): The Intentional Stance, MIT Press, Cambridge (Mass.). Glüer, K. (2000): 'Bedeutung zwischen Norm and Naturgesetz', Deutsche Zeitschrift für Philosophie 48, 2000, 449-468. Grice, P. (1989): Studies in the Ways of Words, Harvard University Press, Cambridge (Mass.). Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.). 8
Rorty, R. (1980): Philosophy and the Mirror of Nature, Princeton University Press, Princeton. Searle, J. (1983): Intentionality, Cambridge University Press, Cambridge. Sellars, W. (1974): ‘Meaning as Functional Classification’, Synthèse 27, 417-437. Sellars, W. (1992): Science and Metaphysics, Ridgeview, Atascadero.
9
The Nature of Meaning: Brandom versus Chomsky Jaroslav Peregrin* Institute of Philosophy, Academy of Sciences of the Czech Republic www.cuni.cz/~peregrin
Philosophy of language: from the picture theory to the toolbox theory The attack in the first half of the XX. century by Russell, Carnap and other prophets of the new, 'scientific' philosophy (consequently called analytic) on the mostly idealistic views of the previous philosophical generation stemmed from the conviction that it is absurd to surmise that the world is made by our minds -- the world is there and we simply picture it in terms of our language and our thoughts. This led these new philosophers to view language as a collection of means for (quite passively) representing things. The passive role of mind and language within the process of acquiring 'knowledge of the external world' was especially stressed by Russell and Moore, as thereby emancipating themselves from the idealism of the previous generation of English philosophers1. The most elaborated version of this 'picture theory' of the language-world relationship was provided by the early Wittgenstein in his Tractatus2. However, Wittgenstein was also rather quick in recognizing the shortcomings of this picture. He realized that if we pay due attention to the way language really functions, then the idea that it is simply a system of names of things serving us to picture more complex things -- viz facts -- fails to be adequate. In particular, Wittgenstein realized that the notion of naming or representing is itself far too problematic to be taken as an 'unexplained explainer', and that we need some deeper level of explanation. To see why, let us consider the seemingly simple-minded question: What does giving a name to a thing consist in in the first place? It may seem that naming is simply something on a par with sticking a label on an object -- an utterly perspicuous move which is in no need of explanation. However, imagine that we literally take a label with a string of letters and stick it on an object, say on a car. Does it mean that we have given a name to the car? Not really: it may count as simply a decoration of the car, or as the indication of the owner of the car etc. What decides whether the string on the label is taken as a name, or as something else? Well, it seems that it is the habits and social practices of the community in question: if sticking
*
Work on this paper was supported by the research grant No. A9009204 of the Grant Agency of the Academy of Sciences of the Czech Republic. 1
See Hylton (1992).
2
Wittgenstein (1922).
1
names on cars is something usual, then it is likely to be taken as a name and hence therefore be a name3; in other cases it may not. Hence something is a name iff it is treated in a certain way. We might perhaps want to say that it is a name of a thing if it serves us to articulate claims about the thing. This is what Wittgenstein points out: ".. naming and describing do not stand on the same level: naming is a preparation for description. Naming is so far not a move in the language-game -- any more than putting a piece in its place on the board is a move in chess. We may say: nothing has so far been done, when a thing has been named. It has not even got a name except in the language-game. This was what Frege meant too, when he said that a word had meaning only as part of a sentence." (1953, §49) However, at the same time Wittgenstein warns us not to assume that there is one principal purpose of language, perhaps describing or representing things and their constellations. There are very various things we can do with words. And as all these things by far cannot be explained in terms of naming (for naming is only one of them, moreover inextricably interwoven with some others), we have to settle for the analysis of the practices. We have to try to understand the 'language games we play'. This was stressed by Coffa in course of surveying Wittgenstein's views (1991, p. 267): "The ultimate explanatory level in semantics is not given by references to unsaturation or to the forms of objects or meanings, but by reference to the meaning-giving activity of human beings, an activity embodied in their endorsement of rules." This is what led Wittgenstein to abandon the picture theory of language in favor of what can be called a toolbox theory: "Language is like a collection of very various tools. In the tool box there is a hammer, a saw, a rule, a lead, a glue pot and glue. Many of the tools are akin to each other in form and use, and the tools can be roughly divided into groups according to their relationships; but the boundaries between these groups will often be more or less arbitrary and there are various types of relationship that cut across one another." (1969, §31). Wittgenstein also realized that what is crucial for our linguistic practices, which draw on the toolbox of language, is the concept of rule -- there is a sense in which the practices are (essentially!) rule-governed. This, however, does not seem to be a great discovery: there are rules associated with almost everything we do. We have the rules of chess, the rules of cooking, even perhaps something like rules for using a hammer ... . But Wittgenstein realized that the rules of language are rather unlike such rules as those of cooking: "Why don't I call cookery rules arbitrary, and why am I tempted to call the rules of grammar arbitrary? Because I think of the concept 'cookery' as defined by the end of cookery, and I don't think of the concept 'language' as defined by the end of language. You cook badly if you are guided in your cooking by rules other than the right ones; but if you follow other rules than those of chess you are playing another game; and if you follow grammatical rules other than such and
3
Of course it can happen that some people hold it for a name by mistake; but to be held for a name generally and regularly simply is what it takes to be a name.
2
such ones, that does not mean you say something wrong, no, you are speaking of something else." (ibid., §133) From this viewpoint, the rules of language are more like the rules of chess: they are constitutive of our language-games in a similar way in which rules of chess are constitutive of chess. At the same time they are very unlike the rules of chess in one crucial respect: they are not explicit. People follow the rules of chess because they have learnt them: they read them (or are explained them by a teacher), interpret them and follow them. This is not possible in the general case of language: if the rules of our language games were all explicit, they would have to be always interpreted, and hence they would presuppose meaningful expressions and hence language. An infinite regress looms. Thus, the rules of language are bound to be non-explicit, they have to be somehow implicit to our language games. But can we make sense of an intrinsically implicit rule at all? It might seem that we could assimilate rules to regularities: we follow a rule in that we act regularly. However, this would erase any distinction between a stone falling ('following the rule of gravitation') and a man playing chess ('following the rules of chess') -- a distinction which appears to be intuitively quite obvious. Moreover, any finite sequence of events is regular -in that it can be seen as the initial segment of a regular infinite sequence (in fact of infinitely many such sequences). Wittgenstein therefore came to the dilemma which Brandom (1994, §I.I.III) depicts as steering between the Scylla of 'regulism', maintaining that rules must be explicit, and the Charybda of 'regularism', maintaining that rules are nothing more than regularities. His considerations have initiated a huge industry of 'rule-following' studies, accelerated especially by Kripke's 1982 book4.
The rules of meaning The later Wittgenstein was surely not the only one to scorn at the picture theory of language and the resulting representational semantics. His natural allies were, of course, pragmatists of all sorts: for a pragmatist wants to consider everything as a means to human ends, and so also linguistic expressions. Hence a pragmatist is hardly tempted to embrace any kind of a picture theory; she is bound to see language as an expedient of certain human doings, and meaning as the "cash-value" of a word (as Ayer, 1978, put it). However, besides pragmatists in the narrow sense of the word (such as James or Dewey), there is a more inclusive range of philosophers (whom we can perhaps call "pragmatists" in a wider sense -- as Brandom, 2002, suggests) who endorse the primacy of the practical (the know-how) over the theoretical (the know-that); and these may also find the toolbox conception plausible. Hence Wittgenstein was far from being the only one to favor this conception and the ensuing use-theory of 4
See, e.g., Baker and Hacker (1984), McDowell (1984), Wright (1989), or Boghossian (1989). An exceptionally illuminating discussion of these matters is given by Blackburn (1984).
3
meaning. What was special about him was his emphasis on the concept of rule. But even in this he was not alone -- there was at least one other philosopher who became famous for his engagement in the enterprise of explaining language via explaining its rules, namely Wilfrid Sellars5. And it was Sellars' legacy that was picked up and has been elaborated by Brandom. Now if it is meaning that we are after, then we need to know which particular kind of rules is constitutive of it -- for there are various kinds of rules associated with our language games, many of which have obviously little to do with semantics. (There are rules determining which strings are 'well-formed', i.e. serving as certain distinguished tokens of linguistic interactions, there are rules concerning what to say at the court and what at an audience with a president etc.) So if we want to get a grip on meaning, we need to single out the semantic ones. Several philosophers have toyed with the idea that these may be the rules of inference. This idea is likely to strike a logician, for many logical words do seem to derive all their significance from the validity of the inferences they figure in: thus it seems that what it takes for a symbol '∧' to be a conjunction is the validity of the inferences from A∧B to both A and B and from A and B together to A∧B6. But Sellars, and Brandom after him, have elevated this to a general paradigm: according to them, the meaning of an expression is, we may say, its inferential role7. Now to make this into a theory of meaning, we have to give an account of how we are to understand the concept of inference in this context. Some statements are clearly inferable from others: thus Fido is a mammal from Fido is a dog. But what does it mean to be inferable? Can it mean that whoever entertains the former thought, or displays the corresponding sentence, goes on to entertain or display the latter? Hardly. Hence that she should go on to entertain or display it? Neither does this seem plausible (not to mention the fact that the nature of the "should" used here would be in as dire need of explanation as the concept of inference itself). The key to understanding the nature of the Sellarso-Brandomian notion of inference lies in understanding that the inferential rules we articulate are neither descriptions of regularities of any human doings, nor prescriptions articulating how the human doings should proceed in an 'optimal' case. They do what the rules of chess do: they restrict the possibilities of our doings thus constituting a space for a new kind of possibilities. They are constraints, or as Sellars was fond of putting it, "rules of criticism". Within chess, we are allowed to move pieces in a certain way only, and if we respect this, we can enter a new space, a space in which we can check, sacrifice a piece for a strong attack etc. Within language, when we accept its rules, we can participate in the new space of meaningfulness, within which we can mean various things previously unavailable to us, we can communicate with others and think propositionally8. 5
See Peregrin (in print).
6
The inferential characterization of logical constants has been elaborated especially within the theory of 'natural deduction' (see Prawitz, 1965).
7
See Peregrin (2001a, Chapter 7).
8
As for the role of propositional thinking, see also Davidson (1999).
4
Thus, if I say that we can correctly infer Fido is a mammal from Fido is a dog, I do not really say what we should do. I rather say what we should not do: namely assert Fido is a dog and at the same time deny Fido is a mammal. In this sense inference is, as Brandom points out, secondary to incompatibility. The rules tell us what we should not do; and we are free to move within the bounds. (We can also investigate the 'outside' of the rules by slightly violating them, which may bring about, in the long run, meager changes of the rules resulting into the development of language.) However, what kinds of entities are rules and where do they reside? Brandom's answer is that they are instituted by our normative attitudes: we treat certain actions of our fellow humans (especially their linguistic utterances) as correct or incorrect. What does it mean to treat something as correct or incorrect? It is underlied by the tendency to call for, support or issue rewards and sanctions; but it is not something which would be translatable into a nonnormative idiom. Therefore, Brandom thinks that to account for meanings, and consequently for thinking and agency, we irreducibly need the normative mode of speech9.
Linguistics: the (unfinished?) revolution Let me now abandon the theme of the rules of language for a while and turn to a different topic: to the revolutionary view of language urged by Noam Chomsky and gradually accepted by a large part of the linguistic (and a slightly smaller part of the philosophical) community. What I take to be the most essential message of this revolution is that language must not be seen as an artifact or an invention like the steam engine or the theory of relativity, but rather as an ability (or an instinct, as Pinker, 1994, puts it) to a large extent 'hard-wired' om our brains. Chomsky implies that the traditional way of seeing language, to which most of us unconsciously (and some philosophers consciously) submit, is severly misleading; and that we cannot truly understand the nature of language if we do not manage to dispose of all such preconceptions. Thus it seems that to follow Chomsky we must forget our 'received wisdoms' and any prejudices which might have come to blur our vision of language10. Doing this and helping ourselves to the usual methods and results of natural sciences, we are, according to Chomsky, bound to arrive at the view of language which he is urging and which is very different from the one Chomsky takes to be standard among contemporary philosophers of language. Now this seems to me to be utterly of a piece with the 'pragmatist turn' of the philosophy of language as just outlined. Chomsky himself claims that semantics in the sense of the picture theory is something close to chimera: "It is possible that natural language has only 9
See Peregrin (to appear).
10
Which are, as Wittgenstein (1953, §103) put it, "like a pair of glasses on our nose through which we see whatever we look at".
5
syntax and pragmatics," he says (1995, 26). This is precisely what the toolbox-theorists urge: semantics cannot but be a matter of the way we employ expressions, hence of what is traditionally referred to by the term "pragmatics". Hence semantics cannot but be somehow parasitic on pragmatics -- or, as Brandom (1994, p. 83) puts it, it must "answer to pragmatics"11. However, instead of a toolbox view of language and the ensuing use-theory of meaning Chomsky votes for an "internalist approach to language", in the course of which he, in effect, reinterprets the word "semantics" so as to refer to peculiar aspects of the components of his reconstruction of the language faculty. Thus Chomsky (1995, pp. 19-20) claims: "[S]ome features of ... expressions ... provide instructions ... for conceptual-intentional systems; this element of the expression is usually called logical form . ... The elements of [the logical form] can be called ... "semantic" features ... . We may take the semantic features S of an expression E to be its meaning." It is not easy to figure out exactly what Chomsky's "logical form" is supposed to be: in his probably most extensive work devoted to the philosophical background of his approach to language he says nothing more explicit than that it is one level of the "structure of language" which constitutes "an interface between language and other cognitive systems" and "yields the direct representation of meaning" (1986, p. 68). What Chomsky appears to be doing here is simply substituting the state of the meaningknowing mind/brain for meaning. Generative grammar, according to him, brought about "an important change of perspective: from the study of behaviour and its products (such as texts), to the inner mechanisms that enter into thought and action" (2000, p. 5). But it should not go without saying that this "change of perspective" is also a change of topic12. The substantiation Chomsky claims for this move is that there is no scientifically respectable way of seizing the old topic (the 'folk concept' of meaning as something that an expression has, can acquire or lose, and that a speaker can learn, grasp or forget). But I cannot see that his response is any more warranted than substituting the study of the brains of the players of chess for the study of the game13. 11
I have discussed this in detail elsewhere (Peregrin, 1999).
12
A related, but separate problem is noted by Stekeler-Weithofer (2001, p. 120): "Cases of 'higher' forms of cognition cannot be identified with a mere physiological, or animal precondition for displaying a competence in particular performances" for these preconditions are "only a condition without not, they do not explain the competence in question." 13
Chomsky (2000, 165-6) contrasts two kinds of generalizations concerning language: the kind exemplified by generalizations concerning "what infants know", which are explained in terms of "inner states", with the one exemplified by the observation that if someone wants X, thinks that obtaining X requires doing Y, and is easily capable of doing X, then he will typically do Y. While Chomsky is certainly right that the latter, in contrast to the former, does not directly lead to a scientific research program, to claim, as he does, that the latter lacks "empirical content" seems preposterous. It seems to me that we do form and verify the latter kind of generalizations all the time - and in fact I cannot imagine how one could understand his fellow agents and predict their behaviour without their help. Thus I agree with Davidson (1970) that though "such accounts of intentional behaviour operate in a conceptual framework removed from the direct reach of physical law", they are none the less not only indispensable, but also theoretically interesting.
6
Of course that when doing theoretical work we are not obliged to take a folk notion of semantics for a dogma -- but the Chomskyan notion does not lead to its revision, but to something having nothing whatsoever to do with it, and it thus becomes doubtful if it deserves to be called "semantics" at all. Hence the trouble, as I see it, is that Chomsky's brilliant analysis of the syntactic aspect of language does not yield an equally acceptable theory of semantics. It would seem that his notion of meaning may make sense only when we utterly strip the term "meaning" of its usual sense. It leads to the dissolution of the concept of meaning within the concept of logical form (which is then sometimes seen as a matter of a "language of thought"14). Moreover, this approach to semantics, apart from not squaring with the intuitive concept of meaning, seems to me to be utterly ill-conceived; in fact it appears to be an instance of what Ryle (1949, p. 18) famously called categorial mistake: The theoretically interesting category-mistakes are those made by people who are perfectly competent to apply concepts, at least in the situations with which they are familiar, but are still liable in their abstract thinking to allocate those concepts to logical types to which they do not belong. ... The representation of a person as a ghost mysteriously ensconced in a machine derives from this argument. Because, as is true, a person's thinking and purposive doing cannot be described solely in the idioms of physics, chemistry and physiology, therefore they must be described in counterpart idioms. As a human body is a complex organized unit, so the human mind must be another complex organised unit, though one made of a different sort of stuff and with a different sort of structure. I think this is precisely what is going on in many contemporary Chomskyan and postChomskyan semantic theories. As we have a profound and very successful theory of syntactic structures and as it is not directly applicable to semantics, it is assumed that we must have a parallel theory of 'semantic structures' -- structures parallel to, but different from, the syntactic ones, and embodied in a quite different stuff. However, while syntax is essentially about structures (of expressions), semantics is about the significances of the expressions thus structured, and also about the way the significances propagate along the syntactic structure (i.e. from compounds to composites or perhaps vice versa)15. Hence urging the explanation of semantics in terms of a structure parallel to the syntactic one is, as I have expressed it elsewhere (Peregrin, 2001a, §10.2), like saying that as the description of boots consists in an account of their cut and the material, we should describe the function of boots by revealing some 'cut and material of the way they are used'. 14
See Pinner (1994, Chapter 3).
15
This is not to rule out the feasibility of something like a semantic structure; but a theory of such structures is not yet semantics, it might be at most instrumental to it. There is no doubt that "logical forms" play a vital role within Chomsky's well established theory of the language faculty -- the objection is that its investigation is not the investigation of meaning.
7
Hence I see the Chomskyan revolution as essentially unfinished, waiting to be carried over from the syntactic aspect of language to the semantic one -- with the same fervor and especially with the same unprejudicedness. How? The answer would appear straightforward: if we are to see linguistic expressions as the means employed by our language faculty, then two kinds of questions arise: which expressions do we use? and how, i.e. for what ends and with what effect, do we use them? The former are the syntactic questions, questions about which expressions are grammatically well-formed. The latter are the semantic -- or we should rather say semantico-pragmatic -- ones, questions concerning significances and meanings16. In this way we by-pass the traditional questions of referring, denoting etc., which Chomsky (I think rightly) sees as artificial17. Hence what seems to me to be continuous with the Chomskyan upheaval is not the kind of semantics which is now current among his followers (which concentrates on various kinds of semantic structures or 'logical forms'), but rather some kind of 'use-theory of meaning'. This is to say that given an expression, we can study how the expression consists of sub-expressions and how it is capable of becoming part of super-expressions; and we can also study what happens when people actually utter it, what kinds of changes its utterances are likely to bring about. This agenda agrees with the Chomskyan one in that it is an enterprise naturalistic through and through; however it is essentially externalist -- we do not study (only) human minds or brains, but interactions between speakers and their environment.
Studying minds (and their language faculties) As made plain in the beginning of this paper, the version of the use-theory of meaning which I follow Brandom in aiming at, is a normative one. What I claim we should aim at is not an account of regularities of linguistic conduct, but of rules 'underlying' it -- which are to be reconstructed on the basis of recording the regularities of our normative attitudes to linguistic conduct. And this might appear to be incompatible with Chomskyan naturalism. But I do not think it really is. (Though Brandom himself does expressly reject naturalism and declares himself to be a rationalist -- see, e.g., his introduction to 2000; and see also my criticism in my 2001b review of the book.) What conceptual resources do we generally need to account for the semantic aspect of language? Thinkers so different as Chomsky and Quine concur in claiming that we do not need anything more than what we need for the rest of the world: hence plus minus the language of physics. But the fact is that in describing the semantic aspect of language we move to the vicinity of describing minds (which is not to be read as involving that meanings 16
See also Peregrin (1999).
17
Again, this is not to rule out the feasibility of these concepts within a theory of language; it is only to say that they should not play the foundational role they play within many contemporary theories (see, e.g., Devitt and Sterelny, 1987).
8
be mental entities! - but the concepts of a language-using creature and a creature with a mind appear to be closely connected), and in describing minds we actually do use a conceptual apparatus different from what we use when describing the mind-less world. We talk about minds as containing thoughts, feelings and intentions and we talk about the actions of creatures with minds as being brought about by these contents of their minds. Whereas Quine appears to hold the concept of physicalism for straightforward and not very problematic, Chomsky (1995) duly stresses the difficulties of specifying what the "language of physics" really is. For him, the concept of "physicalism" is vague to the point of vanity. It would seem to follow that "naturalism" cannot be characterized but very vaguely: as something like 'a devotion to the spirit of the methods of the sciences.' From this viewpoint, the stress Chomsky puts on the divide between naturalism and the rest appears to be overloaded. He insists that there is no reason to approach minds with a conceptual apparatus essentially different from that established by natural sciences; and he urges naturalism as the opposite to "the doctrine that in the quest for theoretical understanding, language and mind are to be studied in some manner other than the ways we investigate natural objects as a matter of principle" (1995, 28). But this is queer: it could hardly be denied that we study some "natural objects" (say, animals) in a manner which is to some extent different from that in which we study other such objects (say, stones). And it is to be expected that studying the specific animals of our own kind, and especially their aspect in connection with which we have come to use terms like "mind", "consciousness", "reason" and also "language", will call for other specific methods and concepts. And why, or at exactly which point, must this incur our departure from naturalism? But perhaps the crux of what Chomsky urges lies in the phrase "as a matter of principle". Well, it is surely possible to agree that if someone insists that language and mind must be studied in a certain way before she pays any attention to their real nature, we would reject it. (Chomsky, 1995, quotes philosophers who appear to be on the verge of saying that a certain approach to language follows from the nature of philosophy alone, irrespectively of the nature of language.) However, in such a case it would be more apt to talk about a prejudice than about a departure from naturalism. Prejudices are to be despised, but I think it is also a prejudice to insist that studying mind and language cannot lead to a legitimate need for methods or concepts not previously employed within the enterprise of the study of nature. Physicalists appear to think that any kind of 'folk psychology', with its feelings, thoughts or intentions should be construed (if not wholly discarded) as a short-cut way of speaking of physical reality, and be translatable into the language of natural sciences. But there are people who disagree: thus, Davidson (1970, and elsewhere) argues that there are good reasons for having two different and mutually irreducible conceptual systems, one for mind-less things and another for minded agents. The reason, according to him, is that we aim at strict natural laws when making sense of the former, whereas for the latter we aim at something a little less cut and dried. We do not expect to be able to predict the behavior of our fellow people in the way we predict the behavior of much simpler things. 9
Dennett (1987) talks about the intentional stance: as people are physical systems too complicated to be understood as such, we have devised a different -- less reliable, but practically very useful -- way of 'understanding' them. We have come to see that if we ascribe them thoughts, intentions etc. and if we assume that they follow such 'laws' as that of the practical syllogism, we are able to make sense of what they do in so far as making very qualified estimates of their behavior (and of course talking and otherwise coping with them). And, Dennett concludes, if having thoughts, intensions etc. is what is needed to explain their behavior, then there is no reason to say that these entities are 'not really there'. We have seen that Brandom takes a related train of thought; but it is nevertheless different. He also thinks that we need specific linguistic resources to account for meanings and minds. But he does not think that this should be seen pimarily as a matter of new concepts; it is rather that of a different mode of speech, namely the normative mode. Hence his idea is that when speaking of minds and meanings we are forced to talk not about what there is, but rather what there should be. This might sound, prima facie, weird. When I say what a word means, in what sense do I not speak about what there is? The key to the answer are the rule-following considerations we tackled in the beginning of this essay. If these are right, then an expression may be meaningful only as a consequence of being subordinated to a collection of rules (just like a piece of wood becomes a chess king by being subordinated to certain rules of chess). Hence to say what an expression means is to spell out certain rules, and citing rules is saying what should (or should not) be done.
Rules and rule-following I doubt that anyone would dream of explaining what it takes to be a chess king without talking about how one should move it or how it is correct to do so. And it should be equally uncontroversial that to say what it takes for a sound- or inscription-type to mean thus-and-so is to say that the item should be used in a certain way. (Let us again stress that these should's do not indicate that people are to do something in a specific way, but rather that they are to avoid certain ways of doing it -- thus delimiting a space between the ensuing crash barriers.) Brandom claims that the should arises from our normative attitudes, the attitudes we assume towards our fellow people's doings and especially utterings. We, as a matter of fact, have come to take our own and others' utterances as correct or incorrect (on various levels and in various senses), we reward those who do the correct and sanction those who do not (again in very various senses -- from arresting oathbreakers to treating those who do not use words in a way which makes sense to us as not members of our linguistic community). The rules and hence the sort of correctness which is distinctively semantic rest, according to Brandom, on the institutions of commitments and entitlements. By making linguistic utterances we commit ourselves to various things -- by making a promise we commit ourselves to doing what we promise, whereas by making an assertion we commit ourselves to 10
justifying it, to giving our reasons for holding it, should it be challenged. We also offer entitlements: by giving permission we entitle the addressee to do what we permit him, whereas by making an assertion we entitle the addressee to reassert the claim deferring its justification to us. Now according to Brandom -- and this may be the most controversial part of his theory -it is asserting and the related cluster of practices of "giving and asking for reasons" which is crucial from the viewpoint of semantics. (That asserting, as 'saying what there is', assumes a prominent position within the picture conception of language is obvious; but it is much less obvious that it should retain it even on the toolbox conception.) Brandom's idea appears to be that what makes our language crucially different from, e.g., the communicating systems of other animals is precisely its suitability for the "giving and asking for reasons", and its consequent inferential articulation, i.e. the fact that each expression must be part of some statements which entail further statements and are themselves entailed by others. This means that from both the Wittgensteinian and the Brandomian viewpoint, to understand language we have to understand the concepts of a linguistic rule and of following a rule. Chomsky (1995, 34ff) appears to think that we do not need any such concept beyond that of a regularity and that we can reduce the concept of rule-following to the concept of regular behavior: he sees no principal difference between a man following grammatical rules hard-wired to his brain and a hypothetical Martian lacking this wiring and following the same rules in force of his explicit mastering of a theory. But it is hard to believe that he would want to claim that there is no difference between a stone's following the rule of gravitation by falling with acceleration g and a person steering her car in a certain way because she consciously follows the rules of traffic (or, for that matter between a person who follows the rules of traffic subconsciously and one thinking about them)18.
Studying language (and meanings of its expressions) From the viewpoint just outlined, studying semantics is studying semantic -- especially inferential -- rules of language. How, then, does the good old concept of meaning fare within this new environment? On any variety of a picture theory (and here I mean both what I called the 'ontologicosemiotic' and the 'psychologico-semiotic' theories elsewhere19), there is a piece of the world to which a typical meaningful expression is -- somehow -- intimately related (though the theories may vary wildly as to the nature of the piece) and meaningfulness consists in this very relationship to the piece. To do semantics, then, is to find out about this very chunk of the world. On the other hand, on the toolbox conception of language we are not tempted to 18
Chomsky might well be right that this difference is not usefully elucidated in terms of an "access to consciousness", but denying it altogether seems to fly in the face of reason.
19
See Peregrin (2000; 2001a, Chapter 2).
11
see meaningfulness as a matter of a relation to an object or a state-of-affairs, for we see it as consisting in the expression's being employed in a certain way, or perhaps being governed by certain rules. I want to claim -- and here I am not sure whether I am still following Brandom -- that even in this setting, we should not let the concept of meaning go by the board; though we should now see it as more of a tool of doing semantic theory than its subject matter. To get an insight into how semantics looks when seen from this perspective, it is good to assume the stance of somebody who wrestles with an utterly unknown language. This is the setting of the notorious thought experiment of radical translation or radical interpretation, proposed by Quine (1960) and elaborated by Davidson (1984). How useful does the concept of meaning appear in this setting? It seems to be clear that when we observe an unknown linguistic community, we gradually come to realize that its members tend to use some kinds of utterances (i.e. sentences) in certain situations for certain purposes, and others in other circumstances for other purposes. The learning and appreciating of these differences is usually described as learning what these sentences mean. Moreover, as we encounter ever new sentences, sentences never heard before, there is no way to master them save by discovering the syntax of the language (a task which we accomplish, if Chomsky is right, almost effortlessly) and disclosing the roles of lexical items -i.e. discovering their general usage. (Reference, for that matter, then comes out as a concept auxiliary to this quest for meaning -- a lot of words play a specific kind of role, which we know very well from our own language and which has come to be called referring to something.) Chomsky does not think that the traditional concept of meaning is of much use. "Communication ..., " he claims (1993, 21), "does not require shared 'public meanings' any more than it requires 'public pronunciations'. Nor need we assume that the 'meanings' (or 'sounds') of one participant be discoverable by the other."20 And: "From the natural language and common-sense concepts of reference and the like, we can extract no relevant 'relation between our words and things in the world'. (1995, 44). Hence Chomsky denies any relevance of the concept of "shared meanings" and also of any other traditional concepts characterizing the language-world relationship. (As for meanings, he again concurs with Quine, 1992, Chapter III, who sees the concept as a "stumbling block cleared away"). My opinion, to the contrary, is that language and communication can exist only insofar as do "shared public meanings". This is, however, not to say that people speaking the same language mean always (or frequently) the same by the same word; it is to say that I can communicate with somebody only insofar as she can interpret me -- in other words share the meanings of my words with me. Chomsky (1993, 21) is certainly right when he stresses that "communication is a more-or-less matter" -- and so is meaning. But I do not think that it follows that we should dispense with the concept of meaning altogether. Let me explain why.
20
The comparison to "pronunciations" and "sounds" seems strange: surely talk about "communication" makes sense only when there are sounds which are acknowledged ("discovered") by an audience, and hence are in this sense "public".
12
The point of departure of radical interpretation is the situation in which we observe the natives and their linguistic behavior, and we set out to make sense of it. We notice that their utterances display certain regularities and some of them also co-variances with the circumstances in which they are produced. We also notice regularities in the ways the natives approve or disapprove of the utterances of their fellow speakers (and we are likely to soon move into the position to suggest utterances ourselves and urge the approval or disapproval -- 'assent' or 'dissent' -- from the natives). On this basis, we try to conjecture the rules of their language games and the roles these rules confer on their sentences and consequently words. Now it is almost certain that we can never capture the whole of the employment of a word -and this is also not our task, for what we are after are circumstance-invariant rules and the roles of words as determined by such rules. We are happy to see some features of the way a word gets actually employed as not directly a matter of these rules, but a matter of either occasional violation of the rules, or of the idiosyncratic strategies of the speakers. What we are aiming at is a kind of e pluribus unum -- a Gestalt which may be seen as ordering our data into something "intelligible". It is clear that the emerging Gestalt is not determined uniquely by the data -- any boundary we posit between circumstance-invariant and hence meaning-conferring rules and the rest of the determinants of linguistic traffic is notoriously vague and fuzzy. And this is not only a matter of our drawing conclusions from restricted evidence -- the boundary between the 'meaning-giving' part of the usage of a word and its 'fact-stating' part is essentially fuzzy. It is, in fact, the notorious boundary between the analytic and the synthetic, which Quine (1952) rightly diagnosed as non-existent. It follows that also meaning itself is fuzzy, a "more-or-less matter". However, it is one thing to say that meaning is a more-or-less matter, and quite another to say that there is no such thing as meaning. After all, everything within the real world is a more-orless matter. There are, for example, no 'true' spheres and no 'true' cubes within the real world, and yet we make great use of geometry, regarding many things around us as spheres or cubes. And similarly it may be useful to see meanings as objects -- despite the fact that this involves an amount of idealization21. True, we must be wary of the notion that meanings are things-in-thewords christened by words; but once we are clear about their real nature, there seems no reason not to bring the objectualizing drift of our minds to bear on them. When a speaker X utters a statement s, then the way we normally perceive this is that X has a belief b and this belief is expressed by s. Hence what we ordinarily -- quite subconsciously -- do is 'decompose' the body of facts about the natives' utterances into a theory of what the speakers believe and a theory of what their words mean -- and we use meanings, as Davidson (1989) stressed, as something like our ‘measuring units’ to account for our findings in a similar way to how a natural scientist would use her's. This engenders that the decomposition is, in fact, stipulated by the interpreter, although she is surely not free to posit it wherever she wants. (She has to draw it so as to create a Gestalt as helpful as
21
See Peregrin (2000; 2001a, Chapter 9).
13
possible for the enterprise of seeing what the natives are saying.) But it seems to me that she truly perceives the linguistic conduct 'in terms' of meanings22.
Conclusion I think that what both Chomsky and the 'toolbox-theorists' urge is a kind of 'pragmatist turn'. We should not see language as a system of names denoting objects, but rather as an expedient or vehicle of a certain human activity. Hence we should see semantics as 'embedded' within pragmatics -- or, as Brandom, puts it, see semantics as answerable to pragmatics. And we should study the semantic(o-pragmatic) aspect of language in a way which is continuous with that in which we study other human activities and the rest of the world in general. However, I do not think Chomsky or his followers have offered a theory of semantics which is truly in this spirit. Instead of trying to explicate "semantics" in terms of structures yielded by the extrapolation of Chomsky's theory of syntax, I think we should settle for a use-theory of meaning, i.e. a theory according to which the meaning of an expression amounts to the role the expression plays within our linguistic transactions. Moreover, I am convinced that our linguistic practices are characterized by the fact that they are rule-governed (in the sense in which a game like chess is rule-governed -- i.e. having constraints of the kind of "you-should-not" which create an 'inner space', in this case the space of meaningfulness). Therefore we have to study the rules of language; and, if what we are after is meaning, especially the rules of inference. And this -- to my understanding -- is the very thing Bob Brandom does.
References Ayer, A. J. (1978): 'Introduction' to W. James: Pragmatism, Harvard University Press, Cambridge (Mass.). Baker, G. P. and P. M. S. Hacker (1984): Scepticism, Rules and Language, Blackwell, Oxford. 22
Seen from this perspective, the character of the pronouncement ‘the meaning of s is such-and-such’ is not unsimilar to that of ‘the (real) price of x is such-and-such’. Just as ‘the price of x is such-andsuch’ is to be understood as a shorthand for ‘the position of x within the selling-and-buying relations among people is such-and-such’, ‘the meaning of s is such-and-such’ should be construed as saying ‘the way s gets employed within the language game to which it belongs is substantially such-andsuch’. Both meaning and price may sometimes be fixed by some explicit act (and in such a case the meaning or the price becomes something which is discoverable by natural-scientific methods); however, in the typical case both are a matter of finding an ‘equilibrium’ of a number of intersecting relationships, i.e. of an interpretation. Thus we can talk about meaning only from the viewpoint of an interpreter, of someone who observes the relevant environment and ‘calculates’ the relevant value out of it. Therefore Sellars (1974) speaks about „meaning as a functional classification".
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Blackburn, S. (1984): Spreading the Word, Clarendon Press, Oxford. Boghossian, P. (1989): 'The Rule-Following Considerations', Mind 98, 507-49. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Brandom, R. (2002): 'Pragmatics and Pragmatisms', in Hilary Putnam: Pragmatism and Realism (eds. J. Conant and U. M. Zeglen ), Routledge, London, 40-59. Chomsky, N. (1986): Knowledge of Language, Praeger, Westport. Chomsky, N. (1993): Language and Thought, Moyer Bell, Wakefield. Chomsky, N. (1995): ‘Language and Nature’, Mind 104, 1-61; reprinted as Chapters 5 and 7 of Chomsky (2000). Chomsky, N. (2000): New Horizons in the Study of Language and Mind, Cambridge University Press, Cambridge. Coffa, A. (1991): The Semantic Tradition from Kant to Carnap, Cambridge University Press, Cambridge. Davidson, D. (1970): 'Mental Events', in Experience and Theory (ed. L. Foster and J. W. Swanson), The University of Massachusetts Press; reprinted in Davidson: Essays on Actions and Events, Clarendon Press, Oxford, 1980, 207-224. Davidson, D. (1984): Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Davidson, D. (1989): 'What is Present to the Mind?', in The Mind of Donald Davidson (eds. J. Brandl, W.L. Gombocz), Rodopi, Amsterdam, 3-18; reprinted in Davidson: Subjective, Intersubjective, Objective, Clarendon Press, Oxford, 2001, 53-68. Davidson, D. (1999): 'The Emergence of Thought', Erkenntnis 51, 7-17; reprinted in Davidson: Subjective, Intersubjective, Objective, Clarendon Press, Oxford, 2001, 123-134. Dennett, D. (1987): The Intentional Stance, MIT Press, Cambridge (Mass.). Devitt, M. and Sterelny, K. (1987): Language and Reality, Blackwell, Oxford. Hylton P. (1992): Russell, Idealism, and the Emergence of Analytic Philosophy, Oxford University Press, Oxford. Kripke, S. (1982): Wittgenstein on Rules and Private Language, Harvard University Press, Cambridge (Mass.). McDowell, J. (1984): 'Wittgenstein on Following a Rule', Synthèse 58, 325-363. Peregrin, J. (1999): 'The Pragmatization of Semantics', in The Semantics/Pragmatics Interface from Different Points of View (ed. K. Turner), Elsevier, North-Hollard, 419-442. Peregrin, J. (2000): 'The "Natural" and the "Formal"', Journal of Philosophical Logic 29, 75101. Peregrin, J. (2001a): Meaning and Structure, Ashgate, Aldershot. Peregrin, J. (2001b): 'Review of Brandom: Articulating Reasons', Erkenntnis 55, 2001, 121127. Peregrin, J. (in print): 'Developing Sellars’ Semantic Legacy: Meaning as a Role', The Philosophy of Wilfrid Sellars (ed. M. Lance), Rodopi, Amsterdam. Peregrin, J. (to appear): 'Brandom and Davidson: what do we need to account for thinking and agency?', to appear in Proceedings of the Pecs' Symposium on Brandom. Pinker, S. (1994): The Language Instinct, W. Morrow, New York. Prawitz, D. (1965): Natural Deduction, Almqvist & Wiksell, Stockholm. Quine, W.V.O (1952): 'Two Dogmas of Empiricism', Philosophical Review 60, 20-43; reprinted in Quine: From a Logical Point of View, Harper and Row, New York, 1953. Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.). Quine, V.W.O. (1992): Pursuit of Truth (revised edition), Harward University Press, Cambridge. Ryle, G. (1949): The Concept of Mind, University of Chicago Press, Chicago. 15
Stekeler-Weithofer, P. (2001): 'Meaning, Intention and Understanding; Formalism and Mentalism in Theories of Communication', in L. Lundsten, A. Siitonen and B. Österman (eds.): Communication and Intelligibility (Acta philosophica Fennica 69), Societas Philosophica Fennica, Helsinky, 113-133. Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London; English translation Routledge, London, 1961. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford; English translation Philosophical Investigations, Blackwell, Oxford, 1953. Wittgenstein, L. (1969): Philosophische Grammatik, Suhrkamp, Frankfurt; English translation Philosophical Grammar, Blackwell, Oxford, 1974. Wright, C. (1989): 'Wittgenstein’s Rule-following Considerations and the Central Project of Theoretical Linguistics', in Reflections on Chomsky (ed. A. George), Blackwell, Oxford .
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What is the logic of inference? Jaroslav Peregrin We may say, with a certain amount of oversimplification, that logicians divide into modeltheoreticians and proof-theoreticians. Model-theoreticians think that it is model theory (or formal semantics) which is primary from the intuitive viewpoint (for it captures directly the meanings expressions have), whereas the latter think that it is proof theory (or axiomatics) which is primary (for it captures our inferential patterns which are constitutive of what our words mean). In this paper I will not take sides (though there is no point in denying that my heart goes with the latter1); but I would rather like to discuss a question relevant not only for those who take special interest in proof theory: "What is the logic of inference?" The question is, to be sure, rather vague; and it is to be expected that to make it reasonably precise can take us a great part of the way to its answer. Moreover, many theoreticians did answer this question in the sense that it is intuitionist logic which is related to inference in the distinguished way. I will not dispute the answer; there is little doubt that it is correct. However, I will present what I believe is a deeper and a more general substantiation for it than the ones which are commonly on offer.
Zucker & Tragesser Let me begin by discussing an example of substantiation of the prominent role of intuitionist logic vis-à-vis inferences and indicating why I am not content with it. Let us consider a rather old, but still influential paper of Zucker and Tragesser (1978), in which the authors claim to establish that it is the intuitionist connectives which are "adequate for inferential logic". What exactly do they mean by this? The authors define that "given a logic L, and a set of logical operators of L, we say that S is adequate for L if every logical operation of L is explicitly definable in terms of S". In this sense we can obviously say that the set {¬, ∧} of classical logical operators is adequate for the whole of the classical, truth-functional propositional calculus. But what about "inferential logic"? They say: "By 'inferential' logic we mean logic formulated in a natural deduction system, in which the meaning of each logical constant is supposed to be given by its set of introduction rules." So the "adequacy problem" involves two sets of logical operators and a notion of equivalence: we show that each of the first set can be equivalently rendered by those of the second set. In case of classical logic the situation is simple: the first set is constituted by all the truth-functionally defined operators, the second one by ¬ and ∧, and the equivalence in question consists in the fact that for any operator c of the first group there is a tautology of the classical propositional calculus of the form c(S1, ..., Sn) ↔ F, where F is a formula containing no other operators than ¬ and ∧. Now how is it with inferential logic? The first group of operators is obviously constituted by those which can be defined by means of the natural deduction rules and Zucker and 1
I put all my cards on the table elsewhere (see Peregrin, 2001).
Tragesser explicitly narrow it down to those which can be defined by means of mere introduction rules. But what is the second group, and, especially, what notion of equivalence should we consider? The authors say that it is "the set of logical constants {→, ∧, ∨, ¬}" without specifying which version of the operators they mean (classical? intuitionist? any other?). This is a bit surprising, especially in view of the fact that the claim is made to summarize the results of the paper; however, as the second class is to be the subclass of the first one, we must assume that what the authors mean are →, ∧, ∨ and ¬ as defined within a natural deduction system, and with some charity we can assume that what is meant is what would be taken as the 'most natural' version of the definitions, i.e. the intuitionist one. (The authors also appear to be willing to use the terms 'inferential logic' and 'intuitionist logic' interchangeably). But what is not so clear is the notion of equivalence we should consider now. What one would expect is that for any operator c definable in terms of the restricted natural deduction is a theorem of inferential logic of the form c(S1, ..., Sn) ↔ F, where F is a formula containing only the intuitionist →, ∧, ∨ and ¬. Again, it is not quite clear what the nature of the ↔ is, but we will again assume that it is the intuitionist operator. Now note that this solves the "adequacy problem of inferential logic" within the medium of the very same logic; hence it compromises the possibility of interpreting the result as proving, in the intuitive sense, that it is intuitionist logic that is the logic of inference. The obvious objection is that by using inferential logic as our metatheory we smuggle in what is to be discovered. Moreover, the authors' argument raises further doubts. First, it is not entirely clear whether there is a deeper reason for their restricting themselves to operators defined by the standard form introduction rules other than ease of treatment. They do not explain why this restriction does not prevent them from going on talking about inferential logic in general. Second, also the way they carry out their proof is not entirely clear. Let us consider the first case they consider within the proof, the case where a propositional operator is defined by a single inferential rule: [A11, ..., A1k1] [An1, ..., Ankn] . . . ...................................... . . . Bn B1 c(A11, ..., A1k1, ..., An1, ..., Ankn, B1, ..., Bn, C1, ..., Cm) In this case they simply state that "the meaning of c -- or more exactly of c(...,Aij, ...,Bi,...,Cl,...) -- is given by (*) (A11→B1)∧ ... ∧(A1k1→B1)∧ ... ∧(An1→Bn)∧ ... ∧(Ankn→Bn)". The only substantiation of this claim they give is the reference to an assumption stated earlier in the paper, namely that "The meaning of the constant c is given by its set of introduction rules". Now if the interpretation we have proposed for the authors' claims earlier are to be 2
accepted, then it is, prima facie, strange why we would need to talk about "the meaning" of c. What we need to prove is that given c is introduced by the rule above, there is a formula F containing as the only operators the intuitionist →, ∧, ∨ and ¬, and such that c(A11, ..., A1k1, ..., An1, ..., Ankn, B1, ..., Bn, C1, ..., Cm) ↔ F. And as F is apparently meant to be (*), what we have to do is to prove, within the natural deduction system for intuitionist logic, the formula c(A11, ..., A1k1, ..., An1, ..., Ankn, B1, ..., Bn, C1, ..., Cm) ↔ ((A11∧ ... ∧A1k1)→B1)∧ ... ∧((An1∧ ... ∧Ankn)→Bn). But surprisingly, they do not do anything like this. Can the reason be that the proof is too obvious? Well, the indirect implication is indeed pretty straightforward, but not so the direct one. It would amount to the transformation of the proof of c(A11, ..., A1k1, ..., An1, ..., Ankn, B1, ..., Bn, C1, ..., Cm) into the proof of ((A11∧ ... ∧A1k1)→B1)∧ ... ∧((An1∧ ... ∧Ankn)→Bn). It is clear how this is supposed to proceed: as the former is provable, and as we assume that the only available proof is based on the introduction rule above (which is what the authors express -oddly, in my view -- by claiming that "the meaning of c is given by its introduction rule"), we must suppose that the premises of the proof are in force, and these premises establish the latter claim. This is a kind of an 'abductive reasoning', which is intuitively clear, which, however, I think should be given some explicit form. Hence I do not think the result presented by Zucker and Tragesser can be taken as establishing an answer to the question within the title of this paper. And I think that one of the important morals which can be drawn from this discussion is that to be able to do it better we need a more explicit framework -- we need a neutral ground on which we can independently delimit both inferential systems and intuitionist logic, and within which we can then show how the latter 'does justice' to the former.
Tennant However, I do not think that filling the gaps I have just diagnosed within Zucker's and Tragesser's paper is too difficult. I think we can do so -- and thereby also get a clearer perspective on the relationship between inference and intuitionist logic -- if we consider another argument, due to Neil Tennant (1997). Tennant's strategy is clearer and more explicit: he first lists the desiderata for inferentially defined logical operators, and then shows that the desiderata are fulfilled by and only by intuitionist operators. (In fact, he votes for a relevant version of intuitionistic logic, but we ignore this here.) The desiderata, according to him, are (p. 319): (1) analytic systematicity (2) separability (3) immediacy (4) harmony of introduction and elimination rules (5) compositionality. 3
(1) amounts to the fact that each operator is introduced by means of a finite number of schematic rules; (2) to the fact that each of the rules contains merely a single dominant occurrence of an operator; (3) establishes that each rule is either an introduction rule, which contains the operator dominant in the conclusion, or an elimination rule, which contains it dominant in a premise; (4) that the introduction and elimination rules are in a precise sense complementary; and (5) that the rules have the subformula property. It is the harmony requirement which is tantamount to Zucker & Tragesser's requirement that "the meaning of a constant is given by its introduction rules". But Tennant spells this out in much clearer terms: The introduction and elimination rules for any logical operator λ should be framed in such a way that (i) the statement of the introduction rule for λ, the conclusion (with λ dominant) should be the strongest that can be inferred under the conditions specified; and (ii) in the statement of the corresponding elimination rule, the major premise (with λ dominant) should be the weakest that can be used in the way specified. (332-3) The principle, as Tennant suggests, "serves to tailor the elimination rule to a previously chosen introduction rule, or vice versa". Hence if we encounter λ within a proof, we know it must have been introduced in its canonical way, and can reason back to the premises of its introduction rules, as done, in effect, by Zucker and Tragesser. Hence Tennant shows that if we want logical operators which are not only inferentially introduced, but also in a certain sense well-behaved, we should restrict ourselves to the very ones Zucker and Tragesser are suggesting, namely those which are introduced by means of the standard form introduction rules (which are taken to 'implicitly contain' also the corresponding elimination rules). And he indicates that any such operators are reducible to the intuitionist ones (which is what Zucker and Tragesser want to prove explicitly). There is nothing to be objected to this whole line of argumentation. It is based on the assumption that we should want a kind of a well-behaved logic -- an assumption which surely is reasonable. But in this paper I would like to indicate that we can see intuitionist logic as the logic of inference even without it. I will argue that it follows from a certain conception of what logic is -- a conception which I believe we should adopt in any case. Hence I believe that there is a sense in which intuitionist logic is not merely the logic of inference, but the logic simpliciter. (I do not mean to dispute the simplicity and consequent vast usefulness of classical logic -- it is clear that from the viewpoint of the needs of a working logician it is largely superior to the intuitionist one. I only want to claim that from a certain genetic viewpoint the former should be viewed as secondary to the latter.) But my argument for intuitionist logic being the logic of inference is independent of the argument that it is the logic; and here I want to concentrate on the former.
What is logic? Many answers to this question have been proposed. (Though given the large body of work being done on the technical side of logic, I do not think it is being paid an excessive attention.) I have argued for my answer elsewhere and will not repeat the argument in detail 4
here. In sum, I think that logic is, as Bob Brandom's (1994; 2000) puts it, first and foremost a tool of making explicit the rules implicit to out treating concepts. According to Brandom, what underlies both human language and logic are inferences. Human language is structured in such a way that commitments to some claims bring about commitments to other claims -i.e. that the former ones entail the latter ones and hence the latter are correctly inferable from the former. In fact, Brandom claims that there is a still deeper layer constituted by the concept of incompatibility. Brandom also claims that what we see as logical vocabulary is first and foremost a means of making explicit the proprieties implicit in our using language. Before we have "if ... then", the inference from "This is a dog" to "This is a mammal" can only be implicitly endorsed (or, as the case may be, violated), but once this connective is at hand, this inference can be explicitly expressed in the form of a claim, viz. "If this is a dog, then this is a mammal" (and with a more advanced logical vocabulary perhaps further transformed into "Every dog is a mammal"), and hence discussed w.r.t. its 'appropriateness' (and perhaps in the end rejected). Thus logical vocabulary helps us make the rules that are intractably implicit in our practices explicit and discussable. This Brandomian view of logic might look if not far-fetched then al least at odds with what logic is usually supposed to be. Is it a mere philosophical jugglery which stems from a lack of sense for what (modern) logic really is? I would like to indicate (without giving a systematic account) that it is not; in particular that it is surprisingly consistent with the way logic was treated by some of its founding fathers2. Thus, consider Frege's (1879, p.5) famous introduction of material implication: Wenn A und B beurtheilbare Inhalte bedeuten, so gibt es folgende vier Möglichkeiten: 1) A wird bejaht und B wird bejaht; 2) A wird bejaht und B wird verneint; 3) A wird verneint und B wird bejaht; 4) A wird verneint und B wird verneint. ▐─┬─ A └─ B bedeutet nun das Urtheil, dass die dritte dieser Möglichkeiten nicht stattfinde, sondern eine der drei andern.3 Hence the implication is introduced to spell out, within his Begriffsschrift, what is otherwise articulable only within a metalanguage -- before we have it, we can only display some kind of combination of assertions and denials of A and B, whereas with it we can articulate the combination in terms of a statement. The situation is similar in respect to quantifiers. Frege (ibid., p.19) writes 2
I do not want to say that Brandom's proposal is unprecedential -- to a certain extent it overlaps with the proof-theoretic approaches to logic (see, e.g. Sundholm, 1983), and especially within the German constructivist tradition (Lorenzen and Schwemmer, 1975). But I think Brandom has given it the most thorough philosophical backing.
3
"If A and B stand for contents that can become judgments, there are the following four possibilities: (1) A is affirmed and B is affirmed; (2) A is affirmed and B is denied; (3) A is denied and B is affirmed; (4) A is denied and B is denied. Now ... stands for the judgment that the third of these possibilities does not take place, but one of the three others does."
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In dem Ausdrucke eines Urtheils kann man die rechts von ▐── stehende Verbindung von Zeichen immer als Funktion eines der darin vorkommenden Zeichen ansehen. Setzt man an die Stelle dieses Argumentes einen deutschen Buchstaben, und giebt man dem Inhaltsstriche eine Höhlung, in der dieser selbe Buchstabe steht, wie in a─Φ(a) ▐─∪ so bedeutet dies das Urtheil, daß jene Function eine Thatsache sei, was man auch als ihr argument ansehen möge.4 In this way, a quantified sentence is not a shortcut for a metalinguistic statement: a statement about the results of replacing a part of an object language sentence by various (suitable) expressions. (Hence also to say that some natural language expressions, such as "something" or "everything", can be regimented by such quantifiers, is to say that these expressions are means of making some metalinguistic pronouncements, which hold about the language, explicit within the language.) Russell (1905, p. 480) takes, in this respect, a very close train of thought: ... everything and nothing and something (...) are to be interpreted as follows: C(everything) means 'C(x) is always true'; C(nothing) means '"C(x) is false'' is always true'; C(something) means 'It is false that "C(x) is false'' is always true.' Hence again, sentences with "everything", "nothing" etc. are taken to express metalinguistic pronouncements; and their presence in a language thus enables us to say in the language what holds about the language and what is otherwise only expressible within a metalanguage. In view of all of this I would like, in this paper, to look at the usual logical operators as a means of explicitating inferential structure. I will claim that the need of such an explicitation leads to a set of 'inferentially native' operators, which, within various logical systems, can mutate into various forms; which, however, find their most straightforward embodiment within the intuitionist calculus.
'Inferentially native' operators We have seen that the important precondition of dealing with the question of the logic of inference adequately is the establishment of a general enough framework. This is what we are about to do now. By inferential structure we shall understand the ordered pair <S, ├─>, where S is a set and ├─ is a relation between finite sequences of elements of S and elements S. We will call the elements of S statements of the system and ├─ the inference relation of the 4
"In the expression for a judgment, the complex symbol to the right of ├─ may always be regarded as a function of one of the symbols that occur in it. Let us replace this argument with a Gothic letter, and insert a concavity in the content-stroke, and make this same Gothic letter stand over the concavity, e.g.: ... . This signifies the judgment that the function is a fact whatever we take its argument to be." (Elsewhere I have discussed the consequences of this Fregean notion of quantification for the concept of variable -- see Peregrin, 2000.)
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system. We will employ the letters A, A1, ..., An, B, C ..., to stand for statements, whereas X, Y, Z, ... to stand for finite sequences thereof. An ordered pair <X,A> will be called an inference; it will be called a valid inference iff X ├─ A. In this case A will be also said to be inferable from X. A statement which is inferable from an empty sequence will be called a theorem. The idea is that an inferential system reconstructs a language, in our particular case especially a (proto-)language in which the inference relation is still only implicit. Within such a (proto-)language we can infer statements from other statements, but we cannot say that a statement is inferable from other statements. Suppose that B is inferable from A, i.e. that A ├─ B. What would it mean to say, within the language regimented by the structure, to make this fact explicit? We need a statement which says that B is inferable from A. But what does it take for a statement of such a structure to say this? Presumably to be true iff B is inferable from A. But the relation ├─ is unchanging and hence the explicitating claim would be true necessarily; and what is the counterpart of necessary truth within the structure is clearly theoremhood. Hence to make the inferability of a statement from another statement explicit is to have, for every pair of statements A and B, a statement which is a theorem iff A ├─ B. Let us form such an 'explicitating' statement by means of the sing , hence let, for every A and B, (*) A ├─ B iff ├─ AB. We will call the operator defined in this way the deductor (for the inferential structure). In this way, to claim AB (as a necessary truth) would be to claim that B is inferable from A. Splitting (*) into its two halves and using "/" as a shorthand for "only if" we have DED: A ├─ B / ├─ AB CODED: ├─ AB / A ├─ B However, allows us to say that a statement is inferable from another statement, but not yet that it is inferable from a sequence of statements. An obvious way how to make this explicit is by way of introducing, for every pair of statements A and B, another new statement, say A⊗B, such that X,A,B,Y ├─ C iff X, A⊗B,Y ├─ C, Let us call the new operator ⊗ the amalgamator. The definition of amalgamator can again be given in the two parts: AMLG: X,A,B,Y ├─ C / X, A⊗B,Y ├─ C DEAMLG: X,A⊗B,Y ├─ C / X,A,B,Y├─ C.
7
If X is the sequence A1 A2 ... An-1 An, then we will write ⊗X as the shorthand for (A1⊗(A2⊗(...(An-1⊗ An))). Now it is obviously the case that X ├─ A iff ├─ ⊗X B However, there is also an alternative to the introduction of the amalgamator; we also can make the deductor recursive, by requiring that (**) X,A ├─ B iff X├─ AB, i.e. DED*: X,A ├─ B / X├─ AB * CODED : X├─ AB / X,A ├─ B. Hence both the deductor and the amalgamator are operators which come quite naturally in case we want to make the relation of inference explicit. Are there some other similarly 'inferentially native' operators? Not directly from the viewpoint of inference (alter all we have seen that the two, or even the deductor alone, can make the whole of inference explicit); but we may want to consider another relationship, namely that of incompatibility. In fact Brandom argues that it is incompatibility which lies the ultimate foundation of 'inferential articulation'. Inference, according to him, is merely a superstructure: for A is inferable from X iff whatever is incompatible with A is incompatible with X. If we write ⊥X, to express that X is incompatible, Brandom assumes that X ├─ A iff for every Y and Z it is the case that if ⊥Y,A,Z then ⊥Y,X,Z. If we want, vice versa, to reduce incompatibility to inference, then the best way appear to be the ex falso quodlibet rule: ⊥X iff for every B it is the case that X ├─ B Now what if we wanted to make also incompatibility explicit? What would it mean to say that a statement A is incompatible? Well, of course again to display a statement which is a theorem iff A is incompatible. Hence let us introduce a new operator, say ∅, such that for every statement A it is the case that ⊥A iff ├─ ∅A. Let us call ∅ the complementator. Again, this enables us to say that a statement is incompatible, but not yet that a sequence of statements is. This can be clearly done with the help of amalgamation -- it is obviously the case that 8
⊥X iff ├─ ∅(⊗X). However, just like in the case of the deductor it may also seem natural to make the definition of the complementator recursive, so that ⊥ X,A iff X ├─ ∅A, i.e. COMP: ⊥ X,A / X ├─ ∅A, DECOMP: X ├─ ∅A / ⊥ X,A. This would clearly yield us ⊥ X,A iff X ├─ ∅A, If we now emulate incompatibility in terms of inference, we have COMP*: X,A├─ ⊥ / X ├─ ∅, DECOMP*: X ├─ ∅A / X,A├─ ⊥. Hence it seems that we can think of at least three kinds of 'inferentially native' operators: deductors, amalgamators and complementators. Our goal now is to see to what extent the usual operators of the common logical calculi can be seen as their embodiments.
Inferential structures We have defined inferential structure simply as a set of statements and any kind of relation between finite sequences of statements and statements. But in fact we should be more restrictive. What is our framework supposed to explicate is the fact that language-using creatures would stipulate (not necessarily by an explicit convention; possibly by an establishment of a praxis) inferential rules and thereby establish inferential relationships. This means that what we should concentrate on are structures generated by (finite collections of) rules. Hence the following refinement of our framework. An inference over a set S is an ordered pair whose first constituent (also called its antecedent) is a finite sequence of elements of S and whose second constituent (called its consequent) is an element of S. (Hence an inferential relation, as defined earlier, is a set of inferences.) A metainference over S is an ordered pair whose first constituent (again called its antecedent) is a finite sequence of inferences over S and whose second constituent (its consequent) is an inference over S. If P is a set ('of parameters'), then a (P-)inferential rule (or simply a rule) over S is an inference over S in which some elements of S are replaced by elements of P. An instance of a rule over S is any inference over S which can be obtained from the rule by a systematic replacement of the elements of P by elements of S. The consequent of an instance of a rule will be also referred to as the value of the rule for the arguments which constitute the antecedent of the instance. Thus, if A and B are parameters, then <,A > is a rule, its 9
value for an argument A⊗B being A. We will often present rules in the form X ├─ A instead of <X,B>. Analogously, a (P-)metainferential rule (or simply a metarule) over S is a metainference over S with some elements of S in its constituent inferences replaced by those of P. An instance of a metarule over S is any metainference over S which can be obtained from the metarule by a systematic replacement of the elements of P by elements of S. And again, the consequent of an instance of an metarule will be also referred to as the value of the metarule for the arguments which constitute the antecedent of the instance. Thus, if A and B are parameters, then <<<,B>, <,C>>, <,C>> is a metarule, its value for the arguments <,B> and <,C> being <,C>. We will usually present rules in the form X1 ├─ A1, ..., Xn ├─ An / X ├─ A instead of <<<X1,A1>, ...,<Xn,An>>,<X,A>>. An inferential basis is an ordered triple F = <S,I,M>, where S is a set, I is a finite set of inferential rules over S, and M is a finite set of metainferential rules over S. The notion of inference valid in F is defined in the following obvious recursive way: (i) an instance of an inferential rule from I is valid; (ii) if <,I> is an instance of a metainferential rule from M and all of I1,..., In are valid, then so is I; (iii) nothing else is an inference valid in F. We will say that A is a theorem of F if <<>,A>, where <> is the empty sequence, is an inference valid in F. We will say that F generates the structure consisting of S plus the relation consisting of all the inferences valid in F. Elements of I will be called the basic rules of the generated structure, whereas those of M are called its basic metarules. As we will deal with inferential structures only via their generating bases, we will often refer to bases directly as structures (meaning the structures generated by them). An inferential rule is a valid rule of the structure F iff all its instances are valid in F. If <X,A> is a valid rule of F, then we will write X ├─F A; and we will leave out the subscript accompanying ├─ where no confusion would be likely to arise. A metainference is valid in F iff either at least one of the inferences in its antecedent is not a valid rule of F, or the inference in its consequent is. A metarule is a valid metarule of the structure F iff all its instances are valid in F. If <,I> is a valid metarule of F, then we will write I1,..., In /F I; and we will again leave out the subscript where feasible. Hence from now on we restrict our attention to finitely generated inferential structures. But this may still not be restriction enough -- for we are not interested in relations generated from the basic rules in just any way, but rather in the definite relation of 'provability by means of the basic rules'. And as is easily seen, the relation of provability by means of the collection I of rules is precisely the closure of I w.r.t. the Gentzenian ('structural') metainferential rules5: (REF) / A ├─ A (EXT) X,Y ├─ A / X,B,Y ├─ A (CON) X,A,A,Y ├─ B / X,A,Y ├─ B (PERM) X,A,B,Y ├─ C / X,B,A,Y ├─ C (CUT) X,A,Y ├─ B; Z├─ A / X,Z,Y ├─ B 5
The reason why we take also the first one as a metarule (with the empty antecedent), rather than a rule, is nothing more that the simplicity of exposition. Nothing relevant for our considerations is affected by this move.
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Again, let us make things more precise. Let S be a set and I a collection of inferential rules over S. A proof of A from X by means of I (where A∈S and X is a finite sequence of elements of S) is a finite sequence of elements of S which ends with A and whose every element is either an element of X or is the value of a rule from I for some arguments which all occur within the sequence earlier. Now we can formulate Theorem 1. A is provable from X by means of I iff X├─ A is derivable from I by means of (REF), (EXT), (CON), (PERM) and (CUT). Proof: Let A be provable from X by means of I. Then there is a sequence A1,...,An of statements such that An = A and every Ai is either an element of X or is inferable by a rule from I from statements which are among A1,...,Ai-1. If n=1, then there are two possibilities: either A∈X and then X├─ A in force of REF and EXT; or A is a consequent of a rule of I with a void antecedent, and then ├─ A and hence X├─ A in force of EXT. If n>1 and An is inferable from some Ai1,...,Aim (where i1). (An inferential structure whose set of basic metarules is a proper superset of {(REF), (EXT), (CON), (PERM), (CUT)} (where it contains at least one metarule with an empty antecedent over and above the structural ones) can be called superstandard; whereas that whose set of basic metarules is its proper subset can be called substandard -- but we will have very little to say about such structures in this paper.) Hence our next task is to try to turn the metainferential rules we used to characterize our native operators into inferential ones.
Deductor Let us start with the deductor. It is easily seen that if we restrict ourselves to standard structures, CODED* becomes equivalent to the modus ponens inferential rule: MP: AB,A ├─ B. This is spelled out by the following theorem: 11
Theorem 2. Within a standard inferential structure, CODED* is valid iff MP is valid. Proof: CODED* ⇒ MP MP ⇒ CODED* 1. AB ├─ AB REF 1. X├─ AB assumption * 2. AB,A ├─ B MP 2. AB,A ├─ B from 1 by CODED 3. X,A ├─ B from 1 and 2 by CUT □ However, things are not that easy with DED*: there does not appear to be an inferential rule or a collection of inferential rules which would be equivalent to it. However, there is nevertheless a possibility of securing the validity of DED* by stipulating inferences. The point is that as what DED* claims is that whenever X,A ├─ B, then also X├─ AB, it would be enough to stipulate X├─ AB for every particular case when X,A ├─ B. And as X,A ├─ B just in case it either is a basic inference, or is derivable from basic inferences by means of the structural metarules, it would be enough to stipulate this for every basic inference and to secure that this property is preserved by all metainferences. Let us call the statement A1(A2(... (AnB))) the internalization of the inference A1,...,An├─ B. Using A1,...,AnB as a shorthand for A1(A2(... (AnB))), we can abbreviate the internalization of A1,...,An├─ B to A1,...,AnB. An inference A1,...,An├─ B will be called internalized if its internalization is a theorem, i.e. iff ├─ A1,...,AnB. Now it holds Theorem 3. Let S be a standard inferential structure in which MP holds. Then DED* holds in it iff every its basic rule is internalized and every its basic metarule preserves internalizedness, i.e. maps internalized inferences always also only on inferences that are internalized. Proof: Let DED* hold. Then, in view of Theorem 2, an inference is internalized iff it is valid. In particular, all basic inferences are internalized, and all inferences on which metarules map internalized, and hence valid, inferences are internalized. Conversely, if all basic inferences are internalized and all basic metarules preserve internalizedness, then every valid inference is obviously internalized. □ What does it take for the structural metarules to preserve internalizedness? The following theorem offers an answer: Lemma 1. (REF), (EXT), (CON), (PERM) and (CUT) preserve internalizedness iff (A1)-(A5) hold: (A1) ├─ AA (A2) ├─ B / ├─ AB (A3) ├─ A,AB / ├─ AB (A4) ├─ X,A,B,YC / ├─ X,B,A,YC (A5) ├─ XA; ├─ AB / ├─ XB Proof: A straightforward transcription of what it takes for (REF), (EXT), (CON), (PERM) and (CUT) to preserve internalizedness yields: (A1) ├─ AA (A2*)├─ X,YB / ├─ X,A,YB (A3*)├─ X,A,A,YB / ├─ X,A,YB (A4)├─ X,A,B,YC / ├─ X,B,A,YC 12
(A5*)├─ XA; ├─ Y,A,ZB / ├─ Y,X,ZB Now it is easy to see that in view of (A4), we can reduce (A2*), (A3*) and (A5*) to the simpler (A2), (A3) and (A5). □ However, from the viewpoint of our aim, namely reduction of metarules to rules, Theorem 4 is not too useful. (A2)-(A5) are still metarules; so we are as yet no better than with DED* itself. The following theorem gives rules which are capable of guaranteeing the validity of these metarules: Theorem 4. If a standard inferential structure fulfills MP plus (1)├─ B(AB) (2)├─ (A(AB))(AB) (3)├─ (CA)((AB)(CB)) then (REF), (EXT), (CON), (PERM) and (CUT) preserve internalizedness in it. Proof: In view of Lemma 1 it is enough to prove that such structure fulfills (A1)-(A5). It is clear that the validity of (A2*), (A3*), (A4), (A5*) is secured by (B2) B├─ AB; (B3) A,AB├─ AB; (B4) X,A,B,YC├─ X,B,A,YC; and (B5) XA, AB├─ XB, and hence by (C2) ├─ B(AB); (C3) ├─ (A,AB) (AB); (C4) ├─ (X,A,B,YC)(X,B,A,YC); and (C5) ├─ (XA)((AB)(XB)). Now it can easily beseen that (C4) can be reduced to (C4*), whereas (C5) to (C5*): (C4*) ├─ (A,BC)(B,AC); (C5*) ├─ (CA)((AB)(CB)); and that (A1) and (C4*) then follow from (C2), (C3) and (C5*).□ Hence if (1), (2) and (3) hold, then DED* holds once all basic inferences are internalized. This means that given a standard inferential structure, we can make into a deductor by stipulating MP, (1), (2) and (3) and by replacing every basic inference X├─ A by ├─ XA. (As ├─ XA yields X├─ A via MP, once we add the former, we can delete the latter.)
Persistence Hence we have seen that within a standard inferential structure, is a deductor if MP, (1), (2) and (3) and the internalizations of all basic inferences hold. However, we cannot say that it is a deductor only if this is the case, for (B2)-(B5) are merely a sufficient, not a necessary condition of (A2)-(A5). Nevertheless as we are going to show in this section, if we strengthen the definition of deductor (in a way which appears to be desirable independently), the sufficient condition becomes also a necessary one. We have said that an inferential structure is to reflect a structure of a real language. However, structures of real languages are often considered on various levels of abstraction. 13
We can, for example, treat statements as indecomposable wholes (as done within propositional logic) and then we may refine the analysis to consider also the structure of statements and inferences resulting from this structure (as within the predicate one). Hence we may want to consider various extensions of a given inferential structure, and consequently may want some properties of the structure not to be destroyed by (well-behaved) extensions. We will call a property of an inferential structure persistent if it pertains to all standard extensions of the structure. It seems that we should require that the deductor (as well as other operators) be persistent in this sense: that if something is a deductor of a structure, it should keep to be a deductor of any refinement of the structure. Now what we are going to show is that (1), (2) and (3) are precisely both necessary and sufficient condition for being a persistent deductor. Again, let us be more precise. A (standard) structure F* is an extension of a (standard) structure F iff every basic rule of F is a basic rule of F* (whereby ├─ F ⊆ ├─F *). A deductor of F is called persistent if it keeps to be a deductor of any standard extension of F. It is easy to see that (A1)-(A5) are not sufficient to make persistent; but we will show that (A1), (B2)(B5) are not only sufficient, but also necessary. Hence if we modify our concept of deductor in such a way that it incorporates persistence, then internalizedness of inferences will be preserved by the structural metarules if and only if (A1), (B2)-(B5), hence iff (1), (2) and (3), are valid; and hence will be a deductor iff MP, (1), (2) and (3) and the internalizations of all basic inferences are valid. Let us first prove an auxiliary statement: Lemma 2. If I is a collection of rules and if I* is I plus the rule ├─ A, then A ├─I B iff ├─I* B. Proof: We have seen that A ├─I B iff there is a proof of B from A by means of I. Now it is easy to see that such a proof exists if and only if there is a proof of A from nothing by means of I*; hence such a proof exists iff ├─I* B. □ Now we are ready to establish the necessary and sufficient conditions for being a persistent deductor: Theorem 5. is a persistent deductor of <S,I> iff (A1), (B2)-(B5) hold within <S,I> (i.e. iff (1), (2) and (3) do). Proof: If (A1), (B2)-(B5) hold in F= <S,I>, then they hold in every extension of <S,I>, for each of them is a rule (rather than a metarule) and hence is not affected by adding other inferences. This means that keeps to be a deductor of every extension, and especially of every standard extension, of F. Now let, conversely, (A1), (B2)-(B5) not hold in <S,I>. We will show that then there is a standard extension <S,I*> of <S,I> in which is not a deductor. Let us assume that it is (B2) which fails in <S,I> (the other cases are analogous). This means that there are some A and B such that it is not the case that B├─ AB. Let I* be I plus the rule ├─* B. Then, if <S, ├─> is the structure generated by <S,I> and <S, ├─*> that generated by <S,I*>, also A├─* B. If were a deductor of <S, ├─*>, it would have to be the case that ├─* AB. However, according to lemma 2├─* A iff B├─ A; so B├─ AB, which contradicts our assumption. □ From now on, we will understand take deductor to mean persistent deductor. (We have seen that this is a reasonable strenghtening of the concept.) Hence we can say that within a structure with no basic rules, is a deductor iff MP plus (1), (2) and (3) hold. 14
What we have reached in this way is the axiomatization of the so called positive logic6. It constitutes an axiomatization of the purely implicative part of the intuitionist propositional calculus; and also of the purely implicative part of the classical propositional calculus -- if “implicative part” is interpreted as referring to as meaning everything provable from purely implicative axioms. (Should it be interpreted as referring to all purely implicative theorems, then the situation would be different, for within classical logic, unlike within the intuitionist one, negation is not conservative over implication, and hence the class of implicative theorems overruns the class of statements provable from purely implicative axioms.)7
Amalgamator Now let us consider the amalgamator. It is not difficult to see that within a standard inferential structure, AMLG becomes equivalent to ICN whereas DEAMLG to ECN1 plus ECN2: ICN: ECN1: ECN2:
A,B ├─ A⊗B A⊗B ├─ A A⊗B ├─ B
Theorem 7. Within a standard inferential structure, AMLG and DEAMLG hold iff ICN, ECN1 and ECN2 do. Proof: ICN ⇒ DEAMLG DEAMLG ⇒ ICN 1. A⊗B ├─ A⊗B REF 1. X,A⊗B,Y ├─ C assumption 2. A,B├─ A⊗B from 1 by DEAMLG 2. A,B├─ A⊗B ICN 3. X,A,B,Y ├─ C from 2, 1 by CUT ECN ⇒ AMLG 1. X,A,B,Y ├─ C assumption 2. A⊗B├─ A ECN1 3. A⊗B├─ B ECN2 4. X,A⊗B,A⊗B,Y ├─ C from 1, 2, 3 by CUT 5. X,A⊗B,Y ├─ C from 4 by CON
AMLG ⇒ ECN1 1. A├─ A 2. A,B├─ A 3. A⊗B├─ A
REF from 1 by EXT from 2 by AMLG
AMLG ⇒ ECN2 1. A├─ A 2. A,B├─ B 3. A⊗B├─ B
REF from 1 by EXT from 2 by AMLG
□ 6 7
Hilbert and Bernays (1939, Supplement III). Suppose we suspend the structural metarule (CON). As the result, we will have only ├─ AZA ├─ BZ(AZB) ├─ (CZA)Z((AZB)Z(CZB))
In this way we reach the implication of what is sometimes called BCK logic (see, e.g., Došen, 1993), and also what constitutes the purely implicative part of Wajsberg's (1931) axiomatization of Łukasiewicz's (1930) three-valued calculus.
15
Now if we have also the deductor, we have a way of making all inferential rules, with the exception of modus ponens, explicit, i.e. of reducing them to axioms. (And, in fact, we must do so in order to keep DED* valid.) As for the ones constitutive of the amalgamator, i.e. ICN, ECN1 and ECN2, they are transformed into8 (⊗1) ├─ A(B(A⊗B)) (⊗2) ├─ A⊗BA (⊗3) ├─ A⊗BB
Complementator The situation is more complicated with the complementator, whose definition given by COMP* and DECOMP* cannot be turned into axioms so straightforwardly. However, we can reduce DECOMP* to DECOMP**: ∅A, A ├─ B: Theorem 8. Within a standard inferential structure, DECOMP** holds iff DECOMP* does. Proof: DECOMP** ⇒ DECOMP* DECOMP* ⇒ DECOMP** 1. X ├─ ∅A assumption 1. ∅A ├─ ∅Α REF 2. X,A├─ B 2. ∅A, A ├─ B from 1 by DECOMP* from 1 and DECOMP** by CUT □
Moreover, in the presence of a deductor, we can reduce COMP* to COMP** (to simplify formulas, we accept the usual assumption that the unary ∅ takes precedence over the binary operators): COMP**: AB, A∅B ├─ ∅ A Theorem 8. Within a standard inferential structure with a deductor, COMP** holds iff COMP* does.
8
If we suspend any structural rule other than PERM, the equivalence between AMLG+DEAMLG and ICN+ECN1+ECN2 will fail. However, as long as REF and CUT hold, the concept of amalgamator will coincide with what has come to be called fussion within the theory of substructural logics (Restall, 2000): IF: X├─ A; Y├─ B / X,Y├─ A⊗B EF: X├─ A⊗B; Y,A,B,Z ├─ C / Y,X,Z├─ C
The proof is easy (and can be found in Peregrin, 2004).
16
Proof: COMP** ⇒ COMP* 1. X,A├─ B from the assumption 2. X├─ AB from 1 by DED* 3. X,A├─ ∅B from the assumption 4. X├─ A∅B from 3 by DED* 5. X,X ├─ ∅A from 1, 3 and COMP** by CUT 4. X ├─ ∅A from 5 by PERM and CON □
COMP* ⇒ COMP** 1. AB,A ├─ B MP 2. A∅B,A ├─ ∅B MP 3. A∅Β,A,AB,A ├─ C from 1, 2 and DECOMP** by CUT 4. AB, A∅B,A ├─ C from 3 by PERM and CON 5. AB, A∅A ├─ ∅A form 4 by COMP*
Finally we must recast DECOMP** and COMP** as (∅1)├─ ∅Α(AB) (∅2)├─ (AB)((A∅Β)∅Α)
Conclusion Let us summarize the inferences which are required to constitute our three native inferential operators within the context of a standard inferential structure. is a (persistent) deductor, ⊗ is an amalgamator and ∅ is a complementator iff the following inferences hold (MP) (1) (2) (3) (⊗1) (⊗2) (⊗3) (∅1) (∅2)
A,AB├─ B ├─ B(AB) ├─ (A(AB))(AB) ├─ (CA)((AB)(CB)) ├─ A(BA⊗B) ├─ (A⊗B)A ├─ (A⊗B)B ├─ ∅A(AB) ├─ (AB)((A∅B)∅A)
It is easy to see that they make up an axiomatization of the intuitionist propositional calculus. Hence what we have shown is that the native inferential operators coincide (within the 'normal' environment, i.e. within standard inferential structures) with the intuitionist ones9. Hence in this sense, it is intuitionist logic which is the logic of inference. What about classical logic? As is well known, it can be derived from intuitionist logic by the addition of an axiom which, in our notation, reads ├─ ∅∅AA.
9
And we have seen, though only in terms of some not quite systematic hints, that within various 'abnormal' (= substructural) environments they acquire more bizarre forms.
17
Is there an inferentialist substantiation of this axiom? I do not see one (which, of course, does not mean that none exists10). However, this axiom appears to provide for an 'alignment' of the intuitionist calculus, which leads to a considerable simplification.
References Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Došen, K. (1993): 'A historical introduction to substructural logics', in K. Došen and P. Schröder-Heister, eds.: Substructural Logics, Clarendon Press, Oxford. Haack, S. (1996): Deviant Logic, Fuzzy Logic, University of Chicago Press, Chicago. Hilbert, D. and P. Bernays (1939): Grundlagen der Mathematik, Springer, Berlin. Lorenzen, P. & Schwemmer, O. (1975): Konstruktive Logik, Ethik und Wissenschaftstheorie, Bibliographisches Institut, Mannheim. Łukasiewicz, J. (1930): ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’, Compt. Rend. Soc. d. Sc. Let. Varsovie, Cl. III 23, No. 51-77. Milne, P. (2002): 'Harmony, Purity, Simplicity and a "Seemingly Magical Fact"', Monist 85, 498-534. Peregrin, J. (2000): 'Variables in Natural Language: Where Do They Come From?', in Variable-free Semantics (eds. ed. M. Böttner & W. Thümmel), Secolo, Osnabrück, 4665. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Peregrin, J. (2004): ‘Logic as “Making it Explicit”’, to appear in The Logica Yearbook 2003, Filosofia, Prague. Prawitz, D. (1965): Natural Deduction, Almqvist & Wiksell, Stockholm. Quine, W.V.O. (1986): Philosophy of Logic, Harvard University Press, Cambridge (Mass.). Restall, G. (2000): Introduction to Substructural Logics, Routledge, London. Russell, B. (1905): 'On denoting', Mind 14, 479-493. Sundholm, B. G. (1983): 'Constructions, Proofs and the Meaning of the Logical Constants', Journal of Philosophical Logic 12, 151-172. Tarski, A. (1930): 'Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I', Monatshefte für Mathematik und Physik 37; English translation in Tarski (1956), 60-109. Tarski, A. (1956): Logic, Semantics, Metamathematics, Clarendon Press, Oxford. Tennant, N. (1997): The Taming of the True, Oxford University Press, Oxford. Wajsberg, M. (1931): 'Aksjomatyzacja trójwartościowego rachunku zdan', Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie, cl.iii 24, No. 126145. Zucker, J.I. & R.S. Tragesser (1978): ‘The Adequacy Problem for Inferential Logic’, Journal of Philosophical Logic 7, 501-516.
10
Cf. the discussion given by Milne (2002).
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MEANING AS AN INFERENTIAL ROLE Jaroslav Peregrin www.cuni.cz/~peregrin
The inferentialist tradition Contemporary theories of meaning can be divided, with a certain amount of oversimplification, into those which see the meaning of an expression as principally a matter of what the expression denotes or stands for, and those which see it as a matter of how the expression is used. A prominent place among the latter ones is assumed by those which identify the semantically relevant aspect of the usage of an expression with an inferential pattern governing it. According to these theories, the meaning of an expression is, principally, its inferential role. Brandom (1985 p.31) characterizes the inferentialist tradition (which, according to him, goes as far back as to Leibniz) in the following way: The philosophical tradition can be portrayed as providing two different models for the significances which are proximal objects of explicit understanding, representational and inferential. We may call ‘representationalism’ the semantically reductive view that inference is to be explained away in favor of more primitive representational relations. ... By ‘inferentialism’, on the other hand, one would mean the complementary semantically reductive order of explanation which would define representational features of subsentential expressions in terms of the inferential relations of sentences containing them. Various degrees of commitment to inferentialism can be found also within the writing of some of the founding fathers of analytic philosophy. Thus, Frege’s first account for the concept of “conceptual content”, which he presents in his Begriffsschrift (1879, p. 2-3), is distinctively inferentialist: The contents of two judgments can differ in two ways: first, it may be the case that [all] the consequences which may be derived from the first judgment combined with certain others can always be derived also from the second judgment combined with the same others; secondly this may not be the case. ... I call the part of the content which is the same in both the conceptual content. Similarly Wittgenstein assumed a distinctively inferentialist standpoint in a particular stage of the development of his thought from the Tractarian representationalism to the more inclusive use theory of meaning of the Investigations. Thus, in his Remarks on the Foundation of Mathematics (1956, pp. 24, 398) we can read: 1
The rules of logical inference cannot be either wrong or right. They determine the meaning of the signs. ... We can conceive the rules of inference – I want to say – as giving the signs their meaning, because they are rules for the use of these signs. Recently, the philosophical foundations of inferentialism have been elaborated especially by Brandom (1994, p. 144): It is only insofar as it is appealed to in explaining the circumstances under which judgments and inferences are properly made and the proper consequences of doing so that something associated by the theorist with interpreted states or expressions qualifies as a semantic interpretant, or deserves to be called a theoretical concept of a content. Hence, according to Brandom (2000, p. 30), the inferentialist semantic explanations beginning with properties of inference, they explain propositional content, and in terms of both go on to explain the conceptual content expressed by subsentential expressions such as singular terms and predicates1. All of this indicates that the idea of identifying meanings with inferential roles is worth investigating. However, its viability has been often challenged (see, e.g. Prior, 1960/61, or Fodor and LePore, 1993). In this paper I will neither argue for inferentialism as a philosophical position, nor discuss its philosophical foundations (I have done so elsewhere – see esp. Peregrin, 2001, Chapter VIII; 2004b). I will rather focus on the question (*) Which kinds of meanings can be conferred on words by means of inferential rules?, and especially I will try to establish a formal framework in which it could be answered. Then I will also offer some fragments of the answer. The intuition behind the question is relatively clear. We can easily furnish ∧ with its standard meaning by stipulating the inferential pattern A∧B ├─ A A∧B ├─ B A,B ├─ A∧B whereas it is far from clear how we could do the same for ∨ or →. Hence there is a suspicion that inferentialism may stop short already before we come to something as platitudinous as disjunction or implication.
1
See also Lance (1996; 2001) and Kaldernon (2001).
2
Note that we do not presuppose that for each word there must be a meaning-conferring inferential pattern independent of those of other words. We do not exclude the possibility that the meaning of a word is specifiable only in a mutual dependence with meanings of other words - i.e. that the pattern constitutive of the meaning of a word involves other words. From this viewpoint it might be better to talk, more generally, about furnishing semantics for a language than about conferring meaning on a single word. (Establishing semantics for a language is conferring meanings on all its words; whereas conferring meaning on one of its words may be inextricable from conferring meanings on other words.) Hence it may be better if we reformulate the above question as (**) Which kinds of semantics are determined by inferential rules? However, to be able to deal with this question a rigorous way, we have to clarify or explicate all the terms which occurs in it; and this is the theme for the upcoming sections. Let us start from the term "inferential".
Inference and incompatibility According to Brandom (1994; 2001), the working of language can be understood in terms of the practice of giving and asking for reasons and of the kinematics of the deontic statuses of commitment and entitlement which this praxis institutes. The most basic speech act, assertion, commits the speaker to giving reasons for what she asserts; and it entitles her hearers to reassert it deferring its justification to the original asserter. From this viewpoint, what we usually called inference is a matter of commitmentpreservation. If I assert ‘Fido is a dog’, I commit myself to maintaining not only this very claim, but also to various other claims, like ‘Fido is a mammal’ or ‘Fido is not a human’ – and this is what it means to say that the latter sentences are entailed by the former one.2 (I will use is entailed and is (correctly) inferable interchangeably.) A (strong) inferential structure is an ordered pair <S, ├─>, where S is a set whose elements are called statements and ├─ is a relation between finite sequences of elements of S and elements of S. If the sequence A1,...,An of statements is in the relation ├─ to the statement A then we will write A1,...,An ├─ A.
2
Brandom (2000, p. 194 ff.) urges that aside of commitment-preservation we should consider the different relation of entitlement-preservation and also what he calls “incompatibility entailment” – but I find his exposition of this theme so unclear (why, for example, should the relation of commitmentpreservation be extensionally different from that of entitlement-preservation?) that I restrict my attention to the first concept.
3
We will use the letters A,A1,A2, ..., B,C for statements, the letters X, Y, Z for finite sequences thereof, and U, V for sets of statements. If X is a sequence of statements, then X* will be the set consisting of all its constituent statements. We define Cn(U) = {A | there is a sequence X such that X*⊆U and X ├─ A} We will say that U is closed if Cn(U) = U. We will say that <S, ├─> is standard iff for every X, Y, Z, A, B, C: (REF) A ├─ A (EXT) if X,Y ├─ A, then X,B,Y ├─ A (CON) if X,A,A,Y ├─ B, then X,A,Y ├─ B (PERM) if X,A,B,Y ├─ C, then X,B,A,Y ├─ C (CUT) if X,A,Y ├─ B and Z├─ A, then X,Z,Y ├─ B The properties of ├─ spelled out by these schemas will be also called reflexivity, extendability, contractibility, permutability and transitivity. (The schemas are also known as identity, thinning, contraction, permutation and cut.) Brandom claims that the level of inferences is still not the ground level, that the concept of inference should be reduced to a more primitive concept, namely the concept of incompatibility. That A is correctly inferable from X means, according to him, that whatever is incompatible with A is incompatible also with X. Hence let us consider the relation of incompatibility. An incompatibility structure is an ordered pair <S, ⊥>, where S is a set of statements and ⊥ is a set of finite sequences of elements of S. If the sequence A1,...,An belongs to ⊥, we will write ⊥ A1,...,An. We will say that a set U of statements is consistent if X*⊆U for no sequence X such that ⊥X. We will say that <S, ⊥> is standard iff for every X, Y, Z, A, B, C (i) if ⊥ X,Y, then ⊥ X,A,Y (ii) if ⊥ X,A,A,Y, then ⊥ X,A,Y (iii) if ⊥ X,A,B,Y, then ⊥ X,B,A,Y Let <S,├─> be an inferential structure. Let us define ⊥ as follows: ⊥ X ≡Def. X ├─ A for every A.
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The resulting incompatibility structure <S, ⊥> will be called induced by <S,├─>. Let conversely <S, ⊥> be an incompatibility structure. Let X ├─ A ≡Def. ⊥ Y,X,Z for every Y and Z such that ⊥ Y,A,Z. The resulting inferential structure <S, ├─> will be called induced by <S, ⊥>. Theorem 1. If an inferential structure is standard, then the incompatibility structure induced by it is standard. If an incompatibility structure is standard, then the inferential structure induced by it is standard. Proof: Most of it is trivial, so let us prove only that if an incompatibility structure is standard, then the induced inference is transitive. Hence we have to prove that if (i) ⊥ W,X,A,Y,W’ for every W and W’ such that ⊥ W,B,W’, and (ii) ⊥ W,Z,W’ for every W and W’ such that ⊥ W,A,W’, then (iii) ⊥ W,X,Z,Y,W’ for every W and W’ such that ⊥ W,B,W’. It is clear that (ii) is equivalent to (ii’) ⊥ W,X,Z,Y,W’ for every W, X, Y and W’ such that ⊥ W,X,A,Y,W’ and hence to (ii’’) for every X and Y it is the case that ⊥ W,X,Z,Y,W’ for every W and W’ such that ⊥ W,X,A,Y,W’. And it is clear that (iii) is a consequence of (i) and (ii’’). □ A generalized inferential structure (gis) is an ordered triple <S, ├─, ⊥>. It is called standard iff <S, ├─> and <S, ⊥> are standard and if, moreover, if ⊥ X, then X ├─ A for every A. if X ├─ A then ⊥ Y,X,Z for every Y and Z such that ⊥ Y,A,Z. A standard gis is called perfect, if moreover
5
if X ├─ A for every A, then ⊥ X (i.e. ⊥ is induced by ├─) if ⊥ Y,X,Z for every Y and Z such that ⊥ Y,A,Z, then X ├─ A (i.e. ├─ is induced by ⊥). Thus, in a perfect structure, incompatibility is reducible to inference (X is incompatible iff everything is inferable from it) and vice versa (A is inferable form X iff everything which is incompatible with A is also incompatible with X). Let us now prove one more general result concerning standard gis's. Theorem 2. Let <S, ├─, ⊥> be a standard gis. Then Cn(X*) is inconsistent only if ⊥ X. Proof: Let Cn(X*) be inconsistent. This means that there exists a sequence Y = A1...An of statements such that Y⊆Cn(X*) and ⊥Y. This further means that there exist X1...Xn so that Xi⊆X* and Xi├─Ai. But due to the extendability and permutability of ├─, it follows that X├─Ai. Thus, whatever is incompatible with A1 must be incompatible with X; hence ⊥ X A2, ..., An, and hence, in force of the permutability of ⊥ it is the case that ⊥ A2,...,An,X. Then, as whatever is incompatible with A2 is incompatible with X, it is the case that ⊥ A3, ..., An,X,X, and so on. Ultimately, ⊥ X,...,X, and in force of the contractibility of ⊥ it is the case that ⊥ X. □
Inference and truth preservation Suppose we have a set V of truth valuations of elements of S. i.e. a subset of {0,1}S. (Thus, valuations can be identified with subsets of S.) The pair <S, V> will be called a semantic system. Then we can define the relation ╞═ of entailment and the property ╨ of incompatibility as follows: X ╞═ A iff v(A) = 1 for every v∈V such that v(B) = 1 for every B∈X ╨ X iff for no v∈V it is the case that v(B) = 1 for every B∈X Then <S, ╞═, ╨> is a gis; and we will say that it is the gis of <S,V>. It is easily checked that this gis is standard. Let us call a gis <S, ├─, ⊥> truth-preservational if there is a V such that the structure is of <S,V>. We have seen that standardness is a necessary condition of truth-preservationality; now we will show that it is also a sufficient condition - hence that a gis is truth-preservational iff it is standard. Theorem 3. A gis is truth-preservational if it is standard. Proof: Let V be the class of all closed and consistent subsets of S. We will prove that then X ├─ A iff X ╞═ A. The direct implication is straightforward: if X ├─ A and X*⊆U for some
6
U∈V, then A∈Cn(U) and hence, in force of the fact that U is closed, A∈U. So we only have to prove the inverse implication. Hence let X ╞═ A. This means that whenever U∈V and X*⊆U, A∈V; i.e. that A∈U for every U such that (i) X*⊆U (ii) U is consistent (i.e. Y*⊆U for no Y such that ⊥Y) (iii) U is closed (i.e. Cn(U) = U). As ├─ is reflexive, X*⊆Cn(X*). As it is transitive, Cn(Cn(X*)) = Cn(X*). This means that Cn(X*), in the role of U, satisfies (i) and (iii), and hence if it is consistent, then A∈Cn(X*). As a consequence we have: either A∈Cn(X*), or Y*⊆Cn(X*) for some Y such that ⊥Y. In both cases it must be the case that Z├─ A for some sequence Z whose all members belong to X*. Due to the extendability and contractibility of ├─, this means that Y ├─ A for some sequence Y with the same elements as X and hence, due to the permutability of ├─, X├─ A. Thereby the proof is finished. □ This means that a structure is truth-preservational if and only if it is standard; and this gives us a strong reason to restrict our attention to standard inferential structures: for is not truthpreservation what logic is about? (For an inferentialist, truth-preservation is not prior to inference, but even he would probably want to have inference explicable as truthpreservation.) Are there some reason to investigate non-standard inferential structures? Well, there is surely an algebraic reason: we are always interested in investigating structures in the neighborhood of an interesting one. But is there a logical reason? Note that far from all statements we use in natural language do have truth-values out of contexts. Some logicians conclude that logic has to restrict itself to those which do have it, or at least to those which can be idealized in this way. But we may also notice that the needed context for a statement can be delivered by another statement; that though statement He is bold does not have a truth value by itself, it acquires one when following The king of France is wise. Hence we may say that The king of France is wise followed by He is bold entails The king of France is bald, but this statement is surely not entailed by He is bold followed by The king of France is wise. This gives us a reason to investigate inferential structures in which permutation fails. We have addressed this issue (though in not more than a sketchy way) elsewhere (see Peregrin, 2004c); for now we will concentrate on standard structures.
Expressive resources of semantic systems From the inferentialist viewpoint, we establish semantics by means of inferences. However, from a more common viewpoint, semantics is something which is here prior to inferences and
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we use inferences only to 'capture' it. From the latter viewpoint, the acceptable truthvaluations of a semantic system delimit what is possible, i.e. represent 'possible worlds', and a statement can be semantically characterized by the class of worlds in which it is true. Hence classes of possible worlds are potential semantic values of statements; and languages may differ as to their 'expressive power'. Take the system <{A,B},{{A,B},{A},{B},∅}>, i.e. a system with two statements and all possible truth-valuations. If we number the valuations in the order in which they are listed, we can see that the statement A belongs to 1 and 2, whereas B belongs to 1 and 3; hence if we switch to the possible-world-perspective, that A expresses {1,2}, whereas B expresses {1,3}. There is no statement expressing {1}, or {1,4}, or, say, {2,3,4}. This can be improved by extending the language: we can, for example, add a statement C expressing {1}: <{A,B,C},{{A,B,C},{A},{B},∅}>3. To make these considerations more rigorous, we need some more terminology. Let F = <S,V> be a semantic system. For every statement A from S, let |A| denote the set of all and only elements of V which contain A; hence let |A| ≡Def. {U∈V | A∈U} A subset U' of U is called expressible in F iff there is an A∈S so that |A| = U'. F is called (fully) expressible iff every element of S is expressible. F is called Boolean expressible iff the complement of any expressible set is expressible and the union of any two expressible sets is again expressible. (It is obvious that if a semantic system is Boolean expressible, then its statements can be looked at as constituting a Boolean algebra.) Fully expressible systems constitute a proper subset of the set of semantic systems; however, in some respects we may want to restrict our attention to them. The point is that it seems that if our ultimate target are natural languages, then we should not take a lack of expressive resources too seriously. A natural language may, for various contingent reasons, lack some words and consequently some sentences, but this does not seem to be a matter of its 'nature' - natural languages are always flexible enough to create new the needed expressible resources in stride4. Therefore it may seem often reasonable to simply presuppose full expressibility, or at least something close to it (like Boolean expressibility). Fully expressible systems have some properties which not all semantic systems have. And example is spelled out by the following theorem:
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To be precise, we now have different truth-valuations, for now we evaluate three instead of two statements. So "expressing (1)" should be read as "expressing a truth-valuation which yields (1) when restricted to {A,B}".
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Consider the recurring discussions about substitutional vs. objectual quantification. It might seem that the substitutional quantification is insufficient for the analysis of language; for why should we assume that all entities (including those not known to anybody) must have names? However, what is plausible is assuming not that every entity is named, but that it is nameable - in the sense that language has the resources to form the name as soon as it becomes needed (cf. Lavine, 2000).
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Theorem 4. The gis of an expressible semantic system is perfect. Proof: Let X ╞═ A for every A. Let ⊥ be an element of S such that |⊥| = ∅. Then X ╞═ ⊥ and hence X cannot be part of any element of V which does not contain ⊥. But as ⊥ belongs to no such element, neither X can, and hence ⊥ X. Let it now be the case that ⊥ Y,X,Z for every Y and Z such that ⊥ Y,A,Z. Now suppose there is a U∈V such that X⊆U, but A∉U. Let B∈S be such that |B| = {U}. Then obviously ╨ A,B, but not ╨ X,B. □
Semantic systems and semantics In the beginning of the paper, we formulated the crucial question (**), and we pointed out that to be able to answer it explicitly, we need an explication of the terms it consists of. We have suggested an explication of the term "inferential" (in terms of the concept of inferential structure). What we want to argue now is that a viable explication for "semantics" is provided by our concept of semantic system. It would be beyond the scope of the present paper to argue for this at length (I have done so elsewhere - see Peregrin, 1997), so I give only a digest. It seems obvious that semantic interpretation goes hand in hand with a truth-valuation of sentences – sentences (or at least some of them), by being semantically interpreted, become true or false. However, this does not necessarily mean that semantic interpretation fixes the truth values of all sentences – surely a sentence such as “The sun shines” does not become definitely true or false by being made to mean what it does. What semantic interpretation generally does is to impose limits on possible truth-valuations: e.g., it determines that if “The sun shines” is true, then “The sun does not shine” must be false; hence that the sentence “The sun shines and the sun does not shine” is bound to be always false etc. This means that semantic interpretation should pose some constraints on the possible truth-valuations of sentences. Moreover, many philosophers of language (most notably Davidson, 1984) have argued that all there is to meaning must consist in truth conditions. Now think about the ways truth conditions can be articulated: we must say something of the form X is true iff Y, where X is replaced by a name of a sentence and Y by a description of the conditions – i.e. in fact a sentence. Hence we need a language in which the truth conditions are expressed – a metalanguage. However, then our theory will work only so long as we take the semantics of the metalanguage at face value – in fact we will merely have reduced the truth conditions of the considered sentence, X, to a sentence of the metalanguage, namely the one replacing Y. And to require that the semantics of the latter be explicated equally rigorously as that of X would obviously set an infinite regress in motion. This indicates that it might be desirable to refrain from taking recourse to a metalanguage and make do with the resources of the object language, the language under investigation. Hence suppose that we would like to use a sentence of this very language in place of Y. Which sentence should it be? The truth conditions of X are clearly best captured by X itself; but using X in place 9
of Y would clearly result in an uninteresting truism. But, at least in some cases, there is the possibility of using a different sentence of the same language. So let us assume that we use a sentence Z in place of Y. Saying “X is true if ...“ or “X is true only if ...” with Z in place of the “...” amounts to claiming that X is entailed by Z and that X entails Z, respectively. (Claiming „’Fido is a mammal’ is true if Fido is a dog” is claiming that “Fido is a mammal” is entailed by “Fido is a dog”.) And claiming that X is entailed by Z in turn amounts to claiming that every truth-valuation which verifies Z verifies also X – or that any truth-valuation not doing so is not acceptable. Hence, the specification of the range of acceptable truth-valuations represents that part of the specification of truth-conditions which can be accounted for without mobilizing the resources of another language5. If we accept this, then the question (**) turns on the relationship between semantic systems (spaces of truth-valuations of sentences) and inferential structures (relations between finite sequence of sentences and sentences), in particular on the way in which the latter are capable of "determining" the former. So let us now turn our attention to this determining.
The inferentializabillity of semantics An inference can be seen as a means of excluding certain truth-valuations of the underlying language: stipulating X ├─ A can be seen as excluding all truth-valuations which contain X and do not contain A. In this sense, every inferential structure determines a certain semantic system (and if we agree that meanings are grounded in truth conditions, thereby also confers meanings on the elements of the underlying language). And hence the question which kinds of meanings are conferrable inferentially is intimately connected with the question which semantic systems can be determined by inferential structures. Now the question might prima facie seem trivial: we have seen that every semantic system has an inferential structure; does this inferential structure not determine this very system? However, the answer to this question is notoriously negative: an inferential structure of a semantic system might determine a different semantic system (though, of course, a system which has the same inferential structure). Let S = {A,B} and let V consist of the two ‘truth-value-swapping’ valuations, i.e. the valuations {A} and {B}. Let us consider all the possible instances of inference for S, and for each of them the valuations we exclude by its adoption: ├─ A ├─ B A├─ A B├─ A
∅, {B} ∅, {A} {B}
5
Of course that if we deal with empirical terms and empirical languages, then we need a way to 'connect them with the world' - hence either an unquestionned metalanguage capable of mediate the connection, or else a direct connection which, however, can be established only practically.
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A,B├─ A A├─ A B├─ A A,B├─ A
{A} -
This means that no combination of the inferences is capable of excluding the valuation {A,B}; and also no combination is capable of excluding ∅ without excluding either {A} or {B}. In other words, no inferential structure determines the system <{A,B},{{A},{B}}>. Now consider, in addition, the possible instances of incompatibility, and the valuations excluded by them: ⊥∅ ⊥A ⊥B ⊥ A,B
∅, {A},{B},{A,B} {A},{A,B} {B},{A,B} {A,B}
With their aid, it becomes possible to exclude {A,B}, by stipulating ⊥ A,B. However, it is still not possible to exclude ∅ without excluding either {A} or {B}. Hence no gis determines <{A,B},{{A},{B}}>. Now this appears alarming: for the semantics we have just considered is precisely what is needed to make B into the negation of A (and vice versa). This indicates that inferentialism might be short of conferring such an ordinary meaning as that of the standard negation. In fact, this should not be surprising at all. What we need to characterize negation is the stipulation that (i) if a statement is true, its negation is false, and (ii) if a statement is false, its negation is true. However, what we can stipulate in terms of inferences is that a statement is true if some other statements are true. In terms of incompatibility we can also stipulate that some statements cannot be true jointly, hence that if some statements are true, a statement is false, which covers (i) - but we still cannot cover (ii). As can be easily observed, the situation is similar w.r.t. disjunction and implication. In the former case, it is easy do exclude the valuations which make one of the disjuncts true and the disjunction false (by the inferences A├─ A∨B and B├─ A∨B), but we cannot exclude all those which make the disjunction true and both disjuncts false. In the latter, we can easily guarantee that the implication is true if the consequent is true, and that it is true only if the antecedent is false or the consequent is true (B├─ A→B and A, A→B ├─ B), but we cannot guarantee that it is true if the antecedent is false. Does this mean that the standard semantics for the classical propositional calculus is not inferential? And if so, how does it square with the completeness of the very calculus - for does not the completeness proof show that the axiomatic (i.e. inferential) delimitation of the calculus coincides with the semantic one? In fact, it is indeed not inferential, which does not contradict its completeness. The axiomatization of the calculus yield us its inferential structure, but this structure does not determine the semantics of the calculus. In fact, it
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determines another semantics, which, however, shares the set of tautologies with the calculus (which is what vindicates the completeness proof).
The Gentzenian generalization Let now us adopt a notation different from the one used so far and write X├─ instead of ⊥X. In this way, we can get inference and incompatibility under one roof - starting to treat ├─ as a relation between finite sequences of statements and finite sequences of statements of the length not greater than one. The ordered pair <S, ├─> with ├─ of this kind will be called a weak inferential structure. Such a structure will be called standard if the following holds (where G is a sequence of statements of the length at most one): (REF) A├─ A (EXT) if X,Y ├─ G, then X,B,Y ├─ G (CON) if X,A,A,Y ├─ G, then X,A,Y ├─ G (PERM) if X,A,B,Y ├─ G, then X,B,A,Y ├─ G (CUT) if X,A,Y ├─ G and Z├─ A, then X,Z,Y ├─ G (EXT') if X├─, then X├─ A If <S, ├─> is a weak inferential structure, then the strong inferential structure which arises out of restricting ├─ to instances with non-empty right-hand sides, will be called its strong restriction. It is obvious that the strong restriction of a standard weak structure is itself standard. If we, on the other hand, restrict ├─ to instances with empty right-hand sides, we get an incompatibility structure, which will be called the incompatibility restriction of the original structure. It is easy to show that if a structure is standard, then its strong restriction, as well as its incompatibility restriction, are standard. Moreover, they make up a standard generalized inferential structure. The condition (EXT') indicates that we can add statements on the right-hand side of ├─, of course unless we make it longer than 1. However, what, then, about relaxing this restriction, i.e. allowing for arbitrary finite sequences on the right side of ├─, and letting the right hand side be freely expandable just as the left hand side is? It is clear that what we reach in this way is in fact Gentzen’s sequent calculus. The ordered pair <S, ├─> with ├─ of this kind will be called a quasiinferential structure. Such a structure will be called standard if the following holds: 12
(REF) A├─ A (EXT) if X,Y ├─ U,V, then X,A,Y ├─ U,V and X,Y ├─ U,A,V (CON) if X,A,A,Y ├─ U, then X,A,Y ├─ U; if X ├─ U,A,A,V then X ├─ U,A,V (PERM) if X,A,B,Y ├─ U, then X,B,A,Y ├─ U; if X ├─ U,A,B,V then X ├─ U,B,A,V (CUT) if X,A,Y ├─ U and Z├─ V,A,W, then X,Z,Y ├─ V,U,W If <S, ├─> is a quasiinferential structure, then the weak inferential structure which arises out of restricting ├─ to instances with right-hand side of the length not greater than 1, will be called its weak restriction. The strong restriction of this restriction, i.e. the strong inferential structure which arises out of restricting ├─ to instances with right-hand side of the length precisely 1, will be called its strong restriction. Again, it is obvious that if a qis is standard, then both its weak restriction and its strong restriction are standard. The problem with this version of an inferential structure is that there does not seem to be any realistic interpretation for ├─ construed in this way6. However, we will use this artificial generalization for the sake of the study of inferentializability of semantic systems. The reason is that quasiinferential structures, in contrast to inferential ones, are capable of determining any system over a finite set of statement. Hence if we call a semantic system <S,V> finite iff S is finite, we claim Theorem 5. Every finite semantic system is determined by a qis. Proof: Let <S,V> be a semantics system and let S be finite (V thereby being also finite). Let S = {A1,...,An} and V = {v1,...,vm}. Let tij be the value vi(Aj). Hence it is the case that v∈V iff v(A1) = t11 and ... and v(An) = t1n, or v(A1) = t21 and ... and v(An) = t2n, or ... v(A1) = tm1 and ... and v(An) = tmn It is easily seen that this disjunction is equivalent to the conjunction of all disjunctions of the form (*) v(Ai1) = t1i1 or ... or v(Aim) = tnim, where i1,...,im all run through 1,...,n. Hence v∈V iff all these disjunctions hold. Now let us sort the indices i1,...,im into two groups according to whether tjij is 1 or 0; let k1,...kl be the former and kl+1,...km the latter. Then (*) holds if and only if Akl+1,..., Akm ├─ Ak1,..., Akl. 6
See Tennant (1997, 319-320).
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This means that all the instances of quasiinference strip down the class of truth-valuations of S to the very elements of V; i.e. <S,├─> determines <S,V>. □ The question which appears to come naturally now is: what about other systems? Are also semantic systems which are not finite determined by qis's? It would seem that many are, but all of them need not be -- but we will not go into these questions here. The reason is that determinedness by a qis is not yet what would make a system inferential in the intuitive sense. Hence we will now try to explicate the intuitive concept of inferentiality more adequately.
Finite bases Let us now return to the enterprise of explication of the question (**): we have dealt with "semantics", "determined" and "inferential", but we have so far not tackled "rules". The point is that the idea behind inferentialism is that it is us, speakers, who furnish expressions, and consequently languages, with their inferential power - we treat the statements as inferable one from others (by taking one to be committed to the former whenever she is committed to the latter) and as incompatible with each other. The idea is that we have a finite number of rules and a statement is inferable from a set of other statements if it can be derived from them with the help of the rules. This means that we should restrict our attention to inferential structures of a specific kind, namely those whose relation of inference is that of inferability by means of a finite collection of inferential rules. What is an inferential rule? Let us call an ordered pair whose first constituent is a finite sequence of elements of S and the second an element of S a (quasi)inference over S. (Hence if <S,├─> is a (quasi)inferential structure, then ├─ is a set of (quasi)inferences.) Now if P is a set (‘of parameters’), then a (P-)(quasi)inferential rule over S will be an inference over S in which some elements of S are replaced by elements of P. An instance of a (quasi)inferential rule over S will be any (quasi)inference over S which can be gained from the rule by a systematic replacement of the elements of P by the elements of S. (REF), for example, is an inferential rule: A ├─ A. Its quasiinferential form, then, is an example of a quasiinferential rule: X ├─ X. However, more interesting rules emerge only when we assume that the set of statements is somehow structured. If, for example, for every two statements A and B there is a statement denoted as A∧B, we can have the pattern A∧B ├─ A 14
A∧B ├─ B A,B ├─ A∧B establishing A∧B as the conjunction of A and B. So let us assume we have fixed some inferential rules, and the relation of inference which interests us is the one which 'derives from' them. How? We obviously need some way of inferring inferences from inferences, some metainferences or metainferential rules. Hence we introduce the concept of meta(quasi)inference over S, which is an ordered pair whose first constituent is a finite sequence of inferences over S and the second one an inference over S. A (P-)meta(quasi)inferential rule over S will be a meta(quasi)inference over S with some elements of S in its constituents replaced by those of P. We will separate the antecedent from the consequent of such a rule by the slash and we will separate the elements of its antecedent by semicolons. Thus, the metainferential rule constituted by (CUT) will be written down as follows: X,A,Y ├─ U; Z├─ V,A,W / X,Z,Y ├─ V,U,W Now what we want is that the inference relation derives from the basic finitely specified inferential rules by means of some finitely specified metainferential rules: A (quasi)inferential basis is an ordered triple <S,R,M>, where S is a set, R is a finite set of (quasi)inferential rules over S and M is a finite set of meta(quasi)inferential rules over S. (Let us assume that all metarules in M have a non-empty antecedent -- for metarules with the empty antecedent can be treated simply as rules and put into R.) The (quasi)inferential structure generated by <S,R,M> is the (quasi)inferential structure whose (quasi)inferential relation is the smallest class of (quasi)inferences over S which contains all instances of elements of R and is closed to all instances of elements of M. A (quasi)inferential structure is called finitely generated iff it is generated by a (quasi)inferential basis. A semantic system is called finitely (quasi)inferential iff it is determined by a finitely generated (quasi)inferential structure. Now it is clear that as far as finite languages are concerned, (quasi)inferentiality and finite (quasi)inferentiality simply coincide. Theorem 6. Every finite (quasi)inferential semantic system is finitely (quasi)inferential. Proof: If the number of statements is finite, then every (quasi)inferential relation over them is bound to have only a finite number of instances. □ The situation is, of course, different in respect to infinite languages. Take the semantic system constituted by the language of PA and the single truth-valuation which maps a statement on truth iff it is true in the standard model. This system is (trivially) inferential: the needed inferential relation consists of all inferences which have the empty antecedent and a statement of PA true in the standard model in the consequent. However, as the class of statements true within the standard model is not recursively enumerable, the semantic system is surely not finitely inferential. 15
But in fact it seems that inferentiality in the intuitive sense amounts to more than delimitation by any kind of a finitely inferential system. The inference relation of the systems we aim at should be derived from the basic inferential rules not by just any metainferential rules, but in a quite specific way. If R is a set of inferential rules, then we want to say that A is inferable, by means of R, from X iff there is a sequence of statements ending with A and such that each its element is either an element of X or is the consequent of an instance of a rule from R such that all elements of the antecedent occur earlier in the sequence. (REF is, strictly speaking, an inferential, rather than a metainferential rule. But we can see it as a metainferential rule with the empty antecedent.) This amounts to M consisting of the five Gentzenian structural rules. Indeed, A is inferable from X by means of R if and only if the inference X├─A is derivable from R by these rules. Theorem 7. A is inferable from X by means of the rules from R (in the sense that there exists a ‘proof’) just in case X├─A is inferable from R by means of REF, CON, EXT, PERM and CUT. Proof: As the proof of the inverse implication is straightforward, let us prove only the direct one. Hence let A be inferable from X. This means that there is a sequence A1,...,An of statements such that An = A and every Ai is either an element of X or is inferable by a rule from R from statements which are among A1,...,Ai-1. If n=1, then there are two possibilities: either A∈X and then X├─ A follows from REF by EXT; or A is a consequent of a rule from R with a void antecedent, and then ├─ A and hence X├─ A in force of EXT. If n>1 and An is inferable from some Ai1,...,Aim by a rule from R, then Ai1,...,Aim ├─ A, where X├─ Aij for j=1,...,m. Then X,...,X ├─ A in force of CUT, and hence X ├─ A in force of PERM and CON. □ This leads us to the following definition: We will call a (quasi)inferential basis standard iff R contains REF and M contains CON, EXT, PERM and CUT (hence if the (quasi)inferential basis is standard, then the (quasi)inferential structure which it generates is standard in the sense of the earlier definition). And we will call it pure iff, moreover, M contains no other rules than CON, EXT, PERM and CUT. A (quasi)inferential structure will be called pure if it is generated by a pure (quasi)inferential basis. (Hence every standard, and especially every pure inferential structure is finitely generated.) It seems that stipulating inferences we implicitly stipulate also all the inferences which are derivable by them by the structural rules -- hence that we should be interested only in structures which are pure, or at least standard. It might seem that it is purely inferential systems which are the most natural candidate for the role of the explicatum of an "inferential semantic system"; however, the trouble is that no finitely inferential system (and hence surely no purely or standardly inferential one) is capable of accommodating the simplest operators of classical logic. Though it is possible to fix the usual truth-functional meaning of the classical conjunction by means of the obvious inferential pattern, the same is not possible, for reasons sketched earlier (page 11), for the classical negation and also the classical disjunction and implication. 16
What is possible is to fix the truth-functional meanings of all the classical operators by means of quasiinferential patterns, e.g. in this way: A,¬A├─ ├─ A,¬A A├─ A∨B B├─ A∨B A∨B ├─ A,B B├─ A→B A, A→B ├─ B ├─ A, A→B Hence (as discovered already by Gentzen) all of classical logic is purely quasiinferential. Nevertheless, it is not purely inferential. For an inferentialist, this situation need not be too embarrassing. I have independent reasons, she might claim, to believe that the only (primordial) way to furnish an expression with a meaning is to let it be governed by inferences; so if there are ‘meanings’ which are not conferrable in this way, they are not meanings worth the name. But things are not this simple. We have seen that many meanings of a very familiar and seemingly indispensable kind fall into the non-inferential category. Classical negation or disjunction; not to speak about the standard semantics for arithmetic. Is the inferentialist saying that these are non-meanings? To be sure, the inferentialist may defend the line that the only ‘natural’ meanings are the inferential ones; and that all the others are late-coming products of our artificial languageengineering. She might claim that the only ‘natural’ logical constants are some which are delimitable inferentially (presumably the intuitionist ones), and that the classical ones are their artificial adjustments available only after metalogical reflections and through explicit tampering with the natural meanings. However, if one does not want to let classical logic by the board in this way, she appears to have no choice but to settle for (purely) quasiinferential systems. The latter are, as we saw, strong enough for the classical operators, but the multiple-conclusion sequents appear to be much less natural than the single conclusion ones. Fortunately there is a sense in which every purely quasiinferential system can be regarded as a (standardly) inferential system, so that we may enjoy some advantages of the quasiinferential systems without officially admitting the multi-conclusion sequents.
The emulation theorem
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What exactly we will show is that for every purely quasiinferential system there exists a standardly inferential system with the same class of tautologies. First, however, we need some more definitions: If X ├─ A1...An is a quasiinference over S, then its emulation will be the metainference YA1Z├─ B; ...; YAnZ├─ B / YXZ ├─ B. An emulation of a quasiinferential basis <S,R,M> will be the inferential basis <S,R',M'> such that R' is the set of all those elements of R which are inferences, and M' is the union of M and the emulations of all elements of R which are proper quasiinferences. Now we are going to prove that an emulation of a pure quasiinferential basis <S,R,M> generates an inferential structure which is identical to the structure which results from taking the qis generated by <S,R,M> and dropping all genuine quasiinferences: Theorem 8. The emulation of a pure quasiinferential structure is its strong restriction. Proof: Let <S,├─> be the quasiinferential structure generated by <S,R,M> and let <S,├─*> be the inferential structure generated by <S,R',M'>. What we must show is that for every sequence X of elements of S and every element A of S it is the case that X ├─ A iff X ├─* A. Let us consider the inverse implication first. As R' is a subset of R, it is enough to show that every metainference from M' which is not a (restriction of an) element of M preserves ├─, i.e. that whenever it is the case that X ├─ A1...An and YA1Z├─ B,..., YAnZ├─ B, then it is always the case that YXZ ├─ B. But this easily follows from the standardness of ├─: YXZ ├─ B A2...An ... Y...YXZ...Z ├─ B...B YXZ ├─ B
(CUT) (CUT) (PERM) and (CON)
Hence if X ├─* A, then X ├─ A. The proof of the direct implication, i.e. of the fact that if X ├─ A, then X ├─* A, is a bit trickier. We will prove that if X ├─ A1...An, then YA1Z├─* B, ..., YAnZ├─* B always entail YXZ ├─* B. Then if X ├─ A, then YAZ├─* B entails YXZ ├─* B; and in particular A├─* A entails X ├─* A; and as A├─* A due to (REF), X ├─* A. To prove that for every X and A1...An such that X ├─ A1...An, YA1Z├─* B, ..., YAnZ├─* B entail YXZ ├─* B we need some notational conventions. If X = A1...An then we will use X ├├* Y as a shorthand for the set of quasiinferences A1├─* Y, ..., An├─* Y. Moreover, [U]X[V] ├├* Y will be the shorthand for UA1V├─* Y, ..., UAnV├─* Y. Now first assume that X ├─ A1...An belongs to R. Then if n≠1, the metainference from UA1Z├─* B, ..., UAnZ├─* B to UXZ ├─* B is an element of M'. If n=1, then X ├─* A, and the fact that UAZ├─* B entails UXZ ├─* B follows by (CUT). Now assume that X ├─ A1...An is the result of an application of a metaquasiinference from M. As <S,R,M> is pure, the only possibilities are CON, EXT, PERM and CUT. We will prove only the less perspicuous case of CUT. Hence assume that X,A,Y ├─ U and Z├─ V,A,W; and that [M]U[N] ├├* B entails M,X,A,Y,N ├─* B and [M]V,A,W[N] ├├* B entails M,Z,N ├─* B. What we want to prove is that then [M]V,U,W[N] ├├* B entails M,X,Z,Y,N ├─* B. However, 18
if [M]V,U,W[N] ├├* B, then especially [M]U[N] ├├* B , and hence M,X,A,Y,N ├─* B; whereby M,X,Z,Y,N ├─* B. □ Corollary: For every purely quasiinferential system there exists a standardly inferential system with the same class of tautologies. Proof: Let <S,V> be semantic system determined by the qis generated by <S,R,M> and let <S,V'> be the system generated by the emulation <S,R',M'> of <S,R,M>. Let <S,├─> be the quasiinferential structure generated by <S,R,M> and let <S,├─*> be the inferential structure generated by <S,R',M'>. Then A is a tautology of <S,V> iff ├─ A; but as <S,├─*> is the restriction of <S,├─>, this holds iff ├─* A and hence iff it is a tautology of <S,V'>.□
An example: classical propositional logic in the light of inferences We saw that not even the classical propositional logic (CPL) is inferential in the sense that there would be inferences which would delimit the very class of truth-valuations which is constituted by the usual explicit semantic definition of CPL. However, the corollary we have just proved tells us that there is a standard inferential structure which, though it does not determine the semantics of CPL, does determine a semantic system which has the same class of tautologies. Which inferential structure it is? It is easy to see that if we base CPL on the primitive operators ¬ and ∧, the semantics of CPL is determined by the following quasiinferences: (1) A∧B ├─ A (2) A∧B ├─ B (3) A,B ├─ A∧B (4) A,¬A├─ (5) ├─ A,¬A What we have to do is to replace the genuine quasiinferences (i.e. those with not exactly a single statement in the consequent, (4) and (5)) with their emulations. This is to say that we must replace (4) and (5) by (4') A,¬A├─ B (5') X,A├─ B; X,¬A├─ B / X├─ B Note that the fact that (1), (2), (3), (4') and (5') determine the tautologies of CPL amounts to the completeness result for the logic. But it is, in a sense, more general than the usual one and it throws some new light on the fact that the axioms of classical logic, despite their completeness, do not pin down the denotations of the operators to the standard truthfunctions. (The point is that the axioms are compatible even with some non-standard
19
interpretations – with negations of some falsities being false and with disjunctions of some pairs of falsities being true. What is the case is that if the axioms hold and if the denotations of the operators are truth functions, then they are bound to be the standard truth functions. But the axioms are compatible with the indicated non-truth-functional interpretation of the constants7.) From our vantage point we can see that classical logic is complete in the sense that its axioms determine a semantics with the class of tautologies which is the same as that of the standard semantics of CPL; that they, however, do not determine this very semantics. Let us give some illustrations of how proofs within (1)-(5) get emulated by those within (1)-(3), (4') and (5'). Consider the inference ¬¬A├─ A, which is valid in CPL. With (4) and (5) it can be proved rather easily: 1. ¬A,¬¬A├─ 2. ├─ A, ¬A 3. ¬¬A├─ A
(4) (5) from 1. and 2. by (CUT)
This gets emulated as follows: 1. ¬¬A,¬A├─ A 2. ¬¬A, A├─ A 3. ¬¬A├─ A
from (4') by (PERM) from (REF) by (EXT) from 1., 2. by (5')
Or consider the proof of the theorem ¬(A∧¬A) 1. A∧¬A├─ A 2. A∧¬A├─ ¬A 3. A,¬A├─ 4. A∧¬A├─ 5. ├─ A∧¬A,¬(A∧¬A) 6. ├─ ¬(A∧¬A)
(1) (2) (4) from 1., 2. and 3. by (CUT) and (CON) (5) from 4. and 5. by (CUT)
The emulation now looks as follows: 1. A∧¬A├─ A 2. A∧¬A├─ ¬A 3. A,¬A├─ ¬(A∧¬A)
(1) (2) (4')
7
This is a fact noted already by Carnap, 1943, but rarely reflected - see Koslow, 1992, Chapter 19, for a discussion.
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4. A∧¬A├─ ¬(A∧¬A) from 1., 2. and 3. by (CUT) and (CON) 5. ¬(A∧¬A) ├─ ¬(A∧¬A) (REF) 6. ├─ ¬(A∧¬A) from 4. and 5. by (5')
Extremality conditions Hence what seems to be a good candidate for the explication of the intuitive concept of "inferential semantics" is the concept of standardly inferential semantic system, i.e. a system generated by a collection of inferential and metainferential rules containing the Gentzenian structural rules. This is obviously of a piece with the ideas of the natural deduction program (Prawitz, 1965; Tennant, 1997; etc.). Hence it seems that the inferentialist agenda should display a large overlap with the agenda of this program. We have also seen that there is a direct way from the natural quasiinferential characterization of structural operators to their superstandly inferential characterization. Let us consider disjunction. A∨B is partly characterized by the inferences A ├─ A∨B B ├─ A∨B but the characterization has to be completed by the genuine quasiinference A∨B ├─ A,B. This quasiinference gets emulated as A ├─ C; B ├─ C / A∨B ├─ C; which yields us the metainferential characterization of disjunction well-known from the systems of natural deduction. Note that the metainferential rule can be looked at as a "minimality condition". Let us say that the statements A,B,C fulfill the condition Φ(A,B,C) iff A ├─ C and B ├─ C. Then A∨B can be characterized in terms of the following two conditions8 (i) Φ(A,B,A∨B); and (ii) A∨B is the minimal statement such that Φ(A,B,C); i.e. if Φ(A,B,C), then A∨B ├─ C.
8
This form is borrowed from Koslow (1992), whose book offers a thorough discussin of the technical side of the issues hinted at in this section.
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Why is this interesting? Because this kind of minimality conditions could be perhaps seen as implicit to the statement of an inferential pattern. When we state that A ├─ A∨B and B ├─ A∨B and when we, moreover, put this forward as an (exhaustive) characterization of A∨B, we would appear to insinuate that there is a sense in which A∨B is the statement which fulfills this condition (i.e. Φ). Imagine I am asked what children do I have -- i.e. to characterize the class of my children - and I answer "I have a son and a daughter". Strictly speaking I did not give a unique characterization of the class -- I only stated that this class contains a boy and a girl. But as it is normally expected that what I say should yield an exhaustive characterization, my statement would be taken to imply (by way of what Grice called a conversational implicature) that the class in question is the minimal one fulfilling the condition I stated. And similar minimality implicature can be seen as insinuated by my stating that Φ is the pattern characteristic of disjunction. More to the point, this train of thought appears to motivate also Gentzen's insistence on that it is only introductory rules which semantically characterize the operators. As Koslow (1992, §2.1) shows, it is natural to see it precisely in terms of extremality conditions: the introduction rule yields the elimination rule via the assumption that the introduction rule gives all what there is to the 'inferential behavior' of the connective. Hence it seems that we can, after all, delimit the classical disjunction by an inferential pattern -- if we assume the minimality implicature. This indicates that instead of allowing for the non-structural metainferential rules (which amounts to passing over from pure to standard inferential structures) we could perhaps admit that stating an inferential pattern involves stating the minimality of the operator fixed by the pattern. Can we see all the other classical logical operators analogously? Well, though here we cannot arrive at this kind of characterization directly via our emulation procedure, we can characterize implication as the maximal operator fulfilling A, C ├─ B. And we can also characterize conjunction as the maximal operator fulfilling C ├─ A C ├─ B. Negation, if we want it to be classical, is unfortunately more fishy. It seems that the only pattern available is A,C ├─ B ¬C ├─ A. which itself contains the negation sign to be determined; and this appears to largely spoil the picture. Is there a remedy? 22
Well we may trade the second part of the negation-pattern, i.e. the law of double negation, for something else -- e.g. for the 'external' assumption that all our operators are truthfunctional. It is clear that the only truth-function which always maps a statement on its maximal incompatible is the standard negation (see Peregrin, 2003, for more details). But a more frank solution would be to simply strike out the law of double negation without a substitute. What would be the result? Of course the intuitionist negation and consequently the intuitionist logic. This indicates, as I have discussed it in detail elsewhere (see Peregrin, 2004a), that it is intuitionist logic that is the logic of inference. On this sense, classical logic is not natural from the inferentialist viewpoint (however, its unnaturalness from this viewpoint is outweighed -- and maybe more than so -- by its simplicity).
Conclusion Arguing for inferentialism, we must first specify what exactly we mean by the term: there are several options. In this paper I have tried to indicate that two of the options can be merged into a single one, which, then, is the hottest candidate for becoming the inferentialism. The winner is the "superstandard inferentialism", capable of 'emulating' and hence treatable as encompassing "standard quasiinferentialism". On the technical side, it comes down to the framework of natural deduction. (Its immediate stricter neighbor, "standard inferentialism" is obviously much too weak; whereas the stronger "quasiinferentialism" appears not to be quite natural.) If we accept this, then we should also see intuitionist logic as the most natural logic. However, as we have taken pains to indicate, this does not preclude the way to classical logic, which is surely natural in some other respects and whose utter inaccessibility would be, I believe, a failure of inferentialism. (Note that inferentialism is a descriptive project concerned with the question what is meaning?; whereas the natural deduction program is more a prescriptive program concerned with the question How should we do logic?. Thus while the latter could perhaps simply ban classical logic if it concluded that one can make do without it, the former is bound to take the extant meanings at face value and face the question If meaning is an inferential matter, then how could there be meanings that are prima facie 'noninferential'?) Hence I think that inferentialism, though it may be 'favoring' some meanings over others, does not result into any kind of unnatural 'semantic ascetism'. I am convinced that the thesis that all meanings are, ultimately, creatures of inferences -- that they are ultimately inferential roles -- is viable.
References Brandom, R. (1985): ‘Varieties of Understanding’, in Reason and Rationality in Natural Science (ed. N. Rescher), University Presses of America, Lanham, 27-51. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). 23
Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Carnap, R. (1943): Formalization of Logic, Harvard University Press, Cambridge (Mass.). Davidson, D. (1984): Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Fodor, J.A. & LePore, E. (1993): ‘Why Meaning (Probably) Isn’t Conceptual Role’, in Science and Knowledge (ed. E. Villaneuva), Ridgeview, Atascadero, 15-35. Frege, G. (1879): Begriffsschrift, Nebert, Halle. Kalderon, M.E. (2001): ‘Reasoning and Representing’, Philosophical Studies 105, 129-160. Koslow, A. (1992): A Structuralist Theory of Logic, Cambridge University Press, Cambridge. Lance, M. (1996): ‘Quantification, Substitution and Conceptual Content’, Noûs 30, 481-507. Lance, M. (2001): ‘The Logical Structure of Linguistic Commitment III: Brandomian Scorekeeping and Incompatibility’, Journal of Philosophical Logic 30, 439-464. Lavine, S. (2000): ‘Quantification and Ontology’, Synthèse 124, 1-43. Peregrin, J. (1997): ‘Language and its Models’, Nordic Journal of Philosophical Logic 2, 123. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Peregrin, J. (2003): 'Meaning and Inference', in T. Childers and O. Majer (eds.): The Logica Yearbook 2002, Filosofia, Prague. Peregrin, J. (2004a): 'Logic as "Making it Explicit"', to appear in The Logica Yearbook 2003, Filosofia, Prague. Peregrin, J. (2004b): ‘Pragmatism and Semantics’, to appear in German in E. Olsson and A. Fuhrmann, eds.: Pragmatismus Heute; available in English from my home page. Peregrin, J. (2004c): 'Semantics as Based on Inference', to appear in The Age Of Alternative Logics (ed. J. van Benthem et al.), Kluwer Dordrecht. Prawitz, D. (1965): Natural Deduction, Almqvist & Wiksell, Stockholm. Prior, A. N. (1960/61): ‘Roundabout Inference Ticket’, Analysis 21, 38-39. Tennant, N. (1997): The Taming of the True, Clarendon Press, Oxford. Wittgenstein, L. (1956): Bemerkungen über die Grundlagen der Mathematik, Blackwell, Oxford.
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ZEN & THE ART OF (DAVIDSONIAN) ANALYTIC PHILOSOPHY Jaroslav Peregrin www.cuni.cz/~peregrin
Hier sieht man, daβ der Solipsismus, streng durchgeführt, mit dem reinen Realismus zusammenfällt. Das Ich des Solipsismus schrumpft zum ausdehnungslosen Punkt zusammen, und es bleibt die ihm koordinierte Realität. Wittgenstein (1922, §5.64)
Donald Davidson and Contemporary Philosophy Since the sixties, when Donald Davidson published his first influential papers, his ideas have slowly, but persistently, been infiltrating philosophical discussions; and his views of philosophical problems have been consistently winning increasing number of adherents. Today, Davidson figures as perhaps the most virtuous representative of contemporary (post)analytic philosophy - drawing freely on his analytic predecessors, continuing to philosophize in their rationalist spirit and sharing their passion for correct argumentation, but meanwhile never hesitating to eschew those aspects of the analytic tradition which he finds misguided. However, Davidson maintains also the air of an obscure philosopher. He seems to belong to that line of philosophers, starting perhaps from Heraclitus (for whom skoteinos, i.e. obscure, became a nickname), whose teaching has been always taken not only as a source of enlightenment, but also (and perhaps even more frequently) as a kind of riddle. This is strange, for Davidson’s writings are neither fragmentary (as are those of Heraclitus), nor enigmatic in the style of, say, Wittgenstein. Nevertheless there appear to be almost as many interpretations of Davidson’s views as there are interpreters. Some take Davidson as, first and foremost, a great semantician; others see him as an inconsequential prophet of pragmatism1; and others compare him to Derrida2 or Heidegger3. In this paper, I would like to point out one aspect of Davidson’s position which I think makes his viewpoint especially hard to grasp. I would like to indicate that one of the things Davidson’s teaching aims at is the overthrowing of the way we commonly see the world (and 1
Notoriously Rorty - see, e.g., Rorty (1998).
2
Wheeler (1986).
3
Malpas (1992).
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especially the relationship of our thought and our language to the world), at making us abandon a conceptual scheme which „holds us captive“4. The trouble is, it seems to me, that his readers often interpret his claims from within the very scheme they are supposed to give up - and this inevitably makes them interpret what Davidson says in a way which is bound to be inadequate (if not absurd). I think that Davidson’s effort to make us abandon the ordinary visual angle is not without precedent. I think that one of the parallels which suggest themselves in connection with it is the parallel with Hegel’s effort to dismantle the Kantian dualism of concepts and intuitions (and as far as I know, nobody has yet explored this parallel in depth) 5. Comparing Davidson to Hegel is, admittedly, weird; and I will refrain. However, not because of the weirdmess, for what I am going to do is to present a comparison which is undoubtedly even weirder - I am going to compare Davidson with a zen buddhist guru. It seems to me that the very weirdness of the comparison may help to pinpoint the radicalness of (what I take to be) Davidson’s stance, to help us see that to understand him properly we must, like when becoming adepts of zen teaching, avoid the „mistake consisting in our splitting into two what is really and absolutely one“ (Suzuki, 1949, p. 15). And besides this ‘therapeutic value’, if it is true that, as Rorty (1991, p. 94) suggests, one of the important tasks of philosophy is „the colligation of hitherto unrelated texts“, such weird treatment of Davidson might be - should it prove enlightening - might have an independent philosophical value.
The Circle To illuminate that aspect of zen to which I am going to allude, I shall employ a metaphor of a contemporary Korean zen master living in the USA, Seung Sahn. In his book Only don’t know (Four Seasons, San Francisco, 1982) he depicts the spiritual way of an adept of zen in terms of going round the following circle:
4
If anyone finds it awkward to use the term „conceptual scheme“ in connection with a philosopher who has devoted much of his philosophical effort to show us that there are no such things, then I refer him to Davidson’s own pronouncements in Borradori (1994).
5
See Pippin (1989, esp. Chap. 2). However, the parallel is most striking if you read Hegel in the way Pinkard (1991) does (Pinkard notes this, on page 303 of his paper, without invoking Davidson’s name).
3
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1800
2700
900
00 = 3600
THE ZEN CIRCLE (ACCORDING TO MASTER SEUNG SAHN): 00 : 900: 1800: 2700: 3600:
Attachment to name & form (1 + 2 = 3) Form is emptiness, emptiness is form (0 = 1, 1= 0) Neither form, nor emptiness (1000 x 0 = 0, 1 x 0 = 0) Freedom with respect to form & emptiness (3 x 3 = 9, 1000 x 100 = 1000 x 9) Form is form, emptiness is emptiness (3 x 3 = 9)
This can be explained, roughly, in the following way. In the zero stage of her development, the adept is bound to name and form, seeing the world ‘uncritically’ ‘as it is’. In the next stage (900), she recognizes that form and emptiness are not insuperably separated, that form in fact is emptiness, and emptiness in fact is form. In the following state (1800), this results into the denial of the very notions of form and emptiness; and subsequently (2700) to one’s freedom with respect to form and emptiness. In the final stage the adept finds herself back at the starting point seeing the world ‘as it is’; however, the actual peregrination of the circle has enabled her to now view the world not with the eyes of a naive adept, but with those of an awakened master.
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My conjecture is that, analogously we can imagine a model adept of the Davidsonian philosophical stance - aiming at liberation from scheme/content dualism - as peregrinating a similar kind of circle. In the beginning, she sees the world as consisting of things, which we somehow manage to mean with our words (00). Then she first experiences the linguistic turn (900): she recognizes that there are no things save things-as-meant, that „to be is to be the value of a variable“, and hence that thingishness and meaningness are two inseparable sides of the same coin. This knowledge is likely to lead her (1800) to doubt whether there are indeed any ‘real’ things (beyond fabrications of our language), and whether there are any ‘real’ meanings (for meaning was supposed to be a kind of ‘linguistic reaching’ to ‘real’, language-external things). This may give rise to the following stage (2700), in which she realizes that if there are no things over and above things fabricated by a language, then she is free to set up the world according to her liking, that it is enough to choose a suitable language. However, the ultimate step (3600) brings her to the realization that if there are no things over and above those fabricated by a language, then these things are as ‘real’ as things can ever be and hence it makes no sense to see them as ‘not thingish enough’ (And, like in zen, this ultimate step is the vital crux). This can be depicted as the following ‘semantic circle’: THE SEMANTIC CIRCLE 00 : 900: 1800: 2700: 3600:
There are things and we mean them by words Thingishness is meaningness, meaningness is thingishness There is no thingishness, and hence no meaningness What I want is thingish, hence what I want is meaningish Thingishness is thingishness, meaningness is meaningness
Continuing to toy with transmutation of the circle, we may come to draw also the ontological and the epistemic versions of the same circular path: THE ONTOLOGICAL CIRCLE 00 :
There is the way the world is (and there are ways in which it appears to us from various parochial viewpoints) 0 90 : Each way the world is is the way it is from a viewpoint 1800: There is no way the world ‘really’ is (in itself) 2700: We are free to choose how the world is 3600: The world is as it is
THE EPISTEMIC CIRCLE 00: 900:
There is the God’s Eye view of the Universe (and there are our parochial views) There is no God’s Eye View, only the plurality of parochial views
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1800: No view is really the view of the Universe 2700: Whatever view I want is a view of the Universe 3600: There is the view of the universe which sees it as it is My present purpose can perhaps be best illustrated with the help of this epistemic variant of the circle: the crucial step from 2700 to 3600 can be seen as consisting in the realization of the fact that if there is no God’s Eye View, then there is no way to deny the God’s eye view - but this does not mean that, after all, there is a view which would be absolute and transcend every parochial view. It means that the whole problem of the absolute vs. parochial viewpoints is, so to say, aufgehoben. Let us try to find some illuminating labels for the ‘ideological’ positions corresponding to the individual stages of the circle. The ideology of the zero stage, 00, corresponds to what is often called metaphysical realism claiming that „things are as they are (in their absolute and for us maybe forever indiscernible way)“. In the first stage (900) we fall into relativism, we see that „things can be seen to be in various different ways“. The second stage (1800) can be dubbed nihilism, claiming that „there are neither things, nor a way how they are“ (in contrast to the other ones, this does not seem to be a stage in which many adepts would acquiesce). The third stage (2700) is the stage of idealism urging that „things are in whichever way I make them be“. The final stage (3600), claiming that „things are as they are (namely as they are for us)“, can then be labeled naturalism. Hence the ‘ideological’ version of the circle: THE IDEOLOGICAL CIRCLE 00 : 900: 1800: 2700: 3600:
Realism Relativism Nihilism Idealism Naturalism
Two Ways of Viewing Language One of the ways to approximate the wisdom which should be gained by going round the circle is in terms of the distinction between two essentially different ways of viewing language, which I have discussed elsewhere (see Peregrin, 1995; 1996). The distinction is that among (i) viewing language as a ‘nomenclature’, i.e. as a means of representing things of the world; and (ii) viewing it as a toolbox, as a means of interacting with the world. (I called the first view the nomenclatural view, and the second view, perhaps unhappily, the structural view;
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and I have insisted that the dividing line between philosophers accepting the first and the second view cuts across standardly recognized philosophical schools). What I now want to suggest is that what one may (or should) gain going round the circle amounts, besides any other benefit, to abandoning the representational view - for it is this abandonment which brings us, into the 3600 stage, dissolving the apparently irreconcilable dualisms in which we are trapped - it frees us from the dualism of the representing and represented and thereby rids us of many philosophical troubles. This is to say that whereas the point of zen training, resulting in satori, is to overcome the dualisms normally present with our making sense of the world and coping with it, the point of the Davidsonian analytic philosophy, resulting into naturalism, is very similar: to overcome harmful dualisms, in this case those which are likely to create philosophical (pseudo)problems. Now just as there are different and mutually independent ways of reaching satori, there are, I think, various, more or less, independent ways of reaching naturalism. One such way, which I am not going to discuss here, was suggested by the late Wittgenstein (leading to the notion of language as a „form of life“ and to the therapeutic treatment of philosophical questions). The American philosophers accepting the analytic tradition chose a different path - for they were not only followers of Carnap and Wittgenstein, but also - at least latently followers of Dewey and William James. Their way was to expel meaning from human minds and relocate it into human behavior. I think that within this path we can further distinguish two partly independent ‘subpaths’, the first of which starting with the Quinean rejection of the analytic/synthetic distinction, while the second issued from the Sellarsian rejection of the ‘Myth of the Given’6 (I think that Rorty, 1980, is right counting Quine and Sellars as the most significant initiators of current ‘post-analytic philosophy’.) In my opinion, while it is the ‘Quinean’ path which culminates in the philosophy of Donald Davidson, the ‘Sellarsian’ path comes into full fruition in the teaching of Sellars’ disciple Robert Brandom. And Brandom’s recent book (1994) also provides an as yet unmatched analysis of the distinction between the two approaches to language mentioned above (although in place of the terms nomenclatural and structural Brandom employs the terms representational and inferential. The crucial element, both in zen and in Davidsonian (post)analytic philosophy, is the completing of the whole circle. For it is tempting to come to rest on the heady doctrine of relativism or idealism; and it is not always easy to see that they are not that which the masters and Davidson urge. True, when Davidson, e.g., praises coherentialism and says such things as „nothing can count as a reason for holding a belief except another belief“ (1986, 123), one may easily get the impression that he is urging a version of solipsistic idealism; but to think so is to mistake the 3600 stage for the 2700. The point of the 2700 stage is that we cannot get outside (of our mind or our language); whereas the point of the 3600 stage is that if there is no getting outside, then there is no outside (for „outside,“ we can say with Wittgenstein, 1953, 6
See Sellars (1956).
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§103, „you cannot breathe“) and hence also no inside. This is what Davidson urges e.g. in the last sentence of his On the Very Idea of a Conceptual Scheme: „In giving up the dualism, we do not give up the world, but re-establish unmediated touch with the familiar objects whose antics make our sentences and opinion true or false“ (Davidson, 1984, 198) And the same point is made, even more vividly, by Brandom: „Discourse practices incorporate actual things. ... They must not be thought of as hollow, waiting to be filled up by things; they are not thin and abstract, but as concrete as the practice of driving nails with a hammer. ... According to such a construal of practices, it is wrong to contrast discursive practice with a world of facts and things outside it, modeled on the contrast between words and the things they refer to“ (332). „Thus a demolition of semantic categories of correspondence relative to those of expression does not involve ‘loss of the world’ in the sense that our discursive practice is then conceived as unconstrained by how things actually are. ... What is lost is only the bifurcation that makes knowledge seem to require the bridging of a gap that opens up between sayable and thinkable contents - thought of as existing self-contained on their side of the epistemic crevasse - and the worldly facts, existing on their side“ (333). We may speak about the ‘world of things’ governed by causal laws; and we may also speak about the ‘world of beliefs’ governed by inferential relations. However, once we take this talk at face value and see the two as literally distinct worlds, especially if we see the former as being ‘outside’ while the latter as being ‘inside’, we must face the unsolvable problem of putting them together, of linking the ‘inside’ to the ‘outside’. We have to invent some relations of correspondence which would guarantee that the inside somehow mirrors the outside (hence the perennial idea of a man as a ‘mirror of nature’7); but in fact we can never succeed in answering the haunting questions of the kind What if, after all, there is NOTHING outside? We may succeed (in the sense that we prohibit the problem from arising) only when we realize that things differ from thoughts not in that they inhabit a different world (or a different part of the world), but rather in that they inhabit the same world in a different way. The ‘world of things’ is the same world as the ‘worlds of thoughts’, just as a ‘land of Eskimos’ may be the same land as a ‘land of low temperatures’: beliefs are not shadowy reflections of facts; facts are simply true beliefs. And once we thus see that beliefs are not ‘inside’, the claim „nothing can count as a reason for holding a belief except another belief“ turns out to be wholly uncontroversial.
What is Wrong with the Idea of Correspondence? The representational model of language is based on the intuitive idea that the working of language can be explained by talking about language as picturing the world similarly to a set of photographs picturing, e.g., members of one’s family. The powerful picture elaborated in 7
Viz Rorty (1980).
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Wittgenstein’s Tractatus views the network of true propositions as reflecting the network of facts. The obvious trouble is the lack of any perceptible similarity between a sentence and a fact which the sentence is said to picture; hence to be able to put the picture to work we have first to explain what we mean by saying that a sentence pictures a fact. And this is arguably not possible without recourse to what we were supposed to be explaining, viz the working of language. To say that a sentence is true iff it corresponds to a fact would be a genuine explanation only if sentences pictured facts in some ‘natural’, self-explicating way; but as this is not the case, we have still to say what we mean when talking about this notion of picturing, and this we cannot do without essentially exploiting the concept of truth. Hnece the trouble is not that the correspondence view of language is necessarily wrong, but that it says nothing worth being said. As Davidson (1984, pp.193-194) puts it, „the trouble is that the notion of fitting the totality of experience, like the notions of fitting the facts, or of being true to the facts, adds nothing intelligible to the simple concept of being true.“ It is not wrong to talk about language corresponding to the world or about its sharing its structure with the world; but doing so we only invent obscure ways of saying things which are otherwise straightforward. So the trouble with saying that words and sentences reflect things and facts is not that one could deny it - on the contrary, the trouble is precisely that this is a triviality no one could deny8. Let us illustrate this by the following two stories. First, imagine Mr. X, who has taken a set of photographs of a certain landscape; suppose that the photographs were taken from a plane, from constant height (so that each depicts a rectangular area of the same dimensions), and suppose that the photos link up with each other in such a way that if we put them side by side in the appropriate way we gain a continuous picture of the whole area of landscape in question. Now Mr. X claims that the set of photographs and the area share a certain structure, namely that the set consists of rectangular snaps analogously to how the depicted landscape consists of rectangular pieces of land. Is he right? Well, in a sense he clearly is. The trouble is that nobody would ever think of such a rectangular structure in connection with that piece of land were it not for the set of photographs. As we normally speak, the area does not have such a structure (by itself) - true, it can be seen as so structured, but it can be seen as structured also in a myriad of other ways. To say that the area and the set of photos share a structure is thus trivial (a more reasonable description of the situation might be that the set of photos causes us to see the area as in a certain way structured) - it is to say nothing, but in a high-spirited way. Similarly, to say that
8
As Blackburn (1984, p. 248) puts it, „there is not really a correspondence ‘theory’ of truth: there is rather an invitation to think of the relation between true belief and whatever it is in the world that makes it true. This invitation can lead to bad developments: to the idea of the mind’s awareness of fact as something which, favourably, is uncontaminated by judgement, and purely passive; or to the idea that each individual judgement has its own identity regardless of its associations with any others in a body of belief, and is in turn made true by one isolated, self-subsistent state of afairs.“
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reality (as we see it) shares a structure with our language is to state nothing but a high-spirited truism. Second, imagine Mr.Y who is learning to play golf. He has a set of clubs, and before each strike he chooses that club which he considers the most appropriate. The criteria according to which he chooses are manifold: they concern the quality of the surface on which the ball rests, the distance of the hole, the weather and perhaps some other aspects. Each club is useful in certain conditions for certain achievements; and Mr. Y qualifies as a true golfer when he is able to select and employ the appropriate ones. Now suppose that Mr. Y describes the situation in terms of correspondence between clubs and strange abstract entities which consist of all those aspects of the environment which are relevant for choosing the club and which Mr. Y proposes to call golfacts - thus, a golfact is some kind of sum of such features as the quality of the surface, the distance of the actual hole, some aspects of the weather etc. Mr. Y says that the reason why a club is appropriate in a certain situation is that it corresponds to an obtaining golfact. Is what he says reasonable? Well, given his explanation, what he says may be right, but in any case it is an immensely awkward way to put things. Why say that a club corresponds to a golfact, when the natural way is to say that, in the current situation, it is the right one, or the most appropriate one, to use? And why say that a statement corresponds to a fact, when the natural way is to say that, in the current situation, it is true? A common way to try to rescue the idea of correspondence is to say that although sentences may not resemble facts in any self-explicable way, propositions do - and, after all, philosophers should be interested in propositions and leave sentences to linguists. And propositions can be simply defined as certain pictures or imprints of facts: whatever facts may be, propositions are to facts as pictures are to what they picture. However, this is a mere sleight of hand: it makes the correspondence relation straightforward only by shifting all the problems to the relation between the sentence and the proposition it expresses. Thus, it is tantamount to saying that although I do not know how far Prague is from Stockholm, I do know the precise distance between Prague and a place Z, which I stipulate to be 400 km north of Prague, and the remainder, namely discovering how far Z is from Stockholm, is not my business.
Language without Correspondence Having eschewed correspondence, we are left with an ‘interactive’ conception of language. However, is this conception not trivial? Does it allow us to illuminate the workings of language in any nontrivial sense? Does it allow for a substantial explication of such traditionally central concepts as meaning and truth? To find out, we should look at language with our new, ‘awakened’ eyes, and see whether (i) we still need the old concepts; and, if so, (ii) which role they now play for us. A
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good starting point might be to put oneself into the shoes of somebody who is confronted with an utterly unknown language and to observe what kinds of concepts she might helpfully employ ‘making sense’ of what the natives say. This engages us in the thought experiment of radical translation or radical interpretation, proposed by Quine (1960) and elaborated by Davidson (1984). Now the concept of truth appears to be, according to Davidson, essential to the enterprise of (radical) interpretation from the start. The point is that we cannot, from Davidson’s viewpoint, interpret somebody - i.e. take him to have beliefs and to make utterances expressing the beliefs (and other things) - unless we take him to be in possession of the concept of truth, in the sense that is capable of taking some sentences to be true, to wish some were true etc. So the essentiality of truth is yielded by almost a ‘transcendental argument’ - truth is, in a sense a precondition of interpretation, and in this sense of language9. This, however, implies a certain ‘elementarity’ of the concept of truth. You cannot define the concept or explicate it in such a way that you could teach it to somebody who lacks it - for if she did not have it, if she did not (implicitly) know what truth is, then no explaining to her would be possible; she, in fact,. would not be a thinking creature. This means that if we see linguistic expressions as tools, then they are tools peculiarly different from other kinds of tools - different in being susceptible to truth (which is according to Davidson, not a species of usefulness10). Thus, for Davidson, to find out how an expression is used (and hence what it means) is to find out when it is held for true. The situation with the concept of meaning is a bit more complicated. It is clear that meaning loses the central place it assumes within the representationalistic theories - the nature of language is no longer considered to consist in ‘grabbing things and making them into the meanings of words’11. However, does it follow that we can dispense with the concept of meaning altogether? Quine seems to be inclined to say yes (see, e.g., Quine 1992, Chapter III); but I think this is premature and I also think Davidson’s notion of radical interpretation does have room for a nontrivial concept of meaning. Let me explain why. The basic setting of the enterprise of radical interpretation is that we see the natives making their utterances and try to grasp the functions of their sentences (and then possibly find their equivalents or approximations among our own sentences12), which, according to Davidson, means that we try to find out when they hold which sentence for true. Now if we do this, there is little hope that we are going to manage to capture all the details of the native employment of an expression (let alone find an expression of our language which is employed in exactly the same way as the alien one) - what we can reasonably expect is some kind of 9
See the helpful discussion of this point given by Ramberg (to appear).
10
And it is this that separates Davidson from all kinds of pragmatists, especially from Rorty, but, it seems, also from Quine.
11 12
See Peregrin (1999).
If we not only grasp the functions, but also see some equivalents, then our radical interpretation yields a radical translation.
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capturing (or agreement) ‘in essential points’. We may figure out that the natives hold gavagai for true usually when there is a rabbit around (and when we hold there is a rabbit around for true), but it is hard to believe that we would encounter no exception, that we will, e.g., never encounter a situation when someone obviously holds gavagai for true where there is - as far as we can see - no rabbit. It should be stressed that the reason for distrusting the likelihood of exact capturing (and/or exact matching) is not only that radical interpretation involves drawing conjectures from restricted empirical evidence and is thus fallible in the way any empirical theory is. In addition to this, it seems to be simply a plain fact that no expression is likely to be employed in exactly the same way by different speakers (nor are any two expressions of different languages likely to be employed in exactly the same way by corresponding communities). If we want to speak about meanings, then what seems to be necessary is to isolate something as ‘substantial parts’ of the ways expressions are employed. However, here we must be careful: this way of presenting the problem may easily lead us to a disguised version of the notion of language as a set of labels simply stuck to meanings - only with meanings now conceived of as ‘substantial parts of their employments’. Any sharp boundary between the substantial aspects of an expression’s functioning (which determines its semantics) and the remainder is nothing but the old analytic/contingent boundary in disguise - for if such a boundary existed, it would again divide language into the meaning-determining (‘analytic’) part, and the factsstating (‘synthetic’ or ‘empirical’) part of language. Could we then simply take the entire employment of an expression as constitutive of its meaning (even at the cost of accepting that we could then rarely learn a whole meaning)? Clearly not, for this would imply that every single attitude of every single speaker would coconstitute meaning; hence that there would be no room for the concept of error within our interpretation of the natives. Having relinguished the analytic/contingent distinction, a follower of Davidson cannot accept that there is a principal division of language into the meaning-determining and facts-expressing parts. One possible reaction to this is the Quinean one, namely assigning the concept of meaning into the philosophical junkyard; but this would render much of what we really do with language nonsensical. It seems to be obvious that he who tries to understand an unknown language is bound to engage in the process of sorting out the ‘central’ and ‘substantial’ aspects of others’ linguistic behavior from the ‘peripheral’ and ‘casual’ ones - i.e. those amounting to meaning from those amounting to mere peculiar standpoints of individual utterers. Is this not inseparable from the very concept of interpretation? It is one thing to recognize that there is no such absolute boundary, and another thing to realize that people do indeed posit such a boundary when they want to „make sense“ of what their fellow humans say. When a speaker X utters a statement s, then the ‘normal’ way to perceive this is that X has a belief b and that this belief is expressed by s, it is the meaning of s. However, Davidson warns us before considering beliefs as something which could be found by opening X’s head or indeed by any other move available to a ‘natural scientist’ - for the belief is something we stipulate to ‘bring an order’ into X’s utterances, analogously to a natural scientist using meters
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or kilograms to bring an order into what she wants to understand13. We start from the facts about speakers’ utterances and we ‘decompose’ the body of these facts into a theory of what the speakers believe and a theory of what their words mean - and we use meanings as our ‘measuring units’ to account for our findings in a similar way as a natural scientist would use hers. This engenders that the decomposition is, in fact, stipulated by the interpreter, although she is surely not free to posit it wherever she wants. She is to draw it so as to create a Gestalt as helpful as possible for the enterprise of seeing what the natives are saying . Seen from this perspective, the character of the pronouncement ‘the meaning of s is such-and-such’ is similar to that of ‘the (real) price of X is such-and-such’. Just as ‘the price of X is such-and-such’ is to be understood as a shorthand for ‘the position of X within the selling-and-buying relations among people is such-and-such’, ‘the meaning of s is such-andsuch’ should be construed as saying ‘the way S gets employed within the language game to which it belongs is substantially such-and-such’. Both meaning and price may sometimes be fixed by some explicit act (and in such a case the meaning or the price becomes something which is discoverable by natural-scientific methods); however, in the typical case both are the matter of finding an ‘equilibrium’ of a number of intersecting relationships, i.e. of an interpretation. Thus we can talk about meaning only from the viewpoint of an interpreter, of someone who observes the relevant environment and ‘calculates’ the relevant value out of it14. This vantage point can also help to illuminate Davidson’s often discussed, and for many people controversial, claim that it is impossible to interpret someone as having overall false beliefs. For what does it mean, from this viewpoint, for a speaker X to have a false belief? It means to hold a sentence s, meaning something false, for true. However, to say that a sentence s of a foreign language is false is to say that the sentence s’ which we would see as its equivalent in our language is false - that we hold s’ for false. So we employ s’ in the ‘substantially same way’ as X employs s (in general, this need not mean direct matching of employment, it can also mean that s’ consists of parts we employ in the ‘substantially same’ way as X uses the parts of s), but in this particular case we differ. In this sense, to have a false belief means to use certain words in a way which we see as deviant, which does not seem to us to cohere with the way they are used otherwise. This implies that we simply cannot interpret someone’s beliefs as overall false - this would be tantamount to claiming that something always looks otherwise than usually. Thus, for Davidson, meanings are not things labeled by expressions; they are rather the measures of the peculiar kinds of roles the expressions play within the ‘economy of truth’. Hence Davidson resolutely rejects the picture on which expressions are essentially mere substitutes for things of the world, mere labels through which we see the things. Indeed, 13
Cf. Wittgenstein (1953, §132): „We want to establish an order in our knowledge of the use of language: an order with a particular end in view; one out of many possible orders; not the order.“ 14
Cf. Willfrid Sellars’ (1974) notion of „meaning as a functional classification“.
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Davidson stresses that the only reasonable way in which we can talk about ‘seeing the world through language’ is „seeing the world by dint of having language“ (see Davidson, 1997, p. 18). Language is not a prism through which we see, but it is a part of our habiliments without which we would not see what we do.
Conclusion The aim of zen is „the unfolding of a new world hitherto unperceived in the confusion of a dualistically-trained mind“ (Suzuki, 1956, p. 84). From a more practical point of view, the aim is to do away with one’s striving, which is considered to be the source of all suffering. However, the point of zen, which makes it so different from other spiritual doctrines, is that the aim is to annihilate the whole framework of striving - not to replace one kind of striving by another, say the striving for money and power by striving for some spiritual values. For zen, to strive for non-striving is as bad as to strive for anything else. Similarly, it seems to me, the naturalistic turn of analytic philosophy, as represented especially by Davidson, does not aim at a change within the traditional representational paradigm - at saying that words do not represent things. It rather urges us to forget about representing and look at language from a wholly different angle. I think that we should follow this advice. Zen masters often help their disciples by making them contemplate koans, short riddles which are apparently unsolvable, but which get dissolved (i.e. cease appearing riddlesome) once the disciple undergoes a required change of mind. The most famous of such conundrums seems to be the task to clap a single hand; but koans are often presented also in the form of a yes/no question supplemented by the comment that both possible answers are bad and will lead to punishment. Thus, for an adept of the Davidsonian analytic philosophy I would like to propose the following koan:
Do words represent things? If you say ‘yes’, you will get thirty blows with a rod; but if you say ‘no’, you will also get thirty blows with a rod!
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References Blackburn, S. (1984): Spreading the Word, Clarendon Press, Oxford. Borradori, G. (1994): The American Philosopher (Conversations with Quine, Davidson, Putnam, Nozick, Danto, Rorty, Cavell, MacIntyre, and Kuhn), University of Chicago Press, Chicago. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Davidson, D. (1986): ‘A Coherence Theory of Truth and Knowledge’, in Truth and Interpretation (ed. E. LePore), Blackwell, Oxford, 307-319; reprinted in and quoted from Reading Rorty (ed. A.R. Malachowski), Blackwell, Oxford, 1990, 120-138. Davidson, D. (1984): Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Davidson, D. (1997): ‘Seeing Through Language’, in Thought and Language (ed. J. Preston), Cambridge University Press, Cambridge, 15-27. Malpas, J. (1992): Donald Davidson and the Mirror of Meaning, Cambridge University Press, Cambridge. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J. (1997): ‘Structure and meaning’, Semiotica 113, 71-88. Peregrin, J. (1999): ‘The Pragmatization of Semantics’, in The Semantics/Pragmatic Interface from Different Points of View (ed. K. Turner), Elsevier, Amsterdam. Pinkard, T. (1991): ‘The Successor to Metaphysics: Absolute Idea and Absolute Spirit’, Monist 74, 295-328. Pippin, R.B. (1989): Hegel’s Idealism, Cambridge University Press, Cambridge. Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.). Quine, W.V.O. (1969): Ontological Relativity and Other Essays, Columbia University Press, New York. Quine, W. V. O (1992): Pursuit of Truth, revised edition, Harvard University Press, Cambridge (Mass.) Ramberg, B. (to appear): ‘Post-ontological Philosophy of Mind: Rorty vs. Davidson’. Rorty, R. (1991): Objectivity, Relativism and Truth (Philosophical Papers, vol. 1), Cambridge University Press, Cambridge. Rorty, R. (1998): ‘Is truth a goal of Inquiry? Donald Davidson vs. Crispin Wright’, Truth and Progress (Philosophical Papers, vol. 3), Cambridge University Press, Cambridge (Mass.). Rorty, R. (1980): Philosophy and the Mirror of Nature, Princeton University Press, Princeton.
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Sellars, W. (1956): ‘The Myth of the Given: Three Lectures on Empiricism and the Philosophy of Mind’, in The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minnesota Studies in the Philosophy of Science 1; eds. Feigl, H. & Scriven, M.), University of Minnesota Press, Minneapolis; reprinted in Sellars: Science, Perception and Reality, Routledge, New York, 1963. Sellars, W. (1974): ‘Meaning as Functional Classification’, Synthèse 27, 417-437. Suzuki, D. T. (1949): ‘The Sense of Zen’, Essays in Zen Buddhism (First Series); reprinted in and quoted from Zen Buddhism (Selected Writings of D.T. Suzuki; ed. W. Barrett), Doubleday, New York, 1996. Suzuki, D. T. (1956): ‘Satori, or Enlightment’, Zen Buddhism (Selected Writings of D.T. Suzuki; ed. W. Barrett), Anchor Books; quoted from the 1996 edition, Doubleday, New York. Wheeler, S.C. III (1986): ‘Indeterminacy of French Interpretation: Derrida and Davidson’, in Truth and Interpretation: Perspectives on the Philosophy of Donald Davidson (ed. LePore, E.), Blackwell, Oxford, 477-494; reprinted (with added ‘Afterthoughts’) in Reading Rorty (ed. R.Malachowski), Blackwell, Oxford, 1990, pp. 120-138. Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford.
Is Compositionality an Empirical Matter? Jaroslav Peregrin www.cuni.cz/~peregrin
The Status of the Principle of Compositionality The principle of compositionality of meaning is often seen as a kind of a ‘natural law’ of semantics: we, finite being, so the story goes, cannot grasp an infinite stock of meanings otherwise than as composed out of a finite stock of primitive building blocks. Therefore we are restricted, out of the many possible kinds of languages, to the compositional kind. Hence although there might be non-compositional languages, they would not be intelligible for us. This received wisdom has not been substantially shattered by periodically appearing attempts at showing that, as a matter of fact, our factual natural language is not compositional. However, in 1983 there appeared a different kind of challenge which was taken more seriously: Janssen (1983) presented a proof of a theorem which suggested that the principle cannot be considered as a real law because it is simply vacuous. The theorem stated, in effect, that any range of expressions can be mapped on any range of entities in a compositional way, and hence appear to imply that the principle is not capable of excluding any kinds of meanings. Recently, an elaborated version of the same argument was presented by Zadrozny (1994), and it gave rise to a wave of criticism (Kazmi and Pelletier, 1998; Westertåhl, 1998; Szabó, 2000). I think that the critical papers clearly indicate that the inference from the mathematical results of Janssen and Zadrozny to the conclusion that the principle is void is flawed; however, I also think that there is still an important aspect of compositionality which is not reflected by the current discussion and which, when scrutinized, poses another kind of challenge to the ‘natural law’ understanding of the principle of compositionality. Therefore I am convinced that despite the recent discussion, which may appear to scrutinize compositionality in an exhaustive way, the standard, pervasive understanding of the principle is much less indisputable than it may prima facie appear. I think this is partly caused by
1
underestimating the distinction between matters of pure mathematics and matters of (mathematical treatment of) empirical phenomena1. By now, the situation is relatively perspicuous on the mathematical side of the problem: it seems that since the seminal work of Montague (1970) it has become clear that the proper mathematical counterpart of the intuitive concept of compositionality is the concept of homomorphism. And given this, two (trivial) facts become relevant (see Westertåhl, ibid., for a more detailed discussion): FACT 1. Not every mapping of the carrier of an algebra A on the carrier of an algebra B is a homomorphism from A to B. FACT 2. Every mapping of the carrier of an algebra A on the set S is a homomorphism from A to an algebra with the carrier S. Now with a certain amount of oversimplification we can say that what Janssen and Zadrozny pointed out was, in effect, Fact 2 (and more elaborated variations on it); while what their critics urge is that the interesting issue is not Fact 2, but rather 1. On the empirical side, the situation might seem similarly clear: the question whether natural languages are compositional (whether their meaning assignments can be reconstructed as homomorphisms) seems to be bound to be a matter of empirical investigation. Turning our attention from formal to natural languages, and hence to the realm of empirical phenomena, we must obviously switch from the deductive, mathematical reasoning to inductive, empirical investigations; hence claims about the compositionality of natural languages appear to have to be empirical findings. Thus, Szabó (ibid., p. 478) writes: “Not all languages are compositional: surely a hypothetical language where the meanings of complex expressions are influenced by the weather, while their structure and the meanings of their constituents are not, would be non-compositional by definition.” This seems a straightforward argument: as natural languages are empirical entities, finding out whether they all have a property, e.g. whether they are compositional, is bound to be an empirical enterprise. But is it really? Consider the following analogous claim: since bachelors are empirical individuals, finding out whether they all have a property, e.g. whether they are 1
See Peregrin (2000a) for a general discussion of this methodological problem.
2
married, is bound to be an empirical enterprise. Of course that this is far from plausible: surely we have not come to be sure that bachelors are unmarried by empirical generalization! We know that bachelors are unmarried for being unmarried is the constitutive property of bachelorhood – to be a bachelor simply is to be unmarried, this is what the word “bachelor” means. Now is the conviction that the compositionality of natural languages is bound to be an empirical issue any more plausible? Compositionality perhaps is not the constitutive property of languagehood, but could it not be one of a cluster of properties which together are constitutive of it? Prima facie, this may not be apparent: why on earth could there not be a language with meanings distributed in a non-compositional manner? Why could we not, as Szabó suggests, create such distribution by definition? The question we have to ask here is whether we can make sense of the concept of meaning without relying on the principle of compositionality (hereafter PC) in the first place. The point is that the possibility of creating a non-compositional language stipulatively makes sense only provided the relation between the concept of meaning (and hence between that of language) and the concept of compositionality is contingent, i.e. empirical – for if it were the case that compositionality were (co-)constitutive of the concept of meaning, and thereby of the concept of language, this possibility would be simply precluded. Understanding PC as an empirical thesis clearly presupposes the prior distinct knowledge of what meanings are, and hence is impossible if PC is what takes part in constitution and individuation of meanings. Compare this status of PC with that of the principle of extensionality of sets, stating that two sets with identical elements are identical: it makes no sense to try to discover whether sets are extensional, for to be extensional is part of what it takes to be a set; and I claim that to be compositional is part of what it takes to be a meaning. Of course, if you assume that you are free to tamper with senses of words (like good old Humpty Dumpty), you can ‘make’ some meanings noncompositional – just like you can ‘make’ some bachelors married by letting the word “bachelor” refer to, say, butchers. Exactly here is where I think the pernicious conflation of ‘the formal’ and ‘the natural’ takes place. Within mathematics, ‘natural’ meanings are usually not crucial: what is crucial are definitions. You can take, say, the word “group” and furnish it with a meaning quite different from its “natural” one: if you do it by means of a correct definition (and if you prevent any confusion of the old and the new sense of “group”), everything is in order. Similarly you can redefine the term “meaning” so that it means, say, ‘any object assigned to an expression and 3
called meaning’ – but then you must keep in mind that studying meanings in this new sense is something different from studying meaning in the intuitive sense. Clearly studying, say, the lengths of expressions is something far removed from studying their meanings – but nothing prevents us from creating a new, artificial sense of “meaning” by calling the mapping of expressions on their lengths meaning assignment. Szabó obviously feels that he should block the possibility of reinterpreting the term “meaning” too weirdly, and hence he stipulates (p. 480): “The meaning of an expression is something that plays a significant role in our understanding of the expression.” But this is odd. What is a significant role? I would assume, for example, that almost everybody’s understanding of the word “brontosaurus” has been significantly aided by a picture – but could such a picture be taken as what the word means? To make Szabó’s postulate at least minimally plausible, we would need to change it to something like “The meaning of an expression is what one grasps when one understands the expression”, but even so it would be of little use, for the concept of “grasping” in this context is just as obscure as the concept of meaning itself. As far as I can see, our predicament is the following: either we can satisfactorily explicate the concept of meaning without the help of PC, and then we are vindicated in taking the principle as an empirical thesis, or we cannot do this, and then the compositionality of meaning, and hence of language, is a conceptual, ‘a priori’ matter. And what I am going to argue for is that there are some good reasons (going back to the work of the ‘founding father’ of modern semantics, Gottlob Frege) to take this possibility seriously. I will discuss Frege in greater detail shortly, but here I would like to provide an illustration of what I have just suggested. Consider, as an example, Frege’s semantic terminology: each expression has a sense (Sinn) and a meaning (Bedeutung). Which of the two values (if any) is the meaning, in the intuitive sense of the word, of the expression? If we hold the Humpty Dumpty view, the our answer is bound to be: obviously the one which Frege calls meaning! But could we hold that to know the meaning of ‘the president of the USA’ is to know its Bedeutung, namely (now, in 2001) G.W. Bush? It is clear that she who understands ‘the president of the USA’ need not know the actual president, but rather needs only to know a criterion (which, as a matter of fact, picks up Bush right now). It thus seems better to say that what Frege calls meaning is in fact not meaning in the ordinary sense – and that it is Frege’s sense, which in this case provides the more plausible counterpart of the intuitive concept of meaning. 4
The Story of Frege’s Bedeutung So far I have argued that we should think twice before taking for granted that PC is an empirical matter, and that we should consider the possibility of PC being an ‘analytic’ principle taking part in shaping the concept of meaning. Now let me present a more explicit story about how the role of PC within the enterprise of this shaping can be seen; it is the story of what Frege called the ‘Bedeutung’ of an expression. In this section I will use the term ‘meaning’ as the equivalent of ‘Bedeutung’. Let me briefly review Frege’s account. First, Frege pointed out that meaning cannot be a matter of what is going on in somebody’s head, that it cannot be a matter of psychology2. The argument he gave was, in effect, the following: (i) What is true and what entails what is an objective matter, independent of what anybody thinks is true or entailed – truth and entailment are not a matter of anybody’s subjective psychology (hence also logic is to be sharply separated from psychology)3. (ii) Whether a sentence is true is generally a matter of how things are in the world, but also of what the words of which it consists mean. (It is clear that the sentence “Snow is white” would not be true if, say, “is” meant has beaten in chess.) (iii) Hence meanings cannot be a matter of subjective psychology – in pain of what is true being influentiable by subjective views. Frege thus concluded that meanings must be some objective entities; and he also concluded that the meanings of at least some expressions cannot but be extracted from claims in which the expressions occur. Let us consider Frege’s analysis of the meaning of a predicate. We know that a predicate can combine with a name to form a sentence. So if we take a predicate p, we know that together with the name n1 it yields a sentence s1, with n2 it yields s2 etc.: p + n1 = s1
“to have wings” + “Frege” = “Frege has wings”
p + n2 = s2
“to have wings” + “Batman” = “Batman has wings”
...
...
Hence we can see p as a means of assigning s1 to n1, s2 to n2 etc.
2
See especially his 1918/9 paper Der Gedanke.
3
I think this holds even for what is nowadays called cognitive psychology.
5
p:
n1 → s1
“to have wings”:
“Frege” → “Frege has wings”
n2 → s2
“Batman” → “Batman has wings”
...
...
Now suppose we know what the meanings of both names and sentences are. If we denote the meaning of a as ║a║ (with quotes, if any, omitted), we can transfer the whole consideration to the level of semantics: ║p║: ║n1║ → ║s1║
║to have wings║:
║Frege║ → ║Frege has wings║
║n2║ → ║s2║
║Batman║ → ║Batman has wings║
...
...
Now suppose further that the meanings of names are the objects named by them (the meaning of “Frege”, ║Frege║, is the person Frege, that of “Batman”, ║Batman║, is Batman) and that the meanings of sentences are their truth values (║Frege has wings║ being the falsity, F, that of ║Batman has wings║ being the truth, T). (Let us, for the sake of the argument, disregard the obvious implausibility of this kind of explication of the concept of meaning of sentences.) Given this, we have ║to have wings║: Frege → F Batman → T ... In other words, ║to have wings║ is what must be added to Frege to yield F, to Batman to yield T and so on for all conceivable individuals. Frege’s ingenious idea was to explicate this entity as a function in the mathematical sense, so that ║to have wings║(Frege) = F ║to have wings║(Batman) = T ...
6
Now can we take PC to vindicate Frege’s proceeding here? Not really. The principle does not tell us more than that if the meanings of Frege is a philosopher and Frege has wings are different, then so must be those of to be a philosopher and to have wings. (The point is that the principle of compositionality implies that if the meanings of two wholes differ, then the meanings of at least one of their corresponding parts are bound to differ too.) It excludes some possibilities of capturing the meanings, but it far from pins their range down to the very one entertained by Frege, namely to identifying the meanings of predicates with functions mapping meanings of names on meanings of sentences. It thus seems that to conclude what Frege did requires some additional principles. However, what I suggest is that Frege’s conclusion can be seen as the result of his realization of the fact that there is no additional meaning-characterizing principle which would narrow down the scope of functions available; and that hence we are justified in taking recourse to a kind of a general ‘minimality maxim’4 – i.e. to see meaning assignment as the simplest of the available functions. From this viewpoint, the meaning of an expression is the simplest thing which must be added to the meanings of the expressions accompanying it to reach the meaning of the complex expression they form together. Hence the meaning of to have wings is the simplest thing which must be added to the meaning of Batman to reach the meaning of Batman has wings, to that of Frege to reach the meaning of Frege has wings etc. From this angle, the problem of finding the meaning of to have wings, given we have the meanings of both Frege, Batman etc. and Frege has wings, Batman has wings etc., is the problem of ‘subtracting’ the meanings of names, individuals, from the meanings of sentences containing the names, i.e. from certain truth values. This is the problem Frege solved by bringing in functions mapping the former on the latter (which eventually provided for the divorce of semantics from psychology and its marriage to mathematics). What has been just summarized is, of course, far from the whole story of Frege’s semantics. And the way the story continued is instructive precisely from the viewpoint advocated here: Frege soon discovered that he could not make do with his Bedeutung, because it is not compositional. (Frege, for instance, realized that to determine the truth value of a sentence like Napoleon, who realized the danger for his right wing, himself led his guards against the position
4
In Peregrin (1984) I called it the principle of Occam’s razor, but this term might perhaps be misleading. What I had in mind was the general maxim governing every scientific explanation, namely to take for extant only the minimal forces needed to bring about an observable effect.
7
of the enemy we may need more than just the truth value of the relative clause5.) In general, if Bedeutung were compositional, any two sentences with the same truth value would have to be interchangeable within any complex sentence salva veritate; and it is easy to see that this is not the case. Frege’s conclusion is that in certain contexts a sentence, or indeed any expression, can come to stand for something else than its usual meaning. This was why Frege postulated, in addition to its Bedeutung, his other semantic value, the Sinn. Hence the corrected story says that the meaning of a complex expression is not always the function of the (ordinary) meanings of its parts, but sometimes the function of their senses – or, expressed differently, that it is always the function of the meanings, but that the place of the meaning of an expression can sometimes come to be occupied by its sense.
Meanings as ‘Contributions’ There is one more aspect of the Fregean method which is worth investigating in some detail. The mechanism by which Frege established the Bedeutungen of predicates seems to be applicable only to establish meanings of one kind of words (e.g. predicates) only given we know those of other kinds of words (names, sentences) – in pain of an infinite regress. We can obtain meanings in this way only if we have some meanings to subtract from and some meanings to subtract. But this is not necessarily the case – there is a principle which stops the regress, and this is the principle stating that sentences are bound to differ in meaning if they differ in truth value6. This means that when subtracting meanings from meanings, we ultimately subtract meanings from truth values. To see what is meant by this, consider the following reformulation of the principle of compositionality: If meanings of two wholes differ, they cannot be composed in the same way, or else the meanings of all their corresponding components cannot be the same. While the standard formulation of PC claims that sameness of meaning projects from parts to wholes, this one reverses the angle and states that difference of meaning projects from wholes to 5
See Frege (1892).
6
For lack of a better name, I called it the principle of verifoundation elsewhere (see Peregrin, 1994). Cresswell (1982) considers it to be the most certain principle of semantics.
8
parts. Let us refer to this reformulation of the principle as PC* (and let us postpone the discussion of the question whether we are right in taking it as nothing more than a reformulation of PC). However, if we see the differences of meanings of parts as grounded in the differences of meanings of wholes which contain them, the question which comes naturally is what are the differences of meanings of the wholes grounded in? In the differences of meanings of larger wholes? But is this not an infinite regress? And here what I suggest the Fregean answer is, is that at some ultimate point the difference of meanings is grounded in the difference of truth values. If the meanings of some two expressions can be seen as differing in force of the fact that the meanings of some expressions containing them differ, then the latter difference, and thereby the former, must be traceable to the fact that some sentences containing the expressions differ in truth value. (And truth is, as Frege stresses, the ultimate subject matter of logic and semantics.) Frege accounted for this fact by simply identifying meanings of sentences with their truth values, but this worked only for the ‘extensional’ core of our language – once we take into account also the variety of modal sentences present within natural language (not to mention propositional attitude reports), it results into a noncompositional mapping. We can see the situation in such a way that although truth is the ultimate source of meaning7, the meaning of a sentence is not a matter of merely its truth value, but also of the sentences of which it is a part8. Thus, the meaning of Batman has wings cannot be its truth value, because it contributes by more than its truth value to the truth value of such sentences as Because Batman has wings, he can fly. (Surely we could not replace Batman has wings in this complex sentence by any other true sentence without changing the truth value of the whole!) So on the ‘upper’ end, the regress-stopper is truth - but still we seem to need a regress-stopper on the ‘lower’ one. Even if what we subtract from, ultimately, are truth values, to do away with the regress completely, we appear to need some ultimate subtrahends. From what we have said about Frege it seems that the role of this opposite regress-stopper can be played by individuals which may appear to be the natural meanings for names. But in fact the story may be more complicated. Consider Frege’s (1884, p. 73) famous claim that “It is only in the context of a proposition that words have any meaning.” This indicates that Frege was not willing to establish meanings of any words, not even names, independently of investigating the truth values of the
7
A point urged forcefully especially by Davidson (1984).
8
See Peregrin (2001, Chapter 8) for more details.
9
sentences in which they occur9. Hence it would suggest that we do not really subtract, but rather decompose. We seem to subtract the meanings of names from the meanings of sentences to reach the meanings of predicates, but this is only because, as a matter of methodology, we do not want to deal with all meanings in one sweep. We simply analyze the meaning of one part of a sentence (predicate) relatively to another part (name). In principle, we could have done it the other way around – namely have taken the meaning of the predicate as primitive and analyzed that of a name as a function assigning truth values to objects of these kinds10. This leads to the conclusion that the meaning of an expression is the contribution which the expression brings to the truth of the sentences in which it occurs. This is, to repeat, the result of the thesis that what is constitutive of the concept of meaning is (i) the principle of compositionality; (ii) the principle of verifoundation (stating that difference in truth value forces difference in meaning); and (iii) the maxim of ‘minimality’. This also indicates that from the Fregean viewpoint, another constitutive characteristic of meaning, besides compositionality, is its interlinkedness with truth – nothing can be the meaning of a sentence unless it determines its truth conditions11. Now consider Szabó’s (ibid.) example of “Crypto-English” which arises from English by the interchange of meanings of “Elephants are gray” and “Julius Caesar was murdered on the Ides of March”. If we were to subscribe to the Fregean attitude just outlined, then if we change the meaning-assignment of English in the suggested way, what results is no longer a meaning assignment – not that it is not a meaning assignment for English, but it is not a meaning assignment for any language. The reason is that if the meaning of a word is the contribution it brings to meanings of the wholes in which it occurs, there is no way of changing the meanings of a sentence without changing the meaning of some of its parts.
Compositionality as a Regulative Principle I anticipate the objection that rewriting PC as PC*, which played a substantial role in our above considerations, is not legitimate. If PC is understood as an empirical thesis about the way 9
See Peregrin (2000b) for a discussion of why Frege took meanings of sentences as constitutive of the meanings of their parts and not vice versa.
10
While in the first case a ‘property’ comes to be analyzed as a class of ‘individuals’, in the latter an individual comes to be analyzed as a bundle of ‘properties’.
11
This was urged also by David Lewis (1972) in course of his criticism of Katz.
10
meanings of parts add up to the meaning of a whole consisting thereof, then turning PC into PC* will probably appear as a mutilation – for from this empiricist perspective it is part and parcel of what PC says that parts are primary to the whole composed of them. And what I suggest is that we give up such a metaphysical reading of PC in favor of the ‘structural’ reading which takes it to * say nothing more than what also PC says. This is, I think, an integral part of Frege’s
‘contextualism’, exhibited by the above quotation and also by his repeated insistence that concepts which add up to a proposition are not primary to the proposition12. I do not expect everybody to accept the particular Fregean story about meaning just outlined; and it is not my purpose to agitate for it here (although I do believe there is much to be endorsed in it). What I do want to argue for is that we should see that there are good reasons for taking PC not as an empirical, but rather as a conceptual, and hence regulative principle: as a principle which is needed to single out the level of meanings from the many level of semantic values which may be associated with meaningful expressions13. The analogy between the concept of meaning and that of set is worth pursuing here. Sets are, we may say, one peculiar kind of the species the other kinds of which are collections, groups, herds etc. What distinguishes sets from the others? It is especially that they are individuated solely by their elements – you can possibly have two different groups of people consisting of the very same individuals, but never two such different sets. Hence, what is distinctive of the concept of set is the principle of extensionality.
12
Hence I think that this view, according to which PC states nothing over and above of what is stated also by PC*, can throw some light on the apparent tension between Frege’s (implicit) endorsement of PC and his contextualism. (See Janssen, 2001, and Pelletier, 2001, for recent contributions to the discussion.) What I suggest is that what we have here is not an incompatibility, but rather a certain kind of complementarity. While the contextualism leads to the view that subsentential expressions are to be seen as based on those of sentences, PC (together with the principle of verifoundation and the ‘minimality’ maxim) instructs us how to individuate their meanings: namely as the contributions these expressions bring to the meanings of sentences (and eventually to their truth values). See Peregrin (2001, Chapter 4) for more details. The point that there is a sense in which compositionality is not incompatible with – but rather complementary to – contextuality is pointed out also by Hodges (2001, p. 19). That compositionality and contextuality are complementary aspects of Frege’s viewpoint is argued also by Rott (2000). A thoughtful general discussion of the possibilities of reading PC as compatible with a holistic view of language (necessitated by contextualism) is given by Pagin (1997). 13
Of course what really is the meaning of the word ‘meaning’ is largely a truly empirical question: a question to be answered by an investigation of the ways in which we actually employ the term. Thus although I suggest that ‘meaning is compositional’ is an analytic truth, that this is so is at most an empirical one – hence I do not think there can be a knock-down argument here. What I put forward are rather ‘pragmatic’ arguments indicating that PC, if endorsed as analytic, can smoothly do us a job which needs to be done.
11
Now meanings are, analogously, a peculiar kind of the entities accompanying expressions, entities other kinds of which are mental images, referents etc.; and we need a way of singling them out. And what I suggest is that it is PC which can plausibly perform this task. Of course, our expressions are also often associated with various kinds of entities which do concern their semantics and which are not compositional. (Fregean Bedeutungen or extensions are an example.) Now it is hard to prevent somebody from calling some of these non-compositional entities “meanings” – just as it is hard for a set-theoretician to prevent somebody from using the term “set” for something which is nonextensional. But if the term meaning is not to be ‘free-floating’ in the sense of being freely applicable in the Humpty Dumpty way, there must be a criterion for singling out the kind of entities to which it applies properly. Consider the language of first-order logic. The language is taken to be extensional (i.e. based on semantic values of the kind of Frege’s Bedeutungen) and hence it might seem that what its formulas should be taken to mean are their truth values. However, if we work on the level of the predicate calculus (rather than the propositional one), we know that there is more than truth values and their combinatorics in play; we know that to account for the semantics of the formulas of the language we need the machinery of satisfaction. Now what is to be seen as the meaning of such a formula? When check the origins of model theory, we can see that the reason why Tarski introduced the concept of satisfaction was precisely because he could not build a compositional semantics out of merely truth values14; and that the level at which the semantics becomes compositional is the level where the formulas are taken to denote classes of valuations of variables (or, equivalently, classes of sequences of objects, or functions from the valuations or the sequences into truth values etc.). It is, then, from our perspective, only the knowledge of which valuations satisfy a given formula which can be taken as amounting to knowledge of what the formula means. Take an example of an allegedly non-compositional language: Hintikka’s independencefriendly logic (IFL)15, in which formulas are interpreted as representing certain games. The author repeatedly claims that the language is not compositional, and indeed the game associated with a formula can generally not be built out of those associated with its subformulas. On the other hand, even the exponents of IFL do not doubt that there must be some level at which the language is compositional: “Various compositional semantics for IF logics have been designed
14
See Kirkham (1992, Chapter 5), and also Peregrin (2001, Section 6.2).
15
See Hintikka (1996) and Hintikka and Sandu (1997).
12
(...). There was no reason to expect otherwise, since the syntax of IF logic is finitely generated. The reason why it is still worth insisting on non-compositionality is that non-compositional semantics by means of games of imperfect information may turn out to be simpler and easier to apprehend than compositional ones; not to mention the philosophical virtues of game-theoretical semantics such as its being a representation of the human activities of seeking and finding connected with quantifiers, decision-making at connectives, and a responsibility-shift at game negation.”16 From our viewpoint, this simply shows nothing more (and nothing less) than that the game introduced as represented by a formula cannot be seen as its meaning; that the real meaning is to be sought on a semantic level which is compositional (and the specification of which appears not to be trivial). Note, however, that to descend to this level is required only if what we are after are the meanings of individual formulas or symbols, i.e. in the individualized contributions these bring to the truth of formulas in which they occur as components – if we are interested in more holistic semantic aspects of the language we may indeed find it preferable to operate at the noncompositional level in which the language is specified. Or consider Kamp’s DRT17, which associates sentences and larger units of discourse with the discourse representations structures (DRS’s). Some sentences (like A farmer owns a donkey) link readily with DRS’s (like <x; y | farmer(x); donkey(y); owns(x,y)>), and hence it might seem that these DRS’s are what the sentences mean. But for sentences containing anaphoric elements (like He beats it) there is no self-standing DRS to be associated with: they need an input DRS (like the above one) and they use it to produce an output one (like <x; y | farmer(x); donkey(y); owns(x,y); beats(x,y)>). This indicates that generally a sentence can be associated not directly with a DRS, but rather with a mapping of DRS’s on DRS’s (which in the ‘non-anaphoric’ cases amounts to trivial addition). It is only this level which can aspire to being compositional. But even if it is, its aspirations to being the level of meanings would be frustrated, from the Fregean viewpoint, by the fact that it does not provide for the interlinkage with truth. This might lead us to seek for the true semantics of DRT, as Kamp himself does, at the level of the embeddings of DRS’s into models of reality and their classes.
16
Pietarinen and Sandu (2000, p. 153).
17
See Kamp (1981) and Kamp and Reyle (1993).
13
Conclusion To summarize, I think that it is simply mistaken to assume that PC must be a kind of a naturalistic generalization, which some or all empirical languages obey, but which languages we produce artificially need not obey. I think we should seriously consider the possibility that the connection between meaning (and hence language) and compositionality is not an empirical (but rather a conceptual, an analytic) issue. And we should be aware that if we take compositionality as coming ‘after meaning’, then we must have another principle to enable us to single out the level of meanings from the various levels of semantic accessories of expressions. I believe that we should follow Frege and realize that it is precisely PC which is particularly suitable for this task. If we do so, we start to see PC as a regulative principle, a principle which helps us settle controversies about what meaning is and what it is not. (Moreover, I believe that if we do this, then we are further to follow Frege to the conclusion that the meaning of an expression is best seen as a kind of contribution, namely the contribution that the expression brings to the truth of sentences in which it occurs – but this is, so to say, ‘another story’.) In any case, I am convinced that the idea of creating a non-compositional language by definition may be no more plausible than so creating a married bachelor.
References Cresswell, M.J.: 1982, 'The Autonomy of Semantics', Processes, Beliefs and Questions (ed. by S.Peters and E.Saarinen), Reidel, Dordrecht. Davidson, D. (1984): Inquiries into Truth and Interpretation, Clarendon Press, Oxford. Dummett, M. (1976): ‘What is a theory of meaning? (II)’, in Truth and Meaning (ed. by G. Evans & J. McDowell), Oxford University Press, Oxford, 67-137; reprinted in and quoted from Dummett: The Seas of Language, Oxford University Press, Oxford, 1993, 34-93. Frege, G. (1880/81): ‘Booles rechnende Logik und die Begriffsschrift’, printed in Frege (1969), pp. 9-52. Frege, G. (1884): Grundlagen der Arithmetik, Koebner, Breslau; English translation Foundations of Arithmetic, Blackwell, Oxford, 1953. Frege, G. (1892): ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und philosophische Kritik 100, 25-50. Frege, G. (1918/9): ‘Der Gedanke. Eine logische Untersuchung’, Beiträgen zur Philosophie des deutschen Idealismus I, 58-77. Frege, G. (1969), Nachgelassene Schriften (ed. Hermes, H., Kambartel, F. and Kaulbach, F.), Meiner, Hamburg; English translation Frege: Posthumous Writings, Blackwell, Oxford, 1979. Hintikka, J. (1996): The Principles of Mathematics Revisited, Cambridge University Press, Cambridge. 14
Hintikka, J., and Sandu, G. (1997): ‘Game-theoretical Semantics’, in Handbook of Logic and Language (eds. J. van Benthem and A. ter Meulen), Elsevier / MIT Press, Oxford / Cambridge (Mass.). Hodges, W. (2001): ‘Formal Features of Compositionality’, Journal of Logic, Language and Information 10, 7-28. Janssen, T.M.V. (1983): Foundations and Applications of Montague Grammar, dissertation, Mathematical Centre, Amsterdam. Janssen, T.M.V. (2001): ‘Frege, Contextuality and Compositionality’, Journal of Logic, Language and Information 10, 115-136. Kamp, H. (1981): ‘A Theory of Truth and Semantic Representation ’, in Formal Methods in the Study of Language (eds. Groenendijk, J. & Janssen, T. & Stokhof, M.), Mathematical Centre, Amsterdam. Kamp, H. & Reyle, U. (1993): From Discourse to Logic, Kluwer, Dordrecht. Kazmi, A. and Pelletier, F.J. (1998): ‘Is Compositionality Vacuous?’, Linguistics and Philosophy 21, 629-633. Kirkham, R.L. (1992): Theories of Truth, Cambridge (Mass.), MIT Press. Lewis, D. (1972): ‘General Semantics’, in Semantics of Natural Language (eds. Davidson, D. & Harman, G.), Reidel, Dordrecht. Montague, R. (1970): ‘Universal Grammar’, Theoria 36, 373-398; reprinted in Montague: Formal Philosophy: selected papers of R.Montague (ed. Thomason, R.), Yale University Press, New Haven, 1974. Pagin, P. (1997): ‘Is Compositionality Compatible with Holism?’, Mind & Language 12, 1133. Pelletier, F.J. (2001): ‘Did Frege Believe Frege’s Principle?’, Journal of Logic, Language and information 10, 87-114. Peregrin, J. (1994): ‘Interpreting Formal Logic’, Erkenntnis 40, 5-20. Peregrin, J. (2000a): ‘The “Natural” and the “Formal”’, Journal of Philosophical Logic 29, 75-101. Peregrin, J. (2000b): ‘“Fregean” Logic and “Russellian” Logic’, Australasian Journal of Philosophy 78, 557-575. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Pietarinen, A. and Sandu, G. (2000): ‘Games in Philosophical Logic’, Nordic Journal of Philosophical Logic 4, 143-173. Quine, W.V.O. (1992): Pursuit of Truth, revised edition, Harvard University Press, Cambridge (Mass.). Rott, H. (2000): ‘Words in Contexts: Fregean Elucidations’, Linguistics and Philosophy 23. Stekeler-Weithofer, P. (1986): Grundprobleme der Logik, de Gruyter, Berlin. Szabó, Z.G. (2000): ‘Compositionality as Supervenience’, Linguistics and Philosophy 23, 475-505. Westertåhl, D. (1998): ‘On Mathematical Proofs of the Vacuity of Compositionality’, Linguistics and Philosophy 21, 635-643. Zadrozny, W. (1994): ‘From Compositional to Systematic Semantic?’, Linguistics and Philosophy 17, 329-342.
15
FORMAL LOGIC AND THE PURSUIT OF MEANING Jaroslav Peregrin
We want to establish an order in our knowledge of the use of language: an order with a particular end in view; one out of many possible orders; not the order. Wittgenstein
1. Introduction The linguistic turn that occurred in the minds of various philosophers during the first half of our century has led to the conclusion that to resolve the traditional philosophical problems means to dissolve them by means of the logical analysis of the language in which they are formulated. The spread of this insight, which presents something truely novel, is probably the most significant event in the history of twentieth-century philosophy; at the same time, however, it is the source of profuse misunderstandings and misinterpretations. Many of the proponents and followers of the linguistic turn have come to the conclusion that this turn amounts to the ultimate word on philosophy, meaning the end of philosophy in the traditional sense and the rise of a new kind of scientifico-philosophical thinking shaped by Cartesian rigor. Formal logic, viewed as the means to uncover the "true structure" of language and consequently the "true structure" of the world, has moved to the center stage of philosophy. In this paper I would like to show that however great the significance of the linguistic turn and of the employment of logical means in philosophical analysis, this idolization is unwarranted. I would like to show that the import of logic for philosophy is neither that it lets us get hold of meanings in an explicit way, nor that it shows us the "real structure" of the world, a structure otherwise obscured by language; rather, its value is that it provides what can be called perspicuous representations. It offers us vantage points from which we can comprehend the vast variety of language, and consequently of the world that we cope with by means of language, allowing us to better understand their nature. I would like also to indicate in which way awareness of the limitations of logical analysis is what distinguishes philosophers like Frege, Wittgenstein or Quine from Tarski, Carnap and the majority of the contemporary analytic philosophers and theoreticians of language.
2. The Linguistic Turn of Philosophy The linguistic turn, as Rorty puts it, is based on "the view that philosophical problems are problems which may be solved (or dissolved) either by reforming language, or by understanding more about the language we presently use."1. 1
Richard Rorty: The Linguistic Turn (Chicago: The University of Chicago Press, 1967), p. 3. 1
Driving the turn, roughly speaking, is the replacement of problematic questions of the form (1) by corresponding questions of the form (2): what is (an) X? what is the meaning of 'X'?
(1) (2)
This means that we stop asking what is consciousness (matter, evil, etc.)? and instead begin to ask what is the meaning of 'consciousness' ('matter', 'evil', etc.)? (The latter question can then possibly be further transformed into how do we use the word 'consciousness' ('matter', 'evil', etc.)?) If 'X' has a meaning, then (1) and (2) turn out to express one and the same question. However, if 'X' has no meaning, then (1) and (2) are essentially different: (1) contains a meaningless word and is thus itself meaningless, whereas (2) is perfectly meaningful. In such case we have trouble answering (1) because it seems to deserve an answer of the form (An) X is ..., which is bound to be meaningless; but we have no trouble answering (2) as we can give the perfectly meaningful and appropriate answer 'X' has no meaning. (1) is misleading in that it presupposes the existence of X - we are forced to assume the existence of X as soon as we accept (1) as a question, independently of how we might answer it. The point of the shift from (1) to (2), from talk about objects to talk about words, is that it dispenses with the misleading presupposition. Quine aptly calls this shift one of "semantic ascent".2 Russell, Carnap and other exponents of the linguistic turn pointed out that the problem with language is that expressions which appear to stand for an object may well not do so. In his pathbreaking paper "On Denoting", Russell showed by logical analysis that, despite appearances, expressions such as someone, everyone or the king of France are not names;3 and Carnap demonstrated how such kind of logical analysis can be used to elucidate the nature of "names" such as God, being or nothingness and so to reveal the emptiness of many classical philosophical problems. Such considerations resulted in the conclusion that the surface or apparent structure of natural language is not the structure which is relevant for the semantics of language, that the relevant structure is hidden, and that the task of the philosopher is to bring it to light. This then led to the view that natural language is only an imperfect embodiment of a perfect ideal structure which can be disclosed by an analysis; and logic was promoted as the general tool for this kind of analysis. In this way logic launched on its triumphant campaign in the realm of philosophy, conquering or exterminating its parts one after another.
3. The Formalistic Turn of Logic At approximately the same time at which the linguistic turn was finding expression in the writings of Russell, Wittgenstein, Carnap and others, another important event, closely connected with it, took place as well. This was the birth of formal logic in the strict sense.
2
Willard Van Orman Quine: World and Object (Cambridge, Massachussets: The MIT Press, 1960), p. 271. 3
Bertrand Russell: "On denoting", Mind, XIV (1905), 479-493. 2
To avoid misunderstanding, let us stress the difference between formal logic and logic that is merely symbolic. Both formal and symbolic logic are based on the substitution of symbols for natural language statements and expressions; however, whereas within the merely symbolic approach symbols are employed solely for the purpose of regimentation (in Quine's sense), i.e. of suppressing those aspects of natural language expressions which are considered irrelevant for the analysis of consequence, within the truly formal approach the resulting systems of symbols logical calculi - are taken to be abstract algebraic structures. Aristotle used letters to represent unspecific terms; hence he could be considered an early symbolic logician. Frege and Russell were symbolic logicians par excellence; but neither of them was a formal logician.4 It was Hilbert who for the first time viewed logic as a strictly formal matter; however, a tendency towards such a conception of logic is clearly recognizable already in the writings of the logical school of Boole and Schröder. For Frege, a symbolic formula represents a definite statement, a definite "thought". There are situations in which it may be reasonable to disregard the particular statement a formula represents; but there is no way to detach the latter from the former completely. For Hilbert, on the other hand, a formula is first and foremost an abstract object, an object which we are free to interpret in various alternative ways. The nature of the difference between Fregean symbolic and Hilbertian formal logic becomes clear when we consider the controversy between the two logicians about the nature of axioms and implicit definitions. For Frege, as for the Ancients, an axiom is a statement the refutation of which is beyond the scope of human imagination; therefore there can hardly be a discussion on whether something is or is not an axiom. For Hilbert, on the other hand, an axiom is a statement which differs from other statements only in that we choose it as foundation; we are free to choose axioms according to our liking. It was the formal approach to logical calculi which allowed logicians to develop metalogic and model theory, to prove theorems about logical calculi. The work of Löwenheim, Skolem, Gödel, Tarski and others who entered the vast new world of "liberated signs" elevated logic to a new paradigm. Tarski's model theory then presented the next step in the takeover of philosophy by logic: after the logical analysis of language as pursued by Frege, Russell and Carnap eliminated the old metaphysics, model theory slowly moved to fill the gap. A volume of selected papers on model-theoretic semantics of one of the most influential twentieth-century theoreticians of meaning, simply bears the title Formal Philosophy.5
4. Correspondence What makes something an expression, be it an expression of natural language or of a symbolic regimentation thereof, is the fact that it is understood as a mere way of presenting its meaning. Normally we do not perceive an expression as such, but rather look "through it" at its meaning. An expressions and its meaning are inseparably connected; they are, as de Saussure puts it, two
4
For both Frege and Russell, symbols were, as Tichý puts it, "not the subject matter of their theorizing but a mere shorthand facilitating discussion of extra-linguistic entities." (Pavel Tichý: Foundations of Frege's Logic, Berlin: de Gruyter, 1988, p. ii). 5
Richmond Thomason (editor): Formal Philosophy: Selected Papers of Richard Montague (Yale: Yale University Press, 1974). 3
sides of a single sheet of paper.6 Thus, the connection between an expression and its meaning can be said to be a priori, not to be found in the world, but rather being constitutive of our grasp of the world. The linguistic turn requires intentional suppression of this normal perception; we look at an expression as an object and check whether there really is something it points at. Similarly for the formalistic turn of logic: to be able to consider a system of logical formulas as something self-contained, something that can be interpreted in various alternative ways, we must regard formulas as not having a priori meanings, but rather as mere strings of letters. This means that both the linguistic turn of philosophy and the formalistic turn of logic require us to view the relation between an expression and its meaning as a contingent, a posteriori, fact. However, we cannot adopt this view universally without undermining our ability to use language and engage in argumentation. If we held the relation between an expression and its meaning to be always contingent, then no statement could be true necessarily, in every possible world (and hence we would not be able to articulate any universally valid argument) - for even for a statement that means (in the actual world) something which is true in all possible worlds there would be possible worlds in which the statement would be false, namely each possible world in which the statement would mean something false (in that world).7 To say that the relation between a statement and its meaning is contingent is to say that there are possible worlds in which the statement means something false; and for the sake of rational argumentation we need statements which have meanings independently of possible worlds (which are, so to say, "about" possible worlds).8 Hence we need both the perspective which allows us to look at the relation between an expression and its meaning as something contingent (a posteriori, "within the world") and the one which allows us to look at this relation as something necessary (a priori, "about the world"). Let us call the former the expression-as-object perspective and the latter the expression-asmedium perspective. The expression-as-object perspective is the perspective of a foreigner trying to figure out how to translate our expressions into those of his own language (or that of a linguist intentionally reflecting upon our usage of language); the expression-as-medium perspective is that of our fellow speakers chatting away without any awareness of their use of language. As philosophers we need both perspectives, and, moreover, we need to go back and forth between them. The need to switch between the two perspectives is quite obvious when we try to state explicitly what a given expression means. Let us consider the statement (3), or its Tarskian variant (4). Snow is white' means that snow is white Snow is white' is true if, and only if, snow is white
(3) (4)
Such articulations of the correspondence between language and the world, which are at the heart of the foundation of the Tarskian correspondence theory of truth, have initiated a broad and 6
Ferdinand de Saussure: Cours de linguistique générale (Paris: Payot, 1931).
7
Contingency of meaning thus makes for a double-dependence of the link of a statement to its truth value on possible worlds: not only that a proposition can be true in some possible worlds and false in others, but also that a statement can mean different propositions in different possible worlds. 8
This is to say that possible worlds cannot be used to explain language, because they themself make sense only on the background of a language. 4
still continuing discussion.9 The central issue in this discussion is the status of sentences articulating correspondence: is (4) a necessary or a contingent truth? If it is necessary, then the correspondence theory manages to state the truth conditions of a contingent statement without telling us anything factual, which seems absurd. If, on the other hand, it is contingent, then how is it possible that we directly see its truth? If we consider the sentence snow is white as a priori equipped with its meaning (i.e., if we use the expression-as-medium perspective), then to say either (3) or (4) is to utter a truism. If, on the other hand, we were to look at snow is white as a string of letters whose meaning (if any) is a matter of empirical investigation (hence adopting the expression-as-object perspective), we would make the intelligibility of (3) and (4) itself an empirical issue. In other words, understanding this sentence presupposes knowledge of its truth. What we need to do is use the expression-as-object perspective for the first occurrence of snow is white in (1) or (2) and the expression-as-medium view for the second; only then are we able to see the statement as a nontrivial piece of information, on a par with 'Schnee ist weiß' is true if and only if snow is white. (This swithc of perspective is, of course, what the apostrophes are employed to effect.) This example illustrates that it is only through the ability to treat meanings as detachable and to switch between the expression-as-medium and the expression-as-object view (comparable with switching between perceiving a window and looking through it) that we can make sense of correspondence. More generally, it is this ability that underlies both the linguistic turn of philosophy and the formalistic turn of logic. The ability to view logical formulas both as selfcontained objects and as mere ways of pointing to their meanings is what makes it meaningful to consider alternative interpretations of formulas. It is this ability which made possible the development of genuine formal logic and model theory. And it is the same ability, applied to expressions of natural language, that makes it possible to understand truth as correspondence and to complete the 'semantic ascent'. However, the art of playing hide and seek with meanings can be deceptive: we may delude ourselves into thinking that we have gained everything when in fact we have lost everything because we have lost firm ground beneath our feet.
5. Two Faces of Language It may be helpful to use the spatial metaphor and to speak about "inside language" and "outside language". To be inside means to use language as the medium of grasping the world; to be outside means to perceive language as a thing among other things of our world. To be inside is to take an expression as inseparably and unquestionably connected with its meaning, while to be outside is to perceive the connection between an expression and its meaning as an empirical fact. If we are inside a house and perceive the sky through a hole in the house's roof, then it makes no sense to ask whether we really do perceive the piece of sky we do; whereas if we are outside the house, then the question whether an inside observer can perceive this or that piece of the sky is meaningful and nontrivial. The perspective from inside is the expression-as-medium perspective; whereas that from outside is the expression-as-object perspective. If we are inside English, then we perceive what statements like (3) and (4) say as a priori; if we are outside, we perceive it as a posteriori. Hence we may draw the conclusion that to be 9
See especially Donald Davidson: "The Structure and Content of Truth", Journal of Philosophy, 87 (1990), 279-328, and the literature quoted there. 5
inside a language prevents us from seeing the language in the unprejudiced way, and that therefore we should try to stay outside every language. However, this seems to be simply impossible for a human being, and it is surely impossible for a theorizing human being. There is no necessity in adopting a particular language, but it is necessary to adopt some language, and adopting a language means to approve the necessity of its necessary statements. As Wittgenstein puts it, "the must corresponds to a track which I lay down in language."10 If we do not speak German, then finding out that the sentence 'Schnee ist weiß' means that snow is white is like finding out that, say, Hamburg is a port; if, on the other hand, it is German that is the language we use to cope with the world, then we cannot find out anything of this kind, because the knowledge of it underlies the very possibility of 'finding out'. Moreover, the replacement of the study of the mind and of the world by the study of language which underlies the linguistic turn is meaningful only due to the fact that language acts as our universal means of coping with the world, that it is a medium. In other words, the linguistic turn makes sense only when related to the language we are inside.11 To sum up: our language is a Janus-faced being; it may be "in the world" (when we are outside it), or it may be "about the world", i.e., "transcendent to the world" (when we are inside it).12 We can move in and out; but we cannot be both in and out in the same time. However, to realize all of this means to question the philosophical significance of the correspondence theory, and of the idea of linguistic turn as resting on this theory. The point is that to make the theory of correspondence nontrivial, we need to be outside the language in question; but to make the theory into a path-breaking piece of philosophy we would have to be inside it. If we are inside, then the theory of correspondence is trivial, whereas if we are outside, then it is one of the numerous hypotheses of natural science to be tested by field scientists. What we can get hold of and thus use to articulate correspondence is the language-as-object; but the notion of correspondence is philosophically significant only when related to language-as-medium. 10
Ludwig Wittgenstein: Bemerkungen über die Grundlagen der Mathematik (Oxford: Blackwell, 1956), II, §30. 11
In fact, the problem of the two faces of language is nothing new; it is only the modern reincarnation of the much more traditional problem of the ambiguity of subjectivity. The subject, the ego, can be considered either as a thing on a par with other things of the world ("psychological" subject) or as something that is transcendent to the world, that is, in Wittgenstein's words, not a part of the world, but rather its boundary ("transcendental" subject). If we want, as Husserl did, to use an analysis of the subject as a step toward the analysis of the world, we must consider the subject in the latter sense, as a transcendental ego; we do not need subjectivity as "Seelenleben", but rather subjectivity as "Geltungsgrund aller objectiven Geltungen und Gründe" (Cartesianische Meditationen, Hamburg: Meiner, 1977, p. 27). The linguistic turn then means only the replacement of the subject by language. The opposition between the psychological and transcendental subject reappears as the opposition between the notion of language-as-object ("grammatical language") and the notion of language-as-medium ("transcendental language"), only the latter being able to underlie ontological considerations. 12
In fact, this kind of "Janus-facedness" is the general characteristic of everything that is normdriven (and thus rational). Any norms are bound to be outgrowths of human communities and viewed as such they appear as contingent products of factual historic developments; but we, as rational beings, are characterized by the ability to obey norms; in other words, to assume the viewpoint from which they appear to us as necessary. An extensive analysis of this range of problems has been given by Robert Brandom (Making it Explicit, Cambridge, Mass.: Harvard University Press, 1994). 6
In contrast to Tarski, Wittgenstein was clear about this predicament from the beginning. Like Tarski, the Wittgenstein of the Tractatus was convinced that correspondence was the key concept, but unlike Tarski he immediately realized its essential deceptivity. He clearly saw that if we understand language in terms of picturing reality, then we question all necessary statements (tautologies and contradictions), because these are not pictures. Thus he was led to the seemingly counterintuitive conclusion that the statements of philosophy cannot be in fact meaningful - the reason is that although truth may be indeed considered reducible to correspondence, no theory of correspondence that would imply the reduction can be consistently articulated.
6. Two Kinds of Logic The primary aim of logic is to summarize basic instances of consequence, basic patterns of our reasoning used in arguments and proofs. Thus, logic is inseparably linked to natural language the medium of expression in which arguments and proofs are originally formulated. The use of symbolic and formal devices within logic arises from recognition of the fact that such patterns are easier to summarize if we do not take natural language at face value but reconstruct it instead as a strictly rule-based grammatical system. This leads us to the concept of a formal calculus, a calculus consisting of a formal grammar determining the class of well-formed expressions, plus axioms and rules of inference determining the relation of consequence and hence providing the needed criterion of validity of proofs. Formal logic suspends the relationship between natural language and its formal reconstructions in order to permit the undisturbed analysis of properties of formal calculi. However, once formal calculi began to be studied independently of their relationship to natural language, they slowly came to be seen as languages of their own - not as reconstructions of natural language, but rather as alternatives. Taken in this way they turned out to be substantially incomplete: whereas it is essential for natural language expressions that they be linked to their extralinguistic denotations, expressions of the formal calculi lack such links. This was the point at which Tarski entered the scene: his model theory appeared to provide precisely what was needed, namely extralinguistic entities to which expressions of formal calculi could be linked. Thus the parallelism between natural language and languages of formal logic seemed to be complete; and scholars like Montague began to deny any real difference between the two kinds of languages. Notice that the original aim of logic is compatible with the language-as-medium perspective. We need not speak about language, we only need to replace natural language statements and arguments by their formal regimentations which allow us to ignore all irrelevant idiosyncrasies and so to see the relevant patterns. Thus, it allows us to capture the unity of sense within the multiplicity of surface forms and to account for the infinite class of valid instances of consequence by finite means.13 However, if interpreted formal calculi are seen as alternatives to natural language rather than as its regimentation, logical analysis might be seen not as a schematization of natural language 13
The intention to use symbolic means precisely to this effect has been clearly formulated in the introduction of Frege's Begriffsschrift. See Gottlob Frege: Begriffsschrift (Halle/Saale: Nebert, 1879), p. V. 7
sentences, but rather as a way of making their meanings explicit by furnishing them with modeltheoretical interpretations14. The problem of explicating meaning has come to be understood as the problem of finding a model theory adequate for natural language. Many theoreticians have settled for so-called "representational semantics", claiming that we must first develop adequate set-theoretic representations of what the world is like and what it could be like, and only then study the relations of sentences to these representations.15 However, to develop an explicit semantics means to step outside natural language and hence to demote it to a mere object among other objects of our world. From the vantage point of the basic aim of logic this whole approach is disputable. We can, of course, consider a formal calculus as a self-contained whole, study various relations between its formulas, and talk about some of these relations as relations of "consequence"; but doing so means doing algebra, not logic in the genuine sense of the word. Algebraic theories resulting from the autonomous study of logical calculi are respectable as such, and provide useful tools to the logician; however, they are not as yet logic; similarly as the theory of solving differential equations, surely indispensable for a physicist, is not as yet physics. Axiomatic systems were introduced to characterize and explicate the pre-theoretical notion of consequence; their basic aim was to characterize the infinite number of instances of consequence by finite means (by reconstructing them as potentially inferable by means of a finite number of inference rules from a finite numbers of axioms), i.e. to deliver a criterion of consequencehood. Model theory is merely another such method of characterization (and it is in fact questionable as a method, in that it does not restrict itself to finite means and hence need not provide a real criterion). Thus the formal completeness of a logical calculus does not prove its axiomatization to be "right" (i.e., to adequately capture the consequence relation as it is "directly" presented model-theoretically), rather, it shows that two alternative formal characterizations of consequence, the axiomatic and the model-theoretic one, coincide (thus corroborating the essentially formally unprovable - claim that they both adequately capture the pre-theoretical notion of consequence). We have distinguished between two notions of language, the notion of language as an object among other objects of our world, and the notion of language as a medium of presentation of the whole world. Given the basic dependence of logic on language, we can draw a similar distinction for logic: a logical calculus can either be taken as a mere object within our world, or it can be understood as a regimentation of language in its transcendental capacity. This is tantamount to the distinction between logic as calculus and logic as language introduced by van Heijenoort.16
14
This institutes an important ambiguity of the term 'interpretation'; see Peregrin: "Interpreting Formal Logic", Erkenntnis, 40 (1994), 5-20. 15
Thus John Etchemendy: The Concept of Logical Consequence (Cambridge, Massachusetts: The Harvard University Press, 1990). In fact, this means a return to metaphysics, although to metaphysics in a set-theoretical disguise. Brilliant samples of systems of such a set-theoretical metaphysics can be found, e.g., in M.J. Cresswell's Logic and Languages (London: Methuen, 1973) or in Jon Barwise and John Perry's Situations and Attitudes (Cambridge, Massachusetts: MIT Press, 1983). 16
Jean van Heijenoort: "Logic as Calculus and Logic and Language", Synthese, 17 (1967), pp. 324-30. 8
There is little doubt that our medium of reasoning is not a language "within the world", but rather a language "about the world"; i.e., that it is the medium view of language that must form the ultimate basis of logic. Calculi of formal logic, if they are not to be understood simply as algebraic structures on a par with groups, rings or vector spaces, must be seen as "regimentations" of our language in its transcendental capacity. Thus, if a logician proposes that "we simply put the logic which we are studying into one compartment, and the logic we are using to study it in another",17 then he is stepping on thin ice, because unless the logic we study is the same as the one we use, it is in fact no logic at all in the authentic sense of the word. Neurath's classic metaphor seems to be particularly apt here: we cannot step out of the boat of our language, we have to rebuild it while staying aboard. We can make logical calculi to capture and to explicate important points of the way we use language, but we cannot throw away our language and put a calculus in its place.
7. The Ineffability of Semantics So is there any way at all to make sense of the linguistic turn and use logic for philosophical purposes? Do we not, as soon as we begin to speak about language, eo ipso adopt the languageas-object perspective and hence do "mere" linguistics? And are we not doing "mere" mathematics as soon as we set out to do model theory? An answer to this question is indicated in Wittgenstein's Tractatus: one can create a picture, articulating correspondence in such a way as to feature language-as-object in the role of language-as-medium, and hope that the reader will get it and yet not take the picture literally. This is why Wittgenstein says that his Tractatus offers no learnable truths, but rather a kind of ladder that should be kicked away once the reader has used it to climb higher. Hence the difference between linguistics and model theory, on the one hand, and a philosophical account of correspondence, on the other, is not that the former speak about language-as-object and the latter about language-as-medium - whenever we speak about language, we eo ipso speak about language-as-object. Language-as-medium can be used, but not fully spoken about. The difference is that in the framework of linguistic or logical discourse we take speech about language literally, whereas in the framework of philosophical discourse we take it as a metaphor, as a picture. In doing Wittgensteinian philosophy we may make use of language-as-object to the extent that it can serve as a vehicle of metaphor; but we must avoid taking the metaphor literally, mistaking language-as-object for language-as-medium. We should devise a theory which permanently reminds us of its metaphorical character. Let us return to (4). The sentence Snow is white is true if and only if snow is white. We may be tempted to say that it is true due to the fact that "out there in the world" or possibly in a model structure which is considered to offer a faithful representation of the world the entity snow instantiates the property of being white, or that snow is an element of the set of white things, or that there exists a fact of the coincidence of snow and whiteness. However, all of this is rather tricky: we can either consider snow is white as a mere string of letters (from the object perspective, i.e., from outside English), and then it is in itself neither true nor false; or we can take for granted that it has its usual meaning (using the medium perspective, i.e., staying inside
17
S.C. Kleene: Mathematical Logic (New York: John Wiley, 1967), pp. 2-3. 9
English), and then (4) turns out to be self-evident18. To say that the entity snow has the property of being white is not an explanation for the truth of the sentence snow is white; it is only its cumbersome paraphrase19. It is, in fact, as Rorty puts it, like explaining why opium makes you sleepy by talking about its dormitive power20. If we realize that our language is the "universal" (the illuminating German word unhintergehbar unfortunately has no exact English equivalent) medium, then we must conclude that its semantics is in a certain sense fixed. Moreover, we must conclude that this semantics is essentially ineffable - to be able to grasp it we would have to step outside language, and this is essentially impossible. "There is no outside;" as Wittgenstein puts it, "outside you cannot breathe."21 By providing a model-theoretical interpretation for a formal calculus or for a natural language we offer a new perspective which may help us perceive patterns and regularities which would remain hidden to our eyes otherwise; however, it is inadequate to see this act as the act of going from the words to what the words are about.
8. Formal Logic as 'perspicuous representation' The exclusive acceptance of the logic-as-calculus notion prevalent now, and the mistaking of this notion for the notion of logic-as-language, has led many philosophers to misguided conclusions. However, there is also a more or less continuous tradition exhibiting awareness of the limitations of this notion in philosophical contexts. As Jaakko Hintikka and Merill B. Hintikka showed, the notion of logic-as-language has been central not only for Frege, but also for some of the most outstanding analytic philosophers of this century, especially Wittgenstein and Quine.22 The employment of formal logic for philosophical purposes is justified only to the extent that it helps capture language in its transcendental capacity. In other words, formal logic, and especially model theory, is not as yet philosophy; it is a device which can be utilized (correctly or incorrectly) by philosophers. Moreover, there is no rule for correct usage. This was clearly 18
If I say that the entity denoted by 'snow' instantiates the property denoted by 'is white', then I speak about English and I hence treat of English from outside. However, that such a statements really requires the perspective from outside means that it says something more than every statement that could be made from inside, especially that it says something over and above the statement that snow is white. This is just the case when I insist that the fact of snow's instantiation of whiteness is a fact independent of, and casually determining the fact of the truth of snow is white. 19
Some paraphrases of such kind, if carried out systematically, may have a purpose, namely helping us see a relevant structure of language; however, this has little to do with the language-world relationship and with the question of what makes sentences true. 20
Richard Rorty: Contingency, Irony and Solidarity (Cambridge: Cambridge University Press, 1989), p. 8. 21
Ludwig Wittgenstein: Philosophische Untersuchungen (Oxford: Blackwell, 1953), §103.
22
See Jaakko Hintikka: "A Hundred Years Later: the Rise and Fall of Frege's Influence in Language Theory", Synthese, 59 (1984), pp. 27-49; Jaakko Hintikka and Merill B. Hintikka: Investigating Wittgenstein (Oxford: Blackwell, 1986); Jaakko Hintikka: "Quine as a Member of the Tradition of the Universality of Language", Perspectives on Quine (ed. by R.Barett and R.F.Gibson, Oxford: Blackwell, 1990). 10
recognized by Frege, Wittgenstein and Quine23, but largely ignored by Tarski, Carnap, Montague and many other philosophers and semanticists. The purpose of formalization is to help us to see certain aspects of language and its functioning more clearly, to achieve what we might call, borrowing from Wittgenstein, an übersichtliche Darstellung, perspicuous representation.24 It is justified to the extent, and only to the extent that it fulfils this function; and it must be constantly evaluated from this point of view. Stekeler-Weithofer has described the situation as follows25: With the development of the functio-logical semantics one constructs a (mathematical) 'object of comparison', a logico-mathematical 'picture' or 'model', and compares certain aspects or regularities of our common language usage, especially of our usual talk about the meanings of linguistic expressions and of normal judgements of correctness (adequacy) and hence truth of statements, with aspects and regularities in the picture. Such a comparison can yield certain keys to understanding how language 'functions' and it can help us formulate explicit rules of meaing sensibly, ... . One should never forget, however, that what is at stake are constructed pictures, perspectives which can be varied, and not descriptions adequate in general, nor generally approvable criteria governing correct speech and argumentation. Russell, Carnap and other scholars were convinced that the structure of language, although something quite definite, is hidden inside language or behind it, and that we need logical analysis to bring this structure to light. In this view, doing logical analysis can be compared to opening the lid of a complicated machine, thereby revealing the machine's inner workings. This metaphor is misguided, however: there is nothing about language that is hidden and can only be made visible by opening a lid. Language is accessible to us in all its aspects; our problem is to comprehend it - to command, as Wittgenstein puts it, a clear view of it. If language is to be seen as a machine, then it is a machine with all its wheels and gears in full view. Thus, the use of logical formulas to analyze language is more felicitously compared to drawing up a scheme to facilitate comprehension of the operating principle of an engine that is itself fully accessible to inspection but too complicated to be understood26. No logical calculus is the scheme which would guarantee understanding language, it is at most one of many possible schemes that may contribute to it.27
23
It is instructive to see how Frege understands the role of formal logic in his Begriffsschrift. For him, his concept script is like microscope: it is a tool excellent for some purposes (namely for the purposes of science demanding extraordinary acuity and differentiation), but useless for others. 24
See Wittgenstein (cited in n. 21, above, §122).
25
Pirmin Stekeler-Weithofer: Grundprobleme der Logik (Berlin: de Gruyter, 1984), pp. 141-2.
26
Cf. Wittgenstein (cited in n. 20, above, §122ff.).
27
For a further elaboration on these themes see Peregrin: Doing Worlds with Words, Kluwer, Dordrecht, 1995. 11
9. Conclusion The linguistic turn is based on the fact that whatever we can speak about is the meaning of an expression of our language and that ontology is thus in a sense reducible to semantics. Model theory, as developed within the framework of modern formal logic offered means for the explicit capturing of semantics; hence it is tempting to promote model-theoretical semantics as ontology. However, this might be really misguiding. If we look at our language "from inside" and if we understand logic "as language", then model theory can be at most one of the formal ways of summarizing ways of using language; and as such it cannot be an explanation over and above being a summarization and making language more comprehensible. On the other hand, if we look at language from outside and if we pursue logic as calculus, then there is no immediate philosophical relevance of model theory; model theory is simply a part of mathematics and model-theoretical semantics is a part of empirical linguistics. Such enterprises can be considered philosophically relevant only as metaphors; metaphors which may (and do) help us see how is our language related to the world, which are nevertheless no direct theories thereof.
12
DEVELOPING SELLARS' SEMANTIC LEGACY: MEANING AS A ROLE Jaroslav Peregrin Dept. of Logic, Institute of Philosophy Academy of Sciences of the Czech Republic Jilská 1, 110 00 Praha 1 Czech Republic
www.cuni.cz/~peregrin
Sellars on Meaning and Abstract Entities In his seminal article 'Abstract entities', Wilfrid Sellars claims that the meaning of a linguistic term is best construed as sort of role played by the term within the drama of language (and that, consequently, abstract entities, which are normally seen as expressed by such terms are linguistic in nature). "Redness," writes Sellars (1963, 627), "as a first approximation, is the word •red• construed as a linguistic kind or sort which is capable of realization or embodiment in different linguistic materials, e.g., *red*, *rot*, and *rouge*, to become the English word 'red', the German word 'rot', and the French word 'rouge'. ... Thus •red• is a type which is shared by the English word 'red', the German word 'rot', and the French word 'rouge'."1 Crucial for this conception is the distinction between what can be called 'generics' and 'true universals'. When I say "The lion is tawny" and use "the lion" as expressive of the generic, my claim is reasonable and true. On the other hand, should I construe it as standing for the universal ("Lionhood is tawny"), what I say would be false (if not utterly meaningless) - a universal surely is not tawny (ibid.). Now what Sellars claims is that universals are in fact a kind of 'second-order' generics - they are generics whose instances are linguistic expressions. Thus the universal lion is in fact the role played by the word "lion" in English (and by other words in other languages). Hence Sellars may be seen as distinguishing between generalization, which is manifested by the step from particular statements to a general one ("All lions are ..." resp. "The lion is 1
Here the asterisks are supposed to form names referring to expressions as mere shapes, quotation marks form names referring to expressions qua expressions of a particular language, whereas the names formed by the dots refer to the roles instnatiable by different expressions in different languages.
1
...") and abstraction, which is a matter of a further step to a particular statement about a universal, which is the notorious "e pluribus unum" ("Lionhood is ..."). And while philosophers had traditionally wanted to analyze the abstraction as a matter of a peculiar knack of human mind to put many things into a nutshell of one, Sellars proposes to analyze it as a capacity into which the capacity of generalization mutates when language sets in. For language itself provides us with a certain "one over many" (one word "lion"2 over many particular lions); and when we apply our capacity of generalization to it, we find the abstract universal (lionhood) as the one over the many words playing the role assumed by "lion" in English within other - actual or possible - languages. Hence the peculiar ability of our minds to abstract appears as secondary to its ability to (generalize and) "to language about languagins" (1974, p. 425). This leads Sellars to the conclusion that the senses of our expressions (or their intensions) are best seen as certain roles of the expressions. What is a role? Sellars invokes probably the most frequented simile within modern philosophy of language and compares language to chess3. It is rather clear what the role of, say, pawn is: it is delimited by the rules saying how the pawn moves, how it interferes with other pieces etc. Sellars points out that to be a pawn usually takes to fulfill two kinds of criteria: descriptive criteria ('to look thus and so') and prescriptive criteria ('to be correctly moved thus and so'). Now the role in his sense is what remains if we subtract the descriptive part. Sellars (1963) invites us to imagine an alternative game, "Tess", which is played in Texas and which is just like chess, save for the fact that the various kinds of pieces are replaced by various types of cars and the fields of the chessboards are replaced by counties. The same role which is assumed within chess by our pawn is assumed in Tess by, say, their Volkswagen; and the role is what these two utterly dissimilar objects share4. Now if we accept that language is something like a 'rule-governed activity' (as urged not only by Sellars, but of course also by the later Wittgenstein and his followers), the idea that also meanings might be sort of roles comes quite naturally. But what kind of rules is 2
The word as such is, of course, itself an abstract item, one shape over many particular inscriptions or sounds. However, it seems that Sellars takes this kind of abstraction as somehow straightforward and unproblematic.
3
In this way he queues up into the long line of linguists and philosophers already containing Husserl, Frege, de Saussure, Wittgenstein, Jakobson, and who not ...
4
These themes are further elaborated especially in Sellars (1950; 1954; 1956; 1963; 1969; 1974; 1992). See also Harman (1970), Marras (1973) and Seibt (1990).
2
constitutive of this kind of roles? There appears to be an abundance of linguistic rules: we have, for example, the rule that it is correct to change "read" in the third person singular to "reads", or the rule that claiming that somebody stole something commits the claimer also to the claim that the very somebody is a thief; or the rule that it is impolite to say "Ciao" to the Queen of England. Which of them, if any, represent the ones that are responsible for meaning? Sellars indicates (see esp. 1953; 1992) that the rules which are crucial in this respect are the rules of inference (Sellars, 1992, p. 114, calls them "consequential rules"), i.e. the rules exemplified by the middle one of the three examples.
What is an inferential rule? Inferential rules are often taken as kind of prescriptions for passing from an antecedent to a consequent: perhaps from asserting the former to asserting the latter, or from believing the one to believing the other. But this is clearly ill-conceived: when we say that it is correct to infer Fido is a mammal from Fido is a dog, then we surely do not want to say that whoever asserts the former is obliged to assert the latter - if we were always to assert all consequences of everything we assert, we could never stop making assertions. And also we surely cannot want to say that one is obliged to believe all the consequences of what she believes - we do not control our own beliefs in such a way that this kind of obligation would make more sense than, say, the obligation to have luck. To understand the Sellarsian notion if inference, we consider his distinction between what he calls the "rules of doing" and the "rules of criticizing" (1999, p. 76). Inferential rules are basically the latter - they do not tell us what to do, but rather what to despise - and hence are really more the rules of what not to do. They delimit a space of what is approvable: if you assert that Fido is a dog, then you should not deny that Fido is a mammal; and if you do deny it, you are a legitimate target of criticism. Hence the rules of inference - which are constitutive of conceptual content - are not a matter of regularities of linguistic conduct, of "passing over" from some utterances or beliefs to other ones (whatever the "passing over" might mean). They are a matter of what Brandom (1994) calls normative attitudes: attitudes to one's own and others' linguistic utterances, of "languaging about languagins". This is an important thought: conceptual content is not a 3
matter of regularities of linguistic behavior, but rather of (regularities of) a 'metalinguistic behavior' (see Sellars', 1950, criticism of "regulism"). This is closely related to the Sellarsian 'dialectics' of ought-to-do and ought-to-be: whereas an ought-to-do in a sense directly is also an ought-to-be (and hence a rule of criticism directly piggybacks on every rule of doing) an ought-to-be bears an ought-to-do via a kind of a 'practical syllogism' (see Sellars, 1969): via the conviction that we ought to do what brings about that which ought to be. (But of course that if roles of inference are really rules of what ought not to be, they lead us to what not to do.)
Can we 'get semantics out of syntax'? One of the main reasons why many people would reject the proposal to see semantics as a matter of inferential rules right off stems from the combination of two 'received wisdoms': (i) inferential rules are a matter of syntax; and (ii) syntax is not enough for semantics. I think that for each of the two claims there is a sense of "syntax" in which it is true. The trouble is, however, that the two needed senses are different; and it is impossible to take the two claims to be true jointly (without taking the two occurrences of "syntax" in them to be mere homonyms). And in particular, it is impossible to infer that inferential rules are not enough for semantics. What does the word "syntax" mean? In one sense, syntax is the theory of well-formedness: it specifies rules for composing more complex expressions out of simpler ones, thus delimiting the space of all expressions, and especially all sentences, of a language. It is clear that if we understand syntax in this way, (ii) will be true, but (i) not. But syntax is sometimes understood in a wider sense, as amounting to every property of expressions (and every relation between them) specified without reference to anything else than the shapes of the expressions (this is the sense put into circulation by Carnap)5. Taken in
5
Elsewhere (Peregrin, 1999) I argued that once we abandon the view that language is a collection of labels stuck to things and see it as a means of an activity, the Carnapian concept of syntax/semantics boundary is no longer available. For then the only perceptible boundary is that between the question
4
this way, (i) becomes true, but at the same time the falsity of (ii) ceases to be a matter of course. Let us see why. Let us consider a list of names, such as a telephone directory. We can say that the names it contains have two kinds of properties: those which they have 'by themselves' (and which thus can be 'read off' them), and those which they have as a consequence of being names of particular persons (and which are thus not 'visible'). Examples of the former properties are to begin with N or to consist of more than ten characters; an example of the latter is to be a name of a bald man. Let us call the former 'syntactic', and the latter 'semantic'. Now suppose somebody finds out that, just by chance, all and only bald men have names beginning with "N". With this knowledge, it becomes possible to read the property to be a name of a bald man off the name - it is enough to check whether it starts with "N". Does this mean that we have managed to change the property in question from a 'semantic' to a 'syntactic' one? Surely not, we have only found its 'syntactic symptom' and thereby a way to 'represent' it by a 'syntactic' property. Notice that the same holds if we are not so improbably lucky as to find out that the names of all bald men share such a simple property. We achieve the same by simply making the list of all the relevant names: then it will be again possible to find out whether a name is that of a bald man by simply inspecting its shape - namely by comparing it with a given list of shapes. If our directory is finite, then we can 'syntacticize' any property in this way; if it is infinite, then perhaps not any, but surely we still can do so with many. Now it seems to be clear that correct inerrability is basically a semantic relation - of course that we cannot see whether a sentence is correctly inferable from other sentences by merely inspecting the shapes of the sentences (i.e. without knowing what they mean). However, we can make a list of correct inferential patterns (such as the one assembled by Frege in his Begriffsschrift as well as by the host of his followers), and given we know this is a (complete) list of valid patterns, we can recognize inerrability simply from the shapes6. (In fact, this is what appears to have been behind the famous Hilbert's program: once we are able to show that whenever A follows from X, it is derivable from it in terms of our rules and vice
which strings are employed (syntax) and the question how they are employed (semantics-cum pragmatics). 6
Of course that as the number of such patterns is unlimited, the list can be given only 'potentially', via a finite list of some basic patterns (axioms) and some derivation rules. A valid proof is then the decomposition of a given inference into a chain of the axiomatic patterns.
5
versa, we may reduce truth to provability and hence to a 'mechanical manipulation' of symbols.) This means that inference, though its rules are articulated in syntactic terms, is not syntactic in the other - more important - sense; and this removes the absurdity from taking inferences as constitutive of meanings.
A Digression: Keepers, Joiners, Crashers & comp. To throw more light on the nature of roles which we will claim underlie meanings, let us make a digression. Suppose we have an alphabet and consider strings which can be formed by its means. Suppose we divide the space of all such potential strings into two groups: the 'good' and the 'bad' ones; and we are interested in the ways of building the good strings. When we toy with putting such strings together, we may come to notice that various letters and strings display specific kinds of 'behavior'. We may, for example, discover that some letters/strings have the property that when concatenated with any strings they always produce bad strings; and we may conceive of calling such strings crashers. Other letters/strings may do the same thing when they are concatenated with other strings in a particular way: we may, for example call a string an l-crasher iff it produces a bad string whenever it is prefixed to a string. Still other strings may have the property that they produce a bad string whenever they are prefixed to good strings – we may call such a string l-killers. Or we may come to call a string a joiner iff it produces a good string if and only if it is put in between two good strings. Here are some examples of the basic categories of strings we may discover when studying buildability of good strings: S is a crasher: any string containing S as a substring is bad S is a l-crasher (r-crasher): any string starting (ending) with S is bad S is a l-killer (r-killer): any string consisting of S followed (preceded) by a good string is bad S is an l-inverter (r-inverter): a string consisting of S followed (preceded) by another string is good iff the other string is bad. (Every l-inverter is obviously an l-killer.) S is a l-keeper (r-keeper): any string consisting of S followed (preceded) by a good string is good 6
S is a l-superkeeper (r-superkeeper): any string consisting of S followed (preceded) by a good string is good iff the string is good. (Every l-superkeeper is an l-keeper.) S is a joiner: any string consisting of two good strings with S between then is good S is a superjoiner: any string consisting of two strings with S between is good iff the two strings are good. (Every superjoiner is a joiner.) Consider, for the sake of illustration, what we will call Language 1. The vocabulary is a, b, c, d, e and the delimitation of the good strings over it is as follows: (i) a, b and c are good strings; (ii) if s is a good strings, then d∩s (i.e. the symbol d followed by the string s) is a good string; (iii) if s1 and s2 are good strings, then s1∩e∩s2 is a good string; (iv) no other string is good. The words of this language can be clearly classified into three categories. First of them is constituted by the letters a, b and c; the second one by the letter d, which is an l-superkeeper; and the third by the letter e, which is a superjoiner. Now suppose we distinguish a further subset of the set of good strings; call its elements 'perfect' and call those good strings which are not perfect 'imperfect'. We can, of course, again start to classify the roles of letters and strings with respect to the new set. Returning to our earlier example, imagine that a and b are perfect strings, while c is imperfect. Further imagine that the string d∩s is perfect if and only if s is imperfect; and that s1∩e∩s2 is perfect iff both s1 and s2 are. It is easy to see that from the viewpoint of the delimitation of the perfect strings, then, d comes to be an l-inverter, whereas e is again a superjoiner. Now consider a little more sophisticated kind of language, Language 2, based on the letters a, b, c, d and e. Suppose that a string over this alphabet is good if it is of the shape s1∩e∩s2 where the strings which are usable in the position of s1 and s2 are defined recursively: a is usable, any usable string preceded by b is again usable, and any string which is of the 7
shape s1∩c∩s2 is s1∩d∩s2 with s1 and s2 usable is again usable. Thus the examples of good strings are aea bbaea baebbbbba acbaeba acaebbbadbbbba To characterize the roles of the strings of this language we do not make do with the simple concepts of joiners, inverters etc. We first have to characterize the strings which may surround e in a good string – call them e-joinables. a is an e-joinable. b is what could be called e-joinable-l-keeper, whereas c is what we could think of calling e-joinablesuperjoiner. Now imagine that the perfect strings of this language are defined in the following way: aea is perfect; if s1∩e∩s2 is perfect, then so is b∩s1∩eb∩s2; s∩cae∩s is perfect; if s1∩c∩s2∩e∩s3 is perfect, then so is s1∩cb∩s2∩eb∩s3; s∩daea is perfect; if s1∩d∩s2∩e∩s3 is perfect, then so is s1∩db∩s2∩e∩s3∩c∩s1; nothing else is a perfect string. Again, these stipulations furnish the strings with roles, which are now rather intricate. Call two e-joinable strings s1 and s2 e-comparable, if s1∩e∩s2 is a perfect string. We may say that a string cancels s from the left in the end position (in short, is a s-l-e-canceller) if a string ending with c∩s is e-comparable to a string without this ending. And we may say that it shifts s from the left to the beginning (is a s-l-b-shifter), if a string containing c∩s is e-comparable with a string starting with s and continuing by something e-comparable with the original string with s excerpted. Using this terminology, c is a a-l-e-canceller and a b-l-b-shifter.
8
Meanings as Roles We have seen that any delimitation of a subset of the set of strings over an alphabet furnishes the strings with certain roles, which are a matter of their 'behavior' with respect of the distinguished set and which may be quite complicated. Especially any language and any axiomatic system (which also delimits a set of sentences, namely the set of its theorems) does so. Now the meaning which an axiomatic system confers on an expression of its language can be seen as precisely this kind of role. Consider formal languages of the kind of those which we can formulate within the framework of the propositional and the predicate calculus, with good strings being the wellformed formulas. It is, for example, easy to see the usual syntax of the propositional calculus is only a minor variation on our Language 1. We also have three categories of signs: the propositional letters, the l-superkeeper ¬ and the superjoiners ∧, ∨ and →7. More generally, taking the set of good strings as the set of well-formed formulas, the roles expressions thus acquire are usually called syntactic categories. (We would not say that, ∧, ∨ and → are superjoiners, but rather that they are propositional connectives.) Now, moreover, take the set of perfect strings to be the set of true sentences (under some interpretation). What, then, do the resulting roles correspond to? Consider conjunction: we have noticed that with respect to the perfect – i.e. true – sentences it acts as a superjoiner. This means that it produces a perfect output iff the two strings which are appended to it from the sides are both perfect, i.e. it maps perfect and perfect on perfect, whereas perfect and imperfect on imperfect etc. However, as perfect is true and imperfect is false, we have the function mapping T and T on T and any other combination of the truth values on F. This is to say that the truth-table usually thought of as the meaning of conjunction can be plausibly seen as the recapitulation of the role of a superjoiner. This indicates that to be the conjunction simply is to be the superjoiner (w.r.t. the class of true sentences), to be the negation is to be an l-inverter etc. Thus our Language 1 amounts to nothing else than the classical propositional calculus.
7
This is an oversimplification for if we consider the standard form of the syntax, we need brackets; and if we consider the Polish variant, the connectives will not be literally superjoiners but rather something which could be called, say, l-bikeepers.
9
Now look at language Language 2. If we replace the signs a, b, c, d and e with 0, Succ, +, ×, and =, respectively, we can see that Language 2 constitutes a kind of rudimentary arithmetic. In this case, the roles are much more complicated and they are mutually entangled (so that it is not possible to specify the role of, say + independently of that of 0), but nevertheless they are still roles in the very sense as before. Hence, we may say, the meaning arithmetic confers on the sign + is the role of a '0-l-e-canceller' and a 'Succ-l-b-shifter'. (Expressed in a more prosaic way, this means that to add zero is to add nothing and to add a successor of a number is to produce the successor of the addition of the number.) However, as Lakatos (1976) reminds us, this "Euclidean" (may be we should better say "Hilbertian"8) way of viewing an axiomatic system (according to which any meanings the signs have are conferred on them by the axioms) is only one possible way; and in fact such systems are often understood in a quite different way, which Lakatos calls "quasi-empirical". From this point of view, such a system is seen as codifying some pre-existing, prototheoretical knowledge, and hence is responsible to it: if it produces theorems whose interpretation is at odds with what we take for granted within the proto-theory, it is revised. Let us take a formula of PA. The answer to the question whether it is true can be sought in two quite different ways: we may check whether it is deducible from the axioms; or we may check whether the 'informal' mathematical sentence which is taken to be 'captured' by it (in view of the fact that the basic vocabulary and grammatical rules of PA more or less copy the basic vocabulary and rules of our pre-formal arithmetical language) is true. If we see the system in the Euclidean way, then a possible discrepancy between the two results need not worry us - for the latter is completely irrelevant for us. However, if we see it in the quasiempirical way, then the discrepancy might lead us to revise the axioms9. Hence we should qualify the claim we made above: any axiomatic system furnishes the expressions of the underlying language with Euclidean meanings. But is it not so that such an 8
In a letter to Frege, Hilbert writes: "You write: 'From the truth of the axioms it follows that they do not contradict one another'. I was very interested to read this particular sentence of yours, because for my part, ever since I have been thinking, writing and lecturing about such matters, I have been accustomed to say just the reverse: if the arbitrarily posited axioms are not in mutual contradiction with the totality of their consequences, then they are true - the things defined by the axioms exist." Hence, according to Hilbert, each delimitation of a set of strings constitutes an Euclidean semantics and this is the only semantics we could possibly require; whereas according to Frege, Euclidean semantics of the systems we assemble is not the true semantics, for the true semantics is what expressions of our natural language have and what we cannot freely create.
9
I have discussed the consequences of these two radically different aspects of logical systems elsewhere (see Peregrin, 2001).
10
Euclidean meaning is restricted to formal systems only, and that the meaning in normal languages is something quite different which is perhaps 'incommensurable' with it? Observe, first, that we can reconstruct, and in some cases even replace, parts of natural language by means of formal languages constituted in the axiomatic way. The axiomatic Peano arithmetic is not only a commonly accepted codification of pre-formal arithmetic, but in the eyes of many mathematicians it simply replaces it. This means that Euclidean meanings can be not only seen as suitable explications of the corresponding 'natural' meanings, but sometimes even as their descendants. What makes this possible? The Sellarsian answer is that natural languages are themselves sort of Euclidean systems, only with axioms and rules not being explicit in the form of formulas or statements, but implicit within the praxis of using language. Our arithmetical praxis is governed by the rule that adding zero to x means letting x unchanged just as the axiomatic PA is - though it need not be written anywhere explicitly10. Children learn to add in this way by being corrected the "ought-to-do" imposed on them by their teachers thus becoming their "ought-to-be" (cf. Sellars, 1969). And just like within the Peano arithmetic there is nothing to semantics save the explicit rules (axioms and derivation rules), there is, in natural language, nothing to semantics save the implicit rules of our 'language games'. Thus, the claim is that every meaning is on some level a sort of Euclidean meaning. Having shown that some meanings are explicitly Euclidean and that other meanings may be implicitly Euclidean and can be explicated in terms of Euclidean meanings, we may want to consider the following two general questions: (1) Is every Euclidian meaning a meaning worth its name? (2) Is every natural meaning reconstructible as an Euclidean one?
10
Of course the difference between a game like chess (or, for that matter, a formal language), the rules of which are explicit, and our natural language games with their rules implicit to our linguistic conduct is not unimportant. In fact, Wittgenstein's Philosophical Investigations (1953) can be seen as to a great deal wrestling precisely with the question what does it mean for a rule to be implicit in behavior and what does it take to follow such a rule. But precisely the fact that it is sometimes possible to explicate parts of natural language in terms of formal systems constituted by means of explicit rules gives further credence to the conviction that the rules are implicitly constitutive of our language games. Brandom (1994) claims that this kind of making the implicit rules explicit is the crucial force behind the development of our language, behind logic, and in general behind our reason.
11
The answer to the first question is clearly no. We may have a formal language with a single sentence; or a language without axioms, or such that any sentence is inferable from any other. None of such systems obviously yields the rich variety of roles characteristic of what we would be willing to call language, and hence furnishes the expressions with what we would be willing to call meanings. (The last example covers the case of Prior's, 1960/61, tonk, which is a grand example of the fact that nontrivial axioms need not lead to a nontrivial semantics. This is important, but it does not show, pace Prior, that axioms or inference rules are generally short of conferring meaning.11) The other question is more intricate; and prima facie there seem to be at least two counterexamples. First, meanings of empirical terms; and, second, meanings of some common non-empirical terms which do not seem to be conferrable by means of rules. Let us deal with the two topics in the sequel.
Empirical terms Of course that Sellars does not claim that the meaning of an empirical term is a matter of nothing else than inferences in the usual sense on the term. What he claims (1974, pp. 423-4) is that, in general, the ways in which expressions are used (and hence their meanings) are a matter of three kinds of regularities or patterns of linguistic activity12: (1) Language Entry Transitions: The speaker responds to objects in perceptual situations, and in certain states in himself, with appropriate linguistic activity. (2) Intra-linguistic moves: The speaker's linguistic conceptual episodes tend to occur in patterns of valid inference (theoretical and practical), and tend not to occur in patterns which violate logical principles. (3) Language Departure Transitions: The speaker responds to such linguistic conceptual episodes as 'I will now raise my hand' with an upward motion of the hand, etc.
11
See Peregrin (2001, §8.5) for more details.
12
A more elaborated (if not overelaborated) version of this classification is presented by Sellars (1992, p. 114).
12
Hence if we want to restrict the term "inference" to the language-language moves, we cannot claim that meaning is in general the inferential role, but rather a role w.r.t. more inclusive kinds of rules. (The term inference is sometime stretched to cover even Sellars' (1) and (3), which then allows for equating meaning and inferential role quite generally, but this appears to be at odds with how the term "inference" is most often used.) We must, however, notice that it is only the intra-linguistic moves which we can nontrivially capture when we are doing a theory of semantics. The trouble with the other two is that they consist of a language-part and a world-part, and when we want to talk about the latter, we have no choice but to put it in words, by which it becomes identical with the former. We can practically teach somebody to recognize the circumstances in which it is appropriate to assert "A dog has just run by", but when we want to capture them within a theory, we end up with the disquotational claim "It is appropriate to assert 'A dog has just run by' iff a dog has just run by". As Wittgenstein (1977, p.27) remarked "The limit of language manifests itself in the impossibility of describing the fact that corresponds to (is the translation of) a sentence without simply repeating the sentence." True, we need not use the very same sentence to describe the circumstances, but if we do not do so, then the ensuing nontriviality can be seen as a matter of the intra-linguistic relationship between the two sentences. This means that insofar we take semantics to deal exclusively with language, we cannot expect that it completely elucidates the meanings of non-empirical terms: part of their meanings is a matter crucially depends on their relationship to the extralinguistic world. After all, the delimitation of the meaning of "dog", i.e. of what a dog is, should not be expected of a semanticist, but rather of a biologist. The situation is, however, different, as far as non-empirical words are concerned. Their meaning can be usually seen as a matter of inferences alone. There is nothing to the meaning of "and" than the inferences which govern the term: the inferences from X and Y to both X and Y, from X and Y to X and Y and perhaps some more.13 This means that while the meaning of an expression is, from the current viewpoint, the role of the expression within the relevant 'language game', the theoretically articulable part of the meaning (which may in some cases, like that of "and", wholly exhaust it) consists in its inferential role. Hence, we may say, syntax, in the form of inferential rules in some cases can alone be sufficient for semantics. (The meaning of an non-empirical term, such as "and" can be 13
The proviso is here because the meaning of the English "and" does not utterly coincide with that of the classical conjunction operator, it, e.g. often behaves non-symmetrically.
13
directly seen as consisting in its inferential role.) And there is an asymmetry between inferences (the language-language kind of moves) and the other two kinds of moves in this respect: while inferences alone are (sometimes) capable of conferring a meaning, neither the world-language moves, nor the language-world moves alone are. An expression cannot become truly meaningful by being subjected to merely language entry rules, i.e. by becoming a correct reaction to something extralinguistic: it must also be subject to inferences. The reason is that language is a holistic, interconnected structure - an expression which would not be connected, by the rules of language, to other expressions would be the notorious "idle wheel"14. Just like there cannot be a chess piece without a potential interaction with all other pieces, there cannot be an expression which would be connected only to something extralinguistic and not to other expressions. And it is only inferences which are capable of interlinking all its elements. Hence syntax, in the form of inferential rules, is in all cases essential for semantics: for something can have meaning only if it is subject to inferences, i.e. if it is in inferential relations to other expressions.
Classical and 'standard' semantics There seem to be some limits to what kinds of meaning can be conferred by rules; and it is sometimes claimed that we need also meanings of the kind which is beyond the reach of inferentially induced semantics. The fact is that if we understand the term inference as we did so far, we even do not get classical propositional connectives in the inferential way. For though there is an inferential pattern which inferentially characterizes the classical conjunction, there is none which would so characterize the classical disjunction, negation or implication. For though we can inferentially express that the conjunction is true if one of its conjuncts is true, we cannot express that it is false whenever both the disjuncts are false15.
14
Cf. Wittgenstein (1953, §271).
15
We must not think that the completeness proof for the classical logic shows that the usual truthfunctional meanings of the logical operators are conferred on them by the standard axioms: the axioms are compatible even with certain non-truth-functional interpretations.
14
However, as I have pointed out elsewhere (see Peregrin, to appear), this obstacle can be solved by reconsidering the concept of inferential pattern, namely by admitting that an inferential pattern can be read as providing a kind of an exhaustive listing. Claiming A ├─ A∨B B ├─ A∨B we claim that A∨B follows from A and from B; but presenting the two inferences as a selfcontained pattern indicates that this list of premises be exhaustive; i.e. that there is nothing else which would entail A∨B and at the same time be entailed by both A and B. This can be accounted for by letting inferential patterns incorporate an implicit exhaustivity claim: if I give an enumeration of premises each entailing a given conclusion, then we assume that the list is to be exhaustive. (Why? For this is how we usually understand enumeration: if I say "I own an VW Beetle and a Pontiac ", the normal understanding is that these are all my cars.) As Koslow (1992) has shown, the admission of such 'extremality conditions' into the concept of inferential pattern leads directly to an inferential capturing of all of classical logic. And as I have indicated elsewhere (see Peregrin, to appear a), it could even lead to the inferential accommodation of what is usually called the standard semantics of higher-order logics (as contrasted to the Henkin one - see, e.g., Shapiro, 1991). Its peculiarity can be located into the incompleteness of arithmetic - whichever axioms and rules satisfiable by the standard number sequence we accept, they will be also satisfied by some nonstandard ones. (The existence of the non-standard interpretations of higher-order logics then can be seen as a matter of the fact that the corresponding semantic structures are bound to incorporate the natural numbers sequence pattern.) Now if we construe the 'exhaustivity condition' which we take to be implicit to inferential pattern broadly enough, we might even think of it as blocking the nonstandard models: the standard axiomatization of Peano arithmetic says that zero is a number and the successor of a number is a number, and the exhaustivity says that nothing else is a number, hence that we intend only the minimal model. Of course the hints I have just given cannot establish that any kind of semantics can be seen as ultimately grounded in inferences (They are clearly in the need of further elaboration, which is, however, beyond the scope of the paper and which I hope to produce in the future.) What I hope they can is shatter the preconception that some common varieties of semantics 15
are - as a matter of principle - unavailable for the inferentialist. This means that nothing appears to stand in the way of our embracing Sellars' role semantics, i.e. the view that meanings are, as a matter of principle, roles of the expressions which are said to express them. However, in contrast to Sellars I do not believe that this would prevent us from engaging within the enterprise of Carnapian semantics. We only have to realize that the set-theoretical denotations with which the systems of formal semantics work are not to be regarded as representations of entities on which expressions of natural language really stuck, but rather as recapitulations of the inferential roles of the expressions (see Peregrin, 2001; to appear b).
References Brandom, R. (1994): Making It Explicit, Harvard UP, Cambridge (Mass.). Harman, G. (1970): 'Sellars' Semantics', Philoisophical Review 79, 404-419. Koslow, A. (1992): A Structuralist Theory of Logic, Cambridge University Press, Cambridge. Lakatos, I. (1976): 'A Renaissance of Empiricismin in the Recent Philosophy of Mathematics?', in Lakatos: Mathematics, Science and Epistemology, Cambridge University Press, Cambridge; reprinted in Tymoczko (1998), pp. 29-48. Marras, A. (1973): 'Sellars on Thought and Language', Nous 7, 152-163. Peregrin, J. (1999): 'The Pragmatization of Semantics', in The Semantics/Pragmatics Interface from Different Points of View (ed. K. Turner), Elsevier, North-Hollard, 419442. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Peregrin, J. (to appear a): 'Semantics as Based on Inference', in Proceedings of 'Philosophical Insights into Logic and Mathematics' (ed. J. van Benthem et al.), Kluwer, Dordrecht. Peregrin, J. (to appear b): 'Pragmatism and Semantics' to appear in What is Pragmatism? (ed. P. Ghirardelli, jr.) Prior, A. N. (1960/61): 'Roundabout Inference Ticket', Analysis 21, 38-39. Seibt, J. (1990): Properties and Processes (A Synoptic Study of Wilfrid Sellars' Nominalism), Ridgeview, Atascadero. Sellars, W. (1950): 'Language, Rules and Behavior', in S. Hook (ed.): John Dewey: Philosper of Science and Freedom, Diall Press, New York, 289-315. Sellars, W. (1953): "Inference and Meaning," Mind 62, 313-38. Sellars, W. (1954): 'Some Reflections on Language Games', Philosophy of Science 21, 204228. Sellars, W. (1956): 'The Myth of the Given: Three Lectures on Empiricism and the Philosophy of Mind', in The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minnesota Studies in the Philosophy of Science 1) (eds. Feigl, H. & Scriven, M.), University of Minnesota Press, Minneapolis; reprinted in Sellars (1963) . Sellars, W. (1963): 'Abstract Entities', Review of Metaphysics 16, 627-71. Sellars, W. (1969): 'Language as Thought and as Communication', Philosophy and Phenomenological Research 29, 506-527. 16
Sellars, W. (1974): 'Meaning as Functional Classification', Syntèhse 27, 417-437. Sellars, W. (1992): Science and Metaphysics, Ridgeview, Atascadero. Shapiro, S. (1991): Foundations without Foundationalism, Clarendon Press, Oxford. Tymoczko, T., ed. (1998): New Directions in the Philosophy of Mathematics (revised edition), Princeton University Press, Princeton. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford; English translation Philosophical Investigations, Blackwell, Oxford, 1953. Wittgenstein, L. (1977): Vermischte Bemerkungen (ed. von Wright, G.H.), Suhrkamp, Frankfurt; English translation Culture and Value, Blackwell, Oxford, 1980.
17
Other worlds What is the use of worlds beyond the real one? Jaroslav Peregrin www.cuni.cz/~peregrin Wenn es aber Wirklichkeitssinn gibt, und niemand wird bezweifeln, daß er seine Daseinsberechtigung hat, dann muß es auch etwas geben, das man Möglichkeitssinn nennen kann. Wer ihn besitzt, sagt beispielsweise nicht: Hier ist dies oder das geschehen, wird geschehen, muß geschehen; sondern er erfindet: Hier könnte, sollte oder müßte geschehn; und wenn man ihm von irgend etwas erklärt, daß es so sei, wie es sei, dann denkt er: Nun, es könnte wahrscheinlich auch anders sein. So ließe sich der Möglichkeitssinn geradezu als die Fähigkeit definieren, alles, was ebensogut sein könnte, zu denken und das, was ist, nicht wichtiger zu nehmen als das, was nicht ist. Robert Musil: Der Mann ohne Eigenschaften
Worlds we enjoy: literature and the arts In a story called “Long journey’s goal”, Michael Ende (the author of the famous Neverending story) tells a story of an eccentric aristocrat who comes across a painting of an otherworldly landscape with a strange valley and a dreamy castle above it. He at once realizes that this is the place he really belongs to, and guided by the advice of a Jewish sage, sets out to find it cost what it may. The point of the story is that his longing for the place has the power to bring it into being. I think that this story illustrates the uncanny passion we humans have for other worlds. As the products of the long history of our literature and other arts document, we are eager world-builders. Literature appears to be traditionally torn apart by two opposite drifts: the drift toward realism, prohibiting any additiions to what is really here, in our actual world, and the drift toward imagery, creating alternatives to our world, sometimes even wildly dissimilar to it. How is this achieved? In a paper on possible worlds in literary fiction, Lubomír Doležel (1989) proposes that the way we create worlds by literary texts is akin to that in which we create reality by means of what Austin calls performative speech acts. The idea is as follows: If I say „I promise to give you money“ or „Hereby I establish the Brotherhood of Other Worlds Explorers“ I do not describe an existing fact, namely that I am obliged to give you the money or that there is a brotherhood being established, but rather I bring it into being, i.e. create it. So there are linguistic performances, which are not descriptive of the world, but rather constitutive of it. And, Doležel argues, we can see literary fiction as constitutive of literary worlds in an analogous way. Whatever the mechanism is, ordinary literary worlds are worlds only by courtesy. As Doležel points out, they are usually incomplete and inhomogeneous. We can say that they are fragmentary, flat and non-interactive: they can be perceived only in part and from one fixed perspective (namely the one offered by the text), and there is no way that they can respond to us, readers. This is what distinguishes them from our real world: for our world can be, in principle, investigated anywhere and from any viewpoint, and it responds to what we do with 1
it, i.e. it wields resistance (which allows us to learn its laws - what is and what is not possible within it). How can a fictional world come closer to the real one? Well it has to be elaborated in reasonable detail, but even more importantly, it must lend itself to being viewed from more than one perspective. The simplest way to achieve this is to let the world be presented from the viewpoint of more than one individual, thus making it display varying aspects just as the real one does. Moreover, it may be even granted existence independent of any individual text, by letting it migrate between different texts. Take William Faulkner’s novel The Sound and the Fury: it consists of four, quite different individuals’s descriptions of the same course of events from each of their parochial viewpoints. Moreover, the world in which these events take place, the Yoknapatawpha county1, underlies many of Faulkner’s stories and novels. Tolkien’s Arda2 has fared even better: it has been liberated not only from the bounds of the single text, but also from the bounds of an individual author: it has been cloned into the numerous fantasy worlds underlying both a great deal fiction and the various fantasy worlds of “role playing games” games such as Dungeons and Dragons. And it seems to me that nowadays the firmness of, say, the fact that elven ears are pointed (i.e. its independence of anyone’s will or subjective imagination) is comparable to that of the facts of the real world. Moreover, some post-literary fictious worlds have even overcome the most conspicuous shortfall of ordinary fictional worlds, namely their lack of interactivity. It is not that I cannot imagine myself as turning up within Yoknapatawpha county and face, say, a bear - but the world would not act independently of my will. If I want to kill the bear, I will; and if not, I will not. The problem of true interactivity is the problem of exempting the world from the legislation of any single mind and enabling it to wield resistance of its own; and this can be achieved by letting it be sustained by the collective collaboration of many. An instance of this occurs in Dungeons and Dragons: their elaborate rules make their world behave in its own independent way, being fuelled by throwing dice. Computers, of course, make all of this much easier, for they can throw dice and project the results into the behavior of the world much more easily than a human arranger. Computers have also provided for an amplification of the idea of disguising a part of the real world as another one (i.e. the idea behind theater) by means of the Internet to produce “collaborative theaters” in which everybody can play a role. (The pioneering software environment is that of the so-called “multi-user dungeon” or MUD, originally a means of implementing RPG’s, but then used for many other purposes – the development continues3). Why do we fabricate the imaginary worlds? I would say there are at least three reasons. The first of them is strictly speaking, not a reason: We contrive and explore other worlds simply for the hell of it – we plainly enjoy it. Second, wayfaring into strange lands and strange worlds ‘deepens our perspectives’ – the more we experience, the smarter we can become. Just as Bilbo Baggins was never the same person after he accomplished his journey ‘there and back’, we are never the same persons after we find the way to Middleearth and back4. Third, contriving worlds different from our own may be a means of checking how things work, and especially how people fare, in counterfactual circumstances. Thus there are a host of hobbit-like people within our world: people who live their parochial lives, enjoy good food and good drinking and never do anything brave. But what if times get so bad that one must simply either be brave or perish (like when the ring of power appears and is desperately sought by its master)? Mr. Tolkien thought that then ‘hobbits’ might fare better than we would expect ...
2
Worlds we exploit: sciences It may seem that science, in this respect, is to side with the realistic wing of literature: it appears to put premium not on fantasy but on austere realism, and concentrates on what is ‘tangibly’ here, in our real world. However, is this true? Is science possible without invoking other worlds? As it turns out, the picture of a scientist as dealing immediately with our tangible reality is problematic in more than one respect. It seems, to start with, that one of the most important achievements of science is that it furnishes us, in Karl Popper’s often quoted dictum, with the possibility of letting “our hypotheses die in our stead”. And this possibility is clearly inseparably connected with modeling and simulating situations and courses-of-events which may (but need not) come to pass. Thus, science is inseparably connected with ways of representing, if not whole non-actual worlds, then surely non-actual situations. This means that at least the last of the three reasons for making fictional worlds, which we listed above, maintains validity also when we turn our attention from the arts to science. However, I think that also the second still holds force. To help us predict and master the behavior of our environment is not the sole purpose of science (although it may sometimes seem so) – for surely there is still also the old-fashioned reason of becoming wiser (where wiser means not only more intellectually skilled). Hence the experience gained by wayfaring into the ‘worlds’ of scientific theories should again return us to our ordinary, everyday world with new eyes. However, I do not have very much to say about how science actually works; and I would prefer to turn my attention to philosophy.
Worlds which sustain meaning: philosophy As an illustrious example of the employment of other worlds within philosophy let us consider Hilary Putnam’s (1975) famous ‘thought experiment’ featuring the so called Twin Earth5. Twin Earth is a world different from ours, but in quite a minimal respect. It is almost just like our world, including that each of us has an atom-by-atom identical copy there; the only difference being that our water, H2O, is replaced by a different liquid with the formula XYZ on the Twin Earth. The point of the thought experiment is to find out what the term ‘meaning’ means: in particular whether the Twin Earth word ‘water’ could mean something different from our ‘water’. Putnam’s conclusion was that, indeed, it not only could, but really would; and that hence, in view of the fact that the heads of our twins are identical to our ones, “meaning ain’t in the head”. Independently of whether we agree with Putnam’s conclusion, this illustrates a particular way of usage of other worlds in philosophical deliberation: we check an argument of the kind of A is determined by B (meaning is determined by the contents of human heads) by inspecting worlds in which B (people’s heads) is kept fixed while other things (environment) are varied – and check whether A also keeps fixed. (In fact this is structurally very much like a scientific experiment: if we want to find out whether an event A is caused by an event B, we keep producing B while varying collateral events to check whether A keeps obtaining. The difference is that philosophy, unlike science, often checks connections which are conceptual rather than causal.) Some naturalistically minded philosophers have invested much effort into devising pictures on which our knowledge is a matter of a non-mediated contact between our selves and the real world. A prototypical example is Bertrand Russell’s (1914; 1918/9) philosophy of his logical atomism period: according to this, the world consists of things, properties and 3
relations, which together constitute facts (which are again only complex objects), and our thinking as well as our language is a means of picturing them. A truth is simply a thought or a sentence that succeeds in picturing a fact. Hence even such truths as those of logic, which prima facie do not seem to directly treat of the real world, must also be pictures of facts, although of more general ones than those pictured by the truths of natural sciences. The problem with this Russellian picture is that it does not provide for a distinction between the content of our thoughts or pronouncements and what they are about. The phrase the current king of France refers to nothing in our real world, but it has content – we understand it. This indicates that we may need, as Frege suggested, a kind of a Platonist ‘world’ harboring meanings, which in contrast to the referents of our phrases, do not seem to be locatable within our real world. The Polish philosopher Kazimierz Twardowski once put the problem of meaning in the following illuminating way6: take a picture of a landscape and consider the term ‘painted landscape’. The term is obviously ambiguous: it might either mean the virtual landscape of the painting (irrespectively of whether there is a real landscape which it would reproduce), or a real landscape which the painting depicts. While there always is a ‘painted landscape’ in the first sense of the word (insofar as the picture can be reasonably said to be one of a landscape), there need not be one in the second. And analogously, it is something different for the current king of France to have a content in the sense of meaning something, and to have a content in the sense of there being an individual being referred to by it. (Just as in the case of the painting, the second kind of content presupposes the first one, but not vice versa.) In fact, it was largely this very observation which led the young Wittgenstein to reject the austerity of Russell’s logico-atomistic picture and to offer his own less austere version of the man-world relationship in the Tractatus (1922). This picture allowed for possible, but nonactual states-of-affairs’ to give the content to false sentences. (The popular anecdote about Wittgenstein’s coming to entertain the Tractarian picture relates to just this aspect of his doctrine: allegedly he saw, in a newspaper, pictures of what could happen during a traffic accident and how to deal with it – i.e. pictures of possible, but non-actual situations.) This indicated that if we want to have meanings, we cannot make do without a Platonist world of ideas or a Fregean third realm over and above our physical world. In fact, Russell himself not only admitted, but obviously hailed this kind of a supernatural world. In The Problems of Philosophy (1912) he is quite explicit about the existence of two realms: The world of universals ... may also be described as the world of being. The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life. The world of existence is fleeting, vague, without sharp boundaries, without any clear plan or arrangement, but it contains all thoughts and feelings, all the data of sense, and all physical objects, everything that can do either good or harm, everything that makes any difference to the value of life and the world. According to our temperaments, we shall prefer the contemplation of the one or of the other. The one we do not prefer will probably seem to us a pale shadow of the one we prefer, and hardly worthy to be regarded as in any sense real. But the truth is that both have the same claim on our impartial attention, both are real, and both are important to the metaphysician. Indeed no sooner have we distinguished the two worlds than it becomes necessary to consider their relations.
4
Hence should we be content with having our physical world plus the Platonist world? As a matter of fact, recent developments in the analysis of meaning have prompted the conclusion that we should see the Platonist world more as a metaworld, somehow ‘comprising’ the real world along with all kinds of its alternatives.
Worlds we use building semantic models: logic When philosophers condemned Russell’s naturalistic picture for not making room for meanings, W.V. Quine launched a counterstrike: if there is no room for meanings in the real world, then the worse for them7. We must simply learn to live without them – take them for mere chimeras. Everything what truly is, according to Quine and his followers, is describable in the extensional, physicalist language of natural science. However, as Bob Brandom (2001, p. 600) has pointed out, independently of whether we approve of the concept of meaning or not, to describe the world we need properly meaningful words, and words cannot get the proper kind of meaning, pace Quine and Kripke (1972), from merely their contact with the actual world: The incompatibility of the property of being green with that of being red, is part of the content of the concept green. To call something ‘rigid’ is to commit oneself to various consequences concerning what would happen if one applied a potentially deforming force. That an object has a finite mass means that the application of force is necessarily to accelerate it. Apart from their involvements in material incompatibilities, counterfactual dependencies, and necessitations of the sorts these examples illustrate, bits of ordinary descriptive empirical vocabulary cannot have the meaning they do. This is reminiscent of (though certainly not reducible to) Rudolf Carnap’s objections to identifying meanings with extensions. If the meaning of a term, says Carnap (1947; 1955), were determined solely by the objects to which the terms applies in the actual world, then the terms ‘human’ and ‘featherless biped’, or the terms ‘creature with heart’ and ‘creature with kidneys’ would be deemed synonymous, which intuitively they are not. Hence we must not restrict ourselves to actual individuals, but consider also potential ones – i.e. individuals which do not come to occur in the actual world, but which inhabit other possible worlds. There surely is a conceivable individual which is a featherless biped, but not human (actually a hobbit is a case in point, not to speak about dwarves, elves etc.), and there is a conceivable creature which has a heart, but no kidney. This indicates that (i) we cannot have the vocabulary we do without having at least an implicit grasp of counterfactuality (and possibility and necessity); and (ii) whereas this is made primarily explicit by means of the adverbs ‘possibly’, ‘necessarily’ and especially by the overt counterfactual conditional ‘if it were the case that ... it would be the case that ...’, these locutions are further illuminatingly explicated in terms of possible worlds. To be necessary is to be true in every possible world, to be possible is to be true in some possible world, and to be true if such and such conditions obtained is to be true in every world in which the conditions do obtain. Saul Kripke (1963) then came up with a more sophisticated theory. He realized that at least in some senses of ‘necessary’, what is necessary in our world need not be necessary in another world. (Take, for example, physical necessity – what is physically necessary here need not be necessary within a world with quite different physical laws.) Thus he proposes to 5
enrich the universe of possible worlds with an accessibility relation reflecting which worlds should be taken as conceivable alternatives of a given world. This idea started an important revolution in logic: it has transpired that semantics of this kind can be set up for a very wide class of logical calculi. Kripke himself has shown that it is usable not only for almost any kind of modal logic, but also, with only a minor fix-up, for intuitionist logic. (The fix-up consists in the fact that within a possible world interpretation of intuitionist logic, a proposition true in a world cannot fail to be true in an accessible world. As a result, the worlds come to behave more like ‘information states’ than like worlds in the literal sense, and the accessibility relation starts to resemble something like an ‘information growth’.) More recently, Kripkean model theory was adapted for the needs of dynamic logic. (In this case, ‘worlds’ are again seen as ‘information states’, but sentences are no longer seen as denoting sets of worlds, but rather relations between worlds – i.e., in effect, ‘accessibility relations’). And the possible-world framework is exploited also by many other logical systems. (Thus, e.g., paraconsistent logic assumes that each world comes with a counterpart world, its alter mundus, such that a negation of a sentence is true iff the sentence itself is false in the counterpart world. If every world is identical with its counterpart, we have simply the classical logic, but if this is not the case, we may have worlds which violate the law of contradiction.)8
Can we live without other worlds? Hence are other worlds just as indispensable for philosophy, logic and the sciences as they are for the arts and literature? I think that the answer to this is complex and crucially depends on what we mean by ‘indispensable’. I am convinced that Quine’s project of ‘ontological asceticism’ is of fundamental importance – as I agree that one of the most basic tasks of philosophy is to discern which kinds of entities we are able to live without, and to reduce as much as possible what we take as existing to a minimum of basic building blocks. On the other hand, I do not believe that this reductionist enterprise should be understood as the pursuit for some ultimate ‘furniture of the universe’. I do not believe there is one true reduction of everything to a unique absolutely primitive basis. Therefore I think we should understand the Quinean physicalist project as a project not of finding what there really is, but rather of finding how far we can go in describing reality by means of the particularly austere final vocabulary (in Rorty’s, 1989, term) of physics. This means that once we are clear about how possible worlds could be ‘explained away’, there is no reason not to use these kinds of entities if we find them useful for scientific or philosophical purposes. (My long-term project, for example, has been to indicate how we can do ‘formal semantics without formal metaphysics’ – how we can reconcile rejecting the metaphysics of possible worlds with a (‘non-committal’) employment of possible worlds within semantics9.) And in this latter sense, ‘other worlds’ are, I think, utterly indispensable. There is no way of furnishing our basic empirical vocabulary with the appropriate semantics without settling the behavior of words within a rich repertoire of counterfactual claims – or, in ontological terms, without invoking worlds alternative to our own; and nor is there a comparable way of making perspicuous semantics for the common logical calculi without employing possible worlds. After the idea of possible worlds has come to be commonly accepted, excited discussions atarted about whether the worlds other than our own ‘really exist’10. This seems to me to be a pseudoquestion. (I can see only two clear senses of ‘exists’, on both of which the answer to 6
the question is trivial. In the one sense, to exist is to assume some spatiotemporal part of our world – and in this sense other worlds surely do not exist. In the other sense, to exist means to be denoted by subjects of true subject-predicate claims – and in this very relaxed sense possible worlds surely do exist.) I think that the truly crucial question regarding possible worlds is the following: how do we learn about what is going on within the other worlds? Some logicians and philosophers seem to assume tacitly that we can as if ‘look’ into them: they assume that we can reduce ‘to be necessary’ to ‘to be true in every (accessible) possible world’, for the latter claim is independently meaningful in the very same way as, say, ‘in every supermarket in New York’. I think that this is a misguided view, which produces both vicious circularities and the very kind of sterile metaphysics against which the fathers of analytic philosophy raised their arms11. The question, then, is whether the claims of a metaphysics of possible worlds should precede and found the claims of a logic which employs possible worlds semantics. My response is that they should not: just as the literary worlds should rightly be seen as brought into being by the literary texts, the possible worlds employed by logic should be seen as brought into being by the laws of logic. Thus, in the Wittgensteinian turn of phrase, not: something is necessary because it is true in every possible world, but rather: that it is necessary reveals that there cannot be a possible world where it would fail to hold.
7
References Brandom, R.: 2001, ‘Modality, Normativity, and Intentionality’, Philosophy and Phenomenal Research 63, 587-609. Carnap, R.: 1947, Meaning and Necessity, University of Chicago Press, Chicago. Carnap, R.: 1955, ‘Meaning and Synonymy in Natural Languages’, Philosophical Studies 7, 33-47. Coffa, J.A.: 1991, The Semantic Tradition from Kant to Carnap, Cambridge University Press, Cambridge (Mass.). Doležel, L.: 1989, ‘Possible Worlds and Literary Fictions’, in S. Allén (ed.): Possible Worlds in Humanities, Arts and Sciences, de Gruyter, Berlin. Kripke, S.: 1963, ‘Semantical Considerations on Modal Logic’, Acta Philosophica Fennica 16, 83-94. Kripke, S.: 1972, ‘Naming and Necessity’, in Semantics of Natural Language (eds. Davidson, D. & Harman, G.), Reidel, Dordrecht, 253-355. Lewis, D.: 1986, On the Plurality of Worlds, Blackwell, Oxford. Neubauer Z.: 1992, Do světa na zkušenou čili O cestách tam a zase zpátky. Praha, Doporučená četba Jůza a Jůzová [in Czech]. Peregrin, J.: 1995, Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J.: 1998, ‘Linguistics and Philosophy’, Theoretical Linguistics 25, 245-264. Peregrin, J.: 2001a, Meaning and Structure, Ashgate, Aldershot. Peregrin, J.: 2001b, ‘Metaphysics as an Attempt to Eat One's Cake and Have it’, in Metaphysics in the Post-Metaphysical Age (ed. U. Mexiner), Hölder-Pichler-Tempsky, Wien, 349-357. Pessin, A., ed.: 1996, The twin earth chronicles: twenty years of reflection on Hilary Putnam's "The meaning of meaning", Sharpe, New York. Priest, G.: 2001, An Introduction to Non-Classical Logics, Cambridge University Press, Cambridge. Putnam, H.: 1975, ‘The Meaning of “Meaning”’, in Language, Mind and Knowledge (Studies in the Philosophy of Science 7; ed. K. Gunderson), University of Minnesota Press, Minneapolis; reprinted in Putnam (1975). Putnam, H.: 1975, Mind, Language and Reality (Philosophical Papers, vol. 2), Cambridge University Press, Cambridge. Quine, W.V.O.: 1992, Pursuit of Truth, revised edition, Harvard University Press, Cambridge (Mass.). Rorty, R.: 1989, Contingency, Irony and Solidarity, Cambridge University Press, Cambridge. Russell, B.: 1912, The Problems of Philosophy, Home University Library. Russell, B.: 1914, Our Knowledge of the External World, Allen and Unwin, London. Russell, B.: 1918/9, ‘The Philosophy of Logical Atomism’, Monist 28/29, 495-527/32-53,190222,345-380. Wittgenstein, L.: 1922, Tractatus Logico-Philosophicus, Routledge, London.
8
Notes 1
See http://www.mcsr.olemiss.edu/~egjbp/faulkner/glossaryy.html#Yoknapatawpha.
2
See http://www.glyphweb.com/arda/.
3
A striking indication of how close the ‘reality’ of such worlds has come to the reality of the actual world is offered by Dibell’s story ‘Rape in Cyberspace’ (http://www.levity.com/julian/bungle.html).
4
This is brought out by Neubauer’s (1992) tiny Czech ‘manual’ to the Tolkien’s saga.
5
See also Pessin (1996).
6
See Coffa (1991) for a discussion of this example.
7
See, e.g., Quine (1992, Chapter III).
8
See Priest (2001) for a variety of logics which can be furnished with possible worlds semantics.
9
See Peregrin (1995; 2001a, Part III).
10
See, e.g., Lewis (1986).
11
See Peregrin (1998; 2001b).
9
PRAGMATISM & SEMANTICS Jaroslav Peregrin, Institute of Philosophy, Academy of Sciences of the Czech Republic* www.cuni.cz/~peregrin
1. Two paradigms for a theory of semantics Theories of language in the twentieth century tend towards one of two radically different models. One paradigm holds that expressions ‘stand for’ entities and their meanings are the entities stood for by them. According to the other, expressions are rather tools of interaction and their meanings are their functions within the interaction, their aptitudes to serve it in their distinctive ways. The first paradigm was elaborated especially by Russell and the young Wittgenstein; and reached blossomed especially in the hands of Rudolf Carnap and his followers, ‘formal semanticians’, represented most famously by Richard Montague. Russell analyzed the ways our names come to represent objects, and attempted to generalize his findings to the whole of language: we get acquainted with entities of our world and we let our expressions represent them, and it is in this way that our expressions come to have their meanings. As Russell (1912, Chapter V) puts it: We must attach some meaning to the words we use, if we are to speak significantly and not utter mere noise; and the meaning we attach to our words must be something with which we are acquainted. This idea was seductively perfected in Wittgenstein’s Tractatus, where language is portrayed as representing the world by means of sharing its form: In propositions thoughts can be so expressed that to the objects of the thoughts correspond the elements of the propositional sign. ... In the proposition the name represents the object. ... The proposition is a picture of reality. (Wittgenstein, 1922, §§ 3.2, 3.22, 4.01) This prompted Carnap to isolate semantics as that part of the theory of language which has to do with expressions’ denoting objects: When we observe an application of language, we observe an organism, usually a human being, producing a sound, mark, gesture, or the like as an expression in order to refer by it to something, e.g. an object. Thus we may distinguish three factors involved: the speaker, the expression and what is referred to, which we shall call the designatum of the expression. ... If we abstract from the user of the language and analyze only the expressions and their designata, we are in the field of semantics. ... Semantics contains *
Work on this paper has been supported by a research grant of the Grant Agency of the Czech Republic No. 401/99/0619.
1
the theory of what is usually called the meaning of expressions ... (Carnap, 1942, pp. 8-10) In the concise formulation of Reichenbach (1947, p. 4): Language consists of signs. ... What makes them signs is the intermediary position they occupy between an object and a sign user, i.e., a person. The person, in the presence of a sign, takes account of an object; the sign therefore appears as the substitute for the object with respect to the sign user. However, in parallel to this, also the other paradigm flourished. At the beginning of the century, it pervaded especially the philosophy of language of the American pragmatists. Thus, Dewey (1925, p. 179) famously claimed that “meaning ... is not a psychic existence, it is primarily a property of behavior”; but a more representative articulation of this paradigm is offered, e.g., by the social anthropologist G.H. Mead (1934, p. 75-76): Meaning arises and lies within the field of the relation between the gesture of a given human organism and the subsequent behavior of this organism as indicated to another human organism by that gesture. If that gesture does so indicate to another organism the subsequent (or resultant) behavior of the given organism, then it has meaning. ... Meaning is thus a development of something objectively there as a relation between certain phases of the social act; it is not a physical addition to that act and it is not an “idea” as traditionally conceived. This is, needless to say, a very different conception of language and consequently a very different concept of meaning. Language is not conceived of as a set of substitutes for entities and meanings are not the entities substituted; language is rather a means of interaction and meaning is the ability to bring the interaction to a kind of ‘resonance’. Also the later Wittgenstein rejected his earlier ‘picture theory’ of language in favor of such ‘pragmatist’ view of language. Within his post-tractarian writings, language ceases to be seen as a set of pictures, and starts to be conceived of as “a collection of very various tools”: Language is like a collection of very various tools. In the tool box there is a hammer, a saw, a rule, a lead, a glue pot and glue. Many of the tools are akin to each other in form and use, and the tools can be roughly divided into groups according to their relationships; but the boundaries between these groups will often be more or less arbitrary and there are various types of relationship that cut across one another. (Wittgenstein, 1969, p. 67) It follows that the meaning of an expression is not a thing pictured by it, but rather a kind of usefulness of the expression, its aptitude to serve our ends: A meaning of a word is a kind of employment of it. Compare the meaning of a word with the ‘function’ of an official. And ‘different meanings’ with ‘different functions’ (Wittgenstein, 1984, §64) 2
Let me call these two paradigms of approaching language the Carnapian and the Deweyan paradigms, respectively. Prima facie it seems that these two approaches are exemplary antipodes: while the former is based on the assumption that language is primarily a system of names or representations of things, the latter assumes that it is a collection of means for doing ‘things’ – and these two conceptions of language do not appear to be compatible. The Carnapian picture has, ultimately, led to the mathematical reconstruction of language as a system of expressions homomorphically mapped on a system of denotations (Montague, 1970); whereas the Deweyan stance seems to lead to the lore of evasive ‘language games’, thought experiments with ‘radical translation’ and ‘interpretation’, and to an overall ‘pragmatization of semantics’ (Peregrin, 1999), which does not seem to lend itself to any such mathematization. However, in this paper I would like to propose that the gap between these two views of language need not be totally unbridgeable. True, the two conceptions of language apparently underlying them are hardly reconcilable – but what I am suggesting is that the technical apparatus engendered by the Carnapian approach, with its wealth of results, can be put into the service of the Deweyan paradigm – if we liberate it from the Carnapian representationalist ideology. Therefore I will argue that subscribing to the Deweyan paradigm need not divorce us from Carnapian and Montagovian formal semantics.
2. Meaning and Rules Assuming the Deweyan or the late Wittgensteinian view of language, we see the meaning of an expression as first and foremost a semantically relevant role or function of the expression. But what is ‘the semantically relevant function’ of an expression? We may start by considering the function of a non-linguistic item we employ, e.g. a hammer. Generally speaking, it is employed in various ways, some of which we see as a matter of the function of the hammer qua hammer, while others we see as casual and not really related to its being a hammer. A paradigm example of the former kind of things we do with the hammer is driving nails; a paradigm example of those of the latter kind might be, e.g., using it as a paperweight. We can clearly draw no sharp boundary between these two kinds of hammer-employing activities (their relationship resembles that between the center and the periphery of a town) – this, nevertheless does not prevent us from thinking that there is a function characteristic of a hammer. The fact that the employment of an expression represents a similar continuum of cases spanning from those which are felt as clearly central for the expression – qua the expression meaning what it does – to those which are held to be peripheral has led some thinkers to a chain of conclusions which appear to be destructive of the Carnapian paradigm: We cannot separate semantics from pragmatics. Therefore we cannot delimit the boundaries of meaning. Therefore there is no meaning worth its name. Therefore there is nothing to investigate for semantics. I think that somewhere along this line of argumentation we start to throw the baby out with the bathing water. After all, there are no ‘pure’ triangles in our real world; and despite this, geometry, which deals exclusively with such ‘pure’ shapes, is undoubtedly an enterprise 3
which is not only respectful, but also useful for our understanding of the real world and for our dealing with it. And in a similar way semantic theory dealing with crisply delimited meanings can be quite useful in the world where we can encounter only fuzzy ones. However, how to get a grip on the fuzzy meanings to reconstruct? Meanings are not visible like shapes of things; so where can we encounter them? It was the late Wittgenstein who urged that there is no other way to approach them than via the rules governing their expressions. There is, Wittgenstein noticed, a characteristic difference between the way we use a thing like a hammer and the way we use an expression: the point is that the usage of the expression is ‘rule-governed’ in a sense in which that of the hammer is not. There are, to be sure, rules for using hammers – but they differ intrinsically from those which govern the usage of linguistic expressions. The rules for using hammers simply spell out how to use them to achieve the end to which hammers are supposed to be a means; whereas the rules of language are of a different ilk: they do not spell out an efficient usage of language, they are rather constitutive of language. This brought Wittgenstein (1969, 184-5) to the conclusion that using language is more like playing a game such as chess than like using a hammer or, for that matter, cooking: Why don’t I call cookery rules arbitrary, and why am I tempted to call the rules of grammar arbitrary? Because I think of the concept ‘cookery’ as defined by the end of cookery, and I don’t think of the concept ‘language’ as defined by the end of language. You cook badly if you are guided in your cooking by rules other than the right ones; but if you follow other rules than those of chess you are playing another game; and if you follow grammatical rules other than such and such ones, that does not mean you say something wrong, no, you are speaking of something else. Hence the parallel between the function of the hammer and the meaning of an expression is limited: meaning is not that kind of function which characterizes the hammer. It is a function which an item acquires by being subordinated to rules of the kind games have. Thus the meaning of an expression can be compared to the role of a chess piece, which acquires its role of, say, a ‘knight’ by being handled according to the rules of chess (Wittgenstein as quoted by Waisman, 1967, p. 105): For Frege, the choice was as follows: either we are dealing with ink marks on paper or else these marks are signs of something, and what they represent is their meaning. That these alternatives are wrongly conceived is shown by the game of chess: here we are not dealing with the wooden pieces, and yet these pieces do not represent anything – in Frege’s sense they have no meaning. There is still a third possibility; the signs can be used as in a game. All in all this resulted into the conclusion that if we are to understand the semantics of our language, we have to concentrate on its rules, and we also have to explicate what it means to ‘follow a rule’ of the kind language has. This was the problem to which Wittgenstein famously devoted most of his Philosophical Investigations (and which was later revived
4
within the discussion initiated by Kripke, 19821). However, despite this being an important step on the way towards isolating the semantically relevant function of an expressions, it still does not provide us with any definite lead: the problem of finding a criterion for distinguishing semantically relevant rules from other rules governing our language (such as rules of syntax, rules of social conduct etc.) is still far from trivial.
2. Meaning and Inference If Wittgenstein is right, the meaning of an expression is to be sought via the investigation of the rules which govern the ‘semantically relevant’ usage of the expression. However, is there a non-circular way of specifying what kind of rules they are? Or do we end up with a kind of ‘semantic agnosticism’? The approach to answering these questions which we would like to follow here is due to another recent philosopher who stressed the key role of rules within human linguistic conduct, namely Willfrid Sellars. Operating in the intersection of the influences of Viennese logical empiricism and American pragmatism, he developed his own theory of the constitutive role of rules with respect to language and meaning2. Moreover, he managed to be more explicit about the nature of linguistic rules which are constitutive of semantics, locating them as the rules of (broadly conceived) inference. Prima facie, this may seem strange: why inference? Do we not employ language for many other important purposes other than drawing inferences? Does this not lead to an overintellectualized conception of language? Well, the Sellarsian claim is not that drawing inference is the most frequent activity for which we need our language – the claim is just that inference is the activity which confers content on our expressions. Why? We can reach the conclusion in three steps. First, we realize the semantic primacy of sentences over the meanings of sub-sentential expressions, and hence of the meanings of sentences, propositions, over meanings of other expressions: the meaning of any expression is derived from the meanings of sentences. Second, we realize that that to be a proposition is to be located within a logical space, i.e. to entail other propositions and to be entailed by them: the meaning of a sentence is an inhabitant of a logical space. Third, we realize that a proposition can acquire these properties only in force of the rules which we let govern a sentence which expresses it: to make a sentence acquire such meaning, it must be embedded into a network of inferences. Let us consider these three steps in sequence. 1. The semantic primacy of a sentence over its parts has been urged by a number of philosophers, including Frege (“It is only in the context of a proposition that words have any meaning”3) and Wittgenstein (“Only propositions have sense; only in the nexus of a proposition does a name have meaning.”4). The reason is clear: it is only a sentence which can serve a self-standing communication purpose and which is in this sense independently 1
See Baker & Hacker (1984) for a different perspective on the problem.
2
See Sellars (1963; 1974); and see Marras (1978) for an overview.
3
Frege (1884, p. 73).
4
Wittgenstein (1922, §3.3).
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significant (and hence it is also only a sentence which can be true or false) – sub-sentential expressions gain significance only via their acting within sentences. There is an obvious counterargument to this: basic units of language must be finite in number, and this is the case of words, not of sentences. However, the list of sentences we really encounter during our acquisition of language is in fact also finite. True, we learn to decompose sentences into worlds, which we then use as our basic linguistic building blocks – as we learn, in the inverse process, to recompose the words to produce new sentences5. And once we start to compose complex sentences, there is no upper limit – the class of sentences becomes (potentially) infinite. However, this does not contradict the assumption of the primacy of the basic sentences – no more than the fact that we can use the roles of mother, uncle etc. to put together an unlimited number of potential family structures contradicts the fact that we have abstracted the roles from the structures of some already existing families6. 2. What, if anything, is a proposition? It is not easy to say; but if there be anything essential to propositions, then it would seem to be the fact that a proposition must have a negation, be conjoinable with other propositions, and especially entail and be entailed by other propositions. This indicates that just as the modus vivendi of physical objects is their causal interaction within physical space (and time), the modus vivendi of propositions should be their logical interrelationships. Thus, propositions appear to exist in a logical space structured by logical relationships, especially by the relation of entailment7, which finds its expression in inferential rules. 3. However, unless we wish to subscribe to an unwieldy metaphysics, we must take the talk about propositions as more of a façon de parler than of a description of a platonist realm. (The problem is not the platonism itself, but the assumption that our sentences gained their inferential properties via a process of naming of entities from such a realm – how could we have achieved this?) And this leads us to the conclusion that our sentences do not have the inferential properties they have because we have used them to christen some ever-existing propositions with ever-existing entailment relations, but rather that we caused the sentences to express the propositions they express (or perhaps better: made them describable as expressing the propositions) via letting them be governed by inferential rules, which establish a certain network providing for the needed logical space. This idea of Sellars has been picked up and elaborated by Brandom (1994, 2000). For him, rules of inference are crucial because the principal, ‘content-conferring’ enterprise giving the point to our language is the enterprise of giving and asking for reasons. Thus something is a language worth the name only insofar as it can provide for this enterprise; and hence something can be said to be a truly meaningful expression only insofar as it participates in it. As Brandom (1994, p. 144) puts it: “It is only insofar as it is appealed to in explaining the circumstances under which judgments and inferences are properly made and the proper consequences of doing so that something associated by the theorist with interpreted states or 5
See Quine (1960, p. 9).
6
See Peregrin (2001, Section 4.4.) for a more extensive discussion.
7
The relations usually seen as engendered by the logical operators – negation, conjunction etc. – can be seen as reducible to the entailment structure, which turns out to be a Boolean algebra. They can be seen as the algebraic relationships of complement, join etc.
6
expressions qualifies as a semantic interpretant, or deserves to be called a theoretical concept of a content.” This means that to get a grip on semantics is to get a grip on the inferential structure; and thus a grip on the meaning of a particular expression is a grip on the inferential rules governing the particular expression. Hence the ‘inferentialist’ semantic explanations: “beginning with properties of inference, they explain propositional content, and in terms of both go on to explain the conceptual content expressed by subsentential expressions such as singular terms and predicates” (Brandom, 2000, p. 30).
3. Meaning as an inferential role How can we see meaning as a matter of inferential role? Let us first consider an inferentialistically simple expression, the logical connective ‘and’. It is clear that its semantics can be characterized in either of the following alternative ways: (i) in terms of denotation: ‘And’ denotes the function f∧ mapping pairs of truth-values on truth-values in such a way that f∧(T,T)=T, while f∧(F,T)=f∧(T,F)=f∧(F,F)=F. (ii) in terms of inferences: ‘And’ joins sentences A and B into the complex sentence A∧B such that the following inferences hold: (∧1) A, B ⇒ A∧B (∧2) A∧B ⇒ A (∧3) A∧B ⇒ B The following two obvious facts show that these two articulations are really equivalent: Fact 1. If “∧” denotes f∧, (∧1) - (∧3) are correct (truth-preserving) Fact 2. If (∧1) - (∧3) hold, then “∧” can be correctly treated as denoting f∧. ((∧1) says that if A and B are true, A∧B is bound to be true too; whereas (∧2) and (∧3) state that A∧B is false if either A, or B is – and f∧ correctly recapitulates this.) Now given the nature of language, is one of the two facts somehow primary to the other? Should we reduce the holding of the inferences to the presence of the denotation (“conjunction sustains (∧1) - (∧3) because it denotes f∧”) or rather the other way around (“we can explicate the meaning of conjunction as f∧, because it is governed by (∧1) - (∧3)”)? While in a formal language the former may be the case (for we may define conjunction by means of denotation assignment), for natural language the claim “‘and’ denotes such and such entity” is clearly in need of further clarification. (It is obviously not to be taken to mean that we sometimes in the past introduced ‘and’ into our language by means of christening f∧. But if not this, then what is it supposed to mean?). In contrast to this, the claim “to assert two sentences joined by means of ‘and’ is correct iff it is correct to assert each of the two individually” is entirely perspicuous. I think that even many philosophers not generally sympathetic with inferentialism may be willing to grant that the meaning of a logical particle like ‘and’ may consist in its inferential 7
role. (A well-known opponent of this possibility is Prior, 1960/61, with his tonk8. But even granted that not every set of inference rule can be reasonably seen as constitutive of a meaning, it is not clear why this should be incompatible with the claim that the particular set of rules listed above does constitute the meaning of ‘and’.) However, they would insist that this cannot be generalized to a substantial part of our language. Admittedly, the situation is less favorable for the inferentialist when we pass over from logical vocabulary to empirical terms. However, there still remains a sense in which inferentialism is viable even here. Take the term ‘rabbit’. Does it not mean what it does solely in force of its having come to refer to rabbits? Well, what does it mean “to refer to rabbits”? To utter “rabbit” in the presence of rabbits? But why, then, do we not say that a particular kind of dog bark, ejaculated by a dog on scenting a rabbit, refers to rabbits? There appear to be two kinds of answers to the question what distinguishes an expression referring to a thing from a mere reaction prompted by the thing (which can be displayed by non-human animals or even inanimate things9). The first of them is that it is our mind which makes the magic, by somehow accompanying the word with a ‘homing in’ on rabbit(hood), or by summoning the idea of rabbit. This is the answer which has been put forward, in different guises, by philosophers so different as John Locke, Edmund Husserl, or John Searle; and which also appears to concur with common sense. The trouble with this answer, as Blackburn (1984, §II.3) puts it, is that it is a “dog-legged theory”: a theory which instead of solving the problem merely shifts it at one remove. Its answer to the question how do our expressions manage to refer? is by expressing mental contents which refer; which, of course, immediately invites the follow-up question well, but how do they manage it?. The last question, then, is held by this theory as being somehow self-answering.
Hence, if we are not content with this kind of answer, we may consider the alternative; a kind which is prima facie less appealing and far less popular, but which, I suggest, is worth pursuing, especially for the pragmatist. The answer is that an expression comes to refer by being employed in a certain way within a specific kind of human (normative) practice. Variants of this answer were tabled by Immanuel Kant, Ludwig Wittgenstein, and recently by Robert Brandom. Brandom, in particular, suggested that even in the case of empirical terms and their referential capacities, it is still the inferential articulation which is crucial. Hence the idea is that even the content of empirical expressions is in some important sense conferred on them by inferential rules governing their usage. Of course, in such cases we cannot restrict ourselves to inferences in the standard, narrow sense – i.e. moves from assertions to assertions –, because we have to admit also ‘inferences’ from situations to assertions and vice versa. Anyway, the idea is that even for empirical terms we still have the two possibilities of characterizing their semantics: (i) in terms of denotation: ‘Rabbit’ denotes some kind of ‘rabbithood’, explicated perhaps as a functions mapping possible worlds on the classes of their rabbits (ii) in terms of inference: ‘Rabbit’ constitutes a part of sentences so that the following inferences hold: 8
See Peregrin (2001, Chapter 8) for a discussion of Prior’s attitude.
9
Consider, for example, a piece of metal reacting to water by rusting, or a thermometer reacting to the temperature of its environment.
8
(r1) X is a rabbit ⇒ X is a mammal (r2) X is a rabbit ⇒ X is not an elephant ... The equivalence of these two articulations would now be seen as a matter of the following two facts: Fact 1. If “rabbit” denotes ‘rabbithood’, then (r1), ... are correct (for rabbithood is supposed to include mammalhood, to be incompatible with elephanthood ...) Fact 2. If (r1), ... hold and if it is appropriate to assert “This is a rabbit” when pointing at a rabbit, then “rabbit” can be correctly treated as denoting ‘rabbithood’. It is clear that here the situation is incomparably more problematic than in the case of a logical connective. For one thing, the collection of inferences is too complex to be easily specifiable. (It is this very complexity, as Sellars argues, which prevents us from being able to give the inferences constituting the inferential role of such an expression explicitly. Therefore we usually only illustrate the role with the help of a known word having the same or similar role – which is what usually happens in radical translation10.) For another thing, the inferential roles must involve not only the inferences in the narrow sense (listed as our (r1) ...), but also what Sellars called ‘world-language transitions’ (together, as the case may be, with ‘language-world’ ones). This last point might engender the suspicion that the inferential standpoint is only much ado about nothing. Do not the ‘inferences in the wide sense’, which we claim to co-constitute the meaning of an empirical word like ‘rabbit’, amount to the relation of reference? And are we thus not returned to the denotational paradigm which we wanted to circumscribe? Not really. The ‘world-language transitions’ do not amount to a relation between things and words, but rather to ones between facts and statements. Moreover, and this is crucial, they can never exhaust an expression’s inferential role: according to the inferentialist, nothing can be contentful without being capable of occurring both in the premises and in the conclusions of inferences. Thus a sentence ‘This is a rabbit’ (and consequently the term ‘rabbit’) cannot be contentful solely in force in its being a correct response to the presence of a rabbit; it must moreover, be capable of serving as a premise of further inferences (viz. ‘Hence, it is not an elephant’, ‘Hence, it is a mammal’ etc.).
5. The circularity of explanations provided by formal semantics All of this seems to suggest that we should abandon the Carnapian paradigm in favor of the Deweyan one; that we should turn our backs on the denotational view of semantics and on its outgrowth, formal semantics. And indeed, there has been much criticism of formal semantics 10
Thus although it is not difficult to specify the role, and hence the meaning, of the aliens’ equivalent of our ‘and’ without invoking our own word, when we want to specify the role, and hence the meaning of their equivalent of our ‘rabbit’, we can hardly do it otherwise than by pointing out that the word means what our ‘rabbit’ does.
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from philosophers of language (myself included). However, from the viewpoint of this paper it is crucial to distinguish between the (substantiated) criticism of formal semantics’ aspirations to become itself a philosophy of language, and the (less substantiated) criticism of formal semantics as a technical tool. As an example of the criticism of the first kind, consider Davidson’s (1999, p. 689) objection to Barwise and Perry’s (1983) situation semantics. We cannot, Davidson claims, reduce “is true” to “expresses an actual situation”, for we cannot specify which situations are actual otherwise than via specifying what is true: Barwise and Perry’s situations are set-theoretical constructs. Called “abstract situations”, they are defined in terms of (real) objects and properties and relations. Truth can’t be explained by reference to abstract situations, for abstract situations correspond to false sentences as well as true. Among the abstract situations are “actual” situations, which do correspond in some sense to true sentences. So far this defines actual situations in terms of truth and not vice versa. Actual situations, however, “represent” real situations, which are said to be “parts” of the world. Barwise and Perry never try to define “real situation”; they say that if you don’t think there are such things, they admit they don’t see how to persuade you. It is easy to specify when a particular abstract situation is actual: the abstract situation that I will call “Sam, mortality” is actual if and only if Sam is mortal (that is, Sam instantiates mortality). Having determined what makes [Sam, mortality] actual, we can now “explain” what makes the sentence “Sam is mortal” true by saying it is true because “Sam is mortal” corresponds to an actual situation. That situation is actual because Sam is mortal. It is obvious that we can retain the contents of this explanation, everything that “relates language to the real world”, by saying “Sam is mortal” is true if and only if Sam is mortal; the apparatus of situations has done no work. The reason it has done no work is that truth must be brought in to explain the relation between Sam and mortality, something the semantics of situations fails to do. Myself I raised a similar kind of objection (Peregrin, 1995; 2000): we cannot reduce “is necessary” to “holds in every possible world”, for we cannot specify which worlds are possible otherwise than via specifying what is necessary. For suppose somebody wants to know why a statement such as ¬(P(a)∧¬P(a)) is logically true. We can hardly answer ‘It holds in all (the model structures capturing) the possible states of the world – I have inspected them carefully and have not encountered a single one in which it does not hold’. Our answer would have to be something akin to ‘nothing can be P and simultaneously not-P’, or perhaps ‘[“]to be P and not-P[”] makes no intelligible sense’. This indicates that it is hard to maintain that a logical truth is true because it is valid in all possible structures – instead it is much more adequate to view the situation the other way around: because something is a logical truth, there cannot be a structure in which it does not hold11. Such objections point out that if we start to treat formal semantics as the basis for a philosophy of language, we are likely to run into a vicious circle: we reduce philosophically 11
We can, to be sure, sometimes discover that something holds in all structures of a certain class – but unless the class is finite, we can hardly do so by going through all the structures; we have to somehow deduce it from various properties constitutive of the class.
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problematical concepts to the seemingly perspicuous formal semantic concepts, which, however, ultimately rest on the obscure concepts to be explicated. Thus, if we want to clarify the concept of truth by the notion of actuality of situations, we face the fact that the latter notion cannot be made intelligible other than via direct or indirect recourse to the concept of truth; and if we want to reduce necessity to holding in every model or every possible world, we must see that the delimitation of the relevant space of models or possible worlds must ultimately rest on the concept of necessity. Hence formal semantics, by itself, is not usable as formal metaphysics, it is not a means of achieving a correspondence theory of truth, nor any other philosophical goals. However, I do not think that it follows that to try to explicate meanings in terms of possible worlds and situations is futile, nor that formal semantics is a misguided enterprise. In particular, I do not think that any kind of employment of the apparatus of formal semantics presupposes the subscription to the denotational approach to meaning, or to the correspondence theory of truth with a ‘formal metaphysics’, or to any other ‘ideology’. I think that the results of formal semantics can be well put into the services of the inferentialist philosophy of language.
6. Formal semantics as envisaging inferential roles Consider possible worlds, the probably most discussed creatures of formal semantics. What are they? There exists what we could possibly call the ‘stock answer’: Possible worlds are entities whose existence is to be established, and whose nature is to be reported, by a theory independent of a theory of language (‘metaphysics’); and formal semantics then can build on this, i.e. explicate meanings of statements by pointing out that they denote sets of such worlds12. However, we have also an inferentialist alternative which runs as follows: Due to the presence of the standard logical operators (negation, conjunction, ...) with their inferential properties the statements of our language are inferentially structured into a Boolean algebra. Hence they can be represented as denoting subsets of a certain set (in force of Stone’s representation theorem13). Moreover, due to the presence of modal and counterfactual operators and locutions, the underlying set cannot be the most trivial, one-element one14. Hence statements denote subsets of a nontrivial set – and as elements of this sets can be seen as “what truth is relative to”15, it is plausible to call them possible worlds. In this way, we can say that possible worlds – just like other entities entertained by formal semanticians – can be seen as means of envisaging inferential patterns. It seems that for some peculiar reasons having to do with the way our minds have developed, the relation of containment is somehow more perspicuous for us than relations of other kinds. And possible 12
Elsewhere (Peregrin, 1998) I have pointed out how this answer may lead to a philosophical deadlock: while linguists would like to leave the explanation of the nature of possible worlds to philosophers, philosophers having undergone the linguistic turn tend to think that the way to the explanation of the nature of such ‘metaphysical’ entities is in the linguistic analyses of our talk which invokes them (explicitly or implicitly).
13
The theorem states that every Boolean algebra is isomorphic to the algebra of subsets of a set.
14
The powerset of which has two elements identifiable with the two truth values.
15
Cf. Stalnaker (1986).
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worlds allow us to turn inference precisely into containment: namely the containment of the intersection of the classes of possible worlds denoted by the sentences in the antecedent of the inference within the class of worlds denoted by that in the consequent. Moreover, inferential patterns usually involve more than one expression, whereas in semantics we are often interested in the meaning of a single one, i.e. the very contribution this single expression brings to the patterns which govern it. And explicating this contribution as an abstract, typically set-theoretical, object appears to be a good way to make it intelligible. Remember, as a paradigmatic example, the logical conjunction: the inferential pattern governing it is fairly simple and perspicuous, but nevertheless it is still helpful to encapsulate it as the truthfunction. Thus, I suggest, formal semantics can be of some help even for the inferentialist. The inferentialist denial of the claim that meanings are essentially objects which have come to be stood for by expressions is not incompatible with the claim that meanings can be accounted for or modeled - as objects. And I think that the inferentialist should realize that such modeling is a very useful thing. Thus I think that although language is not literally a nomenclature or a code (as the Carnapian paradigm has it) it remains useful, at times, to see it as a code, just as it is often useful to see atoms as cores orbited by electrons. Of course that viewed from this angle, denotation ceases to be a subject matter of semantic theory, and rather becomes its tool. The object which an expression is envisaged as denoting is not to be taken as a (reconstruction of a) real entity denoted by the expression in the real world, but rather as an encapsulation of the expression’s inferential role. (One important consequence of this is that there is no one correct semantics, just as there is no one correct model of the inside of an atom, nor one correct plan of an unknown city.) Sellars (1992, p. 109n.) comes to the following verdict: [Carnap’s formalization of semantic theory in terms of a primitive relation of designation which holds between words and extralinguistic entities] commits one to the idea that if a language is meaningful, there exists a domain of entities (the designata of its names and predicates) which exist independently of any human concept formation From the perspective entertained here, we have to disagree: what Sellars disregards here is the possibility of understanding the Carnapian “formalization of semantic theory” not as a straightforward description, but rather as a ‘creative’ kind of an explication of the semantic aspect of language. What we thus suggest is that buying technologies of formal semantics need not necessitate buying the ideology of those who take formal semantics for a self-standing philosophy of language16. The moral, then, for the inferentialist, is that the Carnapian way of reconstructing the semantic aspect of language should not be despised, but rather understood in the Deweyan way: as a way to single out and materialize each expression’s contribution to the inferences in which it occurs.
16
See Peregrin (2001, Chapter 8).
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References Baker, G. P. & Hacker, P. M. S. (1984): Scepticism, Rules and Language, Blackwell, Oxford. Barwise J. & Perry, J. (1983): Situations and Attitudes, MIT Press, Cambridge (Mass.). Blackburn, S. (1984): Spreading the Word, Clarendon Press, Oxford. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Davidson, D. (1999): ‘Reply to J. Higgenbotham’, in L.E. Hahn (ed.): The Philosophy of Donald Davidson (Library of Living Philosophers), Open Court, La Salle.
Frege, G. (1884): Grundlagen der Arithmetik, Koebner, Breslau; English translation Foundations of Arithmetic, Blackwell, Oxford, 1953. Kripke, S. (1982): Wittgenstein on Rules and Private Language, Harvard University Press, Cambridge (Mass.). Marras, A. (1978): ‘Rules, Meaning and Behavior: Reflections on Sellars’ Philosophy of Language’, in The Philosophy of Willfrid Sellars: Queries and Extensions (ed. J.C. Pitt), Dordrecht, Reidel, 163-187. Montague, R. (1970): ‘Universal Grammar’, Theoria 36, 373-398; reprinted in Montague: Formal Philosophy: selected papers of R.Montague (ed. Thomason, R.), Yale University Press, New Haven, 1974. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J.: ‘Linguistics and Philosophy’, Theoretical Linguistics 25, 1998, 245-264. Peregrin, J. (1999): ‘The Pragmatization of Semantics’, in The Semantics/Pragmatics Interface from Different Points of View (ed. K. Turner), Elsevier, North-Hollard, 419-442. Peregrin, J. (2000): ‘The “Fregean” logic and the “Russellian” logic’, Australasian Journal of Philosophy 78, 557-575. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. Prior, A. N. (1960/61): ‘Roundabout Inference Ticket’, Analysis 21, 38-39. Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.). Reichenbach, H. (1947): Elements of Symbolic Logic, Free Press, New York. Russell, B. (1912): The Problems of Philosophy, Home University Library. Sellars, W. (1963): Science, Perception and Reality, Routledge, New York. Sellars, W. (1974): ‘Meaning as Functional Classification’, Syntèhse 27, 417-437. Sellars, W. (1992): Science and Metaphysics, Ridgeview, Atascadero. Stalnaker, R. (1986): ‘Possible Worlds and Situations’, Journal of Philosophical Logic 15, 109-123. Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London; English translation Routledge, London, 1961. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford; English translation Philosophical Investigation, Blackwell, Oxford, 1953. Wittgenstein, L. (1969): Philosophische Grammatik, Suhrkamp, Frankfurt; English translation Philosophical Grammar, Blackwell, Oxford, 1974.
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Pragmatismus und Semantik Jaroslav Peregrin* Institut für Philosophie, Akademie der Wissenschaften der Tschechischen Republik www.cuni.cz/~peregrin
1. Zwei Paradigmen für eine Theorie der Semantik Sprachtheorien im zwanzigsten Jahrhundert tendieren zu jeweils einem von zwei völlig verschiedenen Modellen. Im ersten Paradigma wird die Auffassung vertreten, daß Ausdrücke Entitäten vertreten und daß ihre Bedeutungen diejenige Entitäten sind, die von ihnen vertreten werden. Dem zweiten Paradigma zufolge handelt es sich bei Ausdrücken eher um Werkzeuge für eine Interaktion, wobei die Bedeutungen der Ausdrücke ihre Funktionen innerhalb der Interaktion, bzw. ihre darin zum Tragen kommende Eignung zum distinktiven Gebrauch, sind. Das erste Paradigma wurde besonders von Russell und dem jungen Wittgenstein ausgearbeitet, und seine Blütezeit erreichte es dann vornehmlich in den Arbeiten Rudolf Carnaps und seiner Anhänger, den „formalen Semantikern“, deren berühmtester Vertreter Richard Montague war. Russell hatte die Art und Weise analysiert, wie unsere Namen dazu kommen, Objekte zu repräsentieren, woraufhin er dann dazu überging, seine Ergebnisse auf die Sprache als Ganzes zu übertragen: Wir machen Bekanntschaft mit den Entitäten in unserer Umgebung und repräsentieren diese mit Hilfe unserer Ausdrücke. Und auf diese Art gelangen unsere Ausdrücke zu ihren Bedeutungen. Wie Russell (1912:V) auch schreibt: We must attach some meaning to the words we use, if we are to speak significantly and not utter mere noise; and the meaning we attach to our words must be something with which we are acquainted. [Wenn wir bedeutungsvoll sprechen wollen und nicht nur bloße Geräusche produzieren, dann müssen wir unseren Wörtern Bedeutungen zuweisen; und die Bedeutung, die wir unseren Wörtern zuweisen, muß etwas sein, mit dem wir vertraut sind.] Diese Idee wurde in Wittgensteins Tractatus auf raffinierte Weise vervollkommnet. In ihm stellt sich Wittgenstein die Sprache so vor, daß sie die Welt darstellt, indem sie deren Formen teilt: Im Satze kann der Gedanke so ausgedrückt sein, daß den Gegenständen des Gedankens Elemente des Satzzeichens entsprechen. ... Der Name vertritt im Satz den Gegenstand. ... Der Satz ist ein Bild der Wirklichkeit. (Wittgenstein 1922, §§ 3.2, 3.22, 4.01) Carnap veranlaßte dies dazu, die Semantik als den Teil der Sprachtheorie abzugrenzen, der sich mit den von Ausdrücken denotierten Objekten befaßt: When we observe an application of language, we observe an organism, usually a human being, producing a sound, mark, gesture, or the like as an expression in order to refer by it to something, e.g. an object. Thus we may distinguish three factors *
Die Arbeit an diesenm Aufsatz wurde von der Alexander von Humboldt Stiftung gefördert. Übersetzt aus Englischem von Dirk Saleschus und dem Author.
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involved: the speaker, the expression and what is referred to, which we shall call the designatum of the expression. ... If we abstract from the user of the language and analyze only the expressions and their designata, we are in the field of semantics. ... Semantics contains the theory of what is usually called the meaning of expressions ... (Carnap, 1942:8-10) [Wenn wir Anwendungen von Sprache beobachten, dann beobachten wir einen Organismus, für gewöhnlich einen Menschen, der Laute, Zeichen, Gesten oder ähnliches als Ausdruck hervorbringt, um damit auf etwas zu referieren, z.B. auf ein Objekt. Wir können demnach drei Faktoren unterscheiden: Den Sprecher, den Ausdruck und das, auf was referiert wird und was wir das Designatum des Ausdrucks nennen werden. Wenn wir nun vom Benutzer der Sprache abstrahieren und uns nur mit der Analyse der Ausdrücke und ihrer Designata befassen, dann befinden wir uns im Bereich der Semantik. ... Die Semantik umfaßt die Theorie dessen, was für gewöhnlich die Bedeutung der Ausdrücke genannt wird. ... ] Und Reichenbach (1947:4) formuliert kurz und bündig: Language consists of signs. … What makes them signs is the intermediary position they occupy between an object and a sign user, i.e., a person. The person, in the presence of a sign, takes account of an object; the sign therefore appears as the substitute for the object with respect to the sign user. [Sprache besteht aus Zeichen. ... Was sie zu Zeichen macht, ist die Zwischenposition, die sie zwischen einem Objekt und einem Zeichenbenutzer, d.h. einer Person, einnehmen. Durch das Vorhandensein eines Zeichens wird der Person das Objekt vermittelt; für den Zeichenbenutzer erscheint das Zeichen also als Ersatz für das Objekt.] Zur gleichen Zeit gedieh allerdings auch das andere Paradigma vorzüglich, und zu Beginn des Jahrhunderts durchdrang es besonders die Sprachphilosophie der amerikanischen Pragmatisten. Anzeichen dafür ist die berühmte Behauptung von Dewey (1925:179), „meaning ... is not a psychic existence, it is primarily a property of behaviour“ [daß also Bedeutung keine psychische Existenz besitze, sondern eher eine Eigenschaft des Verhaltens sei]. Eine charakteristischere Darstellung dieses Paradigmas findet sich jedoch u.a. bei dem Sozialanthropologen G.H. Mead (1934, 75-76): Meaning arises and lies within the field of the relation between the gesture of a given human organism and the subsequent behaviour of this organism as indicated to another human organism by that gesture. If that gesture does so indicate to another organism the subsequent (or resultant) behaviour of the given organism, then it has meaning. ... Meaning is thus a development of something objectively there as a relation between certain phases of the social act; it is not a physical addition to that act and it is not an „idea“ as traditionally conceived. [Bedeutung entsteht und befindet sich innerhalb des Bereiches der Relation zwischen den Gesten eines bestimmten menschlichen Organismus und dem sich daran anschließenden Verhalten des Organismus, das durch eben diese Geste einem anderen Organismus angezeigt wird. Wenn diese Geste einem anderen Organismus tatsächlich das nachfolgende (oder daraus resultierende) Verhalten anzeigen kann, dann hat sie auch Bedeutung. ... Bedeutung ist daher die Herausbildung von etwas objektiv Vorhandenem als eine Relation zwischen bestimmten Phasen des sozialen Handelns; es ist keine physische Ergänzung dieses Handelns und es ist ebensowenig eine „Idee“ im Sinne der traditionellen Auffassung.] Man braucht nicht gesondert darauf hinzuweisen, daß hier eine völlig andere Vorstellung von Sprache und folglich auch eine völlig andere Auffassung von Bedeutung vorliegt. Die 2
Sprache wird nicht als eine Menge von Ersatzes für Entitäten betrachtet, und die Bedeutungen sind keine ersetzten Entitäten. Vielmehr ist hier die Sprache ein Mittel zur Interaktion, und Bedeutung ist die Fähigkeit, diese Interaktion gewissermaßen zur „Resonanz zu bringen“. Auch der spätere Wittgenstein verwarf seine frühere „Abbildtheorie“ der Sprache zugunsten solch einer „pragmatistischen“ Auffassung. In seinen dem Traktat nachfolgenden Schriften ist die Sprache nun nicht mehr eine Menge von Abbildern, sondern es entsteht die Vorstellung von einer „Sammlung verschiedener Werkzeuge“: Die Sprache ist eben eine Sammlung sehr verschiedener Werkzeuge. In diesem Werkzeugkasten ist ein Hammer, eine Säge, ein Maßstab, ein Lot, ein Leimtopf und der Leim. Viele der Werkzeuge sind mit einander durch Form und Gebrauch verwandt, man kann die Werkzeuge auch beiläufig in Gruppen nach ihrer Verwandtschaft einteilen aber die Grenzen dieser Gruppen werden oft, mehr oder weniger, willkürlich sein; und es gibt verschiedenerlei Verwandtschaften, die sich durchkreuzen. (Wittgenstein, 1969, 67) Aus diesem folgt, daß die Bedeutung eines Ausdrucks kein Ding ist, das von ihm abgebildet wird, sondern daß sie vielmehr eine Art Nutzbarkeit dieses Ausdrucks darstellt, eine Art Fähigkeit, entsprechend unserer Absichten eingesetzt zu werden: Eine Bedeutung eines Wortes ist eine Art seiner Verwendung ... Die Bedeutung eines Worts vergleiche mit der ‘Funktion’ eines Beamten. Und ‘verschiedene Bedeutungen’ mit ‘verschiedenen Funktionen’. (Wittgenstein, 1984, §§ 61, 64) Diese zwei Paradigmen zur Herangehensweise an die Sprache will ich hier entsprechend das Carnapsche und das Deweysche Paradigma nennen. Prima facie sieht es so aus, als ob diese zwei Ansätze exemplarische Gegensätze bildeten: Während das erste auf der Annahme basiert, daß Sprache primär ein System von Namen oder Repräsentationen von Dingen bildet, nimmt das zweite an, daß Sprache eine Ansammlung von Mitteln ist, Dinge zu tun – und diese beiden Auffassungen von Sprache erscheinen als unvereinbar miteinander. Das Carnapsche Bild von Sprache hat letztendlich zur mathematischen Rekonstruktion von Sprache als einem System von Ausdrücken geführt, die homomorph auf ein System von Denotationen abgebildet werden (Montague, 1970), wohingegen die Deweysche Einstellung eher zu einer Lehre von ausweichenden „Sprachspielen“, Gedankenexperimenten mit „radikaler Übersetzung“ und „Interpretation“ zu führen scheint und schließlich zu einer vollständigen „Pragmatisierung der Semantik“ (Peregrin, 1999), die sich für keinerlei mathematische Formalisierung zu eignen scheint. In diesem Aufsatz möchte ich jedoch aufzeigen, daß die Kluft zwischen beiden Auffassungen von Sprache nicht vollkommen unüberbrückbar sein muß. Es stimmt zwar, daß die Konzeptionen von Sprache, die beiden Ansichten zugrundeliegen, kaum miteinander versöhnbar scheinen – was ich aber vorschlage, ist, daß der vom Carnapschen Ansatz hervorgebrachte technische Apparat mitsamt seiner Fülle von Resultaten in den Dienst des Deweyschen Paradigmas gestellt werden kann – wenn wir es denn von der Carnapschen repräsentationellen Ideologie befreien. Daher werde ich dafür argumentieren, daß der Beitritt zum Deweyschen Paradigma nicht zur Aufgabe von Carnaps und Montagues formaler Semantik führen muß.
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2. Bedeutung und Regeln Wenn wir Deweys Auffassung von Sprache oder auch die des späteren Wittgenstein als vorausgesetzt betrachten, dann sehen wir in der Bedeutung eines Ausdrucks in erster Linie eine semantisch relevante Rolle bzw. Funktion dieses Ausdrucks. Was genau ist aber „die semantisch relevante Funktion“ eines Ausdrucks? Wir sollten damit anfangen, uns die Funktion eines nichtlinguistischen Gebrauchsgegenstandes anzusehen, z.B. die eines Hammers. Allgemein gesprochen kann ein Hammer auf vielfältige Art und Weise verwendet werden, wobei wir einige dieser Verwendungsweisen als dem Hammers inhärente Funktionen betrachten, während uns andere eher zufällig und weniger „hammerspezifisch“ vorkommen. Ein Musterbeispiel für die erstere Art von Dingen, die wir mit einem Hammer tun, ist das Einschlagen von Nägeln, wohingegen es ein Musterbeispiel für die letztere Art von Verwendungen wäre, ihn z.B. als Briefbeschwerer zu benutzen. Natürlich können wir zwischen beiden Arten der Benutzung keine scharfe Grenze ziehen (ihre Beziehung ist vergleichbar mit der zwischen dem Zentrum und den Randbezirken einer Stadt) – doch deswegen lassen wir uns nicht davon abhalten zu denken, daß es eine für den Hammer charakteristische Funktion gibt. Die Tatsache nun, daß die Benutzung eines Ausdrucks ein ganz ähnliches Kontinuum von Fällen umfaßt, von solchen, die ganz klar als zentral für diesen Ausdruck aufgefaßt werden – durch eben jene Bedeutung, die der Ausdruck mit sich führt –, bis zu denen, die eher als peripher betrachtet werden, hat einige Denker zu einer Kette von Schlüssen geführt, die sich destruktiv auf das Carnapsche Paradigma auszuwirken scheinen: Wir können Semantik nicht von Pragmatik trennen. Deswegen können wir keine Grenzen von Bedeutungen angeben. Folglich gibt es für eine Semantik nichts zu erforschen. Ich denke, irgendwo in dieser Argumentation fangen wir an, das Kind mit dem Bade auszuschütten. Letztendlich gibt es keine „reinen“ Dreiecke in unserer realen Welt, doch trotzdem ist die Geometrie, in der man sich ausschließlich mit solchen „reinen“ Formen befaßt, zweifellos nicht nur ein ehrerbietiges Unternehmen, sondern auch nützlich für unser Verständnis von der Welt und für unser Handeln in ihr. Und auf ähnliche Art und Weise kann sich eine semantische Theorie, die mit klar abgegrenzten Bedeutungen arbeitet, als nützlich erweisen auch in einer Welt, wo man nur unscharfen Bedeutungen finden kann. Doch wie geht man die Rekonstruktion von unscharfen Bedeutungen an? Bedeutungen sind ja nichts Sichtbares, so wie die Formen von Dingen; wo also sind sie anzutreffen? Es war der spätere Wittgenstein, der nachdrücklich darauf hingewiesen hat, daß es wohl keinen anderen Weg der Annäherung an Bedeutungen gibt, als über die Regeln, die deren Ausdrücke regulieren. Es gibt, wie Wittgenstein bemerkte, einen charakteristischen Unterschied zwischen der Benutzung z.B. eines Hammers und der Benutzung eines Ausdrucks: Der Unterschied liegt darin, daß der Gebrauch eines Ausdrucks, im Gegensatz zu dem eines Hammers, regelgeleitet ist. Sicherlich gibt es auch Regeln für die Benutzung eines Hammers – aber diese unterscheiden sich doch wesentlich von denen, die für einen Ausdruck bestimmt sind. Die Gebrauchsregeln für Hämmer geben lediglich an, wie man sie benutzen muß, um das Ziel zu erreichen, für das sie konzipiert wurden. Sprachregeln hingegen sind von ganz anderer Art: Sie geben nicht den effektiven Gebrauch der Sprache an, sondern sind vielmehr der Sprache zugrundeliegend. Dies brachte Wittgenstein (1969, 184-5) zu der Schlußfolgerung, der Gebrauch der Sprache sei eher mit einem Schachspiel zu vergleichen als mit dem Benutzen eines Hammers oder, wie in dem Fall, mit dem Kochen: Warum nenne ich die Regeln des Kochens nicht willkürlich; und warum bin ich versucht, die Regeln der Grammatik willkürlich zu nennen? Weil ich den Begriff ‘Kochen’ durch den Zweck des Kochens definiert denke, dagegen den Begriff ‘Sprache’ nicht durch den 4
Zweck der Sprache. Wer sich beim Kochen nach andern als den richtigen Regeln richtet kocht schlecht; aber wer sich nach andern Regeln als denen des Schach richtet, spielt ein anderes Spiel; und wer sich nach andern grammatischen Regeln richtet, als etwa den üblichen, spricht darum nichts Falsches, sondern von etwas Anderem. Die Parallelen zwischen der Bedeutung eines Ausdrucks einerseits und der Funktion eines Hammers andererseits sind also begrenzt: Die Funktion von Ausdruck ist von anderer Art als die eines Hammers. Bei ersterem handelt es sich um eine Funktion, die einer Sache dadurch zukommt, daß sie sich Regeln ähnlich denen von Spielen unterordnet. Die Bedeutung eines Ausdrucks kann daher mit der Rolle einer Schachfigur verglichen werden. Eine Schachfigur, wie vielleicht ein Läufer, erhält ihre Rolle durch ihre Einbindung in die Regeln des Schachspiels (Wittgenstein, hier zitiert von Waisman, 1967, 105): Für Frege stand die Alternative so: Entweder haben wir es mit Tintenstrichen auf Papier zu tun, oder diese Tintenstriche sind Zeichen von etwas, und das, was sie vertreten, ist ihre Bedeutung. Daß diese Alternativen nicht richtig ist, zeigt gerade das Schachspiel: Hier haben wir es nicht mit den Holzfiguren zu tun, und dennoch vertreten die Figuren nichts, sie haben in Freges Sinn keine Bedeutung. Es gibt eben noch etwas drittes, die Zeichen können verwendet werden wie im Spiel. Alles zusammengenommen führte dies nun zu dem Schluß, daß wir uns auf die Regeln unserer Sprache konzentrieren müssen, wenn wir die Semantik der Sprache verstehen wollen, und daß wir ebenso herauszuarbeiten haben, was genau es eigentlich heißt, „einer solchen Regel innerhalb einer gegebenen Sprache zu folgen“. Diesem Problem widmete Wittgenstein den größten Teil seiner Philosophischen Betrachtungen (und später wurde es durch die in Kripke (1982)1 angestoßenen Diskussionen erneut ins Leben gerufen). Doch obwohl dies auch ein bedeutender Schritt in Richtung auf die Herausarbeitung semantisch relevanter Funktionen von Ausdrücken war, so bietet sich uns damit noch kein konkreter Anhaltspunkt: Das Problem, ein Kriterium zur Unterscheidung semantisch relevanter Regeln von anderen in der Sprache vorkommenden Regeln zu finden (wie z.B. Syntaxregeln, Regeln des sozialen Handelns usw.) ist keineswegs trivial.
3. Bedeutung und Inferenz* Wenn Wittgenstein recht hat, dann ist die Bedeutung eines Ausdrucks in den Regeln zu suchen, die den „semantisch relevanten“ Gebrauch des Ausdrucks bestimmen. Falls dem so ist, gibt es dann eine nicht zirkuläre Möglichkeit festzustellen, um was für eine Art von Regeln es sich dabei handelt? Oder enden wir vielleicht in einer Art von „semantischem Agnostizismus“? Der von uns vorgeschlagene Ansatz zur Beantwortung dieser Fragen geht auf einen anderen modernen Philosophen zurück, der die Schlüsselrolle von Regeln beim menschlichen Sprachverhalten hervorgehoben hat, nämlich Wilfrid Sellars. In seinen Arbeiten, die sich zwischen den Einflüssen des Wiener logischen Empirismus einerseits und denen des amerikanischen Pragmatismus andererseits bewegen, entwickelte er seine eigene Theorie zur
1
Für eine andere Sichtweise dieses Problems siehe auch Baker & Hacker (1984). Anmerkung des Übersetzers: Im Folgenden wird der in der englischen Version des Aufsatzes benutzte Ausdruck „inference“ einfach mit „Inferenz“ übersetzt. *
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grundlegenden Rolle von Regeln in Bezug auf Sprache und Bedeutung.2 Darüber hinaus erfaßte er mit größerer Präzision die Natur der für die Semantik so grundlegenden linguistischen Regeln, indem er diese Regeln als Regeln der Inferenz (im weiteren Sinne) beschrieb. Prima facie scheint das etwas schräg auszusehen: Warum Inferenzen? Benutzen wir Sprache denn nicht für viele wichtige Zwecke andere als Inferieren? Führt dies am Ende noch zu einer zu intellektualisierten Auffassung von Sprache? Nun, Sellars behauptet keineswegs, daß das Inferieren die Tätigkeit ist, bei der wir unsere Sprache am häufigsten gebrauchen – er behauptet nur, daß das Inferieren diejenige Aktivität ist, durch die unsere Ausdrücke mit Bedeutungen versehen werden. Warum? Für diesen Schluß benötigen wir drei Teilschritte. Zuerst einmal stellen wir fest, daß Sätze semantischen Vorrang gegenüber subsententiellen Ausdrücke besitzen und daß folglich die Bedeutungen von Sätzen, die Propositionen, Vorrang gegenüber den Bedeutungen anderer Ausdrücke haben: Die Bedeutung jedes Ausdrucks leitet sich her aus der Bedeutung eines Satzes. Zweitens beobachten wir, daß eine Proposition etwas ist, was sich im logischen Raum befindet, d.h., woraus sowohl andere Propositionen folgen, als auch aus anderen Propositionen folgt: Die Bedeutung eines Satzes ist im logischen Raum angesiedelt. Drittens können wir beobachten, daß eine Proposition diese Eigenschaften nur kraft der Regeln erhält, die den die Proposition zum Ausdruck bringenden Satz regulieren: Damit ein Satz solch eine Bedeutung erhalten kann, muß er in ein Netzwerk von Folgerungen eingebettet werden. Diese Teilschritte sollen nun im Einzelnen durchgegangen werden. 1. Auf den semantischen Vorrang eines Satzes gegenüber seinen einzelnen Teilen ist nachdrücklich von einer Anzahl von Philosophen hingewiesen worden, darunter Frege („Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas.“3) als auch Wittgenstein („Nur der Satz hat Sinn; nur im Zusammenhang des Satzes hat ein Name Bedeutung.“4). Der Grund dafür ist klar: Nur ein Satz kann für einen eigenständigen Kommunikationszweck gebraucht werden, und in diesem Sinn verfügt er über eine unabhängige Bedeutung (und folglich kann auch nur ein Satz wahr oder falsch sein) – subsententielle Ausdrücke hingegen erlangen ihre Bedeutung nur über ihre Einbindung in einen Satz. Nun gibt es ein augenfälliges Gegenargument: Die Grundbausteine der Sprache müssen von endlicher Zahl sein, und dies trifft zwar für Wörter zu, nicht aber für Sätze. Allerdings ist die Zahl der Sätze, denen wir während des Aneignens der Sprache begegnen, in der Tat nur endlich. Es stimmt, daß wir Sätze in Wörter zu zergliedern lernen, die wir dann wiederum als linguistische Grundbausteine benutzen – als wir in dem umgekehrten Prozeß Wörter zu neuen Sätzen zusammensetzen.5 Und haben wir einmal mit der Bildung komplexer Sätze begonnen, gibt es keine Obergrenze mehr – die Menge der Sätze wird (potentiell) unendlich. Doch dies alles widerspricht keinesfalls der Annahme, daß die Basissätze Vorrang haben – jedenfalls nicht mehr als die Tatsache, Rollen wie Mutter, Onkel usw. zu einer unbegrenzten Anzahl von möglichen Familienstammbäumen zusammensetzen zu können, der Tatsache widerspricht, daß wir diese einzelnen Rollen durch Abstraktion von einigen existierenden Stammbäumen gewonnen haben.6 2. Was ist dann eine Proposition (wenn überhaupt etwas)? Dies ist nicht leicht zu sagen, aber wenn sich Propositionen durch etwas auszeichnen, dann sollten es Eigenschaften sein, wie eine Negation zu haben, mit anderen Propositionen konjunktiv verbudnen werden zu können und besonders, andere Propositionen zu implizieren und ihrerseits durch andere Propositionen impliziert zu werden. Dies läuft darauf hinaus, daß ebenso, wie der modus 2
Siehe Sellars (1963; 1974); und für einen Überblick auch Marras (1978). Frege (1884:73). 4 Wittgenstein (1922, § 3.3.). 5 Siehe auch Quine (1960:9). 6 Für eine ausführlichere Diskussion siehe Peregrin (2001, Abschnitt 4.4). 3
6
vivendi physikalischer Objekte darin besteht, in kausale Zusammenhänge in Raum (und Zeit) zu treten, so auch der modus vivendi von Propositionen in ihren logischen Relationen besteht. Propositionen scheinen sich demnach im logischen Raum zu befinden, dessen Struktur aus den logischen Relationen besteht, darunter besonders aus der Relation der Folgerung, die ihren Ausdruck in den Inferenzregeln findet. 3. Wie auch immer, bevor wir in eine unhandliche Metaphysik hineingeraten, sollten wir die Rede von Propositionen eher als façon de parler auffassen, als daß wir sie mit der Beschreibung einer platonischen Sphäre gleichsetzen. (Das Problem liegt nicht im eigentlichen Platonismus, sondern in der Annahme, daß unsere Sätze durch die Benennung von Entitäten aus einer solchen Sphäre zu ihren inferentiellen Eigenschaften gelangen – wie konnten wir so etwas erreichen?) Und dies führt uns zu dem Schluß, daß unsere Sätze nicht deshalb über ihre inferentiellen Eigenschaften verfügen, weil wir sie als Namen für einige immerwährende Propositionen mit immerwährenden Folgerungsrelation benutzen. Vielmehr haben wir die Sätze dazu gebracht, diese Propositionen auszudrücken, indem wir sie von Inferenzregeln regulieren lassen, welche ein bestimmtes Netzwerk ausbilden und damit den nötigen logischen Raum bereitstellen. Die Idee von Sellars wurde von Brandom (1994, 2000) aufgenommen und weiterentwickelt. Die Inferenzregeln sind für ihn aus folgendem Grund entscheidend: Das hauptsächliche und „inhaltgebende“ Unterfangen, das unserer Sprache den grundsätzlichsten Sinn gibt, besteht letzten Endes darin, Gründe zu finden und zu suchen. Daraus ergibt sich, daß Sprache nur dann ihren Namen verdient, wenn sie zu diesem Unterfangen beitragen kann, und folglich kann etwas auch nur insofern ein wahrer bedeutungsvoller Ausdruck genannt werden, als es eben dazu beiträgt. Bandom selbst (1994:144) beschreibt es so: „It is only insofar as it is appealed to in explaining the circumstances under which judgements and inferences are properly made and the proper consequences of doing so that something associated by the theorist with interpreted states or expressions qualifies as a semantic interpretant, or deserves to be called a theoretical concept of a content.“ [„Was der Theoretiker mit interpretierten Zuständen oder Ausdrücken verbindet, eignet sich nur dann als semantischer Interpretant bzw. verdient es nur dann, das theoretische Konzept eines Inhaltes genannt zu werden, insofern es zum Erklären der Umstände benutzt werden kann, unter denen man passende Urteile und Schlüsse sowie die sich daraus ergebenden passenden Konsequenzen zieht.“] Die Semantik in den Griff zu bekommen heißt also, die inferentielle Struktur in den Griff zu bekommen; und folglich ist der Zugang zur Bedeutung eines bestimmten Ausdrucks der Zugang zu den Inferenzregeln, die diesen Ausdruck bestimmen. Daher auch die „inferentialistischen“ semantischen Erklärungen: „beginning with properties of inference, they explain propositional content, and in terms of both go on to explain the conceptual content expressed by subsentential expressions such as singular terms and predicates“ (Brandom, 2000:30). [Die inferentiellen Eigenschaften erklären den popositionalen Inhalt, und mit beide zusammen erklärt man die konzeptuellen Inhalte, die durch subsententielle Ausdrücke wie Singularterme und Prädikate ausgedrückt werden.]
4. Bedeutung als inferentielle Rolle Wie kann man Bedeutung als Sache von inferentiellen Rollen auffassen? Betrachten wir zuerst einen bezüglich Inferenzen einfachen Ausdruck, nämlich den logischen Junktor „und“. Es ist klar, daß seine Semantik durch eine der beiden folgenden Alternativen charakterisiert werden kann:
7
i) ii)
In Form von Denotation: „Und“ denotiert die Funktion f∧, die Paare von Wahrheitswerten auf Wahrheitswerte abbildet derart, daß f∧(T, T) = T, wobei f∧(T, F) = f∧(F, T) = f∧(F, F) = F. In Form von Inferenzen: „Und“ verbindet die Sätze A und B zu komplexen Sätzen A∧B derart, daß folgende Inferenzen gelten: (∧1) A, B ⇒ A∧B (∧2) A∧B ⇒ A (∧3) A∧B ⇒ B
Die folgenden zwei offensichtlichen Fakten zeigen, daß beide Formulierungen äquivalent sind: Fakt 1. Wenn „∧„ die Funktion f∧ denotiert, dann sind (∧1) - (∧3) korrekt (wahrheitserhaltend) Fakt 2. Wenn (∧1) - (∧3) gelten, dann kann „∧„ so aufgefaßt werden, daß es f∧ denotiert. ((∧1) besagt, daß wenn A und B wahr sind, A∧B auch wahr sein muß; wohingegen (∧2) und (∧3) besagen, daß A∧B falsch ist, wenn entweder A oder B falsch ist – und dies wird von f∧ korrekt erfaßt.) Kann man nun sagen, daß aus dem Gesichtpunkt der Natur der Sprache grundlegender als der andere ist? Sollten wir die Gültigkeit der Folgerungen auf das Vorkommen von Denotationen reduzieren („die Konjunktion stützt (∧1) - (∧3) weil sie f∧ denotiert“) oder eher andersherum („wir können die Bedeutung der Konjunktion als f∧ angeben, weil sie von (∧1) - (∧3) bestimmt wird)? Während der erste Fall in einer formalen Sprache durchaus möglich ist (da wir die Konjunktion durch Denotatszuweisung definieren können), hat die Behauptung „‘und’ denotiert diese oder jene Entität“, bezogen auf die natürliche Sprache, weiteren Klärungsbedarf. (Offensichtlich ist ja nicht gemeint, daß wir „und“ irgendwann in der Vergangenheit in unsere Sprache eingeführt haben, indem wir damit der Funktion f∧ einen Namen gaben. Aber wenn nicht dies, was ist dann damit gemeint?) Im Gegensatz dazu ist die Behauptung „zwei durch ‚und’ verbundene Sätze zu behaupten ist korrekt, gdw. es korrekt ist jeden einzelnen der Sätze zu behaupten“ völlig einsichtig. Ich denke, daß viele Philosophen, die dem Inferentialismus nicht unbedingt zustimmen, trotzdem zugestehen werden, daß die Bedeutung einer logischen Partikel wie „und“ in seiner inferentiellen Rolle besteht. (Ein wohlbekannter Gegner dieser Auffassung ist Prior, 1960/61, mit seinem tonk.7 Doch selbst wenn man zugesteht, daß nicht jede Menge von Inferenzregeln auf vernünftige Art und Weise als einer Bedeutung zugrundeliegend betrachtet werden kann, ist trotzdem nicht klar, warum dies mit der Behauptung unvereinbar sein sollte, daß diese bestimmte Menge der oben aufgelisteten Regeln der Bedeutung von „und“ zugrundeliegt.) Allerdings würden sie darauf bestehen, daß diese Generalisierung nicht für die wesentlichen Teile der Sprache möglich ist. Zugegeben, für den Inferentialisten wird die Sache unbequemer, wenn wir vom logischen Vokabular zu den empirischen Wörter übergehen. Doch sogar hier läßt sich der Inferentialismus aufrechterhalten. Nehmen wir z.B. das Wort „Kaninchen“. Gelangt er zu seiner Bedeutung ganz allein kraft seiner ihm zugekommenen Fähigkeit, auf Kaninchen zu referieren? Nun, was genau heißt es eigentlich, „auf Kaninchen zu referieren“? Im Beisein eines Kaninchens „Kaninchen“ verlautbaren zu lassen? Aber warum sagen wir dann nicht auch, daß das Bellen eines Hundes nach der Witterung eines Kaninchens auf Kaninchen referiert? 7
Zur Diskussion von Priors Auffassung siehe Peregrin (2002, Kapitel 8).
8
Es scheint zwei mögliche Antworten auf die Frage zu geben, was einen auf einen Gegenstand referierenden Ausdruck von einer bloßen Reaktion unterscheidet, die durch einen Gegenstand hervorgerufen wurde (und welche sowohl von Tieren als auch von unbelebten Gegenständen an den Tag gelegt werden kann8). Die erste Antwort ist, daß das Wunder wird von unserem Geist vollgebracht, der es irgendwie fertigbringt, das Wort so oder so mit Kaninchen(heit) zu verbinden, bzw. durch das Aufrufen der Vorstellung eines Kaninchens. Dies ist die Antwort, die in verschiedener Gestalt von so unterschiedlichen Philosophen wie John Locke, Edmund Husserl oder auch John Searle vorgetragen wurde, und sie stimmt wohl am ehesten mit dem „common sense“ überein. Diese Antwort hat das Problem, daß sie, wie Blackburn (1984, § II.3) es ausdrückt, eine „dog-legged theory“ ist: Anstatt das Problem zu lösen, schiebt sie es bloß einen Schritt weiter. Ihre Antwort auf die Frage Wie schaffen es unsere Ausdrücke zu referieren? besteht in der Angabe mentaler Inhalte, die referieren, was natürlich gleich die nächste Frage provoziert: Fein, aber wie bringen die das fertig?. Diese letzte Frage sollte sich nach der Theorie dann irgendwie von selbst beantworten. Wenn wir uns mit dieser Art von Antwort nicht zufriedengeben, können wir noch die andere Alternative betrachten. Diese ist prima facie weniger anziehend und bei weitem nicht so populär, aber nach meinem Dafürhalten durchaus der Betrachtung wert, besonders für den Pragmatiker. Nach dieser Antwort referiert ein Ausdruck dadurch, daß er innerhalb einer speziellen Form menschlicher (normativer) Praxis auf bestimmte Art und Weise benutzt wird. Varianten dieser Antwort wurden von Immanuel Kant, Ludwig Wittgenstein und kürzlich auch von Robert Brandom vorgelegt. Besonders Brandom hat angenommen, daß es sogar im Falle empirischer Begriffe und ihrer referentiellen Kapazitäten immer noch die inferentielle Artikulation ist, die entscheidend bleibt. Die Idee ist hier also, daß in bestimmter Hinsicht sogar der Inhalt empirischer Ausdrücke durch inferentielle Regeln, die ihren Gebrauch bestimmen, auf diese Ausdrücke übertragen wird. In solchen Fällen können wir uns natürlich nicht auf Inferenzen im üblichen, engeren Sinn beziehen – d.h., Schritte von Assertionen zu Assertionen – da wir auch „Inferenzen“ von Situationen zu Assertionen und umgedreht zulassen müssen. Wie dem auch sei, angenommen wird, daß uns auch für empirische Begriffe immer noch die beiden Möglichkeiten zur Charakterisation der Semantik bleiben: i) ii)
In Form von Denotation: „Kaninchen“ denotiert eine Art von ‚Kaninchenheit’, vielleicht anzugeben als eine Funktion, die mögliche Welten auf die Klassen ihrer Kaninchen abbildet. In Form von Inferenzen: „Kaninchen“ bildet Teil eines Satzes, derart daß die folgenden Inferenzen gelten: (r1) X ist ein Kaninchen ⇒ X ist ein Säugetier (r2) X ist ein Kaninchen ⇒ X ist kein Elephant ...
Die Äquivalenz dieser beiden Artikulationen könnte nun als Angelegenheit der folgenden beiden Fakte betrachtet werden: Fakt 1. Wenn ‚Kaninchenheit’ durch „Kaninchen“ denotiert wird, dann sind (r1), ... korrekt (da angenommen wird, daß Kaninchenheit Säugetierheit enthält, unvereinbar mit Elephantheit ist, …).
8
Wenn man z.B. ein Stück Metall betrachtet, dessen Reaktion auf Wasser durch den Rost sichtbar wird, oder auch ein Thermometer, das auf seine Umgebungstemperatur reagiert.
9
Fakt 2. Wenn (r1), … gelten und wenn die Aussage „Dies ist ein Kaninchen“ beim Zeigen auf ein Kaninchen angebracht ist, dann ist es korrekt anzunehmen, daß „Kaninchen“ Kaninchenheit denotiert. Es ist klar, daß die Situation hier unvergleichbar problematischer ist als im Fall des logischen Junktors. Zum einen ist die Ansammlung von Inferenzen zu komplex, um auf einfache Art spezifizierbar zu sein. (Es handelt sich dabei um eben jene Komplexität, die, wie Sellars argumentiert, uns von der Fähigkeit abhält, die Inferenzen, die der inferentiellen Rolle eines solchen Ausdrucks zugrundeliegen, explizit anzugeben. Deshalb geben wir für gewöhnlich die Rolle nur mithilfe eines bekannten Wortes an, das dieselbe oder zumindest eine ähnliche Rolle besitzt – was genau das ist, was normalerweise auch in der radikalen Übersetzung geschieht9). Zum anderen dürfen die inferentiellen Rollen nicht nur Inferenzen im engeren Sinn beinhalten (bei uns unter (r1), ... aufgelistet), sondern auch das, was Sellars „worldlanguage transitions“, also Übergänge von der Welt zu der Sprache genannt hat (zusammen mit den „language-world transitions“, also Übergänge von der Sprache zu der Welt). Dieser letzte Punkt könnte den Verdacht erwecken, daß der inferentielle Standpunkt eigentlich viel Lärm um nichts ist. Laufen denn die „Inferenzen im weiteren Sinne“, die für uns die Bedeutung eines empirischen Wortes wie „Kaninchen“ mitbegründen, nicht letzten Endes auf die Relationen der Referenz hinaus? Und sind wir dann nicht wieder beim denotationellen Paradigma angelangt, das wir eigentlich umschreiben wollten? Nicht wirklich. Die „Übergänge Welt-Sprache“ laufen nicht auf eine Relation zwischen Gegenständen und Wörtern hinaus, sondern vielmehr auf eine zwischen Tatsachen und Aussagen. Darüberhinaus ist entscheidend, daß sie die inferentielle Rolle eines Ausdrucks niemals vollständig erschöpfen können: Dem Inferentialisten zufolge kann nichts einen Inhalt haben, was nicht sowohl in den Prämissen als auch in den Konklusionen von Inferenzen vorkommen kann. Daher ist ein Satz wie „Dies ist ein Kaninchen“ (und folglich auch das Wort „Kaninchen“) nicht in der Lage, einen Inhalt ganz allein dadurch zu tragen, daß er eine richtige Antwort auf die Anwesenheit eines Kaninchens ist, und darüber hinaus muß der Satz als Prämisse für weitere Inferenzen dienen können (nämlich für „Also ist dies kein Elephant“, „Also ist dies ein Säugetier“ usw.).
5. Die Zirkularität der von der formalen Semantik bereitgestellten Erklärungen All dies scheint nahezulegen, daß wir das Carnapsche Paradigma zugunsten des Deweyschen aufgeben sollten, daß wir der denotationellen Sicht von Semantik und ihrem Auswuchs, der formalen Semantik, den Rücken kehren sollten. Und tatsächlich hat es seitens der Sprachphilosophen (mich selbst eingeschlossen) viel Kritik an der formalen Semantik gegeben. Entscheidend für den vorliegenden Aufsatz ist aber der Unterschied zwischen einerseits der (gerechtfertigten) Kritik an den Bestrebungen der formalen Semantik, eine Sprachphilosophie zu begründen und andererseits der (weniger gerechtfertigten) Kritik der formalen Semantik als technischem Instrument. Als Beispiel der ersten Art von Kritik können wir Davidsons (1999, 689) Einwand zur Situationssemantik von Barwise und Perry (1983) betrachten. Davidson behauptet, daß wir den Ausdruck „ist wahr“ nicht auf den Ausdruck 9
Es ist eigentlich nicht schwer, die Rolle und damit die Bedeutung eines unbekannten Äquivalents zu unserem „und“ zu spezifizieren, ohne uns dabei auf unser eigenes Wort zu berufen. Doch wenn wir die Rolle und damit auch die Bedeutung des fremden Äquivalents zu unserem „Kaninchen“ spezifizieren wollen, können wir das kaum anders tun, als indem wir anzeigen, daß das Wort dasselbe wie unser „Kaninchen“ bedeutet.
10
„drückt eine aktuelle Situation aus“ reduzieren können, da wir nicht in der Lage sind anzugeben, wann eine Situation aktuell ist, ohne daß wir angeben, was wahr ist: Barwise and Perry’s situations are set-theoretical constructs. Called „abstract situations“, they are defined in terms of (real) objects and properties and relations. Truth can’t be explained by reference to abstract situations, for abstract situations correspond to false sentences as well as true. Among the abstract situations are „actual“ situations, which do correspond in some sense to true sentences. So far this defines actual situations in terms of truth and not vice versa. Actual situations, however, „represent“ real situations, which are said to be „parts“of the world. Barwise and Perry never try to define „real situation“; they say that if you don’t think there are such things, they admit they don’t see how to persuade you. It is easy to specify when a particular abstract situation is actual: the abstract situation that I will call „Sam, mortality“ is actual if and only if Sam is mortal (that is, Sam instantiates mortality). Having determined what makes [Sam, mortality] actual, we can now „explain“ what makes the sentence „Sam is mortal“ true by saying it is true because „Sam is mortal“ corresponds to an actual situation. That situation is actual because Sam is mortal. It is obvious that we can retain the contents of this explanation, everything that „relates language to the real world“, by saying „Sam is mortal“ is true if and only if Sam is mortal; the apparatus of situations has done no work. The reason it has done no work is that truth must be brought in to explain the relation between Sam and mortality, something the semanics of situations fails to do. [Die Situationen von Barwise und Perry sind mengentheoretische Konstrukte. Sie werden „abstrakte Situationen“ gennant und in Form von (realen) Objekten, Eigenschaften und Relationen definiert. Wahrheit kann nicht durch Bezug auf abstrakte Situationen erklärt werden, da abstrakte Situationen sowohl falschen als auch auf wahren Sätzen entsprechen. Unter den abstrakten Situationen gibt es weiterhin „aktuelle“ Situationen, die in gewisser Hinsicht den wahren Sätzen ensprechen. Andererseits „repräsentieren“ aktuelle Situationen die wirklichen Situationen, die ihrerseits als „Teile“ der Welt aufgefaßt werden. Barwise und Perry machen niemals den Versuch einer Definition von „wirklichen Situationen“, eher müssen sie eingestehen, daß sie niemanden, der nicht an sie glaubt, von deren Existenz überzeugen können. Man kann leicht angeben, wann eine Situation aktuell ist: Die abstrakte Situation, die ich „Sam, Sterblichkeit“ nennen werde, ist aktuell genau dann, wenn Sam sterblich ist (d.h., Sam verkörpert Sterblichkeit). Nach der Angabe dessen, was [Sam, Sterblichkeit] aktuell macht, können wir nun erklären, was den Satz „Sam ist sterblich“ wahr macht, indem wir sagen, er sei wahr, weil „Sam ist sterblich“ einer aktuellen Situation entspricht. Diese Situation ist deswegen aktuell, weil Sam sterblich ist. Offensichtlich können wir den Gehalt dieser Erklärung, alles was sich „auf die wahre Welt bezieht“, beibehalten, indem wir sagen, „Sam ist sterblich“ ist wahr genau dann, wenn Sam sterblich ist; die Vorgabe von Situationen leistet dazu keinen Beitrag. Der Grund, warum sie keinen Beitrag leistet, liegt darin, daß man Wahrheit zur Erklärung der Relation zwischen Sam und der Sterblichkeit einbringen muß, etwas, was die Situationssemantik nicht schafft.] Ich selbst habe einen ähnlichen Einwand erhoben (Peregrin, 1995; 2000): Wir können „ist notwendig“ nicht auf „gilt in allen möglichen Welten“ reduzieren, da wir nicht angeben können, welche Welten möglich sind, ohne daß wir angeben, was notwendig ist. Denn angenommen, jemand möchte wissen, warum eine Behauptung wie ¬(P(a)∧¬P(a)) logisch wahr ist. Wir können dann schlecht antworten „Sie gilt in allen (Modellstrukturen erfassenden) möglichen Weltzuständen – ich habe sie alle sorgfältig untersucht und keinen 11
einzigen Weltzustand gefunden, in dem sie nicht gilt“. Unsere Antwort müßte eher von der Art „nichts kann zugleich P und nicht-P sein“ sein, oder vielleicht auch „[‚]P und nicht-P[‘] ergibt keinen Sinn“. Damit zeigt sich die Schwierigkeit daran festzuhalten, eine logische Wahrheit sei wahr, weil sie in allen möglichen Strukturen gültig ist – statt dessen wäre es angebrachter, die Sache genau andersherum zu betrachten: Weil etwas eine logische Wahrheit ist, kann es keine Struktur geben, in der sie nicht gilt10. Diese Einwände zeigen, daß wir mit aller Wahrscheinlichkeit in einen Zirkelschluß geraten, wenn wir die formale Semantik als Grundlage einer Sprachphilosophie verwenden wollen: Wir reduzieren philosophisch problematische Begriffe auf scheinbar glasklare Begriffe der formalen Semantik, welche letzten Endes jedoch auf den zu erklärenden und obskuren Begriffen beruhen. Wenn wir daher den Begriff der Wahrheit durch die Vorstellung der Aktualität von Situationen klären wollen, sehen wir uns mit der Tatsache konfrontiert, daß letzteres nicht anders als durch direkten oder indirekten Rückgriff auf den Begriff der Wahrheit verständlich gemacht werden kann; und wenn wir Notwendigkeit auf Gültigkeit in jedem Modell oder in allen möglichen Welten reduzieren wollen, müssen wir feststellen, daß die Einschränkung des Raumes relevanter Modelle oder möglicher Welten letztendlich auf dem Begriff von Notwendigkeit beruht. Für sich allein ist formale Semantik demzufolge nicht als formale Metaphysik zu verwenden, sie ist kein Mittel, um damit zu einer Korrespondenztheorie der Wahrheit zu gelangen, noch zu irgendwelchen anderen philosophischen Zielen. Und trotzdem ergibt sich für mich daraus weder die Nutzlosigkeit des Versuches, Bedeutungen mit Hilfe von möglichen Welten und Situationen zu rekonstruieren, noch die Überzeugung, daß die formale Semantik ein mißglücktes Unterfangen ist. Speziell bin ich der Überzeugung, daß jegliche Art von Gebrauch der Instrumentarien der formalen Semantik uns weder zum denotationellen Ansatz zur Bedeutung, noch zur Korrespondenztheorie der Wahrheit mit einer ‚formalen Metaphysik’, noch zu irgendeiner anderen ‚Ideologie’ verpflichtet. Ich denke, daß die Ergebnisse der formalen Semantik durchaus in den Dienst der inferentiellen Sprachphilosophie gestellt werden können.
6. Formale Semantik als Darstellung inferentieller Rollen Richten wir unser Augenmerk nun auf die möglichen Welten, die wahrscheinlich am häufigsten diskutierten Geschöpfe der formalen Semantik. Was genau sind sie eigentlich? Da gibt es zum einen, was wir die „Standardantwort“ nennen könnten: Mögliche Welten sind Entitäten, deren Existenznachweis und Zustandsbeschreibung durch eine von der Sprachtheorie unabhängige Theorie (‚Metaphysik’) erbracht werden muß; und darauf kann dann die formale Semantik aufbauen, d.h., sie kann die Bedeutungen von Aussagen mit dem Hinweis darauf angeben, daß sie Mengen solcher Welten denotieren11. Es gibt allerdings auch eine inferentialistische Alternative, die folgendermaßen aussieht: Durch die logischen 10
Sicherlich können wir manchmal auch entdecken, das etwas in allen Strukturen einer bestimmten Klasse gilt – aber wenn diese Klasse nicht von endlicher Größe ist, ist uns dies kaum durch das Betrachten aller Strukturen möglich; wir müssen es dann auf irgendeine Art und Weise von verschiedenen konstitutive Eigenschaften der Klasse deduzieren. 11 An anderer Stelle (Peregrin, 1998) habe ich darauf hingewiesen, daß diese Antwort in eine philosophische Sackgasse führen kann: Während die Linguisten die Natur möglicher Welten gerne von den Philosophen erklärt hätten, denken wiederum diejenigen Philosophen, die die lingustische Wende vollzogen haben, daß die Möglichkeit zu einer Erklärung dieser „metaphysischen“ Entitäten in einer linguistischen Analyse unserer Sprache liegt, von welcher sie (explizit oder implizit) hervorgebracht werden.
12
Standardoperatoren (wie Negation, Konjunktion, …) mit ihren inferentiellen Eigenschaften werden die Aussagen unserer Sprache inferentiell in einer Booleschen Algebra angeordnet. Damit sind sie nun auch so darstellbar, daß sie Teilmengen einer bestimmten Menge denotieren (kraft des Repräsentationstheorems von Stone12). Weiter gilt, daß diese zugrundeliegende Menge durch das Vorhandensein modaler und kontrafaktischer Operatoren und Ausdrücke nicht die einfachste, also die einelementige Menge, sein kann.13 Folglich denotieren Aussagen Teilmengen einer nicht einfachen Menge – und als man sich die Elemente dieser Menge als „das, wovon die Wahrheit abhängig ist“14 vorstellen kann, ist es ganz plausibel, sie mögliche Welten zu nennen. Auf diese Art können wir sagen, daß mögliche Welten – genau wie andere Entitäten der formalen Semantik auch – als Darstellung inferentieller Muster betrachtet werden können. Aus bestimmten Gründen, die mit der Entwicklung unseres Geistes zu tun haben, scheint uns die Relation des Enthaltens zugänglicher als andere Relationen zu sein. Und mögliche Welten erlauben uns, Inferenzen in ‚Enthalten’ umzuwandeln: Nämlich das Enthalten der Schnittmenge von Klassen möglicher Welten, welche durch die Sätze im Antezedenten einer Inferenz denotiert werden, in der Klasse von Welten, die durch den Konsequenten der Inferenz denotiert werden. Weiter gilt, daß inferentielle Muster für gewöhnlich mehr als nur einen Ausdruck enthalten, wohingegen wir in der Semantik üblicherweise an der Bedeutung eines einzelnen Ausdrucks interessiert sind, d.h., am wirklichen Beitrag dieses einzelnen Ausdrucks zu dem Muster, von dem er bestimmt wird. Und wenn wir diesen Beitrag als ein abstraktes, typischerweise mengentheoretisches Objekt herausarbeiten, so scheint uns dies ein guter Weg zu sein, uns eben diesen Beitrag verständlich zu machen. Erinnern wir uns an das Paradebeispiel der logischen Konjunktion: Das inferentielle Muster, durch das sie bestimmt wird, ist ziemlich einfach und durchsichtig, doch nichtsdestoweniger ist es hilfreich, wenn man die Konjunktion als Wahrheitsfunktion wiedergibt. Formale Semantik kann daher, wie ich denke, selbst für den Inferentialisten von einiger Hilfe sein. Wenn der Inferentialist die Behauptung zurückweist, Bedeutungen seien im Grunde genommen Objekte, die durch Ausdrücke wiedergegeben werden, so muß dies nicht gleich unvereinbar mit der Behauptung sein, daß die Bedeutungen als Objekte explizierbar oder modellierbar sind. Und der Inferentialist sollte sich nach meinem Dafürhalten klar darüber werden, daß solch eine Nachbildung eine äußerst nützliche Sache sein kann. Für mich ist es daher von Nutzen, wenn man sich Sprache hin und wieder als einen Code vorstellt, auch wenn sie rein wörtlich keine Nomenklatur oder kein Code (wie es das Carnapsche Paradigma annimmt) ist, genauso wie es oftmals von Nutzen ist, sich Atome als von Elektronen umkreiste Kerne vorzustellen. Von dieser Seite betrachtet hört die Denotation natürlich auf, Gegenstand einer semantischen Theorie zu sein und wird eher zu ihrem Werkzeug. Das Objekt, welches durch einen Ausdruck denotiert werden soll, wird nicht als (rekonstruierte) echte Entität aufgefaßt, die durch diesen Ausdruck in der wirklichen Welt denotiert wird, sondern vielmehr als Wiedergabe der inferentiellen Rolle dieses Ausdrucks. (Als wichtige Konsequenz daraus ergibt sich, daß es nicht „die“ richtige Semantik gibt, genauso wie es weder das richtige Modell vom Innern eines Atoms gibt, noch der richtige Grundriß einer unbekannten Stadt). Sellars (1992:109 f.) gelangt zu folgendem Urteil:
12
Dieses Theorem sagt aus, daß jede Boolesche Algebra isomorph zur Algebra von Teilmengen einer Menge ist. 13 Deren Potenzmenge zwei Elemente enthält, die mit den beiden Wahrheitswerten identifiziert werden können. 14 Vgl. Stalnaker (1986).
13
[Carnap’s formalization of semantic theory in terms of a primitve relation of designation which holds between words and extralinguistic entities] commits one to the idea that if a language is meanigful, there exists a domain of entities (the designata of its names and predicates) which exist independently of any human concept formation. [(Carnaps Formalisierung der semantischen Theorie in Form von primitiven Relationen der Bezeichnung, die zwischen Wörtern und extralinguistischen Entitäten besteht) verpflichtet uns zu der Idee, daß es für den Fall, daß ein Ausdruck bedeutungstragend ist, einen Bereich von Entitäten gibt (die Designata ihrer Namen und Prädikate), die unabhängig von jeder menschlichen Begriffsbildung existieren.] Aber aus der hier in Betracht gezogenen Perspektive müssen wir widersprechen: Was Sellars ignoriert, ist die Möglichkeit, die Carnapsche „Formalisierung der semantischen Theorie“ weniger als unmittelbare Beschreibung, sondern vielmehr als eine „kreative“ Explikation der semantischen Aspekte von Sprache zu betrachten. Wir vertreten also den Standpunkt, daß es die Anschaffung von Instrumenten aus der formalen Semantik nicht erforderlich macht, sich eine Ideologie ins Haus zu schaffen, die formale Semantik zugleich als eine eigenständige Sprachphilosophie ansieht.15 Die Moral für den Inferentialisten dabei ist, daß man die Carnapsche Methode der Rekonstruktion der semantischen Aspekte von Sprache nicht verachten, sondern daß man sie vielmehr in Deweys Sinn verstehen sollte: Als Methode der Aussonderung und Darstellung der Beiträge eines jeden Ausdrucks, die von ihnen zu den Inferenzen, in denen sie vorkommen, beigesteuert werden.
Referenzen Baker, G. P. & Hacker, P. M. S. (1984): Scepticism, Rules and Language, Blackwell, Oxford. Barwise J. & Perry, J. (1983): Situations and Attitudes, MIT Press, Cambridge (Mass.). Blackburn, S. (1984): Spreading the Word, Clarendon Press, Oxford. Brandom, R. (1994): Making It Explicit, Harvard University Press, Cambridge (Mass.). Brandom, R. (2000): Articulating Reasons, Harvard University Press, Cambridge (Mass.). Davidson, D. (1999): ‘Reply to J. Higgenbotham’, in L.E. Hahn (ed.): The Philosophy of Donald Davidson (Library of Living Philosophers), Open Court, La Salle.
Frege, G. (1884): Grundlagen der Arithmetik, Koebner, Breslau. Kripke, S. (1982): Wittgenstein on Rules and Private Language, Harvard University Press, Cambridge (Mass.). Marras, A. (1978): ‘Rules, Meaning and Behavior: Reflections on Sellars’ Philosophy of Language’, in The Philosophy of Willfrid Sellars: Queries and Extensions (ed. J.C. Pitt), Dordrecht, Reidel, 163-187. Montague, R. (1970): ‘Universal Grammar’, Theoria 36, 373-398; reprinted in Montague: Formal Philosophy: selected papers of R. Montague (ed. R. Thomason), Yale University Press, New Haven, 1974. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Peregrin, J.: ‘Linguistics and Philosophy’, Theoretical Linguistics 25, 1998, 245-264. Peregrin, J. (1999): ‘The Pragmatization of Semantics’, in The Semantics/Pragmatics Interface from Different Points of View (ed. K. Turner), Elsevier, North-Hollard, 419-442. Peregrin, J. (2000): ‘The „Fregean” logic and the „Russellian” logic’, Australasian Journal of Philosophy 78, 557-575. Peregrin, J. (2001): Meaning and Structure, Ashgate, Aldershot. 15
Siehe Peregrin (2001, Kap. 8).
14
Prior, A. N. (1960/61): ‘Roundabout Inference Ticket’, Analysis 21, 38-39. Quine, W.V.O. (1960): Word and Object, MIT Press, Cambridge (Mass.). Reichenbach, H. (1947): Elements of Symbolic Logic, Free Press, New York. Russell, B. (1912): The Problems of Philosophy, Home University Library. Sellars, W. (1963): Science, Perception and Reality, Routledge, New York. Sellars, W. (1974): ‘Meaning as Functional Classification’, Syntèhse 27, 417-437. Sellars, W. (1992): Science and Metaphysics, Ridgeview, Atascadero. Stalnaker, R. (1986): ‘Possible Worlds and Situations’, Journal of Philosophical Logic 15, 109-123. Wittgenstein, L. (1922): Tractatus Logico-Philosophicus, Routledge, London. Wittgenstein, L. (1953): Philosophische Untersuchungen, Blackwell, Oxford. Wittgenstein, L. (1969): Philosophische Grammatik, Suhrkamp, Frankfurt.
15
WHAT DOES ONE NEED, WHEN SHE NEEDS "HIGHER-ORDER LOGIC"? Jaroslav Peregrin1 www.cuni.cz/~peregrin [Proceedings of LOGICA'96, FILOSOFIA, Praha, 1997, 75-92.]
1. First-Order Predicate Calculus Formal languages, which are the medium of modern formal logic, have reached their definitive form during the first part of this century. The most substantial of them, the language of predicate
calculus, is characterized by three types of syntactic rules. Two of these rules reflect, quite straightforwardly, basic syntactic structures of natural language: (i) the fact that an elementary natural language sentence typically consists of a verbal phrase complemented by several nominal phrases, and (ii) the fact, that sentences can be negated and joined into more complex sentences by means of certain connectives. Hence the rules (i): An n-ary predicate plus n terms yield an (elementary) statement. (ii) An n-ary logical operator plus n statements yield a statement. The third kind of rule is of a different sort: it reflects no such general syntactic structure of natural language, but rather the structure of certain specific judgments which we can make about language. If we turn an expression into a „matrix“ (a general scheme) by substituting variables - in the role of gap-markers - for some of its parts, we can then imagine the gaps filled with various concrete things (and the matrix thus being turned back into a statement) and scrutinize in which cases the resulting statement holds, and in which it does not not. On the basis of such observations we formulate judgments such as „the matrix M yields a true statement independently of whatever we fill its gaps“ and „the gaps in M can be filled by something to yield a true statement“; or simply „for every x, N“ and „for some x, M“2. This leads to an intuition (which is, in contrast to the previous two, very specific), that a statement might consist of a quantifier (stating whether we are talking about all possibilities, about the existence of at least one possibility, or, as the case may be, about some other pattern of existence of possibilities), a variable (stating which kind of gap the assertion is about), and a matrix: (iii) A quantifier plus a variable plus a statement yield a statement (where we suppose that a variable is also a term and hence can enter the rule (i)). This last intuition, however, has not always been understood and explicated quite uniformly: differences have occurred especially in whether variables can legitimately replace only terms, or also other kinds of expressions, particularly predicates. This ambiguity lead to the constitution
1
Thanks to V. Švejdar, P. Kolář a P. Materna for inspiring comments on a previous draft of the paper. 2
This view of quantification is discussed in Peregrin (1995).
of first-order predicate logic as a self-contained subsystem of general predicate logic3. The firstorder paradigm means not only that we have no other variables than those substitutable for terms, but also that we cannot have a mechanism which would allow for a higher-order quantification in an indirect way.4 Due to all the nice properties of this subsystem (completeness, compactness, Löwenheim-Skolem property), many logicians are convinced, that logic should concentrate on this system alone. Besides the exponents of first-order logic, however, there are also many logicians regarding this as too restrictive. Such voices are heard not only from the camp of those involving themselves with the analysis of natural language (and thus requiring a far richer repertoire of syntactic means than can be offered by the first-order predicate calculus), but also from among those who are concerned with the foundations of mathematics (Barwise a Feferman, 1985; Shapiro, 1991). Unfortunately it seems that many discussions around this theme suffer both from the fact that their participants sometimes fail to acknowledge all the facts about the relationship between first-order and higher-order logic, and also from the fact that terms like higher-order logic are employed in various different senses. The aim of this contribution is to clarify what is, and what is not, a legitimate subject to such discussions: it thus presents nothing essentially new, but it assembles some available relevant facts in a way which the author finds useful and which he has found lacking in the current literature.
2. Beyond the border of first-order logic When you ask someone why she needs a logic of an order higher than one, why she cannot make do with first-order logic, she is likely to respond that she needs a higher expressive power, than first order-logic can offer. However, this may be ambiguous. It may be interpreted as expressing (i) the need of a syntactically richer language; which can, in turn, represent either (i.i) the simple need of higher-order predicates; or (i.ii) the need to quantify over the corresponding variables. However, the response might also be interpreted amounting to (ii) the need of a logic which would allow us to articulate some of the concepts which are not expressible within firstorder logic, e.g. the concept of finiteness. These motifs, however interconnected they may be, cannot be identified. Let us discuss them in greater detail. The need (i.i) may be motivated by an effort to apply a straightforward logical analysis to natural language statements like (1) or (2); the difficulty being that in order to form the formula (1'), we need the second-order predicate La, and in order to form (2'), we need the ‘predicate’ Qu, whose application to a predicate yields again a predicate. To be brave is laudable
(1)
3
See Moore (1988).
4
Such mechanism might be e.g. a rule of comprehension, which guaranteed that each predicate (or its denotation) possesses its unique ‘objectual correlate’: then we could emulate the quantification over predicates via quantification over their correlates. It was precisely this kind of mechanism which made the first systematic articulation of formal logic, presented by Frege, into more than first-order logic. (In fact, it was also one of the reasons why Frege’s system was contradictory, as pointed out by Russell.).
Charles runs quickly La(Br) (Qu(Ru))(Ch)
(2) (1') (2')
Similarly we may want to capture some mathematical concepts by means of second-order predicates; we may, e.g., want to regiment (3) as (3'), where Pr is a unary predicate. There are infinitely many primes Inf(Pr)
(3) (3')
The need (i.ii) may be motivated by the effort of capturing the statements like (4) and (5) as something like (4') and (5'): Charles and Peter share a property To do something quickly is not to do it slowly ∃p.p(Ch)&p(Pe) ∀p∀ ∀x.(Qu(p))(x)→ →¬(Sl(p))(x)
(4) (5) (4') (5')
Similarly, the Dedekind definition of an infinite range, (6), can be straightforwardly regimented as (6'), while the axiom of induction, (7), as (7'). The range P is infinite, if it can be injectively mapped on its proper subset. (6) Inf(P)≡∃ ≡∃f.∀ ∀y(fx=fy→ →x=y)&∀ ∀x(P(x)→ →P(fx))&∃ ∃y(P(y)&∀ ∀x(P(x)→ →fx≠y)) (6') ≡∃ ∀x∀ If 0 has a property, and if, moreover, a successor n' of every number n has the property if n has the property, then every number has the property. (7) ∀p.(p(0)&∀ ∀n(p(n)→ →p(n')))→∀ →∀n.p(n) (7') →∀ In contradistinction, the need (ii) then can be motivated simply by the desire for a logic in which it would be possible to express concepts which are arguably not expressible within first-order logic (infinity, for instance), but not necessarily by means of an explicit definition in the object language (such as (6')). If we accept the model-theoretic notion of logic (viz. Barwise & Feferman, 1985), we can legitimately define, for example, the quantifier ∃∞ by the following metalinguistic prescription (where ║...║[x←d] is that interpretation which differs from ║...║ at most in that ║x║[x←d]=d): ║∃ ∃∞xP(x)║ = 1 iff there is an infinite number of such objects d that ║P(x)║[x←d] = 1 Such a definition does not necessarily carry us over the borders of the syntax of firstorder logic - ∃∞ is an expression of the very same syntactic category as ∃ and ∀, and to introduce new expressions of this category does not cause any paramount problems for the firstorder framework (viz. ∃!). This indicates that (i) and (ii), however interconnected, differ substantially in character. In the case of (i) the requirement is a larger repertoire of syntactic means, which alone need not involve any nontrivial step beyond the boundaries of first-order logic. For there is a strategy for both directly "emulating" such means within the framework of first-order logic, or alternatively for extending the language of first-order logic so as to bring us the needed syntactic means
without crossing the boundaries of first-order logic. Let me now sketch two variants of the strategy for reducing higher-order quantification to first-order one.
3. Predicates as individuals The first of the variants stems from the conviction that a subject of predication is always necessarily an individual. Frege (1892, s.197) says: "[der Begriff] kann wegen seiner prädikativen Natur nicht ohne weiteres [als Bedeutung des grammatischen Subjekts] erscheinen, sondern muβ erst in einen Gegenstand verwandelt werden, oder, genauer gesprochen, er muβ durch einen Gegenstand vertreten werden." This means that what we may see as a property of properties applied to a property should be viewed as a property of individuals applied not to a property, but rather to some kind of its "objectual correlate" (its extension, within Frege’s framework). In natural language the situation is indeed such that a predicate can be conjoined only with a nominal form (nominalization) of another predicate (typically a verbal noun, an infinitive, or a gerund). This consideration leads to the situation that, for example, the statement (1) is regimented as the application of the predicate La to the term BBr, which denotes the „objectual correlate“ of the predicate Br. Obviously, however, the systematic relationship between predicates and their objectual correlates is logically significant: the inference such as (8) is generally valid. Charles is brave To be brave is laudable ──────────────── hence Charles has a laudable property
(8)
Such inferences are nevertheless easy to render: it is only necessary to take the predicate to have a property seriously (to regiment it as a binary predicate constant), and further to understand BBr not as a primitive term, but rather as a „nominalizing“ operator B applied to the predicate Br - i.e. to understand BBr as B(Br). Then we can articulate the general inference rule P(T) ──────── hence HasPr(T,B(P));
(9)
and regiment the inference (8) with its help: Br(Ch) La(B(Br)) ───────────── hence HasPr(Ch,B(Br))&La(B(Br)) and hence ∃x.HasPr(Ch,x)&La(x)
(8')
An operator such as B cannot, of course, be squeezed directly into the first-order framework; but nevertheless its introduction requisites a modification of the framework other than the introduction of higher-order predicates. Problems connected with operators of this kind, and,
more generally, problems concomitant with logical analysis of the phenomenon of nominalization in natural language are discussed in detail by Chierchia (1982) and Turner (1983).5 A variation on the same theme is the Davidsonian approach to the regimentation of sentences of type (2): Davidson (1980) proposes to enrich each predicate by a new, in natural language covert, argument place, fillable with something like „events“: the statement (2) will thus be understood as There is an "event of running" the agent of which is Charles and this event is quick (see also Parsons, 1990). ∃e.Ru(e,Ch)&Qu(e).
(2'')
In a certain sense, both model theory and set theory can be seen as general expressions of this strategy: a model-theoretic interpretation of a formal language can be seen as a sort of a translation of this language into the language of set theory - hence into a first-order language.6 For the usual model theory can be seen as - in effect - a means of translating, e.g., the statement P(T) into the "metastatement" ║T║∈ ∈║P║ (similarly, with some insubstantial complications, for predicates of higher arities); and hence the reduction of the truth of the former to the truth of the latter - if we understand ║...║ simply as a nominalizing device (so that ║T║=T, for T is nominal in itself, and ║P║=B(P)), and if we write HasPr instead of ∈, we turn ║T║∈ ∈║P║ into our 7 familiar HasPr(T,B(P)).
4. Henkinian Understanding of Higher-Order Logics The second variant of this strategy is based on the idea of accepting the syntactic means of higher-order logics without any limitations, but semantically interpreting them in the spirit of first-order logic; i.e. taking them as mere ‘notational variants’ of first-order means. If we concentrate on second-order logic, this strategy amounts, informally speaking, to taking relations as a peculiar kind of individuals (the relations will hence be elements of the domain of individuals). This causes the expression P(T) to be seen as expressing a relation between two individuals: the relation-taken-as-individual ║P║ and the ("classical") individual ║T║. In this way, quantification over relations becomes quantification over a certain kind of individuals. The interpretation of a second-order language consists of a universe U and an interpretation function which maps individual constants on the elements of U and predicate constants on the relations over U; the range of the individual variables is then U and the ranges of predicate variables are the corresponding sets of relations (subsets of Cartesian powers of U). Hence the difference between a first-order interpretation and a second-order one is that the latter works not only with the range U, but also with the ranges Pow(U), Pow(U2), ... . Nevertheless, multiple ranges can also be readily accommodated even within first-order semantics: directly within sorted first-order logic (which is a straightforward and formally unproblematic variety of 5
See also Peregrin (1990).
6
There is, of course, also a higher-order set theory, but model theory is usually considered as based on the first-order version. 7
See also Peregrin (1992).
standard first-order logic, in which we have terms of multiple categories and hence multiple universes of individuals, or multiple compartments of the single universe), and indirectly even within standard (unsorted) first-order logic; the idea is that of „modeling“ the various ranges as various parts of the single universe. This can be done in such a way that the quantification over a specific range is replaced by the quantification over the whole universe, but each quantified formula is interpreted as a conditional whose antecedent restricts the quantification to that part of the universe which models the range in question: ∀p.p(x) is thus interpreted as ∀y.P(y)→ →PR(y,x), where P is the characteristic function of that part of the universe which models the range of unary predicate variables, and PR is the binary predicate which renders predication as the relation between a pair of individuals. Each second-order interpretation thus straightforwardly „induces“ a certain first-order interpretation, and the relation between the inducing and the induced preserves satisfiability. We may single out a certain class of first-order interpretations which are of the kind of those induced by second-order interpretations and call them quasisecond-order interpretations. Such a class can be characterized by a certain first-order theory; however, doing this suscitates the problem that although each second-order interpretation induces a quasisecond-order interpretation, not every quasisecond-order interpretation is induced by a second-order interpretation. Hence there will be no guarantee that each formula valid under every secondorder interpretation will be valid also under every quasisecond-order interpretation; and Gödel’s theorem implies that there will indeed be formulas, which are second-order valid, but not quasisecond-order valid. It is interesting that instead of interpreting second-order logic in the first-order way we can equally accept second-order interpretations in which the ranges of the predicate variables need not necessarily contain all the relations of the corresponding arities. The point is that it can be easily proved that there is a one-to-one correspondence between these so-called Henkinian interpretations8 and quasisecond-order interpretations. It is also worth noting that the difference between this kind of „reducing“ second-order logic to the first-order one, and the procedure discussed in the previous section is in fact only an „ideological“ one: while in the previous case we first translated the second-order language into a first-order language which we then interpret in the normal way, in the present case these two steps get mingled together: second-order logic is directly interpreted in the first-order way, and there is no intervening first-order language. (Neither do we introduce the problematical nominalization operator of the kind of B of the previous section.) In the previous section we said that we first „translate“ predicate into terms and then interpreted these by individuals; now we say that we interpret predicates directly by individuals - the difference is obviously not substantial.
5. The Principles Of Second-to-First-Order Translation Let us analyze more closely how higher-order logic is reduced to lower-order logic - we sketch an algorithm for transforming any second-order language into a first-order one and each secondorder theory into a first-order one. For simplicity we restrict ourselves to monadic second-order 8
According to Henkin (1950).
logic, i.e. to second order logic the language of which contains no predicates of any arity greater than 1; and we shall further limit ourselves to languages not containing functors. The language of monadic second-order predicate calculus (MPC2) thus consists of individual and predicate constants (ic, pc), individual and predicate variables (iv, pv), logical operators and quantifiers. The language of two-sorted first-order predicate calculus (PC1(2)) has no predicate variables, but rather only individual ones, and its individual constants and variables are divided into two sorts (hence we have ic1, ic2 and iv1, iv2). Let us now have a language L1 of MPC2. Let us construct the language L2 of PC1(2) in such a way that: - the set of ic1 of L2 is identical with the set of ic of L1 - the set of ic2 of L2 is identical with the set of pc of L1 - the set of iv1 of L2 is identical with the set of iv of L1 - the set of iv2 of L2 is identical with the set of pv of L1 - the set of pc of L2 contains a single expression, the binary pc PR of the type <2,1> (i.e. such that it yields a statement together with a term of the sort 2 and a term of the sort 1). Let us define, by induction, the translation of expressions of L1 into those of L2 - if X is an expression of L1, let us denote its translation into L2 as X*: X* (P(T))* (F1 & F2)* (F1 ∨ F2)* (F1 → F2)* (¬F)* (∀ ∀xF)* (∀ ∀pF)*
= X if X is ic, pc, iv or pv = PR(P*,T*) = F1* & F2* = F1* ∨ F2* = F1* → F2* = ¬(F*) = ∀x*F* = ∀f*F*
Let us first note that the translation defined in this way could be interpreted as simply an introduction of new notation for MPC2 - as the trivial replacement of the notation p(t) by the notation PR(p,t). From this vantage point, PR is nothing more than an auxiliary symbol on par with brackets. What is going to change, then, when we begin to see these notational variants of formulas of MPC2 as formulas of PC1(2) (and so also PR as a fully-fledged binary predicate)? The specific axioms of MPC2 concerning quantification over predicates obviously emerge as instances of the axioms of PC1(2) concerning quantification over terms of the sort 2; and similarly the rule for second-order generalization. As we include also the instances of the rule of comprehension (i.e. statements of the form ∃p∀ ∀x(p(x)↔ ↔F, where x is the only variable free in F) among the axioms of MPC2, we have to add also the translation of this rule to the axioms of PC1(2). The translation is as follows (where x is a variable of the sort 1, y a variable of the sort 2, and x is again the only variable free in F): ∃y∀ ∀x(PR(y,x)↔ ↔F)
(Compr)
If, in L2, the identity sign = is applicable only to terms of the sort 1, the translation thus defined will be a one-to-one function (to each formula of L1 there corresponds a unique formula of L2 and vice versa), and, moreover, it will obviously be the case that a formula of L1 is provable in MPC2 just where its translation in L2 is provable in PC1(2)+(Compr). If we admit the identity
sign = between the terms of the sort 2, we will obtain formulas in L2 which are not translations of any formulas of L1 (the translation relation will not be surjective); and it will be reasonable to add the following axiom: ∀y∀ ∀z(∀ ∀x(PR(y,x)↔ ↔PR(z,x))→ →(y=z))
(Ext)
Clearly, it will hold that a formula of L1 is provable in MPC2 just where its translation in L2 is provable in PC1(2)+(Compr)+(Ext). Let now T be a theory in L1; we define the theory T* in L2 in such a way that it contains the translation A* of every axiom A of T, plus (Ext) and (Compr). Let I= (where U is a set and P is an assignment of elements of U to ic’s of L1 and subsets of U to pc’s of L1) a model of the theory T. Let U1=U, U2=Pow(U) and let P* be such minimal extension of the interpretation function P that P*(PR)={│x∈y}. Then, I*= is obviously an interpretation of L2. It is easy to check that I satisfies a statement F of L1 if and only if I* satisfies the translation F* of F into L2; and as I* obviously satisfies both (Ext) and (Compr), I* is a model of T*. It follows that to every interpretation of a theory in MPC2 there corresponds a certain unique interpretation of the translation of the theory into PC1(2); especially to each interpretation of MPC2 there corresponds some unique interpretation of PC1(2)+(Ext)+(Compr). Considering now the inverse case, let us take I*= to be a model of T*. Let every element y of the set U2 be assigned the subset m(y) of U1, so that m(y)={x∈U1│∈P*(PR)}. (In this way we see the elements of U2, informally speaking, as ‘objectual correlates’ of the subsets of U1 - the element y is the objectual correlate of the set m(y), or, we may say, it is ‘this-set-understood-as-an-object’. The axiom (Ext) guarantees that m is injective, i.e. that every element of U2 is the objectual correlate of at most one subset of U1). Let now P be such a function that P(i)=P*(i*) for every ic i of L1 and P(p)=m(P*(p*)) for every pc p of L1; then I= is an interpretation of the language L1. Let us distinguish two cases: first, if the range of the function m is the whole set Pow(U1) (i.e. if every u⊆U1 is the value of m(y) for some y∈U2), then it is again clear that every statement F of L1 is satisfied by I if and only if F* is satisfied by I*, and especially that I is the model of T; and the interpretations I and I* correspond to each other in this sense. Second, if this is not the case, i.e. if the range of m is the proper part of Pow(U1) (i.e. if there exists an u⊆U1 which is m(y) for no y∈U2), then we cannot exclude the possibility of the existence of a statement F of L1 which is satisfied by I, although F* is not satisfied by I*, or vice versa. (Such a statement could, for example, state the existence of just such a subset of the universe, the objectual correlate of which is not in U2.) Hence: to some, but not necessarily to every, interpretation of a theory in PC1(2)+(Ext)+(Compr) there corresponds a unique interpretation of the translation of the theory into MPC2; especially to some, but not necessarily to every, interpretation of PC1(2)+(Ext)+(Compr) there corresponds some unique interpretation of MPC2. Let us further show that a theory in sorted first-order logic can be straightforwardly translated into a theory in unsorted first-order logic. For that purpose, let us construct the language L3 of PC1 so that - the set of ic of L3 is identical with the union of the set of ic1 and the set of ic2 of L2 - the set of iv of L3 is identical with the union of the set of iv1 and the set of iv2 of L2 - the set of pc of L3 is constituted by the binary predicate PR and the unary predicates S1 and S2. We define the translation of L2 into L3 by induction - if X is an expression of L2, we shall denote its translation into L3 as X+:
X+ = X if X is ic or iv + PR(T,T') = PR(T+,T'+) + (F1 & F2) = F1+ & F2+ = F1+ ∨ F2+ (F1 ∨ F2)+ (F1 → F2)+ = F1+ → F2+ + (¬F) = ¬(F+) (∀ ∀xF)+ = ∀x+(Si(x+)→ →F+), where i is the sort of the variable x in L2 However, this translation is surely not surjective: hence there exist formulas of L3 which translate no formula of L2, viz. formulas which quantify over the whole universe, rather than over one of its parts modeling the sorts L2 (i.e. formulas of the shape ∀xF or ∃xF, where F is not →F'), or formulas, which contain the predicate Si elsewhere than within the of the shape Si(x)→ antecedent of a quantified implication. Consider a formula F of L2 which is an axiom of PC1(2), and its translation F+ into L3. If F is an axiom of the propositional calculus, then F+ is obviously an axiom of PC1; and if F is an axiom of quantification, F+ will be a direct consequence of the corresponding general axiom of quantification of PC1 if we accept, for every ic1 X occurring in F, the postulate S1(X+)
(IC1)
and for every such ic2 the postulate S2(X+).
(IC2)
Let thus T2 be a theory in L2; we define the theory T3 in L3 in such a way that it contains the translation A+ of every axiom A of the theory T2, plus the corresponding instance of the axiom (IC1) resp. (IC2) for every ic1 resp. ic2 of L2, plus the following axioms (which concern exclusively such formulas of L3 which translate no formulas of L2): ∃x.S1(x) ∃x.S2(x) ∀x.S1(x)∨ ∨S2(x) 1 ¬∃ ∃x.S (x)&S2(x) PR(y,x) → S2(y) & S1(x)
(NEmpt1) (NEmpt2) (Exhst) (Disj) (PR)
The theory T3 is obviously a first-order theory and it holds that a formula F of L2 is provable in T2 just where its translation F+ into L3 is provable in T3. Let now I= be a model of T2. Let U=U1∪U2 and let P+ be such function that P+(X)=P(X) whenever X is ic or pc of L2, and P(Si)=Ui for i=1,2; then I+= is clearly an interpretation of L3 and it is easy to check that a formula F of L2 is satisfied by I if and only if F+ is satisfied by I+. Moreover, as I+ obviously satisfies (NEmpt), (Exhst), (Disj), (PR) and all instances of (IC1) and (IC2), I+ is a model of T3. Conversely, let I+= be a model of T3. Let Ui=P(Si) for i=1,2, and let P- be the restriction of the function P to the set of all ic and pc of L2; then I= is obviously an interpretation of L2 and it holds that the formula F+ of L3 is satisfied by I+ just where F is satisfied by I; and thus I is also a model of T2. Hence: a formula F
is satisfied by a model of a theory T just where F+ is satisfied by a model of T+; and F is satisfied by every model of T just where F+ is satisfied by every model of T+. Assembling our results so far, we can conclude that there exists a class of first-order interpretations (namely those which satisfy the axioms (NEmpt), (Exhst1), (Exhst2), (Disj), (PR) and the translations (Ext+) and (Compr+) of the axioms (Ext) a (Compr)), which „model“ - in a certain, exactly specified sense - second-order interpretations within first-order logic. ∀y.S2(y)→∀ →∀z.S →(∀ ∀x(S1(x)→ →(PR(y,x)↔ ↔PR(z,x)))→ →(y=z)) →∀ 2(z)→ 2 1 ∃y.S (y)&∀ ∀x.S (x)→ →(PR(y,x)↔ ↔F)
(Ext+) (Compr+)
Let us call these interpretations quasisecond-order. If we call the first order theory which is constituted by the axioms (NEmpt1), (NEmpt2), (Exhst), (Disj), (PR), (Ext+) and (Compr+) quasisecond-order predicate calculus (QPC2), a quasisecond-order interpretation will turn out to be a (first-order) interpretation of QPC2. The conclusion reached above then reads that there exists a one-to-one correspondence between the set of all second-order and a subset of the set of all quasi-second order interpretations such that a second-order interpretation I is a model of a second-order theory T just in case the corresponding quasisecond-order interpretation I' is a model of the translation of T into first-order logic; hence there is a one-to-one correspondence between the set of all second-order and the set of certain quasisecond-order interpretations such that it preserves satisfaction - modulo translation. However, there exist also such quasisecond-order interpretations which correspond to no second-order interpretation. This means that every formula of MPC2 the translation of which is valid in QPC2 is valid in MPC2; but it is generally not the case that the translation of every formula valid in MPC2 is valid in QPC2. The reason is that a formula of QPC2 may be satisfied by every interpretation which corresponds to an interpretation of MPC2, but in the same time not be satisfied by an interpretation which corresponds to no interpretation of MPC2. This possibility could be excluded only if we were able to restrict the set of quasisecond-order interpretations in such a way that it contained only those interpretations corresponding to second-order interpretations. In case of monadic second-order logic this is arguably possible - it has been proved that the set of formulas valid in MPC2 is recursive (see, e.g., Dreben and Goldfarb, 1979, chapter 8.3). Hence there exist quasisecond-order interpretations to which there correspond no second-order interpretations - and these cause the fact that the translations of some formulas which are second-order valid are not quasisecond-order valid. In this sense, the translation of second-order logic into first-order logic will never be ‘perfect’. The standard second-order logic is thus not in general reducible to first-order logic. However, the situation is different if we do not define second-order interpretations in the way we have done it above, i.e. standardly - if we allow for such interpretations in which the ranges of predicate variables are proper subsets of the sets of all the corresponding relations, i.e. if we allow for Henkinian interpretations. There is a one-to-one correspondence between quasisecond-order interpretations and Henkinian secondorder interpretations which preserves satisfaction (viz, e.g., Shapiro, 1991, chap. 4.3) - each Henkinian interpretation can thus be seen as a quasisecond-order interpretation and vice versa. Therefore second-order logic interpreted in the Henkinian way is reducible to first-order logic.
6. Summary of Translatability We have indicated how to translate monadic second order logic into first-order logic; the translation of the full (non-monadic) second-order logic into first-order logic is analogous. We only need to add other sorts (or „quasisorts“) for the predicates with arities higher than 1. The case of nonmonadic second-order logic, however, is provably different in that it is no longer possible to delimit the range of quasi-second order interpretations in such a way that secondorder validity implies quasisecond-order validity: it follows from Gödel’s incompleteness theorem that the set of all second-order valid formulas is not recursively enumerable (and hence axiomatizable); hence there exists a formula valid in second-order logic the translation of which is not valid within quasisecond-order logic. (This follows directly from the fact that there is a finite categorical axiomatization of Peano arithmetic within second-order logic: if PA is the conjunction of the axioms and G Gödel’s undecidable formula, then the formula PA→G is obviously second-order valid, whereas its translation into first-order logic is not valid). Thus, we can summarize: 1. There exists a translation of second-order logic into first-order logic such that it generally holds that if F is a formula of second-order logic and F' its translation into first-order logic, then if F' is valid, F is also valid; and, moreover, F is provable in second-order logic if and only if F' is provable in first-order logic. 2. There does not exist a translation of second-order logic into first-order logic such that it would generally hold that if F is a second-order formula and F' its first-order translation, then if F is valid, F' would be also valid. In a similar way we can define the translation of any logic of order n into a logic of an order lower than n. However, once we begin to investigate the translation of third-order logic into second-order logic, we discover a fact which may surprise us: third-order logic, and in general any logic of an order higher than 2, is perfectly reducible to second-order logic; hence passing from second-order logic to a higher-order one does not provide, in contrast to passing from first-order logic to second-order one, an increase in ‘expressive power’. The only substantial difference is between the first and the second order - any logic of an order higher than 2 can be, without any loss of generality, considered a mere ‘notational variant’ of secondorder logic. Why this is so can be seen if we return to our considerations of the impossibility of reducing second-order logic to first-order one. We have reached the conclusion that the problem is that the set of quasisecond-order interpretations, as we managed to define it, contains also some interpretations which have no equivalents among second-order interpretations; we noted, that this problem would be solved if we managed to characterize just the set of those quasisecond-order interpretations which do have such counterparts. Returning to the terminology of the previous chapter, we can say that they are such interpretations for which each subset of the universe has its ‘objectual correlate’; that is if is a quasisecond-order interpretation, then it has a second-order equivalent if and only if for every subset u of U1 there exists a y from U2 so that u={x∈U1│∈P*(PR)}. The desired subset of the set of quasisecond-order interpretations therefore could be delimited if we added the following axiom to the axioms of QPC2 ∀p∃ ∃y.S2(y)&∀ ∀x.S1(x)→ →(PR(y,x)↔ ↔p(x))
The reason why we could not do this was that this is a second-order formula - p is a predicate variable (hence we were able to accept only the weaker axiom schema (Compr+)). However, the situation would be different if the language into which we translate were second-order - then such an axiom could be accepted. If what we desired were, as before, the reduction of secondorder logic, then our effort would, of course, be futile (we would find ourselves ‘reducing’ second-order logic to second-order logic); nevertheless the procedure can be nontrivially employed if what we reduce to second-order logic is a logic of an order higher that 2. For details see Shapiro (1989, Chap. 6).
7. Discussion and Conclusion The question now is to what extent we really need full second-order predicate calculus with all its power, i.e. with all its not-first-order-reducible validities, and to what extent we can make do with that part of it which is first-order reducible - in other words, how far we need construe the semantics of second-order logic in the standard way, and how far we can construe it in the Henkinian way. It seems plausible that if what we are pursuing is the analysis of natural language with its pronouncements like (1) or (4), then nothing stands in the way of accepting the Henkinian semantics and thus construing higher-order logics as mere notational variants of firstorder logic. The situation is, of course, more involved if what we are investigating are the foundations of mathematics. Let us take the definition of infinity, as it is expressed in (6'). It is clear that this definition can be articulated as soon as we have the syntactic means of secondorder logic. This definition also always (respectable of whether we interpret the language in the standard, or in the Henkinian way) delimits those sets which can be mapped on their own proper subsets, and declares these very sets as infinite. It is usually assumed that such a definition is correct where we are interpreting the language standardly, but incorrect if our interpretation is Henkinian; i.e. that to define the ‘real’ infinity (and hence also finiteness) we need the fullyfledged second-order logic. This is because under the standard interpretation our ‘nonexistence’ necessarily means real nonexistence, and our definition thus delimits just those sets which can be really mapped onto their own proper parts and are thus really infinite; whereas under the Henkinian interpretation our ‘nonexistence’ can mean only ‘nonexistence within the range of the interpretation considered’, and a set which cannot be, apparently, mapped on its proper part may quite possibly be infinite - for it may be the case that the mapping of the set on its proper subset does indeed exist, but happen not to be included into the range of the interpretation considered. Without attempting at a real analysis of this state of affairs, let us note that this construal of the difference between the standard and the Henkinian second-order logic, although often taken for granted, is not wholly unproblematic. The difficulty is that it presupposes the picture according to which we apprehend directly the mathematical reality, and use the languages of logic only to describe it (let us stress that to assume this is more than to construe mathematical reality realistically, to assume that it exists independently of mathematicians). For the standard construal of second-order logic would not make good sense if we did not take such concepts as all subsets of a given set for granted. Against such a picture, a different picture can be opposed the picture which was analyzed, for the first time, by Skolem (esp. 1958), and which is based on the view that mathematical concepts are inherently relative - that they make sense only within
the context of a particular theory. If we thus say that a set is infinite, we have to ask in which theory - for the set can be finite according to one theory (perhaps according to the Henkinian second-order logic), whereas be infinite according to another theory (the standard second-order logic). The fact that from the viewpoint of a standard model the corresponding Henkinian model may appear as lacking something does not yet mean that the former is complete, and the latter incomplete, in an absolute sense. However, this view makes the very notion of a standard interpretation problematic: for ‘to be standard’ means ‘to include all subsets’, and something thus can be called standard only from an absolute vantage point, from which it is possible to decide, when the subsets are all, and when not. Standard interpretations therefore cannot be delimited otherwise than by recoursing to a further unanalyzed concept of all subsets (which is quite straightforward for finite sets, but less so for infinite ones) - in contrast to Henkinian interpretations they cannot be gerrymandered via a recursive specification. It follows that to the same extent to which secondorder logic is more plausible as the foundation of mathematics, it is - in a certain sense - more trivial. Exaggerating somewhat we can say that whereas we lack the means to characterize, e.g., infinite sets within first-order logic, in second-order logic we are enabled to do so, but with little more significance than when we simply said that infinite sets were those which were (‘really’) infinite. Although the relationship between first-order and second-order logic may present problems, it is sure that the relation between second-order and higher-order logic is not problematic - any logic of an order higher than two can be considered as a ‘notational’ variant of second-order logic (which does not, of course, preclude this very notational variant from being sometimes quite useful). I think that discussions of the relationship between first-order and higher-order logic often suffer both from the fact that its participants do not satisfactorily specify what exactly they mean by higher-order logic, and from the fact that they do not consider the whole depth of the problematicity of the relationship. This paper is meant to summarize some facts by which this problematicity is characterized.
References Barwise, J., Feferman, S., eds. (1985): Model-theoretic logics, Springer, New York. Davidson, D. (1980): Essays on Actions and Events, Clarendon Press, Oxford. Dreben, B. and W. Goldfarb (1979): The decision problem: solvable classes of quantificational formulas, Addison-Wesley, London. Frege, G. (1892): 'Über Begriff und Gegenstand', Vierteljahrschrift für wissentschaftliche Philosophie 16, pp. 192-205. Henkin, L. (1950): 'Completness in the Theory of Types', Journal of Symbolic Logic 15, pp. 81-91. Chierchia, G. (1982): ‘Nominalization and Montague Grammar’, Linguistics and Philosophy 5. Church, A. (1940): 'A Formulation of the Simple Theory of Types', Journal of Symbolic Logic 5, pp. 56-68. Kripke, S. (1963): 'Semantical Considerations on Modal Logic', Acta Philosophica Fennica 16, pp. 83-94. Moore, G.H. (1988): 'The Emergence of First-Order Logic', History and Philosophy of Modern Mathematics (ed. W. Aspray and P.Kitcher), University of Minnesota Press, Minneapolis. Parsons, T. (1990): Events in the Semantics of English, MIT Press, Cambridge (Mass.). Peregrin, J. (1990): Theory of Types: Good Servant Bad Master, Prague Studies in Mathematical Linguistics 10, 159-176. Peregrin, J. (1992): Meaning, Truth and Models, From the Logical Point of View 2/92, 67-75. Peregrin, J. (1995): Doing Worlds with Words, Kluwer, Dordrecht. Shapiro, S. (1991): Foundations without Foundationalism, Clarendon Press, Oxford. Skolem, T. (1958): 'Une relativisation des notions mathématiques fondamentales', Colloques internationaux du Centre de la Recherche Scientifique, Paris, 13-18; reprinted in Skolem: Selected Works in Logic (ed. J.E. Fenstadt), Universitetsforlaget, Oslo, 633-38. Turner, R. (1983): ‘Montague Semantics, Nominalization and Scott's Domains’, Linguistics and Philosophy 6.
Is propositional calculus categorical? Jaroslav Peregrin Department of Logic, Institute of Philosophy, Academy of Sciences of the Czech Republic www.cuni.cz./~peregrin
Can axioms confer meanings on classical logical connectives? According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result. Hence, it seems, we can make do with the axiom system alone, and hence the axiom system is capable of confering the right meanings on the logical operators. But this is wrong. As is well known, there are theories which do justice to all the axioms and inference rules of the standard logic, which are nevertheless incompatible with the standard truth-functional interpretation of the operators: e.g. theories containing disjunctions together with the negations of both disjuncts2. To clarify the situation, we have to delimit the concepts and the framework of out investigation with some rigor. We will take language to be simply a set of sentences with a delimited set of acceptable truth-valuations of the sentences. (We refrain from the discussion of whether this definition is too broad – form the viewpoint of the present paper it is not important3). Then we can see an axiomatic system as a way of delimiting the space of acceptable truth-valuations: an axiom is a 1
We can, of course, switch to another logic, say the intuitionist one, and have a different kind of negation, but then again the semantics of the negation-sign will be fixed 'a priori' (i.e. prior to the framweork being put to use) rather than being up for grabs. 2 The first one to note it was probably Carnap (1943). See Koslow (1992, Chapter 19) for more details. 3 I have discussed this questions in detail elsewhere (Peregrin, 1997), where I argued that any kind of semantics should be seen as a tool of such a delimitation
1
sentence which must be be mapped on 1 by every acceptable valuation, whereas a rule gives a sentence which must be mapped on 1 by every valuation which maps some other sentences on 1. The general question now is which kinds of spaces of acceptable valuations are delimitable in this way, i.e. in terms of axiomatic systems; and in particular whether we can carve the space in such a way that some connectives get pinned down to the standard truth-functions. Let us be a little bit more precise. Let L be a set and O a function mapping L×L on L. Let C be a finite set of 'constraints' on valuations of the elements of L, which are of the shape [if A1,...,An (are true), then also] A (is true). Where v is a function from L to B={0,1}, we will say that v respects C if it does justice to all the constraints of C. Where F is a function from B×B to B, we will say that C sets the denotation of O to F the following two conditions are equivalent (i) v respects C; (ii) v(O(A1, A2)) = F(v(A1),v(A2)) for every A1, A2∈L. Now let F¬, F∧¸ F∨ and F→ be the truth-functions standardly assigned to the classical negation, conjunction, disjunction and implication, respectively. Then it is obviously the case that the three constraints if A1, A2, then O(A1, A2); if O(A1, A2), then A1; and if O(A1, A2), then A2 set the value of O to F∧. On the other hand, there is no finite set of constraints which would set the value of O to F¬, F∨, or F→. To see this, consider, for instance, the case of F∨. We may clearly help oursevles to the constraints if A1, then O(A1, A2); and if A2, then O(A1, A2). By them we exclude all valuations which map either A1 or A2 on 1, and O(A1, A2) on 0; but what we need in addition to this is exclude also all those which map both A1 and A2 on 0 and O(A1, A2) on 1. Suppose, for the sake of simplicity, that the language we are considering has no other sentences than A1, A2 and O(A1, A2) (hence that now there is only a single valuation left to be excluded). Then the exhaustive listing of all possible nontrivial constraints, clearly, is the following: 1. A1 2. A2 3. O(A1, A2) 4. if A2, then A1 5. if O(A1, A2), then A1 6. if A2, O(A1, A2) then A1
2
7. if A1, then A2 8. if O(A1, A2), then A2 9. if A1, O(A1, A2) then A2 10. if A1, then O(A1, A2) 11. if A2, then O(A1, A2) 12. if A1,A2, then O(A1, A2) Out of these, 3., 4., 6., 7., 9., 10., 11., 12. do not exclude the unwanted valuation at all, whereas the other do exclude it, but only at the cost of excluding also some of those valuations which should remain unexcluded. Of course this is not very surprising: the only thing our constraints can stipulate is that if some sentences are true, also some other sentences are true; but not, e.g., that if some sentences are false, other sentences are false. But it is only constraints of this very kind which are available to a builder of an axiomatic system. This indicates that many possible spaces of acceptable valuations (and especially some truth-functions) are not categorically delimitable in the axiomatic way. How does this square with the soundness and completness of the classical, truth-functional logic? To see it, let us first give some more definitions. Let V be a class of mappings of L on B. We define Pos(V) = {A∈L | v(A) = 1 for every v∈V} Neg(V) = {A∈L | v(A) = 0 for every v∈V} Two classes V1 and V2 of valuations are called positively equivalent iff Pos(V1) = Pos(V2); they are called negatively equivalent iff Neg(V1) = Neg(V2). Let L be a language such that for every its sentence A there is a sentence ¬A and for every two sentences A1 and A2 there is a sentence A1∧A2. We will call its valuation v classical iff v(¬A) = F¬(v(A)) v(A1∧A2) = F∧(v(A1),v(A2)) We will call it noncontradictory if for no A it is the case that v(¬A)=v(A)=1; and we will call it full iff for no A it is the case that v(¬A)=v(A)=0. (Hence every classical valuation is both noncontradictory and full.) Calling now the valuation quasi-classical iff it respects the axiom system of the classical propositional claculus, what we are going to prove in this paper is the following: (1) The class of all quasi-clasical valuations is positively equivalent to the class of all classical valuations. (2) The class of all quasi-clasical valuations which are noncontradictory is positively and negatively equivalent to the class of all classical valuations. (3) The class of all quasi-clasical valuations which are full and noncontradictory conicides with the class of all classical valuations. It is clear that (1) is nothing else than the standard completeness theorem for classical propositional logic (and hence proving merely it would be no achievement). From the current perspective, however,
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it is only one of the series of results which characterize the relationships between the class of all quasi-clasical valuations and the class of all classical valuations, other ones being (2) and (3). Can these results be obtained by a straightforward generalization of the classical completness proof? I do not think so. How do we usually prove (1)? The idea of the most common proof, which Mendelson (1964) attributes to Kalmár (1936), can be construed as follows. For a truth table, let us call a formula (X1→(...(Xn→X)...)) the internalization of its i-th row of the table iff the following holds: (i) if the value in the i-th row and j-th column is 1, then Xi is Ai, whereas in the opposite case it is ¬Ai; (ii) if the value in the i-th row and the value column is 1, then X is A, whereas in the opposite case it is ¬A. Now Kalmár’s proof can be seen as showing that (i) the internalization of any row of the truth table of any formula is a theorem; and (ii) if a formula is a tautology, then it is derivable from the internalizations of the rows of its truth table. As a consequence, a tautology is derivable from theorems, and hence is itself a theorem. There does not seem to be a way of generalizing this proof to cover not only (1), but also (2) and (3). And what we are going to do within the rest of the paper is to develop a more general framework, within which we can not only prove the theorems sketeched above, but also reach a helpful vantage point to oversee an aspect of the landscape of elementary logic.
Constraints and their internalization Hence the problem, as we have articulated it in the previous section, is to try to find an axiomatic delimitation of the space of acceptable valuations which would pin down the denotations of some operators to the usual truth-functions. More generally, the problem is that of finding an axiomatic delimitation of a space of valuation delimited in some more general way. Before we turn directly to this task, we give some definitions and prove some simple preparatory results. DEFINITION 1. A language is a set (whose elements are called sentences) plus a set of its mappings on B={0,1} (called acceptable valuations). A sentence S of L is called verified by a valuation v iff v(S) = 1; it is called falsified by v iff v(S) = 0; it is called valid iff it is verified by every acceptable valuation and it is called countervalid iff it is falsified by every one. A class C of sentences is called to entail a sentence S iff S is verified by every acceptable valuation that verifies all sentences of C. If {S1,...,Sn} entails S, we will write S1,...,Sn |= S. It is clear that the concept of validity is reducible to that of entailment: a sentence is valid simply iff it is entailed by an empty set. Now we are going to characterize a class of languages for which there is also an inverse reduction. DEFINITION 2. Let ► be a binary function from the set of sentences of a language L to the same set; we will write S1►S2 instead of ►(S1,S2)4. ► is called implication iff for every sentences S1,...,Sn, S of L 4
Note that we do not see implication as a sign, but rather as a function. Of course that in a typical case the function will map any pair of sentences on a sentence built out of them with the help of an implication-sign; but we do not care about the syntax. In this sense, our treatment of implication and negation is close to that of Koslow (1992).
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(i) S1►S2, S1 |= S2. (ii) if S1,...,Sn |= S (where n>0), then S1,...,Sn-1 |= Sn►S, If such a function exists, then L is said to have implication. THEOREM 1. Let ► be a implication. Then S1►(...(Sn►S)...) is valid iff S1,...,Sn entails S. PROOF: We first prove the direct implication, by induction. Let first n=1. If S1►S is valid, then, according to (i) of the definition of implication, S must be true whenever S1 is. Hence if S1►S is valid, then S1 entails S. Let now the theorem hold for n=m; we will show that it holds for n=m+1. So let S1►(...(Sm+1►S)...) be valid. Then, according to the inductive assumption, S1,...,Sm |= Sm+1►S. However, according to (i) of the definition of implication, Sm+1►S, Sm+1 |= S. But as S1,...,Sm,Sm+1 entails both Sm+1►S and Sm+1, then, due to the obvious transitivity of entailment, S1,...,Sm,Sm+1 |= S. Now we prove the inverse implication, again by induction. Let first n=1. If S1 |= S, then S1►S is valid according to (ii) of the definition of implication. Let now the theorem hold for n=m; we shall show that it holds for n=m+1. Let S1,...,Sm+1 entail S. Then, according to (ii) of the definition of implication, S1,...,Sm |= Sm+1►S; and S1►...(Sm+1►S)...) is valid according to the inductive assumption. The most perspicuous species of implication is constituted by the well-known material implication: DEFINITION 3. A binary function f from the set of sentences of a language L to the same set is called material implication iff, for every sentences S1, S2 of L, every acceptable valuation verifies f(S1,S2) just in case it either falsifies S1 or verifies S2. It is clear that a material implication is an implication; but it can be shown that not every implication is material. We will need to consider also a relation between sentences which is more general than entailment. We will say that a distribution of truth values among the sentences S1, ..., Sn forces a truth value of a sentence S iff any acceptable valuation which distributes the truth values among S1, ..., Sn in the former way assigns the latter value to S: DEFINITION 4. Let L be a language, let S1, ..., Sn, S be its sentences, and let V1, ..., Vn, V ∈{0,1}. We will say that the assignments of Vi to Si (for i=1 to n) forces the assignment of V to S, and will abbreviate this to S1V1, ..., SnVn |= SV, iff v(S)=V for any acceptable valuation v such that v(Si)=Vi (for i=1 to n). An instance of forcing will be called positive if the only truth value mentioned in it is 1, i.e. if it is of the form S11, ..., Sn1 |=S1. An instance of forcing will be called absolute if n = 0, i.e. if it is of the form |= SV. It is clear that every instance of entailment can be seen as a (positive) instance of forcing (S11, ..., Sn1 |= S1 is the same as S1, ..., Sn |= S); hence entailment can be seen as a special case of forcing. Let us now characterize a class of languages for which entailment is equivalent with forcing.
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DEFINITION 5. An unary function # from sentences to sentences is called negation iff no acceptable valuation verifies both S and #S. It is called a standard negation if, moreover, every acceptable valuation which does not verify S verifies #S, i.e. if every acceptable valuation verifies S if and only if it falsifies #S. THEOREM 2. Let # be a negation. Then: (i) if S1V1,...,SnVn |= #Sn+11, then S1V1,...,SnVn |= Sn+10 (in particular, if |= #S1, then |= S0); and (ii) if S1V1,...,Si0,...,SnVn |= Sn+1Vn+1, then S1V1,...,#Si1,...,SnVn |= Sn+1Vn+1. PROOF: (i) Suppose that S1V1,...,SnVn |= #Sn+11. Then #Sn+1 is verified by every acceptable valuation which assigns Vi to Si for i=1 to n. But then every such valuation is bound to falsify Sn+1. Hence S1V1,...,SnVn |= Sn+10. (ii) Suppose that S1V1,...,Si0,...,SnVn |= Sn+1Vn+1. Then every acceptable valuation which assigns Vj to Sj for j=1,...,i-1,i+1,..., n and 0 to Si assigns Vn+1 to Sn+1. However, as every truth valution which verifies #Si is bound to falsify Si, and hence every acceptable valuation which assigns Vj to Sj for j=1,...,i-1,i+1,..., n, and 1 to #Si, assigns Vj to Sj for j=1,...,i-1,i+1,..., n, and 0 to Si. Hence S1V1,...,#Si1,...,SnVn |= Sn+1Vn+1. THEOREM 3. Let # be a standard negation. Then (i) if S1V1,...,SnVn |= Sn+10, then S1V1,...,SnVn |= #Sn+11 (in particular, if |= S0, then |= #S1); and (ii) if S1V1,...,#Si1,...,SnVn |= Sn+1Vn+1, then S1V1,...,Si0,...,SnVn |= Sn+1Vn+1. Hence for every sentences S1,...,Sn,Sn+1 and every truth values V1,...,Vn,Vn+1 it holds that S1V1,...,SnVn force Sn+1Vn+1 iff X1,...,Xn entail Xn+1, where Xi is Si if Vi is 1 and it is #Si if Vi is 0. Hence in a language which has a standard negation, any instance of forcing is expressible in the form of an instance of entailment. PROOF: As a sentence S of L is verified by a acceptable valuation iff #S is not, the requirement that S is not verified is equivalent to the requirement that #S is. DEFINITION 6. A language is called normal if it has a standard negation and an implication. A language is called strongly normal if it has a standard negation and a material implication. Negation is directly characterized in terms of constraints. The same is not true of implication in general, but it is true about its material version: THEOREM 4. g is a standard negation iff (N1) S1 |= #S0 , and (N2) S0 |= #S1; f is a material implication iff (MI1) S10 |= (S1►S2)1, (MI2) S21 |= (S1►S2)1, and (MI3) S11,S20 |= (S1►S2)0. PROOF: Obvious.
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Characterizing validity What we are after can now be generally restated as delimiting the set of acceptable valuations of sentences of a language in terms of positive constraints. If we assume that the set is delimited by (not necessarily positive) constraints (which is the case as far as our main problem is concerned, for the space of acceptable truth valuations of the classical propositional calculus is delimited by the definition of its semantics), our task is that of turning non-positive constraints into positive ones with the smallest possible tampering with their effect. We have seen that if a language has an implication and a standard negation, every its constraint is expressible as positive and absolute – i.e., in effect, as a valid sentence. Let us call a sentence expressing a constraint in this way the internalization of the constraint: DEFINITION 7. Let f and g be a binary resp. a unary function from the set of sentences of a language L to the same set. An (f,g)-internalization of a constraint S1V1, ..., SnVn |= Sn+1Vn+1 is the sentence f(X1,...f(Xn,Xn+1)), where Xi is Si if Vi is 1 and is g(Si) if Vi is 0. (Hence the internalization of an absolute constraint |= Sn+1Vn+1 is Xn+1; and that of a positive absolute constraint |= Sn+11 is Sn+1.) THEOREM 5. If ► is an implication and # a standard negation, then S1V1, ..., SnVn forces Sn+1Vn+1 if and only if the (►,#)-internalization of S1V1, ..., SnVn |= Sn+1Vn+1 is valid. PROOF: Due to Theorem 1, S1V1, ..., SnVn |= Sn+1Vn+1 iff X1,...,Xn entail Xn+1 (where Xi is Si or #Si according to whether Vi is 1 or 0); and due to Theorem 3, X1,...,Xn entail Xn+1 iff X1►(...(Xn►Xn+1)...) is valid. The space of acceptable valuations of a language is normally specified in a metalanguage. However, if the language we are considering is normal, it allows us to express the needed constraints also in the object language itself – hence normal languages can be seen as displaying certain 'self-explicitating' capabilities. Moreover, the fact that having an implication and a standard negation we can turn every constraint into a positive and absolute one may seem to imply that any set of valuations which is delimitable in terms of constraints at all is delimitable in terms of positive and absolute constraints (namely of the internalizations of the original constraints). This would solve our task. But unfortunately it is hopeless. The reason is that constraints are ‘internalizable’ (i.e. expressible in the form of valid sentences) only if there is a negation and an implication, where the negation and implication are themselves defined in terms of (non-positive and non-absolute) constraints – and these constraints’ ability to constitute negation and implication does not survive their internalization. To see this, consider the definition of negation: S1 |= #S0 , and S0 |= #S1. These constraints are internalized to S►##S , and #S►#S , respectively. However, it is clear that the validity of the internalizations alone does not make # into a negation: this can be seen, e.g., from the fact that they are obviously compatible with # being the identity mapping.
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This means that a set of constraints delimiting the space of acceptable valuations cannot be always turned into a set of positive constraints. (Of course not: positive constraints can characterize acceptable valuations only in terms of the setences they must verify, and consequently a valuation which verifies more sentences than some acceptable one is bound to be acceptable too. In contrast to this, non-positive constraints can characterize accpetable valuations also in terms of sentences which are to be falsified; and hence they can render a valuation unacceptable even if the set of sentences it verifies contains the set of sentences verified by an acceptable valuation). However, despite of the fact that the possibility of delimiting the space of acceptable valuations of sentences in terms of positive constraints does not obtain in general, and does not obtain even for strongly normal languages, for a language of the latter kind we can define a language with its space of acceptable valuations delimited by means of merely positive constraints and such that its set of valid sentences coincides with that of the original one. This is what we are going to show now. But let us first adopt some definitions which will allow us talk more concisely: DEFINITION 8. A language is called (positively) delimited iff the space of the acceptable valuations of its sentences is the set of all and only valuations satisfying a set of (positive) constraints. Hence what we are after now is to prove that to every strongly normal and delimited language there exists a positively delimited language with the same class of valid sentences. First we will show that for every strongly normal and delimited language L there exists a certain language L∗ with the following two properties: (*) Whenever S1V1, ..., SnVn |= Sn+1Vn+1 in L, the internalization of this constraint is valid in L∗. (**) All constraints delimiting the space of acceptable valuations of L∗ are in force in L. It is easy to see that (*) implies that the set of valid sentences of L is contained in that of L∗; whereas (**) implies that, vice versa, the set of valid sentences of L∗ is contained in that of L. To be able to prove (*), we first need to precisely characterize the class of constraints mentioned in it: DEFINITION 9. Let L be a language and S1, ..., Sn, Sn+1 its sentences and V1, ..., Vn, Vn+1 truth values. If S1V1, ..., SnVn forces Sn+1Vn+1, we will say that the constraint S1V1, ..., SnVn |= Sn+1Vn+1 is in force for L. Note that if the space of accepable valuations of L is delimited by a set C of constraints, then all constraints from C are in force for L, but not necessarily every constraint which is in force for L belongs to C. For example the constraint S |= S is in force for every language, independently of how its space of acceptable valuations is delimited. Or if the set of delimiting constraints contains S1V1, ..., SnVn |= SV then also the constraint S1V1, ..., SnVn, Sn+1Vn+1 |= SV is thereby in force. Etc. (Hence constraints ‘entail’ other constraints.)
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Now we are ready to formulate what we will call the Internalization Theorem and what spells out that for a strongly normal and delimited language there is a language which fulfills the above requirement (*). Because its proof is somewhat complicated, we will postpone it to the next section. THEOREM 6. (‘Internalization Theorem’) Let L be a language, ► and # be a binary resp. a unary function from the set of sentences of L to the same set, and let the space of acceptable valuations of L be delimited by (MI1), (MI2), (MI3), (N1), (N2) and a set C of other constraints (which implies that the language is strongly normal). Let L∗ be a language which is just like L with the exception that its space of acceptable valuations is delimited by the set of constraints containing the (►,#)-internalizations of all constraints from C plus the following positive constraints: (A1) |= S2►(S1►S2) (A2) |= (#S2►#S1)►(S1►S2) (A3) |= (S1►S2)►((S2►S3)►(S1►S3)) (A4) |= (S1►(S1►S2))►(S1►S2) (MP) S1►S2, S1 |= S2 Then the (►,#)-internalization of every constraint which is in force in L is valid in L∗. PROOF: Postponed to the next section. It is obvious that (A1)–(A4) together with (MP) constitute an axiomatization of the classical propositional calculus (this very one being proposed, e.g., by Tarski, 1965); and the theorem resembles the completness theorem for this calculus. However, as we have already indicated, whereas the concept of completness aims at mere coincidence of valid sentences, our aim is more ambitious: to inspect the conditions of coincidence of the entire spaces of acceptable valuations. Classical completness then falls out of this as a special case. Consider the language of the classical propositional calculus based on the primitive connectives ¬ and → (hereafter CPC). The space of acceptable valuations of its sentences of is obviously delimited by the following constraints S1 |= ¬S0 S0 |= ¬S1; S10 |= (S1→S2)1 S21 |= (S1→S2)1 S11,S20 |= (S1→S2)0 Hence the function mapping S on ¬S is a standard negation, whereas that mapping S1 and S2 on S1→S2 is a material implication (and hence CPC is strongly normal). Now take the language CPC∗ which is just like CPC with the single exception that the space of acceptable valuations of its sentences is delimited by the following (positive) constraints: (A1PC) |= S2→(S1→S2) →(S2→S1) (A2 PC) |= (¬S1→¬S2)→ PC →((S2→S3)→ →(S1→S3)) (A3 ) |= (S1→S2)→ PC (A4 ) |= (S1→(S1→S2))→ →(S1→S2) PC (MP ) S1→S2, S1 |= S2
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According to the theorem just stated, the internalizations of all constraints which are in force for CPC are valid sentences of CPC∗. Hence, as, e.g., S1→S2, ¬S1→S2 |= S2 is a constraint which is in force in CPC (as is easily computed), its internalization (S1→S2)→ →((¬S1→S2)→ →S2) is a valid sentence of CPC∗. In particular, every sentence which is valid in CPC is valid in CPC∗. Note that it would be quite easy to find some positively determined language in which all the internalizations of the constraints being in force in L would be valid – take the trivial language which results from L by replacing its delimitation of the space of acceptable valuations by the single constraint |= S1 (i.e. which would render all sentences valid). (It would be easy to find a complete axiomatization of a logic, if we were not to require soundness.) The following lemma says that L∗ fulfils not only (*), but also (**); namely that all the constraints delimiting L∗ are in force in L. LEMMA 1. Let L be a strongly normal language with I being its material implication and # being its standard negation. Then the constraints (A1)-(A4) and (MP) are in force in L. PROOF: An easy computation. The desired result, namely that the set of valid sentences of L∗ is the same as that of L is now forthcoming: THEOREM 7. The sets of valid sentences of the languages L and L∗ from Theorem 6 coincide. PROOF: Let S be valid in L. This means that the constraint |= S is in force in L, and hence its internalization is valid in L∗. But the internalization of |= S is S and hence S is valid in L∗. Let now conversely, S be valid in L∗. As all the costraints of L∗ are in force in L (as demonstrated by the previous lemma), everything which is valid in L∗ is a fortiori valid in L – hence S is valid in L. This implies that the set of valid sentences of CPC coincides with that of CPC∗. However, note that this does not mean that the classes of acceptable valuations of the two languages coincide! Indeed, there are valuations which are not acceptable in CPC, but are acceptable in CPC∗: for example the valuation which maps every sentence on 1 is not acceptable in CPC (because it violates the constraint S1 |= ¬S0), whereas it is acceptable in CPC∗ (as the constraints of CPC are positive, it can force no sentence to be false). Theorem 7 says that if we are interested only in the validity of sentences, we are free to replace L with ∗ L . Now we prove that if we ‘manually’ tamper with the space of the acceptable valuations of L∗ so that # becomes a negation, the equivalence extends to countervalidity. LEMMA 2. If (A1)-(A4) and (MP) are in force in L, then no acceptable valuation verifies both S and #S, unless it verifies every sentence whatsoever; i.e. the constraint #S1, S1 |= S2 is in force in L.
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PROOF: In force of (A1), every sentence of the form #S1►(#S2►#S1) is valid. However, the antecedent of a sentence of the shape (#S1►(#S2►#S1))►(((#S2►#S1)►(S1►S2))►(#S1►(S1►S2))), which is valid in force of (A3), is just of this form; hence, in force of (MP), every sentence of the form ((#S2►#S1)►(S1►S2))►(#S1►(S1►S2)) is valid. However, the antecedent of such sentence is valid in force of (A2), hence, in force of (MP), every sentence of the form #S1►(S1►S2) is valid. This means that if #S1 is true, also S1►S2 must be true; hence that if #S1 and S1 are true, S2 must be true. THEOREM 8. Let L and L∗ be as before. Let L∗∗ be a language which is just like L∗ with the single possible exception that its space of acceptable valuation excludes the valuation which falsifies every sentence. Then both the sets valid and the sets of countervalid sentences of L and L∗∗ coincide. PROOF: It follows from (N1) that the space of acceptable valuations of L does not contain the valuation falsifying all sentences. Hence every acceptable valuation of L is an acceptable valuation of L∗∗; and hence every sentence countervalid in L∗∗ is countervalid in L. From the other side, all the internalizations of the constraints of L clearly keep to be valid in L∗∗, and hence if a sentence S is countervalid in L, #S is bound to be valid in L∗∗. Moreover, no valuation of L∗∗ verifies both S and #S (due to the previous lemma) – i.e. # is a negation. This means that if #S valid, S is bound to be countervalid (cf. Theorem 2(i)). It follows that everything which is countervalid in L is countervalid in L∗∗. Let CPC∗∗ be like CPC∗ with the single exception that its space of acceptable valuations does not contain the function mapping everything on 1. Then both the spaces of valid and countervalid sentences of CPC∗∗ and CPC coincide. Note that this still does not mean that the classes of acceptable valuations of the two languages coincide. For example, a valuation which falsifies both S1 and ¬S1 is not acceptable in CPC (because it violates the constraint S0 |= ¬S1), but it is acceptable in CPC. If we now, moreover, guarantee that # is a standard negation, the equivalence becomes complete: THEOREM 9. Let L and L∗∗ be as before. Let L∗∗∗ be a language which is just like L∗∗ with the single possible exception that its space of acceptable valuations excludes all valuations which, for some sentence S, falsity both S and #S. Then the spaces of acceptable valuations of L and L∗∗∗ coincide, and hence L coincides with L∗∗∗. PROOF: It follows from (N2) that no valuation for which there exists a sentence S such that both S and #S are falsified is acceptable in L. Hence none of the valuations which are acceptable in L∗∗, but not in L∗∗∗ are acceptable in L, and hence every acceptable valuation of L is an acceptable valuation of L∗∗∗. From the other side, all the internalizations of the constraints of L still keep to be valid in L∗∗∗. Moreover, every acceptable valuation of L∗∗∗ now verifies S if and only if it falsifies #S and vice versa – i.e. # is a standard negation. This means that the validity of an internalization implies the internalized constraint’s being in force (cf. Theorem 3(i)); and hence that every constraint which is in force in L is in force also in L∗∗∗. It follows that every acceptable valuation of L∗∗∗ is an acceptable valuation of L. Let CPC∗∗∗ be like CPC∗∗ with the single exception that its space of acceptable valuations does not contain any function falsifying any sentence together with its negation. Then the spaces of acceptable valuations of CPC∗∗∗ and CPC coincide, and hence so do the two languages.
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Proof of the ‘Internalization Theorem’ The proof of the basic theorem of the previous section is somewhat complex, and hence in order not to blur the coherence of the exposition it had to be postponed. We are going to present it in this section. Remember that what we want to prove is that for every strongly normal language whose valuations are delimited by a class C of constraints there is a certain language with merely positive constraints such that the internalizations of all constraints from C are valid in it. In order to be able to prove this, we must first characterize the class of all constraints which are in force in a language whose space of acceptable valuations is delimited by C. Hence the question which is before us is the following: given that the space of acceptable valuations of a language L is delimited by the class C of constraints, what is the set of all constraints which are in force for L (which are ‘entailed’ by C)? The strategy we are going to employ answering it is based on the fact that what a constraint S1V1, ..., SnVn |= SV effects is the exclusion of all valuations of the sentences of L which map Si on Vi, but do not map S on V – in other words which (being subsets of the Cartesian product of the set of sentences of L and the set B of the two truth values) contain the set {<S1,V1,>,...,<Sn,Vn,>,<S,VC,>} where VC is the truth-value complementary to V. Now we are first going to characterize the set of all valuations not containing any of a given set of subsets of S×B, and then go on to transform this characterization, in several steps, into the desired characterization of all constraints ‘entailed’ by a given class of constraints. DEFINITION 10. Let S be the set of sentences of a language. A truth-valuative set (over S) (tvs) will be any subset of S×B. If Y = <S,V> is an element of S×B, then YC will denote the pair <S,VC>, where VC is the truth value complementary to V. A tvs is called an elementary contradiction if it is of the form {Y, YC}. A tvs is called a (truth-)valuation (tv) (of S) if it is a total function. A tvs is called inconsistent iff it is contained in no tv. LEMMA 3. A tvs is inconsistent iff it contains some Y together with YC; hence the set of all inconsistent tvs’s is the smallest set containing all elementary contradictions and closed to forming supersets. PROOF: Obvious. DEFINITION 11. If C is a set of tvs’s, then a tvs is C-inconsistent if it is contained in no tv which does not have an element of C for its subset. It might seem that C-inconsistent tvs’s are precisely those which contain either an elementary contradiction or an element of C. But a more careful consideration reveals that this is not the case. Consider a tvs X and imagine we want to extend it to a tv not containing an element of C. The extension is clearly impossible if X contains an elementary contradiction or an element of C. Otherwise we can imagine that the extension proceeds in the following way: we take one of the elements of S which does not belong to the domain of X (i.e. such that <S,V> does not belong to X for any V), form the pair <S,V>, where V is an arbitrarily chosen truth value, and add it to X; and we repeat this for all the elements of the complement of the domain of X in S. Could such a procedure fail to provide the desired extension? Indeed it could: it might happen that both X∪{<S,1>} and X∪{<S,0>} belong to C (or an analogous thing
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may happen at some later step). In such a case X is not extendable to a tv not containing an element of C; and hence is C-inconsistent. In other words, we have another closure condition for the set of Cinconsistent sets: X is C-inconsistent if both X∪{<S,1>} and X∪{<S,0>} are C-inconsistent. DEFINITION 12. The tvs X is a resolution of the tvs’s Y1 and Y2 iff there is a sentence S and a truth value V so that Y1=X∪{<S,V>} and Y2=X∪{<S,VC>}. LEMMA 4. The set of C-inconsistent tvs’s is the smallest set containing all elementary contradictions and all elements of C, and closed to forming supersets and resolutions. PROOF: We have seen that the only tvs which cannot be extended to a tv is one which either contains an elementary contradiction, or contains an element of C, or is the resolution of two C-inconsistent tvs’s. DEFINITION 13. A quasiconstraint (qc) is an ordered pair <X,Y> where X is a tvs and Y is an element of S×B. A tv is said to conform to the qc if it either does not contain X or contains Y. A qc is said to be valid if it is conformed to by every tv. If C is a set of qc’s, then a qc is said to be C-valid if every tv which conforms to every element of C conforms to it. The qc <X,YC> is said to be a representation of the tvs X∪Y; X∪YC is said to be the projection of <X,Y>. Two qc’s are equivalent iff they have the same projection. LEMMA 5. A qc is valid iff its projection is inconsistent. It is C-valid iff its projection is C*-inconsistent (where C* is the set of all projections of elements of C). PROOF: Obvious. It is clear that a set contains all C-valid qc’s iff: it contains representations of all C*-inconsistent sets and it is closed to the equivalence of qc’s (i.e. iff it always contains all constraints equivalent to a constraint it contains). LEMMA 6. A set contains all C-valid qc’s iff: (i) it contains <{Y},Y> for every Y (ii) it contains all elements of C (iii) it contains <X∪{Y},S> whenever it contains <X,S> (iv) it contains <X,S> whenever it contains <X∪{Y},S> and <X∪{YC},S> (v) it contains <{X1,...,Xi-1,YC,Xi+1,...,Xn},XiC> whenever it contains <{X1,...,Xn},Y> PROOF: (i) guarantees that the set contains representations of all elementary contradictions; (ii) guarantees that it contains representations of all projections of C. (iii) guarantees that it contains a representation of a tvs, then it contains a representation of every its superset; whereas (iv) guarantees that if it contains representations of two tvs’s which have a resolution, then it contains a representation of the resolution. (v) then guarantees that if the set contains a representation of a tvs, then in contains all other representations of the same tvs. DEFINITION 14. A constraint (c) is an ordered pair <X,Y> where X is a n-tuple of elements of S×B and Y is an element of S×B. A c <<X1,...,Xn>,Y> is said to represent the qc <{X1,...,Xn},Y>; whereas the latter is said to be a projection of the former. Two constraints are said to be equivalent if their projections coincide.
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LEMMA 7. A set contains all C-valid c’s iff: (i) it contains <,Y> for every Y (ii) it contains all elements of C (iii) it contains <<X0,X1,...,Xn>,Y> whenever it contains <<X1,...,Xn>,Y> (iv) it contains <<X2,...,Xn>,Y> whenever it contains <<X1,...,Xn>,Y> and <<X1C,...,Xn>,Y> (v) it contains <,X1C> whenever it contains <<X1,...,Xn>,Y> (vi) it contains <<Xπ(1),...,Xπ(n)>,Y> whenever it contains <<X1,...,Xn>,Y>, for every permutation π of {1,...,n} (vii) it contains <<X1,...,Xn>,Y> whenever it contains <<X1,X1,...,Xn>,Y> PROOF: A c is C-valid iff it is a representation of a C*-valid qc, where C* is the set of projection of the elements of C. Hence a set contains all C-valid c’s iff it contains a representation of every C*-valid qc and it contains every c equivalent to a c it contains. The former condition is guaranteed by (i)-(v) which straightforwardly correspond to the clauses of the previous lemma. The latter is guaranteed by (vi) and (vii) (that the set contains <<X1,X1,...,Xn>,Y> whenever it contains <<X1,...,Xn>,Y> follows from (iii)). Now we can pass over from constraints to sentences (remember that the (►,#-)internalization of the constraint S1V1, ..., SnVn |= Sn+1Vn+1 is the sentence (X1►...(Xn►Xn+1)), where Xi is Si if Vi is 1 and is #Si if Vi is 0) LEMMA 8. A set of sentences contains the ►,#-internalizations of all C-valid c’s iff: (i) it contains (S►S) for every sentence S (ii) if c∈C, then it contains the ►,#-internalization of c (iii) it contains X0►(X1►...(Xn►Y)...) whenever it contains X1►(...(Xn►Y)...) (iv) it contains X2►(...(Xn►Y)...) whenever it contains X1►(X2►...(Xn►Y)...) and #X1►(X2►...(Xn►Y)...) (v) it contains X1►(...(#Y►#Xn)...) whenever it contains X1►(...(Xn►Y)...) (vi) it contains (Xπ(1)►(...(Xπ(n)►Y)...) whenever it contains X1►(...(Xn►Y)...), for every permutation π of {1,...,n} (vii) it contains X1►(...(Xn►Y)...) whenever it contains X1►(X1►...(Xn►Y)...) and vice versa. PROOF: Each of the clauses of the lemma obviously guarantees that the set contains the internalizations of all c’s specified in the corresponding clause of the previous lemma. LEMMA 9. A set of sentences contains the internalizations of all C-valid c’s iff it contains all sentences of the following shapes: (i) X►X (ii) the ►,#-internalization of c, where c∈C (iii) (X1►...(Xn►Y))►(X0►(X1►...(Xn►Y)...)) (iv) (X1►(X2►...(Xn►Y)))►((#X1►(X2►...(Xn►Y)))►(X2►...(Xn►Y))) (v) ((X1►...(Xn►Y)))►(X1►...(#Y►#Xn)))) (vi) (X1►...(Xn►Y))►(Xπ(1)►...(Xπ(1)►Y)) for every permutation π of {1,...,n} (vii) ((X1►(X1►...(Xn►Y))))►(X1►...(Xn►Y)))) and (viii) it contains Y whenever it contains (X►Y) and X
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PROOF: Given (viii), (iii)-(vii) obviously have the power of the corresponding clauses of the previous lemma. LEMMA 10. Let a set of sentences contain all sentences of the following shapes: (ii) Y►(X►Y) (iii) (#Y►#X)►(X►Y) (iv) (X1►X2)►((X2►X3)►(X1►X3)) (v) (X►(X►Y))►(X►Y) and let it (vi) contain Y whenever it contains X►Y and X. Then the set contains all sentences of the following shapes: (a) Y►Y (b) X1►((X1►X2)►X2) (c) (X1►(X2►X3))►(X2►(X1►X3)) (d) #X1►(X1► X2) (e) ##X1►X1 (f) X1►##X1 (g) (X2►X3)►((X1►X2)►(X1►X3)) (h) (X►Y)►(#Y►#X) (i) (X1►X2)►((#X1►X2)►X2) (j) X1►(#X2►#(X1►X2)) PROOF: As the statements (ii)-(v) and the rule (vi) straightforwardly correspond to an axiomatization of the classical propositional calculus (the Tarski's one mentioned above), the proof is straightforwardly analogous to the proof of the fact that statements corresponding to (a) – (j) are provable within the system. In order to make it more comprehensible, let us adopt the following notational conventions: A formula following by the symbol n in brackets is a shorthand for ‘F belongs to the set for it is of the form n’ (where n is either a letter marking a formula proved earlier or a number formula occuring earlier in the proof). A formula followed by two symbols n1 and n2 in brackets is a shorthand for ‘F belongs to the set in force of (vi) applied to the formulas n1 and n2’. Moreover, we will make use of the fact that if the set contains a formula (X1►...(Xn►Y)), then if it contains its ‘antecedents’ X1, ..., Xn, it is, in force of (iv) bound to contain Y: hence a formula followed by a letter L and symbols n1 through nn in brackets will be a shorthand for ‘F belongs to the set in force of the set containing L and its ‘antecedents’ n1 through n2’. Now we can give the proofs of the clauses (a)-(j) in the following concise form: a: 1 (ii) (Y►(Y►Y)) 2 (v 1) (Y►Y) b: 1 (ii) X1►((X1► X2)► X1) 2 (iv) ((X1► X2)►X1)►(X1►X2)►((X1► X2)►X2) 3 (iv 1,2) X1►(X1►X2)►((X1► X2)►X2) 4 (v) (X1►X2)►((X1►X2)► X2)►((X1►X2)►X2) 5 (iv 3,4) X1►((X1►X2)►X2) c:
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1 (iv) (X1►(X2►X3))►(((X2►X3)►X3)►(X1►X3)) 2 (iv) X2►((X2►X3)►X3)►((((X2►X3)►X3)►(X1►X3))►(X2►(X1►X3))) 3 (a) X2►((X2►X3)►X3) 4 (2,3) ((((X2►X3)►X3)►(X1►X3))►(X2►(X1►X3))) 5 (iv 1,4) ((X1►(X2►X3))►(X2►(X1►X3))) d: 1 (ii) (#X1►(#X2► #X1)) 2 (iii) ((#X2►#X1)►(X1► X2)) 3 (iv 1,2) (#X1►(X1► X2)) e: 1 (c) (##X1►(#X1►#X2)) 2 (iii) ((#X1►#X2)►(X2►X1)) 3 (iv 1,2) (##X1►(X2►X1)) 4 (c 3) (X2►(##X1►X1)) 5 (4) ((X2►(##X1►X1))►(##X1►X1)) 6 (4,5) (##X1►X1) f: 1 (d) (###X1►#X1) 2 (iii) ((#X1►#X1)►(X1►X1)) 3 (1,2) (X1►##X1) g: 1 (iv) (X1►X2)►((X2►X3)►(X1►X3)) 2 (b) (X1►X2)►((X2►X3)►(X1►X3))►(X2►X3)►((X1►X2)►(X1►X3)) 3 (1,2) (X2►X3)►((X1►X2)►(X1►X3)) h: 1 (iv) (##X1►##X2)►((##X2►X2)►(##X1►X2)) 2 (c 1) (##X2►X2)►((##X1►##X2)►(##X1►X2)) 3 (e) (##X2►X2) 4 (2,3) ((##X1►##X2)►(##X1►X2)) 5 (iv) (X1►##X1)►((##X1►X2)►(X1►X2)) 6 (f) (X1►##X1) 7 (5,6) ((##X1►X2)►(X1►X2)) 8 (iv 4,7) ((##X1►##X2)►(X1►X2)) i: 1 (iv) (#X2►#X1)►((#X1►(X1►X3))►(#X2►(X1►X3))) 2 (c 1) (#X1►(X1►X3))►((#X2►#X1)►(#X2►(X1►X3))) 3 (d) (#X1►(X1►X3)) 4 (3,2) ((#X2►#X1)►(#X2►(X1►X3))) 5 (iv) (#X2►X1)►((X1►(#X2►X3))►(#X2►(#X2►X3))) 6 (c 5) (X1►(#X2►X3))►((#X2►X1)►(#X2►(#X2►X3))) 7 (c) ((#X2►(X1►X3))►(X1►(#X2►X3))) 9 (iv 6,7) (#X2►(X1►X3))►((#X2►X1)►(#X2►(#X2►X3))) 10 (v) ((#X2►(#X2►X3))►(#X2►X3))) 11 (iv) ((#X2►X1)►(#X2►(#X2►X3)))►(#X2►(#X2►X3))►(#X2►X3)),((#X2►X1)►(#X2►X3)))) 12 (iv 9,11) (#X2►(X1►X3))►((#X2►X1)►(#X2►X3))
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13 (iv 4,12) (#X2►#X1)►((#X2►X1)►(#X2►X3)) 14 (h) ((X1►X2)►(#X2►#X1)) 15 (iv 14,15) (X1►X2)►((#X1►X2)►(#X3►X2)) 16 (15) (X1►X2)►((#X1►X2)►(##(X1►X1)►X2)) 17 (a) (##(X1►X1)►X2)►(##(X1►X1)►X2) 18 (c 17) ##(X1►X1)►((##(X1►X1)►X2)►X2) 19 (a) (X1►X1) 20 (f 19) ##(X1►X1) 21 (20,18) ##(X1►X1)►((##(X1►X1)►X2)►X2) 22 (18,21) ((##(X1►X1)►X2)►X2) 23 (g 22) ((#X1►X2)►(##(X1►X1)►X2))►((#X1►X2)►X2) 24 (iv 15,23) (X1►X2)►((#X1►X2)►X2) j: 1 (a) ((X1►X2)►(X1►X2)) 2 (c) ((X1►X2)►(X1►X2))►(X1►((X1►X2)►X2)) 3 (1,2) X1►((X1►X2)►X2) 4 (h) (((X1►X2)►X2)►(#X2►#(X1►X2))) 5 (iv 3,4) (X1►(#X2►#(X1►X2))) LEMMA 11. A set of sentences contains the representations of all C-valid c’s iff it contains all sentences of the following shapes:: (i) the f,g-internalization of c, where c∈C (ii) Y►(X►Y) (iii) (#Y►#X)►(X►Y) (iv) (X1►X2)►((X2►X3)►(X1►X3)) (v) (X►(X►Y))►(X►Y) and (vi) it contains Y whenever it contains (X►Y) and X. PROOF: Let us prove that given (i)-(vi), the individual clauses of the previous lemma are fulfilled. I will distinguish the numbers of the clauses of the previous lemma from those of the current one by putting them into square brackets. The clauses [ii], [iii], [vii] and [viii] are unproblematic, they follow directly from (i), (ii), (v) and (vi). Hence we are left with proving [i], [iv], [v] and [vi]. [i] is Lemma 10(a). [iv] is an instance of Lemma 10(h). As for [v], it follows from (iii) that the set contains (Xn►Y)►(#Y►#Xn). Now as it follows from Lemma 10(g) that it contains (X1►X2)►(X1►X3) whenever it contains (X2►X3), it must contain (Xn-1►(Xn►Y))►(Xn-1►(#Y►#Xn)), and, by repetitive application of the same step, Now consider [vi]: as the set contains ((X1►...(Xn►Y)))►(X1►...(#Y►#Xn)). ((Xi►Xi+1►...(Xn►Y)))►(Xi+1►(Xi►...(Xn►Y))) which is an instance of the formula in Lemma 10(c), and as it contains (Y►X1)►(Y►X2) whenever it contains (X1► X2) (which follows from Lemma 10(g)), it must contains ((Xi-1►(Xi►Xi+1►...(Xn►Y))))►(Xi-1►(Xi+1►(Xi►...(Xn►Y)))); and, by the reperitive application of the same step, it must contain also (X1►...(Xi►(Xi+1►...(Xn►Y)))))►(X1►...(Xi+1►(Xi►...(Xn►Y)))). And as any permutation of a finite sequence is the result of a finite number of exchanges of consequent elements of the sequence, it contains (X1►...(Xn►Y))►(Xπ(1)►(...(Xπ(n)►Y)...)) for every permutation π of {1,...,n}.
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LEMMA 10. Let (A1)-(A4) and (MP) be in force for a language L. Then the ►,#-internalizations of (N1), (N2), (MI1), (MI2), and (MI3) are valid in L. PROOF: The internalizations are: (N1I) S►##S (N2 I) #S►#S (MI1I) #S1►(S1►S2) (MI2 I) S2►(S1►S2) (MI3 I) S1►(#S2►#(S1►S2)) (N1I) is an instance of Lemma 10(f); (N1I) is is an instance of Lemma 10(a); (MI1I) is an instance of Lemma 10(d); (MI2 I) is an instance of (ii); and (MI3 I) is an instance of Lemma 10(i). Now we are ready to present the proof of the ‘Internalization Theorem’: PROOF OF THEOREM 6. Let L be a language, ► and # be a binary resp. a unary function from the set of sentences of L to the same set, and let the space of acceptable valuations of L be delimited by (MI1), (MI2), (MI3), (N1), (N2) and a set C of other constraints. Let L∗ be a language which is just like L with the exception that its space of acceptable valuation is delimited by the set of constraints containing the ►,#-internalizations of all constraints from C plus the constraints (A1)-(A4) and (MP). As accodring to the previous lemma the ►,#-internalizations of (MI1), (MI2), (MI3), (N1), (N2) are valid in L∗, the internalizations of all constraints of delimiting the space of acceptable valuations of the sentences of L are valid in L∗. Let C* be C∪{(MI1), (MI2), (MI3), (N1), (N2)}; then to be in force in L is to be C*-valid; and it follows from Lemma 11 that the internalizations of all C*-valid constraints are valid in L∗.
Conslusion The upshots of the above considerations is that the relstionship between the semantics and the axiomatics of the predicate calculus is more intricate then it might prima facie seem; and that we are not able to fix the denotations of the standard logical constants in terms of axioms. (This is of course, one of the sources of the relative popularity of the intuitionistic logic, whose relationship to axioms and rules is much more straightforward – see Peregrin, 2004). What we have gained as a by-product is a story about why the axioms of an axiomatic system of the classical propositional logic are such as they are: we can read them as outcomes of a proces of articulation of restrictions of acceptability of truth valuations.
References Carnap, R. (1943): Formalization of Logic, Harvard University Press, Cambridge (Mass.). Kalmár, L. (1936): 'Zurückführung des Entscheidungsproblems auf den Fall von Formeln mit einer einzigen binären Funktionsvariablen', Comp. math. 4, 137-144. Koslow, A. (1992): A Structuralist Theory of Logic, Cambridge University Press, Cambridge. Mendelsohn, E. (1964): Introduction to Mathematical Logic, Wadsworth & Brooks, Pacific Grove. Peregrin, J. (1997): 'Language and its Models', Nordic Journal of Philosophical Logic 2, 1-23.
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Peregrin, J. (2004): 'Logic as "making it explicit"', The Logica Yearbook '03, Filosofia, Praha. Tarski, A. (1965): Introduction to Logic and to the Methodology of Deductive Sciences, Oxford University Press, Oxford.
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