Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox (Logic, Epistemology, and the Unity of Science, 8)

UNITY,\TRUTH\AND THE\LIAR

LOGIC, EPISTEMOLOGY, AND THE UNITY OF SCIENCE VOLUME 8

Editors Shahid Rahman, University of Lille III, France John Symons, University of Texas at El Paso, U.S.A. Editorial Board Jean Paul van Bendegem, Free University of Brussels, Belgium Johan van Benthem, University of Amsterdam, the Netherlands Jacques Dubucs, University of Paris I-Sorbonne, France Anne Fagot-Largeault Collège de France, France Bas van Fraassen, Princeton University, U.S.A. Dov Gabbay, King’s College London, U.K. Jaakko Hintikka, Boston University, U.S.A. Karel Lambert, University of California, Irvine, U.S.A. Graham Priest, University of Melbourne, Australia Gabriel Sandu, University of Helsinki, Finland Heinrich Wansing, Technical University Dresden, Germany Timothy Williamson, Oxford University, U.K. Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light no basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity.

For other titles published in this series, go to www.springer.com/series/6936

Unity,\Truth and\\the\Liar The\Modern\Relevance\of\Medieval Solutions\to\the\Liar\Paradox Edited by

Shahid\Rahman University\Charles-de-Gaulle\-\Lille\III,\France

Tero\Tulenheimo University\of\Helsinki, Finland and

Emmanuel Genot University Charles-de-Gaulle\-\Lille\III,\France

ABC

Emmanuel Genot U.F.R. de Philosophie ,, Domaine Universitaire Pont de Bois Universite´ Lille III , 59653 Villeneuve d Ascq France [email protected] ,,

Shahid Rahman U.F.R. de Philosophie ,, Domaine Universitaire Pont de Bois Universite´ Lille III , 59653 Villeneuve d Ascq France [email protected] ,,

Tero Tulenheimo Department of Philosophy University of Helsinki FI-00014 Helsinki Finland [email protected]

Cover image: Adaptation of a Persian astrolabe (brass, 1712–13), from the collection of the Museum of the History of Science, Oxford. Reproduced by permission.

ISBN 978-1-4020-8467-6

e-ISBN 978-1-4020-8468-3

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Preface

And in my haste, I said: “All men are Liars” —Psalms 116:111

The Original Lie Philosophical analysis often reveals and seldom solves paradoxes. To quote Stephen Read: A paradox arises when an unacceptable conclusion is supported by a plausible argument from apparently acceptable premises. [...] So three different reactions to the paradoxes are possible: to show that the reasoning is fallacious; or that the premises are not true after all; or that the conclusion can in fact be accepted.2

There are sometimes elaborate ways to endorse a paradoxical conclusion. One might be prepared to concede that indeed there are a number of grains that make a heap, but no possibility to know this number. However, some paradoxes are more threatening than others; showing the conclusion to be acceptable is not a serious option, if the acceptance leads to triviality. Among semantic paradoxes, the Liar (in any of its versions) offers as its conclusion a bullet no one would be willing to bite.3 One of the most famous versions of the Liar Paradox was proposed by Epimenides, though its attribution to the Cretan poet and philosopher has only a relatively recent history. It seems indeed that Epimenides was mentioned neither in ancient nor in medieval treatments of the Liar 1 Jewish

Publication Society translation. [1]. 3 Save, perhaps, for the dialetheist, who would be willing to accept that the Liar sentence is both true and false. 2 Read

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Paradox.4 One source of this attribution is most probably Paul of Tarsus, who in his Epistle to Titus (1:12) writes: One of themselves, a prophet of their own, said, Cretans are always liars, evil beasts, idle gluttons.5

This saying, which Paul does not explicitly mention as paradoxical, but only as ‘true’6 has been later identified with a fragment of Epimenides’ writings (though we do not dare to enter the scholarly disputes around the identification of the proper fragment).7 It seems that Epimenides was condemning the widespread Cretan religious belief of Zeus’ mortality (for whom the Cretans had allegedly erected a tomb), though once again this is a disputed conjecture.8 That this utterance was not considered paradoxical, though widely discussed and commented on by the early Church Fathers, may at first be puzzling. But we may suppose (applying some version of the Charity Principle) that if this was so, it was not because they failed to notice its paradoxical nature, but because they considered that Epimenides’ saying was not meant to apply to itself. Such a proposal might indeed be seen as a way out of the paradoxes arising from self-reference. Excluding Epimenides from the set of intended Cretans would offer a kind of solution, since if this was how Epimenides’ claim should be interpreted, there would be no paradox left. The same result would be obtained by proposing that one should simply exclude Epimenides’ utterance of the sentence ‘The Cretans are always liars’ from the set of utterances made by the Cretans, all the members of which Epimenides wishes to claim are false. Another famous paradox, namely the Barber Paradox, has a simpleminded solution in the very same spirit. Assume that the barber shaves all and only those who do not shave themselves; assume also (or, on a suitable view of presupposition, presuppose) that we are only talking about those who actually need to shave, thus excluding women from the range of the quantifier ‘all and only those who . . . ’. Now, one way to avoid paradoxical conclusions regarding who shaves the barber, while maintaining 4 It is a matter of scholarly dispute as to who was the first to attribute to Epimenides the paternity of the Liar as a paradox. Since Russell and Prior this attribution is a matter of common knowledge among logicians and philosophers (see [2, 3]). 5 American Standard Version translation. 6 He indeed adds: ‘This testimony is true’ (1:13). 7 The identification of the anonymous Cretan with Epimenides is, however, ancient, and dates back to Clement of Alexandria (Stromata, 14) who himself does not consider the saying as paradoxical, but rather finds the attribution of a true statement to a pagan surprising. 8 It was Origen (Against Celsus, 3:53) and Athenagoras (A Plea for the Christians, 30) who connected the Liar Paradox to this religious belief, though the quote ‘Cretans, always Liars’ stems from a hymn to Zeus attributed to Callimacchus of Cyrene who, living in a Greek Libyan colony, was hence not uttering a paradox.

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that the description of the barber is correct, would be to suggest that the barber lies outside the range of the relevant quantifier: the barber is a woman. Such a dissolution strategy simply excludes those individuals from the range of the relevant quantifier whose inclusion therein would give rise to a paradox. It may be a disappointment to the reader that Epimenides’ original statement was not, in its most likely rendering, a paradox after all. In any event, there is a lesson to be learned from Epimenides’ hasty statement: in some cases at least, one may reasonably attempt to rid oneself of a paradox by suitably restricting the range of the relevant quantifiers. As history teaches us, this is the way Russell wished to avoid the paradox of the set of all sets. Tarski acknowledges Russell’s solution as a source of his own systematic distinction between language and metalanguage. But such a solution may have unintended consequences when one wishes to model the semantics of a natural language employing a formal one.

Why Paradoxes Matter Semantics deals with meaning – even though one of the tasks of semantics, as an object of study of the philosopher, is to explain away ‘meanings’ conceived as abstract entities attached to our words by some sort of metaphysical glue. Since the work of Frege, Wittgenstein and Tarski, the notion of truth has been seen by many as a relatively well understood notion, in terms of which the notion of meaning could be explicated. Adopting this perspective, one might suggest that to grasp the meaning of a sentence can be equated with grasping its truth conditions; and to propose that the meaning of a word consists of the contribution it makes to determine the truth condition of a sentence in which it occurs. To give but two examples, Carnap’s replacement of the Fregean distinction sense/denotation by the distinction intension/extension, or Davidson’s ‘bold conjecture’ that a theory of meaning for a language is a theory of truth for that language, are both proposed moves from meaning to truth conditions. Truth-conditional semantics is often linked with the idea that questions about meaning are best interpreted as questions about understanding. To understand the meaning of a sentence, it is said, is to know its truth conditions. However, it is by no means philosophically unproblematic to attempt reducing considerations about meaning to considerations of truth conditions. Michael Dummett, who has delved into the notion of understanding, has launched the so-called ‘anti-realist’ (or ‘justificationist’) theory of meaning, according to which the understanding of a sentence consists of an ability to recognize, in appropriate circumstances,

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whether it is true or false – grasping the sense of a statement consists of an ability to recognize evidence for the statement when presented with it.9 Under this view, if a putative truth condition of a sentence S is of such a kind that it could not even in principle be recognized whether the condition prevails or not, then S will not qualify as true or false in any circumstances. Jaakko Hintikka’s game-theoretical semantics can be seen – as Hintikka himself argues – as a semantic theory that is at the same time verificationist (hence arguably doing justice to the anti-realists’ concerns) and truth-conditional: truth conditions are themselves defined in terms of activities of verification and falsification.10 The notions of meaning and truth are intimately related, be one’s overall position that of an anti-realist or that of a proponent of (some variant of) truth-conditional semantics. In a semantic theory of truth, the truth conditions of sentences of the formal language considered are specified inductively. This may happen in various ways. In Tarski’s truth definition, one can compute the satisfaction conditions of a complex formula on a model from the satisfaction conditions of its simpler constituents and the way in which these are put together, while, for instance, in Hintikka’s game-theoretical semantics, two-player games are inductively associated with pairs of sentences and models, and truth conditions are specified by reference to the existence or non-existence of a winning strategy for a certain player. Generally, one faces the question of rationally choosing a semantic theory of truth to serve as a basis of a theory of meaning. Tarski’s theory is fitting when attention is restricted to a formal language (such as first-order logic), but cannot unproblematically and without further ado be applied in connection with natural languages. Once again, to mention but two phenomena ubiquitous in ordinary language, selfreference and indexicality cannot be dealt with on the object language level in Tarski’s theory.11 Tarski saw in the Liar sentence a possibility of arguing generally for the undefinability of truth. No sufficiently strong language (one capable of speaking of its own semantics) can define its own truth. For if it was possible, the sentence ‘This sentence is false’ could be formulated within such a language. Now this sentence would either be true, or else fail to be true. It would follow that the sentence is neither true nor false, contrary to the law of excluded middle, which Tarski took for granted. Tarski’s 9 Cf.

[4]. For more recent expositions of the justificationist standpoint, see [5, 6]. (For an approach to the anti-realist theory of meaning, based on Martin-Löf’s intuitionistic type theory, see [7].) 10 Hintikka argues that Wittgenstein implicitly held such a view on meaning. See, e.g., [8, 9]. 11 S. Read’s target paper (Chap. 1) reminds us of some of the well-known failures of Tarski’s T-scheme in similar cases (but see also Jan Woleński’s defense, Chap. 12).

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famous theorem that there can be no explicit first-order truth definition for a first-order language of arithmetic is a precise mathematical result whose proof makes use of a formalized version of the Liar sentence. If there was an arithmetical predicate τ (·) true of all and only the Gödelnumbers
ψ of sentences ψ true in the standard model N of arithmetic, it could be shown that there is a sentence φ true in N if and only if the sentence ¬τ (
φ) is true in N . But this is impossible, for if τ (·) is a truth predicate, ¬τ (
φ) in effect asserts of φ that it is not true.12 Given Tarski’s assumptions, no explicit truth definition of a language in that language itself is, then, possible. On the other hand, there is no obstacle to giving a truth definition for the object language in a metalanguage that is essentially stronger than the object language. This is why the systematic distinction between an object language and a metalanguage is crucial for Tarski’s semantic theory of truth.13 If not all Tarski’s assumptions are accepted, it even becomes possible to find a language explicitly defining its own truth predicate. Examples are furnished by Kripke’s three-valued interpretation (to which we will return soon) and Hintikka’s independence-friendly logic.14 In both cases a truthdefinition is made possible by the fact that not all instances of the law of excluded middle hold. For Kripke, this happens because the semantics of first-order logic is suitably redefined at the outset (the union of the extension and antiextension of a unary predicate need not exhaust the domain), while for Hintikka this is due to the way he extends first-order syntax and interprets the resulting language utilizing game-theoretical semantics. Let us spend some more words on Tarski’s theory, as it is what most contributions in this volume are concerned with. In Tarski’s theory, it is possible to define truth while avoiding the Liar paradox, precisely by making a rigid distinction between an object language and a metalanguage. Since there is no such distinction in connection with bare natural languages, no such remedy to the paradox seems open to them. Given Tarski’s assumptions (notably the law of excluded middle, and compositionality of semantics), the option that remains is of course to accept 12 It is useful to bear in mind that Tarski proved — in addition to the result that first-order arithmetical truth is not explicitly definable in first-order logic — that first-order arithmetical truth is implicitly definable in first-order logic! That is, if the arithmetical vocabulary considered is L, there is a first-order formula Ψ(x) of a larger vocabulary L ∪ {SAT } such that for any first-order L-sentence ψ, we have N |= ψ if and only if there is an interpretation S ⊆ N2 of the binary predicate SAT such that (N , S) |= Ψ(
ψ). Intuitively, SAT xy says that x is the Gödel-number of an L-sentence ψ, and y encodes a variable assignment γ such that N , γ |= ψ. 13 See [10]. 14 See [11, 8].

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non-definability of truth in natural languages. As truth is a key semantic notion, Tarski’s result has been widely taken as supporting the idea that natural language semantics cannot be expressed in these languages.15 On the other hand, even if no explicit truth predicate can be formulated in natural languages (supposing Tarski’s assumptions hold for them), still one may of course put into use, in studying semantics, various aspects of the Tarskian framework. Hence for instance compositionality has become, for better or worse, virtually a standard in much semantic theorizing. Formal semantics plays an important role in the study of natural language: Montague semantics, Hintikka’s game-theoretical semantics, Kamp’s Discourse Representation Theory, as well as Groenendijk and Stokhof’s Dynamic Predicate Logic – all make essential use of formal model-theoretic tools to analyze natural language phenomena which were beyond the scope of formal semantics as originally formulated by Tarski. Against this background, semantic paradoxes appear to be of general interest: as in the case of the Liar paradox, they may reveal important limitations with which a semantic theory of truth is bound to comply.

Medieval Solutions. . . Medieval logicians, for reasons of their own, developed several lines of response against the Liar Paradox. Some advocated restrictionism, or the idea that what a significant term stands for in the context of a proposition (its suppositio) is determined via an appropriate contextual restriction imposed on the precontextually given extension of the term. From this view, self-reference was ruled out by suitable postulates concerning restrictions. Others attempted to provide a modified account of meaning, based on the idea that meaning is closed under implication and entailment (closure postulate of meaning). In fact, in the target paper of this volume, Stephen Read develops a solution to the Liar that was first formulated by Thomas Bradwardine (1295–1349) who, like William Heytesbury (before 1313 – ca. 1373), opposed the solution proposed by the restrictionists and who explicitly stated the closure postulate of meaning.16 It is important to note that both restrictionists and their opponents assumed a rather precise theory of truth conditions for categorical 15 Tarski himself seems to have endorsed the view that natural language semantics was ineffable (see [12]). 16 S. Read’s reconstruction of Bradwardine’s theory is not uncontroversial, however, for it rests on a ‘charitable’ reading of Bradwardine’s closure principle (discussed and defended against other interpretations, including those of many contributors to this volume, at the end of his response, Chap. 13, Sect. 1).

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sentences, and that this theory was based on the notion of suppositio. Thus the medieval discussions on the Liar were centered around the link between truth and meaning – just as the discussion in contemporary (analytic) philosophy. Because of this fact it is hardly surprising that medieval solutions, once recast in modern parlance and formalism, still offer valuable insights. They even point at an alternative conception (or definition) of truth, which can be made use of in attempting to escape the Liar tangle. It may be particularly useful when moving beyond formal languages. One issue, addressed repeatedly in the present volume – though not often explicitly – is a strong assumption made by Tarski, namely the uniqueness of meaning.17 If sentences have exactly one meaning, or (to use S. Read’s expression) ‘say’ exactly one thing, then the Liar sentence and its variants are paradoxical because they seem to have an indeterminate meaning. What a sentence18 says is at the very heart of the Aristotelian conception: as Ross translates Metaphysics IV, 7: To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false.19

In interpreted formal languages, as normally conceived, the uniqueness assumption comes of course for free, since extensions are assigned to non-logical symbols by functions, therefore, applying usual definitions of logical operators, a given sentence says exactly one thing (there is, up to logical equivalence, one and only one metalanguage truth condition for the sentence). If one wishes to entertain a broader conception of meaning, perhaps with the motivation of doing justice to the complexity of everyday speech, the question can be raised whether (an utterance of) a sentence can say more than one thing. A positive answer to this last question immediately raises another one: how will quantification over those things that (an utterance of) a sentence says enter into an account of its truth conditions? According to S. Read, the heart of Bradwardine’s solution is this: once it is admitted that a sentence possibly says a great many things,20 and 17 Or,

as C. Dutilh Novaes names it (Chap. 3) the Principle of Uniformity. a given utterance of a type sentence, or a given speaker by uttering a given occurrence of a given type of sentence. 19 Tarski mentions explicitly this passage (quoted without reference in Tarski (1944), op. cit., Sect. 3), and introduces it writing that he ‘should like [his] definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth – intuitions which find their expression in the well-known words of Aristotle’s Metaphysics’. 20 But not generally of itself that it is true, on pain of trivializing the account of truth conditions (see S. Read’s target paper, Chap. 1, Sect. 6). 18 Or

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given a reasonably accurate account of ‘saying that’, the Liar sentence turns out to be simply false: among what the Liar sentence in effect ‘says’, given suitable closure postulates (e.g. that a sentence says whatever is implied by what it says), there is a contradiction.21 This volume is an attempt at evaluating the consequences of this solution, once it is recast in ‘modern’ terms, and given a precise (formal) expression.

. . . And Modern Approaches This modern aggiornamento of Bradwadine’s solution may be more telling if contrasted with the (nearly) dominant approach to paradoxes – one we have already mentioned – namely the Kripke (or Kripke– Feferman) theory. The Kripke theory is very famous, having dominated most discussions of the Liar for the past 30 years, but has still not been universally accepted as a solution. Kripke’s basic idea was to drop Tarski’s assumption of bivalence (or, as we said, assumption of the excluded middle), and give a partial interpretation to the truth predicate. Strictly speaking, there are still two values, true and false, but not every proposition need take either value. Kripke’s theory proceeds defining the truth predicate recursively from the set of well-formed sentences not containing it. Unlike Tarski’s hierarchy of self-contained truth predicates, Kripke allows for the truth being defined as the union of all the definition stages. Yet paradoxes are avoided since, after denumerably many stages, the recursive procedure reaches a fixed point. Some sentences may not be assigned any value at those fixed points, sentences that are not “grounded” in the (original) set of sentences (not containing the truth predicate). Neither the Liar (“This sentence is not true”) nor the Truthteller (“This sentence is false”) can be assigned any value, true or false, in any fixed point, since they are ungrounded. That way, Kripke can retain semantic closure, so that each language contains its own truth predicate, without contradiction nor (usual) paradoxes. Kripke is also concerned to retain (the spirit of) Tarski’s T-scheme, in that a sentence s and the sentence ‘s is true’ have always the same value, provided they have a value at all.22 This is at the expense of bivalence, for ungrounded sentences do not receive any truth-value. Though widely popular nowadays, Kripke’s theory

21 Indeed,

as conceded by S. Read in his response (Chap. 13), that at least some sentences (namely, those saying of themselves that they are false) necessarily say more than one thing is a necessary condition for Bradwardine’s (and hence, Read’s) solution to the Liar to work. 22 However, instances of the T-scheme are not expressible in the original Kripkean theory, since the three-valued Kleene semantics it is based on does not contain an adequate conditional connective.

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has not imposed itself as a solution, any more than Tarski’s. Appealing as it is, it still has the same consequences as Tarski’s theory: some semantic concepts cannot be explicitly defined in the object language. Just as Tarski would have denied the question “Is this sentence true?” any assignable meaning,23 so the question “Is this sentence grounded?” cannot be answered, and for analogous reasons: its presupposition fails. Without some story, yet to be told, about which sentences are grounded and which are not, it is unclear whether some conceptual gain has been obtained moving from Tarski’s theory to Kripke’s. Yet the motivation to maintain semantic closure clearly relates Bradwardine’s solution to Kripke’s, though it falls short of uniting them under the same banner. Bradwardine’s approach is very different, for he rejects the kind of equivalence Tarski frames in the T-scheme. Rather, Bradwardine defines truth in terms of signification: s is true iff things are only as s signifies, and this is a universally quantified formula, not a schema as in Tarski. And signification is (allegedly) a simpler concept to grasp than ‘groundedness’, or at least, it has some intuitive pedigree. Yet Bradwardine’s (and Read’s) move is likely to close the door to an extensional theory of meaning, since it is usually understood in terms of a grasp of truth conditions, and this would be blatantly circular. A recent offspring of the family of solutions spawned by Kripke’s initial theory is due to Tim Maudlin.24 In one sense, it is more relevant to the present discussion than Kripke’s original theory, once rephrased in terms of one of its commentators, Hartry Field [14]. Maudlin presents his theory as one where so called T-inferences – the ‘upward’ inference from s to ‘s is true’, and the ‘downward’ inference from ‘s is true’ to s – are both valid, but where the Excluded Middle fails. Field argues that this theory is better viewed as one where upward T-inference fails inferentially, though not semantically, because, according to Field, Maudlin’s notion of validity is primarily inferential. It is not our intention to evaluate the relative merits of Maudlin’s and Field’s formulations, but the latter has the advantage of allowing a clearer comparison with Bradwardine’s solution. Indeed summarized by Field, When Maudlin’s theory is put in terms of inferential validity, it turns out to be (an extension of) what’s often called the Kripke–Feferman theory (KF): a classical logic theory that allows Downward T-Inference but not Upward T-Inference (but contains the weaker rule that T [‘A’ is true] implies T> [‘ ‘A’ is true’ is true]). 23 We assume that the meaning of a yes-or-no question is given through the truth conditions of its potential answers, and that it is meaningful in a given situation (model, fixed point, etc.) if the disjunction of its potential answers is true in this situation (in the model, at the fixed point). 24 See [13].

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According to Field’s account, distinguishing what he calls ‘semantic validity’ and ‘inferential validity’, while Upward T-Inference is semantically valid, it is not inferentially valid. Indeed, Maudlin’s own account is that the Liar sentence, L, is false, and that it is false that L is false. From someone’s belief that L is false, one cannot then infer the belief that it is true that L is false (even though it is valid, i.e., semantically valid, but from this it does not follow that it also is inferentially valid). Since however belief is norm-dependent, in that the property of ‘being correct to believe’ is governed by norms none of which is uniquely best, it is (according to Maudlin, and in his own terms) permissible to believe that it is true that L is false, though Upward T-Inference is not what warrants this belief. There are other fascinating features of Maudlin’s theory (such as his account of groundedness), but for the purpose of drawing a comparison to Bradwardine’s theory, the overall picture is already sufficiently clear. Describing Maudlin’s theory in Field’s terms makes conspicuous the possible analogies and disanalogies with Bradwardine: both reject Upward T-Inference (though Maudlin does not describe it that way), but while Maudlin is in the strange (Moorean) position of believing that L is false and that it is false that L is false (though it is permissible to believe that it is true that L is false), Bradwardine distinguishes the statement, L, that L is false, which is false, from his own statement that L is false. Bradwardine’s explanation is that the first is self-referential while the second is not, and this is in essence his way out of the so-called revenge problem – that is, for any account of the Liar according to which L is false, to distinguish between the judgment made from the theory’s standpoint, and the Liar sentence itself.25 So the two theories, Bradwardine’s and Maudlin’s, have something in common, but differ in their explanation of why Upward T-Inference is to be rejected: for Maudlin it is (as described by Field) semantically valid but not inferentially so. For Bradwardine it is simply not valid, being a fallacy secundum quid et simpliciter, moving from a (true) statement which says only that L is false to (a false) one that says both that L is false and that L is true. Once again, the heart of the matter is that L says too many things, and if we follow Bradwardine and Read, what it says (of itself) is contradictory. Once we are in the Bradwardinean position to talk about L without self-reference, because we say that what L says is contradictory, hence false, without uttering the (self-referential) L, we need not be in the Maudlin–Moorean position which severs the link between doxastic permissibility and inferential validity. According 25 On

this, see B. Armour-Garb’s contribution, and S. Read’s response.

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to Bradwardine, one can infer that it is true that L is false from L being false, but not thanks to Upward T-Inference, but because what L says is a contradiction. Subsequently one can believe that it is true that L is false, not because of some norm-relative reason, but because of logic being a standard of doxastic health.

Content of this Volume This volume consists of two parts. The first part is somewhat reminiscent of medieval regimented controversies, and named after them Disputatio, with Stephen Read as Proponent. His reconstruction of Bradwardine’s solution (Chap. 1), as attempting both to revive a historical proposal, and to give it a formal reconstruction, can be discussed on several grounds: the adequacy of the interpretation of Bradwardine’s theory it offers, the nature of the formal apparatus used to express it, and its alleged merits (or flaws) with respect to other – historical or formal – proposals. The other contributors of this part, acting as Opponents, may be grouped into (somewhat unevenly between) those who: (1) discuss the theory itself; (2) discuss the nature of its formal reconstruction; (3) do both. The second part provides a comprehensive background for apprehending the conceptual contribution of medieval theories of truth conditions. The ‘restrictionist’ theory is especially discussed; this theory is fully compatible with uniqueness of meaning, as opposed to the ‘manifold’ theory of meaning. Bradley Armour-Garb (Chap. 2: “Read and Indirect Revenge”) calls into question whether Read’s account is able to cope with the problem of ‘indirect revenge’: finding a way to express that the Liar sentence is false without uttering the Liar sentence itself. Along the way he discusses Read’s position against the background of contemporary theories of truth (including deflationism and dialetheism). Central to this discussion is the notion of assertion, and the fact that a truth predicate is critical in expressing assertive commitments (such as ‘I believe that what X just said is true’). Catarina Dutilh Novaes (Chap. 3: “Tarski’s Hidden Theory of Meaning”) discusses one of the core features of S. Read’s account: the eventual plurality of meaning of a sentence. Giving independent grounds for denying the ‘Principle of Uniformity’ (of meaning), she explores the consequences of adopting a quantificational conception of truth. Important issues that arise in this connection are the need for a proper way to define falsity (a recurring concern in several other contributions of this volume), and the doubt that meaning, as a semantic notion, could be explained in non-semantic terms.

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Laurence Goldstein (Chap. 4: “Doubting Thomas: From Bradwardine back to Anon”) – while challenging S. Read’s solution to the Liar Paradox on the grounds that it may not be able to answer other semantic paradoxes – also attempts to articulate a solution of his own. He does so following a line rejected by Bradwardine (and hence by Read himself), the cassationist view, according to which some utterances of a given type of sentence can ‘say’ nothing, while others do indeed say something. Goldstein claims that his solution fares better as regards other semantic paradoxes (especially ‘iterative’ paradoxes) than Read’s. In return, Read argues against this claim in his response.26 But the success of Goldstein’s argument against Read’s position is independent of the relative merits of the two solutions. Namely, Read assumes explicitly that each sentence (to which his schema (A) applies, thus including any form of the Liar) says something,27 and it is this general contention that Goldstein challenges. Gyula Klima’s contribution (Chap. 5: “Logic Without Truth: Buridan on the Liar”) manifests the same type of interest for medieval solutions as Goldstein’s, in attempting to reconstruct Buridan’s solution. Klima argues, contra Read, that Buridan’s solution is independent of a specific theory of truth, since Buridan’s account of logical consequence is not defined in terms of truth. According to this reconstruction, Buridan aims at dealing with the problem of consistent use of semantic predicates in the presence of semantic closure – not at providing a solution to a problem pertaining to the theory of truth. Klima’s paper shows that Buridan’s theory nevertheless fails for other reasons, though he eventually suggests that a somewhat charitable interpretation of the theory might be defensible. In the body of papers of the present volume, Eugene Mills’ article (Chap. 6: “Scheming and Lying: Truth-Schemas, Propositions, and the Liar”) presents the peculiarity of attempting to argue against Read from the standpoint of a theory having so much in common with Read’s that one might expect any argument presented against the former to ‘bounce back’ to the latter. But the crucial point relates, according to Mills, to linguistic − and ultimately, logical − ontology (though Mills does not himself use these terms). Attacking Read’s ‘unacknowledged commitment to semantic essentialism’ (p. 122), Mills argues that his own solution is unhindered by this commitment. Read eventually ‘plead[s] guilty’ of this essentialism in his response (Sect. 2, p. 210). Mills’ contribution forces

26 Another feature of Goldstein’s position is that it considers utterances as bearers of truthvalues, just as the rival (to both the Bradwardine–Read and Anon–Goldstein) ‘weak restrictionism’ considered (and defended) by Panaccio (Chap. 14) does. 27 See Chap. 1, p. 8.

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anyone evaluating the eventual merits of the proposed solutions to pay due attention to their ‘ontological commitments’. Terence Parsons (Chap. 7: “Comments on Stephen Read’s ‘The Truthschema and the Liar’ ”) mounts a short but sharp attack against Read’s solution. He notes that Read does not accept the principle according to which every sentence says that it itself is true; and that Read’s argument for this conclusion is that if this principle indeed held, a theory incorporating this principle could determine no truth-value of any sentence. Parsons argues, however, that Read’s argument is not sound; according to Parsons, it merely follows from the acceptance of the principle every sentence says that it itself is true that a theory complying with it could determine no sentence to be true (while nothing would prevent determining some sentences to be false). Parsons goes on to argue that the failure of the stronger entailment leads Read’s original view into difficulties: he attempts to show that by the same token Read’s very theory cannot be used in determining any sentence to be true. Greg Restall’s contribution (Chap. 8: “Models for Liars in Bradwardine’s Theory of Truth”) proposes a thorough analysis of the kind of conditional by means of which the “Bradwardine Axiom” (that every proposition signifies everything which follows from it) should be formalized. After having shown the difficulties associated with the material conditional reading, Restall explores two ways: a modal reading of the conditional, and a relevant reading (within the Anderson, Belnap and Dunn tradition of relevant logic). Restall argues that Bradwardine’s argument fails in some models of both families (modal and relevant): the Liar sentence fails to assert its own truth, and then to signify a contradiction. Restall proposes several options to develop a “rich theory of signification” along the lines advocated by Bradwardine, none of which leaves intact the original argument. José M. Sagüillo’s contribution (Chap. 9: “On a New Account of the Liar”) attempts to compare ‘logics’ underlying, respectively, the orthodox Tarskian T-schema and the principle by which Read wishes to replace it. Thereby Sagüillo is led to discuss the nature of the ‘saying that’-relation as involving a ‘semantic closure’ postulate. Read addresses these issues in his response (Sects. 2 and 3 of Chap. 1, respectively).28 György Serény’s (Chap. 10: “The Liar Cannot be Solved”) detailed argument receives an extensive response from Read, and leads Read to 28 It is to be noted that Sagüillo’s contribution – even if more sympathetic to Read’s approach than Jan Woleński’s (see below) – partially overlaps with the latter’s criticisms. Namely, both authors want to locate the difference between (T) and (A) (respectively, Tarski’s T-schema and Read’s analysis of truth) – and subsequently, the difficulties in applying (A) – in the fact that the former is extensional, while the latter is intensional.

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clarify his way of reformulating Bradwardine’s theory (nearly the whole Sect. 3 of Chap. 13, and a substantial part of Sect. 4, are devoted to meet Serény’s arguments). Nowhere in the present volume does the controversy resemble more closely a medieval dispute than in Serény’s attempt to refute Read’s solution, and Read’s subsequent attempt at defending it. Hartley Slater, in his paper (Chap. 11: “Out of the Liar Tangle”), tries to restate Read’s solution within a more parsimonious logical framework: he diagnoses the difficulties in logically representing propositional referring phrases as being responsible for the paradoxical appearance of the Liar sentence, proposes how to properly carry out such representation, and criticizes Read for appealing to Church’s type theory in his proposed solution.29 Jan Woleński is the only contributor to the present volume who (Chap. 12: “Read about T-scheme”) attempts to defend Tarski’s semantic definition of truth. In doing so, he reminds us of its scope as well as of its limits. He also raises important questions concerning the relationship between the notion of ‘saying that’ and the notion of interpretation. Woleński wishes to stress that Read’s schema is intensional, while Tarski’s semantic definition of truth is extensional and operates with respect to a fixed interpretation. A further important point to which Woleński calls attention is the behavior of Read’s truth predicate in the presence of negation. In the second part of the volume, consisting of the contributions of Claude Panaccio and Fabienne Pironet, the reader will find a comprehensive account of the medieval conceptual landscape with respect to the question of insolubilia – against the background opposition between uniqueness vs. plurality of meaning. Panaccio’s contribution (Chap. 14: “Restrictionism: A Medieval Approach Revisited”) is an attempt to carefully expound and reconstruct the medieval restrictionist approach. Panaccio argues that not only may a specific (weak) form of restrictionism, when properly reconstructed, be defensible even from a modern viewpoint, but that it actually enjoys appealing features, when compared with other approaches to truth – such as the Tarskian and Kripkean analyses. Focusing on the debate on the fourteenth century, Pironet (Chap. 15: “William Heytesbury and the Treatment of Insolubilia in FourteenthCentury England”) offers a more general contribution than the title of her article might suggest: Heytesbury’s solution is indeed ‘Bradwardinian’ 29 It is also worth noticing that Slater states briefly the very same point as C. Dutilh Novaes, when mentioning that “ ‘singleness of saying’ [is required] before there can be truth assessments of sentences in the traditional T-schema form” (p. 194).

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in nearly all respects, and was historically of considerable influence, as Pironet’s thorough paper demonstrates. Furthermore, Pironet presents a critical edition of three medieval treatises inspired by Heytesbury (and hence, indirectly, by Bradwardine), which testify to the long-standing popularity and conceptual robustness of such a type of solution.

Acknowledgments The editors express their gratitude to Stephen Read, who not only provided the impetus for this volume, but supplied substantial and continuing help throughout its preparation. Many thanks to those who, through the careful and time-consuming process of refereeing the papers, certainly helped with their comments improving the quality of this volume, namely: Joël Biard (CNRS, University of Tours) Boudewijn de Bruin (University of Groningen) Jean Celeyrette (University of Lille 3) Simo Knuuttila (Academy of Finland, University of Helsinki) Ulrich Nortmann (University of Saarlandes) Jaroslav Peregrin (Academy of Sciences of the Czech Republic) Graham Priest (Universities of Melbourne and St Andrews) Philippe de Rouilhan (CNRS, University of Paris I) Kevin Scharp (Ohio State University) Paul Vincent Spade (Indiana University) John Symons (University of Texas, El Paso) Shahid Rahman expresses his thanks to the MSH–Nord-Pas-de-Calais, and to Fabienne Blaise and Philip Miller, director and subdirector of the Laboratory UMR 8163 ‘Savoirs, Textes, Langage’ (Lille 3) for fostering an efficient and collegial research environment. Shahid dedicates this book to Laura-Milena and Djamal-Alexandre. Tero Tulenheimo expresses his gratitude to Shahid Rahman and his logic group for the inspiring research atmosphere in Lille during the spring terms of 2005 and 2006. Emmanuel Genot thanks the authors for their patience with his difficulties in maintaining a proper e-mail address, and their cooperation at the last stage of producing this volume; and Cédric Dégremont, for helping in the LATEX-ing process.

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References [1] S. Read (1995) Thinking about Logic, Oxford: Oxford University Press, p. 150 [2] B. Russell (1908), “Mathematical Logic as Based on the Theory of Types,” American Journal of Mathematics, 30 [3] A.N. Prior (1908) “Epimenides the Cretan,” Journal of Symbolic Logic, 23 [4] M. Dummett (1976), “What is a Theory of Meaning? (II)”. In G. Evans and J. McDowell (editors), Truth and Meaning, Oxford: OUP, pp. 67–137 [5] M. Dummett (2004), Truth and the Past, New York: Columbia University Press [6] M. Dummett (2006), Thought and Reality, Oxford: Oxford University Press [7] G. Primiero (2008), Information and Knowledge, Berlin Heidelberg New York: Springer, LEUS [8] J. Hintikka (1996), The Principles of Mathematics Revisited, Cambridge: Cambridge University Press, pp. 22–3 [9] M.B. Hintikka and J. Hintikka, Investigating Wittgenstein, Oxford: Basil Blackwell, Chap. 8 [10] A. Tarski (1944), “The Semantic Conception of Truth and the Foundation of Semantics,” Philosophy and Phenomenological Research, 4, Sect. 5 [11] S. Kripke (1975), “Outline of a Theory of Truth,” Journal of Philosophy, 72: 690–716 [12] Hintikka J. (1989), “Is Truth Ineffable?” in Les formes actuelles du vrai: Entretiens de Palermo 1985, no ed. indicated, Palermo: Enchiridion, pp. 89–120; reprinted in J. Hintikka (1996), Lingua Universalis vs. Calculus Ratiocinator: An Ultimate Presupposition of Twentieth-Century Philosophy, Selected Papers II, Dordrecht: Kluwer, pp. 20–45 [13] T. Maudlin (2004) Truth and Paradox: Solving the Riddles, Oxford: Oxford University Press [14] H. Field (2006) “Maudlin’s Truth and Paradox,” Philosophy and Phenomenological Research, 73, pp. 713–720

Contents

Preface Acknowledgments Contributing Authors

v xix xxiii

Part I Disputatio 1 The Truth Schema and the Liar Stephen Read

3

2 Read and Indirect Revenge Bradley Armour-Garb

19

3 Tarski’s Hidden Theory of Meaning: Sentences Say Exactly One Thing Catarina Dutilh Novaes

41

4 Doubting Thomas: From Bradwardine Back to Anon Laurence Goldstein

65

5 Logic Without Truth Gyula Klima

87

6 Scheming and Lying Eugene Mills

113

7 Comments on Stephen Read’s “The Truth-schema and the Liar” Terence Parsons

129

xxii

Contents

8 Models for Liars in Bradwardine’s Theory of Truth Greg Restall

135

Acknowledgments

147

9 On a New Account of the Liar Jose ´ M. Sagu ¨ illo Acknowledgments

149 157

10 The Liar Cannot be Solved Gyo ¨ rgy Sere ´ ny Acknowledgments

159 185

11 Out of the Liar Tangle B. Hartley Slater

187

12 Read About T-Scheme Jan Wolen ´ ski Acknowledgments

199 203

13 Further Thoughts on Tarski’s T-Scheme and the Liar Stephen Read

205

Part II Historical Background: Restrictionism versus the Manifold Theory of Meaning 14 Restrictionism: A Medieval Approach Revisited Claude Panaccio

229

Acknowledgments

250

15 William Heytesbury and the Treatment of Insolubilia in Fourteenth-Century England Followed by a Critical Edition of Three Anonymous Treatises De insolubilibus Inspired by Heytesbury Fabienne Pironet

255

Index Nominum

335

Index Rerum

337

Contributing Authors

Bradley Armour-Garb Department of Philosophy, University at Albany – SUNY, State University of New York, 1400 Washington Ave., Albany, NY 12222, USA [email protected]

Catarina Dutilh Novaes Leiden University, P.O. Box 9500, 2300 RA Leiden, The Netherlands and Fordham University, Rose Hill Campus Bronx, NY 10458 (718) 817-1000, USA [email protected]

Laurence Goldstein University of Kent, Canterbury, England Fife KY16 8RA, Scotland, UK [email protected]

Gyula Klima Fordham University, Rose Hill Campus, Bronx, NY 10458(718)817-1000, USA [email protected]

Eugene Mills Virginia Commonwealth University, Richmond, Virginia 23284, USA [email protected]

Claude Panaccio University of Quebec at Montreal, Montreal, Quebec, Canada H3C 3P8 [email protected]

Terence Parsons University of California, Los Angeles, CA, USA [email protected]

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Contributing Authors

Fabienne Pironet Département de Philosophie, University of Quebec at Montreal, Montreal, Quebec, Canada H3C 3P8 [email protected]

Stephen Read Department of Logic and Metaphysics, University of St. Andrews, Fife KY16 8RA, Scotland, UK [email protected]

Greg Restall Department of Philosophy, The University of Melbourne, VIC, Australia [email protected]

José M. Sagüillo Department of Logic and Moral Philosophy, University of Santiago de Compostela, Spain [email protected]

György Serény Department of Algebra, Budapest University of Technology and Economics, Budapest, Hungary [email protected]

B. Hartley Slater Department of Philosophy University of Western Australia, Perth, Australia [email protected]

Jan Woleński Jagiellonian University, Cracow, Poland [email protected]

I

DISPUTATIO

Chapter 1 THE TRUTH SCHEMA AND THE LIAR Stephen Read∗ Department of Logic and Metaphysics University of St. Andrews Fife KY16 8RA Scotland, UK [email protected]

Abstract

Since Tarski published his study of the concept of truth in the 1930s, it has been orthodox practice to suppose that every instance of the T-schema is true. However, some instances of the schema are false. These include the paradoxical instances exemplified by the Liar sentence. It is shown that a better schema allows a uniform treatment of truth in which the semantic paradoxes turn out to be simply false.

Keywords: Truth, Liar paradox, Correspondence platitude, Tarski, Bradwardine.

1.1

Tarski’s T-scheme

Carnap, in his Autobiography ([1], p. 60) tells an anecdote about a meeting with Tarski in Vienna in the early 1930s. Both Carnap and Tarski had been struggling with the problem of providing a satisfactory definition of truth. Carnap asked Tarski to explain the basic idea: how can one define the truth of an empirical sentence like ‘This table is black’. “‘This table is black’ is true,” replied Tarski, “if and only if this table is black.” Coffa ([2], p. 304) recalls that Carnap later described this as the moment when “the scales fell from my eyes.” Over the years since Tarski published his study of the concept of truth [3], it has been an unquestioned orthodoxy that every instance of the T-scheme: x is true if and only if p (T) ∗ Proponent.

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is true, where what replaces ‘x’ is a name of a sentence whose translation into the metalanguage replaces ‘p’. Indeed, in the homophonic case of identity (in sound or in fact) of object language and metalanguage, what replaces ‘x’ is a quotation-name of the sentence which replaces ‘p’. Nonetheless, not all instances of (T) are true, even where the very same sentence replaces ‘x’ and ‘p’, the first in quotation-marks. For example, ‘I am tired’ is true if and only if I am tired

(1)

and ‘That book was stolen’ is true if and only if that book was stolen (2) are not true unless the person who utters (1) is the same as the utterer of the sentence cited, and unless the book indicated by the demonstrative pronoun in (2) is the very same as that demonstrated in the sentence whose truth is at stake, as Davidson [4] pointed out. Moreover, even in the absence of indexicals, the validity of the T-scheme is not assured. Hintikka’s example [5] was ‘Any man is mortal’ is true if and only if any man is mortal.

(3)

‘Any’ always takes widest scope, so that whereas ‘any’ in the quoted sentence acts as a universal quantifier within it, the occurrence of ‘any’ on the right-hand side functions as a universal quantifier with wide scope over the whole of (3), and so as an existential quantifier with scope restricted to the right-hand side. Thus homophonic translation fails. Despite such counterexamples, (T) is often taken as a truism. For example, in his paper ‘The Establishment of a Scientific Semantics’, Tarski wrote: “Quite generally, we shall accept as valid every instance of the form the sentence x is true if and only if p where ‘p’ is to be replaced by any sentence of the language under investigation and ‘x’ by any individual name of that sentence.” ([6], p. 404)

More recently, Paul Horwich wrote: “for any declarative sentence ‘p’ our language generates an equivalent sentence ‘The proposition that p is true’.” ([7], pp. 4–5)

and “the concept of truth is entirely captured by stipulating the equivalence schema, ‘The proposition that p is true if and only if p’ – where p can be replaced by any declarative sentence.” [8]

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5

Lastly, Aladdin Yaqub writes: “The concept of truth is completely and correctly defined by the Tarskian scheme, whose instances are all the biconditionals obtained from the phrase ‘x is true if and only if p’ by substituting any sentence for ‘p’ and any expression which stands for that sentence for ‘x’.” ([9], p. 3)

Yaqub invokes Tarski, Ayer, Horwich and Gupta in support of his claim that “the Tarskian schema is the most fundamental intuition about the concept of truth” (p. 27). Tarski did not propose (T) as a definition of truth, though others, e.g., Horwich, have done so since. They all describe (T) as a truism, as something so obviously correct as to need no argument, at least, provided the obvious counterinstances, such as (1), (2) and (3) are excluded. Tarski himself presented (T) as a condition of what he called “material adequacy”. It was a necessary condition on any adequate definition of truth that it generate all instances of (T) as theorems: A formally correct definition of the symbol ‘Tr’ . . . will be called an adequate definition of truth if it has the following consequence: all sentences which are obtained from the expression ‘x ∈ T r if and only if p’ by substituting for the symbol ‘x’ a structural-descriptive name of any sentence of the language in question and for the symbol ‘p’ the expression which forms the translation of this sentence into the metalanguage. ([3], pp. 187–188)

Tarski’s account was for many years described as a version of the correspondence theory of truth – an acceptable version of a theory relieved of its suspect ontology of facts and correspondence. The (T)-scheme presents in a stark and nominalistically acceptable form the correlation of language, on the left-hand side, and world on the right-hand side. What replaces ‘x’ refers to a sentence or other truth-bearer, and what replaces ‘p’ refers to, or describes, how things are. Thus the (T)-scheme can be summed up as expressing the idea that a sentence is true if things are as it says they are: x is true if and only if things are as x says they are.

(CP)

Crispin Wright describes (CP) as “the correspondence platitude” ([10], p. 27), and Julian Dodd agrees: “a platitude it certainly is . . . it says nothing that would be disagreed with by a correspondence theorist, coherence theorist or deflationist.” ([11], p. 75). In this paper, I want to show that (CP) is at best equivocal, and (T) is false. This is not just a matter of showing that not all instances of (T) are true. Many proponents of (T) admit that. They nonetheless claim that (T) offers a good account (partial or complete) of the truth-conditions of those sentences for which it is true. However, this is not so. Where it

6

S. Read

is true, (T) is true by lucky chance. Where true, its instances are also instances of a better scheme, the right scheme of truth. (T) is very far from being a truism, or a good test of the adequacy of a truth-theory.

1.2

How Things Are

How can things be as x says they are unless x is true? Clearly, ‘snow is white’ says that snow is white, and ‘snow is white’ is true just when snow is as ‘snow is white’ says it is, that is, when snow is white. But ‘Every even number greater than 2 is the sum of two primes’ not only says that every even number greater than 2 is the sum of two primes, but also in particular that, e.g., 4 is the sum of two primes. So 4’s being the sum of two primes is as Goldbach’s Conjecture says things are. But Goldbach’s Conjecture is not true just because 4 is the sum of two primes. 4’s being the sum of two primes is necessary for the truth of Goldbach’s Conjecture, but it is not sufficient. So (CP) needs more careful expression. What was meant was x is true if and only if things are wholly as x says they are,

(S)

x is true if and only if however x says things are, they are.

(S’)

or

A sentence says many things, and however it says they are, they must be, for the sentence to be true. Anything of the form ‘p and q’ says inter alia that p, but the truth of p does not suffice for that of ‘p and q’. Indeed, the phrase ‘how x says they are’ or even ‘however x says they are’ is imprecise in another way. ‘All men are mortal’ says that if Socrates is a man then Socrates is mortal. But Socrates was a man. So if ‘All men are mortal’ is true, Socrates was mortal. Whether the universal sentence actually says that Socrates was mortal is, perhaps, unclear. If Socrates was not mortal, then ‘All men are mortal’ is false – unless Socrates was not a man after all. If x is true, things must be however x implies they are. Conversely, if things are however x implies they are, x will be true. “Safety first” would support the identification of ‘however x says they are’ with ‘what x implies’. This may make ‘how x says they are’ more precise than is natural, but in the context of (CP) it is welcome precision. We find such a conception of ‘saying that’ as closed under implication in Wittgenstein’s Tractatus ([12], Sects. 5.122 and 5.124): “A proposition affirms every proposition that follows from it . . . If p follows from q, the sense of ‘p’ is contained in the sense of ‘q’.” By ‘follows from’, Wittgenstein here clearly intended the logical truth of the conditional (as John Corcoran pointed out to me). Elementary

The Truth Schema and the Liar

7

propositions are logically independent and p follows from q only if the truth-conditions, or truth-grounds (Sect. 5.101), of q contain those of p. For Wittgenstein, the conditional itself was a material one. I will follow him in this, for the purposes of this paper. Thus logical implication is presently strict implication, that is, the necessitation (or logical truth) of the material conditional. Certain moves in the argument to follow work only for such an implication. Let us represent ‘x says that p’ by ‘x:p’, where ‘x’ designates a sentence, strict implication by ‘⇒’ and the material conditional by ‘→’. Then we can represent the closure of ‘saying that’ under logical implication by (∀p, q)((p ⇒ q) → (x:p ⇒ x:q)). (K) The context x:p is an intensional context for p, allowing substitution only of logical equivalents (and then, only if we accept the closure of x:p (saying that) under ‘⇒’).1 The truth-condition given by (S) or (S ) is then represented by Tx ⇔ ∀p(x:p → p) (A) Here, ‘⇔’ represents strict equivalence. For Tx and ∀p(x:p → p) do not just happen to be equivalent. The right-hand side of (S) gives a philosophical analysis, or explication, or truth. (A) contrasts with (T) in two important ways. First, (A) is an analysis of truth; (T) is merely, for Tarski, a condition of material adequacy, a test of the accuracy of his and others’ theories. In particular, (A) is not the final analysis of truth: that awaits an analysis of ‘saying that’. Thus (A) need not conflict with Tarski’s requirement that all semantic notions be defined and eliminated, any more than does his own account of truth: Tx ⇔ ∀s(s is a sequence → s satisfies x). Tarski defined truth in terms of satisfaction, but then proceeded to define satisfaction in non-semantic terms. Secondly, (A) is a logical equivalence, whereas (T) was for Tarski merely a material equivalence. For Horwich, the partial definitions whose form is (T) are given as material equivalences, but, he says, their a priori role allows us to derive the corresponding strict equivalences (see [7], Chap. 2, Sect. 3) That is fitting for his use of (T) in defining truth, a use 1 Indeed, if ‘⇒’ were ‘→’ and x said anything, say p, then since (p → q) ∨ (p → ¬q) is a tautology, we would have for all q, x:q ∨ x:¬q, which would trivialize the whole project and render x:p extensional. What one would like for ‘⇒’ is some enthymematic conditional, which includes the case of Socrates above without trivializing the whole notion. But identifying such a conditional is, presently, I believe, an unrealized project.

8

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which Tarski did not make. Accordingly, if (A) is a logical equivalence in providing the correct analysis of truth, we must assume that what a sentence says, it says of necessity too. Of course, any particular form of words expresses what it does only contingently. When I speak of the truth of sentences, and what they say, in what follows, I will be speaking of a sentence taken as meaning what it does – of a proposition, in Horwich’s terminology, or of a sentence in a context of speaker, place and time. If we take the more cautious line of rejecting (K), we can still capture (S) and (S ) by Tx ⇔ ∀p(∃q(x:q ∧ (q ⇒ p)) → p).2

(A )

Whereas (A) expresses the thought that a sentence is true if things are however it says they are, (A ) spells out that it is true if things are however what it says (logically) implies them to be. They are equivalent just if we identify what a sentence says with what it implies. Of course, taking ‘→’ to be material implication, (A) and (A ) will make true all sentences which say nothing. One solution is to conjoin (∃p)x:p to the right-hand side of (A) and (A ). But for simplicity, I will assume in what follows that each sentence to which we apply (A) and (A ) does say something. Another possible objection to (A) – and to (S) – concerns ambiguity. A different way for a sentence to say more than one thing is for it to be ambiguous. But if x has two meanings, e.g., ‘Visiting relatives can be boring’, we do not require that both senses be the case for x to be true – either will suffice. In reply, remember that the point of formalization is to avoid ambiguity. Accordingly, we must remove ambiguity from x and require that all sentences governed by (A) be unambiguous. For example, we might deem ‘Visiting relatives can be boring’ to be a pair of homophones, as we do with ‘I have a pair/pear’. Nonetheless, an unambiguous sentence, like ‘All men are mortal’, can say many things – that Socrates is mortal, that Plato is mortal and so on. To be true, all the things it says must hold – things must be however it says they are. The crucial advantage of (A), or (A ), over (CP) is that it makes clear that everything which a sentence says or implies must be the case for the sentence to be true. (CP) is equivocal: x may say things are in many ways. They are all severally necessary, but only jointly sufficient. Things 2 Note that (K), (A) and (A ) should be thought of as expressions in type theory, where x has type ι (of individuals) and p has type o (of propositions). See, e.g., [14].

The Truth Schema and the Liar

9

must be not only how x says they are, but however x says they are, in order that x be true. Tarski’s later discussion [13] helps us see where he went wrong. Denoting ‘snow is white’ by ‘S’, he writes: “What do we mean by saying that S is true . . . ? The answer to this question is simple: . . . by saying that S is true we mean simply that snow is white . . . By eliminating the symbol ‘S’ we arrive at the following formulation: (1) ‘snow is white’ is true if and only if snow is white.” (p. 64)

But he generalizes this later on the same page: “Partial definitions of truth analogous to (1) . . . can be constructed for other sentences as well. Each of these definitions has the form: (3) ‘p’ is true if and only if p, where ‘p’ is to be replaced on both sides of (3) by the sentence for which the definition is constructed.”

That snow is white is, arguably, all that ‘snow is white’ says. So the right-hand side of (1) gives a sufficient, as well as necessary condition for its truth. But not all cases are as straightforward as this. In general we need to do more than simply repeat ‘p’ on the right-hand side. (T) is a special case of (A) where, as for ‘snow is white’, a sentence reveals explicitly all that it says. The crucial importance of using (A), and the disastrous consequences of using (T) in its place, will be demonstrated in Sects. 1.3 and 1.4.

1.3

The Liar Paradox

Tarski and others exclude the Liar paradox from the scope of the T-scheme. Tarski did it by forbidding semantic closure, and so making the semantic paradoxes inexpressible ([3], Sect. 1; [6], p. 402). Horwich ([7], Sect. 10) simply excludes the paradoxical instances of (T) from the infinite conjunction of its instances which he takes to define truth. The consequence is that neither author can offer an account of the semantics of the Liar paradox and its ilk. For the one, the material test of the correctness of his theory of truth is that the right instances of (T) be derivable in the theory; for the other, the theory of truth is constituted by the conjunction of the correct instances. Each implicitly concedes, therefore, that not all instances of (T), even the homophonic ones where what replaces ‘x’ is a quotation-name of the sentence which replaces ‘p’, are true. Nonetheless, they persevere in keeping (T) as the cornerstone of their theory of truth, rather than looking for a better scheme. A better scheme would be one all of whose instances are true, one which did not need an ad hoc exclusion clause, excluding any counterexamples by fiat.

10

S. Read

There are many ways to formulate the Liar paradox, but the general idea is to find a sentence-name L such that L and ‘L is not true’ are interchangeable in light of some fact about L (e.g., where L is written, or through the correlations of some system of Gödel-numbering, and so on). Recall that we are assuming that a sentence (or proposition) necessarily says what it says, even, for example, in the case of contingent paradox (see [15], p. 691). Then since L implies itself, it implies that L is not true (by substitution), and indeed, if ‘says that’ is closed under implication (as proposed in Sect. 1.2) or if the connection is close enough, L says that L is not true. Thus we can substitute ‘L is not true’ for ‘p’ and ‘L’ for ‘x’ in (T), or (CP), to obtain the paradoxical instance: L is true if and only if L is not true.

(P)

Without serious and damaging revisions to logical principles, (P) is false, for it entails: L is true and L is not true. (P ) Those who reject Ex Falso Quodlibet (the principle that a contradiction implies every sentence) may attempt to maintain that (P ) is true without collapsing into triviality. But the Curry paradox, finding for an arbitrary sentence q, a sentence Cq which is interchangeable with ‘If Cq is true then q’, produces the paradox of triviality without recourse to contradiction. Substituting ‘Cq ’ for ‘x’ in (T), we obtain Cq is true iff if Cq is true then q, and by familiar steps of Modus Ponens and Contraction, we get a proof of q. Since q was arbitrary, the truth-theory based on (T), and (CP), is shown to be unacceptable. (T) is simply false, for it abounds in false instances.

1.4

Truth-Theory Without Paradox

What we have recalled in Sect. 1.3 is that if we look to (T) for an account of the semantics of L and Cq , the result will be trivial and paradoxical – in a word, wrong. The mistake encapsulated in (T) is to suppose that the expression replacing ‘p’ on the right-hand side fully expresses its own truth-condition. This may seem a plausible assumption. But at least in the case of the semantic paradoxes, as we will see, it is a mistaken assumption. For sentences imply other things which they do not themselves explicitly say. By the lights of (A), they say them nonetheless. By the lights of (A) or (A ), if those other things are not the case, then what the original sentence says cannot hold either, and so the original sentence is false. Reading (CP) as (T) misses that implication.

11

The Truth Schema and the Liar

(T) identifies the truth of a sentence with things being how it says they are; but not with however it says they are. Reading (CP) as (S), and so as (A) or (A ), includes in the truth-conditions of a sentence all its consequences. Consider L, for example. L says that L is not true, that is, L:¬TL. It may say more: call it q, that is, suppose L:(¬TL ∧ q). (If ¬TL is all that L says, we can simply let q be ¬TL, or a tautology.) By (A), TL ⇔ ∀p(L:p → p) i.e., TL ⇔ (¬TL ∧ q) since ¬TL ∧ q is all L says, whence ¬TL ⇒ ¬(¬TL ∧ q) i.e., ¬TL ⇒ (TL ∨ ¬q). If saying that is closed under ‘⇒’, as (K) says, then, since L:¬TL, L:(TL ∨ ¬q) and so (since L:q)

L:((TL ∨ ¬q) ∧ q)

whence L:TL since ((TL ∨ ¬q) ∧ q) ⇒ TL. Thus, if L says ¬TL, it says TL as well. That is, L:(¬TL ∧ TL). So by (A), TL ⇔ (¬TL ∧ TL ∧ . . .) whence ¬TL. The argument can be repeated using (A ) in place of (A), and dropping the assumption that saying that is closed under ‘⇒’. Assume again that L:(¬TL ∧ q). By (A ), since ¬TL ∧ q is all that L says, TL ⇔ ∀p(((¬TL ∧ q) ⇒ p) → p), so ¬TL ⇒ ∃p(((¬TL ∧ q) ⇒ p) ∧ ¬p), i.e., ¬TL ⇒ ∃p(¬p ∧ (¬p ⇒ (TL ∨ ¬q))),

12

S. Read

so ¬TL ⇒ ∃p(¬p ∧ (TL ∨ ¬q)), whence ¬TL ⇒ (¬TL → ¬q), which by Contraction yields, ¬TL ⇒ ¬q. Contraposing, we have q ⇒ TL. Using

(A )

once again, TL ⇔ (¬TL ∧ TL ∧ . . .)

whence ¬TL. It may not be immediately obvious what a breakthrough this is. Of course L is not true. By (A) TL ⇔ (L:¬TL → ¬TL) ∧ . . . whence TL ⇒ (L:¬TL → ¬TL). Since L:¬TL,

TL ⇒ ¬TL

and so ¬TL. But the longer argument concerning L shows more than that L is not true. In Sect. 1.3, using the (T)-scheme, we concluded not only that L was not true but also that L was true, and so landed in paradox. This reasoning fails when the truth-conditions of L (and all sentences) are given by (A), or (A ). To use (T) to infer that L is true, all we need to show is that L is not true, as in Sect. 1.3. To use (A) or (A ) to show that L is true we need to show both that L is not true and, in addition, that L is true. But that is impossible. L cannot be true, for to be true, it would have to be both true and not true. Nothing can be both true and not true. So L cannot be true. L is not only not true, it cannot (also) be true, as (A) reveals. Note that I have not shown that ¬TL ⇒ TL. The closure principle (K) implies that L says whatever is implied by what L says, and, by the argument above, it follows from the fact that L:¬TL that L:TL. But

13

The Truth Schema and the Liar

¬TL ⇒ TL is in fact false. It entails ¬TL → TL, which is equivalent to TL. But TL is false, as we have seen, and consequently, so is ¬TL ⇒ TL. The problem with (T), as a condition of truth, is that it implies the impossible, for it declares (unless, of course, suitably restricted) that L is both true and not true. Thus (T) is false. (T) has the false instance: TL ⇔ ¬TL.

(†)

The mistake made by Tarski, Horwich and others is to think that a suitably restricted version of (T) can serve as a determiner of the right theory of truth. It cannot. Either for no good reason other than to preserve the (universal) truth of (T), the counterinstances are deemed ill-formed, a thoroughly ad hoc manoeuvre; or the counterinstances are simply excluded from the theory of truth by fiat, again ad hoc and involving the further consequence that there are apparently well-formed sentences (such as L) for which there is no theory of truth. How, unless we can apply our theory of truth to L can we discover that it is paradoxical? The solution is ready to hand. Abandon (T) and realise that the correct theory of truth is given by (A) – or (A ), depending on one’s preferred theory of meaning – governing all well-formed sentences in a semantically closed language. As applied to L, we obtain the correct truth-condition: TL ⇔ (¬TL ∧ TL).

(††)

I said that (CP) equivocates, and perhaps (T) does so too, or at least has been misunderstood. Tarski requires that any adequate theory of truth for a language must generate all instances of (T) where what replaces ‘p’ is a translation of the sentence whose name replaces ‘x’. But a proper translation must spell out all that the sentence says. So arguably, (†) is not a correct instance of (T), for the right-hand side does not say all that L says. Rather, the correct instance of (T) is (††), whose right-hand side says all that L says.

1.5

Paradox Without Truth

Tarski ([3], p. 161) observed that one can repeat the Liar reasoning in the absence of a truth-predicate, simply with a quotation-functor. Even after replacing (T) by (A), there is a danger that we might similarly find paradox has returned, through use of the ‘says that’ predicate. For consider ∀p(C:p → ¬p) (C) Clearly C:∀p(C:p → ¬p). Suppose that ∀p(C:p → ¬p). By Universal Instantiation, C:∀p(C:p → ¬p) → ¬∀p(C:p → ¬p)

14

S. Read

so by Modus Ponens, ¬∀p(C:p → ¬p) and so by reductio, ¬∀p(C:p → ¬p), whence ∃p(C:p ∧ p).

(∗)

We might be tempted to infer ∀p(C:p → ¬p) from (*), since that is what C seems to say. But we have already shown ¬∀p(C:p → ¬p), and so contradiction would have returned, without the involvement of (T) or (A) and the truth-predicate. However, although C:∀p(C:p → ¬p), this may not be all that C says. Suppose C also says that q, that is, C: (∀p(C:p → ¬p) ∧ q). Then by (A), TC ⇔ (∀p(C:p → ¬p) ∧ q) whence (¬∀p(C:p → ¬p) ∨ ¬q) ⇒ ¬TC and so ¬∀p(C:p → ¬p) ⇒ ¬TC, whence, since ∀p(C:p → ¬p) ⇒ ¬∀p(C:p → ¬p), ∀p(C:p → ¬p) ⇒ ¬TC and so C:¬TC. Thus it is true that ∃p(C:p ∧ p), for C:¬TC ∧ ¬TC. So from (*) we cannot legitimately infer that whatever C says is the case. Some of what C says is the case, namely, ¬TC. But (*) is no ground for inferring that whatever C says is the case. In particular, ∀p(C:p → ¬p) is not the case, since C is false. Although this frees C from the threat of immediate paradox, it raises a further worry. For it is crucial to the present solution to the paradoxes that not every proposition say of itself that it is true. The reason that L, C and Cq turn out not to be paradoxical is that they say of themselves that they are true, like the Truth-teller; but since they also directly or indirectly say that they are not true, things cannot be as they say they are, and so by (A) they are not true but false. But if we could show that every sentence said (albeit indirectly) of itself that it was true, the truth-condition (A) would be crippled. For (A)

The Truth Schema and the Liar

15

says that a sentence is true only if things are as it says they are, and if part of what it said was that it was true, (A) would, correctly but unhelpfully, endorse a sentence as true if it was true, but would not reveal whether it was true or not. Every sentence (bar the paradoxical ones) would become a Truth-teller, true if true and false if not, but of undetermined truth-value. What induces this anxiety that every sentence is a truth-teller? The thought is this: implicit in endorsing (A) as the right truth-condition (in place of (T)) was that ‘saying that’ is closed under implication – every sentence says (implicitly, or indirectly) whatever is implied by what it says. But by (A), things being as it says they are implies that a sentence is true. It seems to follow that every sentence says that it is true. That it is true is implied by whatever it says, so if it says whatever it implies, it says that it is true. Fortunately, this is a specious argument, which we can see from our formalization of it. (A) says Tx ⇔ ∀p(x:p → p) and the closure of ‘saying that’ under implication, (K), is (∀p, q)((p ⇒ q) → (x:p ⇒ x:q)). These do not entail x:Tx. For a counterexample, read x:p as p ∈ X, where X is a theory, that is, a collection closed under logical implication. Then the closure assumption is true, i.e., (∀p, q)((p ⇒ q) → (p ∈ X ⇒ q ∈ X)). We want x:Tx to be false. (A) identifies Tx with ∀p(p ∈ X → p). So we need to show that it is possible that ∀p(p ∈ X → p) ∈ / X. What ∀p(p ∈ X → p) says is that X is correct – that X contains only truths. But theories can be mistaken, and it would be an arrogant theory which included the statement ∀p(p ∈ X → p), which says that everything X says is true. Thus x:Tx does not follow from (A) together with the closure principle. Thus the argument that every sentence says of itself that it is true is sophistical. A sentence says whatever it says, but it does not say that whatever it says is the case – it does not say that things are as it says they are. The Truth-teller says that. But that is what is wrong with the Truth-teller. To say that things are as it says they are is to say nothing. Things must be as a sentence says they are for it to be true, but it does not (in general) say that things are as it says they are. It follows that paradox, with or without truth, is prevented by the adoption of the correct account of truth, namely, the truth-condition (A).

16

1.6

S. Read

Some History

The argument in Sect. 1.4 to the effect that L says that L is true since it says that L is not true is due to Thomas Bradwardine, briefly Archbishop of Canterbury in 1349 before succumbing to the Black Death. His treatment of the semantic paradoxes [16] dates from some 25 years earlier, when he was a young teacher at the University of Oxford. His solution was taken up and adapted by John Buridan and Albert of Saxony at the University of Paris, and in Buridan’s version was discussed in the 1950s and sixties by Ernest Moody [17] and Arthur Prior [18]. However, both John and Albert omit the proof from Sect. 1.4, John ([19], p. 969) simply asserting that every sentence implies its own truth, Albert ([20], f. 43rb ) inferring that every sentence says that it itself is true from the assumptions that every (affirmative) sentence says that its subject and predicate are identical, and that every such sentence whose subject and predicate are identical is true (and likewise for negatives). It was this reasoning of Albert’s which inspired the sophistical reasoning at the close of Sect. 1.5. But if every sentence says that it itself is true, or implies its own truth, then no sentence is true, or at least no sentence can be shown, using (A) or (A ), to be true. (A) simply says that any such sentence is true if it is true (and whatever else it says or implies is the case). John and Albert are shown to have no adequate theory of truth at all.3 The same problem does not affect Bradwardine’s treatment. He is not committed to saying that every sentence says of itself that it is true, only that paradoxical sentences such as L (and C and Cq , if q is not true) do so. His theory shows that they are simply not true, and so false, since he accepts Bivalence.

References [1] R. Carnap, Intellectual autobiography, in P. Schilpp, The Philosophy of Rudolf Carnap. La Salle, IL: Open Court, 1963, 3–84. [2] J.A. Coffa, The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press, 1991. [3] A. Tarski, The concept of truth in formalized languages, tr. in [22], 152–278. 3 A fuller treatment of John’s, Albert’s and Thomas’ attempts at a solution was given in [21]. I am grateful to John Corcoran and Ignacio Jané, among others at a workshop at the University of Santiago, Spain, for helpful comments on an earlier version of this paper and to Thomas Schmidt and others at the University of Göttingen, Germany, for further helpful comments.

The Truth Schema and the Liar

17

[4] D. Davidson, True to the facts, Journal of Philosophy 66, 1969, 304–323, reprinted in his Inquiries into Truth and Interpretation. Oxford: Clarendon Press, 1984, 37–54. [5] J. Hintikka, A counterexample to Tarski-type truth-definitions as applied to natural languages, Philosophia 5, 1975, 207–212. [6] A. Tarski, The establishment of a scientific semantics, tr. in [22], 401–408. [7] P. Horwich, Truth, second edition. Oxford: Clarendon Press, 1998. [8] P. Horwich, In the truth domain, Times Literary Supplement no. 4711 (July 16, 1993), 28. [9] A. Yaqub, The Liar Speaks the Truth. Oxford: Oxford University Press, 1993. [10] C.J.G. Wright, Truth and Objectivity. Cambridge, MA: Harvard University Press, 1992. [11] J. Dodd, Is truth supervenient on being?, Proceedings of the Aristotelian Society 102, 2001–2002, 69–86. [12] L. Wittgenstein, Tractatus Logico-Philosophicus, tr. D. Pears and B. McGuinness. London: Routledge and Kegan Paul, 1961. [13] A. Tarski, Truth and proof, Scientific American 220, no. 6, June 1969, 63–77. [14] A. Church, A formulation of the simple theory of types, Journal of Symbolic Logic 5, 1940, 56–68. [15] S. Kripke, Outline of the theory of truth, Journal of Philosophy 72, 1975, 690–716. [16] Insolubilia, in M.-L. Roure, La problématique des propositions insolubles au XIIIe siècle et au début du XIVe , suivie de l’édition des traités de W. Shyreswood, W. Burleigh et Th. Bradwardine, Archives d’Histoire Doctrinale et Littéraire du Moyen Age 37, 205–326. [17] E.A. Moody, Truth and Consequence in Medieval Logic. New York: Greenwood Press, 1953. [18] A. Prior, Some problems of self-reference in John Buridan, Proceedings of the British Academy 48, 1962, 281–296. [19] J. Buridan, Summulae de Dialectica, tr. G. Klima. New Haven, CT: Yale University Press, 2001. [20] Albert of Saxony, Perutilis Logica, Venice 1522, repr. Olms, 1974. [21] S. Read, The liar paradox from John Buridan back to Thomas Bradwardine, Vivarium 40, 2002, 189–218. [22] A. Tarski, Logic, Semantics, Metamathematics, second edition, J. Corcoran, tr. J. Woodger (eds.). Indianapolis: Hackett, 1983.

Chapter 2 READ AND INDIRECT REVENGE Bradley Armour-Garb∗ Department of Philosophy University at Albany – SUNY State University of New York 1400 Washington Ave. Albany, NY 12222, USA [email protected]

. . . is it not a great result when the scientist after much hesitation and careful inquiry, can finally say ‘what I supposed is true’ ? The meaning of the word ‘true’ seems to be altogether unique. —[Frege 1956] [18, p. 291]

Keywords: Deflationism, Semantic closure, Indirect revenge, Dialetheism.

2.1

Introduction

Stephen Read provides a solution to the liar and a theory of truth, by putting forward a novel definition of truth. Read’s theory attempts to capture certain facts about truth and is, in turn, supported by its attempt

∗ Opponent.

20

B. Armour-Garb

to resolve paradoxes that challenge it. Why adopt Read’s favoured theory of truth (which theory I will discuss below)? Are the extant theories inadequate, or otherwise incorrect? Read claims that they are. One reason for this is that none of the relevant theories of truth appear adequate to resolve the truth-involving semantic paradoxes.2 The problem is paradox and, in particular, a central feature of truth that does not seem directly bound to the concept of truth. Read claims that a general problem for a number of theories of truth is an insistence on a proper reading of Tarski’s condition of adequacy, “the truth schema”, X is true if, and only if P

(T)

where what is replaced by ‘X’ is a mention sentence whose translation is used by the sentence that replaces ‘P ’. But this is not the only problem.3 While a deflationist (or, more broadly, one who endorses a deflationary conception of truth) takes the instances of a (T)-like scheme to provide a definition of truth, others (e.g., Kripke, ultimately), following Tarski, take its import and its generality to rule out certain features of the language – semantic closure, universality, expressibility, etc. – which features are thus restricted, eradicated or, at any rate, in some sense avoided.4 In response to these problems, Read maintains two things: First, that (T) (or its instances) does not capture our conception (or, per [27, p. 5], our idea) of truth and, second, that the proper conclusion to draw from the paradoxes is not the denial of certain features of our language; it is, rather, the denial of the unquestioned status that has been assigned to (T).5 Read takes the assumption that every instance of (T) is true to be a bit of unquestioned orthodoxy, which he questions and ultimately rejects. He concludes that unrestricted (T) is false (or invalid), though many of its instances are true. Moreover, he sees their true instances as deriving from a more satisfying conception of truth – a definition of truth that, 2 The other reason that Read suggests regards the proper treatment of the (T) scheme, given mentioned sentences that include indexicals, demonstratives, and the like. Read is right that a theory of truth must have something to say about how they yield true instances of (T). But it would be incorrect to say that such cases pose a problem (say) for deflationism (or for the status they attribute to (T)). This becomes clear when we recognise that any workable, deflationary, theory of truth will take the instances of (T) to apply only to interpreted sentences. 3 For what follows, I shall be somewhat loose about use and mention, save for cases where context is not able to do the job. 4 I clarify (a bit) the notion of a ‘deflationary conception of truth’ in Sect. 2.4. 5 Different versions of deflationism will endorse different variants of (T). For what follows, I will not dwell unduly on deflationism. For a fairly comprehensive discussion of the varieties of deflationism, see [4].

Read and Indirect Revenge

21

in principle, agrees on the status of all of the uncontentious instances of (T), without threat of impending inconsistency. If successful, Read will provide a novel theory of truth which, following [13], promises a diagnosis of the liar paradox, and a treatment that avoids truth’s inconsistency, without compromising our conception of that predicate. Of course, dialetheists, who hold that some sentences are true with true negations, also provide a response that purports to resolve the paradoxes without losing certain prima facie plausible principles – certain key features of truth. Read has no patience for dialetheism, but his argument here is not dogmatic. What he argues, in effect, is that a dialetheist’s proposed treatment of Curry’s Paradox will suffer from the very problems that he aimed to avoid: The loss of certain important, and otherwise plausible, principles. Accordingly, he (ibid., p. 10) sets out a dilemma for the dialetheist: Either the dialetheist cannot resolve Curry’s or he can, but only by giving up contraction and absorption, in which case he will find that he faces the very problems that he attributed to consistent solutions to the ‘aletheic paradoxes’, which problems he sought to avoid. While I grant that the dialetheist will give up such features, it does not follow that he will be hoisted by his own petard. The reason regards the dialetheist’s general diagnosis of all such paradoxes as making evident the presence of logical, as opposed to semantic, pathology.6 Once the diagnosis is procured, it will be evident that, at a certain level of abstraction, the dialetheist’s response to Curry’s is formally identical to his response to the liar paradox writ large. More generally, while I do have some worries about dialetheism (cf. [7], for the worries), my worries are not Read’s, for I am not convinced that a ‘Curry worry’ has the consequences that Read and others have assumed. Rather than dwell on the differences, I will grant that a dialetheic treatment of Curry’s Paradox constitutes some sort of loss, small though I take it to be. It is important to note, however, that a consistentist need 6 For more on the difference between logical and semantic pathology, see [2, 3, 7]. Briefly, consistent theorists (viz., ‘consistentists’) see certain of our naïve semantic concepts as pathological – indeed, as semantically pathological – because of their apparent resistance to semantic characterisation. Properly diagnosing such pathology consists in identifying its symptoms and explaining its aetiology – and treating it rids the language of its pernicious disorder. By contrast, dialetheists reject a consistentist’s diagnosis of any such case as making evident semantic pathology. Of course, the dialetheist, like the consistentist, sees the threat of trivialism as pathological (because patently unacceptable), but his diagnosis of the problematic component of this combination is not to class a liar sentence as semantically pathological. Rather, it is to class the relevant logic that gives rise to the disaster as pathological – as an instance of logical pathology. Thus, his recommended treatment of the logical pathology that he identifies – the misfiring of the standard functioning of certain principles – is to provide a logical means for blocking explosion and obviating disaster, while still allowing for the warranted assertability of some contradictions.

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not poke many holes in dialetheism in order to motivate his solution to the liar paradox. The reason is as follows. All dialetheists affirm a general (norm-involving) preference for consistency. More specifically, no dialetheist would motivate his solution to paradoxes by arguing for the acceptance of dialetheia. To the contrary, he would argue for a dialetheic position by dissolving objections to true contradiction, and by highlighting insuperable problems that arise from whatever proposed, consistent treatment of a paradox is up for discussion. Accordingly, far from evaluating Read’s proposed solution to the liar paradox by contrasting it with dialetheism, I shall be concerned to investigate Read’s solution, given only assumptions that most, if not all, consistentists would be prepared to accept. The plan is as follows: Section 2.2 sets out Read’s theory of truth (Sect. 2.2.1), and his solution to the liar paradox (Sect. 2.2.2). Section 2.3 discusses Read’s treatment of inference rules. And Sect. 2.4 introduces ‘indirect revenge’. Section 2.4.1 discusses assertion and truth; Sect. 2.4.2 discusses the role of truth; Sect. 2.4.3 demonstrates the indirect revenge problem, and Sect. 2.5 concludes by imagining Read’s response to indirect revenge.

2.2

Read’s Theory of Truth and the Liar

A solution to a paradox should, ideally, provide both a diagnosis of the paradox and a treatment aimed at excising the life-threatening virus. In general, we expect a treatment, and hope for a diagnosis that will serve to illuminate the assumptions that we previously, and erroneously, accepted.7 For a consistentist, a treatment of the liar will be a solution that holds onto as many as possible of the features of language and logic – e.g., prima facie sacrosanct logical principles such as the Law of Non-Contradiction, and prima facie empirically verifiable (or anyway sacrosanct) semantic and linguistic features such as semantic closure and semantic expressibility. Of course, one may jettison or revise certain beliefs – as Kripke does (cf. [21]), when he denies that an ungrounded sentence expresses a proposition – but, in general, if the putative bedrock principles are to be revised or jettisoned, a reason for doing so is always demanded. To Read’s credit, he takes on one of the more recent, apparently sacrosanct, truths about truth – the Tarski-inspired equivalence schema: X is true if, and only if P

7 For

more on this, see [13].

(T)

Read and Indirect Revenge

23

where ‘X’ is replaced with a name of the sentence whose translation (e.g., in the metalanguage) replaces ‘P ’.8 As Read notes (following Tarski), (T) itself is not a theory of truth; at best, it serves as a method for evaluating the correctness of proposed theories of truth. So, for example, the truth of the instances of (T) may serve to support a proposed theory of truth because, say, we take those instances to capture certain facts about the concept of truth and we want our definition to line up with our concept, lest we be accused of putting forward an artificial theory that, while interesting, can hardly serve as a theory of truth. In fact, Read denies that (T) supports a tenable theory of truth. As he suggests, when our conception of truth is properly characterised – via Read’s Bradwardine-inspired account of that notion – we will find both that (T) is false (or, generally, invalid, depending on its exact formulation) and that the correct definition of truth renders the liar sentence to be false (that is, as not true), and without contradiction.

2.2.1

Read’s Theory of Truth

So what, per Read, is our concept of truth? One answer that Read (p. 5) considers is Crispin Wright’s ‘correspondence platitude’, x is true if, and only if, things are as x says they are

(CP)

This scheme is platitudinous because it appears to be consonant with any (reasonable) theory of truth, whether distinctly deflationary or distinctly inflationary. While Read does not reject (CP) out of hand, he does demote it, for he takes it to be too vague to be put to work (conceptually speaking). What (CP) is meant to capture – what Read takes to render it platitudinous – is the thought that a sentence (or expression) is true if and only if whatever it says to be the case is the case.9 Read maintains that we need a more accurate expression of our conception of truth, which expression, in turn, will support a novel theory 8 I leave out Read’s claim that the translation on the right-hand side of (an instance of) (T) is in the metalanguage. For what follows, nothing shall turn on the question as to whether or not we grant the existence of a metalanguage. 9 I prefer a different reading to Wright’s platitude, viz.,

A sentence (or expression) x is true iff whatever it (actually) expresses actually is (the case) (CP*) which seems to better capture Read’s point, especially since it leaves out the putatively irrelevant ‘says’. Moreover, one might take (CP*) to further support Read’s (K). For present purposes, the reader is encouraged to render Wright’s platitude in terms of either (CP) or (CP*).

24

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of truth. His proposed conception of truth can be formulated in either of the following two ways: x is true if an only iff things are wholly as x says they are

(S)

or x is true if an only iff however x says things are, they are

(S’)

which schemata Read codifies in his proposed definition of truth. Reflection on (S) and (S’) allows us to see Read’s problem with (CP). Under (CP), the truth of a truth bearer, x, (viz., a meaningful sentence, x, or, as I will usually assume, the proposition that it expresses) depends on whether things are as x implies them to be. But that is not sufficient, according to Read, to capture our conception of truth, for truth depends on more than just whether things are as x implies them to be; if x is true then things must be however it implies that they are. (S’) and (S) capture this thought, which cannot be culled from (CP) In order to capture (S) and (S’), Read proposes a general schema that he takes to characterise the notion of truth, Tx ⇔ ∀p(x:p → p)

(A)

where ‘⇔’ is strict equivalence, ‘→’ is the material conditional, and ‘x : p’ symbolizes ‘x says that p’. In order for (A) properly to capture both (S) and (S’), Read assumes Bradwardine’s view of ‘saying that’, which holds that a sentence says everything that it entails. The closure of ‘saying that’ under strict implication is represented by the schema (∀p, q)((p ⇒ q) → (x:p ⇒ x:q)).

(K)

where ‘⇒’ is strict implication. Modulo (K), then, (A) provides Read’s Bradwardine-inspired definition of truth, one that supports both (S) and (S’). For, given (K), (A) says that a sentence is true if things are however it says they are. Of course, one might be inclined to accept something like (A), but might (rightly, in my opinion) have worries about (K). In order to avoid the problem, Read proposes an alternative schema, Tx ⇔ ∀p(∃q(x:q ∧ (q ⇒ p)) → p).

(A )

which relies on a weaker entailment principle than that which (K) defines. Unlike (A), (A’) says that a sentence is true if things are however what it says (logically) implies them to be, where says is taken as identical to logical implication.

Read and Indirect Revenge

25

Before moving from Read’s proposed account of truth to its application, I offer a convention, and a putatively helpful clarification. The convention: Read takes the ‘say’ relation to be logical implication – more specifically, strict implication. I have no objection to the usage but, so as to make clear that this implication relation may be distinct from the meaning relation, I will represent Read’s notion of ‘say’ as ‘say’, or, where context (and grammar) demands, as ‘said’. There is also a colloquial reading of ‘say’, where what is said is just what is expressed. To mark the difference between what is expressed and what is (strictly) implied, I will represent ‘say’, with its colloquial reading, as ‘say’ (and ditto for ‘said’).10 Given the convention, the putatively helpful clarification is as follows. What a meaningful (viz., an interpreted) sentence says – what is expressed by an assertion of it – is a proposition, which is what we express, when we assert what we do.11 What is a proposition? Well, since Read (Sect. 1.2) follows Horwich’s use of proposition-talk (cf. [20]), I will follow suit (more or less) and claim that a proposition will be whatever is expressed by a (meaningful) that-clause. I briefly dwell on these details, in order to make clear that even if we grant Read’s notion of ‘saying’, as captured by (K), it does not follow that the proposition expressed by an interpreted sentence, ρ, contains all that ρ says. That is, what we say will not always be whatever our interpreted sentences say, save for cases where we say no more than it says (e.g., as we and it would do, in the case of, say, ‘snow is white’, which is true just when snow is as the sentence says that it is – namely, when snow is white).12

2.2.2

Read on the Liar

Read’s definition of truth, as captured by (A) or (A’), rules out the unrestricted status of (T), since, under (A) or (A’) (as Read notes, the 10 I mark the difference because Read has not proposed an accompanying theory of meaning – viz., an account of what a sentence (or a word) says, or of what the meaning-relation is such that things say – that is, express – what they do. 11 I shall not quibble with the motivation that Read ([27], Sect. 1.1) musters for dethroning the conventional status of (T). That said, I should point out that the requirement that only meaningful – that is, interpreted – sentences (including, of course, sentences that contain context sensitive expressions) are appropriately ‘truth apt’ rules out the non-paradoxical cases that Read enlists. 12 Of course, one may have worries about whether ‘snow is white’ says more than what it strictly speaking says, for example, that snow is not pink. I shall leave aside the obvious constellation of issues that this raises, noting, however, that more should be said about the say relation (as Read is of course well aware).

26

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two are equivalent if we assume (K)), a sentence’s truth condition may not always be provided only by the proposition that it expresses (viz., what it says). According to Read, this is all for the better. As he notes, with few exceptions (cf., [10, 16], for such an exception), consistent solutions to the liar paradox have seemed, inevitably, and thus unacceptably, to attempt to limit the range of (T), in ways that are aimed at excluding its liar-like instances. Given a supposed pattern of failures, Read suggests that the resultant ad hockery of such consistent solutions to the liar shows that this way of attempting to resolve the paradox is simply on the wrong track. He (p. 10) holds that (what I shall call) ‘the consistentist’s mistake’ was to assume that the truth conditions for a sentence were exhaustively provided by the right-hand side of (T)’s instances. In order to right the philosophical (and methodological) wrongs, Read proposes his Bradwardine-inspired account of truth, the goal of which is to provide truth conditions that capture our conception of truth, without the threat of impending paradoxicality. To see this, consider a standard ‘semantic spandrel’, λ, such that λ = ¬Tλ, and where λ says that ¬Tλ, viz., λ:¬Tλ – that what is labelled ‘λ’ is not true.13 Given (A) and our spandrel, λ, the relevant instance of λ yields: Tλ ⇔ ∀p(λ:p → p)

(a)

λ ⇔ (¬Tλ ∧ q)

(b)

viz., where ‘q’ is all and only what λ says other than ¬Tλ. Given (b), closure, some obvious juggling, and the assumption that λ:¬Tλ ∧ q, we get: ¬Tλ ⇒ (Tλ ∨ ¬q) (c) from which, given (K) and what λ says, we get λ:(Tλ ∨ ¬q) ∧ q)

(d)

And that, given ((Tλ ∨ ¬q) ∧ q) ⇒ Tλ, yields: λ:Tλ

(e)

13 A spandrel is a necessary by-product of something else, as for example in the spandrels of the San Marco basilica in Venice (cf. [18]), which are the triangular spaces that emerge from the basilica’s rounded arches. The spandrels of the basilica have been beautifully painted, suggesting that they were intentionally created. They were not, but the arches that inevitably yield them were intentionally created. In other work [5], I have called a liar sentence a ‘semantic spandrel’ – a necessary by-product of certain features of a natural language, which features, akin to the arches, were (in a sense) intentionally created.

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So, since λ:Tλ when λ:¬Tλ, it follows that λ says ¬Tλ ∧ Tλ, from which, given (A) and a bit more juggling, we derive ¬Tλ

(f)

i.e., that λ is not true, which is what most theorists (of whatever logical stripe) wish to say about the liar – to the effect that λ is at least untrue. What Read has shown is quite ingenious. For λ to be true, it would have to be both true and not true. But nothing can be both true and not (given another orthodox view that is quite different from that which delivers (T)14 ), so, in particular, λ cannot be true. Accordingly, λ says both that it is true and that it is not true, but, without accepting inconsistency, it cannot show that λ is true and not true, for nothing is both true and not. Moreover, since Read maintains that any meaningful sentence says its own truth condition, and since λ’s truth condition, viz., ¬Tλ∧Tλ, cannot consistently be satisfied, we shall have to conclude that λ is false, and, given bivalence, that it is therefore not true.

2.3

Read and Inference-Rule Aletheism

A liar sentence says the impossible and is false for so saying. Under (K), λ says whatever is implied by what λ says. According to Read, λ says that λ is not true, but, being false (and, thus, failing to be true), it does not thereby follow that λ is true. Inference-rule deflationists would be horrified at the thought that we cannot infer the truth of a statement from a statement that seems, by all lights, to be true. Of course, such deflationists have not considered (A). What they have considered, and what they accept (unrestrictedly speaking) are the inference rules, true-in and true-out, viz., α ⇒ Tα

(TI )

Tα ⇒ α15

(TO )

These inference rules are generally thought to be valid, in which case, given an assumption of classical logic, the inference from (to) that p 14 For

a discussion of how one might plausibly go about revising the LNC, see [9]. might also opt for ‘intersubstitution’, which is the rule according to which Tα and α can be inter-substituted in all extensional – that is, non-opaque – contexts [15] and others have accepted ‘intersub’, taking it to be a central feature of deflationism. The points that I make below go through if we substitute the rule of intersubstitution for the relevant inference rules.

15 One

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to (from) that p is true is truth preserving.16 That is, the unrestricted inference rules (usually) yield (T), along with its relevant modal status. Read denies that (T) has that status, so, as we already knew, Read is no inference-rule deflationist (if he is a deflationist at all). Suppose, though, that Read were to accept the conjunction of (TI ) and (TO ). Would that enable Read to yield (A’) (or its twin, (A), given (K))? If (as happens to be correct) he does not affirm that conjunction, does he reject both inference rules, or does he deny one while affirming the other? To all appearances, Read affirms (TO ). Indeed, under (A), Tα trivially implies α, for if α is true then all of α’s consequences are the case – that is, everything α says is correct – including, of course, α itself. Suppose then that α is the case. Can we then conclude that α is true, that is, that (TI ) is valid? According to Read, we cannot, for, if we replace λ, i.e., ¬Tλ, for α then, since λ says Tλ (and, of course, since it also says ¬Tλ), λ will provide a false instance of (TI ). Hence, unlike standard deflationists, though in line with Kripke’s outline, Read denies the (general) validity of (TI ), and, with it, that of Tα ⇔ α (T∗ ) for all meaningful instances, α. Perhaps surprisingly, I will not directly question Read’s diagnosis. Rather, I will question it indirectly, by providing reasons for thinking that it will be difficult for Read to provide a consistent solution to the liar, without losing a language’s impressive expressive power. Simply put, there is more behind (T) than Read has considered.

2.4

Read and Indirect Revenge

Since Read maintains that λ is not true, he must also accept that it is impossible for λ to be true – impossible, given that λ = ¬Tλ, that that λ is not true is true. Kripke [21] maintains that λ is not true.17 But he is not prepared to assert that λ is not true, for he believes that we cannot assert how things are for λ, within a language, L, in which it was constructed (and Soames [28] agrees). More specifically, he maintains that, without paradox, a semantically closed language cannot contain its 16 Of course, not all theorists will affirm the validity of both inference rules, especially when confronting the liar paradox. For example, [21] can be seen as affirming (TO ) while rejecting (TI ), given classical logic. 17 The term ‘strengthened liar’ was introduced by van Frassen [29, p. 147], and was “designed especially for those enlightened philosophers who are not taken in by Bivalence.” As is familiar, if the consistentist declares that a simple liar sentence, L = ‘L is false’, is gappy, without a truth-value, etc. then a strengthened version of L, viz., λ, will create the resulting paradox. As is evident, given Read’s commitment to Bivalence, λ is no ‘stronger’ than L.

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own untruth predicate, or, what amounts to the same, strong negation cannot be expressed in that language.18 Read has no need for strong negation, since he thinks that all meaningful sentences that are not true are false, and that all false sentences are also not true.19 Under Kripke’s scheme, a ‘gappy’ sentence like λ may (in some sense) say that it is not true, but only in a non-threatening, non-proposition expressing, way. Moreover, it will, in a sense, have a ‘semantic status’ – it will be gappy, or valueless, or something of that sort. But, under Kripke’s scheme (and without paradox) we (or, more accurately, our semantically closed language) cannot express that λ has the status that it appears to have. This is not to say that λ’s status is inexpressible. Indeed, Kripke maintains that one can very well consistently express that λ is not true; the problem is that he can do that only by appeal (or, more accurately, by ascent) to a metalanguage.20 The putative revenge problem that λ presents (and that leads to Kripke’s ascent) may be seen as evincing a tension between semantic universality, according to which we can semantically characterise all of the expressions of a given (natural) language, and a sort of semantic incompleteness, according to which we cannot, as some of the putative concepts of the language are locally inexpressible.21 The apparent split between the pull of semantic universality and the push to semantic incompleteness highlights the unstable status of semantic closure, modulo standard consistent solutions to the liar paradox. It is a purported virtue of Read’s account of truth that he can consistently express that λ is not true, given a semantically closed language that appears to be semantically complete, thereby respecting semantic universality. If it should end up that Read cannot utter ‘λ is not true’ – cannot, that is, express that λ is not true – this would raise a substantial 18 Semantic closure is a technical term that is defined over formal languages. For present purposes, I call a language, L, semantically closed if:

1 All expressions of L possess a semantic status; and 2 Each of these semantic statuses can be expressed within L. If we assume (1) and (2), we will conclude that all (well-formed) sentences of L have semantic statuses – that each admits of a semantic characterisation – and that, within L, we can thus express the semantic status as it is assigned to each expression (e.g., sentence) of L. 19 Of course, some theorists (e.g., [24]) deny that what is false is, thus, not true. Such theorists must thereby deny intersub. 20 Recall [21], ‘the ghost of the Tarski hierarchy is still with us.’ 21 In order for a language to be semantically universal, it must be able to explain the proper use of each of the expressions of that language, thereby distinguishing ‘semantic universality’ from an apparent conception of general semantic universality according to which all concepts are expressible in a language. One might endorse general semantic universality, but it is not entailed by semantic universality, and, indeed, will play no role, for what follows.

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worry about the prospects for Read’s proposed account of truth. As I will now show, there is reason to worry. Baldly stated, my worry is that Read will either be stuck with an untenable quietism – he will accept that λ is not true, but he will not be in a position directly to assert that λ is not true – or he will find himself having to deny certain claims that, on the face of it, do not have anything to do with the considerations that drove him to his Bradwardine-inspired account of truth.

2.4.1

Truly Asserting What We Believe

I begin with the first disjunct, by reflection on the act of assertion. For what follows, I will assume – what is, anyway, very hard to reject – that the act of asserting is essentially rule-governed, viz., that there is at least one norm that governs the act.22 For Williamson [30], the rule of assertion is constitutive of the speech act of assertion. He does not, however, take this rule to provide a necessary condition for successful assertion, for he does not take the rule to be necessary for an act to count as an assertion. Rather, he seems to claim that one can be said to participate in the act of asserting only if he has come somehow to internalise the relevant rule of assertion.23 So, rather than taking the rule to govern what is to count as an act of assertion, it seems that Williamson takes it to be a constitutive feature of a participation in the act of asserting.24 My present concern does not compel me to argue for one or another formulation of this rule. But there does seem to be one consequence that we can draw from Williamson’s general framework, whatever we ultimately wish to say about his favoured rule of assertion. The consequence: Any acceptable way of filling in the rule for ‘C’ (of ‘C(P )’) will strictly prohibit the combination whereby a putative asserter performs the act of asserting that p when P is false, and when its falsity is taken to be so by the prospective asserter.

22 Here I follow [30], though I will not commit myself to his preferred ‘Knowledge Rule’; viz., that one must: assert that p only if one knows that p [schematically: One must: assert that p only if C(p), where ‘p’ is a schematic sentence letter and ‘C(p)’ expresses a condition (knowledge, truth, or what have you).] While I am sceptical about the tenability of Williamson’s claim that this rule is correct, I shall not be concerned to raise issues for it in this paper. Indeed, my concern here regards only a consequence of adopting a norm of assertion; whether the norm is (is not) captured by the Knowledge Rule is beside the point. 23 The knowledge rule will then go as follows: One who aims to perform the speech act of assertion, when he is actually asserting, must perform the act of asserting that p only if he knows that p. Thanks to Jon Adler for helpful (and ongoing) discussion. 24 For an application of the rule to a proposed solution to the liar, see [6].

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More directly, we can extract what we might call ‘The FA norm’: That one should refrain from asserting what is not true.25 This norm is really the flipside of a truth-involving norm (the ‘TA norm’), to the effect that the truth is (at least part of) what we ought to assert, the untruth being something that we ought not to assert.26 A version of the TA norm has been widely supported (e.g., by Brandom [11, 12], Dummett, Priest and Williamson [11, 14, 23, 30], and a host of other competent theorists), and is sometimes taken to highlight a particular feature of assertion: That, whatever else we wish to say about the speech act, truth is its telos.27 My aim is not to review or extend the arguments in favour of the respective norms, TA and FA; nor is it to consider what Read should say about these norms, given his proposed account of truth. Rather, I wish to point out that whether we go with Read’s account of truth, or perhaps with some other account, it will follow from Read’s proposed solution to the liar, together with FA and TA, that we ought not to assert that λ is not true (deontically speaking). Of course, in a certain sense, Read can utter the sentence, λ. After all, he takes λ to name a meaningful, proposition-expressing, truth-apt sentence. Even so, he cannot assert that λ is not true, at least not without clearly, and obviously, violating one or another of the truth-involving norms. And since he is aware of the norm, for all intents and purposes, it seems that he must embrace quietism, which indirectly threatens the assumption of semantic completeness. I have claimed that Read cannot (in a suitable, deontic sense) assert that λ is not true, while abiding a norm that governs that practice. But there is a further point, which seems to be in line with the TA norm (in fact, the two seem mutually supportive) – that assertion, like belief, aims at truth.28 One venerable way of cashing this out is to maintain that, in sincerely asserting, we are putting forward what is asserted as true. Of course, this does not constitute an attack on Read’s denial of the claim that every sentence says of itself that it is true. What it raises,

25 I shall assume that the untrue and the false are the same, given Read’s commitments. Of course, nothing turns on this; I could reformulate two analogous rules, should one (say) deny bivalence. 26 In order for the TA norm to be sanctioned, we must assume that it is sometimes OK to assert, of course. 27 The claim that truth is the telos of assertion is compatible with Williamson’s knowledge rule. 28 Thus, there seems to be something wrong, in the Moore-like way, with a sentence like ‘I believe that ‘p’ but ‘p’ is not true’, which is what Read seems compelled to accept. The problem is obvious if we replace the liar sentence, λ, for ‘p’, and provided we are careful about use and mention.

32

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however, is the thought that when we assert, what we say is being put forward by us, by (sincere, honest) asserters, as true. Read may well attempt to deny this claim, for if he does not, he will be forced to maintain (and without ensuing inconsistency) that a liar sentence (that is, what it expresses) is thus, at least in this sense, not assertible. Neither prospect – denying the claims about assertion, or denying the warranted assertibility of the status of λ – looks promising. For one thing, there is nothing implicit in his theory of truth that tells against either the norm of assertion or the related point about its aim. Thus, one might worry that his response will be ad hoc.29 If, instead, he were to deny the general assertibility of λ, it would seem that he would inherit the objections to standard consistent treatments of the liar paradox, both threatening his solution to the liar (for further worries about unintended ad hocery), and raising the question of why his solution is to be preferred to the myriad other consistent solutions, all of which also seem to have similar expressibility-related problem. As I hope is clear, rather than constructing a direct revenge problem for Read’s proposed solution to the liar, I have argued for (what I will call) an indirect revenge problem, which is so-called because it raises some of the very worries that Read was attempting to avoid. So what can we draw from the going observations? What we cannot draw is that Read lands in a contradiction, if he says, or even believes, that a liar-like spandrel is not true. In this sense, these worries do not yield (what we might call) ‘direct revenge problems.’ But indirect revenge problems emerge when we see what we must give up, in order to square Read’s account of truth with the aletheically infused concepts, norms, etc. that, at least prima facie, we once had reason to endorse.

2.4.2

Aletheic Neutralism and the Role of Truth

I mentioned (Sect. 2.3) that it did not seem that Read endorsed deflationism, since he accepted neither unrestricted (T) nor the meaningconstituting inference rule, (TI ). Of course, I was only half serious. I am not concerned in this paper to determine whether an account of truth that is governed by (A) or (A’) should properly be said to be a deflationary theory of truth. Rather, for what follows, I will focus on the 29 He might point out that, while it is not the case that λ is true, the means by which he can express this fact calls for an invocation of tokens, as the plausible bearers of truth and falsity. He might then go on to insist that the employment of tokens is not ad hoc, as it is consonant with Bradwardine’s own semantic theory (as I believe that it is) – the one that he employed, when he set out his treatment of the liar. I shall have a bit to say about this defence, in Sect. 2.5. For now, I assume that Read will not rely on a tokenist view.

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33

expressive role of the truth predicate, which we might see as in some sense definitive of a deflationary conception of truth.30 As Quine noted [26], the truth predicate performs an important logical role as a device for forming generalizations. This logical role, in turn, supports truth’s expressive role as a device that enables the expression of blind endorsement, where blind, truth-involving, endorsement regards the expression of a commitment to (the general assent to; the general acceptance of) claims that can be specified by descriptions (e.g., ‘The last thing that Read said’, ‘Everything the Pope said’, etc.) or by names (e.g., ‘Dorothy’s theory of conditionals’).31 It seems that the truth predicate is especially useful when we find that we wish to endorse a large class of claims (e.g., everything Einstein said, for example), or when we wish to endorse something, but cannot quite recall what it was (e.g., what Tom said two years ago, just before he died). In those cases, the truth predicate facilitates the expression of acceptance, thereby enabling us to express our commitments to claims that we accept, without endorsing them directly. Whereas the truth predicate facilitates the expression of blind endorsement (e.g., for a claim or set of claims), the falsity predicate (or, for what follows, the untruth predicate),32 truth’s expressive dual, facilitates the expression of blind denial, enabling us to express our commitments against claims that we reject (or, at least, that we do not accept), without denying them directly. Like the truth predicate, the falsity predicate is especially useful when we find that we wish to deny a large class of sentences, or when we wish to deny a single sentence, Given these roles, we might go on to distinguish a deflationary conception of truth from a non-deflationary conception of truth, by holding that it is only in the case of the former that the point of the truth predicate is exhausted by its expressive role.33 We could then say that a non-deflationary conception of truth (or, generally, of our aletheic notions) would deny that the expressive role is the unique role for truth. This would not be to deny the import of truth’s expressive role; instead, it would be to demote its status – as the role for truth. 30 Thanks to Jody Azzouni for useful (and enjoyable) discussion of this section. For more on this, see [8]). 31 Quine did not engage in this exact sort of ‘truth talk’, though he (and later Field, Horwich and Leeds [15, 19, 22]) pointed out that truth performed this important – indeed, practically speaking, indispensable – role. 32 I shall assume the equivalence between falsity and untruth, for purposes of this discussion only (i.e., given Read’s assumption of bivalence). The question of their link is important (and contentious), but we can safely ignore it, given the issues that are raised in this paper. 33 ‘Deflationism’ is then to be seen as, in a sense, second-order, as it applies to theories of truth, each such theory aiming to explain truth, and to explain how it is able to perform its expressive role.

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These last points are worth emphasising (briefly). As I see it, the question as to whether or not we endorse a deflationary conception of truth turns crucially on the question of whether truth (or the truth predicates) performs any role beyond its expressive one. That the truth predicate performs this role – that it facilitates the expression of acceptance and denial – is not up for debate. What may be up for debate is how it is able to perform the role, and, additionally, whether there is any other sort of role for the truth predicate to perform. I take these two questions to be integral to a debate about the nature and structure of truth, i.e., about what sort of theory of truth one ought to adopt, but will leave them aside, for what follows. Prior to applying the expressive role of truth to Read’s solution to the liar, let us briefly return to the worry about assertion. The worry there was not that Read’s theory was internally inconsistent; it was, rather, that we got a tension when we attached his theory of truth to what seemed like some fairly uncontentious claims about the act of asserting (or believing). When we turn to the expressive role of the truth predicate, we see a related worry – but one that emerges even if we do not affirm the apparent platitude about assertion and belief (namely, that in sincerely asserting what one accepts, one is indirectly expressing his commitment to what he believes and, thus, in at least some sense, is putting forward what he has asserted as true). In order to see the worry, suppose (as Read must) that Read can assert that λ is not true, and let us further suppose (as we all must) that truth performs its expressive roles. As we will see, a problem emerges when we consider how Read can satisfy both of these conditions, given the commitments that follow from his proposed account of truth.

2.4.3

Indirect Revenge

If Frege said that no whales are mammals, and if Dr. Gustav Lauben accepted what he said, then the good doctor could either have said that no whales are mammals or he could have said that what Frege said was true, both of which would have enabled him to express his commitment to what Frege said, to the effect that no whales are mammals. If, alternatively, Gustav came to believe that some whales are mammals, and, thus, came to reject Frege’s claim, he could either have said that some whales are mammals or he could have said that what Frege said was false (or, at least, that it was not true), both of which would have enabled Dr. Lauben to express his denial of Frege’s claim (that is, to express his commitment against what Frege said), to the effect that no whales are mammals.

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As is familiar, this is by no means restricted to single claims. If Frege proposed a novel theory, and if Gustav came to accept that theory, then Gustav could express his acceptance of Frege’s novel theory, simply by remarking that it is true. In so doing, he indirectly expressed his commitment to the statements that comprise that theory, which statements, we may assume, he could not or would not endorse directly (e.g., because there are too many of them). And, of course, the same would go for Gustav’s rejection of Frege’s theory, the denial of which he could express by asserting that not all of Frege’s theory is true, or by asserting that Frege’s theory is false, or, more generally, by indirectly expressing his denial of the theory, without directly denying any of the statements that are to be rejected. These (familiar) examples make evident the import of truth’s (falsity’s) expressive role. With the truth predicate (together, for example, with a name that denotes Frege’s new theory and a suitable quantifier), Gustav is able to express his endorsement (denial) of Frege’s novel theory, simply by asserting that Frege’s novel theory is true (not true). When we turn to Read, we see that if his theory of truth is to be sufficient, it must be able to accommodate truth’s logical and expressive role. And yet, (TI ) is a central feature of that role, for without it (or, more generally, without (T*), or accompanied principles), Dr. Lauben could not have concluded that Frege’s novel theory is true. That is, without something like (T*), it seems that this use of the truth predicate is unavailable. Suppose that you wish to deny a part of Read’s account of truth, but that you cannot recall which parts of the account you intended to deny. It seems that you should be able to express your denial by saying something like ‘Read’s account of truth is not true’ (or, instead, ‘Not all of Read’s account is true’), which is meant to have the effect of expressing your commitment against Read’s account of truth. But, without (TI ), it seems that this denial cannot be expressed, for, as we have seen, at least some of what Read wants to say, and what he takes to be the case, is not true (e.g., that λ is not true). We get a related problem if we attempt indirectly to express our commitment to claims that we accept. Let us assume that Read can assert that λ is not true, and that you accept what Read last said, but that you cannot quite recall what it was that he had said (recall: It was that λ is not true).34 Given the expressive role of the truth predicate, you 34 If

Read were to concede that we cannot directly assert that λ is not true then the game would be over. Accordingly, for what follows, I assume that there is a means by which we can assert what is expressed by the sentences that we utter.

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can blindly endorse Read’s claim by asserting that what Read last said is true. That is, assuming that Read asserts (i) λ is not true, you, accepting what he has said, can respond by asserting (ii) What Read last said is true. In so asserting, you thereby indirectly express your endorsement of the claim that Read made. All of this is, of course, routine: Since (ex hypothesis) Read can assert (i), then if his account of truth is to be adequate to capture truth’s logical and expressive role, if you affirm Read’s claim, it should follow that either you can assert (i), too, or you can achieving roughly the same result, by asserting (ii). In so asserting, it seems that you thereby blindly endorsed what Read last said. Have you indirectly expressed your commitment to the claim that λ is not true? Unfortunately, under Read’s account of truth, it seems that the answer must be no. One way of seeing this is by considering the question of how it is that the truth predicate is able to perform its expressive role. What facilitates the expression of blind endorsement is the contribution that the truth predicate adds to the formulation of generalisations that we could not otherwise – or, at least, could not so easily – make. And at least part of what secures this expressive role is that the truth predicate (plus logic) enables us to deduce general claims from the things to which we directly express our commitment. At least prima facie it is for this reason that it seems that we can infer (ii) from (i): If we are told that (i) is what Read last said, and if we accept whatever it was that he said, then, given a suitable quantifier, the truth predicate enables us to conclude (ii). Analogously, if we accept (and, thus, can assert) (ii), and if we are told what Read last said, then we can deduce (i), which we thereby accept. The problem with Read’s account of truth is now clear. Even if Read accepts, and so can assert, (i), it does not follow that he can conclude (ii). In fact, he cannot infer (ii) from (i) (together with plain facts), without employing something – for example, (T*), or perhaps a related principle (e.g., one that sanctions intersub) – that would allow him to generalise over instances of the form Read last said that p and p is true, from instances of the form Read said that p and p. But these principles (or rules) will be ones that Read will reject, in order to provide an account of truth that is adequate to resolve the liar. Indeed, we know that Read must reject these principles, because they will be the ones that will inevitably yield (T), the very schema that he aimed to demote.

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2.5

37

Conclusion: Read and Indirect Revenge

I have attempted to show that Read can resolve the liar paradox only by insisting on an account of truth that indirectly yields the sorts of expressive problems that are standardly attributed to consistent solutions to that paradox. How might Read respond to these worries? Although Read has not explicitly assumed a token view, it might be that he (along with Bradwardine) will end up being a tokenist, since he wants to say that λ is false, and he wants his solution to be true. Of course, token theorists face revenge problems of their own. Consider, for example, a standard revenge problem, Every token that is co-typical with (UL) is false.

(UL)

Or consider the following revenge problem put forth by Quine35 , which appears to yield that a certain result is itself not true, (Z) For any context c, the token that results in c by putting the quotation of ‘For any context c, the token that results in c by putting the quotation of w for the variable ‘w’ in w is not true’ for the variable ‘w’ in ‘For any context c, the token that results in c by putting the quotation of w for the variable ‘w’ in w is not true’ is not true.

As Quine reminds us, a variant of (Z) tells us how to construct the result that, to all appearances, (Z) itself possesses. Rather than attempting to show that Read will have the same worries that any tokenist has, let us assume that Read will avail himself of a token view and that, when properly fleshed out (and integrated into his account of truth), all of the emergent, putatively paradoxical (revengerelated) tokens are rendered false, thereby defusing the threat of direct revenge. What I wish to show is that even under those circumstances, the threat of indirect revenge remains. As we saw, there is a sense in which one (following Soames [28]) can see Kripke’s solution to the liar as defusing the threat of direct revenge, without the threat of indirect revenge. The idea was that indirect revenge could be avoided, but at the cost of ‘local expressibility’: λ would be inexpressible, relative to the object language; we could not assert (within that language) that λ has the status that we take it to have. We can leave aside the issues that this raises, for it is clear that Read cannot afford to give up expressibility. After all, Read wishes to hold onto Bivalence, and he does not seem inclined towards an epistemicist 35 See [25, pp. 307–308]. (In the cited pages, Quine is constructing a protosyntactical version of the G¨ odel-style sentence ‘(1) is not a theorem’, where ‘(1)’ names that very sentence.) Variants of (Z) are possible – for example, one beginning ‘For any possible context c, . . . ’.

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solution to the liar paradox. Moreover, he says, explicitly (and more than just once), that a liar sentence like ‘λ’ is not true. Thus (as Read will freely admit), he cannot ignore the threat of indirect revenge. In turn, we can ask: Assuming that Read will attach a tokenist view to his account of truth (thereby rendering false all of the putatively paradoxical tokens), what would it take for him to resolve the extant, and indirect, revenge problems? The answer seems plain: In order to avoid the problem of indirect revenge that I have presented, Read would have to find an expression, t, where ‘t’ may or may not be a token, which can truly be said to characterise a liar sentence as Read takes it to be – that is, as false. Now, let us leave aside the worry that Read will not be in a position to find such a t, given the problems that arise from (UL), (Z), and myriad other such cases (this may well be a serious worry, but it is not one that we need to dwell on for what follows). If Read could find such a t for a semantically closed language, which is, in addition, expressively complete, then he would have no reason to insist on his Bradwardineinspired account of truth, for the token solution to the paradox would suffice, provided he was prepared to deny bivalence.36

References [1] Armour-Garb B. (2004) Minimalism, The generalization problem and the liar, Synthese 912. [2] Armour-Garb B. (2004) Diagnosing dialetheism. In G. Priest, J.C. Beall and B. Armour-Garb (eds.) New Essays in the Law of Non-Contradiction. Clarendon: Oxford University Press. [3] Armour-Garb B. (2005) Wrestling with (and without) dialetheism, Australasian Journal of Philosophy 83. [4] Armour-Garb B. and Beall J.C. (2005) Deflationism: the basics. In B. Armour-Garb and J.C. Beall (eds.) Deflationary Truth. Chicago: Open Court Press. [5] Armour-Garb B. and Beall J.C. (2005) Minimalism, epistemicism and semantic paradox. In J.C. Beall and B. Armour-Garb (eds.) Deflationism and Paradox. [6] Armour-Garb B. and Priest G. (2005) Against analetheism. Analysis 65. 36 Thanks

to Stephen Read for helpful comments.

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[7] Armour-Garb B. and Woodbridge J. (2006) Dialetheism, semantic pathology, and the open pair, Australasian Journal of Philosophy 83 (3). [8] Azzouni J. (2006) Tracking Reasons. Oxford: Oxford University Press. [9] Azzouni J. and Armour-Garb B. (2005) Standing on common ground, Journal of Philosophy, CII Number 10. [10] Beall J.C. (2004) True and false – as if. In G. Priest, J.C. Beall and B. Armour-Garb (eds.) The Law of Non-Contradiction: New Philosophical Essays. Clarendon: Oxford University Press. [11] Brandom R. (1983) Asserting, Noûs 7. [12] Brandom R. (1983). Making it Explicit. Cambridge, MA: Harvard University Press. [13] Chihara C. (1979). The semantic paradoxes: a diagnostic investigation, The Philosophical Review 88. [14] Dummett M. (1959) Truth, Proceedings of the Aristotelian Society 59. [15] Field H. (1994) Deflationist views of meaning and content, Mind 103. [16] Field H. (2008) Solving the paradoxes, escaping revenge. In JC Bell (ed.) Revenge of the Liar—New Essays on the Paradox. Oxford: Oxford University Press. [17] Frege G. (1957). The thought: a logical inquiry, translated by A. and M. Quinton, Mind 65: 289–311. [18] Gould S.J. and Lewontin R. (1978) The spandrels of San Marco and the Panglossian paradigm: A critique of the adaptationist programme, Royal Society of London 205. [19] Horwich P. (1990) Truth. Clarendon: Oxford University Press. [20] Horwich P. (1998) Truth, 2nd edition. Clarendon: Oxford University Press. [21] Kripke S. (1975) Outline of a theory of truth, Journal of Philosophy 72. [22] Leeds S. (1978) Theories of truth and reference, Erkenntnis 13: 111–129. [23] Priest G. (1979) Logic of paradox, Journal of Philosophical Logic 8: 219–41. [24] Priest G. (1987) In Contradiction. Dordrecht: Martinus Nijhoff.

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[25] Quine W.V.O. (1965), Mathematical Logic. Cambridge, MA: Harvard University Press, revised ed. [26] Quine W.V.O. (1970), Philosophy of Logic. Englewood Cliffs, NJ: Prentice Hall. [27] Read S. (2008) Truth schema and the liar. This volume. [28] Soames S. (1998) Understanding Truth. New York: Oxford University Press. [29] van Frassen B. (1966) Singular terms, truth value gaps, and free logic, Journal of Philosophy 63 (17). [30] Williamson T. (1996) Knowing and asserting, The Philosophical Review 105 (4).

Chapter 3 TARSKI’S HIDDEN THEORY OF MEANING: SENTENCES SAY EXACTLY ONE THING Catarina Dutilh Novaes∗ Leiden University, P.O. Box 9500, 2300 RA Leiden The Netherlands and Fordham University Rose Hill Campus Bronx NY 10458 (718) 817-1000, USA [email protected]

Abstract

I argue that Tarski’s truth schema (T) concerns meaning just as much as it concerns truth. Better said, (T) implicitly presupposes a theory of meaning; it rests on a simple and yet contentious assumption, namely what Andjelković and Williamson [1] have termed the principle of uniformity (U): every sentence of a language says exactly one thing. I here challenge the validity of (U), and, insofar as (U) is a necessary condition for the validity of (T), I conclude that (T) is false.

Keywords: Truth schema, Meaning, Principle of uniformity.

3.1

Introduction

The influential position occupied by Tarskian theories of truth in the last decades is at least partially explained by their (purported) strictly extensional nature. Tarski intended his conception of truth to be purely ‘semantic’, that is, on a par with other concepts such as designation, denotation, satisfaction, etc. which concern exclusively the relation between language and the world.

∗ Opponent.

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C. Dutilh Novaes Semantics is a discipline which, speaking loosely, deals with certain relations between expressions of a language and the objects (or “states of affairs”) “referred to” by those expressions [12, p. 345].

Typically, no appeal is made to dubious entities such as the meanings of sentences. This is made patent in Tarski’s condition of material adequacy for theories of truth, usually expressed by the schema: (T) x is true if and only if p where what replaces ‘x ’ is a name of a sentence whose translation into the metalanguage replaces ‘p’. For its simplicity and (apparent) indisputability, it is often viewed as a truism (cf. [8, p. 4]). In fact, it is not only (presumed to be) a true schema; it is also a very convenient one, since it “presents in a stark and nominalistically acceptable form the correlation of language, on the left-hand side, and world on the right-hand side” [8, p. 5]. (T) seems to imply that the correlation between language and world can be completely explained without mediating entities. Granted, in formalized languages, (T) establishes a relation between an expression in the object language and its counterpart in higher-order languages; but, as can be inferred from Tarski’s comparison of the two sides of (T) to formal and material supposition (cf. [12, p. 343]), the gist of (T) is to establish a connection between entities belonging to the linguistic realm and entities belonging to the physical realm. However, Tarski’s truth schema is arguably just as much about meaning as it is about truth – perhaps one of Read’s most important insights in [8]. To be precise, I am not referring to the uses of (T) to develop theories of meaning, most notably by Davidson [3]; rather, what I mean here is that (T) is based on an implicit and unjustified assumption concerning the meaning of sentences. In other words, (T) implicitly presupposes a theory of meaning. But not only is this theory of meaning not explicitly discussed; its validity is also questionable, as I will attempt to show. In [8], Read seems to avoid the term ‘meaning’ of a sentence in favor of the perhaps more neutral formulation ‘what a sentence says’ presumably on account of the controversial status of the former.1 But clearly, what is at stake with his conception of ‘saying that’ is what is often referred to as the meaning of sentences. What Read’s analysis shows is that, while Tarski may appear to have circumvented this thorny notion in his formulation of (T), this appearance is quickly unraveled by a further examination of the matter. Indeed, (T) rests on a simple and yet contentious assumption, namely what Andjelković and Williamson have 1 But it must be granted that some may find the phrase ‘what a sentence says’ even more controversial than the term ‘meaning’.

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termed the principle of uniformity – henceforth (U): every sentence of a language says exactly one thing, or in any case “everything said by a given sentence in a given context has the same truth-value” [1, p. 225]. One of Read’s main goals in his paper is to show that (T) is false. He starts with a list of cases in which (T) recognizably does not hold; naturally, this is not sufficient, as the very proponents of (T) admit that not all instances of (T) hold. He then goes on to propose an alternative truth schema, (S) or (S ), which differs from (T) most notably by the presence of a universal quantifier ranging over what a sentence says to be the case. In order to show that (S) is a better truth schema than (T), Read examines one specific case, that of Liar sentences, and argues that, while (T) fails to offer a satisfactory account of such sentences (except for the ad hoc stipulation that they should be banned from the language), (S) gives a plausible and compelling account of them, simply by accepting that some sentences, Liar ones in particular, say more than one thing, with different truth-values – i.e. (U) does not hold in such cases. Given the significance of Liar sentences as semantic ‘pathologies’, the fact that (S) yields a satisfactory account of it, while (T) does not, is in itself convincing evidence to the effect that (S) is a better truth schema than (T). Thus, Read’s analysis suggests that the ‘weak spot’ of (T) is its tacit commitment to (U). In this paper, I develop this insight, which was, to my knowledge, put forward for the first time in [8]. To be precise, I will not deal with Liar sentences in particular; rather, I am interested in the role of (U) as an essential underlying assumption to (T). I argue, as suggested in [8], that (U) is simply false, and insofar as it is essential to (T), this fact alone is sufficient proof of the falsity of (T). This is particularly manifold in Liar sentences, but, clearly, for any sentence that says more than one thing (and there are many of them, as I will argue), (T) simply fails. By contrast, since (S) (and similar schemata) is not committed to (U), it provides a satisfactory account of the truth of these sentences that say more than one thing (in addition to giving a correct account of sentences that say one single thing as well). I first discuss the role of (U) as a hidden assumption on the meaning of sentences in Tarskian theories of truth; I then argue that a quantificational truth-schema avoids the pitfalls of the (tacit) commitment to this assumption, and conclude that there is an asymmetry between the concepts of truth and falsity; after that, I investigate the principle of uniformity in more detail, and, in the final part of the paper, I argue that, also on the basis of some influential views in recent philosophy of

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language, (U) is simply false. My analysis is largely inspired by Read [8] and Andjelković and Williamson [1].2

3.2

A Theory of Truth or a Theory of Meaning?

In standard versions of Tarskian theories of truth, (T) is usually taken in its minimal form, i.e. as stating a direct connection between a linguistic entity and the world. But some – proponents as well as opponents of the Tarskian conception of truth – have felt that the schema thus stated gives an incomplete account of the matter, as one important element is left out: what the sentence on the left-hand side of (T) says. True enough, the exclusion of this element from (T) was one of Tarski’s main insights, as it seemed to simplify the discussion considerably; but, in light of the difficulties discussed by Read, this appears to be a vicious kind of simplification, a facile solution, rather than welcome parsimony. Ockham’s razor tells us that entities are not to be multiplied beyond necessity; but excluding what a sentence says from an analysis of truth appears to be a misuse of this principle. From a historical point of view, the appeal to the notion of what a sentence says is in fact rather natural. Indeed, Andjelković and Williamson [1, p. 214] motivate their introduction of the notion of saying referring to Plato’s and Aristotle’s formulations of what Tarski had called “the classical conception of truth”, where mention is made of what a sentence says. Similarly, in [8], Read first introduces the notion of what a sentence says in connection with the traditional correspondence theory of truth, which Tarski’s account is supposed to be a version of, according to Tarski himself [12, p. 342].3 But in both cases, for Read as well as for Andjelković and Williamson, the introduction of the notion of saying, a deviation from the Tarskian orthodoxy, is ultimately justified by the considerable sharpening of the analysis that may follow from it. The orthodox Tarskian may have reservations towards this move at first, since the very core of Tarski’s conception of truth is that it ought to be purely ‘semantic’, but both papers promise to shed new light on the concept of truth by taking into account the notion of what a sentence says. To my mind, this goal is fully attained. Andjelković and Williamson present a series of principles and definitions relating the notions of truth, falsity and ‘saying’, which are discussed in particular in connection with the phenomena of vagueness and 2I

owe the reference to Andjelković and Williamson [1] to Read as well, in [7]. should like our definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth [...]”. 3 “We

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bivalence. They start out with the usual Tarskian framework for truth, which becomes progressively refined; the first step is the introduction of the notion of what a sentence says. ‘Saying’ is relativized to contexts, but this is immaterial for the present investigation; accordingly, I will disregard the mention to contexts in the definitions presented below. Moreover, as Read represents ‘x says that p’ by ‘x : P’, I will follow his notation rather than Andjelković and Williamson’s. The truth schema proposed by the latter authors, which will undergo scrutiny throughout their paper but which will eventually prevail, is the following (I name it (TAW), referring to the authors’ names – → represents the material implication, ≡ the corresponding material biconditional and T x the truth predicate applied to x ): (TAW) ∀x∀P[x : P → (T x ≡ P)] Its dual, the falsity schema, is the following (the authors provide an interesting discussion of whether the negation is indeed the most appropriate device to state the falsity schema, but they finally settle for what they view as “the natural principle about falsity” [p. 217] – F x represents the falsity predicate applied to x )): (FAW) ∀x∀P[x : P → (F x ≡ ¬P)] (TAW) and (FAW) have an advantage over the usual disquotational truth schema, namely to outline the significance of the notion of saying. But there is a crucial problem with them: in each schema, one of the directions – (TAW←) and (FAW→) – presupposes that the sentence in question says exactly one thing. Andjelković and Williamson recognize that (TAW) and (FAW), and their corresponding variations, are not definitions properly speaking, because they are creative: they both imply what the authors call the principle of Uniformity (U). A deeper problem is that (T) [here, (TAW)] and (F) [here, (FAW)] are too strong to be mere definitions of True and False, because they are creative. [...] More specifically, each of them entails a principle of uniformity to the effect (at least in the classical context) that everything said by a given sentence in a given context has the same truth-value [1, p. 225].

Their formulation of (U) (adapted to the notation being used here) is: (U) ∀x∀P∀Q[[(x : P) ∧ (x : Q)] → (P ≡ Q)]

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Clearly, (TAW) implies (U), for the antecedent of (U) yields T x ≡ P and T x ≡ Q by instances of (TAW), from which P ≡ Q follows (the same reasoning can be applied to (FAW)). Therefore, (U) is a necessary condition for the validity of (TAW) and (FAW); appropriately, Andjelković and Williamson conclude that these schemata “behave more like theories than definitions” [p. 226]. And what are they theories of? Certainly not only of truth: what becomes clear from their argument is that these schemata are just as much theories of meaning, since they presuppose a non-trivial assumption concerning the meaning of sentences. Of course, the biconditional in the consequent of (U) can be interpreted in two ways: as (mere) material implication, or as strict implication. Under the first reading, (U) states that a sentence may even say more than one thing, but that all the things it says have the same truth-value (in a given context). Under the second reading, the biconditional can be understood as equality between the ‘things’ that a sentence says. This equality can be viewed as equality of truth-conditions, which is a stronger condition than equality of truth-values, or else in even stricter, intensional ways. Be that as it may, even if one takes the weakest reading of (U), which seems to be the one intended by Andjelković and Williamson, it is still a non-trivial and contentious assumption concerning the meaning of sentences, hidden in Tarskian theories of truth. At this point, I am not concerned with whether (U) is indeed true or false. What matters now is that (U) be recognized as a strong contention about the meaning of sentences, such that a theory of truth that assumes the validity of (U) must be viewed as a theory of meaning as much as it is a theory of truth. What is particularly problematic concerning (U) and Tarskian theories of truth is that (U) is usually only tacitly assumed; it is almost never argued for, and one may go as far as wondering whether the proponents of such theories are aware of their commitment to (U). Since Andjelković and Williamson are themselves aware of the significance of the notion of saying for theories of truth, and of the fact that (U) is a necessary condition for the validity of (T), they propose to investigate what a theory of truth where (U) does not hold would be like (even though they are partisans of (U)). This leads them (and us) to quantificational truth-schemata, which will be investigated in the next section (where it will also become clear why (TAW←) and (FAW→) are the invalid directions in these schemata). Of course, it goes without saying that a proper analysis of the notion of ‘saying’ as applied to sentences still remains to be given. Clearly, accepting or rejecting the validity of (U) (and thus of (T)) depends entirely on the notion of ‘saying’ that one is prepared to accept. But this fact only shows that the assumptions concerning the meaning of sentences under-

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lying theories of truth are a very important matter indeed, contrary to what the Tarskian orthodoxy seems to imply.

3.3

Quantificational Schemata

If we seek to exclude the assumption of uniqueness of what sentences say from a truth-schema, a natural approach is to assume, even if only for the sake of the argument, that there may be several different things that a sentence says, and to consider the possibility of quantifying over these different things. We therefore need to characterize truth and falsity in some alternative way. Within the present framework, the natural idea is to define truth and falsity explicitly by quantifying into sentence position [1, p. 226].

The first candidates for the positions of truth and falsity schemata to be considered must be schemata as close as possible to (TAW) and (FAW); and since these schemata imply that a sentence says only one thing, it seems reasonable to consider existential quantification first. Indeed, the first two quantificational schemata proposed by Andjelković and Williamson make use of existential quantifiers (I maintain the names they give to the schemata to facilitate cross-reference): (TDEF1) ∀x[T x ≡ ∃P[(x : P) ∧ P]] (FDEF1) ∀x[F x ≡ ∃P[(x : P) ∧ ¬P]] But clearly, there is a problem with these definitions; they are not mutually exclusive. Suppose that a sentence x says two things (remember that we are now supposing that (U) does not hold), namely P and Q, and that P is the case but Q is not (and thus ¬Q is the case). Is x then true or false? According to the definitions, it is both (to the joy of the dialethist). Even for those who do not seek to keep bivalence at all costs (intuitionists, for instance), this is too hard a pill to swallow, indicating that these are not (at least not jointly) valid schemata for truth and falsity; they are (jointly) too strong. Andjelković and Williamson do mention this awkward consequence of (TDEF1) and (FDEF1) for all sentences of which (U) does not hold, but they again seem to imply that this occurs only in borderline cases (in particular, cases of vagueness – p. 228). However, remember that at this stage of the analysis, the hypothesis is that (U) does not hold not only in special borderline cases; but by considering the phenomenon of vagueness, the authors somehow hint on their bias towards the validity of (U).

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If (TDEF1) and (FDEF1) are not (jointly) the right schemata, the next alternative to be considered is, obviously, universal quantification. Andjelković and Williamson put forward the following schemata: (TDEF2) ∀x[T x ≡ ∀P[(x : P) ∧ P]] (FDEF2) ∀x[F x ≡ ∀P[(x : P) ∧ ¬P]] The first obvious difficulty with these schemata is that they can be vacuously true and would attribute truth to things that say nothing, such as lakes and mountains. To remedy this, a natural solution is to add a clause to the effect that a sentence x must say something: (TDEF2*) ∀x[T x ≡ [∃P(x : P) ∧ ∀P[(x : P) ∧ P]]] (FDEF2*) ∀x[F x ≡ [∃P(x : P) ∧ ∀P[(x : P) ∧ ¬P]]] Clearly, if one wants to safeguard bivalence, (TDEF2*) and (FDEF2*) are (jointly) not valid schemata for truth and falsity, respectively, as every sentence which says things with different truth values would be, according to these schemata, neither true nor false. They are jointly too weak. Since (TDEF1) and (FDEF1) are jointly too strong and (TDEF2*) and (FDEF2*) are jointly too weak, an intermediate alternative is certainly worth investigating. In other words, we may consider two different pairs, (TDEF1) and (FDEF2*), and (TDEF2*) and (FDEF1), as candidates for the position of truth- and falsity-schemata. Andjelković and Williamson do consider the possibility of ‘mixing’ quantifiers, but they dismiss it as counterintuitive. They [these pairs] undermine the natural conception of truth and falsity as somehow parallel notions: one of the defining conditions is quantified universally, the other existentially [1, p. 229].

In fact, their only argument against the use of different quantifiers in each schema is that it goes against the ‘natural’ conception of truth and falsity; but as much as they are parallel notions, the idea that they are opposite notions is, to my mind, just as natural. Therefore, if the truth-schema and the falsity-schema are not contradictories of each other (if they both correspond to existential/universal quantification), an important aspect of their ‘naturalness’ seems to be disregarded. Accordingly, it seems quite reasonable to consider the possibility that the valid schemata for truth and falsity would be a pair of universal-existential quantification.

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(TDEF1)-(FDEF2*) and (TDEF2*)-(FDEF1) are in effect both pairs of contradictories; but which schema should be seen as a universal quantification, while the other is an existential quantification: truth or falsity? After one has read [8], it seems quite obvious that truth must be viewed as universal quantification, and falsity as an existential quantification. But here is another argument to corroborate the asymmetry between truth and falsity. Consider the case of someone testifying in a court of law. Her testimony typically consists of several assertions. Clearly, it is intuitive to speak of quantification into what is said in such cases, as there are explicitly several things being said. Now, when would we say that a testimony is true? Would we be prepared to call it a true testimony if, among the many things said, only one was the case? Obviously not; typically, we would call it a true testimony if and only if each and every single thing said was the case. Similarly, if a testimony is largely composed of statements that do hold, but with the exception of one or two, we would call it a false testimony. Hence, as Read has argued, and as the example above suggests, truth corresponds to universal quantification and falsity to existential quantification. Therefore, the right truth schema is (TDEF2*) – which is equivalent to Read’s (A) –, and the right falsity schema is (FDEF1). They are mutually exclusive and jointly exhaustive (bivalence is preserved). Moreover, (TDEF2*) and (FDEF1) do not imply (U), i.e. they are not creative (not with respect to (U)), but, from them together with (U), (TAW) and (FAW) follow. Hence, the proponents of (TAW) and (FAW) need not reject the quantificational approach; what the present analysis shows is that the latter is in fact more fundamental than the Tarskian approach, since no assumption concerning what sentences say is made. If one is willing to accept (U) as valid, then (TAW) and (FAW) are derived theorems within this framework; by themselves, they are not the proper definitions of truth and falsity, as they also implicitly state a thesis concerning the meaning of sentences.4 At this point, it also becomes clear why (TAW←) and (FAW→) are not valid (by themselves): they are valid iff there is exactly one thing which x says, i.e. if (U) holds. (TAW←) states that, if x says P and P is the case, then x is true, thus disregarding the possibility that the other things that x might say may not be the case (in which case x would be

4 Andjelković and Williamson correctly state that the schema where ‘saying that’ occurs is more basic that the disquotational biconditional, where it does not occur (p. 216). But this analysis shows that the quantificational schema is even more basic than their schemata (TAW) and (FAW).

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false). Likewise, (FAW→) states that, if x says P and x is false, then P is not the case, thus disregarding the possibility that what falsifies x is something else that it says (and is not the case), other than P. As for Read’s account of the Liar, it is clear why it is a false sentence on the basis of the quantificational schemata. The Liar says two things, that it is true and that it is not true; even though we do not know beforehand which one is the case, we known that these two things together cannot obtain. Thus, the Liar is primarily an impossible sentence, since what it says is not only not the case, it is in fact impossible. But if a sentence says something impossible, then afortiori it says something that is not the case; therefore, it satisfies the falsity-schema (FDEF1), and thus is a false sentence. Admittedly, quantifying over suspicious entities such as ‘what a sentence says’ is not a straightforward matter, as their ontological status is rather problematic. Andjelković and Williamson let the variable x range over names of sentences, while the variable P is seen as ranging over sentences themselves, presumably the very sentences that are named by substitutional instances of x. However, it seems to me that the relation of saying does not take place between the name of a sentence and a sentence, as Andjelković and Williamson seem to imply. In fact, this relation thus defined is not very different from what Tarski tried to accomplish with his strictly ‘semantic conception of truth’; it is roughly tantamount to the disquotational relation between the name of a sentence on the left-hand side of (T) and the sentence on its right-hand side. But as already argued, this relation, thus defined, does not tackle the hart of the matter, as it does not properly characterize the notion of saying. The relation of saying involves sentences (or the names thereof) and entities corresponding to what sentences say. The determination of the latter is, of course, the controversial part of the issue: they may correspond to sentences, to possible states of affairs, to truth-conditions, or to abstract entities such as propositions or contents. This is not the time and place to discuss this ontological issue, but it seems that sentences and names of sentences alone cannot exhaust the nature of the relation of saying; more is needed to explain the phenomenon of sentences saying ‘things’, and the main problem is precisely that of defining what these ‘things’ would amount to. In any case, it seems that the notion of truth simply cannot be analyzed in purely ‘semantic’ terms, as Tarski had intended: the meaning of sentences cannot be circumvented.5 5 To be exact, strictly semantic theories of truth may be possible (take for example Ockham’s account of the truth-conditions of sentences exclusively in terms of the supposition of their

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The Principle of Uniformity

After having granted, for the sake of the argument, that sentences might say more than one thing and thus that (U) might not hold in all cases, Andjelković and Williamson move on to argue in favor of this principle; they offer a (purported) proof of its validity. But even before this argument is given, it is clear that the authors have a bias towards (U). They contend that, even if it does not always hold, the cases where it might not hold are at best complicated, borderline cases. They consider as borderline cases, where one might think that (U) does not hold, cases of (semantic) ambiguity and of vagueness. They also briefly consider the notion of saying closed under implication as a possible objection to (U), but quickly dismiss it: That notion of saying [closed under implication] is clearly irrelevant to the present problem, for it would yield counterexamples to (U) even in unproblematic, non-borderline cases. [...] Since the relevant instances of (T) and (F) should hold in such unproblematic cases, so should their consequences, the relevant instances of (U) [1, p. 231] (emphasis added).

In other words, a notion of saying that challenges (U) in non-borderline cases is dismissed precisely for this reason, in what appears to be a question-begging move; the possibility that (U) may not hold in nonborderline cases is never seriously considered. Of course, which ones are ‘normal’ cases and which ones are ‘borderline’ cases is to a large extent a matter of subjective judgement; but to define normal cases as those that comply with (U) would be question-begging. Similar reasons appear to be at the root of Tarski’s tacit commitment to (U): he would have taken for granted that sentences (must) say exactly one thing, on the basis of a generalization from what he viewed as quintessential sentences to all cases. Indeed, Read argues that Tarski’s faux-pas when proposing (T) was to generalize the case of a rather simple sentence, ‘Snow is white’ to all cases. That snow is white is, arguably, all that ‘snow is white’ says. So the right-hand side of (1) [‘snow is white’ is true if and only if snow is white] gives a sufficient, as well as a necessary condition for its truth. But not all cases are as straightforward as this. In general we need to do more than simply repeat ‘p’ on the right-hand side [8, p. 9].

In other words, Tarski’s generalization is unwarranted, as it implicitly assumes that all sentences behave as simple cases like ‘snow is white’. terms), but it seems to me that a strictly semantic correspondentist theory of truth, as Tarski intended his theory to be, is an impossibility.

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Andjelković and Williamson seem to be well aware that one cannot simply assume all cases to be as straightforward. From their considerations, three possibilities seem to emerge: (i) (U) always holds; (ii) (U) fails only in borderline cases; (iii) (U) fails in ‘normal’ cases as well. If either (i) or even (ii) occurs, then the Tarskian approach to truth seems to be a good candidate as the correct analysis of this concept; if however (iii) is the case, then clearly this approach is insufficient.6 In sum, it all depends on the notion of ‘saying that’ that one is prepared to accept, and consequently on the status of (U). The position defended by Andjelković and Williamson is (i): (U) holds in all cases, even in borderline ones. Their main argument in favor of (U) is a ‘proof’ of its validity, which takes as its main assumption what they call [...] a plausible principle of compositional semantics. Roughly, since E expresses ≡ Est says the biconditional of what s says and what t says: [1, p. 233]

(E1) ∀s∀t∀P∀Q[(s : P) ∧ (t : Q) → (Est : P ≡ Q)]7 They also note that “if a sentence says something then its biconditional with itself is true” [p. 233]: (E2) ∀s∀P[(s : P) → T Ess] They argue that to deny the validity of (U) would amount to denying the validity of these principles, which, according to them, “is too high a price to pay, for it destroys our conception of what biconditionals say” [p. 234]. Indeed, their proof of (U) based on (E1) and (E2) is compelling, but it does not take long before one realizes that (E2), as much as (T), (TAW) and (FAW), already assumes the validity of (U). For suppose that (U) does not hold. Then, presumably, P is only one of the things that s says, and the same holds of Q and t. As I understand (E1), the biconditional of two sentences s and t which say different things says different things as well, namely the different combinations of what each of them says. So, if (s : P) and (s : R), and (t : Q) and (t : T), for example, then Est says several different things, such as P ≡ Q, P ≡ T, R ≡ Q, R ≡ T, etc. 6 Naturally, we are still left with the problem of demarcating ‘normal’ from ‘borderline’ cases, but let us assume for the time being that this demarcation is unproblematic. 7 Again, I disregard the relativization to contexts in the principles borrowed from Andjelković and Williamson.

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Now, consider the case of (s : P∧¬P). Clearly, there is a sense in which (s : P) and (s : ¬P), which goes beyond the notion of saying defined as closure under implication – conjunctions are particularly convincing cases of sentences that say more than one thing (and remember that, for now, we are explicitly denying that (U) holds, since the principles (E1) and (E2) should hold also if (U) does not). The fact that s is contradictory is also not to be seen as a problem, as there is no ban against saying contradictory things. According to (E1), Ess says (at least) these four things: Ess : P ≡ P Ess : P ≡ ¬P Ess : ¬P ≡ P Ess : ¬P ≡ ¬P Clearly (E2) does not hold in the second and third cases. Hence, if (U) does not hold, then (E2) does not hold, which means that (U) is a necessary condition for the validity of (E2); but if (E1) and (E2) are meant to be non-contentious principles, they must not assume (U) to be the case. So either (E1) does not provide the correct analysis of what a biconditional says, or (E2) does not hold. I am inclined to think that the problem is with (E2), as (E1) is not contentious towards (U): it states that the biconditional of s and t says P ≡ Q, but it does not state that it only says that. Of course, it can be argued that what Ess says is in fact (P ∧ ¬P) ≡ (P ∧ ¬P), and in this case (E2) does hold. I can gladly accept that Ess says (P ∧ ¬P) ≡ (P ∧ ¬P); what I reject is the contention that Ess only says (P ∧ ¬P) ≡ (P ∧ ¬P) and nothing else (and once again, remember that the assumption now is that (U) does not hold). It can also be argued that, if (s : P∧¬P) then s is a borderline case, a contradiction. But since Andjelković and Williamson claim that (U) holds in all cases, including borderline ones, this line of argumentation is obviously not of much help to safeguard the validity of their proof. In sum, Andjelković and Williamson’s principle (E2), which they see as a “classical tautology”, is not valid, or it is valid only provided that (U) is valid, which clearly undermines their proof of (U) on the basis of (E1) and (E2). In other words, their ‘proof’ of (U) presupposes the validity of (U). The fact that they hold (E2) to be a classical tautology also indicates that their understanding of the notion of what a sentence says is not adequate; it seems to be exclusively extensional, since classical tautologies are said to govern its behavior, whereas clearly the subtleties involved in this notion demand a non-extentional treatment.

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Therefore, contrary to what they claim, Andjelković and Williamson have not proved that (U) holds in all cases. As (U) is a universal quantification, to disprove it it is sufficient to provide counterexamples; if we find counterexamples, we will either conclude that (U) does not hold only in borderline (and insignificant) cases, or we will conclude that (U) does not hold in many entirely ‘normal’ cases as well. In the next section, I defend the latter view.

3.5

(U) Does Not Hold in Too Many (Important) Cases

What I have done so far is to argue that the validity of (T) and related truth-schemata depends on the validity of (U), and to challenge a purported proof of the validity of (U). I have also argued that a more adequate approach to the notion of truth seems to be through quantificational schemata, insofar as they avoid the presupposition that (U) is valid. In this section, I argue that (U) is not valid. Naturally, (U) is prima facie an easy prey to catch, as it is a universal quantification and therefore one single counterexample would suffice to prove its invalidity. However, proponents of (U) typically argue that, even if (U) does not hold in all instances (as much as the proponents of (T) recognize that the same happens with (T)), the cases where it fails (if any) are at best borderline, pathological cases. Again, I doubt that a clear-cut demarcation dividing pathological from normal cases can be given; but in this section I will discuss some cases which (I presume) would not easily be viewed as pathological or borderline. I reckon that most people would consider them to be as ‘normal’ as ‘snow is white’ and the like, and yet (U) appears not to hold of them, or in any case not in a straightforward way. Of course, the validity of (U) ultimately depends on one’s understanding of what a sentence says – in other words, on one’s theory of sentential meaning. It is not the purpose of the present paper to put forward a general theory of meaning, so a satisfactory analysis of the notion of what a sentence says is not to be found here.8 For my purposes, it is sufficient to present a few sentences (in English, but similar examples could presumably be found in most if not all languages, formalized as well as so-called ‘natural’ ones) that seem to challenge the validity of (U).

8 Note that, while this notion is crucial for Andjelković and Williamson, they say very little about it in their paper, as noticed by Read in [7]; by contrast, in [8], Read does examine this notion more closely (in particular in its ‘closed under implication’ variant), but in the end his argumentation does not require a full-fledged analysis of what a sentence says, as long as it is recognized that a sentence may say more than one thing.

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I should add that my objections to the validity of (U) do not apply to Tarski himself, but rather to applications of the Tarskian framework going beyond the realm of interpreted languages. In Tarski’s case, the validity of (U) simply follows from the definition of interpretation as a function. But in cases where the meaningfulness of a given language is not established by the (highly regimented) procedure of interpretation, the validity of (U) must not be taken for granted, and in fact often seems to fail. None of this would have had any importance if uses of (T) (and consequently, assumptions of the validity of (U)) had remained restricted to cases of interpreted languages; but, as we know, many have found it suitable to apply (T) much more widely, yielding the misconceptions discussed here. For (U) to be invalid means that at least one sentence does not say exactly one thing. It may say nothing, but these are arguably uninteresting cases. Particularly relevant for the present analysis are cases of sentences that seem to say more than one thing, possibly with different truth-values, even in the same context. It seems to me that there are two main ways in which a sentence may say more than one thing: what I call the conjunctive and the disjunctive modes. As the terms imply, in the conjunctive mode a sentence may say more than one thing conjointly, while in the disjunctive mode it may say different things, but it typically says one of them rather than the others at a given occasion.

3.5.1

Conjunctive Mode

Discrepancy of surface and deep structures. It is by now almost unanimously accepted that some sentences do not present their actual logical forms in their surface forms. Some sentences that may appear to say only one thing on the surface in fact may say more than one thing in their deep structure. Russell’s account of sentences that seem to be of the predicate-argument form, but which fail to be meaningful from this perspective on account of having an empty subject [9], is a famous example of this phenomenon. He holds that a sentence such as ‘The present king of France is bald’ says not one but rather three things: that there is a king of France (now), that there is only one king of France (now), and that this man is bald. Such cases certainly raise suspicion concerning the validity of (U). Of course, the proponent of (U) may reply that such sentences say exactly one thing, namely the conjunction of these three clauses; if one of them fails to obtain, then that the whole sentence is false can be inferred simply from the usual properties of conjunctions. But I have already noted in the foregoing that conjunctions seem one of the most convincing cases of sentences that say more than one thing: isn’t it natural to think

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that, if I say ‘It will rain and I will stay home’, I have said that it will rain and I have said that I will stay home?9 Moreover, Russell’s claim that such sentences must be analyzed as compound existential sentences touches upon another assumption concerning the meaning of sentences involved in (T), which I have not discussed so far, namely that “a sentence reveals explicitly all that it says” [8, p. 9]. How far can we go applying (T) to sentences such as ‘The present king of France is bald’ ? The appropriate instance of (T) would be: (T1) ‘The present king of France is bald’ iff the present king of France is bald. What does this tell us about the truth of this sentence in the case that there is currently no king of France? In such cases, i.e. (if Russell’s analysis is correct) cases in which what a sentence says is not explicit from its surface structure, a Tarskian analysis of truth does not even get off the ground. By contrast, the quantificational analysis, aided by a suitable notion of saying, would have it that this sentence says (at least) three things, and that, if it is true, then all of them obtain (but not the converse, as it may say other things as well). Interestingly, the traditional late-medieval approach to such cases is very similar to Russell’s. According to Ockham, for instance, an affirmative sentence (be it existential, indefinite or universal) has two causes of falsity, namely if the subject does not supposit for anything, or if the predicate does not supposit for what the subject supposits for (cf. [13] II, Chap. 3). If either of these two conditions fail, then the sentence is false, since such a sentence seems to say two things, namely that there is at least one suppositum for the subject, and that this entity is also a suppositum of the predicate (these two conditions are necessary for its truth); for it to be true, all it says must obtain.10 Presuppositions. Another group of sentences which appear to say more than one thing, some of which are said explicitly while others implicitly, are sentences involving presuppositions. Of course, we here encounter again the issue of the notion of saying one is to adopt. Is it reasonable to consider that a sentence says what it presupposes? At least according to some views on presuppositions (chiefly the pragmatic approach – [11] 9 In

other words, the following axiom would hold of the relation of saying: ∀x∀P∀Q[(x : P ∧ Q) → ((x : P) ∧ (x : Q))]. 10 Notice also that an important medieval topic was that of exponibilia, i.e. sentences that appeared to be saying only one thing, but which in fact had to be unfolded into several others in order to be analyzed. Cf. for example [13, pp. 279–315].

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(part II)), when one utters a sentence which has a presupposition, one is also communicating one’s belief in the truth of what is presupposed, thus somehow also ‘saying’ the presupposition. In other words, if I utter ‘John stopped smoking’, I am arguably also saying that John used to smoke – i.e. I am communicating my belief in this fact. How does the Tarskian approach fare in the analysis of the truth of such cases? What does the instance of (T): (T2) ‘John stopped smoking’ iff John stopped smoking. have to say about the truth or falsity of this sentence if John has never smoked? Not much, it seems to me. Alternatively, if one accepts the notion of saying as including presupposition, then this sentence says that (i) John smoked in the past, and (ii) John does not smoke now, and under the quantificational approach to truth, if either of them fails, then the sentence is false. If they both obtain (and as there seems to be no other salient things that the sentence says), then it is true. Of course, some would prefer to say that, if John has never smoked, then ‘John stopped smoking’ is not false, as the quantificational approach would have it, but rather that it lacks a truth-value (the so-called logical view on presupposition – cf. [10]), or even fails to say anything at all (the so-called expressive view – cf. [10]). This is again a matter of the notion of saying and of presupposing that one is prepared to adopt; at any rate, it is clear that, given a suitable notion of saying, the quantificational approach to truth can give an account of the truth of such sentences, while the Tarskian approach seems to have nothing to say on this matter, precisely because (U) arguably does not hold in these cases. Implicatures. Another linguistic phenomenon that seems to disprove the validity of (U) is the phenomenon of implicature, as defined by Grice (cf. [4]). When one breaks a conversational maxim, it in practice means that more is being ‘said’ than what the sentence in question explicitly says. In the famous ‘letter of recommendation’ example, if in such a letter it is said that X, who is applying for a position at a philosophy department, is always on time and has good handwriting, since the maxim of relation is being broken, this statement says not only what it says, but arguably also that X is not a good philosopher. As for the truth-values of sentences in such cases, matters seem more complicated than what both the Tarskian and the quantificational approach can handle. On the one hand, the Tarskian approach would deem such a statement true iff X is always on time and has good handwriting; but this account would miss an important aspect of the statement, as

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its purported truth-value is indifferent to whether X is a good philosopher. On the other hand, the quantificational approach, accompanied by a suitable notion of what a sentence says, would deem the same statement true iff X is always on time, has good handwriting and is not a good philosopher (assuming that this is all that the statement says), and false if either of them fail (so it would be false if X is a good philosopher, even though he is also punctual and has good handwriting). Neither approach seems to yield an entirely satisfactory analysis of this phenomenon, but the quantificational approach at least is able to take implicatures into account. In any case, my main point here concerning the phenomenon of implicature is that it does seem to offer evidence against the validity of (U), regardless of what an appropriate approach to truth and implicature would be like. Closure under implication. In both [8] and [1], the most-lengthily discussed notion of saying that does not comply with (U) is the one defined by closure under implication/entailment. Elsewhere [7] Read calls it the ‘Carnapian’ notion of what a sentence says. It can be formulated as follows (⇒ represents strict implication): (K) ∀x∀P∀Q[(P ⇒ Q) → (x : P ⇒ x : Q)]11 Read adds that, even if one does not want to commit to this rather strong notion of saying, one may still recover the same results with respect to the truth of sentences by using (A ) instead of (A) (here (TDEF2*)) and (K): (A’) ∀x[T x ⇔ ∀P[∃Q(x : Q ∧ (Q ⇒ P) → P)]] As already mentioned, Andjelković and Williamson dismiss this notion of saying as “irrelevant”, but, as argued by Read, the very opposite seems to be the case; it is, to say the least, a very plausible and compelling view on what a sentence says. When one makes a statement, one is also committing oneself to all its logical implications, as the antecedent of a valid implication cannot be true while its consequents are false. Of course, this view raises the problem of logical omniscience, as it is unlikely that one is aware of all the logical implications of one’s statements. Nevertheless,

11 In

[8], Read leaves the variable x free, but this is an insignificant modification, brought about only to keep coherence with the other schemata in this paper. The same applies to the next schema, (A ).

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whether one is aware of it or not, one is indeed committed to the truth of the logical implications of one’s statements, and in this sense it seems natural to think that a sentence ‘says’ all its logical implications as well. Notice that Read uses strict implication in his definition of what a sentence says closed under implication. Indeed, mere material implication would not be a sufficiently fine-grained approach to the notion of saying; a few pages ago I reproached Andjelković and Williamson for adopting too extensional a notion of saying, so naturally material implication would also not be an adequate way of capturing this notion. It is not sufficient that Q just happens to be true in a situation where P is true to warrant the claim that, if (x : P) then (x : Q): a connection tighter than material implication between P and Q is required to characterize the notion of saying. To my mind, Read has argued very convincingly in favor of this notion of saying, so I don’t feel that I have anything much to add on this matter. In truth, I take it to be the most important notion of saying to challenge (U), even beyond the aforementioned phenomena (discrepancy of surface/deep structure, presupposition and implicature), insofar as it concerns strictly formalized contexts as well, where pragmatic elements are regimented or excluded. Indeed, a typical argument in favor of (U) is that the circumstances where (U) does not hold are at best related to the ‘imperfect’ nature of the language in question; but this argument obviously does not hold against the notion of saying closed under implication, as the systematic study of (logical) implication is arguably the very purpose of the regimented and formalized languages of logic. In sum, there are (at least) a few plausible ways in which a sentence may be seen as saying more than one thing simultaneously or conjunctively, as I put it. It seems to me that these phenomena are significant enough to be taken seriously as important counterexamples to (U).

3.5.2

Disjunctive Mode

Besides the cases of sentences that appear to say more than one thing all at once (which, if we take saying as closed under implication, is the case of virtually every sentence, since virtually every sentence implies other sentences), there are also important cases of sentences that say either one thing or another, i.e. which can potentially convey more than one meaning. Usually, elements from the context help the hearer determine the meaning intended by the utterer, so it can be argued that such cases are only caused by an imperfection of the relevant languages, as ultimately there is only one ‘correct’ meaning intended (either semantically determined, or established by the intention of the utterer). Still, these are

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significant phenomena, which, it seems to me, are usually overlooked in discussions concerning truth. In this section, I discuss some of the ways in which a sentence may say several things in a disjunctive way. Ambiguities. Ambiguous sentences are the cases par excellence of sentences saying more than one thing disjunctively. Ambiguous sentences say either one thing or another (or yet another...), but it does not seem accurate to view them as stating the disjunction 12 or even less so the conjunction of its possible meanings (cf. [6]). Does (U) hold in such cases? I think that it does not, or in any case not in its crude formulation. Take Read’s example, ‘Visiting parents can be boring’: both the reading that relatives who visit can be boring and the reading that the act of visiting relatives can be boring are legitimate, and it seems reasonable to view this sentence as somehow saying both things. Of course, one could object that, when thoroughly interpreted in a given context, uses of such sentences are usually disambiguated, and thus that, relative to a context, to a utterer and a hearer, an ambiguous sentence says only one thing. Nevertheless, this phenomenon indicates that a suitable notion of saying must be far more fine-grained than what proponents of (U) usually take it to be. True enough, Andjelković and Williamson do include the notion of context in their framework, which I have excluded from my analysis for the sake of simplicity; still, the notion of context must not be seen as a ‘magic’ trick to allow for disambiguation in all cases, as it is very well conceivable that even within a given context, taking into account all its salient elements, ambiguity persists. While logicians and most philosophers are notoriously wary of ambiguities13 , and, as noticed by Read, “the point14 of formalization is to avoid ambiguity” [8, p. 8], the philosophical significance of the phenomenon of ambiguity must not be underestimated; moreover, the resolution of ambiguity is a lively topic in computational semantics and linguistics. Several taxonomies of different kinds of ambiguities have been proposed (lexical ambiguity, syntactic ambiguity, semantic ambiguity, pragmatic ambiguity, etc.), as much as the adequate apparatus to handle them in formal contexts. In other words, the point I wish to make here is that ambiguity should not be dismissed as an unimportant aspect of our uses of languages, as an imperfection, such that it would not count as a worthy 12 On the medieval debate on whether ambiguous sentences corresponded to a disjunction, see [2]. 13 I take Kripke’s critique of Donnellan’s distinction between referential and attributive uses of definite descriptions in [5] to be representative of this general attitude. 14 I prefer to think that to avoid ambiguity is one of the points of formalization.

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counterexample to (U); in fact, it seems to me that it is essential for a suitably refined notion of what a sentence says to take the phenomenon of ambiguity into account. What is clear in any case is that (T) and similar schemata cannot handle ambiguous sentences, and, admittedly, neither can the quantificational truth schema (for it to deem an ambiguous sentence as true, all its readings would have to obtain, which is clearly too strong a condition). Nevertheless, taking the phenomenon of ambiguity seriously, as I do, is surely a good reason to doubt the validity of (U). Conventionality of language. Another good reason to be suspicious of (U) is the essentially conventional character of language, as noted by several influential philosophers (ranging from Buridan to Wittgenstein). While (U) arguably concerns sentences with determined meaning, sentences can obviously change their meaning if, for example, one of their terms changes meaning. In other words, sentences do not necessarily say what they say; the things they say may change if the conventions regulating the use of a language change, so that the intricacies of its expressive power often go far beyond what (U) seems to attribute to all languages indiscriminately. Context-dependence. Context-dependence can also be seen as a phenomenon undermining the validity of (U). Its importance is increasingly recognized in current philosophy of language, and a number of theoretical apparatuses15 have been proposed to deal with the role of context for the meaning and truth of expressions. The proponents of (T) themselves have recognized that the application of the schema in languages that are context-dependent must take contextual elements into account, in particular with respect to pronouns and indexicals (Read mentions Davidson’s observation of this fact – [8, p. 4]). But even though the notion of what a sentence says can be relativized to a context, as Andjelković and Williamson have done, in such a way that (U) seems to be safeguarded, the fact that the meaning of so many expressions and sentences of the languages we speak and write are contextdependent strikes me as an indication that the thesis of uniformity of meaning contained in (U) may be unnatural and far-fetched. To be precise, I do not take the cases where sentences seem to say more than one thing disjunctively, related to the phenomena of ambiguity, context-dependence and conventionality of language, as full-blown 15 Among these, the several variations of two-dimensional semantics are perhaps the most notable example, cf. the issue of Philosophical Studies 118: 1–2, dedicated to this framework.

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arguments against the validity of (U); rather, what I suggest is that these cases indicate that the notion of what a sentence says is far more complex than (U) seems to imply it is. Of course, it may be argued that, in formalized languages, these phenomena, as well as the phenomena described in the previous section (presupposition, implicature and discrepancy of surface and deep structure) do not (should not!) occur, and that (U) does (should) hold in such cases, insofar as these languages must be transparent and express explicitly and uniquely what is said. But when it comes to the application of (T) to account for the concept of truth in so-called ‘natural’ languages, these phenomena cannot be simply disregarded, as they often seem to be. While to ensure the validity of (U) may be one of the very purposes of devising formalized languages16 , this understandably desired property simply does not hold in non-regimented contexts, as I hope to have shown. Even though Tarski himself intended (T) to apply most and foremost to formalized languages, whose meaningfulness is established by means of the technical notion of interpretation, this fact obviously undermines the uses of (T) to account for the notion of truth in so-called ‘natural’ languages, as has been done extensively (cf. [8]).

3.6

Conclusion

From these considerations, the conclusions to be drawn appear to be quite evident. (T) and related truth-schemata presuppose the validity of (U) and therefore are not suitable to account for the truth of sentences and languages of which (U) does not hold. But (U) does not hold universally: Andjelković and Williamson’s purported proof of (U) is fallacious, as it presupposes its validity; moreover, an examination of a few perfectly ‘normal’ cases suggests that the notion of what a sentence says goes much beyond (U). Accordingly, an alternative truth schema that does not presuppose the validity of (U), and which is based on universal quantification over what a sentence says, is not only more suitable in cases where (U) does not hold; it is also conceptually more fundamental than (T) and similar schemata. Equally important is the realization that a theory of truth cannot exist without a theory of meaning (at least not in the case of correspondence theories of truth), and thus that meaning is arguably a more fundamental notion than that of truth, contrary to what the Tarskian approach to truth suggests.

16 Even though, again, if one adopts the notion of saying closed under implication, then (U) will not hold in formalized languages either.

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In sum, Read’s analysis, inspired by Bradwardine’s solution of the Liar paradox, has the outstanding merit of revealing some of the hidden and false assumptions contained in (T), which I here have attempted to disclose even further.

References [1] M. Andjelković and T. Williamson (2000) Truth, falsity and borderline cases. Philosophical Topics 28(1). [2] E.J. Ashworth (1991) Nulla propositio est distinguenda: La notion d’equivocatio chez Albert de Saxe. In J. Biard (ed.) Itinéraires d’Albert de Saxe. Paris, Vrin. [3] D. Davidson (1997) Truth and meaning. In P. Ludlow (ed.) Readings in the Philosophy of Language. Cambridge, MA: MIT Press. [4] H.P. Grice (1961) The causal theory of perception. Proceedings of the Aristotelian Society, Supplementary Volume 35. [5] S. Kripke (1997) Speaker’s reference and semantic reference. In P. Ludlow (ed.) Readings in the Philosophy of Language. Cambridge, MA: MIT Press. [6] G. Priest and S. Read (1981) Ockham’s rejection of ampliation. Mind 90(358). [7] S. Read (2006) Symmetry and paradox. History and Philosophy of Logic 27(4). [8] S. Read (2008) The Truth Schema and the Liar. In S. Rahman, T. Tulenheimo & E. Genot (eds.) Unity, Truth and the Liar. Berlin: Springer. [9] B. Russell (1905) On denoting. Mind 14(56). [10] S. Soames (1989) Presuppositions. In D. Gabbay and D. Guenther (eds.) Handbook of Philosophical Logic, vol. 4. Dordrecht, Reidel. [11] R. Stalnaker (1999) Context and Content. Oxford: Oxford University Press. [12] A. Tarski (1944) The semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research 4(3): 342–376. [13] William of Ockham (1974) Opera Philosophica I – Summa Logicae St. Bonaventure, Franciscan Institute.

Chapter 4 DOUBTING THOMAS: FROM BRADWARDINE BACK TO ANON Laurence Goldstein∗ University of Kent, Canterbury, England Fife KY16 8RA, Scotland, U.K. [email protected]

Abstract

Stephen Read, following an idea of Thomas Bradwardine’s, undertakes to repair the Tarski T-scheme. One happy result is that he is able to solve the Liar Paradox without departing from ordinary language. The upshot of Read’s reasoning is that someone uttering a Liar sentence fails to speak the truth. But does this mean that that person utters a falsity, as Bradwardine claims, or that the speaker fails to utter a falsity as well as failing to utter a truth? An author writing more than a century before Bradwardine thinks the latter, holding that the Liar utterance does not express a proposition. Like Read, Bradwardine repudiates this view. But criticisms of his own theory keep leading us in the direction of that earlier theory. It is the earlier theory that is the right one.

Keywords: Bradwardine, Cassatio, Contradiction, Liar, Mediaeval, Paradox, Quotation, Read, Tarski, Truth, Yablo.

4.1

An Instructive Read

The scholarly community is in debt to Stephen Read [23, 25] for rediscovering a princely solution to the Liar paradox, thereby reinstating its author, Thomas Bradwardine, to his rightful position as one of the greatest logicians of the mediaeval period. Various writers have attributed to Bradwardine a bastard progeny of his actual theory, less interesting and open to decisive objection. Read exposes these misattributions. He supplies, so far as I can tell, a faithful account of Bradwardine’s fascinating reasoning which supports a solution to the Liar that seems extremely powerful: the Liar sentence is shown to be not true, and Bradwardine ∗ Opponent.

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provides the resources for showing that, from the assumption of its untruth, we cannot validly infer its truth. In other words, the Liar sentence is simply not true. Further, as Read shows (2002: 214-7), Bradwardine can employ the same resources to deliver a simple, clean solution to Curry’s Paradox. I do not have any knockdown rebuttal of Bradwardine (otherwise this paper would be titled “Refuting Thomas”) but what I do have is a number of considerations that should cause us to doubt his solution and to dampen Read’s enthusiasm for it. Bradwardine opens his innings with an assault on some rival theories and, in the course of this, he dismisses a solution that seems to me more promising and deeper than his own. Read thinks that Buridan distorted Bradwardine’s view, and, as the title of his [23] indicates, he wants to take us back from Buridan to Bradwardine. I think that Bradwardine ridiculed and twisted a view defended by an anonymous author writing over 100 years before him. In Section 4.4, I want to take us back, briefly, from Bradwardine to that author. Unlike Read, who only presents Bradwardine’s view without himself endorsing it, I take Anon to have provided essentially the correct solution to the Liar, even though completing the details would prove to be a devil of a task for later centuries. The structure of this paper is that, first (Section 4.2), I shall compare Read’s improved account of truth with Tarski-inspired accounts, and assess how well it handles paradoxes in the Liar family. In Section 4.3, I look at a kind of paradox to which the Bradwardine/Read theory seems particularly vulnerable. Then comes the promised examination of whether Bradwardine’s view represents an advance on the Anon theory that he dismisses. In the final section, I suggest that Read’s repair of the Tarski T-scheme only papers over the cracks, and that a more radical overhaul is required if we are to have a satisfying solution of the Liar.

4.2

Readefining Truth

It is common to define logical connectives by means of rules for their introduction and elimination. Any connective can be properly so defined, as long as the rules are in harmony (Read [22], [24]). What more natural (and harmonious), then, than to define a truth operator by the rules TI From A infer TA TE From TA infer A and to define a truth predicate similarly, making the adjustment, for syntactic propriety, that the predicate attach not to a sentence, but to a name or description of that sentence? It is not therefore surprising

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that writers, including Paul Horwich [17], have sought to parlay Tarski’s T-scheme into a definition of truth. Horwich avoids Tarski’s mistake of predicating truth of type-sentences,1 but, according to Read, commits a mistake of a different sort. In Horwich’s account, for all substitution instances of p, the proposition that p is true if and only if p. But, according to Read, the proposition that p is true if and only if everything that p says (or implies) is true. In symbols, and adopting Read’s notation (writing “says that” as “:”) Tx ⇔ ∀p(x:p → p)

(A)

How do the Horwich and Read definitions differ? One of the things that “Snow is white” implies is that snow is colored, and it implies many things besides. So, for Read, “Snow is white” is true if and only if snow is white and snow is colored and. . . Horwich would surely agree, and would (presumably) agree to modify his scheme accordingly: The proposition that p is true if and only if p and q (where “q” holds the place for the conjunction of all the propositions implied by the proposition that p). So (with the important proviso that Horwich defines truth for propositions, Read for type-sentences, as we shall discuss in Section 4.5) Read and Horwich would be agreed on the truth-conditions for “Snow is white”. It is only with Liar-paradoxical sentences (Horwich washes his hands of these) that Read (following Bradwardine) is apparently able to do better. Read considers the Strengthened Liar sentence L, where L:L is not true. Closely following Bradwardine’s proof of Thesis 2 in his Insolubia,2 Read then proves ([25], p.11) that L says that it itself is true (L:TL). Now, in reasoning about the Liar sentence, the first thing that most people notice is that if it were true, what would be true is that it is not true, so they conclude that it is not true. But the next step in the reasoning goes (informally) “But that it is not true is just what L says, so things are just as L says they are. . . so it is true”. It is this second step of the reasoning that Bradwardine and Read block. For they deny that “L is not true” is just what L says. As we noted, they take themselves to have proved that, while L overtly says that it itself is not true (L:¬TL), what L also covertly says is that it is true (L:TL). So, by their improved definition (A) of truth, L is true iff ¬TL ∧ TL∧ anything else that L covertly says. 1 See Tarski [29], p. 156, footnote 1, where Tarski makes clear that by “sentence” he does not mean particular concrete series of signs “but whole classes of such series which are of like shape with the series given”. 2 For an English translation of this proof, see Read [23], pp.192–3.

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Since the RHS of this biconditional is a contradiction, we can infer the negation of the LHS, viz., ¬TL. What we cannot do is to infer L’s truth from its non-truth. L is not true, so the implicit, or covert, claim that it makes that it is true is simply false. That is no more paradoxical than me claiming that I am best served fried with bacon for breakfast. Crucial to the above reasoning is the proof (or alleged proof) that L covertly says that (i.e. strictly implies that) it is true. One immediate problem is that there is no comparable proof showing what the Truthteller V (where V :TV ) covertly says; in particular, we cannot show that while overtly saying of itself that it is true, it covertly says of itself that it is not true, so we have no proof comparable to the one for L, to show that V is false. But a theorist who offers a solution to the Liar ought to be able to offer more on V than a shrug of the shoulders and the comment “If it’s true, it’s true; if it’s false, it’s false”. Does Bradwardine have the resources for determining a unique truth-value for V ? There appear to be two options available to him. He could either insist that V has a definite truth-value that we are in principle unable to discover, or he could say that V is not the sort of thing that can have a truth-value at all. The first option would, of course, be anathema to an anti-realist, but it ought to be anathema to all of us. V purports to say that V is true. A follower of Read might urge that it says something more besides, but any suggestion as to what more it says (e.g. that pigs can fly) is likely to seem arbitrary and implausible, for there is no proof nor the slightest reason to believe that that is what more it says. But, if V says just that V is true, then we seem justified in saying that it has no content, for want of a grounding, and that is tantamount to saying that it says nothing and hence has no truth-value. The conclusion that the Truth-teller tells nothing is one that Read himself embraces ([25], p.15). There is something perverse about Read’s attitude to the Truth-teller. He holds that V says that things are as it says they are. But since V , he assures us, says nothing, it says what some nonsensical sentence such as “The Nothing noths” says. So V says that things are as “The Nothing noths” says they are. But, since nothing could be as “The Nothing noths” says, it seems that Read should hold that V is false. That looks to be the best bet for someone who, like Bradwardine, subscribes to the Principle of Bivalence. Read does not take that route, but holds that V says nothing, so he owes us an explanation of why he departs from the line that Bradwardine, presumably, would take. All Read says, however, is that “[t]o say that things are as it says they are is to say nothing”. But then one wonders why he thinks that to say that things are not as it says they are (as L purports overtly to do) is to say other than, or more than, nothing.

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As earlier indicated Read’s (A) can be regarded as a welcome refinement of Horwich’s definition of truth, its advantage only showing up in the special case of the paradoxes. But Read regards it as a fundamental departure. He motivates it with the example of Goldbach’s Conjecture that says, inter alia, that 4 is the sum of two primes. “Inter alia” because, of course, Goldbach’s Conjecture also says that 6 is the sum of two primes, 8 is the sum of two primes, and so on. Now, Goldbach’s Conjecture is a universal generalisaton and we can spell out what it says (as we began to do in the previous sentence), and arguably, when we have spelled out the infinite conjunction of its clauses, we have said all that it says. So, by Read’s (A) Goldbach’s conjecture is true iff 4 is the sum of two primes and 6 is the sum of two primes and 8 is the sum of two primes. . . . (where the “. . . ” is the conjunction of clauses of the form “2n is the sum of two primes” for all integers n > 4.) Let us take a look, then, at Yablo’s Paradox which, as Read recognizes ([23], p. 201), belongs in the Liar family. Each sentence in the Yablo sequence is a universal generalization. It begins: Y1 : For all integers n > 1, Yn is not true Y2 : For all integers n > 2, Yn is not true Y3 : For all integers n > 3, Yn is not true ............ A typical member Yk of the Yablo sequence is “For all integers n > k, Yn is not true”. One can prove formally or informally (Read [23], pp.201–2) that the Yablo sequence is paradoxical. As with the Goldbach sentence, each of the Yi can be spelled out in such a way that what you see is all it says. Y3 , for example, says “Y4 is not true and Y5 is not true and Y6 is not true . . . . . . ”. There seems no room here for Read’s “trick” of positing a covert conjunct, and hence no means available to Bradwardine for dissolving this paradox. It is a serious strike against any proposed solution of the Liar that it cannot deal with other paradoxes in the same family. Even if we allow that each of the Yi says something more than what it overtly says, this does not help matters. For, in line with the secret message that Read found hidden in L, we might suggest that that Yk also says that Yk is true. Thus, the first three now fully explicit sentences in this New Yablo are: N Y1 : For all integers n > 1, N Yn is not true and N Y1 is true N Y2 : For all integers n > 2, N Yn is not true and N Y2 is true N Y3 : For all integers n > 3, N Yn is not true and N Y3 is true

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This infinite chain has a closely related finite circular counterpart:3 S1 : S2 is not true ∧ S1 is true S2 : S1 is not true ∧ S2 is true As the reader may verify, even after this modification, paradox (of the “truth-teller” variety) remains. The circular paradox has the peculiarity that one can consistently assign opposite truth-values to S1 and S2 , yet considerations of symmetry indicate that we should assign the same truth-value to both, because S1 says of itself and S2 exactly what S2 says of itself and S1 . Bradwardine might wish to insist that all of the N Yi are false. Fine. But does he have any good reason for preferring this assignment of truth-values to any of the other consistent assignment of truth-values to the N Yi ? We could, for example, consistently assign truth to N Y1 and nontruth to all the other N Yi . So the suggestion that, like L, each of the sentences in the Yablo series covertly asserts its own truth does not seem to be of much help to someone who, like Bradwardine, wants to demonstrate that each paradoxical sentence has an unproblematic unique truth-value. If instead we suggest that the implicit “bit” in any Yi is a second conjunct that declares true each proposition that is declared in the first conjunct to be not true, then each member of this version of the Yablo sequence is a contradiction. In this case, no paradox ensues but we face the charge of changing the subject in an uninteresting way – if all the premises of an argument are contradictions, then it is hardly paradoxical that one can derive from them a contradictory conclusion. Finally, if, following Read, we simply add an unknown to represent what is covertly said by a member of the Yablo sequence, then, most generally, we obtain the variant sequence beginning GY1 : For all integers n>1, GYn is not true and p GY2 : For all integers n>2, GYn is not true and q GY3 : For all integers n>3, GYn is not true and r If GY1 is true, then all the GYi following it are not true; in particular, GY3 , GY4 etc. are not true. But this is just what the first conjunct of GY2 states. Therefore this conjunct is true. But our starting assumption, that GY1 is true implies that GY2 is not true. Therefore its second conjunct, p, must be false etc.. We can easily ensure that all the GYi are false by 3 For

the general relation between infinite paradoxes and circular variants, see Goldstein [14].

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ensuring that q, r, etc. are all falsehoods. So, again, a consistent set of values can be assigned to the members of this variant sequence and it is therefore non-paradoxical. But, again, we face the charge of merely changing the subject. Of course, the paradoxical pair The sentence below is true The above sentence is false can be transformed into a non-paradox by adding “and pigs can fly” to the first sentence. But nobody would think that, by this manoeuvre, the original pair paradox is solved. There is a much more satisfactory way with Yablo’s Paradox, and this can be illustrated by comparing two infinite series – the reversed variant of the Yablo and a variant of the mathematical Fibonacci – series in which the nth term is obtained not by summing the preceding two terms, but by summing all the preceding terms.

All All All All All

the the the the the

Reverse Yablo ............... ............... preceding statements preceding statements preceding statements preceding statements preceding statements

are are are are are

untrue untrue untrue untrue untrue

Total Hindsight Fibonacci ............... ............... The sum of all preceding numbers The sum of all preceding numbers The sum of all preceding numbers The sum of all preceding numbers The sum of all preceding numbers

The standard Fibonacci is specified by “F ib[i] = F ib[i−2]+F ib[i−1]”. But this is a partial (incomplete) specification. Clearly we cannot assign any numerical values to any member of the series unless we first specify F ib[1] and F ib[2], or at least first stipulate a value for at least two members of the series. Without the specification of values for at least two members, the series is ungrounded. In the Total Hindsight Fibonacci (above) the apparently number-designating expressions designate no numbers, because of underspecification. And similarly, in the Reverse Yablo and in the Standard Yablo, each sentence in each sequence is syntactically perfectly in order, yet none yields a statement with any truth-value. That is what the modern cassationist would say. We saw, above, that applying Read’s (A)-scheme to the Strengthened Liar sentence L gives the biconditional TL iff ¬TL ∧ TL ∧ anything else that L covertly says Read is impressed that the RHS is a contradiction, and he concludes that the LHS must be false. But another way of reading, or parsing, the biconditional is TL iff ¬TL ∧ anything else that L covertly says

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This is supposed to inform us of the conditions necessary and sufficient for L’s being true. But it does no such thing. For comparison, think of various attempts to define what it is for a person to be intelligent (I). They each take the form Ix iff F x ∧ Gx ∧ . . . and any serious such attempt can be assessed for plausibility. But imagine an attempt at defining necessary and sufficient conditions for intelligence that took the form Ix iff ¬Ix ∧ . . . This is just as iffy4 as: Ix iff Ix It fails to set out any conditions for being I (which is not the same as merely setting out wrong ones). The mistake in thinking otherwise is closely related to the fallacy definiens per definiendum. Now, how does one specify the content of a statement? Donald Davidson [8] gave the reasonable answer that one does so by indicating those possible states of affairs (if any) which, were they actualized, would render the statement true and those possible states of affairs (if any) which, were they actualized, would render the statement false. In other words, its content is specified by setting out the truth-conditions of a statement. But, using Read’s (A) to specify the truth-conditions of the Strengthened Liar L results in L is true iff L is not true ∧ . . . . As the comparison with spurious definitions for “I” shows, this is merely a failed attempt to set out truth-conditions, hence what we should conclude is that L has no locateable content; it is content-less (“sinnlos” was the epithet that Wittgenstein used in the Tractatus [31]) and hence has no truth-value. Once again, we arrive at the verdict of the cassationist. Let us take stock. If we assume that the Liar sentence is either true or false then, by familiar simple reasoning, we arrive at the result that the sentence, if true, is false; if false, true and hence, unless we are willing to accept “serious and damaging revisions to logical principles” (Read [25], p.10) we reach a contradiction. To anyone except a dialetheist, this will seem an unsatisfactory outcome. Now, we could simply deny the original 4 For the benefit of those unused to idiomatic English, I must explain that “iffy” means “dodgy”, “not kosher”. [In deference to a referee who made a number of other helpful observations, and who thought that non-native speakers of English might not be happy with these synonyms, I should say that they all mean “dubious”.]

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assumption that the Liar sentence is either true or false. But (quite apart from the problem of revenge Liars) a denial without a rationale is not respectable; it is merely petulant. The derivation of a contradiction can be avoided by various means. There is the “ambiguity” manoeuvre which argues, roughly speaking, that the “familiar simple reasoning” commits the fallacy of equivocation – that the Liar sentence is ambiguous between two sentences (see Slater [27]), or that the meanings of different tokens occurring at different stages of the reasoning, differ – an option explored by Tyler Burge [5] amongst others. There is the Bradwardine manoeuvre of claiming that the Liar sentence says more than it appears to say; in particular, that it covertly says the opposite of what it overtly says, so obviously what it says must be false. There is the cassationist position of claiming that that Liar says nothing and that therefore nothing can be derived from what it says. These proposals (and the many others that have been made) need to be assessed not just for their internal coherence, but also for their rationales and their power to cope with other paradoxes in the Liar family. What we have attempted to show, so far, is that the Bradwardine solution does not do particularly well on this last count. It also does not do well with the kind of example discussed in the next section. But what seems to have happened in the preceding discussion is that, in working through our objections to Bradwardine, we keep coming back to the cassationist alternative. Let us put that thought on the back burner, for a while.

4.3

Read My Lips

The “says that” operator “:” is embeddable. We render “Joe says that Max says that Andrew says that p” as “J:(M:(A:p))”. George W. Bush says “What I am now saying is not true and, read my lips, I am not saying that it is true”. The formal version of this, that we shall name “B” in George W’s honour, is B:(¬TB ∧ ¬B:TB) Again following Read, let us assume that a sentence generally does not wear all that it says on its sleeve and that even George W’s says something more than it seems to say; call this extra bit “q”. Then, using Read’s (A)-scheme, we can write the truth-condition for B as follows TB ⇔ ¬TB ∧ ¬B:TB ∧ q Looking at the left-to-right implication, we see that the LHS cannot be true, i.e., B is not true. Let us, then, work with the assumption that B

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is not true,¬TB. From the above biconditional, we learn ¬TB ⇒ ¬(¬TB ∧ ¬B:TB ∧ q) i.e. ¬TB ⇒ TB ∨ B:TB ∨ ¬q So, since ¬TB has to be true, at least one of the disjuncts on the RHS has to be true. We already know that the first disjunct is not true. Nor is the second – read my lips. Therefore there has to be something else that is said by B and is true. But what else? There is, apparently, no particular proposition; any proposition (so long as it is a true one) will serve. Yet this is an uncomfortable conclusion. The claim is that B says something in addition to what it appears to be saying, but we do not, and cannot know what this additional proposition is; all we know is that, whatever it is, it is true! This is rather like the difficult position into which we seem to be forced by the Cretan Liar. Epimenides (a Cretan) says that all Cretan utterances, including the current one that he is in the process of making, are false. What he says cannot be true, so it must be false. And that means that either the utterance that we called “the current one” is true or some other utterance made by some Cretan is true. We have already established that “the current one” is not true. Therefore it seems that we have established, by pure logic, that a true statement must have been made either by Epimenides or by some other Cretan. A true Cretan utterance seems to have been conjured into existence even though common sense tells us that St. Paul may just have been right in his suggestion that all utterances made by all Cretans are false.5 Rather than accepting that logic can work such magic, we would be better off retracing our steps to the starting point and reconsidering whether Epimenides’ utterance has any truth value at all. Similarly with B. To a normal observer, uncontaminated by theory, that utterance of Bush’s does seem to be entirely vacuous. “Read my lips”, he mouths exaggeratedly, “What I am now saying is not true and I am not saying that it is true”. But just 5 If this seems historically far-fetched, consider, instead (an adaptation of) Jonathan Cohen’s example of the policeman who, in court, asserts that everything that the prisoner says in court will be false. The prisoner then takes the stand and swears that what the policeman just said was true. Obviously what the prisoner said cannot be true, so it is false – in other words, it is false that everything said by the prisoner will be false – the prisoner must say something true, and this will have to be something other than what the prisoner has already said since that, as we know, was false. But although this conclusion has classical logic’s seal of approval, it must be wrong – if only because the prisoner could drop dead immediately after making his opening remark (see Cohen [7]). A similar example can be found in Kripke [19].

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what is George here saying? Nothing. He’s all mouth. (Despite our best intentions, the cassationist alternative put on the back burner at the end of the last section has not stayed there for long.) We said, in the penultimate paragraph, that B does not say that it itself is true. The reason we said that is because we took George W’s word for it – he invited us to read his lips as he enunciated a denial that he was saying that his own utterance is true. Now we could (and this is reminiscent of a problem with the Epistemicist solution to the phenomenal Sorites) claim that, unknown to himself, Bush is actually saying what he sincerely and explicitly denies that he is saying. And certainly someone who says something of the form “In saying that p, I’m not implying that q” might simply be wrong. But someone who says something of the form “I am not saying that p” speaks the truth, because what he is saying is not that p, but that he is not saying that p. And what he is saying does not imply p. So, if we are disinclined to accept that there is an arbitrary truth lurking as a hidden conjunct in George’s utterance, we ought to conclude that his utterance was really all just hot air, with no content and no truth-value. Bush, as a devotee of the “resolutist” interpretation of Wittgenstein’s Tractatus, might feel honoured to agree.6

4.4

Re-Reading Anon

Bradwardine argues that the Liar utterance is not true. Since he accepts the Principle of Bivalence, his claim is that the Liar utterance is just false. In the early mediaeval period, the cassatio type of solution was popular but, probably as a result of its harsh condemnation by Bradwardine and other influential writers, it almost died out. The cassantes would agree with Bradwardine that the Liar sentence is not true. Their reason, however, would be that the Liar utterance fails to express a proposition and hence is not truth-apt, is not the sort of thing that could be either true or false. Note that this view too respects the Principle of Bivalence: every proposition is either true or false. William Shyreswood gives short shrift to the cassatio approach and the shrift that Bradwardine gives it is even shorter.7 He seems to think 6 Near the end of the Tractatus ([31], 6.54), Wittgenstein wrote: “My propositions are elucidatory in this way: he who understands them eventually recognizes them as nonsensical (unsinnig)”. So-called “resolute” interpreters take him to be meaning that all the preceding propositions in the book, which look as if they are candidates for truth-value, are pure gibberish – hot air. This is clearly a foolish interpretation if only because hot air does not have the substance of a ladder and, as 6.54 continues, “when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly”. 7 See, respectively Roure ([26], p.250 (2.01-2.06) and p.295 (5.05-5.06)).

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that expressing a proposition really is no more than just opening one’s mouth and uttering a sentence (“homo potest aperire os suum et formare istas voces: ego dico falsum”) and hence is able to dismiss the cassatio position summarily: “Hec autem positio falsa est quia Sortes [when he says “Sortes dicit falsum”] sic dicens dicit litteras et sillabas, dictiones et orationes, ut sensui et omnibus affatibus satis patet. Preterea Sortes auditur loqui, ergo Sortes aliquid dicit.” (Roure [26], p. 295)

He ends with a jibe: “Oppinio ergo cassantium satis est cassata” – roughly “Thus the nullifiers are nicely nullified”. It should be obvious that this criticism is hardly worthy of a “prince of modern natural philosophers” 8 Bradwardine is attacking a straw man. No cassationist would deny that an utterance is made up of syllables and words or that, if the utterance is heard, something must have been spoken. The cassationist wishes to insist that a proposition or statement evaluable as true or false is more than just a syntactically correct sentence. A proposition is fully ert, whereas a syntactically correct string of signs is inert. A later (15th century) writer, Hieronymous Savonarola, put it like this: An insoluble proposition is to a genuine proposition as a dead man is to a live one (see Ashworth [1], p.37). (Of course, a proper defence of cassationism must get beyond these metaphors.) The anonymous author (Anon) of the first known mediaeval tract defending cassationism, the Insolubilia Monacensia, himself considers an objection to his claim that you say nothing (“nil dicis”) when making a Liar utterance. He writes: “Sed contra. Dicere nil aliud est quam prius asserere et postea proferre. Sed assertio non est cassanda, cum ex ipsa non sequitur inconveniens. Dato enim quod asseratur et non preferatur, inpune potest sustineri quod sit falsum. Similiter prolatio non est cassanda nec solet cassari. Et ita cum assertio non sit cassanda nec prolatio et ista duo sint dicere, dictio non est cassanda. Et ita si aliquid est cassandum, et assertio non sit cassanda, cum prior sit et ex ipsa non sequitur inconveniens, restat, ut videtur, quod prolatio non sit cassanda. Et hoc non solet fieri.” (Munich C.L.M. 14.458, f.39va; De Rijk [11], p.106) Translation:9 “But on the contrary. To speak is nothing other than first to think and 8 This

is the description bestowed on Bradwardine by Ralph Strode. See (Read, 2002:189). of this and the following passage proved to be a nightmare. It is very difficult to say what the author means by key words such as “asserere” and “proferre”. The former seems to connote forming a thought that one would be prepared to assert; the latter to the uttering of it. One can glean some understanding of what the author is about, but unclarity predominates. I am deeply grateful to Fabienne Pironet, Jenny Ashworth, and especially to Stephen Read for advice on the translation. In Buridan’s fifth sophism (Hughes, 1982: 54-7) one also finds the distinction between words uttered and proposition expressed. 9 Translation

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then to utter. But a thought is not nullified when nothing unacceptable follows from it. Given, then, that it is thought and not uttered, one can maintain without any problem that it is false. Likewise, an utterance is not nullified nor should it be nullified. And thus, since neither the thought nor the utterance are to be nullified, and these two constitute speaking, then speaking is not to be nullified. And thus, if anything is to be nullified and the thought is not to be nullified since it comes first and from it nothing unacceptable follows, it remains, it appears, that the utterance is not to be nullified. And this should not happen.”

However, Anon immediately provides a “Solutio”: “Sicut hec definitio “animal rationale” removetur a quodlibet alio ab homine, tamen partes eius non removentur, similiter assertio cum prolatione [non] removetur ab hoc enuntiabili, tamen partes eius non removentur. Unde assertio non est cassanda simpliciter, sed habito respectu ad prolationem. Unde neutra est cassanda simpliciter, sed una respectu alterius.” (De Rijk [11], p.106) Translation: “Just as the definition of ‘rational animal’ is withheld from anything other than a man, though its parts [may] still apply, so too a thought which is uttered [may] not apply to something that is spoken, though its parts [may] still do so. Hence thought is not to be nullified absolutely but [only] in respect of its being uttered. Hence neither is to be nullified absolutely, but one with respect to the other.”

A bunch of words, such as “I am not talking to you” is not the kind of thing that can be nullified, destroyed, cassated. And one might form the thought “I am not talking to you”. But of course, there is something funny, something self-defeating about telling someone that you are not talking to them, since in telling you are talking. Something of this sort occurs, according to the cassationist, when you utter a Liar sentence. Anon is not denying that, when a Liar utterance is made, something is said in the sense that words – in fact, a syntactically correct sentence – is spoken, but nothing is said in what J.L. Austin ([2], p.92) called the full sense of “say”. And this is surely the crux of the cassationist position. For the reasoning within a semantical paradox typically proceeds from the assumption that the paradox-generator says something true or false. If one can show that, contrary to appearances, the paradox-generator says nothing, then the paradox-reasoning cannot even get started. Postulate 3 of Bradwardine’s Insolubilia is “The part can supposit for its whole and for its opposite and for what is equivalent to them”. Read claims that this postulate asserts Bradwardine’s opposition to his predecessors, the restringentes [restricters – those who countenance only non-self-referential sentences] and the cassantes ([23], p. 191). Now, while it is clear that that the postulate records a rejection of the restringentes’ view, for Thomas will allow, for example, “This proposition is not very

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interesting” or “The negation of this proposition is true” to count as genuine propositions, it is not clear that Postulate 3 stands opposed to the cassantes’ position. Their position is that some sentences, though grammatically in perfect order, fail to express propositions. A word in a sentence can have signification, but supposition is a property of terms as they occur in propositions. When Thomas says “The part can supposit. . . ” he means “The part of a proposition can supposit. . . ”, so his Postulate 3 does not impinge on the cassantes, who argued that the Liar sentence is a non-proposition. Bradwardine, as we have seen, rubbishes the cassantes’ position, but if he himself regarded Postulate 3 as indicating a rejection of that position, this could only be because he question-beggingly assumes that paradoxical sentences like the Liar’s express genuine propositions. Read follows Bradwardine in his disrespect for cassatio, simply assuming that the approach is wrong. He writes “for simplicity I will assume. . . that each sentence to which we apply (A) does say something” ([25], p.8). Without that assumption, his discussion would leave open the possibility that certain sentences say nothing. He establishes that the Liar sentence strictly implies a contradiction, and another way of stating that result would be to say that, contrary to surface appearances, the Liar sentence is a contradiction. A dialetheist, who is a glutton for truthvalues, will say that some contradictions, including the Liar are both true and false. A cassationist, exercising a prudent parsimony and with a proper respect for generality, should say just the opposite, that a contradiction is neither true nor false. This view of contradictions, though not mainstream, has been advocated by several philosophers including Wittgenstein and possibly even Aristotle (see Goldstein [16]). Bradwardine’s Postulate 2 is “Every proposition signifies or means contingently or necessarily everything which follows from it contingently or necessarily”. As Read acknowledges ([23], p.191) this postulate plays a crucial role in Bradwardine’s diagnosis of the Liar. The familiar reason for being suspicious of this postulate is that a contradiction logically implies (for Bradwardine, strictly implies) any proposition, yet we should be loathe to grant that a contradiction signifies or means every proposition. But, from a non-proposition, nothing follows. If the cassationist view that the Liar sentence says nothing can be supported, then, among its other advantages, it would offer an explanation for Bradwardine’s own conclusion that the Liar sentence is not true. So the intriguing question arises as to whether Bradwardine was unfairly dismissive of a solution that digs deeper than his own. This is not the occasion to mount a fullscale defence of cassationism, but it is hard to keep a good theory down, and it will pop up again in the next section.

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Bashing Away at that old Type-Reader

There are, as Read points out, well-known counterexamples to Tarski’s T-scheme. How devastating are these? Hintikka’s attempted counterexample, that Read discusses ([25], p.4) is “Any man is mortal” is true if and only if any man is mortal. This is clever, but is hardly a counterexample. It merely highlights that “any” is ambiguous between “every” and “some”, and so encourages reality to intrude on the logician’s fantasy that each of the syncategoremata has a unique logical rôle. Even “a” and “the” do not. Tarski needs only insist that the “any” on the RHS is an “every”. Indexical utterances are often cited as being the clearest counterexamples to Tarski, but they are not really, given that Tarski was concerned to provide a definition of truth for a formal language in which there are no ambiguities and all the sentences are eternal. Read says that the following instance of the scheme: “That book was stolen” is true if and only if that book was stolen is not true, unless the book indicated by the demonstrative pronoun is the same in both its occurrences. But, since Tarski was attempting to define truth for type sentences, considerations about who uttered a particular token sentence when, and what book he or she intended to talk about are entirely misplaced. The correct criticism to make of Tarski is that, in taking type sentences as truth-bearers he ignored indexicals, and, insofar as he was interested in providing a solution to the Liar Paradox, did so at his peril, since indexicality is so intimately implicated in Liar paradoxicality. This can be obscured by the fact that, in many discussions of the Liar, no explicit indexicals or demonstratives feature in the problematic sentences; instead sentence letters are used. What is not generally appreciated is that, since these letters are standing in for indexicals, type-identical tokens containing them may differ in truth-value from one context to the next. In Tarski’s own discussion of the paradox, it is the geographical location of a certain token sentence (its position on a certain page of Tarski’s article) that is crucial to generating the contradiction ([29], p.158, line 5). Placed elsewhere, it would be entirely innocuous. How, then, should we state truth-conditions for utterances containing indexicals? Well, supposing that somebody says “It’s raining here”. Obviously a “straight” Tarski T-scheme will not give the right result, but the following modification of Tarski seems appropriate: “It’s raining here” is true if and only if it was raining in the vicinity of the quoted speaking when that speaking occurred.

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A word of explanation is in order. In this formulation, we refer, perhaps rather inelegantly, to “the quoted speaking”. It would be incorrect to replace this phrase by “the quoted speaker ”, for we need to make clear that we are indicating an utterance made by that speaker at a particular time and place. The quotation marks therefore refer to the content of a particular token utterance that, if written down in the appropriate font, would look like what lies between the quotation marks. The rôle of these quotation marks is not therefore (as per Tarski) to form a name of the string of signs between them ([29], p.156), nor (as per Davidson) to refer to a typographical or phonetic shape ([9], p. 27).10 Quotation marks can be used for various purposes, but their behavior, when we are using them to indicate the content of utterances, is quite subtle. It can be explained by attending to the dialect of the Valley girls in Southern California. Mediaeval authors did not have quotation marks at their disposal and instead used the particle “ly” to mark the beginning of a quotation. Valley girls, no doubt deriving their inspiration from this mediaeval precedent, use the particle “like” and, in speech, mimic the pronunciation of the quoted speaking. This has an advantage over its mediaeval counterpart in that it is easy for a hearer to identify the end of the quoted fragment, for it is here that the speaker stops mimicking and switches back to her own voice. A typical Valley girl utterance is: Kitty-Jo was like My boyfriend Donny is so kinda gross, but, huh, she is still dating him. Here the italicized portion is spoken in a voice resembling that of Kitty-Jo’s at the time when she made that remark about Donny, while the rest is spoken in propria voce of the speaker. What is clear is that the quotation mechanism (the particle “like” plus the voice-mimicking) serves to indicate a token – a particular historical utterance. Should we be interested in the truth-value of what Kitty-Jo reportedly said, we can ignore the way she said it, but concentrate on the content of her indicated utterance. Quoting is like pointing. You point to a flamingo but, in certain contexts, you do so merely as a means of indicating its color. You point, by means of a quotation device, to a particular token utterance or inscription but, for certain purposes, you do so merely as a means of indicating its content – the proposition expressed by it. Identification of the content of most non-learnèd utterances demands attention to the context of utterance. We predicate truth or falsity of what 10 For an excellent discussion of quotation, see the Stanford Encyclopaedia entry by Cappelen and LePore [6].

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is said when a particular utterance is uttered or inscription inscribed. In the case of eternal sentences, it does no harm to ascribe truth or falsity to the type sentence simply because the statements resulting from each use of any token of that type all have the same truth-value. This does not detract from the fact that the primary bearer of truth or falsity is the proposition expressed by the use of a token sentence. We can agree with Read that, to provide the truth-condition of an utterance, we need look beyond the sentence used by the utterer to everything that is said in the use of that sentence. But, more importantly, we need to recognize what both Tarski and Read failed to recognize – that it is token utterances (specifically, what is said in the issuing of any such token in a given context) that are the truth bearers. Once we are clear about this, we see that the use of one token of a certain type sentence may yield a true proposition; another token of the same type sentence may be used, in a different situation, to yield a false proposition and, most importantly in the province of the paradoxical, another use of a token of that very same type may yield no proposition at all, rendering otiose the question as to its truth or falsity.11 A token of the sentence “That statement is false” succeeds in making a statement only when its demonstrative clause succeeds in demonstrating another statement. The recognition that truth and falsity are properly predicated of the content of tokens is the vital addition that modern discussion has made to traditional cassationism (see Goldstein [15], Gaifman [13]). The cassationist holds that L cannot say that L is not true, the most obvious reason being that if A has a truth-value, it cannot say the same as the proposition that A is not true, for the latter would have the opposite truth-value. Therefore, since the latter is discernibly different with respect to truth-value from the former, then, by Leibniz’ Law, they cannot be identical. Likewise, with the standard strengthened pair paradox. We can assign the name S1 to the proposition that S2 is not true, so S1 : S2 is not true. Therefore S1 and S2 have different truth values. This means that the name S2 is no longer free for assigning to a proposition with the same truth-value as S1 ; in particular, we cannot simply accept that S2 : S1 is true. So what we should say of the paradox consisting of the pair S1 and S2 is that neither sentence expresses a proposition; the assumption that each has a truth-value (from which a contradiction follows) can therefore be rejected and the paradox dissolves. We could attempt to write out the Valley girls variant of this paradox as follows:

11 Various techniques are on the market for identifying those sentence-tokens that fail to yield statements with either of the classical truth-values. See (Sommers [28], Tennant [30]), (Maudlin [21]) and – closest to my own approach – (Gaifman [12], [13]).

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Mary-Beth: Kitty-Jo was like What Mary-Beth said is true and that’s not true. Kitty-Jo: Mary-Beth was like: Kitty-Jo was like What Mary-Beth said is true and that’s not true and that’s true. But, if we look back to Mary-Beth’s utterance, we can see that she is not accurately quoting Kitty-Jo’s; the latter’s utterance is obviously different to the one attributed to her by Mary-Beth. For starters, Kitty-Jo refers to one of her own utterances (she has to mimic – possibly even to caricature – herself!) But, if we change Mary-Beth’s utterance so that she is now accurately referring to Kitty-Jo’s, then we see that our formulation of Kitty-Jo’s utterance is no longer correct, for it should refer to Mary-Beth’s amended utterance, and so on ad inf.. This is but a new illustration of a familiar point concerning the impossibility of identifying any content for paradoxical utterances. More picturesquely, one can think of Mary-Beth trying to make a ’phone call to Kitty-Jo. She goes through the usual motions of dialing Kitty-Jo’s number, a procedure that normally results in a successful connection (the analogy is: she utters a perfectly good sentence), but, if, on this occasion, Kitty-Jo is engaged in making a call of her own (the analogy is: engaged in attempting to assign a truth-value to another’s utterance) then the line will be dead; no call will result. No call will of course result if, while Mary-Beth is attempting to call Kitty-Jo, Kitty-Jo is attempting to call her. Updating Savonarola, “An insoluble proposition is to a genuine proposition as a dead line is to a live one”. A similar objection can be made to Read’s rebuttal of the idea that paradoxical utterances are neither true nor false. He asks us to suppose that he, Stephen Read, utters “What I am saying is not true”. Cassationists will want to say that his performance is like some loser dialing his own telephone number – no connection is made; no proposition is yielded. Because Read’s utterance fails to yield a proposition, it is not true. So cassationists say “What Steve says is not true”, and take themselves to have told the truth. But, Read counters, “The problem for [them] is to explain why I [Steve Read] did not speak truly when I anticipated them and said “What I am now saying is not true”.” (Read [23], p.190). Suppose, to make life even more difficult for the cassationist, Steve, in regal mood says “What Steve is saying is not true”. It seems that the cassationist wants to condemn that utterance as neither true nor false, and wants to do so by saying exactly the same thing – only asking us to believe that in the mouth of a cassationist, it is a truth! Doesn’t Kant have some sort of imperative forbidding that kind of behavior, requiring us to treat others just as we ask to be treated ourselves?

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In order to get out of this apparent bind, all the Valley girl cassationist need do is, first, to remind us that truth and falsity are predicated of the content of token utterances and, second, switching into her own lingo, to give her verdict on Steve’s utterance as follows: Steve was like What Steve is saying is not true, and, hey, that’s not true; it’s radically truth-valueless.12 This is unproblematically true. The “that” points to Steve’s utterance (italicized) which the cassationist deems to convey no content and hence to be neither true nor false. Suppose that Steve, in an uncharacteristically Heideggerian funk, blurts out “The Nothing noths”, then the cassationist will sensibly comment: Steve was like The Nothing noths, and that’s not true; it’s junk. As we can see, the cassationist can retain calm and dignity, commenting on non-propositions and nonsense without slipping into either herself. Only under the profound influence of prohibited substances would she say something like “Steve’s wrong – the Nothing doesn’t noth”. (If she said that, then she would be talking junk, not speaking the truth.) So, even if I uttered the words “What I am now saying is either false or radically truth-valueless”, if my utterance is void of content (hence radically truth-valueless) it is quite possible for an outsider to truly say so: “What Laurence was then uttering was <either false or> radically truth-valueless”. Bradwardine will agree with the cassationist that Liar utterances like Steve’s, above, are not true, but will go on to make the unwarranted claim that they are false. If Bradwardine subscribes to the mediaeval adage “Veritas est adaequatio rei et intellectus”, this commits him to saying that Steve has expressed a bona fide proposition that somehow fails to square with (adaequatio) the way things are. Though failing to pierce its heart, I have made various stabs at showing that this position is moribund.

Reading List [1] Ashworth, E.J., 1972, “The treatment of semantic paradox from 1400 to 1700”, Notre Dame J. of Philsophy 13: 34–52. [2] Austin, J.L., 1975, How to do Things with Words, 2nd ed., Oxford University Press, Oxford. 12 The

phrase ‘radically truth-valueless’ is due to Tennant ([30], p.378).

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[3] Bottin, F., 1976, Le Antinomie Semantiche nella Logica Medievale, Editrice Antenore, Padova. [4] Bradwardine, T., 1970 (originally composed in early 1320’s) Insolubilia in [26]: 285–326. [5] Burge, T., 1979, “Semantical paradox”, J. of Philosophy 76: 169–198, reprinted in [20], pp.83–117. [6] Cappelen, H. and LePore, E., 2005, “Quotation”, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), . [7] Cohen, L. J., 1957, “Can the logic of indirect discourse be formalised?”, J. of Symbolic Logic 22: 225–232. [8] Davidson, D., 1967, “Truth and meaning”, Synthese 17: 304-323, reprinted in [10], pp.17–36. [9] Davidson, D., 1979, “Quotation”, Theory and Decision 11: 27–40, reprinted in [10], pp.79–92. [10] Davidson, D., 2001, Inquiries into Truth and Interpretation, 2nd ed., Oxford University Press, Oxford. [11] De Rijk, L.M., 1966, “Some notes on the mediaeval tract De Insolubilibus, with the edition of a tract dating from the end of the twelfth century”, Vivarium 4: 83–115. [12] Gaifman, H., 1992, “Pointers to Truth”, The Journal of Philosophy 89: 223–61. [13] Gaifman, H., 2000, “Pointers to Propositions”, in: Circularity, Definition, and Truth, A. Chapuis and A. Gupta, eds, Indian Council of Philosophical Research, New Delhi, pp.79–121, also available at . [14] Goldstein, L., 1999a, “Circular queue paradoxes – the missing link”, Analysis 59: 284–290. [15] Goldstein, L., 1999b, “A unified solution to some paradoxes”, Proceedings of the Aristotelian Society 100: 53–74. [16] Goldstein, L., 2004, “The Barber, Russell’s Paradox, Catch-22, God, contradiction and much more”, in: The Law of Non-Contradiction: New Philosophical Essays, G. Priest, JC Beall and B. Armour-Garb, eds, Oxford University Press, Oxford, pp.295–313. [17] Horwich, P., 1998, Truth, 2nd ed., Clarendon Press, Oxford. [18] Hughes, G., 1982, John Buridan on Self-Reference: Chapter Eight of Buridan’s Sophismata, Cambridge University Press, Cambridge. [19] Kripke, S., 1975, “Outline of a theory of truth”, J. of Philosophy 72: 690–716, reprinted in [20], pp.53–81.

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[20] Martin, R.L., 1984, Recent Essays on Truth and the Liar Paradox, Oxford University Press, Oxford. [21] Maudlin, T., 2004, Truth and Paradox: Solving the Riddles, Oxford University Press, Oxford. [22] Read, S., 2000, “Harmony and autonomy in classical logic”, J.Phil. Logic 29: 123–154. [23] Read, S., 2002, “The liar paradox from John Buridan back to Thomas Bradwardine”, Vivarium 40: 189–218. [24] Read, S., 2004, “Identity and harmony”, Analysis 64: 113–119. [25] Read, S., 2008, “The truth schema and the liar”, This Volume, pp.3–17 [26] Roure M.-L., 1970, “La problématique des propositions insolubles au XIIe siècle et au début du XIVe , suivie de l’édition des traits de W. Shyreswood, W. Burleigh et Th. Bradwardine”, in: Archives d’Histoire Doctrinale et Littéraire du Moyen Age 37: 205–326. [27] Slater, B., 1991, “Liar Syllogisms and Related Paradoxes”, Analysis 51: 146–53. [28] Sommers, F., 1969, “The concept of truth in natural languages”, Review of Metaphysics 23: 259–86. [29] Tarski, A., 1983, “The concept of truth in formalized languages”, in: Logic, Semantics, Metamathematics, 2nd ed., Hackett Publishing Company, Indiana, pp.152–278. [30] Tennant, N., 2004, “An anti-realist critique of dialetheism”, in: The Law of Non-Contradiction: New Philosophical Essays, G. Priest, JC Beall and B. Armour-Garb, eds, Oxford University Press, Oxford, pp.355–84. [31] Wittgenstein, L., 1922, Tractatus Logico-Philosophicus, Routledge and Kegan Paul, London. [32] Yablo, S., 1993, “Paradox without self-reference”, Analysis 53: 252–253.

Chapter 5 LOGIC WITHOUT TRUTH Buridan on the Liar Gyula Klima∗ Fordham University Rose Hill Campus, Bronx NY 10458(718)817-1000, USA [email protected]

Abstract

Stephen Read’s criticism of Buridan’s solution of the Liar Paradox is based on the charge that while this solution may avoid inconsistency, it does so at the expense of failing to provide a theory of truth. This paper argues that this is one luxury Buridan’s logical theory actually can afford: since Buridan does not define formal consequence in terms of truth (and with good reason), his logic simply does not need it. Therefore, Buridan’s treatment of the paradox should be regarded as an attempt to eliminate a problem concerning the possibility of the consistent use of semantic predicates under the conditions of semantic closure, rather than as an attempted solution of a problem for a theory of truth. Nevertheless, the concluding section of the paper argues that Buridan’s solution fails, because it renders his logical theory inconsistent. A postscript, however, briefly considers an interpretation that may quite plausibly save the consistency of Buridan’s theory.

Keywords: Nominalism, Insolubilia, Truth, Correspondence, Validity, Virtual implication, Consequences, Signification, Supposition, Syncategoremata, Complexe significabilia, Token-sentence, Bradwardine, Buridan, Albert of Saxony.

5.1

Read, Bradwardine and Buridan

In a couple of recent, extremely intriguing papers [1, 2], Stephen Read has successfully revived Thomas Bradwardine’s ingenious treatment of the Liar Paradox, along with his theory of truth and propositional signi∗ Opponent.

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fication, after being nearly completely forgotten and generally unappreciated for almost seven centuries. In the course of this resuscitation process, Read has also argued against contemporary “infatuation” with another, already quite successfully resuscitated medieval treatment of the Liar, namely, John Buridan’s, and for the superiority of Bradwardine’s solution, which (or rather, a significantly modified version of which) Buridan had abandoned. Despite possible (and even actual) appearances to the contrary, I am not one of those who are “infatuated” with Buridan in general or his treatment of the Liar in particular.1 Nevertheless, I believe fairness demands that we acknowledge Buridan’s genuinely good reasons for abandoning his own earlier solution within its own theoretical framework. Indeed, we should realize that the charges leveled against Buridan’s solution coming from the demands of a different theoretical framework are not quite justified, if we consider the role of his final solution within its own theoretical context. Therefore, given the importance of the different theoretical contexts in which these solutions are proposed, I believe I should begin by clarifying some points concerning the relationships between Bradwardine’s and Buridan’s positions within their respective theoretical contexts. As Stephen Read has carefully pointed out, there is a significant difference between Bradwardine’s solution and Buridan’s early solution, despite the fact that they are both framed with reference to the signification of propositions, as opposed to Buridan’s final solution, which is framed with reference to the “co-supposition” of the terms of a “virtually implied” proposition. The fundamental difference between the two solutions provided in terms of propositional signification is that whereas Buridan’s early solution involves the thesis that all propositions signify their own truth, Bradwardine’s solution restricts this claim to propositions signifying that they are false, i.e., according to Bradwardine, it is only such propositions that signify their own truth (and so, signifying both their own falsity and truth, they must be false). But the difference between their solutions is not restricted to the different scopes of these two theses: the authors provide radically different reasons for these theses. Bradwardine’s thesis is based on an elaborate argument, specifically designed to deal with propositions signifying themselves not to be true or to be false (see [1], p. 192). Buridan’s, on the other hand, is based on what he considers to be the general feature of the signification of all propositions based on their 1 In fact, I consider Buridan my worthiest philosophical opponent on some fundamental issues in metaphysics. This is precisely the reason why I spend considerable time and effort on reconstructing his genuine positions.

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form (i.e., the meaning of their copula; see [1], pp. 193–202, esp. p. 195). Therefore, it is actually not quite clear whether Buridan’s early solution was directly influenced by Bradwardine’s, or rather by just the formula in general circulation that a proposition is true because things are in [all] the way[s] it signifies them to be (qualiter[cumque] significat [rem esse] ita [res] est ). I cannot, and I do not want to, decide this historical question here. But because of their theoretical differences, I do want to distinguish Bradwardine’s own solution from Buridan’s early solution,2 both provided in terms of propositional signification, but involving claims of different generality, as well as from Buridan’s final solution framed in terms of the requirement of a “virtual implication”. Thus, I will refer to “Bradwardine’s solution”, as the one involving the claim that propositions signifying their own falsity signify themselves to be true; I will also talk about Buridan’s early, “Bradwardinian solution” (allowing for the possibility that it was actually influenced by Bradwardine), as the one involving the different claim that all propositions signify their own truth; and I will finally talk about “Buridan’s final solution”, as the one framed in terms of a “virtual implication”, and involving the rejection of Buridan’s own “Bradwardinian solution”. Given these distinctions among these three solutions, I am going to argue for the following four theses. 1 Buridan was justified in abandoning his own “Bradwardinian solution” within his own logic, for in Buridan’s logic a crucial thesis of that solution cannot be expressed by a true sentence. 2 Bradwardine’s own solution could still be maintained in Buridan’s framework, at least for a certain class of cases of the paradox, provided it is sustainable in that framework at all. 3 The demands on Buridan’s final solution, requiring it to provide a theory of truth, coming from a different theoretical framework, are unjustified, given the theoretical role this solution plays in Buridan’s logic. 4 Buridan’s solution nevertheless fails, because it renders his theory inconsistent.

5.2

The Liar Paradox and Buridan’s Solutions

The Liar Paradox emerges for Buridan as a natural consequence of his conception of logical theory, treating logic as primarily a (practical) science of inferential relations among token-sentences of human languages 2 Indeed, I want to do so especially because I treated these solutions indistinctly elsewhere (where their distinction, however, was not relevant to my argument). See [3], esp. p. 103.

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(propositiones – propositions), whether spoken, written, or mental.3 Accordingly, the languages to which his theory applies are semantically closed: they contain semantic predicates and means of referring to items they contain. Therefore, in these languages, any proposition claiming its own falsity is well-formed, and given Buridan’s unrestricted endorsement of the principle of bivalence, must be either true or false. However, apparently, such a proposition would have to be both true and not true. For if it is true, then, given that it (truly) claims itself to be false, it is false. So, if it is true, then it is false; therefore it is false. On the other hand, if it is false, then things are the way it says they are; therefore, it is true. But then, if it is true, then it is false, and if it is false, then it is true, whence it is true if and only if it is false, which, given bivalence, leads to the explicit contradiction that it is true and it is not true. As has been discussed in a number of papers including Read’s,4 Buridan’s solution to the paradox accepts the proof of the falsity of Liarsentences, but blocks the reverse implication from their falsity to their truth. The fundamental point of the solution, namely, blocking the reverse implication, which Buridan shares with Bradwardine, Albert of Saxony and other medieval philosophers, is the claim that things being the way a Liar-sentence claims they are is not sufficient for its truth. So, given that its truth entails its falsity, it is false, but its falsity will not entail its truth, for even if things are the way it claims them to be (for it claims itself to be false and it is indeed false), this much is not sufficient for its truth. For its truth some further condition would have to be met, which the Liar-sentence fails to meet. That further condition in Bradwardine’s and Buridan’s early, “Bradwardinian” solution was formulated in terms of the signification of the Liar-sentence. Buridan, however, in his later works changed his mind about the viability of stating this further condition in terms of the signification of propositions, and formulated it with reference to the terms of a “virtually implied” proposition. This is a fundamental departure from both Bradwardine’s and Buridan’s “Bradwardinian” solution, which finds its explanation in Buridan’s nominalist theory of propositional signification. Therefore, to understand Buridan’s reasons, we first need to take a closer look at this theory.

3 Henceforth, I am going to use the term ‘proposition’ in this medieval sense, referring to sentence-tokens, whether spoken, written, or mental. For Buridan, inferential relations hold primarily among mental propositions, given his conception of language in general, according to which any semantic features of conventional spoken or written languages are derivative, and dependent on the primary, natural semantic features of the language of human thought. 4 See again the papers referred to in [1, 2], and the “classic” treatments provided by Spade, Hughes, Scott, Moody and Prior referred to in those papers.

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Buridan’s Theory of Propositional Signification

Buridan’s nominalist ontology is a world of individuals: individual substances and their individualized qualities and quantities.5 In this ontology, therefore, there is no place for another type of entities, say, facts, or states of affairs, or their late-medieval counterparts famously endorsed by Adam Wodeham and Gregory of Rimini, the so-called complexe significabilia, for propositions to signify.6 Buridan’s semantics maps all items of any language it concerns (spoken, written, or mental) ultimately onto this parsimonious ontology. But this ontology, since it encompasses all entities there are, includes also items of these languages: conventionally significative individual inscriptions and utterances, and naturally significative acts of thought (which are just certain naturally representing individualized qualities of thinking substances). Thus, in assigning semantic values to the items of these languages, one has to take into account not only how things other than items of a language are, but also how things that are items of the language under evaluation are. In dealing with the semantic evaluation of propositions, therefore, Buridan has to heed two demands of his nominalist metaphysics: 1. propositional signification can only be provided in terms of individuals permitted by his ontology, and 2. special care needs to be taken of those propositions whose semantic values depend not only on individuals that are other than items of the language under consideration, but also on individuals that are items of the language in question. For example, the proposition ‘No proposition is negative’, being itself a negative proposition, cannot be true in a situation in which it is actually formed. Still, it is an obviously possible scenario in which there are no negative propositions in the world. (Indeed, this was certainly the case before the first human being formed the first negative proposition in the history of the universe, assuming we are only talking about negative propositions formed by human beings and disregard the issue of non-human intelligences.) Therefore, this proposition is clearly true of that scenario, even if it cannot be true in that scenario. So, in evaluating this proposition (and especially its modal versions), Buridan clearly has to take into account the existence or non-existence of this proposition itself in the situation in which its truth-value is assigned. 5 I should also add “and their modes” but those need not detain us in this context. For more on this aspect of Buridan’s ontology, see [4, 5]. 6 See Buridan’s arguments against positing such quasi-entities, based primarily on the observation that they would not fit into any broad and jointly exhaustive ontological categories (for they cannot be substance or accident, or God or creature). See [6, 7].

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Given these theoretical demands, Buridan constructs a two-tiered semantics for propositions, namely, one that contains a “fine-grained mapping” from spoken and written propositions to mental propositions and a “coarse-grained mapping” from mental propositions (and by their mediation from spoken and written propositions) to things in the world, where the world itself contains also all items of the languages to which these propositions belong. The first mapping, from conventional spoken and written languages to mental language, maps token-sentences of conventional languages to corresponding mental propositions, where the corresponding mental propositions are those token-acts of singular minds that are compositionally dependent for their semantic values on the semantic values of those concepts that are signified in these minds by the syntactical parts of the conventionally signifying sentences.7 It must be noted here that this mapping is not one-to-one. In the case of synonymous sentences (say, in the case of strictly matching translations or sentences containing synonymous terms), it is many-to-one.8 Still, this mapping is sufficiently fine-grained to provide the semantic distinctions one needs to make, especially in intentional contexts [9]. But when the mental propositions concern things other than items in a language, they cannot further be mapped onto some sort of propositional entities, given the demands of Buridan’s nominalist ontology. So, the second mapping has to be “coarser”: a mental proposition concerning things in the world can only signify things that are signified by its categorematic terms (the terms flanking its copula), whence even non-synonymous propositions that share the same terms will end up signifying the same things. It is for this reason that Buridan explicitly draws a number of apparently rather counterintuitive conclusions concerning the extra-mental (ad extra) signification of written and spoken propositions, as opposed to their intra-mental (apud mentem) signification. For example, a result of this conception is that although the written propositions ‘God is God’ and ‘God is not God’ signify different (indeed, contradictory) mental propositions, they signify the same ad extra, namely, what their categorematic terms signify, i.e., God. But this result is counterintuitive only 7 For the issue of compositionality in the mental-language tradition in general, see the excellent historical survey provided by C. Panaccio, Le discours intérieur de Platon à Guillaume d’Ockham, Éditions du Seuil, Paris, 1999. For Buridan’s conception in particular, see my Introduction to Buridan’s Summulae, esp. [6], SD, pp. xxxvii–xliii. 8 One would think that, correspondingly, in the case of ambiguous sentences the mapping should be one-to-many. However, in his Questiones Elencorum, Buridan argues that ambiguous sentences need not be distinguished, for they express their different senses disjunctively. So, apparently, an ambiguous written or spoken proposition would then be mapped onto a single disjunctive mental proposition. But Buridan seems to have abandoned this strong position in his later works. See [8].

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if extra-mental signification is thought to determine truth-conditions; for example, under the assumption that the truth of a proposition consists in the actual extra-mental existence of its significatum. But, as we shall see, for Buridan their signification has no role in determining the truth of propositions (it is rather determined by the supposition of their terms), while their synonymy-relations are adequately accounted for even in accordance with his parsimonious ontology. For the contradictory written and spoken propositions, although they signify the same thing ad extra, are not synonymous, given that they signify distinct mental propositions apud mentem. And the mental propositions, even if they also signify the same thing, are not synonymous either, for they signify the same thing, but not in the same way, on account of their different compositional structure (the one being negative and the other affirmative; see [6], SD, pp. 10–14, 232–234, 825–826, 841–843). So the extramental significata of propositions can be identified without trouble with the significata of their categorematic terms, without any need for specific, extra-mental propositional significata, which Buridan, therefore, happily eliminates from his ontology.

5.4

The Semantics of Sentential Nominalizations

In accordance with this conception, then, sentential nominalizations, such as “that-clauses” or infinitive constructions, which by supporters of special propositional entities, i.e., dicta, enuntiabilia, real propositions or complexe significabilia, were taken to name what the corresponding propositions signify, cannot have this function in Buridan’s semantics. Instead, Buridan explains their function with reference to his semantic theory of categorematic terms.9 Categorematic terms are terms that can be the subject or predicate of a syntactically well-formed proposition, i.e., terms that can suitably flank the copula of a proposition.10 Propositional nominalizations can obviously do so (as in ‘That a man walks is possible’ or ‘For a man to walk is possible’).11 Therefore, Buridan is clearly entitled to his move of treating these as complex common terms with the same type of semantic

9 The best monographic survey of the history of medieval theories of propositional signification is still in [10]. The best source materials for early medieval theories can be found in [11], where one can find elaborate theories of the referring function of sentential nominalizations, called appellationes dicti. 10 There is more to the distinction, but the details need not detain us here. For more, see my article in [12]. Buridan’s discussion of the discussion can be found in [6], SD, pp. 232–234. 11 The corresponding constructions in Latin are actually more natural. In English, the corresponding ‘It is possible that a man walks’ or ‘It is possible for a man to walk’ are “smoother”, but syntactically more complicated.

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functions that ordinary complex common terms (such as ‘wise man’ or ‘braying donkey’) have. The basic semantic functions of such common terms are signification (roughly, meaning) and supposition (roughly, reference). Common terms of spoken and written languages immediately signify in the mind common concepts, i.e., individualized, naturally representative qualities of the mind, which in turn naturally signify or represent individuals of the same kind. The common terms of spoken and written languages, therefore, ultimately signify the individuals naturally represented by the concepts they immediately signify. So, signification is a property of a spoken or written term that renders it a meaningful utterance or inscription, as opposed to some meaningless noise or scribble. This is the property that makes an utterance or inscription part of a spoken or written human language. But in their actual use in that language, these terms take on another property, namely, supposition, or reference.12 Buridan’s theory of supposition is designed to describe the various ways in which terms are used to refer to or stand for (supponere pro) various things in different propositional contexts. The primary division of the kinds of supposition spoken or written terms can have is that between personal and material supposition. A term in personal supposition is used to stand for individuals it ultimately signifies. A term in material supposition is used to stand not for its ultimate, but for its immediate significata, the concepts it signifies in individual human minds, or for token terms of the same type, including itself.13 For example, in the proposition ‘Man is an animal’, insofar as this proposition is true, both terms are taken to stand in personal supposition, i.e., for individual humans and individual animals, respectively, and what renders the proposition true is the identity of some of the individuals referred to (or, using the coinage by now standard in the secondary literature, supposited for ) by both terms. By contrast, in ‘Man is a species’, insofar as this proposition is true, the term ‘man’ obviously cannot be taken to supposit for its ultimate significata, namely, individual humans, but it can be taken to stand for the specific concept of humans in this or that individual human mind (i.e., the individual acts of these minds that represent human beings indifferently, in abstraction from their individual

12 I will deal here only with Buridan’s theory. For a brief survey of the varieties of the theory, as well as references to the vast secondary literature, see [13]. 13 Medieval authors commonly distinguished personal, simple, and material supposition, reserving simple supposition for the case where the spoken or written term is used to refer to the concept to which it is subordinated (or the simple, common nature grasped by that concept). But Buridan simply lumps together all “non-significative” uses of terms under the heading of material supposition, i.e., uses, when the term is not taken to stand for its (ultimate) significata. Cf. [6], SD, tr. 4, c. 3, Sect. 2, especially, p. 253.

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differences, but as being specifically distinct from other animals), and for token utterances and inscriptions that signify these concepts in these minds, including itself. But then, clearly, if ‘man’ is taken in material supposition in ‘Man is a species’ (and ‘species’ is taken in personal supposition, for its ultimate significata), then this proposition is true on account of the co-supposition of its terms, for at least some (indeed, all) of the material supposita of its subject are identical with some of the (personal) supposita of its predicate. Now, applying this doctrine to propositional nominalizations, Buridan claims that these can also be taken either materially or personally. Taken materially, they have the function of standing for the corresponding token-propositions, whether written, spoken, or mental. Taken personally, however, they stand for those significata of the corresponding propositions of which the terms of these propositions are co-verified, i.e., for which these propositions are true.14 For example, in the proposition ‘For Socrates to love God is good’, the subject term, taken personally, supposits for what the terms of the corresponding proposition ‘Socrates loves God’ co-supposit. Thus, if Socrates does in fact love God, then the terms of this proposition co-supposit for him, namely, Socrates loving God, and so the subject of the original proposition supposits for the same. On the other hand, if Socrates does not in fact love God, then the terms of the proposition ‘Socrates loves God’, i.e., ‘Socrates is a lover of God’, do not co-supposit, and so the corresponding sentential nominalization supposits for nothing, and then the original proposition is false.15

5.5

Buridan’s Rejection of His Own Bradwardinian Solution

After these preliminaries, we are in a better position to appreciate Buridan’s reasons for rejecting his own earlier, “Bradwardinian” solution to the Liar Paradox, provided in terms of propositional signification.

14 I am restricting this discussion now to present tense affirmative propositions, as Buridan does in his corresponding remarks in the Sophismata. Whether and how this account could be generalized to provide a full-fledged Buridanian theory of propositional signification and sentential nominalizations is a further issue that is not directly relevant to our present concern with Buridan’s treatment of insolubilia. 15 Possible intuitions to the contrary, according to which the proposition ‘For Socrates to love God is good’ is true even if Socrates actually does not love God, might be accounted for by saying that these intuitions are based on the consideration that it would be good for Socrates to love God even if he does not; in this case, however, the proposition to be considered would have to be ‘For Socrates to love God would be good’, when not the actual, but possible cosupposition of terms is required for truth, because of the ampliative force of the subjunctive copula. For a reconstruction of Buridan’s theory of ampliation, see [14].

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In a crucial passage in his Sophismata, discussing the problem-sentence (sophisma) ‘Every proposition is false’, positing the case that all true propositions are eliminated (see [6], SD, Sophismata, Chap. 8, 7th sophism, pp. 965–971). Buridan first briefly recapitulates his earlier solution as follows: For some people have said, and so it seemed to me elsewhere [15], that although this proposition does not signify or assert anything according to the signification of its terms other than that every proposition is false, nevertheless, every proposition by its form signifies or asserts itself to be true. Therefore, every proposition asserting itself to be false, either directly or implicitly, is false, for although things are as it signifies, insofar as it signifies itself to be false, nevertheless, things are not as it signifies insofar as it signifies itself to be true. Therefore, it is false and not true, since for its truth it is required not only that things be as it signifies but also that they be in whatever way it signifies [them to be]. But this response does not seem to me to be valid, in the strict sense (see [6], SD, p. 968).

The solution is clear enough, and seems to be pretty much in line with Bradwardine’s solution, as presented by Read. But it is important to note here that Buridan employs in this “Bradwardinian” solution the crucial thesis that every proposition signifies itself to be true, which Bradwardine’s original solution restricts to propositions that signify themselves to be false. The importance of this point is that since Buridan bases his rejection of this “Bradwardinian” solution on the rejection of his own unrestricted claim, the argument he employs for this rejection may not affect Bradwardine’s solution.16 The argument is presented in the following passage: [. . . ] I [am going to] show that it is not true that every proposition signifies or asserts itself to be true. For you take the expression ‘itself to be true’ either materially or significatively. If materially, then the proposition ‘A man is an animal’ does not signify or assert itself to be true, for then the sense [of your claim] would be that it would signify the proposition “The proposition ‘A man is an animal’ is true,” and this is false, for this second proposition is already of second intentions, and the first, since it was purely of first intentions, did not signify second intentions.17 But if you say that ‘itself to be true’ is taken significatively, 16 This is because the rejection of a more universal claim does not in and of itself entail the rejection of a more restricted, less universal claim. For example, rejecting the claim ‘All intelligent beings are material beings’ does not commit one to rejecting the claim ‘All human beings are material beings’, even if one accepts that all human beings are intelligent beings and not vice versa. But then, of course, it may turn out that the reason for rejecting the more universal claim is also compelling against the less universal one, but that is a separate question. 17 Second intentions are concepts by means of which we conceive of concepts (or other signs) insofar as they are concepts (or signs). For example, the concept to which the term ‘species’

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then the proposition ‘A man is a donkey’ does not signify itself to be true, for just as that a man is a donkey is nothing, because a man cannot be a donkey, so also that the proposition ‘A man is a donkey’ is true is nothing, nor can it be anything, for it [namely, the proposition ‘A man is a donkey’] cannot be true.18 But it is not true to say of that which is nothing, nor can be anything, that it is signified or understood or asserted, as was sufficiently discussed elsewhere.19 For if you say that that the proposition ‘A man is a donkey’ is true is signified or asserted or understood, then you say something false, for this proposition is affirmative and its subject supposits for nothing.20 And the case is similar here, for the proposition ‘Every proposition is false’ cannot be true; therefore, that it is true is not, nor can it possibly be; hence, it is neither signified nor understood, and so it does not signify itself to be true.21

The point of the argument is that the fundamental claim of Buridan’s “Bradwardinian” solution, namely, that every proposition signifies itself to be true, cannot be true. For if we analyze this claim, we can see that whether we take the sentential nominalization, i.e., the infinitive construction, in it in material or in personal supposition, the universal claim cannot be true. To see this in more detail, consider the universal proposition ‘Every proposition signifies itself to be true’. From this, by eliminating the infinitive construction in favor of the more transparent corresponding “that-clause”, we get ‘Every proposition signifies that it is true’, where ‘it’ is ranging over token-propositions (written, spoken, or mental). Now consider the sentential nominalization in this sentence: ‘that it is true’. According to Buridan’s theory, this can be taken either materially or personally. Taken materially, it is a common term suppositing for propositions of the form ‘it is true’, in which ‘it’ refers to some proposition. is subordinated is a second intention. First intentions are concepts by means of which we conceive of things other than concepts (or other signs), or perhaps concepts, but not insofar as they are concepts (or signs). Such is, e.g., the concept to which the term ‘man’ is subordinated, by which we conceive of human beings, who are not concepts or the concept to which the term ‘being’ is subordinated, by which we conceive of both things that are not concepts and things that are concepts; however, by this concept we conceive of the latter not insofar as they are concepts but insofar as they are entities, regardless of their representative function. See [16]. 18 This is because the sentential nominalization “that the proposition ‘A man is a donkey’ is true” should refer to things of which the terms of the corresponding proposition, namely, “The proposition ‘A man is a donkey’ is true”, are jointly true. But the subject of this proposition refers to any proposition of the form ‘A man is a donkey’, which is necessarily false; therefore, the predicate ‘true’ cannot be true of any of these, whence the two terms cannot be jointly true of anything, and so the corresponding nominalization can refer to nothing. 19 Sophismata, c. 1, Fourth sophism, conclusion 5. 20 Namely, “that the proposition ‘A man is a donkey’ is true,” which is the subject of the proposition “That the proposition ‘A man is a donkey’ is true is signified”, supposits for nothing. 21 The notes referenced inside this passage come from my translation of Buridan’s Summulae. See [6], SD, pp. 968–969.

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However, in that case an instance of the original universal proposition would be ‘The proposition ‘a man is a donkey’ signifies the proposition ‘the proposition ‘a man is a donkey’ is true”. But any proposition of the form ‘a man is a donkey’ signifies men and donkeys, and not propositions. Therefore, this instance of the universal proposition is false, and so the universal proposition is false. Indeed, it would be false for any proposition whose terms are terms of first intention, as opposed to terms of second intention, just as Buridan claims. For if ‘S’ and ‘P’ can be replaced by terms of first intention, then a proposition of the form ‘S is P’ signifies all the things signified by ‘S’ and all the things signified by ‘P’. But since ‘S’ and ‘P’ are terms of first intention, their significata are things that are not items of any language, and so they are things that are not propositions, whence they cannot be true or false. Accordingly, ‘The proposition ‘S is P’ signifies the proposition ‘the proposition ‘S is P’ is true” will always be false for all such terms, since ‘S is P’ will never signify any proposition, let alone a proposition of the form ‘the proposition ‘S is P’ is true”. On the other hand, if we take the “that-clause” in personal supposition, then it would have to supposit for everything of which the terms of the corresponding proposition are jointly true. But in this case, an instance of the universal proposition would be ‘The proposition ‘a man is a donkey’ signifies everything that is both the proposition ‘a man is a donkey’ and is true’. But since any proposition of the form ‘a man is a donkey’ is impossible, nothing can be both a proposition of this form and true. So, the original universal proposition is false on this interpretation as well. Therefore, given Buridan’s own theory of propositional signification and sentential nominalizations (or rather, the few principles he lays down of a would-be theory), he is compelled to reject his own “Bradwardinian” solution, given the fact that he has to reject the universal proposition that every proposition signifies itself to be true, which is the foundation of this solution. And this was the point of the first thesis that I proposed to argue for in the first section. However, since Bradwardine’s own solution does not rest on this universal claim, the truth of the second thesis is still an open question.

5.6

Bradwardine’s Solution in the Buridanian Framework

Indeed, it is easy to see that Bradwardine’s more restricted thesis, according to which every proposition signifying itself to be false signifies itself to be true could be maintained even on Buridan’s theory of propositional signification and sentential nominalizations, if those sentential nominalizations are taken materially, according to Buridan’s own rules.

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Consider again a more transparent version of the thesis, using “thatclauses”: ‘Every proposition signifying that it is false signifies that it is true’, where the pronouns refer to some proposition, written, spoken, or mental. Let such a proposition be ‘C is false’, and let ‘C’ be the name of this proposition. On Buridan’s rules for propositional signification, this proposition signifies everything its terms signify; so it signifies C and it signifies all false propositions. (Indeed, since on Buridan’s solution C is false, the terms of this proposition co-supposit for C.) Therefore, Bradwardine’s thesis has the following instance: ‘C signifying that C is false signifies that C is true’. In this proposition, the string ‘that C is false’, taken materially, stands for all propositions equiform to this: ‘C is false’. Such a proposition, according to Buridan’s rules, signifies C and all other false propositions. The string ‘that C is true’, again, taken materially, stands for any proposition of the form ‘C is true’. But then, Buridan’s objection to taking his own, unrestricted universal claim with its “thatclause” in material supposition does not apply here. For the point of that objection was that the claim would not be true for any proposition with terms of first intention. But Bradwardine’s restricted claim only concerns propositions with terms of second intention, i.e., terms that signify propositions. Indeed, if we substitute token-propositions referred to by the “that-clauses” in Bradwardine’s thesis as stated above, we get: ‘C signifying ‘C is false’ signifies ‘C is true”. This, given that C does signify the original token equiform to ‘C is false’ in this paragraph, reduces to ‘C signifies ‘C is true”. But then, since the predicate term of C signifies all false propositions and C is not true, the sentence ‘C signifies ‘C is true” is true on Buridan’s principles. Thus, apparently, on Buridan’s principles we can find no falsifying instance to Bradwardine’s original claim, at least among versions of the paradox formed with terms of second intention. Therefore, Buridan could have kept it, if he had wanted to use it, at least for these cases. And of course this was the point of the first half of my second thesis in the first section. On the other hand, it has to be noted that other versions of the Liarparadox, involving terms of first intention could still not be said to signify their own truth. For when I say ‘I am saying something false’, i.e., ‘I am someone saying something false’, then the terms of my proposition supposit for me and signify me and everybody saying something false. But none of these things is a proposition, so none of these things can be supposited for by the relevant sentential nominalization taken in material supposition, standing for propositions. Therefore, Bradwardine’s restricted claim could not have been maintained as universally true in Buridan’s framework.

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Further complications would arise from assuming, as is plausible to assume, that ‘What I am saying is false’ and ‘I am saying something false’ are equivalent, at least ut nunc. Indeed, since any proposition is formed by someone, any proposition that can be referred to directly by means of terms of second intention can also be referred to indirectly, referring to the person forming it, by means of terms of first intention. But then, for any proposition claiming itself to be false apparently there should be an equivalent proposition claiming that the person forming it forms something false. And so in those cases Buridan’s objection would apply again. Therefore, if we maintain the equivalence of Liar-sentences with first intention-terms with those of second-intention terms, Bradwardine’s solution may not hold up in Buridan’s framework at all. And this was the point of the second half of the second thesis of the first section. In any case, Buridan clearly did not maintain Bradwardine’s thesis in his final solution, even if he could have done so at least for cases involving only terms of second intention. In fact, it is quite possible that Buridan was simply not directly influenced by Bradwardine, and he did not consider Bradwardine’s more restricted thesis at all. Or he may have considered it, but thought that it entailed the more general thesis.22 However, even aside from these subtle considerations, given Bradwardine’s strong “entailment principle” concerning signification, according to which a proposition signifies whatever it entails simpliciter or ut nunc, one might argue that any proposition trivially signifies (what is signified by) any other proposition. For the proposition p with the assumption that q, trivially entails q, ut nunc. But I do not want to pursue this idea here. Or, as it seems more likely to me, he just found the universal claim that all propositions signify their own truth intuitively clear on the basis of the meanings of the words involved (as he explicitly states on several occasions), and realized only later its untenability within his own theory of propositional signification and sentential nominalization. Indeed, since Buridan abandoned the idea of a direct link between propositional signification and truth altogether as well as the idea of a direct link between truth and logical validity, he did not have to feel any pressing theoretical need to pursue the ideas involved in Bradwardine’s solution, even if he considered it in any detail at all.

22 At least, he may have thought that his more general thesis was entailed by Bradwardine’s thesis and Bradwardine’s other, explicit or implicit postulates, or some other intuitive principles, as did Paul Spade. Spade’s recent response to Read’s criticism of his argument against Bradwardine can be found in [17].

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Truth Without Complexe Significabilia

In question 9 of his question-commentary on book 6 of Aristotle’s Metaphysics, Buridan raised the question whether every proposition is true because the thing/s signified by it is/are all the ways it signifies it/them to be.23 In typical scholastic fashion, after arguing against the affirmative answer, he provides the main motivation for it in the following passage: Many people commonly hold the opposite based on the authority of Aristotle, who in the Categories says that a proposition is true or false because the thing [signified by it] exists or does not exist. [. . . ] And truth is also commonly described in this way, namely, that it is the adequation or conformity of the understanding and the things understood. But this sort of adequation or conformity cannot obtain except because things are in this way; therefore, etc.

Buridan never really “bought into” the conception described here, even if he never abandoned this manner of speaking either. In any case, the formula “a proposition is true or false because the thing [signified by it] exists or does not exist” expresses a semantic conception radically different from his own; indeed, a radically different way of constructing logical semantic theory. Therefore, Buridan could only keep it by filling it with radically different content, making it eventually in principle entirely eliminable. The sort of logical semantics required by the original conception, even if it may never have been spelled out in this way in the Middle Ages, should first provide the significations of simple terms, both categorematic and syncategorematic, then a compositional semantics for the significata of complex terms and propositions based on the significations of simple terms, specifying the rules of how the actuality of the significata of the complex expressions depends on the actuality or non-actuality of the significata of their components (for example, a simple rule could specify that if the significatum of a proposition is actual, then the significatum of its negation is non-actual, or that for the actuality of the significatum of a conjunction the actuality of the significata of all of its members is required, etc.), and then it could provide a simple criterion for truth for all kinds of propositions in terms of the actuality of their significata, just as Aristotle’s formula requires. Finally, with this criterion of truth in hand, logical validity could be defined as truth for all possible interpretations, i.e., for all possible assignments of significata as specified by these rules. 23 QM, lb. 6, q. 10: Utrum omnis propositio ex eo est vera quia qualitercumque significat ita est in re significata vel in rebus significatis.

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Buridan’s conception is radically different. In the first place, he does not have rules to specify the unique, extramental significata of whole propositions as a function of the semantic values of their components. In fact, as we could see, he denies that propositions extramentally signify anything as a whole, over and above what their categorematic terms signify. But then, extramental propositional signification as Buridan conceives of it is unable to distinguish even contradictories, so it obviously cannot serve for specifying their truth-conditions. Therefore, truthconditions are to be specified in terms of the supposition of their terms, and hence also their signification, presupposed by their different modes of supposition in different contexts, as well as the signification of syncategorematic terms, providing the formal structure of different types of proposition (affirmative, negative, universal, particular, indefinite, pasttense, future-tense, modal, categorical, hypothetical, etc.). And so, since the truth conditions of these different types of propositions have to be specified differently for each type, the Aristotelian formula can at best serve as an abbreviation, a quick reference to the specification of these different types of truth-conditions. In fact, this is precisely how Buridan proceeds in his most mature treatment of the issues of truth and validity, in his Sophismata. In the first place, he declares that (on the basis of his theory of propositional signification), propositional signification cannot provide a criterion of truth: . . . every true affirmative proposition about actuality [de inesse] and about the present [de praesenti] is not true on the ground that whatever and howsoever it signifies as being, so it is, for [. . . ] whatever and howsoever is signified as being by the two propositions ‘A man is a man’ and ‘A donkey is a donkey,’ that also is signified as being in the same way by the proposition ‘A man is a donkey’, as is clear from what has been said. But the latter is false, and the former two were true. And thus, it seems to me that in assigning the causes of truth or falsity of propositions it is not sufficient to deal with significations, but we have also to take into account the suppositions concerned (see [6], SD, p. 854).

Buridan then proceeds in his subsequent “conclusions” (conclusions 9–14; see [6], SD, pp. 854–859.) to specify the truth-conditions of various types of propositions in terms of the supposition of their terms in the various types of contexts provided by the syncategorematic terms of these propositions. Significantly, however, after recapitulating these truth-conditions at the end of this discussion, he adds the following remark: But in the end we should note – since we can use names by convention [ad placitum], and many people commonly use this way of putting the

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matter – that with respect to every true proposition we say: ‘It is so’, and with respect to every false one we say: ‘It is not so’, and I do not intend to eliminate this way of speaking. But for the sake of brevity I may use it often intending by it not what it signifies on account of its primary imposition, but the diverse causes of truth and falsity assigned above for diverse propositions, as has been said (see [6], SD, p. 859).

So, for Buridan, the Aristotelian formula apparently becomes just a moniker, an inappropriate expression serving to remind us what he really means when he uses it.

5.8

Logic Without Truth

But, as it turns out in Buridan’s subsequent discussion, the formula with its changed, Buridanian meaning has a deeper significance for Buridan’s logic. For while in the context of c. 2 of the Sophismata it may appear that Buridan is after all providing the clauses of a complex definition of truth (which he will then just inappropriately indicate by means of the Aristotelian formula) in order to use it for the definition of logical validity, in the context of his discussion of logical validity, in c. 8, he argues that validity cannot properly be defined in terms of truth. The gist of the argument (which, quite importantly, he also uses in his systematic treatise on consequences) [18]. I provide a detailed discussion of Buridan’s argument in [3, 19]. is that an obviously invalid consequence with a self-falsifying antecedent would on a definition of validity in terms of truth turn out to be trivially valid, whence such a definition cannot be correct. For example, take the consequence: ‘No proposition is negative; therefore there is a stick in the corner’.24 This consequence is obviously invalid, for it is a quite possible situation in which there are no negative propositions and no stick in the corner either, as was certainly actually the case before the first negative proposition was formed by a human being (and when that stick – probably Buridan’s walking stick left in the corner of his classroom – did not yet exist). But on the proposed definition of validity, according to which a consequence is valid if and only if it is impossible for its antecedent to be true and its consequent not to be true when they are both formed together, this consequence would have to be valid, since the antecedent, being a negative proposition, always falsifies itself whenever it is formed; thus it cannot be true, and so it is

24 Buridan’s example with the stick had ‘no proposition is affirmative’ as its antecedent, and the reason why that consequence has to be deemed valid on the proposed definition is that the affirmative consequent, formed together with the antecedent, always falsifies the antecedent. But this version, presenting a consequence with a self-falsifying antecedent, which will also be featured in the next example, will better serve our present purposes.

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indeed trivially impossible for it to be true while the consequent is not true.25 Therefore, Buridan proposes a different definition of validity, not in terms of truth, but in terms of the Aristotelian formula, as he interpreted it in c. 2. As he writes: The fifth conclusion is that for the validity of a consequence it does not suffice for it to be impossible for the antecedent to be true without the consequent if they are formed together, as has been correctly argued above about the stick in the corner. And this is also obvious from another example, for this is not valid: ‘No proposition is negative; therefore, no proposition is affirmative’. And this is clear because the opposite of the consequent does not entail the opposite of the antecedent. Yet, the first cannot be true without the truth of the second, for it cannot be true. Therefore, something more is required, namely, that things cannot be as the antecedent signifies without being as the consequent signifies. But in connection with this it has been determined that this is not the proper expression of the point, but we use it in the sense given there, for we cannot generally express in a single expression covering all true propositions a reason why they are true, nor concerning all false propositions a reason why they are false, as has been said elsewhere (see [6], SD, pp. 955–996).

So, as it turns out, Buridan’s logic as such has simply no use for a theory of truth. What it really needs is just the set of “correspondenceconditions” briefly indicated by the Aristotelian formula. Indeed, as this argument shows, the notion of truth is not only unnecessary, but it leads to paradoxical results if used in the definition of validity; therefore it had better be abandoned in considerations concerning the validity of inferences. But why does this situation arise, and what does Buridan gain by this further move? The situation obviously arises from the semantic closure of the languages for which Buridan devises his theory. Under conditions of semantic closure, self-falsifying propositions can naturally occur. But in their case we have examples of propositions that cannot be true, despite the fact that they describe situations that are obviously possible, or using 25 In fact, Buridan might have come up with a further, unrelated reason to reject the definition of validity in terms of truth. For as he sees it, truth is a property of propositions; but the clauses of a consequence are not propositions. So, one could not strictly speaking talk about the truth or falsity of the antecedent and the consequent, but at most about the truth or falsity of equiform proposition tokens formed in all possible situations in which their truth values need to be checked to check the validity of the consequence formed in the actual situation. But Buridan obviously does not want to go into these complications, and allows the improper way of talking about the clauses of a consequence as propositions. However, strictly speaking, with a definition of validity based on truth, he would have to consider the existence of equiform propositions in possible situations, and not just the clauses of the consequence formed in the actual situation. For more on this issue, see [19].

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the Aristotelian formula, things can be the way they signify them to be, even if they can never be true. So such propositions provide the primary examples of the possibility of a divergence between correspondence and truth under the conditions of semantic closure: they can obviously correspond to a possible situation, in which, however, they cannot be true, for if they are formed in that situation, then their existence immediately falsifies them in the same situation. What Buridan gains, therefore, by returning to the (re-interpreted) Aristotelian formula is a way of expressing the satisfaction of the correspondence conditions of a proposition in a given situation, independently from its truth, indeed, independently from its existence in that situation. This is most obvious in Buridan’s discussion leading to his final definition of logical validity. The issue is whether the consequence ‘No proposition is negative; therefore, some proposition is negative’ is valid (or as Buridan says, ‘true’, but he makes clear that he means the same by a ‘true’ consequence and by a ‘valid’ or even a ‘good’ consequence). Buridan here directly argues against even his improved definition of validity, provided in terms of the (re-interpreted) Aristotelian formula: Again, it is not possible for things to be as the first [proposition, i.e., the antecedent] signifies without their being as the second [the consequent] signifies; therefore, the consequence is valid. The consequence seems to be manifest from what we said a valid consequence was in the previous sophism, and you cannot otherwise express the reason why a consequence is said to be valid. But I prove the antecedent: for it follows that if things are as it signifies, then it signifies; and it follows that if it signifies, then it is; and, if it is, then things are as is signified by the second (see [6], SD, pp. 956–967).

In his reply to this objection, Buridan draws a very important distinction between two possible ways of understanding his improved definition of validity: To the second, which seems to be troublesome, I reply that a consequence is never true or false unless it is; and thus the validity or truth of a consequence requires that its antecedent and consequent exist. And then, with this assumption, we give the rule that a consequence is valid if it is impossible for things to be as the antecedent signifies without their being as the consequent signifies. And this rule can be understood in two ways: first, that it is one proposition about impossibility in the composite sense, in the way that this is commonly used, and its sense then is that this is impossible: ‘When it is formed, things are as the antecedent signifies and not as the consequent signifies’. And taken in this way the rule is not valid, for according to this rule it follows that the sophism is true. And it is according to this false rule that the argument proceeded. Taken in the other way, the rule is understood as a proposition about impossibility in the divided sense, so that its sense is: a consequence is valid if in whatever way the antecedent signifies

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So, the final definition of validity understood in the divided sense provides a clear criterion for judging the validity of a consequence, regardless of the existence of the antecedent and consequent in the possible situations in which the satisfaction of their correspondence conditions needs to be checked in order to determine the validity of the consequence in which they actually occur. Thus, by means of the re-interpreted Aristotelian formula, as summarizing the correspondence conditions of propositions Buridan laid out in terms of the supposition of their terms, he finds a way of identifying a “possible state of affairs”, the way things are as signified by a proposition in a possible situation regardless of whether the proposition in question exists in that situation. Yet, spelling out “the ways things are” signified by a proposition in terms of the conditions concerning the supposition of its terms, he can do so without reifying that “state of affairs” in the form of some ontologically suspect entity, a complexe significabile, distinct from the ordinary things admitted in his nominalist ontology. But then, understanding the issue of validity in this way, as definable without any reference to the truth-values of the antecedent and consequent which they can only have in those situations in which they exist, Buridan has a logic without truth, a logical theory that works for determining the validity of inferences, and yet one that can do so without checking the truth-values of propositions in any situation. Thus, Buridan’s logic does not have and does not need a definition of truth.

5.9

Correspondence Without Truth and Truth Without Paradox

The only thing Buridan’s logic needs to do with truth is to eliminate the Liar-type puzzles that are bound to crop up under the conditions of semantic closure. But this is exactly what Buridan does in the remainder of c. 8 of the Sophismata, already in possession of the logical devices he needs for doing so, in particular the logical devices needed to handle the above-mentioned possibility of divergence between correspondence and truth.

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As we could see in connection with ‘No proposition is negative’, under the conditions of semantic closure it is quite possible that the correspondence conditions of a proposition are satisfied in a possible situation, even if the proposition cannot be true in that situation, for its very existence in that situation would falsify it. In the case of Liar-type propositions, the situation is quite similar. Given the fact that they are false, their correspondence conditions are satisfied. But since the satisfaction of their correspondence conditions means precisely that they fall under the term ‘false’, given bivalence, they cannot be true. However, Buridan has already shown that the satisfaction of correspondence conditions need not be sufficient for the truth of a proposition. In the case of ‘No proposition is negative’, the existence of the proposition in a possible situation would falsify it in that situation, although, if it does not exist in that situation, its correspondence-conditions may be satisfied in the same situation. In the case of a Liar-type proposition, the existence of the proposition in the actual situation is assumed, and the problem is assigning its truth-value in that situation. Since the assumption of its truth entails its falsity, i.e., given bivalence, it entails its own contradictory, it cannot be true. But that is precisely what it says. So, its correspondence conditions are satisfied: its subject supposits for the proposition itself, which falls under the term ‘false’; hence, its terms cosupposit. But given the possibility of divergence between the satisfaction of correspondence conditions and truth, it should come as no surprise in this context that the proposition is not true, despite the satisfaction of its correspondence-conditions. Therefore, Buridan merely has to specify that further condition the failure of which prevents the proposition from being called ‘true’, i.e., he has to specify what would constitute the sufficient conditions for a proposition to be called true. He finds this further condition in the trivial “virtual entailment principle”: any proposition “virtually” entails another proposition that claims the original proposition to be true (where the point of “virtuality” seems to be that the relevant consequence need not actually be formed). With this principle at hand, Buridan can now claim an easy victory over the paradox. The Liar-sentence is simply false, for despite the fact that it corresponds to the actual, real situation (namely, to the situation that it is false), its correspondence to that real situation need not entail that it is true. Indeed, that correspondence is insufficient for its truth, for it fails to meet another, trivially required condition, namely, the correspondence of the virtually implied proposition to the same situation. This further, trivial requirement is no more ad hoc than the general, trivial requirement that a proposition can only be true if all proposi-

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tions it validly entails are true as well, as required by modus ponens. And this trivial requirement will not render Buridan’s “theory of truth” nonsensical, for as I claimed above, he does not have a theory of truth, and does not need one. As far as checking validity is concerned, all his logic needs is checking whether the correspondence conditions laid out in c. 2 of the Sophismata that satisfy the antecedent in any possible situation will also satisfy the consequent in the same situation. For this, he will only have to invoke the supposition of terms in those situations, of course, occasionally, the supposition of the terms ‘true’ and ‘false’ as well. But upon seeing that the terms of an affirmative proposition can co-supposit in a possible situation without placing the proposition itself among the supposita of the term ‘false’, he can be sure that the proposition in that situation is true, provided it exists in that situation. On the other hand, if the co-supposition of its terms places the proposition itself among the supposita of the term ‘false’, Buridan can be sure that the virtually implied proposition cannot be true, and hence the original proposition cannot be true either. This procedure is entirely effective, without any circularity, i.e., without requiring us to see first whether the proposition is true so we can know whether it is true.26 But then, if the paradox is effectively dispelled without any need for a general theory of truth, Buridan can apparently rest satisfied. He did all that he could reasonably be asked to do with his logic. And this was the point of the third thesis of the first section.

5.10

The Failure of Buridan’s Solution

At any rate, these are the things one can say in defense of Buridan’s solution against the charges of adhockery, circularity, and in general, its failure to provide a theory of truth. Nevertheless, this is not to say that Buridan’s approach is immune to all criticism (unless one is truly “infatuated”). In fact, I will now argue that the solution cannot work, because it renders Buridan’s theory inconsistent. As we could see, a fundamental claim of the solution is that every proposition virtually implies another proposition claiming that the original proposition is true. If the terms of the implied proposition do not co-supposit in a given situation, I will say that the “virtual implication condition” (VIC) of the original proposition is not satisfied in that situation. Another fundamental claim of the solution is that an affirmative Liar-sentence is false, and so, since its subject refers to the proposition itself and its predicate is the term ‘false’, its terms co-supposit. In general,

26 For

this charge, see especially [1], p. 201.

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I will say that when the terms of an affirmative proposition co-supposit (and, correspondingly, if the terms of a negative proposition do not cosupposit), then its “co-supposition condition” (CSC) is satisfied. Next, we should recall that Buridan defined the validity of a consequence in terms of howsoever the antecedent and the consequent signify things to be, and he reminded us that this “Aristotelian formula” should be understood as an abbreviation of the “conclusions” he gave us in c. 2 of the Sophismata. In discussing the issue of validity, I somewhat loosely referred to the satisfaction of the conditions specified by those “conclusions” as the satisfaction of the “correspondence-conditions” of the relevant kinds of proposition. But now we should more specifically ask whether those “correspondence-conditions” include both the VIC and the CSC or only the latter (other possibilities being naturally excluded). If only the latter, then, despite Buridan’s claim, the virtual implication of a Liar-sentence cannot be valid by his own criterion of validity. If both, then, despite Buridan’s claim, the consequence ‘No proposition is negative; therefore, some proposition is negative’ will turn out to be valid. So, either way, Buridan cannot maintain all his claims together; his theory is inconsistent. To see this in more detail, consider first the Liar-sentence: (A) (A) is false This, allegedly, virtually implies a sentence claiming (A) to be true: (B) (A) is true Suppose the “correspondence-conditions” involve only CSC. In that case, since the subject and the predicate of (A) co-supposit for (A), the CSC of (A) is satisfied. But then the CSC of (B) cannot be satisfied. Therefore, (A) cannot entail (B), despite what Buridan says, on his own account of validity.27 Now suppose the “correspondence-conditions” involve both the CSC and the VIC. In that case, since the VIC of (A) is not satisfied, (A) may validly entail (B), although, of course, in that case both (A) and (B) are false and their correspondence conditions are not satisfied (for 27 Note that this argument is based on the assumption that the point of Buridan’s talking about a virtual implication is to assure that he can invoke this requirement even if the consequence expressing this implication is not actually formed. But in all cases when the question is whether a Liar-sentence satisfies the VIC it is assumed that the Liar-sentence itself exists, and that it, or rather a proposition equiform to it, would figure in the antecedent of the consequence expressing the “virtual implication” if it were formed. So, a defense to the effect that it is not only (A), but (A) and a proposition ‘(A) exists’ would be required for the implication probably would not work.

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although the CSC of (A) is satisfied, its VIC is not, because the CSC of (B) is not satisfied). But in this case, if their VIC is supposed to be among the correspondence-conditions of all propositions, then Buridan’s solution will not work for ‘No proposition is negative; therefore, some proposition is negative’. The reason is that if the VIC is supposed to be part of the correspondence-conditions of all propositions, then, on Buridan’s final definition of validity, this consequence will be valid if the VIC of its antecedent cannot be satisfied. But this is precisely the case here. The VIC of that antecedent could only be satisfied in a possible situation in which that antecedent is true, and so it exists. But if it exists in that situation, then the situation contains a negative proposition, whence that antecedent (stating that no proposition is negative) cannot correspond to that situation (because its obvious supposition-condition, namely, that its terms do not co-supposit, would have to fail). Therefore, the correspondence-conditions of the antecedent cannot be satisfied, and hence the correspondence conditions of the antecedent cannot be satisfied without the satisfaction of the correspondence-conditions of the consequent; whence the consequence must be deemed valid on Buridan’s definition, despite what he says. Ergo, Buridan’s solution fails within the context of his own logical theory, for his theory in the end is rendered inconsistent by this solution. And this was the point of the fourth thesis of the first section.

5.11

Postscript

Upon re-reading the argument of the previous section (a couple of months after I thought I had completed this paper), it appears to me that there is a plausible way to save the consistency of Buridan’s theory. For concerning his “virtual implication” Buridan may plausibly claim that it is not a formally, but merely materially valid consequence, depending for its validity not on the logical form of the propositions involved, but on the meaning of their terms. Thus, the argument that if the correspondence conditions of a proposition included only its CSC, then the “virtual implication” of its truth would not be a formally valid consequence by Buridan’s own criterion of formal validity would simply miss the mark: on this understanding of Buridan’s “virtual implication”, it is not even supposed to be formally valid; it is just valid on account of the meaning of the terms of the propositions involved, in particular, the meaning of the term ‘true’. This defense may actually work, for on this interpretation Buridan may claim without inconsistency that the “correspondence conditions” of propositions are nothing but their CSC, and so his treatment of Liar-

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sentences is satisfactory, given that even if their CSC is satisfied (since they are false) their VIC, which would be required for their truth, cannot be satisfied precisely for this reason. Still, the validity of the virtual implication involved in the VIC need not be judged in terms of Buridan’s criterion for formal validity, because this implication is not supposed to be formally valid in the first place. To be sure, Buridan justifies his “virtual implication” with reference to the meaning of the copula, which he takes to be “the formal part” of any categorical proposition; so, its copula is part of the “logical form”, rather than the “matter” of a proposition. Still, it is precisely this “formal part” of the antecedent of the “virtual implication” that is supposed to justify the application of the predicate ‘true’ in its consequent, given the meaning of ‘true’. Indeed, perhaps this is all Buridan has to say about the meaning of ‘true’, by way of a (strongly deflationist) “theory of truth”.

References [1] S. L. Read (2002) The liar paradox from John Buridan back to Thomas Bradwardine, Vivarium, 40: 189–218. [2] S. L. Read (2008) The Truth Schema and the Liar, c. 1 [3] G. Klima (2004) Consequences of a closed, token-based semantics: the case of John Buridan, History and Philosophy of Logic, 25: 95–110. [4] C. Normore (1985) Buridan’s ontology, in: J. Bogen and J.E. McGuire (eds.) How Things Are. D. Reidel Publishing Company: Dordrecht-Boston-Lancaster, pp. 189–203. [5] G. Klima (1999) Buridan’s logic and the ontology of modes, in: S. Ebbesen and R.L. Friedman (eds.) Medieval Analyses in Language and Cognition. Copenhagen: The Royal Danish Academy of Sciences and Letters, pp. 473–495. [6] J. Buridan (2001) Summulae de Dialectica (henceforth: SD), an annotated translation with a philosophical introduction by Gyula Klima. New Haven: Yale University Press, pp. 829–831. [7] J. Buridan (1964) In Metaphysicen Aristotelis Questiones Argutissimae (henceforth: QM), Paris 1588 (actually 1518). Reprinted as Kommentar zur Aristotelischen Metaphysik, Minerva, Frankfurt a. M, 1964, lb. 6, q. 8. [8] J. Buridan (1994) Questiones Elencorum (henceforth: QE), R. van der Lecq and H.A.G. Braakhuis (eds.), Nijmegen, Introduction, Sect. 3.2.

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[9] G. Klima (1993) ‘Debeo tibi equum’: A reconstruction of Buridan’s treatment of the sophisma, in S.L. Read (eds.) Sophisms in Medieval Logic and Grammar: Acts of the 9th European Symposium for Medieval Logic and Semantics. Dordrecht: Kluwer, pp. 333–347. [10] G. Nuchelmans (1973) Theories of the Proposition: Ancient and Medieval Conceptions of the Bearers of Truth and Falsity. Amsterdam: North-Holland. [11] L.M. De Rijk (1962–1967) Logica Modernorum: A Contribution to the History of Early Terminist Logic, 3 vols. Assen. [12] Buridan (2006) Syncategoremata, in: K. Brown (ed.) Elsevier’s Encyclopedia of Language and Linguistics, vol. 12, 2nd ed. Oxford: Elsevier, pp. 353–356. [13] S. L. Read (2002) Medieval theories: properties of terms, in: E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy (Spring 2002 Edition), URL = http://plato.stanford.edu/archives/ spr2002/entries/medieval-terms/. [14] G. Klima (2001) Existence and reference in medieval logic, in: A. Hieke and E. Morscher (eds.) New Essays in Free Logic. Dordrecht: Kluwer, pp. 197–226. [15] J. Buridan. Quaestiones in primum librum Analyticorum Posteriorum, q. 10 (unpublished edition by H. Hubien). [16] Albertus de Saxonia (1974) Perutilis Logica (Venice, 1518; reprint). Hildesheim: Georg Olms Verlag, f. 4, va. [17] P.V. Spade (2005) Insolubles, in: E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy (Fall 2005 Edition), URL = http://plato. stanford.edu/archives/fall2005/entries/insolubles/. [18] J. Buridan (1976) Tractatus de Consequentiis, in: H. Hubien (ed.) Philosophes Médiévaux, vol. 16. Louvain: Publications universitaires, pp. 21–22. [19] G. Klima (2004) John Buridan and the force-content distinction, in: A. Maierú and L. Valente (eds.) Medieval Theories On Assertive and Non-Assertive Language, Acts of the 14th European Symposium on Medieval Logic and Semantics. Rome: Olschki, pp. 415–427.

Chapter 6 SCHEMING AND LYING Truth-Schemas, Propositions, and the Liar Eugene Mills∗ Virginia Commonwealth University, Richmond Virginia 23284, USA

Abstract

Stephen Read argues correctly that Tarski’s T-schema cannot underpin an adequate account of truth, because it yields neither an analysis of truth nor a solution to the Liar paradox. Read offers two schemas intended to do what the T-schema cannot. I argue first that Read’s schemas share the failings of the T-schema, offering neither an explication of truth nor a solution to the Liar. I briefly compare Read’s solution to the Liar with my own (previously published) solution. On both accounts, the Liar-sentence turns out to be simply false, but my account avoids the pitfalls that beset Read’s.

Keywords: Liar paradox, Truth, T-schema, Propositions.

6.1

Introduction

All philosophy worthy of the name aims at truth. The Liar paradox mocks philosophy by seeming to expose truth as incoherent. Thus all philosophers had better believe that the Liar fails, even if they can’t say how. Those focusing on truth itself should make the Liar a chief concern.1 Stephen Read [13]2 commendably recognizes the centrality of the Liar and makes a serious effort to solve it. I’ll argue here, regrettably, that Read’s effort falls short. (Regrettably, because there’s much in that effort with which I sympathize.) Read’s ∗ Opponent.

1 Though many don’t. A small random sample of works that, given their ambit, culpably omit discussion of the Liar comprises [6, 7, 10]. 2 Otherwise unspecified page numbers refer to this work of Read’s.

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solution to the Liar rests on a theory of truth, which rests in turn on a pair of truth-schemas. I’ll argue in Sect. 2 that these schemas don’t undergird a theory of truth. In Sect. 3, I argue that Read’s argument for the falsity of the Liar-sentence is fallacious. In Sect. 4, I briefly compare Read’s solution to the Liar with my own (previously published) solution, arguing that mine avoids the pitfalls that beset Read’s.

6.2

Truth and Truth-Schemas

Read offers his truth-schemas as replacements for Tarski’s T-schema x is true is and only if p.

(T)

(T)’s advocates take it that every instance of (T) is true, when a name of a sentence replaces ‘x’ and the translation of that sentence into English replaces‘p’. Many take this alleged fact about (T) to be a definition of, or at least a necessary truth about, truth. Read argues correctly that they’re wrong. (T)’s advocates take it to capture the ‘correspondence platitude’ x is true if and only if things are as x says they are.

(CP)

But (CP) is equivocal, Read argues, and true only on one rendering; and the true rendering isn’t the one that (T) captures. (T) fails both as a basis and as a test for a theory of truth. Read rightly points out that (T) fumbles some pedestrian sentences involving indexicals and quantifiers, and it can’t handle the Liar paradox. Hence it can’t tell the whole story about truth. Either it yields false instances, or else its application must be restricted ad hoc, so that it’s mum on the truth-values of sentences that clearly have them. In the first case, (T) obviously doesn’t tell the truth about truth. In the second, it may be spared false instances, but it doesn’t tell the whole truth about truth, and so it can’t serve to analyze it. Read’s criticisms of (T) aren’t new.3 They are, however, both decisive and generally unappreciated, so they well deserve repetition and emphasis. Read offers two schemas of his own that are supposed to satisfy (T)’s ambitions while evading its problems. The two schemas correspond to two views on meaning between which he remains neutral. These two views differ on whether the expression-relation, holding between sentences and propositions, is closed under strict implication. This difference is supposed to turn on whether all instances of the schema (∀p, q)((p ⇒ q) → (x:p ⇒ x:q))

(K)

3 Read says it’s ‘been an unquestioned orthodoxy that every instance of the T-scheme. . . is true’ (p. 3). He’s half right: it’s been orthodoxy, but James Cargile has long railed against it; for a small sample of his published heresy, see his [1], p. 86 and [2, 3].

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are true (where ‘x’ designates a meaningful sentence, ‘x:p’ means ‘x says that p’ and ‘⇒’ and ‘→’ designate strict implication and the material conditional, respectively). I’ll shortly give reasons for thinking that (K) is a matter of some obscurity, but I’ll suppose for now that it succeeds in capturing the intended distinction between two views of meaning. Taking ‘Tx’ to mean that x is true and ‘⇔’ to represent strict equivalence, Read’s first schema – intended to apply in case (K) holds – is Tx ⇔ ∀p(x:p → p)

(A)

If (K) fails, no matter; in that case (says Read) we can make do with Tx ⇔ ∀p(∃q(x:q ∧ (q ⇒ p)) → p)

(A’)

(A), according to Read, ‘is an analysis of truth’ (p. 7),4 and presumably (A’) is as well. Read clearly means that each is a correct analysis of truth, given only the correctness of the corresponding verdict on (K). To avoid needless repetition, I’ll focus henceforth on (A), but everything I say about it applies, mutatis mutandis, to (A’) as well. Some will criticize Read for buying propositions, intensional entities distinct from the sentences that express them. I don’t; I share his commitment to such presumably abstract entities. I do, however, see problems with his schemas. (A) doesn’t provide an analysis of truth. Read doesn’t say why he thinks it does. I’m guessing that it’s because its right side contains no term meaning ‘true’. This absence wouldn’t make (A) an analysis, though, even if (A) were otherwise unflawed. Analysis demands an analysans giving two or more individually necessary and jointly sufficient conditions. (This isn’t all it demands.) Furthermore, it’s hard to see how a mere schema can analyze anything, since schemas say nothing. These points pose serious challenges to Read’s account. They don’t, however, form my main complaint against it. My main complaint is twofold. First, (A) isn’t well formed. Roughly, anyway: a schema isn’t ill or well formed in isolation, but only relative to a syntax, and Read doesn’t supply details. But given natural syntactic assumptions, (A) isn’t well formed, and I can think of no alternatives to these assumptions that would help with (A)’s interpretation. (A) is reparable, but – my second point – the only repair in view succeeds by inserting ‘true’ (or some synonym) on its right side. Whether or not the absence of such a term qualifies (A) or an (A)-analogue as an analysis, 4 Though it’s not ‘the final analysis’, he says, because it leaves ‘saying that’ unanalyzed. A final analysis must invoke no unanalyzed semantic notions.

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the presence of one certainly disqualifies it. To summarize the forthcoming argument: given natural assumptions about the relevant syntax, (A) isn’t well formed, and once repaired, it’s no analysis; absent those assumptions, it’s obscure. With or without those assumptions, then, Read hasn’t given an analysis of truth. I assume that (A) is supposed to be understood according to the standard syntax of the predicate calculus, with ‘∀p’ a restricted universal quantifier ranging over propositions. Given this assumption, (A) isn’t well-formed because its right-hand side ∀p(x:p → p)

(AR)

isn’t well formed. (AR) isn’t well formed because it equivocates between treating ‘p’ as a schematic variable and treating it as a nominal variable. There’s no uniform treatment on which (AR) is coherently instantiable. To see this, try instantiating. ‘Instantiation’ means one thing for a schema (such as (AR)), something quite different for a universally quantified sentence. Schemas don’t express propositions (though their instances do, given standard background assumptions). Instantiating them is a matter of substituting suitable expressions for schematic variables, but there’s no general requirement that these expressions be denoting terms. Instantiating an interpreted (restricted) universally quantified sentence, on the other hand, is a matter of removing the quantifier and replacing the variables thereby freed with names (or at least denoting terms). Attention to the distinction between these two kinds of instantiation reveals (AR) as ill formed. First instantiate the schema (AR) by replacing the schematic letter, ‘x’, with a particular sentence to yield, for example, ∀p(‘Snow is white’:p → p)

(AR1)

So far, so good. (AR1) is not a schema but (it seems) a general sentence. Now try universal instantiation on (AR1). The variable ‘p’ ranges over propositions, and so its substituends must (it seems) be terms denoting propositions. One proposition expressed by ‘Snow is white’ is, I assume, the proposition that snow is white. The italicized ‘that’-clause in the previous sentence is standardly taken to denote this proposition, and there’s no reason we can’t incorporate this clause into the lexicon of our language. Instantiating (AR1) by substituting this clause for ‘p’ yields ‘Snow is white’:that snow is white → that snow is white

(AR2)

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In English (or perhaps ‘English’): If ‘Snow is white’ says that snow is white, then that snow is white.

(AR2E)

These can be used perfectly sensibly: take ‘that snow’ as a demonstrative phrase indicating some particular bit of snow. This usage and its attendant interpretation, however, obviously aren’t what Read intends. ‘Snow is white’ isn’t intended as synonymous with ‘that snow is white’, where ‘that snow’ denotes a particular bit of snow. On any other interpretation, including the apparently intended one, (AR2) and (AR2E) make no clear sense. The arrow, ‘→’, is a sentential connective, but what it connects in (AR2) are not sentences, if ‘that snow is white’ is interpreted as a propositional denoter. (AR-1) is nonsensical if ‘that’-clauses denote propositions. For if a universally quantified formula yields nonsense when instantiated by proper substituends, then the formula is nonsensical. And if (AR1) is nonsensical, then so is (A). Read might reply that I’ve ignored his stipulation that ‘:’ means ‘says that’. I’ve put ‘that’ at the leading edge of propositional denoters, whereas he builds it into the expression-relation. So instead of substituting for ‘p’ a ‘that’-clause denoting a proposition, we should (goes this Readian reply) substitute instead a sentence that expresses a proposition. Instantiating (AR1) accordingly, substituting ‘snow is white’ for ‘p’ and translating into English, yields the distinctly intelligible If ‘Snow is white’ says that snow is white, then snow is white. (AR3E) Thus, it seems, my complaint is disarmed. Not so fast. Given that ‘snow is white’ expresses the proposition that snow is white, either it also denotes that proposition, or it doesn’t. If it doesn’t denote the proposition, then Read’s logic remains mysterious. ‘Universal instantiation’ along the lines suggested isn’t a matter of dropping the quantifier and replacing the freed variable with a term denoting a proposition.5 What, then, is it? Lacking a satisfactory answer,6 I turn to the other alternative: that ‘snow is white’ both expresses and denotes the proposition that snow is 5 Or,

presumably, anything else. You might go Fregean and say that proposition-expressing sentences denote truth-values, so that the quantifier ranges over (not propositions but) truthvalues; see [4]. Aside from the difficulties of the view that sentences refer to truth-values, this interpretation makes (A) obviously inadequate to Read’s purposes. 6 I take it as established that substitutional quantification is a dead end rather than an avenue of escape; see [8], p. 380 and [12], pp. 277–278.

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white. This line may seem more promising. For it does allow instantiation by substituting terms for variables. Ordinary usage licenses this line, too. For we idiomatically read John believes snow is white

(1)

as synonymous with – expressing the same proposition as – John believes that snow is white

(2)

If the ‘that’-clause in (2) denotes a proposition, then (it seems) the occurrence of ‘snow is white’ in (1) must do likewise. I grant that the occurrence of ‘snow is white’ in (1), on its natural reading, denotes a proposition. This fact, however, doesn’t blunt my criticism, for at least two reasons. The first is that the idiomatic synonymy of (1) and (2) rests on taking the ‘that’-clause as the fundamental propositional denoter and the ‘that’-less form as implicit, context-relative shorthand for it. If I utter ‘Snow is white’ with assertive intent, I assert that snow is white.7 If I utter ‘That snow is white’ with assertive intent, either I assert that some particular bit of snow is white or else I fail to assert anything. In neither case do I assert the proposition that snow is white. Hence ‘snow is white’ can’t generally be replaced with ‘that snow is white’ without change of truth or meaning. On the other hand, wherever a ‘that’-less sentence denotes a proposition – as I grant it does in (1) – we can substitute the corresponding ‘that’-clause without change of truth or meaning. Thus if ‘snow is white’ as it occurs in (AR3E) denotes the proposition that snow is white, then it’s replaceable without loss of truth or meaning by ‘that snow is white’. It clearly isn’t: the replacement would once again yield (AR2E), which is no synonym of (AR3E). Hence ‘snow is white’ as it occurs in (AR3E) doesn’t denote the proposition that snow is white. The second alternative is blocked, and the obscurity of (AR) remains. Read might simply deny the asymmetry of replaceability that I assert. In some contexts – such as that of (AR) – a ‘that’-less sentence denotes a proposition but isn’t replaceable (he might insist) by a corresponding ‘that’-clause. Absent other clear cases of this phenomenon, such an insistence is ad hoc. But grant it for argument’s sake. The second reason why my criticism stands is that one ad hoc insistence requires another; the implausibilities mount. Here’s why. If we can denote a proposition with a sentence that expresses it, as I’m granting we can denote the proposition that snow is white using ‘snow is white’, then nothing prevents us 7 This

is no necessary truth, and it doesn’t rest on the T-schema. It just happens to be true.

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from dubbing that proposition with a name – another name, if you like. I hereby dub the proposition that snow is white ‘Snedley’. Now, I’m granting for argument’s sake that ‘snow is white’ denotes Snedley, and so I’m granting that (AR3E) is a correct translation of a correct instantiation of (AR). It’s also true, however, that ‘Snedley’ denotes Snedley. Hence the following must also be a correct translation of a correct instantiation of (AR): If ‘Snow is white’ says that Snedley, then Snedley

(AR4E)

(AR4E) is, however, nonsensical. So if it’s a correct translation of a correct instantiation of (AR), then (AR) is again nonsensical. Thus (AR) is nonsensical, and so (A) is as well. Given natural syntactic assumptions, then, (A) isn’t a well formed schema.8 Perhaps some other syntax would render it intelligible while also vindicating Read’s claims about its function as an analysis. It’s obscure, though, what such a syntax might be. Read hasn’t succeeded in using (A) to convey a theory of truth. I sympathize with Read’s claim that what it is for a sentence to be true is for things to be wholly as it says they are. The only way I see of repairing (A) to capture this idea is along the following lines. Let ‘:’ mean the same as ‘says’ (or ‘expresses’) rather than ‘says that’, and let substituends for ‘p’ be terms denoting propositions – including relevant ‘that’-clauses. Then the following (A)-like schema is well formed: Tx ⇔ ∀p(x:p → Tp)

(AT)

(K) is problematic for reasons exactly parallel to those adduced against (A). Repairing it in parallel form yields (∀p, q)((Tp ⇒ Tq) → (x:p ⇒ x:q))

(KT)

If (KT) is true, then all of (AT)’s instances are true. If (KT) isn’t true, then modify (AT) in the obvious way to yield (AT’); all of ((AT’)’s instances are true. Plainly, though, neither (AT) nor (AT’) is an analysis of truth, since the truth-predicate ‘T’ appears on both flanks of both schemas.9 8 My argument on this score echoes Tarski’s own ([14], Sect. 15) criticism of a suggestion by von Juhos. 9 Although (AT)’s acceptability (given the acceptability of (KT)) doesn’t point to an analysis of truth per se, it does show that truth must be properly understood as applying to propositions as well as sentences, and it points to a definition of sentential truth in terms of propositional. A sentence is true just in case (a) it says something, and (b) the proposition which is all of what it says is true. No such definition of propositional truth in terms of sentential is available, for there’s no guarantee that every proposition is expressed by some sentence or other. Hence sentential truth presupposes propositional, but not the reverse. Lewy [9] makes this point in connection with the T-schema.

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While (AT) doesn’t analyze truth, it doesn’t follow that no other repair of (A) could analyze truth (or at least contribute to such an analysis; recall my other points about (A)’s shortcoming as an analysis). I simply can’t think of any other repair that would do the job.10 If Read can produce one, well and good. As it stands, though, his attempt at analyzing truth fails.

6.3

Read’s Solution to the Liar

Read’s Bradwardinian solution to the Liar appeals merely to the unrestricted truth of (A)’s instances, not to (A)’s putative status as an analysis of truth. That solution shares (A)’s problems of intelligibility. It adapts smoothly, though, to (AT), and (AT)’s failure to analyze truth doesn’t show that it can’t handle the Liar. Nevertheless, I’ll argue, a repaired version of Read’s argument, using (AT) instead of (A), still fails. I start by quoting the salient part of Read’s argument. Following Read, I’ll use ‘L’ to name the sentence ‘L is not true’.11 Read says (pp. 10–11): L says that L is not true, that is, L:¬TL. It may say more: call it q, that is suppose L:(¬TL ∧ q). (If ¬TL is all that L says, we can simply let q be ¬TL, or a tautology.) By (A), TL ⇔ ∀p(L:p → p) i.e., TL ⇔ (¬TL ∧ q) since ¬TL ∧ q is all L says. . . .

The argument continues, but a fallacy in this initial segment derails it. Schema-sentence conflations of the sort I’ve already discussed run throughout. I won’t catalog and correct them. Correction requires the insertion at various places of ‘that’ and of ‘T’, and it requires interpreting ‘∧’ not (in the usual way) as a sentential connective but rather as a term-forming operator on propositional denoters.12 I skip the technical 10 Another possible repair differs technically from the one I discuss above but is in the same mold and suffers the same defect. Take the usual sentential connectives and reinterpret them as term-forming operators: letting substituends for ‘P ’ and ‘Q’ denote propositions, interpret ‘P → Q’ not as a sentence but as a term denoting a proposition, namely the proposition to the effect that if P is true, then Q is. On this interpretation, writing ‘P → Q’ – or for that matter, writing ‘P ’ – is merely writing the name of a proposition, not an assertion of it. To assert it, you’d write ‘T(P → Q)’, which then must be taken as a sentence rather than a propositional denoter. Other problems aside, capturing the intent of (A) will, on this interpretation, once again require the insertion of a truth-predicate where Read doesn’t want it. 11 Whether ‘L’ is supposed to name a token or a type won’t matter for this section, though it will for the next. 12 As I’ve shown, intelligibility requires interpreting ‘:’ as short for ‘says’ rather than ‘says that’. So the intelligibility of ‘L:¬TL’ requires that ‘¬TL’ abbreviates not the sentence ‘it

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corrections here, for they don’t bear on the real problem with Read’s argument. The following recasting of Read’s passage captures its argumentative intent while avoiding its technical glitches. (As before, I use ‘:’ to mean ‘says’ rather than ‘says that’). L says that L is not true, that is, L:¬TL. It may say more: call it ‘q’, that is suppose that L expresses the conjunction of the proposition that L is not true and (the proposition) q. (If that L is not true is all that L says, we can simply let ‘q’ be replaced by ‘that L is not true’, or a term denoting a tautologous proposition.) By (AT), TL ⇔ ∀p(L:p → Tp)

(α)

TL ⇔ T[the conjunction of that ¬TL and q]

(β)

i.e., since the conjunction of (the proposition) that ¬TL and (the proposition) q is all L says. . .

The problem lies in the ‘i.e.’. (β) doesn’t follow from (α) alone. Read’s ‘since’ seems to acknowledge this: the move from (α) to (β) hinges, apparently, on the premise that the conjunction of the proposition that ¬TL and the proposition q is all L says; that is, ∀p{L:p → [(p = that¬TL) ∨ (p = q) ∨(p = the conjunction of that ¬TL and q)]}(γ)

(γ)

I have no gripe with (γ): it’s true, given the stipulation about ‘q’. Conjoining it to (α), however, still doesn’t yield (β). It’s a straightforward is not the case that L is true’ but rather the propositional denoter ‘that it is not the case that L is true’, and ‘q’ must likewise denote (and not merely express) a proposition. But if ‘∧’ is read in the usual way as ‘and’, then ‘¬TL ∧ q’ must be read as ‘that it is not the case that L is true and q’, which neither expresses nor denotes a proposition. Thus ‘L:(¬TL ∧ q)’ cannot intelligibly be read as saying that there is a single proposition, denoted by ‘¬TL ∧ q)’, expressed by L. Compare: ‘Alice loves Bob and Carol’ doesn’t say that Alice loves a single person, denoted by ‘Bob and Carol’; there’s no such person. It goes instead into logician’s English as the conjunction ‘Alice loves Bob and Alice loves Carol’, asserting that Alice stands in the loves-relation to two (presumably distinct) people. Similarly, ‘Snow is white and grass is green’ says that snow is white and that grass is green doesn’t say that the quoted sentence expresses a single proposition, denoted by ‘that snow is white and that grass is green’; there is no such proposition. It says instead that the quoted sentence stands in the says-relation to two distinct propositions. There is also, of course, the single proposition that snow is white and grass is green, and the quoted sentence does say this. But that it does cannot intelligibly be formulated by ‘Snow is white and grass is green’:(that snow is white ∧ that grass is green) for the parenthetical clause doesn’t denote a proposition, if ‘∧’ connects sentences. See note 10 above.

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modal fallacy to suppose otherwise. Avoiding this fallacy requires boxing in (γ) with a necessity-operator to yield 2∀p{L:p → [(p = that¬TL) ∨ (p = q) ∨(p = the conjunction of that ¬TL and q)]}

(γ )

(β) does follow from the conjunction of (α) and (γ ). Unfortunately, though, (γ ) isn’t true. It says, in effect, that L has its semantic properties necessarily – that among other things, L necessarily says that L is not true. But ‘is not true’ doesn’t necessarily predicate untruth. It could just as easily have predicated the property of comprising four words, in which case L would have been straightforwardly true. Read’s argument has a faulty inference if it appeals to (γ), a false premise if it appeals to (γ ). It’s a non-starter unless it appeals to one or the other (or to something equivalent to one or the other). So it fails. The only response is to insist that L does, in fact, bear its semantic properties essentially, and so to embrace (γ ). This is just what Read seems to do: ‘if (A) is a logical equivalence in providing the correct analysis of truth,’ he says, ‘we must assume that what a sentence says, it says of necessity too.’ Read performs modus ponens rather than modus tollens on this conditional while acknowledging that ‘any form of words expresses what it does only contingently’. His reason is this: When I speak of the truth of sentences and what they say, in what follows, I will be speaking of a sentence taken as meaning what it does-of a proposition, in Horwich’s terminology, or of a sentence in a context of speaker, place, and time.

This passage is at best obscure as a basis for semantic essentialism. To speak of a sentence ‘taken as meaning what it does’ is not to make its meaning essential to it. Compare: I might speak of Read ‘taken as having the career that he has’ that is, taken as a philosopher; but speaking of him this way does nothing to establish that he has the property of being a philosopher necessarily, which he clearly doesn’t. In any case, it’s clear that Read doesn’t mean to say that a sentence per se has its meaning essentially. He means instead that certain particular event-tokens, those which involve the assertion of certain sentencetokens, bear meaning-properties essentially. This claim, however, is no more plausible or well motivated than the analogous claim about sentences themselves. Any particular use of a sentence-token, by a particular speaker, in a particular place and time, could have meant something other than what it does. (‘There is a possible world in which it means something other than. . . ’, if you like world-talk.). Or so it seems: at the least, a semantic essentialist owes us a compelling argument against this possibility.

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For Read’s argument to go through, we must find an interpretation on which (γ ) is true. It’s clear that taking ‘sentences’ as sentence-tokens or as assertion-tokens won’t do. The more common and perhaps more natural rendering of semantic essentialism casts it in terms of sentencetypes. Though this is apparently not Read’s view, I’ll consider it to see whether it offers him any succor. An immediate difficulty is that types, as normally construed, are dubious entities. Neither fish nor fowl, they’re supposed to be abstracta and so occupy no space while having formal properties whose possession entails the occupation of space. You can run from this contradiction, but you can’t hide: there are no types as usually understood. There is a perfectly good property of being an ‘L is not true’-token (or the property of being a token of the L-type). All tokens of L share this property. Properties aren’t types, though. Genuinely abstract, they lack form. They also lack truth-values. For now, I’ll set aside my scruples and talk types. On the most natural construal of the claim that ‘L’ refers to a type, in saying that ‘L is not true’ is not true, one would mean that every token of this type is not true. Now, our options subdivide. Is the type merely syntactic? If so, then (γ ) is clearly false. For it’s clear that many tokens of ‘L is not true’ say things that are true – among them, by Read’s lights, the tokens13 that he uses in saying “the longer argument concerning L shows more than that L is not true” (p. 12, my emphasis). Thus we come to the idea that ‘L’ refers to a syntactic-cum-semantic type. There’s more than one way to flesh out this idea. One is this: a token counts as a token of the relevant type just in case (a) it’s of the syntactic ‘L is not true’-type; (b) it has semantic properties such that its embedded token of ‘is not true’ predicates untruth of the referent of its token of ‘L’; and (c) the embedded token of ‘L’ refers to the very sentence token in which it’s embedded. Supposing (dubiously, even granting types) that this stipulation specifies a unique sentence-type, I’ll call it ‘the SStype’. Now, it’s a necessary truth that every token of the SS-type says of itself that it’s not true. This fact, however, provides no license for (γ ). Here’s why. We’re supposing that ‘L’, as it occurs in (γ ), refers not to any token of the SS-type, but to the type itself. So from the fact that tokens of the SS-type contain self-referential ‘L’-tokens, nothing follows about the referent of the ‘L’-type embedded in the SS-type. After all, each token of this type has an embedded ‘L’-token whose referent is distinct from that of every other token of the SS-type. It’s completely opaque, in fact, how to make sense of ‘the referent of the ‘L’-type embedded in 13 ‘Tokens’

rather than ‘token’ because there are multiple tokens of Read’s paper.

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the SS-type’, unless it simply has none. From the definition of ‘SS-type’ that I’ve given, there’s no basis for saying that the type (as opposed to its tokens) contains anything that refers to anything. It won’t help, either, to let ‘L’ designate a particular token of the SS-type. Suppose we do so. Now while it’s necessarily true (given my concessions) that L is of the SS-type only if it says of itself that it’s not true, and it’s true that L is of the SS-type, you’d once again commit a modal fallacy by inferring that it’s necessarily true that L says of itself that it’s not true. (The fallacy is the same as that involved in inferring from the premises that (a) necessarily, all bachelors are unmarried, and (b) Bob is a bachelor, the conclusion that (c) necessarily, Bob is a bachelor.) To get the conclusion that it’s necessarily true that L says of itself that it’s not true, you’d need a premise to the effect that L – a concrete physical object – is not merely of the SS-type but necessarily of the SS-type. Any such premise is as flatly absurd as the claim that Bob is necessarily a bachelor. You might try, then, providing a type as the referent of ‘L’ that doesn’t come via a stipulation regarding tokens (unlike the stipulation for the SS-type). The idea is that there’s a unique sentence-type of the form ‘L is not true’ such that the embedded letter-type ‘L’ necessarily refers to that very sentence-type and such that the embedded predicate-type ‘is not true’ necessarily predicates untruth of the referent of the embedded letter-type ‘L’. This type is (on the current hypothesis) what our uses of ‘L’-tokens denote. Hence our uses of those tokens are not self-referential, for they refer not to themselves but rather to an abstract type. This construal of the relevant type casts doubt on the idea that paradoxical uses of tokens of ‘L is not true’ derive in some way from selfreference, since on the present construal no such self-reference occurs in those uses. Furthermore, it’s enough for present purposes to point out that on this construal of the relevant type, Read would have no solution on offer to the token-indexical version of the Liar presented by L1 , which is this very sentence-token (the one you are now reading), is not true.

(L1 )

For L1 , which is a physical object on this page, clearly doesn’t have its semantic properties necessarily. So no analogue of (γ ) is even remotely plausible here. There’s more to be said against the type here suggested. Saying it would, however, be an unnecessary diversion. Read’s solution to the Liar hinges on a commitment to semantic essentialism. Semantic essentialism is, however, unmotivated; some versions of it are plainly false; and it doesn’t solve the Liar on its best construal. Hence Read’s solution to the Liar fails.

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6.4

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My Solution to the Liar

I’ve elsewhere [11] presented a solution to the Liar that resembles Read’s in some respects. Like Read, I make essential use of abstract propositions. Like Read, I argue (roughly) that a Liar sentence says not only that it’s false but also more than this, and that it’s true only if all of what it says is the case. Like Read, I argue that a Liar sentence is false because contradictory and that there’s no good argument from its falsity to its truth. Given these central points of agreement, you may well wonder whether my criticisms of Read’s solution boomerang to wound my own. Thus prudence urges me to point out the features of my solution that armor it against the criticisms that doom his. In doing so, I’ll also rebut a criticism that Read aims at a view central to my solution. I can’t rehash or defend my solution here in detail; here’s a quick sketch. The Liar paradox gets off the ground only when the unit of assessment is a particular use of a particular sentence-token. (Having said this, I’ll occasionally make concessions to readability in what follows by talking about what ‘the sentence says’ or ‘the sentence-token says’. Any such verbiage should be read as shorthand for more cumbersome talk about uses of tokens.) Now, the truth or falsity of a use of a sentence-token is that (if any) of the proposition thereby expressed. Like Read, I distinguish between part of what a sentence says and all of what it says, and I say explicitly that what I mean by ‘the proposition expressed’ is all of what the sentence says. (I also assume bivalence for propositions – equating their untruth with their falsity – though not for sentences.) A paradoxical use of a Liar-sentence-token expresses a proposition that says of itself both that it’s true and also that it’s false. Hence such a use expresses a self-contradictory proposition, and so that proposition and the (use of the) token that expresses it are both false. As a separate matter, I diagnose the fallacy in arguing from the falsity of such a use to its truth. Neither (A) nor its deficiencies have any parallel in my discussion. My argument for the falsity of a paradoxical use of a Liar sentence presupposes the propositionalist view that a sentence (strictly, a use of a sentence-token) is false just in case the proposition it expresses (which is all of what it says) is false. Beyond this, it hinges on two premises ([11], p. 202): The first premise is that every proposition entails its own truth. The second is that the proposition expressed by a self-referential use of L entails its own falsity.

Neither premise rests on a theory of truth. I do offer a schema to distinguish the intended de dicto reading of the first premise from an unintended but possible de re reading ([11], p. 202):

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E. Mills [T]he proposition that grass is green does not entail that the only proposition expressed by Greg on Monday is true, even if the only proposition expressed by Greg on Monday is that grass is green. I assume, of course, a different reading. The point of the first premise is captured by the claim that substitution instances of the following schema express necessary truths, so long as the two sentence-tokens which replace the two occurrences of ‘p’ express one and the same proposition: p if and only if the proposition that p is true

(5)

My claim about (5), I now see, allows misinterpretation. Let me clarify. That claim is supposed to ‘capture’ the point of the first premise only in the sense of snaring something elusive: if the de re/de dicto ambiguity of the claim that ‘every proposition entails its own truth’ eludes you, my claim about (5) should help you to see it and to distinguish the intended from the unintended disambiguation. That’s all: my claim about (5) is certainly no analysis or synonym of the premise it’s supposed to capture. (My claim about (5) is a contingent, metalinguistic claim; the premise is neither.) Furthermore, I never suggest that my claim about (5) does anything to analyze truth. It doesn’t, for multiple reasons. One reason is that the restriction I impose on substitution-instances of (5) can’t be met for all propositions, most notably the propositions expressed by Liar sentences. Since I rely on no analogue of (A), the first problem with Read’s account simply has no parallel in mine. The second problem with Read’s account is his faulty argument for the falsity of the Liar sentence. The fallacy occurs in the course of an argument for the claim that L says of itself (not just that it’s false but also) that it’s true. (Read takes pains to disavow Buridan’s view that every sentence says of itself that it’s true. So do I; see my [11], p. 210, note 6.) This fallacy, too, has no parallel in my argument; indeed, this argument has no parallel in mine. I neither argue nor claim that the Liar sentence says of itself that it’s true.14 I argue that it (suitably used) expresses a proposition which says of itself – of the proposition, not the sentence expressing it – both that it’s true and that it’s false. Hence the Liar sentence expresses a contradictory proposition. Since the truth-value of a sentence derives from that of the proposition it expresses, the Liar sentence is false, but not on the ground that it says of itself that it’s true and that it’s false. There’s no oddity in this. After all, ‘Snow is white, and it’s not the case that snow is white’ is false by way of expressing 14 Though I don’t deny this, either. My solution is inspired by but distinct from that of [1]. It doesn’t require Cargile’s theory of assertion and predication, which underlies his approach to a broader range of paradoxes than the simple versions of the Liar that Read and I discuss. I do accept that theory, though, and it arguably – though not obviously – has the consequence that Liar-sentences say of themselves that they’re true.

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a contradictory proposition, but it doesn’t say of itself that it’s true, as Read would agree. I argue that the Liar sentence is false by way of expressing a contradictory proposition, but I don’t argue that it says of itself that it’s true. This point leads, finally, to an argument that Read gives in opposition to a thesis I endorse. In support of my first main premise, I argue that every proposition attributes truth to itself, in the sense that (for example) the proposition that grass is green is strictly identical with the proposition that it is true that grass is green. Read argues (p. 14) that it’s “crucial to [his] solution to the paradoxes that not every proposition say of itself that it is true” (my emphasis). His reason is this: [I]f we could show that every sentence said (albeit indirectly) of itself that it was true, the truth-condition (A) would be crippled. [. . . ] Every sentence (bar the paradoxical ones) would become a Truth-teller, true if true and false if not, but of undetermined truth-value. (14–15; my emphasis)

(A) is worse than crippled in any case, as I’ve argued. But this isn’t the main point. I agree entirely with Read that sentences that say of themselves that they’re true are the exceptions, not the rule. I agree that to insist otherwise – and especially to adopt Buridan’s claim that every sentence attributes truth to itself – is to incur dire consequences, though not quite the ones that Read fears. The problem for Read is this: his insistence that not every proposition say of itself of that it’s true is utterly unconnected by argument with his allegation of dire consequences if every sentence says of itself that it’s true. I assert that every proposition says of itself that it’s true. This assertion simply doesn’t entail that every sentence says of itself that it’s true. Read seems mistakenly to assume otherwise. Furthermore, my claim that every proposition says of itself that it’s true doesn’t saddle every proposition with the Truth-teller’s difficulties. What the claim comes to in a particular case is, again, the claim that (e.g.) the proposition that grass is green is the very same proposition as the proposition that it’s true that grass is green. No problem hinders a determination of the truth-value of this proposition beyond that of determining whether grass is green, which just is determining the truthvalue of the proposition. The faults that undo Read’s solution to the Liar have no counterparts in my own solution. You might attack my solution on other points. Whatever the outcome of such hypothetical attacks, though, they hold

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no succor for Read. His solution to the Liar, while admirable in ambition and in attitude, does not succeed.15

References [1] Cargile, James (1979) Paradoxes. Cambridge: Cambridge University Press. [2] Cargile, James (1986) Critical Notice of Recent Essays on Truth and The Liar Paradox ed. by R. L. Martin, Mind 95, 116–126. [3] Cargile, James (1995) Paradoxes, in The Oxford Companion to Philosophy ed. by Ted Honderich, pp. 642–644. Oxford: Oxford University Press. [4] Church, Alonzo (1978) The need for abstract entities in semantic analysis, in [5]. Reprinted from Daedalus 80 (1951). [5] Copi, Irving and James Gould, Eds. (1978) Contemporary Philosophical Logic. New York: St. Martin’s Press. [6] Davidson, Donald (1984) Inquiries into Truth and Interpretation. Oxford: Clarendon Press. [7] Devitt, Michael (1991) Realism and Truth, 2nd ed. Oxford: Blackwell. [8] Kripke, Saul (1976) Is there a problem with substitutional quantification?, in Truth and Meaning ed. by G. Evans and J. McDowell. Oxford: Clarendon Press. [9] Lewy, Casimir (1976) Meaning and Modality. Cambridge: Cambridge University Press. [10] Lynch, Michael (1998) Truth in Context. Cambridge, MA: MIT Press. [11] Mills, Eugene (1998) A simple solution to the liar, Philosophical Studies 89, 197–212. [12] Quine, Willard (1978) Reply to Professor Marcus’, in [5]. Reprinted from Synthese 27 (1962), 323–330. [13] Read, Stephen (2008) The Truth Schema and the Liar, in this volume (Unity, Truth, and the Liar, ed. Shahid Rahman, Tero Tulenheimo , and Emmanuel Genot, vol. 8 of series on Logic, Epistemology, and the Unity of Science, Springer), 3–17. [14] Tarski, Alfred (1944) The semantic conception of truth and the foundations of semantics, Philosophy and Phenomenological Research 4, 341–375. 15 Thanks

to James Cargile and Donald Smith for comments and related conversation.

Chapter 7 COMMENTS ON STEPHEN READ’S “THE TRUTH-SCHEMA AND THE LIAR” Terence Parsons∗ University of California, Los Angeles, CA, USA [email protected]

Abstract

This is an interesting and valuable essay. In it, Read takes a semantical analysis of truth developed by some of the best medieval writers, and formulates it using the resources of modern logic. His formulation seems to me to be impeccable. He also gives a pointed criticism of another medieval view (the view that every sentence says of itself that it is true). Attempts to provide a definition of truth for a language that contains a truth predicate that applies meaningfully to the sentences of that very language usually lead either to inconsistency (because of some version of semantic paradox) or else to a theory that fails to classify some sentences as true or not true. I will argue that Read’s account of truth falls into the latter category.

Keywords: Liar paradox, Truth, Medieval.

7.1

A Definition of Truth

Read begins by criticizing Tarski’s theory of truth; these criticisms are apt, and I will not go into them. He then gives what he proposes to be a better account – a definition of truth that avoids criticisms of Tarski’s account. A medieval formulation of Tarski’s idea is: A sentence is true iff things are as it says they are. The intent of this formulation is to legitimize the following schema: T‘S’ ↔ S where ‘things are as it says they are’ is interpreted as duplicating the content of the sentence itself. Medieval authors were aware that such a ∗ Opponent.

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schema leads to various semantic paradoxes, and some of them suggest improving the schema to take into account the possibility that a sentence might say several things. This necessitates a slightly different account of truth: A sentence is true iff things are howsoever it says they are. The ‘-soever’ suffix is to convert the Tarski schema into one in which a sentence is true iff things are every way that it says things to be. To formalize this requires universally quantifying a sentence letter. This is what Read does, using substitutional quantification. First, he formulates: sentence x says that p as x:p where the colon abbreviates ‘says that’. The revised account of truth can then be stated as: x is true iff for any p, if x says that p, then p and formalized as1 : Tx ⇔ ∀p(x:p → p)

(A)

A general assumption is in force to the effect that any sentence says what it explicitly states; for any sentence S: ‘S’:S

(E)

As a result, the theory entails the following corollary for each sentence S: T‘S’ → S

(C)

This is the safe half of the Tarski biconditional. The reverse implication does not follow, and this is how the Liar paradox is avoided.

7.2

Saying, and How the Liar Paradox is Avoided

Read’s theory relies on the notion of a sentence being able to say more than one thing. In some cases this seems like a natural thing to say. For example, not only does ‘A ∧ B’ say that A ∧ B, it also says that A and it 1 The symbol ‘⇔’ means strict co-implication, so that ‘A ⇔ B’ is true iff ‘A’ and ‘B’ are logically equivalent, or are equivalent by definition. Thus (A) can be taken as a definition of the truth predicate ‘T’.

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says that B. Read holds that the notion he has in mind is one that allows a sentence to say additional things that it does not logically imply. He illustrates this with the example: ‘All men are mortal’ says that Plato is mortal, and says that Socrates is mortal, etc. This example is effective in loosening up our intuitions about what might affect the truth of a sentence. However, my sense is that the particular example he has picked is not crucial for his enterprise. If you disagree with him about this example, this will have no effect on understanding or assessing his proposals. What he needs for a solution to the Liar paradox is that the liar sentence itself says more than one thing. Suppose that we have that L is the sentence: ¬TL So L is a sentence that says of itself that it is not true. It is possible to argue that because of the meanings of the terms in L, it also says more than this. Read’s theory does this; it entails that L also says of itself that it is true. This view is well worth considering. It is easy to show that L is not true. It is usually easy to argue that L is also true. But the revised definition of truth under discussion (principle (A) above) says that in order for a sentence to be true everything it says must hold. And the liar sentence L says of itself both that it is true and that it is not true. These don’t both hold (this is a claim that is argued for, not something that is apparent by observation), and so one can’t use (T) to prove that L is true. So paradox is avoided. (This sketch of the situation is also faithful to the medieval authors that Read references.)

7.3

Principles of the Theory

We have already seen two of the axioms of the theory. Read holds that saying is closed under strict implication: If a sentence says something, p, then it also says whatever is strictly implied by p2 : (∀x∀p∀q)((p ⇒ q) → (x:p ⇒ x:q))

(K)

2 In (K), ‘⇒’ is strict implication, and ’→’ is material implication. Read shows that if (K) is not assumed, then a slightly more complicated version of the theory with the same virtues may also be stated. I’ll assume that (K) holds, and not discuss that other version.

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With principles (A), (E), and (K), Read shows by the usual argument that the liar sentence is not true. He then shows how the argument that it is also true breaks down because this theory does not validate the Tarskian principle; in particular it does not validate: S ⇒ T‘S’ So far as I can see, all of this is plausible.

7.4

Does Every Sentence Say of Itself that it is True?

Many medieval philosophers who gave the above account of truth also endorsed an additional principle; the principle that every sentence says of itself that it is true3 : (BRAG) Every sentence says that it itself is true. ∀x x:Tx Or, schematically, the following holds for any sentence S: ‘S’:T‘S’ Read claims: (Claim) If (BRAG) were true, then no truth-value of any sentence could be determined by the theory. The theory would be too weak to establish regarding any sentence at all whether it is true or not. This is because for any sentence, in order to show it true, you need to show that everything that it says holds. But if (BRAG) were true then in order to show any sentence to be true, you would first have to show it to be true. You would face a regress in every case. There is an easily repaired inaccuracy in the statement of (Claim). Perhaps no sentence could be determined to be true, for the reason he gives, but nothing would prevent sentences from being determined to be false. For example, ‘giraffes are fish’ is easily shown not to be true since it says, perhaps among other things, that giraffes are fish – and giraffes aren’t fish. A single application of (A) then shows that ‘giraffes are fish’ is not true. So the principle (Claim) must be confined to determinations of truth; it needs to be revised to: 3 Read points out that Albert of Saxony argues for this view, though his argument is question begging.

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Claim’ If (BRAG) were true, then no sentence could be shown to be true by the theory. I suspect that (Claim’) is correct,4 and this is useful.

7.5

Weakness in the Original Theory

(Claim’), however, presages a difficulty with Read’s own view. This is because if (Claim’) were correct, one could easily establish that his own theory is subject to the same shortcoming – you can’t use it to show any sentence to be true. Here is why. One needs to be clear about what the theory currently under discussion consists of. It is the principles (A), (E), (K), and (BRAG), which are to be used together with all unproblematic assertable sentences, such as ‘There are giraffes’. It is not clear exactly what the unproblematic assertable sentences would be, but we know enough to discuss certain features of the theory. Suppose that (Claim’) is true, so that (BRAG) together with (A), (K), (E), etc are not sufficient to derive ‘True s’ for any sentence s. Now Read’s background logic is classical. So if you remove (BRAG) from the set of assumptions (thus going back to Read’s own theory), you still won’t be able to derive ‘True s’ for any sentence s. Removing BRAG merely results in having less resources in the theory. It means that you can prove less, not more. So if (Claim’) is right (as I suspect it is), Read’s original system is also not sufficient to derive ‘True s’ for any sentence s. This might seem implausible. After all, if you can show that things are not true, then if you want to show that a sentence s is true, just show that its negation isn’t true. Wouldn’t this do the trick? No, it wouldn’t. This strategy would have to employ the assumption that if the negation of a sentence isn’t true, then the sentence itself is true: removing an initial negation from a sentence reverses its truth-value. Call this principle (UNNEG): ¬T‘¬S’ ⇒ T ‘S’

(UNNEG)

(UNNEG) does not always hold. For example, it leads to inconsistency in the case of the liar paradox. For Read has shown: ¬TL 4 I say ‘suspect’ here because it would take a substantive argument to show this. For example, one would need to establish the consistency of the theory, since if it were inconsistent you could establish anything. I think that Read’s theory is consistent, but I don’t have a proof.

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that is: ¬T ‘¬TL’ Applying the questionable principle (UNNEG), we could infer: T ‘TL’ But by (C) we could then infer: TL which contradicts what Read has proved. So (UNNEG) is an unacceptable principle, and we cannot use it to show a sentence true by showing its negation false. What to do? It seems plausible that many simple sentences say only what they explicitly state. We could massively increase the number of sentences that can be determined to be true by classifying simple atomic sentences whose predicates are not semantical as sentences which say only what they say explicitly. (That is, they say only what logically follows from what they say explicitly.) For example, we could stipulate: ∀p[(‘Snow is white’:p) → (Snow is white ⇒ p)] Each such sentence S is easily shown to satisfy the Tarski biconditional: T‘S’ ⇔ S This massively increases the number of sentences that can be shown to be true. But it is only a step in the right direction. There are hosts of sentences that are not atomic, and hosts of harmless sentences with semantical predicates (such as “ ‘Grass is green’ is true”). I think that the odds of being able to specify which sentences say only what they explicitly say are low. (I think that this would be like the effort to define ‘grounded sentence’ in Kripke’s truth theory.) A good account of truth still eludes us, lying somewhere between the inconsistent (T) and Read’s too-weak (A).

Chapter 8 MODELS FOR LIARS IN BRADWARDINE’S THEORY OF TRUTH

Greg Restall∗ Department of Philosophy, The University of Melbourne, VIC, Australia [email protected]

Abstract

Stephen Read’s work on Bradwardine’s theory of truth is some of the most exciting work on truth and insolubilia in recent years [4, 5]. In this paper, I give models for Read’s formulation of Bradwardine’s theory of truth, and I examine the behaviour of liar sentences in those models. I conclude by examining Bradwardine’s argument to the effect that if something signifies itself to be untrue then it signifies itself to be true as well. We will see that there are models in which this conclusion fails. This should help us elucidate the hidden assumptions required to underpin Bradwardine’s argument, and to make explicit the content of Bradwardine’s theory of truth.

Keywords: Liar Paradox, Model Theory, Modal Logic, Relevant Logic.

As has been made clear in many of the papers in this volume, the crucial feature in Bradwardine’s theory of truth is the notion of signification. Expressed by a ‘connecticate’, which I shall write with the simple infix colon “:”, whenever t is a singular term and p is a sentence t:p is another sentence, to be read ‘t signifies that p’, or simply ‘t says that p.’ Bradwardine uses signification to define predicates of truth and falsehood: t is false if and only if it signifies something that is not the case, and it is true if and only if it signifies something, and everything it signifies is the case. Truth and falsity are defined notions, where the definitions ∗ Opponent.

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utilise signification and what we now call propositional quantification.1 F t is (∃p)(t:p & ¬p) – t is false if and only if it says something that is not the case. (Notice the syntax. The colon for “says that” binds more tightly than the conjunction, so “t:p & ¬p” is the conjunction of “t:p” and “¬p”.) Similarly, T t is (∃p)(t:p) & (∀p)(t:p → p), where ‘&’ expresses some kind of conjunction and ‘→’ expresses some notion of implication. (For smooth exposition, I will introduce yet one more definition: Dt is (∃p)(t:p). This says that t is declarative: it says something. So, T t is Dt & (∀p)(t:p → p).) The distinctive feature of Bradwardine’s approach is not merely this definition of truth. It is what I will call Bradwardine’s axiom: Definition 1 (Bradwardine’s Axiom). Every proposition signifies or means contingently or necessarily everything which follows from it contingently or necessarily [4]. We may render the condition in the following way: If t:p then if (if p then q) then t:q Rearranging the conditionals, we might have another formulation If (if p then q) then (if t:p then t:q). The crucial issue in understanding Bradwardine’s axiom is what form of conditional expression might be used in formulating it. What conditionals feature?

8.1

Classical Collapse

If all of these conditionals are material, then we have (p ⊃ q) ⊃ (x:p ⊃ x:q) which we might call the material Bradwardine Axiom. The material axiom collapses almost all distinctions concerning signification. Fact 2 (Bradwardine’s Collapse). Under the material Bradwardine axiom • The non-declarative objects (the x such that ¬Dx) say nothing • The false declarative objects say everything • The true declarative objects say all and only what is the case. 1 The vocabulary may cause some confusion. What Bradwardine calls a ‘propositio’ is the denotation of the singular term ‘t’ in t:p. The propositio signifies. Propositional quantification, in modern terminology, is quantification into sentence position. I will attempt to avoid talk of ‘propositions’ in any substantial sense which might ask us to choose between modern and medieval terminology here.

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In other words, what an object signifies is completely determined by whether or not it is declarative, and, if declarative, whether it is true or false. Proof. The proof turns on the behaviour of the material conditional. We have q ⊃ (x:p ⊃ x:q) since q entails p ⊃ q. It follows from this that if q is the case, then if x says anything (if x is declarative) then x says q. All declarative objects say everything that is the case. On the other hand, we have ¬p ⊃ (x:p ⊃ x:q) In other words, if there is some p that x says, that is not the case, then x says that q too. But q is arbitrary. In other words, if x is false, then x says everything. All that remains are the non-declarative objects, but by definition, these are those that say nothing at all. It follows that in the presence of the material Bradwardine’s axiom, the falsehoods say everything, the truths say all and only what is the case, and the non-declaratives say nothing. What is said collapses into this tripartite division. Notice that the principles grounding this collapse are that q entails p ⊃ q and that ∼q entails p ⊃ q. Both of these principles are valid in intuitionistic logic as well, so an intuitionistic understanding of Bradwardine’s axiom fares little better that the classical one. An intuitionist cannot conclude (on the basis of logic alone), that t is either true or false: the argument for this relies on the intuitionistically invalid law of the excluded middle. However, even intuitionistically, given the material Bradwardine axiom, all truths say the same thing (namely, everything that is the case), and all falsehoods say the same thing (namely everything). The extra ‘wriggle room’ provided by the failure of the law of the excluded middle may provide more discrimination in some things may say, but this collapse of signification among the truths and the falsehoods is nonetheless crippling for Bradwardine’s programme. So, we must move further afield in logical space to find an appropriate conditional to express the connection required in Bradwardine’s axiom. It seems to me that there are two major options on the table for retaining Bradwardine’s axiom without collapse. The first is to move to a kind of relevant implication. This is Read’s preferred option, and it has the virtue of explicitly allowing for what Bradwardine draws to our attention: both necessary and contingent consequences [3]. Relevant implication may be very robustly contingent without being material or extensional. A crucial principle in strong relevant logics is the principle of assertion: p → ((p → q) → q)

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which cannot hold if the conditional has any modal force. The fact that p is the case does not mean that in other circumstances where p → q is the case, then q is the case, if those other circumstances need not be ones where p is true.2 In relevant logics such as Anderson and Belnap’s R, the principle of assertion holds and the conditional expressed is contingent. The other option is to ignore contingent consequences to concentrate on necessary ones: we require merely that if p strictly implies q, then t:p strictly implies t:q. In this paper, I will consider models which encompass both choices. We shall look at models in which p → q entails t:p → t:q where ‘→’ expresses some kind of non-truth-functional conditional, whether relevant or strict. These are intensional conditionals, and so, modelling them will require intensional models.

8.2

Intensional Models

The models of this section will be structures in which we can interpret sentences in the language in which Bradwardine’s theory is expressed. That is, sentences in a language containing the connecticate ‘:’, the conditional ‘→,’ quantification over objects ‘(∀x)’, and quantification into sentence position ‘(∀p)’. (We will later consider conjunction and negation, but the current suite of items of the language will suffice for this section.) The models we will consider will allow us to interpret sentences in the language. Definition 3 (Frames). A frame is a structure P, R, O, {Da : a ∈ P }, {Sd : d ∈ D} with • A set P of points • A ternary relation R on P , interpreting the conditional • A non-empty set O of objects • subsets Dp of O of objects declarative at point p • A binary relation Sd on P for each d ∈ Dp 3 The points are points of evaluation in a frame: sentences are evaluated as holding or not relative to a point in the frame. In modal models, these 2 So, this rules out temporal understandings of the conditional, in which p → q holds if whenever p holds, q holds. We may have p true now. It does not follow that whenever p → q holds, q holds, since later, p may fail to hold. 3 We will only evaluate S from point a if d is in D , so we can add the restriction that aS b a d d only if d ∈ Da , but we need not do so, as it makes no difference to do without it.

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points are consistent and complete evaluations of the language (they model “possible worlds” if you like to think of it that way). In models for relevant logics, the points also evaluate formulas in the same way, but the requirement of consistency and completeness (with respect to negation) is not imposed. Formulas with free variables are also evaluated at points, but this must take into account not only the point at which evaluation occurs, but also the value of the variable. The value of a variable in term position will be a member of the domain O of objects, since this is the possible semantic value for a term. The value of a variable in sentence position will be a set of points, since this is the possible semantic value for a sentence. (If two sentences hold at exactly the same set of points, they are indistinguishable as far as a model is concerned.) So, an assignment α of values to the variables in our language will assign to each object variable an element of O and each propositional variable a subset of P . Now we have enough structure to define evaluations on our frames. Definition 4 (Models). Given a language with a number of atomic predicates, a model M on a frame P, R, O, {Da : a ∈ P }, {Sd : d ∈ D} is determined by the interpretation of the non-logical vocabulary. • For an n-place predicate, [[F ]] : P → P(On ). The extension of an n-place predicate is a function returning a set of n-tuples of objects for each point in the domain. • For a constant term t, [[t]] ∈ O. The extension of a term t is a choice of an object in O. We extend the notion of the denotation of a term to be relative to an assignment of variables [[t]]α to include the interpretation of variables: [[x]]α is the value that α assigns to the variable x. For uniformity, we write “[[t]]α ” for all terms t, even for constant terms where the extension does not depend on the assignment α. Given this information, a model M defines a relation  of satisfaction, evaluating formulas relative to assignments α and points a ∈ P : • M, α, a  p iff the point a is in the value that α assigns p. • M, α, a  F t1 · · · tn iff the n-tuple [[t1 ]]α , . . . , [[tn ]]α  is in the extension [[F ]]a . • M, α, a  A ∧ B iff M, α, a  A and M, α, a  B. • M, α, a  ¬A iff M, α, a  A.4 4 This clause treats negation as boolean. In a wider range of models for relevant logics, we would rather do without boolean negation in favour for a negation that goes further in respecting relevance considerations. Here, however, boolean negation will suffice.

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• M, α, a  A → B iff whenever Rabc, if M, α, b  A then M, α, c B. • M, α, a  (∀x)A iff M, α , a  A for every x-variant α of α. • M, α, a  (∀p)A iff M, α , a  A for every p-variant α of α. • M, α, a  t : A iff [[t]]α ∈ Da and for each a where aS[[t]]α,a a , we have M, α, a  A. The novelty in this definition is the last clause, evaluating signification. A claim of the form t : A holds at point a if and only if (1) the denotation of the term t is an object that is declarative at point a, and (2) in every point accessible from a, by way of the binary relation S[[t]]α (the relation determined by the denotation of t) the formula A holds. This makes t : A act as a normal modal operator when the denotation of the term t is declarative. I have discussed the choice of this interpretation elsewhere [7]. The choice of this interpretation is motivated by two factors. First, it allows us to simply construct models in which we can investigate what does not follow from the Bradwardine axiom. Given a model in which the Bradwardine axiom holds, if something else does not hold, then it is not a consequence (relative to the background logic of the model, at least) of the Bradwardine axiom. Second, the novel consequences of this particular interpretation are not, in themselves, overly problematic. For example, one consequence of this interpretation is as follows. If we interpret A ∧ B as holding at a point if and only if A and B both hold at that point,5 then it follows that t : A and t : B entails t : A∧ B. If t : A and t : B both hold at a point a, then [[t]]α is declarative at a, and A and B both hold at all of the points S[[t]]α -accessible from a. It follows that A ∧ B holds at all of these points, and hence that t : A ∧ B holds at a. So, the closure of signification under conjunction is a consequence of how we have interpreted it in these models. I do not know whether or not Bradwardine explicitly or implicitly assumes this condition, but it does not seem unduly implausible. So, models in which it holds do not (on this account at least) look problematic. The reader with a little experience of models of modal logic may think that I have skewed these models in favour of relevant logics by using a three-place relation to interpret the conditional, instead of a two-place relation. It is true that the generality of a three-place relation is used to model relevant logics such as R (in which logical truths such as p → p need not hold at every point, since we want to find counterexamples to 5 This is the case in standard models for modal logics, and also in ternary relational models for relevant logics, so it is not a particularly controversial assumption.

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the validity of the argument from q to p → p). However, they may be used to interpret strict conditionals from modal logic. For example, if we set Rabc to hold if and only if b = c, then the resulting conditional “→” interpreted by R is the strict implication of the logic s5, where necessity is interpreted as truth at every point.6 Now, we have enough information to interpret the language, and to define entailment as preservation of holding at points in our models. These models make satisfaction closed under entailment (if all A points are B points, then all points at which t : A holds are points at which t : B holds), but the Bradwardine axiom does not necessarily hold. It is not necessarily the case that A → B entails t : A → t : B. For this, we need to impose one condition connecting the relation R and the relations Sd . Definition 5 (Bradwardine Frames). A frame is a Bradwardine frame if and only if the following conditions hold between R and Sd : • For all points a, b, c, if Rabc and d ∈ Db then d ∈ Dc too. • For all points a, b, c, c , if Rabc, d ∈ Db and cSd c , then there is some point b where bSd b and Rab c . This suffices to ensure that if A → B holds at a then so does t : A → t : B, so A → B entails t : A → t : B in our models, as required. Fact 6 (Bradwardine’s Axiom in Bradwardine Frames). In any model on a Bradwardine frame, if A → B holds at a, then so does t : A → t : B. Proof. Suppose that A → B holds at a in our model. To show that a  t : A → t : B (we suppress mention of α and M since these do not vary in this proof) we consider b and c where Rabc, and b  t : A. We wish to show that c  t : B too. Since b  t : A we have [[t]]α ∈ Db . By the first condition on Bradwardine frames, we have [[t]]α ∈ Dc too.7 Now that [[t]]α ∈ Dc , we can ask the second part of the question concerning t : B at c. Suppose that cS[[t]]α c . Does c  B? The second condition on Bradwardine frames tells us that since Rabc and cS[[t]]α c , we have some b where bS[[t]]α b and Rab c . Since b  t : A and bS[[t]]α b , we have b  A and since Rabc and a  A → B we have c  B as desired. This concludes the proof.8 6 These

are the models discussed in my earlier paper [7]. reasoning would have failed had we allowed the denotation of t to vary from point to point, or we would have had to impose a more complex condition connecting R and the declarative objects. 8 This result is straightforwardly extended to a correspondence result. Suppose that a frame is not a Bradwardine frame. Then it is not difficult to construct a model such that there is a point a at which p → q holds but x:p → x:q does not. 7 This

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Many frames are Bradwardine. In fact, all frames for strict conditionals are Bradwardine. A conditional is strict if the ternary relation modelling it is essentially a binary relation. We have A → B at a if and only if all of the points accessible from a are such that if A holds there, so does B. As ternary relational frames, Rabc only when b = c. Fact 7 (Strict Implication Frames Are Bradwardine). If Rabc only if b = c, then the frame satisfies the Bradwardine condition. Proof. Immediate consequence of the definition of the condition. If R is a genuinely ternary relation, the Bradwardine condition has some bite. Not every frame on a ternary relation is Bradwardine. Example 8. Let P be the set of positive natural numbers {1, 2, 3, . . .}, and define Rlmn if and only if n divides both l and m evenly.9 If we have some object d that is declarative at 4 but not 2 then the first part of the Bradwardine frame condition fails, since R6 4 2 (2 divides 6 and 4) but d is declarative at 4 but not 2. If we have an object d that is declarative everywhere, and a relation Sd such that and nSdm if and only if n = m, then we have, for example, R2 2 2 and 2Sd 3, but there is no number m such that R2 m 3 since 3 does not divide 2. So, many frames do not satisfy the Bradwardine condition. However, it is not too difficult10 to construct frames for the relevant logic R which satisfy the Bradwardine condition. These frames will occupy us for the next sections.

8.3

Liars and Bradwardine’s Argument

Bradwardine’s most interesting contribution to the discussion of the Liar paradox is the argument to the effect that if something says of itself it is false, then it also says of itself that it is true. The argument, as discussed by Read, is as follows: . . . suppose s : F s, that is, suppose some proposition, s, says of itself that it is false, and suppose that it is false. By [the definition of F ], it follows that something s says fails to obtain: (∃p)(t:p & ¬p), if not that s is false then something else s says, call it q. Then if it’s not q that fails to hold, it must be F s that fails to hold, i.e., F s ⇒ (q → ¬F s) (∗), indeed, by Residuation and Bivalence, (F s & q) ⇒ T s. But s : F s, and

9 This

10 Not

is a model for the positive fragment of the relevant logic R. too difficult with the aid of a computer, at least.

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s:q, so by [the Bradwardine condition], s : T s. Thus any proposition which says of itself that is not true (or false), also says of itself that it is true. [5, page 311]11

This argument essentially uses a notion of conjunction, expressed by “&,” and a notion of entailment, expressed by “⇒,” together with negation. The argument uses a number of principles: the definition of the falsity predicate F , and a strong version of the Bradwardine axiom – we infer from s : F s and s:q, with (F s & q) ⇒ T s to s : T s. This is not only the closure of signification under entailment, but also the kind of conjunction expressed by “&.” This principle seems properly stronger than Bradwardine’s axiom as I have stated it, but (given a reasonable interpretation of “&”), it seems not unreasonable. The other significant step in the argument is the inference to (∗): the conclusion that F s entails q → ¬F s, for the particular choice of q – the “something else” said by s. I will show that this step does not follow from the Bradwardine condition, by constructing models in which it fails. Example 9 (A Modal Model). Let P be the two points a and b. Let Rabc if and only if b = c, so the logic of implication is the strict implication of the modal logic s5. Let O, Da and Db contain the object l. We set xSl y if and only if x = y. That is, aSl b and bSl a, but neither aSl a nor bSl b. Let λ be a term whose denotation is the object l. We will show that λ : F λ holds at a and at b, but that λ : T λ fails at both a and at b. This model provides a counterexample to Bradwardine’s argument in the background logic s5. First, let p hold at b but not a. Then λ:p holds at a, since p is true at all of the points Sl accessible from a (namely, b). However, ¬p also holds at a, so we have (λ:p) ∧ ¬p, so (∃p)(λ:p ∧ ¬p) holds at a. In other words, F λ holds at a. By symmetric reasoning, F λ holds at b too, since here, (λ : ¬p) ∧ ¬¬p holds. Therefore, F λ holds both at a and at b. Therefore, since l is a declarative object at both a and b, λ : F λ holds at a and b too, since F λ holds at every point accessible from a, and at every point accessible from b. So, the object l is a liar: it signifies of itself that it is false. Does it signify of itself that it is true? No. Since F λ holds at a and b, T λ fails at both a and b. Therefore, λ : T λ fails at a (since at the Sl accessible point, b, T λ fails), and it fails at b (since at the Sl accessible point, a, T λ still fails). So, in this model the Bradwardine axiom holds, we have an object that signifies itself to be false (in each point of the model), it is false 11 I

have harmonised the notation with that used in this paper, and marked a step with ‘(∗)’ for later reference.

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(in each point of the model), yet it does not signify itself to be true. This is a counterexample to Bradwardine’s argument. Step (∗) in the argument fails, since the other principles used hold in our model. What this means we will consider soon. Before that, however, we will show that counterexamples may be constructed in relevant models as well. Example 10 (A Relevant Model). This is more difficult construction, since models for relevant logics are more complicated. I will not go through the details of models for the relevant logic R here. Appropriate texts to read are numerous and widely available [1–3, 6]. Instead, I will sketch some simple models of a logic stronger than R, the Boolean relevant logic KR. These models are simple, and they do not force us to answer difficult questions concerning the interaction between signification, Truth, Falsity and an intensional (non-Boolean) notion of negation. A relation R is a KR relation if and only if • There is some point 0 such that R0ab if and only if a = b. • Raaa for every point a. • Rabc iff Rbac iff Racb for all points a, b, c. • If R(ab)cd then Ra(bc)d. That is, if there is some e where Rabe and Recd then there is some f where Rbcf and Raf d. The conditions are motivated as follows: the first, R0ab if and only if a = b tells us that there is a point at which the conditionals that hold are those that are valid on the model. In particular, at 0, A → A holds. (It may fail at other points: if Rabc where b =  c, and A holds at b but not c then A → A fails at a.) For the second, Raaa tells us that if A and A → B hold at a then B holds at a too. In other words, A, A → B entails B. For the third, the first component (Rabc iff Rbac) tells us that if A holds at a then (A → B) → B holds at a too. The second component (Rabc iff Racb) tells us that if A → B holds at a, then ¬B → ¬A holds at a too, if we interpret “¬” in the usual Boolean manner: ¬p holds at a point iff p fails at that point. The final and most complicated condition tells us that if A → B holds at a, then (C → A) → (C → B) holds at a too. These models interpret the whole of the classical relevant logic KR. Now we can construct our particular KR model in which Bradwardine’s conclusion fails. The domain of points of this model is the set {0, 1, 2}. We interpret the ternary relation R in the following way: R 0 1 2

0 1 2 0 1 2 1 012 12 2 12 012

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where Rabc holds if and only if the number c is found in the a-row and b-column of the table. In other words, we have R110, R111 and R112 (this is the “012” in the middle of the table), but R121 and R122 but not R120 (since we have “12” in the 1-row and 2-column), and so on. This is a model for KR. Verifying this (especially the last condition) is a non-trivial matter. Now, take λ to be a term in our language, and let its denotation be an object l that is declarative at every point. Let the relation Sl be defined by setting 0Sl 1, 1Sl 2 and 2Sl 1. We can think of Sl as a function, where 0sl = 1, 1sl = 2 and 2sl = 1. When we model “λ :” we can then say that λ : A holds at a if and only if A holds at asl . The first thing to verify is that the Bradwardine condition holds in this frame. In this case, we need to verify that if Rabc then Rabsl csl . This condition is satisfied in our model, and it is not tedious to check. If a = 0, then R0bc iff b = c and then, R0bsl csl too, since bsl = csl . When a = 0 if Rabc, we have Rabsl csl , since a = 0 and bsl = 0 (sl sends each point to 1 or 2 but not 0) and in this case, Rabsl 1 holds and Rabsl 2 holds. We know that csl is either 1 or 2, and hence Rabsl csl . So, the condition holds. It is a Bradwardine frame. Now T λ is false at every point, and F λ is true at every point. Choose a point b. We choose the extension of the proposition p so that it is true at every point other than b. Then at b, λ:p is true, since p holds at bsl . Yet at b, ¬p is true (negation is Boolean). So, at b we have (∃p)(λ:p∧¬p), where we treat ∧ extensionally in the usual fashion. Indeed, if we treat conjunction intensionally, by setting A & B true at a point c when there are a and b where Rabc, A holds at a and B holds at b (which is required for the residuation condition A & B → C if and only if A → (B → C)), then we also have, at b, (∃p)(λ:p & ¬p) since Rbbb. With either definition of falsity (using the intensional conjunction & or the extensional conjunction ∧) F λ holds at b. Similarly, since λ:p → p fails at b (we have Rbbb), it follows that T λ fails at b. So, at every point, T λ fails, ¬T λ holds and F λ holds too. So, from any point b, ¬T λ is true at bsl , so at b, both λ : F λ and λ : ¬T λ holds. On the other hand, from b, we have T λ failing at bsl , so λ : T λ fails at b. So, we have a relevant counterexample to the conclusion to Bradwardine’s argument to the effect that if λ : F λ then λ : T λ. We have an object that says of itself that it is not true (at each point of the model). It doesn’t say of itself that it is true (at any point of the model). What do these counterexamples mean for the Bradwardine’s argument? The argument appeals to a principle – the step marked (∗) – that fails in these models. Consider the relevant model, though what I write holds in the s5 model as well. What is curious in this model is the way

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that what λ signifies varies from point to point. Let p hold at 2 only, and let q hold at 1 only. Then it follows that from the perspective of point 0, we have λ:p, but we do not have λ:p at point 2 itself. At point 2 we have λ:q instead, where q and p are jointly inconsistent (they hold together nowhere). So, at point 0 we have both λ:q

q → (λ:p ∧ ¬(λ:q))

which, when you think about it, is a very odd combination. At point 0, λ says that q (it says “the world is like point 2”), but if that is the case, then λ is not true, since in that circumstance, λ no longer says that q, it says that p. To consider the step (∗) in Bradwardine’s deduction. At 0, in the model, we have λ : F λ, and indeed, F λ holds at 0. It does follow that something λ says fails to obtain. In the argument, we are asked to “call it q.” There are two different interpretations for any q that fails to obtain at 0 and is said by λ at 0. It can hold at {2} or at {1, 2}. Let’s take {2}, as it is the most specific statement, from which the most follows. Our sentence q is true at 2 only. We do have (λ:q ∧ ¬q) at 0, and also (λ:q & ¬q). Then the argument continues: Then if it’s not q that fails to hold, it must be F s that fails to hold, i.e., F s ⇒ (q → ¬F s) (∗)

This is the step that fails in our model. At 0 we have F λ. But we do not have q → ¬F λ at 0, since R022 and q holds at 2, yet ¬F λ fails at 2, since F λ holds at 2 as it does everywhere in our model. Why does this step fail in our models? It seems to me that I cannot say that if q then ¬F λ, since we have that if q then λ would not have said that q. It would have said something else, had q been the case. In other words, we have a counterexample to the principle λ:q → (q → λ:q) in this model. The failure of this condition seems crucial, and it is quite possibly a principle one might implicitly assume – since it is valid when “→” is read materially. However, the principle seems suspicious to a relevantist. If λ signifies q, then how does this fact follow (relevantly) from q? If the only argument is an appeal to p → (q → p), this will cut no ice for the relevantist, since it is relevantly invalid. Can this principle be motivated on relevantist grounds, without collapsing distinctions of relevance? There seem to be three options on the table, if we wish to keep a rich theory of signification. 1 Endorse a modal Bradwardine axiom but not a material Bradwardine axiom. 2 Keep the relevant Bradwardine axiom, but reject Bradwardine’s conclusion that liars signify their own truth.

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3 Motivate a principle, such as t:p → (p → t:p), which might undergird step (∗). I do not know which choice ought to be made. I hope it suffices to present them, to allow us all to investigate the options, for Bradwardine’s intriguing and fruitful theory of truth.

Acknowledgments Thanks to Catarina Dutilh Novaes and to the audience at the 1st gpmr Conference in Medieval and Applied Logic for feedback on some of the ideas in this paper. My most hearty thanks to Stephen Read for introducing me to Bradwardine’s theory of truth, and for many emails and discussions about the details. This research is supported by the Australian Research Council, through grant dp0343388, and Keith Jarrett, through his recording of Shostakovich’s 24 Preludes & Fugues.

References [1] A.R. Anderson, N.D. Belnap, and J.M. Dunn. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992 [2] J.M. Dunn and G. Restall. “Relevance Logic”. In D.M. Gabbay, editor, Handbook of Philosophical Logic, volume 6, pages 1–136. Kluwer, Dordecht, Second edition, 2002 [3] S. Read. Relevant logic: a philosophical examination of inference. Basil Blackwell, Oxford, 1988 [4] S. Read. The Liar Paradox from John Buridan back to Thomas Bradwardine. Vivarium, 40(2):189–218, 2002 [5] S. Read. Symmetry and Paradox. History and Philosophy of Logic, 27:307–318, 2006 [6] G. Restall. An Introduction to Substructural Logics. Routledge, New York, 2000 [7] G. Restall. Modal Models for Bradwardine’s Theory of Truth. To appear in the Review of Symbolic Logic, special issue on Mathematical Methods in Philosophy, edited by Richard Zach, Alasdair Urquhart and Aldo Antonelli http://consequently.org/writing/ bradwardine

Chapter 9 ON A NEW ACCOUNT OF THE LIAR Comments on Stephen Read José M. Sagüillo∗ Department of Logic and Moral Philosophy University of Santiago de Compostela, Spain [email protected]

Abstract

The aim of this paper is to discuss some of the subtleties in the distinct way Stephen Read responds, in his paper “The Truth-Schema and the Liar” to the Liar paradox using an improved formulation of Tarski’s T-scheme. Examining the orthodox and the new account of the Liar shows two different ways of tackling the problem of modelling the underlying logic of Liar-type discourse. Specific discussion is provided of Read’s key notions of saying that and strict implication, which he uses to articulate his new intensional proposal.

Keywords: Argumentation, Consequence, Fallacy, Liar, Paradox, Read, Reasoning, Implication, Tarski, Truth, T-scheme.

9.1

Introduction

The aim of this paper is to identify and discuss some of the subtleties in the distinct way Stephen Read responds (using an improved formulation of Tarski’s T-scheme) to the Liar paradox in his paper “The TruthSchema and the Liar”. With this in mind, it may be useful to contrast the views of Tarski and Read on the Liar, showing thereby two different accounts of both the different problems identified under the same paradox-symptom and the corresponding different solutions proposed. A closer look at these two lines of response to the Liar paradox enables an assessment of some of the divergent key characteristics of Read’s new account. ∗ Opponent.

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Let us begin with Read’s assertion in his abstract that the Liar sentence falsifies the T-schema. This will strike orthodox interpreters of Tarski as puzzling since in Tarski’s account Liar strings – strings which are not well-formed – are out of the picture,1 Tarski’s definition of the truth predicate implies all of the so-called Tarski biconditionals, which are instances of the T-scheme. According to Read and others, some of these biconditionals are false, in particular those exhibiting the structure of Liar-type sentences. Read’s attempt to improve Tarski’s proposal stems from the observation that avoiding semantic closure of the language for which the truth predicate is defined precludes modelling Liar-type sentences. Read considers Tarski’s proposal to be an ad hoc exclusion. Thus, it is important to indicate from the beginning that Read’s use of the word ‘sentence’ departs from Tarski’s. Tarski [9, p. 155] says that in order to be true or false it is necessary for a sentence to say something, to be fully interpreted. In other words, Tarski uses the word ‘sentence’ to refer to interpreted or meaningful sentences, i.e., sentences expressing a proposition in a given interpretation, of which truth and falsehood are (coherently) predicable.2 We are thus faced with two ways of tackling the problem of modelling the underlying logic3 of Liar-type discourse. For the Tarskian it is prima facie incoherent to consider a Liar string a sentence which could be one of the substitution-instances of the T-scheme. In this context, the Liar was precisely invoked to introduce the need for a hierarchy of languages thus avoiding confusion of Liar-type strings or seemingly Liar-type sentences with sentences. It should not be overlooked that Tarski’s primary interest was in formalized or scientific languages; languages whose sentences are free from use-mention mistakes and free from ambiguity, whether arising from indexical (pace Davidson [4]) or from unstated ranges or domains of quantification (pace Hintikka [5]). In other words, Tarski-type truthdefinition does not apply to natural, or semantically closed, languages. This is considered by philosophers to be a shortcoming of the Tarskian analysis. It is this perceived shortcoming which Read has undertaken to overcome. Thus, Read demands “an account of the semantics of the Liar paradox and its ilk” involving a “better scheme” (the right one) 1 Tarski uses the word ‘sentence’ not in the contemporary sense of simply a string of characters, which in and of itself has no meaning, but rather he uses the word ‘sentence’ for what would properly be called a proposition-sentence, i.e., a sentence expressing a proposition. 2 Definition 12 of Tarski [9] establishes that a (meaningful) sentence of the language of the calculus of classes is a sentential function of that language with no free variables. 3 The expression ‘underlying logic’ is used in the sense of Church [1, pp. 57–58] and Corcoran [2, Sect. 2]. They regard it as a three-part system involving a grammar, a deductive subsystem and a semantics, without involving specific primitive terms of the domain to be axiomatized.

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without excluding Liar-type counterexamples by fiat. Moreover, under his proposal these counterexamples for the T-scheme “turn out to be simply false” if viewed in the light of his improved truth-scheme, every instance of which is true.

9.2

Inquiring into the Underlying Logic of the Liar

These two divergent accounts of the issue at hand are worth investigating. They stem from different diagnoses of the problem involved in the Liar paradox. In establishing a common point of departure, Read’s definition of “paradox” is useful. According to Read [8, p. 150] “[a] paradox arises when an unacceptable conclusion is supported by a plausible argument from apparently acceptable premises.” There are several key terms or concepts involved in this definition that are debatable, but for the sake of argument, I take this conception of paradox as one that involves human beings arguing from accepted premises to an unacceptable conclusion via a plausible argument or apparently cogent chain of reasoning. This paraphrase of Read’s definition of paradox is clearly doxastic. It involves people who may or may not have accepted correctly the premises or the conclusion or the chain of reasoning leading to the paradox. As Read [8, p. 150] indicates “[. . . ] three different reactions to the paradoxes are possible: to show that the reasoning is fallacious; or that the premises are not all true after all; or that the conclusion can in fact be accepted.” As will be shown below, in these terms, Tarski and Read offer different forms of reaction to the Liar. In the Epimenides version of the Liar, Read [8, p. 150–151] correctly indicates the unacceptable conclusion that is reached: namely, that Epimenides was both telling the truth and not telling the truth. Of course, if the premises were all taken to be meaningful by the intended audience, and provided that the other conditions are fulfilled, a paradox (in the foregoing sense) results. It is useful to point out that Tarski thought the Liar string was ill-formed due to a breach of the object/metalanguage boundary. Such breaches are “responsible” for the family of liar-type paradoxes. According to Tarski’s diagnosis, the problem lies in taking something that looks like an (interpreted) sentence to be a sentence, when actually it is not a sentence at all because it expresses incoherence rather than a proposition. It is well known that Tarski’s solution amounts to a paradox-avoiding restriction, based on a suitable hierarchy of languages. In this hierarchy, predicating truth or falsity of a given sentence of a certain n-level language must be expressed by a sentence belonging to an n + 1 level language, the first language being a sublanguage of the second. In my understanding of Tarski’s way out

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of the paradox, his solution amounts to considering reasoning on which the Liar paradox rests to be fallacious. The thinker doing the reasoning from premises to conclusion mistakenly employs as a premise a sentence that does not express a proposition. Hence, no cogent or correct reasoning is actually developed since no proposition is available in the first place to allow cogent step-by-step deduction. There is no first link in the chain of reasoning. Hence there is no (cogent) reasoning to the paradox but, rather, an inferential gap. Notice in this connection that rejecting or dismissing the illegitimate premise should not be taken as meaning “changing the previous belief (acceptance or rejection) with respect to the premises to the opposite belief” but rather as meaning “there was no real belief to begin with”.4 As Kripke [6, p. 695] correctly indicates, Tarski was inclined to dismiss natural language altogether from the definition. Moreover, the decision explicitly taken in Tarski [10, p. 349] was just “not to use any language which is semantically closed”. This shows that Tarski found reasons to be wary on this issue, adopting the simplest way of dealing with the problem without dwelling on the philosophical debate of the consistency5 of everyday language. Read proposes a different diagnosis of the same paradoxical symptom. It is that not all the meaningful sentences in the argumentation are true after all. In other words, his diagnosis focuses on the premises underlying the paradox rather than on its chain of reasoning. On this account, once the subject realizes his mistake he modifies his beliefs and now accepts that at least one of the premises is actually false. The paradox vanishes or is re-classified as an indirect proof of the falsity of at least one of the premises, namely the self-referential sentence. The procedure exemplifies a successful application of the hypothetic-deductive method: no false proposition can be deduced 6 from all true premises. Hence, no fallacy is thereby invoked. 4 Nonetheless, Read [8, p. 153] demands “a positive account of meaning” in order to explain the meaninglessness of the premise involved and he declares such an account unavailable. The question whether a set of sentences of a given language is consistent presupposes meaningfulness and Tarski says that “the problem of consistency has not exact meaning” with respect to everyday language. 5 Tarski (ibid.) says: “The problem arises as to the position of everyday language with regard of this point. At first blush it would seem that this language satisfies both assumptions (1) [semantically closed] and (2) [sound ordinary logic], and that therefore it must be inconsistent. But actually the case is not so simple. Our everyday language is certainly not one with an exactly specified structure. We do not know precisely which expressions are sentences, and we know to an even smaller degree which sentences are to be taken as assertible. Thus the problem of consistency has no exact meaning with respect to this language. We may at best only risk the guess that a language whose structure has been exactly specified and which resembles our everyday language as closely as possible would be inconsistent”. 6 ‘Deduced’ and ‘deduction’ are here used as ‘success terms’. Hence, no deduction contains a fallacy.

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In the final analysis the philosopher of logic faces the question of which model for the underlying logic of Liar-type discourse is more promising as an investigation of a logic, whether real or putative, underlying that type of discourse. On the one hand, if linguistic stratification is really necessary then the Liar is based on a fallacy and hence no contradiction is really generated. On the other hand, if the reasoning leading from premises to paradoxical conclusion is coherent and non-fallacious, then linguistic stratification is not after all necessary (at least for present purposes). The first is Tarski’s strategy, focusing on a grammatical deficiency of semantically closed languages. The second is Read’s strategy, which grants meaningfulness to the sentences held to generate the Liar. Of course, this is simplifying things somewhat. In order to show the Liar sentence to be false, and its falsity not in turn to imply its truth, Read needs to replace (T) by his improved schema.

9.3

Understanding Read’s Reformulation of the Truth-Scheme

Since whenever (T) is not properly restricted it generates false instances, its modified version should generate no false instances and render Liar-sentences simply false. Notice first that Tarski’s T-scheme (x is true if and only if p) involves two side conditions worth mentioning.7 The first is that ‘p’ is to be filled in with a [meaningful] sentence and ‘x’ is to be replaced by a name of that sentence. The second is that ‘if and only if’ is to be taken in its extensional or truth-functional sense. This second side condition clearly points to a non-modal or a de-modal conception of truth.8 As Read argues “The (T)-scheme presents in a stark and nominalistically acceptable form the correlation of language, on the left-hand side, and the world on the right-hand side”. Indeed Tarski’s scheme does not say necessarily, x is true if and only if p. According to the Correspondence Platitude (CP), ‘x is true if and only if things are as x says they are’. Read, on the other hand, holds that what a sentence (including a Liar-type string) says is not just that things are as the given sentence says they are, but rather that things are as however the given sentence says they are: “A sentence says many things, and however it says they are, they must be, for the sentence to

7 For

a discussion of the ontological and epistemological presuppositions of the use of schemata, see Corcoran [3]. 8 Tarski thought that modalities were mere “vestiges” of a psychological attitude. It may be worth recalling in this connection his purely quantificational characterization of the concept of logical consequence: a certain conclusion c is a logical consequence of a set of premises P if and only if every model of P is also a model of c.

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be true”. Displaying the many things a given sentence says involves – according to Read – considering not just what the given sentence says but also what each of its implications9 says, in a sense of ‘implication’ to be specified below. In order to articulate this idea Read proposes to replace (CP) with a more careful expression (S): x is true if and only if things are wholly as x says they are,

(S)

x is true if and only if however x says things are, they are.

(S’)

or

Truth-conditions given by (S) and (S’) are then represented by: Tx ⇔ ∀p(x:p → p)

(A)

where ‘x:p’ means ‘x says that p’ and ‘x’ designates a sentence. In addition, principle (K) (∀p, q)((p ⇒ q) → (x:p ⇒ x:q)) represents the closure of ‘saying that’ under strict implication. The move here is to identify what a given sentence says with what each of its strict implications says. According to Read, strict implication is understood as “the necessitation (or logical truth) of the material conditional”. Identifying what a sentence says, i.e., the proposition expressed by it, with what each of its implications says – a suggestion made by Read with his conception of “saying that” as closed under strict implication – may not seem intuitively right to someone not taking comfort from Wittgenstenian quarters.10 The idea here is that two sentences say the same when the truth-conditions of each are contained in the truth conditions of the other. My point is that intensional equivalence, i.e., extensional equivalence in every index or possible world, does not necessarily imply saying the same if what a sentence says in a given interpretation is the proposition it expresses. By holding a “robust” identity criterion for propositions two sentences may have the same “meaning” (intensional equivalence in the Wittgenstenian sense) without expressing the same proposition. If what a sentence says in a given interpretation is the proposition it expresses, where ‘proposition’ is taken in its “robust” sense, then that sentence does not say what any of its non-identical implications say. For example, no sentence in the premise set composed of the sentence 9 The word ‘implication’ is here used as a common noun and not as a relational object-language connective. 10 Compare Wittgenstein’s paragraph quoted by Read on page 6: “A proposition affirms every proposition that follows from it [. . . ] If p follows from q, the sense of ‘p’ is contained in the sense of ‘q’.”

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‘The morning star is The moon and The moon is the Evening star’ says what the implied sentence ‘The morning star is The evening star’ says. In fact, this implied sentence is not in that premise set. Moreover, the sentence expressing the proposition “No even number is odd”, does not say “It is not the case there is an even and odd number”, which is the proposition expressed by the sentence ‘It is not the case there is an even and odd number’. The two sentences are composed of different logical expressions and the two propositions expressed are composed of different logical concepts. Both different but logically equivalent sentences accurately express different propositions composed of different concepts.11

9.4

Understanding Read’s implication

Read takes implication to be strict implication (or some species of it). The following comments are intended to illuminate the kind of semantics behind his modal object-language connective, which Read does not specify, despite telling us that he thinks of it as “logical truth of the material conditional”. Read holds that the sentence ‘Every even number greater than 2 is the sum of two primes’ not only says that every even number greater than 2 is the sum of two primes, but also that 4 is the sum of two primes. In other words, 4’s being the sum of two primes is as Goldbach’s hypothesis says things are. 4’s being the sum of two primes is necessary, but of course not sufficient, for the truth of the hypothesis. Note that Read’s strict implication is enthymematic and not topicneutral if viewed from a purely formal viewpoint, and hence so are his intensional biconditionals.12 Thus, the sentence ‘4 is the sum of two primes’ is a material or enthymematic consequence of the sentence ‘Every even number greater than 2 is the sum of two primes’ in the following sense: given that the universe of discourse of the interpretation is the class of natural numbers and that the numerals and the predicates denote natural numbers and numerical properties in the way they do, it is impossible for the premises all to be true without the conclusion being true. Clearly this is a feature of the intended interpretation of the sentences.13 Similarly, there is also a sense in which the propositions expressed are implied: as long as the numbers are what they are, i.e., by virtue of the nature of the natural numbers, it is impossible for the premise to be true without the conclusion being true. In other words, it is by virtue of the subject matter of the propositions, not the notation used to express them, that 11 The point can also been made about (A’) Tx ⇔ ∀p(∃q(x:q ∧ (q ⇒ p)) → p), which Read incorporates regarding relevant context. 12 Read’s first footnote explicitly recognizes that his strict conditionals are enthymematic. 13 Mutatis mutandis with Read’s example of Socrates.

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the conclusion follows from the premise. In this sense the previous argument is “arithmetically valid”, i.e., it is arithmetically necessary for the conclusion to be true where the premise is true and it is arithmetically impossible for the premise to be true yet the conclusion false.14 Read does not explicitly identify the range of hidden premises he is willing to take into account, over and above a given sentence p, when p’s implicational closure is to be identified. This amounts, in turn, to his employing an unorthodox sense of logical truth for the semantics of strict implication. In fact, Read’s strict implication is not purely formal and it does not seem to pass the test of being truth-preserving in every domain of interpretation. Read summarizes the conceptual advantages of his intensional proposal by pointing out its two fundamental features. First, (A) provides a more perspicuous analysis of truth than (T). (A) requires that everything a sentence says must be the case for the sentence to be true whereas (T) establishes merely a condition of material adequacy. Second, (A) requires a logical equivalence: what a sentence says, it says of necessity. (T), on the other hand, requires merely a truth-functional equivalence.

9.5

Conclusion

The Liar is a key perennial topic prompting critical assessment of our material beliefs as well as of our favourite logical conceptions. Tarski’s orthodoxy takes a Liar string to be ungrammatical expressing no proposition. The outcome of the paradox on his account is a fallacy since a gap vitiates its chain of reasoning, and hence, no contradiction is actually deduced. Read considers Liar strings meaningful and shows them to be false by using his richer scheme, each of whose instances is true. Paradox vanishes when a meaningful Liar sentence L is taken as instance of the new scheme. This new premise, whose right-hand side says all that L says, serves to deduce ¬TL through a cogent chain of reasoning. 14 It is tempting to think that perhaps material validity in the previous sense might be related to the mediaeval concept of consequentia materialis. This consequence relation, which goes back at least as far as Peter Abailard (1079–1142), is not a truth-value relation like “material implication”. In his Dialectica, Abailard distinguished between complexional consequentia (“valid” secundum complexionem) and topical or local consequentia (“valid” ex natura rerum). Roughly, in this tradition a complexional consequentia is a logically true conditional statement, i.e., no statement in the same form has true antecedent and false consequent. Perhaps it would be more accurate to recall Abailard’s examples of complexional consequentia ‘Therefore-sentences’ or ‘So-sentences’ since the consequent is normally preceded by the expression ‘ergo’. A topical or local conditional statement is a true conditional whose consequent “follows” (perhaps “is inferred”) topically, by virtue of the subject matter, from the antecedent. As was already indicated, topical consequentia in this sense resembles enthymematic argumentative discourse. See in this context Footnote 1 of Read’s article. In Read [7] there is a discussion of Etchemendy’s views on representational semantics, a conception resembling material or enthymematic implication which Read seems to endorse, as opposed to Tarskian interpretational or formal semantics.

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Stephen Read’s new breakthrough elicits deductive comprehension of the Liar sentence by making it amenable to logic once his more perspicuous truth-condition is adopted. Undoubtedly the new scheme is intuitive but also philosophically loaded. My previous comments were intended to suggest some of the subtleties involved in the clarification of the intensional ‘saying that’ and its enthymematic implicational closure. These key-notions require further discernment to serve present purposes in full extent. Read is well aware of them and I look forward to his forthcoming insights to settle on their own grounds.

Acknowledgments This paper is part of research projects HUM2006-04955 and HUM200500365 of DGES (Ministry of Science and Technology of Spain). I am especially indebted to three referee reports for their observations. Thanks also to Shahid Rahman for comments and to Stephen McLeod for comments and advice on style.

References [1] Church, A. (1956/1996): Introduction to mathematical logic. 10th edition. Princeton, New Jersey: Princeton University Press [2] Corcoran, J. (1973): Gaps between logical theory and mathematical practice. In [1], 23–50 [3] Corcoran, J. (2005): Schemata: The concept of schema in the history of logic. Bulletin of Symbolic Logic 12, 219–240 [4] Davidson, D. (1969): True to the facts. Journal of Philosophy 66, 304–323 [5] Hintikka, J. (1975): A counterexample to Tarski-type truthdefinitions as applied to natural languages. Philosophia 5, 207–212 [6] Kripke, S. (1975): Outline of a theory of truth. The Journal of Philosophy 72, 690–716 [7] Read, S. (1994): Formal and material consequence. Journal of Philosophical Logic 23, 247–265 [8] Read, S. (1995): Thinking About logic. Oxford, New York: Oxford University Press [9] Tarski, A. (1936): On the concept of logical consequence. In Tarski (1983), 409–420 [10] Tarski, A. (1969): Truth and proof. Scientific American 220, 63–77

Chapter 10 THE LIAR CANNOT BE SOLVED György Serény∗ Department of Algebra Budapest University of Technology and Economics Budapest, Hungary [email protected]

Abstract

Attempting to resolve the Liar paradox in a simple and straightforward way, Stephen Read suggests a modification of Tarski’s famous truthschema that he claims turns the Liar paradox into an innocent sentence, which is simply false. In my note I examine the way Read tries to solve the paradox and argue that some doubts can be raised regarding the successfulness of his attempt. I analyze his project from two independent points of view. First, I try to shed light on some weak points in Read’s argument, then I present my own arguments to the effect that any revision of Tarski’s truth-schema can, in principle, be only a part of the solution to the Liar paradox.

Keywords: Truth, Truth schema, Liar paradox.

As a part of a bold attempt to resolve the Liar paradox in a simple and straightforward way, in a recent paper entitled The Truth Schema and the Liar, Stephen Read suggests a modification of Tarski’s famous truth schema (cf. [10]). After describing (two versions of) the new schema in a semi-formal way and arguing that it gives a more faithful account of our intuitive notion of truth than the one proposed by Tarski, Read purports to show that the application of his schema turns the Liar paradox into an innocent sentence, which is simply false.1 In this paper we try to convince the reader that Read looks for the solution to the Liar in the wrong direction. In fact, we shall demonstrate ∗ Opponent.

1 As he puts it, “It is shown that a better schema allows a uniform treatment of truth in which the semantic paradoxes turn out to be simply false.” (see the abstract of [10]).

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that, even if there existed a universally acceptable solution to the semantic paradoxes at all, a possibility that we shall argue can almost certainly be excluded, simply tampering with the truth schema cannot in principle take us closer to such a solution. Our paper consists of three main parts. In the first one, we describe briefly the way Read suggests to modify Tarski’s famous truth schema and touch on some questions related to the important fact that Read’s schemas are really only frameworks for creating new truth schemas. We discuss one of these questions in a little more detail and show that there seems to be no plausible way to obtain an essentially new notion along the lines of Read’s suggestions since the only promising realization of his schemas turns out to give back the original concept. In the second part, we turn to the analysis of the purported solution to the Liar. We analyze Read’s argument in three steps. After reconstructing Read’s derivation in a more acceptable form, we point out the main fallacy in his argument. We shall argue that his train of thought purporting to solve the Liar is fundamentally flawed. Indeed, it is essentially a reductio ad absurdum argument, which cannot be applied to prove consistency; this kind of argument is a completely inappropriate tool in the context of paradoxes. The reason is that to argue by contradiction, in just these cases, is simply begging the question. The second step of our analysis will be an indirect reinforcement of our claim that Read could not have proved what he claims he has proved. Actually, we shall supplement Read’s proof of the fact that the Liar is not true with a proof of the fact that the Liar is true using the most obvious concrete realization of his truth schemas. It follows that Read’s claim that the application of his truth schemas is in itself enough to eliminate the paradox is not justified. We finish our detailed examination of Read’s argument by an attempt to describe the role the close analogy between the truth conditions of the paradox of Epimenides (in its original form)2 and those of the Readian Liar (in which truth is interpreted according to one of Read’s truth schemas) could play in creating Read’s illusion of solving the age old paradox. The results of our analysis described above show that Read has failed to accomplish what he claims he has accomplished. On the other hand, these results do not exclude the possibility of solving the paradox along the lines of his suggestions, that is, of finding a realization of his modified truth schemas the application of which does eliminate the paradox. Now, in the third major part of our paper, we shall show that Read has not only not succeeded in solving the Liar, but his attempt to achieve 2 Not

to be confused with the Eubulidean Liar, see footnote 15.

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this goal was doomed to failure. Indeed, as we shall see, taking into consideration some minimal and very plausible assumptions concerning the notion of truth, one can, in principle, exclude the possibility of solving the paradox the way Read suggests. But there is even more to it than that. We conclude our paper by presenting a simple theorem (which can be considered as a possible abstract version of Tarski’s theorem on the undefinability of arithmetical truth) showing that all attempts to solve the Liar paradox (formulated, of course, in a semantically closed language) without a radical revision of the most fundamental laws of our logic are bound to fail. Where it was possible, we have retained the notation of the original article. On the other hand, we often omit the explicit reference to it or give only partial one (e.g. ‘Sect. 2’ instead of ‘[10], Sect. 2’) since the complete reference will always be clear from the context. Further, since Read’s basic definitions and statements are formulated in terms of strict implication, for the sake of definiteness, we need a system for handling strict implication (though the core of our criticism does not at all depend on the specific character of the system we work in). We choose it to be S4, the weaker of the two best-known basic systems introduced by Lewis.3 Nevertheless, everything we shall do could be done in much weaker systems as well. In the course of our various derivations, we shall make extensive use of results presented in [6], the fundamental work on strict implication. Therefore, for the sake of brevity, we refer to these results simply by the numbers that serve to identify them in the original (e.g. ‘11.6’ stands for ‘[6], 11.6’).

10.1

Read on Truth

In the introductory part of his paper, Read characterizes as equivocal the generally accepted principle according to which (CP) x is true if and only if things are as x says they are (where what replaces ‘x’ is a name of a sentence), since the expression ‘x says’ in it can be interpreted in different ways. On the one hand, we can interpret it in a narrow sense taking into consideration only what x says explicitly. On the other hand, the phrase ‘x says’ also has a wider meaning. What a sentence says might indeed be considered to include all those states of affairs that can, in some way or other, be associated with the one that is actually described by it explicitly. Read claims that only this latter interpretation yields an adequate truth schema and considers Tarski’s schema 3 Cf. e.g.

[6], p. 501; [1], p. 5; [2], p. 22.

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(T) x is true if and only if p (where what replaces ‘x’ is a name of the sentence which replaces ‘p’) as being false since it corresponds to the narrow interpretation. In fact, according to Read, a sentence may say more than it says explicitly; generally, x says not only that p, but much more than that. He therefore suggests a suitable modification of (CP) that excludes the narrow interpretation: (S) x is true if and only if things are wholly as x says they are, or (S’) x is true if and only if however x says things are, they are. Read then goes on to give two possible formal schemas corresponding to these informal definitions. The first takes the following form: (A) Tx ⇐⇒ (∀p)(x : p → p), where ‘x : p’ represents the phrase ‘x says that p’, strict implication is denoted by ‘⇒’ and the material conditional by ‘→’.4 Further, the set of all p’s such that x : p is required to be (strictly) closed under strict implication: (K) (∀p)(∀q)((p ⇒ q) → (x : p ⇒ x : q)). The alternative formal definition is a single schema:
 (A’) Tx ⇐⇒ (∀p) (∃q)(x : q ∧ (q ⇒ p)) → p . A very important point is to be made here. As Read himself is ready to admit, in the absence of any exact definition of x : p, neither of the schemas (A) and (A’) above can be considered a complete definition of truth (cf. Sect. 2): 5 4 It is clear from the context that T denotes the truth predicate (even though this notation is not fixed). 5 In other words, for a given x, the requirements expressed by the formulas above do not characterize the set S(x) consisting of all p’s such that x : p (and hence the truth-conditions of Tx) in a unique way even if we suppose the self-evident (but not explicitly stated) assumption (let it be denoted by (C)) that every sentence does say what it explicitly says. Indeed, let us suppose that x is given and s is what x explicitly says. In the case of (A’), apart from the condition that s ∈ S(x), there is no restriction imposed on S(x), thus every set S of sentences containing s can be considered as S(x). As far as the other case is concerned, the condition (K) is in itself not enough to define S(x) uniquely. For example, if q is a sentence such that it is not the case that s ⇒ q (generally, such a q obviously exists), then there are two different interpretations of x : p that satisfy both (C) and (K). Actually, if we interpret x : p as s ⇒ p and s ∧ q ⇒ p, respectively, then (C) and (K) are satisfied in both cases (cf. 12.1, 11.2, and the proof of Proposition 4 (ii)), but the two interpretations are different since, under the first interpretation, it is not the case that x : (s ∧ q) (cf. 16.33) while, under the second one, it is the case that x : (s ∧ q) (cf. 12.1).

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(A) is not the final analysis of truth: that awaits an analysis of ‘saying that’.

But he does not give such an analysis. Thus there are some fundamental questions that remain open. Is there any consistent way to make the definitions above complete? Will the result of such a completion, if it exists at all, turn out to be an acceptable theory of truth? Will it be in concert with our fundamental common sense intuitions concerning the notion of truth? Will it, among others, retain the basic properties of our intuitive concept? And even if the answers to these questions turned out to be in the affirmative, a further question inevitably arises. Will the resulting notion of truth prove to be really new, that is, will it be essentially different from the original Tarskian one? Read seems to be convinced that the doubts expressed by these questions can be put to rest. His remarks and examples suggest that his conviction is based on the assumption that, in case one could not find a better candidate, strict implication would fill the gap. Among others, he makes the following remarks (see Sect. 2): If x is true, things must be however x implies they are. Conversely, if things are however x implies they are, x will be true. “Safety first” would support the identification of ‘however x says they are’ with ‘what x implies’. This may make ‘how x says they are’ more precise than is natural, but in the context of (CP) it is welcome precision.

Let us, then, follow this route and try to make ‘however x says they are’ precise. To this end, let us introduce the following notion: Definition. x explicitly says that s if and only if what replaces ‘x’ is a name of the sentence which replaces ‘s’. Now, defining ‘x says that p’ as ‘s strictly implies that p’, that is, taking x : p to stand for s ⇒ p, where s is what x explicitly says, we can turn the formulas above into formally complete truth schemas. Nevertheless, some confusion may arise as to exactly what Read has in mind. On the one hand, all but one of his four examples of ‘what x says’ seem to reinforce the assumption that this phrase is to be interpreted as ‘what x implies’. Indeed, the three illustrations to this effect can be found at the beginning of Sect. 2 (the emphasis is in the original): ‘Every even number greater than 2 is the sum of two primes’ not only says that every even number greater than 2 is the sum of two primes, but also in particular that, e.g. 4 is the sum of two primes. ‘All men are mortal’ says that if Socrates is a man then Socrates is mortal. Anything of the form ‘p and q’ says inter alia that p [. . . ].

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On the other hand, the fourth example, which is given more emphasis since it is mentioned twice, seems to contradict the hypothesis above. Actually, Read makes the following remarks at the end of Sect. 2 (the emphasis is mine): That snow is white is, arguably, all that ‘snow is white’ says. [. . . ] But not all cases are as straightforward as this. [. . . ] (T) is a special case of (A) where, as for ‘snow is white’, a sentence reveals explicitly all that it says.

But, obviously, there are a lot of sentences that ‘snow is white’ implies. To mention but a few ones, ‘something is white’, ‘snow is white or grass is green’, or ‘it is not the case that snow is not white’ etc.6 Therefore, assuming that a sentence says everything it implies, ‘snow is white’ says not only that snow is white; that snow is white is not all that ‘snow is white’ says. In any case, no plausible alternative to interpreting the expression ‘saying that’ as ‘implying that’ is in sight. Thus it is worth showing that, under this interpretation, Read’s schemas do not yield a new notion of truth. As a matter of fact, as we shall show below, if we interpret ‘x says that p’ as ‘what x explicitly says strictly implies that p’, then the truth schemas (A) and (A’) provide a notion of truth that is logically equivalent to the original Tarskian one. Of course, this result does not come as a great surprise since, taking truth in the old sense, a sentence is true just in case all of its consequences are true, which turns out to be essentially the same concept of truth that is defined by Read’s schemas provided ‘says’ means ‘implies’. We first need some simple facts. Proposition 1. 
(i) (s ⇒ s)
→s ⇒s (ii) (∃q) (s ⇒ q) ∧ (q ⇒ p)) ⇐⇒ (s ⇒ p) Proof. (i) We apply a simple semantic argument since, on the one hand, the modal logic system K is complete with respect to the class of all frames (cf. [1], pp. 78–80 or [2], p. 25) and K ⊆ S4 (cf. [1], p. 5), on the other hand, the modal logic formula
[(
(A → A) → A) → A] can easily be shown to be valid in all frames.7 In order to prove this latter claim, let 6 Actually, ‘if snow is white, then something is white’ is just the application of the existential generalization, formally ϕ(c) → (∃x)ϕ(x), which is the basic existential inference corresponding to, and (as a formula schema) being logically equivalent to, the universal specification, formally (∀x)ϕ(x) → ϕ(c). But the first two of Read’s examples are just different applications of the universal specification! 7 Recall the definition of strict implication: A ⇒ B =
(A → B) (cf. e.g. [1], p. xviii or [9], p. 81).

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us denote by M, w |= A the fact that the formula A is true at the world w ∈ W in the model M = W, R, V  (where W = ∅ is the set of possible worlds, R ⊆ W 2 is the accessibility relation and V is the valuation on the set of atoms, that is, the mapping of the set of atoms into the power-set of W ). Now, let M = W, R, V  and w ∈ W
be arbitrary and let  x be such that wRx. We should prove that M, x |=
(A → A) → A → A, that is, we should infer M, x |= A from M, x |=
(A → A) → A. To this end, let us suppose that M, x |=
(A → A) → A. This means that M, x |= A follows from M, x |=
(A → A). Since M, x |=
B for all propositional tautologies B, we have M, x |=
(A → A). Therefore M, x |= A, which is what was to be proved. (ii) can be proved in a purely syntactic way as follows:8 (1) (s ⇒ p) ⇐⇒ (s ⇒ (s ∧ p)) (2) (s ⇒ p) ⇒ (s ⇒ (s ∧ p)) (3) (s ⇒ (s ∧ p)) ⇒ (s ⇒ s) ∧ (s ⇒ p) (4) (s ⇒ p) ⇒ (s ⇒ s) ∧ (s ⇒ p) (5) (s ⇒ p) ⇒ (∃q)(s ⇒ q) ∧ (q ⇒ p) (6) (s ⇒
q) ∧ (q ⇒ p) ⇒ (s⇒ p) (7) (∃q)
(s ⇒ q) ∧ (q ⇒ p) ⇒ (s ⇒ p) (8) (∃q) (s ⇒ q) ∧ (q ⇒ p) ⇐⇒ (s ⇒ p) Proposition 2.
Let s be an arbitrary sentence. Then  (i) s ⇐⇒ (∀p)
(s ⇒ p) → p ,  (ii) s ⇐⇒ (∀p) (∃q)((s ⇒ q) ∧ (q ⇒ p)) → p . Proof. (i) (1) s ∧ (s ⇒ p) ⇒ p
 (2) [s ∧ (s ⇒ p) ⇒ p] ⇒ [s ⇒ (s ⇒ p) → p ]
 (3) s ⇒ (s ⇒
p) → p  (4) s ⇒
(∀p) (s ⇒ p)→ p
 (5) (∀p) (s ⇒ p) → p ⇒ (s ⇒ s) → s
(6) (s ⇒
s) → s ⇒ s (7) (∀p) (s ⇒
p) → p ⇒ s  (8) s ⇐⇒ (∀p) (s ⇒ p) → p

16.33 (1) 19.62 (2), (3) (4), Exist Gener 11.6 (6) (5), (7)

11.7 14.26 (1), (2) (3), Univ Gener Univ Spec Proposition 1 (i) (5), (6) (4), (7)

8 In the derivations below and in all subsequent ones, the column on the right contains the references to the inference rules and/or the theorems justifying the actual steps. Recall that the numbers without brackets are those identifying a theorem in [6]. Further, of course, we follow Read in assuming that, in the logical system in which his formulas are given and his derivations take place, the most basic rules of quantification hold also in the case of quantification over propositions. On the other hand, we usually omit the explicit reference to the most fundamental inference rules and theorems (such as e.g. Substitution or the Principle of the Syllogism cf. [6], p. 125).

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(ii) follows from (i) and Proposition 1 (ii) by substitution ([6], p. 125). This proposition immediately yields Corollary. The truth predicates corresponding to (T), (A), and (A’) respectively are all logically equivalent if x : p is interpreted as s ⇒ p, where s is what x explicitly says.9 What we have just shown is that the most plausible possible revision of Tarski’s truth schema along the lines of Read’s suggestions is simply not a revision at all since the revised schemas give back the original notion of truth. Later we shall see in detail how this fact manifests itself in the specific case of the Liar paradox. On the other hand, no matter how implausible it may seem, we cannot, in principle, exclude the possibility that there exists an exactly definable, meaningful, and intuitively acceptable interpretation of the common language phrase ‘x says that p’ that, through Read’s schemas, turns out to yield a proper modification of Tarski’s truth schema. Thus it may be of some interest to examine, in a general way, the consequences of defining truth according to Read’s schemas independently of the specific features of any possible realization. In fact, as his treatment of the Liar paradox suggests, Read, in effect, purports to establish just the claim that, independently of the exact definition of ‘saying that’, the application of any notion of truth that fits into the framework given by one of his truth schemas solves the Liar paradox. In a little more detail, his tacit claim seems to be that, independently of the concrete way we might succeed in defining a modified truth schema in exact formal terms, merely defining the truth of a sentence in such a way that its truth conditions include something more than what the sentence explicitly asserts is, in itself, enough to solve the Liar paradox. That might be the reason why Read is satisfied with his incomplete definition of truth and makes no attempt to improve it. On the one hand, being convinced that, at the worst case, the strict implication would do the job, he assumes the existence of well-defined truth schemas satisfying his requirements. On the other hand, he is simply not interested in the concrete realizations of his schemas since he thinks he can prove that the solution to the Liar does not depend on their specific nature. We have already seen that his first assumption is a mistaken one; strict implication does not provide a proper revision of Tarski’s truth schema. Let us now see what we can say about his second (tacit) assumption that any application of his schemas leads to a straightforward solution to the Liar paradox. 9 And, of course, the phrase ‘if and only if’ in (T) is interpreted in the sense of strict equivalence.

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10.2

Read on the Liar

The central part of Read’s paper is the section entitled Truth-theory without Paradox. In this section, Read presents the main argument purporting to justify his suggested modifications of Tarski’s truth schema. Actually, he makes an attempt to show that, if we interpret the notion of truth according to one of his previously defined truth schemas, then the Liar sentence L will prove not to be paradoxical at all. What is more, and this fact will later turn out to be crucial, his argument is essentially independent of the exact definition of what L ‘says’. First we analyze his argument and point out its shortcomings. Then we show that the paradox can be reconstructed even if we define truth according to Read’s schemas. Finally, we make an attempt to explain the logical motives for his trying to solve the Liar paradox in just the way he has chosen to do it.

10.2.1

The Argument

In order to show that the application of his truth schemas really turn the Liar into an ordinary false sentence, Read presents an argument consisting of two parts. In the first part he gives two formal derivations of the formula

(α) TL ⇐⇒ (¬TL ∧ TL ∧ . . .) from the results of applying his truth schemas to L, and draws the final conclusion that L is not true. In the second part, he argues that (α) implies not only that L is not true, but also that L cannot (be proved to) be true. Let us examine the two parts of this argument separately.

Proofs of (α). As far as the application of his first schema is concerned, Read writes (cf. Sect. 4): Consider L, for example. L says that L is not true, that is, L : ¬ TL. It may say more: call it q, that is, suppose L : (¬ TL ∧ q). (If ¬ TL is all that L says, we can simply let q be ¬ TL, or a tautology.) By (A), TL ⇐⇒ (∀p)(L : p → p) i.e., TL ⇐⇒ (¬TL ∧ q) since ¬TL ∧ q is all L says, whence ¬TL ⇒ ¬(¬TL ∧ q) i.e., ¬TL ⇒ (TL ∨ ¬q).

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L : ((TL ∨ ¬q) ∧ q)

whence L : TL since ((TL ∨ ¬q) ∧ q) ⇒ TL. Thus, if L says ¬TL, it says TL as well. That is, L : (¬TL ∧ TL). So by (A), TL ⇐⇒ (¬TL ∧ TL ∧ . . .) whence ¬TL.

Obviously, this is not a completely sound argument. Indeed, since the truth of L is defined in terms of what L says, what L says has to be given independently of the truth conditions of L. This means that, in any derivation concerning the truth of L, all what L says should be completely defined before we apply a truth schema (formulated in terms of what L says) to L. Otherwise, what L says becomes dependent on its truth, which in turn has been defined just in terms of what it says. Thus both the truth of L and what L says are determined in a circular, and hence a vacuous way. The derivation above commits this fallacy since what L says (TL, among others) is derived from its second formula
 TL⇐⇒(¬TL ∧ q) , which, in turn, being the definition of the truth of L, is based on what L says, and hence depends heavily on what L says. Consequently, what L says is derived from a formula that depends on the result of this derivation. An analogous objection can be formulated regarding the second derivation.10 On the other hand, this criticism does not affect the essence of the argument. Indeed, as the proof of the proposition below shows, the same result can straightforwardly be obtained from both (A) and (A’). First we need a simple fact connecting the results of the application of these truth schemas. We accept Read’s approach of applying his truth-schemas to the Liar (see Sect. 4). Thus, depending on which schema is used, we assume that, for some q such that L : q, (B) or (B’)

TL ⇐⇒ (¬TL ∧ q)
 TL ⇐⇒ (∀p) ((¬TL ∧ q) ⇒ p) → p .

10 Nevertheless, that is a sound argument until the last but one step. Thus the derivation of the formula q ⇒ TL is unobjectionable. This fact will be used later in the proof of Proposition 4.

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Lemma. (B) can be derived from (B’). Consequently, everything that can be derived from (B) can also be derived from (B’). Proof. Suppose that (B’) holds, that is,
 TL ⇐⇒ (∀p) ((¬TL ∧ q) ⇒ p) → p . Now, applying Proposition 2 (i) to s = ¬TL ∧ q, we have
 (¬TL ∧ q) ⇐⇒ (∀p) ((¬TL ∧ q) ⇒ p) → p . Since the right-hand sides of the last two equivalences are identical, we obtain TL ⇐⇒ (¬TL ∧ q), which is what we had to prove. Proposition 3. Suppose that truth is defined by (A) or (A’). Then TL ⇐⇒ (¬TL ∧ TL ∧ . . .) and ¬ TL. Proof. Suppose that truth is defined by (A). Applying it to the Liar, we have (1) TL ⇐⇒ (¬TL ∧ q) (2) TL ⇒ (¬TL ∧ q) (3) TL ⇒ TL (4) TL ⇒ (¬TL ∧ TL ∧ q) (5) (¬TL ∧ TL ∧ q) ⇒ TL (6) TL ⇐⇒ (¬TL ∧ TL ∧ q) (7) (¬TL ∧ q) ⇒ ¬TL (8) TL ⇒ ¬TL (9) (TL ⇒ ¬TL) ⇒ ¬TL (10) ¬TL

(B) (p. 168) (1) 12.1 (2),(3), 19.61 11.1, 11.2, 11.4 (4), (5) 11.2 (2), (7) 12.87 (8), (9)

As far as the case of (A’) is concerned, it is enough to refer to the previous lemma. So, at last, we have (α) at our disposal. Therefore we can turn to the second part of the argument.

The Basic Fallacy. Since Read’s main claim that the Liar is simply a false sentence is based on (α), the key question is: what, in fact, does (α) imply? Let us see Read’s answer. After finishing his derivation of (α) (and ¬TL), he writes (cf. Sect. 4, all emphases are mine):

170

G. Serény It may not be immediately obvious what a breakthrough this is. Of course L is not true. [. . . ] But the longer argument concerning L shows more than that L is not true. [. . . ] To use (A) or (A’) to show that L is true we need to show both that L is not true and, in addition, that L is true. But that is impossible. L cannot be true, for to be true, it would have to be both true and not true. Nothing can be both true and not true. So L cannot be true. L is not only not true, it cannot (also) be true, as (A) reveals.

Unfortunately, this is mere hand waving. Indeed, the statement that nothing can be both true and not true holds only if we assume that the logical system concerned is consistent. Actually, the reductio ad absurdum argument above tacitly presupposes the consistency of our logical system containing the Liar sentence. To be consistent is to be free of contradictions, that is, a system is consistent just in case no absurdity, no logical impossibility can be derived within it. But the aim of Read’s whole endeavour is to show just this fact, i.e., to demonstrate that – unlike the original Tarskian truth concept, the application of which (as witnessed by the Liar paradox) leads us to inconsistency – his modified notion of truth makes it possible to retain the assumed consistency of our logical system even in the presence of the Liar sentence! Consequently Read’s argument simply begs the question. In fact, his claim that the application of his truth schemas leads to the solution of the paradox would only be justified if he could prove just the very fact he takes for granted, i.e., that no logical impossibility can be derived within our logical system containing the Liar sentence. In the absence of such a proof the argument is unfounded. Thus the statement that it is impossible to derive the truth of the Liar and, together with it, the claim that the paradox has been solved turns out to be a mere declaration. While it is true that, if we apply Read’s truth schemas to the Liar, then the paradox does not manifest itself as openly as in the classic case, this fact alone is not enough to logically exclude the possibility that it lurks, as yet unnoticed, somewhere in the background and will show up some day. A further remark is in order here. The weakness of Read’s argument can simply be tested by applying it to the original Tarskian Liar. Actually, it is easy to show that the counterpart of (α) can be derived in the original case as well (see the derivation in footnote 14), thus nothing excludes the possibility of repeating the concluding part of Read’s above quoted argument word-by-word with the same conclusion, having in mind, in this second case, the original Liar. Thus, if we accepted Read’s argument as a solution to the Readian Liar, we would have to accept it as a solution of the original Liar as well. Therefore, the fact that nobody questions the paradoxicality of the original Liar can be taken as indirect evidence that Read’s argument must be inconclusive.

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We have therefore shown that Read has not succeeded in proving that his modifications of Tarski’s truth schema, in the general form in which they are given, can in themselves lead to the solution of the Liar paradox. But, of course, the possibility of solving the paradox along the lines of Read’s suggestions has still remained open. Let us then examine this possibility.

10.2.2

The Classic Liar Reappears

Now, since we are in the position to see Read’s entire argument as a whole, we can recognize an important feature of his reasoning that stems directly from the fact that he defines his truth schemas in very general terms, that is, he imposes virtually no restriction on the notion of ‘saying that’. Consequently, the line of thought presented by Read should be, and indeed is, essentially independent of the definition of ‘saying that’. To put it in another way, what Read in effect does is not what it appears to be at first sight. In fact, what he gives is not a pair of two single well-defined truth schemas (A) and (A’). Depending on the concrete choice of x : p in them, these are rather two schemas of truth schemas. Therefore, his derivations based on these schemas are just schemas of derivations. Thus what his argument, in effect, purports to show is that any concrete realization of these schemas (if there is any at all) would lead to the solution of the Liar paradox. Actually, apart from the self-evident assumption that every sentence does say what it explicitly says, the only condition used in his argument is that ‘saying that’ is closed under strict implication.11 Consequently, if Read’s (tacit) assumption that any concrete realization of his schemas leads to the solution of the Liar paradox is correct, then TL cannot be proved no matter how the notion of ‘saying that’ is defined as long as it satisfies the following conditions: (i) every sentence does say what it explicitly says (ii) ‘saying that’ is closed under strict implication. Now we shall show that if we interpret ‘saying that’ as ‘strictly implies that’, then the conditions above are satisfied and yet TL can be proved even if T is defined according to Read’s schemas (A) or (A’). This result refutes Read’s argument, showing not only that he has not succeeded in establishing the impossibility of proving TL, but also that his attempt was bound to fail: to prove (in those general terms as he tried to) that it is impossible to prove TL is itself impossible.

11 Moreover, this sole non-trivial condition only appears in the first derivation of (α) (based on (A)), and it can be omitted even from there as is witnessed by our version of this derivation appearing in the proof of Proposition 3.

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Proposition 4. Let us define the relation x : p in the following way: x : p if and only if s ⇒ p, where p and s are arbitrary sentences and x is the name of s (that is, x explicitly says that s). Then (i) Every sentence does say what it explicitly says : if y explicitly says that q, then y : q. (ii) ‘Saying that’ is closed under strict implication : (∀p)(∀q)((p ⇒ q) → (x : p ⇒ x : q)). (iii) Let T be the truth predicate defined by either (A ) or (A’) and let L be the name of the Liar sentence. Then TL. Proof. (i) q ⇒ q by 12.1. (ii) The modal logic system K4 is complete with respect to the class of transitive frames (cf. [1], pp. 78–81) and K4 ⊆ S4 (cf. [1], p. 5). Thus it is enough to prove that the modal logic formula

(A → B) →

(C → A) →
(C → B) is valid in all transitive frames, since, by the definition of strict implication, this implies what we need, i.e., that the formula
 (p ⇒ q) → (s ⇒ p) ⇒ (s ⇒ q) is a theorem of S4. We shall use the notation of Proposition 1. Let M = W, R, V  and w ∈ W be arbitrary such that R is transitive. What we have to prove is
 M, w |=
(A → B) →

(C → A) →
(C → B) . Let us, therefore, suppose that (1)

M, w |=
(A → B)

and let x be such that wRx. We have to prove that M, x |=
(C → A) →
(C → B). Let us suppose, to this end, that M, x |=
(C → A) and let y be arbitrary such that xRy. Then (2)

M, y |= C → A.

On the other hand, wRy since R is transitive and wRx, xRy. Consequently, (1) implies that M, y |= A → B, which in turn, by (2),

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yields M, y |= C → B. Since y was arbitrary such that xRy, we have M, x |=
(C → B), which is what was to be proved. (iii) (a) Suppose that truth is defined by (A). Applying it to the Liar, we have (1) (2) (3) (4) (5) (6) (7)

TL⇐⇒(¬TL ∧ q), L : q (¬TL ∧ q) ⇒ TL ¬TL ⇒ q ¬TL q ¬TL ∧ q TL

(B) (p. 168) (1) L explicitly says that ¬TL, L : q by (1) Proposition 3 (3), (4) (4), (5) (2), (6)

(b) Suppose that truth is defined by (A’). Then the result of the application of this schema to the Liar, i.e., the formula (B’) defined on p. 168 holds. On the other hand, by our Lemma, (B) can be derived from (B’). Therefore, TL follows from (a) above. Alternatively, making use of Read’s intermediate result that q ⇒ TL can be derived from (B’), we can obtain TL using (B’) itself:
 (1) TL⇐⇒(∀p) ((¬TL ∧ q) ⇒ p) → p , L : q (B’) (p. 168) (2) q ⇒ TL [10] Sect. 4 (see also our footnote 10) (3) ¬TL ⇒ q L explicitly says that ¬TL, L : q by (1) (4) ¬TL ⇒ TL (2), (3) (5) ¬TL Proposition 3 (6) TL (4), (5) The proposition we have just proved demonstrates the fact that Read’s argument is fundamentally wrong. It is not only that it is incomplete; it cannot be made complete at all. If Read’s line of reasoning to establish the impossibility of proving TL were correct, then his argument would have to go through with ‘saying that’ being defined as ‘strictly implies that’. The reason is simple: ‘strictly implies that’ satisfies all the conditions that the argument makes use of; in other words, nothing in the argument excludes the possibility of applying it to the particular case in which ‘L says that q’ is interpreted as ‘what L explicitly says strictly implies that q’. But this application, clearly, leads to a contradiction. In the light of this proposition, we can recognize the way the errors in Read’s argument are connected. In his derivations of ¬TL, q is declared to be such that L : q. But this specification is empty. Actually, it remains a vacuous declaration since in the absence of any definition that characterizes it in a unique way, ‘L : q’ is simply an undefined symbol. The derivation in which it occurs is only a meaningless framework, which becomes a meaningful train of thought only in case L : q is interpreted in some way or other. Such an interpretation, however, in addition to

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filling in the frame and thus turning it into a definite and meaningful argument, also widens the set of premises the possible derivations can be built on. Actually, it describes some kind of connection between L and q in terms of notions that have already been defined. This connection manifests itself in a condition that is satisfied by a sentence p just in case L : p. Therefore, q should satisfy this condition. Obviously, this fact is a proper restriction imposed on q.12 This restriction, in turn, may be added to the set of premises that we can use to prove TL, and hence, naturally, might affect the provability of TL. Consequently, it should be taken into account when one analyzes the impact that Read’s modification of Tarski’s truth schema exerts on the solution to the Liar. (Actually, what we did in Proposition 4 above was just to show that the restriction corresponding to a possible interpretation of L : q does indeed make it possible to prove TL.) Leaving out of consideration the possible consequences of this restriction is a logical fallacy. In the particular case of the Liar, failing to draw a possible conclusion might be a logical mistake.13 But Read is convinced that his reductio ad absurdum argument conclusively excludes the possibility of proving TL, and therefore he is satisfied with being able to prove ¬TL without using any additional conditions. That might be the reason why he overlooks the obvious unexploited possibility of proving TL by imposing some restrictions on q. On the other hand, of course, the fact that interpreting ‘saying that’ as ‘implying that’ leads us back to the original Liar paradox is not strikingly unexpected since we have already proved that the Tarskian and Readian truth concepts are, under this interpretation, essentially equivalent (see the corollary to Proposition 2 on p. 166). It is interesting to observe that the illusion that the application of his truth concept solves the paradox makes Read blind to the proper requirements that are needed to eliminate the paradox. Actually, he writes (see the end of Sect. 4): [. . . ] the correct theory of truth is given by (A) – or (A’) [. . . ]. As applied to L, we obtain the correct truth-condition: TL ⇐⇒ ( ¬TL ∧ TL )

What he does not realize is that, as we have already noted earlier, the formula he characterizes as ‘the correct truth-condition’ can, without any essential modifications, also be derived straightforwardly if the notion of truth is interpreted in the classic Tarskian way, the only (and obviously, 12 Clearly,

we can exclude the trivial case in which L : q for every sentence q. imagine someone who, in the course of his or her analysis of the original Liar, being satisfied with the conclusion that the Liar cannot be true, does not go on to show that, nevertheless, it cannot be not true either. 13 Indeed,

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from the given point of view, entirely immaterial) difference being that the strict equivalence is replaced by its material counterpart.14 If the truth concepts of Tarski and Read are different at all, this difference does not consist in the fact that TL is equivalent to an absurdity only if we use the latter one. But then, what does this difference really consist in? Actually, as we shall see, just what Read supposes this difference implies may account for the surprising fact that he is unable to recognize the obvious weaknesses of his argument. In fact, the most immediate question raised by our analysis, if it is correct at all, concerns the reasons why Read has fallen into the trap of illusion of solving the Liar paradox.

10.2.3

The Epimenidean Trick

In order to locate the source of Read’s mistaken belief that the application of his truth schemas turns the Liar into an innocent false sentence, let us see again the way he treats this sentence (cf. Sect. 4): . . . By (A), TL ⇐⇒ (∀p)(L : p → p) i.e., TL ⇐⇒ (¬TL ∧ q) since ¬TL ∧ q is all L says, . . .

Clearly, the crucial move is to insert a sentence q into the formula TL ⇐⇒ ¬TL

(λ)

(describing the classic Tarskian truth conditions of the Liar in terms of strict implication) to obtain the truth conditions of the Liar in the case when truth is interpreted according to Read’s first truth schema: TL ⇐⇒ (¬TL ∧ q)

(B)

for some q such that L : q. The beneficial effects of this modification (which we may call the Epimenidean trick since (B) shows a striking 14 Indeed, let T be the Tarskian truth predicate. Applying, in this case, the rules of ordinary elementary propositional calculus, we obtain the counterpart of the proof of Proposition 3. Actually, L is true if and only if what it (explicitly) says is the case, and L (explicitly) says that L is not true:

(1) (2) (3) (4) (5) (6)

TL ←→ ¬TL TL → ¬TL TL → TL TL → (¬TL ∧ TL) (¬TL ∧ TL) → TL TL ←→ (¬TL ∧ TL)

(1) (2), (3) (4), (5)

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analogy to the formula describing the truth conditions of the paradox of Epimenides15 ) are, at first glance, obvious. On the one hand, as we have already seen, it follows from (B) that ¬TL. Consequently, L has indeed turned out to be not true. On the other hand, in a very sharp contrast to (λ), it does not seem obvious at all that TL can also be derived from (B). The mere existence of q, breaking the symmetry of the formula (λ), excludes the possibility of obtaining the paradox the way it was obtained in the original case. Thus the paradox has apparently been eliminated. It seems to be solved independently of any realization of the modified truth schema as long as the truth of a sentence is defined in such a way that its truth conditions include something more than what a sentence explicitly asserts.16 There is, however, a fact that our analysis in the previous section showed to be crucial. The move to insert q into the formula (λ) has to be justified. And this justification does not come free. We have to 15 Apart from the fact that strict implication replaces the material one, (B) is nothing other than the formula describing the truth conditions of the paradox of Epimenides, the famous close relative of the Liar paradox. The relation between the two paradoxes is so close that there is a widespread terminological confusion. Actually, the Liar (invented, in fact, by Eubulides) is very often mistakenly identified with an equally perplexing but essentially different statement attributed to Epimenides. (Eubulides of Miletus, a Megarian philosopher, was a pupil of Euclid of Megara and lived in the fourth century bc. When asking “A man says that he is lying. Is what he says true or false?”, he was fully aware of the paradoxical nature of his question. Epimenides, on the other hand, was a Cretan seer, author of religious and poetical writings, who lived (probably) in sixth century bc. There is no evidence that he was interested in logic.) In fact, the sentence ‘The Cretans are always liars’ said by Epimenides, himself a Cretan, though, at first sight, seems paradoxical, is not paradoxical at all. More precisely, it is not paradoxical just in case we assume that some other Cretan some time uttered a true sentence (see also footnote 17). Of course, as was noticed by Church and Prior, in this case the unacceptable alternative forces us to consider our assumption as a logical necessity. Therefore we run into a different kind of awkwardness: we can establish a contingent fact by pure logic (see [8]). As Kleene puts it (cf. [4], p. 39): “It is logically unsatisfactory that we should escape paradox only through the historical accident that some Cretan existed who sometime told the truth.” This shows that the Epimenides paradox is also a close relative of the Curry paradox (cf. [3]). The Epimenidean Liar is what Epimenides, in effect, said:

(E) What I am saying now is not true and nothing ever uttered by another Cretan is true. Now, apart from the fact that strict implication is replaced by its material counterpart, the formula expressing the Tarskian truth conditions for E is of the same form as that expressing the Readian truth conditions for L: TE ←→ (¬TE ∧ q) for some q expressing the fact that nothing ever uttered by another Cretan is true. 16 It would not be, of course, a strikingly unexpected result if the paradoxicality of the Liar could indeed be eliminated along this lines. Actually, it is well known that breaking its circularity may lead to the solution of the Liar since, obviously, if we anchor it to the body of the (logical) world in some way or other by an additional statement, which we suppose it (tacitly) makes, then it loses the very property (i.e., its ungroundedness) that makes it paradoxical (cf. [5], p. 66, [11], p. 118). The central question is whether we can find an appropriate justification for grounding the Liar. As we shall see, this is highly improbable.

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pay for it. The price is the requirement guaranteeing that q is not an arbitrary sentence but a sentence such that L : q. And this requirement, like it or not, is reflected by a restriction imposed on q describing a necessary connection between L and q. Now as we have already seen, this restriction, breaking the connection (provided by q) with the (logical) word outside the narrow ungrounded word of the Liar can, in some cases, repair its lost circularity.17 That is, there are cases in which the beneficial effects of the insertion of q are eliminated, and we get back again our paradox, witnessing the fact that it cannot be solved in those general terms in which Read formulated his argument. On the other hand, of course, what we have done so far does not exclude the other possibility. There might, in principle, exist interpretations of ‘saying that’ not affecting fundamentally the beneficial effects of the Epimenidean trick. Therefore, while we have proved that Read’s argument to solve the Liar paradox in terms of his very general truth conditions has irreparably failed, this fact, of course, does not exclude the possibility of finding a solution along the lines of Read’s suggestions. Indeed, the question whether or not we could find concrete truth conditions satisfying one of Read’s general schemas in terms of which the Liar paradox can be solved has still remained open.

10.3

The Robustness of the Liar

So far we have examined the arguments Read has presented. We have shown that these arguments were fallacious, and hence they have failed to prove that if we accept Read’s truth schemas instead of Tarski’s one, then we can circumvent the Liar paradox. From now on, we take another route and present our own arguments in order to show that Read’s effort was doomed to failure since it is, in principle, impossible to solve the Liar along the lines of his suggestions. First we shall answer the question we formulated at the end of the previous section, and show that one cannot hope to find a suitable interpretation of ‘saying that’ that, when used to define truth through Read’s schemas, does not give rise to paradox. There is no such interpretation. No concrete realization of Read’s truth schemas leads to the solution of the paradox. Secondly, we shall prove a more general theorem 17

We can see again the analogy with the paradox of Epimenides here. In fact, as Kleene characterizes the relation between the paradox of Epimenides and the Liar ([4], p. 39, the emphasis is mine): “The falsity of the statement [made by Epimenides] requires that there has been, or will eventually be, a Cretan who at some time tells the truth. Had the quoted statement been the only one any Cretan makes, we should have a paradox.” That is, just as in the case of the Readian Liar, an additional assumption (which affects the very statement that cuts apart the vicious circle), healing its lost circularity, repairs the broken paradox.

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that we hope will convince the reader that what Read tries to do, that is, to eliminate the Liar paradox without giving up at least one of the fundamental intuitive requirements concerning the notion of truth, is completely hopeless. Tarski formulated the dilemma created by the Liar in the following way (see [13], pp. 348–349), the emphasis is mine): If we now analyze the assumptions which lead to the antinomy of the liar, we notice the following: (I) We have implicitly assumed that the language in which the antinomy is constructed contains, in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term “true” referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called “semantically closed”. (II) We have assumed that in this language the ordinary laws of logic hold. [. . . ] [T]he assumptions (I) and (II) prove essential. Since every language which satisfies both of these assumptions is inconsistent, we must reject at least one of them. It would be superfluous to stress here the consequences of rejecting the assumption (II), that is, of changing our logic (supposing this were possible) even in its more elementary and fundamental parts. We thus consider only the possibility of rejecting the assumption (I). Accordingly, we decide not to use any language which is semantically closed in the sense given.

Actually, confronted with the Liar, we must make a choice. What the Liar shows is that the possibility of talking about the truth of sentences within the same language that these sentences belong to, contradicts the most fundamental syntactic laws of formal logic, the laws of contradiction and that of excluded middle, according to which, for any sentence18 s, the sentences ¬ (s ∧ ¬s) and s ∨ ¬ s are logical theorems.19 The way out of the paradox, therefore, seems to be to give up either at least one of these fundamental laws, or else the possibility of talking about the truth of sentences within the same language that they belong to. Tarski chose to completely exclude this second possibility, Kripke has suggested a less radical restriction (see [5]), while the advocates of paraconsistency have been looking for the solution by trying to do without one of the so far unquestionable logical laws (cf. [7]). Read seems to be convinced to 18 Of

course, what we have in mind are well-formed sentences of some (at least implicitly given) logical system. 19 Indeed, the paradox just consists in the fact that the validity of the formula TL ←→ ¬TL (describing the truth conditions of the Liar) logically excludes at least one of these laws. For, let s = TL. Then it follows from s ←→ ¬s that s → (¬s ∧ s) and ¬s → (¬s ∧ s) thus (s ∨ ¬s) → (s ∧ ¬s), and hence ¬ (s ∨ ¬s) ∨ (s ∧ ¬s).

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have found a third way out of the dilemma by modifying the notion of truth itself. But he has failed to do the first thing anyone ought to do when modifying a classic notion. What are the effects of the modification? Would the new notion be such that it can indeed replace the old one, that is, would the new notion retain the defining features of the old concept? The new notion, in order for it to be accepted as a viable alternative to the classic one, ought to be in concert with our basic common sense conception of truth. Now, the concept of truth is, above all, characterized by the semantic counterparts of the syntactic laws we referred to above. Actually, the semantic laws of contradiction and of excluded middle are the most central laws governing the notion of truth. It was not pure coincidence that, after introducing the formal definition of truth, the first thing Tarski himself did was to check whether the notion he had defined obeyed these laws (cf. [14], Theorem 1 and 2, p. 197). As he wrote in [13] (p. 354, all emphases are in the original): Moreover, we can deduce from our definition various laws of a general nature. In particular, we can prove with its help the laws of contradiction and of excluded middle, which are so characteristic of the Aristotelian conception of truth; i.e., we can show that one and only one of any two contradictory sentences is true. These semantic laws should not be identified with the related logical laws of contradiction and excluded middle; the latter belong to the sentential calculus, i.e., to the most elementary part of logic, and do not involve the term “true” at all.

Let us explain these words in a little more detail. First of all, we need some notation, which will also be useful later. Notation. (i)
s denotes the name of the sentence s. (ii) S is the name of the negation of the sentence whose name is S. In symbols,
s =
¬ s for any sentence s. Now, according to the law of contradiction, for any sentence and its negation, at most one of them is permitted to be true. Using the notation introduced above, this law takes the following form: for any sentence s,
¬ T
s∧T
¬ s . The law of excluded middle,20 on the other hand, is, in some sense, a dual of the former one. Actually, for an arbitrary sentence and its negation, at least one of them has to be true. In symbols, for any sentence s, T
s ∨ T
¬ s. What the two laws together express is, no doubt, the most basic feature of the notion of truth, which provides the following characterization of this concept: for any sentence and its 20 In order for it not to be confused with its syntactic counterpart, this semantic version of the law is often referred to as the principle of bivalence.

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negation, exactly one of them is true, or in other words, the negation of a sentence is true just in case the sentence itself is not true: T
¬ s ←→ ¬ T
s.

(τ )

The question, therefore, Read should have answered is, whether or not his new truth concept preserves the basic connection between negation and truth described by (τ ). Read has not examined this question. The reason may be again that he is convinced that the answer to this question can simply be obtained and it will turn out to be in the affirmative. Consequently, he must have considered the problem not to be worth examining. As Theorem 1 below shows, he was wrong. His whole project is endangered by the fact that, assuming (τ ), the application of his truth schemas to the Liar leads back to the paradox again. The reason is that “one sentence’s falsehood is another sentence’s truth”.21 In the propositions below, of course, L is, as before, the name of the Liar sentence. Proposition 5. Suppose that T is the truth predicate defined by either (A) or (A’). Then there is a q such that T L ⇐⇒ (TL ∧ q) in the case of (A), and
 T L ⇐⇒ (∀p) ((TL ∧ q) ⇒ p) → p in the case of (A’). Proof. The Liar is the following sentence (L)

L is not true.

Thus, obviously, its negation (L)

It is not the case that L is not true.

Now, we should only apply the truth schemas to L in a completely analogous way as they were applied to L. Indeed, the only thing we should do is to replace ¬ TL by ¬ ¬ TL in the first lines of the derivations in [10] Sect. 4, and then apply the rule of double negation. The result will be the following valid argument. 21 Indeed, as we shall see, the symmetry of the relation between the Liar and its negation, together with (τ ), makes it possible to transform a proof of ¬ TL into that of TL and vice versa by simply interchanging these formulas in the original proof. Therefore, if we accept (τ ), then either both TL and ¬ TL or else neither of them can be shown to hold. We can reject the proof of TL, but only at the expense of rejecting the proof of ¬ TL as well.

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L says that it is not the case that L is not true, that is, (L) : ¬ ¬ TL. It may say more: call it q, that is, suppose L : (¬ ¬ TL ∧ q). (If ¬ ¬ TL is all that L says, we can simply let q be ¬ ¬ TL, or a tautology.) (i) Let us first consider the schema (A). Then TL ⇐⇒ (∀p)(L : p → p) i.e. TL ⇐⇒ (¬ ¬ TL ∧ q) since ¬ ¬ TL ∧ q is all L says. Now, by the rule of double negation (12.3), ¬ ¬ TL ⇐⇒ TL therefore TL ⇐⇒ (TL ∧ q). (ii) The case of the schema (A’) is treated in the same way. Corollary. Suppose that T is the truth predicate defined by either (A) or (A’) and, for any sentence s, T
¬ s ⇐⇒ ¬T
s. Then there is a q such that, interchanging TL with ¬TL in the results of the application of these schemas to the Liar, we obtain valid formulas, i.e., for some q, ¬TL ⇐⇒ ( TL ∧ q) in the case of the schema (A), and
 ¬TL ⇐⇒ (∀p) ((TL ∧ q) ⇒ p) → p in the case of the schema (A’). Proof. Since T L ⇐⇒ ¬TL, the claim follows from the previous proposition. Corollary. Suppose that T is the truth predicate defined by either (A) or (A’) and, for any sentence s, T
¬ s ⇐⇒ ¬T
s. Then ¬ TL and TL. Proof. The other case being analogous, we only consider the case of the schema (A). We have already proved ¬ TL (see Proposition 3). On the other hand, as can easily be checked, if we interchange the formulas TL and ¬TL in it, our proof of Proposition 3 becomes a valid derivation of TL from ¬TL⇐⇒( TL ∧ q).22 By the preceding corollary, the proof is complete. 22 The only adjustment we have to make is that, at the last but one step, we should either refer to 12.88 (instead of 12.87), or use the rule of double negation (12.3). Of course, those who, despite our criticism, accept Read’s proof of ¬TL, may make use of that proof to obtain the derivation of TL in a similar way.

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Now we can prove one of our main claims that any concrete realization of Read’s truth schemas can lead to the solution of the Liar paradox only at the too high price of giving up the most fundamental requirements concerning the concept of truth. Actually, it is highly questionable, to say the least, that eliminating the Liar by abandoning the defining features of truth could be considered as a solution to the paradox at all. Theorem 1. Suppose that T is the truth predicate defined by either (A) or (A’) and, for any sentence s, T
¬ s ←→ ¬T
s. Then ¬ TL and TL. Proof. In view of our Lemma (cf. p. 169), it is enough to consider the case in which truth is defined by (A).23 Using the notation introduced on p. 168, the application of (A) to the Liar yields: (B)

T L ⇐⇒ (¬ TL ∧ q).

Further, by Proposition 5, there is a q ∗ such that (1)

T L ⇐⇒ (TL ∧ q ∗ ).

Now, obviously, material implication follows from the strict one in both the strict and the material sense of consequence. Indeed, all sentences of the form
A → A are axioms of the modal logic system T and T ⊆ S4 (cf. [1], p. 5). Therefore, (p ⇒ q) → (p → q) is a theorem of S4. So is its strict counterpart (p ⇒ q) ⇒ (p → q) by 14.1. Consequently, by (B) and (1), we have (2)

T L ←→ (¬ TL ∧ q)

(3)

T L ←→ (TL ∧ q ∗ ).

The application of our assumption T L ←→ ¬TL to (3) yields (4)

¬ TL ←→ (TL ∧ q ∗ ).

Now let us consider the derivations of ¬TL and TL from the formulas TL⇐⇒(¬TL∧q) and ¬TL⇐⇒(TL∧q ∗ ), respectively, that we referred to in the proof of the preceding corollary. Replace strict implication by its material version in these derivations. Thus we convert valid derivations given in the system S4 to valid derivations of ordinary propositional calculus. What we obtain are valid derivations of ¬ TL and TL from the valid formulas (2) and (4), respectively. 23 Again, for those who accept any one of Read’s derivations of ¬ TL, we could have given a completely analogous argument built on that original derivation.

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We conclude our paper by a simple theorem showing that Read’s attempt to solve the Liar by revising Tarski’s truth schema was, in principle, bound to fail: no alternative truth schema can lead to the solution of the Liar paradox since the paradox does not depend on the definition of truth. More precisely, it does not depend directly on the definition of truth itself but instead on some properties of truth that can be inferred from this definition. No matter how the notion of truth is defined, as long as it can be considered as a concept of truth at all (that is, if it complies with our most basic common sense principles concerning such a notion), it leads to a contradiction. In fact, in any system in which our most fundamental logical assumptions concerning the behavior of equivalence, negation and truth hold and (some equivalents of) a sentence asserting its own non-truth can be formulated, a contradictory statement can also be derived. The theorem itself covers the cases of those logical systems in which the Liar sentence can only be formulated implicitly (e.g. through some kind of coding such as a Gödel numbering), while its corollary is about those systems, like our informal common sense one based on natural language, where both the truth predicate and the self-reference is directly available.24 The intended meaning of the functions T,
, ¬ and the relation  should be clear from the context. They represent the truth predicate, a naming function, the negation, and any kind of equivalence, respectively. Theorem 2. Let L = (S,  , ¬, T, N,
) be a system such that ¬ is a unary function mapping the non-empty set S (of sentences) to itself, T is a unary function mapping the set N (of names) to S, and
 is a unary function mapping S onto N in a one-one way. Further, let  be a binary relation on S such that, for any σ, τ, ρ ∈ S, (a) σ  τ and τ  ρ imply σ  ρ (b) σ  τ implies ¬ σ  ¬ τ . Assume that, for any σ, τ ∈ S, (c) T(
¬σ)  ¬T(
σ) (d) T(
T(
σ))  T(
σ) (e) σ  τ implies T(
σ)  T(
τ ). Suppose that there is a sentence Λ such that Λ  ¬T(
Λ). 24 Indeed, though various common language formulations of the Liar were subject to heavy criticism, there are, in fact, absolutely unobjectionable ways to achieve self-reference, and hence to formulate the paradox, in terms of natural languages, cf. [12].

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Then T(
Λ)  ¬T(
Λ). Proof. (1) (2) (3) (4)

T(
Λ)  T(
¬T(
Λ)) T(
¬T(
Λ))  ¬T(
T(
Λ)) ¬T(
T(
Λ))  ¬T(
Λ) T(
Λ)  ¬T(
Λ)

Definition of Λ, (e) (c) (d), (b) (1),(2),(3),(a)

Corollary. Assume that the conditions of the preceding theorem hold and, further, for any σ, τ ∈ S, σ = τ implies σ  τ . Suppose that there is a sentence Λ such that Λ = ¬T(
Λ). Then T(
Λ)  ¬T(
Λ). The theorem above is a possible abstract generalization of Tarski’s theorem on the undefinability of arithmetical truth. It shows that if we liked to retain the basic characterizing properties of equivalence, negation, and truth, then we should give up the common sense requirement of the possibility of talking, without any further restriction, about the truth of sentences. In fact, this result explains the remarkable phenomenon that, in spite of the innumerable attempts to solve it since its invention two and a half millennia ago, the Liar paradox has not yet been solved. The reason is simple enough. It cannot, in principle, be solved. As our theorem shows, if we permit talking about the truth of our sentences in an unconstrained way, then some characterizing properties of truth simply lead to a contradiction. Truth and the Liar are two sides of the same coin. You can only eliminate the contradiction by giving up at least one of our most basic intuitive requirements concerning the logic of truth. The answer to the question “Which is the basic logical law that we can do without?” is, of course, a matter of taste. But de gustibus non est disputandum. Therefore, there is no way out of the dilemma that everybody would accept; there exists no universally acceptable solution to the Liar paradox. Clearly, tastes cannot be the subject of rational philosophical or logical discussion. In the light of this theorem and its corollary, therefore, it is entirely clear that any attempt to solve the Liar paradox is tantamount to an argument trying to convince the community of logicians that one (or more) of our basic logical notions such as truth, equivalence, or negation can be modified in such a way that, on the one hand, the modification leads to the elimination of the paradox, on the other, the most important logical laws are affected only to such an extent that can be considered a reasonable price for our getting rid of

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the paradox. An obvious example of such a partial revision of our logic that has proved to be completely successful is the epoch-making classic theory of truth worked out by Tarski. To put it briefly, the theorem and its corollary show that what Read would like to do, that is, trying to eliminate the Liar paradox without giving up at least one of the fundamental intuitive requirements concerning our basic logical notions is completely hopeless. In order to solve the paradox, it is not enough to revise the definition of truth, our whole logic needs a revision. Such a revision, however, would be an entirely different and a much greater endeavour.

Acknowledgments This work was supported by Hungarian NSF Grant No. T43242

References [1] Boolos, G. (1995). The Logic of Provability. Cambridge: Cambridge University Press. [2] Goldblatt, R. (1992). Logics of Time and Computation. Stanford: CSLI. [3] Goldstein, L. (1986). Epimenides and curry. Analysis 46(3): 117– 121. [4] Kleene, S.C. (1964). Introduction to Metamathematics. Princeton: D. Van Nostrand Company, Inc. [5] Kripke, S. (1984) Outline of a theory of truth, in: R.L. Martin (ed.) Recent Essays on Truth and the Liar Paradox. Oxford: Clarendon Press, pp. 53–81. [6] Langford, C.H. and Lewis, C.I. (1959). Symbolic Logic (2nd edition). New York: Dover. [7] Priest, G., R. Routley, and J. Norman (eds.) (1989). Paraconsistent Logic: Essays on the Inconsistent, München: Philosophia Verlag. [8] Prior, A.N. (1958). Epimenides the Cretan. The Journal of Symbolic Logic 23(3): 261–266. [9] Read, S. (1988). Relevant Logic. Oxford: Basil Blackwell. [10] Read, S. (2005). The Truth Schema and the liar. [11] Sainsbury, R.M. (1989). Paradoxes. Cambridge: Cambridge University Press. [12] Serény, G. (2003). Gödel, Tarski, Church, and the Liar. The Bulletin of Symbolic Logic 9(1): 3–25.

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[13] Tarski, A. (1944). The semantic conception of truth. Philosophy and Phenomenological Research 4. [14] Tarski, A. (1956) The concept of truth in formalized languages, in: A. Tarski (ed.) Logic, Semantics, Metamatematics. Oxford: Clarendon Press.

Chapter 11 OUT OF THE LIAR TANGLE B. Hartley Slater∗ Department of Philosophy University of Western Australia, Perth, Australia [email protected]

Abstract

There are some seemingly small points to be made, first of all, about use-mention confusions in Stephen Read’s paper ‘The Truth Schema and the Liar’. But underlying them is a grammatical point that has much wider repercussions. For it generates, on its own, a more straight forward way of understanding what gets people into a tangle with Liar and Strengthened Liar sentences, and that leads to a much fuller, critical assessment of the line of approach to these matters that Read derives from Bradwardine. Difficulties with representing propositional referring phrases of the form ‘that p’ are shown to be what have made Liar and Strengthened Liar sentences seem paradoxical. Using “ ‘p’ ” as an alternative confuses syntactic expressions with their immediate readings, and leads to misunderstandings about the necessity of the T-schema. Frege’s content stroke, i.e. the horizontal line that he used to indicate the thought expressed by a sentence, has not been incorporated into the generality of logic texts which have followed his formal work. I set out here how provision of an explicit content marker gets one out of Liar and Strengthened Liar ‘paradoxes’, and what similarities and differences there are with Read’s account of truth, as a result.

Keywords: ‘That’-clauses, Propositions, Indirect speech.

11.1

Tarski’s Truth Scheme versus Horwich’s

When discussing Tarski’s T-schema, at the beginning of his paper, Read also mentions a similar schema in Horwich’s work. He even thinks of them as the same, saying at one point “Tarski did not propose (T) as ∗ Opponent.

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a definition of truth, though others, e.g. Horwich, have done so since”. But there is a crucial difference between Tarski’s and Horwich’s schemas. Horwich’s is a propositional truth schema, viz ‘the proposition that p is true if and only if p’, whereas Tarski’s is a sentential schema: x is true if and only if p, when what replaces ‘x’ is a name of a sentence whose translation into the metalanguage replaces ‘p’. The difference is most pointed in the homophonic sentential case, which parallels very closely the propositional one. For what replaces ‘x’ then is a quotation-name of the sentence that replaces ‘p’, not that sentence itself. As we shall see, there is a formal difficulty with distinguishing the two schemas clearly, since there is no agreed, distinct symbolisation for predications of the form ‘that p is true’, and something more like the sentential predication “ ‘p’ is true” is often used in place of it. Certainly there would be no need to make the distinction if all sentences were unambiguous and non-indexical, i.e. had just one interpretation, since then facts about propositions could be mapped 1–1 onto facts about sentences. But, as we shall see later, the central question is whether Liar sentences are indeed univocal in the required way. It has to be said that the difference between ‘that p is true’, and “ ‘p’ is true” may not be completely appreciated even in Horwich’s informal work, since he thinks there are still paradoxical cases of his propositional schema. But ‘it is true that’ is the null, or vacuous modality in the modal system T, i.e. an ‘L’ for which it is necessary that Lp ≡ p, and so one cannot have p ≡ ¬Lp, since T is consistent. Going with that, there is no way to construct self-referential propositions parallel to sentential constructions that, for instance, produce identities like t = ‘t is not true’. For ‘t’ is there the name of the sentence, and not the sentence itself, and one cannot have anything of the form ‘p’ = ‘that p is not true’, since nothing can be a proper part of itself. This shift in grammar already is revealing about Read’s core concerns. So we must first look directly at problems with certain Liars in the light of it, before turning to a wider assessment of Read’s paper.

11.2

Self-reference is not an Extensional Matter

The main question, of course, centres on what is wrong with the following kind of argument. From: t = ‘t is not true’,

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there follows: t is true ≡ ‘t is not true’ is true But from the T-schema there follows: ‘t is not true’ is true ≡ t is not true, so it seems we can derive: t is true ≡ t is not true. There is no difficulty finding true identities like that expressed on the first line, and the second line follows necessarily from the first, the last from the preceding two, hence the problem must be with the third line itself. Read agrees, but what is causing the problem? As we shall see, it is the lack of a clear expression for the thought that t is not true which is keeping people gripped with the third line. One realises there is a problem with getting a clear expression for such a thought not just by attending to the sort of grammatical difficulties in Read’s paper discussed above. The difficulty is much more widespread, as can be seen if one attends even to the common description of sentences like t as ‘self-referential’. One must remember, first of all, that the original identity does not itself show that some sentence is about itself, since it does not entail that: (∃x)(x = ‘x is not true’), by existential generalisation, and neither can ‘t’ be replaced throughout by a quotation name for the supposedly self-referential sentence, since nothing of the form: ‘p’ = “ ‘p’ is not true” is possible: again, nothing can be a proper part of itself. By contrast, if one has a statement about the content of t, in a sentence where there is not direct quotation, but indirect speech, such as t says that t is not true, there follows unproblematically that (∃x)(x says that x is not true), and so that something is self-referential. And there is an equal possibility of providing a quotation name for a sentence of the required kind, since there is nothing against cases of the following form being true: ‘p’ says that ‘p’ is not true.

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So properly there is no syntactic self-reference: no sentence in itself refers to itself. That, indeed, may be granted quickly by many, saying that, of course, a sentence needs an interpretation before it can be said to be about anything, since it itself is just a syntactic string. But the rapidity of such an admission hides the crucial difficulty. The difficulty comes in various forms, and is at least threefold. First, in the non-professional area, people commonly do not attend closely to use-mention issues, so the syntactic string ‘t is not true’ gets conflated with its prima facie interpretation, that t is not true, with the consequence that the expression “ ‘t is not true” ’ gets used both for the uninterpreted string, and for the interpreted expression. Second, in the professional area of semantics, this conflation gets institutionalised when there is talk about sentences being ‘true in models’. Is ‘P a’ true if ‘a’ refers to Socrates and ‘P ’ expresses the property of being wise? Why not say instead that what is true in the specified model is simply that Socrates is wise? The answer to that leads to the third, and major point enunciated before: in the professional area of formal logic there is considerable difficulty with saying the latter, since there is no standard expression, in the symbolism coming down from Frege, for the word ‘that’ in such expressions as ‘that Socrates is wise’. Some notable individuals have followed Frege, who himself used a ‘horizontal’, and introduced a symbol for this purpose: thus Kneale used ‘§’, and Cocchiarella ‘λ’ ([1], p. 217; [4]). But the generality of logic texts contain no such expression. It is all this that combines to be the cause of the problem people have had with those Liar problems which involve direct quotation. There is another class of Liar problems involving indirect speech, which we shall look at later, but first we can clear up the problem with ‘t is not true’ is true ≡ t is not true. For, because of the above, a question now arises about the referent of the “ ‘t is not true” ’ on the left hand side. Either the quoted expression is a syntactic object, or it is an interpretation of that sentence which is meant, in which case the quoted expression is not a Tarskian syntactic object, but a Horwichian semantic one. In the Horwichian case, there is then an ambiguity, since in t is true ≡ ‘t is not true’ is true, the right hand side definitely involves a syntactic object, which means one cannot go on to get the traditional contradiction t is true ≡ t is not true,

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by combining the two equivalences. The matter is even clearer if one removes the use-mention conflation in this case, and writes the former equivalence as the indubitable that t is not true is true ≡ t is not true. Then it is very evident that there is no way to get the contradiction. But in the Tarskian case, i.e. if it is a syntactic object in the T-schema, that again shows that we do not get a contradiction. For we would need in addition, as was pointed out before, a proof of the univocality of the sentence ‘t is not true’, showing there is no other interpretation of it possible beyond the superficial one. Homophonic examples of the sentential T-schema are in the required way disquotational, but they cannot arise with semantically ambiguous sentences like ‘There is a bank’, or indexical ones like ‘She is pretty’. So the question first has to be answered whether the same holds with ‘t is not true’ in the special case where t = ‘t is not true’. Once they are separated from their prima facie interpretations, however, it becomes clear that such sentences can have more than one interpretation. In the case of any sentence like ‘t is not true’ what I have called above its prima facie interpretation, drawn from reading what is in its face, is that t is not true. But that depends on taking the ‘t’ in it to refer to t, which, while certainly possible, is not necessary – as we saw before, there is no such thing as syntactic self-reference. Maybe at the top of some page there is the sentence ‘the sentence at the top of the page is not true’, but if a preceding sentence was, say, ‘Once upon a time a wise old man could be found reading a certain page in a book’, then the reference of the following subject term ‘the sentence at the top of the page’ is not the sentence at the top of the page that sentence is on, but a sentence in another possible world. And it is no good replacing that sentence with anything like ‘the sentence at the top of the actual page this sentence is on is not true’ since that involves an indexical, which could be given a variety of references, in different possible worlds. In these, and similar cases there is, or can be, therefore, a second semantic reading of the subject phrase, beyond the superficial one, showing no semblance of a paradox need be around. Of course, there will be if the quoted ‘t’ is given the same reference as the unquoted ‘t’, but in that case, we must primarily remember that giving it that reference is a matter of choice. There is no syntactic selfreference, but also any semantic self-reference is not obligatory. Hence there is no way to get the T-schema absolutely, when the quoted expression on the left is a syntactic object, with the result that a contradiction does not necessarily follow this way, either. Isn’t, at least, the sentence t,

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on the self-referential interpretation paradoxically both true and false? No, for what one is involved with, if one chooses the self-referential interpretation is not directly the syntactic identity, but the kind of statement about content which was introduced before, namely t says that t is not true. This is clearly true only on a certain semantic interpretation, and so it is not just an extensional remark about the syntactic object ‘t is not true’. The distinction makes it plain that what is true or false is not the sentence ‘t is not true’, but the proposition that t is not true. As was pointed out before, the inability of the logical tradition to represent such a propositional referring phrase as ‘that t is not true’ has made it seem that what is true or false on the given interpretation is still the (mentioned) sentence, but only the sentence in use, preceded by ‘that’, refers to the item that has the truth value. In fact, as a result, sentences, in themselves, are neither true nor false, so that t is not true is definitely true, in this case. It follows that there is no longer any problem with Strengthened Liars. Certainly the question naturally arises about what to say in ‘strengthened’ cases, where, for instance, s = ‘s is neither true nor false’. But first of all there is no need to take the sentence thus defined to speak about it itself being neither true nor false. For sentences, by themselves, have no voice. If anyone chooses to interpret this sentence self-referentially that is therefore an additional (intensional) matter beyond the direct speech identity. And what is true in that self-referential case is simply that s is neither true nor false, which is not paradoxical in any way, since to say that is not to say that what is true is s, i.e. ‘s is neither true nor false’. The self-referential case, moreover, is expressed with s says that s is neither true nor false, so it is to a study of such expressions in indirect speech, which form a large part of Read’s paper, that we now turn.

11.3

Differences from Read’s Account

The above supports very forcibly Read’s arguments against the Tarskian T-schema, but, as it stands, it leads to a positive account of truth that, while very similar to Read’s, is crucially different from it. Read’s case (C), for instance, is one of those covered, in a very similar way, in Goodstein’s original paper on the formalisation of indirect speech

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[2], which led directly to Prior’s theory [5], and then eventually to its subsequent modification using Kneale’s ‘that’. But, for one thing, there are some grammatical problems with Read’s symbolisation of this same kind of material. More importantly, he also introduces, or at least relies on, a further axiom that rules out the kind of ambiguity we have seen to be necessary to allow an escape from the above syntactic paradoxes. What are the grammatical difficulties when we move over to Liar problems arising with the indirect speech form ‘x says that p’, if this is symbolised as Read symbolises it: ‘x : p’ ? For a start, the previous difficulties with formalising ‘that p is true’ affect this style of symbolisation, in formulas such as Read’s (A), i.e. Tx ⇔ ∀p(x : p → p) (A) This expression employs quantification over propositions, which Read glosses with a reference to Church’s type theory, in footnote 2. This kind of symbolism was used at length by Prior, and indeed Read’s thesis (A) is to be found in Prior’s book [5], on p. 104. But there are major problems with reading Prior’s propositional quantifiers, which he himself struggled with in Chap. 3 of his book (see also [3] p.130), and these are also present in Read’s account. Centrally, there is no provision, within such a system of propositional quantification, for truth predications on propositions. In ‘x : p ∧ p’, for example, the ‘p’ is not a referential phrase, and so any ‘p’ in a quantifier before it equally would have to be not a referential phrase; also when reading ‘x : p ∧ p’ as ‘x says that p and it is true that p’, there is no representation of the ‘it is true that’ in the formalism. What is wanted, first of all (c.f. [6]), is a nominaliser, which will produce, in a formalisation of ‘that p’, a referring phrase to a proposition expressed by ‘p’. In addition one then needs a truth predicate of such nominalised phrases, and, when generalising expressions like ‘x says that p, and that p is true’, one must employ variables which range not only over ‘that’ clauses, but also other propositional referring phrases, like ‘what x says’, for instance. The form of the given case is then ‘x says r, and r is true’ over which one can quantify objectivally, with quantifiers which read quite straightforwardly ‘for some/for all propositions r’, etc. The analysis of Read’s (C), i.e.: ∀p(C : p → ¬p)

(C)

then proceeds as follows. First C says that everything that C says is false, i.e. Sc§(∀r)(Scr ⊃ ¬Tr) so suppose that everything C says was indeed false, i.e. (∀r)(Scr ⊃ ¬Tr), then, contradictorily, something C says would be true, i.e. (∃r)(Scr.Tr),

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since from the propositional T-schema there follows in particular that: (∀r)(Scr ⊃ ¬Tr) ≡ T§(∀r)(Scr ⊃ ¬Tr) Hence not everything C says can be false, i.e. ¬(∀r)(Scr ⊃ ¬Tr), and so, equivalently, (∃r)(Scr.Tr), but then something C says is false, namely §(∀r)(Scr ⊃ ¬Tr), and so, also, (∃r)(Scr.¬Tr). It follows that C must say (indirectly) at least two distinct things, one true and the other false. Read says similarly: “However, although C : ∀p(C : p → ¬p), this may not be all that C says”, obtaining, as well, (∃p)(C : p ∧ p). Indeed, there is a close similarity with another matter raised at the start, since from his (A), i.e. Tx ⇔ ∀p(x : p → p) there follows a conditionalised T-schema, requiring ‘singleness of saying’, i.e. something like univocality, or uniqueness of interpretation, before there can be truth assessments of sentences in the traditional T-schema form, viz: [x : p &(q)(x : q → q = p)] ⇒ (Tx ≡ p) So both the necessity of this conditionalised T-schema, and its consequences, are endorsed by Read, and they are to be accepted, once grammatical corrections have been made. What is there to be said, by contrast, in substantive criticism of Read’s treatment of Liar paradoxes arising in indirect speech? The main point which needs to be made relates to this issue of univocality, since the contrary line of analysis allows for ambiguous sentences, and even requires them in certain places, while Read explicitly dismisses ambiguity, and holds that if x both says that p and says that q, then x says that p and q. Certainly this latter rule, namely Adjunction, is not made explicit in Read’s axioms, but he still employs it, making, for instance, his transposition, in Sect. 4, from: L : (TL ∨ ¬q) and so L : ((TL ∨ ¬q) ∧ q) given L : q, not something actually enforced by his stated rules. Read uses Adjunction, implicitly, in this derivation in Sect. 4, and in his discussion of (C) in Sect. 5. Thus, after the remark about C : ∀p(C : p → ¬p) possibly not being all that C says, quoted above, he goes on: “Suppose C also says that q, that is C : (∀p(C : p → ¬p) ∧ q)”, adding the extra ‘q’ as a conjunct to the other thing C says. But, for a start, given

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that so much can be obtained without the use of this rule (see e.g. [7]), Ockham’s Razor, if nothing else, suggests strongly that it should be dispensed with. One is reminded, on that score, of the discussion of Moore’s Paradox in the ‘Budget of Paradoxes’ chapter of [5], pp. 81–84. This is first analysed using Hintikka’s axioms for belief, but then an entirely adequate explanation of its ‘logical oddity’ is produced using nothing more than quantification theory. There is a stronger argument against Adjunction, however, than the need for axiomatic economy. For we have seen, in Sect. 2, when dealing with Liar paradoxes arising in direct speech, that it is necessary to grant ambiguity a place, in connection with identities like t = ‘t is not true’. So why is Read against ambiguity? Read does debate the matter, but only briefly, and his conclusions are not well supported by his arguments. Thus he is concerned that the point of formalisation is to remove ambiguity, but seems to forget that concern still allows entirely precise statements to be made about the ambiguity of certain sentences. If x is ambiguous, it can still be completely unambiguous that x has two meanings, i.e. means that p and means that q without meaning that p and q. More specifically, Read says, about ambiguous expressions such as ‘Visiting relatives can be boring’, that “we do not require that both senses be the case for x to be true – either will suffice”. But such a sentence is then true in one sense and false in another, not true simpliciter either time, making Read’s ‘Tx’ better read ‘x is true in all senses’, not ‘true simpliciter’. Only starting from the conditionalised truth schema identified above is the truth of sentences limited to truth simpliciter, and so to truth as more generally understood within the Tarskian tradition. And there is another notion that is more generally part of that tradition. For the discussion in Sect. 2, for instance, involved the notion of meaning, which is a notion not found in Read’s paper, but crucial to the Davidsonian tradition following on from Tarski. Read glosses his ‘x : p’ in various ways, such as ‘x says that p’, ‘x implies that p’, but there is another expression in the area, ‘x means that p’, which behaves rather differently. Thus ‘John is a bachelor’ means that John is an unmarried male of marriageable age, and the latter implies that John is unmarried. So the meaning of something is its entire content, or, at least, something from which its entire content can be deduced. Writing ‘x means that p’ as ‘M x§p’, and using ‘I’ for the relation of implication, we can in fact then define ‘x says that p’ as ‘something x means implies that p’, i.e. (∃r)(Mxr.Ir§p).

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Given this, the other main axiom Read employs, (K), i.e. (∀p, q)((p ⇒ q) → (x : p ⇒ x : q)) becomes, with appropriate grammatical corrections, just a matter of definition also. Thus we can get ‘x says whatever is implied by what it says’, as the schema: (p ⇒ q) → [(∃r)(Mxr.Ir§p) ⇒ (∃r)(Mxr.Ir§q)] This follows from the transitivity of implication, using some quantification theory, and attending to the appropriate grammar. It must be remembered, in this connection, that ‘implies’ is a verb and relates two propositional subjects given by referring phrases, as in ‘that John is a bachelor implies that he is unmarried’ and ‘what Peter stated implies that John is unmarried’. The symbol which Read uses, ‘⇒’, is, by contrast, a propositional connective, which can be defined via the equivalence: I§p§q if and only ifp ⇒ q Lack of a propositional nominaliser, of course, is what inclines many (indeed most) to read the connective as ‘implies’. That confusion only deepens if one says, like Read, that sentences imply things; indeed one is then involved in the full grammatical confusion with which we started.

11.4

Conclusion

In conclusion, I have supported Read’s arguments against Tarski, pinpointing more accurately the fallacy that holds people to his schema – a certain conflation of syntax and semantics. However, the core grammatical insight needed to be clear of Tarski is still widely ignored in Read’s system, and there are superfluous and restrictive axioms there that either can be reduced to definitions, or do not allow for the full extent of the cases that can be encountered. Logic is an exact science, but there are many problems with being as exact as it requires, and so Logic must allow for inexact and ambiguous expressions. Only quantification theory, at the propositional, i.e. indirect speech level, is required to work through the ‘paradoxical’ cases that result. Read’s rejection of ambiguity and his consequent endorsement of Adjunction, are the central features of his account which prevent him from seeing that only quantification theory is needed. But his lack of appreciation of the place of ambiguity is also directly linked to his confusion over the sentential versus propositional nature of the T-schema. And we have seen, at length, that a clarification of that distinction is what gets one out of standard Liar paradoxes in direct speech.

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References [1] Cocchiarella, N. (1986) Logical Investigations of Predication Theory and the Problem of Universals, Bibliopolis, Naples. [2] Goodstein, R.L. (1958) On the formalisation of indirect discourse, Journal of Symbolic Logic, 23, 417–419. [3] Haack, S. (1978) Philosophy of Logics, C.U.P. Cambridge. [4] Kneale, W. (1972) Propositions and truth in natural languages, Mind, 81, 225–243. [5] Prior, A.N. (1971) Objects of Thought, O.U.P. Oxford. [6] Slater, B.H. (2001) Prior’s analytic revised, Analysis, 61, 86–90. [7] Slater, B.H. (2004) Ramseying liars, Logic and Logical Philosophy 13, 57–70.

Chapter 12 READ ABOUT T-SCHEME Jan Woleński∗ Jagiellonian University, Cracow, Poland [email protected]

Abstract

This note argues that Read confuses provability and truth in his interpretation of T-sentences as related to the T-scheme. Moreover, his proposal is much more intensional than that of Tarski. This circumstance has a relevance for defining falsity.

Keywords: T-sentences, Provability, Intensionality.

Read considers the orthodoxy view to be that every instance of the formula x is true if and only if p (T) is true. However, Tarski focused on the provability of T-sentences, but not on their truth. Of course, the question whether T-sentences are true or false is legitimate, but it is not of the utmost importance for the semantic definition of truth. If we assume that the metatheory of truththeory is ω-complete, we obtain (see [1], p. 80) that false T-sentences are not available. It is sufficient to show that every provable instance of (T) is true. That T-sentences are assumed to be provable is important for their status. Read says that according to Tarski T-sentences are only material equivalences. Although Tarski was not quite explicit at this point, it is obvious that provable theorems of the form A ↔ B are something more than material equivalences. In particular, one cannot replace A or B by their material equivalences (this does not mean that Read says that they can). ∗ Opponent.

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The role of the provability of T-sentences has its best illustration in the problem of the definability of the T-predicate. Combining the fix-point theorem and the Tarski undefinability theorem, leads to the unprovability of some instances of (T). This fact concerns the Liar sentence, independently of whether we formulate it as ‘this sentence is false’ or ‘this sentence is not true’. The situation is clear in formal arithmetic, but the results that hold for arithmetic cannot be directly applied to natural language. However, pace Tarski, excluding the Liar sentence from the stock of admissible formulas plays a role similar to showing that some formalized instances of (T) are not provable. Both moves justify the thesis that truth is undefinable. In this perspective, it is not essential whether the Liar sentence is inconsistent or, as Read prefers, simply false. Incidentally, Read’s analysis of antinomies is interesting in itself. Read neglects the fact that the concept of truth is defined for interpreted languages only. Taking that into account one can easily demonstrate that all of the counterexamples given by Read fail. This concerns the sentences: ‘I am tired’ is true if and only if I am tired; (1) ‘That book was stolen’ is true if and only if that book was stolen; (2) ‘Any man is mortal’ is true if and only if any man is mortal. (3) The argument is that the words ‘I’, ‘that’ and ‘any’ have different meanings on the left and right sides of the respective equivalences. However, this cannot happen by definition in the case of interpreted languages. The valuation is a function and ‘I’, ‘that’ and ‘any’ have to have the same interpretations in all their occurrences in (1)–(3). This is particularly clear for indexicals, like ‘I’ and ‘that’. An ostensive specification of references (for example, supplementing the act of using ‘I’ or ‘that’ by a gesture) fixes the ascribed objects. On the other hand, if the references of such words are not made precise, they function as variables and the problem of truth (falsehood) of (1)–(3) does not arise at all. I do not argue that ambiguities do not occur in ordinary language, but only point out that the semantic definition of truth assumes that an interpretation is fixed. Read proposes to replace (T) by Tx ⇔ ∀p(x:p → p)

(A)

This scheme codes an intuition expressed by x is true if and only if things are wholly as x says they are

(S)

Since what a sentence says is covered by all its implications, (A) formalizes (S), provided that the symbol ⇔ denotes strict equivalence. However,

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(A) holds for every interpreted language, because if the formula A ↔ B is a theorem, what A and B says is recorded by their logical consequences (however, see below). This means that (T), if properly interpreted, serves the same purpose as (A). Moreover, (A) has an implausible consequence. Consider (I do not assume that Peano arithmetic is first-order) ‘Peano axioms’ ⇔ ∀p(‘Peano axioms’:p → p)

(4)

Peano axioms are true in the standard model as well as in the nonstandard model of arithmetic. Let N be the standard model and N’ a non-standard one. Take the sentence (*) ‘all natural numbers have finitely many predecessors’, which is true in N, but false in N’. Intuitively speaking, Peano axioms say (*) in N, but its negation in N’. Thus, ‘says that’ requires a relativisation to a model, but if not relativized to a model (T) has false instances. This also shows that defining the context x:p by the consequences of p may be insufficient in some cases, because a reference to models is required. (T) and (A) differ because the former is intensional, but the latter extensional. Read takes this fact as an advantage of (A), but I see a problem here. The definition of F (‘is false’) corresponding to (A’) can be recorded as Fx ⇔ ∃p((x:p) ∧ ¬p) (A”) As an example we have F(‘Warsaw is the capital of France’) ⇔ ∃p((‘Warsaw is the capital of France’:p) ∧ ¬p)

(5)

However, (5) has its instantiation in F(‘Warsaw is the capital of France’) ⇔ ((‘Warsaw is the capital of France’: the greatest Polish city is the capital of France) ∧the greatest Polish city is not the capital of France)

(6)

Now assume that someone knows that a person O knows that the sentence ‘the greatest Polish city is the capital of France’ is false and that the right side of (6) is a correct (true) instantiation of the right side of (5). Hence, he knows that the sentence ‘Warsaw is the capital of France’ is also false; we use here the principle ‘if an instantiation of A is false, A is false too’. However, to justify that, one must assume that ‘Warsaw’ and ‘the greatest Polish city’ are co-denotative (I neglect that the latter is a description). This consideration shows that we do not need to worry whether the sentences ‘Warsaw is the capital of France’

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and ‘the greatest Polish city is the capital of France’ say the same or, eventually, in which circumstances they cover the same content, because it is sufficient to known the values of nominal expressions. Although it is true, as Read indicates, that logically equivalent sentences say the same thing, this observation does not close the issue, because it can happen, as in the case of (5) and (6) that sentences are equivalent in theories or some language systems modulo denotative conventions, although they are not equivalent on purely logical grounds. Thus the interpretation of a language has a crucial importance for establishing what sentences say and when they are true or false. Further, the negation of (x:p) can be interpreted either as ¬(x:p) or as (x:¬p) (the latter is stronger than the former). It matters in the case of negative sentences because T(‘Warsaw is not the capital of France’) ⇔ ∀p(¬p → ‘Warsaw is not the capital of Poland’:¬p)

(7)

looks more plausible than T(‘Warsaw is not the capital of France’) ⇔ (8) ∀p(¬p → it is not the case that‘Warsaw is the capital of Poland’:¬p) Perhaps the most important critical observation concerns metalogical properties of T. The law of the excluded middle can be stated as ∀p(x:p → p) ∨ ∃p((x:p) ∧ ¬p)

(9)

which is an instance of TA ∨ ¬TA

(10)

However, the relation of ¬TA and T¬A is not clear under Read’s definition. In general we have T¬A → ¬TA

(11)

and in the classical bivalent (two-valued) case the equivalence T¬A ⇔ ¬TA ⇔ FA

(12)

holds. Now T¬A has its interpretation (according to Read’s definition) in ∀p(x:¬p → ¬p) (13) However, (13) does not implies ∃p((x:p) ∧ ¬p)

(14)

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although we have (or should have) (x:¬p) → ¬(x:p)

(15)

I do not claim that (12) is indispensable, but only note that important relations are unclear under Read’s proposals. In particular, the functor of negation commutes with truth (in its semantic understanding), but not with ‘says that’ Summing up, I think that Read’s did not show that his approach is better than the orthodox view. In particular, ‘says that’ does not work better than interpretations of a language in a model. Of course, if someone is not satisfied with the extensional approach to truth, Read’s proposals seem to be an interesting starting point, although I am skeptical how far the intensional way goes. In particular, metalogical issues must be clarified at the beginning.

Acknowledgments I am indebted to Philippe de Rouilhan and Tero Tulenheimo for valuable remarks.

Reference [1] J. Woleński, In defence of the semantic definition of truth, Synthese 126: 67–90 (2001).

Chapter 13 FURTHER THOUGHTS ON TARSKI’S T-SCHEME AND THE LIAR Stephen Read Department of Logic and Metaphysics, University of St. Andrews, Fife KY16 8RA, Scotland, UK [email protected]

Abstract

In ‘The Truth Schema and the Liar’, I criticised Tarski’s formulation of the T-schema and proposed a revised truth-condition which promises to solve the semantic paradoxes by rendering them all false. I have left that paper unchanged in the light of the objections raised in the other papers in this volume, and respond to those criticisms in the present paper. I also elaborate further the historical background to the proposal.

Keywords: Semantic closure, Saying that, Liar paradox, Convention T, Type theory, Negation, Bradwardine, Tarski, Prior, Epimenides, Roger Swyneshed.

13.1

Semantic Closure

In his penetrating study of the Epimenides paradox, Arthur Prior noted: “What is really paradoxical about the Epimenides . . . [is] that if any Cretan [such as Epimenides] does assert that nothing asserted by a Cretan is true, then this cannot possibly be the only assertion made by a Cretan.” ([22], p. 261)

Prior recognises that what is paradoxical here is that we seem to settle an empirical question on logical grounds, namely, that not only is this Cretan’s utterance false, but therefore something true must have been asserted by a Cretan. Prior’s response is to conclude that if there was no other Cretan utterance (or at least no true one) then Epimenides in fact failed to say

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anything at all. Asserting must be more difficult than we thought ([22], p. 264). This is a hard conclusion, and one cannot help thinking that Prior has extricated himself from one paradox only to land in another. We understood what Epimenides said, and inferred correctly that it could not be true. Was this just an illusion, as Prior claims (loc.cit.)? Thomas Bradwardine, thinking about this issue in the early 1320s, drew a different but not dissimilar conclusion. As Laurence Goldstein notes ([11], p. 76), he was very short with the cassationists, or nullifiers, those who claim that in such cases nothing has been said. There is a grammatical utterance containing meaningful words, so something must have been said: “But because these nullifiers of the first and second kinds appear so very asinine,” he wrote, “we do not need to argue with them any further, as Aristotle said: ‘It is not worthwhile to consider seriously the oversubtleties of myth’ (Metaphysics Γ 4, 1006a13).” ([6], Sect. 5.6)

Prior’s modus tollens was for Bradwardine a modus ponens: Epimenides clearly said something, so since that what he said was false could not, on logical grounds as we have seen, be the only thing said by a Cretan, Epimenides (if no one else) must have said more than he seemed to say: as well as saying that nothing asserted by a Cretan was true, he also said that what he himself said was true. Bradwardine spelled out the assumption on which this result of his was based, and showed how the result followed. There were two crucial assumptions made by Bradwardine. The first is that signification, or what an utterance says, is closed under implication. That is, an utterance says anything which what it says implies. In symbols: ∀p, q(s : p ∧ p ⇒ q → s : q) (K) where ‘s : p’ abbreviates ‘s says that p’ (written as ‘Say(s, p)’ in [3]), ‘∧’ is conjunction, ‘→’ is implication and ‘⇒’ is entailment. In my paper, ‘The Truth schema and the Liar’ [28], I assumed that ∧ was extensional, truth-functional conjunction, that → was material implication, ⇒ was strict implication, and ran the argument in classical logic. I now realise that the argument should run just as well in non-classical logic, so here I will take → to be any implication, material, intuitionistic, relevant, linear or whatever, for which modus ponens holds, and that it is the residual of ∧, that is, (R) p ∧ q → r iff p → (q → r).1 1 I will also assume that ¬ (negation) obeys the principle: (p → q) ⇔ ¬(p ∧ ¬q). Intensional conjunction is usually denoted by ‘◦’ and called ‘fusion’: see, e.g., [2] Sect. 27.1.4, and [30] Sect. 8.3.

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∧ is then intensional iff → is, and ⇒ is the modal correlate of →, so that p ⇒ q expresses the logical truth of p → q. It is easy to misunderstand Bradwardine’s closure assumption, as do several contributors to this volume. So too did Paul Spade in his several discussions of Bradwardine’s argument. Indeed, Bradwardine himself states (K) very loosely; he writes: “Every utterance signifies or means everything which follows from it.” ([6], Sect. 6.3).

But his actual use of this postulate shows that he does not mean quite what he says here. Moreover, as put, this postulate would lead immediately to contradiction. Charity suggests, therefore, the more subtle reading: every utterance signifies everything which follows from what it signifies – that is the principle which Bradwardine follows in practice. We can rewrite (K) to show this explicitly: ∀q(∃p(s : p ∧ p ⇒ q) → s : q). Before we can understand the significance of this distinction, however, we need to review Bradwadine’s argument, and before we can do that we need to look at his diagnosis of the fallacy in the semantic paradoxes (what he called the “insolubles”) like Epimenides’. The other crucial assumption he made lies in his definition of truth and falsehood. For if an utterance can mean more than one thing, we need to consider how this multiple signification connects with the criterion for truth. For this particular conception of multiple signification is not one of ambiguity or equivocation: it is conjunctive. For example, if s says that p and s says that q then s says that p and q. So if either p or q fails to eventuate, s is false; truth is a limit and any imperfection results in falsehood. Consequently, Bradwardine lays down that s is true iff things are wholly as it says they are, and false if anything it signifies fails to eventuate: Ts ⇔ ∀p(s : p → p) and

(A) Fs ⇔ ∃p(s : p ∧ ¬p)

We can now give Bradwardine’s argument. He states and proves a bold thesis: any utterance saying of itself that it is false, or not true, also says of itself that it is true and is false. The proof is as follows: Suppose s says of itself that it is not true. This may not be all it says: suppose what else it says is that q, i.e., s : (¬Ts ∧ q). First, suppose s is not true. Then by (A), ∃p(s : p ∧ ¬p), i.e., something s says fails to

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hold, either ¬Ts or q. So if it’s not q which fails to hold, it must be ¬Ts which fails, that is, q → Ts. In sum, we have shown ¬Ts ⇒ (q → Ts) so by Residuation (R), ¬Ts ∧ q ⇒ Ts. Since s : (¬Ts ∧ q), it follows by (K) that s : Ts. Given Bradwardine’s endorsement of Bivalence, embedded in the definition of truth and falsehood in (A), it immediately follows that if s : Fs then s : Ts by the same reasoning. Indeed, by (A), ¬Ts and Fs are equivalent. Finally, it follows straightaway that if s : ¬Ts or s : Fs then s is false, for in either case we have shown that s : (¬Ts ∧ Ts), and ¬Ts ∧ Ts cannot hold: one or other of the conjuncts must fail, so something s says fails to hold, so by (A), s is false. At the heart of this reasoning is an application of Disjunctive Syllogism, when we inferred that Ts from the assumption that q, given that either Ts or ¬q. That application of Disjunctive Syllogism is relevantly acceptable provided the disjunction is intensional (the relevant connective usually denoted by ‘+’), which it will be if (and only if) the conjunction of things said is similarly intensional. Depending on one’s other logical preferences and convictions, one can endorse Bradwardine’ reasoning by suitable interpretation, whether classical, intuitionistic, relevant or whatever. It is clear that the closure principle (K) plays a crucial role in Bradwardine’s argument. But, as I said, the principle can be readily misunderstood, as it was by Armour-Garb ([4], p. 24 – but corrected on p. 27), Goldstein ([11], pp. 67 and 73), Klima ([14], p. 100), Panaccio ([19], note 22, p. 235), Sagüillo ([32], p. 155) and Serény ([33], pp. 163 and 173). Take the Liar paradox, L, where L : ¬TL. Bradwardine does not claim that ¬TL ⇒ TL, but that for some q, q ⇒ TL where L : q. If he had claimed that ¬TL ⇒ TL, he would be in serious trouble. For his conclusion is that L is false, and so not true. If the truth of L followed from its falsehood, he would be committed to saying that L is true as well as false, and paradox would have returned by the standard revenge move: if one claims that ‘L is false’ is false, has one not shown that it is true?2 No, for the truth of L requires more than that L be false. That is only part of what L says. We have shown that L must say more than 2 Cf.

[16], p. 4. Cf. below, p. 220.

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that, as Prior implicitly realised. No utterance can say only that it itself is false, by Bradwardine’s argument. Paul Spade was similarly misled by Bradwardine’s statement of postulate (K) (which Spade called ‘Bradwardine’s Principle’, BP) and accordingly misdescribed Bradwardine’s reasoning so badly that he was led to attribute to Bradwardine a principle which undermines his whole approach, the notorious ‘Converse Bradwardine Principle’, CBP. CBP states that “whatever a sentence signifies follows from it”. Spade writes: “Although Bradwardine does not explicitly state CBP as one of the bases of his theory, it is presupposed in some of his reasoning” ([39], p. 120).3 For an author who is so meticulous to set out his assumptions clearly and fully, and who spells out his reasoning with such care, this is a hard judgment. Spade is led by this misunderstanding to claim (p. 122) that one of Bradwardine’s steps is idle, and that a later one goes through only by invoking CBP. A closer reading of the text shows that following the first step avoids the need for the second. Admitting CBP would be disastrous for Bradwardine. For it would again yield ¬TL ⇒ TL, and revenge would once more be wrought. L does not entail its own truth. Rather, that L is true follows from what L says, namely, ¬TL ∧ TL. Despite Bradwardine’s unfortunate statement of his closure postulate, it is clear from his application of it in the proof (and again and again in similar proofs in later chapters of his treatise4 ) that he intends to endorse what I have formulated as (K). Only then is his project viable.

13.2

Type Theory

Dutilh-Novaes ([10], p. 58) comments that I have elsewhere called the conception of signification embodied in (K) “Carnapian”. Rightly so, but as with much in Carnap, it arguably goes back to Frege. In his Logical Syntax of Language ([7], Sect. 49), Carnap defines the content (or sense) of a proposition to be the set of its non-analytic consequences. The point of excluding the analytic consequences is not just that, given a classical conception of consequence, the analytic truths are consequences of every proposition and so useless in distinguishing senses. The exclusion serves also to articulate the Tractarian idea that analytic truths are empty. 3 Cf.

[40], Sect. 3.1 and Supplementary Document. of Goldstein’s objections to Bradwardine’s account is that it does not extend to other semantic paradoxes, like Yablo’s ([11], p. 69). This is not so. Bradwardine himself extended it to the ‘yes’–‘no’ (or postcard) paradox, the ‘no’–‘no’ and other paradoxes in subsequent chapters of his treatise. I showed in [25] how Bradwardine’s solution extends to Curry’s paradox, and in [26], Sect. 4 how it extends to Yablo’s and other interative paradoxes. Bradwardine’s solution does not, as Goldstein suggests (p. 69), proceed by “simply adding” an extra conjunct, but by deriving that conjunct using the method outlined. 4 One

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But the central question goes back to Frege’s Begriffsschrift Sect. 3 ([5], p. 54), where Frege dismisses any aspect of the content of a judgment which does not affect its consequences. Judgments had, for him, the same content if and only if they had the same consequences. It does not follow immediately from this that those consequences make up the content. However, this seems to have been Frege’s view, which Carnap may have learned at Frege’s lectures at Jena.5 For Frege wrote in his letter to Husserl of 9th December 1906: if A and B are equivalent, “then nothing can belong to the content of A . . . which does not also belong to the content of B” ([5], p. 306), and vice versa. Thinking of the content of an utterance as a collection of thoughts between which logical relations can hold we are led naturally to the closure principle (K). But note that, pace Carnap, the consequences which form the content of an utterance are not consequences of the utterance itself, but of elements of its content, as we saw in Sect. 13.1. Its content is deductively closed. Hartley Slater [34] and others claim that the expression ‘s : p’ is not well formed. Not so. ‘s says that p’ is, as Prior remarked ([23], Chap. 2, Sect. 2, p. 19), a relation at one end and a connective at the other; that is, what replaces ‘s’ and ‘p’ to yield a particular instance are respectively a name and a sentence. Thus ‘s : p’ is an expression of type theory, where s is of type ι (iota, for individuals) and p is of type o (omicron, for propositions).6 o and ι are the basic types, and if α and β are types, so is αβ, the type of functions from objects of type β to objects of type α. Thus ‘s : p’ is of type ooι (bracketing to the left), i.e., a map from objects of type o and ι respectively to one of type o (expressed by a sentence: the claim that s says that p). What exactly are the objects of type ι, however? Eugene Mills accuses me of semantic essentialism ([17], p. 122). I plead guilty. Utterances, as ready to take their place in (A), have their content essentially. They are interpreted sentences. For (A) is intended as an analysis of the concept of truth, contra Tarski, who presented the T-scheme (T) as a material condition on the correctness of a theory.7 The equivalences in (A) are logical equivalences, which they could not be if the referents of the terms replacing ‘s’ were uninterpreted, or variously interpretable marks. The string of letters ‘snow is white’ might have meant that snow is black. So that assertion cannot be logically equivalent to the claim that what it actually says is the case, if that change of meaning could take place (we suppose) without any change in the colour of snow. But given the interpreted utterance ‘snow is white’, the equivalence does hold of necessity. 5 Unfortunately,

there seems to be no mention of this in [29]. [8]. 7 See, e.g., [41], p. 404. 6 See

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Mills ([17] p. 123) considers whether ‘L’ refers to a type or a token. It’s become the orthodoxy to describe the objects of the semantic theories of nominalist logicians of the fourteenth century like Ockham and Buridan as tokens, since they treat their existence as contingent.8 Catarine DutilhNovaes ([9], p. 279) showed this interpretation was mistaken. The utterances that Bradwardine and Buridan discuss are repeatable: e.g., the sentence at the top of p. 6 occurs at the top of p. 6 of every copy of the book in question. What is important is to spot the right equivalence relation uniting the tokens into a type. Applying (A) to ‘snow is white’, for example, does not mean applying it to occurrences of ‘snow is white’ which mean that snow is black. It means applying it to all occurrences which have the meaning that ‘snow is white’ actually has. This ‘snow is white’-type has its meaning essentially, even though the meaning of ‘snow is white’, that is, the ‘snow is white’-type embracing all strings of these letters, is a contingent matter. Consider now a contingent paradox, such as Epimenides’. If there were some true Cretan utterance it would not be paradoxical. So suppose every other Cretan utterance is false. Epimenides says that all Cretans are liars, that is, he utters E such that E : ∀x(Cx → ¬Tx), where ‘Cx’ abbreviates ‘x was uttered by a Cretan’. (Note that ‘x’ is a variable of type ι.) There are two contingent assumptions, namely, CE and (‡) ∀x(Cx ∧ x = E → ¬Tx). Rehearsing Bradwardine’s argument, we show that ∀x(Cx → ¬Tx) ∧ q ⇒ (CE ∧ ‡ → TE) where q abbreviates whatever else E says. Thus by (K) E : CE ∧ ‡ → TE. But furthermore, by ∀E, E : CE → ¬TE so again by (K), E : CE → (CE → ¬‡) whence by contraction (and (K) again) E : CE → ¬ ‡ .9 8 See,

e.g., [18], pp. 207–209. that all the applications of (K) are legitimate, since they proceed by entailments: ∀x(Cx → ¬Tx) ∧ q ⇒ (CE ∧ ‡ → TE) ∀x(Cx → ¬Tx) ⇒ (CE → ¬TE) (CE ∧ ‡ → TE) ∧ (CE → ¬TE) ⇒ (CE → (CE → ¬‡) and (CE → (CE → ¬‡) ⇒ (CE → ¬‡).

9 Note

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But this conditional is false, for it has, ex hypothesi, a true antecedent and false consequent. Hence something E signifies fails to hold, and so by (A), E is false. But we cannot infer that E is true, for E says more than that ¬TE, indeed, more than that ¬TE ∧ ‡, which does indeed hold. Things are not wholly as E says they are. Terence Parsons ([20], p. 130) attributes a substitutional interpretation of quantification to me. In the same vein, Slater objects ([34] p. 193): “‘p’ [in (A)] is not a referential phrase.” He insists on reading ‘s : p ∧ p’ as ‘s says that p and it is true that p’ just as Mills did ([17], p. 119) when inserting ‘T’ in the schema to produce the apparent circularity ‘Ts ⇔ ∀p(s : p ∧ Tp)’.10 What none of them seems to realise is that all quantification is substitutional, and so the distinction objectual/substitutional is pointless. One can take any expression of mixed categories, e.g., ‘My utterance says that snow is white’ and replace the subject ‘my utterance’ by a variable s of type ι, and the subordinate clause ‘snow is white’ by the variable p of type o, to obtain the form ‘s says that p’, which is not, pace Dutilh-Novaes ([10], p. 50), a relation, but a mixed relation/connective of type ooι, as we saw in reference to Prior. Ajdukiewicz [1] in his theory of categories had a less elegant but more transparent notation. In place of o and ι, he used ‘s’ for the category of sentences and ‘n’ for that of names, so that ‘s says that p’ has category x s ns . In general, y is the category of expressions which take expressions of category y to form an expression of category x.11 One can then quantify over expressions of any of these categories or types, governed by the usual quantifier generalization (or introduction) and specification (or elimination) rules. For example, take ‘(∀s)(∃p)s : p’. Eliminating the universal quantifier, one replaces the variable ‘s’ in ‘s : p’ by a constant expression of the same type (viz ι or n), that is, one substitutes some constant, e.g., ‘my utterance’, for ‘s’, to obtain ‘(∃p)my utterance says that p’. This follows by existential generalization from the result of substituting any constant of the same type as the variable ‘p’ (viz o or s) for it in ‘My utterance says that p’, e.g., it follows from ‘My utterance says that snow is white’. In both cases, in the case of the name ‘my utterance’ and in the case of the sentence ‘snow is white’, the constant expression is substituted for the variable;12 similarly, in composing the form ‘s : p’ one substitutes a variable of the appropriate type for some grammatical expression. 10 Cf.

[11], p. 67. x that yz = ( xy )/z. 12 Note that it must be substituted uniformly. 11 Note

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Although in my original paper ([28], p. 8, n. 2) I explicitly said that (K) and (A) were formulae of type theory, Mills construes (A) “according to the standard syntax of the predicate calculus” ([17], p. 116), and concludes it is not well formed. The predicate calculus is a restricted subsystem of type theory, where one can quantify only over expressions of type ι, though it contains expressions of other types. Predicates are of type ox, x ≥ 0, defined recursively as follows: o0 = o,

o(x + 1) = (ox)ι,

so that ox is the type of maps from x individuals to a proposition, that is, an x-place predicate. One sometimes adds function symbols, but again one may not quantify over functions (at first order). Function terms have type ιx, x ≥ 0, where ι0 = ι,

ι(x + 1) = (ιx)ι,

that is, the type of maps from x individuals to an individual. But why not allow quantification over these other types? – one then obtains secondorder logic. Then why restrict the types in this way? Allowing all types αβ, as above, and permitting quantification over them, one obtains type theory, a language better suited to the expression of philosophical ideas. (A) does not place such a heavy demand. It can be formulated in secondorder logic, in which one can quantify over (expressions of) both types o and ι.13

13.3

Negation

So far, I have done little more than expand on what I wrote in my original paper [28], and attempt to correct some misunderstandings. I have commented that I wish to retract my claim ([28], p. 7) that Bradwardine’s analysis works only for material and strict implication, that is, in a truth-functional logic such as classical logic. As noted above, this is not true. Neither was my repeated comment ([28], p. 15) that the Truthteller says nothing. Bradwardine does not discuss it, but it does respond to a Bradwardinian analysis: it is false.14 There are two responses to my paper, however, which do give me pause. One is the extended critique by György Serény, the other the model-theoretical counterexample to Bradwardine’s thesis presented by 13 Note,

pace Woleński ([42], p. 201), that the Peano axioms are categorical in (second- and higher-order) type theory. See, e.g., [24], p. 87. 14 Goldstein ([11], p. 68) was right to castigate me here. In fact, and surprisingly, the Truthteller also says that it is false, in addition to saying it is true. See [26].

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Greg Restall. First, consider Serény’s argument. Let L be the (or at least a) Liar sentence such that L : ¬TL. Now consider what appears to be the negation of L, L, such that L : TL. A natural principle about the negation, s, of an utterance s is that Ts ⇔ ¬Ts.

(N)

TL ⇔ ¬TL.

( )

Putting ‘L’ for ‘s’, we obtain

Since L : TL, we have by (A) TL ⇔ TL ∧ q where q is whatever else L may say, whence TL ⇒ TL

( )

so from ( ) and ( ), ¬TL ⇒ TL and we are back in paradox. Bradwardine’s core idea is to be able to claim that L is false, and so not true, without being ensnared by the usual revenge problem into conceding that L is therefore true, since it asserts what Bradwardine wants to assert, namely, that L is not true. But L seems to have destroyed that hope. The operative principle is clearly (N), which bears some similarity to Serény’s principle (τ ) ([33], p. 180). But there is something very odd about (τ ), or at least about the notation Serény uses in it. He writes (p. 180): “Notation (i)
s denotes the name of the sentence s (ii) S is the name of the negation of the sentence whose name is S. In symbols,
s =
¬s for any sentence s.”

Slater, Mills and others complained about my use of ‘p’ as a variable over propositions, where the substituends for ‘p’ were not names (of propositions). They were mistaken, as we saw in Sect. 13.2, for Church and Ajdukiewicz give sense to expressions like ‘s : p’ in type theory. But Serény’s language here really is a nonsense. If what replaces ‘s’ in the schemata in (i) and (ii) are names, then ‘¬s’ in (ii) is ill formed. One cannot prefix ‘¬’ to a name. But if what replaces ‘s’ is a sentence (as it must be for ‘¬s’ to be well formed), then (i) is ill formed, where ‘s’ (and what replaces it) sits in apposition to ‘the sentence’.

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Bradwardine’s language is much more careful. Consider, for example, the derivation in Sect. 1.4 of my original paper [28], and in Sect. 13.1 above, that L : TL. Consequently, I concluded, L : TL ∧ ¬TL. But where did we show that L : ¬TL? One might suppose that there is a further postulate missing from Bradwardine’s presentation telling us what, e.g., ‘¬TL’ says. Serény in fact tries to supply it ([33], p. 163): Definition. x explicitly says that s if and only if what replaces ‘x’ is a name of the sentence which replaces ‘s’.”

But (in contrast with his successors like William Heytesbury, discussed by Pironet [21]) Bradwardine explicitly rejects such a distinction between what is explicitly and what is only implicitly or consequentially said.15 There is no place for it in his reasoning. I did not need to show that L : ¬TL. Following Bradwardine, I assumed that L : ¬TL, and such a supposition does not necessarily single out L uniquely. Bradwardine’s main thesis is that “if a proposition signifies itself not to be true or to be false, it signifies itself to be true and is false.” 16 This is a general and hypothetical claim, about any proposition which says of itself that it is not true. It is we who then add the observation (as Bradwardine does too in Chap. 7) that, e.g., ‘This utterance is not true’ signifies that that utterance is not true. But the general theory simply takes any utterance which says that it itself is not true and concludes that it also says of itself that it is true. There is no requirement, or suggestion, that one thing said is explicit and the other implicit. I may have suggested such a distinction in my original paper, when, for example, I wrote ([28], p. 9): “‘snow is white’ reveals explicitly all that it says”, and later (p. 15): “every sentence says (implicitly or indirectly) whatever is implied by what it says”. But as Serény rightly observes ([33], p. 164; cf. [11], p. 67), my concession that even ‘snow is white’ might say only that snow is white was too generous. Given (K), much that any utterance says will be inexplicit. And the later talk of ‘saying implicitly’ was in the mouth of a mistaken opponent, suggesting a reductio ad absurdum of the theory which I proceeded to show was based on a misunderstanding. But now take the sentence ‘¬TL’, call it L. Given that L : ¬TL, then by Bradwardine’s argument, L : TL and L is false, that is, ¬TL. Prefix L by ‘¬’ and we obtain ‘¬¬TL’, which clearly says that L is true. But L is false. So ‘TL’ must be false too. So both ‘TL’ and ‘¬TL’ are false. We appear to have proved that two contradictory sentences, ‘TL’ and ‘¬TL’ are false, which is surely impossible. Contradictories not only cannot be true together, they cannot be false together either. 15 See 16 [6],

[6], Sect. 7.2.5. Sect. 6.4.

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Not so, declared Roger Swyneshed, about ten years after Bradwardine wrote his ground-breaking work. Swyneshed’s observation was one of three iconoclastic theses: “1. Some false proposition signifies principally as things are 2. In some formally valid inference, falsehood follows from truth 3. Two mutual contradictories are both false.” ([38], pp. 188–189)

Swyneshed did not accept Bradwardine’s theory, in particular he presents a simplistic theory whereby the insolubles are taken to falsify themselves. His treatise lacks the meticulous argument which pervades Bradwardine’s, though he maintains a semblance of Bradwardine’s mathematical style. His example supporting thesis 1 is the ‘no’–‘no’ paradox, instancing two propositions A and B where A says (principally) that B is false and B says that A is false. By a non sequitur ([38], p. 182) he argues that A and B are both false, so A (and B) is indeed a false proposition signifying principally as things are, namely, that B is false. The second thesis is demonstrated by supposing that B is the conclusion of this inference: (♣) B is false So B is false. The conclusion is false, since it falsifies itself (according to Swyneshed’s doctrine) but the premise is simply true, for it does not falsify itself, but signifies as things are (viz that B is false). And the third thesis takes ‘L is false’ (i.e., ‘¬TL’) and ‘L is not false’ (i.e., ‘TL’). These are both false, but they seem to be contradictories. These arguments are all bad, and instructively so. The inference (♣) is indeed invalid. The self-referential conclusion is false, for the reasons we noted in Sect. 13.1, for it says of itself both that it is false and (by Bradwardine’s argument) that it is true. The premise is true, for it does not say of itself that it is false, but says that of B, and B is, as we have seen, false. It is yet another example of the “revenge” problem and Bradwardine’s answer to it. Swyneshed’s paradox depends, however, on accepting that (♣) is a formally valid argument, the reason being that its form is ‘p. So p’. But that is a wholly superficial analysis, which does not reveal its true form. To take an analogy, consider Quine’s famous argument: (♠) 9 is necessarily greater than 7 9 is the number of planets So the number of planets is necessarily greater than 7. That appears to have the form ‘φa, a = b. So φb’, which is a valid form, yet the premises of (♠) are true and the conclusion false. The solution is to realise that the form of (♠) is more subtle, since ‘the number

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of planets’ is not a name, and so not properly represented by a constant, ‘b’.17 So too with (♣). Its logical form is not a case of identity, for the premise says that ¬TB, whereas the conclusion says that ¬TB ∧ TB, so does not follow (formally or otherwise) from the premise. What, then, of the third thesis, that contradictories can be simultaneously false? Once again, Swyneshed has been misled by the superficial form. ‘L is not false’ is not the contradictory of ‘L is false’, any more than, e.g., ‘Some man is not running’ is the contradictory of ‘Some man is running’. Very often, one forms the contradictory of a proposition by prefixing a negation-sign to it (or negating its main verb), but there are many cases where this fails, and it is here shown to fail by the very fact that ‘L is false’ and ‘L is not false’ are both false, just as ‘Some man is running’ and ‘Some man is not running’ are seen not to be contradictories by the fact that both can be true. ‘L is false’, being self-referential, says that ¬TL ∧ TL, so to deny it, one must deny one or other part, that is, find a sentence which says that TL ∨ ¬TL (where ‘∨’ is the extensional or intensional correlate of ‘∧’, as appropriate). The same diagnosis defuses Serény’s objection. There is nothing wrong with principle (N): for any sentence s, there is another sentence s such that Ts ⇔ ¬Ts. The challenge is to identify s correctly. Serény naively took it that if L : ¬TL then L : TL. That does not follow. That ¬TL is only part of what L says. To repeat, L : ¬TL ∧ TL. So L : TL ∨ ¬TL. To deny L one must find a sentence which says either that L is true or that it is false. ‘L is either true or false’ will do. Then L, that is, ‘L is either true or false’, is true whereas L is false, and (N) is preserved.18 What, however, of Restall’s counterexample [30]? The real problem with Restall’s models is, I believe, that in them, negation is Boolean (that is, a  ¬A iff a  A). The consequence is that any contradictory proposition, such as the Liar, L, says nothing (or put another way, everything) – discrimination goes. L is still a declarative object, provided Lα ∈ Da , a = {a | aSLα a }, but the proposition expressed by L at a, that is UL α is empty. Hence every contradictory proposition says everything, since a , if that class is empty. So to M, α, a  A for all A and all a ∈ Us α show that if L : ¬TL then L : TL one would have to show that L : q for all q. Quite why Bradwardine’s argument for his second thesis fails in Restall’s models is unclear to me. Restall speculates that it hinges on the possibility that L might have said something else. Yet it is crucial to Bradwardine’s solution that it deal with contingent paradox such as 17 See,

e.g, [35].

18 Bradwardine

made the point himself: see [6], Sect. ad 7.3.

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Epimenides’ utterance, which is contradictory only if Epimenides utters it and no other Cretan utterance is true, as we saw in Sect. 13.2. There is great need for further work here, but Restall’s construction of models for Bradwardine’s theory will be an important technique for deeper understanding of Bradwardine’s proposal.

13.4

The Truth Schema

My original paper focussed on Tarski’s T-schema, rejecting it for two reasons. First, as is commonly recognised, it does not offer an analysis of the concept of truth, but at best provides an extensional test of the adequacy of a putative analysis.19 Secondly, it fails even as an extensional test, for, to adapt Etchemendy’s terminology (as applied to Tarski’s analysis of the concept of logical consequence), it overgenerates: it deems some sentences true, e.g., L, which are not. For that very reason, Tarski excludes those sentences – from the analysis as well as from the test. They should indeed be excluded from the test, by revising the test to specify the extension of ‘T’ properly. By making the correct test the analysis, we settle both failings at once. The correct analysis of truth, I argued, is given by Ts ⇔ ∀p(s : p → p) (A) requiring for truth that things be wholly as s says they are. Serény claims ([33], Corollary, p. 166) that the truth predicates given by (T), (A) and (A ) (an alternative to (A) which I offered to those who found (K) unappealing as a principle of ‘saying that’) “are logically equivalent if x : p is interpreted as s ⇒ p, where s is what x explicitly says”.20 We have already noted the problem with this interpretation of ‘saying that’: e.g., L : TL but we do not have ¬TL ⇒ TL. But there are deeper problems with this identification. Take, e.g., Serény’s derivation in footnote 14 ([33] p. 175) of TL ↔ (¬TL ∧ TL) (the application of (A) to L) from TL ↔ ¬TL (the application of (T) to L). This is trivial: TL ↔ ¬TL is logically false and so, given the classical logic in which Serény is working, it entails everything. Again, take his “proof” on p. 169 that TL ⇔ (∀p)(((¬TL ∧ q) ⇒ p) → p) (B ) (the application of (A ) to L) can be derived from TL ⇔ (¬TL ∧ q)

(B)

19 See, e.g., [13], Chap. 1, especially p. 37. Jan Woleński [42] is right to note that Tarski required that the instances of (T) should be derivable from any adequate theory. But he clearly imposed that requirement because he thought those instances were true. 20 Cf. [4], p. 25.

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(the application of (A) to L). This is nonsense, for (A ) is the analysis of truth I proposed for the case where (K) is rejected, and so where there need be no q, additional to ¬TL, such that L : q. Finally, what of Serény’s charge that I have not established my “claim that, independently of the exact definition of ‘saying that’, the application of any notion of truth that fits into the framework given by one of [my] truth schemas solves the Liar paradox” ([33], p. 166) and accordingly “that L cannot (be proved to) be true” (p. 167)? I make, and made, no such claim. It could easily be shown wrong by, e.g., defining ‘s says that p’ in such a way that, say, L means only that ¬TL. Then (A) would give us TL ⇔ ¬TL (†) and we would be back in paradox. What I claimed ([28], Sect. 4) was that, given (K), so that L does not mean only that ¬TL, and using (A) as the account of truth, then the standard reasoning using the (T)-schema, applied to the proof that ¬TL and concluding that TL, fails. Actually, this point shows that my olive branch, offering (A ) without (K), works only if, for some other reason, L does not mean only that ¬TL. For if L says only that ¬TL, then (A ) yields TL ⇔ ∀p((¬TL → p) → p), which at least classically entails (†). Whether or not one accepts (K), the solution works only if no proposition says only that it itself is false, as Prior realised. There is, however, a further challenge to my rejection of (T), which seems to me to constitute the hardest challenge to date. It can be seen to arise from reflecting on Kripke’s famous outline of a theory of truth permitting semantic closure, and was prompted by hearing a talk by Beall. Kripke, unlike Bradwardine, rejects Bivalence: L, for example, is for Kripke neither true nor false. The T-predicate admits gaps, and in fact “if ‘k’ abbreviates a name of the sentence A, T(k) is to be true, or false, respectively iff A is true, or false” ([15], p. 80). Accordingly, Kripke cannot endorse “Convention T” as originally given as a biconditional, since p ↔ q is, at least on the Kleene tables, undefined if either p or q is undefined. Rather, Kripke claims to retain Tarski’s “intuition that T(k) is to have the same truth conditions as A itself” (loc.cit.). Now suppose L : ¬TL and consider ‘TL’, call it L . What does L say? It is plausible to suppose that it says that L is true. But moreover, by (A), TL ⇒ ∀p(L : p → p), so since L : ¬TL, TL ⇒ ¬TL. Consequently, by (K), L : ¬TL and thus L : TL ∧ ¬TL. But that is what L says. So L and L appear to say exactly the same. The Principle of Intersubstitutivity (PI) says that, if s and t say the same thing, then they should

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be intersubstitutable salva veritate in all contexts. So L and L should be intersubstitutable. Take the Law of Identity ∀p(p ⇔ p) and apply it to L :

(I)

TL ⇔ TL.

Now replace the right-hand occurrence of L by L by (PI): TL ⇔ ¬TL

(†)

and the paradoxical instance of (T) has returned. Contradiction and paradox seem inevitable.21 The diagnosis should be clear, however, from the discussion of Swyneshed’s second thesis in Sect. 13.3 above. The instance of ‘B is false’ in the premise of (♣) is true, whereas that in the conclusion is false. Similarly in the above derivation of (†), the occurrence of ‘¬TL’ on its right-hand side is not an instance of L and not self-referential as L is. Hence it does not say the same as does L occurring on its left-hand side, and so the application of (PI) was illegitimate. This leaves us with a final puzzle. It is crucial to Bradwardine’s solution to the Liar and other paradoxes that we distinguish the false and self-referential occurrences of ‘¬TL’ from the true and non-self-referential ones. Bradwardine and I want to say that L is false.22 When we do so, we utter an instance of ‘¬TL’ – our utterance is equiform with L. No one can deny that different utterances of equiform sentences can have different truth-values: that was the point of Davidson’s counterexamples to (T) rehearsed in Sect. 1.1 of my original paper [28]. Graham Priest therefore challenged me (in conversation): I want to say that L is false. My statement has the same subject and predicate as L. So even if they are different tokens, or tokens of different types (as suggested in Sect. 13.2 above), how can one be self-reflexive and the other not? Why is my utterance not (an instance of) L too? This is Armour-Garb’s challenge of “indirect revenge” ([4], Sect. 2.4.3). In his ‘Introduction’ to his translation of and commentary on Buridan’s chapter on insolubles and other self-referential propositions in his Summulae de Dialectica, George Hughes added a personal remark in a footnote ([12], p. 25): “I thought at one time that this argument was irrefutable and that it showed Buridan’s theory to be inconsistent. My 21 Armour-Garb [4] obtains the same result, p. 28) by (TI ) α ⇒ Tα and (T0 ) Tα ⇒ α. Clearly, (TI ) has to be rejected in the context of (A) and (K). 22 Contra [11], p. 66, I do want to endorse Bradwardine’s solution.

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enthusiasm for him dates largely from the moment when it struck me that he could reply to it along the lines suggested here.” The argument was this: Buridan wants to accept ‘L is false’ and reject ‘L is true’. But Buridan’s own principle (“every proposition virtually implies its own truth”) seems to allow us to infer from ‘L is false’ that “L is false” is true’ and so conclude that L is true. In my paper, ‘The Liar paradox from John Buridan back to Thomas Bradwardine’, I observed ([25], p. 213) that where Hughes had to conjecture how Buridan would respond to this challenge, I did not need to conjecture what Bradwardine’s response would be, for he tackled the objection directly himself. He devotes Chap. 7, nearly 3,000 words, to it: suppose Socrates utters L and Bradwardine utters something which has the same subject, predicate and copula as L. Different utterances of the same sentence can have different truth-values, as we saw from Davidson’s examples. Consequently, when we group together different cases of equiform utterances and ascribe a truth-value to the consequential types, we must not group together cases whose accidental differences affect what they say and so what truth-value they have, on pain of contradiction. L says of itself that it is false, and so Socrates’ utterance is false because it is self-referential. Bradwardine is not referring to his own utterance, but to Socrates’. So Bradwardine’s utterance is not self-referential, and so his utterance is true, for it does not say of itself that it is true (or false). These accidental features (as Bradwardine called them: what Socrates and Bradwardine each intended) explain why their equiform utterances do not say the same thing and so are not instances of a common type fit for evaluation as true or false. Contra Armour-Garb, Bradwardine did not utter L, but a different proposition with the same subject, predicate and copula. They are only syntactically the same; they are semantically different, for one is self-referential and the other is not.23

13.5

Conclusion

Requiring conformity with Tarski’s T-scheme has become the catechism of late twentieth-century semantics. Scott Soames speaks for the orthodox when he writes: “. . . if conveying the meaning of the truth predicate of English sentences involves conveying the acceptability of all instances of schema T, then it is hard to see how anyone competently employing the predicate could reject any such instance.” ([36], p. 51)

The T-scheme is the sacred cow whose questioning is unthinkable. 23 I

have discussed the problem of revenge at greater length in [27].

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Yet the T-scheme is the result of a rash generalization. It seems so obviously right for ordinary, straightforward cases that it becomes an idée fixe, an obsession which blinds us to its flaws. Problems of indexicality, scope and idiolect are brushed aside. They must adapt to its demands, rather than expect it to adapt to their needs. In particular, when the T-scheme meets the needs of semantic closure “our global truth-concept is shattered into infinitely many hierarchically arranged fragmentary concepts” ([37], p. 598). Is the T-scheme so clearly valid as to underpin such a consequence? What the T-scheme fails to recognise is that our utterances may have implicit consequences which are not fully explicit. In evaluating the truth of those utterances, all its consequences must be satisfied. (As ArmourGarb notes – [4], p. 35 – that makes (A) hard to apply in practice.) Things must be wholly as an utterance says they are for it to be true, as Bradwardine and his contemporaries put it. Bradwardine himself adopted a bold closure principle for what an utterance says: it says whatever follows from what it says. Most remarkable among those consequences is encapsulated in his conclusion about semantic closure: any utterance which says of itself that it is false, or not true, also says of itself that it is true and is false. In my original paper, I presented this idea in the modern dress of symbolic logic. To that end, I used the formalism of type theory to capture the semantic notion, ‘s says that p’, where ‘s’ is a variable over individuals (utterances, of type ι) and ‘p’ is a variable over propositions (things said, of type o). In the present paper, I have responded to a number of challenges mounted to the coherence of that semantic theory and its success in dealing with the semantic paradoxes. Some were based on a misunderstanding; others hit at the heart of the proposal. Responding to both types has clarified the proposal and allowed me to explore it further, and for that I am very grateful.24 I believe the proposal is still viable, but also is open to further discussion and elaboration, which I hope it will receive.

References [1] Ajdukiewicz, K. (1967) Syntactic connexion, in Polish Logic: 1920– 1939, ed. and tr. S. McCall. Oxford: Clarendon Press, pp. 207–231. [2] Anderson, A. and Belnap, N.D. (1975) Entailment: the Logic of Relevance and Necessity, vol. 1. Princeton: Princeton University Press. 24 I

am also grateful to J.C. Beall and Graham Priest for insightful comments and questions.

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[3] Andjelković, M. and Williamson, T. (2000) Truth, falsity and borderline cases, Philosophical Topics 28, 211–244. [4] Armour-Garb, B. (2008) Read and Indirect Revenge, this volume, pp. 19–40. [5] Beaney, M. ed. (1997) The Frege Reader. Oxford: Blackwell. [6] Bradwardine, T. Insolubilia, ed. S. Read (Dallas Medieval Texts and Translations). Leuven: Peeters, forthcoming. (An earlier but unreliable text is found in [31].) [7] Carnap, R. (1937) The Logical Syntax of Language, tr. A. Smeaton. London: Routledge. [8] Church, A. (1940) A formulation of the simple theory of types, Journal of Symbolic Logic 5, 56–68. [9] Dutilh-Novaes, C. (2005) Buridan’s Consequentia: consequence and inference within a token-based semantics, History and Philosophy of Logic 26, 277–297. [10] Dutilh-Novaes, C. (2008) Tarski’s Hidden Theory of Meaning: Sentences Say Exactly One Thing, this volume, pp. 41–63. [11] Goldstein, L. (2008) Doubting Thomas, this volume, 65–85. [12] Hughes, G.E. (1982) John Buridan on Self-Reference. Chapter Eight of Buridan’s ‘Sophismata’, translated with an Introduction, and a philosophical commentary. Cambridge: Cambridge U.P. [13] Kirkham, R. (1992) Theories of Truth. Cambridge, MA: MIT Press. [14] Klima, G. (2008) Logic Without Truth, this volume, pp. 87–112. [15] Kripke, S. (1975) Outline of a theory of truth, Journal of Philosophy 72, 690–716; reprinted in [16], 53–82. [16] Martin, R.L. (1984) Recent Essays on Truth and the Liar Paradox. Oxford: Clarendon Press. [17] Mills, E. (2008) Scheming and Lying, this volume, pp. 113–128. [18] Nuchelmans, G. (1982). The semantics of propositions, in The Cambridge History of Later Medieval Philosophy, ed. N. Kretzmann, A. Kenny and J. Pinborg. Cambridge: Cambridge University Press, pp. 197–210. [19] Panaccio, C. (2008) Restrictionism: A Medieval Approach Revisited, this volume, pp. 229–253. [20] Parsons, T. (2008) Comments on Stephen Read’s The Truth-Schema and the Liar ’, this volume, pp. 129–134. [21] Pironet, F. (2008) William Heytesbury and the Treatment of Insolubilia in Fourteenth-Century England, this volume, pp. 255–333.

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[22] Prior, A.N. (1958). Epimenides the Cretan, The Journal of Symbolic Logic 23, 261–266. [23] Prior, A.N. (1971) Objects of Thought. Oxford: Clarendon Press. [24] Read, S. (1997). Completeness and categoricity: Frege, Gödel and model theory, History and Philosophy of Logic 18, 79–93. [25] Read, S. (2002) The Liar paradox from John Buridan back to Thomas Bradwardine, Vivarium 40, 189–218. [26] Read, S. (2006). Symmetry and paradox, History and Philosophy of Logic 27, 307–318. [27] Read, S. (2007) Bradwardine’s revenge, in Revenge of The Liar ed. J.C. Beall. Oxford: Oxford U.P., pp. 250–261. [28] Read, S. (2008) The Truth Schema and the Liar, this volume, pp. 3–17. [29] Reck, E.H. and Awodey, S. ed. and tr. 2004. Frege’s Lectures on Logic. Carnap’s student notes 1910–1914, ed. G. Gabriel. Chicago: Open Court. [30] Restall, G. (2008) Models for Liars in Bradwardine’s Theory of Truth, this volume, pp. 135–147. [31] Roure, M.-L. (1970) La problématique des propositions insolubles au XIIIe siècle et au début du XIVe , suivie de l’édition des traités de W. Shyreswood, W. Burleigh et Th. Bradwardine, Archives d’Histoire Doctrinale et Littéraire du Moyen Age 37, 205–326. [32] Sagüillo, J.M. (2008) On a New Account of the Liar, this volume, pp. 149–157. [33] Serény, G. (2008) The Liar Cannot be Solved, this volume, pp. 159– 186. [34] Slater, B.H. (2008) Out of the Liar Tangle, this volume, pp. 187–197. [35] Smullyan, R. (1948) Modality and description, Journal of Symbolic Logic 13, 31–37. [36] Soames, S. (1999) Understanding Truth. New York: Oxford University Press. [37] Spade, P.V. (1975) On a conservative attitude toward some naive semantic principles, Notre Dame Journal of Formal Logic 16, 597– 602. [38] Spade, P.V. (1979). Roger Swyneshed’s Insolubilia: edition and comments, Archives d’Histoire Doctrinale et Littéraire du Moyen Age 46, 177–220. [39] Spade, P.V. (1981) Insolubilia and Bradwardine’s theory of signification, Medioevo 7, 115–134.

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[40] Spade, P.V. (2005) ‘Insolubles’, in The Stanford Encyclopedia of Philosophy (Fall 2005 Edition), ed. Edward N. Zalta. URL = http://plato.stanford.edu/archives/fall2005/entries/insolubles/. [41] Tarski, A. (1953) The establishment of a scientific semantics, tr. by J. Woodger, in Logic, Semantics, Metamathematics. Oxford: Clarendon, pp. 401–408. [42] Woleński, J. (2008) Read About T-Scheme, this volume, pp. 199–203.

II

HISTORICAL BACKGROUND: RESTRICTIONISM VERSUS THE MANIFOLD THEORY OF MEANING

Chapter 14 RESTRICTIONISM: A MEDIEVAL APPROACH REVISITED Claude Panaccio University of Quebec at Montreal, Montreal, Quebec, Canada H3C 3P8

Abstract

Restrictionism is the doctrine that self-reference is to be banned somehow, at least in some sort of cases. As a solution to the Liar paradox, restrictionism has been popular in the thirteenth and early fourteenth century, and then rapidly lost ground to rival theories such as Bradwardine’s. The aim of this paper is twofold: (1) to explain what medieval restrictionism amounted to as a matter of historical fact and why it came to be abandoned; (2) to provide a modern reformulation of the approach that still seems promising as a solution to the Liar and related paradoxes.

Keywords: Adhocness, Restriction (restrictio), Restrictionism, Supposition (suppositio), Ungroundedness.

Restrictionism is the doctrine that self-reference is to be banned somehow, either in all cases (strong restrictionism) or in some cases only (weak restrictionism). As a solution to the Liar paradox, the strong version was popular in the thirteenth century and the weak one was adopted by several authors in the first quarter of the fourteenth century. Restrictionism, both weak and strong, then came under heavy attack and rapidly lost ground to rival approaches, until finally nobody was interested in the problem anymore, as happened in the early modern period. The twentieth century, of course, spectacularly revived the discussion on the Liar, and the last two or three decades, in particular, have even intensified it.

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Yet, restrictionism has been nearly forgotten in the recent debates.1 My aim here is twofold. First, I will explain what medieval restrictionism amounted to as a matter of historical fact, both in its strong and weak versions, and why it came to be abandoned by most logicians in the fourteenth century. And second, I will sketch a modern reformulation for a (weak) form of restrictionism that still seems promising to me insofar as it avoids altogether, at a reasonable price, the well-known threatening contradictions, the improbable Tarskian hierarchy of languages, and the Kripkean truth-value gaps, while requiring – in a nominalistic spirit – only token utterances as the bearers of truth-values.2

14.1

Historical Survey

Paul Vincent Spade, who provided the best available examination of the medieval ‘insolubilia’ literature,3 distinguished three periods: (1) from the beginnings of the medieval debate in the middle or late twelfth century to ca. 1320; (2) from ca. 1320 to ca. 1350 – the most creative period; and (3) from ca. 1350 to the end of the Middle Ages, as logicians discussed and refined the ideas of their predecessors without introducing much novelty [47, p. 246]. Restrictionism emerged very early in the first period as one of the favourite solutions. Many texts have been lost unfortunately, but the position is already discussed as a current one around 1225 in an anonymous Parisian Insolubilia treatise (see [14, Sect. 3, pp. 93–98]); it is positively defended by the mid-thirteenth in at least three of the rare treatments of the question that we still have from that period;4 it is also found in a revised version at the end of the century in such renowned authors as Simon of Faversham and (probably) John Duns Scotus;5 and two of the most prominent logicians of the early fourteenth century, Walter Burley and Richard Campsall, resolutely subscribed to weak restrictionism.6 1 One noteworthy exception is the work of Keith Simmons, whose “Singularity Theory of Truth” is partly inspired by what he calls the “Ockham–Burley–Pseudo-Sherwood solution” to the Liar. See in particular [40]. 2 Former attempts of mine to discuss various aspects of the matter are to be found in Panaccio [28–30]. With respect to these, the present paper is intended to be both more embracing and more systematic. 3 Most of these papers are reprinted in [49]. See also [47, 48], and Spade’s very useful ‘Catalogue’ on the medieval Liar literature [45]. 4 Those are to be found in: (1) the anonymous Insolubilia edited by Braakhuis [5]; (2) the Insolubilia of the Pseudo-Sherwood [57] edited by Roure [38]; and (3) the Summulae logicales of Lambert of Auxerre [23] edited by Alessio [1]. 5 See Simon of Faversham [41], quest. 25, ed. by Ebbesen et al. [16, pp. 165–167]; and John Duns Scotus [39], quest. 52–53, ed. by Wadding [55, vol. 2, pp. 73–76] (the authenticity of this text, however, has been questioned by some commentators; see [45, pp. 65–66]). 6 See Walter Burley [9], Insolubilia, ed. by Roure [38, pp. 262–284]; and Richard Campsall [10], quest. 10, ed. by Synan [50, pp. 162–177].

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In Spade’s second period – from ca. 1320 to ca. 1350 – as the Insolubilia came to be more intensely investigated, William of Ockham still adopted — without much discussion – Burley’s form of weak restrictionism7 and was followed in this by a few authors such as Roger Roseth.8 And at least one important English philosopher, Robert Holkot, endorsed strong restrictionism.9 But the most salient relevant development of the period was that the approach was severely criticized by two of the most influential thinkers of the time, Thomas Bradwardine and John Buridan;10 and very few authors after that returned to it during Spade’s third period.11 In order to understand what the approach amounts to, the first thing to be noticed – even though it is not frequently stressed by the commentators – is that the basic concept at the heart of restrictionism – the very idea of restrictio – is a technical term of the medieval theory of supposition (or reference, more or less). The main point of this theory, taken as a whole, is that we should systematically distinguish two sorts of semantical properties for each significant term: the noncontextual – or precontextual – properties, such as signification, and the referential properties the term acquires when it is used in the context of a proposition, supposition mainly. The realist logicians of the thirteenth century, for instance, would say that a categorematic term such as “horse” noncontextually signifies a common nature, horseness in this case, and that taken within a sentence such as “horses are mammals” or “some horses are running”, it typically stands for – or “supposits for” (supponit pro) – individual horses.12 A nominalist logician, on the other hand, such as William of Ockham who unequivocally rejects common natures, would say that the term “horse” non-contextually signifies all horses, whether past, present or future (and even merely possible ones), while taken in the context of a given proposition, it normally supposits for some of them only.13 Both groups, at any rate, would agree that every simple 7 See William of Ockham [56], Expositio super libros Elenchorum 10, ed. by Del Punta [13], pp. 267–268; and Summa l ogicae III-3, 46, ed. by Boehner et al. [4], pp. 744–746. 8 On Roseth’s position, see [45, pp. 101–102]. 9 On Holkot’s position, see [45, pp. 98–99]. 10 See Thomas Bradwardine [6], Insolubilia, esp. Chaps. 2–5, ed. by Roure [38, pp. 286–296]; and John Buridan [8], soph. 7, ed. by Pironet [34, pp. 152–153] (Engl. Transl. in [19, p. 966]). 11 Exceptions include Peter of Ailly [32], who subscribed to some form of restrictionism in his Insolubilia written around 1372 (Engl. transl. in [46, pp. 35–94]); and Henry of England, who wrote at the very end of the fourteenth century, and about whom virtually nothing is known (see [45, pp. 55–56]). 12 See in particular Peter of Spain [33], Tractatus VI, ed. by De Rijk [15, pp. 79–83]. 13 See William of Ockham [56], Summa logicae I, 4: “Thus, the term ‘man’ signifies all men; the term ‘animal’, all animals; and the term ‘whiteness’ all whitenesses” (Transl. [24, p. 55]). For Ockham’s theory of supposition, see Summa logicae I, 63–77.

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categorematic term has by itself a non-contextual extension, and that when taken as subject or predicate of a proposition, it acquires a – usually different, and variable – contextual extension. For the realists, the non-contextual extension of a term was identified with all the individuals of any time that exemplify the signified common nature, while the nominalists directly equated the non-contextual extension with the (singular) significata of the term. The contextually variable extension, on the other hand, was identified by both groups with what the term supposits for in the relevant context. In most cases, the contextual extension was thought to be arrived at by a “restriction” on the precontextual one. This is exactly the technical notion of restrictio that was used in the discussion of the insolubilia. In a sentence such as “all cherry trees are blooming”, for example, it was typically said that the present tensed verb restricts the subject to stand for presently existing cherry trees. In “the cherry trees were blooming last year”, the past tense verb (plus the temporal specification in this case) can restrict the subject to supposit for the cherry trees that existed last year. And so on. As a solution to the Liar paradox, what restrictionism holds is precisely that a certain type of contextual restriction in this technical sense operates in such contexts to modify the extensions of the terms in such a way that the threatening contradictions are blocked. Strong restrictionism thus holds that a term in context can never supposit for a complex expression – a sentence in particular – of which it is a part, while weak restrictionism has it that the relevant restriction applies only to truth-predicates (“true” and “false”) and only in the cases where contradictions would otherwise follow. Neither doctrine considered that the significations of the terms in themselves – nor, consequently, their precontextual extensions – were in any way modified by such special restrictions. The second thing to pay attention to is that a rather precise theory of truth-conditions for categorical sentences was standardly presupposed by restrictionists. It corresponds (more of less) to the one that has been explicitly stated by William of Ockham in Part II of his Sum of Logic.14 The view admits of only two truth-values – True and False – thus subscribing to what we call today the Bivalence principle; and the truth-conditions for each sort of categorical sentences (whether singular, universal or particular, on the one hand, and whether affirmative or negative, on the other hand) are formulated in terms of what the subject supposits for with respect to what the predicate supposits for. A universal affirmative 14 See

William of Ockham, Summa logicae II, 2–5.

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proposition such as “all horses are mammals”, for example, will be true if and only if the predicate supposits for whatever the subject supposits for.15 A negative universal proposition such as “no horse is running” will be true if and only if the subject supposits for nothing that the predicate supposits for. And so on, mutatis mutandis, for singular and particular propositions.16 Given such truth-conditions, restrictionism tackles the Liar in the following way. Suppose that a man named Socrates utters only one sentence: “What Socrates says is false” (where the term “Socrates” refers to this very man). The standard paradoxical reasoning in such a case is that if what Socrates says is true, then it is false, and if what Socrates says is false, then it is true, with no apparent way out of the contradiction. But restrictionism, whether strong or weak, states that the predicate “false” is restricted in the sentence under consideration in such a way that the very sentence of which it is then a part cannot be admitted among its supposita in this context. If this is so, the Liar sentence (“what Socrates says is false”) turns out to be false, and the Strengthened Liar sentence (“what Socrates says is not true” – uttered by Socrates under the same conditions as before) turns out to be true, without the familiar contradictions ensuing. To see this, assume first that the restriction against self-reference is applied only to the truth-predicates (“true” and “false”), as in weak restrictionism. The subject of the Liar sentence (“what Socrates says”) then supposits for something for which its restricted predicate (“false”) does not supposit, namely the sentence itself. Being affirmative, the Liar sentence, consequently, must be counted as false. And similarly, the subject of the Strengthened Liar sentence (“what Socrates says” again) supposits for something for which its restricted predicate (“true” in this case) does not supposit, namely the sentence itself. Being negative, the sentence, consequently, must be counted as true. In neither cases, however, does a contradiction ensue. Take the Liar sentence “what Socrates says is false” (as uttered by Socrates). Its predicate being restricted in the required way, what that sentence means in this context is not merely that what Socrates says is false (which is indeed the case according to the restrictionist’s diagnosis), but (as Ockham, for one, strongly insists) that what

15 Ockham does indeed speak of “sufficient and necessary conditions” for the truth of propositions (“quid sufficit et requiritur ad veritatem propositionum”), which warrants my use of “if and only if” in this context. 16 Note that according to such truth-conditions every affirmative proposition in which the subject supposits for nothing comes out false (since there is nothing in such cases for which the subject and the predicate both supposit), and every corresponding negative proposition comes out true (for the same reason).

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Socrates says is a falsehood other than “what Socrates says is false”. Since Socrates says nothing else – and a fortiori no other falsehood — this utterance, thus interpreted, is plainly and unproblematically false. And similarly, the Strengthened Liar sentence (as uttered by Socrates) does not merely mean in this context that what Socrates says is not true, but that what Socrates says is not a truth other than “what Socrates says is not true”, which, in the case under consideration, is plainly and unproblematically true, for (ex hypothesi) he says nothing else. If, on the other hand, both the subject-term (“what Socrates says”) and the predicate are restricted, as required by strong restrictionism, the analysis gets a bit more complicated, admittedly, but the same results are reached in the end: the Liar comes out false and the Strengthened Liar true. Take the former. If both terms are restricted, “what Socrates says is false” comes to mean that what Socrates says in addition to “what Socrates says is false”, is a falsehood other than “what Socrates says is false” which – under the medieval truth-conditions – turns out to be unproblematically false if Socrates says nothing else. And the reasoning is similar, mutatis mutandis, for the Strengthened Liar.17 These solutions, in short, do succeed in avoiding the contradictions without renouncing bivalence. Why is it, then, that they were finally abandoned in the fourteenth century? As to strong restrictionism, the main reason is straightforward: it is much too strong. As Walter Burley remarked, if the only thing Socrates says is “something is said by Socrates” instead of “what Socrates says is false”, then what he says should come out unambiguously – and unproblematically – true, while the strong restrictionist’s diagnosis in this case as in the previous one would have to be that the sentence is false, for both the subject-term (“something”) and the predicate (“said by Socrates”) would be restricted

17 Most medieval authors, I must say, would have presented the case a little differently, by formulating the Liar sentence as “Socrates says a falsehood” (Sortes dicit falsum) rather than “what Socrates says is false”, and the Strengthened Liar as “Socrates does not say a truth” (Sortes non dicit verum) rather than “what Socrates says is not true”. Their analysis, then, would be that the proper name “Socrates” is the subject of both sentences, the verb is “to say” (with or without a negation) and the predicate is “false” in the Liar and “true” in the Strengthened Liar. Whether in strong or in weak restrictionism, only the predicates would be restricted under this analysis and the relevant sentences would thus be taken to mean, respectively, that Socrates says a falsehood other than “Socrates says a falsehood”, which is clearly and unproblemetically false in the context, and that Socrates does not say a truth other than “Socrates does not say a truth”, which is clearly and unproblematically true in the context. These analyses admittedly are simpler and more intuitive than what I propose in the text, but they have the drawback of concealing the connection of their diagnosis with an explicit – and very simple – theory of truth-conditions for categorical sentences with the verb “to be” as a copula, such as the one we find in Part Two of William of Ockham’s Sum of Logic.

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in exactly the same way as in the Liar sentence.18 And, of course, indefinitely many other counterexamples could be adduced, such as “what Socrates says is in English” (supposing he does speak English) or even – as Bradwardine suggests – “a sentence is presently uttered” (supposing no other sentence is uttered by anybody at the same moment).19 In addition, it must be said that the arguments adduced in favour of strong restrictionism were usually very bad, and even “shockingly bad”, as Paul Spade once put it (see [44, p. 299]). The basic argument, in particular, had to do with the part-whole relation: what a discourse is about, says a thirteenth-century anonymous Tractatus de insolubilibus, is a part of this discourse; so if the subject term supposits for the sentence of which it is itself a part, the part will come to be equal with the whole, which is impossible (see [5, p. 132]). Walter Burley disparagingly reports a couple of similar arguments for strong restrictionism, which badly conflate, in the same vein, semantical relations with part-whole relations.20 Other objections have been addressed both to strong and weak restrictionism simultaneously, the main one being that we can very well, after all, think and speak about whatever we want to. As John Buridan remarks, I can obviously say that the sentence I am presently uttering is affirmative, and since I do understand what “false” means, why couldn’t I say that it is false as well?21 Such difficulties, as formulated mainly by Thomas Bradwardine in the 1320s and John Buridan in the 1330s, eventually led to the abandonment of restrictionism by most scholars of the fourteenth and fifteenth centuries. Since Bradwardine and Buridan, in addition, were able to put forward a completely different solution of their own to the Liar paradox, the restrictionist approach came to look like an artificial and ad hoc device, or, as Bradwardine himself put it, as “a mere invention of those who know of no other way of responding to the insolubles”.22 18 See

Walter Burley, Insolubilia 2.05, in [38, p. 270]. Thomas Bradwardine [6], Insolubilia 3.04, in [38, p. 288]. 20 See Walter Burley, Insolubilia 2.04, in [38, pp. 269–270]. Burley’s correct rebuttal of these arguments occurs just a little further down in Sect. 2.06 (p. 270). 21 See John Buridan, Summulae de practica sophismatum 8, soph. 7, in [34], p. 152 (Engl. Transl. in [19, p. 966]). Thomas Bradwardine has a similar argument in his own Insolubilia 3.06 [38, pp. 288–289]. He also stresses that since the mental concept “falsehood” naturally signifies every falsehood, there is no reason why it couldn’t supposit for any particular one of them in any given context (ibid. 3.02 and 3.10, in [38, pp. 288, 290]). 22 Thomas Bradwardine, Insolubilia 3.04, in [38, p. 288] (my translation). Bradwardine and Buridan’s solutions to the Liar were based (roughly) on the idea that the truth of a sentence s requires not only that what s says should be the case, but also that s should not itself entail its own falsity. For an in-depth analysis of this approach – especially in Bradwardine’s version – see [37]. 19 See

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Despite this pessimistic outcome of the medieval debate, however, it cannot be said that restrictionism in its weak form has ever been conclusively refuted. It is true, of course, as Bradwardine and Buridan pointed out, that self-reference is neither psychologically nor linguistically impossible, and insofar as some medieval restrictionists ever thought so, they were wrong. But couldn’t a weak restriction, of the sort Burley and Ockham considered, be admitted as a conventional rule for certain languages if its consequence was to neutralize the paradoxes in these languages at a reasonable cost? This is the suggestion I would like to explore in the rest of this paper.

14.2

Theoretical Perspectives

2.1 There are today four main schools of thought with respect to the Liar and related paradoxes: The hierarchists, who think that the paradoxes are best avoided if the truth-predicate that is applied to the sentences of a certain language belongs not to that language itself, but to a richer metalanguage;23 The “gappers”, who invoke truth-value gaps;24 The so-called “glutters”, who admit that certain sentences do have more than one truth-value;25 The indexicalists, according to whom the extension of the truthterms can vary with the context (see [2, 7, 20, 40]). Those are not logically exclusive: indexicalists can be gappers of some sort, for example;26 and many authors today are ready to admit, anyway, that more than one solution might be manageable. But the general 23 The seminal texts for this approach are Tarski [51, 52]. An anonymous referee for the present paper interestingly remarked that Tarski-like “hierarchy” theories might be viewed in effect as a form of restrictionism. This is not technically true, however, and for an instructive reason: Tarski did not draw any systematic distinction between the precontextual and the contextual extensions of the truth-predicate, as the restrictionists did. Which is why, exactly, he ended up multiplying truth-predicates and hierarchizing them. 24 The most salient “gapper” proposal is to be found in [22]. See also, among others, Van Fraassen [54], McGee [27], Soames [42]. 25 See in particular Priest [35]. 26 Burge [7] and Simmons [40] both admit of relative truth-value gaps with respect to contextually restricted truth-predicates: having identified a restricted truth-predicate “truei ” in a certain context, they hold that it is possible for a given sentence s in this context to be neither truei nor falsei , although the same sentence does have a truth-value with respect to a (contextually) different truth-predicate, e.g. “truej ”. This is a feature which saliently distinguishes Burge and Simmons’s approach from the one that will be proposed here.

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tendency in the last decades has been to move away from the hierarchist approach which had dominated the scene from the 1940s to the 1960s. The new consensus is that the heavily artificial and bothersome Tarskian hierarchies should be eschewed if possible, and the search for alternatives is what motivated the exciting recent explorations and discussions of the gapper, the glutter, and the indexicalist hypotheses. With respect to this general picture, weak restrictionism displays some prima facie interesting virtues. First of all, it does eschew language hierarchies: the truth-predicate it uses belongs to the language itself for which it is used, which is both more natural and more economical. Secondly, contrary to the gapper’s approach, it respects the Bivalence principle: every apparently paradoxical sentence is taken to be definitely true or false, and the solution, consequently, introduces no special element of indetermination. Thirdly, contrary to the glutters’s approach, it efficiently avoids contradictions, as seems intuitively desirable. And fourthly, it requires no special ontological eccentricity, since it has to do only with the contextual extensions of the truth-predicate, and is therefore compatible with any independently acceptable ontology for truth-bearers. Weak restrictionism, of course, is a variety of indexicalism, and it can find some support in the arguments that have been adduced in favour of the general approach.27 With respect to other varieties of indexicalism, its interest, if it can be worked out, would be to limit the contextual variations of “true” and “false” to a bare minimum, in conformity with what Keith Simmons calls the Principle of Minimality [40, pp. 106–108], without dropping bivalence in the process. The main worry, of course, is that no conventional restriction could plausibly have such virtues without being intolerably ad hoc. This is a point I will discuss in Sect. 2.5, when the rule I propose for exploration will have been explicitly stated. 2.2 The challenge, first and foremost, is to formulate a weak restriction rule that will adequately neutralize the paradoxes, at least in certain important fragments of natural languages, without being too costly. I will limit myself here to paradoxes involving simple truth-predicates, such as “true” and “false”, occurring in what the medievals called “categorical sentences”. A categorical sentence in this sense is a complex sequence of terms capable in principle of being true or false, and composed of a (possibly quantified) subject-term, a non modal copula (with or without negation) and a predicate, both the subject and the predicate being either 27 Burge [7], for example, convincingly argues that it is quite natural in a typical Liar situation to invoke some sort of truth-value shift between the evaluation of the Liar himself about his own utterance (e.g. “what Socrates says is false” as uttered by Socrates) and the evaluation of an external observer using the very same sentence.

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simple terms such as “Socrates” or “horse”, or complex terms, such as “black horse” or “what Socrates says”, and so on. Some well-known paradoxes, of course, do not involve truth-predicates at all (such as Grelling’s, for example), while others (such as Curry’s) spring from noncategorical sentences, but space being limited, those will not be addressed here. Even with these limitations in view, however, certain familiar puzzles that are related to the Liar, but somewhat different from it, will need to be taken into consideration. One of them is the so-called “Truthteller” problem. Suppose Socrates says only one thing: “What Socrates says is true”. In this situation, by contrast with the Liar, no contradiction threatens: if the sentence is true, it is true; and if it is false, it is false, period; and the case, consequently, did not bother the medievals. Ockham, for example, defines an “insoluble” as a sentence from which its opposite can apparently be inferred by necessary rules,28 which is plainly not what we have with the Truthteller sentence. Yet, as many recent authors have insisted, the Truthteller is just as much a problem as the Liar is, since, even though no contradiction threatens, nothing whatsoever seems to determine which truth-value the sentence is supposed to be attributed, and this seems just as irrational as deriving a plain contradiction.29 Because the medieval weak restrictionists such as Burley and Ockham did not think of the Truthteller as an insoluble, their own restriction rule was too weak to deal with it.30 But once we are alerted to the problem, it is fairly easy to strengthen their approach in consequence. Bradwardine and Buridan’s solution, by contrast, cannot be extended to deal with the Truthteller, since its truthvalue attribution heavily rests on the fact that the Liar sentence entails its own falsehood, which the Truthteller does not. A further group of puzzles that our tentative restriction rule will have to face are the ones that involve several speakers, whether they lead to contradiction as in the Liar case, or to mere indetermination, as in the Truthteller case. Suppose, for example, that the only thing Socrates says is: “What Plato says is false”; and the only thing Plato says is: “What Socrates says is true”. A contradiction, then, seems to follow on any hypothesis and what we have, therefore, is a complexified Liar situation. If, on the other hand, Socrates says: “What Plato says is true” and Plato says: “What Socrates says is true”, what we get is a complexified 28 William

of Ockham, Summa l ogicae III-3, 46, ed. by Boehner et al. [4, p. 744]. point is clearly made, for example, by Mackie [25, pp. 240–241]. 30 Burley’s restriction rule is explicitly meant to apply only in the cases where a negative self-reflection threatens (with either the predicate “true” occurring in a negative sentence, or the – intrinsically negative – predicate “false” occurring in an affirmative sentence). See Burley’s Insolubilia 3.04, in [38, p. 272]. And Ockham on this simply follows Burley. 29 The

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Truthteller. And, of course, related puzzles can be devised with any number of speakers and any combination of “true” and “false” within their statements. Insolubilia involving three speakers, for example, were considered by medieval logicians as early as in the twelfth century,31 and it is to be presumed that Burley’s restriction rule, for one, was intended to cover such cases in principle by treating them as examples of indirect self-reflections. If we want to be explicit about it, however – as we certainly should –, special complications arise, as we will see, that medieval restrictionists do not seem to have suspected. 2.3 What we are looking for is a reasonably simple rule that will neutralize in a single blow the Liar, the Strengthened Liar, the Truthteller, and all their variants involving several speakers. A certain number of choices need to be made for this. First, should the restriction apply only to the truth-predicates (“true” and “false”), or only to the other term in the sentence (to “what Socrates says”, for example, in “what Socrates says is false”), or to both? The Principle of Minimality instructs us to exclude the latter option and restrict only one term per sentence since this is sufficient in all cases. Since, on the other hand, the responsibility for the threatening contradictions and indeterminations in the cases under consideration obviously lies with the truth-predicates, it seems prima facie reasonable to follow the medieval weak restrictionists and to restrict in a special way the contextual extensions of precisely these predicates. A second – and more delicate – choice that faces us is whether the restriction should have to do with the relevant sentence type or only with the particular token involved. Should “false” in any given occurrence of “what Socrates says is false”, for example, be prevented from suppositing for this sentence itself considered as a type (or for any occurrence of it, if we prefer), or should it be prevented only from suppositing for the single utterance token of which it is then a part (e.g. Socrates’s own utterance in the original case under consideration)? According to the first option, not only will Socrates’s utterance of the sentence be false, but so will any other utterance of it, since, whoever the speaker is, the sentence, thus interpreted, will always mean that what Socrates says is a falsehood other than “what Socrates says is false”, which is plainly false in the imagined case. According to the other interpretation, by contrast, the truth-value of the sentence will vary with the tokens, each token

31 See the Insolubilia Monacensis edited by De Rijk [14], : “If there are three men, Socrates, Plato and Cicero; and Socrates says that Plato lies; and Plato says that Cicero lies; and Cicero says that Socrates lies; then a circular deduction follows” (my translation). Other references to such medieval Liar trios are found in Ebbesen and Spade [17, esp. pp. 193–194].

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being taken to mean something like: what Socrates says is a falsehood other than the present token, which, ex hypotheses, is false in the case of Socrates’s own utterance, but true in Plato’s mouth, or in ours for that matter. A close reading of Burley’s and Ockham’s texts reveals that they both favoured the type interpretation of their restriction rule.32 But this seems to be unnecessarily strong, thus offending in a new way the Principle of Minimality. As I see it, one powerful motivation in favour of restrictionism is that it is prima facie compatible with a nominalistic ontology, in which neither linguistic types nor abstract propositions enjoy any independent existence, and only singular tokens are accepted as the bearers of truth-values. Such an ontology, admittedly, does not rule out by itself the type interpretation of the restriction rule (the rule, thus interpreted, could still be taken as having to do with all the occurrences of a certain phrase, rather than with some special abstract entity), but it makes it inappropriately strong. In Ockham’s approach, for example, we are entitled to say “this is false” when pointing at Socrates’s utterance of “what Socrates says is false” (since this utterance is indeed false according to the restrictionist diagnosis), and we are entitled to say, of course, “what Socrates says is this” (pointing at the same utterance again), but we are not entitled to infer from these two premises “what Socrates says is false”, since Socrates’s utterance of the same sentence would then be excluded from the extension of our own use of “false” in this context. Which seems unnecessarily cumbersome. My preference will be to treat the truth-predicates as indexical terms, restricted only, in each context, from suppositing for certain singular token utterances. A third choice, finally, must concern the situations where more than one utterance is involved, as when Socrates says “what Plato says is false”, and Plato says “what Socrates says is false” (which yields no contradiction in this particular case, but a bothering indetermination of the Truthteller sort: they cannot both be right, but nothing in the situation seems to determine which one is). The restrictionist policy in such cases will be to restrict at least one of the two truth-terms involved from suppositing for the other utterance. But the question is: should both predicates thus be restricted, or only one of them? The Principle of Minimality here favours the latter option: if one of the two predicates is restricted, there is no point anymore for restricting the other one. Suppose that “false” in Socrates’s utterance is prevented somehow from suppositing for Plato’s utterance; Socrates’s utterance, then, is ipso facto false, and there seems

32 Ockham holds, for example, that “what Socrates says is false” is equivalent to “what Socrates says is a falsehood other than ‘what Socrates says is false’ ”, even when we – rather than Socrates – utter it. See his Summa logicae III-3, 46, ed. by Boehner et al. [4, p. 746].

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to be no point, consequently, for preventing Plato from saying so. But the only way to successfully restrict only one of the two predicates in such a situation is to postulate an order between the two utterances. Our restriction rule could then prevent the predicate of the lower-ranked utterance, say, from suppositing for the upper-ranked utterance, and the resulting truth-values for both utterances would thus turn out to be relative to the chosen order. This might appear to be quite strange at first sight, and to offend the “Principle of Justice” proposed by Tyler Burge with respect to such cases: “One should not give one statement truthconditions instead of another without some reason”.33 As far as I can see, however, opting for such an asymmetrical restriction in these situations is the more economical and less damaging approach available to weak restrictionism and it is, consequently, the one I propose to endorse here for exploratory purposes. The apparently peculiar character of this choice will be further discussed in section 2.5 along with the general question of adhocness. 2.4 With these constraints and choices in mind, the general idea will be to coin a rule that prevents any token t of “true” or “false”, or an equivalent, from having in its extension in a certain context the sentential token utterance s i of which t is a part in this context, as well as any other (upper ranked) elementary sentential utterance s j that purports to attribute a truth-value to s i . In view of this, let me first introduce a few preliminary definitions that will be useful. Definition (Df1). Any token of a categorical sentence in the medieval sense (as described above in Sect. 2.2) will be called an elementary sentential token. Example: Any token of “Socrates is a man”, “Socrates’s father was not Roman”, “all horses are mammals”, “some horses are black”, “no black horse is presently running”, “some of Socrates’s horses are not black”, and so on, is an elementary sentential token.

Definition (Df2). Let us say that an elementary sentential token s 1 directly T-refers to a token s 2 if and only if: (1) The subject or the predicate of s 1 is a simple truth-term such as “true”, “false” or an equivalent in whatever language; (2) the other term of s 1 (i.e. the subject if the truth-term is the predicate, and conversely) supposits for s 2 in the context of utterance (i.e. has s 2 in its extension in this particular context).

33 Burge

[7], in [26, p. 110].

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Definition (Df3). An elementary sentential token s 1 will be said to T-refer to a token s 2 if and only if: Either s 1 directly T-refers to s 2 , Or there is a sentential token s 3 which is such that s 1 T-refers to s 3 and s 3 T-refers to s 2 . Example: In the following trio: (a) whatever Plato says is false (uttered by Socrates), (b) whatever Cicero says is true (uttered by Plato), (c) it is raining (uttered by Cicero), (b) T-refers to (c) [since it directly T-refers to it], and (a) T-refers both to (b) [since it directly T-refers to it] and to (c) [since it T-refers to the token (b) that T-refers in turn to (c)]. Note: (Df3), being a recursive definition, can be applied to any similar chain with n elements for any number n.

Definition (Df4). A T-reference path from an elementary sentential token s 1 to a token s n is the sequence of tokens <s 1 . . . s n > such that for every i in the sequence, s i directly T-refers to s i+1 . Example: The T-reference path from (a) to (c) in the above example is the sequence <(a), (b), (c)>; the T-reference path from “what Socrates says is false”, uttered by Socrates to this utterance itself in the medieval Liar situation is the sequence whose only member is Socrates’s utterance (nothing prevents in effect that for any i and j in such a sequence, si = sj ).

To these definitions, let us add the following assumption: Assumption (A) The set of all linguistic tokens (in a given context) is well-ordered (i.e.: any relevant token has a distinctive rank with respect to any other). (Note: For any set of tokens, any conceivable ordering will do. The tokens, in other terms, can be ordered in whichever way we like as long as each one ends up occupying a distinctive rank). Given assumption (A) and definitions (Df1)–(Df4), the proposed restriction rule can be formulated in the following way: Rule (RR) A simple truth-term occurring within an elementary sentential token s1 cannot supposit for (i.e. have in its contextual extension)

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any token s2 which is such that s2 T-refers to s1 through a T-reference path containing only sentential tokens of a rank which is greater than or equal to the rank of s1 (in the chosen order). It is clear, for example, how (RR) solves our original Liar situation. In Socrates’s utterance of “what Socrates says is false”, the predicate “false” is prevented by (RR) from suppositing for Socrates’s own utterance, since this utterance T-refers to itself through a T-reference path that contains only one member, namely Socrates’s own utterance, which, of course, is of the same rank as itself. Being affirmative, the utterance, consequently, is false, since its predicate does not supposit for what its subject supposits for. From this falsehood, however, the utterance cannot be inferred to be true (for the reasons explained in section 1 above) and no contradiction ensues. For a more sophisticated case, consider the trio imagined by the twelfthcentury Insolubilia Monacensis:34 (d) what Plato says is false (being the only thing uttered by Socrates), (e) what Cicero says is false (being the only thing uttered by Plato), (f) what Socrates says is false (being the only thing uttered by Cicero). Supposing that the order in the relevant situation is: (d) = 1, (e) = 2, (f ) = 3, the predicate “false” in (e) will not be prevented from suppositing for (f) by (RR), since the T-reference path from (f) to (e) contains one step which is of a lower rank than (e) (namely (d)); and the predicate “false” in (f) will not be prevented from suppositing for (d), since the T-reference path from (d) to (f) contains two steps which are of a lower rank than (f). But the predicate “false” in (d) will be prevented from suppositing for (e) since the T-reference path from (e) to (d) contains only steps of an upper rank than (d). Consequently, (d) is to be assessed as false (since it is affirmative, and its predicate does not supposit for what its subject supposits for, namely (e)). Consequently, (f) is to be assessed as true, since (d) is indeed false as (f) says it is, and the predicate “false” in (f) is not restricted by (RR) from suppositing for (d). And consequently, (e) is to be assessed as false, since although its predicate is not prevented by (RR) from suppositing for what its subject supposits for (namely (f)), it turns out to be so prevented anyway by the mere fact that (f) is not false after all (as we have just established). Each utterance in the trio thus receives one and only one truth-value, without any contradiction or indetermination ensuing. And the same result, obviously, would be reached for whatever similar circular chain,

34 See

above note 31.

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in which every member T-refers to every other. (RR), therefore, does presumably solve all the cases of the Liar type, the Strengthened Liar type and the Truthteller type, with any number of utterances involved in the relevant circular chains. 2.5 Is this solution unacceptably ad hoc? The concern has been strongly expressed by Paul Spade, in particular, with respect to medieval weak restrictionism (Ockham’s version especially) (see [44, 46], p. 5) and the reconstruction of the last section makes it even more pressing apparently. For one thing, the rule (RR) was devised with the express goal of blocking certain familiar paradoxes, but it has not been derived from any independently plausible principles, nor has it been extracted in any way from a conceptual investigation into our ordinary understanding of the truth-predicates. And what is even worse, we have been led to introduce the obviously artificial idea of an order among the linguistic tokens, in such a way that the arbitrary choice of a particular order rather than another one decisively determines the truth-values of certain utterances in cases such as the Socrates–Plato–Cicero situation considered above. Many will be tempted to conclude, no doubt, that such adhocness bluntly disqualifies the approach without further ado. Yet, the question, I want to say, is not so easily settled. Although it is not always realized, one lesson we need to draw from the discussions of the last decades on the Liar and related paradoxes, is that some sort of adhocness and artificiality is simply inevitable in such matters. The reason for this is that the paradoxes directly arise from our ordinary use of the truth-predicates. Insofar as we do have a reasonably clear and relevant ordinary notion of truth, it seems to correspond indeed to what Tarski took to be the sole criterion for the material adequacy of any definition of truth, namely that a sentence “p” should be admitted to be true if and only if “p” itself is admitted. But the problem is, as Tarski insisted, that any truth-predicate satisfying this condition leads to paradoxes if it is unrestrictedly applied to the sentences of the very language to which it belongs, as our ordinary notion of truth certainly requires. No solution will be found, as many have hoped, in a deeper investigation into our intuitive understanding of truth, as if a natural antidote was hidden somewhere in our innate mental apparatus. Plain naturalness in this regard simply leads to contradiction. What the paradoxes reveal in the end is that our spontaneous intuitions about truth need to be held back in some ways, as Tarski himself very well understood. In order to decide which artificiality is acceptable in these circumstances, and which not, we have to resort to a crucial distinction, that was proposed, in particular, by Charles Chihara, between two different problems with respect to paradoxes: the “diagnostic problem” (where

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does the paradox arise from?) and the “preventative problem” (how can the paradox be avoided?) [11].35 As to the diagnostic problem, the discussion of the last 30 years strongly suggests that the sort of paradoxes we are interested in here arise from the “ungrounded” use of semantical terms such as “true” and “false” somehow. Saul Kripke has proposed to say that a sentence involving the concept of truth (or falsehood) is grounded if its own truth-value can be ascertained on the basis of the truth-value of other sentences not involving the concept of truth (or falsehood), while it is ungrounded if its truth-value does ultimately depend on the truth-values of sentences involving it.36 The current consensus is, to use Keith Simmons’s words, that “the pathological sentences are those that are ungrounded” in Kripke’s sense [40, p. 100]. The recently discovered Yablo’s paradox, for example, involves no circularity or selfreflection of any sort, yet it does entail a contradiction just as plainly as the Liar by postulating an infinite series of ungrounded utterances (see [43, 58]). This sort of diagnosis, admittedly, still requires further investigation if we are to reach a satisfactory understanding of what is going on exactly. But however fully it is finally developed, it will never impose by itself one and only one precise cure for the paradoxes. What it suggests is that ungrounded utterances should be neutralized somehow, but it does not point by itself to a unique way of doing it. Weak restrictionism, on the other hand, is merely a partial solution to the preventative problem. What it purports to do is to put a certain salient group of ungrounded sentences under control: those, namely, that prima facie result in (direct or indirect) self-reflection. If, as I believe, there is no natural way of doing this, then a solution of this sort for the preventative problem with respect to the Liar and related paradoxes, must involve some degree of artificiality and counterintuitiveness. And the best way to limit this artificiality is not to hide it behind a superficial – and inevitably misleading – appearance of naturalness, but to confine it to the smallest possible number of cases. Which is why I relied so heavily on the Principle of Minimality in the construction of the restriction rule (RR). Bradwardine and Buridan were certainly right, after all, that nothing naturally prevents us from thinking or saying of any utterance that it is true or false. The point of restrictionism, in today’s context, is not to dig out a rule that looks natural, but to identify the smallest possible restrictions on what we are allowed to say if we are to avoid the Liar-related contradictions. As I see it, the problem with Tarski’s 35 A similar distinction was also proposed by Gupta [18] between what he calls the “descriptive problem” and the “normative problem”. 36 Kripke [22], in [26, p. 57].

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hierarchies of languages was not that they were artificial, but that the artificial constraints they imposed were much too strong. From this point of view, even the appeal to orderings among linguistic tokens is fully justified, insofar as it is called for by the Principle of Minimality, as we concluded. Under any version, restrictionism implies that the truth-predicates should be treated as indexicals of a special sort. What is added by taking token ordering into account is that the contextual extension of a truth-predicate will be relative (among other factors) to a given order among linguistic tokens, and that, as a consequence, the truth-values of some ungrounded sentences might turn out to be relative to such orders. In the Socrates–Plato–Cicero case considered above, for example, Socrates’s utterance is false relative to the token ordering in which it is of a lower rank than the other two, but it is true relative to the token ordering in which it occupies the higher rank. In this approach, then, there is no absolute answer as to whether Socrates’s utterance is true or false. It is false relative to a certain order, and true relative to another one, just as “Socrates is now sitting” is true relative to a certain time and false relative to another one. The difference, of course, is that the taking of time (or place) into account in the evaluation of certain sentences is something quite natural and it is in fact incorporated in our natural languages, while this is obviously not so for token orderings. But this is precisely the sort of artificiality we might be well-advised to accept in order to solve the Liar-related paradoxes, insofar as its effects are felt only in the pathological cases. We don’t really care, after all, about who is right in the Socrates–Plato–Cicero triangle. What we do care about is that we should not be dragged into contradiction by this infernal trio. Since there is no independent fact of the matter anyway, there is no harm in adopting a rule that efficiently avoids the contradiction, even at the cost of making the truth-values relative to an otherwise indifferent factor, if our doing so has no effect whatsoever upon the truth-values of the grounded sentences, which, after all, are the ones we really care about. (RR), I believe, achieves just that. 2.6 A worry of another sort about weak restrictionism might be that it is too intimately associated with the medieval theory of “supposition” to be helpful anymore in today’s context. How could an obsolete preFregean framework be of any interest for tackling problems that have been addressed in the last decades with the most sophisticated semantical tools available? To which there are two elements of reply. First, the weak restrictionist approach as I developed it is easily adaptable to any semantical theory in which truth-value terms such as “true” and “false” have singular utterances in their extensions, whatever the logical forms of the sentences in which they occur are taken to be. Should we turn,

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for example, to a quantificational representation of our original Liar sentence as something like “there is a u such that Socrates uttered u and u is false”, a restriction very much like the one we considered could still be applied to the predicate “false” in this new scheme, with similar results: the sentence would turn out to have one and only one truth-value without any contradiction ensuing. It might mean, for instance, in this particular case, that Socrates says something which is a falsehood other than this very utterance, which is plainly and unproblematically false just as in our previous analysis. The main point, however, is that supposition theory, in its general outline, is not at all obsolete as a semantical representation for the fragments of natural languages we are interested in here. Non modal categorical sentences can be represented as having one subject and one predicate, and their truth-conditions can normally be given in terms of what the subject and the predicate supposit for.37 Supposition theory, actually, yields a much better description of natural languages with respect to such sentences than quantificational logic, for instance. The drawbacks of the medieval theory come out when we get to more complex sentences and try to rigorously evaluate which inferences are valid among them and which not. Supposition theory then quickly becomes cumbersome. But as long as we stick to simple subject-predicate sentences, it does remain an appropriate tool for semantical analysis. One salient advantage it has for our present purpose is that it is entirely based upon a crucial distinction that is often neglected by more recent approaches: the distinction, namely, between precontextual and contextual semantical properties. It is precisely this distinction which is applied to “true” and “false” in the restrictionist approach to the paradoxes. Just like any other general term, “true” and “false” are attributed variable extensions according to the utterances in which they occur. Their special distinction in this regard is that the contextual restrictions that affect them are supposed to involve extra factors which play no role in the other cases (those factors, namely, that are identified as such in the formulation of a restriction rule such as [RR]). But this comes out in supposition theory as just a special case of a much more general phenomenon. The overall framework thus makes it intelligible how a conventional rule can be introduced in a given language for artificially restricting the extensions of certain terms in certain contexts without changing their

37 This, of course, includes the cases where either the subject or the predicate supposits for nothing. As William of Ockham insists: “ ‘the term has a supposition, therefore it supposits for something’ is not a valid inference” (Summa logicae I, 72, ed. by Boehner et al. [4, p. 219]; my translation).

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meanings. The noncontextual meanings – or significations – of terms are still taken to play a decisive role in the determination of their contextual extensions, as they certainly should, but once we see that various other factors always intervene anyway, nothing prevents us in principle from coining conventional rules in this respect if this should be useful. Once such a special restriction rule is introduced for “true” and “false”, of course, the resulting semantics for the relevant sentences becomes a bit awkward. For one thing, an inference schema such as “p is true, therefore p is true” will not be universally valid anymore (since it might happen in some context that the second occurrence of “true”, but not the first, should be prevented by [RR] from suppositing for one or more of the utterances for which the subject-term supposits). And the familiar Tarskian equivalence schema between “p is true” and p will not be universally acceptable either (since p might contain a truth-term relevantly restricted in one occurrence but not in the other). But this is not due to any special inadequacy of supposition theory. It is the price to pay for treating the extension of our truth-terms as being indexically sensitive to the occurrence of certain token utterances in the relevant contexts. After all, the inference from “the present token is the first one of its type in the present book” to “the present token is the first one of its type in the present book” is not universally valid either; and nor is the equivalence between “the present token is the first one of its type in the present book” and “ ‘the present token is the first one of its type in the present book’ is true”. What matters for our semantics is that these inference and equivalence schemas remain valid when no indexicals of this sort are involved, and this is something that is preserved in weak restrictionism. In the end, the main impact of resorting to supposition theory for the treatment of paradoxes is to suggest that the problem can be addressed by minimally regulating the contextual extensions of certain terms. Given that the Tarskian equivalences do lead to contradiction if they are left unchecked, this medieval strategy does not seem unreasonable after all.

14.3

Conclusion

What was provided here is merely a sketch. The main point is that if some degree of artificiality is deemed acceptable for solving the preventative problem with respect to paradoxes – as I think it should be – then medieval weak restrictionism is still a good option despite the criticisms it met with in the fourteenth and fifteenth centuries. The reconstruction that was explored here, with the rule (RR) as its main component, stands out in the end as a promising way of dealing with a salient group of paradoxes: those paradoxes, namely, that involve simple truth-terms

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occurring within elementary predicative sentences in such a way that (direct or indirect) self-reflection loops are produced. These include the Liar paradox, the Strengthened Liar, the Truthteller, and a wide variety of variants involving loops with several utterances. Many other puzzles having to do with ungroundedness would need to be tackled by a complete semantical theory. But insofar as the restrictionist strategy has revealed itself fruitful in the basic cases, it seems likely that it can be adapted to the more complex ones. The core of the approach is to impose minimal ad hoc restrictions on the contextual extensions of certain terms so as to neutralize the ungrounded sentences from leading to contradiction or to undesirable indetermination, without affecting the grounded ones in so doing. Whenever it is possible to identify exactly which terms should be restricted to yield the required result, and which utterances should be removed from the contextual extensions of these terms, this sort of solution could in principle be made to work. Whether this is possible in all puzzling cases of ungroundedness remains to be seen, but the prospect so far seems encouraging. Take Yablo’s paradox, for instance: each speaker in an infinite series says only one thing: “somebody after me in the series says something false”.38 There is no self-reflective truth assessment in this situation, even indirectly, but, just as in the original Liar case, a plain contradiction is quickly derived if we try to attribute a truth-value to any one of the utterances involved, as can easily be checked. Our rule (RR) leaves the problem utterly untouched, since any particular utterance in the series T-refers only to the ones that follow it, without any T-reference path leading back to it from any of them. Yet, a similarly inspired restriction rule can very likely be devised for this new sort of cases if we so wish.39 And as many such rules as necessary could be successively introduced to cover the various pathological cases, provided, as before, that the truthvalues of grounded sentences are left unchanged. The strategy does not require that the speakers of the language should all be consulted about the introduction of such rules, or even that they should be publicly advised of it. These rules are meant to cover exceptional situations that are of merely theoretical interest. Should the general program succeed, the resulting gain would not be that the real

38 See

Yablo [58]. is not the place, obviously, to engage in a detailed discussion of the point, but something like the following formulation – which closely parallels that of (RR) – might do the job: A simple truth-term occurring within an elementary sentential token s 1 cannot supposit for any token s 2 which is such that s 2 T-refers to every member of an infinite series of ungrounded utterances through a T-reference path containing only sentential tokens of a rank which is greater than or equal to the rank of s 1 (in the chosen order). 39 This

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truth-values of the paradoxical utterances would have been found, at long last; but that an economical – if artificial – device would be available in principle for avoiding all contradictions and unwanted indeterminations specifically due to ungroundedness, without dropping bivalence and without having to move to a richer metalanguage in the process. That the approach, in addition, should be compatible with a nominalistic ontology for truth-bearers is – in my view, at least – a welcome bonus to the package deal.

Acknowledgments Thanks are due to the Social Sciences and Humanities Research Council of Canada, the Killam Foundation, and the Canada Research Chairs Foundation for their generous support to the research that led to the present paper.

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[13] Del Punta, F. (1979). Guillelmi de Ockham Expositio super libros Elenchorum (= Opera Philosophica III). St. Bonaventure, NY: The Franciscan Institute [14] De Rijk, L.M. (1966). “Some Notes on the Medieval Tract De Insolubilibus, with the Edition of a Tract Dating from the End of the Twelfth Century”. Vivarium 4, 83–115 [15] De Rijk, L.M. (1972). Peter of Spain. Tractatus. Assen: Van Gorcum [16] Ebbesen, S. et al. (1984). Simon of Faversham. Quaestiones super libro Elenchorum. Toronto: Pontifical Institute of Mediaeval Studies [17] Ebbesen, S. and P.V. Spade (1988). “More Liars”. Cahiers de l’Institut du Moyen Âge Grec et Latin 56, 193–227 [18] Gupta, A. (1982). “Truth and Paradox”. Journal of Philosophical Logic 11, 1–60. Reprinted in [26], 175–235 [19] Klima, G. (2001). John Buridan. Summulae de Dialectica. Engl. transl. New Haven: Yale University Press [20] Koons, R.C. (1992). Paradoxes of Belief and Strategic Rationality. Cambridge: Cambridge University Press [21] Kretzmann, N., A. Kenny and J. Pinborg eds. (1982). The Cambridge History of Later Medieval Philosophy. Cambridge: Cambridge University Press [22] Kripke, S. (1975). “Outline of a Theory of Truth”. The Journal of Philosophy 72, 690–716. Reprinted in [26], 53–81 [23] Lambert of Auxerre. See [1] [24] Loux, M.J. (1974). Ockham’s Theory of Terms. Part I of the Summa Logicae. (Engl. transl.). Notre Dame, IN: University of Notre Dame Press [25] Mackie, J.L. (1973). Truth, Probability and Paradox. Oxford: Clarendon [26] Martin, R.L. ed. (1984). Recent Essays on Truth and the Liar Paradox. Oxford: Clarendon [27] McGee, V. (1991). Truth, Vagueness, and Paradox. Indianapolis: Hackett [28] Panaccio, C. (1993). “Solving the Insolubles: Hints from Ockham and Burley”. In [36], 398–410 [29] Panaccio, C. (2001). “Le Menteur: plaidoyer pour une solution ad hoc”. In [31], 87–105 [30] Panaccio, C. (2005). “Le paradoxe du menteur et le langage mental: réflexions sur l’approche restrictionniste”. In [3], 57–71

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[31] Pelletier, Jérôme and Joëlle Proust eds (2001). La normativité. Caen: Presses Universitaires de Caen [32] Peter of Ailly. See [46] [33] Peter of Spain. See [15] [34] Pironet, F. (2004). Johannes Buridanus. Summulae de practica sophismatum. Groningen: Ingenium [35] Priest, G. (1987). In Contradiction. A Study of the Transconsistent. Dordrecht: Martinus Nijhoff [36] Read, S. ed. (1993). Sophisms in Medieval Logic and Grammar. Dordrecht: Kluwer [37] Read, S. (2002). “The Liar Paradox from John Buridan back to Thomas Bradwardine”. Vivarium 40, 189–218 [38] Roure, M.-L. (1970). “La problématique des propositions insolubles au xiiie siècle et au début du xive , suivie de l’édition des traités de W. Shyreswood, W. Burleigh et Th. Bradwardine”. Archives d’histoire doctrinale et littéraire du Moyen Âge 37, 205–326 [39] Scotus, J.D. See [55] [40] Simmons, K. (1993). Universality and the Liar. Cambridge: Cambridge University Press [41] Simon of Faversham. See [16] [42] Soames, S. (1999). Understanding Truth. Oxford: Oxford University Press [43] Sorensen, R.A. (1998). “Yablo’s Paradox and Kindred Infinite Liars”. Mind 107, 137–155 [44] Spade, P.V. (1974). “Ockham on Self-Reference”. Notre Dame Journal of Formal Logic 15, 298–300 [45] Spade, P.V. (1975). The Mediaeval Liar: A Catalogue of the Insolubilia-Literature. Toronto: Pontifical Institute of Mediaeval Studies [46] Spade, P.V. (1980). Peter of Ailly. Concepts and Insolubles (Engl. transl.). Dordrecht: Reidel [47] Spade, P.V. (1982). “Insolubilia”. In [21], 246–253 [48] Spade, P.V. (1987). “Five Early Theories in the Mediaeval Insolubilia-Literature”. Vivarium 25, 24–46 [49] Spade, P.V. (1988). Lies, Language and Logic in the Late Middle Ages. London: Variorum Reprints [50] Synan, E.A. (1968). The Works of Richard of Campsall. Vol. I: Questiones super Librum Priorum Analeticorum. Toronto: Pontifical Institute of Mediaeval Studies

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[51] Tarski, A. (1944). “The Semantic Conception of Truth and the Foundations of Semantics”. Philosophy and Phenomenological Research 4, 341–375 [52] Tarski, A. (1956a). “The Concept of Truth in Formalized Languages” (Engl. transl.). In [53], 152–278 [53] Tarski, A. (1956b). Logic, Semantics, Metamathematics. Oxford: Oxford University Press [54] Van Fraassen, Bas C. (1972). “Inference and Self-Reference”. In [12], 695–708 [55] Wadding, L. (1968). Iohannes Duns Scotus Opera omnia. (Reprint from the 1639 edition). Hildesheim: Georg Olms [56] William of Ockham. See [4, 13, 24] [57] William of Sherwood (Pseudo-). See [38] [58] Yablo, S. (1993). “Paradox without Self-Reference”. Analysis 53, 251–252

Chapter 15 WILLIAM HEYTESBURY AND THE TREATMENT OF INSOLUBILIA IN FOURTEENTH-CENTURY ENGLAND FOLLOWED BY A CRITICAL EDITION OF THREE ANONYMOUS TREATISES DE INSOLUBILIBUS INSPIRED BY HEYTESBURY Fabienne Pironet Département de Philosophie University of Quebec at Montreal Montreal, Quebec, Canada H3C 3P8 [email protected]

Keywords: Heytesbury, Insolubles (insolubilia), Restriction (restrictio), Supposition (suppositio).

15.1

Introduction

When one considers the history of the Liar paradox’s treatment in medieval logic, it is evident that the solution offered by William Heytesbury (before 1313–ca. 1373) was not only very original, although he himself denies it was, but also very influential. Indeed, it was more or less taken up again by John of Holland (second half of fourteenth century), Robert Fland (between 1335 and 1370), the author of a treatise preserved in Ms. Vat. Lat. 674 (ca. 1368), John Hunter(?) (1390 at the latest), the “Pseudo-Heytesbury” (after 1335) and the author of a treatise preserved in Ms. Oxford, Lat. Misc. e.79 (after 1335). The three first texts mentioned have already been edited;1 I here offer a critical edition of the last 1 See

[1–3].

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three. But before studying those texts, I should proceed to some recollections and remarks about Heytesbury’s position on insolubles. These preliminary remarks will indeed allow us to understand, first, to what extent English logicians after Heytesbury followed him or not and, second, how they solved several questions the Mertonian master left aside, using for that, for the most part, the text of one other slightly earlier mertonian master, Thomas Bradwardine (1295–1349).2 Through this journey through the texts we will be able to recount the outlines of the development of the debate on insolubles in fourteenth-century England.

15.2 15.2.1

Heytesbury on Insolubles Heytesbury’s Position

In the first chapter of his Regulae solvendi sophismata, specifically dedicated to insolubles,3 Heytesbury presents and criticizes four opinions on this problem. According to him, the first three opinions share one and the same error: those who defend these positions erroneously believe that an insoluble proposition can signify only or precisely (praecise) as its terms commonly pretend (significat praecise sicut verba illius communiter praetendunt). He will then adopt the last opinion even if, as he himself recognizes, it is not fully satisfactory.4 This last opinion relies on the following postulate: when a proposition is set in a casus, a context, which makes it insoluble, it is impossible that it signify only (praecise) as its terms commonly pretend – note that I will from now on talk about primary signification to replace the expression “to signify as the terms commonly pretend”. From this initial assumption the three main rules Heytesbury proposes follow:5 (1) When a casus is proposed to you and it is not specified how the terms of the insoluble proposition signify, you have to answer as if no casus had been posited.6 2 Bradwardine’s text has been edited by M.-L. Roure in her paper “La problématique des propositions insolubles au XIIIe siècle, suivie de l’édition des traités de W. Sherwood, W. Burleigh et Th. Bradwardine”, in Archives d’histoire doctrinale et littéraire du moyen âge 37, 205–326, 1970. 3 See [4]. Heytesbury shares this position, see his Sophismata asinina, Soph. 6[2], 15[2], 27[2] et 30[3] [5]. 4 “Multae possent fieri hujusmodi objectiones contra istam responsionem quibus esset difficile vel impossibile in toto satisfacere”, fol.6rb. 5 See text below, Sect. 48 sqq. 6 As a proposition like “Socrates is saying what is false” can be either true or false, according to the casus it is set in but also according to the signification of its terms, another answer in this case could be: how do the proposition and the terms signify? We shall see that this is exactly what one of Heytesbury’s successors suggests.

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(2) When a casus is proposed to you where it is assumed that the terms of the insoluble proposition only (praecise) signify according to their primary signification, you have to deny the casus, because it is impossible.7 (3) When a casus is proposed to you where it is assumed that the terms of the insoluble proposition do not signify according to their primary signification, but according to a secondary signification (that is not specified and that the respondent does not have to specify), you have to admit the casus (because it is possible), you have to concede the insoluble proposition when it is proposed to you (because it logically follows from what precedes), but you have to deny that this proposition is true (because a proposition cannot be true if it is not true according to its primary signification).

15.2.2

Remarks on Heytesbury’s Position

At this stage, two observations can be made. First, according to Heytesbury, no proposition can be called insoluble if it is not set in a context, a casus that makes it insoluble. Second, the vocabulary Heytesbury uses as well as the solution he gives to insolubles are directly linked to the theory of obligations, and this last point is generally regarded as his original contribution in the history of the De insolubilibus treatises. Concerning the first point, whereas Heytesbury is really the first to integrate the notion of casus into the definition of an insoluble proposition, this necessary link between insoluble proposition and casus was nevertheless implicit in most authors before him. Indeed, when we examine some standard examples of insoluble propositions, it is immediately clear that they cannot be insolubles unless they are uttered or written in a particular context: “hoc est falsum”: “hoc” is a demonstrative pronoun that signifies only according to the designation that will be given to it,8 and this designation will determine the primary signification of the pronoun and, therefore, of the proposition it is a part of. If “hoc” refers to any other true or false proposition than the one it is a part of (let us call it A), the proposition “hoc est falsum” will primarily signify 7 As

a proposition like “Socrates is saying what is false” can be either true or false, according to the casus it is set in but also according to the signification of its terms, another answer in this case could be: how do the proposition and the terms signify? We shall see that this is exactly what one of Heytesbury’s successors suggests. 8 See [4]. Heytesbury shares this position, see his Sophismata asinina, Soph. 6[2], 15[2], 27[2] et 30[3] [5].

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that the proposition ‘A’ is false and will thus be either true or false, but in no way insoluble. If, on the other hand, “hoc” refers to the proposition it is a part of, the primary signification of the proposition “hoc est falsum” will be that the proposition “hoc est falsum” itself is false and, if it is false, it is true, and, if it is true, it is false. The proposition “hoc est falsum” will thus be insoluble if and only if it is assumed that the pronoun “hoc” refers to the proposition it is a part of. “omnis propositio est falsa”: if there is at least one true proposition, then the universal proposition that signifies that every proposition is false, is false, but in no way insoluble. To make it insoluble, it is necessary to posit the casus either that there exist only false propositions or that this universal proposition is the sole existing proposition, so that it would be equivalent to the singular one “haec propositio est falsa”, whose subject, given the context, could not refer to anything else than the proposition it is a part of. “Socrates dicit falsum”: this proposition will be insoluble if and only if it is assumed that it is uttered by Socrates and that it is the only proposition Socrates utters. “Socrates non habebit denarium”: unlike previous ones, this proposition has nothing in itself that could make us think that it is or could be insoluble. It is evident here that it is necessary to set this proposition in a specific situation or casus to make it insoluble. If we now consider conjunctive and disjunctive insolubles like “hoc est falsum et deus est” or “hoc est falsum vel homo est asinus”, it is clear that, given the way they are constructed, it is the truthvalue of the insoluble component of it (“hoc est falsum”) that will determine the truth-value of the whole hypothetical proposition. It should however be noticed at this point that the different conditions mentioned here for each of the examples are not sufficient to make those propositions really insoluble; they are only potentially insoluble. Indeed, the ultimate necessary condition for them to be really insoluble is that, while posited in such casus, their terms also signify according to their primary signification. This point will be developed later. Concerning the second point, expressions like casus, admittere, conceditur tamquam sequens, negatur tamquam repugnans, etc. clearly call to mind obligational disputations, this very specific kind of logical “game” whose aim was to examine logical links between propositions. This is most probably what led Spade to write that

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– “Heytesbury formulates his position in five rules governing a respondent’s replies when insolubles emerge in a disputation de obligationibus”.9

I think that this claim is not entirely correct, because it reduces the application field of those rules to the treatises and exercises De obligationibus alone, while they are also applicable and applied elsewhere. Indeed, the Sophismata literature is very rich in casus, and it is not rare to read there a sentence like “the proposition is conceded (or denied)” where one would rather expect to read “the proposition is true (or false)”. To explain this, we can recall that, although casus and positio are not strictly synonymous – Ockham has written some paragraphs on this subject10 – some authors, among them Kilvington and Heytesbury,11 are inclined to use one for the other. This is understandable if we consider the way they conceive them. A casus is a fictional situation, what we would today call a possible world. If someone admits such a situation, and it can be admitted only if it is not impossible (i.e. if it does not imply a formal contradiction), the only thing he is obliged to do is to consider that this situation is the real state of the world during the time of the disputation. Of course, the aim of the game is to see what would logically follow from such an hypothesis. A casus is posited and, once posited, is admitted or denied; it is formulated in the following ways: “pono casum quod Socrates sit niger” or “pono Socratem esse nigrum”, which can be paraphrased by “pono quod ista propositio ‘Socrates est niger’ est possibilis”. The positio is a specific variety of obligation by which the opponent obliges the respondent to regard a proposition as true during the time of the disputation. Here the aim of the game is either to train oneself to juggle with relations of relevance between propositions or, if the posited proposition is for the time being conceded as true although it is manifestly false or impossible,12 to see what logically follows from it. The 9 See

[6]. [7]; see also [8]. The author analyses the theories of Walter Burley, Richard Kilvington and Roger Swyneshed. 11 Especially in the Sophismata asinina, Soph. 17[2]: – “Pono tibi hunc casum ‘tu es asinus vel iste casus est impossibilis’ ”. In the arguments that follow, the opponent says that this casus is a disjunctive proposition, that it is true and that its contradictory is not necessary, above all qualities that would be rigorously reserved for propositions, not for situations or casus. 12 About the treatises called De falsi positione or De impossibili posiitone, see [9]. The author of an anonymous treatise, who in passing underlines the difference between the way practicians (lawyers and moralists) and theoreticians (physicists) do respectively proceed, explains the three reasons why we posit casus: – “Pro primo notandum quod casus causa triplici supponuntur. Prima, ut temptetur respondens numquid artem habeat. Secunda, ut dispensatio habeatur, ut accidit inter conferentes ob exercitii causam. Tertia causa est ut falsum possi10 See

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respondent can or has to deny a posited proposition only if it is logically impossible, i.e. if it contains or implies a formal contradiction. The positing of a proposition is formulated this way: “pono istam propositionem ‘Socrates est niger’ ”, which can be paraphrased by “pono quod ista propositio ‘Socrates est niger’ est vera”; a posited proposition is conceded or denied. Let us now show the difference between casus and positio in the case of the propositions “omnis propositio est falsa” and “hoc est falsum”: “pono casum quod omnis propositio est falsa” is paraphrased by “pono quod ista propositio ‘omnis propositio est falsa’ est possibilis”; the casus that this proposition is possible has to admitted, as it is possibile to imagine a situation where every proposition would be false. “pono istam propositionem ‘omnis propositio est falsa’ ” is paraphrased by “pono quod ista propositio ‘omnis propositio est falsa’ est vera”; if the terms of the proposition signify according to their primary signification, the casus that this proposition is true has to be denied, because it makes the proposition insoluble. “pono casum quod hoc est falsum” is paraphrased by “pono quod ista propositio ‘hoc est falsum’ est possibilis”; the casus that this proposition is possible has to be admitted, as it is possible to imagine a situation where this proposition would be true. “pono istam propositionem ‘hoc est falsum’ ” is paraphrased by “pono quod haec propositio ‘hoc est falsum’ est vera”; if the terms of the proposition signify according to their primary signification, the casus that this proposition is true has to be denied, because it makes the proposition insoluble. We note that, in an obligational disputation, the proposition “nulla propositio est tibi posita” will not be insoluble unless it is the first proposition proposed by the opponent.13 This condition plays the role of a casus: bile admittamus inquirendo quid sequitur, ut cum res ita de facto se habeant ut falsus casus denotat, sciamus per prius suppositum et rationem quid sumus acturi et responsuri. Iuristae enim et morales philosophi per possibiliter casus antequam in re accidat de observandis inquirentes veritatem in exercitiis statuendam explorant. Sed philosophi naturales scientiam speculativam quae circa conceptus et intentus hominum versatur principalius contemplatur”, see [10]. 13 As its aim is to examine the logical relations between different propositions, it would not be very interesting to introduce an insoluble proposition that would have no logical relation with what precedes it. Indeed, if the respondent answers correctly, he will deny it and the dispute will end at that point. If, on the other side, the proposed insoluble proposition should have a logical relation with what precedes its introduction, it would be necessary that the casus that makes it insoluble has already been posited earlier.

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given that it is the first proposition during the time of the disputation, it is the only existing proposition at this moment, which is one necessary condition for it to be insoluble. Moreover, if its terms signify according to its primary signification, it will be effectively insoluble and will have to be denied. The obligational disputations are also the occasion to construct insoluble cases that are more difficult to perceive at first sight: let us posit a situation where there are only two agents, Socrates and Plato, where the one who is saying something true is the only one that answers correctly, where the one who is saying something false is the only one that answers badly, and where Socrates says: “one of us (Socrates and Plato) answers badly” while Plato says “Socrates answers correctly”. . . But let us leave aside these subtle variations to go back to Heytesbury’s theories of signification and truth. They can be summarized as follows:14 Some terms have a signification by themselves (the term ‘man’ signifies a man), other terms do not have any signification by themselves (the term ‘A’ is a terminus ignotus, it does not signify anything by itself). The primary signification of a term that has a signification by itself is determined by the common or habitual use of this term (the term ‘man’ primarily signifies a man, the term ‘Oxford’, the town of Oxford, etc.). The primary signification of a term that does not have any signification by itself depends on the first imposition it will receive (if I posit that the term ‘A’ signifies every thing that the term ‘man’ signifies, the term ‘A’ will primarily signify a man). Spoken and written languages are arbitrary; it is thus possible and allowed to modify the signification of any spoken or written term, always and everywhere (and especially during the time of logical disputations). The signification of a term can be modified in two ways: (1) By attaching to it a secondary signification (I posit that the term ‘man’ primarily signifies a man and, secondarily, a stone) (2) By completely replacing its primary signification by a new one (I replace the signification of ‘man’ by the signification 14 On

these theories, see [5], analysis of sophism 6, pp. 148–163.

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of ‘donkey’, so that the term ‘man’ will exclusively (praecise) signify everything that the term ‘donkey’ signifies) The signification of a proposition is determined by the signification of the terms that compose it (ex compositione terminorum). The signification of a proposition can be modified in two ways: (1) By modifying the signification of one of its terms (in the proposition ‘homo est animal’, the term ‘animal’ will signify everything that the term ‘donkey’ signifies) (2) By modifying its global signification, which can be done afresh in two ways: (a) By attaching to it a secondary signification (the proposition ‘homo est animal’ will signify that a man is an animal and that you are a donkey) (b) By completely replacing its primary signification by a new one (I replace the signification of the proposition ‘homo est animal’ by the signification of the proposition ‘tu es asinus’, so that the proposition ‘homo est animal’ will exclusively (praecise) signify that you are a donkey) A proposition is true if and only if it signifies only or precisely as is the case (praecise sicut est ), for example the proposition ‘homo est animal’, at least if its terms signify only or precisely according to their primary signification. A proposition is false if it does not signify only or precisely as it is the case, i.e.: (1) If it signifies otherwise than is the case (aliter quam sicut est), for example the proposition ‘homo est asinus’, at least if its terms signify only or precisely according to their primary signification, or (2) If it signifies as is the case and otherwise than is the case, for example the proposition ‘homo est animal’ when it is assumed that the term ‘animal’ primarily signifies an animal and secondarily a donkey It is not possible for an insoluble proposition in an insoluble casus to signify only or precisely according to its primary signification (significat praecise sicut verba illius communiter praetendunt ), because, should it signify according to its primary signification, it would then follow that it would be both true and false, which is

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impossible. For it to be admitted, a casus de insolubili will then have to contain a clause about the signification of the potentially insoluble proposition, and this clause will have to state that the terms of this proposition do not signify or do not signify only or precisely according to their primary signification. Contrary to many of his contemporaries, who establish the criteria for the truth-value of a proposition in terms of supposition, we see that Heytesbury adopts a definition of truth in terms of signification, so that it is the primary signification of a proposition that determines its truthvalue. This is why, when he gives his third rule, Heytesbury says that (1) One has to concede an insoluble proposition that does not signify praecise according to the primary signification of its terms, (2) To deny that it is true and that (3) Even if it is asked of him, the respondent does not have to specify how this proposition signifies otherwise than according to its primary signification. Many authors have considered that this last affirmation was a weakness in Heytesbury’s theory. While others will try to find a remedy for that (cf. infra), an anonymous author addressed very caustic criticisms to him: - “Fundamentum illius viae est quod ad omne insolubile concernat certa obligatio, et quod insolubile propter casum factum nullo modo aliter significat, licet oporteat concedi ipsam aliter significare... Sed contra illam viam instant aliqua. Primo sic: nam responsio illius viae videtur esse fuga, eo quod non vult dicere qualiter aliter insolubile significat. Sic ergo facile esset respondere ad omnia vel solvere omnia. Quia dicerem quod homo currit, et numquam dicerem quis esset ille homo. Secundo: nam aliquid insolubile esset sine obligatione; igitur oporteret hominem in tali casu certificare quaestionem quale insolubile aliter significat”.15

Spade rightly noticed that Heytesbury’s silence was justified by the strict rules of obligations [6]: indeed, it does not belong to the respondent to specify what is the signification of propositions, a fortiori if the opponent does not do it himself. But there is also another more fundamental reason why, I think, Heytesbury was not “obliged” to specify this other signification of insoluble propositions inside a casus. Let us take the case where Socrates is the only existing Socrates and that he only says this sole proposition “Socrates dicit falsum” which signifies that Socrates is saying what is false, but also signifies something else: 15 Anonymous,

Ms. Vienna, Nat. Bibl. 4953, f.90v. Quoted from P.V. Spade [11].

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The only reason why a casus has to be denied is that it is impossible, i.e. implies a logical contradiction. In our example, what would make the casus impossible is that the proposition “Socrates dicit falsum” signifies praecise according its primary signification, i.e. signifies that Socrates is saying something false and does not signify anything else than that. As it is not so, the casus has to be admitted. Then, when the proposition “Socrates dicit falsum” is proposed, it has to be conceded, because it logically derives from what precedes: as Socrates says only one proposition and as this proposition is false, whatever its secondary signification may be, it follows from that that Socrates is saying something false. Then, when the proposition “ ‘Socrates dicit falsum’ est vera” is proposed, it has to be denied, because, whatever its secondary signification may be, it will never be true. Indeed, the secondary signification does not have any influence on the truth-value of the proposition “Socrates dicit falsum”, because, as the terms that constitute this proposition have a signification by themselves, and the secondary signification has not completely replaced the primary signification, but has been added to it, the total signification of the proposition “Socrates dicit falsum” is a conjunction of its primary signification, according to which it is false (it implies both that it is true and that it is false; thus it does not signify praecise sicut est; hence it is false), with its secondary signification that, true or false, will not change anything, as every conjunctive proposition at least one of whose parts is false is false. Such then is the way Heytesbury solved the problem of insoluble casus. By demanding that an insoluble casus can be admitted if and only if the terms of the insoluble proposition signify according to their primary signification, but not praecise, he could escape the vicious circle (if it is false, it is true and if it is true, it is false, etc.) as follows: we have seen that the proposition “Socrates dicit falsum” cannot be true. Now, if it is false, could we still derive from that that it is true? Here is the way to answer: The proposition “Socrates dicit falsum” is false, And since Socrates is the only speaker and since he only utters this sole proposition, the proposition “Socrates dicit falsum” signifies that it is false (and it also signifies something else, i.e. what it signifies according its secondary signification, whatever that may be). Hence it is false, and this is what it signifies (i.e. that it is false). From this one is not allowed to conclude that it is

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true, because for that it should be the case that it is the only thing it signifies; but it also signifies something else. The point discussed here above, probably very early considered as a weakness in Heytesbury’s solution, as well as the fact that Heytesbury himself recognizes that his solution is not fully satisfactory, led some of his successors to try to answer the criticisms Heytesbury’s text prompted. It is noteworthy that for most of them several elements of their response were inspired by another famous Mertonian logician, slightly earlier than Heytesbury: Thomas Bradwardine. In Sect. 15.3, I will briefly present the three anonymous authors whose texts are edited in this paper, followed by a comparative analysis of the treatises of eight fourteenth-century English logicians (the six mentioned in the introduction plus Heytesbury and Bradwardine) that led me to propose, as a conclusion, an hypothesis about the evolution of the debate on insolubles in fourteenth-century England.

15.3 15.3.1

Heytesbury’s Successors Brief Presentation of the Three Authors Whose Text is Edited as an Annex

The “Pseudo-Heytesbury”. This text has been preserved in only one manuscript, where it has been copied between Heytesbury’s Sophismata asinina and Juxta hunc textum. As this manuscript also contains Heytesbury’s Regulae solvendi sophismata and Sophismata, it was quite natural, as Weisheipl did, to attribute to Heytesbury a text whose explicit only tells us that it is “secundum Heytesbury”. But, as Spade already showed [11, pp. 35–36], a more careful study of this text allows one to conclude that, despite many textual similarities, this text is not by Heytesbury nor does it follow Heytesbury’s exact principles. The identity of the author is still a mystery. John Hunter(?). Of John Hunter (Johannes Venator) we know very few things for certain. In the 1380s he wrote a Logica [12] that would have been one of the direct sources for Paul of Venice [13]. Has the treatise on insolubles edited here been written by this same author? It is difficult to answer to this question. Three things only are certain: (1) As, according to the indications given inside the manuscript, our text dates from 1390 at the latest, the dates would correspond, (2) As John Hunter’s Logica16 does not contain any chapter on insolubles, it is impossible to proceed to a comparison with our text, 16 See

Johannes Venator Anglicus. Logica [14].

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(3) It is clear that our text could not have been a source for Paul of Venice’s chapters on insolubles, neither in his Logica magna nor in his Logica parva.17 The only argument in favour of an attribution of this text to John Hunter is what is written in the Oxford manuscript as an explicit: “Expliciunt insolubilia more Exonie compilata edita ab eximio doctore magistro Johanne Venatore”. But is it enough? In the Erfurt manuscript, the text is attributed to Heytesbury himself: “Et sic de insolubilibus Hesbri et dicta sufficienter” and the Vatican manuscript does not give any indication about the author: “Explicit brevis tractatus de insolubilibus secundum usum Heusonie”? Since nothing is assured, it is more cautious to talk here about “John Hunter(?)”.

The Author of a Treatise Copied in Ms Oxford, Lat. Misc. As was the case for the two previous treatises, we find an ine.79. dication about the author in the explicit: “Explicit sumula sumularum magistri Iohannis Ecaf ”. As the name “Ecaf” does not appear in any catalogue of medieval logicians, de Rijk suggested that it was perhaps designating Wyclif, the only known author whose name is more or less similar to “Ecaf” [19]. This hypothesis is all the more attractive since “Ecaf” in the manuscript could also be read as “Eclif”. But here again we have to give up this hypothesis. It is clear indeed that Wyclif’s authenticated texts18 do not have anything in common with our text.

15.3.2

Analysis of the Texts of the Six Successors of Heytesbury’s in Comparison with Heytesbury’s and Bradwardine’s

I have chosen to deal with the seven main topics in those texts: definition of the insoluble casus, definition of the insoluble proposition, definition of the true proposition, division of insolubles, discussion of other opinions, list of the rules and obligations for the respondent. For each of these topics I will proceed in two parts: (1) Quotation or summary of the texts (2) Provisional conclusions

17 See 18 See

[15–18]. [20, 21].

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Definition of the Insoluble Casus. Heytesbury Casus de insolubili est ille in quo fit mentio de aliqua propositione quae, si cum eodem casu significet praecise sicut verba illius communiter praetendunt, ad eam esse veram sequitur eam esse falsam. Pseudo-Heytesbury Casus insolubilis est ille in quo fit mentio de aliqua propositione quae, si cum eodem casu significat praecise sicut verba illius praetendunt, sequitur eamdem esse veram et falsam. John Hunter(?) Casus de insolubili est talis casus in quo fit mentio de aliqua propositione quae, si cum eodem casu significet praecise sicut verba praetendunt, sequitur eamdem esse veram et eamdem esse falsam. John of Holland Casus autem insolubilis est aliquis casus in quo fit mentio de aliqua propositione, quae si cum eodem casu ponatur praecise significare sicut termini praetendunt, sequitur istam propositionem esse veram et falsam. Anonymous Ms. Vat. Lat. 674 —– Robert Fland—— Anonymous Ms. Oxford Lat. Misc. e.79 —– Bradwardine —– As mentioned in the introduction, Heytesbury is the first that systematically and explicitly considered the (potentially) insoluble proposition within the context of a casus that makes it (really) insoluble; he thus had to define the insoluble case. On this point, he has been followed almost word for word by three of the six authors studied here. We shall see later that this similarity is not sufficient to conclude that they literally followed Heytesbury’s position. Like Bradwardine and most of the logicians before Heytesbury, the three others do not define the insoluble case; but this is not sufficient to draw any conclusion about their dependence upon Bradwardine on this point either. This first topic is in fact quite superficial; we only get here some initial clues that should be refined subsequently.

Definition of the Insoluble Proposition. Heytesbury Propositio insolubilis est de qua fit mentio in casu insolubili quae, si cum eodem casu significet praecise sicut verba illius communiter praetendunt, ad eam esse veram sequitur eam esse falsam, et e converso. Pseudo-Heytesbury Propositio insolubilis est illa de qua fit mentio in aliquo casu quae, si cum eodem casu sic significet praecise, sequeretur ipsam esse veram et falsam. John Hunter(?) Propositio insolubilis est talis propositio de qua fit mentio in aliquo casu quae, si cum eodem casu significet praecise sicut verba praetendunt, sequitur eamdem esse veram et eamdem esse falsam.

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John of Holland Propositio insolubilis vel insolubile est propositio de qua fit mentio in tali casu, quae si cum eodem casu ponatur praecise significare sicut termini praetendunt, sequitur eam esse veram et falsam. Anonymous Ms. Vat. Lat. 674 After two general and classical definitions proposed at the very beginning of the treatise: “insolubile est propositio vel argumentum non de facili solubile, vel sic: insolubile est propositio vel argumentum de difficili solubile, ita quod non dicitur ‘insolubile’ quod nullo modo possit solvi, sed cum difficultate solvitur”, the author proposes a little bit further on two more precise definitions: (1) ‘Insolubile’ quandoque accipitur pro toto paralogismo, et tunc definitur sic: ‘insolubile’ est difficilis paralogismus secundum quid et simpliciter ex reflexione alicujus actus super se cum dictione privativa. (2) Quandoque accipitur ‘insolubile’ pro propositione insolubili, et tunc definitur sic: ‘insolubile’ est difficilis propositio quae, si cum ponitur significare sicut termini praecise praetendunt, sequitur ipsa <m> fore falsam et veram. Robert Fland Insolubile est propositio significans primo et principaliter sicut est et ex consequenti aliter quam est, deducta nova impositione. Anonymous Ms. Oxford Lat. Misc. e.79 Insolubile est oratio quae, in casu posito secum, dum solum significat secundum communem institutionem idiomatis in quo est disputatio, simul verificat et falsificat ipsammet vel ejus qui taliter imposuit ad significandum. Bradwardine Insolubile est difficilis paralogismus secundum quid et simpliciter ex reflexione alicujus actus supra se cum determinatione privativa proveniens. On this second topic, we observe that, very logically, those who took up Heytesbury’s definition of an insoluble casus also took from him the definition of an insoluble proposition almost word for word, but this still does not indicate that their position will correspond exactly to his. The two essential characteristics of an insoluble according to Heytesbury can be found in the text preserved in Ms. Oxford Lat. Misc. e. 79, but expressed with other words. Concerning the last two authors, the influence of Bradwardine is perceptible only in the text preserved in Ms. Vat. Lat. 674. It is nevertheless noteworthy that the text reveals a certain duality: indeed, the author makes a distinction between the insoluble as a paralogism arising from self-reference (like Bradwardine) and the insoluble as a proposition posited in a certain casus that makes it insoluble (following the spirit but not the letter of Heytesbury). We shall see later that this duality Bradwardine/Heytesbury is also to be found by Robert Fland, who in his definition of an insoluble does not identify any cause for the insolubility, neither self-reference nor a casus.

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Definition of the True Proposition. Heytesbury. Implicit definition: propositio vera significat praecise sicut est; propositio falsa significat aliter quam est.19 Pseudo-Heytesbury. Implicit definition: propositio vera significat praecise sicut est; propositio falsa significat aliter quam est. John Hunter(?). Implicit definition: propositio vera significat praecise sicut est; propositio falsa significat aliter quam est. John of Holland: implicit definition: propositio vera significat praecise sicut est; propositio falsa significat aliter quam est. Anonymous Ms. Vat. Lat. 674. Implicit definition: propositio vera significat praecise sicut est; propositio falsa significat aliter quam est. Robert Fland. Explicit definition: propositio vera significat praecise sicut est; propositio falsa significat aliter quam est. Anonymous Ms. Oxford Lat. Misc. e.79. The definition is given in the margin20 : propositio vera significat praecise sicut est; propositio falsa significat praecise primarie sicut non est. Bradwardine. Explicit definition: propositio vera est oratio significans tantum sicut est; propositio falsa est oratio significans aliter quam est. This topic is not the most interesting for my intention in this paper. We indeed observe that, whether it is explicitly or implicitly given, everyone agrees about the definition of a true proposition. From an historical point of view, one should note that this fact confirms the general giving up in England of a definition of a true proposition in terms of supposition in favour of a definition in terms of signification. We can note here that by those who are closer to Heytesbury, the definition is implicit, as it is for him, while by those who will proove closer to Bradwardine, the definition is explicit, as it is for him. 19 See also Heytesbury, Sophismata, Soph. 5, Sophismata asinina, Soph. 6 and Juxta hunc textum, concl. 9 et 10. 20 In the text, the second supposition refers to the definitions of true and false propositions probably given in a first definition which is missing, hence this addition in the margin by another scribe.

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Division of Insolubles. Heytesbury. He does not make an explicit distinction between simple insolubles and complex insolubles, but the rules 1–3 are about simple insolubles whereas rules 4 and 5 are about complex insolubles, i.e. those insolubles that signify conjunctively (rule 4) or disjunctively (rule 5). Pseudo-Heytesbury. Explicit distinction between simple insolubles (rules 1–3) and complex insolubles, i.e. those that signify conjunctively (rule 4) and those that signify disjunctively (rule 5), to which he adds the disjunctive, conjunctive, exclusive and exceptive insolubles. One can also find in this text explanations about the way to construct disjunctive or conjunctive insolubles. John Hunter(?). Explicit distinction between simple insolubles (rules 1–3) and complex insolubles, divided into disjunctive, conjunctive, exclusive and exceptive insolubles. One can also find in this text explanations about the way to construct disjunctive or conjunctive insolubles. John of Holland. Explicit distinction between simple insolubles (rules 2–3) and complex insolubles, divided into conjunctive, disjunctive, exclusive and exceptive insolubles. No explanation about the way to construct disjunctive or conjunctive insolubles. Anonymous Ms. Vat. Lat. 674. Explicit distinction between simple insolubles and complex insolubles, divided into insolubles constituted by a conjunctive, disjunctive, conditional, causal, temporal, exclusive or exceptive proposition. One can also find in this text explanations about the way to construct any kind of composite insolubles. Robert Fland. Explicit distinction between simple insolubles and complex insolubles, divided into disjunctive and conjunctive insolubles. No explanation about the way to construct disjunctive or conjunctive insolubles. Anonymous Ms. Oxford Lat. Misc. e.79. Whereas he does not make any distinction between insolubles according to their form (he nevertheless mentions propositions that signify conjunctively or disjunctively), the author of this text makes a distinction between insolubles according to their origin: insolubles proceeding from an internal act, an external act or from properties of terms.

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Bradwardine. Explicit distinction between simple insolubles and complex insolubles, divided into disjunctive, conjunctive, exclusive and exceptive insolubles. He adds a distinction between insolubles according to their origin (insolubles proceeding from an internal act, an external act or from properties of terms) and according to their quality (affirmative and negative insolubles). No explanation about the way to construct disjunctive or conjunctive insolubles. On this topic, we observe that four elements appear in the successors’ texts that did not exist in Heytesbury’s. The source of three of them can be found in Bradwardine, but my aim here is not to determine whether Bradwardine is the real source for them but rather to determine if each of these elements can or has to be considered as a lack or a weakness in Heytesbury’s position. Let us examine them one by one. (1) The distinction between simple insolubles and complex insolubles is not explicit in Heytesbury, but, as I already mentioned, it is possible to infer it from his text, because he proposes rules that are applicable to each kind of insoluble, simple and complex. (2) Heytesbury only deals with insolubles that signify conjunctively or disjunctively while some other authors make a distinction between insolubles that signify conjunctively (the proposition “Socrates dicit falsum” signifies that Socrates is saying what is false and that God exists) and conjunctive insolubles (the proposition “Socrates is saying what is false and God exists”). One could say that Heytesbury did not make this distinction in his concern to be brief, because what goes for insolubles that signify conjunctively of course also goes for conjunctive insolubles. One could also imagine that Heytesbury rather insisted on insolubles that signify conjunctively or disjunctively because they are obviously easier to detect. Similarly, one could again call upon his concern to be brief to explain the fact that he does not deal with exclusive, exceptive and conditional insolubles, because the general rules for insolubles are obviously applicable to them by extrapolation. (3) The three authors who, as we have seen, are closer to Heytesbury’s text do explicitly state the principles guiding the construction of conjunctive and disjunctive insolubles. This shows their pedagogical concern towards students that will have to play an active role in disputations, the role of respondent as well as the role of opponent. It thus seems that these texts, unlike Heytesbury’s, not only make them able to recognize and efficiently respond to insolubles (role of the respondent), but also able to create insolubles and to propose

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them to their adversary (role of the opponent). If the absence of such explanations in Heytesbury’s text has been considered as a lack, it is evidently not a theoretical weakness. Perhaps we should just conclude that the texts that propose such principles are rather more practical than their model. Moreover we remark that two of those three authors, the Pseudo-Heytesbury and John Hunter(?) completely omit the theoretical discussion about other opinions on insolubles, which accentuates further the practical aspect of their texts. (4) The fourth and last element we find in one of our texts that is completely absent from Heytesbury’s is the distinction between insolubles according to their origin: internal act, external act and properties of the terms. It is only found in the Ms. Oxford Lat. Misc. e. 79 and probably goes back to Bradwardine. But, even if, like about the previous point, one could have considered it as an omission for which he had to find a remedy, this cannot be considered as a theoretical weakness either. Indeed, whatever may be the origin of an insoluble, what matters to Heytesbury is that his young students are able to recognize them and to respond to them efficiently so that they will never be forced into a logical contradiction. To sum up, the different additions we find in other texts on this topic are clearly not actual theoretical pieces of information susceptible to modify, invalidate or improve Heytesbury’s position, they are rather additional technical or practical pieces of information.

Discussion of Other Opinions. Heytesbury. Four opinions are discussed and he strongly argues against the first three, that all admit that an insoluble proposition can signify only (praecise) according to its primary signification. He does not discuss at all the solution by restriction. Pseudo-Heytesbury. ——– John Hunter(?). ——– John of Holland. ——– Anonymous Ms. Vat. Lat. 674. Two opinions are mentioned, the first one, according to which a part cannot supposit for the whole (solution by restriction), and the second one, according to which a part can supposit for the whole. The author notes that the second opinion has many arguments in favour of it, but, without giving

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any explanation, he adds that both opinions can be saved if the expression “a part supposit for the whole” is interpreted in one or another way. Robert Fland. ——– Anonymous Ms. Oxford Lat. Misc. e.79. The solution by restriction is discussed and the author admits its basic rule for singular insoluble propositions. Bradwardine. Nine opinions are discussed, including the solution by restriction. He strongly argues against it because, according to him, the rule that says that a part cannot supposit for the whole is not sufficient, it is irrelevant in fact, to solve the problem raised by insolubles. I mentioned earlier a reason why several of the texts probably omit discussion of the value of the different positions that can be held on insolubles: it could be due to the fact that their aim is rather practical than theoretical. However, two points should be underlined here. First, the only two texts in which, like Heytesbury’s, we find a discussion of other opinions are precisely those that depart the most from the letter of Heytesbury’s text. Second, the only solution that is discussed in these two texts is restriction, a solution that Heytesbury completely passes over in silence although it had some famous defenders in England, among them Burleigh and Ockham. It thus seems that while Bradwardine had to argue against the solution of his two famous contemporaries, especially Burleigh, after only about ten years Heytesbury did not feel the same, all the more so since the solution by restriction had lost followers in England, as also did the definition of a true proposition in terms of supposition. But later texts, like the anonymous one found in the Oxford manuscript and the one in the Vatican manuscript, nevertheless show that the solution by restriction, which they did not adhere to, did not completely fall into disuse.

List of Rules.

Heytesbury:

(1) Si fiat casus de insolubili, aut ponitur qualiter illud insolubile debeat significare aut non. Si non, proposito illo insolubili, respondendum est ad illud omnino sicut responderetur non supposito illo casu. (2) Si ponatur casus de insolubili, et cum hoc ponatur quod illud insolubile praecise significet sicut termini illius communiter praetendunt, casus ille nullatenus admittatur.

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(3) Si fiat casus de insolubili, et cum hoc supponatur quod illud insolubile significet sicut termini ipsius praetendunt, non tamen sic praecise, admisso illo casu, concedendum est illud insolubile quocumque loco proponatur tamquam sequens, et negandum est illud est verum tamquam repugnans. (4) Si fiat casus de insolubili, et cum hoc supponatur illud insolubile significare copulative praecise sicut verba illius, . . . , si oppositum illius possit stare cum toto casu, negandum est casus tamquam claudens contradictoria formaliter. (5) Si fiat casus de insolubili, et cum hoc supponatur quod illud insolubile significet praecise disjunctive sicut verba praetendunt, . . . , nisi illud disjunctum cum illo insolubili possit stare cum toto isto casu non est casus admittendus. Pseudo-Heytesbury: (1) Si ponatur casus de insolubili, et non ponatur qualiter illud insolubile debeat significare, supposito illo insolubili, respondendum est sicut ad impertinens. (2) Si ponatur casus de insolubili, et cum hoc ponatur quod illud insolubile significet praecise sicut verba illius communiter praetendunt, tunc nullo modo admittendus <est> casus. (3) Si ponatur casus de insolubili, et cum hoc ponitur quod illud insolubile significat sicut termini ejus communiter praetendunt, et non tamen sic praecise, admittendus est casus ille, et est concedendum illud insolubile quando proponitur tamquam sequens, et negandum est illud fore verum. (4) Si ponatur casus de insolubili, et cum hoc ponatur illud insolubile significare praecise sicut verba illius praetendunt copulative, si oppositum illius copulati non potest stare cum toto casu, negandus est casus ille tamquam claudens opposita. (5) Si ponatur casus de insolubili, et cum hoc supponatur illud insolubile significare copulative praecise sicut verba illius praetendunt, si oppositum copulati potest stare cum toto casu sic significando primarie, admittitur casus, et concedendum est illud insolubile quando proponitur, et negandum est illud fore verum, et negandum est illud copulatum et quodlibet antecedens ad illud.

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John Hunter(?): (1) Si ponatur casus de insolubili, et cum hoc non ponatur qualiter illud insolubile debeat significare, proposito illo insolubili, respondendum est ad illud ac si nullus casus poneretur. (2) Si ponatur casus de insolubili, et cum hoc ponatur quod illud insolubile significet praecise sicut termini praetendunt, nullo modo est casus admittendus. (3) Si fiat casus de insolubili, et cum hoc ponatur quod illud insolubile significet sicut termini praetendunt, et non praecise, est admittendus casus, et concedendum est illud tamquam sequens quando proponitur, et negandum est ipsum esse verum tamquam repugnans. John of Holland: (1) Si in casu insolubili ponatur insolubile precise significare sicut termini pretendunt, casus est negandus. Eo quod sequitur contradictio, videlicet quod eadem propositio est vera et falsa. (2) Si in casu insolubili ponatur insolubile significare sicut termini communiter pretendunt, et non precise, tunc casus est admittendus. Sed si non limitetur qualiter significabat, tunc ad insolubile est dubie respondendum. Eo quod dubitatur qualiter insolubile significat. Anonymous Ms. Vat. Lat. 674 : (1) Numquam insolubile debet admitti quando cum insolubili ponitur quod propositio significet <praecise> sicut termini praetendunt, quia tunc de necessitate oporteret concedere contradictoria. (2) Insolubile semper est admittendum quando ponitur cum casu quod propositio significet sicut termini communiter praetendunt. Et, si postea proponatur, debet admitti. (3) Insolubile positum sine casu debet admitti ac si poneretur quod propositio significet sicut termini communiter praetendunt. (4) Quaelibet propositio insolubilis aequivalet uni propositioni copulativae, sicut ista “Socrates dicit falsum” aequivalet huic “Socrates dicit falsum et haec est vera ‘Socrates dicit falsum’ ”. (5) Oppositum cujuslibet propositionis insolubilis aequivalet vel debet esse una disjunctiva. Et haec patet ex superiori.

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Robert Fland : ——– Anonymous Ms. Oxford Lat. Misc. e.79 : (1) Quilibet insolubilis casus est negandus. (2) Numquam terminus singularis supponit pro seipso vel pro tota propositione cujus est pars. (3) Numquam est positio vel propositio admittenda sine nova impositione ex qua, praecise significante sicut grammatice verba praetendunt, sequeretur contradictio. (4) Quandocumque in aliquo casu imponitur aliqua propositio ad copulative significandum praecise sicut verba praetendunt et quod tu curris vel aliquod tale, tunc si contradictorium illius copulati, scilicet ‘tu non curris’, stat cum casu, casus est admittendus; et si non stat, non est admittendus. (5) Quandocumque ponitur propositio de insolubili praecise significare disjunctive sicut ejus verba praetendunt vel quod deus est sive aliquid aliud, tunc si istud disjunctum stat cum toto casu, casus est admittendus; et si non stat, est negandus. Unsurprisingly, we observe here (1) the great closeness of four of the texts to Heytesbury’s, although three of them omit Heytesbury’s rules 4 and 5, (2) the complete absence of Heytebury’s five rules from Robert Fland and (3) the originality of the text preserved in Ms. Oxford Lat. Misc. e.79: the way he expresses the rules is original as well as their content, in particular the restrictive rule for singular propositions. Does the Respondent have to Specify the Secondary Signification of an Insoluble Proposition? Heytesbury. No, see rule 3. Pseudo-Heytesbury. The question is irrelevant, because the author judges that any insoluble proposition conjunctively signifies what it signifies according to the primary signification of its terms and that it is true. John Hunter(?). The question is irrelevant, because the author judges that any insoluble proposition conjunctively signifies what it signifies according to the primary signification of its terms and that it is true. The proposition ‘Socrates is saying what is false’ uttered by Socrates is not completely similar to the same proposition

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uttered by anyone other than Socrates: they are similar vocally, but not in signification. John of Holland. The question is irrelevant, because he judges that any insoluble proposition conjunctively signifies what it signifies according to the primary signification of its terms and signifies also something else. He adds that according to some logicians (secundum aliquos, saltem ut plures dicunt) this other thing it signifies is that it is true (Bradwardine). The proposition ‘Socrates is saying what is false’ uttered by Socrates is not completely similar to the same proposition uttered by anyone other than Socrates: they are similar vocally, but not in signification. Anonymous Ms. Vat. Lat. 674. As we have seen above, this author considers as a rule (rule 4) the fact that every insoluble proposition conjunctively signifies what it signifies according to the primary signification of its terms and that it is true. Robert Fland. The author mentions two possibilities: either (1) an insoluble proposition signifies aliter quam est and thus the respondent has to specify what its secondary signification is (it conjunctively signifies that it is false and that it is true – Bradwardine), or (2) an insoluble proposition signifies aliter quam est and the respondent does not have to specify what its secondary signification is (Heytesbury). Robert Fland leaves the respondent free to choose any of those positions (“Eligat ergo respondens unam istarum pro sua solutione ad insolubilia”). Anonymous Ms. Oxford Lat. Misc. e.79. The question is irrelevant, because the author judges that, unless the opponent and the respondent did agree to modify the signification of a proposition, any proposition only (praecise) signifies according to its primary signification. In support of his position, he mentions a very practical reason: should this principle not be admitted in the course of disputes, the respondent would have to doubt (i.e. to answer ‘I doubt’ to) any proposition proposed to him. Bradwardine. The question is irrelevant, because he judges that any insoluble proposition conjunctively signifies what it signifies according to the primary signification of its terms and that it is true. The proposition ‘Socrates is saying what is false’ uttered by Socrates is not completely similar to the same proposition uttered by anyone other than Socrates: they are similar vocally, but not in signification. It is on this topic that Bradwardine’s influence can

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be felt the most in our six texts, including those that are generally closer to Heytesbury’s. Indeed, it is in Bradwardine’s text that our authors found the answer to the criticism of Heytesbury’s third rule. By two of them, the distinction in voce/in significatione probably also comes from Bradwardine’s text. We should note here (1) the duality in Robert Fland’s text: he leaves his reader the choice between two possible answers, (2) the fact that the only proposal John of Holland has for making explicit the expression “and signifies also something else” is Bradwardine’s one, (3) the remark of good sense made in the text preserved in Ms. Oxford Lat. Misc. e. 79.

15.4

Conclusions

We have seen that our six authors of treatises on insolubles written after 1335 more or less follow Heytesbury: four of them follow his rules and principles rather closely and, where he left some points unsolved or obscure, it is in Bradwardine’s text that they found an answer or an alternative. There are very few differences between those four texts, and these are not so important on a theoretical point of view, so that one may claim that they make up an homogeneous group. It is nevertheless difficult to determine whether one of them would have served as a model for the others. If we list them by decreasing order of nearness to Heytesbury’s text, this group includes the Pseudo-Heytesbury, John Hunter(?), John of Holland and the Anonymous of Ms. Vat. Lat. 674. The two remaining texts clearly distinguish themselves from this first group. Robert Fland is the most distant from Heytesbury’s text: it is evident that he knows Heytesbury’s position, but he seems to prefer Bradwardine’s. The anonymous text preserved in Ms. Oxford Lat. Misc. e. 79 is the most distant from all the others and the most original in content as well as in form. What is striking in this text is the author’s insistence on the notion of imposition (or institution) of the terms: this notion is so important to him that a third of the treatise is exclusively dedicated to the three different modes of imposition. Although this part of the text has no equivalent in Heytesbury’s, it is not without any relation to Heytesbury’s doctrine. Indeed, we have seen that the signification of the terms, and hence their imposition, is crucial for him to solve the problem of insoluble propositions. Besides, it is perhaps by reaction to a basic supposition by Heytesbury that our author was brought to develop the topic of the imposition of the terms. Indeed, Heytesbury supposes that, by default, any proposition of which one did not explicitly specify that it signifies only (praecise) according to its primary signification can have a secondary signification, and this is why insoluble casus can sometimes be admitted.

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Such a supposition leaves the door open to many possibilities concerning the secondary signification of propositions, and, as we have seen, Heytesbury considered that the secondary signification did not need to be specified by the respondent. Whereas the majority of the others went on to draw the specification of the secondary signification of an insoluble proposition from Bradwardine (it secondarily signifies that it is true), our anonymous author proposes a more radical answer. He supposes for himself that, by default, any proposition signifies only (praecise) according to the primary signification of its terms. And this supposition leads him to a very succinct formulation of his first rule: any insoluble case has to be denied (rule 1) – unless the participants to the disputation come to an agreement to give the terms a new imposition (rule 3). At the end of this survey, it is possible to trace the outlines of the debate on insolubles in fourteenth-century England. One can distinguish four stages: (1) The solution by restriction in its weak version (the rule “a part cannot supposit for the whole it is a part of” has to be applied to insoluble propositions only21 ) was supported by Burleigh (ca. 1320) and Ockham (1324–1327). It was highly criticised, mostly because it was a purely ad hoc solution. (2) Bradwardine (1321–1324) was one of those who argued against the “restrictors” and he then proposed a new solution that also marked an evolution concerning the definition of the truth-conditions of a proposition: from there the definition in terms of supposition was supplanted by the definition in terms of signification. For recall, Bradwardine’s solution is as follows: a proposition is true if and only if it signifies only as is the case (tantum sicut est); a proposition is false if it signifies otherwise than is the case (aliter quam est). Since it signifies at the same time that it itself is true and that it itself is false, and since every proposition is either true or false (principle of bivalence), an insoluble proposition does not signify “tantum sicut est”; it is therefore false. (3) Probably motivated by theoretical as well as pedagogical reasons, Heytesbury, while admitting the same truth-conditions of a proposition as Bradwardine, moved the debate into the terrain and vocabulary of obligational disputations. Indeed, it was not sufficient to say, as did Bradwardine, that an insoluble proposition is false, because, as only the impossible has to be denied, it should then have to be conceded when proposed in a disputation. Now an 21 In the strong version, the rule of restriction is valid for any proposition, including those that would not be problematic at all, like “This sentence is written in English”.

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insoluble proposition such as “Socrates is saying what is false” will also signify that it is true (secondary signification) only if it signifies only (praecise) according to its primary signification. We thus see that, in addition to the fact that Bradwardine’s solution led to serious problems inside obligational disputations, Heytesbury pinpointed the central theoretical weakness of Bradwardine’s doctrine and of all those who thought that an insoluble proposition can signify only (praecise) as its terms commonly pretend. Here is the difficulty: following them, the insoluble proposition “Socrates is saying what is false” has a primary signification (Socrates is saying what is false) and also a secondary signification (the proposition “Socrates is saying what is false” is true); however, this proposition is insoluble only if it signifies only (praecise) according to its primary signification, which implies that it cannot have any secondary signification. . . This leads me to conclude that what is most original in Heytesbury’s solution is not so much the fact that he discussed insolubles in the strict context of obligational disputations, but rather the importance he gave to the signification of terms in an insoluble proposition. We must moreover note that, in a certain way, Heytesbury goes back to a more ancient solution called “cassatio”. His own version of the solution by cassation is nevertheless less abrupt than the earlier ones. He does not claim that the one who says a proposition such as “I am saying what is false” does not produce any act (cassation of the power of the act), nor that the one who says such a sentence does not say anything (cassation of the act); he just claims that if one admits an insoluble proposition, one admits something impossible that will lead one to a logical contradiction. Of course, the criticism that immediately comes to mind concerning this kind of solution is that although it allows one, from a practical point of view, to avoid a contradiction, it does not offer a satisfactory solution to the theoretical problem of insolubles. Heytesbury was well aware of that, and ended his treatise by acknowledging it and by exhorting his readers to proceed to more useful study. (4) Even if, as we have seen, Heytesbury’s and Bradwardine’s solutions are not compatible, it is interesting to note that most of the later English authors mentioned in this paper were under the combined influence of these two Mertonian masters. Some of them adopted a hybrid solution that completely missed the core of Heytesbury’s solution (Pseudo-Heytesbury, John Hunter(?), the Anonymous of

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Ms. Vat. Lat. 674), while two others left the door more or less open to both solutions (John of Holland and Robert Fland). From this point of view, it is the text that is the most distant from Heytesbury’s in form that is the closest to it in content (the Anonymous of Ms. Oxford Lat. Misc. e. 79). The texts written by Heytesbury and Bradwardine would continue to influence the debate on insolubles until the end of the Middle Ages in England, of course, but also in Italy where Paul of Venice, for example, adopted exactly the same hybrid solution as did the three authors just mentioned.

15.5

The Edition of the Texts

15.5.1

The Manuscripts

Pseudo-Heytesbury P = Padua, Biblioteca Universitaria 1123, 22vb-24rb. This manuscript dates from the third quarter of the fourteenth century. It has been described by L.M. de Rijk, in Some 14th Century Tracts on the Probationes terminorum, Nijmegem, Ingenium Publishers, 1982, p. 31. John Hunter(?) E = Erfurt, Ampl. 4ř 270, 37r1–39r30. F = Florence, Biblioteca Laurenziana, Gaddianae.188, 4v–6v. O = Oxford, Bodleian Library, Canon. Misc. 219, 7ra-9rb. V = Rome, Vat. Lat. 3065, 28r–30v. The Erfurt manuscript, dating from 1390 at the latest, has been described in W. Schum, Beschreibendes Verzeichniss der Amplonianischen Handschriften-Sammlung zu Erfurt, Berlin, Weidemannsche Buchhandlung, 1887, pp. 513–514. It contains Chaps. 2–6 of Heytesbury’s Regulae solvendi sophismata. The Florence manuscript, dating from 1473–1474, has been described in [5, pp. 616–619]. It contains Heytesbury’s Regulae solvendi sophismata as well as the treatise Juxta hunc textum in what I elsewhere22 called the ‘Heytesbury Version’. The Oxford manuscript, dating from the end of the fourteenth century or from the beginning of the fifteenth century, has been described in A. Maierù, in “Le M.S.. 22 See

[22].

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F. Pironet

Oxford, Canonici Misc. 219”, in English Logic in Italy in the 14th and 15th Centuries, A. Maierù (ed.), Napoli, Bibliopolis, 1982, pp. 98–103. It contains Heytesbury’s De sensu composito et diviso as well as Bradwardine’s De insolubilibus. The Rome manuscript has been described in J. Ruysschaert, Bibliotheca Apostolica Vaticana, Codices Vaticani Latini (Vatican City 1902–1985). See also A.M. Piazzoni et P. Vian, Manoscritti Vaticani Latini 14666–15203, Catalogo sommario (Vatican City 1989); O.P. Kristeller, Iter Italicum II, pp. 310–388 et 581–588; VI, pp. 318– 354. It contains Heytesbury’s De sensu composito et diviso as well as Bradwardine’s De insolubilibus. Stemma EFVO EFV O E FV FV Since O was very independent from the other manuscripts, I have chosen the EFV branch as a base for the edition. Generally speaking and unless another manuscript evidently had the right reading, I favoured the following groupings: EFV:O; FOV:E; EFO:V; EO:FV; EO:F:V; FV:E:O; FV:E; FV:O; FO:EV; FO:E:V; EV:F:O; EV:F; EV:O; OV:EF. Anonymous Ms. Oxford, Lat. Misc. e.79 O = Oxford, Bodleian Library, Lat. Misc. e.79, 43ra–45vb. S = Sevilla, Biblioteca capitular y Colombina, Cod. 5-1-12, 52r–54v. The Oxford manuscript, dating from the second half of the fifteenth century has been described in L.M. de Rijk, in “Logica Oxoniensis. An Attempt to Reconstruct a Fifteenth Century Oxford Manual of Logic”, in Medioevo III, 121–164, 1977, pp. 150–155. It has not been copied all along by the same scribe. The first part of the text (ff. 43v–44r) is written by a first hand (O1) and shows some traces of a second hand (O2) that made some corrections and completed the copying of the treatise. The Seville manuscript, probably dating from the second half of the fifteenth century has been described in [5, 625–627]. These two manuscripts contain exactly the same treatises, among which the Sophismata asinina in a version that is not Heytesbury’s as well as the treatise Juxta hunc textum in Robert Alyngton’s version. For this edition I only used the Oxford manuscript for two reasons: first, I did

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not have the Seville manuscript at hand any more and, second, as I noticed when I first used these manuscripts for my edition of Heytesbury’s Sophismata asinina, the Seville manuscript is clearly a copy of the Oxford one and does not bring any other information than additional errors. Heytesbury L = Incunabulum Venice, Bonetus Locatellus, 1494, 4va–7rb. As there are so many manuscripts containing this text, I decided to propose a transcription based on the most commonly used incunabulum; this text is good enough to justify this choice. I have omitted the first part of the text, dedicated to the discussion of different opinions (ff. 4va–6rb) because the authors I am studying here, although influenced by Heytesbury, nevertheless omitted this part of his text.

15.5.2

Signs in the Edition

The apparatus is positive and complete. expunxi < > supplevi alb. blank in the text add. addidit corr. correxi om. omisit in mg. in margine LAC lacuna ## erased or expuncted portion of text ? uncertain reading

15.6

William Heytesbury De insolubilibus Partial Transcription Venice, Bonetus Locatellus, 1494

(4va) Incipiunt Regulae solvendi sophismata praeclarissimi Magistri Gulielmi Hentisberi omnium logicorum subtilissimi. Et primo de insolubilibus. <...> 4423 (6rb) Restat igitur videre aliquas regulas superius 23 Numbers

in bold correspond to paragraph numbers in the English translation by Spade.

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F. Pironet

memoratas. Et primo praemittendae sunt duae suppositiones pro intellectu duorum terminorum. Una est quod casus de insolubili est ille in quo fit mentio de aliqua propositione quae, si cum eodem casu significet praecise sicut verba illius communiter praetendunt, ad eam esse veram sequitur eam esse falsam, et e converso. 45 Alia est ista: propositio insolubilis est de qua fit mentio in casu insolubili quae, si cum eodem casu significet praecise sicut verba illius communiter praetendunt, ad eam esse veram sequitur eam esse falsam, et e converso. 46 Verbi gratia, si ponatur quod Socrates dicat talem propositionem “Socrates dicit falsum” et nullam aliam, vel talem “Socrates non dicit verum”, vel talem “Socrates dicit aliter quam est”, aut quod talis propositio “falsum est” sit omnis propositio, aut quod haec sit quaelibet propositio “nulla propositio est vera”, et sic de talibus, quilibet talis casus dicitur casus de insolubili, et ista propositio “Socrates dicit falsum” in tali casu dicitur propositio insolubilis: quia si illa cum tali casu significaret praecise quod Socrates dicit falsum, ad eam esse veram sequitur eam esse falsam, et e converso. 47 Aliter etiam posset fieri casus de insolubili ponendo quod Socrates solummodo diceret talem propositionem “deus est” quae praecise significet quod Socrates dicit falsum, aut quod ista propositio “chimaera est” sit quaelibet propositio et quod illa significet praecise quod falsum est, vel quod nulla propositio sit vera, et sic de consimilibus. Sed ille non est communis modus supponendi (6va) hujusmodi casus, et etiam illi expresse convertuntur cum casu de insolubili ubi insolubile significet praecise sicut communiter termini illius praetendunt, et ita in effectu idem accidit sicut prius. Ut igitur ad propositum redeamus fiat talis divisio. 48 Si fiat casus de insolubili, aut ponitur qualiter illud insolubile debeat significare aut non. Si non, proposito illo insolubili, respondendum est ad illud omnino sicut responderetur non supposito illo casu. Verbi gratia, supposito quod Socrates dicat hanc propositionem “Socrates dicit falsum” et nullam aliam, et cum hoc non ponatur qualiter significare debeat illa propositio “Socrates dicit falsum” cum ipsa proponitur, primo loco est respondendum ad ipsam omnino sicut responderetur extra casum. Stat

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enim cum casu quod illa sit vera et etiam quod24 illa sit falsa; et ideo stat cum casu quod Socrates dicat verum in dicendo illam, et etiam stat quod dicat falsum; et ideo cum illa propositio insolubilis sit impertinens illi casui propter casum illum, proposita, primo loco non est aliter respondendum ad illam quam ante casum. Et eodem modo est faciendum posito illo casu, scilicet quod ista propositio “falsum est” sit quaelibet propositio non supposito qualiter illa significet, cum illa tibi proponatur, respondendum est ad illam primo loco consimiliter omnino sicut faceres absque casu. <Secunda regula> 49 Secundo est advertendum quod si ponatur casus de insolubili, et cum hoc supponatur quod illud insolubile praecise significet sicut termini illius communiter praetendunt, casus ille nullatenus admittatur. Sicut cum supponitur quod ista propositio “falsum est” sit quaelibet propositio et ipsa significet praecise quod falsum est, statim negandus est ille casus et quilibet convertibilis cum eodem. Verbi gratia, si supponatur quod Socrates dicat talem propositionem solummodo “Socrates dicit falsum”, et cum hoc etiam supponatur quod illa propositio praecise significet quod Socrates dicit falsum, nullatenus talis casus est admittendus nec aliquis convertibilis cum eodem propter multa impossibilia quae sequuntur. 50 Tertio si fiat casus de insolubili, et cum hoc supponatur quod illud insolubile significet sicut termini ipsius praecise praetendunt non tamen sic praecise, admisso illo casu, concedendum est illud insolubile quocumque loco proponatur tamquam sequens, et negandum est illud esse verum tamquam repugnans. Verbi gratia, supposito quod Socrates solummodo dicat istam propositionem “Socrates dicit falsum” et quod illa sic significet non tamen praecise, tunc concedenda est illa cum proponitur tamquam sequens, et negandum est illam esse veram tamquam repugnans: sequitur enim “Socrates dicit hanc propositionem ‘Socrates dicit falsum’ quae sic significat; igitur Socrates dicit falsum”, sed cum nullam propositionem dicat Socrates nisi istam “Socrates dicit falsum”, sequitur quod illa est falsa, et ex hoc sequitur illam non esse veram. Concedendum est igitur quod Socrates dicit falsum tamquam sequens ex casu, et negandum est istam propositionem “Socrates dicit falsum” esse veram tamquam eidem repugnans. Et idem accidit universaliter in quocumque casu consimili, sicut discurrenti per singulos poterit apparere. 24 etiam

quod corr. Spade: quod etiam L.

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51 Si autem quaeratur in illo casu quod significavit illa propositio dicta a Socrate aliter quam quod Socrates dicit falsum, huic dicitur quod respondens non habet istam quaestionem solvere sive determinare: quia ex casu sequitur quod illa propositio aliter significat quam quod Socrates dicit falsum, sed casus non certificat quid illud sit; ideo non habet respondens quaesitum illud ulterius determinare. 52 Quoniam autem admissi sunt hujusmodi casus de insolubilibus ubi ponitur quod illa insolubilia significent sicut verba praetendunt cum illa non sic praecise significent, sed etiam aliter quam verba illa solent communiter intelligi, posset forte aliquis supponere quod Socrates dicat solummodo (6vb) hanc propositionem “Socrates dicit falsum” et quod illa praecise significet quod Socrates dicit falsum et quod deus est et quod homo est animal vel aliquid hujusmodi necessarium. Pro quo est advertendum et tamquam pro regula observandum quod si fiat casus de insolubili, et cum hoc supponatur illud insolubile significare copulative praecise sicut praetendunt verba illius, ut A et B esse, vel aliquid aliud quidquid fuerit, si oppositum illius copulati non possit stare cum toto casu, negandum est casus tamquam claudens contradictoria formaliter. Verbi gratia, supposito quod Socrates dicat solummodo hanc propositionem “Socrates dicit falsum” et quod illa praecise significet quod Socrates dicit falsum et quod deus est, ex quo oppositum illius copulati, scilicet istud “nullus deus est”, non potest stare sic significando cum toto illo casu supposito, sequitur quod casus ille mere est impossibilis: quia ex casu illo formaliter sequitur quod ista propositio “Socrates dicit falsum” sic dicta a Socrate esset vera et etiam esset falsa, sicut statim apparet cuilibet intuenti. Supposito etiam quod Socrates solummodo dicat hanc propositionem “Socrates dicit falsum” et quod illa praecise significet quod Socrates dicit falsum et quod Socrates loquitur, quia oppositum hujus copulati, scilicet hoc “Socrates non loquitur”, non stat cum illo casu, non est ille casus nec aliquis consimilis admittendus. Et in proposito dico universaliter illa non stare simul quorum unum formaliter repugnat alteri eorumdem, aut quorum aliquid per se est impossibile. Multa enim sunt qui non stant simul qui tamen formaliter non repugnant. Ista enim “tu es asinus” et “tu non es capra” seu “rex sedet” non simul stant, et tamen non formaliter repugnant. Sicut enim duo contradictoria non stant simul cum aliquo tertio, sic nec propositio per se impossibilis stat cum aliquo alio: quia illa quasi universaliter clauditur opposita. 53 Si autem fiat casus de insolubili et cum hoc supponatur quod illud insolubile significet praecise disjunctive sicut verba praetendunt vel quod

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A est B vel aliquid hujusmodi, nisi illud disjunctum cum illo insolubili possit stare cum toto isto casu, non est casus admittendus. Unde, posito quod Socrates solummodo dicat hanc propositionem “Socrates dicit falsum”, et quod illa praecise significet quod Socrates dicit falsum vel quod homo est asinus, quia hoc disjunctum “homo est asinus” non potest stare cum illo casu; ideo ille casus est impossibilis. Similiter, si fiat talis casus quod Socrates dicat solummodo hanc propositionem “Socrates dicit falsum”, et quod illa praecise significet quod Socrates dicit falsum vel quod Socrates non dicit propositionem vel quod Socrates non dicit verum, non admittatur ille casus propter inconveniens dictum prius. 54 Et sic25 in omni casu ubi ponitur cum casu de insolubili quod illud copulative significet sicut termini illius communiter praetendunt, et sic esse vel sic, ubicumque propositum fuerit, est concedendum, et negandum est illud esse verum. Ita e converso, cum ponitur casus de insolubili et cum hoc ponitur quod illud insolubile significet disjunctive praecise sicut verba praetendunt vel sic esse vel sic, negandum est illud insolubile quotienscumque proponitur, et concedendum est illud esse verum. Verbi gratia, posito quod haec propositio “falsum est” sit quaelibet propositio, et quod illa praecise significet quod falsum est vel quod deus est, proposita illa “falsum est”, illa est neganda, et concedendum est illam esse veram. Unde semper in tali casu oportet concedere illud disjunctum esse verum, sicut in alio oportet negare copulatum esse verum. Facto enim tali casu quod haec propositio “falsum est” sit quaelibet propositio, et quod illa praecise significet quod falsum est vel quod tu curris in bello monte, proposita ista “tu curris in bello monte”, oportet eam concedere tamquam sequens quantumcumque scias ipsam esse falsam, et in omni consimili casu accidit (7ra) illud idem. 55 E contrario autem, supposito quod haec propositio “falsum est” sit quaelibet propositio, et quod illa praecise significet quod falsum est et quod tu es homo, proposita ista “tu es homo”, habes illam negare, et quodlibet aliud antecedens ad illam, cujus causa statim apparere poterit intuenti. 56 Aliquando autem, cum fiat casus de insolubili, difficile est cito videre quae est propositio insolubilis, sicut posito tali casu, scilicet quod omnis homo dicens verum bene respondeat, et solummodo talis, et quod quilibet homo dicens falsum male respondeat, et solummodo talis, et quod Socrates solummodo dicat hanc propositionem “aliquis istorum male respondet”, demonstratis Socrate et Platone, et quod illa sic significet, et quod Plato dicat hanc propositionem “Socrates bene respon-

25 sic

corr.: sicut L.

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F. Pironet

det”, et quod illa sic significet, tunc in casu illo utraque illarum propositionum potest esse insolubilis. Posito autem quod illa propositio sic dicta a Socrate significet praecise cum toto casu quod aliquis istorum male respondet, tunc illa dicta a Platone est insolubilis et significabit aliter tunc quam quod Socrates bene respondet: non enim staret cum toto casu quod ipsa significaret tunc praecise quod Socrates bene respondet. Si autem in casu primo praecise significet illa dicta a Platone quod Socrates bene respondet, sequitur quod illa dicta a Socrate sit insolubilis et quod illa non solummodo significat quod aliquis istorum, demonstrato Socrate et Platone, male respondet, sed etiam aliter sicut falsum est esse. Quod enim illa tunc praecise significaret quod aliquis istorum male respondet non stat cum casu, sicut cum nullo casu stat quod illa propositio quae insolubilis est significet praecise sicut verba praetendunt, sicut ab initio dictum est; et ideo in tali casu oportet respondentem diligenter advertere quae sit propositio insolubilis et quae non. 57 Sed forte quia tamquam pro fundamento dictum est quod non est possibile quod aliqua propositio insolubilis significet praecise sicut verba illius communiter praetendunt cum toto casu ejusdem insolubilis, arguet adversarius cavillatorie contra hoc sic: possibile est quod Socrates solummodo dicat hanc propositionem “Socrates dicit falsum” et quod ista propositio “Socrates dicit falsum” significet praecise sic sicut verba illius praetendunt communiter: quia in principio pono tibi hanc copulativam “haec propositio ‘Socrates dicit falsum’ et quaelibet talis propositio significat praecise sicut verba illius communiter praetendunt et dant intelligere”. Haec copulativa, ut notum est, est possibilis: quia possibile est quod quaelibet talis propositio significet ex communi intellectu terminorum quod Socrates currit in bello monte. Quo posito, ponatur quod Socrates dicat istam solam et non aliam “Socrates dicit falsum”, et quod illa significet praecise sicut verba illius communiter praetendunt, et sequitur oppositum fundamenti. 58 Sed illa objectio et ratio nimis cavillatoria est, et bene potest dici sophistica: quia vadit solummodo ad verba et non ad intellectum, cum intelligantur omnia superius posita in sensu diviso. Arguitur autem ille cavillator contra ista in sensu composito: nimis enim esset prolixum in verbis tantum instare, ut nihil diceretur quod cavillatorie non posset impugnari; ideo non tantum ad verba nuda, sed ad sententiam referas argumentum, et videbis quam potenter concludit. 59 Consimiliter posset objici eo quod prius frequenter est admissum quod Socrates solummodo dicat hanc propositionem [et] “Socrates dicit falsum”, cum tamen de virtute sermonis est valde impossibile. Probo

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quod haec propositio est impossibilis sic significando de praesenti “Socrates dicit hanc propositionem”: nulla enim est propositio vel esse potest ita modica quam Socrates possit subito et simul dicere secundum se totam aut aliquid sui partem: quia cum omnis vox sit divisibilis, nullus subito aut in instanti profert (7rb) aliquam. 60 Similiter, dato quod Socrates dicat hanc propositionem totam “Socrates dicit falsum”, sequitur quod Socrates diceret hanc, scilicet “Socrates dicit”, quae est pars prioris, et sic Socrates non solummodo diceret illam primam. 61 Verumtamen, quia non faciunt hujusmodi cavillationes ad propositum, sicut nec hujusmodi aliae objectiones litterales circa eas, non est ulterius instandum. 62 Frequenter autem fit casus de insolubili per terminos qui non immediate concernunt veritatem vel falsitatem propositionis, sed mediate, sicut posito quod Socrates solummodo credat hanc propositionem “aliquis homo decipitur”, et quod illa sic [praecise] significet, et quod quilibet alius a Socrate solummodo credat sicut est, tunc notum est quod illa propositio “aliquis homo decipitur” quam credit Socrates est insolubilis et quod illa significat aliter quam quod aliquis homo decipitur: quia si aliquis homo decipitur, ipse credit aliquam propositionem significantem5 aliter quam est: quia si aliquis homo decipitur, ipse credit aliter quam est, et si aliquis credit aliter quam est, ipse credit propositionem falsam: quia propositionem significantem aliter quam est. Nisi vellet aliquis ponere vel dicere quod qualitercumque credat aliquis homo esse non sequitur ipsum credere aliquam propositionem, et qualitercumque sciat esse non requiritur aliquam propositionem esse scitam aut scientiam illius esse, quod, ut puto, posset probabiliter sustineri. Sed quia vix nobis apparet illud verum supposito casu priori respondeas ut praemissum est et in quolibet casu de insolubili consimili consimile judicium elicias per praedicta. 63 Multi etiam fiunt casus in hac parte prolixi nimis et inutiles in quibus oportet diligenter calculare et discurrere ab una propositione ad aliam quousque appareat quae illarum sit insolubilis. Nec turbetur aliquis posito tali casu, et si prima vice non videat quid sit respondendum aut qualiter: quia nec hoc faceret opponens in casu simili vel eodem paululum variato. 64 Haec igitur de insolubilibus dicta sunt ad introductionem et exercitium juniorum, ut viso quod insolubilia, juxta significationem nominis, absque evidenti objectione solvi non possunt, ea cito transeat unusquisque utiliori studio attentius insistendo et cetera.

290

15.7

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Pseudo-Heytesbury, Insolubilia Padua, Biblioteca Universitaria 1123, ff. 22vb–24rb

Incipiunt insolubilia optima. Insolubile est propositio affirmativa vel negativa aliqualiter esse vel aliqualiter non esse significativa quam impossibile est significare praecise, si foret in conceptu. Jam sequuntur regulae de insolubilibus. Primo sciendum est quid est insolubile et postea propositio de insolubili. Et sciendum est quod casus insolubilis est ille in quo fit mentio de aliqua propositione quae, si cum eodem casu significat praecise sicut verba illius praetendunt, sequitur eamdem esse veram et falsam. Verbi gratia, pono quod unus Socrates sit omnis Socrates, et quod Socrates dicat talem propositionem et nullam aliam ‘Socrates dicit falsum’, et sit una talis quaelibet26 propositio, et significet illa praecise quod Socrates dicit falsum, tunc illa propositio dicta a Socrate est propositio insolubilis: quia ex casu sequitur quod illa est vera et similiter falsa. Probo, et quaero an ista sit vera vel falsa. Si dicatur quod illa est vera; ergo illa significat praecise sicut est. Tunc sic: illa significat praecise sicut est; et illa significat praecise quod Socrates dicit falsum; ergo ita est quod Socrates dicit falsum. Tunc sic: ita est quod Socrates dicit falsum; et nihil dicit nisi illam; ergo illa est falsa, et per te est vera; ergo et cetera. Si dicatur quod illa est falsa, tunc sic: illa est falsa; ergo significat aliter quam est. Tunc sic: illa significat aliter quam est; et illa significat praecise quod Socrates dicit falsum; ergo non est ita quod Socrates dicit falsum; et nihil dicit nisi istam propositionem ‘Socrates dicit falsum’; ergo illa est vera; ergo illa non est falsa. Et sic sequitur ex casu quod illa est vera et falsa; ideo iste casus est insolubilis. Et eodem modo potest fieri casus de insolubili respectu istam ‘Socrates dicit aliter quam est’, ‘Socrates non dicit verum’, ‘Socrates legit falsum’, ‘Socrates scribit falsum’, et ita de aliis. Et pono quod ista propositio ‘falsum est’ sit omnis propositio, vel quod sit quaelibet propositio ‘nulla propositio <est> vera’, et ita de consimilibus. Sed propositio insolubilis est illa de qua fit mentio in aliquo casu quae, si cum eodem casu sic significet praecise, sequeretur ipsam esse veram et falsam. Exemplum sicut prius: in casu quod ista propositio ‘Socrates dicit falsum’ sit quaelibet propositio, tunc sequitur quod ista est propositio 26 quaelibet:

talis add.P.

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insolubilis: quia ex casu sequitur quod illa sit vera et similiter falsa, sicut argutum est prius. Et sic dicendum est in omnibus propositionibus consimilibus. Tertium notandum est quod propositio insolubilis dividitur: quia quaedam est propositio insolubilis simplex et quaedam composita. Insolubile simplex est sicut ista ‘Socrates dicit falsum’, et ita de aliis. Sed insolubile compositum dicitur multis modis, scilicet copulative, disjunctive, sicut postea patebit. De insolubili dantur tres regulae. Prima regula est haec quod si ponatur casus de insolubili, et non ponatur qualiter illud insolubile debeat significare, supposito illo insolubili, respondendum est sicut ad impertinens. Verbi gratia, supposito quod Socrates dicat talem propositionem ‘Socrates dicit falsum’ et nullam aliam, tunc proposita ista propositione ‘Socrates dicit falsum’, respondendum est ad istam dubitative sicut ante casum. Et causa est haec quia stat illam esse veram et similiter falsam; ideo cum casu stat quod Socrates dicit verum et falsum; et ideo cum illa sit impertinens casui, non est aliter respondendum ad illam quam fuerat ante casum. Et eodem modo respondendum est ad istam ‘falsum est’, posito quod ista sit quaelibet propositio, non supposito qualiter debeat significare, et ita dicendum est in omnibus casibus consimilibus. <Secunda regula> Secunda regula est: si ponatur casus de insolubili, et cum hoc ponatur quod illud insolubile significet praecise sicut verba illius communiter praetendunt, tunc nullo modo admittendus <est> casus. Verbi gratia, pono quod ista propositio ‘falsum est’ sit quaelibet propositio, et quod illa sic praecise significet, negatur casus: quia ex illo sequitur quod illa est vera et similiter falsa. Probo, et quaero an sit vera vel falsa. Si dicatur quod est vera, tunc sic: illa est vera; ergo illa significat praecise sicut est; et illa significat praecise quod falsum est; ergo ita est quod falsum est. Tunc sic: ita est quod falsum est; ergo propositio falsa est. Et ultra: ergo non est vera. Et si dicatur quod illa est falsa, tunc sic: illa est falsa; ergo significat aliter quam est. Tunc sic: illa significat aliter quam est; et illa significat praecise quod falsum est; ergo non est ita quod falsum est. Tunc sic: non est ita quod falsum est; ergo non est ita quod propositio falsa est. Tunc

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sic: non est ita quod propositio falsa est; et illa est quaelibet propositio; ergo illa est vera. Et ultra: ergo illa non est falsa. Sic ex casu illo sequitur quod illa est vera et similiter falsa; et ideo negatur casus et quilibet consimilis. Tertia regula est ista: si ponatur casus de insolubili, et cum hoc ponitur quod illud insolubile significat sicut termini ejus communiter praetendunt, et non tamen sic praecise, admittendus est casus ille et est concedendum illud insolubile quando proponitur tamquam sequens et negandum est illud fore verum. Verbi gratia, posito quod iste Socrates sit omnis Socrates, et quod Socrates dicat istam propositionem ‘Socrates dicit falsum’ et nullam aliam, et quod illa sic significet, admittitur casus et conceditur quod Socrates dicit falsum tamquam sequens. Nam sequitur: Socrates dicit istam propositionem ‘Socrates dicit falsum’; et illa est falsa; ergo Socrates dicit falsum. Illa consequentia est bona; antecedens est verum; ergo <et> consequens; et consequens est ista propositio ‘Socrates dicit falsum’; ergo ista est vera ‘Socrates dicit falsum’. Tunc sic: ista est vera ‘Socrates dicit falsum’: quia Socrates dicit istam propositionem ‘Socrates dicit falsum’; ergo illa propositio est vera vel falsa. Si vera; ergo ita est totaliter sicut illa significat, et illa significat quod Socrates dicit falsum; ergo Socrates dicit falsum. Et si illa sit falsa, et Socrates dicat illam; ergo Socrates dicit falsum; ideo ista propositio est concedenda ‘Socrates dicit falsum’, et negandum est illam fore veram: quia si ista sit vera ‘Socrates dicit falsum’, et Socrates dicit illam; ergo Socrates dicit verum; et nullam dicit nisi illam; ergo illa est vera; ergo eadem est propositio vera et falsa. Ideo in quolibet casu ubi proponitur talis propositio insolubilis, conceditur, et negatur illud insolubile fore verum propter causam praedictam. Et eodem modo dicendum est de ista ‘Socrates audit falsum’, et ‘Socrates legit falsum’, ‘Socrates dicit aliter quam est’, ‘Socrates non dicit verum’, et ‘falsum est’ significat similiter copulative, videlicet quod falsum est et quod eadem propositio est vera, et ejus oppositum significabit8 hoc disjunctive, scilicet quod non falsum est vel quod illa non est vera, et hoc est verum. Et ita dicendum est in omnibus consimilibus. Unde si sic arguitur sicut in priori: ista propositio est falsa ‘Socrates dicit falsum’; ergo illa significat aliter quam est, conceditur consequentia. Tunc sic: illa significat aliter quam est; et illa significat quod Socrates dicit falsum; ergo non est ita quod Socrates dicit falsum, negatur consequentia: quia ad hoc quod consequentia valeat, oportet addere in minori istam dictionem ‘praecise’, tunc foret minor neganda.

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Aliter arguitur sic forte: illa est quod Socrates dicit falsum; ergo Socrates dicit falsum, conceditur consequentia. Tunc sic: illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est propositio dicta a Socrate; ergo illa propositio est vera, quod falsum est. Pro illo dicitur quod consequentia non est bona et formalis: quia repugnans. Et similiter dicitur in aliis, ut pono quod non sit aliqua propositio nisi ista ‘omnis propositio est falsa’ sic significando, admittitur. Deinde proponitur “omnis propositio est falsa”, admittitur per tertiam regulam. “Illa est vera”, negatur: quia repugnans. Tunc sic: “illa est falsa”, conceditur. Tunc: “illa est falsa; et est omnis propositio; ergo omnis propositio est falsa”, concedo consequentiam. Tunc sic: “illa consequentia est bona et formalis; et antecedens est verum; ergo <et> consequens; et consequens est ista ‘omnis propositio est falsa’; ergo omnis propositio est falsa”, negatur antecedens: quia antecedens non est verum: quia repugnat casui quod antecedens sit verum. Simile <est> de universali negativa, posito quod ista propositio negativa ‘nulla propositio est vera’ sit quaelibet propositio, et quod illa sic significet, admittitur casus. Tunc proponitur ista ‘nulla propositio est vera’, illa est concedenda per tertiam regulam. Tunc sic “illa est vera”, negatur: quia repugnans est. Tunc: “ista est falsa; et ista est quaelibet propositio; ergo nulla propositio est vera; ergo omnis propositio est falsa. Illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est illa propositio; ergo illa propositio est vera”, quod erat negatum. Ideo negatur quod antecedens sit verum: quia repugnat casui quod antecedens sit verum: quia per casum non est alia propositio nisi ista propositio ‘nulla propositio est vera’, et sic respondendum est de omnibus aliis consimilibus. Aliter fiat casus respectu insolubili, ut hic suppono istam suppositionem quod omne dicens verum habebit denarium, et quod nihil dicens falsum habebit denarium, et quod unus Socrates sit omnis Socrates, et dicat ille istam propositionem et nullam aliam ‘Socrates non habebit denarium’, admittitur casus. Tunc proponitur “Socrates non habebit denarium”, illa est dubitanda per primam regulam: quia non ponitur qualiter illud insolubile significat; ideo stat quod sit verum et similiter quod sit falsum; et ideo dubitanda est illa.

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Sed si ponatur cum toto casu quod illa sic significet praecise quod Socrates non habebit denarium, tunc negatur casus per secundam regulam: quia ex casu sequitur quod illud sit verum et similiter falsum; ideo sequitur ex casu quod Socrates habebit denarium et quod Socrates non habebit denarium, et hoc est impossibile. Et ideo si ponatur cum toto casu quod illud insolubile significabit sicut termini illius communiter praetendunt, non tamen praecise, admittendus est casus per tertiam regulam. Tunc quando proponitur “Socrates non habebit denarium”, conceditur tamquam sequens. Nam sequitur: Socrates dicit falsum; et nihil dicens falsum habebit denarium; ergo Socrates non habebit denarium. Illa consequentia est bona, et minor patet per suppositionem. Et probo majorem: quia Socrates dicit istam ‘Socrates non habebit denarium’; ergo ista est vera vel falsa. Si vera; et dicit illam; ergo dicit verum; et quilibet talis habebit denarium; ergo Socrates habebit denarium; et Socrates dicit ‘Socrates non habebit denarium’; ergo dicit falsum. Si ista sit falsa ‘Socrates non habebit denarium’; et Socrates dicat illam; ergo Socrates dicit falsum. Ergo ex utraque parte sequitur majorem esse veram; ideo conceditur insolubile tamquam sequens, et negandum est illud fore verum tamquam repugnans. Et si arguitur sic: ista propositio est falsa ‘Socrates non habebit denarium’; ergo significat aliter quam est; et illa significat quod Socrates non habebit denarium; ergo non est ita quod Socrates non habebit denarium, negatur consequentia: quia ad hoc quod consequentia valeat oportet addere in minori quod illa significat praecise. Et tunc foret minor neganda: quia significat copulative et ejus oppositum disjunctive, ut prius dictum est. Aliter arguitur sic: ita est quod Socrates dicit falsum; ergo Socrates dicit falsum; illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est <propositio> dicta a Socrate; ergo Socrates dicit verum. Et ultra: ergo habebit denarium. Solutio: negatur quod illa consequentia sit bona et formalis tamquam repugnans, ut dicitur in casu praecedenti. Et consimiliter dicendum est in omni materia consimili, ut posito quod omne dicens verum sit aeger, et sic ille Socrates <sit> omne Socrates, et dicat ille istam propositionem et nullam aliam ‘Socrates non est aeger’, et quod illa sic significet, non tamen praecise, admittitur casus et conceditur quod Socrates non sit aeger tamquam sequens, et negandum est illud insolubile fore verum tamquam repugnans. Simile est posito quod omne dicens verum pertransibit [a] pontem, et nihil dicens falsum pertransibit [a] pontem, et illa sic significat sic,

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non tamen praecise, admittitur casus et conceditur quod Socrates non pertransibit [a] pontem et negatur quod illa est vera. Consimiliter sic dicendum est in omnibus casibus consimilibus et cetera. Dicendum est de insolubilibus compositis. Primo de a regulis, <et prima est ista> quod si ponatur casus de insolubili, et cum hoc ponatur illud insolubile significare praecise sicut verba illius praetendunt copulative, si oppositum illius copulati non potest stare cum toto casu, negandus est casus ille tamquam claudens opposita. Verbi gratia, posito quod iste Socrates sit omnis Socrates, et dicat ille istam propositionem et nullam aliam ‘Socrates dicit falsum’, praecise significando quod Socrates dicit falsum et quod deus est, oppositum illius copulati est hoc ‘nullus deus est’, et hoc non potest stare cum toto casu sic significando praecise; ideo casus est impossibilis: quia ex casu sequitur quod propositio dicta a Socrate sit vera et similiter falsa. Probo: nam quaero an illa sit vera vel falsa. Si vera, tunc sic: illa est vera; et illa significat praecise quod Socrates dicit falsum et quod deus est; ergo ita est quod Socrates dicit falsum et quod deus est; ergo ita est quod Socrates dicit falsum; et nihil dicat nisi illam; ergo illa est falsa. Et ultra: ergo illa non est vera. Et si dicatur quod illa est falsa; et illa significat praecise quod Socrates dicit falsum et quod deus est; ergo non est ita quod Socrates dicit falsum et quod deus est. Tunc sic: non est ita quod Socrates dicit falsum; et ipse aliquid dicit; ergo dicit verum; et nihil dicit nisi illam; ergo illa est vera. Et ultra: ergo non est falsa. Et sic sequitur quod illa est vera et similiter falsa; et ideo casus est impossibilis: quia oppositum copulati est impossibile, et nullum impossibile potest stare cum isto casu; et ideo negatur casus. Simile est hoc, posito quod ista propositio ‘falsum est’ sit quaelibet propositio, et quod illa significet praecise quod falsum est et quod nullus deus est. Et simile est hoc, posito quod Socrates dicat istam propositionem et nullam aliam ‘Socrates dicit falsum’, significando praecise quod Socrates dicit falsum et quod Socrates loquitur, et sic de aliis. Alia regula est ista: si ponatur casus de insolubili, et cum hoc supponatur illud insolubile significare copulative praecise sicut verba illius praetendunt, si oppositum copulati potest stare cum toto casu sic significando primarie, admittitur casus, et concedendum est illud insolubile quando proponitur, et negandum est illud fore verum, et negandum est illud copulatum et quodlibet antecedens ad illud, ut posito quod haec

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propositio ‘falsum est’ sic significet precise quod falsum est et quod tu curris, admittitur casus, et conceditur ista ‘falsum est’, et negatur quod illa est vera: quia illa est falsa, et negatur tunc quod tu curris, utrum sit ita quod tu curris vel non. Simile est de ista propositione ‘Socrates dicit falsum’, et ita de omnibus aliis consimilibus et cetera. Nunc sequuntur regulae de disjunctivis, quarum prima est haec: si ponatur casus de insolubili, et cum hoc ponatur illud insolubile significare disjunctive sicut verba ejus communiter praetendunt, nisi illud disjunctum posset stare cum toto casu sic significando praecise, non admittitur casus. Verbi gratia, ponatur quod unus Socrates sit omnis Socrates, et quod ille dicat istam propositionem et nullam aliam ‘Socrates dicit falsum’ sic significando praecise quod Socrates dicit falsum vel quod homo est asinus27 , hoc disjunctum ‘homo est asinus’ non potest stare cum toto illo casu sic significando praecise; ideo casus est impossibilis: quia ex casu sequitur quod illa est vera et similiter falsa. Probo28 : quaero an illa sit vera vel falsa. Si vera, tunc sic: illa est vera; et illa significat praecise quod Socrates dicit falsum vel quod homo est asinus; ergo ita est quod Socrates dicit falsum vel quod homo est asinus; sed non est ita quod homo est asinus; ergo ita est quod Socrates dicit falsum; et Socrates nihil dicit nisi illam propositionem; ergo illa propositio est falsa; ergo illa non est vera. Si sit falsa, tunc sic: illa est falsa; et illa significat quod Socrates dicit falsum vel quod homo est asinus; ergo non est ita quod Socrates dicit falsum vel quod homo est asinus; sed est ita quod Socrates dicit aliquid; ergo ita est quod Socrates dicit verum vel quod homo est asinus; sed non est ita quod homo est asinus; ergo est ita quod Socrates dicit verum; et Socrates nihil dicit nisi illam propositionem; ergo illa propositio est vera. Et ultra: ergo illa non est falsa. Et sic sequitur quod illa est vera et similiter falsa; et ideo ille casus est impossibilis, et ita est dicendum in omnibus aliis consimilibus. Secunda regula est haec: si ponatur casus de insolubili, et cum hoc supponatur illud insolubile significare disjunctive sicut verba ejus praetendunt, si illud disjunctum potest stare cum toto casu significando primarie, admittatur casus, et negandum est illud insolubile quando proponitur, et concedendum est illud fore verum tamquam sequens et concedendum est illud disjunctum utrum fuerit verum vel falsum. Verbi gratia, 27 asinus: 28 probo

nam add.P. corr. primo P.

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posito quod illa propositio ‘falsum est’ sit omnis propositio sic significando quod falsum est vel quod tu non es homo: quia hoc disjunctum est possibile, admittitur casus, et negatur quod falsum est, et conceditur quod illa est vera tamquam sequens quod tu non es homo, et negandum est illud insolubile fore verum. Et ita est dicendum in omnibus casibus consimilibus. Jam sequuntur regulae de insolubilibus compositis, videlicet de copulativis, de disjunctivis, de exceptivis et ita de aliis. Primo dicendum est de disjunctivis. Et ad faciendum insolubile de disjunctivis, sumenda est una propositio falsa et disjungenda est illa cum una propositione quae significat totam disjunctivam esse falsam. Verbi gratia, ponatur quod unus Socrates sit omnis Socrates et quod ille dicat istam propositionem et nullam aliam ‘homo est asinus vel disjunctiva dicta a Socrate est falsa’, et significet29 illa disjunctiva ex compositione terminorum, et utraque ejus pars sumatur. Deinde proponatur “disjunctiva dicta a Socrate est falsa”, si conceditur, tunc sic: illa est vera; et illa est secunda pars disjunctivae dicta a Socrate; ergo secunda pars disjunctivae dicta a Socrate est vera; et illa disjunctiva non significat aliter quam suae partes; ergo illa disjunctiva est vera. Et ultra: ergo illa non est falsa, quod est oppositum concessi. Si negatur quod illa disjunctiva est vera, tunc illa est falsa. Si conceditur, tunc: illa est falsa; et illa est disjunctiva dicta a Socrate; ergo disjunctiva dicta a Socrate est falsa; illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est secunda pars disjunctivae dicta a Socrate; ergo secunda pars disjunctivae dicta a Socrate est vera; et illa disjunctiva non significat aliter quam suae partes; ergo illa disjunctiva est vera, quod erat negatum. Solutio: admitto casum, et concedo quod disjunctiva dicta a Socrate est falsa, et negatur quod illa est vera. Et ad aliud argumentum “illa disjunctiva est falsa; et illa disjunctiva est disjunctiva dicta a Socrate; ergo disjunctiva dicta a Socrate est falsa”, conceditur consequentia, et negatur quod illa consequentia est bona et formalis: quia repugnans est. Et si arguitur sic: illa disjunctiva a Socrate est falsa30 ; et illa non significat aliter quam suae partes; ergo utraque ejus pars est falsa, conceditur consequentia et similiter consequens. Tunc sic: secunda pars 29 significet: #illa disjunctiva dicta a Socrate est falsa, sed conceditur tunc illa est vera ...# add.P. 30 falsa: conceditur consequentia dicta a Socrate est falsa add.P.

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disjunctivae est falsa; et secunda pars disjunctivae significat quod disjunctiva dicta a Socrate est falsa; ergo non est ita quod disjunctiva dicta a Socrate est falsa, negatur consequentia: quia ad hoc quod consequentia esset bona, oportet addere in minori quod illa sic significat praecise, et tunc casus foret impossibilis: quia illa pars significat copulative, videlicet quod disjunctiva dicta a Socrate est falsa et quod eadem propositio est propositio vera, et sic illa propositio est vera, et oppositum ejus significat disjunctive, ut prius dictum est. Et eodem modo dicendum est de disjuncto negativo, ut posito quod Socrates dicat istam propositionem et nullam aliam ‘homo est asinus vel disjunctiva dicta a Socrate non est vera’ sic significando. Et similiter respondendum est ad illam disjunctivam et ad omnes consimiles sicut ad illam praecedentem. Jam sequuntur regulae de copulativis, et ad faciendum insolubile de copulativis sumenda est una propositio vera et copulanda est illa cum una propositione significans31 totam copulativam esse falsam. Verbi gratia, ponatur quod Socrates dicat istam et nullam aliam ‘deus est et ista copulativa dicta a Socrate est falsa’, et significet ista copulativa sicut suae partes, et significet utraque pars juxta compositionem terminorum. Quaero utrum illa copulativa sit vera vel non. Si vera, tunc sic: illa est vera; et illa non significat aliter quam suae partes; ergo utraque ejus pars est vera. Et ultra: ergo secunda pars est vera. Et tunc arguo sic: secunda pars est vera; et illa significat quod copulativa dicta a Socrate est falsa; ergo ita est quod copulativa dicta a Socrate est falsa; et nulla est copulativa dicta a Socrate nisi illa; ergo illa est falsa. Et ultra: ergo illa non est vera, quod est oppositum concessi.32 Ideo si dicatur quod illa est falsa, tunc sic: illa est falsa; et illa non significat aliter quam suae partes; ergo altera ejus pars est falsa; et non prima; ergo secunda. Tunc sic: secunda pars ejus est falsa; et secunda pars significat quod copulativa dicta a Socrate est falsa; ergo non est ita quod copulativa dicta a Socrate est falsa; ergo33 sic est quod copulativa dicta a Socrate est vera; ergo illa non est falsa, quod est oppositum concessi. Solutio: admitto casum, et negatur quod illa copulativa est vera, sed illa est falsa. Tunc quando arguitur “ista est falsa; ergo34 altera ejus pars est falsa; et non prima; ergo secunda”, conceditur quod secunda pars est 31 significans

corr.: significare P. #solutio# add.P. 33 ergo corr.: sed P. 34 ergo corr.: tunc P. 32 concessi:

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falsa. Tunc ad argumentum “illa est falsa; et illa significat quod copulativa dicta a Socrate est falsa; ergo non est ita quod copulativa dicta a Socrate est falsa”, negatur consequentia: quia ad hoc quod consequentia valeret, oportet addere in minori quod illa sic significet praecise, et tunc negatur casus: quia tunc sequeretur quod illa copulativa foret vera et similiter falsa, quod est impossibile. Et causa est quia illa propositio quae est secunda pars significat copulative et ejus oppositum significat disjunctive, ut prius dictum est. Et consimiliter dicendum est ad istam necessariam: posito quod haec copulativa proferatur a Socrate ‘homo est animal et copulativa prolata a Socrate non est vera’ et nulla alia, et sic de omnibus consimilibus copulativis. Tunc sequuntur regulae de exclusivis. Et sit iste casus: suppono quod unus Socrates sit omnis Socrates, et dicat ille istam propositionem et nullam aliam ‘deus est’ sic significando praecise, et dicat Plato istam et nullam aliam ‘tantum Socrates dicit verum’. Isto posito, quaero an illa sit vera vel falsa. Si dicatur quod illa est vera, tunc sic: illa est vera; et Plato dicit illam et nullam aliam; ergo Plato dicit verum; et Plato est alius a Socrate; ergo alius a Socrate dicit verum. Et ultra: ergo non tantum Socrates dicit verum, quod est oppositum concessi. Si dicatur quod illa est falsa, tunc sic: illa est falsa; et Plato dicit illam; ergo Plato dicit falsum; et nullus alius dicit aliquid nisi Socrates et Plato; ergo tantum Socrates dicit verum, quod fuit negatum. Pro illo dicitur admittendo casum. Et quando quaeritur an illa sit vera vel falsa, dicitur quod est falsa. Tunc ad argumentum “illa est falsa; ergo significat aliter quam est”, conceditur. Tunc sic “illa significat aliter quam est; et illa significat quod tantum Socrates dicit verum; ergo non est ita quod tantum Socrates dicit verum”, negatur consequentia: quia ad hoc quod valeret, oportet addere in minori quod illa significat praecise, tunc casus est impossibilis. Aliter fiat casus respectu insolubili de exclusivo, posito quod A sit ista ‘deus est’ sic significando praecise, et B <sit> ista ‘tantum A est verum’, et pono quod non sint plures propositiones quam A et B. Admitto casum, deinde proponitur ista “tantum A est verum”. Si conceditur, tunc sic: illa est vera; et illa est B; ergo B est verum; et B est aliud ab A; ergo non tantum A est verum, quod est oppositum concessi. Si negatur ista ‘tantum A est verum’, tunc sic: illa est falsa; et illa est B; ergo B est falsum. Tunc sequitur: B est falsum; et A est verum; et non

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sunt plures propositiones quam A et B; ergo tantum A est verum, quod fuit negatum. Pro illo dicitur admittendo casum, et concedendo quod tantum A est verum, et negatur quod illa est vera: quia illa est falsa. Et eodem modo respondendum est ad omnes consimiles. Jam sequuntur regulae de exceptivis. Sit iste casus quod unus Socrates sit omnis Socrates, et dicat ille istam propositionem et nullam aliam ‘deus est’ sic significando, et Plato dicat istam et nullam aliam ‘nullus praeter Socratem dicit verum’ sic significando, et ponatur quod nullus dicat aliquid nisi Socrates vel Plato, deinde proponatur “nullus praeter Socratem dicit verum”. Si conceditur, tunc sic: illa est vera; et Plato dicit illam; ergo Plato dicit verum35 ; et Plato est alius a Socrate; ergo alius a Socrate dicit verum; ergo non nullus praeter Socratem dicit verum, quod est oppositum concessi. Si negatur ista ‘nullus praeter Socratem dicit verum’, tunc <sic>: illa est falsa; et Plato dicit illam; ergo Plato dicit falsum. Tunc arguo sic: Plato dicit falsum; et Socrates dicit verum; et nullus dicit verum nisi Socrates vel Plato; ergo nullus praeter Socratem dicit verum, quod fuit negatum. Ideo dicitur admittendo casum. Et quando proponitur ista “nullus praeter Socratem dicit verum”, concedo, et nego quod illa est vera: quia illa est falsa. Et si arguitur sic: “illa est falsa; ergo illa significat aliter quam est; et illa significat quod nullus praeter Socratem dicit verum; ergo non est ita quod nullus praeter Socratem dicit falsum”, negatur consequentia: quia ad hoc quod valeret oportet addere in minori quod illa sic significat praecise, et tunc casus foret impossibilis. Aliter arguitur sic: Socrates dicit verum; et nullus alius a Socrate dicit verum; ergo nullus praeter Socratem dicit verum; illa consequentia est bona; et antecedens est verum; ergo <et> consequens; et consequens est hoc ‘Plato dicit verum’; ergo ita est quod Plato dicit verum; ergo aliud a Socrate dicit verum, quod est oppositum concessi. Pro illo dicitur admittendo casum, et nego quod illa consequentia est bona et formalis. Et eodem modo36 respondendum est in omnibus casibus consimilibus. Item arguitur sic: haec est falsa; ergo suum contradictorium est verum, negatur consequentia, et dicitur quod repugnat casui quod suum contradictorium est. 35 verum: 36 modo:

#Plato dicit verum# add.P. #repugnantia# add.P.

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Et sicut per omnia respondendum est de universali negativa. Verbi gratia, ponatur quod non sit alia propositio nisi ista ‘nulla propositio est vera’, et quod illa significet praecise sicut termini pretendunt, tunc quaero an illa sit vera vel falsa. Si dicatur quod illa est vera, ; ergo significat praecise sicut est; et illa significat praecise quod nulla propositio est vera; ergo ita est quod nulla propositio est vera. Tunc sic: ita est quod nulla propositio est vera; et illa est aliqua propositio; ergo illa est falsa. Si dicatur quod illa est falsa, tunc sic: illa est falsa; ergo significat aliter quam est. Tunc sic: illa significat aliter quam est; et illa significat praecise quod nulla propositio est vera; ergo non est ita quod nulla propositio est vera; et aliqua propositio est; ergo aliqua propositio est vera; et nulla est propositio nisi illa; ergo illa est vera. Et sic sequitur ex casu quod eadem propositio est vera et similiter falsa; ideo casus non est admittendus. Et si ponatur quod illa significet sicut termini praetendunt, et non praecise, casus est admittendus, et respondendum est omnino similiter in proximo insolubili praecedenti. Et eodem modo respondendum est ad omnia talia. Et eodem modo potest fieri casus de insolubili respectu illius propositionis ‘hoc est falsum’, ponatur quod non sit nisi una propositio talis respectu illius ‘hoc est falsum’ seipsa demonstrata per subjectum, et ad omnia talia respondendum est per praedicta sufficienter. Et sicut dictum est ad illa insolubilia, sic dicendum est ad omnia consimilia. Expliciunt insolubilia valde utilia secundum Hetysbery.

15.8

Johannes Venator-John Hunter(?),

Insolubilia

Jam sequuntur regulae37 de insolubilibus. Et primo sciendum est quid sit casus de insolubili, et postea38 quid sit propositio insolubilis. Est sciendum quod casus de insolubili39 est talis casus in quo fit mentio de aliqua propositione quae, si cum eodem casu significet praecise sicut verba praetendunt, sequitur eamdem esse veram et eamdem esse falsam. 37 regulae

EFO: om.V. postea EFV: secundo O. 39 de insolubili FOV: insolubilis E. 38 et

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Sed propositio insolubilis est talis propositio de40 qua fit mentio in aliquo casu quae, si cum eodem casu significet praecise sicut verba praetendunt, sequitur eamdem41 esse veram et eamdem42 esse falsam. Verbi gratia, ponatur quod iste Socrates sit omnis Socrates43 , et quod Socrates dicat istam propositionem44 ‘Socrates dicit falsum’ et nullam aliam, et quod illa significet praecise quod Socrates dicit falsum, tunc illa propositio dicta a Socrate est insolubilis: quia ex casu45 sequitur quod ipsa est vera46 et similiter falsa47 . Probo48 , et quaero a te49 utrum50 illa51 sit vera vel falsa. Si dicas quod illa sit52 vera, tunc arguo sic: haec est vera; ergo53 illa praecise significat sicut est. Tunc ego arguo ex consequente54 : illa praecise significat sicut est; et illa significat praecise quod Socrates dicit falsum; ergo ita est quod Socrates dicit falsum. Tunc sic: ita est quod55 Socrates dicit falsum56 ; et Socrates nihil dicit nisi istam propositionem57 ‘Socrates dicit falsum58 ’; ergo ipsa est falsa. Et ultra: ergo ipsa non est vera; et tu dixisti quod fuit vera; ergo male. Si dicatur quod illa propositio sit59 falsa, tunc arguo sic: illa est falsa; ergo illa significat aliter quam est; et illa significat praecise60 quod Socrates dicit falsum; ergo non est ita61 quod Socrates dicit falsum. Tunc sic: non est ita quod Socrates dicit falsum62 ; et Socrates dicit aliquam propositionem; ergo Socrates dicit propositionem veram; et nullam dicit nisi istam propositionem63 ‘Socrates dicit falsum’; ergo illa64 est vera65 ; 40 de

O: in EFV. EO: eam FV. 42 eamdem EO: eam FV. 43 omnis Socrates OV: unus Socrates et omnis E: omnis homo F. 44 propositionem EFV: om.O. 45 ex casu EFV: casu stante O. 46 est vera FOV: sit falsa E. 47 et similiter falsa F: et similiter quod est falsa V: et etiam est falsa O: et quod sit vera E. 48 probo EFV: quod probo O. 49 a te FOV: om.E. 50 utrum FOV: an E. 51 illa: propositio add.E. 52 sit O: est FV dicas quod illa sit: om.E. 53 haec est vera ergo EF: om.OV. 54 ego arguo ex consequente FOV: sic E: si illa significat praecise sicut est; ergo add.F. 55 ita est quod FO: om.E. 56 Socrates dicit falsum], tunc sic ... Socrates dicit falsum] LAC V. 57 istam propositionem OV: quod E. 58 Socrates dicit falsum] et Socrates nihil ... Socrates dicit falsum] LAC F. 59 sit EO: est FV. 60 illa significat praecise FOV: Socrates dicit E. 61 non est ita FOV: aliter quam E. 62 falsum] tunc sic ... falsum] LAC E. 63 propositionem EFO: om.V. 64 illa FOV: Socrates dicit falsum E. 65 vera: et ultra add.V. 41 eamdem

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ergo non est falsa. Et sic, ex isto casu sequitur quod illa propositio dicta a Socrate est vera, et, consimiliter66 , quod illa est falsa; ideo iste casus est impossibilis. Et eodem modo potest fieri casus de insolubili respectu aliquarum67 propositionum68 : ‘Socrates non69 dicit verum’, ‘Socrates dicit aliter quam est’, ‘Socrates intelligit falsum’, ‘Socrates scribit falsum’, et sic de consimilibus70 . Unde notandum est quod insolubile est duplex71 : quoddam est simplex et aliud72 compositum. Insolubile simplex est propositio categorica, sicut ‘Socrates dicit falsum’, et sic de aliis73 . Sed insolubile compositum dividitur multis74 modis, scilicet in copulativam, disjunctivam, exceptivam et exclusivam, sicut postea patebit. Sed de insolubilibus simplicibus tres dantur regulae quarum prima est haec: si ponatur casus de insolubili75 , et cum hoc non ponatur qualiter illud insolubile76 debeat significare77 , proposito illo insolubili, respondendum est ad illud ac si nullus casus poneretur. Verbi gratia, suppono78 quod iste Socrates sit omnis Socrates, et quod ipse dicat istam propositionem et nullam aliam79 ‘Socrates dicit falsum’, tunc proponatur istam ‘Socrates dicit falsum’. Ad illam respondendum est dubitative80 , et causa est quia stat cum casu quod illa sit vera et quod illa sit falsa81 : quia stat quod Socrates dicat82 verum dicendo illam et, consimiliter, quod ipse dicat83 falsum 66 consimiliter

O: est similiter F: in primo casu V: om.E. FO: aliarum V. 68 aliquarum propositionum FOV: om.E. 69 non FOV: om.E. 70 consimilibus EF: similibus O: aliis V. 71 est duplex FOV: om.E. 72 aliud OV: quoddam EF. 73 aliis FOV: similibus E. 74 multis: aliis add.V. 75 casus de insolubili FOV: insolubilis E. 76 insolubile FOV: om.E. 77 significare EFO: om.V. 78 suppono EO: supposito V: ponatur F. 79 aliam EFV: om.O. 80 dubitative EO: dubie F: dupliciter V. 81 falsa: et quod illa sit falsa add.O: #dubia# add.E. 82 dicat FOV: dicit E. 83 dicat O: dicit E. 67 aliquarum

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dicendo illam.84 Ideo cum ipse tamen non dicat utrum dicat85 verum vel falsum, non est aliter respondendum intra casum quod extra casum.86 <Secunda regula> Secunda regula est ista: si ponatur casus de insolubili, et cum hoc ponatur quod illud insolubile87 significet praecise sicut termini88 praetendunt, nullo modo89 est casus admittendus, sicut patet in primo casu. Tertia regula est ista: si fiat90 casus de insolubili, et cum hoc ponatur quod illud insolubile significet sicut termini praetendunt, et non praecise, est admittendus casus, et concedendum est illud tamquam sequens91 quando proponitur, et92 negandum est ipsum esse.93 verum tamquam repugnans.94 Verbi gratia, pono95 quod ille96 Socrates sit omnis Socrates97 , et quod ille Socrates dicat istam propositionem ‘Socrates dicit falsum’ et nullam aliam, et quod illa significet sicut termini praetendunt, et non praecise, admittendus est casus. Et quando proponitur quod Socrates dicit falsum,98 concedendum est quod Socrates dicit falsum; et quando dicitur “haec est vera”, demonstrando illam dictam a Socrate, negatur, et conceditur99 quod est falsa. Et si arguatur100 sic: illa est falsa; ergo significat aliter quam est; et illa significat quod Socrates dicit falsum; ergo non est ita quod Socrates 84 et consimiliter quod ipse dicat falsum dicendo illam O: quia Socrates dicit Socrates dicendo illam E: om. FV. 85 cum ipse tamen non dicat utrum dicat O: cum illo casu utrum dicit F: cum ipso casu utrumque V: cum ipso casu unum E. 86 extra casum: No quod oppinio hujus in isto tractatu est quod pars supponat pro toto cujus est pars, quod clare patet inferius ex dictis suis add. in mg. O. 87 quod illud insolubile FOV: ille casus quod E. 88 termini FOV: verba E. 89 nullo modo FOV: non E. 90 fiat OV: ponatur EF. 91 sequens: et add.FV. 92 et EO: om.FV. 93 ipsum esse FV: ipsum fore O: quod sit E. 94 repugnans: #verbi gratia, si dicatur quod illa propositio est falsa, tunc arguitur sic: illa propositio est falsa; ergo significat aliter quam est, et illa significat quod Socrates dicit falsum; ergo non ita est quod Socrates dicit falsum, tunc non ita est quod Socrates dicit falsum, et Socrates dicit aliquam propositionem# add.F. 95 pono FOV: ponatur E. 96 ille FOV: unus E. 97 Socrates EOV: homo F. 98 quod Socrates dicit falsum FOV: om.E. 99 conceditur EFV: concedatur O. 100 arguatur FOV: arguo E.

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dicit falsum101 , negatur consequentia. Et causa est quia numquam valet talis consequentia nisi addatur102 in minori quod illa significat praecise103 sic vel sic, sicut in arguendo104 : illa est falsa; ergo significat aliter quam est; et illa praecise significat quod Socrates dicit falsum; ergo non est ita quod Socrates dicit falsum. Illa consequentia est bona et concedenda; sed minor est neganda in casu illo, videlicet quod illa significat praecise quod Socrates dicit falsum.105 Aliter arguitur106 sic, et probo illam propositionem esse veram quam Socrates dicit, et pono quod A sit illa. Tunc arguitur sic: A est falsum; Socrates dicit A; ergo Socrates dicit falsum. Illa consequentia est bona, et antecedens est verum; ergo et consequens; et Socrates dicit consequens; ergo Socrates dicit verum. Solutio. Negandum est quod Socrates dicit107 consequens, sed dicit unam aliam sibi consimilem in voce et non in significatione: quia illa quae dicta est a Socrate significat quod Socrates dicit falsum, et sic, quod ipsamet est falsa, cum Socrates nihil aliud dicat108 , et quod ipsa eadem est vera. Sed quaecumque alia109 consimilis quae non est dicta a Socrate significat praecise quod Socrates dicit falsum.110 Sed si ponatur quod non sit aliqua propositio talis ‘Socrates dicit falsum’ nisi111 illa quae dicta est a Socrate vel quod omnis talis112 significet eodem113 modo sicut illa dicta a Socrate114 , admittatur. Et tunc arguitur115 sicut prius sic: A est falsum; Socrates dicit A; ergo Socrates dicit falsum. Illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est illa propositio dicta a Socrate vel convert101 dicit

falsum] ergo non ... dicit falsum] LAC E. FOV: adm?tur E. 103 quod illa significet praecise EO: quod illa significat F: om.V. 104 sic vel sic, sicut in arguendo E: sic vel sic arguendo F: sic et idem sic, sicut in arguendo O: sic arguendo V. 105 in casu ... dicit falsum EFV: om.O. 106 arguitur FOV: arguo E. 107 dicit: #quod Socrates dicit# add.O. 108 et sic quod ipsamet est falsa cum Socrates nihil aliud dicat O: om.EFV. 109 alia: sit huic add.V. 110 falsum: Ex oppinione illa sequitur quod propositio dicta <supponit> pro toto cujus est pars, quod multi negant, et, ex illo, sequitur differentia inter propositionem dictam a Socrate et ab aliis: nam in aliis quaelibet talis <significat se> esse veram et Socrates dicere falsum, ita quod ly ‘falsum’ supponit pro propositione dicta a Socrate et non pro seipsa; illa autem quam profert Socrates significat Socratem dicere falsum et seipsam esse veram, sed ly ‘falsum’ <supponit> pro seipsa dicta a Socrate add. in mg.O. 111 nisi EFO: nec V. 112 talis: dicta a Socrate add.O. 113 significet eodem: alb. O. 114 a Socrate] vel quod ... a Socrate] LAC E. 115 arguitur EFV: arguo O. 102 addatur

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ibilis116 cum illa117 dicta a Socrate; ergo illa dicta a Socrate118 est vera vel119 propositio120 convertibilis cum illa121 est vera122 ; et si123 propositio124 convertibilis cum illa est vera; ergo illa est vera. Neganda125 est ista consequentia “A est falsum; Socrates dicit A; ergo Socrates dicit falsum”: quia126 consequens plus significat quam antecedens: quia consequens significat quod Socrates dicit falsum et quod ipsamet est vera, et antecedens non sic127 significat; ideo non valet consequentia talis128 . Aliter arguitur sic: A est falsum; et suum contradictorium est129 ; ergo suum contradictorium est verum; et suum contradictorium est ‘Socrates non dicit falsum’; ergo illa est vera ‘Socrates non dicit verum’130 . Tunc ego arguo131 sic: ista est vera ‘Socrates non dicit falsum’132 ; ergo illa praecise significat sicut est; et illa significat quod Socrates non dicit falsum133 ; ergo ita est quod Socrates non dicit falsum; et si ita est quod Socrates non dicit falsum, et Socrates dicit aliquid; ergo Socrates dicit verum; et134 nihil dicit nisi A; ergo A est verum. Solutio. Quando dicitur quod135 A est falsum136 , concedo137 , “igitur suum oppositum est verum”, concedo. Et quando dicitur quod138 suum oppositum est istud139 ‘Socrates non dicit falsum’, nego; sed suum op116 convertibilis

FO: propositio V: universalis E. FOV: ea E. 118 dicta a Socrate: om.E. 119 vel EO: quia V. 120 propositio OV: om.E. 121 cum illa EV: om.O. 122 est vera] quia propositio ... est vera] LAC F. 123 et si FOV: sed E. 124 propositioFOV: om.E. 125 neganda EFV: responsio neganda O. 126 quia FO: nam V: et E. 127 sic EFO: om.V. 128 talis: Potest tamen aliter responderi: quia, facta impositione illa, concedam consequentiam, ut prius concedebam ante casum. Ipsam tamen negabo esse bonam: quia antecedens est verum, consequens vero falsum ex impositione facta. Consequens enim significat ut primo significabat secunda pars copulativae, et secunda pars erat propositio se falsificans, quare et hoc consequens erit similiter add. in mg.O talis OV: illa F: om.E. 129 est: verum add.F. 130 ‘Socrates non dicit falsum’]; ergo ista est vera ‘Socrates non dicit falsum’] LAC V. 131 ego arguo O: arguitur V: om.E. 132 ‘Socrates non dicit falsum’ V: om.EO. 133 ‘Socrates non dicit falsum’]; ergo illa praecise ... Socrates non dicit falsum] LAC V ‘Socrates non dicit falsum’] Tunc ego arguo ... Socrates non dicit falsum] LAC F. 134 et: Socrates add.E. 135 quod FOV: om.E. 136 falsum EO: verum FV. 137 concedo: et quando dicitur add.FV. 138 quod: ista est add.E. 139 est istud FV: est hoc O: om.E. 117 illa

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positum est istud140 ‘non Socrates dicit falsum’, et illa significat disjunctive, videlicet quod Socrates non141 dicit falsum vel quod haec non142 est vera ‘Socrates dicit falsum’. Et haec143 est causa quare illa significat disjunctive: quia illa propositio dicta a Socrate144 significat copulative, videlicet quod Socrates dicit falsum et quod haec est vera ‘Socrates dicit falsum’; et ideo suum contradictorium significat disjunctive.145 Et sicut ille casus ponitur de illo insolubili, eodem modo potest poni de quolibet consimili, et eadem est responsio in toto.146 Verbi gratia, supposito quod unus Socrates sit omnis Socrates, sicut prius147 , et quod dicat istam propositionem ‘Socrates non dicit verum’ et nullam aliam, et quod illa significet sicut termini praetendunt, et non praecise, ille casus est admittendus per tertiam regulam. Sed si poneretur quod illa significaret praecise sicut termini praetendunt148 , negandus esset casus per secundam regulam. Et si in illo casu proponitur149 “Socrates non dicit verum”, dicitur concedendo illam, et dicitur quod haec est vera, sed, demonstrando illam dictam a Socrate, negatur quod illa est vera, et si demonstretur aliqua alia150 sibi consimilis, conceditur quod151 aliqua talis est vera. Et si dicatur sic “illa est falsa152 , demonstrata propositione dicta a Socrate; ergo ipsa significat aliter quam est; et illa significat quod Socrates non dicit verum; ergo non est ita quod Socrates non dicit verum153 ”, neganda est ista consequentia: quia ad hoc ut illa154 consequentia valeret155 , oporteret156 in minori addere quod illa significat157 praecise quod Socrates non dicit verum, et tunc neganda es<se>t minor.

140 suum

oppositum est istud] ‘Socrates ... suum oppositum est istud’] LAC V. om.E. 142 non: om.F. 143 haec EFO: hujus V. 144 Socrate: videlicet quod Socrates dicit falsum add.E. 145 disjunctive: Sed contra hoc potest argui autoritate Aristotilis primo Perihermeneias et aliorum multorum dicentorum? quod in terminis innumeratis non refert praeponere negationem et ideo, si secundum erit suum contradictorium, et primum. Huic dicendum quod illa regula tenet in propositionibus non se falsificantibus add. in mg.O. 146 Et sicut ... in toto: om.E. 147 sicut prius: om.F. 148 sicut termini praetendunt] et non praecise, ille casus est admittendus per tertiam ... sicut termini praetendunt] LAC V. 149 proponitur EV: proponatur F: poneretur O. 150 alia FOV: om.E. 151 conceditur quod corr.: concedenda est quod O: concedenda est quando E: concedenda est et quando F: conceditur et quando V. 152 illa est falsa EFO: hoc est falsum V. 153 non dicit verum] ergo ... non dicit verum] LAC EV. 154 ut illa F: ut talis O: quod illa V: quod E. 155 valeret O: valeat EFV. 156 oporteret EFO: oportet V. 157 significat EV: significaret FO. 141 non:

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Aliter arguitur sic: ita est quod Socrates non dicit verum; ergo Socrates non dicit verum, conceditur consequentia. Tunc sic: illa consequentia est bona; et antecedens est verum; ergo et consequens; et Socrates dicit consequens; ergo Socrates dicit verum, negatur quod Socrates dicit consequens, sed dicit unam aliam158 consimilem in voce et non in significatione. Et si arguitur159 de ejus contradictorio sic: haec est falsa160 ‘Socrates non dicit verum’; et suum oppositum est; ergo suum oppositum est verum, conceditur. Et quando dicitur quod suum oppositum est ista ‘Socrates dicit verum’, negatur161 , sed suum oppositum162 est una propositio significans disjunctive163 quod Socrates dicit verum vel quod haec non est vera ‘Socrates non dicit verum’. Et illa propositio sic164 significans disjunctive est vera pro altera parte, videlicet pro ista165 “haec non est vera ‘Socrates non dicit verum’ ”.166 Et sicut responsum est et argutum167 ad illa duo insolubilia, eodem modo respondendum est ad omnia168 alia169 consimilia insolubilia simplicia.170 Sequitur de insolubili composito et primo de disjunctiva. Et sciendum est quod disjunctiva insolubilis171 isto modo fit: capias unam propositionem impossibilem, et facias172 unam173 disjunctivam ex illa propositione impossibili174 et ex una175 alia propositione quae significabit176 totam illam disjunctivam fore falsam. Verbi gratia, capio istam disjunctivam177 ‘homo 158 aliam

FOV: om.E: propositionem add.V. EFO: argueretur V. 160 falsa FOV: vera E. 161 conceditur ... negatur EO: et suum oppositum est illud ‘Socrates non dicit verum’, negandum est quod suum oppositum sit illud ‘Socrates non dicit verum’ V: negandum est F. 162 suum oppositum EFV: sua opposita O. 163 disjunctive: videlicet add.FV. 164 sicFV: om.E. 165 pro ista FV: om.E. 166 Socrates non dicit verum] et illa propositio ... Socrates non dicit verum] LAC O. 167 et argutum FOV: ad argumentum et E. 168 omnia FV: om.EO. 169 alia FOV: om.E. 170 insolubilia simplicia EFV: om.O. 171 disjunctiva insolubilis FOV: ista duo insolubilia disjunctiva insolubilia E. 172 facias FOV: sumenda? E. 173 unam FOV: om.E. 174 impossibili FOV: insolubili E. 175 una EFV: om.O. 176 significabit FO: significet V: significat. 177 disjunctivam EFV: om.O. 159 arguitur

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est asinus vel ista disjunctiva est falsa’, et demonstro per subjectum178 secundae partis illam totam disjunctivam, et suppono quod illa179 disjunctiva significet180 praecise sicut termini praetendunt. Si admittatur casus181 , tunc quaero an182 illa propositio sit vera an183 falsa. Si dicas quod illa sit184 vera, tunc arguo185 sic: illa disjunctiva186 est vera; ergo altera ejus pars est vera; sed non prima,187 scilicet ista ‘homo est asinus’; ergo188 secunda ejus pars est vera.189 Tunc sic: secunda pars est vera; ergo significat praecise sicut est; et illa significat praecise quod disjunctiva est falsa; ergo ita est quod illa disjunctiva est falsa; et si sic190 ; ergo non est vera. Si dicatur quod illa191 disjunctiva sit falsa, tunc arguitur192 sic: illa disjunctiva est falsa; ergo utraque ejus pars est falsa. Et ultra193 : ergo secunda pars est falsa. Tunc sic: secunda pars est falsa; ergo significat aliter quam est; et illa significat praecise quod illa disjunctiva est falsa; ergo non est ita quod illa disjunctiva est falsa; et si non est ita quod illa194 disjunctiva est falsa; et ipsa est; ergo195 ita est quod illa disjunctiva196 est vera. Et ultra: ergo non est falsa. Et sic197 , ex isto casu sequitur quod illa disjunctiva sit vera et quod ipsa sit198 falsa, quod est impossibile; ergo casus ille non est admittendus. Sed si ponatur quod sit talis disjunctiva, et quod ipsa199 significet sicut termini praetendunt, et non praecise, admittatur casus. Et quando dicitur “aut illa disjunctiva est vera aut falsa”, dicitur quod illa est falsa. Et

178 subjectum

EFV: istam O. FV: secunda O: ista tota E. 180 significet FOV: significat E. 181 casus FO: om.EV. 182 an FOV: aut E. 183 an OV: aut E: vel F. 184 sit EFO: est V. 185 arguo EO: arguitur FV. 186 disjunctiva OV: propositio EF. 187 non prima OV: prima pars non est vera EF. 188 ergo EV: restat ergo quod O: tunc F. 189 vera: scilicet ‘ista disjunctiva est falsa’ add.E. 190 si sic: quod illa est falsa, si sic quod illa est falsa add.E. 191 illa FV: ista E: dicta O. 192 arguitur FV: ego arguo O: om.E. 193 et ultra OV: om.EF. 194 illa FV: ista E: dicta O. 195 et ipsa est EOV: om.F. 196 vera: scilicet ‘ista disjunctiva est falsa’ add.E. 197 sic: ergo patet quod add.V. 198 quod ipsa sit EFO: om.V. 199 et quod ipsa FOV: sic E. 179 illa

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tunc ad200 argumentum “ipsa est falsa; ergo utraque pars ipsius est falsa”, concedo201 . Et cum ulterius arguitur quod secunda pars est falsa202 , concedo. <Et quando arguitur> “ergo significat aliter quam est203 ”, concedo204 ; “et illa significat quod illa disjunctiva est falsa; ergo non est ita quod illa disjunctiva est falsa”, negatur consequentia: quia, ad hoc quod illa205 consequentia206 esset bona207 , oporteret208 addere in minori209 quod illa significat praecise sicut termini praetendunt, et tunc minor es<se>t neganda. Similiter potest argui sic: utraque pars illius disjunctivae est falsa; ergo illa210 disjunctiva est falsa, concedo consequentiam. Tunc sic: illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est altera pars disjunctivae; ergo altera pars disjunctivae est vera, illud negatur: quia211 consequens non est212 altera pars disjunctivae, sed consequens est una propositio sibi similis in voce et non in significatione. Et si arguitur sic “secunda pars est falsa; et suum oppositum est; ergo suum oppositum est verum”, conceditur consequentia. Et quando dicitur213 “suum oppositum est ista propositio ‘ista disjunctiva non est falsa’ ”, dicitur quod non est suum oppositum214 , sed suum oppositum est una propositio significans disjunctive, videlicet quod illa disjunctiva non est falsa vel quod haec non215 est vera ‘ista216 disjunctiva217 est falsa218 ’. Et causa219 quare suum oppositum est significans220 disjunctive est illa:

200 ad

EFV: om.O. FV: conceditur O: mot coupé dans E. 202 est falsa FOV: om.E. 203 aliter quam est EFV: om.O. 204 concedo: illa significat aliter quam est add.FV. 205 illa FV: dicta O: om.E. 206 consequentia FOV: om.E. 207 esset bona FO: foret bona V: valeat E. 208 oporteret V: oportet EFO. 209 in minori FOV: ad minorem E. 210 illa: tota add.E. 211 illud negatur quia EFO: negatur quod V. 212 non est EFO: sit V. 213 dicitur: quod add.EV. 214 dicitur quod non est suum oppositum FV: dicitur quod non E: negatur hoc O. 215 non FV: om.E. 216 est vera ‘ista EF: om.V. 217 disjunctiva: non add.E. 218 falsa: vel quod haec non est vera ‘ista disjunctiva est falsa’ add.E: vel quod haec disjunctiva est falsa add.V. 219 causa: est add.E. 220 est significans FV: illud significat E. 201 concedo

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quia secunda pars significat copulative, videlicet quod haec disjunctiva est falsa221 et quod haec est vera ‘ista disjunctiva est falsa222 ’. Similiter capiatur ista disjunctiva223 ‘homo est asinus vel nulla disjunctiva est vera’, et ponatur quod nulla disjunctiva sit nisi illa. Si224 ponatur quod illa225 significet praecise sicut termini praetendunt, negandus226 est casus: quia ex hoc sequitur quod ista disjunctiva est vera et227 falsa, sicut breviter potest probari sicut in prima disjunctiva. Sed si ponatur quod illa228 significet sicut termini praetendunt, et non praecise, admittendus est casus. Et quando quaeritur aut229 illa disjunctiva est vera aut230 falsa, dicitur quod est falsa. Et si arguitur sic231 : illa est falsa; et illa est omnis disjunctiva; ergo omnis disjunctiva est falsa, conceditur consequentia. Et si tunc arguitur232 sic: omnis disjunctiva est falsa233 ; ergo nulla disjunctiva est vera. Illa consequentia est bona; et antecedens est verum; ergo et consequens; et consequens est altera pars disjunctivae; ergo altera pars disjunctivae est vera. Et ultra: ergo tota disjunctiva est vera, pro illo negatur quod altera pars disjunctivae est consequens, sed est una alia234 propositio235 sibi similis in voce et non in significatione. Sed si fuerunt alia argumenta facta, sicut in alia236 respondendum est consimiliter237 sicut responsum est ad illa238 . Sequitur de copulativa insolubili, et fit hoc modo: capiatur propositio necessaria et fiat una239 copulativa ex ea240 et241 una242 alia propositione 221 est

significans FV: illud significat E. est falsa et quod haec est vera ‘ista disjunctiva est falsa’ LAC F. 223 disjunctiva EFV: om.O. 224 Si FO: Sed si EV. 225 illa EFV: ista disjunctiva. 226 negandus FOV: non admittendus E. 227 et: quod ipsa est add.O. 228 illa EFV: secunda disjunctiva. 229 aut EO: an FV. 230 aut EOV: an est F. 231 sic EV: quod FO. 232 tunc arguitur O: om.EFV. 233 omnis disjunctiva est falsa EF: om.OV. 234 alia EFV: om.O. 235 propositio: om.E. 236 alia: disjunctiva add.EF. 237 respondendum est consimiliter OV: consimiliter respondendum est E: om.F. 238 responsum est ad illa OV: ad illa, et eodem modo respondendum est ad omnia similia EF. 239 una FOV: om.E. 240 ex ea FOV: om.E. 241 et FOV: ex illa E. 242 una FV: cum E: om.O una copulativa ex ea corr.: ex ea una copulativa FOV. 222 disjunctiva

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quae significabit243 totam copulativam esse falsam. Verbi gratia, capiatur ista copulativa244 ‘deus est et ista copulativa est falsa’, et pono quod illa copulativa significat praecise sicut termini praetendunt, tunc quaero aut245 ista est vera aut246 falsa. Si dicatur quod illa copulativa247 est248 vera, tunc arguo249 sic: illa copulativa est vera; ergo utraque pars illius250 est vera; ergo secunda pars est vera. Tunc sic: secunda pars est vera; ergo illa significat praecise sicut est251 ; et ipsa252 praecise significat quod illa253 copulativa est falsa254 ; ergo ita est quod illa255 copulativa est falsa256 . Et ultra257 : ergo illa copulativa non est vera, quod fuit prius a te concessum.258 Si dicatur quod illa copulativa est falsa, tunc arguitur sic: illa est falsa259 ; ergo illa significat aliter quam est; et illa significat praecise quod deus est et quod ista260 copulativa est falsa; ergo non est ita quod deus est et quod ista copulativa est falsa261 ”. Tunc sic: non est ita quod deus est et quod ista copulativa est falsa; et ita est quod deus est; ergo non est ita quod ista copulativa est falsa262 ; ergo ita est quod ista copulativa est vera. Vel aliter potest argui sic: illa copulativa est falsa; ergo pro altera ejus263 parte; non264 pro prima; ergo pro secunda. Sed tunc arguitur265 sic266 : secunda pars est falsa; ergo significat aliter quam est; et illa significat praecise quod ista copulativa267 est falsa; ergo non est ita quod ista copulativa est falsa. Et ultra: ergo ista copulativa est vera.

243 quae

significabit FOV: significante E. FOV: propositio E. 245 aut EFO: an V. 246 aut: illa est add.E. 247 ista copulativa: om.E. 248 est FO: sit EV. 249 arguo EO: arguitur FV. 250 illius EF: ipsius O: ejus V. 251 illa significat praecise sicut est FV: taliter est qualiter ipsa significat O. 252 ergo secunda pars ... ipsa significat et ipsa FOV: ergo copulativa est vera. Si dicatur quod ista est vera. Tunc sic: ista pars est vera; igitur significat sicut est, et illa E. 253 illa EFV: nulla O. 254 falsa EFV: vera O. 255 illa EFV: nulla O. 256 est falsa EV: non est vera O. 257 ultra V: si sic O: om.E. 258 quod fuit prius a te concessum: om.EV et ultra ... concessum: om.F. 259 est falsa] tunc arguo sic illa est falsa] LAC F. 260 ista: tota add.E. 261 copulativa est falsa] ergo non est ita ... copulativa est falsa] LAC F. 262 est falsa] tunc sic ‘non est ita ... est falsa’] LAC E. 263 altera ejus O: altera E: secunda V. 264 non OV: sed non E. 265 arguitur EO: om.V. 266 argui sic] illa copulativa ... arguitur sic] LAC F. 267 copulativa FOV: om.E. 244 copulativa

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Ex268 illo casu sequitur quod ipsa est vera et269 quod ipsa est falsa; et ideo casus non270 est admittendus. Sed si ponatur quod sit271 talis272 copulativa, et quod illa significet sicut termini praetendunt, et non praecise, admittendus est casus. Et si quaeritur273 a te274 aut275 illa est vera aut illa est falsa276 , dicatur quod illa est falsa. Et si tunc arguitur sic: illa est falsa; ergo pro altera ejus parte; sed non pro prima; ergo pro secunda, concedo: quia277 pro secunda parte.278 Tunc arguo279 sic: secunda pars280 est falsa; ergo illa significat aliter quam est; et illa significat quod ista copulativa est falsa; ergo non est ita quod ista copulativa est falsa, negatur consequentia: quia ad hoc quod consequentia281 foret282 bona, oporteret addere in minori quod illa significat283 praecise quod ista copulativa est falsa, et tunc minor es<se>t neganda: quia284 falsa.285 Et286 arguo sic: altera pars illius copulativae est falsa; ergo illa copulativa est falsa, conceditur287 consequentia.288 Tunc289 sic: illa consequentia est bona; et antecedens est verum, ergo et consequens; et secunda pars illius copulativae est consequens; ergo secunda pars illius copulativae est290 vera, huic dicitur negando quod consequens est291 secunda292 pars dictae293 copulativae, sed est una propositio sibi similis in voce et non in significatione.

268 Ex

OV: Et sic ex EF. sequitur add.V. 270 non FOV: om.E. 271 quod sit FOV: om.E. 272 talis: disjunctiva add.V. 273 quaeritur FOV: arguitur E. 274 a te O: om.EFV. 275 aut OV: an F: om.E. 276 falsa: igitur significat add.E aut illa est falsa EOV: an falsa F. 277 quia OV: om.EF. 278 parte: et si add.O. 279 arguo FV: arguitur O: om.E. 280 secunda pars FOV: illa E. 281 consequentia EFV: illa copulativa O. 282 foret: donec esse add.E. 283 significat EFO: significaret V. 284 neganda quia: om.E. 285 falsa] et tunc ... falsa] LAC O. 286 et EFV: sed ulterius O. 287 conceditur EFV: concedo O. 288 consequentia EFV: consequentiam O. 289 tunc: arguitur add.E. 290 illius copulativae est] consequens ... illius copulativae est] LAC EF. 291 est EFO: sit V. 292 secunda EFO: altera V. 293 dictae OV: om.EF. 269 et:

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Et si arguatur de contradictorio secundae partis, dicitur294 similiter sicut295 prius: quia ejus contradictorium est una propositio significans disjunctive: quia296 secunda pars hujus significat copulative, videlicet quod297 illa copulativa est falsa et quod haec298 est vera ‘ista299 copulativa est falsa’; ergo300 suum oppositum significat opposito modo. Et eodem modo potest fieri casus insolubilis de ista copulativa301 ‘deus est et nulla302 copulativa303 est vera’. Pono quod nulla talis sit nisi illa, et quod ipsa significet praecise sicut termini praetendunt, casus non est admittendus; sed, si ponatur quod ipsa304 significet sicut termini praetendunt, et non praecise305 , casus est admittendus. Et ad argumenta306 respondendum est similiter sicut in prima copulativa, et similiter ad omnia consimilia.307 Sequitur de exclusivis.308 Suppono309 quod Socrates dicat istam et nullam aliam ‘deus est’, et Plato dicat istam ‘tantum Socrates dicit verum’, et pono quod utraque istarum310 significet praecise311 sicut termini praetendunt, et quod non sint plures homines loquentes quam312 Socrates et Plato, tunc propono tibi istam “tantum Socrates dicit verum”. Si conceditur313 , quaero a te314 aut315 illa est vera aut illa est316 falsa. Si dicatur quod illa sit317 vera, tunc sic: illa est vera; ergo significat praecise sicut est; et illa significat praecise quod tantum Socrates dicit verum; ergo ita est quod tantum Socrates dicit verum318 ; et si ita est 294 dicitur

EO: dicatur FV. EFV: ut O. 296 quia EFV: illud quod O. 297 quod EFO: om.V. 298 haec EFV: illa O. 299 ista EFV: et haec O. 300 ergo OV: ideo E: huic F. 301 copulativa EFV: propositione O. 302 nulla FOV: ista E. 303 copulativa: non add.E. 304 ipsa OV: ista propositio non tamen praecise EF. 305 et non praecise OV: om.EF. 306 ad argumenta FO: om.EV. 307 respondendum ... ad omnia similia EFV: dicendum est ut in praecedenti O. 308 exclusivis FO: exclusiva insolubili EV. 309 suppono EFV: supposito O. 310 istarum FOV: om.E. 311 praecise FOV: om.E. 312 quam EFV: nisi O. 313 conceditur EFV: concedatur O. 314 a te FV: om.EO. 315 aut EOV: an F. 316 aut illa est EOV: om.F. 317 sit EFO: est V. 318 Socrates dicit verum FOV: om.E. 295 sicut

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quod tantum Socrates dicit verum319 ; Plato est alius320 a Socrate; ergo Plato non dicit verum. Et tunc ultra321 : ‘Plato322 non dicit verum; et323 dicit324 aliquam propositionem; ergo Plato325 dicit falsum; et Plato nihil aliud326 dicit327 nisi328 istam ‘tantum Socrates dicit verum’; igitur ista est falsa ‘tantum Socrates dicit verum329 ’; et per te est vera; ergo propositio falsa330 est propositio vera331 . Si dicatur quod illa est falsa, tunc arguo332 sic: illa est falsa; ergo illa significat aliter quam est; et illa significat praecise quod tantum Socrates dicit verum; ergo non est ita quod tantum Socrates dicit verum; et si non est ita quod tantum Socrates dicit verum333 ; ergo est ita quod alius a Socrate dicit verum; et nullus alius a Socrate quam Plato loquitur; ergo Plato dicit verum; et Plato nihil dicit nisi istam propositionem334 ‘tantum Socrates dicit verum’; ergo illa est vera. Et ultra: ergo illa non est falsa; ideo335 casus est impossibilis et non admittendus. Sed si ponatur talis casus dictus336 , cum hoc337 quod illa propositio dicta a Platone significet quod tantum Socrates dicit verum, et non praecise338 , conceditur.339 Et quando dicitur quod340 haec est vera, demonstrando illam341 dictam a Platone, negatur quod illa est vera.342 Et quando dicitur343 “haec est falsa”, conceditur344 Et quando arguitur sic: “illa est falsa; ergo significat aliter quam est; et345 illa significat quod 319 tantum

Socrates dicit verum] et si ita quod tantum Socrates dicit verum] LAC V. EO: aliud V. 321 ultra O: sic si V: om.E. 322 Plato] est alius ... Plato] LAC F. 323 non dicit verum et FOV: om.E. 324 dicit EO: Plato dicit FV. 325 Plato EFV: om.O. 326 aliud EFO: om. 327 dicit: om.E. 328 nisi EFV: quam O. 329 Socrates dicit verum] igitur ista est falsa ‘tantum Socrates dicit verum’] LAC OV. 330 falsa EO: vera FV. 331 vera EO: falsa FV. 332 arguo EFO: om.V. 333 verum] et si non ... verum] LAC V. 334 propositionem FOV: om.E. 335 ideo FOV: igitur E. 336 talis casus dictus OV: talis casus F: om.E. 337 cum hoc EFV: et cum hoc ponatur O. 338 et non praecise FO: om.EV. 339 conceditur EV: concedatur O: admittendus est casus et quando ponitur quod tantum Socrates dicitur verum, conceditur F. 340 quod FOV: om.E. 341 illam EFV: jam O. 342 quod illa est vera: om.V. 343 dicitur: quod add.E. 344 conceditur EFV: concedatur O. 345 et: #quando# add.O. 320 alius

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F. Pironet

tantum Socrates dicit verum; ergo non est ita quod tantum Socrates dicit verum”, negatur consequentia: quia ad hoc quod dicta consequentia valeret346 , oporteret addere in minori quod illa significat347 praecise quod tantum Socrates dicit verum, et tunc minor es<se>t neganda. Et si ulterius348 arguitur sic: “Plato dicit falsum; et Socrates dicit verum; et nullus alius349 ab istis loquitur350 ; ergo tantum Socrates dicit verum”, conceditur351 consequentia. Et quando dicitur “illa consequentia est bona; et antecedens est verum; ergo et consequens352 ; et Plato dicit consequens; ergo Plato dicit verum”, negatur353 quod Plato dicat consequens, sed bene354 dicit unam aliam355 propositionem sibi similem in voce et non in significatione.356 Sed si ponatur quod non sit nisi una talis propositio357 ‘Socrates dicit verum’, et fiat haec358 consequentia: Socrates dicit verum; et nullus alius a Socrate dicit verum359 ; ergo tantum Socrates dicit verum, negatur consequentia: quia exclusiva plus significat quam suae exponentes360 : quia illa copulative361 significat quod tantum Socrates dicit verum et quod haec est vera ‘tantum Socrates dicit verum’, et suum362 oppositum363 significat disjunctive, videlicet quod <non> tantum Socrates dicit verum364 vel quod haec non est vera ‘tantum Socrates dicit verum.365 ’ Sequitur de exceptivis.366 Verbi gratia, supposito367 quod Socrates tantum368 dicat istam propositionem369 ‘deus est’ et nullam aliam, et quod 346 ad

hoc quod consequentia valeret O: om.EFV. V: significaret O: significet EF. 348 si ulterius O: quando V: om.EF. 349 nullus alius EFO: nihil aliud V. 350 ab istis loquitur O: a Socrate dicit verum EFV. 351 conceditur FV: concedatur O. 352 consequens: est verum add.F: et consequens est verum add.E. 353 negatur: hoc add.O. 354 bene OV: om.EF. 355 unam aliam EFV: om.O. 356 significatione: vel significato add.O. 357 talis propositio FOV: propositio talis tantum E. 358 haec O: talis V: una talis E. 359 verum] et nullus ... verum] LAC V verum] et fiat haec ... verum] LAC F. 360 exponentes: et fiat quartum argumentum de contradictorio: dicendum est negando quod illud sit suum contradictorium add.O. 361 copulative O: copulativa EFV. 362 et suum FV: ejus autem O. 363 oppositum: hoc add.O. 364 tantum Socrates dicit verum] ejus autem ... tantum Socrates dicit verum] LAC E. 365 videlicet quod... verum’ FV: om.O verum: ergo et cetera add.V. 366 exceptivis FO: exceptiva insolubili EV. 367 supposito EO: pono V: om.F. 368 Socrates tantum O: tantum Socrates EFV. 369 propositionem FOV: om.E. 347 significat

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Plato dicat istam ‘nullus homo praeter Socratem dicit verum’ et nullam aliam, et quod non sint plures homines loquentes quam Socrates et Plato. Si,370 cum dicto371 casu, ponatur quod dicta propositio dicta a Platone372 significet373 praecise sicut termini praetendunt, casus non est admittendus: quia ex casu sequitur quod374 propositio dicta a Platone est vera et375 similiter376 falsa; ideo ille casus est impossibilis. Et si377 ponatur quod378 significet379 sicut termini praetendunt, et non praecise380 , casus est admittendus. Et quando proponitur “nullus homo praeter Socratem dicit verum”, concedo. Et quando dicitur “haec est vera”, negatur, et conceditur381 quod haec est falsa. Et si arguitur sic: “haec est falsa; ergo significat aliter quam est; et illa significat quod nullus homo praeter Socratem dicit verum; ergo non est ita quod nullus homo praeter Socratem dicit verum”, negatur consequentia propter causam praedictam, videlicet quod ad hoc quod valeat382 , oportet383 addere in minori ly384 ‘praecise’, sed tunc minor est falsa. Et si fiant alia385 argumenta, sicut de exclusiva386 , respondendum est sicut ibi387 , et eodem modo388 arguendum389 est390 ad omnem391 casum de insolubili392 de exclusiva et de exceptiva. Aliter si fiat casus de insolubili, et hoc393 de universali affirmativa. Verbi gratia394 , supposito quod non sit aliqua propositio nisi ista ‘omnis propo370 Si

O: Similiter EFV. OV: isto EF. 372 Platone FV: Socrate EO. 373 significet EFV: ponatur O. 374 quod: illa add.EV. 375 et EFV: om.O. 376 similiter: quod est add.O. 377 si: non add.EFV. 378 quod: illa propositio add.E. 379 significet: ita praecise add.V: praecise add.F. 380 et non praecise EO: om.FV. 381 conceditur EFV: concedatur O. 382 ad hoc quod valeat O: om.EFV. 383 oportet EFO: oporteret V. 384 ly FV: om.EO. 385 alia EFO: om.V. 386 exclusiva: eodem modo add.EF. 387 sicut ibi O: sicut responsum est E: om.FV. 388 eodem modo: om.F. 389 arguendum EFO: responsum V. 390 est: alibi et eodem modo add.V. 391 omnem FV: consimilem O: dictum E. 392 de insolubili FOV: est alibi E. 393 Aliter .. et hoc EO: sequitur V. 394 Verbi gratia FOV: om.E. 371 dicto

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sitio est falsa’, et quod illa significet praecise sicut termini praetendunt, tunc quaero an395 illa sit vera an396 falsa. Si dicatur quod illa est vera; ergo significat praecise sicut est; et illa significat praecise quod omnis propositio est falsa397 ; ergo ita est quod omnis propositio est falsa; et si ita est quod omnis propositio est falsa398 ; et illa est aliqua propositio; ergo illa399 est falsa400 ; ergo illa non est vera. Si autem401 dicatur quod illa est falsa; ergo illa significat aliter quam est; et illa significat praecise402 quod omnis propositio est falsa; ergo non est ita quod omnis propositio est falsa; ergo ita est quod aliqua propositio est vera; et nulla est propositio nisi illa; ergo illa est vera. Et ultra: ergo illa non est falsa. Et sic ex casu403 sequitur quod aliqua propositio404 est vera et similiter405 falsa406 ; ideo casus non est admittendus. Sed si ponatur quod non sit aliqua propositio nisi illa, et quod illa significet sicut termini praetendunt, et non407 praecise, admittendus est casus. Et quando408 quaeritur an409 illa sit vera an410 falsa, dicitur quod illa sit falsa. Tunc ad argumentum “illa est falsa; ergo significat aliter quam est; et illa significat quod omnis propositio est falsa; ergo non est ita quod omnis propositio est falsa”, negatur consequentia: quia oportet addere in minori quod ipsa significat411 praecise quod omnis propositio est falsa412 ; et tunc neganda est minor. Aliter arguitur sic: illa propositio est falsa; et illa propositio est omnis propositio413 ; ergo omnis propositio est falsa, conceditur414 consequentia. Tunc sic: illa consequentia est bona; et antecedens est verum; ergo et consequens, huic dicitur negando illam consequentiam esse415 bonam416

395 an

FOV: aut E. FO: vel V: aut illa sit E. 397 falsa: et si ita est quod omnis propositio est falsa add.V. 398 omnis propositio est falsa] et si ita ... omnis propositio est falsa] LAC E. 399 illa: propositio add.E. 400 falsa: et ultra add.EF. 401 autem FOV: om.E. 402 praecise OV: om.EF. 403 ex casu FOV: om.E. 404 aliqua propositio FV: ipsa O: illa E. 405 similiter EFV: quod ipsa est O. 406 falsa: ex isto casu add.E. 407 et non OV: non tamen EF. 408 quando EFO: si V. 409 an FOV: aut E. 410 an FO: aut illa sit E: vel V. 411 significat corr.: significet FOV. 412 omnis propositio est falsa] negatur ... omnis propositio est falsa] LAC E. 413 propositio: est falsa add.V. 414 conceditur EFV: concedatur O. 415 esse FO: fore V. 416 negando ... bonam FOV: negando quod illa consequentia sit bona E. 396 an

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et similiter antecedens417 et consequens esse verum418 : quia repugnat419 quod aliqua consequentia sit420 : quia ponitur in casu421 quod non sit nisi una propositio, et si422 non sit423 nisi una propositio, non est424 aliqua consequentia: quia si consequentia425 est426 ; antecedens et consequens sunt, et si consequentia est, plures propositiones sunt427 , quod repugnat casui428 ; ideo repugnat429 casui quod aliqua consequentia sit. Item, si arguitur sic: “ista est falsa; ergo suum oppositum est verum”, negatur consequentia: quia430 repugnat casui quod habeat contradictorium. Consimiliter431 per omnia respondendum est de universali negativa. Verbi gratia, ponatur quod non sit aliqua propositio nisi ista ‘nulla propositio est vera’, et quod ipsa significet praecise432 sicut termini praetendunt, tunc quaero an433 illa sit434 vera an435 falsa. Si dicatur quod illa est436 vera, tunc arguo437 sic: illa est vera; ergo438 significat praecise sicut est; et ipsa significat praecise439 quod nulla propositio est vera; ergo ita est quod nulla propositio est vera440 ; et si ita est quod nulla propositio est vera; et illa est441 aliqua propositio; ergo442 illa non est vera443 ; ergo ipsa est falsa. 417 antecedens:

est verum add.EF. verum OV: om.EF. 419 repugnat: casui add.E. 420 sit: #bona# add.E. 421 in casu FOV: om.E. 422 si: sic add.E. 423 sit FOV: est E. 424 non est] nisi una ... non est] LAC E. 425 consequentia EO: antecedens FV. 426 est FOV: igitur E. 427 plures propositiones sunt FOV: om.E. 428 et si consequentia ... repugnat casui EFO: om.V casui EF: nobis O. 429 repugnat OV: est repugnans EF. 430 quia EFO: et dicitur quod V. 431 consimiliter: est arguendum add.E. 432 praecise FOV: om.E. 433 an FOV: aut E. 434 sit EFV: est O. 435 an FO: vel V: aut E. 436 est FO: sit EV. 437 arguo O: arguitur F: om.EV. 438 est vera; ergo FOV: om.E. 439 praecise FOV: om.E. 440 vera] ergo ita ... vera] LAC FV. 441 ita est] quod nulla ... illa est] LAC E. 442 ergo FOV: om.E. 443 vera: et ultra add.V. 418 esse

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Et si dicatur quod illa est444 falsa445 , tunc sic: illa est falsa; ergo illa significat aliter quam est; et illa significat praecise quod nulla propositio est vera; ergo non est ita quod nulla propositio est vera; et si sic; ergo est ita quod aliqua propositio est vera; et nulla propositio446 est nisi illa; ergo illa447 est vera. Et sic ex casu posito448 sequitur quod eadem propositio est vera et falsa449 ; ideo casus450 non est admittendus. Sed si ponatur quod illa significet sicut termini praetendunt, et non praecise, admittendus est casus, et respondendum est omnino451 sicut452 in proximo praecedenti453 insolubili,454 et eodem modo455 ad omnia talia.456 Eodem modo potest fieri casus de insolubili457 respectu istius propositionis ‘hoc est falsum’, ponendo quod non sit nisi una talis propositio458 ‘hoc est falsum459 ’, seipsa460 demonstrata per subjectum. Et ad omnia talia461 respondendum est sufficienter per praedicta.462 Aliter fit casus de insolubili sic: et suppono istam suppositionem sic quod463 omnis dicens verum habebit denarium464 , et pono465 quod466 unus Socrates sit omnis Socrates, et quod ipse dicat istam propositionem et nullam aliam ‘Socrates non habebit denarium’, et quod illa quam 444 est

FO: sit E. et si dicatur ... falsa] LAC. 446 propositio FOV: om.E. 447 illa: propositio add.E. 448 posito FV: proposito E: om.O. 449 sequitur quod eadem propositio est vera et falsa FV: sequitur quod illa propositio est vera E: sequitur ipsam fore veram et ipsam fore falsam O. 450 ideo casus EFV: quare O. 451 omnino FOV: per omnia E. 452 sicut: responsum est add.FV. 453 praecedenti OV: om.EF. 454 insolubili EFV: sophismate O. 455 eodem modo FOV: similiter E: respondendum est add.FV. 456 talia EFV: alia consimilia O. 457 casus de insolubili EFV: insolubile O. 458 propositio: respectu istius add.OV. 459 hoc est falsum] ponendo ... hoc est falsum] LAC E. 460 seipsa EO: ipsamet FV. 461 omnia talia EFV: omnes tales propositiones O. 462 per praedicta EO: per dicta V: om.F. 463 sic quod OV: om.EF. 464 denarium EFO: bagatium? V (everywhere in this insoluble). 465 pono FOV: om.E. 466 quod EFO: om.V. 445 falsa]

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Socrates dicit467 significet praecise quod Socrates non habebit denarium.468 Isto posito,469 quaero an470 illa propositio471 sit vera an falsa.472 Si dicatur quod sit vera, tunc arguo sic: si473 illa est474 vera, tunc475 ipsa significat praecise sicut est; et illa significat praecise quod Socrates non habebit denarium; ergo ita est quod Socrates non habebit denarium; et si ita est quod Socrates non habebit denarium,476 Socrates dicit falsum; et Socrates nihil dicit nisi istam ‘Socrates non habebit denarium’; ergo ipsa est falsa. Si dicatur quod illa est falsa, tunc arguitur sic: ipsa est falsa; ergo significat aliter quam est; et illa significat praecise quod Socrates non habebit denarium; ergo non est ita quod Socrates non habebit denarium; sed477 si non est ita quod Socrates non habebit denarium, sequitur quod478 Socrates dicit verum; et nihil dicit nisi istam propositionem ‘Socrates non habebit denarium’; ergo illa est vera. Et ultra: ergo ipsa non est falsa. Et sic sequitur ex casu quod illa propositio dicta a Socrate est vera et falsa; ideo casus non est admittendus. Sed si ponatur ille casus cum hoc quod propositio dicta a Socrate479 non significet praecise480 sicut termini praetendunt481 , est admittendus casus. Et si quaeratur aut482 illa est483 vera aut484 falsa,485 dicatur486 quod illa est falsa. Et tunc ad argumentum487 “illa est falsa; ergo illa significat aliter quam est; et illa significat quod Socrates non habebit denarium488 ; ergo non est ita quod Socrates non habebit denarium,489 ” 467 quam

Socrates dicit EV: prolata a Socrate O. non habebit denarium] et quod illa quam Socrates dicit significet praecise quod Socrates non habebit denarium] LAC F. 469 isto posito FOV: tunc E. 470 an FOV: aut E. 471 propositio FV: propositio Socratis E: prolata a Socrate O. 472 an falsa F: vel falsa V: aut falsa E: an non O. 473 si FOV: om.E. 474 est FV: sit EO. 475 tunc FO: ergo EV. 476 et si ita est quod Socrates non habebit denarium EF: et si ita est quod Socrates non habebit? ogatinum; ergo V: ergo per casum O. 477 sed F: et E: om.OV. 478 sequitur quod EFO: ergo V. 479 a Socrate FV: om.O. 480 non significet praecise FV: significet O. 481 praetendunt: sed non praecise add.O. 482 aut FV: an O. 483 est FV: sit O. 484 aut FV: an O. 485 falsa: si add.V. 486 dicatur OV: dicendum est F. 487 ad argumentum FO: arguitur V. 488 non habebit denarium] ergo illa est vera et ultra ... non habebit denarium] LAC E. 489 denarium EO: om.V denarium] ergo non est ita quod ... denarium] LAC F. 468 Socrates

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negatur consequentia: quia oportet priu490 addere in minori491 quod illa significat praecise quod Socrates non habebit denarium,492 et493 tunc minor est falsa.494 Et si495 arguitur sic: “nullus dicens falsum habebit denarium; Socrates dicit falsum; ergo Socrates non habebit denarium”, conceditur496 consequentia. Et si arguitur497 “illa consequentia est bona; et498 antecedens est verum; ergo et consequens; sed Socrates dicit consequens499 ; ergo Socrates dicit verum”, huic dicitur negando quod Socrates dicat500 consequens, sed dicit unam501 sibi consimilem in voce et non in significatione502 : quia illa dicta a Socrate significat copulative, videlicet quod Socrates non habebit denarium et quod haec est vera ‘Socrates non habebit denarium,503 ’ et suum contradictorium significat disjunctive,504 opposito modo,505 quod Socrates [non] habebit denarium vel quod haec non506 est vera ‘Socrates non habebit denarium’. Et eodem modo respondendum est507 posito quod quilibet dicens verum pertransibit pontem508 , et quod unus Socrates sit omnis Socrates, et quod Socrates dicat istam propositionem509 ‘Socrates non pertransibit510 pontem’ et nullam aliam. Tunc, illo casu posito,511 respondendum est sicut in proximo512 insolubili. Et sic sit dictum de insolubilibus quantum spectat ad praesens.513 Amen. 490 prius

F: plus O: om.EV. F: plus O: om.EV. 492 denarium EFO: ?ogatinum aut denarium V. 493 et EF: sed O: om.V. 494 falsa FO: neganda EV. 495 Et si EO: Aliter V: sed F. 496 conceditur EFV: concedatur O. 497 et si arguitur O: sed quando arguitur F: tunc sic EV. 498 illa consequentia est bona et EV: om.FO. 499 Socrates dicit consequens EF: consequens dicit Socrates V: consequens dicitur a Socrate O. 500 dicat EFV: dicit O. 501 unam: aliam add.E: aliam propositionem add.F. 502 significatione EFV: significato O. 503 quia illa dicta a Socrate ... denarium EOV: om.F. 504 disjunctive OV: scilicet E: om.F. 505 modo: disjunctive, videlicet add.EF. 506 non EFV: om.O. 507 est: ad illud insolubile add.V. 508 pontem: et nullus alius pertransibit pontem add.E. 509 propositionem FOV: om.E. 510 pertransibit EFV: transibit O. 511 posito: eodem modo add.EF. 512 proximo EO: praecedenti FV. 513 quantum spectat ad praesens: om.FV. 491 prius

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Expliciunt insolubilia more Exonie514 compilata edita ab eximio doctore magistro Johanne Venatore.515 Casus Insolubilium Suprascriptorum.516 Primum: ‘Socrates dicit falsum’, posito quod Socrates sit omnis Socrates et dicat istam et nullam aliam, et cetera. Secundum: ‘Socrates non dicit verum’, eodem casu posito. Tertium: ‘homo est asinus vel ista disjunctiva est falsa’, istam demonstrando. Quartum: ‘homo est asinus vel nulla disjunctiva est vera’, cum eodem casu. Quintum: ‘Deus est et ista copulativa est falsa’, ipsa met demonstrata. Sextum: ‘Deus est et nulla copulativa est vera’. Septimum: ‘Deus est et tantum Socrates dicit verum’, posito quod Socrates dicat primam, et Plato secundam, et nullus ipsorum aliud dicat. Octavum: ‘Deus est et nullus homo praeter Socratem dicit verum’, primo posito. Nonum: ‘omnis propositio est falsa’, posito quod nulla alia sit in mundo. Decimum: ‘nulla propositio est vera’, posito quod haec sola sit in mundo. Undecimum: ‘hoc est falsum’, ipsa demonstrata et nulla alia sit in mundo. Duodecimum: ‘Socrates non habebit denarium’, posito primo casu, et quod omnis dicens verum habebit denarium et nullus alius. Tertium decimum: ‘Socrates non transibit pontem’, posito quod Socrates dicat istam et nullam aliam, et quod omnis dicens verum pertransibit pontem, et dicens falsum non. 514 Exonie:

Oxoniae? ... Venatore O: Explicit brevis tractatus de insolubilibus secundum usum Heusonie V: Et sic de insolubilibus Hesbri et dicta sufficienter. E: om.F. 516 Only in manuscript O. 515 Expliciunt

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Expliciunt casus insolubilium suprascriptorum. Anonymus, De insolubilibus Oxford, Bodl. Libr., Lat. misc. e.79, ff. 43ra–45vb. Insolubile est oratio quae, in casu posito secum, dum solum significat juxta communem institutionem idiomatis in quo est disputatio, simul verificat et falsificat ipsammet517 vel cujus taliter imposuit ad518 significandum. Pro quo notandum est quod omne insolubile aut provenit ex actu hominis, qualis est intelligere, scire, credere, dubitare, et sic de aliis quae concernunt actum animae interiorem; aut ex actu exteriori, qualis est dicere, videre, audire, et sic de aliis actibus sensitivis519 ; aut ex proprietate vocis, ut hic ‘intelligo falsum’, ‘dico falsum’, ‘audio falsum’, ‘video falsum’, et sic de aliis. Primo quaedam est opinio520 probabilis, licet forte non in toto vera, quae evadit omnia insolubilia per hoc quod dicit “nullum actum potest521 reflecti super se ipsum”, sicut hic ‘intelligo impossibile’, ‘audio falsum’, semper actus animae respicit ad aliud quam522 ad illud idem numero quod tunc dicitur vel cogitatur. Consequenter etiam potest quod numquam pars potest supponere pro toto cujus est pars nec pro ejus contrario vel subcontrario vel contradictorio vel subalterno vel convertibili cum aliquo illorum, sicut in istis ‘hoc est falsum’, seipso demonstrato, ‘omnis propositio est falsa’, et sic de aliis, subjectum talis propositionis non supponit pro ipsa propositione cujus est pars.523 Sed quia tam multae auctoritates quam fortes responsiones sunt contra istam opinionem524 ; ideo probabiliter525 dici potest quod quamvis sit extortus et inconsuetus modus loquendi sive intelligendi in propositionibus ubi actus reflectitur super seipsum vel ubi pars supponit pro 517 ipsammet

corr.: semet ipsam O: secundum ipsam De Rijk. imposuit ad O2: #tale hypotheticum ponitur taliter# O1. 519 sensitivis: sensuum De Rijk. 520 opinio corr.: oppositio O. 521 potest: in mg.O2. 522 quam corr.: quin O. 523 pars: cujus est pars add.O. 524 opinionem corr.: oppinionem O. 525 probabiliter corr.: probabilis O. 518 taliter

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toto cujus est pars, tamen hoc est possibile ubi illud quod supponit pro suo toto est terminus communis sive universalis, ut hic ‘omnis propositio est falsa’, sed numquam ubi talis terminus est singularis, sicut ‘hoc est falsum’, eodem demonstrato, ut hic ‘tu es asinus vel haec disjunctiva est falsa’, vel ‘haec consequentia est bona; ergo tu es asinus’, eadem consequentia demonstrata, et sic de aliis. Unde per responsionem ad insolubilia primo suppono quod quaelibet propositio526 in latina lingua527 disputationis praecise significet secundum communem institutionem grammaticae nisi528 in aliqua nova impositione529 conveniant opponens et respondens. Haec suppositio patet eo quod in omni secta et in omni alia facultate disputabili praesupponi grammatica exprimens significatum vocum530 illius idiomatis in quo fit disputatio. Aliter enim talis propositio ‘deus est’ vel ‘aliquid est’ vel quaecumque alia vera proposita semper foret disputanda: quia nesciretur quo modo significat, quod est inconveniens et absurdum. Confirmatur sic: aliquando eadem propositio diversimode significat in diversis idiomatibus; ergo vel oportet respondentem quemlibet scire omnes linguas vel dubitare quamlibet propositionem sibi proposita<m> vel solum intelligere propositum juxta communem institutionem idiomatis in quo disputat ubi non fit sibi nova impositio termini. Sed duae primae partes communiter sunt falsae, ut patet; ideo tertia pars est vera, quae convertitur cum prima suppositione. <Secunda suppositio> Secundo suppono hic descriptiones propositionis verae et propositionis falsae prius positas in secunda diffinitione.531 Tertia suppositio sequitur ex his, videlicet talis: quaelibet talis est impossibilis ‘propositio vera est propositio falsa’. 526 propositio:

#latina# add.O. in mg.O2. 528 nisi corr.: ubi O. 529 impositione corr.: propositione O. 530 vocum corr.: vocalem O. 531 This definition of true and false propositions lacks in the text; it has been added at the top margin of fol. 42b by the second scribe (O2). 527 lingua:

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Ex his tamen suppositionibus et descriptione insolubilium prius data sequuntur tres conclusiones quarum prima est ista: quilibet insolubilis casus est negandus. Haec conclusio probatur sic: ex quolibet insolubili, juxta descriptionem ejus, sequitur eamdem propositionem esse veram quae est falsa; sed hoc est impossibile, sicut patet per tertiam suppositionem; ergo quilibet alius casus talis est impossibilis et negandus. <Secunda conclusio> Secunda conclusio est haec: numquam terminus singularis supponit pro seipso vel pro tota propositione532 cujus est pars. Haec probatur sic: unusquisque experiri potest in se quod in tali propositione ‘hoc est intellectus’ numquam intelligit illud idem ad quod533 sic dicit vel cogitat singulariter, sed aliquid aliud. Nam tunc idem singulariter et distincte significaret seipsum, et, per consequens, cum intellectu534 prius naturaliter intelligit significans quam significatum, prius naturaliter intelligetur ‘hoc est’ distincte quam intelligeretur ‘hoc est’, quod est impossibile. Confirmatur sic: subjectum talis propositionis tantum et tam distincte significat sicut ipsa tota; sed prius naturaliter intelligitur subjectum; ergo totum prius intelligitur quam ipsum totum, distincte semper: quia non est inconveniens quod mediato termino communi aliquid prius intelligatur universaliter sive confuse quam distincte: nam immediate subjecto hujus propositionis ‘omnis propositio est’ intelligitur eadem propositio confuse, et postea mediate in tali propositione ‘haec est demonstrata’ illa universali<s> intelligitur distincte. Item, si possibile sit subjectum hujus ‘hoc est falsum’ supponere pro eadem propositione, petatur535 ergo536 contra: vel nunc est vera vel falsa. Si vera, ergo ista juxta totalem significationem suam complexam praecise significat sicut est, ut patet per descriptionem propositionis verae; et, per suppositionem primam, illa juxta totalem significationem suam complexam praecise significat hoc esse falsum; ergo sic est quod hoc est falsum, et, per consequens, idem est verum et537 falsum. Si dicitur quod ista propositio sit falsa, tunc si haec est falsa, et haec juxta totalem significationem suam complexam praecise significat sicut

532 propositione

O2: suppositione O1. in mg.O2. 534 intellectu O2: intellectum O1. 535 petatur O2: petiatur? O1. 536 ergo: in mg. O2. 537 et corr.: vel O. 533 quod:

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est; ergo haec est vera, et, per consequens, simul vera et falsa, quod est impossibile. Tertia conclusio est haec: numquam est positio vel propositio admittenda sine nova impositione ex qua, praecise significante sicut grammatice verba praetendunt, sequeretur contradictio. Haec conclusio patet per primam suppositionem: nam quaelibet talis propositio deducta nova impositione praecise significat sicut ejus verba grammatice praetendunt; ergo si tunc ex ea sequitur contradictio, ista est impossibilis nec admittenda. Verbi gratia, ponatur quod haec propositio ‘falsum est’ sit quaelibet propositio, tunc quaero numquid haec sit vera vel falsa. Si sit vera, ergo juxta totalem significationem suam complexam ista praecise significat sicut est; sed ipsa praecise significat quod falsum est; ergo sic est quod falsum est; et nullum falsum est nisi haec propositio ‘falsum est’; ergo haec est falsa ‘falsum est’, et, per consequens, cum haec praecise significat quod falsum est, sequitur quod haec est vera. Similiter, ponatur quod Socrates dicat istam ‘ego dico falsum’ et nullam aliam, vel, sub aliis verbis, quod idem sonat, dicat538 hanc ‘Socrates dicit falsum’, et sit unus Socrates omnis Socrates. Tunc aliter sic: si Socrates dicit verum, et solum dicit illam; ergo haec est vera; ergo haec juxta totalem significationem <suam> praecise significat sicut est; sed totalis significatio sua est quod Socrates dicit falsum; ergo sic est quod Socrates dicit falsum; et Socrates non dicit nisi hanc; ergo haec est falsa. Si dicatur primo quod haec sit falsa, tunc sequitur per descriptionem propositionis falsae quod haec juxta totalem significationem suam complexam significat sicut non est; sed illa praecise sic significat quod Socrates dicit falsum; ergo non est ita quod Socrates dicit falsum; et Socrates dicit aliquam propositionem; ergo Socrates solum dicit verum, et, per consequens, cum dicit hanc, sequitur quod haec est vera. Dicendum est ergo quod quilibet talis casus est insolubilis et non admittendus, sicut ex suppositionibus et conclusionibus positis satis patet. Sicut ex eisdem patet quod nulla talis demonstratio est admittenda ‘hoc est falsum’, ‘hoc non est verum’, sicut non est ‘homo est asinus et tota copulativa est falsa sive non vera’, ‘tu es asinus vel haec disjunctiva est falsa’, ‘haec consequentia est bona; ergo tu es asinus’, et sic de similibus. Similiter nullus talis casus est admittendus, videlicet quod aliqua talis propositio est ‘omnis propositio est falsa’, ‘omnis propositio est negativa sive particularis’, ‘nulla propositio est vera’, ‘aliqua propositio non est 538 dicat

corr.: dicerat O.

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vera’, ‘aliqua hypothetica non est categorica’, “si omnis consequentia est bona, [haec539 est bona] ‘tu es asinus’ ”, et sic de similibus. Consimiliter patet quod non est admittendum aliquod tale ‘Socrates solum jurat quod ipse perjurat’, “ille solum dicit istam ‘ego non pertransibo pontem’ ”, “omnis homo aeger dicit falsum et solum talis, et Socrates solum dicit istam ‘ego sum aeger’ ”, “omnis homo praeter Socratem dicit quod deus est et Socrates solum dicit istam ‘omnis homo praeter Socratem dicit verum’ ”, “omnis homo qui non est ille homo dicit falsum et ille Socrates solum dicit istam ‘tantum haec propositio est vera’, demonstrando primam”, “dicat Socrates solum hanc ‘homo est asinus vel aliqua disjunctiva dicta a me non est vera’ ”, “dicat Socrates solum hanc ‘deus est et aliqua copulativa dicta a me non est vera sive est falsa’ ”, “sit solum tibi propositum ‘homo est asinus vel nullum tibi propositum est verum’ ”, “audies tu solum istam per aurem sinistram ‘tu audis propositionem falsam per aurem sinistram’ ”, “videas tu solum istam per oculum dextrum ‘tu vides propositionem falsam per oculum dextrum’ ”, “credat Socrates solum istam ‘ego credo falsum’ ”, “intelligat Socrates solum istam ‘intelligo intelligibilem sive falsum’ ”, et sic de similibus casibus qui in infinitum posset excogitari de humanis actibus interioribus, sensitivis vel de proprietate vocis, diligenter advert<ent>i patet solum. Nam semper in talibus casus est negandus nisi omittatur aliqua particula ratione cujus casus est insolubilis, sicut in casu est haec particula ‘unus Socrates sit omnis Socrates’ vel ‘una propositio est omnis propositio’ vel ‘Socrates dicat sive audiat istam et nullam aliam’, et sic de aliis. Nam si talis particula omittatur ratione cujus fuit insolubile, residuum potest admitti, et patet responsio ad idem per regulas obligationum. Sed quia [p]interdum opponens imponit terminum sive propositionem ad significandum aliter quam ex grammatica significat; ideo ad hoc expedit breviter pertractare. Nova impositio solet fieri tribus modis. Aut enim terminus sive propositio imponitur ad significandum aliquam ultra communem institutionem et non certificatur respondens qualis ista significatio erit, sicut hic540 : significet ista ‘falsum <est>’ sicut communiter verba praetendunt, non tamen praecise. Aut de tali significatione certificatur [et] respondens sicut hic541 : significet ista ‘falsum est’ praecise quod falsum est et quod deus est, vel significet disjunctive praecise quod falsum est vel quod tu 539 haec

O2: om. O1. corr.: haec O. 541 hic corr.: haec O. 540 hic

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curris. Aut mutatur terminus sive propositio totaliter in aliam significationem, sicut hic542 : significet quaelibet talis ‘homo est’ praecise quod asinus currit, vel significet talis terminus ‘A’ –sive aliquis alius qui non prius significavit aliquid praecise– hominem, sive convertatur cum tali termino ‘homo’. De primo modo impositionis habet respondens quaerere totalem significationem termini sive propositionis et cognoscere eam et aliter non admittere casum, sicut patet per primam suppositionem: quia tunc talis propositio non foret sufficienter intellecta per respondentem.543 Antecedens patet: quia si aliquid sibi intelligitur, illud probatur sic: proposita tibi tali propositione ‘hoc est homo.’544 cujus subjectum non est intellectum a te; ergo tota propositio est sufficienter intellecta per respondentem.545 Antecedens patet: quia si aliquid distincte intelligis per subjectum sine demonstratione; ergo scis numquid istud sit homo vel non homo, et, per consequens, haec foret scita a te esse vera vel scita a te esse falsa. Consequens falsum; ergo et cetera. Consimiliter sic: posito quod altera istarum ‘deus est’ et ‘homo est asinus’ <sit A>, et lateat te quae illarum sit A, tunc tu non intelligis subjectum istius ‘A est verum’. Nam si sic, tunc intelligis per subjectum illam ‘deus est’ praecise vel ‘homo est asinus’ praecise vel utramque istarum copulative vel utramque illarum disjunctive. Si primo modo vel secundo vel tertio intelligis, patet quod habes concedere vel negare illam ‘A est verum’, quod est falsum: quia tu nescis quid est A. Si quarto modo intelligis per subjectum, tunc illa ‘A est verum’ apud tuum intellectum convertitur cum ista ‘altera illarum est vera’, et, per consequens, haec est concedenda ‘A est verum’. Restat ergo [verum est] quod nulla impositio est admittenda nisi respondens de totali significatione certificetur eo quod aliter non sufficienter intelligit terminos vel propositiones. Pro secundo modo impositionis, eo quod aliter non sufficienter, dandae sunt duae regulae quarum prima est ista: quandocumque in aliquo casu imponitur aliqua propositio ad copulative significandum praecise sicut verba praetendunt et quod tu curris vel aliquod tale, tunc si contradictorium illius copulati, scilicet ‘tu non curris’, stat cum casu, casus est admittendus; et si non stat, non est admittendus. Verbia gratia, <posito> 542 hic

corr.: haec O. corr.: responsionem O. 544 homo corr.: hoc O. 545 respondentem corr.: responsionem O. 543 respondentem

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quod haec <sit> quaelibet propositio ‘falsum est’, et quod praecise significet quod falsum est et quod deus est, vel546 quod praecise significet quod falsum est et quod nullus homo est asinus, tunc neuter casus est possibilis: quia nec ista [est] ‘nullus deus est’ nec ista ‘homo est asinus’ stat cum casu. Sed si ponatur quod ‘[hoc] falsum est’ sit quaelibet propositio, et quod illa praecise significet quod falsum est et quod tu curris, tunc quia oppositum istius copulati ‘tu curris’, scilicet ‘tu non curris’ stat cum casu; ideo talis impositio est admittenda. Et haec propositio ‘falsum est’ est concedenda, et negandum est illam fore veram, et negandum est copulatum ‘tu curris’ tamquam repugnans, et sic de omnibus quaestionibus. Secunda regula est ista: quandocumque ponitur propositio de insolubili praecise significare disjunctive sicut ejus verba praetendunt vel quod deus est sive aliquid aliud, tunc si istud disjunctum stat cum toto casu, casus est admittendus; et si non stat, est negandus. Verbi gratia, ponitur quod unus Socrates sit omnis Socrates, et quod ipse solum dicat istam ‘Socrates dicit falsum’, quae praecise significet disjunctive quod Socrates dicit falsum vel quod nullus Socrates dicit aliquid, tunc quia hoc disjunctum ‘nullus Socrates dicit aliquid’ non stat cum casu, casus est negandus. Si tamen ponitur [notaretur] quod ista ‘Socrates dicit falsum’ praecise disjunctive significet quod Socrates dicit falsum vel quod deus est, casus est admittendus, et hoc disjunctum ‘deus est’ semper est concedendum tamquam sequens, et ista ‘falsum est’ negandum est tamquam repugnans. Concedendum tamen est illam esse veram, et sic de consimilibus quibuscumque. Pro tertio modo impositionis notandum est quod nec terminus nec propositio in voce vel in scripto intelligitur vel conceditur vel negatur vel dubitatur rationabiliter nisi per modum suum significandi ex impositione; ideo quando idioma mutatur vel fit nova grammatica, noviter est respondendum. Verbi gratia, posito quod quaelibet talis ‘homo est’ praecise significet lapidem esse, haec consequentia non valet ‘si homo est, animal est’, quae tamen in idiomate nostro valet. Confirmatur sic: ponitur, sicut forte est verum de fato, quod talis ‘si homo est, animal est’, quae in idiomate latino est necessaria –et tam antecedens quam consequens est necessarium–, in aliquo alio idiomate praecise significet quod si asinus currit, lapis currit, tunc patet quod eadem propositio ab uno est concedenda tamquam necessaria et neganda ab alio tamquam impossibilis. Et hoc non est nisi propter diversam significationem ex 546 vel

corr.: et O.

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prima voluntaria significatione; ergo, si per novam institutionem idiomatis Socratis mutaretur in idioma Platonis et e contra, secundum hoc ipsi deberent suam responsionem variare, et, per consequens, consimiliter ubicumque est nova impositio, pro tanto est responsio varianda. Haec praemissa quamvis quibuscumque cavillationibus multis non placuerunt, tamen pueris propter malam doctrinam non infectis maxime proficient547 quod ipsa concedat trinitas benedicta quae mihi hanc sumulam incipere gratiam tribuit et finire. Explicit sumula sumularum magistri Iohannis Eclif.

References [1] E.P. Bos, John of Holland. Four Tracts on Logic. Nijmegen, Ingenium, 1985, pp. 125–146 [2] P.V. Spade, “Robert Fland’s Insolubilia: an Edition with Comments on the Dating of Fland’s Work”, in Medieval Studies 40, 56–80, 1978 [3] P.V. Spade, “An Anonymous Tract on Insolubilia from Ms Vat. Lat. 674. An Edition and Analysis of the Text”, in Vivarium IX, no. 1, 1–18, 1971 [4] Priscien, Institutiones grammaticorum, Book XVII, Ed. Hertz, Grammatici latini, Vol. 3. Georg Olms Verlag, Hildesheim-NewYork, p. 149, ll.11-12 [5] F. Pironet, Guillaume Heytesbury, Sophismata asinina, une introduction aux disputes médiévales. Présentation, édition critique et analyse. Paris, Vrin, Coll. Sic et non, 1994 [6] P.V. Spade, “Insolubilia”, in N. Kretzmann, A. Kenny, et J. Pinborg (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100–1600. Cambridge, Cambridge University Press, 1982, p. 252 [7] Ockham, Summa logicae, pars tertia, chap. 41: “De tertia specie obligationis, quae dicitur positio. Quomodo differt a casu?”, pp. 735– 736 [8] M. Yrjönsuuri, “The Role of Casus in Some Fourteenth Century Treatises on Sophismata and Obligations”, in K. Jacobi (ed.) Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns. Leiden, New York, Köln, E.J. Brill, 1993, pp. 301–321 547 proficient:

proficiet De Rijk.

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[9] L.M. de Rijk, “Some Thirteenth Century Tracts on the Game of Obligation”, in Vivarium XII, no. 2, 94–123, 1974 and XIII, no. 1, 22–54, 1975 [10] N. Kretzmann et E. Stump, “The Anonymous de arte obligatoria in Merton College Ms. 306”, an Edition and a Translation, in E.P. Bos (ed.) Mediaeval Semantics and Metaphysics, Studies Dedicated to L.M. de Rijk. Nijmegen, Ingenium Publishers, Artistarium Supplementa, II, 1985, pp. 243–244 [11] P.V. Spade, The Medieval Liar: A Catalogue of the InsolubiliaLiterature. Toronto. The Pontifical Institute of Mediaeval Studies, 1975, pp. 24–25 [12] L.M. de Rijk, “Semantics in Richard Bilingham and Johannes Venator”, in A. Maierù (ed.) English Logic in Italy in the 14th and 15th Centuries. Acts of the 5th European Symposium of Medieval Logic and Semantics, Rome 10–14 November 1980. Napoli, Bibliopolis, History of Logic, 1, 1982, p. 168 [13] Fr. del Punta et M. McCord Adams, Paulus Venetus: Logica Magna. Secunda pars. Fasc. 6. Tractatus de veritate et falsitate propositionis. Tractatus de significato propositionis. Edition with Notes on the Sources. Translation in English with Explanatory Notes. London, Oxford University Press, The British Academy Classical and Medieval Logic Texts, 1978, pp. x–xii [14] Johannes Venator Anglicus. Logica. Edited by de Rijk Lambertus Marie. Stuttgart, Bad Cannstatt: Frommann-Holzboog 1999. Vol. I: Tractatus I–II, Vol. II: Tractatus III–IV. Grammatica speculativa [15] E.J. Ashworth, “The Treatment of Semantic Paradoxes from 1400 to 1700”, in Notre Dame Journal of Formal Logic 13, 34–52, 1972; “Will Socrates Cross the Bridge? A Problem in Medieval Logic”, in Franciscan Studies 14, 75–84, 1976 [16] I.M. Bochenski, A History of Formal Logic, Translated and Edited by I. Thomas. Notre Dame, Indiana, 1961, pp. 238–251 [17] Fr. Bottin, Le antinomie Semantiche nella logica medievale. Padova, Editrice Antenore, 1976, pp. 144–151 [18] A.R. Perreiah, “Insolubilia in the Logica parva of Paul of Venice”, in Medioevo IV, 145–171, 1978 [19] L.M. de Rijk, “Logica Oxoniensis. An Attempt to Reconstruct a Fifteenth Century Oxford Manual of Logic”, in Medioevo III, 1977, p. 153 [20] M.H. DZIEWICKI, Johannis Wyclif Tractatus de logica. London, Trübner for the Wyclif Society, 1893–1899, vol. 2, pp. 194–227

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[21] P.V. Spade and G.A. Wilson, Wyclif Summa insolubilium. Binghamton, NY: Medieval and Renaissance Texts and Studies, 1986 [22] F. Pironet, Les traités “Juxta hunc textum” de Guillaume Heytesbury et Robert Alyngton. Edition critique précédée d’une introduction historique et paléographique, Université de Genève, Projet Sophismata, 2003, xliii + 302p. http://www.unige.ch/lettres/philo/ sophismata/publications.htm

Index Nominum

A Ajdukiewicz, K., 212, 214 Albert of Saxony, 16, 87, 90, 132 Anderson, A. R., 138 Andjelković, M., 41, 42, 44–54, 58–62 Aristotle, 44, 78, 101, 206 Ashworth, E. J., 76 Austin, J. L., 77 Azzouni, J., 33 B Barwise, J., 250 Beall, J. C., 219, 222 Beaney, M., 223 Beets, F., 250 Belnap, N. D., 138 Bochenski, I. M., 332 Boolos, G., 185 Bottin, F., 84 Bradwardine, T., 3, 16, 23, 24, 26, 30, 32, 37, 38, 63, 65–70, 73, 75–78, 83, 87–90, 96, 98–100, 135 Brandom, R., 31 Burge, T., 73, 236, 237, 241 Buridan, J., 16, 61, 66, 76, 87–97 Burley, W., 230, 231, 234–236, 238–240, 259

De Rijk, L. M., 76, 77, 231, 239, 266, 281, 282, 324, 331 Devitt, M., 128 Dodd, J., 5 Dummett, M., 31 E Ebbesen, S., 230, 239 Etchemendy, J., 156, 218. See also Barwise, J. F Field, H., 19, 33 Fland, R., 255, 267–270, 273, 276–278, 281 G Gaifman, H., 81 Gavray, M.-A., 250 Goldblatt, R., 185 Goldstein, L., 65–83, 206, 208, 209, 213 Goodstein, R. L., 192 Gould, J., 128 Gould, S. J., 39 Grice, H. P., 57 Gupta, A., 5, 245

C Campsall, R., 230 Cappelen, H., 80 Cargile, J., 114, 126, 128 Carnap, R., 3, 209, 210 Chihara, C., 244 Church, A., 150, 176, 193, 214 Cocchiarella, N., 190 Cohen, L. J., 74 Copi, I., 128 Corcoran, J., 6, 16, 150, 153

H Haack, S., 197 Harman, G., 250 Heytesbury, W., 215, 255–257, 259, 261, 263–269. See also PseudoHeytesbury Hintikka, J., 4, 79, 150, 195 Horwich, P., 4, 5, 7–9, 13, 25, 33, 67, 69, 122, 187, 188, 190 Hughes, G. E., 76, 90, 220, 221 Hunter, J., 255, 265–267, 269, 270, 272, 275, 276, 278, 280, 281, 301

D Davidson, D., 4, 42, 61, 72, 80, 150, 220, 221 Del Punta, F., 231

J Johannes Venator. See Hunter, J. John of Holland, 255, 267–270, 272, 275, 277, 278, 281

336 K Kirkham, R., 223 Klima, G., 87–111, 208 Kneale, W., 190, 193 Koons, R. C., 251 Kretzmann, N., 251, 332 Kripke, S., 19, 20, 22, 28, 29, 37, 60, 74, 134, 152, 178, 219, 245 L Lambert of Auxerre, 230 Langford, C. H., 185 LePore, E., 80 Lewis, C. I., 161 Lewontin, R., 39 Lewy, C., 119 Loux, M.J., 251 Lynch, M., 128 M Mackie, J. L., 238 Martin R. L., 85, 223, 251 Maudlin, T., 81 McGee, V., 236 Mills, E., 113–128, 210–214 Moody, E. A., 16 O Ockham. See William of Ockham P Panaccio, C., 92, 208, 229–250 Paul of Venice, 265, 266, 281 Pelletier, J., 252 Perreiah, A. R., 332 Peter of Ailly, 231 Peter of Spain, 231 Pironet, F., 76, 215, 231, 255–331 Priest, G., 220, 222, 236 Prior, A. N., 16, 90, 176, 193, 205, 206, 209, 210, 212, 219 Proust, J., 252 Pseudo-Heytesbury, 255, 265, 267, 269, 270, 272, 274, 276, 278, 280, 281, 290 Q Quine, W. V. O., 33, 37, 216 R Rahman, S., xix, 157 Read, S., 3–16, 19–38, 42–45, 49–51, 54, 58–60, 63 Roure, M. -L., 75, 76, 230, 231, 256 Russell, B., 55, 56

Index Nominum S Sagüillo, J. M., 149–157, 208 Sainsbury, R. M., 185 Scotus, J. D., 230 Sereny, G., 159–185, 208, 213–215, 217–219 Simmons K., 230, 236, 237, 245 Simon of Faversham, 230 Slater, B. H., 73, 187–196, 210, 212, 214 Smullyan, R., 224 Soames, S., 28, 37, 221, 236 Sommers, F., 81 Sorensen, R. A., 252 Spade, P. V., 90, 100, 209, 230, 231, 239, 258, 263, 265, 283, 285 Stalnaker, R., 63 Sten. See Ebbesen, S. Swyneshed, R., 205, 216, 217, 220, 259 Synan, E.A., 230

T Tarski, 3–5, 7–9, 13, 20, 22, 23, 29, 41–47, 49–52, 55–57, 62, 65–67, 79–81, 113, 114, 119, 129, 130, 132, 134, 149–153, 156, 159–161, 163, 164, 166, 167, 170, 171, 174–179, 183–185, 187, 188, 190–192, 195, 196, 199, 200, 205, 210, 218, 219, 221, 230, 236, 237, 244, 245, 248 Tennant, N., 81, 83

V Van Fraassen, Bas C., 236

W Wadding, L., 230 William of Ockham, 231, 232, 234, 238, 247 Williamson, T., 41, 42, 44–54, 58–62 Wittgenstein, L., 6, 7, 61, 72, 75, 78 Woleński, J., 199–203 Wright, C. J. G., 5

Y Yablo, S., 69–71, 209, 245, 249 Yaqub, A., 5

Index Rerum

A Adhocness, 241, 244 Argumentation, 53, 54, 152

F Fallacy, 72, 73, 120, 122, 124–126, 152, 153, 156, 160, 168, 169, 174, 196, 207

B Biconditional. See Equivalence: material and strict Bivalence, 16, 27, 28, 31, 33, 37, 38, 45, 47–49, 68, 75, 107, 125, 142, 179, 190, 208, 219, 232, 234, 237, 250, 279

I Implication material, 8, 45, 46, 59, 131, 156, 182, 206 relevant, 137 strict, 7, 24, 25, 46, 58, 59, 114, 115, 131, 141–143, 149, 154–156, 161–164, 166, 171, 172, 175, 176, 182, 206, 213 virtual, 89, 108–111 Implicatures, 57, 58 Insolubilia, 76, 77, 87, 95, 135, 230–232, 235, 238, 239, 243, 255, 266, 277, 286

C Cassatio, 75, 76, 78, 280 Closure, 7, 9, 12, 15, 20, 22, 24, 26, 29, 53, 58, 87, 104–107, 140, 143, 150, 154, 156, 157, 205, 207–210, 219, 222 under implication, 6, 10, 15, 51, 53, 54, 58, 59, 62, 206 semantic (see Semantic closure) Complexe significabilia, 93, 101–103 Conditional. See Implication: material and strict Context-dependency, 61 Contradiction (Law/Principle of), 72, 78, 81, 90, 123, 153, 156, 160, 170, 173, 178, 179, 183, 184, 190, 191, 207, 220, 221, 230, 232– 234, 237–240, 243–250, 259, 260, 264, 272, 280 Correspondence, 5, 23, 44, 62, 104–110, 114, 153

L Language* semantically closed language, 13, 28, 29, 38, 150, 153, 161 M Meaning* Principle of uniformity, 41, 43, 45, 51–54 Minimality (Principle of), 237, 239, 240, 245, 246 N Negation*, 21, 29, 45, 68, 78, 101, 133, 134, 138, 139, 143–145, 179–181, 183, 184, 201–203, 205, 206, 213–218, 234, 237

D Dialetheism, 21, 22

O Obligatio, 263

E Equivalence material, 7, 199 strict, 7, 24, 115, 166, 175, 200

P Paradox* curry, 10, 21, 66, 176, 209, 238 liar* epimenidean liar, 176

338 strengthened liar, 28, 67, 71, 72, 81, 187, 192, 233, 234, 239, 244, 249 Yablo, S., 65, 69–71, 209, 245, 249 Paraconsistency, 178. See also Dialetheism Presupposition, 54, 56–57, 59, 62, 153 Propositions* vs. Sentences, 196 Q Quantification over propositions, 165, 193 Quotation, 37, 65, 80, 188–190, 266 R Reasoning, 46, 65, 67, 68, 72, 73, 77, 141, 143, 149, 151–153, 156, 157, 171, 173, 208, 209, 215, 219, 233, 234 Restrictio, 126, 138, 151, 162, 171, 174, 177, 178, 184, 229, 231–233, 236–242, 245, 247–249, 255, 272, 273, 279 Restrictionism, 229–237, 240, 241, 244–246, 248 Restriction. See Restrictio S Saying that explicitly, 38, 49, 53, 56, 57, 62, 75, 100, 125, 130, 134, 161–164, 166, 171–173, 175, 176, 209, 213, 215, 218, 237, 271, 278 and implying, 75, 164, 174 Self-reference, 124, 183, 188–191, 229, 233, 236, 268

Index Rerum Semantic closure. See Language: semantically closed Sentences token-sentence, 79, 81, 87, 89, 92 Signification, 78, 87–96, 98–102, 135–137, 140, 143, 144 Suppositio, 229, 255, 325 Syncategoremata, 79, 87 T Truth*, 3–16, 19–38, 41–62, 65–75, 79–83 truth bearers, 79, 81, 237, 250 truthteller, 68, 213, 238–240, 244, 249 T-schema*, 3, 113, 114, 118, 119, 150, 187, 189, 191, 192, 194, 196, 218 as criterion of material adequacy, 5, 7, 42, 156, 244 as definition of truth, 3, 5, 20, 23–25, 67, 69, 79, 103, 129–131, 136, 162, 166, 179, 183, 185, 188, 199, 200 Type theory, 8, 193, 205, 209–214, 222 Type/Token, 81, 94, 120, 123, 124 U Uniqueness of meaning. See Meaning: Principle of Uniformity Ungroundedness, 176, 229, 249, 250 V Validity*, 4, 28, 41, 42, 46, 47, 51–55, 57, 58, 61, 62, 87, 100