Handbook of Philosophical Logic 2nd Edition Volume 3
edited by Dov M. Gabbay and F. Guenthner
CONTENTS Editorial Pref...
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Handbook of Philosophical Logic 2nd Edition Volume 3
edited by Dov M. Gabbay and F. Guenthner
CONTENTS Editorial Preface
vii
Dov M. Gabbay
Basic Modal Logic
1
R. A. Bull and K. Segerberg
Advanced Modal Logic
83
M. Zakharyaschev, F. Wolter and A. Chagrov
Quanti cation in Modal Logic
267
J. Garson
Correspondence Theory
325
J. van Benthem
Index
409
PREFACE TO THE SECOND EDITION It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the rst edition and there have been great changes in the landscape of philosophical logic since then. The rst edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the rst edition as `the best starting point for exploring any of the topics in logic'. We are con dent that the second edition will prove to be just as good.! The rst edition was the second handbook published for the logic community. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983{1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and arti cial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisation on the one hand and to provide the theoretical basis for the computer program constructs on the other. The result was that the Handbook of Philosophical Logic, which covered most of the areas needed from logic for these active communities, became their bible. The increased demand for philosophical logic from computer science and arti cial intelligence and computational linguistics accelerated the development of the subject directly and indirectly. It directly pushed research forward, stimulated by the needs of applications. New logic areas became established and old areas were enriched and expanded. At the same time, it socially provided employment for generations of logicians residing in computer science, linguistics and electrical engineering departments which of course helped keep the logic community thriving. In addition to that, it so happens (perhaps not by accident) that many of the Handbook contributors became active in these application areas and took their place as time passed on, among the most famous leading gures of applied philosophical logic of our times. Today we have a handbook with a most extraordinary collection of famous people as authors! The table below will give our readers an idea of the landscape of logic and its relation to computer science and formal language and arti cial intelligence. It shows that the rst edition is very close to the mark of what was needed. Two topics were not included in the rst edition, even though
viii they were extensively discussed by all authors in a 3-day Handbook meeting. These are:
a chapter on non-monotonic logic
a chapter on combinatory logic and -calculus
We felt at the time (1979) that non-monotonic logic was not ready for a chapter yet and that combinatory logic and -calculus was too far removed.1 Non-monotonic logic is now a very major area of philosophical logic, alongside default logics, labelled deductive systems, bring logics, multi-dimensional, multimodal and substructural logics. Intensive reexaminations of fragments of classical logic have produced fresh insights, including at time decision procedures and equivalence with non-classical systems. Perhaps the most impressive achievement of philosophical logic as arising in the past decade has been the e ective negotiation of research partnerships with fallacy theory, informal logic and argumentation theory, attested to by the Amsterdam Conference in Logic and Argumentation in 1995, and the two Bonn Conferences in Practical Reasoning in 1996 and 1997. These subjects are becoming more and more useful in agent theory and intelligent and reactive databases. Finally, fteen years after the start of the Handbook project, I would like to take this opportunity to put forward my current views about logic in computer science, computational linguistics and arti cial intelligence. In the early 1980s the perception of the role of logic in computer science was that of a speci cation and reasoning tool and that of a basis for possibly neat computer languages. The computer scientist was manipulating data structures and the use of logic was one of his options. My own view at the time was that there was an opportunity for logic to play a key role in computer science and to exchange bene ts with this rich and important application area and thus enhance its own evolution. The relationship between logic and computer science was perceived as very much like the relationship of applied mathematics to physics and engineering. Applied mathematics evolves through its use as an essential tool, and so we hoped for logic. Today my view has changed. As computer science and arti cial intelligence deal more and more with distributed and interactive systems, processes, concurrency, agents, causes, transitions, communication and control (to name a few), the researcher in this area is having more and more in common with the traditional philosopher who has been analysing 1 I am really sorry, in hindsight, about the omission of the non-monotonic logic chapter. I wonder how the subject would have developed, if the AI research community had had a theoretical model, in the form of a chapter, to look at. Perhaps the area would have developed in a more streamlined way!
PREFACE TO THE SECOND EDITION
ix
such questions for centuries (unrestricted by the capabilities of any hardware). The principles governing the interaction of several processes, for example, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based e ective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. I believe the day is not far away in the future when the computer scientist will wake up one morning with the realisation that he is actually a kind of formal philosopher! The projected number of volumes for this Handbook is about 18. The subject has evolved and its areas have become interrelated to such an extent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupings of chapters. I would like to thank our authors are readers for their contributions and their commitment in making this Handbook a success. Thanks also to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers for their continuing support for the Handbook.
Dov Gabbay King's College London
x Logic
IT Natural language processing
Temporal logic
Expressive power of tense operators. Temporal indices. Separation of past from future
Modal logic. Multi-modal logics
Algorithmic proof Nonmonotonic reasoning
Probabilistic and fuzzy logic Intuitionistic logic
Set theory, higher-order logic, calculus, types
Program control speci cation, veri cation, concurrency
Arti cial intelligence
Logic programming
Extension of Horn clause with time capability. Event calculus. Temporal logic programming.
generalised quanti ers
Action logic
Planning. Time dependent data. Event calculus. Persistence through time| the Frame Problem. Temporal query language. temporal transactions. Belief revision. Inferential databases
Discourse representation. Direct computation on linguistic input Resolving ambiguities. Machine translation. Document classi cation. Relevance theory logical analysis of language Quanti ers in logic
New logics. General theory Generic theo- of reasoning. rem provers Non-monotonic systems Loop checking. Intrinsic logical Non-monotonic discipline for decisions about AI. Evolving loops. Faults and comin systems. municating databases
Procedural approach to logic
Montague semantics. Situation semantics
Non-wellfounded sets
Expressive power for recurrent events. Speci cation of temporal control. Decision problems. Model checking.
Real time systems Constructive reasoning and proof theory about speci cation design
Expert systems. Machine learning Intuitionistic logic is a better logical basis than classical logic
Negation by failure and modality
Negation by failure. Deductive databases
Semantics for logic programs Horn clause logic is really intuitionistic. Extension of logic programming languages Hereditary - -calculus exnite predicates tension to logic programs
PREFACE TO THE SECOND EDITION
xi
Imperative vs. declarative languages
Database theory
Complexity theory
Agent theory
Special comments: A look to the future
Temporal logic as a declarative programming language. The changing past in databases. The imperative future
Temporal databases and temporal transactions
Complexity An essential questions of component decision procedures of the logics involved
Temporal systems are becoming more and more sophisticated and extensively applied
Dynamic logic
Database up- Ditto dates and action logic
Possible tions
ac- Multimodal logics are on the rise. Quanti cation and context becoming very active
Types. Term Abduction, rel- Ditto rewrite sys- evance tems. Abstract interpretation Inferential Ditto databases. Non-monotonic coding of databases
Agent's implementation rely on proof theory. Agent's rea- A major area soning is now. Impornon-monotonic tant for formalising practical reasoning
Fuzzy and Ditto probabilistic data Semantics for Database Ditto programming transactions. languages. Inductive Martin-Lof learning theories
Connection with decision theory Agents constructive reasoning
Semantics for programming languages. Abstract interpretation. Domain recursion theory.
Ditto
Major area now Still a major central alternative to classical logic More central than ever!
xii Classical logic. Classical fragments
Basic back- Program syn- A basic tool ground lan- thesis guage
Labelled deductive systems
Extremely useful in modelling
Resource and substructural logics Fibring and combining logics
Lambek calculus Dynamic syn- Modules. tax Combining languages
A unifying framework. Context theory. Truth maintenance systems Logics of space and time
Fallacy theory
Logical Dynamics Argumentation theory games
Widely applied here Game semantics gaining ground
Object level/ metalevel
Extensively used in AI
Mechanisms: Abduction, default relevance Connection with neural nets
ditto
Time-actionrevision models
ditto
Annotated logic programs
Combining features
PREFACE TO THE SECOND EDITION
xiii
Relational databases
Linear logic
Logical com- The workhorse The study of plexity classes of logic fragments is very active and promising. Labelling Essential tool. The new unifyallows for ing framework context for logics and control. Agents have limited resources Linked Agents are The notion of databases. built up of self- bring alReactive various bred lows for selfdatabases mechanisms reference Fallacies are really valid modes of reasoning in the right context. Potentially ap- A dynamic plicable view of logic
Important feature of agents Very important for agents
A new theory of logical agent
On the rise in all areas of applied logic. Promises a great future Always central in all areas Becoming part of the notion of a logic Of great importance to the future. Just starting A new kind of model
ROBERT BULL AND KRISTER SEGERBERG
BASIC MODAL LOGIC Historical Part 1 HISTORICAL OVERVIEW It is popular practice to borrow metaphors between di erent elds of thought. When it comes to evaluating modal logic it is tempting to borrow from the anthropologists who seem to agree that our civilisation has lived through two great waves of change in the past, the Agricultural Revolution and the Industrial Revolution. Where we stand today, where the world is going, is diÆcult to say. If there is a deeper pattern tting all that is happening today, then many of us do not see it. All we know, really, is that history is pushing on. The history of modal logic can be written in similar terms, if on a less global scale. Already from the beginning|corresponding to the stage of hunter-gatherer cultures in anthropology|insights into the logic of modality has been gathered, by Aristotle, the Megarians, the Stoics, the medievals, and others. But systematic work only began when pioneers found or forged tools that enabled the to plough and cultivate where their predecessors had had to be content to forage. This was the First Wave, and as with agriculture it started in several places, more or less independently: C. I. Lewis, Jan Lukasiewicz, Rudolf Carnap. These cultures grew slowly, from early this century till the end of the sixth decade, a period of more than 50 years. Then something happened that can well be described as a Second Wave. What brought it out spectacularly was the achievements of the teenage genius of Saul Kripke, but he was not alone, more strictly speaking the rst of his kind: the names of Arthur Prior, Stig Kanger, and Jaakko Hintikka must also be mentioned, perhaps also those of J. C. C. McKinsey and Alfred Tarski. Now modal logic became an industry. In the quarter of a century that has passed since, this industry has seen steady growth and handsome returns on invested capital. Where we stand today is diÆcult to say. Is the picture beginning to break up, or is it just the contemporary observer's perennial problem of putting his own time into perspective? For a long while one attraction of modal logic was that it was, comparatively speaking, so easy to do|now it is becoming as diÆcult as the more mature branches of logic. And the sheer bulk of published material is making it diÆcult to survey. But there is also the increasing di erentiation of interests and the subsequent tendency
2
ROBERT BULL AND KRISTER SEGERBERG
towards fragmentation. In addition to more traditional pursuits we are now seeing phenomena as diverse as the application of modal predicate logic to philosophical problems at a new level of sophistication (Fine [1977; 1977a; 1980]), the analysis of conditionals started by Stalnaker [1968], Lewis [1973], the generalisation of model theory with modal notions (Mortimer [1974], Bowen [1978]), in-depth studies of the so-called provability interpretation (see Boolos [1979]; see also Craig Smorynski's Chapter in this Handbook), the advent of dynamic logic (see Pratt [1980] and David Harel's Chapter in this Handbook) and Montague grammar (see Montague [1974]). This is not the place to go deeply into the history of modal logic, even though we will say something about it in the next few sections. A reader who would like to know more about the beginnings of the discipline is referred to Prior [1962], Kneale and Kneale [1962], and Lemmon [1977]. For the discipline itself, as distinct from its history, the reader may consult a number of textbooks or monographs, from E. J. Lemmon's and Dana Scott's fragment Lemmon [1977], and Hughes and Cresswell [1996]. Schutte [1968], Makinson [1971], Segerberg [1971], Snyder [1971], Zeman [1973], and Gabbay [1976] to the recent and very readable Rautenberg [1979] and Chellas [1980]. Notable journal collections of papers on modal logic include `Proceedings of a colloquium on modal and many-valued logics' (Acta Philosophica Fennica, 16, 1963), `In memory of Arthur Prior' (Theoria, 36, 1970), and `Trends in modal logic' (Studia Logica, 39, 1980). Good bibliographies of early work are found in Feys [1965], Hughes and Cresswell [1996] and Zeman [1973]. Among survey papers from the last few years we recommend Montague [1968], Belnap [1981], Bull [1982; 1983], and F ollesdal [1989]. All writing of history is to some extent arbitrary. The historian, in his quest for order, imposes structure. A favourite stratagem is the imposition of n-chotomies. As long as the arbitrary element is recognised, the procedure seems perfectly legitimate. This admitted we should like to impose a trichotomy on early modal logic: modern modal logic derives from three fountain-heads which may be classi ed according to their relation to semantics. The syntactic tradition is the oldest and is characterised by the lack of explicit semantics. Then we have the algebraic tradition with a semantics of sorts in algebraic terms. Finally there is the model theoretic tradition, the youngest one, whose semantics is in terms of models. Possible worlds semantics is the dominating kind of model theoretic semantics, perhaps even, if we take advantage of the vagueness of this term and stretch it a little, the only kind. In the next few sections we propose to give a brief account of each of the three traditions.
BASIC MODAL LOGIC
3
2 THE SYNTACTIC TRADITION Modern modal logic began in 1912 when C. I. Lewis led a complaint in Mind to the e ect that classical logic fails to provide a satisfactory analysis of implication, `the ordinary \implies" of ordinary valid inference', [Lewis, 1912]. Roughly it is the paradoxes of material implication that Lewis worries about, but his subtle argument goes beyond the vulgar objections, implication is not the only connective that worries him. In fact, his very rst analysis concerns disjunction. Consider, he says the following two propositions: 1. Either Caesar died, or the moon is made of green cheese. 2. Either Matilda does not love me, or I am beloved. If we disregard the complication that there is also an exclusive reading of `or', classical logic will consider that both these propositions are of the form (i) A _ B . Yet, Lewis argues, there are more important di erences between the two. For example, we know that (1) is true since we know that, as it happens, Caesar is dead, but we know that (2) is true without knowing which of the disjuncts is true. Thus (2) exhibits a `purely logical or formal character' and an `independence of facts' that is lacking in (1). This much all can agree. But disagreement arises over how to account for the di erence between (1) and (2). One possibility would be to hold that while both (1) and (2) are of the same form, viz. (i) they di er in that only (2) satis es the further condition (ii)
` A _ B,
where the turnstile ` stands for assertability or provability in some suitable system. But Lewis embraces another possibility. The di erence between (1) and (2), he feels, is a di erence in meaning. More speci cally, he feels that there is a connection between the disjuncts of (2) which is part of the meaning of (2). On this view, the `or' of (1) and the `or' of (2) are di erent kinds of disjunction, and Lewis proposes to call the former extensional and the latter intensional. While extensional disjunction is rendered by the traditional, truth-value functional operator _, a novel sort of operator is needed to render intensional disjunction. Lewis himself never introduced a symbol for it, but E. M. Curley, in a recent historical study, uses the symbol _ [Curley, 1975]. Thus, while (1) is of the form (i), we may say that, according to Lewis, (2) is of the form (iii) A _ B .
4
ROBERT BULL AND KRISTER SEGERBERG
The same problem also concerns other connectives. In the case of implication there is, according to Lewis, an extensional kind which is adequately rendered by the `arrow', !, the material implication of ordinary truth- value functional logic. But there is also an intensional kind of implication, called strict implication` by Lewis, and for this he introduces a new symbol, the ` sh-hook', 3 . The latter is not found, nor de nable, in classical logic, and so Lewis proposes to develop a calculus of strict implication. Thus there is a triad corresponding to (1){(iii), viz., (i0 ) A ! B , (ii0 ) ` A ! B , (iii0 ) A 3 B .
(The condition A ` B is logically equivalent to (ii0 ); Lewis would also have regarded the condition ` A 3 B as equivalent to (ii0 ).) The reader should notice the di erence in theoretical status between ! and 3 on the one hand, and ` on the other. In both cases the rst two are, or name, operators belonging to the object language, while the turnstile is part of the metalanguage, standing for provability or deducibility. (Provability may of course be seen as a special case of deducibility, viz. deducibility from the empty set of premises.) Evidently the crucial question is whether the logical di erence between (1) and (2) should be expressed in the object language or not|is it a feature about logic or in logic? Gerhard Gentzen is often regarded as having opted for the former alternative (although see [Shoesmith and Smiley, 1978, p. 33f] concerning the historicity of this view). It is hard to say whether Lewis was aware that there was a choice. However, looking back on his work we must represent him as having favoured (iii) over (ii) and (iii0 ) over (ii0 ) as the logical form of certain propositions. he has been much criticised for this. It has been maintained that his whole enterprise rests on a violation of the use/mention distinction and is hopelessly confused. this is not the place to go into that discussion, all we can do is to refer the reader to [Scott, 1971] which contains what is probably the deepest discussion of this matter and certainly the most constructive one. The method chosen by Lewis in his search for a calculus of strict implication was the axiomatic one. Lewis' intuitive understanding of logical necessity, logical possibility and related notions was of course (at least) as good as any man', but he never tried to give it direct systematic expression; what there is, is what is implicit in the axiom systems, plus scattered informal remarks. In other words, there is no formal semantics in Lewis' work; semantics is left at an informal level. In mathematics, there is an important and time-honoured way to proceed, ultimately going back on Euclid. In the case of logic the method may be described as follows. A formal language
BASIC MODAL LOGIC
5
is de ned. Formulas from this language are understood to be meaningful. A number of them are somehow selected for testing against one's intuition. Some are accepted as valid, some are rejected as nonvalid, some may be diÆcult to decide. The valid ones one tires to axiomatise so as to give a nite description of an in nite scene. In Lewis' case, the rst e ort was presented in [Lewis, 1918], a calculus which has since become known as the Survey System. however, if your semantics is only intuitive, as Lewis' was, and consequently vague, then you have a completeness problem: even if you are satis ed that the theses of your system are acceptable, how do you know that your axiom system captures as theses all the formulas that you would nd acceptable? The answer is that you do not, and it did not take long for other systems to emerge with, apparently, as good a claim as the Survey System to the title conferred upon it in [Lewis, 1918] as the System of Strict Implication. In [Lewis and Langford, 1959] several more were de ned and others hinted at. here Lewis himself de ned ve systems called S1, S2, S3, S4, and S5, the survey system coinciding with S3. Later S6 was introduced by Miss Alban and S7 by Hallden, but in e ect there were contemplated already by Lewis [Alban, 1943; Hallden, 1949]. The series of S-systems has been extended even further, but those mentioned are the principal ones. Of modal logicians working in the same vein as Lewis, Oskar Becker is remembered for his early treatise [Becker, 1930], but perhaps it is g. H. Von Wright who should be named the second most important author in the syntactic tradition. In his in uential monograph [von Wright, 1951] he remarks that, strictly speaking, modal logic is the logic of the modes of being. In this work and the related paper [von Wright, 1951a], Von Wright sets out to explore modal logic in a wider sense, the logic of the modes of knowledge, belief, norms and similar concepts; this wider sense of the term has since gained currency. These two works marked the beginning of much work in epistemic, doxastic, and deontic logic. Some studies of the same kind had already been published, such as [Mally, 1926] and [Hofstadter and McKinsey, 1955] (see [Follesdal and Hilpinen, 1971] or Von Wright [1968; 1981] for more of the prehistory of deontic logic), but Von Wright's work becomes seminal, especially in deontic logic. (For epistemic and doxastic logic the real trigger was a book written some ten years later by Von Wright's one time student Jaakko Hintikka, but this work [Hintikka, 1962] was written in what we call the model theoretic tradition and so does not belong in this section.) There are two other subtraditions that should be mentioned under the present heading. One is the development of entailment and relevance logic associated with the names of Alan Ross Anderson and Nuel D. Belnap. This movement concentrated on C. I. Lewis' concern to develop a logic of strict implication, that is, to give a syntactic characterisation of `the ordinary \implies" of ordinary valid inference'. Early contributions in the axiomatic style were given by [Church, 1951a] and [Ackerman, 1956], but it was only
6
ROBERT BULL AND KRISTER SEGERBERG
with Anderson and Belnap and their many students that the project got o the ground. Algebraic and model theoretic semantics came later to this kind of logic than to modal logic, and it is perhaps fair to say that the e orts towards nding an explicit semantics have led to results that are less natural than in modal logic. This may have to do with the fact that while model logicians aim at improving classical logic, entailment/relevance logicians wish to replace it. Students interested in this subtradition will nd the powerful tome [Anderson and Belnap, 1975] a rich source of information. (Cf. also Dunn, in a later volume of this Handbook.) The other subtradition that should be mentioned is that of proof theory. Gentzen methods have never really ourished in modal logic, but some work has been done, mostly on sequent formulations. Early references are [Curry, 1950; Ridder, 1955; Kanger, 1957; Ohnishi and Matsumoto, 1957/59]. A monograph in this tradition is [Zeman, 1973]. In the eld of natural deduction [Fitch, 1952] would seem to be the pioneer with [Prawitz, 1965] the classical reference. the recent interest in the provability interpretation of modal logic has spurred renewed interest in the proof theory of particular systems (for example [Boolos, 1979; Leivant, 1981]). In Section 9 we return to this topic. Finally, let it be remarked that the syntactic tradition in Lewis' spirit is by no means dead. For a recent declaration of allegiance to it by a distinguished logician, see [Grzegoczyk, 1981].
3 THE ALGEBRAIC TRADITION That classical logic is truth-functional is enormously impressive! As shown by the existence of intuitionistic and other dissenting logics, it is by no means self-evident that it should be possible to understand the usual propositional operators in terms of simple truth-conditions (the familiar truth-tables). But given the success of classical logic it is natural to ask if the same treatment can be extended to other operators of interest, for example, modal ones. It is immediately clear that such an extension is not straight-forward, if it exists at all. There are four unary truth-functions (identity, negation, tautology, and contradiction), so if necessity or possibility is to be truthfunctional, it would have to be one of them, which is absurd. But if one insists, nevertheless, that it must be possible to give a truthfunctional analysis of `necessary' and `possible'? Bright idea: perhaps there are more truth-values than the ordinary two|three, say. This idea occurred to Jan Lukasiewicz around 1918. His rst e ort was to supplement the ordinary truth-values 1 (truth) and 0 (falsity) with a third truth-value 21 (possibility (of some kind)). his new truth-tables were as follows:
BASIC MODAL LOGIC
^
1
1 2 1 21 2
0
0 0 0 0 0 0 1 1 1 2
1 2
:
1 0
_
1 half 0
1 1 1 1 2 0 1
1
1 21 2
1 1 2
0
7
!
1
1 2
0
1 1 12 0 1 1 1 1 2 2 0 1 1 1
1 1 1 1 1 0 1 1 2 2 0 1 0 0 0 0 With 1 singled out as the sole designated truth value, the concept of validity is clear: a formula is valid if and only if it takes the value 1 under all (three- valued) truth-value assignments to its propositional letters. Let the resulting logic be called L3 . it is an immediate corollary that L3 is a subsystem of the classical propositional calculus; for if everything to do with the new truth-value 12 is deleted from the truth-tables, then we get the old, classical ones back. Exactly what sort of possibility would 21 represent? the inspiration for his new logic Lukasiewicz had got from Aristotle's discussion of the theoretical status of propositions concerning the future. It is an interesting suggestion that a new truth-value is needed to analyse propositions of type `there will be a sea-battle tomorrow'; for it might be held that there are points in time when such propositions are meaningful, yet neither true nor false. In other words, if one is not a determinist|and Lukasiewicz de nitely was not one| then one will agree that there spare propositions P such that, today, P is possible and also :P is possible; that is, that both P and :P are true. This is in agreement with Lukasiewicz' matrix, for if P has value 12 , then P and :P take the value 1. So far, so good, but here a diÆculty lurks. For under the matrix (P ^:P ) gets the value 1 which is absurd intuitively: whatever the future may bring, it will not be both a sea-battle and not a sea-battle tomorrow. The counter-example is agrant, and it is interesting that Lukasiewicz was not moved by it. What is at issue is evidently whether one can accept a modal logic which validates all instances of the type A ^ B ! (A ^ B ): Our counter-example would appear to settle this question in the negative| cf. [Lewis and Langford, 1959, p. 167]|but Lukasiewicz was not impressed. In a paper published only a few years before his death he states that he cannot nd any example that refutes the schema in question: `on the contrary, all seem to support its correctness' [Lukasiewicz, 1953]. He goes on to intimate that when people disagree over questions of this sort, they have di erent concepts of necessity and possibility in mind. 1 2
1 2
8
ROBERT BULL AND KRISTER SEGERBERG
Once invented, this game admits of endless variation. Even among threevalued logics, L3 is not the only possibility, and there is literally no end to how many truth-values you may introduce. Lukasiewicz himself extended his ideas rst to n-valued logic, for any nite n, and then to in nitelyvalued logic, where in nite could mean either denumerably in nite or even non-denumerably in nite. In this way the notion of matrix was developed. ([Malinowski, 1977] is a compact and informative reference on Lukasiewicz and his work. For Lukasiewicz's own papers non-Polish speaking readers are referred to the collections [Lukasiewicz, 1970] and [McCall, 1967].) A matrix is given if you have (i) a set of objects, called truth-values, (ii) a subset of these, called the designated truth-values, and (iii) for every n-ary propositional operator ? in your object language, a truth-table for ? (essentially, an n-place function from truth-values to truth-values). In tuple talk, if ?0; : : : ; ?k 1 are all your propositional operators, the matrix can be thought of as a (k + 2)-tuple hA; D; M(?0 ); : : : ; M(?k 1 )i, where A is a non-empty set, D a non-empty subset of A, and, for each i < k; M(?i ) is a function from the Cartesian product Ani to A, where ni is the arity of ?i . It is easy to see how this can be generalised to any number of operators. Opinions may be divided over what philosophical importance to attach to the logics that Lukasiewicz introduce. However, there can be no doubt that he started or tied in with a line of development which is of great mathematical importance. the matrices that he invented became generalised in two steps. the rst one seems like a mere change of terminology: the introduction of the concept of an algebra as a tuple hA; f0 ; : : : ; fk 1 i, where A is a non-empty set and f0 ; : : : ; fk 1 are operations on A; that is, for each i < k there is a non-negative number ni such that fi is a function from Ani to A. As before, the generalisation to in nitely many functions is obvious. The connection with the concept of matrix is patent. Roughly speaking, it is only the set of designated elements that has been omitted; and as far as logic is concerned, that concept is needed for the de nition of validity, not for the assignment of values of A to formulas. The most important thing about the new de nition of algebra is perhaps that it encourages the study of these structures independently of their connection with logic. The second step of generalisation was to consider classes of algebras rather than one matrix or algebra at the time. Thus, whereas at rst algebraic structures (matrices) were introduced in order to study logic, later on logic was used to study algebra. The person who more than anyone deserves credit for this whole development is Alfred Tarski, a student and collaborator of Lukasiewicz. Some papers by Tarski written jointly with J. C. C. McKinsey or Bjarni Jonsson rank with the most important in the history of modal logic. Among early results stemming from the algebraic tradition are that Lewis' ve systems are distinct [Parry, 1934]; the analysis of S2 and S4 along with a proof that they are decidable [McKinsey, 1941]; that no logic between S1
BASIC MODAL LOGIC
9
and S5, inclusively, can be viewed as an n-valued logic, for any nite n [Dugundj, 1940]; that even though S5 is not a nitely-valued logic, all its proper extensions are [Scroggs, 1951]. It does not seem as if anyone had ever worked out exactly what the relation is between abstract algebras and the intended applications. But the idea must have been something like this. We are told to think of the elements of a matrix as truth-values, but in the case of an algebra one should perhaps rather think of the elements as propositions (identifying propositions that are logically equivalent). The class of all propositions, if it exists, would presumably form one gigantic, complicated, universal algebra. But in a given context only a subclass of propositions are at issue, and they will form a simpler, more manageable algebra. A particularly interesting paper with implications for modal logic is [Jonsson and Tarski, 1951]. If it had been widely read when it was published, the history of modal logic might have looked di erent. the scope of the paper is quite broad, but we should like to mention one or two results of particular relevance to modern modal logic. First, according to M. H. Stone's famous representation theorem, every Boolean algebra is isomorphic to a set of algebra. In other words, if A = hA; 0; 1; ; \; [i is any Boolean algebra, then there exists a certain set U and a set B of subsets of U , closed under the Boolean operations, such that A is isomorphic to the Boolean algebra B = hB; ?; U; ; \; [i. (See [Rasiowa and Sikorski, 1963] for a good presentation of this and related results.) Jonsson and Tarski extend this result to Boolean algebras with operations (that is, functions from An to A, for any n). If this does not sound too exciting, wait. Suppose that U is any non-empty set, and let F be a family of subsets of U closed under the Boolean operations. Let l; m : F ! F be functions satisfying the following conditions: (l1) lU = U; (m1) m? = ?; (l2) l(X \ Y ) = lX \ lY; (m2) m(X [ Y ) = mX [ mY; (lm) mX = U l(U X ); (ml) lX = U m(U X ): Then, according to Jonsson and Tarski, there exists a uniquely de ned binary relation R on U |that is R U U |such that (lR) lX = fx 2 U : 8y(xRy ) y 2 X )g; (mR) mX = fx 2 U : 9y(xRy&y 2 X )g; moreover, of the following conditions, (i1), (i2), and (i3) are mutually equivalent, for i = r; s; t: (r1) (8X 2 F )(lX X ), (r2) (8X 2 F )(X mX ),
10
ROBERT BULL AND KRISTER SEGERBERG
(r3) R is re exive with eld U ; (s1) (8X; Y
2 F )(Y [ lX = U i X [ lY = U ), (s2) (8X; Y 2 F )(Y \ mX = ? i X \ mY = ?), (s3) R is symmetric; (t1) (8X 2 F )(lX llX ), (t2) (8X 2 F )(mmX mX ), (t3) R is transitive. Conversely, if R is any binary relation on U , then (lR) and (mR) de ne functions l; m : F ! F such that again (i1), (i2), and (i3) are mutually equivalent, for i = r; s; t. Putting all this together we arrive at the following picture. If we are analysing a class of propositions satisfying certain conditions, then we may try to cast them as an algebra B = hB; 0; 1; ; \; [l; mi where hB; 0; 1; ; \; [i is a Boolean algebra and l and m are two additional unary operations. (If an element a 2 B is taken to represent a proposition, then la and ma would represent the propositions `a is necessary and `a is possible', respectively.) By the representation theorem, there exists a set U such that B is isomorphic to an algebra A = hA; ?; U; ; \; [l; mi, where A is a set of subsets of U and ; \; [, are the usual set theoretical operations. Note that it is not claimed that every subset of U corresponds to a proposition, but that the converse claim is made: to every proposition a 2 B a subset kak U corresponds. Under the intended interpretation it seems reasonable that l and m should satisfy conditions (l1), (l2), (lm) and (m1), (m2), (ml) above. Consequently Jonsson's and Tarksi's result applies, and so l and m are completely determined by a certain binary relation R. Thus A is completely determined by U; R, and P , where P is the set of elements kP k such that P is an atomic proposition. In this sense, A is equivalent to the triple hU; R; P i. Moreover, in the special case that the closure of P under l and m equals Bu; A is in the same sense equivalent to the pair hU; Ri. In view of later developments this is a striking result. The reader is asked to keep the following observations in mind when readings Sections 4 and 10 below: for all a; b 2 B and x 2 U ,
x 2 k ak if x 62 kak; x 2 ka \ bk i x 2 jjak and x 2 kbk; x 2 ka [ bk i x 2 kak or x 2 kbk; x 2 klak i 8y 2 U (xRy ! y 2 kak); x 2 kmak i 9y 2 U (xRy&y 2 kak):
BASIC MODAL LOGIC
11
4 THE MODEL THEORETIC TRADITION If algebraic semantics is discounted, then Rudolf Carnap was the rst to provide a semantics for modal logic. Three of the all time greats came together in him. From Frege he got his interest in semantics and, more speci cally, learnt to distinguish between intension and extension; and he attributes to Leibniz the notion that necessity is to be analysed as truth in all possible worlds. Moreover, he credits Wittgenstein with some ideas that formed the starting point for part of his own work (Carnap [1942; 1947]). By a state-description let us understand a set of atomic propositions (propositional letters). If S is a state-description, then we may say what it means that a formula A holds in S , which in symbols we write S A:
S P i P 2 S; if P is an atomic proposition; S :A i not S A; S A ^ B i S A and S B; S A _ B i S A or S B; S A ! B i if S A then S B: If one is considering a de nite collection C of state-descriptions, then also the following conditions become meaningful:
S A i , for all T 2 C; T A; S A i , for some T 2 C; T A: Let us say that a formula is valid in C if it holds in every state description in C , and simply valid if it is valid in every collection of state-descriptions. this de nition singles out a well-de ned subset from the set of all formulas. Interestingly enough, this subset is the same as the set of theses of Lewis' system S5. Is this a coincidence? On the surface of it, Carnap's characterisation of S5 looks very di erent from the original one due to Lewis. This still does not look like modern modal logic: possible worlds are missing. According to Hintikka [1975], `Carnap came extremely close to the basic ideas of possible-worlds semantics, and yet apparently did not formulate them, not even to himself'. this is drawing a very ne line, at least on the level of propositional logic. Carnap does talk about possible worlds. He is quite clear that he wants to latch on to Leibniz' suggestion that a necessary truth is one that holds in all possible worlds. Moreover, he says that his state-descriptions `represent' possible worlds, which would seem to indicate that the former are (partial) descriptions of the latter. Thus from a formal point of view|Hintikka agrees with this|instead of the collections of state-descriptions that appear in the preceding paragraph, we could just as well have collections of possible worlds, provided only that we nd a way of dealing with the rst clause in the de nition of `holds in'. One virtue of state- descriptions, not shared by possible worlds, is that it is at once
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ROBERT BULL AND KRISTER SEGERBERG
clear what it means that a given atomic proposition hold in a given statedescription. What we need, it seems, is a new primitive to perform this service. This leads us to re-cast Carnap's semantics in the following terms. We call hU; V i a Carnap-model if U is any set (of possible worlds) and V (the valuation) is a function assigning to each atomic proposition P and possible world x a truth-value V (P; x) which is either T (truth) or F (falsity). In the de nition of `holds at' the rst clause is replaced by this condition:
x P
i V (P; x) = T; if P is an atomic proposition.
The other conditions are changed accordingly. In particular, those concerning the modal formulas become x A i 8y 2 U y A; x A i 9y 2 U y A: All this is no improvement on Carnap, but it brings us into line with modern terminology. It should be added that the picture of Carnap given here is a pale one since so much of importance in his work is found at the level of predicate logic, which is not considered in this article. The next step of importance within the semantic tradition was taken by Arthur Prior. both Lewis and Carnap had been concerned with the analysis of modal concepts in the strict sense, but, as remarked in Section 2, some authors have also tried to model concepts which are called modal in the wide sense (imperative, deontic, etc.). The e orts of the latter had been syntactic, but Prior, whose interests lay in temporal notions, gave an algebraic avoured analysis which in e ect was a model theoretic one. In his book, Prior [1957], he models time as the set ! of natural numbers. Thus instead of Carnap models we now meet with structures h!; V i which we might call Prior models and in which the unspeci ed collection U of possible worlds of a Carnap model hU; V i is replaced by the special set ! representing a set of points of time. With the help of Prior models many new operators are de nable. In [Prior, 1957] attention is focused on the operators de ned by the conditions t A i 8u = t u A; t A i 9u = t u A: Later Prior was to consider also the related operators de ned by the conditions t A i 8u > t u A; t A i 9u > t u A: There is almost no end to the number of new operators thus de nable. Already in [Prior, 1957] one nds conditions like t A i t A and t+1 A; t A i t A or t+1 A;
BASIC MODAL LOGIC
13
and later developments have seen a host of others. Once Prior had shown how to do tense logic, much activity followed. For example, it is natural to study Prior models in which the set ! of natural numbers is replace by the set of all integers, or the set of rational numbers, or the set of real numbers. Much attention was also devoted to studying the interaction of several temporal and other operators in multimodal systems. (One among many good references in tense logic is [Rescher and Urquhart, 1971].) Prior's work paved the way for Kamp [1968] where for the rst time exact de nitions of the notion of tense were o ered. For example, according to Kamp, an n-place tense in discrete time is a function f from (B )n to B ; and an n-ary operators ? will express this tense if, for all t 2 ,
t ?(A0 ; : : : ; An
1)
i t 2 f (fu :u A0 g; : : : ; fu :u An 1 ):
With Kamp [1968] tense logic achieved a new level of sophistication. However, much of the early interest concerned more basic problems, for example, that of characterising the operators de ned by the rst of the three de nitions given above. This logic, the so-called Diodorean logic, is not as strong as S5, yet stronger than S4, as pointed out by Hintikka, Dummett and others. Its true identity was nally settled by S. A. Kripke and R. A. Bull, independently [Bull, 1965]. For an entertaining account of this, see [Prior, 1967, Chapter 2]. All of this is sorted out in the chapter on tense logic (see the chapter by Burgess in a later volume of this Handbook. What is important here is that Prior replaces Carnap's unordered set of possible worlds (actually, state-descriptions) by an ordered set of possible worlds (actually, points of time). In order to stress this di erence we should perhaps have introduced the Prior models as triples h!; 5; V i, where 5 is the ordinary less-thanor-equal-to ordering of the natural numbers. Thus in retrospect it seems that Carnap and Prior between them supplied all the necessary ingredients for modal logic as we know it at present. Already Jonsson and Tarski had explored the mathematics that is needed, and in Carnap and Prior there was suÆcient philosophical underpinning to get modern modal logic going. The modern notion of a model is a triple hU; R; V i, where U is a set (of possible worlds, or, more neutrally, indices, or even just points), R a binary relation on U (the accessibility relation (Geach) or the alternativeness relation (Hintikka)), and V a valuation. As we say the elements U and V were contributed by Carnap, and the relation R is obtained by generalising ever so slightly over Prior: instead of working with his special cases, we keep as the one general requirement that R is a binary relation, not necessarily an ordering. But this is not the way history is usually written. So-called possible worlds semantics or Kripke semantics is commonly attributed to S. A.
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ROBERT BULL AND KRISTER SEGERBERG
Kripke, who laid down the foundations of modern propositional and predicate modal logic in several in uential papers (Kripke [1959; 1963; 1963a; 1965]). Relatively less in uential were the papers by Jaakko Hintikka and Stig Kanger (Hintikka [1957; 1961; 1963]; Kanger [1957; 1957a; 1957b; 1957c]). Actually the three seem to have been independent of one another; but Kanger published rst. Kanger's writings are diÆcult to decipher, and this fact, paired with the unassuming mode of their publication, may have been what has deprived him of some of the recognition due to him (cf. Hintikka's generous review, [Hintikka, 1969a]). Hintikka has had more impact, especially on the philosophers. The reason his work has been less important for the formal development of modal logic than that of Kripke is perhaps his style of presentation which tones down mathematical aspects and skips proofs. 5 OTHER TRADITIONS In the preceding sections we have described what seems to us to be the main developments in early modal logic. no history is ever complete, and starts not recorded here have been made without their developing into what we regard as a major tradition. In this section we will brie y mention ve or six such starts. First there is the so-called provability interpretation(s) of modal logic, the embryo of which is found in [Godel, 1933]. In view of recent development one may perhaps say that this is expanding into a new tradition right now. Via Montague [1963], Friedman [1975] and Solovay [1976] it has begun to generate a literature of its won. For more information on this, see [Boolos, 1979] and Smorynski's chapter in a later volume of this Handbook. Another start, more suggestive than seminal, was made by J. C. C. McKinsey who described what is now known as McKinsey's syntactic interpretation of modal logic [McKinsey, 1945]; McKinsey's idea was perhaps foreshadowed in Fitch [1937; 1939], it is taken up again in [Morgan, 1979]. A third start was made by Alonzo Church in a series of papers ([1946; 1951; 1973{ 74]); recent contributions to this area are Parsons [1982] and C. A. Anderson [1980]. (Cf. also his chapter in volume 4 of this Handbook.) A fourth start worth mentioning was made with the appearance of Arthur Prior's threevalued modal logic Q. many-valued modal logic is not a vast eld and in any case mainly falls under what we have called the algebraic tradition, but Q, rst de ned in [Prior, 1957], seems to be of particular philosophical interest; see, for example, [Fine, 1977]. Finally there ought to be a tradition called intuitionistic modal logic, but it is debatable whether today even a subtradition can be found under that heading. Perhaps Ditch [1948], Curry [1950] and Prawitz [1965] can be regarded as starts, but they are not very illuminating as analyses of
BASIC MODAL LOGIC
15
modality; and work on semantics has, to date, been in the classical spirit (Bull [1965a], Fischer Servi [1977; 1981]). Why intuitionistically minded logicians have not been attracted to this area is not clear, and surely it would be interesting to see an intuitionistic-logical analysis of knowledge (including extra-mathematical knowledge), obligation, imperative, perception, and other notions which are modal in the wide sense.
Systematic Part 6 LOGICS AND DEDUCIBILITY RELATIONS In the preceding sections our primary concern has been historical. It is now time to being a more systematic exposition. In this section we will give a number of concepts which are useful when it comes to classifying modal logics. First we give a family of (more or less) traditional de nitions, and then we develop similar de nitions of a slightly more general nature. Modal logics are often de ned as sets of formulas of a certain kind. One might begin by de ning a logic as a set L of formulas satisfying the following conditions:
A 2 L, whenever A is a tautology in the sense of classical propositional logic; (mp) if A ! B 2 L and A 2 L, then B 2 L; (sb) if A 2 L, then sA 2 L, if sA is the result of uniform substitution of formulas for propositional letters in A. (tf)
Then one might perhaps go on to say that a logic L is classical modal if it contains the formulas K. (P ! Q) ! (P ! Q),
.
T ,
(where P; Q are two propositional letters and T is either primitive or some chosen tautology) and in addition is closed under replacement of tautological equivalents: (rte) If A and B are tautologically equivalent and C and C are identical except that one occurrence of A in C has been replaced by an occurrence of B to give C , then C 2 L i C 2 L. This is a very weak conception of classical modal logic (incidentally, di ering from that in [Segerberg, 1971]), and usually one would require much more, for example, closure under congruence (cgr), monotonicity (mon), or necessitation (nec):
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ROBERT BULL AND KRISTER SEGERBERG
(cgr) if A $ B 2 L, then A $ R 2 L; (mon) if A ! B 2 L, then A ! B 2 L; (nec) if A 2 L, then A 2 L. A modal logic satisfying (cgr) ((mon), (nec)) would be called congruential (regular, normal). Moreover, a modal logic would be quasi-congruential (quasi-regular, quasi-normal) if it contained some congruential (regular, normal) modal logic. (A logic containing a classical modal logic is of course itself classical modal.) Notice that normality implies regularity implies congreuentiality. If is the only non-Boolean operator, then congruentality implies replacement of tautological equivalents. (Our terminology is not completely standard, but at lest the de nitions of `logic', `regular', `normal', and `quasi-normal' appear to be.) So far tradition. however, there is also a more roundabout way to arriving at similar de nitions which begins with deducibility relations instead of with logics. It may be instructive to o er these slightly more general de nitions as well. In this paper|and here we o er less than full generality|a deducibility relation R is a set of ordered pairs h ; Ai, where is a set of formulas and A is a formula. If h ; Ai 2 R we say that yields A and write `R A, or even ` A when suppression of the subscript does not lead to confusion. If ` A and = ? we write ` A and say that A is a thesis of R. The set of theses of R is denoted by Th R. We usually write A0 ; : : : ; An 1 ` B instead of fA0 ; : : : ; An 1 g ` B ; also A0 ; : : : ; An 1 ; ` B instead of fA0 ; : : : ; An 1 g; ` B . If A ` B and B ` A we write A a` B . Common conditions on deducibility relations re re exivity (RX), (left) monotonicity (LM), cut (CUT), and substitutivity (SB): (RX) A ` A; (LM) if ` A and , then ` A; (CUT) if ` C and C; ` A, then ` A; (SB) if ` A, then s ` sA, if s and sA are the result of uniform substitution in and A, respectively, of formulas for propositional letters. A deducibility relation is Boolean if it also satis es the conditions in Table 1 (we assume a truth-value functionally complete set of Boolean operators). A deducibility relation is compact if, wherever ` B , there are some A0 ; : : : ; An 1 2 , for some n = 0, such that A0 ; : : : ; An 1 ` B . Notice that two compact Boolean deducibility relations coincide if they agree on their theses: ThR = ThR0 implies that R = R0 . The concepts de ned above for logics may now be given analogous definitions in the context of deducibility relations. rst, let us say that a deducibility relation is n-modal if
BASIC MODAL LOGIC (n-M) if
tautologically implies A, then n 6 ?. =
17
` n n A, provided that
Table 1. (^ E) If ` A ^ B , then ` A and ` B . (^ I) If ` A and ` B , then ` A ^ B . (_ E) If ` A _ B and A; ` C and B; ` C , then (_I) If ` A or ` B , then ` A _ B . (!E) If ` A ! B and ` A, then ` B . (! I) If A; ` B , then ` A ! B . (:E) If ` :A and ` A, then ` B . (:I) If A; ` :A, then ` :A. (RAA) If :A; ` A, then ` A.
` C.
(Here nA is the formula consisting of the formula A preceded by a string of n occurrences of , while n = fnB : B 2 g. Let us say that a Boolean deducibility relation is modal if it is 1-modal, and strongly modal if it is n- modal for all n.) Next, let us say that a deducibility relation is classical if it is closed under the following condition of replacement under tautological equivalents: (RTE) If A and B are tautologically equivalent, and C and C are identical except that one occurrence of A in C has been replaced by an occurrence of B to give C , then C a` C . Finally, let us say that a deducibility relation is congruential (regular, normal) if it satis es (CGR)((SC1), (SC2)): (CGR) If A a` B , then A a` B ; (SC1) If
` A, then ` A, provided that 6= ?; ` A, then ` A.
(SC2) If (Conditions (SC1) and (SC2) are due to Dana Scott, whence the notation.) Let us now review the situation. It is readily seen that every Boolean deducibility relation R determines a unique logic, viz. Th R. Conversely, every logic L determines a compact Boolean deducibility relation Rel L in a natural manner: ` B i there are A0 ; : : : ; An 1 2 , for some n = 0, such that ((A0 ^ : : : ^ An 1 ) ! B ) 2 L. Note that L= Th Rel L, for every logic L, R= Rel Th R, for every compact, Boolean deducibility relation R. Moreover, note that if L is classical modal (and also congruential, regular, or normal, respectively), in the sense of logics, then so is Rel L, in the sense
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ROBERT BULL AND KRISTER SEGERBERG
of deducibility relations; and if a compact Boolean deducibility relation is classical modal (and also congruential, regular, or normal, respectively), in the sense of deducibility relations, then so is Th R, in the sense of logics. In view of a preceding remark we know that Rel L is the only compact deducibility relation with L as its set of theses. Therefore, evidently, if, as in this paper, one is only interested in compact deducibility relations, it is harmless to restrict oneself to the study of logics; which is what one has usually done traditionally. For some recent works in which deducibility is seen as primary, rather than thesishood, see [Scott, 1971; Kuhn, 1977; Shoesmith and Smiley, 1978; Gabbay, 1981; Segerberg, 1982]. Ultimately this approach seems to derive from two quite di erent sources, Gentzen and Tarski.
7 A CATALOGUE OF MODAL LOGICS Almost all recent work in modal logic has been concerned with normal logics. At least from a technical point of view, non-normal, regular or quasi-regular logics|a class which includes S2, S3, S6 and S7|seem to o er little of interest beyond what normal logics o er, and for that reason we will not treat them here but refer the reader to [Kripke, 1965] and [Lemmon, 1957; Lemmon, 1966]. Among logics that are not even quasiregular, the congruential merit some attention, and in Section 21 below some are implicit. But with this exception the purview of this paper is normal modal logics. Over the years an almost astronomical number of modal logics have been put forward. Under such circumstances, naming or identifying logics becomes a problem. The best nomenclature is perhaps the one proposed by E. J. Lemmon in [Lemmon, 1977], and here we will usually employ a variant of it. The smallest normal logic we designate by `K' (in honour of Kripke who, curiously enough, seems never to have dealt with this particular logic). If `Xo ', . . . , `Xm 1 ' name any formulas, then `KX0 ; : : : ; Xm 1 ' is the Lemmon code for the smallest normal logic that contains X0 ; : : : ; Xm 1 . Note that, by de nition, this logic is closed under substitution. Lemmon's convention presupposes that formulas have names. Here is a list of formulas with names that either are more or less standard, or else in the opinion of the authors deserves to be:
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19
D. P ! P , T. P ! P , 4. P ! P , E. P ! P , B. P ! P , Tr. P $ P , V. P , M. P ! P , G. P ! P , H. (P ^ Q) ! ((P ^ Q) _ (P ^ Q) _ (Q ^ P )), Grz. ((P ! P ) ! P ) ! P , Dum. ((P ! P ) ! P ) ! (P ! P ), W. (P ! P ) ! P . the following remarks will make it easier to remember these names. `D' stands for deontic, `T' comes from `t', a name invented by Feys, 4 is the characteristic axiom of Lewis' S4, `E' stands for Euclidean, `B' for Brouwer, `Tr' for trivial, `V' for verum, `M' for McKinsey, `F' for Geach, `H' for Hintikka, `Grz' for Grzegoczyk, `Dum' for Dummett, and `W' for (anti)well-ordered. The strangest of these names is perhaps `B' for Brouwer, as the father of mathematical intuitionism was never known to harbour much sympathy for logic, let alone modal logic. The name hails back to Oskar Becker who saw a similarity between the logic KTB and intuitionistic logic [Becker, 1930]. Of the many logics that can be de ned in terms of the above formulas we list the following: KT = T = the Godel/Feys/Von Wright system, KT4 = S4 KT4B = KT4E = S5 KD = deontic T, KD4 = deontic S4, KD4E = deontic S5, KTB = the Brouwer system (`the em Brouwersche system'), KT4M = S4.1, KT4G = S4.2, KT4H = S4.3, KT4Dum = D = Prior's Diodorean logic, KT4Grz = KGrz = Grzegoczyk's system, K4W = KW = Lob's system, KTr = KT4BM = the trivial system, KV = the verum system. There is no upper bound to the number of normal modal logics, and many| perhaps too many|have found their way into the literature. But the given catalogue includes many of the most studied systems.
20
ROBERT BULL AND KRISTER SEGERBERG
If the inconsistent logic, the set of all formulas, is accepted as a normal modal logic|and under the de nition given here it must be|then the set of all normal modal logics forms a distributive lattice under the operations g.l.b. (L, L0 ) = the greatest normal logic to be contained in both L and L0 (which is the same as L \ L0 ) and l.u.b. (L; L0 ) = the smallest normal logic to extend both L and L0 (which is not the same as L [ L0 ). Much e ort has gone into exploring the nature of this enormously complicated lattice. Early contributions were made by Scroggs who mapped out all the extensions o f S5 [Scroggs, 1951]; by Bull who did the same for the extensions of S4.3 [Bull, 1966]; by Makinson who showed that the trivial system and the verum system are the two dual atoms of this lattice [Makinson, 1971]; and by McKinsey and Tarski who showed that there are non-normal extensions of S4 [McKinsey and Tarski, 1948]. Kit Fine and Wim Block have done more than anyone else to complete the picture, and some of their work is described below. Schumm [1981] sums up some of the things that are known about the elements of the big lattice. Readers interested in the geography of modal logic are also referred to Hansson and Gardenfors [1973].
8 SEMANTIC TABLEAUX AND HINTIKKA SYSTEMS The deductive systems given in the preceding sections are of so-called Hilbert type, strict on rules and soft on axioms. Most of the deductive systems in the modal logic literature are of this type. From a metamathematical point of view such systems have much to o er. But if one's interest lies in proving theorems in a system rather than about it, then they are not terribly accommodating. Yet in modal logic they have had relatively little competition from other kinds of deductive systems. The most common system of a di erent kind is no doubt the procedure due to Hintikka and Kripke (similar ideas in a less developed form are found in [Guillaume, 1958]). Hintikka's work on model system [1957; 1961; 1962; 1963] and Kripke's on semantic tableaux [1963; 1963a] were independent, and even though the two methods are equivalent they are not identical. It would take us too far here to discuss both, and here we will follow Hintikka. For classical logic the general references are the classic works [Beth, 1959] and [Hintikka, 1955] as well as the later monograph [Smullyan, 1968]. an elementary and particularly readable account is given in [Je rey, 1990]. We de ne a set of formulas as downward saturated if it satis es the following conditions:
BASIC MODAL LOGIC (C:) (C^) (C_) (C!) (C::) (C:^) (C:_) (C: !)
21
If :A 2 , then A 62 Sigma. If A ^ B 2 , then A 2 and B 2 Sigma. If A _ B 2 , then A 2 or B 2 , If A ! B 2 , then A 2 only if B 2 . If ::A 2 , then A 2 . If :(A ^ B ) 2 , then :A 2 or :B 2 . If :(A _ B ) 2 , then :A 2 and :B 2 . If :(A ! B ) 2 , then A 2 and :B 2 .
The seven last conditions de ne an e ective procedure: given any nite set it is possible to add a nite number of new formulas to it to obtain a set which satis es all the conditions except perhaps (C:); this would be to embed in . Notice that is downwards saturated only if also (C:) holds. The latter condition is evidently of a di erent character from the others: they prescribe membership under some conditions, whereas (C:) proscribes it under all. That is to say, (C:) is a consistency condition. We are now able to de ne a deducibility relation as follows: ` B if and only if the set [f:B g cannot be embedded in a downwards saturated set. Speci cally, if is nite, (*)
A0 ; : : : ; An 1 ` B i , for every downwards saturated set , if A0 ; : : : ; An 1 2 , then :B 62 .
The reason this deducibility relation is of interest is that it coincides with classical logic: ` A i tautologically implies A. Furthermore, by the compactness theorem of classical propositional logic, ` B only if for some n = 0 and some A0 ; : : : ; An 1 2 we have A0 ; : : : ; An 1 ` B . The question arises, how to extend this analysis to modal logic. From a syntactic point of view, all that would be needed is two additional rules, (C) and (C:) of a similar kind. By `similar' is meant that the rules would have to be such that the Augmented set of rules would again de ne a (not necessarily e ective) procedure. It turns out that in order to do this we have to widen the perspective. What both Hintikka and Kripke did was to consider not just downward saturated sets (respectively, semantic tableaux) but systems of such sets (respectively, tableaux). Let us call a triple h0 ; U; Ri a Hintikka system if the following is true. First, U is a set of downward saturated sets of which 0 is one; and R is a binary relation over U (called the alternativeness relation by Hintikka) which generates U from 0 in the sense that, for each 2 U , there are some sets 1 ; 2 ; : : : ; k 2 U , for some k = 0, such that i Ri+1 , for all k < k, and k = . Second, for every 2 U the following conditions are satis ed: (C) If A 2 , then A 2 0 , for all 0 2 U such that R0 . (C:) If :A 2 , then :A 2 0 , for some 0 2 U such that R0 .
22
ROBERT BULL AND KRISTER SEGERBERG
We are now able to de ne a deducibility relation for modal logic: ` A i the set [ f:Ag cannot be embedded in a Hintikka system (in the obvious sense: there is no Hintikka system h0 ; U; Ri such that [ f:Ag 0 ). As Hintikka and Kripke proved (and, in e ect, Kanger had proved before them), the deducibility relation thus introduced will coincide with the famous modal logics T, S4, and S5, respectively, if special conditions are placed on the alternativeness relation, viz. re exivity; re exivity and transitivity; re exivity, transitivity, and symmetry; respectively. These are no doubt the most celebrated of all results in modal logic, and much of the success of the new semantics is probably due to the fact that the three most important systems of modal logic can be given such a simple characterisation in these new terms. Other conditions than those mentioned can also be considered, and it turns out that for practically all systems in the literature that have been proposed for their philosophical virtues, a similar model theoretic characterisation is possible. What we have so far is just a procedure. Primarily it is a disproof procedure (successful if an appropriate Hintikka system is found). Secondarily it is also (the beginning of) a proof procedure (successful if it can be shown that no appropriate Hintikka system can be found). In general neither procedure need be e ective, though, for the new rule (C:) may introduce new formula sets, and the implicit procedure may therefore not terminate. In other words, given some conditions on the alternativeness relation and formulas A0 ; : : : ; An 1 ; B , there is no guarantee that one will ever be able to settle the question whether A0 ; : : : ; An 1 ` B (even though, as it turns out, in many cases such a guarantee can be given). From a philosophical point of view it should be noted that what we have above is not yet a semantics in any but a combinatorial sense of the word. As in the case of Carnap|there is of course a close connection between statedescriptions and a downward saturated set|a real semantics is obtained if possible worlds are postulated and downward saturated sets are identi ed as partial descriptions of them. We shall append two observations which are of some interest. Let us say that a set of formulas is upward saturated if the converses of the above C conditions for the classical operators are satis ed, and maximal consistent if it is saturated both upward and downward. The rst observation is a familiar one: we again get classical propositional logic by stipulating that ` B i [ f:B g cannot be embedded in a maximal consistent set. Speci cally, if is nite, (x)
A0 ; : : : ; An 1 ` B i , for every maximal consistent set , if A0 ; : : : ; An 1 2 , then B 2 .
This statement, which is nothing but the famous Lindenbaum's Lemma, should be compared to (*) above.
BASIC MODAL LOGIC
23
Suppose now that we call a set h0 ; U; Ri of maximal consistent sets a Henkin system if U is a set of maximal consistent sets of which 0 is one, and R is a binary relation on U such that (C ) and (C :) as well as their converses are satis ed by every 2 U . Then once again we get a deducibility relation by stipulating that ` A i [f:Ag cannot be embedded in a Henkin system (in the obvious sense: there is no Henkin system h0 ; U; Ri such that [g:Ag 0 ). This suggests the second observation, viz. that the relation between downward saturated sets and maximal consistent sets in classical logic is, in some sense, the same as that between Hintikka systems and Henkin systems in modal logic. In fact, Henkin systems have been more used than Hintikka systems in the study of modern modal logic. They were introduced independently by Makinson [1966], Cresswell [1967], Schutte [1968] and perhaps others. Dana Scott had similar ideas a little earlier and exerted a powerful in uence even though he did not publish; cf. Kaplan [1966]and Lemmon [1966; 1977]. Another early reference in this context is [Bayart, 1959]. 9 NATURAL DEDUCTION IN MODAL LOGIC Seen in a grand perspective, the Hintikka/Kripke deductive technique is an extension to modal logic of ideas introduced into the study of classical logic by P. Hertz and G. Gentzen. However, some have proposed a more straightforward extension of those ideas. In this section we will consider to what extent such an e ort is likely to succeed. Perhaps the most important work in the latter tradition is Prawitz [1965]. We will begin by giving a standard system of natural deduction for classical propositional logic which is similar to one found there. First there are the inference rules listed in Table 2. here `E' and `I' stand for `elimination' and `introduction' respectively, while `RAA' is short for `reductio ad absurdum'. Next we should give the deduction rules, that is, rules which legislate how inference rules may be used to produce deductions. But deduction rules are cumbersome to state in full detail. Therefore we will make a short-cut. (Readers who are led stray by this short-cut should consult [Prawitz, 1965].) As usual, ` A is de ne to mean that there is a deduction where A is the conclusion (`the bottom formula') and where contains all premises (`undischarged top formulas'). It is immediate that the deducibility relation ` will satisfy the common conditions (RX), (LM), (CUT), and (SB) de ned in Section 6. Now we declare|this is the short-cut|that the deduction rules are exactly what it takes to make certain that the conditions of Table 1 of the same section to be satis ed; thus ` is a Boolean deducibility relation. Notice that there is a one-to-one correspondence between the conditions of Table 1 and the inference rules of Table 2. In order to stress the connection we have used the same name for both condition and inference rule: in e ect
24
ROBERT BULL AND KRISTER SEGERBERG Table 2. A B A^B A^B (^I) (^E) A B A^B (A) (B ) A B A_B C C (_E) (_I A_B A_B c (A) A!B A B (! E) (! I) B A!B (A) :A A :A (:E) (:I) B :A (RAA)
(:A) A A
the condition explains how the inference rule is to be applied. This is needed, especially in the case of the so-called improper inference rules, that is, those containing parentheses: (_E) (!I), (:I), (RAA). What is at issue here is on exactly what premises a conclusion depends, and this can be gathered from the observations. The interest in the system thus presented is that the deducibility relation it de nes coincides with that of classical logic: ` A i tautologically implies A. In order to generalise it to modal logic, the most direct course is to try and devise rules for of the same kind as those governing the classical operators; in other words, to force the classical pattern on the modal operator. Thus one elimination and one introduction rule are called for, and their form is obvious:
A
A A A This is what Prawitz does. he considers ( E) a proper rule, which means that
( E)
(I)
(E) If
` A, then ` A. By contrast, (I) is very much improper:
taking it as a proper rule would literally trivialise modal logic. That is, if one accepts ( I) If
` A, then ` A,
BASIC MODAL LOGIC
25
then the resulting deducibility relation coincides with the trivial system de ned in Section 7. Thus in all interesting cases the deduction rule for (I) will have to contain some proviso if the trivial system is to be avoided. Prawitz discusses two possibilities. In one case every premise must be of the form A, in the other of the form either A or :A. If we adopt the convention according to which ?n = f?nA : A 2 g, where ? is any unary propositional operator, then we can give Prawitz's rules the following formulation: ( I)S4 If (I)S5
` A, then ` A, provided that, for some set , = . If ` A, then ` A, provided that, for some sets 0 and 1 , = 0 [ :1 .
The indexing of the rules is not fortuitous: Prawitz's two systems really coincide with Lewis' S4 and S5. However, it has proved diÆcult to extend this sort of analysis to the great multitude of other systems of modal logic. it seems fair to say that a deductive treatment congenial to modal logic is yet to be found, for Hilbert systems are not suited for the purpose of actual deduction, and in Hintikka/Kripke systems the alternativeness relation introduces an alien element which, moreover, can become quite unmanageable in special cases. The situation has given rise to various suggestions. One is that the Gentzen format, which works so well for truth-functional operators, should not be expected to work for intensional operators, which are far from truthfunctional. (But then Gentzen works well for intuitionistic logic which is not truth-functional either.) Another suggestion is that the great proliferation of modal logics is an epidemy from which modal logic ought to be cured: Gentzen methods work for the important systems, and the other should be abolished. `No wonder natural deduction does not work for unnatural systems!' We will now present a deductive system which explores a third alternative: trying to achieve generality at the expense of modifying the Gentzen format (there will be no special E- or I-rules for ). As far as we know, this system is new; there is a forerunner for some special cases in Segerberg [Segerberg, 1989]. Let us begin by trying to learn from the success of the Hintikka/Kripke venture. This success can perhaps be attributed to a certain division of labour: n Hintikka systems of downward saturated sets the classical conditions govern the relationship between the sets. How can this feature be imitated in the setting of natural deduction? The crux of the matter seems to be that any classically valid argument should remain valid in any modal context; the diÆculty is to explicate the italicised phrase. The solution seems to be to require that whenever tautologically implies A, then also n ` n A. This condition we recognise from Section 6 where it was introduced as the condition that the deducibility relation be strongly modal.
26
ROBERT BULL AND KRISTER SEGERBERG
The condition of strong modality may of course be adopted as a new rule in a sequent formulation of our logic. But as a proof-theoretic analysis such a move would not go very far: sequent theories, it would appear, are most naturally understood as meta-logics( theories about deductive systems). However that may be, here is the promised system. First there are the inference rules list in Table 3. For each rule in the old system there are now in nitely many rules. It is almost as if each power of would be an independent operator. As before, we do not state the deduction rules but are content to make a number of observations from which they can be reconstructed. We introduce the convention np
= fA : n A 2 Table 3.
(^E)n (_E) (! E)n (:E)n
n (A ^ B ) n(A ^ B ) n A n B (a)n (b)n n (A _ B ) C C n B n (A ! B )n A n B n (:A)n A n B (RAA)n
(^I)n (_I)n (! I)n (:I)n (:A)n A n A
g:
n An B n(A ^ B nA n B n(A _ B ) n(A _ B ) (A)n B n (A ! B ) (A)n :A n :A
Notice that the new rules (Table 3) have `( )n ', where the old (Table 2) have `( )'. this new notation also is explained by the observations listed in Table 4. It is easy to check that the deducibility relation de ned by this system is classical if is the only non-Boolean operator. Nor is it diÆcult to prove that it also satis es Scott's Rule (SC2): if ` A, then ` A. In fact, the system coincides with the minimal normal system K. The given system looks more complicated than the Hilbert type formulation of K in Section 6. But for deductive purposes it may be an alternative. If one would like to general modal logic within this framework, di erent logics would have to be characterised by special axioms. This means giving up the idea of nding characteristic rules for those systems. This is perhaps
BASIC MODAL LOGIC (^E)n (^I)n (_E)n (_In (! E)n (! I)n (:E)n (:I)n (RAA)n
27
Table 4. If ` n (A ^ B ), then ` n A and ` nB . n If ` n A and ` n B p, then ` (nAp^ B ). n n If ` (A _ B ) and ; A ` C and ; B ` C, then ` n C . If ` n A or ` n B , then ` n (A _ B ). If p` n (A ! B ) and ` n A, then ` nB . If n ; A ` B , then ` n (A ! B ). If p` n (:A) and ` n A, then ` nB . If n p ; A ` :A, then ` n :A. If n ; :A ` A, then ` n A.
a price worth paying, for|as remarked before|only exceptional systems would seem to be characterisable in terms of reasonably simple rules. The same point can perhaps be put in the following way. When we go to systems of traditional modal logic stronger than K, we should like to preserve classicalness, usually also Scott's Rule. The best way to do this appears to be to add more in the way of axioms rather than rules. In this manner, modal propositional logics become a bit like theories of ordinary predicate logic. Let be any set of modal formulas closed under substitution (that is, A 2 whenever A is a substitution instance of some A 2 ). Then we de ne L() as the logic got by adopting as a set of new axioms: ` A in L() i [ ` A in the basic system. It is obvious that L() will always be classical. Moreover, if is closed also under necessitation (that is, if ), then L() is a normal logic. In this fashion we preserve more of the Gentzen/Prawitz avour than the Hintikka/Kripke procedure does, while retaining full generality. 10 MODAL ALGEBRAS, FRAMES, GENERAL FRAMES The sections which follow survey the mainstream of technical modal logic. It is felt that the major results have been fairly represented. However, the selection of secondary results has been decidedly subjective, and another writer might well have chosen di erent topics. The best uni ed and detailed presentation in the area is [Goldblatt, 1976], which extends his PhD thesis of 1974 to account for the work of other logicians of that period. A good picture of an earlier stage is given in [Segerberg, 1971]. The startling di erence of content between these two `monographs' re ects the great increase of mathematical sophistication in technical modal logic at that time. This trend was led by Kit Fine, S. K. Thomason and R. I. Goldblatt. A more recent exploitation of algebra in the work of W. J. Blok will not be discussed in detail in this survey.
28
ROBERT BULL AND KRISTER SEGERBERG
A modal algebra A = hA; 0; 1; ; \; [; l; mi consists of a set A including 0 and 1, with functions ; \; [; l; m on it which satis es the conditions that hA; ; 1; ; \; [i is a Boolean algebra and
l1 = 1; l(a \ b) = la \ lb; ma = l a;
or, equivalently, that
m0 = 0; m(a [ b) = ma [ mb; la = m a: A valuation v on A is a function from the propositional formulas to the elements of the algebra which satis es the conditions
v(:A) = v(A); v(A ^ B ) = v(A) \ v(B ); v(A _ B ) = v(A) [ v(B ); v(A) = lv(A); v(A) = mv(A):
An algebraic `model' hA; vi is a modal algebra with a valuation on it, and A is true or veri ed in this `model' i v(A) = 1 A formula is true in a modal algebra i it is true in all `models' on that algebra (cf. Section 3). A frame F = hW; Ri consists of a set W and a binary relation R on W . A valuation V on F is a function such that V (A; x) 2 fT; F g for each propositional formula A and x 2 W , which satis es the conditions
V (:A; x) = T i V (A; x) = F; V (A ^ B; x) = T i V (A; x) = T and V (B; x) = T; V (A _ B; x) = T i V (A; x) = T or V (B; x) = T; V (A; x) = T i 8y(xRy ! V (A; y) = T ); V (A; x) = T i 9y(xRy ^ V (A; y) = T ):
A model hF; V i is a frame with a valuation on it, and A is satis ed in it i
V (A; x) = T for some x 2 W;
and is true or veri ed in it i
V (A; x) = T for each x 2 @: A formula is true or veri ed in a frame i it is true in all models on that frame. (Cf. Section 4.) A modal logic is normal i it includes all tautologies and the axiom
` (P ! Q) ! (P ! Q); and is closed under the rules of substitution for variables, modus ponens, and necessitation, if ` A then ` A:
BASIC MODAL LOGIC
29
An alternative to this axiom and necessitation is to take
` (P ! P ) ` (P ^ Q) ! (P ^ Q) and the rule from which
if
` A ! B then ` A ! B; ` (P ^ Q) ! (P ^ Q)
is derivable. (Cf. Section 6.) The minimal normal modal logic is called K, and its formulas are true in every modal logic and frame. Well-known formulas which are true in every modal algebra satisfying a corresponding equation, and every frame satisfying a corresponding rst-order condition on its relation, are shown in Table 5. Here a b is an abbreviation for a \ b = a or a [ b = b. It is convenient to label the extension of K with certain axioms by concatenating K with their labels, so that the extension of K with T and 4 is KT4, except that KT has usually been replaced by S. (Cf. Section 7.) Note that the modal algebras verifying S4 satisfy la and lla = la, being the closure algebras or interior algebras of McKinsey and Tarski [1944]. When added to K4, the formulas in Table 4 are true in every transitive frame satisfying the corresponding condition on its relation. (Here the condition for 3 is known as connectedness, and the condition for M asserts that after each point x there is a `second last' point y.) (Of these formulas, M was introduced in [McKinsey, 1945], 3 in [Dummett and Lemmon, 1959], and Grz in [Sobincinski, 1964], where it is shown that T and M are derivable in K4G4z. In fact 4 is derivable in KGrz by [van Benthem and Blok, 1978].) A frame F = hW; Ri determines a modal algebra F+ with carrier B(W ), where 0 = ; and 1 = W; ; \; [ are the usual set-theoretic operations, B(W ) is the set of subsets of W , and
lR a = fx : 8y(xRy ! y 2 a)g; mRa = fx : 9y(xRy ^ y 2 a)g: Writing v(A) for fx : V (A; x) = T g, each valuation V on F determines a subset fv(A) : A a formulag of B(W ). This subset is in fact the carrier of a subalgebra of F+ . For many purposes this is the most important point of a valuation, so that it is often preferable to consider general frames hW; R; P i, where P is the carrier of a subalgebra of hW; Ri+ . A formula is true or veri ed in a general frame hW; R; P i i it is true in each model hW; R; V i for which v is a function into P . (General frames were introduced in [Thomason, 1972], though they are foreshadowed in [Makinson, 1970] and in the secondary models of [Bull, 1969; Fine, 1970] and [Kaplan, 1970] for modal
30
ROBERT BULL AND KRISTER SEGERBERG
logics with propositional quanti ers.) The construction + can be extended to general frames F = hW; R; P i by taking the carrier of F+ to be P instead of B(W ). Label Formula T P ! P B P ! P 4 P ! P
Table 5. Equation Condition on R la a 8x(xRx) mla a 8x8y(xRy ! yRx) la lla 8x8y8z ((xRy ^ yRz ) ! xRz ) Table 6.
Label Formula Condition on R 3 (P ! Q) _ (Q ! P ) 8x8y8z ((xRy ^ xRz ) ! (yRz _ zRy)) M P ! P 8x9y(xRy ^ 8z 8w((yRz ^ yRw) ! z = w)) Grz ((P ! P ) ! P ) ! P There is no in nite chain x0 ; x1 ; x2 ; : : : with xi Rxi+1 and xi 6= xi+1 , for all i. A modal algebra A determines a general frame A+ = hWA ; RA ; PA i, where WA is the set of ultra lters of A,
xRA y i 8a(a 2 y ! ma 2 x) or, equivalently,
xRA y i 8a(la 2 x ! a 2 y); PA = ffx : a 2 xg : a 2 Ag; i.e. for each element of the modal algebra we take the set of ultra lters x containing it. (The lters of A are the subsets F of A which satisfy the conditions 1 2 F and not 0 2 F; if a; b 2 F then a \ b 2 F; if a 2 F and a b then b 2 F; and the ultra lters F also satisfy for each a 2 A; either a 2 F of
a2F
note that also not both a 2 F and a 2 F .) Here we write A] for the underlying frame hWA ; RA i. Note that if A is nite then PA is B(WA ), and A+ and A] coincide.
BASIC MODAL LOGIC
31
Clearly a formula is true in a model hF; V i i it is true in the algebraic `model' hF+ ; vi and hence true in F i it is true in F+ , since they have the same valuations. It can also be shown that a formula is true in an algebraic `model' hA; vi i it is true in hA] ; V i, where
V (A; x) = T i v(A) 2 x: (These constructions and results are due to Lemmon [1966], though they would also have been easy consequences of [Jonsson and Tarski, 1951].) In fact, each modal algebra A is isomorphic to (A+ )+ by similar arguments. Let us consider the properties of A+ . A set X A has the f.i.p. ( nite intersection property) i
a1 \ : : : \ an 6= 0; for each a1 ; : : : ; an 2 X: Each set X with the f.i.p. can easily be extended to a lter, which can in turn be extended to a maximal lter by Zorn's Lemma. Conversely each subset of a lter has the f.i.p. As a lemma, if X has the f.i.p. but X [ f ag does not, then a 2 F , for each lter F with X F . It follows immediately that each maximal lter is an ultra lter. As a second lemma following from the rst, b 2 F , for each ultra lter F with X F , i
a1 \ : : : \ an b; for some a1 ; : : : ; an 2 X: In both the results above we are concerned with the function : A ! PA with (a) = fF : F an ultra lter on A with a 2 F g: The crucial point is to show that
9G(F RA G ^ G 2 (a)) i F 2 (ma); in order to establish the properties of V (A; x) on A+, and the properties of mRA in (A+ )+ . This is immediate from left to right, using the de nition
F RA G i 8b(b 2 G ! mb 2 F ):
Going from right to left, suppose that the left-had side is false, so that
8G(F RA G ! a 2 G); for the ultra lter F . Using the alternative de nition
F RA G if f8b(lb 2 F
! b 2 G)
and taking X = fb : lb 2 F g, each ultra lter G with X G has a 2 G. Applying the second lemma above to X it is easy to show that l( a) 2 F , and hence not F 2 (ma), as required.
32
ROBERT BULL AND KRISTER SEGERBERG
However, (F+ )+ is not in general `isomorphic' to F, for a general frame F. Therefore we need a subclass of the general frames which will include all the general frames A+ and be closed under this pair of operations. In the terminology of [Goldblatt, 1976], given a general frame hW; R; P i write
P x = fS 2 P : x 2 S g; MP x = fmRS : x 2 S ^ S 2 P g: Then Thomason [1972] de nes the conditions if P x = P y then x = y (1-re nement); if MP y P x then xRy (2-re nement); and calls a general frame re ned when it satis es both of them. In e ect a general frame hW; R; P i has enough propositions in P to determine W when it is 1-re ned, and enough propositions in P to determine R when it is 2-re ned. (Kit Fine independently introduced analogous conditions di erentiated, tight, and natural for models.) Clearly each general frame A+ determined by a modal algebra A is re ned. As Thomason [1972] shows, for each general frame hW; R; P i there is a re ned general frame for which precisely the same formulas are true. One rst replaces R by R0 with xR0 y i (8S 2 P )(y 2 S ! mRS 2 x); so that hW; R; ; P i+ is the same as hW; R; P i+ but 2-re nement is satis ed. Then an equivalence relation w is de ned on W by taking
x w y i (8S 2 P )(x 2 S y 2 S ): This is a congruence on hW; R0 ; P i in the sense that
if x1 w x2 and y1 w y2 then x1 R0 y1 = x2 R0 y2 : Now the quotient general frame hW= w; R0 = w; P= wi with
W= w= f[x] : x 2 W g; [x]R; = w [y] i xR0 y; P= w= ff[x] : x 2 S g : S 2 P g; is re ned, and hW= w; R0 = w; P= wi+ is isomorphic to hW; R0 ; P i+ . Thus these two steps yield a re ned general frame with an associated modal algebra which is isomorphic to that for the given general frame. Fine [1975] introduces saturation or compactness conditions on models analogous to \F 6= ?, for each ultra lter F of hW; R; P i+ , and
\fmRS : S 2 F g mR(\F ) (2-saturation):
BASIC MODAL LOGIC
33
Since each x 2 W generates an ultra lter P x, this rst condition is equivalent to F = P x; for some xW (1-saturation) for each ultra lter F of hW; R; P i+ . Note that applying 2-saturation to the ultra lter P x yields if MP y P x then 9z (xRz ^ P z = P y) (20 -saturation): In Goldblatt [1976] it is shown that 20 -saturation is equivalent to 2-saturation in the presence of 1-saturation, and equivalent to 2-re nement in the presence of 1-re nement. Goldblatt [1976] then introduces the descriptive general frames as the re ned general frames which also satisfy 1-saturation and, hence, 2-saturation. For each modal algebra A the general frame A+ is descriptive. To see that 1-saturation is satis ed we must consider each ultra lter F of hWA ; RA ; PA i+ , i.e. of PA with members
(a) = fF : F an ultra lter of A with a 2 F g; for each a 2 A. The required x 2 WA with F = PA x is fa : (a) 2 F g. It can also be shown that each descriptive general frame F is `isomorphic' to (F+ )+ , so that the descriptive frames are the required `duals' of the modal algebras. In Goldblatt [1976] this duality is expressed in terms of category theory, which involves the appropriate morphisms between structures as well as the structures themselves. The appropriate frame morphisms are a slight extension of the pseudo-epimorphisms of Segerberg [1968], which have to be onto. Given frames F = hW; Ri and F0 = hW 0 ; R0i; : W ! W 0 is a frame morphism i if xRy then (x)R0 (y); if (x)R0 z then 9y(xRy ^ (y) = z ): Frame morphisms are extended to models hW; R; V i and general frames hW; R; P i by taking
v(P ) = 1 [v0 (P )] = fx 2 W : (x) 2 v0 (P )g; for each propositional variable P , if S 2 P 0 then 1 [S ] = fx 2 W : (x) 2 S g 2 P: As in Segerberg [1968],
V (A; x) = T i V 0 (A0 ; (x)) = T; by an easy induction on the construction of A. The induction basis uses the condition above on V 0 . For the step on , the rst condition on frame
34
ROBERT BULL AND KRISTER SEGERBERG
morphisms shows that if V (B; x) = F , then V 0 (B; (x)) = F , and the second condition shows that if V 0 (B; (x)) = F then V (B; x) = F . Now the descriptive frames F and (F+ )+ can be shown to be frame isomor+ phic. For each descriptive frame F = hW; R; P i, the function : W ! W F with (x) = P x; for each x 2 W; is a one-one frame morphism from F onto (F+ )+ . To see this, is oneone because F is 1-re ned, and not because F is 1-saturated. Also, by the de nition of lR and 2-re nement, xRy i (8S 2 P )((S 2 P y ! mRS 2 P x) i P xRF+ P y i (x)RF+ (y). To complete the proof that F and (F+ )+ are frame isomorphic, i.e. that and 1 are general frame morphisms, it can be shown that S 2 P i [S ] 2 P F+ . To establish the category-theoretic contravariant duality, correspondences must be established between homomorphisms of modal algebras and general frame morphisms of descriptive general frames, with the functions applied in opposite directions. Given general frames F = hU; R; P i; G = hV; S; Qi and a general frame morphism : F ! G, de ne + : G+ ! F+ by
+ (S ) = 1 [S ]; for each S 2 Q; where 1 [S ] 2 P by the third condition. It is easy to show that + is a homomorphism. Given modal algebras A; B and a homomorphism : A ! B, de ne + : B+ ! A+ by + (x) = fa 2 A :
(a) 2 xg; for each x 2 WB :
This set is an ultra lter in WA , and + satis es the conditions on general frame morphisms. For the rst condition, if xRB y and la 2 + (x) then a 2 + (y). For the second condition, if + (x)RA z then fa : Bla 2 xg [ f (b) : b 2 z g can be shown to have the f.i.p. Therefore it can be extended to an ultra lter y, which satis es xRB y and + (y) = z . For the third condition, if
S = fF : F an ultra lter of A with a 2 F g in PA , then +
1 [S ] = fG : G
an ultra lter of B with (a) 2 Gg
in PB . The category of modal algebras is a variety, and varieties are characterised by being closed under homomorphic images, subalgebras and direct products. So what are the corresponding constructions in the contravariantly dual category of descriptive frames? Frame-morphic images correspond to sub- algebras.
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Subframes correspond to homomorphic images, where hW 0 ; R0 ; P 0 i is a subframe of hW; R; P i i W 0 is a subset of W satisfying the condition if x 2 W 0 and xRy then y 2 W 0 ;
R0 is the restriction of R to W 0 , and P 0 is fS \ W 0 : S 2 P g. The generated submodels hWx ; Rx; Vx i of Segerberg [1970] are a special case of subframes. Here, for x 2 W , Wx = fyn : XRy1 ^ : : : ^ yn 1 Ryn; for some y1 ; : : : ; yn
1 g;
and Rx ; Vx are the restrictions of R; V to RWx . (In the context of Segerberg [1970] R is transitive, so that it suÆces to take Wx = fy : xRyg.) Clearly a formula is true in hW; R; V i i it is true in all the generated submodels hWx ; Rx ; Vx i, a surprisingly important fact as we shall see. Note that if hW; R; P i is re ned or descriptive, then so is each hWx ; Rx ; Px i. For 1-saturation use the fact that the ultra lters of hWx ; Rx ; Px i+ are the restrictions of the ultra lters of hW; R; P i+ to subsets of Wx . Disjoint unions correspond to direct products, in which we consider a set of general frames hWi ; Ri ; Pi i, for i 2 I , for which each Wi and Wj are disjoint. (This can always be achieved by attaching indices.) The disjoint union hW; R; P i then has W = [i2I Wi ; R = [i2I Ri , and
S 2 P i S \ Wi 2 Pi ; for each i 2 I: It is easy to show that if each hWi ; Ri ; Pi i is re ned, then so is their disjoint union. Goldblatt [1976, Section 9] shows that the disjoint union preserves 1-saturation if I is nite, but not if it is in nite. The attempt to characterise the class of descriptive frames in terms dual to the usual characterisation of varieties fails in view of this point. (Category-theoretic duality is not always as good as it might sound!) Section 12 of [Goldblatt, 1976] solves this problem by using another characterisation of varieties, as being closed under homomorphic images, subalgebras, nite direct products, and unions of chains. Onto inverse limits correspond to unions of chains, where the inverse limit of a directed set of descriptive frames is a complex construction set out in Section 11 of [Goldblatt, 1976]. Another important construction in varieties is Birkho 's subdirect product, A being a subdirect product of the modal algebras Ai with i 2 I i it is isomorphic to a subalgebra of their direct product which has the following property. Since A is a subalgebra of i2I Ai , there is a one{one homomorphism from A into i2I Ai . For each i 2 I there is a projection i from i2I Ai onto Ai . The condition on the subdirect product is that the homomorphisms i Æ from A into each Ai be onto, so that each Ai is a homomorphic image of A. Using this condition it is easy to show that a
36
ROBERT BULL AND KRISTER SEGERBERG
formula is true in A i it is true in each Ai . Each homomorphic image of a modal algebra A is isomorphic to a quotient A=F , where F is an open lter of A, i.e. a lter satisfying the condition if a 2 F then la 2 F: The quotient is de ned by taking the equivalence relation
a ' b i ( a)[) \ (a [ ( b)) 2 F and then taking A=F to be f[a] : a 2 Ag with l[a] = [la], etc. In view of this we can restrict attention to Ai 's of the form A=Fi for Fi an open lter of A. Birkho de ned a modal algebra A to be subdirectly reducible i it is a subdirect product of quotients A=Fi with Fi nontrivial, and showed that every modal algebra is subdirectly reducible to subdirectly irreducible algebras. If some nonunit element a of A is in every nontrivial open lter F then [a] = [1] in each A=Fi , so that A cannot be a subalgebra of i2I A=Fi . Thus v is subdirectly irreducible already. Otherwise each non-unit member a of A lies outside some nontrivial open lter, and applying Zorn's Lemma yields a (nontrivial) maximal open lter Fa among those not containing a. Now A is subdirectly reducible to the A=Fa's, noting that if b 6= c and a = (( b) [ c) \ (b [ ( c)) 6= 1 then [b] 6= [c] in A=Fa . Here each A=Fa is subdirectly irreducible, since [a] 2 F for each nontrivial lter F of A=Fa by the maximality of Fa among the open lters of A not containing a. In view of Birkho 's theorem, we can restrict attention to modal algebras with some nonunit element in every nontrivial open lter, when verifying formulas in a modal logic. (The importance of this result in modal logic lies in its use in the recent work of W. J. Blok.) In a closure or interior algebra, an open lter is determined by its open elements, so that a closure or interior algebra is subdirectly irreducible i it has a maximum nonunit open element, or equivalently, a minimum nonzero closed element. In such an algebra, if la [ lb = 1 then la = 1 or lb = 1; a condition we shall use later. It is easy to see that a modal algebra hW; Ri+ is subdirectly reducible to the algebras hWx ; Rxi+ for x 2 W , which are subdirectly irreducible. In view of the contravariant duality between modal algebras and descriptive general frames, what theorem for the latter corresponds to Birkho 's Theorem? Note that the lack of a disjoint union of in nitely many descriptive frames will block a dualisation of Birkho 's proof. Let us say that a general frame F is the subdirect sum of general frames Fi with i 2 I i it is a frame-morphic image of their disjoint union i2I Fi which has the following property. Since F is a frame- morphic image of i2I Fi there is a frame
BASIC MODAL LOGIC
37
morphism from i2I Fi onto F. For each i 2 I there is embedding frame morphism i from Fi into i2I Fi . The condition on the subdirect sums is that the frame morphisms Æ i from each Fi into F be embedding, so that each Fi is isomorphic to a subframe of F. In view of this we can restrict attention to Fi 's which are subframes of F. Again it is easy to show that a formula is true in F i it is true in each Fi . Say that a general frame is subdirectly reducible i it is a subdirect sum of its proper subframes. Then it is clear that a general frame is subdirectly reducible to its generated subframes, and that these are subdirectly irreducible. So although the disjoint union of descriptive frames is not usually descriptive, Birkho 's deep result for modal algebras is analogous to the easy, known result that a formula is true in a descriptive general frame i it is true in its generated subframes, which are again descriptive! 11 CANONICAL STRUCTURES So far we have not constructed any modal algebras or frames. given a normal modal logic L, de ne an equivalence relation 'L on formulas by taking
B 'L C i
`L B C:
Then the canonical modal algebra AL is constructed by taking
AL 0 [B ]L [B ]L \ [C ]L [B ]L [ [C ]L l[B ]L m[B ]L
= = = = = = =
f[B ]L : B a formulag; [P ^ :P ]L and 1 = [(:P ) _ P )]L ; [:B ]L ; [B ^ C ]L ; [B _ C ]L ; [B ]L ; [B ]L :
That AL is indeed a modal algebra is easily shown using the de ning axioms and rules of normal modal logics. De ning a valuation vL by
vL (B ) = [B ]L ; for each formulaB; we have
vL (B ) = 1 i B 2 L; so that the canonical algebraic `model' hAL ; vL i characterises the normal modal logic L. Further, for each valuation v on AL ; v(B ) is [C ]L for some substitution instance C of B , so that B is true in AL i it is in l. Given a normal modal logic L, a set X of formulas is inconsistent i `L :(A1 ^: : :^An ), for some A1 ; : : : ; An 2 X , and is consistent otherwise. (Note the analogy between consistency and the f.i.p. The existence of maximal
38
ROBERT BULL AND KRISTER SEGERBERG
consistent sets is proved with Zorn's Lemma, just as for that of maximal lters. However, if L has only countably many propositional variables, then a more elementary construction due to Henkin can be used.) De ne the canonical frame hWL ; RL i by taking WL to be the set of maximal consistent set of formulas, and taking
F RL G i 8A(A 2 G ! A 2 F ) or, equivalently,
F RL G i 8A(A 2 F ! A 2 G): Note the analogy with the construction of the frame A] from a modal algebra A. De ne a valuation VL by taking VL (B; F ) = T i B 2 F; for each formula B; a de nition which is shown to be sound by an induction on the construction of B . For the induction step on B = C it must be shown that
9G(F RL G ^ G 2 vL (C )) i F 2 vL (C ): This proof is exactly analogous to the one used when showing that (A+ )+ is isomorphic to A, using the de ning axioms and rules of normal modal logics. Now
vL (B; F ) = T; for each F
2 WL ;
i B 2 L;
since each consistent set of formulas can be extended to a member of WL , so that the canonical model hWL ; RL ; VL i characterises the normal modal logic L. Taking PL = fvL(B ) : B a formulag gives the canonical general frame hWL ; RL ; PL i. For each valuation V on this frame, v(B ) is vL (C ), for some substitution instance C of B , so that B is true in hWL ; RL ; PL i i it is in L. In fact hWL ; RL ; PL i is AL+ , so that it has a descriptive general frame characterising l. It does not follow that the canonical frame hWL ; RL i itself characterises the normal modal logic L. Nonetheless, in a number of cases it can be shown that RL satis es some condition for frames to verify l, so that hWL ; RL i does characterise L. In particular, the canonical frames for KT, KB, K4, and the logics obtained by combining these axioms, satisfy the rst-order conditions on R given in Section 10. (These completeness proofs were given independently in [Lemmon, 1977], written in 1966, and in [Makinson, 1966].) These partial results suggest a number of important problems which have provided the main motivation for modal logic in the 1970s. Under what conditions is a formula true on the underlying frame hW; Ri when it is true on a model hW; R; V i or a general frame hW; R; P i? Are there logics which
BASIC MODAL LOGIC
39
are not characterised by the ordinary frames which verify them? What is the relationship between modal axioms and rst-order conditions on R in the frames hW; Ri? Are there formulas not characterised by the class of frames satisfying some rst- order condition? Generalising the problem of completeness, often a problem can be easily solved for descriptive general frames by their duality with the variety of modal algebras, an the diÆculty lies in transferring the problem to the underlying frames. We shall return to answers to these questions after studying various particular logics which have attracted attention. 12 THE F. M. P. AND FILTRATIONS A logic L is said to have the f.m.p. ( nite model property) i , for each formula ; `L A i A is true in each nite modal algebra or frame which veri es the formulas of L. Thus in showing that L has the f.m.p. we must nd, for each nonthesis A, a nite modal algebra or frame which veri es L but does not verify A. Note that modal algebras and frames are interchangeable here. For if F is a nite frame, then of course F+ is a nite modal algebra, and if A is a nite modal algebra, then A] = A+ is a nite frame. The f.m.p. is important, among other reasons, for giving decidability to a nitely axiomatised normal modal logic. For as Harrop pointed out, we can construct the countably many nite models in some order, checking each one for verifying the nitely many axioms and the given formula A. Again a problem of independence is raised, which will be considered in a later section: are there logics which are characterised by frames, but not by the nite frames which verify them? (The position of the logics characterised by one nite model in the lattice of modal logics is investigated in detail in [Blok, 1980]. The normal modal logics immediately below these, which also have the f.m.p., are the subject of [Block, 1980a].) We now consider a pair of methods for constructing nite modal algebras and frames from given structures, both known as ltration. Consider an algebraic `model' hA; vi and a formula A with v(A) 6= 1. Let fA1 ; : : : ; An g be a nite set of formulas including A and closed under subformulas, and let hB; 0; 1; ; \; [i be the subalgebra of hA; 0; 1; ; \; [i generated by fv(A1 ); : : : ; v(An )g, noting that it is non-trivial and nite. (Usually A1 ; : : : ; An are A and its subformulas, but sometimes some larger set is preferable.) This Boolean algebra is extended to a nite modal algebra B = hB; 0; 1; ; \; l0; m0i by taking l0 b = [fla 2 B : a 2 B ^ a bg; m0b = \fmc 2 B : c 2 B ^ b cg; (In the case of a closure or interior algebra A; m is determined by the closed elements of A and l by the open elements. Therefore it suÆces to take l0b
40
ROBERT BULL AND KRISTER SEGERBERG
to be the union of the open elements of B contained by b, and take m0b to be the intersection of the closed elements of B containing b.) In particular, if lb 2 B then l0 b = lb; if mb 2 B then m0b = mb; for each b 2 B . Now B is indeed a modal algebra, satisfying l0 1 = 1 and l0 (a \ b) = l0a \ l0b; m00 = 0 and m0(a [ b) = m0a [ m0b;
using distibutivity and the fact that A satis es these conditions. Construct a valuation w on B by taking w(P ) = v(P )\B , for each propositional variable P in A1 ; : : : ; An , and applying the de ning conditions for valuations. We now have a(Ai ) = v(Ai ) for i = 1; : : : ; n; so that w(A) 6= 1 in the ltered algebraic `model' hB; wi. It is not in general true that hB; wi, let alone B, veri es a logic L veri ed by A. Nonetheless, in a number of cases it can be shown that each ltration B of A satis es some condition for modal algebras to verify L. In particular, ltrations of algebraic `models' verifying KT, KB, Kr, and the logics obtained by combining these axioms, again satisfy the equations given in Section 10. It follows that these logics have the f.m.p. and are decidable, being characterised by the ltrations of their canonical modal algebras. (This technique was introduced in [McKinsey, 1941], and extended in [Lemmon, 1966], to establish many decidability results.) Now consider a model hW; R; V i and a formula A with v(A) 6= W . Again let fA1 ; : : : ; An g be a nite set of formulas including A and closed under subformulas. De ne an equivalence relation ' on W by taking
x ' y i V (Ai ; x) = V (Ai ; y); for i = 1; : : : ; n; so that W is partitioned into a nite set W 0 of equivalence classes [x] under '. Consider nite frames hW 0 ; R0 i satisfying the conditions if xRy then [x]R0 [y]; if [x]R; [y] then [if V (Ai ; x) = T; for Ai = Aj ; then V (Aj ; y) = T ]; for i = 1; : : : ; n:
(A suitable condition in terms of could equally well be used.) There are a number of relations R0 on W 0 which satisfy these conditions, e.g. R with [x]R[y] i [if V (Ai ; x) = T; for Ai = Aj ; then V (Aj ; y) = T ]; for i = 1; : : : ; n: This relation satis es the rst conditions, since if xRy then the right-hand side of the de ning condition holds for all formulas B = C . This is in fact
BASIC MODAL LOGIC
41
the largest such relation R0 . The smallest is the intersection R of all such relations, which again satis es the two conditions. Construct a valuation V 0 on hW 0 ; R0 i by taking V ; (P; [x]) = V (P; x) for each propositional variable P in A1 ; : : : ; An , and applying the de ning conditions for valuations. It can now be shown that V 0 (Ai ; [x]) = V (Ai ; x); for i = 1; : : : ; n; by induction on the construction of formulas, so that v0 (A) 6= W 0 in the ltered model hW 0 ; R0 ; V 0 i. for the induction step on , consider Ai = Aj . If V (Aj ; x) = T and [x]R0 [y] then V (Aj ; y) = T by the second condition on R0 , and V 0 (Aj [y]) = T by the induction hypothesis. Applying this to each [y] we have V 0 (Aj ; [x]) = T . If V 0 (Aj ; [x]) = T and xy, then [x]R0 [y] by the rst condition on R, so that V 0 (Aj ; [y]) = T and V (Aj ; y) = T by the induction hypothesis. Applying this to each y we have V (Aj ; x) = T . Again it is not in general true that hW 0 ; R0; V 0 i, let alone hW 0 ; R0 i, veri es a logic L veri ed by hW; R; V i. Nonetheless, in a number of cases it can be shown that R0 satis es some condition for frames to verify L. In particular V i of models verifying KT, KB, K4, and the logics ltrations hW 0 ; R; obtained by combining these axioms, again satisfy the rst- order conditions on R given in Section 10. This gives alternative proofs of the decidability V i was introduced in [Lemmon, of these logics. (The construction hW 0 ; R; 0 0 1977] and was generalised to hW ; R ; V 0 i in [Segerberg, 1968].) In many more cases a further step after ltration, or a variation on the construction V i to suit the axioms involved, will yield a nite frame hW 0 ; R0 i hW 0 ; R; verifying the logic concerned. We shall see some of these techniques in the following sections. 13 UNRAVELLING AND BULLDOZING (The technique of unravelling was introduced in [Dummett and Lemmon, 1959] and used extensively in [Sahlqvist, 1975], apparently without knowledge of the earlier paper.) Consider a frame hW; Ri which is generated by w0 2 W , so that w0 Rw1 ; : : : ; wn 1 Rwn , for some w1 ; : : : ; wn 1 , for each other wn 2 W . Construct a new frame hW ; R i by taking hw0 ; : : : ; wn i 2 W i w1 ; : : : ; wn 2 W and w0 Rw1 ; : : : ; wn 1 Rwn ; hw0 ; : : : ; wm iR hw0 ; : : : ; wn i i hw0 ; : : : ; wn = hw0 ; : : : ; wm i hwn i: Thus R has been unravelled in the sense that if un 1 Rwn and vn 1 Rwn then wn is replaced by hw0 ; : : : ; un 1; wn i and hw0 ; : : : ; vn 1 ; wn i with hw0 ; : : : ; un 1iR hw0 ; : : : ; un 1 ; wn i and hw0 ; : : : ; vn 1 iR hw0 ; : : : ; vn 1 ; wn i.
42
ROBERT BULL AND KRISTER SEGERBERG
Unravelling is extended to models hW; R; V i by taking V (P; hw0 ; : : : ; wn i) = V (P; wn ) for each propositional variable P , and applying the de ning conditions for valuations. It is easy to show that V (A; hw0 ; : : : ; wn i) = V (A; wn ); for each formula A; by induction on the construction of A. Since K is characterised by the nite frames using ltrations, it is now characterised by the unravelled frames. Note that these unravelled frames are irre exive, asymmetrical, and intransitive. Therefore none of these conditions characterise a proper extension of K. A frame hW; Ri could be de ned to be a tree i there is w0 2 W and a relation S on W satisfying the conditions, for each wn 2 W other than w0 , only one wn 1 2 W with wn 1 Swn , for some w1 ; : : : ; 2wn 1 2 W ; there is only one wn 1 2 W with wn 1 Swn ' and wm Rwn if wm Swm+1 ; : : : ; wn 1 Swn , for some Rwm+1 ; : : : ; wn 1 2 W . A tree could be re exive or irre exive. Then trees cold be obtained by taking the transitive closures of unravelled frames, with or without the re exive closure as required. (Sahlqvist [1975] uses a more general notion of tree, and proves a number of results concerning them.) The clusters of a transitive frame hW; Ri are de ned in [Segerberg, 1971] to be the equivalence classes of W under the equivalence relation
x ' y i (xRy ^ yRx) _ x = y: Clusters are divided into three kinds: proper, with at least two elements, all re exive; simple, with one re exive element; and degenerate with one irre exive element. Note that when a nondegenerate cluster is unravelled, it will give rise to many branches of hW ; R i in which the members of the cluster are repeated. Thus unravelling imposes asymmetry on frames, sometimes without losing the property of characterising a given logic. Another technique for removing nondegenerate clusters and so imposing asymmetry is the bulldozing of Segerberg [1970]. Let us suppose that the logic concerned is an extension of K4 which has countably many propositional variables P0 ; P1 ; P2 ; : : : and consider a generated transitive frame hW; Ri. Construct a new frame hW 0 ; R0 i by rst replacing each nondegenerate cluster C of W by
C 0 = fhx; ii : x 2 C ^ i = 0; 1; 2; : : :g; and replacing each degenerate cluster C = fxg of W by fhx; 0ig, to obtain W 0 . De ne R0 on W 0 by taking
hx; iiR0 hy; j i i either not x ' y and xRy orx ' y and i < j or x ' y and i = j and xrC y;
BASIC MODAL LOGIC
43
where rC is an arbitrary strict ordering of the proper cluster C with x; y 2 C . Thus each nondegenerate cluster C of W is `bulldozed' into an in nite set C 0 on which R0 is a strict linear ordering. In hC 0 ; R0i a copy hy; j i of y occurs after each copy hx; ii of x, for each x; y 2 C . If hW; Ri is re exive, so that there are no degenerate clusters, modify the construction as follows to make hW 0 ; R0 i re exive as well. Form C 0 as above only for proper clusters C , and replace simple clusters C = fxg by C 0 = (hx; 0i); and add the clause `or x = y' to the right- hand side of the de nition of R00 . In this case each proper cluster C is `bulldozed' into an in nite set C0 on which R0 is a linear ordering. Bulldozing is extended to models hW; R; V i by taking V 0 (pj ; hx; ii) = V (Pj ; x); for j = 0; 1; 2; : : : ; and applying the de ning conditions for valuations. Now V 0 (A; hx; ii) = V (A; x); for each formula A by induction on the construction of A. (For the induction step on ; V 0 (B; hx; ii) = F i V 0 (B; hy; j i) = F , for some hy; j i 2 W 0 with hx; iiR0 hy; j i, i V (B; y) = F , for some y 2 W with hx; iiR0 hy; j i, (by the induction hypothesis) i V (B; y) = F , for some y 2 W with xRy, (by the de nition of R0 if not x ' y, and by a remark above if x ' y) i V (B; x) = F .) Now consider any normal modal logic L containing S4.3. First we shall use `L (A ! B ) _ (B ! A) to show that the canonical frame hWL ; RL i is connected with 8x8y8z ((xRLy ^ xRL z ) ! (yRL z _ zRLy)): Let us suppose that we have maximal consistent sets F; G; H of L with F RL G; F RL H but not GRL H and not HRL G, and obtain a contradiction. Since not GRL H there is some A 2 G with not A 2 H , and since not HRL G there is some B 2 H with not B 2 G. Just as maximal lters are ultra lters, it can be shown that a maximal consistent set F satis es A 2 F or :A 2 F; for each formula A: It is easy to deduce that if A _ B 2 F then A 2 F or B 2 F; for all formulas A; B: Therefore `L (A ! B ) _ (B ! A) implies (A ! B ) 2 F or (B ! A) 2 F implies A; A ! B 2 G or squareB; B ! A 2 H implies B 2 G or A 2 H implies B 2 G or A 2 H
44
ROBERT BULL AND KRISTER SEGERBERG
(since `L P ! P )|the required contradiction. The canonical frame for L is also re exive and transitive. Clearly its generated subframes hWL ; RLx i satisfy 8y8z (yRz _ zRy), and bulldozing adds 8y8z (y 6= z ! :(yRz ^ zRy))
to these conditions in hWl0z ; RL0 x i, so that RL0 x is a linear ordering in the full sense. Often such frames still verify L, so that they characterise it, in particular when L is S4.3 itself. (Segerberg [1970] proves the analogous result for extensions L of K4.3, using ltrations of the canonical frame which are connected although the canonical frame itself is not. Many other results along these lines are obtained in [Segerberg, 1970; Segerberg, 1971] and [Sahlqvist, 1975].) 14 S4.1 AND S4GRZ (K4.1 = K4M and S4.1 = KT4M were shown to be characterised by frames satisfying the appropriate conditions in [Lemmon, 1977], written in 1966, and S4.1 was shown to be characterised by the appropriate nite frames in [Segerberg, 1968]. Independently Bull [1967] gave an algebraic proof of the f.m.p. for S4.1, and described a characteristic frame for it. The extension S4 Grz of S4.1 was shown to be characterised by the appropriate nite frames in [Segerberg, 1971].) Bull [1967] begins by showing that S4.1 can also be axiomatised by extending S4 with either of the rules if if
` A; ` B then ` (A ^ B ); ` A; then ` A:
Although a ltration B of the canonical modal algebra for S4.1 may not verify these rules, an extension B+ of B an be constructed which does. (Thinking in terms of hW; Ri+ , where R satis es the conditions in Section 10 for verifying S4.1, we need to isolate the R-last points of W . This is achieved by the following trick.) Taking aB = [f(mb b) : b 2 B g, where the join and m are that of AS4:1, we shall consider separately what happens in aB and what happens in aB (the set of R-last points, in e ect). Let hB + ; 0; 1; ; \; [; l0; m0i be the ltration of AS4:1 generated by B [ faB g, and de ne l+ b = (l0 b \ aB ) [ (b aB ); m+b = (m0 b \ aB ) [ (b aB );
for each b 2 B+ . The required modal algebra B+ is hB + ; 0; 1; ; \; [; l+; m+i. The canonical modal algebra AS4:1 and the ltrations of it are closure or interior algebras, and it can be shown that B+ is as well. Using the fact
BASIC MODAL LOGIC
45
that AS4:1 veri es the rst rule above, it can be shown that l+aB = 0. From this it follows that if l= b = 0 then l+ m+b = 0; so that the second rule above is indeed veri ed by B+ . Finally it can be shown that l+ b = lb and m+b = mb if these are in B , so that B+ rejects the given formula A rejected by B. Thus S4.1 is characterised by these nite closure or interior algebras B+ . For the re exive and transitive frames which verify S4, the condition given in Section 10 for hW; Ri to verify M becomes
8x9y(xRy ^ 8z (yRz ! y = z )); i.e. that each point x has an R-last point y after it. For nite frames it suÆces that each nal cluster be simple. It is well-known that in S4 the only non-equivalent formulas obtained by applying :; ; to P are P itself, P; P; P and P; P; P , and the negations of these. Thus in S4.1 there are only 10 of these `modalities'. In forming a ltration V i let us take fA1 ; : : : ; An g to be the nite closure of A and its hW 0 ; R; subformulas under these modalities of S4.1. Now these ltrations of the canonical model hWS4:1 ; RS4:1; VS4:1i have all their nal clusters simple, 0 in a nal cluster of and so characterise S4.1. For consider [F ]; [G] 2 WS4 :1 0 such a frame hWS4:1; RS4:1i, with Ai 2 F . Since [F ] is in a nal cluster, for each [H ] with [F ]RS4:1[H ] we have [H ]RS4:1[F ], and so Ai 2 H . Therefore Ai 2 F , as well as Ai ! Ai 2 F , so that Ai 2 F . Now there must be an H with [F ]RS4:1[H ] and Ai 2 H . But since R is transitive and this is a nal cluster, [H ]R S4:1[G] and so Ai 2 G. We have shown that if Ai 2 F then Ai 2 G, so that extending the argument yields
Ai 2 F i Ai 2 G; for i = 1; : : : ; n; i.e. [F ] = [G], as required. For nite re exive and transitive frames, to satisfy the condition given in Section 10 for hW; Ri to satisfy Grz, it suÆces that each cluster be simple. V i of the canonical model for S4 Grz may Unfortunately ltrations hW 0 ; R; not have this property, and it is necessary to replace R by a suitable asym0 ; R Grz ; VGrz i, say metric R0 . Given a cluster C of re exive, transitive hWGrz that x 2 C is `virtually last' in C i there is some Fx 2 x with
8G((Fx RGrz G ^ [G] 2 C ) ! x = [G]): It is clear that the member of a simple cluster of this frame is virtually last. In [Segerberg, 1971, Chapter II, Section 3], it is shown by a diÆcult argument that each proper cluster has a virtually last element as well. 0 on W 0 by taking xR0 y i either Assuming this result, de ne RGrz Grz Grz not x ' y and xRGrz y or x ' y and xrC y, where rC is an arbitrary ordering
46
ROBERT BULL AND KRISTER SEGERBERG
of C in which the rC -last member of nite C is virtually last in C . Now 0 RGrz , and hW 0 ; R0 i has only simple clusters and so veri es RGrz 0 onGrz 0 Grz 0 i by taking V 0 (P; [F ]) = VGrz (P; F ) S4Grz. De ne VGrz hWGrz ; RGrz Grz for each propositional variable P in fA1 ; : : : ; An g, and applying the de ning conditions for valuations. It can be shown that 0 (A; [F ]) = VGrz (Ai ; [F ]); for i = 1; : : : ; n; VGrz 0 ; R0 ; V 0 i rejects the by induction on their construction, so that hWGrz Grz Grz given formula as well. For the induction step on , consider Ai = Aj , one 0 RGrz . For the diÆcult direction take x direction being easy with RGrz to be a cluster C with y virtually last in C , and then VGrz (Aj ; x) = F implies VGrz (Aj ; y) = F implies VGrz (Aj ; Fy ) = F and 8G((Fy RGrz G ^ [G] 2 C ) ! y = [G]) implies VGrz (Aj ; G) = F; for some G with either Fy RGrz G and not [G] 2 C or y = [G] 2 C; 0 (Aj ; [G]) = F and either not y ' [G] implies VGrz and yRGrz [G] or y ' [G] and yrC [G] 0 (Aj ; [G]) = F and xR0 y and yR0 [G] implies VGrz Grz Grz 0 (Aj ; x) = F: implies VGrz With what natural axiom can S4.1 be extended to S4Grz? Clearly we need a formula A such that S4A is characterised by the nite re exive-andtransitive frames in which all but the nal clusters are simple. Segerberg [1971, Chapter II, Section 3] shows that Dum:P ! (((P ! P ) ! P ) ! P ) (i.e. P ! Grz) has this property, so that S4Grz is S4.1Dum. 15 THE TRANSITIVE LOGICS OF FINITE DEPTH Given a frame hW; Ri, say that x1 ; : : : ; xr 2 W form a chain i xi Rxi+1 and xi 6= xi+1 and not xi+1 Rxi , for i = 1; : : : ; r 1. (Thus x1 ; : : : ; xr come from a chain of distinct clusters. We include hx1 i as a chain.) Say that x1 has a rank r in hW; Ri i there is a chain hx1 ; : : : ; xr i but no chain hx1 ; : : : ; xr ; xr+1 i. And say that hW; Ri itself has rank r i each element in it has a rank which is r, and some element in it has rank r. In this section (which is derived from work in [Segerberg, 1971]) we study normal extensions of K4 with characteristic frames of nite depth in this sense. De ne formulas Bn , for n = 1; 2; 3; : : : by taking B1 = B = P1 ! P1 ; Bn+1 = (Pn+1 ^ :Bn ) ! Pn+1 :
BASIC MODAL LOGIC
47
Then transitive hW; Ri veri es Bn i it has rank n. For it is easy to show that hW; R; V i rejects Bn at x0 2 W i there exists x1 ; : : : ; xn 2 W with xi Rxi+1 and V (Pn i ; xi ) = F; v(Bn i ; xi ) = F; v(Pn i ; xi+1 ) = T; for i = 0; : : : ; n 1, by induction from n 1 to 0. And it can be checked that these conditions can hold i x0 ; : : : ; xn satisfy the conditions for being a chain. We shall see that any normal logic L which contains K4Bn has the f.m.p. Consider a formula A with propositional variables from P1 ; : : : ; Pm , and take r to be maximum of m and n. Taking Lr to be the restriction of L to P1 ; : : : ; Pr , it is clear that `L A i `Lr A. Suppose that A is a nonthesis of both logics. The canonical general frame hWLr ; RLr ; PLr i veri es L and rejects A, and we shall see that it is nite. Firstly hWLr ; RLr i has rank n. For if it has a chain F0 ; : : : ; Fn then there must be formulas A1 ; : : : ; An with
An 1 2 Fi+1 and not An 1 2 Fi ;
for i = 0; : : : ; n 1: Then it is easy to show that the formula Bn0 obtained from Bn by substituting Ai for Pi ; i = 1; : : : ; n, has not Bn0 2 F0 , in contradiction to the properties of WLr . Now WLr has nitely many maximal consistent sets of rank i, by induction from i = 1 to i = n. Say that a formula is modally atomic i it is a propositional variable or of the form C or C . Since a maximal consistent set F , like an ultra lter, satis es the conditions
:A 2 F i not A 2 F; A ^ B 2 F i A 2 F and B 2 F; A _ B 2 F i A 2 F or B 2 F; it is determined by its modally atomic formulas. Note that if F is a maximal consistent set in WLr of rank i then C 2 F i
C 2 \fG : F
' G _ (F RLr G ^ G has rank < i)g
and C 2 F i
C 2 [fG : F
' G _ (F RLr G ^ G has rank < i)g:
By the induction hypothesis there are nitely many sets of maximal consistent sets G with (F RLr G ^ G has rank < i). There are nitely many ways of allocating P1 ; : : : ; Pr to the maximal consistent sets G with F ' G. Once these items are xed, the members of each maximal consistent set in the
48
ROBERT BULL AND KRISTER SEGERBERG
cluster including F are determined (by an easy induction on the construction of formulas). In particular the number of maximal consistent sets in the cluster is at most the number of ways of allocating P1 ; : : : ; Pr to those sets. It follows that there are nitely many possible sets of modally atomic formulas for F , and hence nitely many maximal consistent sets F of rank i in hWLr ; RLr i. 16 THE NORMAL EXTENSIONS OF S4.3 (Bull [1966] gives an algebraic proof that every normal extension of 4.3 has the f.m.p. Fine [1971] gives a frame-theoretic proof, together with a description of the lattice of these logics. Both proofs are rather elegant.) Let L be any normal modal logic containing S4.3. by what we have seen in Section 10, l is characterised by the subdirectly irreducible closure or interior algebras which verify it. Let A be such an algebra. Since A veri es (P ! Q) _ (Q ! P ) and satis es the condition if la [ lb = 1 then la = 1 or lb = 1; it is well-connected in the sense that
la lb or lb la: It also satis es the condition if la < lb then l(a [ ( lb)) = la; where la < lb is (la lb) ^ la 6= lb. This is shown by rst applying the same argument to (P ! Q) _ ((P ! Q) ! Q), which can be shown to be a thesis of S4.3, so that lb la or l(( lb) [ la) la. But if la < lb then not lb la, and in any interior algebra it can be shown that la l(( lb) [ la) = l(( lb) [ a). dualising these results, we have
ma mb or mb ma; if mb < ma then m(a mb) = ma; for each a; b 2 A. Given a nonthesis A of l and an algebraic `model' hA; vi which rejects it, let A1 ; : : : ; Am be A and its subformulas and let B = hB; 0; 1; ; \; [i be the nite subalgebra of hA; 0; 1; ; \; [i generated by fv(A1 ); : : : ; v(Am )g. Take W to be the set fb1 ; : : : ; bn g of atoms of the atomic Boolean algebra B and de ne R on W by taking
bi Rbj i bi mbj : Now hW; Ri+ is a nite closure or interior algebra, such that there is an isomorphism from B onto the underlying Boolean algebra of hW; Ri+ on
BASIC MODAL LOGIC
49
B(W ). (Note that hW; Ri+ is not a ltration of A in the usual sense.) De ne a valuation V 0 on hW; Ri by taking
v; (P ) = v(P ); for each propositional variable P in A; and applying the conditions on valuations. We have v0 (Ai ) = v(Ai ); for i = 1; : : : ; n; because is a Boolean isomorphism and
x 2 (mb) i 9y(xRy ^ y 2 (b)); for each b 2 B. For taking b = x1 [ : : : [ xr for atoms x1 ; : : : ; xr of B, we have x 2 (mb) i x m(x1 [ : : : [ xr ) i x mx1 [ : : : [ mxr i x m(x1 or : : : x mxr i xRx1 or : : : or xRxr i 9y(xRy ^ y 2 (b)): + In particular hW; Ri rejects A. To show that hW; Ri+ veri es L, it is suÆcient to construct an embedding homomorphism from hW; ri+ into A. Suppose that b1 ; : : : ; bn are indexed so that, in their indexed order, mbk)1) = : : : = mbk(2) 1 < : : : < mbk(s) = : : : = mbk(s+1) 1 in A, where 1 = k(1) < : : : < k(s + 1) = n + 1. Set bk(0) = 0 and note that mbk(1) mbk(0); : : : ; mbk(s) mbk(s 1) is a disjoint cover of 1. De ne by taking
( ) = 0; for i = 1; : : : ; s,
(fbk(i) g) = mbk(i) = bk(i)+1 [ : : : [ bk(i+1)
1
mbk(i
1) ;
for i = 1; : : : ; s and k(i) + j = k(i) + 1; : : : ; k(i + 1) 1,
(fbk(i)+j g) = bb(i)+j mbk(i+1); (fbi(1 ; : : : ; bi(r)g) = (fbi(1) g) [ : : : [ (fbi(r) g): It is clear that is an embedding homomorphism of the underlying Boolean algebras. It can also be shown that
m(fbk(i) g) = m(bl(i) mbk(i 1) ); m(fbk(i)+j g) = (fb1 ; : : : ; bk(i+1) 1 g); for i = 1; : : : ; s and k(i) + j = k(i); : : : ; k(i + 1) 1. (The second result uses the rst and the lemma of the rst paragraph.) But fb1; : : : ; bk(i+1) 1 g
50
ROBERT BULL AND KRISTER SEGERBERG
is the closure of fbk(i)+j g in hW; Ri+ , so that is now easily seen to be a homomorphism w.r.t. m as well. Alternatively, L is characterised by the generated submodels hWLx ; RLx ; VLx i of its canonical model. We know from Section 13 that these satisfy the condition 8y8z (yRLxz _ zRLxy): So, given a nonthesis A of L, let hW; R; V i be a model which satis es this condition and rejects A. Let fA1 ; : : : ; An g be A and its subformulas, and V i determined by this set of formulas. consider the ltration hW ; R; Let us rst try to prove that nite hW 0 ; R i veri es each formula veri ed by hW; R; V i and, hence, L, which would establish the f.m.p. for L. We must V 00 i to hW 0 ; R; V i. Say a subset of W 0 is rst reduce any model hW 0 ; R; de nable in hW; R; V i i it is v(B ), for some formula B ; that hW 0 ; PR; v00 i V i i v00 (P ) is de nable in hW 0 ; R; V i, is a de nable variant of hW 0 ; R; 0 for each propositional variable P ; and that hW ; R; V i is di erentiated i f[w]g is de nable, for each [w] 2 W 0 (cf. 1- re nement). It is easy to show V i is di erentiated; that therefore each hW 0 ; R; V 00 i is a that nite hW 0 ; R; 0 de nable variant of it' and that therefore if hW ; R; V i veri es L then so V 00 . To show that hW 0 ; R; V i veri es L, it would clearly does each W 0 ; R; suÆce to show that if xRy then [x]R [y]; if [x]R [y] then 9z (xRz ^ z 2 [y]): The rst condition is of course true, but unfortunately it is quite possible that the second could fail. In view of this set-back, let us try to eliminate elements for which the second condition fails. given ; 2 W 0 , de ne sub to hold i
9x(x 2 ^ 8y(y 2 ! :xRy)): . Say Note that if this holds then yRx, since hW; Ri is connected and so R and sub . (Note the that is eliminable` i there is some with R similarity of the conditions `virtually last' and `eliminable' on the members of a cluster in a ltration.) Take U to be the set of noneliminable elements V i by restricting R; V to U . It is easy to show that of V , and form hU; R;
V (Ai ; [x]) = T i Ai 2 x; for i = 1; : : : ; n and each [x] 2 U; V i once the lemma of the following paragraph is proved. It follows that hU; R; rejects the given formula A and is di erentiated. The lemma is that, for each formula B in fA1 ; : : : ; An g, if B 2 x then there is some y with B 2 y such that [x]R [y] and y is not eliminable. This is done by constructing a sequence 0 ; 1 ; 2 ; : : : in W 0 by taking 0 = [x],
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51
and for each i = 1; 2; 3; : : :,
2i 1 is some [z ] with B 2 z and not [z ] sub 2i 2 ; 2i 1 and 2i 1 sub [z ]: 2i is some [z ] with [z ]R It is easy to see that B 2 2i 1 and B 2 2i , for i = 1; 2; 3; : : :. It can be shown that this sequence must terminate, but that it cannot terminate at any 2i . the required y is the z with B 2 z such that the sequence terminates at 2j 1 = [z ]. To complete the argument it will suÆce to set up a frame morphism V i. Fro then hU; R; V i from some de nable variant of hW; R; V i onto hU; R; will verify L, as shown in Section 10, and so will each variant of di erentiated V i, as in the original `proof'. De ne : W ! U by taking hU; R;
(x) = [x]; if [x] 2 U; = the rst element in some arbitrary ordering of U which is g; otherwise R- rst in f : [x]R (y) yield (x)R (y). |noting that is onto U . If xRy then [x]R [y] and [y]R If (x)R (y) then we must have some z 2 (y) with xRz , otherwise (y) sub (x) and (y) would be eliminable. Now (y) = (z ) and xRz as required. Thus is an onto frame morphism. De ne a valuation V 0 on hW; Ri by taking V 0 (P; x) = V (P; (x), for each propositional variable P , and applying the conditions on valuations. Then it is easy to show that hW; R; V 0 i is a de nable variant of hW; R; V i and to extend to a morphism of models. (What is the relationship between these two proofs? Take hW; R; V i to be a generated sub model of the canonical model of L, and take hA; vi to be hhW; Ri+ ; vi, for the same valuation. Thus A is indeed a subdirectly irreducible closure or interior algebra verifying L. Relabelling the nite frame hW; Ri of the rst proof as hW 0 ; R0 i; W 0 is the usual set obtained from fv(A1 ); : : : ; v(An )g in a ltration, but from R0 _ i 8x(x 2 ! 9y(xRy ^ y 2 )): Since a one{one homomorphism from hW 0 ; R0 i+ into hW; Ri+ is the dual of a frame morphism from hW; Ri onto hW 0 ; R0 i, we would expect that all the elements in W 0 are noneliminable. To see that this is indeed true, suppose that R0 and sub and try to obtain a contradiction. In this case there is some x 2 with 8y(y 2 ! :xRy) by the de nition of sub . then the de nition of R0 give us some y 2 with xRy| the required contradiction. Unfortunately, the other condition on frame morphisms, that if xRy then [x]R0 [y], is not satis ed by this construction. and indeed the frame morphism of which i the dual, is not (x) = [x], for each x 2 W , but a more complicated function which can be constructed from the de nition of above.)
52
ROBERT BULL AND KRISTER SEGERBERG Say that a nonempty sequence of positive integers is a list. A nite frame
hW; Ri which veri es S4.3 must consist of a nite chain of nite clusters, so
that it is described by the list of numbers of elements in successive clusters. Say that a list t contains a list s = hA1 ; : : : ; am i when there is a subsequence hbi1 ; : : : ; bim i of t with a1 bi1 ; : : : ; am bim . And that t = hb1 ; : : : ; bn i covers s i t contains s and am bn. Given nite frames hW; Ri and hU; S i which verify S4.3, described by lists t and s, it is easy to show that if t covers than in each in nite sequence t1 ; t2 ; t3 ; : : : of lists there is an in nite subsequence ti1 ; ti2 ; ti3 ; : : :, such that if h < k then tih is covered by tik . From this it is easy to deduce that there is no in nite increasing sequence L1 L2 L3 : : : of normal modal logics containing S4.3. For take Ai to be a formula in Li+1 but not in Li , and take ti to be the list describing a suitable nite frame which rejects Ai . Then the result yields a tj with i < j which covers ti , and now Ai is also not in Lj with i +1 j , a contradiction. 17 THE PRETABULAR EXTENSIONS OF S4 (A normal modal logic is said to be tabular i it is characterised by a single nite structure, and to be pretabular i all its proper extensions are tabular. Thus the well-known [Scroggs, 1951] shows that S5 = S4B is a pretabular logic. Maksimova [1975] and [Esaia and Meskhi, 1977] independently prove the very pretty result that there are precisely ve pretabular extensions of S4. The work of the last four sections provides the background needed for [Esaia and Meskhi, 1977]. The pretabular extensions of K4 are a much more diÆcult topic, dealt with by [Block, 1980a]. This paper takes as its starting point the very strong results of [Jonsson, 1967] on the subdirectly irreducible algebras in a variety.) Consider the nite, generated, re exive-and-transitive frames hW; Ri. Which parameters of these frames can be left unrestricted by the formulas that they verify? It turns out that there are precisely ve of them. 1. The maximum number of points in any nal cluster. 2. the maximum number of points in any non- nal cluster. A cluster [z ] is a successor of [x] i xRz but [x] 6= [z ], and an immediate success for i , further, there is no cluster [y] such that [z ] is a successor of [y] and [y] is a successor of [x]. Say that the external branching of a cluster is the number of nal clusters which are immediate successors of it. And that the internal branching of a cluster is the number of non- nal clusters which are immediate successors of it. 3. The maximum of the external branching of the clusters. 4. The maximum of the internal branchings of the clusters.
BASIC MODAL LOGIC
53
5. The maximum number of clusters in any chain of cluster, i.e. the rank of hW; Ri in the sense of Section 15. It is clear that once all ve parameters are bounded, the class of re exiveand- transitive frames satisfying those bounds is nite. Thus if L is determined by such a class of frames then it is determined by a single nite frame, namely the nite disjoint union of these nite frames. For each of the ve parameters, given a nite frame hW; Ri of the kind being considered, a frame hWi ; Ri i of a certain kind can be constructed, which has the same value of that parameter. The constructions needed are subframes and frame-morphic images. We saw in Section 10 that a class of frames verifying a normal modal logic L is closed under them. The ve kinds of simple frames and their constructions are as follows. 1. hW1 ; R1 i has one cluster. Take the largest nal cluster of hW; Ri, which is a subframe and has the required properties.
2. hW2 ; R2 i has two clusters, of which the nal one is simple. Take the largest non nal cluster [x] of hW; Ri and form hWx ; Rx i. Take W2 = [x] [ f!g and de ne R2 on it by taking xR2 y i x ' y _ y = !. De ne a frame morphism 2 from Wx onto W2 by taking 2 (y) = y if x ' y; 2 (y) = ! otherwise.
3. hW3 ; R3 i has W3 = f0; 1; : : : ; ng with xR3 y i x = y _ x = 0. Take [x] to have the maximal external branching in hW; Ri with nal clusters [y1 ]; : : : ; [yn] immediately succeeding it. Form hWx ; Rx i and de ne a frame morphism 3 from WX onto W3 by taking 3 (y) = 0 if y 2 [x]; 3 (y) = i if y 2 [yi ], for i = 1; : : : ; n; 3 (y) = 1 otherwise. 4. hW4 ; R4 i has W4 = f0; 1; : : : ; n; !g with xR4 y i x = y _ x = 0 _ y = !. Take [x] to have the maximal internal branching in hW; Ri, with non nal clusters [y1 ]; : : : ; [yn ] immediately succeeding it. For hWx ; Rx i and de ne a frame morphism 4 from Wx onto W4 by taking 4 (y) = 0 if y 2 [x]; 4 (y) = i if y 2 [yi ], for i = 1; : : : ; n; 4 (y) = ! otherwise. 5. hW5 ; R5 i has W5 = f2; : : : ; ng with iR5 if i j . Suppose that hW; Ri has rank n, with a maximal chain hx1 ; : : : ; xn i. De ne a frame morphism 5 from W onto W5 by taking 5 (y) = i if xi ' y, for i = 1; : : : ; n 1; 5 (y) = n otherwise. Each of these ve sets of simple frames characterises a normal modal logic, as follows: 1. S4B, known as S5. 2. S4:3B2M 3. S4GrzB2 .
54
ROBERT BULL AND KRISTER SEGERBERG 4. S4GrzB3 plus P 5. S4 3Grz.
! P .
For each of these extensions of S4Bn or S4 3 has the f.m.p. by Sections 15 and 16, and it is easy to check the class of nite generated frames which veri es each logic. Any pretabular extension L of S4 must be one of these logics. For pretabular L must have the f.m.p. with a class of nite frames in which one of the ve parameters is not bounded, as we saw above. Its class of nite frames must therefore include one of the ve sets of simple frames. Therefore L must be contained in one of the ve corresponding logics. But every proper extension of pretabular L must be tabular, so that L has to be identical with one of these logics. Finally it can be shown that any nontabular logic is contained in a pretabular logic, and hence in one of these ve. But these ve logics are pairwise incomparable, so that they must all be pretabular logics. 18 THE TRANSITIVE LOGICS OF FINITE WIDTH (The work of this section is taken from Fine [1974a; 1974b], which extend the ideas of [Fine, 1971] to a wider set of logics.) Given a frame hW; Ri say that points x; y 2 W are incomparable i x 6= y and not xRy and not yRx. The frame hW; Ri is of width n if it has n pairwise incomparable points but does not have n + 1 incomparable points. (In particular, for transitive frames, hW; Ri is connected i it is of width 1.) For i = 1; : : : ; n take In to be the formula n ^ i=0
P
!
_
0i6=j n
(Pi ^ (Pj _ Pj )):
It is easy to see that a generated frame veri es In i it is of width n. Various of the nice properties of the connected frames break down at greater widths. As an example of this, there is an in nite increasing chain of normal extensions of S4I2 . Indeed there are continuum many distinct normal extensions of S4I2 . This is shown by de ning certain frames F1 ; F2 ; F3 ; : : : of width 2, and proving that distinct subsets of this set of frames characterise distinct logics. Each frame Fn = hWn ; Rn i is de ned by taking Wn = f0; : : : ; 2n + 4g and taking Rn to be the restriction to Wn of R with
iRj i either i = 0 or i is odd, j is odd, and i > j or i is odd, j is even, and i > j + 2 or i is odd, j is odd, and i > j + 4 For example, F2 is depicted in Figure 1.
BASIC MODAL LOGIC
55
8 6 4 2 0
7 5 3 1
Figure 1. The result will follow if it can be shown that each Fn rejects a formula
:An which is veri ed by every other Fm . In each case An is taken to be the
frame formula for Fn , in the following sense. The frame formula AF for any nite re exive-and-transitive frame F = hf0; : : : ; rg; Ri generated by 0 is the conjunction of the formulas P0 and (P0 _ : : : _ Pr ); (Pi ! :Pj ); for each i 6= j; (Pi ! Pj ); whenever iRj; (Pi ! :Pj ); whenever not iRj: In general, frame formulas have the property that AF can be satis ed in a frame S = hU; S i i , for some u 2 U , there is a frame morphism from Su onto F. We know from Section 10 that if this condition holds then each formula satis ed in F can be satis ed at u in F. but AF is satis ed in F when V is de ned on f)0; : : : ; rg by taking
V (Pi;j ) = T i i = j; for each i = 0; : : : ; r; which yields V (AF ; 0) = T . For the converse, suppose that there is a u 2 U and a valuation V 0 on S with V 0 (AF ; u) = T . Then de ne a function from Uu into f0; : : : ; rg by taking (x) = i i V 0 (Pi ; x) = T; for each x with uSx and i = 0; : : : ; r. It is straightforward to show, using the construction of AF , that is is an onto frame morphism. Therefore, to show that :An is veri ed by Fm , i.e. An is not satis ed by Fm , it suÆces to show that there is no frame morphism from Fm;k onto Fn unless m = n and k = 0. Clearly, if m < n or 1 k 2n + 6 then Fm;k does not have enough points for there to be a frame morphism from it onto Fn . (Compare F2;k with F0 .) So suppose that m > n and k = 0 or k 2n + 7, and that is a frame morphism from Fm;k onto Fn , and try to obtain a contradiction. Firstly it can be shown that (1) and (2) are distinct nal points of Fn , say (1) = 1 and (2) = 2. Then it can be shown
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ROBERT BULL AND KRISTER SEGERBERG
that (i) = i, for i 1, in Fm;k , by induction on odd or even i = 1; 2; : : :. Now i = 2n + 5 or i = 2n + 6 is in Fm;k but not in Fn , so that Fn does not have enough points for to map Fm;k but not in Fn , so that Fn does not have enough points for to map Fm;k into it. (Compare F1;0 ; F2;7; F2;8 with F0 .) (Check why this argument cannot be used on a connected frame!) Nonetheless, each normal extension of K4In is characterised by the transitive frames of width n which verify it. The proof of this major result is diÆcult, and all that will be given here is a brief glance at the ideas involved. Let L be any normal extension of K4In . the big di erence from the second half of Section 16 is that we are working with in nite hWLr ; RLr ; FLr i instead of with a nite ltration of hWL ; RL ; VL i. (Here Lr is the restriction of L to the propositional variables P1 ; : : : ; Pr .) Therefore the problem comes at a di erent point. It is now immediate that hWLr ; RLr ; VLr i veri es L, but since this di erentiated model is not nite, it is no longer true that each variant of it is de nable. (Note that just as the canonical general frame is re ned, the canonical model is not only di erentiated but natural. That is, it satis es the condition that if V (A; x) = T ! V (A; y) = T , for each formula A, then xRy.) As before it is necessary to eliminate certain points from the given frame. Say x 2 WLr is eliminable i , for each formula A, if V (a; x) = T then 9y(xRLr y ^ :yRLr x ^ VLr (A; y) = T ):
A reduced canonical model is not formed on the noneliminable points. It must be shown that there are enough noneliminable points, i.e. that if VLr (A; x) = T then there is some noneliminable y with xRLr y and VLr (A; y) = T , and that hey are de nable. The proof that the reduced canonical frame veri es L, because the de nable variants of the reduced canonical model do, uses the facts that hWLr ; RLr ; VLr i id natural and that hWLr ; RLr i has no in nite ascending R-chains. (So does the proof of the de nability of the noneliminable points.) So a crucial step in the argument is the lengthy proof that a di erentiated model which is transitive and of nite width has no such chains. 19 THE VEILED RECESSION FRAME The recession frame h!; Ri is de ned on ! = f0; 1; 2; : : :g by taking
mRn i m n + 1 for each m; n 2 !: Thus R is re exive, and transitive for increasing numbers, but is not transitive for decreasing numbers, when only mRn i m = n + 1. For any valuation V on h!; Ri,
v(A) = [m; 1) = fn : m ng and v(A) = [m + 1; 1);
BASIC MODAL LOGIC
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where [m 1; 1) is the `largest unbroken interval in v(A)'. It is easy to verify that the recession frame veri es KT 3. The veiled recession frame h!; R; P i is the general frame de ned on the recession frame by taking P to consist of the nite and co nite subsets of !. (Co nite subsets are the complements of the nite ones.) In fact Blok has shown that it characterises KT 3M plus (P ! P ) ! (P ! P ) and two further axioms, all of which correspond to certain rst-order conditions on frames; see [van Benthem, 1978]. The recession frame was introduced in [Makinson, 1969] to show that a certain logic does not have the f.m.p. the veiled recession frame was introduced in [Thomason, 1974] to show that a certain logic is not characterised by frames. Two similar but sharper examples were produced in [van Benthem, 1978]. These four results are discussed in this section. Thomason [1972a] uses the nite fragments of the recession frame with one point added. It shows that a certain formula (10) is veri ed by any frame verifying a certain in nite set of axioms, of which each nite subset is veri ed by a frame rejecting (10). It follows that whatever nitary rules are used, a logic with these axioms is not characterised by the frames which verify it. Finally Blok [1980] uses variations on the veiled recession frame to show that there is a continuum of distinct extensions of KT which are all veri ed by the same class of frames! This paper takes as its starting point the very strong results of [Jonsson, 1967] on the subdirectly irreducible algebras in a variety. These results are usually described as incompleteness theorems, but they are better thought of as showing the independence of various notions of consequence. In each case we have a logic L and a formula F . Firstly there is modal logical consequence L ` F , using the rules of normal modal logics. then for each class S of structures there is a corresponding notion of semantic consequence, with L F i F is veri ed by each structure in S which veri es L. We know from Sections 10, 11, 12 that nite semantic consequence is as strong as (frame) semantic consequence, which is as strong as general (frame) semantic consequence, which is equivalent to algebraic `semantic' consequence and modal logical consequence. The problem is to show that these relative strengths are strict. The method is to show by example that some formula F is a consequence of L in the rst sense but not in the second sense. In order to show that nite semantic consequence is strictly stronger than semantic consequence, take L to be KT plus (P ^ :2 P ) ! (2 P ^ :3 P ); and take F to be 4. If the recession frame veri es this formula, it will show that 4 is not a semantic consequence of this l. It is clear that if a valuation V on h!; Ri rejects this formula m then
V (P; m) = T; V (2 P; m) = F; V (2 P; m + 1) = F or V (3 P; m + 1) = T:
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ROBERT BULL AND KRISTER SEGERBERG
In the second case, (m + 1)Rm yields V (P; m) = T and a contradiction. The rst case requires some n with m n such that V (P; n) = F , and some k with n < k + 1 such that V (P; k) = F . Now m k + 1 and so V (P; m) = F , another contradiction. To show that 4 is a nite semantic consequence of this L, it is suÆcient to show that if a model hW; R; V i verifying L rejects 4 then W is in nite. But in a model which rejects 4 we have v(2 P ) v(P ), which serves as the induction basis for an inductive proof that v(n+1 P ) v(n P ), for n 1. The induction step uses the fact that if v(k P ) v(k+1 P ) 6= 0; then v(k+1 P ) v(k+2 P ) 6= 0; from the veri cation of (P ^ :2 P ); (2 P ^ :3 P ). The argument can be sharpened to prove the existence of an in nite ascending R-chain if (P ^ 2 Q) ! (Q _ 2 (Q ^ P )) is added to L. For suppose that hW; Ri veri es this formula and rejects 4, having x; y; z 2 W such that xRy and yRz but not xRz . Then taking v(P ) = fxg and v(Q) = fz g we have V (P; x) = T; V (2 Q; x) = T; V (Q; x) = F , so that V (2 (Q ^P ); x) = T . It follows that V (P; z ) = T , which can only hold if zRx. This fact, that if xRy and yRz but not xRz then zRx, can be used to construct an in nite ascending R-chain from the decreasing sequence v(P ); v(2 P ); v(P ); : : : of subsets of W . Note that this additional formula is also veri ed by the recession frame. For if V (P; m) = T; V (2 Q; m) = T; V (Q; m) = F then V (Q; m 2) = T , V (Q ^ P; m 2) = T , and V (2 (Q ^ P ); m) = T . To show that semantic consequence is strictly stronger than general semantic consequence, it only remains to nd a formula A which is veri ed by the veiled recession frame but is rejected by any frame with an in nite ascending R-chain. Thomason [1974] does give a complicated formula A with this property. Now, for each frame verifying the extension of KT with the two formulas of recent paragraphs, rejection of 4 implies the rejection of A, so that veri cation of A requires the veri cation of 4. Taking L to be the extension of KT with the two stated formulas and A; 4 is a semantic consequence of L but not a general semantic consequence of it. Another proof that semantic consequence is strictly stronger than general semantic consequence goes as follows. Take L to be KT 3M plus
(P ! P ) ! (P ! P ); and take F to be P ! P . This formula reduces the modal operators to triviality, with the corresponding condition on R that if xRy then x = y.
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De ne xRn y on a frame hW; Ri, for n 0, taking
xR0 y i x = y; xR1 y i xRy; xRn+1 y i xRz1 ; : : : ; zn Ry; for some z1 ; : : : ; zn 2 W: Given a frame hW; Ri and x; y 2 W such that xRy but not yRn x, for n 0, de ne V on hW; Ri by taking V (P; z ) = T i yRn z , for some n 0. it is easy to show that V ((P ! P ); x) = T , V (P ! P; x) = F . Therefore in any frame hW; Ri which veri es (P ! P ) ! (P ! P ) we have (*) if xRy then yRnx, for some n 0.
It can be shown that any re exive frame hW; Ri which veri es 3 satis es the condition
8x8y8z ((xRy ^ xRz ) ! (8u(yRu ! zRu) _ 8v(zRv ! yRv)): Call this condition strong connectedness, noting that connectedness is the special case with u = y and v = z , and that this condition can be derived from the ordinary one and transitivity. It can be shown that if a re exive, strongly connected frame hW; Ri satis es condition (), then it veri es (P ! P ) ! (P ! P ). As an application of this result, the recession frame veri es this formula. Thus the veiled recession frame veri es L but not P ! P . Suppose that hW; Ri is a re exive, strongly connected frame which satis es condition (*). It can be shown that if hW; Ri also veri es M then xRy implies x = y, so that any frame which veri es L also veri es P ! P . For given any x 2 W , de ne
Sn = fy : yRnx ^ :9m(m < n ^ yRm x)g; for n 0, and de ne V on hW; Ri by taking
V (P; y) = T i 9m(y 2 S2m ); for each y 2 W: Now it can be shown that V (P; x) = T , so that V (P; x) = T by the veri cation of M . From this it can be deduced that V (P; x) = T . Finally we suppose that xRy and x 6= y, and obtain a contradiction. For in this case we have V (P; y) = T , so that y 2 S2m , for some m 1, and there are some z1; : : : ; z2m 1 2 W with yRz1; : : : ; z2m 1Rx and not z1 Rx. Thus xRy; xRx; yRz1 but not xRz1 ; xRx but not yRx|which contradicts strong connectedness when we put x for z; z1 for u, and x for v. A third proof that semantic consequence is strictly stronger than general semantic consequence takes L to be KT plus
((P ! P ) ! 3P ) ! P
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ROBERT BULL AND KRISTER SEGERBERG
and takes F to be 4 again. for it can be shown that the veiled recession frame veri es this axiom of L, but that each frame which veri es it is transitive. The interest of this example lies in the fact that the extension of S4 with this axiom is precisely S4Grz. Given a frame hW; Ri, consider the evaluation of any formula A in any model on hW; Ri. Our de nition of valuations determines V (A; x) in terms of rst-order logic applied to propositions of the form yRz and V (P; y) = T for propositional variables P . Replace each yRz by an atomic proposition R(y; z ), and each V (P; y) = T by an atomic proposition P (y). Now the truth of A in hW; Ri can be expressed by a formula in second-order predicate logic with unary predicate parameters P; Q, etc. and one binary parameter R. This formula is known as the standard translation ST (A) of A. As we have seen, ST (A) is often equivalent to a rst-order predicate formula in R alone, but this is not always the case. If we take some axiom system for second-order predicate logic then we can introduce yet another notion of consequence. Say that F is a second-order logical consequence of L i ST (F ) is derivable from the standard translations of the formulas of L. In fact whenever we have shown that F is a semantic consequence of L, we have used an argument in some unspeci ed, informal second-order logic to show that F is a second-order logical consequence of L. Clearly semantic consequence is as strong as second-order logical consequence, which is as strong as modal logical consequence. Van Benthem [1978; 1979a] discuss whether second-order logical consequence is strictly stronger than modal logical consequence. History added point to this question, in that transitivity was derived from ST (GRz) before 4 was derived in KG4z. Of course the answer will depend on the axiomatisation used for second-order predicate logic. For example, close inspection of the informal argument for P ! P being a second-order logical consequence of KT3M plus (P ! P ) ! (P ! P ), shows that it involves an Axiom of Choice. It turns out that if this is dropped, then a second-order derivation is no longer possible. Consider the axiomatic second-order logic with just the weak second-order substitution axiom
8P A ! SPB (A);
for rst-order formulas B:
(Here SPB (A) is obtained from A by substituting Sxt (B ) for P (t) throughout, under suitable conditions.) the proof that P ! P is not a general semantic consequence of this modal logic used the veiled recession frame, for which the possible values of formulas are the nite and co nite subsets of !. It can be shown that these are precisely the subsets of ! de nable by rst-order formulas with = and R as their only predicate parameters. Since these are the subsets of ! to which the weak second-order substitution axiom applies, the same argument shows that P ! P is not a second-order logical consequence of this modal logic.
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The normal modal logic k plus (P ! P ) ! P is easily shown to be inconsistent. De ne a general frame h! [ f1g; R; P i by taking
xRy i x > y _ x = 1; and taking P to consist of the nite subsets of ! and their complements in ! [ f1g. Then it is easy to show that (P ! P ) ! P is satis ed at 1 by each valuation on h! [ f1g; R; P i (but not of course veri ed). So consider the non-normal logic K plus (P ! P ), from which the rule of necessitation has been dropped. Now P ^ :P is a second- order logical consequence of this logic, but not a modal logical consequence of it. Van Benthem [1979a] shows how to adopt this argument to give a normal modal logic L and a formula F , such that F is a second-order logical consequence of L but not a modal logical consequence of it. 20 INDEPENDENCE RESULTS ABOVE S4 None of the logics used in the previous section is an extension of S4 (though KT 3M plus (P ! P ) ! (P ! P ) is a very strong logic in a sense, with no frames between it and triviality). Further, the methods of that section cannot be applied to extensions of S4, since transitivity reduces the recession frame to a frame verifying S5. For independence results above S4 we turn to a brief description of the complicated constructions of [Fine, 1972; Fine, 1974a]. In showing that nite semantic consequence is strictly stronger than semantic consequence, L is taken to be S4 plus a certain axiom Y ! Z , and F is taken to e :Y . The frame used to show that :Y is not a consequence of S4 plus Y ! Z consists of three chains of points ai ; bi ; ci , for i 0, with R a lattice on them, and a nal related pair of points d; e. This frame is illustrated in Figure 2 with R going from left to right. The points in these chains are described by corresponding formulas Ai ; Bi ; Ci , for i 0, with A0 = P; B0 = Q; C0 = R. Each Ai+1 is
Ai ^ Bi ^ :Ci ; expressing the fact that
ai+1 Rai ^ ai+1 Rbi ^ :ai+1 Rci and similarly for Bi+1 ; Ci+1 . (Remember the frame formulas of the rst half of Section 18.) Because of this construction there are theses of S4 describing the relations between the points. For example, not ai Rbi and not ai Rci , and `S4 (Ai ! (:Bi ^ :Ci )). The formula Y is simply a description of a0 ; b0 ; c0 ; d in these terms, so that if V is de ned on this frame by taking V (P ) = fa0 g; V (Q) = fb0 g,
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ROBERT BULL AND KRISTER SEGERBERG
V (R) = fc0 g; V (S ) = fdg, then V (Y; d) = T . Thus V (:Y; d) = F and :Y is rejected on this frame as required. but it is also true that if V is a valuation on this frame with V (Y; x) = T then x is d or e and V (P ); V (Q); V (R) are a permutation of fai g; fbig; fci g, for some i 0. The formula Z describes a property of four such points, so that again V (Z; x) = T . Thus V (Y ! Z; x) = T , for each x 2 W and each valuation V , so that this frame veri es Y ! Z as required. d
e
Æ :::
Æ
Æ
Æ
a0
:::
Æ
Æ
Æ
b0
Æ :::
Æ
Æ
Æ
c0
Figure 2. These formulas also have the property that any frame hW; Ri which veri es Y ! Z and has a valuation V which satis es Y must be in nite. First it can be shown that if V (Y; x) = T then V (Ai ; x) = T , for i 0, by an induction on i. The induction basis with i = 0 uses V (Y; x) = T , the induction step fro i = 1 uses V (Y ! Z; x) = T , and the other induction steps use theses of S4 as above and V (Y 0 ! Z 0 ; x) = T , for substitution instances Y 0 ; Z 0 of Y; Z . Then it can be shown that `S4 Ai ! Ai j , for each 0 < j < i, by an induction using theses of S4 above. It follows that there must be points ai with xRai and V (Ai ; ai ) = T , for i 0, and with ai 6= aj , for i 6= j . Thus any nite frame which veri es S4 plus Y ! Z must reject :Y , for otherwise it would satisfy Y and be in nite. A similar strategy is used to show that semantic consequence is strictly stronger than general semantic consequence. At rst sight Fine [1974] is not about general semantic consequence at all. Instead hW; R; V i strongly veri es A i all substitution instances of A are true in hW; R; V i. But this is clearly equivalent to A being true on hW; R; P i, where P = fv(B ) : B a formulag. Unfortunately there are a number of omissions and other typographical slips in this paper. See Bull [1982; 1983]. Again L is S4 plus certain axioms E ! F and H , and the other formula is :E . The underlying frame used in showing that :E is not a general semantic consequence of this logic has two descending R-chains of points bm ; cm , for m 0, with R a
BASIC MODAL LOGIC
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lattice on them. It also has a sequence of unrelated points am linked to an ascending R-chain of points dm , for m 0. (Note that because of the unrelated am 's, this frame is not of nite width.) This frame is illustrated in Figure 3 with R going from left to right. (As the page is nite, the ascending and descending parts have been overlapped. Each dn should be linked to its an from the left, so that dm Ran for each m n.) The points in the rst three sequences are described by corresponding formulas Am ; Bm ; Cm , for m 0, with B0 = Q0 ; B1 = Q1 ; C0 = R0 ; C1 = R1 . Each Am is
Bm+1 ^ Cm+1 ^ :B )m + 2 ^ :Cm+2 ; expressing the fact that
amRbm+1 ^ am Rcm+1 ^ :am Rbm+2 ^ :am Rcm+2 ; and so on. Because of this construction there are theses of S4 describing the relations between the points. For example, bi+1 Rbi but not bi+1 Rci , and `S4 (Bi+1 ! (Bi ^ :Ci )):
:::Æ
Æ
Æ Æ b0
:::Æ
Æ
Æ Æ c0
:::Æ
Æ Æ a0
Æ
Æ Æ :::
d0
Figure 3. The formula E is a description, from the viewpoint of d0 , of the frame given in Figure 4, together with the fact that there is an R-chain after it. Thus E is rejected at d0 on this frame by taking
v(P0 ) = fd2m : m 0g; V (P1 ) = fd2m+1 : m 0g; V (Q0 ) = fb0g; V (Q1 ) = fb1g; V (R0 ) = fc0 g; V (R1 ) = fc1g:
But it is also true that if V is a valuation on this frame with V (E; x) = T then V must give the propositional variables values which are points in this
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ROBERT BULL AND KRISTER SEGERBERG
con guration. Thus x must be some dn . The formula F describes the Rchain beginning at d1 from the viewpoint of d0 , so that again V (F; x) = T . Thus V (E ! F; x) = T , for each x 2 W and each valuation V , so that this
b1 d0
Æ Æ b0
Æ Æ a0 c1
Æ Æ c0
Figure 4. frame veri es E ! F . These formulas also have the property that any frame hW; Ri which veri es E ! F and has a valuation V which satis es E at x 2 W must have an in nite ascending R-chain after x. To see this, write En ; Fn for the formulas obtained from E; F by replacing A0 ; A1 with An ; An+1 , and so on. It can be shown by an induction on n that there is an R-chain hx = y0 ; : : : ; yn i such that V (En ; yn ) = T , for n 0. (Think of y0 ; : : : ; yn as dm ; : : : ; dm+n .) The inductions step uses V (En ! Fn ; yn ) = T and these of S4 as above. The crucial point is that
Fn = ((P0 _ P1 ) ^ :An ^ An+1 ) sends us from yn with V (Fn ; yn ) = T to some yn+1 with yn Ryn+1 and V (An+1 ; yn+1 ) = T . Using this in nite ascending R-chain after x, it is easy to reject H = S ^ (S ! ((:S ^ T ) ^ ((:S ^ :T ) ^ S ))) at x with a suitable valuation. Thus any frame which veri es S4 plus E ! F and H must reject :E . Finally, consider again the frame illustrated in Figure 4 above, and the valuation V on it used to satisfy E at d0 . This valuation determines a general frame on it, in which P is the set of values v(B ) of all formulas B . We already know that E ! F is veri ed by this general frame and that :E is rejected by it, so it only remains to show that it veri es H . Suppose then that V (:H 0 ; x) = T , for some x 2 W , and some substitution instance H 0 of H , and try to obtain a contradiction. It is clear that x must be dm , for some m 0, for H can only be rejected on a proper cluster or an in nite ascending R-chain. Note that H 0 is constructed from three incompatible propositions a; :A ^ B; :A ^ :B . Further, A and B are constructed from propositional variables and formulas C1 ; : : : ; Ck with :; ^; _. Note that
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after some dn , the formulas C1 ; : : : ; Ck must have xed truth values V (Ci ; dj ). Consider j1 ; j2 ; j3 with n j1 ; j2 ; j3 and
V (A; dj1 ) = V (:A ^ B; dj2 ) = V (:A ^ :B; dj3 ) = T: At least one pair of these j 's must have an even di erence, e.g. j2 and j3 . In this case V (:A ^ :B; dj2 ) = V (:A ^ B; dj2 ) = T; using the construction of each V (Pi ; dj ) an the fact about each V (Ci ; dj ). But this contradicts the mutual incompatibility of these three formulas. 21 NEIGHBOURHOOD FRAMES A neighbourhood frame hU; N i consists of a set U and a function N : U ! B(B(U )). Thus each value N (x) of N is a subset of B(U ), the subsets of U in N (x) being known as the neighbourhoods of x. Valuations V and models on hU; N i are de ned as for ordinary frames except that
V (A; x) = T i V (A) 2 N (x): The canonical neighbourhood model hUL ; NL; VL i for a logic L is de ned as for ordinary frames except that
S 2 N (F ) i 9A(A 2 F ^ S = fG : A 2 Gg): Satisfaction, veri cation, and neighbourhood semantic consequence are de ned s for ordinary frames. The minimal normal modal logic K is characterised by the class of neighbourhood frames hU; N i in which each N (x) is a lter on U . Such a neighbourhood frame is said to be normal, and determines a modal algebra on B(U ). Each ordinary frame hW; Ri determines a normal neighbourhood frame hW; N i by taking
N (x) = fS : fy : xRyg S g; for each x 2 W: Here hW; N i veri es the same formula as hW; Ri. Also each normal neighbourhood frame hU; N i determines an ordinary frame hU; Ri by taking
xRy i y 2 \N (x); for each x; y 2 U: But here hU; Ri may not be equivalent to hU; N i, so that we must ask whether semantic consequence is strictly stronger than normal neighbourhood semantic consequence. (Neighbourhood frames seem to have been created independently by Dana Scott and Montague. See [Segerberg, 1971] for a full discussion of
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ROBERT BULL AND KRISTER SEGERBERG
them. Gerson [1975] established that normal neighbourhood semantic consequence was strictly stronger than general semantic consequence, while Gerson [1976; 1975a] established that ordinary semantic consequence was strictly stronger than it.) In showing that normal neighbourhood semantic consequence is strictly stronger than general semantic consequence, the arguments of Thomason [1974] and Fine [1974] can be taken over with only slight alterations. These come when showing that each normal neighbourhood frame which veri es the logic concerned also veri es the other formula 4 or E . For the rst case, if hU; N i veri es S. K. Thomason's axiom (P ^ 2 Q) ! (Q _ 2 (Q ^ P )); and there are R; S; T U with R mS and S mT but not R mT , then T \ mR is nonempty. Now the proof that, if hW; Ri veri es Makinson's axiom (P ^ :2 P ) ! (2 P ^ :3 P ) but rejects 4 then it must be in nite, can be sharpened as follows. If hU; N i veri es both these axioms but rejects 4 then U contains an in nite sequence of distinct subsets W1 ; W2 ; W3 ; : : : with Wi mWj if i < j . S. K. Thomason's second axiom A can be rejected on any hU; N i with this property, so that if a normal neighbourhood frame veri es the logic of [Thomason, 1974] then it veri es 4. For the second case, suppose that hU; N i veri es E ! F and has valuation V which satis es RE at u 2 U . Then it can be shown that U contains an in nite sequence of distinct subsets W1 ; W2 ; W3 ; : : : with u 2 Wi , for i 0, and Wi mWj if i < j , taking Wi = v(Ei ), for i 0. Using this nite sequence of sets it is easy to reject :H with V at u, so that if a normal neighbourhood frame veri es S4 plus E ! F and H then it veri es :E . Gerson [1976] uses a minor variation on the logic L of the `noncompactness' proof in [Thomason, 1972a]. A very complicated argument shows that this logic is veri ed by a certain normal neighbourhood frame, which is largely determined by an ordinary frame consisting of all nite fragments of the recession frame, with one point added. A further three points are then added and their neighbourhoods speci ed. Otherwise the argument is like that of [Thomason, 1972a]. Gerson [1975a] uses a version L0 of the logic L of [Fine, 1974], with E ! Fn for n 1. That any ordinary frame which veri es L0 also veri es :E goes as before. A complicated argument shows that L0 is veri ed by a certain normal neighbourhood frame, which is largely determined by an ordinary frame similar to that of [Fine, 1974] illustrated above. The di erence is that the in nite ascending R-chain of dm 's has been replaced by an in nity of nite ascending R-chains hdm;1 ; : : : ; dm;m i for m 1. A further two points are then added and their neighbourhood speci ed. Otherwise the argument is fairly similar to that of [Fine, 1974].
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22 ELEMENTARY EQUIVALENCE AND D-PERSISTENCE Consider the rst-order predicate logic with binary predicate constants = and R. Write F A i the formula A of predicate logic is true of the frame F, and similarly for F , where is a set of predicate formulas. A class X of frames is elementary i
X = fF : F Ag; for some formula A of predicate logic; -elementary i it is an intersection of elementary classes, -elementary i it is a union of elementary classes, and -elementary i it is an intersection of - elementary classes. Note that X is -elementary i it is axiomatic, with
X = fF : F
g;
for some set
of formulas of predicate logic:
And X is -elementary i it is closed under elementary equivalence, where F and G are elementarily equivalent i
F A i G A; for each formula A of predicate logic. The importance of elementarily equivalent frames for modal logic lies in the following lemma. Given a general frame F = hW; R; P i, there is a general frame F0 = hW 0 :R0 ; P 0 i such that F0 is 1- and 20 -saturated (see Section 10), F+ and F0+ are isomorphic, hW; Ri and hW 0 ; R0 i are elementarily equivalent, and there is a frame morphism from hW 0 ; R0 i onto (F+ )] . Alternatively, consider modal logic as usual, again writing hF; V i A i the formula A of modal logic is true in the model hF; V i, and so on. A class X of frames is modal elementary i
X = fF : F Ag; for some formula A of modal logic; and is modal axiomatic i
X = fF : F
g;
for some set
of formulas of modal logic:
Again modal axiomatic is equivalent to modal -elementary. A set of formulas of modal logic is c-persistent i hWK ; RK i (the canonical frame for the normal modal logic K plus ), d-persistent i if hF; P i then F , for each descriptive general frame hv; P i, and r-persistent i if hF; P i then F , for each re ned general frame hF; P i. Note that r-persistent implies d-persistent, implies c-persistent, implies characterised by frames. Many proofs that a logic is characterised by frames involve cpersistence. However K plus (P ! P ) ! P is characterised by frames but is not c-persistent (see [Segerberg, 1971; van Benthem, 1979]). A class of frames veri es a d-persistent set of formulas i it is closed under subframes, frame-morphic images, disjoint unions, and both it and
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its complement are closed under the construction (F+ )] . We know from Section 10 that the class of frames verifying a set of formulas is closed under subframes and frame-morphic images, and that any frame F is frameisomorphic to a subframe of (F+ )] . The latter point can be extracted from the proof hat a descriptive fame F is isomorphic to (F+ )+ , and shows that the complement of a class of frames verifying a set of formulas is closed under the construction (F+ )] . It is easy to show that the class of frames verifying a set of formulas is closed under disjoint unions. If a frame F veri es a set of formulas then so do the modal algebra F+ and the descriptive general frame (F+ )+ , by Section 10. If is a d-persistent set of formulas then (F+ )] also veri es , so that the class of frames verifying a d-persistent set of formulas is closed under the construction (F+ )] . Conversely, suppose that a class X of frames satis es these closure conditions. Consider the class
X + = fF+ : F 2 X g of modal algebras and the set = fA : F+ A; for each F+ 2 X + g of formulas. If F 2 X then F+ and so F by Section 10. For the other direction, suppose that F and so F+ . The set of formulas is closely analogous to the set of equations in modal algebra veri ed by X +, so that the set of all modal algebras verifying is the variety generated by X +. Using a theorem of Birkho 's on varieties, a modal algebra F+ veri es i it is a homomorphic image of a subalgebra of a direct product of modal algebras fF+i : i 2 I g in X +. Checking the de nition of the disjoint union i2I Fi 2 X , the direct product i2I F+i is isomorphic to (i2I Fi )+ . Taking the carrier of the subalgebra to be P , this subalgebra is hi2I Fi ; P i+ . Thus there is a homomorphism from hi2I Fi ;i+ onto F+ . As in Section 10, we can dualise from the category of modal algebras to the category of descriptive frames, with homomorphic images going to subframes and subalgebras going to frame-morphic images. Thus (F+ )+ is frame-isomorphic to a subframe of (hi2I Fi ; P i+ )+ , and (hi2I Fi ; P i+ )+ is a frame-morphic image of ((i2 Fi )+ )+ , and (hi2I F; P i+ )+ is a framemorphic image of ((i2I Fi )+ )+ . Going to the underlying frames, (F+ )] is frame-isomorphic to a subframe of a frame-morphic image of ((i2I Fi )+] . Since i2I Fi 2 X and X is closed under subframes, frame-morphic images, and the construction (F+ )] , we have (F+ )] 2 X . Since the complement of X is also closed under the construction (F+ )] , we have F 2 X . Thus F i F 2 X , so that X is the class of frames verifying . It remains to show that is d -persistent. Supposing that a descriptive frame hF; P i veri es , and repeating the previous argument with hF; P i+ in place of F+ , wills how that (hF; P i+ )] 2 X . But the descriptive frame
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69
hF; P i is frame-isomorphic to (hF; P i+ )+ by Section 10, so that going to the underlying frames yields that F is frame-isomorphic to (hF; P i+ )] . Thus F 2 X and, hence, F veri es , so that is d- persistent.
Consider a set of formulas characterised by the class X of frames which verify it. Then is d-persistent i X is closed under the construction (F+ )] . If is d-persistent then one direction of the result applies, and yields X closed under the construction (F+ )] . If X is the class of frames verifying and is closed under the construction (F+ )] , then the other direction of the result applies. In this case it yields that X is the class of frames verifying some d-persistent set of formulas. Inspection of the proof shows that this is the set of semantic consequences of . But since is characterised by frames, it equals its set of semantic consequences so that is a d-persistent set of formulas. Combining our lemmas elementary equivalence and d-persistence yields two important theorems. Firstly, if a set of formulas is characterised by the class X of frames which verify it and X is closed under elementary equivalence, then is d-persistent. For then X is closed under the construction (F+ )] by the rst lemma, and so is d-persistent by the second lemma. Secondly, given a class X of frames closed under elementary equivalence, X is modal axiomatic i it is closed under subframes, frame -morphic images, disjoint unions, and its complement is closed under the construction (F+ )] . We have already seen that a modal axiomatic class of frames has these closure properties. If X is closed under elementary equivalence and these conditions then it satis es all the closure properties of the theorem on d-persistent sets, using the rst lemma. Thus X is modal axiomatic; indeed it is determined by a d-persistent set of formulas. The presentation here has followed the elegant van Benthem [1979]. The rst paper in this area was the important [Fine, 1975]. It de ned notions of modal saturation and persistence, and introduced the lemma on classes of frames closed under elementary equivalence. (It worked in terms of models rather than of general frames, but the analogy is close.) It proved the slightly weaker result, that if a set of formulas is characterised by the class X of frames which verify it and X is closed under elementary equivalence, then is c-persistent. The theorem giving the closure conditions for a class X of frames, which is closed under elementary equivalence, to be axiomatic, is Goldblatt's contribution to Goldblatt and Thomason [1975]. The proof was roughly similar to the one here but more complicated. It woo started with the duality between varieties of modal algebras and classes of descriptive frames, and used Fine's lemma and the properties of (F+ )] to bridge the gap between the frames and descriptive frames. Fine [1975] used classical modal theory to show that if a set of formulas is r-persistent then the class X of frames which verify is -elementary (and of course characterises the normal modal logic K plus ). It also gives counter-examples to the converse of both its theorems. In the second case the counter-example is
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ROBERT BULL AND KRISTER SEGERBERG
S4 3M. We know that it is characterised by the elementary class of frames determined by certain conditions. and it is veri ed by the re ned general frame h!; ; P i, where P is the set of nite and co nite subsets of !, but h!; i rejects M . 23 MODAL ELEMENTARY AND AXIOMATIC CLASSES The main construction for this topic is the ultraproduct of frames. Consider frames Fi hWi ; Ri i for i 2 I , and an ultra lter G on I . Remember that the members f of the direct product i2I Wi are the functions f : I ! [i2I Wi such that f (i) 2 Wi , for each i 2 I . De ne an equivalence relation ' on i2I Wi by taking f ' g i fi : f (i) = g(i)g 2 G and consider the equivalence classes [f ] under '. The ultraproduct FG = i2I Fi =G = hWG ; RG i is de ned by taking Q
Q
WG = i2I Wi =G = f[f ] : f 2 i2I Wi g; [f ]RG [g] i fi : f (i)Ri g(ig 2 G: To extend this de nition to general frames hFi ; Pi i, for i 2 I , it can rst be shown that if f ' g then fi : f (i) 2 S (i)g 2 G fi : g(i) 2 S (i)g 2 G; S ' T i 8f (fi : f (i) 2 S (i)g 2 G fi : f (i) 2 T (i)g 2 G); for f; g 2 i2I Wi and S; T
2 i2I Pi . This justi es de ning
[S ] = f[f ] : fi : f (i) 2 S (i)g 2 Gg; for each S 2 i2I Pi , and taking (
PG = [S ] : S 2
Y
i2I
)
Pi :
Here the de nition of a general frame requires that PG be a subalgebra of (i2I Fi =G)+ . for the case mRG we need
mRG [S ] = [mS ]; where
(mS )(i) = mRi (S (i)); for each i 2 I;
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for each S 2 i2I Pi . We have [f ] 2 mRG [S ] i [f ]RG[g]; for some [g] 2 [S ]; i fi : f (i)Ri g(i)g 2 G and fi : g(i) 2 S (i)g 2 G; for some g 2 i2I Wi ; i fi : f (i)Ru g(u) ^ g(i) 2 S (i)g 2 G; for some g 2 i2I Wi ; i fi : f (i) 2 mRi (S (i))g 2 G i fi : f (i) 2 (mS )(i)g 2 G i [f ] 2 [mS ]: Given a valuation Vi on each general frame Fi = hWi ; Ri ; Pi i, for each i 2 I , de ne a valuation VG on FG = hWG ; RG; PG i by taking
VG (P; [f ]) = T i [f ] 2 [VG (P )] i fi : Vi (P; f (i)) = T g 2 G; for each propositional variable P , and apply the de ning conditions for valuations. Then the argument like that of the previous paragraph shows that VG (A; [f ]) = T i fi : Vi (A; f (i)) = T g 2 G; for each formula A. It is now easy to show that
FG A i fi : Fi Ag 2 G: Going from left to right, note that if not fi : Fi Ag 2 G then fi : not Fi Ag 2 G, since G is an ultra lter. Now use valuations Vi and points f (i) with Vi (A; f (i)) = F , for each i in the member of G. Note that taking Pi = B(Wi ), for each i 2 I , does not yield PG = B(i2I Wi =G), so that i2I hFi ; B(Wi )i=G is not the same as i2I Fi =G. Therefore this result for ultraproducts of general frames yields only if FG A then fi : Fi = Ag 2 G; for ultraproducts of ordinary frames. (As we shall note later, M is a counterexample to the converse.) It follows that if X is a modal elementary class of frames, then its complement is closed under ultraproducts. Similarly, if X is a modal axiomatic class of frames than its complement is closed under ultrapowers. Here an ultrapower FI =G is the ultraproduct i2I Fi =G for which Fi = F, for each i 2 I . Classical model theory proves the following characterisations of the various kinds of elementary classes. A class X of frames is elementary i X and X are closed under frame isomorphism and ultraproducts. Class X is -elementary i X is closed under frame isomorphism and ultraproducts, and X is closed under ultrapowers. Class X is - elementary i X is closed under ultrapowers, and X is closed under frame isomorphism and
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ultraproducts. Class X is - elementary i X and X are closed under isomorphism and ultrapowers. Combining the results so far, it is easy to show that a modal elementary class of frames is elementary if it is closed under ultraproducts. And a modal axiomatic class is -elementary i it is closed under ultraproducts. Further, a class X of frames closed under frame isomorphism, subframes, disjoint unions, and ultrapowers is also closed under ultraproducts. for, given Fi 2 X , for i 2 I , it is easy to show that i2I Fi =G is isomorphic to a subframe of (i2I Fi )I =G. Now it is easy to show that for a modal elementary class X of frames, all the following conditions are equivalent: X is elementary, X is -elementary, X is -elementary,X is - elementay, X is closed under ultrapowers, X is closed under ultraproducts. For a modal axiomatic class X of frames, the conditions elementary and elementary are equivalent, and the following conditions are equivalent: X is -elementary, X is -elementary.X is closed under ultrapowers, X is closed under ultraproducts. Ultraproducts of frames were introduced in [Goldblatt, 1975], and are described in detail in [Goldblatt, 1976]. Goldblatt [1975] obtained some of the results above, and gave a complicated example of frames which verify M but have an ultraproduct which does not. It follows that the class of frames verifying M is not ( rst-order) axiomatic, although [Fine, 1975] shows that KM is characterised by the class of frames verifying it. (Therefore this class of frames is characterised by some formula of second-order predicate logic, as in the last part of Section 19.) This result was also proved independently in [van Benthem, 1975], by a direct method. Van Benthem [1976] proved more of the results above, the published version using Goldblatt's ultraproducts. The picture was completed in [Goldblatt, 1976], where there is also a more detailed explanation of the ultraproduct of frames which verify M . 24 TWO FURTHER RESULTS We have found closure conditions for a modal axiomatic class of frames, provided that it is closed under elementary equivalence and, hence, includes enough saturated frames. Can closure conditions for axiomatic classes of frames still be found when this condition is dropped? A rather complicated answer is provided in [Goldblatt and Thomason, 1975] (originally part of [Thomason, 1975]). given a frame hW; Ri, choosing a general frame hW; R; P i represents a choice of which `propositions' are to be considered. In then forming hU; S i = (hW; R; P i+ )] , the members of U are the ultra lters on P , representing `states-of-a airs', i.e. maximal consistent sets of `propositions'. The natural de nition of S on these `states-of-a airs' is, as usual, uSv i (8X 2)(X 2 v ! mRX 2 u):
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Under what conditions will hU; si again verify the formulas veri ed by hW; Ri? Firstly, there must be no `new propositions' in hU; S i, i.e. (8Y
U )(9X 2 P )(Y = (X )); where (X ) = fu 2 U : X 2 ug, or (8Y U )(9X 2 P )(u 2 Y ! X 2 u): Secondly, to carry out the necessary induction step on the value of A, we must have (8u 2 U )(8X 2 P )(mRX 2 u ! (9v 2 u)(uSv ^ X 2 v)): If hU; S i satis es these conditions for the carrier P of some subalgebra of hW; Ri+ , then we say that hU; S i is SA-based on hW; Ri. It can be shown, by a fairly diÆcult proof, that hU; S i is frame- isomorphic to a frame SA-based n hW; Ri i hU; S i+ is a homomorphic image of a subalgebra of hW; Ri+ . now a class of frames is modal axiomatic if it is closed under frame isomorphism, nontrivial disjoint unions, and the construction of hU; S i SA-based on hW; Ri. It is easy to show that a modal axiomatic class is closed under these conditions. For the converse, suppose that a class X of frames is closed under these conditions. As in the theorem in Section 23 on the closure conditions for the class of frames verifying a d-persistent set of formulas, we take
X + = fF+ : F 2 X g; = fA : F+ A ^ F+ 2 X + g; and show that X is the class of frames verifying . Again F+ veri es i it is a homomorphic image of a subalgebra for a direct product of modal algebras fF+i : i 2 I g in X +, where the direct product is isomorphic to (i2I Fi )+ for i2I Fi 2 X . by the lemma stated above F must be SA-based on i2I Fi , and so F 2 X . Thus if F then F 2 x, and the converse is clear. We are familiar with the duality between modal algebras and descriptive frames, and with the fact that we must shift from frames to descriptive frames before a duality can be established. Can we, as an alternative, shift to some other kind of algebra and then establish a duality with frames proper? This is done in [Thomason, 1975]. The appropriate algebras are the complete atomic modal algebras, i.e. modal algebras based on complete atomic Boolean algebras with
l \ fbi : i 2 I g = \flbi : i 2 I g; m [ fbi : i 2 I g = [fmbi : i 2 I g:
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ROBERT BULL AND KRISTER SEGERBERG
An atom of a Boolean algebra B = hB; 0; 1; ; \; [i is an element a with a b _ a \ b = 0; for each b 2 B: Then B is atomic i
2B
8b9a(a an atom ^ a b); and is complete i it is closed under the operations \ and [ for arbitrary subsets fbi : i 2 I g of B . In a complete atomic Boolean algebra, each element b is determined by the set of atoms a with a b. the appropriate morphisms for the category of complete atomic modal algebras are the complete homomorphisms, i.e. the homomorphisms with
([fbi : i 2 I g) = [f (bi ) : i 2 I g: this category is dual to the category of frames and frame morphisms. As far as the structures go, for each frame F the usual modal algebra F+ on B(W ) is complete and atomic. For each complete atomic modal algebra A with set of atoms At(A), we take the frame A+ = hAt(A); Ri with
xRy i x my; for each x; y 2 At(A): For the morphisms, given frames F = hW; Ri; F0 = hW 0 ; R0 i and a frame morphism : F ! F0 , de ne + : F0+ ! F+ by taking + (S ) = 1 [S ]; for each S 2 B(W 0 ) as before. In the other direction a new de nition is needed. given complete atomic modal algebras A; B and a complete homomorphism : A ! B, de ne + : B+ ! A+ by taking
+ (y) = x i y (x); for each x 2 At(A; y 2 At(B): To see that this de nition is valid, note that f (x) : x 2 At(A)g is a disjoint cover of B , since At(A) is a disjoint cover of A and is a complete homomorphism. It can be checked that each frame F is `isomorphic' to (F+ )+ , sand that each complete atomic modal algebra A is isomorphic to (A+ )+ , so that these categories are contravariantly dual to each other. ACKNOWLEDGEMENTS This chapter is the result of collaboration on the following terms. Segerberg wrote Section 1{9, Bull Sections 10{24. Although the authors met and together planned the paper, each wrote his part independently of the other will little ex post script discussion.
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Segerberg wishes to thank S. K. Thomason (who conveniently spent part of his sabbatical 1982 at the University of Aukland) for a number of very useful critical comments. Robert Bull University of Canterbury, New Zealand Krister Segerberg University of Uppsala, Sweden BIBLIOGRAPHY
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[Hintikka, 1975] J. Hinktkka. Carnap's heritage in logical semantics. In Rudolf Carnap, Logical Empiricist: Materials and Perspectives, J. Hintikka, ed. pp. 217{242. Reidel, Dordrecht, 1975. [Hofstadter and McKinsey, 1955] A. Hofstadter and J. C. C. McKinsey. On the logic of imperatives. Philosophy of Sciences, 6, 446{457, 1939. [Hughes and Cresswell, 1996] G. E. Hughes and M. J. Cresswell. A New Introduction to Modal Logic. Routledge, 1996. [Je rey, 1990] R. C. Je rey. Formal Logic: Its Scope and Limits, 3rd Edition. McGrawHill, NY, (1st edition 1967), 1990. [Jonsson, 1967] B. Jonsson. Algebras whose congruence lattices are distributive. Mathmatica Scandinavica, 21, 110{121, 1967. [Jonsson and Tarski, 1951] E. Jonsson and A. Tarski. Boolean algebras with operators. Part I. Am. J. Math., 74, 891{939, 1951. [Kamp, 1968] J. A. W. Kamp. On Tense Logic and the Theory of Order. PhD Dissertation, UCLA, 1968. [Kanger, 1957] S. Kanger. Provability in Logic. Disseration, Stockholm, 1957. [Kanger, 1957a] S. Kanger. New Foundations for Ethical Theory, Stockholm, 1957. Reprinted in Hilpinen [1971]. [Kanger, 1957b] S. Kanger. The Morning Star Paradox. Theoria, 23, 1{11, 1957. [Kanger, 1957c] S. Kanger. A note on quanti cation and modalities. Theoria, 23, 131{ 134, 1957. [Kaplan, 1966] D. Kaplan. Review. Journal of Symbolic Logic, 31, 120{122, 1966. [Kaplan, 1970] D. Kaplan. S5 with quanti able propositinal variables, Abstract. Journal of Symbolic Logic, 35, 355, 1970. [Kneale and Kneale, 1962] W. Kneale and M. Kneale. The Development of Logic. Clarendon Press, Oxford, 1962. [Kripke, 1959] S. A. Kripke. A completeness theorem in modal logic. Journal of Symbolic Logic, 24, 1{14, 1959. [Kripke, 1963] S. A. Kripke. Semantical considerations on modal logic. Acta Philosophical Fennica, 16, 83{94, 1963. [Kripke, 1963a] S. A. Kripke. Semantical analysis of modal logic I: Normal propositional calculi. Zeit. Math. Logik. Grund., 9, 67{96, 1963. [Kripke, 1965] S. A. Kripke. Semantical analysis of modal logic II: Non-normal modal propositional calculi. In The Theory of Models, J. W. Adison et al., eds. pp. 206{220. North-Holland, Amsterdam, 1965. [Kuhn, 1977] S. T. Kuhn. Many-sorted Modal Logics. Philosophical studies published by the Philosophical Society and the Department of Philosophy, University of Uppsala, Vol. 35, Uppsala, 1977. [Leivant, 1981] D. Leivant. On the proof theory of the modal logic for arithmetic provability. Journal of Symbolic Logic, 46, 531{538, 1981. [Lemmon, 1957] E. J. Lemmon. New foundations for Lewis modal systems. Journal of Symbolic Logic, 22, 176{186, 1957. [Lemmon, 1966] E. J. Lemmon. Algebraic semantics for modal logics. Journal of Symbolic Logic, 31, 46{65, 191{218, 1966. [Lemmon, 1977] E. J. Lemmon. an Introduction to Modal Logic. In collaboration with D. Scott, Blackwell, Oxford, 1977. [Lewis, 1912] C. I. Lewis. Implication and the algebra of logic. Mind, 21, 522{531, 1912. [Lewis, 1918] C. I. Lewis. A Survey of Symbolic Logic. University of California Press, Berkeley, 1918. [Lewis and Langford, 1959] C. I. Lewis and C. H. Langford. Symbolic Logic. The Century Co, NY, 1932. Second edn, Dover, NY, 1959. [Lewis, 1973] D. Lewis. Counterfatuals. Harvard University Press, Cambridge, MA, 1973. [Lukasiewicz, 1953] J. Lukasiewicz. A system of modal logic. Journal of Computing Systems, 1, 111{149, 1953. [Lukasiewicz, 1970] J. Lukasiewicz. Selected Works, L. BOrkowski, ed. North Holland, Amsterdam, 1970. [McCall, 1967] S. McCall. Polish Logic 1920{1939, Clarendon Press, Oxford, 1967.
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[McKinsey, 1941] J. C. C. McKinsey. A solution of the decision problem for the Lewis systems S2 and S4 with an application to topology. Journal of Symbolic Logic, 6, 117{134, 1941. [McKinsey, 1945] J. C. C. McKinsey. On the syntactical construction of modal logic. Journal of Symbolic Logic, 10, 83{96, 1945. [McKinsey and Tarski, 1944] J. C. C. McKinsey and A. Tarski. the algebra of topology. Annals of Mathematics, 45, 141{191, 1944. [McKinsey and Tarski, 1948] J. C. C. McKinsey and A. Tarski. Some theormes about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic, 13, 1{15, 1948. [Makinson, 1966] D. Makinson. On some completeness theorems in modal logic. Zeit. Math. Logik. Grund., 12, 379{384, 1966. [Makinson, 1969] D. Makinson. A normal modal calculus between T and S4 without the nite modal property. Journal of Symbolic Logic, 34, 35{38, 1969. [Makinson, 1970] D. Makinson. A generalisation of the concept of a relational model for modal logic. Theoria, 36, 331{335, 1970. [Makinson, 1971] D. Makinson. Aspectos de la logica mdoal, Instituto e matematica. Universidad Nacional del Sur, Bahia Blanca, 1971. [Makinson, 1971a] D. Makinson. Some embedding theorems for modal logic. Notre Dame Journal of Formal Logic, 12, 252{254, 1971. [Maksimova, 1975] L. L. Maksimova. Pretabular extensiosn of Lewis' S4. Algebra i logika, 14, 28{55, 1975. (In Russian) [Malinowski, 1977] G. Malinowski. Historical note. In selected Papers on Lukasiewicz Sentential Calculi, R. Wojcicki, ed. pp. 177{187. Polish Academy of Sciences, Wroclaw, 1977. [Mally, 1926] E. Mally. Grundgesetze des Sollens: Elemente der Logik des Willens. Lenscher and Lugensky, Graz, 1926. [Montague, 1963] R. Montague. Syntactical treatments of modality, with corollaries on re extion principles and nite axiomatisability. Acta Philosophica Fennica, 16, 153{ 167, 1963. Reprinted in Montague [1974]. [Montague, 1968] R. Montague. Pragmatics. In Contemporary Philosophy: A Survey, Vol. 1. R. Klibansky, ed. pp. 102{122. La Nuova Editrice, Florence, 1968. Reprinted in Montague [1974]. [Montague, 1974] R. Montague. Formal Philosophy: Selected Papers of Richard Montague. Edited, with an introduction by Richmond H. Thomason. Yale University Press, New Haven, 1974. [Morgan, 1979] C. Morgan. Modality, analogy, and ideal experiments according to C. S. Pierce. Synthese, 41, 65{83, 1979. [Mortimer, 1974] M. Mortimer. Some results in modal model theory. Journal of Symbolic Logic, 39, 496{508, 1974. [Ohnishi and Matsumoto, 1957/59] M. Ohnishi and K. Matsumoto. Gentzen method in modal calculi. Osaka Mathematical Journal, 9, 113{130; 11, 115{120, 1957/1959. [Parry, 1934] W. T. Parry. The postulates for `strict implication'. Mind, 43, 78{80, 1934. [Parsons, 1982] C. Parsons. Intensional logic in extensional language. Journal of Symbolic Logic, 47, 289{328, 1982. [Pratt, 1980] V. R. Pratt. Application of modal logic to programming. Studia Logica, 34, 257{274, 1980. [Prawitz, 1965] D. Prawitz. Natural Deduction: A Proof-theoretic study, Stockholm Studies in Philocopy 3, Almqvist and Wiskell, Stockholm, 1965. [Prior, 1962] A. N. Prior. Formal Logic. Clarendon Press, Oxford, 1955. Second Edition, 1962. [Prior, 1957] A. N. Prior. Time and Modality. Clarendon Press, Oxford, 1957. [Prior, 1967] A. N. Prior. Past, Present and Future. Clarendon Press, Oxford. 1967, [Rasiowa and Sikorski, 1963] H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics, Panstwowe Wydawnictwo Naukowe, 1963. [Rautenberg, 1979] W. Rautenberg. klassische und nichtklassische Aussagenlogik, Bieweg, Braunschweig, Wiesbaden, 1979. [Rescher and Urquhart, 1971] N. Rescher and A. Urquhart. Temporal Logic. SpringerVerlag, NY, 1971.
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[Ridder, 1955] J. Ridder. Die Grntzensschen Schlussverfahren in modalen Aussagenlogiken I. Indagationes mathematicae, 17, 163{276, 1955. [Sahlqvist, 1975] H. Sahlqvist. Completeness and correspondence in the rst and second order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium, S. Kanger, ed. pp. 110{143. North-Holland, Amsterdam, 1975. [Schumm, 1981] G. F. Schumm. Bounded properties in modal logic. Zeit. Math. Logik. Grund., 27, 197{200, 1981. [Schutte, 1968] K. Schutte. Vollstandige Systeme modaler und intuitionistischer Logik. Springer-Verlag, Berlin, 1968. [Scott, 1971] D. Scott. On engendering an illusation of understanding. Journal of Philosophy, 68, 787{807, 1971. [Scroggs, 1951] S. J. Scroggs. Extensions of the Lewis system S5. Journal of Symbolic Logic, 16, 112{120, 1951. [Segerberg, 1968] K. Segerberg. Decidability of S4.2. Theoria, 34, 7{20, 1968. [Segerberg, 1970] K. Segerberg. Modal logics with linear alternative relations. Theoria, 36, 301{322, 1970. [Segerberg, 1971] K. Segerberg. An Essay in Classical Modal Logic. Philosophical studies published by the Philosophical society and the Department of Philosophy, University of Uppsala, Vol. 13, Uppsala, 1971. [Segerberg, 1982] K. Segerberg. Classical Propositional Operators: An Exercise in the Foundations of Logic, Clarendon Press, Oxford, 1982. [Segerberg, 1989] K. Segerberg. Von Wright's tense-logic. In The Philosophy of Georg Henrik von Wright, P. A. Schlipp, ed. 1989. [Shoesmith and Smiley, 1978] D. J. Shoesmith and T. J. Smiley. Multiple-conclusion Logic. Cambridge University Press, Cambridge, 1978. [Smullyan, 1968] R. M. Smullyan. First-order Logic. Springer-Verlag, NY, 1968. [Snyder, 1971] D. P. Snyder. Modal Logic and its Applications. Van Nostrand Reinhold, NY, 1971. [Sobincinski, 1964] B. Sobincinski. Family K of the non-Lewis modal systems. Notre Dame Journal of Formal Logic, 5, 313{318, 1964. [Solovay, 1976] R. S. M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 25, 287{304, 1976. [Stalnaker, 1968] R. Stalnaker. A theory of conditionals. In Studies in Logical Theory, N. Rescher, ed. p. 98{112. Blackwell, Oxford, 1968. [Thomason, 1972] S. K. Thomason. Semantic analysis of tense logics. Journal of Symbolic Logic, 37, 150{158, 1972. [Thomason, 1972a] S. K. Thomason. Noncompactness in propositional modal logic. Journal of Symbolic Logic, 37, 716{720, 1972. [Thomason, 1974] S. K. Thomason. An incompleteness theorem in modal logic. Theoria, 40, 30{34, 1974. [Thomason, 1975] S. K. Thomason. Categories of frames for modal logic. Journal of Symbolic Logic, 40, 439{442, 1975. [van Benthem, 1975] J. F. A. K. van Benthem. A note on modal formulae and relational properties. Journal of Symbolic Logic, 40, 55{58, 1975. [van Benthem, 1976] J. F. A. K. van Benthem. Modal formulas are either elementary or not -elementary. Journal of Symbolic Logic, 41, 436{438, 1976. [van Benthem, 1978] J. F. A. K. van Benthem. Two simple incomplete modal logics. Theoria, 44, 25{37, 1978. [van Benthem, 1979] J. F. A. K. van Benthem. Canonical modal logics and ultra lter extensions. Journal of Symbolic Logic, 44, 1{8, 1979. [van Benthem, 1979a] J. F. A. K. van Benthem. Syntactic aspects of modal incompleteness theorems. Theoria, 45, 67{81, 1979. [van Benthem and Blok, 1978] J. F. A. K. van Benthem and W. Blok. Transitivity follows from Dummett's axiom. Theoria, 44, 117{118, 1978. [von Wright, 1951] G. H. von Wright. An Essay in Modal Logic. North Holland, Amsterdam, 1951. [von Wright, 1951a] G. H. von Wright. Deontic logic. Mind, 60, 1{15, 1951.
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[von Wright, 1968] G. H. von Wright. An essay in deontic logic and general theory of action with a bibliography of deontic and imperative logic. Acta Philosophical Fennica, 21, 1968. [von Wright, 1981] G. H. von Wright. Problems and propsects of deontic logic. A Survey. In Modern Logic|A Survey, ed. Evandro Agazzi, ed. pp. 199{423. Reidel, Dordrecht, 1981. [Zeman, 1973] J. J. Zeman. Modal Logic: The Lewis-Modal Systems. Clarendeon Press, Oxford, 1973.
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
ADVANCED MODAL LOGIC This chapter is a continuation of the preceding one, and we begin it at the place where the authors of Basic Modal Logic left us about fteen years ago. Concluding his historical overview, Krister Segerberg wrote: \Where we stand today is diÆcult to say. Is the picture beginning to break up, or is it just the contemporary observer's perennial problem of putting his own time into perspective?" So, where did modal logic of the 1970s stand? Where does it stand now? Modal logicians working in philosophy, computer science, arti cial intelligence, linguistics or some other elds would probably give di erent answers to these questions. Our interpretation of the history of modal logic and view on its future is based upon understanding it as part of mathematical logic. Modal logicians of the First Wave constructed and studied modal systems trying to formalize a few kinds of necessity-like and possibility-like operators. The industrialization of the Second Wave began with the discovery of a deep connection between modal logics on the one hand and relational and algebraic structures on the other, which opened the door for creating many new systems of both arti cial and natural origin. Other disciplines| the foundations of mathematics, computer science, arti cial intelligence, etc.|brought (or rediscovered1) more. \This framework has had enormous in uence, not only just on the logic of necessity and possibility, but in other areas as well. In particular, the ideas in this approach have been applied to develop formalisms for describing many other kinds of structures and processes in computer science, giving the subject applications that would have probably surprised the subject's founders and early detractors alike" [Barwise and Moss 1996]. Even two or three mathematical objects may lead to useful generalizations. It is no wonder then that this huge family of logics gave rise to an abstract notion (or rather notions) of a modal logic, which in turn put forward the problem of developing a general theory for it. Big classes of modal systems were considered already in the 1950s, say extensions of S5 [Scroggs 1951] or S4 [Dummett and Lemmon 1959]. Completeness theorems of Lemmon and Scott [1977],2 Bull [1966b] and Segerberg [1971] demonstrated that many logics, formerly investigated \piecewise", have in fact very much in common and can be treated by the same methods. A need for a uniting theory became obvious. \There are two main lacunae in recent work on modal logic: a lack of general results and a lack of negative results. This or that logic is shown to have such and such a property, but very little is known about the scope or bounds of the property. 1 One of the celebrities in modal logic|the G odel{Lob provability logic GL|was rst introduced by Segerberg [1971] as an \arti cial" system under the name K4W. 2 This book was written in 1966.
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Thus there are numerous results on completeness, decidability, nite model property, compactness, etc., but very few general or negative results", wrote Fine [1974c]. The creation of duality theory between relational and algebraic semantics ([Lemmon 1966a,b], [Goldblatt 1976a,b]), originated actually by Jonsson and Tarski [1951], the establishment of the connection between modal logics and varieties of modal algebras ([Kuznetsov 1971], Maksimova and Rybakov [1974], [Blok 1976]), and between modal and rst and higher order languages ([Fine 1975b], [van Benthem 1983]) added those mathematical ingredients that were necessary to distinguish modal logic as a separate branch of mathematical logic. On the other hand, various particular systems became subjects of more special disciplines, like provability logic, deontic logic, tense logic, etc., which has found re ection in the corresponding chapters of this Handbook. In the 1980s and 1990s modal logic was developing both \in width" and \in depth", which made it more diÆcult for us to select material for this chapter. The expansion \in width" has brought in sight new interesting types of modal operators, thus demonstrating again the great expressive power of propositional modal languages. They include, for instance, polyadic operators, graded modalities, the xed point and di erence operators. We hope the corresponding systems will be considered in detail elsewhere in the Handbook; in this chapter they are brie y discussed in the appendix, where the reader can nd enough references. Instead of trying to cover the whole variety of existing types of modal operators, we decided to restrict attention mainly to the classes of normal (and quasi-normal) uni- and polymodal logics and follow \in depth" the way taken by Bull and Segerberg in Basic Modal Logic, the more so that this corresponds to our own scienti c interests. Having gone over from considering individual modal systems to big classes of them, we are certainly interested in developing general methods suitable for handling modal logics en masse. This somewhat changes the standard set of tools for dealing with logics and gives rise to new directions of research. First, we are almost completely deprived of proof-theoretic methods like Gentzen-style systems or natural deduction. Although proof theory has been developed for a number of important modal logics, it can hardly be extended to reasonably representative families. (Proof theory is discussed in the chapter Sequent systems for modal logics in a later volume of this Handbook; some references to recent results can be found in the appendix.) In fact, modern modal logic is primarily based upon the frame-theoretic and algebraic approaches. The link connecting syntactical representations of logics and their semantics is general completeness theory which stems from the pioneering results of Bull [1966b], Fine [1974c], Sahlqvist [1975], Goldblatt and Thomason [1974]. Completeness theorems are usually the rst step in understanding various properties of logics, especially those that have semantic or algebraic equivalents. A classical example is Maksimova's
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[1979] investigation of the interpolation property of normal modal logics containing S4, or decidability results based on completeness with respect to \good" classes of frames. Completeness theory provides means for axiomatizing logics determined by given frame classes and characterizes those of them that are modal axiomatic. Standard families of modal logics are endowed with the lattice structure induced by the set-theoretic inclusion. This gives rise to another line of studies in modal logic, addressing questions like \what are co-atoms in the lattice?" (i.e., what are maximal consistent logics in the family?), \are there in nite ascending chains?" (i.e., are all logics in the family nitely axiomatizable?), etc. From the algebraic standpoint a lattice of logics corresponds to a lattice of subvarieties of some xed variety of modal algebras, which opens a way for a fruitful interface with a well-developed eld in universal algebra. A striking connection between \geometrical" properties of modal formuT las, completeness, axiomatizability and -prime elements in the lattice of modal logics was discovered by Jankov [1963, 1969], Blok [1978, 1980b] and Rautenberg [1979]. These observations gave an impetus to a project of constructing frame-theoretic languages which are able to characterize the \geometry" and \topology" of frames for modal logics ([Zakharyaschev 1984, 1992], [Wolter 1996c]) and thereby provide new tools for proving their properties and clarifying the structure of their lattices. One more interesting direction of studies, arising only when we deal with big classes of logics, concerns the algorithmic problem of recognizing properties of ( nitely axiomatizable) logics. Having undecidable nitely axiomatizable logics in a given class [Thomason 1975a; Shehtman 1978c], it is tempting to conjecture that non-trivial properties of logics in this class are undecidable. However, unlike Rice's Theorem in recursion theory, some important properties turn out to be decidable, witness the decidability of interpolation above S4 [Maksimova 1979]. The machinery for proving the undecidability of various properties (e.g. Kripke completeness and decidability) was developed in [Thomason 1982] and [Chagrov 1990b,c]. Thomason [1982] proved the undecidability of Kripke completeness rst in the class of polymodal logics and then transferred it to that of unimodal ones. In fact, Thomason's embedding turns out to be an isomorphism from the lattice of logics with n necessity operators onto an interval in the lattice of unimodal logics, preserving many standard properties [Kracht and Wolter 1999]. Such embeddings are interesting not only from the theoretical point of view but can also serve as a vehicle for reducing the study of one class of logics to another. Perhaps the best known example of such a reduction is the Godel translation of intuitionistic logic and its extensions into normal modal logics above S4 [Maksimova and Rybakov 1974; Blok 1976; Esakia 1979a,b]. We will take advantage of this translation to give a brief survey of results in the eld of superintuitionistic logics which actually were always
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studied in parallel with modal logics (see also Section 5 of Intuitionistic Logic in volume 7 of this Handbook). Listed above are the most important general directions in mathematical modal logic we are going to concentrate on in this chapter. They, of course, do not cover the whole discipline. Other topics, for instance, modal systems with quanti ers, the relationship between the propositional modal language and the rst (or higher) order classical language, or proof theory are considered in other chapters of this Handbook. It should be emphasized once again that the reader will nd no discussions of particular modal systems in this chapter. Modal logic is presented here as a mathematical theory analyzing big families of logics and thereby providing us with powerful methods for handling concrete ones. (In some cases we illustrate technically complex methods by considering concrete logics; for instance Rybakov's [1994] technique of proving the decidability of the admissibility problem for inference rules is explained only for GL.) 1 UNIMODAL LOGICS We begin by considering normal modal logics with one necessity operator, which were introduced in Section 6 of Basic Modal Logic. Recall that each such logic is a set of modal formulas (in the language with the primitive connectives ^, _, !, ?, ) containing all classical tautologies, the modal axiom (p ! q) ! (p ! q); and closed under substitution, modus ponens and necessitation '='.
1.1 The lattice NExtK First let us have a look at the class of normal modal logics from a purely syntactic point of view. Given a normal modal logic L0 , we denote by NExtL0 the family of its normal extensions. NExtK is thus the class of all normal modal logics. Each logic L in NExtL0 can be obtained by adding to L0 a set of modal formulas and taking the closure under the inference rules mentioned above; in symbols this is denoted by
L = L0 : Formulas in are called additional (or extra) axioms of L over L0 . Formulas ' and are said to be deductively equivalent in NExtL0 if L0 ' = L0 . For instance, p ! p and p ! p are deductively equivalent in NExtK, both axiomatizing T, however (p ! p) $ (p ! p) 62 K. (For more information on the relation between these formulas see [Chellas and Segerberg 1994] and [Williamson 1994].)
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We distinguish between two kinds of derivations from assumptions in a logic L 2 NExtK. For a formula ' and a set of formulas , we write `L ' if there is a derivation of ' from formulas in L and with the help of only modus ponens. In this case the standard deduction theorem| ; `L ' i `L ! '|holds. The fact of derivability of ' from in L using both modus ponens and necessitation is denoted by `L '; in such a case we say that ' is globally derivable3 from in L. For this kind of derivation we have the following variant of the deduction theorem which is proved by induction on the length of derivations in the same manner as for classical logic. THEOREM 1 (Deduction). For every logic L 2 NExtK, all formulas ' and , and all sets of formulas , ; ` ' i 9m 0 ` m ! '; L
L
where m = 0 ^ ^ m and n is pre xed by n boxes. It is to be noted that in general no upper bound for m can be computed even for a decidable L (see Theorem 194). However, if the formula tran = n p ! n+1 p
is in L|such an L is called n-transitive|then we can clearly take m = n. In particular, for every L 2 NExtK4, ; `L ' i `L + ! ', where + = ^ . Moreover, a sort of conversion of this observation holds. THEOREM 2. The following conditions are equivalent for every logic L in NExtK: (i) L is n-transitive, for some n < !; (ii) there exists a formula (p; q) such that, for any ', and , ; ` ' i ` ( ; '): L
L
Proof. The implication (i) ) (ii) is clear. To prove the converse, observe rst that (p; q) `L (p; q) and so (p; q); p `L q. By Theorem 1, we then have (p; q) `L np ! q, for some n. Let q = n+1 p. Then (p; n+1p) ` n p ! n+1 p: L
And since p `L n+1 p, (p; n+1 p) 2 L. Consequently, tran 2 L.
REMARK. Note also that (i) is equivalent to the algebraic condition: the variety of modal algebras for L has equationally de nable principal congruences. For more information on this and close results consult [Blok and Pigozzi 1982].
3 This name is motivated by the semantical characterization of ` to be given in L Theorem 19.
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The sum L1 L2 and intersection L1 \ L2 of logics L1 ; L2 2 NExtL0 are clearly logics in NExtL0 as well. The former can be axiomatized simply by joining the axioms of L1 and L2 . To axiomatize the latter we require the following de nition. Given two formulas '(p1 ; : : : ; pn ) and (p1 ; : : : ; pm ) (whose variables are in the lists p1 ; : : : ; pn and p1 ; : : : ; pm , respectively), denote by '_ the formula '(p1 ; : : : ; pn ) _ (pn+1 ; : : : ; pn+m ). THEOREM 3. Let L1 = L0 f'i : i 2 I g and L2 = L0 f j : j 2 J g. Then
L1 \ L2 = L0 fm 'i _ n j : i 2 I; j 2 J; m; n 0g: Proof. Denote by L the logic in the right-hand side of the equality to be established and suppose that 2 L1 \ L2 . Then for some m; n 0 and some nite I 0 and J 0 such that all '0i and j0 , for i 2 I 0 , j 2 J 0 , are substitution instances of some 'i0 and j0 , for i0 2 I , j 0 2 J , we have ^ ^ 0 m '0i ! 2 L0 ; n j ! 2 L0 ; 0 0 i2I j 2J from which ^ (k '0i _ l j0 ) ! 2 L0 i2I 0 ;j2J 0 k;lm+n and so 2 L because k '0i _l j0 is a substitution instance of k 'i0 _l j0 . 0
Thus, L1 \ L2 L. The converse inclusion is obvious.
Although the sum of logics di ers in general from their union, these two operations have a few common important properties. THEOREM 4. The operation is idempotent, commutative, associative and distributes over \; the operation \ distributes over (in nite) sums, i.e.,
L\
M
i2I
Li =
M
i2I
(L \ Li ):
It follows that hNExtL0; ; \i is a complete distributive lattice, with L0 and the inconsistent logic, i.e., the set For of all modal formulas, being its zero and unit elements, respectively, and the set-theoretic its corresponding lattice order. Note, however, that does not in general distribute over in nite intersections of logics. For otherwise we would have (K :?)
\
1n
(K n ?) =
\
1n
(K :? n ?);
which is a contradiction, since the logic in the left-hand side is consistent (D, to be more precise), while that in the right-hand side is not.
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If we are interested in nding a simple (in one sense or another) syntactic representation of a logic L 2 NExtL0 , we can distinguish nite, recursive and independent axiomatizations of L over L0 . The former two notions mean that L = L0 , for some nite or, respectively, recursive , and a set of axioms is independent over L0 if L 6= L0 for any proper subset of . In the case when L0 is K or any other nitely axiomatizable over K logic, we may omit mentioning L0 and say simply that L is nitely (recursively, independently) axiomatizable. It is fairly easy to see that L is not nitely axiomatizable over L0 i thereLis an in nite sequence of logics L1 L2 : : : in NExtL0 such that L = i>0 Li . This observation is known as Tarski's criterion. (It is worth noting that nite axiomatizability is not preserved under \. For example, using Tarski's criterion, one can show that D \ (K p _ :p) is not nitely axiomatizable.) The recursive axiomatizability of a logic L, as was observed by Craig [1953], is equivalent to the recursive enumerability of L. As for independent axiomatizability, an interesting necessary condition can be derived from [Kleyman 1984]. Suppose a normal modal logic L1 has an independent axiomatization. Then, for every nitely axiomatizable normal modal logic L2 L1 , the interval of logics [L2; L1 ] = fL 2 NExtK : L2 L L1 g contains an immediate predecessor of L1 . Using this condition Chagrov and Zakharyaschev [1995a] constructed various logics in NExtK4, NExtS4 and NExtGrz without independent axiomatizations. To understand the structure of the lattice NExtL0 it may be useful to look for a set of formulas which is complete in the sense that its formulas are able to axiomatize all logics in the class, and independent in the sense that it contains no complete proper subsets. Such a set (if it exists) may be called an axiomatic basis of NExtL0 . The existence of an axiomatic basis depends on whether every logic in the class can be represented L as the sum of \indecomposable" logics. A logic L 2 NExtL0 is said to be {irreducible L in NExtL0 if for any family fLi : i 2LI g of logics in NExtL0 , L = i2I Li implies {prime if for any family fLi : i 2 I g, L L = Li for some i 2 I . L is L i2I Li only if there is i 2 L I such that L Li . L It is not hard to see (using Theorem 4) that a logic is {irreducible i it is {prime. This does T T not hold, however, for the dual notions of {irreducible and {prime logics. T T We have only one implication in general: ifTL is {prime (i.e., i2ITLi L only if Li L, for some i 2 I ) then it is {irreducible (i.e., L = i2I Li only if L = LLi , for some i 2 I ). A formula ' is said to be prime in NExtL0 if L0 ' is {prime in NExtL0 . PROPOSITION 5. Suppose a set of formulas is complete for NExtL0 and contains no distinct deductively equivalent in NExtL0 formulas. Then is an axiomatic basis for NExtL0 i every formula in is prime.
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Although the de nitions above seem to be quite simple, inTpractice it L is not so easy to understand whether a given logic is { or {prime, at least at the syntactical level. However, these notions turn out to be closely related to the following lattice-theoretic concept of splitting for which in the next section we shall provide a semantic characterization. A pair (L1 ; L2 ) of logics in NExtL0 is called a splitting pair in NExtL0 if it divides the lattice NExtL0 into two disjoint parts: the lter NExtL2 and the ideal [L0; L1 ]. In this case we also say that L1 splits and L2 cosplits NExtL0 . T THEOREM 6. A logic L1 splits L NExtL0 i it is {prime in NExtL0 , and L2 cosplits NExtL0 i it is {prime in NExtL0 . Moreover, the following conditions are equivalent: (i) (L1 ; L2T) is a splitting pair in NExtL0;T (ii) L1 is L{prime in NExtL0 and L2 = L fL 2 NExtL0 : L 6 L1 g; (iii) L2 is {prime in NExtL0 and L1 = fL 2 NExtL0 : L 6 L2 g. Splittings were rst introduced in lattice theory by Whitman [1943] and McKenzie [1972] (see also [Day 1977], [Jipsen and Rose 1993]). Jankov [1963, 1968b, 1969], Blok [1976] and Rautenberg [1977] started using splittings in non-classical logic. A few standard normal modal logics are listed in Table 1. Note that our notations are somewhat di erent from those used in Basic Modal logic. (A was introduced by Artemov; see [Shavrukov 1991]. The formulas Bn bounding depth of frames are de ned in Section 15 of Basic Modal Logic.)
1.2 Semantics
The algebraic counterpart of a logic L 2 NExtK is the variety of modal algebras validating L (for de nitions consult Section 10 of Basic Modal Logic). Conversely, each variety (equationally de nable class) V of modal algebras determines the normal modal logic LogV = f' : 8A 2 V A j= 'g. Thus we arrive at a dual isomorphism between the lattice NExtK and the lattice of varieties of modal algebras, which makes it possible to exploit the apparatus of universal algebra for studying modal logics. It is often more convenient, however, to deal not with modal algebras directly but with their relational representations discovered by Jonsson and Tarski [1951] and now known as general frames. Each general frame F = hW; R; P i is a hybrid of the usual Kripke frame hW; Ri and the modal algebra
F+ = hP; ;; W; ; \; [; ; i in which the operations and are uniquely determined by the accessibility relation R: for every X 2 P 2W ,
X = fx 2 W : 8y (xRy ! y 2 X )g; X = X:
ADVANCED MODAL LOGIC
Table 1. A list of standard normal modal logics.
D T KB K4 K5 Altn D4 S4 GL Grz K4:1 K4:2 K4:3 S4:1 S4:2 S4:3 Triv Verum S5 K4B A Dum K4BWn K4BDn K4n;m
= = = = = = = = = = = = = = = = = = = = = = = = =
K p ! p K p ! p K p ! p K p ! p K p ! p K p1 _ (p1 ! p2 ) _ _ (p1 ^ ^ pn ! pn+1 ) K4 > K4 p ! p K4 (p ! p) ! p K ((p ! p) ! p) ! p K4 p ! p K4 (p ^ q) ! (p _ q) K4 (+ p ! q) _ (+ q ! p) S4 p ! p S4 p ! p S4 (p ! q) _ (q ! p) K4 p $ p K4 p S4 p ! p K4 p ! p GL p ! (+p ! q) _ (+ q ! p) S4 ((p ! p) ! p) ! (p ! p) V W K4 ni=0 pi ! 0i=6 jn (pi ^ (pj _ pj )) K4 Bn K4 n p ! mp; for 1 m < n
91
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So, using general frames we can take advantage of both relational and algebraic semantics. To simplify notation, we denote general frames of the form F = W; R; 2W by F = hW; Ri. Such frames will be called Kripke frames. Given a class of frames C , we write LogC to denote the logic determined by C , i.e., the set of formulas that are valid in all frames in C ; it is called the logic of C . If C consists of a single frame F, we write simply LogF. Basic facts about duality between frames and algebras can be found in the chapters Basic Modal Logic and Correspondence Theory in this volume. Here we remind the reader of the de nitions that will be important in what follows. A frame G = hV; S; Qi is said to be a generated subframe of a frame F = hW; R; P i if V W is upward closed in F, i.e., x 2 V and xRy imply y 2 V , S = R V and Q = fX \ V : X 2 P g. The smallest generated subframe G of F containing a set X W is called the subframe generated by X . A frame F is rooted if there is x 2 W |a root of F|such that the subframe of F generated by fxg is F itself. A map f from W onto V is a reduction (or p-morphism) of a frame F = hW; R; P i to G = hV; S; Qi if the following three conditions are satis ed for all x; y 2 W and X 2 Q (R1) xRy implies f (x)Sf (y); (R2) f (x)Sf (y) implies 9z 2 W (xRz ^ f (z ) = f (y)); (R3) f 1 (X ) 2 P . The operations of reduction and generating subframes are relational counterparts of the algebraic operations of forming subalgebras and homomorphic images, respectively, and so preserve validity. A frame F = hW; R; P i is di erentiated if, for any x; y 2 W ,
x = y i 8X 2 P (x 2 X $ y 2 X ):
F is tight if
xRy i 8X 2 P (x 2 X ! y 2 X ): Those frames that are both di erentiated and tight are called re ned. A frame F is said to be compactTif every subset X of P with the nite intersection property (i.e., with X 0 6= ; for any nite subset X 0 of X ) has non-empty intersection. Finally, re ned and compact frames are called descriptive. A characteristic property of a descriptive F is that it is isomorphic to its bidual (F+ )+ . The classes of all di erentiated, tight, re ned and descriptive frames will be denoted by DF , T , R and D, respectively. When representing frames in the form of diagrams, we denote by ir re exive points, by Æ re exive ones, and by ÆÆ two-point clusters. An arrow from x to y means that y is accessible from x. If the accessibility relation is transitive, we draw arrows only to the immediate successors of x.
ADVANCED MODAL LOGIC nontransitive
! + 1-!
2
93
transitive -1 -0
Æ Figure 1.
EXAMPLE 7. (Van Benthem 1979) Let F = hW; R; P i be the frame whose underlying Kripke frame is shown in Fig. 1 (! + 1 sees only ! and the subframe generated by ! is transitive) and X W is in P i either X is nite and ! 2= X or X is co nite in W and ! 2 X . It is easy to see that P is closed under \, and . Clearly, F is re ned. Suppose X is a subset of P with Tthe nite intersection property. If X contains a nite set T then obviously X 6= ;. And if X consists of only in nite sets then ! 2 X . Thus, F is descriptive. A frame F is said to be {-generated, { a cardinal, if its dual F+ is a {-generated algebra.4 Each modal logic L is determined by the free nitely generated algebras in the corresponding variety, i.e., by the Tarski{ Lindenbaum (or canonical) algebras AL(n) for L in the language with n < ! variables. Their duals are denoted by FL (n) = hWL (n); RL (n); PL (n)i and called the universal frames of rank n for L. Analogous notation and terminology will be used for the free algebras AL ({) with { generators. Note that hWL ({); RL ({)i is (isomorphic to) the canonical Kripke frame for L with { variables (de ned in Section 11 of Basic Modal Logic) and PL ({) is the collection of the truth-sets of formulas in the corresponding canonical model. Unless otherwise stated, we will assume in what follows that the language of the logics under consideration contains ! variables. An important property of the universal frame of rank { for L is that every descriptive {0 -generated frame for L, {0 {, is a generated subframe of FL({). Thus, the more information about universal frames for L we have, the deeper our knowledge about the structure of arbitrary frames for L and thereby about L itself. Although in general universal frames for modal logics are very complicated, considerable progress was made in clarifying the structure of the upper part (points of nite depth) of the universal frames of nite rank for logics in NExtK4. The studies in this direction were started actually by Segerberg [1971]. Shehtman [1978a] presented a general method of constructing the universal frames of nite rank for logics in NExtS4 with the nite model property. Later similar results were obtained by other authors; see e.g. [Bellissima 1985]. The structure of free nitely generated algebras 4 An algebra is said to be { -generated if it contains a set X of cardinality { such that the closure of X under the algebra's operations coincides with its universe.
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for S4 was investigated by Blok [1976]. Let us try to understand rst the constitution of an arbitrary transitive re ned frame F = hW; R; P i with n generators G1 ; : : : ; Gn 2 P . De ne V to be the valuation of the set of variables = fp1; : : : ; pn g in F such that x j= pi i x 2 Gi . Say that points x and y are -equivalent, x y in symbols, if the same variables in are true at them; for X; Y W we write X Y if every point in X is -equivalent to some point in Y and vice versa. Let d(F) denote the depth5 of F; if F is of in nite depth, we write d(F) = 1. For d < d(F), W =d and W >d are the sets of all points in F of depth d and > d, respectively; W
Proof. (i) follows from the di erentiatedness of F and the obvious fact that precisely the same formulas (in p1 ; : : : ; pn) are true under V at -equivalent points in the same cluster. The proof of (ii) proceeds by induction on d. Let x 2 W >d. Since F is transitive and W d is nite (by the induction hypothesis), there exists a non-empty upward closed in W >d set X (i.e., X = X " \ W >d) such that x 2 X #, points in X see exactly the same points of depth d and either (1)
8u; v 2 X 9w 2 u" \ X w v
or (2)
8u; v 2 X (u v ^ :uRv):
Such a set X is called d-cyclic; it is nondegenerate if (1) holds and degenerate otherwise. One can readily show that the same formulas are true at equivalent points in X . Since F is re ned, X is then a cluster of depth d + 1. Thus, W >d W =d+1 #. The upper bound for the number of distinct 5
In Section 15 of Basic Modal Logic d(F) was called the rank of F.
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clusters of depth d + 1 follows from the di erentiatedness of F and the de nition of d-cyclic sets. To establish (iii), for every point x of depth d + 1 one can construct by induction on d a formula (expressing the de nition of the d-cyclic set containing x) which is true in F under V only at x. For details consult [Chagrov and Zakharyaschev 1997]. < 1 It is fairly easy now to construct the (generated) subframe FK4 (n) of the universal frame of rank n for K4 consisting of nite depth points. Indeed, FK4(n) is n-generated, re ned and so has the form as described in Theorem 8. On the other hand, it is universal and contains any n-generated descriptive frame as a generated subframe, which means roughly that it contains all possible points of nite depth that can exist in n-generated re ned frames. More precisely, assuming that each point is assigned the set of variables in that are true at it, we begin constructing a frame GK4 (n) nby putting at depth 1 in it 2n non--equivalent degenerate clusters and 22 1 non-equivalent non-degenerate clusters with 2n non--equivalent points. d Suppose that G K4 (n) is already constructed. Then for every antichain a of d clusters in GK4 (n) containing at least one cluster of depth d and di erent d from a singleton with a non-degenerate cluster, we add to G K4 (n) copies n n 2 of all 2 + 2 1 clusters of depth 1 so that they would be inaccessible from each other and could see only the clusters in a and their successors. And for every singleton a = fC g with a non-degenerate cluster C , we add to GK4d (n) copies of those clusters of depth 1 which are not -equivalent to any subset of C (otherwise the frame will not be re ned) so that again they would be mutually inaccessible and could see only C and its successors in GK4d (n). Let NK4 (n) = hGK4 (n); UK4 (n)i be the resulting model (the relational component of GK4 (n) is completely determined by the construction and its set of possible values is the collection of the truth-sets of formulas in GK4 (n) under UK4 (n)). It is not hard to show that GK4 (n) is atomic. Moreover, for every point x in this frame one can construct a formula '(p1 ; : : : ; pn) such that x 6j= ' and, for any frame F, F 6j= ' i there is a generated subframe of F reducible to the subframe of GK4 (n) generated by x. It follows in particular d that GK4 (n) is re ned. Thus, every G K4 (n) is a generated subframe of FK4(n). On the other hand, by Theorem 8, FK4 (n) contains no clusters of d <1 depth d di erent from those in G K4 (n) and so FK4 (n) is isomorphic to GK4 (n). It worth noting also that, since K4 has the nite model property, 1 it is characterized by F< K4 (n), and so FK4 (n) is isomorphic to the bidual of < 1 FK4 (n). The universal frame FL(n) for an arbitrary consistent logic L in NExtK4 is a generated subframe of FK4(n). It can be constructed by removing from FK4(n) those points at which some formulas in L are refuted (under
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV p1
ÆP a PQ #~AC PaPQcPaPPaPP ##ÆHQScb@HQcbHb c C S @ A # PP @ H A CCS@ASAQc@QcaQc#Pa#PaPPaPPaPPSPSPPQc@Qc@bHQbHbH ### CCA C S@# Q a SPcQ H A C#A#AS @cQcQ aPaSPaP@PaPc@#PPQcb#PPQbPPHbPHbHH CC AA ##CC AASS@@ cQcQcQ##S#S a@ac@aQcPaQPaPQbPPbPPHbPPHPCPHC A QÆ b~ PHAÆ Æ ~ Æp# ~ CÆp ~ SÆ @Æp ~ QÆ SÆp ~ cÆ a p1 1 1 1 1 FS42 (1)
Figure 2.
VK4 (n)). For example, F<S41 (n) is obtained by removing from F
Fig. 2, where ~ denotes the cluster with two points at one of which p1 is 1 (n) and F<1 (n), we take only simple clusters and true. To construct F
Proof. Since L2 in the splitting pair (L1 ; L2) is a proper extension of L0, there is a nite frame G such that G j= L0 and G 6j= L2 . It follows that LogG L1 . As we shall see later (Corollary 86), every extension of a tabular logic T is also tabular. So L1 = LogF for some nite F j= L0 . And since L1 is {prime, F must be rooted. We say that a frame F splits NExtL0 if LogF splits NExtL0. The logic L2 of the splitting pair (LogF; L2 ) is denoted by L0 =F and called the splitting of NExtL0 by F. This notation re ects the fact that L2 is the smallest logic in NExtL0 which is not validated by F. EXAMPLE 10. We show that D = K=. Recall that D = K > is characterized by the class of serial frames (in which every point has a suc-
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cessor). So if j= L then L Log; otherwise no frame for L has a dead end, which means that > 2 L and D L. The inconsistent logic For can be represented as D=Æ. To illustrate some applications of splittings we require a few de nitions. Given L 2 NExtL0, we say that the axiomatization problem for L above L0 is decidable if the set f' : L0 ' = Lg is recursive. L is strictly Kripke complete above L0 if no other logic in NExtL0 has exactly the same Kripke frames as L. If all frames in a set F split NExtL0 , we call the logic L fL0=F : F 2 Fg the union-splitting of NExtL0 and denote it by L0=F . EXAMPLE 11. Grz is not a splitting of NExtS4. However, it is a union-
Æ ÆÆ ÆÆ6 g. S4:1 = S4=ÆÆ . A frame may split the splitting: Grz = S4=f ; lattice NExtL0 =F but not NExtL0: e.g. Æ splits NExtK= but does not split NExtK. THEOREM 12. Suppose L 2 NExtL0 and L = (: : : (L0 =F1 )= : : : )=Fn , for a sequence F1 ;S: : : ; Fn of sets of nite rooted frames. (i) If F = ni=1 Fi is nite and L is decidable then the axiomatization problem for L above L0 is decidable. More precisely,
f' : L0 ' = Lg = f' 2 L : 8F 2 F F 6j= 'g: (ii) If L is Kripke complete then L is strictly Kripke complete above L0. (iii) The immediate predecessors of L in NExtL0 are precisely the logics L \ LogF, for F 2 F such that F is not a reduct of a generated subframe of another frame in F .
Proof. (i) is left to the reader as an easy exercise. (ii) Let L0 be a logic in NExtL0 with the same Kripke frames as L. Then obviously L0 L. On the other hand, the frames in F do not validate L0 and so L L0 . (iii) If L0 is an immediate predecessor of L in NExtL0 then F j= L0 , for some F 2 F . Therefore, L0 L \ LogF L and so L0 = L \ LogF. Suppose now that F is not a reduct of a generated subframe of another frame in F and L \ LogF L0 L. Then L0 LogF0 for some F0 2 F , and hence F0 = F, L0 = L \ LogF. As follows from Theorem 12 and Example 10, For has exactly two immediate predecessors in NExtK: Verum = Log and Triv = LogÆ (and each consistent normal modal logic is contained in one of them). This result is known as Makinson's [1971] Theorem. Moreover, the axiomatization problem for For is decidable, i.e., there is an algorithm which decides, given a formula ' whether K ' is consistent. Likewise, since D = K > is decidable, there is an algorithm recognizing, given ', whether D = K '.
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We shall see later in Section 4.4 that in fact not so many properties of logics are decidable (e.g. the axiomatization problem for K :> is undecidable; see Theorem 207) and that Theorem 12 (i) provides the main method for proving decidability results of this type. To determine whether a nite rooted frame F = hW; Ri splits NExtL0, we need the formulas de ned below: F =
fpx ! py : x; y 2 W; xRyg [ fpx ! :py : x; y 2 W; :xRyg [ fpx ! :py : x; y 2 W; x 6= yg; ^ _ F = F ; ÆF = F ^ fpx : x 2 W g:
The meaning of ÆF is explained by the following lemma, in which
Proof. ()) Suppose fpr g [
Proof. The implication ()) follows from Lemma 13. Suppose now that n ÆF ! :pr 62 L, for every n < !. Then the set fpr g [ n. In this case L0=F = L0 nÆF ! :pr .
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Proof. ()) Suppose otherwise and consider a sequence fGn : n < !g of frames for L0 such that n ÆF ^ pr is satis able in Gn but m ÆF ^ pr is not T satis ed, for some m > n. By Lemma 14, the former condition implies n
Proof. That frames with cycles do not split NExtK follows from the fact that K is characterized by cycle free nite rooted frames. And the converse is an immediate consequence of Lemma 13 and Theorem 15. An element x 6= 0 of a complete lattice L is called an atom in L if the zero element 0 in L is the immediate predecessor of x, i.e., there is no y such that 0 < y < x. Splittings turn out to be closely related to the existence of atoms in nitely generated free algebras; see [Blok 1976], [Bellissima 1984, 1991] and [Wolter 1997c]. We demonstrate the use of splittings by the following THEOREM 18 (Blok 1980a). The lattice NExtK has no atoms. L
Proof. If a logic L is an atom in NExtK, it is {prime. It follows that L cosplits NExtK and the logic L0 = LogF in the splitting pair (L0 ; L) has no proper predecessor that splits NExtK. Add a new irre exive root to F. By Theorem 17, the resulting frame G splits NExtK, and clearly LogG LogF, which is a contradiction. A logic is linked with its semantics via completeness theorems. The most general completeness theorem states that every consistent normal modal logic is characterized by the class of (descriptive) frames validating it. Or, if we want to characterize the consequence relations `L and `L , we can use the following
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
THEOREM 19. (i) For L 2 NExtK, `L ' i for any model M based on a frame for L and any point x in M, x j= implies x j= '. (ii) For L 2 NExtK, `L ' i for any model M based on a frame for L, M j= implies M j= '. However, usually more speci c completeness results are required. What is the \geometry" of frames for a given logic? Are Kripke or even nite frames enough to characterize it? Questions of this sort will be addressed in the next several sections.
1.3 Persistence The structure of Kripke frames for many standard modal logics can be described by rather simple conditions on the accessibility relation which are expressed in the rst order language with equality and a binary (accessibility) predicate R. (This observation was actually the starting point of investigations in Correspondence Theory studying the relation between modal and rst (or higher) order languages; see Chapter 4 of this volume.) Moreover, in many cases it turns out that the universal frame FL(!) for such a logic L also satis es the corresponding rst order condition . Since says nothing about sets of possible values in PL (!), it follows immediately that the canonical (Kripke) frame FL (!) also satis es and so characterizes L. Thus we obtain a completeness theorem of the form: ' 2 L i F j= ' for every Kripke frame F satisfying . This method of establishing Kripke completeness, known as the method of canonical models, is based essentially upon two facts: rst, that L is characterized by its universal frame FL(!) and second, that L is \persistent" under the transition from FL(!) to its underlying Kripke frame. Of course, instead of FL(!) we can take any other class of frames C with respect to which L is complete and try to show that L is C {persistent in the sense that, for every F = hW; R; P i in C , if F j= L then F = hW; Ri validates L as well. PROPOSITION 20. If a logic is both C {complete and C {persistent, then it is complete with respect to the class fF : F 2 Cg of Kripke frames. It follows in particular that L is Kripke complete whenever it is DF {, or R{, or D{persistent. Since every descriptive frame for L is a generated subframe of a suitable universal frame for L, L is D{persistent i it is persistent with respect to the class of its universal frames. It is an open problem, however, whether canonicity, i.e., FL (!){persistence, implies D{ persistence. Here are two simple examples. THEOREM 21 (van Benthem 1983). A logic is persistent with respect to the class of all general frames i it is axiomatizable by a set of variable free formulas.
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It is easily checked that a Kripke frame validates Altn i no point in it has more than n distinct successors (see [Segerberg 1971]). THEOREM 22 (Bellissima 1988). Every L 2 NExtAltn is DF {persistent, for any n < !.
Proof. The proof is based on the fact that, for any di erentiated frame F = hW; R; P i, any nite X W , and any y 2 X , there is Y 2 P such that X \ Y = fyg. It follows that at most n distinct points are accessible from every point in a di erentiated frame for L; in particular, Altn is DF { persistent. Suppose now that a formula ' 2 L is refuted at a point x under a valuation V in F, F a di erentiated frame for L. Let X be the set of points accessible from x in md(') steps.6 Since X is nite, there is a valuation U in F such that U(p) \ X = V(p), for every variable p. Consequently, ' is false in F at x under U, which is a contradiction. The proof of Fine's [1974c] Theorem that all logics of nite width, i.e., logics in NExtK4BWn , for n < !, are Kripke complete (a sketch can be found in Section 18 of Basic Modal Logic) may also be regarded as a proof of persistence. Recall that a point x in a transitive frame F = hW; R; P i is called non-eliminable (relative to R) if there is X 2 P such that x 2 X but no proper successor of x is in X (in other words, x is maximal in X ); in this case we write x 2 maxR X . Denote by Wr the set of all noneliminable points in F and put Fr = hWr ; Rr ; Pr i, where Rr = R Wr , Pr = fX \ Wr : X 2 P g. (Fine called the frame Fr reduced.) THEOREM 23 (Fine 1985). Let F = hW; R; P i be a transitive descriptive frame and x 2 X 2 P . Then (i) there exists a point y 2 maxR X \ x" and (ii) Fr is a re ned frame whose dual F+r is isomorphic to F+ .
Proof. (i) Suppose otherwise, i.e., there is no maximal point in X \ x". Let Y be a maximal chain of points in X \ x" (that it exists follows from Zorn's Lemma) and X = fZ 2 P : 9y 2 Y y " \ Y Z g. Clearly, X is non-empty and has the nite intersection property (because T X \ x" has no maximal point). By compactness, we then have a point z in X which, by tightness, is maximal in Y , contrary to X \ x" having no maximal point. (ii) is a consequence of (i). It follows that to establish the Kripke completeness of a logic L 2 NExtK4 it is enough to show that it is persistent with respect to the class
RE = fFr : F a nitely generated descriptive frameg: That is what Fine [1974c] actually did for logics of nite width.
6 Here md('), the modal degree of ', is the length of the longest chain of nested modal operators in '.
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THEOREM 24 (Fine 1974c). All logics of nite width are RE {persistent and so Kripke complete. Let us return, however, to the method of canonical models. Having tried it for a number of standard systems, Lemmon and Scott [1977] found a rather general suÆcient condition for its applicability and put forward a conjecture concerning a further extension (which was proved by Goldblatt [1976b]). This direction of completeness (and correspondence) theory culminated in the theorem of Sahlqvist [1975] who proved an optimal (in a sense) generalization of the condition of [Lemmon and Scott 1977]. To formulate it we require the following de nition. Say that a formula is positive (negative ) if it is constructed from variables (negated variables) and the constants >, ? using ^, _, and . THEOREM 25 (Sahlqvist 1975). Suppose ' is a formula which is equivalent in K to a formula of the form k ( ! ), where k 0, is positive and is constructed from variables and their negations, ? and > with the help of ^, _, and in such a way that no 's subformula of the form 1 _ 2 or 1 , containing an occurrence of a variable without :, is in the scope of some . Then one can e ectively construct a rst order formula (x) in R and = having x as its only free variable and such that, for every descriptive or Kripke frame F and every point a in F, (F; a) j= ' i F j= (x)[a]: (Here (F; a) j= ' means that ' is true at a in F under any valuation.)
Proof. We present a sketch of the proof found by Sambin and Vaccaro [1989]. Given a formula '(p1 ; : : : ; pn ), a frame F = hW; R; P i and sets X1 ; : : : ; Xn 2 P , denote by '(X1 ; : : : ; Xn ) the set of points in F at which ' is true under the valuation V de ned by V(pi ) = Xi , i.e., '(X1 ; : : : ; Xn ) = V('). Using this notation, we can say that (F; x) j= '(p1 ; : : : ; pn ) i 8X1; : : : ; Xn 2 P x 2 '(X1 ; : : : ; Xn ): EXAMPLE 26. Let us consider the formula p ! p and try to extract a rst order equivalent for it in the class of tight frames directly from the equivalence above and the condition of tightness. For every tight frame F = hW; R; P i we have: (F; x) j= p ! p i i i
8X 2 P x 2 (X ! X ) 8X 2 P (x 2 X ! x 2 X ) 8X 2 P (x" X ! x 2 X ):
To eliminate the variable X ranging over P , we can use two simple observations. The rst one is purely set-theoretic:
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103
\
8X 2 P (Y X ! x 2 X ) i x 2 fX 2 P : Y X g:
And the second one is just a reformulation of the characteristic property of tight frames: (4)
\
fX 2 P : x" X g = x":
With the help of (3) and (4) we can continue the chain of equivalences above with two more lines: (F; x) j= p ! p i : : : T i x 2 fX 2 P : x" X g i x 2 x": Thus, F j= p ! p i 8x x 2 x" i 8x xRx. The proof of Sahlqvist's Theorem is a (by no means trivial) generalization of this argument. De ne by induction x"0 = fxg, x"n+1 = (x"n )", and notice that in (4) we can replace x" by any term of the form x1"n1 [ [ xk"nk , thus obtaining the equality \
fX 2 P : x1"n [ [ xk"nk X g = x1"n [ [ xk"nk which holds for every descriptive frame F = hW; R; P i, all x1 ; : : : ; xk 2 W and all n1 ; : : : ; nk 0. A frame-theoretic term x1"n [ [ xk"nk with (not necessarily distinct) (5)
1
1
1
world variables x1 ; : : : ; xk will be called an R-term. It is not hard to see that for any R-term T , the relation x 2 T on F = hW; R; P i is rst order expressible in R and =. Consequently, we obtain LEMMA 27. Suppose '(p1 ; : : : ; pn ) is a modal formula and T1 ; : : : ; Tn are R-terms. Then the relation x 2 '(T1 ; : : : ; Tn) is expressible by a rst order formula (in R and =) having x as its only free variable. Syntactically, R-terms with a single world variable correspond to modal formulas of the form m1 p1 ^ ^ mk pk with not necessarily distinct propositional variables p1 ; : : : ; pk . Such formulas are called strongly positive. By induction on the construction of ', one can prove the following LEMMA 28. Suppose '(p1 ; : : : ; pn ) is a strongly positive formula containing all the variables p1 ; : : : ; pn and F = hW; R; P i is a frame. Then one can e ectively construct R-terms T1 ; : : : ; Tn (with one variable x) such that for any x 2 W and any X1 ; : : : ; Xn 2 P ,
x 2 '(X1 ; : : : ; Xn ) i T1 X1 ^ ^ Tn Xn : Now, trying to extend the method of Example 26 to a wider class of formulas, we see that it still works if we replace the antecedent p in p ! p
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with an arbitrary strongly positive formula . As to generalizations of the consequent, let us take rst an arbitrary formula instead of p and see what properties it should satisfy to be handled by our method. Thus, for a modal formula ( ! )(p1 ; : : : ; pn ) with strongly positive and a descriptive frame F = hW; R; P i, we have: (F; x) j=
! i 8X1; : : : ; Xn 2 P (x 2 (X1 ; : : : ; Xn ) ! x 2 (X1 ; : : : ; Xn )) i 8X1; : : : ; Xn 2 P (T1 X1 ^ ^ Tn Xn ! x 2 (X1 ; : : : ; Xn )) i 8X1; : : : ; Xn 1 2 P (T1 X1 ^ ^ Tn 1 Xn 1 ! 8Xn 2 P (Tn Xn ! x 2 (X1 ; : : : ; Xn ))):
(3) does not help us here, but we can readily generalize it to (6)
8X 2 P (Y X ! x \ 2 (: : : ; X; : : : )) i x 2 f (: : : ; X; : : : ) : Y X 2 P g:
So (F; x) j=
! i 8X1\ ; : : : ; Xn 1 2 P (T1 X1 ^ ^ Tn 1 Xn 1 ! x 2 f (X1 ; : : : ; Xn ) : Tn Xn 2 P g):
But now (4) and (5) are useless. In fact, what we need is the equality \
(7)
f (: : : ; X; : : : ) : T X 2 P g = \ (: : : ; fX 2 P : T X g; : : : )
which, with the help of (5), would give us (8)
\
f (: : : ; X; : : : ) : T X 2 P g = (: : : ; T; : : : ):
Of course, (7) is too good to hold for an arbitrary , but suppose for a moment that our satis es it. Then we can eliminate step by step all the variables X1 ; : : : ; Xn like this: (F; x) j=
! i 8X1; : : : ; Xn 1 2 P (T1 X1 ^ ^ Tn 1 Xn 1 ! x 2 (X1 ; : : : ; Xn 1 ; Tn)) i : : : (by the same argument) i x 2 (T1 ; : : : ; Tn):
And the last relation can be e ectively rewritten in the form of a rst order formula (x) in R and = having x as its only free variable. So, nally we shall have F j= ! i 8x (x).
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Now, to satisfy (7), should have the property that all its operators distribute over intersections. Clearly, ! and : are not suitable for this goal. But all the other operators turn out to be good enough at least in descriptive and Kripke frames. So we can take as any positive modal formula. The main property of a positive formula '(: : : ; p; : : : ) is its monotonicity in every variable p which means that, for all sets X , Y of worlds in a frame, X Y implies '(: : : ; X; : : : ) '(: : : ; Y; : : : ). To prove that all positive formulas satisfy (7) in Kripke frames and descriptive frames, recall that distributes over arbitrary intersections in any frame. As to , we have the following lemma in which a family X of non-empty subsets of some space W is called downward directed if for all X; Y 2 X there is Z 2 X such that Z X \ Y . LEMMA 29 (Esakia 1974). Suppose F = hW; R; P i is a descriptive frame. Then for every downward directed family X P ,
\
X 2X
X=
\
X 2X
X:
Using Esakia's Lemma, by induction on the construction of ' one can prove LEMMA 30. Suppose that F = hW; R; P i is a Kripke or descriptive frame and '(p; : : : ; q; : : : ; r) is a positive formula. Then for every Y W and all U; : : : ; V 2 P , \
(9)
f'(U; : : : ; X; : : : ; V ) : Y X 2 P g = \ '(U; : : : ; fX 2 P : Y X g; : : : ; V ):
It follows from this lemma and considerations above that Sahlqvist's Theorem holds for formulas ' = ! with strongly positive and positive . The remaining part of the proof is purely syntactic manipulations with modal and rst order formulas. Notice that using the monotonicity of positive formulas, equivalence (6) can be generalized to the following one: for every F = hW; R; P i, every positive i (: : : ; p; : : : ) and every xi 2 W ,
8X 2 P (Y X ! (10)
_
in
_
in
xi 2 i (: : : ; X; : : : )) i
xi 2
\
f i (: : : ; X; : : : ) : Y X 2 P g:
Say that a modal formula is untied if it can be constructed from negative formulas and strongly positive ones using only ^ and . If (p1 ; : : : ; pn ) is negative then : (p1 ; : : : ; pn ) is clearly equivalent in K to a positive formula; we denote it by (:p1 ; : : : ; :pn ).
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LEMMA 31. Let (p1 ; : : : ; pn ) be an untied formula and F = hW; R; P i a frame. Then for every x 2 W and all X1 ; : : : ; Xn 2 P ,
x 2 (X1 ; : : : ; Xn ) i 9y1 ; : : : ; yl (# ^
^
in
Ti Xi ^
^
j m
zj 2 j (X1 ; : : : ; Xn ))
where the formula in the right-hand side, e ectively constructed from , has only one free individual variable x, # is a conjunction of formulas of the form uRv, Ti are suitable R-terms and j (p1 ; : : : ; pn) are negative formulas. We are ready now to prove Sahlqvist's Theorem. To construct a rst order equivalent for k ( ! ) supplied by the formulation of our theorem, we observe rst that one can equivalently reduce to a disjunction 1 _ _ m of untied formulas, and hence k ( ! ) is equivalent in K to the formula
k ( 1 ! ) ^ ^ k (
m
! ):
So all we need is to nd a rst order equivalent for an arbitrary formula k ( ! ) with untied and positive . Let p1 ; : : : pn be all the variables in and and F = hW; R; P i a descriptive or Kripke frame. Then, for any x 2 W , we have: (F; x) j= k ( ! ) i 8X1; : : : ; Xn 2 P x 2 k ( ! )(X1 ; : : : ; Xn ) (by Lemma 31) i 8X1; : : : ; Xn 2 P 8y (xRk y ! (9y1 ; : : : ; yl (# ^ ^ ^ Ti Xi ^ zj 2 j (X1 ; : : : ; Xn )) ! in j m y 2 (X1 ; : : : ; Xn ))) ^ i 8X1; : : : ; Xn 2 P 8y; y1; : : : ; yl (#0 ^ Ti Xi ^ in ^ zj 2 j (X1 ; : : : ; Xn ) ! y 2 (X1 ; : : : ; Xn )) j m where #0 = xRk y ^ #. Let j (p1 ; : : : ; pn) = j (:p1 ; : : : ; :pn ). We continue this chain of equivalences as follows: ^ i 8y; y1; : : : ; yl (#0 ! 8X1; : : : ; Xn 2 P ( Ti Xi ! in _ zj 2 j (X1 ; : : : ; Xn ))) j m+1 (where m+1 (p1 ; : : : ; pn) = (p1 ; : : : ; pn ) and zm+1 = y) _ i 8y; y1; : : : ; yl (#0 ! zj 2 j (T1 ; : : : ; Tn )); j m+1 as follows from (10), Lemma 30 and equality (5). It remains to use Lemma 27.
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The formulas ' de ned in the formulation of Theorem 25 are called Sahlqvist formulas. It follows from this theorem that if L is a D{persistent logic and a set of Sahlqvist formulas then L is also D{persistent. Moreover, L is elementary (in the sense that the class of Kripke frames for it coincides with the class of all models for some set of rst order formulas in R and =) whenever L is so. Other proofs of Sahlqvist's Theorem were found by Kracht [1993] and Jonsson [1994] (the latter is based upon the algebraic technique developed in [Jonsson and Tarski 1951]). Venema [1991] extended Sahlqvist's Theorem to logics with non-standard inference rules, like Gabbay's [1981a] irre exivity rule. In [Chagrov and Zakharyaschev 1995b] it is shown that there is a continuum of Sahlqvist logics above S4 and that not all of them have the nite model property (above T such a logic was constructed by Hughes and Cresswell [1984]). As we shall see later in this chapter, there are even undecidable nitely axiomatizable Sahlqvist logics in NExtK. It would be of interest to nd out whether such logics exist above K4 or S4. Kracht [1993] described syntactically the set of rst order equivalents of Sahlqvist formulas. To formulate his criterion we require the fragment S of rst order logic de ned inductively as follows. Formulas of the form xRm y are in S for all variables x; y and every m < !; besides, if ; 0 are in S then the formulas 8x 2 y"m ; 9x 2 y"m ; ^ 0 ; and _ 0 are also in S . For simplicity we assume that all occurrences of quanti ers in a formula bind pairwise distinct variables. Call a variable y in a formula 2 S inherently universal if either all occurences of y are free in or contains a subformula 8y 2 x"m 0 which is not in the scope of 9. THEOREM 32 (Kracht 1993). For every rst order formula (x) (in R and =) with one free variable x, the following conditions are equivalent: (i) (x) is classically equivalent to a formula 0 (x) 2 S such that any subformula of the form yRmz of 0 (x) contains at least one inherently universal variable; (ii) (x) corresponds to a Sahlqvist formula in the sense of Theorem 25. Condition (i) is satis ed, for example, by the formula
8u 2 x" 8v 2 x" 9z 2 u" vRz which corresponds to p ! p. On the other hand, (x) = 9y 2 x" 8z 2 y" zR0y does not satisfy (i). In fact, even relative to S4 the condition expressed by (x) does not correspond to any Sahlqvist formula. Notice, however, that
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S4 p ! p is a D-persistent logic whose frames are precisely the transitive and re exive frames validating 8x (x). We conclude this section by mentioning two more important results connecting persistence and elementarity (the idea of the proof was discussed in Section 22 of Basic Modal Logic.) THEOREM 33. (i) (Fine 1975b, van Benthem 1980) If a logic L is characterized by a rst order de nable class of Kripke frames then L is D{persistent. (ii) (Fine 1975b) If L is R-persistent then the class of Kripke frames for L is rst order de nable. It is an open problem whether every D{persistent logic is determined by a rst order de nable class of Kripke frames; for more information about this and related problems consult [Goldblatt 1995].
1.4 The degree of Kripke incompleteness All known logics in NExtK of \natural origin" are complete with respect
to Kripke semantics. On the other hand, there are many examples of \arti cial" logics that cannot be characterized by any class of Kripke frames (see Sections 19, 20 of Basic Modal Logic or the examples below). To understand the phenomenon of Kripke incompleteness Fine [1974b] proposed to investigate how many logics may share the same Kripke frames with a given logic L. The number of them is called the degree of Kripke incompleteness of L. Of course, this number depends on the lattice of logics under consideration. The degree of Kripke incompleteness of logics in NExtK was comprehensively studied by Blok [1978]. In this section we present the main results of that paper following [Chagrov and Zakharyaschev 1997]. By Theorem 12, all Kripke complete union-splittings of NExtK have degree of incompleteness 1. And it turns out that no other union-splitting exists. THEOREM 34 (Blok 1978). Every union-splitting of NExtK has the nite model property.
Proof. Let F be a class of nite rooted cycle free frames. We prove that L = K=F has the nite model property using a variant of ltration, which is applied to an n-generated re ned frame F = hW; R; P i for L refuting a formula '(p1 ; : : : ; pn ) under a valuation V. Since F is di erentiated, for every m 1 there are only nitely many points x in F such that x j= m ? ^ :m 1 ?; we shall call them points of type m. Given Sub', Sub' the set of all subformulas in ', we put m = m if m is the minimal number such that a point in F is of type m
ADVANCED MODAL LOGIC nontransitive x1 -x11 xk1
Æ
6
Æ Æ
1
-xÆ k1
x1
6
109
-x 2 x n -x11 -x 12 x 1n x k1 -x k2 x kn
(a)
(b) Figure 3.
whenever x j= and the formulas in Sub' are false at x (under V); if no such m exists, we put m = 0. Let
k = maxfm : Sub'g;
= Sub(' ^ k ?):
Now we divide F into two parts: W1 consisting of points of type k and W2 = W W1 . For x; y 2 W , put x y if either x; y 2 W1 and x = y or x; y 2 W2 and exactly the same formulas in are true at x and y. Let N = hG; Ui be the smallest ltration (see Section 12 of Basic Modal Logic) of M = hF; Vi through with respect to . Since W1 is nite, G is also nite and, by the Filtration Theorem, (M; x) j= i (N; [x]) j= , for every 2 . So it remains to show that G j= L. Notice that [x] in G is of type m k i x has type m in F. Moreover, there is no [x] of type l > k. For otherwise x 6j= k ? and m = 0 for = f 2 Sub' : x j= g, which means that arbitrary long chains (of not necessarily distinct points) start from [x], contrary to [x] being of type l. Thus G consists of two parts: points of type k, which form the generated subframe hW1 ; R W1 i of F, and points involved in cycles. Since F j= L and frames in F are cycle free, it follows from Lemma 13 and Theorem 17 that G j= L. THEOREM 35 (Blok 1978). If a logic L is inconsistent or a union-splitting of NExtK, then L is strictly Kripke complete. Otherwise L has degree of Kripke incompleteness 2@0 in NExtK.
Proof. That For is strictly complete follows from Example 10 and Theorem 12. Suppose now that a consistent L is not a union-splitting and L0 is the greatest union-splitting contained in L. Since L0 has the nite model property, there is a nite rooted frame F = hW; Ri for L0 refuting some ' 2 L and such that every proper generated subframe of F validates L. Clearly, F is not cycle free. Let x1 Rx2 R : : : Rxn Rx1 be the shortest cycle in F and k = md(') + 1. We construct a new frame F0 by extending the cycle x1 ; : : : ; xn ; x1 as is shown in Fig. 3 ((a) for n = 1 and (b) for n > 1). More precisely, we add to F copies x1i ; : : : ; xki of xi for each i 2 f1; : : : ; ng, organize them into the nontransitive cycle shown in Fig. 3 and draw an arrow from xji to y 2 W fx1; : : : ; xn g i xi Ry. Denote the resulting frame
110
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV nontransitive
b
a
I@
transitive
H6HHx d d d0 -d -m 1 9
transitive a1 -a0
-Æ -i - 6 e e1 e 0 j c
F0 0
@Æ e0j
1
Figure 4. by F0 = hW 0 ; R0 i and let x0 = xkn . By the construction, F is a reduct of F0 . Therefore, for every models M = hF; Vi and M0 = hF0 ; V0 i such that V0 (p) = V(p) [ fxj : xi 2 V(p); j < kg i
and for every x 2 W , 2 Sub', we have (M; x) j= i (M0 ; x) j= . So we can hook some other model on x0 , and points in W will not feel its presence by means of ''s subformulas. The frame to be hooked on x0 depends on whether j= L or Æ j= L. We consider only the former alternative. Fix some m > jW 0 j. For each I ! f0g, let FI = hWI ; RI ; PI i be the frame whose diagram is shown in Fig. 4 (d0 sees the root of F0 , all points ei and e0j and is seen from x0 ; the subframes in dashed boxes are transitive, e0i 2 WI i i 2 I , and PI consists of sets of the form X [ Y such that X is a nite or co nite subset of WI fb; ai : i < !g and Y is either a nite subset of fai : i < !g or is of the form fbg[ Y 0 , where Y 0 is a co nite subset of fai : i < !g. It is not hard to see that the points ai , c, ei and e0i are characterized by the variable free formulas
0 = (Æm ^ (Æm
1 ^ ^ Æ0 ) : : : ) ^ :
m ^ (Æm 1 ^ ^ Æ0 ) : : : );
2 (Æ
i+1 = i ^ :2 i ; = 2 0 ^ : 0 ; 0 = ; i+1 = i ^ :2 i ; 0i+1 = i ^ :+ i+1 ; (in the sense that x j= i i x = ai , etc.), where Æ0 = ?; Æ1 = Æ0 ^ :Æ0 ; Æ2 = Æ1 ^ :Æ1 ^ :+ Æ0 ; Æk+1 = Æk ^ :Æk ^ :+ Æk 1 ^ ^ :+ Æ0 : De ne LI to be the logic determined by the class of frames for L and FI , i.e., LI = L \ LogFI . Since :(0i ^ m+6 :') 2 LJ LI for i 2 I J (' is refuted at the root of F0 ), jfLI : I ! f0ggj = 2@0 .
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Let us show now that LI has the same Kripke frames as L. Since LI L, we must prove that every Kripke frame for LI validates L. Suppose there is a rooted Kripke frame G such that G j= LI but G 6j= , for some 2 L. Since is in L, it is valid in all frames for L, in particular, j= . And since 62 LI , is refuted in FI . Moreover, by the construction of FI , it is refuted at a point from which the root of F0 can be reached by a nite number of steps. Therefore, the following formulas are valid in FI and so belong to LI and are valid in G: (11)
: !
(12)
: !
l _ i=0 l ^ i=0
i ; i ( ! (0(0 p ! p) ! p));
where p does not occur in and l is a suÆciently big number so that any point in FI is accessible by l steps from every point in the selected cycle and every point at which may be false, and 0 = ( 0 ! ). According to (11), G contains a point at which is true. By the construction of , this point has a successor y at which, by (12), 0(0 p ! p) ! p is true under any valuation in G and y j= 0 . De ne a valuation U in G by taking U(p) = y ". Then y j= 0 (0 p ! p), from which y j= p and so y 2 y ". Now de ne another valuation U0 so that U0 (p) = y " fyg. Since y is re exive, we again have y j= 0 (0 p ! p), whence y j= p, which is a contradiction. This construction can be used to obtain one more important result. THEOREM 36 (Blok 1978). Every union-splitting K=F has { @0 immediate predecessors in NExtK, where { is the number of frames in F which are not reducts of generated subframes of other frames in F . Every consistent logic di erent from union-splittings has 2@0 immediate predecessors in NExtK. (For has 2 immediate predecessors in NExtK.)
Proof. The former claim follows from Theorem 12. To establish the latter, we continue the proof of Theorem 35. One can show that L is nitely axiomatizable over LI (the proof is rather technical, and we omit it here). Then, by Zorn's Lemma, NExtLI contains an immediate predecessor L0I of L. Besides, LI LJ = L whenever I 6= J . Indeed, LI LJ = (L \ LogFI ) (L \ LogFJ ) = L \ (LogFI LogFJ )
and if i 2 I
J then, for every 2 L and a suÆciently big l,
:
l _ k=0
k 0i ! 2 LogFI ; :0i 2 LogFJ ;
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
from which 2 LogFI LogFJ and so L LogFI LogFJ . It follows that L0I 6= L0J whenever I 6= J . It is worth noting that tabular logics, proper extensions of D and extensions of K4 are not union-splittings in NExtK. Similar results hold for the lattices NExtD and NExtT, where every consistent logic has degree of incompleteness 2@0 (see [Blok 1978, 1980b]). It would be of interest to describe the behavior of this function in NExtK4, NExtGL, NExtS4, NExtGrz (where Theorem 34 does not hold and where every tabular logic has nitely many immediate predecessors) and other lattices of logics to be considered later in this chapter.
1.5 Stronger forms of Kripke completeness In the two preceding sections we were considering the problem of characterizing logics L 2 NExtK by classes of Kripke frames. The same problem arises in connection with the two consequence relations `L and `L as well. Theorem 19 shows a way of introducing the corresponding concepts of completeness. With each Kripke frame F let us associate a consequence relation j=F by putting, for any formula ' and any set of formulas, j=F ' i (M; x) j= implies (M; x) j= ' for every model M based on F and every point x in F. Clearly, a modal logic L is Kripke complete i , for any nite set of formulas and any formula ', 6`L ' only if there is a Kripke frame F for L such that 6j=F '. Now, let us call L strongly Kripke complete7 if this implication holds for arbitrary sets . In other words, L is strongly complete if every Lconsistent set of formulas holds at some point in a model based on a Kripke frame for L. Another reformulation: L is strongly complete i L is Kripke T complete and the relation fj=F: F is a Kripke frame for Lg is nitary. It follows from the construction of the canonical models that every canonical (in particular, D{persistent) logic is strongly complete, which provides us with many examples of such logics in NExtK. By Theorem 33, all logics characterized by rst order de nable classes of Kripke frames are strongly complete. The converse does not hold: there exist strongly complete logics which are not canonical. The simplest is the bimodal logic of the frame hR; <; >i ; see Example 144 below. By applying the Thomason simulation (to be introduced in Section 2.3) to this logic we obtain a logic in NExtK with the same properties; see Theorem 123. Moreover, in contrast to D{persistence, strong Kripke completeness is not preserved under nite sums of logics (see [Wolter 1996b]). It is an open problem, however, whether such logics exist in NExtK4. 7 Fine [1974c] calls such logics compact, which does not agree with the use of this term by Thomason [1972].
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113
Perhaps the simplest examples of Kripke complete logics which are not strongly complete are GL and Grz (use Theorem 58 and the fact that these logics are not elementary; see Correspondence Theory). It is much more diÆcult to prove that the McKinsey logic K p ! p is not strongly complete; the proof can be found in [Wang 1992]. For other examples of modal logics that are not strongly complete see Section 3.4. It is worth noting also that, as was shown in [Fine 1974c], every nite width logic in a nite language turns out to be strongly Kripke complete, though this is not the case for logics in an in nite language, witness
GL:3 = GL (+ p ! q) _ (+ q ! p): For the consequence relation `L , we should take the \global" version j=F of j=F . Namely, we put j=F ' if M j= implies M j= ' for any model M based on F. A modal logic L is called globally Kripke complete if for any nite set of formulas and any formula ', 6`L ' only if there is a frame F for L such that 6j=F '. L is strongly globally complete if this holds for arbitrary (not only nite) . We also say that L has the global nite model property if for every nite and every ', 6`L ' only if there is a nite frame F for L such that 6j=F '. The global nite model property (FMP, for short) of many standard logics can be proved by ltration. Say that a logic L strongly admits ltration if for every generated submodel M of the canonical model ML and every nite set of formulas closed under subformulas, there is a ltration of M through based on a frame for L. PROPOSITION 37 (Goranko and Passy 1992). If L strongly admits ltration then L has global FMP. V Proof. Suppose that 6`L ', nite. Then
Proof. Let L0 = L and 6`L0 ', nite. Then there exists a ( nite) Kripke frame F for L such that contains no variables, F j= L0.
[ 6`L ' and so [ 6j=G '. Since For n-transitive logics L the global consequence relation `L is reducible to the \local" `L and so L is Kripke complete (has FMP, is strongly complete)
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i L is globally complete (has global FMP, is strongly globally complete). In general the global properties are stronger than the \local" ones. Although L is globally complete (has global FMP) only if L is complete (has FMP), the converse does not hold (see [Wolter 1994a] and [Kracht 1999]). EXAMPLE 39. Let L = Alt3 p ! p (p ^ :p) ! :(q ^ :q). A Kripke frame F validates L i no point in F has more than three successors, F is symmetric, and irre exive points in it have at most one successor. By Proposition 22, L is Kripke complete. The class of Kripke frames for L is closed under (not necessarily generated) subframes. So, by Proposition 59 to be proved below, L has FMP. We show now that it does not have global FMP. To this end we require the formulas:
1 = q1 ^ :q2 ^ :q3 ; 2 = :q1 ^ q2 ^ :q3 ; 3 = :q1 ^ :q2 ^ q3 ; ' = p ^ :p ^ 1 ;
=
^
f i ! i+1 : i = 1; 2g ^ 3 ! 1 :
Let F = hW; Ri, where W = ! and
R = fhm; mi : m > 0g [ fhm; m + 1i : m < !g [ fhm; m 1i : m > 0g: We then have 6j=F :'. In fact, ' is true at 0 and is true everywhere under the valuation V de ned by V(p) = W f0g and V(qi ) = f3n + i : n < !g. Clearly, F j= L and so 6`L :'. Suppose now that (N; x0 ) j= ' and N j= , for a model N based on a Kripke frame G = hV; S i for L. Then we can nd a sequence xj , j < !, such that xj Sxj+1 and x3j+i j= i+1 , for j < ! and i = 1; 2; 3. The reader can verify that all points xj are distinct. Let us consider now the algebraic meaning of the notions introduced above. A logic L is Kripke complete i the variety AlgL of modal algebras for L is generated by the class KrL = fF+ : F is a Kripke frame for Lg. By Birkho 's Theorem (see e.g. [Mal'cev 1973]), this means that AlgL = HSPKrL; (i.e., AlgL is obtained by taking the closure of KrL under direct products, then the closure of the result under (isomorphic copies of) subalgebras and nally under homomorphic images). Clearly, L is globally complete i precisely the same quasi-identities hold in KrL and AlgL. And since the quasi-variety generated by a class of algebras C is SPPU C (where PU denotes the closure under ultraproducts; see [Mal'cev 1973]), L is globally complete i AlgL = SPPU KrL: Goldblatt [1989] calls the variety AlgL complex if AlgL = SKrL, or, equivalently, if AlgL = SPKrL (this follows from the fact that the dual of the disjoint union of a family of Kripke frames fFi : i 2 I g is isomorphic
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Q
to the product i2I F+i ). We say a logic L is {-complex, { a cardinal, if every modal algebra for L with { generators is a subalgebra of F+ for some Kripke frame F j= L. As was shown in [Wolter 1993], this notion turns out to be the algebraic counterpart of both strong completeness and strong global completeness of logics in in nite languages with { variables. THEOREM 40. For every normal modal logic L in an in nite language with { variables the following conditions are equivalent: (i) L is strongly Kripke complete; (ii) L is globally strongly complete; (iii) L is {-complex.
Proof. (i) ) (iii) Suppose the cardinality of A 2 AlgL does not exceed {. Denote by L the algebra of modal formulas over { propositional variables and take some homomorphism h from L onto A. For each ultra lter r in A, the set h 1 (r) is maximal L-consistent. Since L is strongly complete, there is a model Mr = hFr ; Vr i with root xr based on a Kripke frame Fr for L and such that (Mr ; xr ) j= h 1 (r). Without loss of generality we may assume that the frames Fr for distinct r are disjoint. Let F be the disjoint union of all of them. De ne a homomorphism V from L into F+ by taking [ V(p) = fVr (p) : r is an ultra lter in Ag: Then V(L) is a subalgebra of F+ 2 AlgL isomorphic to A. The implication (iii) ) (ii) is trivial. To prove (ii) ) (i), consider an L-consistent set of formulas of cardinality { and put = fpg [ fn(p ! ') : n < !; ' 2
g;
where the variable p does not occur in formulas from . It is easily checked that all nite subsets of are L-consistent, so is L-consistent too. It follows that fp ! ' : ' 2 g 6`L :p. And since L is globally strongly complete, there exists a model M based on a Kripke frame for L such that M j= fp ! ' : ' 2 g and (M; x) j= p, for some x. But then (M; x) j= .
1.6 Canonical formulas The main problem of completeness theory in modal logic is not only to nd a suÆciently simple class of frames with respect to which a given logic L is complete but also to characterize the constitution of frames for L (in this class). The rst order approach to the characterization problem, discussed in Section 1.3 in connection with Sahlqvist's Theorem, comes across two obstacles. First, there are formulas whose Kripke frames cannot be described in the rst order language with R and =. The best known example
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is probably the Lob axiom
la = (p ! p) ! p:
F j= la i F is transitive, irre exive (i.e., a strict partial order) and Noethe-
rian in the sense that it contains no in nite ascending chain of distinct points. And as is well known, the condition of Noetherianness is not a rst order one. The second obstacle is that this approach deals only with logics that are Kripke complete; it does not take into account sets of possible values. There is another, purely frame-theoretic method of characterizing the structure of frames. For instance, a frame G validates K=F i G does not contain a generated subframe reducible to F. It was shown in [Zakharyaschev 1984, 1988, 1992] that in a similar manner one can describe transitive frames validating an arbitrary modal formula. It is not clear whether characterizations of this sort can be extended to the class of all frames (an important step in this direction would be a generalization to n-transitive frames). That is why all frames in this section are assumed to be transitive. First we illustrate this method by a simple example. EXAMPLE 41. Suppose a frame F = hW; R; P i refutes la under some valuation. Then the set V = fx 2 W : x 6j= lag is in P and V V #. It follows from the former that G = hV; R V; fX \ V : X 2 P gi is a frame| we call it the subframe of F induced by V . And the latter condition means that G is reducible to the single re exive point Æ which is the simplest refutation frame for la. Moreover, one can readily check that the converse also holds: if there is a subframe G of F reducible to Æ then F 6j= la. This example motivates the following de nitions. Given frames F = hW; R; P i and G = hV; S; Qi, a partial (i.e., not completely de ned, in general) map f from W onto V is called a subreduction of F to G if it satis es the reduction conditions (R1){(R3) for all x and y in the domain of f and all X 2 Q. The domain of f will be denoted by domf . In other words, an f -subreduct of F is a reduct of the subframe of F induced by domf . A frame G = hV; S; Qi is a subframe of F = hW; R; P i if V W and the identity map on V is a subreduction of F to G, i.e., if S = R V and Q P . Note that a generated subframe G of F is not in general a subframe of F, since V may be not in P . Thus, the result of Example 41 can be reformulated like this: F 6j= la i F is subreducible to Æ. A subreduction f of F to G is called co nal if
domf " domf #:
This important notion can be motivated by the following observation: F refutes > i F is co nally subreducible to (a plain subreduction is not enough).
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THEOREM 42. Every refutation frame F = hW; R; P i for '(p1 ; : : : ; pn ) is co nally subreducible to a nite rooted refutation frame for ' containing at most c' = 2n (cn (1) + + cn (2jSub'j )) points.8
Proof. Suppose ' is refuted in F under a valuation V. Without loss of generality we can assume F to be generated by V(p1 ); : : : ; V(pn ). Let X1 ; : : : ; Xm be all distinct maximal 0-cyclic sets in F. Clearly, m cn (1) but unlike Theorem 8, F is not in general re ned and so these sets are not necessarily clusters of depth 1. However, they can be easily reduced to such clusters. De ne an equivalence relation on W by putting x y i x = y or x; y 2 Xi , for some i 2 f1; : : : ; mg, and x y (as before = fp1; : : : ; pn g). Let [x] be the equivalence class under generated by x and [X ] = f[x] : x 2 X g, for X 2 P . By the de nition of cyclic sets, xRy i [x] [y] #. So the map x 7! [x] is a reduction of F to the frame F01 = hW10 ; R10 ; P10 i which results from F by \folding up" the 0-cyclic sets Xi into clusters of depth 1 and leaving the other points untouched: W10 = [W ], [x]R10 [y] i [x] [y] # and P10 = f[X ] : X 2 P g. (Roughly, we re ne that part of F which gives points of depth 1.) Put V01 (pi ) = [V(pi )]. Then by the Reduction (or P-morphism) Theorem, we have x j= i [x] j= , for every 2 Sub'. Let X be the set of all points in F01 of depth > 1 having Sub'-equivalent successors of depth 1. It is not hard to see that X 2 P10 . Denote by F1 = hW1 ; R1 ; P1 i the subframe of F01 induced by W10 X and let V1 be the restriction of V01 to F1 . By induction on the construction of 2 Sub' one can readily show that has the same truth-values at common points in F01 and F1 (under V01 and V1 , respectively) and so F1 6j= '. The partial map x 7! [x], for [x] 2 W1 , is a co nal subreduction of F to F1 . Then we take the maximal 1-cyclic sets in F1 , \fold" them up into clusters of depth 2 and remove those points of depth > 2 that have Sub'-equivalent successors of depth 2. The resulting frame F2 will be a co nal subreduct of F1 and so of F as well. After that we form clusters of depth 3, and so forth. In at most 2jSub'j steps of that sort we shall construct a co nal subreduct of F refuting ' and containing c' points. It remains to select in it a suitable rooted generated subframe. For the majority of standard modal axioms the converse also holds. However, not for all. The simplest counterexample is the density axiom den = p ! p. It is refuted by the chain H of two irre exive points but becomes valid if we insert between them a re exive one. In fact, F 6j= den i there is a subreduction f of F to H such that f (x") = fag, for no point x in domf " domf , where a is the nal point in H. Loosely, every refutation frame for formulas like la can be constructed by adding new points to a frame G that is reducible to some nite refutation 8
The function cn (m) was de ned in Section 1.2.
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frame of xed size. For formulas like > we have to take into account the co nality condition and do not put new points \above" G. And formulas like den impose another restriction: some places inside G may be \closed" for inserting new points. These \closed domains" can be singled out in the following way. Suppose N = hH; Ui is a model and a an antichain in H. Say that a is an open domain in N relativeVto a formula ' if there is a pair ta = ( a ; a ) W such that a [ a = Sub', a ! a 62 K4 and
2 2
a implies
2
a,
a i a j= + for all a 2 a.
Otherwise a is called a closed domain in N relative to '. A re exive singleton a = fag is always open: just take
ta = (f
2 Sub' : a j= g; f 2 Sub' : a 6j= g):
It is easy to see also that antichains consisting of points from the same clusters are open or closed simultaneously; we shall not distinguish between such antichains. For a frame H and a (possibly empty) set D of antichains in H, we say a subreduction f of F to H satis es the closed domain condition for D if (CDC) :9x 2 domf " domf 9d 2 D f (x") = d". Notice that the co nal subreduction f of F to the resulting nite rooted frame H in the proof of Theorem 42 satis es (CDC) for the set D of closed domains in the corresponding model N on H refuting '. Indeed, every x 2 domf " domf has a Sub'-equivalent successor y 2 domf , and so an antichain d such that f (x") = d" is open, since we can take
td = (f
2 Sub' : y j= g; f 2 Sub' : y 6j= g):
On the other hand, we have PROPOSITION 43. Suppose N = hH; Ui is a nite countermodel for ' and D the set of all closed domains in N relative to '. Then F 6j= ' whenever there is a co nal subreduction f of F to H satisfying (CDC) for D. Moreover, if ' is negation free (i.e., contains no ?, :, ) then a plain subreduction satisfying (CDC) for D is enough.
Proof. If f is co nal and F = hW; R; P i then we can assume domf " = W . De ne a valuation V in F as follows. If x 2 domf then we take x j= p i f (x) j= p, for every variable p in '. If x 62 domf then f (x") 6= ;, since f is co nal. Let a be an antichain in H such that a" = f (x"). By (CDC), a is an open domain in N, and we put y j= p i p 2 a , for every y 62 domf such that f (y ") = f (x"). One can show that V is really a valuation in F and,
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for every 2 Sub', x j= i f (x) j= in the case x 2 domf , and x j= i 2 a , where a is the open domain in N associated with x, in the case x 62 domf . If ' is negation free and f is a plain subreduction then f (x ") may be empty. In such a case we just put x j= p, for all variables p. Now let us summarize what we have got. Given an arbitrary formula ', we can e ectively construct a nite collection of nite rooted frames F1 ; : : : ; Fn (underlying all possible rooted countermodels for ' with c' points) and select in them sets D1 ; : : : ; Dn of antichains (open domains in those countermodels) such that, for any frame F, F 6j= ' i there is a co nal subreduction of F to Fi , for some i, satisfying (CDC) for Di . If ' is negation free then a plain subreduction satisfying (CDC) is enough. This general characterization of the constitution of refutation transitive frames can be presented in a more convenient form if with every nite rooted frame F = hW; Ri and a set D of antichains in F we associate formulas (F; D; ?) and (F; D) such that G 6j= (F; D; ?) (G 6j= (F; D)) i there is a co nal (respectively, plain) subreduction of G to F satisfying (CDC) for D. For instance, one can take
(F; D; ?) =
^
ai Raj
'ij ^
n ^ i=0
'i ^
^
d2D
'd ^ '? ! p0
where a0 ; : : : ; an are all points in F and a0 is its root,
'ij = 'i = 'd = '? =
+(pj ! pi ); ^ +(( pk ^
n ^
pj ! pi ) ! pi ; j =0;j 6=i n ^ _ ^ pj ^ pi ! pj ); +( aj 2d i=0 ai 2W d" n ^ +( + pi ! ?): i=0 :ai Rak
(F; D) results from (F; D; ?) by deleting the conjunct '? . (F; D; ?) and (F; D) are called the canonical and negation free canonical formulas for F and D, respectively. It is not hard to check that if (F; D; ?) is refuted in G = hV; S; Qi under some valuation then the partial map de ned by x 7! ai if the premise of (F; D; ?) is true at x and pi false is a co nal subreduction of G to F satisfying (CDC) for D; and conversely, if f is such a subreduction then the valuation U de ned by U(pi ) = V f 1(ai ) refutes (F; D; ?) at any point in f 1 (a0 ).
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THEOREM 44. There is an algorithm which, given a formula ', returns canonical formulas (F1 ; D1 ; ?); : : : ; (Fn ; Dn ; ?) such that
K4 ' = K4 (F1 ; D1 ; ?) (Fn ; Dn ; ?): So the set of canonical formulas is complete for the class NExtK4. If ' is negation free then one can use negation free canonical formulas. It is not hard to see that K4 ' is a splitting of NExtK4 i ' is deductively equivalent in NExtK4 to a formula of the form (F; D] ; ?), where D] is the set of all antichains in F (in this case K4=F = K4 (F; D] ; ?)). Such formulas are known as Jankov formulas (Jankov [1963] introduced them for intuitionistic logic), or frame formulas (cf. [Fine 1974a]), or Jankov{Fine formulas. Since GL is not a union-splitting of NExtK4, this class of logics has no axiomatic basis. We conclude this section by showing in Table 2 canonical axiomatizations of some standard modal logics in the eld of K4. For brevity we write (F; ?) instead of (F; ;; ?) and ] (F; ?) instead of (F; D] ; ?). Each in the table is to be replaced by both Æ and . For more information about the canonical formulas the reader is referred to [Zakharyaschev 1992, 1997b].
1.7 Decidability via the nite model property Although, for cardinality reason, there are \much more" undecidable logics than decidable ones, almost all \natural" propositional systems close to those we deal with in this chapter turn out to be decidable. Relevant and linear logics are probably the best known among very few exceptions (see [Urquhart 1984], [Lincoln et al. 1992]). The majority of decidability results in modal logic was obtained by means of establishing the nite model property. FMP by itself does not ensure yet decidability (there is a continuum of logics with FMP); some additional conditions are required to be satis ed. For instance, to prove the decidability of S4 McKinsey [1941] used two such conditions: that the logic under consideration is characterized by an e ective class of nite frames (or algebras, matrices, models, etc.) and that there is an e ective (exponential in the case of S4) upper bound for the size of minimal refutation frames. Under these conditions, a formula belongs to the logic i it is validated by ( nite) frames in a nite family which can be e ectively constructed. Another suÆcient condition of decidability is provided by the following well known THEOREM 45 (Harrop 1958). Every nitely axiomatizable logic with FMP is decidable. Here we need not to know a priori anything about the structure of frames for a given logic. This information is replaced by checking the validity of its
ADVANCED MODAL LOGIC
Table 2. Canonical axioms of standard modal logics
D4 S4 GL Grz K4:1
= = = = =
K4 (; ?) K4 () K4 (Æ) K4 () (ÆÆ ) K4 (; ?) (ÆÆ ; ?)
Æ K4 (Æ) ( 6) Æ S4 ( Æ6) K4 ( 6) (4 axioms) 1 2 AK GL ( A ; ff1g; f1; 2gg) 6 AK K4 ( 6; ?) ( Æ6; ?) ( A ; ?) (8 axioms) AK K4 ( A ) (6 axioms) Æ ÆÆ Æ 6) S4 ( AKÆ ) (ÆÆ
Triv
ÆÆ ) ( Æ6) = K4 () (
Verum
=
S5
=
K4B
=
A
=
K4:2
=
K4:3
=
Dum
=
n+1
z }| {
K4BWn =
K4BDn
K4n;m
I @ K4 ( @ ) (2n + 4 axioms) . n
..6 1 = K4 ( 60 ) (2n+1 axioms) . m ..6 1 = K4 ( 60 ; D])
121
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
axioms in nite frames, and the restriction of the size of refutation frames is replaced by constructing all possible derivations: in a nite number of steps we either separate a tested formula from the logic or derive it. Note that unlike the previous case now we cannot estimate the time required to complete this algorithm. The condition of nite axiomatizability in Harrop's Theorem cannot be weakened to that of recursive axiomatizability. For there is a logic of depth 3 in NExtK4 (i.e., a logic in NExtK4BD3 ) with an in nite set of independent axioms; so the logic of depth 3 axiomatizable by some recursively enumerable but not recursive sequence of formulas in this set is undecidable and has FMP. On the other hand there are examples of undecidable logics characterized by decidable classes of nite frames (see e.g. [Chagrov and Zakharyaschev 1997]). Yet one can generalize Harrop's Theorem in the following way. A logic is decidable i it is recursively enumerable and characterized by a recursive class of recursive algebras. However, this criterion is absolutely useless in its generality. In this connection we note two open problems posed by Kuznetsov [1979]. Is every nitely axiomatizable logic characterized by recursive algebras? Is every nitely axiomatizable logic, characterized by recursive algebras, decidable? (That nite axiomatizability is essential here is explained by the following fact: if a lattice of logics contains a logic with a continuum of immediate predecessors then there is no countable sequence of algebras such that every logic in the lattice is characterized by one of its subsequences. For details see [Chagrov and Zakharyaschev 1997].) FMP of almost all standard systems was proved using various forms of ltration (consult Section 12 Basic Modal Logic and [Gabbay 1976]). However, the method of ltration is rather capricious; one needs a special craft to apply it in each particular case (for instance, to nd a suitable \ lter"). In this and two subsequent sections we discuss other methods of proving FMP which are applicable to families of logics and provide in fact suÆcient conditions of FMP. (It is to be noted that the families of Kripke complete logics considered in Section 1.3 contain logics without FMP.) A pair of such conditions was already presented in Basic Modal Logic: THEOREM 46 (Segerberg 1971). Each logic in NExtK4 characterized by a frame of nite depth (or, which is equivalent, containing K4BDn , for some n < !) has FMP. THEOREM 47 (Bull 1966b, Fine 1971). Each logic in NExtS4:3 has FMP and is nitely axiomatizable (and so decidable). The former result, covering a continuum of logics, follows immediately from the description of nitely generated re ned frames for K4 in Section 1.2 and the latter is a consequence of Theorem 52 and Example 54 below. It is worth noting also that since FL (n) is nite for every logic L 2 NExtK4 of nite depth and every n < !, there are only nitely many pairwise
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non-equivalent in L formulas with n variables. Logics with this property are called locally tabular (or locally nite). Moreover, as was observed by Maksimova [1975a], the converse is also true: if L 2 NExtK4 has frames of any depth < ! then the formulas in the sequence '1 = p, 'n+1 = p _ (p ! 'n ) are not equivalent in L. Thus, a logic in NExtK4 is locally tabular i it is of nite depth. For L 2 NExtS4 this criterion can be reformulated in the following way: L is not locally tabular i L Grz:3, where Grz:3 = S4:3 Grz. Likewise, L 2 NExtGL is not locally tabular i L GL:3. Nagle and Thomason [1985] showed that all normal extensions of K5 are locally tabular.
Uniform logics Fine [1975a] used a modal analog of the full disjunctive normal form for constructing nite models and proving FMP of a family of logics in NExtD (containing in particular the McKinsey system K p ! p which had resisted all attempts to prove its completeness by the method of canonical models and ltration). Let us notice rst that every formula '(p1 ; : : : ; pm ) is equivalent in K either to ? or to a disjunction of normal forms (in the variables p1 ; : : : ; pm ) of degree md('), which are de ned inductively in the following way. NF0 , the set of normal forms of degree 0, contains all formulas of the form :1 p1 ^ ^ :m pm , where each :i is either blank or :. NFn+1 , the set of normal forms of degree n + 1, consists of formulas of the form ^ :1 1 ^ ^ :k k ;
where S2 NF0 and 1 ; : : : ; k are all distinct normal forms in NFn . Put W NF = ng [ f0 2 NF : 0
0 < 00 i 0 is a conjunct of 00 ; 0 R 00 i either 0 > 00 or md(0 ) = 0 and 00 = >; V (p) = f0 2 W : p is a conjunct of 0 g: According to the de nition, > is the re exive last point in F and so F is serial. By a straightforward induction on the degree of 0 2 W one can
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readily show that (M ; 0 ) j= 0 . It follows immediately that D has FMP. Indeed, given ' 62 D, we reduce :' to a disjunction of D-suitable normal forms with at least one disjunct , and then (M ; ) j= . It turns out that in the same way we can prove FMP of all logics in NExtD axiomatizable by uniform formulas, which are de ned as follows. Every ' without modal operators is a uniform formula of degree 0; and if ' = ( 1 1 ; : : : ; m m ), where i 2 f; g, md( (p1 ; : : : ; pm )) = 0 and 1 ; : : : ; m are uniform formulas of degree n, then ' is a uniform formula of degree n + 1. A remarkable property of uniform formulas is the following: PROPOSITION 48. Suppose ' is a uniform formula of degree n and M, N are models based upon the same frame and such that, for some point x, (M; y) j= p i (N; y) j= p for every y 2 x"n and every variable p in '. Then (M; x) j= ' i (N; x) j= '. Given a logic L, we call a normal form L-suitable if F j= L. THEOREM 49 (Fine 1975a). Every logic L 2 NExtD axiomatizable by uniform formulas has FMP.
Proof. It suÆces to prove that each formula ' with md(') n is equivalent in L either to ? or to a disjunction of L-suitable normal forms of degree n. And this fact will be established if we show that every D-suitable normal form such that ! ? 62 L is L-suitable. Suppose otherwise. Let be an L-consistent and D-suitable normal form of the least possible degree under which it is not L-suitable. Then there are a uniform formula 2 L of some degree m and a model M = hF ; Vi such that (M; ) 6j= . ForWevery variable p in , let p = f0 2 "m: (M; 0 ) j= pg and let Æp = p (if p = ; then Æp = ?). Observe that for every 0 2 "m we have (M ; 0 ) j= Æp i 0 2 p i (M; 0 ) j= p. Therefore, by Proposition 48, the formula 0 which results from by replacing each p with Æp is false at in M . Now, if md( 0 ) > n then m > n and so Æp = ? for every p in , i.e., 0 is variable free. But then 0 is equivalent in D to > or ?, contrary to F 6j= 0 and L being consistent. And if md( 0 ) n then either ! 0 2 K, which is impossible, since (M ; ) 6j= ! 0 , or ! : 0 2 K, from which 0 ! : 2 K and so : 2 L, contrary to being L-consistent.
Logics with -axioms Another result, connecting FMP of logics with the distribution of and over their axioms, is based on the following LEMMA 50. For any ' and , ' $ 2 S5 i ' $ 2 K4. Proof. Suppose ' ! 62 K4. Then there is a nite model M, based on a transitive frame, and a point x in it such that x j= ' and x 6j= . It follows from the former that every nal cluster accessible from x, if any, is non-degenerate and contains a point where ' is true. The latter means
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that x sees a nal cluster C at all points of which is false. Now, taking the generated submodel of M based on C , we obtain a model for S5 refuting ' ! . The rest is obvious, since p $ p is in S5 and K4 S5. Formulas in which every occurrence of a variable is in the scope of a modality will be called -formulas. THEOREM 51 (Rybakov 1978). If a logic L 2 NExtK4 is decidable (or has FMP) and is a -formula then L is also decidable (has FMP).
Proof. Let = 0 ( 1 ; : : : ; n ), for some formula 0 (q1 ; : : : ; qn ). If '(p1 ; : : : ; pm ) 2 L then there exists a derivation of ' in L in which substitution instances of contain no variables di erent from p1 ; : : : ; pm . Each of these instances has the form 0 ( 01 ; : : : ; 0n ), where every 0i is some substitution instance of i containing only p1 ; : : : ; pm . By Lemma 50 and in view of the local tabularity of S5 (it is of depth 1), there are nitely many pairwise non-equivalent in K4 substitution instances of i of that sort (the reader can easily estimate the number of them). So there exist only nitely many pairwise non-equivalent in K4 substitution instances of containing p1 ; : : : ; pm , say 1 ; : : : ; k , and we can e ectively construct them. Then, by the Deduction Theorem, ' 2 L i 1 ; : : : ; k ` ' i + ( 1 ^ ^ k ) ! ' 2 L L
and so L is decidable (or has FMP) whenever L is decidable (has FMP).
It should be noted that by adding to L with FMP in nitely many -
formulas we can construct an incomplete logic. For a concrete example see [Rybakov 1977]. By adding a variable free formula to a logic in NExtK with FMP one can get a logic without FMP. However, K ', ' variable free, has FMP, as can be easily shown by the standard ltration through the set Sub' [ Sub , where 62 K '. In nitely many variable free formulas can axiomatize a normal extension of K4 without FMP (for a concrete example see [Chagrov and Zakharyaschev 1997]).
1.8 Subframe and co nal subframe logics A very useful source of information for investigating various properties of logics in NExtK4 is their canonical axioms. Notice, for instance, that the canonical axioms of all logics in Table 2, save A and K4n;m , contain no closed domains. Canonical and negation free canonical formulas of the form (F) and (F; ?) are called subframe and co nal subframe formulas, respectively, and logics in NExtK4 axiomatizable by them are called subframe and co nal subframe logics. The classes of such logics will be denoted by SF
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and CSF . Subframe and co nal subframe logics in NExtK4 were studied by Fine [1985] and Zakharyaschev [1984, 1988, 1996]. THEOREM 52. All logics in SF and CSF have FMP.
Proof. Suppose L = K4 f (Fi ; ?) : i 2 I g and ' 62 L. By Theorem 44, without loss of generality we may assume that ' is a canonical formula, say, (F; D; ?). Now consider two cases. (1) For no i 2 I , F is co nally subreducible to Fi . Then F j= L, F 6j= (F; D; ?), and we are done. (2) F is co nally subreducible to (Fi ; ?), for some i 2 I . In this case we have (F; D; ?) 2 K4 (Fi ; ?) L, which is a contradiction. Indeed, suppose G 6j= (F; D; ?). Then there is a co nal subreduction of G to F. And since the composition of (co nal) subreductions is again a (co nal) subreduction, G is co nally subreducible to Fi , which means that G 6j= (Fi ; ?). Subframe logics are treated analogously. The names \subframe logic" and \co nal subframe logic" are explained by the following frame-theoretic characterization of these logics. A subframe G = hV; S; Qi of a frame F is called co nal if V " V # in F. Say that a class C of frames is closed under (co nal) subframes if every (co nal) subframe of F is in C whenever F 2 C . THEOREM 53. L 2 NExtK4 is a (co nal) subframe logic i it is characterized by a class of frames that is closed under (co nal) subframes.
Proof. Suppose L 2 CSF . We show that the class of all frames for L is closed under co nal subframes. Let G j= L and H be a co nal subframe of G. If H 6j= (F; ?), for some (F; ?) 2 L, then (since G is co nally subreducible to H) G 6j= (F; ?), which is a contradiction. So H j= L. Now suppose that L is characterized by some class of frames C closed under co nal subframes. We show that L = L0 , where L0 = K4 f (F; ?) : F 6j= Lg: If F is a nite rooted frame and F 6j= L then (F; ?) 2 L, for otherwise G 6j= (F; ?) for some G 2 C , and hence there is a co nal subframe H of G which is reducible to F; but H 2 C and so, by the Reduction Theorem, F is a frame for L, which is a contradiction. Thus, L0 L. To prove the converse, suppose (F; D; ?) 2 L. Then F 6j= L, and hence (F; ?) 2 L0, from which (F; D; ?) 2 L0 . Subframe logics are considered in the same way. It follows in particular that SF CSF (K4:1 and K4:2 are co nal subframe logics but not subframe ones). One can easily show also that CSF is a complete sublattice of NExtK4 and SF a complete sublattice of CSF .
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EXAMPLE 54. Every normal extension of S4:3 is axiomatizable by canonical formulas which are based on chains of non-degenerate clusters and so have no closed domains. Therefore, NExtS4:3 CSF . The classes SF and CSF SF contain a continuum of logics. And yet, unlike NExtK or NExtK4, their structure and their logics are not so complex. For instance, it is not hard to see that every logic in CSF is uniquely axiomatizable by an independent set of co nal subframe formulas and so these formulas form an axiomatic basis for CSF . The concept of subframe logic was extended in [Wolter 1993] to the class NExtK by taking the frame-theoretic characterization of Theorem 53 as the de nition. Namely, we say that L 2 NExtK is a subframe logic if the class of frames for L is closed under subframes. In other words, subframe logics are precisely those logics whose axioms \do not force the existence of points". For example, K, KB, K5, T, and Altn are subframe logics. To give a syntactic characterization of subframe logics we require the following formulas. For a formula ' and a variable p not occurring in ', de ne a formula 'p inductively by taking
qp = q ^ p; q an atom; ( )p = p p ; for 2 f^; _; !g; ( )p = (p ! p ) ^ p and put 'sf = p ! 'p . LEMMA 55. For any frame F, F j= 'sf i ' is valid in all subframes of F. Proof. It suÆces to notice that if M is a model based on F, M0 a model based on the subframe of F induced by fy : (M; y) j= pg and (M; x) j= q i (M0 ; x) j= q, for all variables q, then (M; x) j= 'p i (M0 ; x) j= '. PROPOSITION 56. The following conditions are equivalent for any modal logic L: (i) L is a subframe logic; (ii) L = K f'sf : ' 2 g, for some set of formulas ; (iii) L is characterized by a class of frames closed under subframes.
Proof. The implication (i) ) (iii) is trivial; (iii) ) (ii) and (ii) ) (i) are consequences of Lemma 55. It follows that the class of subframe logics forms a complete sublattice of NExtK. However, not all of them have FMP and even are Kripke complete. EXAMPLE 57. Let L be the logic of the frame F constructed in Example 7. Since every rooted subframe G of F is isomorphic to a generated subframe of F, L is a subframe logic. We show that L has the same Kripke frames
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as GL:3. Suppose G is a rooted Kripke frame for GL:3 refuting ' 2 L. Then clearly G contains a nite subframe H refuting '. Since H is a nite chain of irre exive points, it is isomorphic to a generated subframe of F, contrary to F 6j= '. Thus G j= L. Conversely, suppose G is a Kripke frame for L. Then G is irre exive. For otherwise G refutes the formula ' = 2 (p ! p) ^ (p ! p) ! p, which is valid in F. Let us show now that G is transitive. Suppose otherwise. Then G refutes the formula p ! (p _ (q ! q)), which is valid in F because ! is a re exive point. Finally, since G j= ', G is Noetherian and since F is of width 1, we may conclude that G j= GL:3. It follows that the subframe logic L is Kripke incomplete. Indeed, it shares the same class of Kripke frames with GL:3 but p ! p 2 GL:3 L. The following theorem provides a frame-theoretic characterization of those complete subframe logics in NExtK that are elementary, D{persistent and strongly complete. Say that a logic L has the nite embedding property if a Kripke frame F validates L whenever all nite subframes of F are frames for L. THEOREM 58 (Fine 1985). For each Kripke complete subframe logic L the following conditions are equivalent: (i) L is universal;9 (ii) L is elementary; (iii) L is D{persistent; (iv) L is strongly Kripke complete; (v) L has the nite embedding property.
Proof. The implications (i) ) (ii) and (iii) ) (iv) are trivial; (ii) ) (iii) follows from Fine's [1975b] Theorem formulated in Section 1.3 and (v) ) (i) from [Tarski 1954]. Thus it remains to show that (iv) ) (v). Suppose F is a Kripke frame with root r such that F 6j= L but all nite subframes of F validate L. Then it is readily checked that all nite subsets of = fpr g [
A similar criterion for the co nal subframe logics in NExtK4 can be found in [Zakharyaschev 1996]. Note, however, that they are not in general universal and certainly do not have the nite embedding property, but (ii), (iii) and (iv) are still equivalent. PROPOSITION 59. Every subframe logic L 2 NExtAltn has FMP. 9 I.e., universal is the class of Kripke frames for L considered as models of the rst order language with R and =.
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Æ 6 6 . .. G Æ
Æ2 6 1 6 F Æ0
(b)
(a)
Figure 5.
Proof. Suppose ' 62 L. By Theorem 22, there is a Kripke frame F for L refuting ' at a point x. Denote by X the set of points in F accessible from x by md(') steps. Clearly, X is nite and the subframe of F induced by X validates L and refutes '. To understand the place of incomplete logics in the lattice of subframe logics we call a subframe logic L strictly sf-complete if it is Kripke complete and no other subframe logic has the same Kripke frames as L. Example 57 shows that GL:3 is not strictly sf-complete. However, the logics T, S4 and Grz turn out to be strictly sf-complete. The following result clari es the situation. It is proved by applying the splitting technique to lattices of subframe logics. THEOREM 60. A subframe logic L containing K4 is strictly sf-complete i L 6 GL:3. All subframe logics in NExtAltn are strictly sf-complete. A subframe logic is tabular i there are only nitely many subframe logics containing it.
1.9 More suÆcient conditions of FMP As follows from Theorem 52, a logic in NExtK4 does not have FMP only if
at least one of its canonical axioms contains closed domains. We illustrate their role by a simple example. EXAMPLE 61. Consider the logic L = K4:3 ] (F; ?) and the formula (F; ?), where F is the frame depicted in Fig. 5 (a). The frame G in Fig. 5 (b) separates (F; ?) from L. Indeed, F is a co nal subframe of G and so G 6j= (F; ?). To show that G j= ] (F; ?), suppose f is a co nal subreduction of G to F. Then f 1 (1) contains only one point, say x; f 1 (0) also contains only one point, namely the root of G. So the in nite set of points between x and the root is outside domf , which means that f does not satisfy (CDC) for ff1gg. On the other hand, if H is a nite refutation frame of width 1 for (F; ?) then H contains a generated subframe reducible to F, from which H 6j= L. Thus, L fails to have FMP. In the same manner the reader can prove that A in Table 2 does not have FMP either.
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We show now two methods developed in [Zakharyaschev 1997a] for establishing FMP of logics whose canonical axioms contain closed domains. One of them uses the following lemma, which is an immediate consequence of the refutability criterion for the canonical formulas. LEMMA 62. Suppose (F; D) and (G; E) ( (F; D; ?) and (G; E; ?)) are canonical formulas such that there is a (co nal) subreduction f of G to F satisfying (CDC) for D and an antichain e domf " is in E whenever f (e") = d" for some d 2 D. Then (G; E) 2 K4 (F; D) (respectively, (G; E; ?) 2 K4 (F; D; ?)). THEOREM 63. L = K4 f (Fi ; Di ; ?) : i 2 I g f (Fj ; Dj ) : j 2 J g has FMP provided that either all frames Fi , for i 2 I [ J , are irre exive or all of them are re exive. Proof. Suppose all Fi are irre exive and (G; E; ?) is an arbitrary canonical formula. We construct from G a new nite frame H by inserting into it new re exive points. Namely, suppose e is an antichain in G such that e 62 E. Suppose also that C1 ; : : : ; Cn are all clusters in G such that e Ci " and e \ Ci = ;, for i = 1; : : : ; n, but no successor of Ci possesses this property. Then we insert in G new re exive points x1 ; : : : ; xn so that each xi could see only the points in e and their successors and could be seen only from the points in Ci and their predecessors. The same we simultaneously do for all antichains e in G of that sort. The resulting frame is denoted by H. Since no new point was inserted just below an antichain in E, H 6j= (G; E; ?). Suppose now that (G; E; ?) 62 L and show that H j= L. If this is not so then either H 6j= (Fi ; Di ; ?), for some i 2 I , or H 6j= (Fj ; Dj ), for some j 2 J . We consider only the former case, since the latter one is treated similarly. Thus, we have a co nal subreduction f of H to Fi satisfying (CDC) for Di . Since Fi is irre exive, no point that was added to G is in domf . So f may be regarded as a co nal subreduction of G to Fi satisfying (CDC) for Di . We clearly may assume also that the subframe of G generated by domf is rooted. Let e be an antichain in G belonging to domf " and such that f (e") = d" for some d 2 Di . If e 62 E then there is a re exive point x in H such that x 2 domf " and x sees only e" and, of course, itself. But then f (x") = f (e") = d" and so, by (CDC), x 2 domf , which is impossible. Therefore, e 2 E and so, by Lemma 62, (G; E; ?) 2 L, contrary to our assumption. In the case of re exive frames irre exive points are inserted. EXAMPLE 64. According to Theorem 63, the logic 1 2 AK L = K4 ( A ; ff1g; f1; 2gg) has FMP. However, Artemov's logic A = L GL does not enjoy this property. So FMP is not in general preserved under sums of logics.
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The scope of the method of inserting points is not bounded only by canonical axioms associated with homogeneous (irre exive or re exive) frames. It can be applied, for instance, to normal extensions of K4 with modal reduction principles, i.e., formulas of the form M p ! N p, where M and N are strings of and (for rst order equivalents of modal reduction principles see [van Benthem 1976]). One can show that each such logic is either of nite depth, or can be axiomatized by -formulas and canonical formulas based upon almost homogeneous frames (containing at most one re exive point), for which the method works as well. So we have THEOREM 65. All logics in NExtK4 axiomatizable by modal reduction principles have FMP and are decidable. One of the most interesting open problems in completeness theory of modal logic is to prove an analogous theorem for logics in NExtK or to construct a counter-example. It is unknown, in particular, whether the logics of the form K mp ! n p have FMP; the same concerns the logics K tran . The second method of proving FMP uses the more conventional technique of removing points. Suppose that L = K4 f (Gi ; Di ; ?) : i 2 I g and = (H; E; ?) 62 L. Then there exists a frame F for L such that F 6j= , i.e., there is a co nal subreduction h of F to H satisfying (CDC) for E. Construct the countermodel M = hF; Vi for as it was done in Section 1.6. Without loss of generality we may assume that domh" = domh# = F and that F is generated by the sets V(pi ), pi a variable in . Actually, the step-wise re nement procedure with deleting points having Sub -equivalent successors, used in the proof of Theorem 42, establishes FMP of L when all Di are empty, i.e., L is a co nal subframe logic. To tune it for L with non-empty Di , we should follow a subtler strategy of deleting points, preserving those that are \responsible" for validating the axioms of L. Suppose we have already constructed a model M0n = hF0n ; V0n i by \folding up" n 1-cyclic sets into clusters of depth n (we use the same notations as in the proof of Theorem 42). Now we throw away points of two sorts. First, for every proper cluster C of depth n such that some x 2 C has a Sub -equivalent successor of depth < n, we remove from C all points except x. Second, call a point x of depth > n redundant in M0n if it has a Sub -equivalent successor of depth n and, for every i 2 I and every co nal subreduction g of (F0n )n to the subframe of Gi generated by some d 2 Di such that d g(x") and g satis es (CDC) for Di , there is a point y 2 x " of depth n such that g(y ") = d". Let X be the maximal set of redundant points in M0n which is upward closed in (Wn0 )>n . We de ne Mn+1 = hFn+1 ; Vn+1 i as the submodel of M0n resulting from it by removing all points in X as well. Since all deleted points have Sub equivalent successors, Mn+1 6j= . And since we keep in Fn+1 points which
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violate (CDC) for Di of possible co nal subreductions to Gi , Fn+1 j= L. So FMP of L will be established if we manage to prove that this process eventually terminates. 2Æ 1 6
Æ Æ AKAÆ , and
EXAMPLE 66. Let L = S4 (G; ff1; 2gg; ?), where G is assume that our \algorithm", when being applied to F, and L, works in nitely long. Then the frame F! = hW! ; R! i, where [ [ W! = W i ; R! = Ri ; Fi = hWi ; Ri ; Pi i ; 0
i
0
i
is of in nite depth. By Konig's Lemma, there is an in nite descending chain : : : xi R! xi 1 : : : R! x2 R! x1 in F! such that xi is of depth i. Since there are only nitely many pairwise non-Sub -equivalent points, there must be some n > 0 such that, for every k n, each point in C (xk ) has a 1 Sub -equivalent successor in Fm m and every point in it has a Sub -equivalent successor in F m . So the only m reason for keeping some x 2 X is that Fm is co nally subreducible to G1 , x sees inverse images of both points in G1 but none of its successors in Fmm does. By the co nality condition, these inverse images can be taken 1 from F 1 . But then they are also seen from xm , which is a contradiction. Thus sooner or later our algorithm will construct a nite frame separating L from , which proves that L has FMP. The reason why we succeeded in this example is that inverse images of points in the closed domain f1; 2g can be found at a xed nite depth in F! , and so points violating (CDC) for it can also be found at nite depth (that was not the case in Example 61). The following de nitions describe a big family of frames and closed domains of that sort. A point x in a frame G is called a focus of an antichain a in G if x 62 a and x" = fxg [ a". Suppose G is a nite frame and D a set of antichains in G. De ne by induction on n notions of n-stable point in G (relative to D) and n-stable antichain in D. A point x is 1-stable in G i either x is of depth 1 in G or the cluster C (x) is proper. A point x is n + 1-stable in G (relative to D) i it is not m-stable, for any m n, and either there is an n-stable point in G (relative to D) which is not seen from x or x is a focus of an antichain in D containing an n 1-stable point and no n-stable point. And we say an antichain d in D is n-stable i it contains an n-stable point in the subframe G0 of G generated by d (relative to D) and no m-stable point in G0 (relative to D), for m > n. A point or an antichain is stable if
ADVANCED MODAL LOGIC 1Æ
6AKA Æ61 3 Æ A Æ2 6AKA A 6 5 Æ A AÆ 4 6AKA A 6 7 Æ A AÆ 6 (a)
1Æ
6AKA Æ61 2 Æ A Æ 2 6AKA A6 3 Æ A AÆ 3 6AKA A 6 4 Æ A AÆ 4 (b)
1Æ 1Æ
6@I @I Æ61 @ 2Æ @ 6@I 2 Æ@I Æ62 3Æ @ 3 Æ @Æ 3 6@I @I 6 4Æ @ 4 Æ @Æ 4 (c)
133 1Æ
6@I@Æ61 3 Æ @Æ 3 6@I@ 6 5 Æ @Æ 5 6@I@ 6 7 Æ @Æ 7 (d)
Figure 6. it is n-stable for some n. It should be clear that if a point in an antichain is stable then the rest points in the antichain are also stable. EXAMPLE 67. (1) Suppose G is a nite rooted generated subframe of one of the frames shown in Fig. 6 (a){(c). Then, regardless of D, each point in G di erent from its root is n-stable, where n is the number located near the point. Every antichain d in G, containing at least two points, is also n-stable, with n being the maximal degree of stability of points in d. (2) If G is a rooted generated subframe of the frame depicted in Fig. 6 (d) and D is the set of all two-point antichains in G then every point in G is n-stable (relative to D), where n stays near the point. However, for D = ; no point in G, save those of depth 1, is stable. (3) If G is a nite tree of clusters then every antichain in G, di erent from a non- nal singleton, is either 1- or 2-stable in G regardless of D. Every antichain containing a point x with proper C (x) is 1- or 2-stable as well, whatever G and D are. (4) Every antichain is stable in every irre exive frame G relative to the set D] of all antichains in G. However, this is not so if G contains re exive points (for re exive singletons are open domains and do not belong to D] ). The suÆcient condition of FMP below is proved by arguments that are similar to those we used in Example 66. THEOREM 68. If L = K4 f (Gi ; Di ; ?) : i 2 I g and there is d > 0 such that, for any i 2 I , every closed domain d 2 Di is n-stable in Gi (relative to Di ), for some n d, then L has FMP. Example 67 shows many applications of this condition. Moreover, using it one can prove the following
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THEOREM 69. Every normal extension of S4 with a formula in one variable has FMP and is decidable. Note that, as was shown by Shehtman [1980], a formula with two variables or an in nite set of one-variable formulas can axiomatize logics in NExtS4 without FMP (and even Kripke incomplete).
1.10 The reduction method That a logic does not have FMP (or is Kripke incomplete) is not yet an evidence of its undecidability: it is enough to recall that the majority of decidability results for classical theories was proved without using any analogues of the nite model property (see e.g. [Rabin 1977], [Ershov 1980]). The rst example of a decidable nitely axiomatizable modal logic without FMP was constructed by Gabbay [1971]. It seems unlikely that the methods of classical model theory can be applied directly for proving the decidability of propositional modal logics. However, sometimes it is possible to reduce the decision problem for a given modal logic L to that for a knowingly decidable rst or higher order theory whose language is expressive enough for describing the structure of frames characterizing L. The most popular tools used for this purpose are Buchi's [1962] Theorem on the decidability of the weak monadic second order theory of the successor function on natural numbers and Rabin's [1969] Tree Theorem. Below we illustrate the use of Rabin's Theorem following [Gabbay 1975] and [Cresswell 1984]. Let ! be the set of all nite sequences of natural numbers and the lexicographic order on it. For x 2 ! and i < !, put ri (x) = x i, where denotes the usual concatenation operation. Besides, de ne the following predicates
x
It follows from [Rabin 1969] that the monadic second order theory S!S of the model h! ; fri : i < !g; f
K m p ! p; K m p ! p;
K mp ! n p; K m p ! n p: By Sahlqvist's Theorem, all these logics are Kripke complete; however, we do not know whether they have FMP. General frames can also be described by means of S!S.
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EXAMPLE 70. The frame F = hW; R; P i constructed in Example 7 can be represented in the language of S!S as follows. Let us encode each n < ! by the sequence h3ni, while ! and ! + 1 by r1 (;) and r2 (;), respectively. Then we have
x 2 W i ; <0 x _ x = r1 (;) _ x = r2 (;); xRy i (; <0 x ^ ; <0 y ^ y x ^ x 6= y) _ (x = r1 (;) ^ ; <0 y) _ x = y = r1 (;) _ (x = r2 (;) ^ y = r1 (;)); X 2 P i 8x (x 2 X ! x 2 W ) ^ ((F in(X ) ^ r1 (;) 2= X ) _ 8Y (8y (y 2 Y $ (y 2 W ^ y 2= X )) ! F in(Y ) ^ r1 (;) 2= Y ));
where x = y means x y ^ y x and
F in(X ) = 9x8y (y 2 X ! y x): It follows that the logic LogF is decidable. Indeed, for every formula '(p1 ; : : : ; pn ), we have ' 2 LogF i the second order formula
8x8X1; : : : ; Xn (X1 2 P ^ ^ Xn 2 P ^ x 2 W ! ST ('(X1 ; : : : ; Xn ))) belongs to S!S. Here ST ('(X1 ; : : : ; Xn )), the standard translation of ', is de ned inductively in the following way (see also Correspondence Theory):
ST (X ) = x 2 X; ST (?) = ?; ST (X Y ) = ST (X ) ST (Y ); for 2 f^; _; !g; ST (X ) = 8y (xRy ! ST (X )fy=xg): Recall that, as was shown in Example 57, LogF is Kripke incomplete. Also, it is not hard to nd examples of applications of this technique for proving the decidability of nitely axiomatizable quasi-normal unimodal and normal polymodal (in particular, tense) logics which do not have Kripke frames at all; perhaps, the simplest one is Solovay's logic S. Sobolev [1977a] found another way of proving decidability by applying methods of automata theory on in nite sequences. Using the results of [Buchi and Siefkes 1973] he showed that all nitely axiomatizable superintuitionistic logics of nite width (see Section 3.4) containing the formula (((p ! q) ! p) ! p) _ (((q ! p) ! q) ! q): are decidable. By the preservation theorem of Section 3.3, this result can be transferred to the corresponding extensions of S4. If a logic is known to be complete with respect to a suitable class of frames, the methods discussed above are usually applicable to it in a rather
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
straightforward manner. A relative disadvantage of this approach is that the resulting decision algorithms inherit the extremely high complexity of the decision algorithms for S!S or other \rich theories" used to prove decidability. On the other hand, the logic S, for instance, turns out to be decidable by an algorithm of the same complexity as that for GL (see Example 75), in particular, the derivability problem in S is P SP ACE -complete. The logic of the frame F in Example 7 is \almost trivial"|it is polynomially equivalent to classical propositional logic, which follows from the fact that every formula ' refutable by F can be also refuted in F under a valuation giving the same truth-value to all variables in ' at all points i such that jSub'j < i < ! (see Section 4.6). Actually, this sort of decidability proofs (ignoring \inessential" parts of in nite frames) was used already by Kuznetsov and Gerchiu [1970] for studying some superintuitionistic logics. Recently more general semantical methods of obtaining decidability results without turning to \rich theories" have been developed. We demonstrate them in the next section by establishing the decidability of all nitely axiomatizable logics in NExtK4:3, which according to Example 61 do not in general have FMP. We show, however, that those logics are complete with respect to recursively enumerable classes of recursive frames in which the validity of formulas can be e ectively checked|it was this rather than the niteness of frames that we used in the proof of Harrop's Theorem. In Section 2.5 this result will be extended to linear tense logics which in general are not even Kripke complete. Our presentation follows [Zakharyaschev and Alekseev 1995].
1.11 Logics containing K4:3 Each logic in L 2 NExtK4:3 is represented in the form L = K4:3 f (Fi ; Di ; ?) : i 2 I g; where all Fi are chains of clusters. So our decidability problem reduces to nding an algorithm which, given such a representation with nite I and a canonical formula (F; D; ?) built on a chain of clusters F, could decide whether (F; D; ?) 2 L. Recall also that, by Fine's [1974c] Theorem, logics of width 1 are characterized by Kripke frames having the form of Noetherian chains of clusters. LEMMA 71. For any Noetherian chain of clusters G and any canonical formula (F; D; ?), G 6j= (F; D; ?) i there is an injective10 co nal subreduction g of G to F satisfying (CDC) for D.
Proof. If G 6j= (F; D; ?) then there is a co nal subreduction f of G to F satisfying (CDC) for D. Clearly, f 1 (x) is a singleton if x is irre exive. 10 That is g (x) = 6 g(y), for every distinct x; y 2 domg.
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Suppose now that x is a re exive point in F. Since G contains no in nite ascending chains, f 1 (x) has a nite cover and so there is a re exive point ux 2 f 1 (x) such that f 1 (x) ux#. Fix such a ux for each re exive x and de ne a partial map g by taking
g(y) =
8 < :
f (y)
if either f (y) is irre exive or f (y) is re exive and y = uf (y) unde ned otherwise.
One can readily check that g is the injective co nal subreduction we need. The converse is trivial. Roughly, every Noetherian chain of clusters refuting (F; D; ?) results from F by inserting some Noetherian chains of clusters just below clusters C (x) in F such that fxg 62 D. We show now that if (F; D; ?) is not in L 2 NExtK4:3 then it can be separated from L by a frame constructed from F by inserting in open domains between its adjacent clusters either nite descending chains of irre exive points possibly ending with a re exive one or in nite descending chains of irre exive points. Let C (x0 ); : : : ; C (xn ) be all distinct clusters in F ordered in such a way that C (x0 ) C (x1 )# C (xn )#. Say that an n-tuple t = h1 ; : : : ; n i is a type for (F; D; ?) if either i = m or i = m+, for some m < !, or i = !, with i = 0 if fxi g 2 D. Given a type t = h1 ; : : : ; n i for (F; D; ?), we de ne the t-extension of F to be the frame G that is obtained from F by inserting between each pair C (xi 1 ), C (xi ) either a descending chain of m irre exive points, if i = m < !, or a descending chain of m + 1 points of which only the last (lowest) one is re exive, if i = m+, or an in nite descending chain of irre exive points, if i = !. It should be clear that G 6j= (F; D; ?). LEMMA 72. If L 2 NExtK4:3 and (F; D; ?) 62 L then (F; D; ?) is separated from L by the t-extension of F, for some type t for (F; D; ?).
Proof. By Lemma 71, we have a Noetherian chain of clusters G for L and an injective co nal subreduction f of G to F satisfying (CDC) for D. By the Generation Theorem, we may assume that f maps the root of G to the root of F. Let G0 be the subframe of G obtained by removing from G all those points that are not in domf but belong to clusters containing some points in domf . The very same map f is an injective co nal subreduction of G0 to F satisfying (CDC) for D, and so G0 6j= (F; D; ?). Since G0 is a reduct of G, G0 j= L. Let C (x0 ); : : : ; C (xn ) be all distinct clusters in G0 such that domf =
n [ i=0
C (xi ); C (x0 ) C (x1 )# C (xn )#:
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
By induction on i we de ne a sequence of frames G0 Gn such that (a) f is an injective co nal subreduction of Gi to F satisfying (CDC) for D, (b) between C (xi 1 ) and C (xi ) the frame Gi contains either a nite descending chain of irre exive points possibly ending with a re exive one or an in nite descending chain of irre exive points, and (c) Gi j= L. Suppose Gi 1 has been already constructed and Ci is the chain of clusters located between C (xi 1 ) and C (xi ). Three cases are possible. (1) Ci is a nite chain of irre exive points. Then we put Gi = Gi 1 . (2) Ci contains a non-degenerate cluster C (x) having nitely many distinct successors in Ci and all of them are irre exive. Then Gi results from Gi 1 by removing from Ci all points save x and those successors. Gi is a reduct of Gi 1 and so conditions (a){(c) are satis ed. (3) Suppose (1) and (2) do not hold. Then Ci contains an in nite descending chain Y of irre exive points accessible from all other points in Ci . In this case Gi is obtained from Gi 1 by removing all points in Ci save those in Y . Clearly, Gi satis es (a) and (b). To prove (c) suppose Gi 6j= (H; E; ?) for some (H; E; ?) 2 L. Then there is an injective co nal subreduction g of Gi to H satisfying (CDC) for E. Consider g as a co nal subreduction of Gi 1 to H and show that it also satis es (CDC) for E. Indeed, (CDC) could be violated only by a point in z 2 Ci Y such that g(z ") = w", for some fwg 2 E. Since g 1 (w) is a singleton and Y z", there is y 2 Y such that g(y") = w" and y 62 domg, contrary to g satisfying (CDC) for E as a subreduction of Gi to H. Thus, a frame separating (F; D; ?) 62 L from L 2 NExtK4:3 can be found in the recursively enumerable class of t-extensions of F, t being a type for (F; D; ?). Moreover, given a formula (H; E; ?) and a type t for (F; D; ?), one can e ectively check whether (H; E; ?) is valid in the t-extension of F. Indeed, let k be the number of irre exive points in H, t = h1 ; : : : ; n i, and G the t-extension of F. Construct a co nal subframe Gk of G by \cutting o " the in nite descending chains inserted in F (if any) just below their k + 1th points, and let X be the set of all these k + 1th points. Clearly, Gk is nite. It is now an easy exercise to prove the following LEMMA 73. G 6j= (H; E; ?) i there is an injective co nal subreduction f of Gk to H satisfying (CDC) for E and such that X \ domf = ;. As a consequence we obtain THEOREM 74. All nitely axiomatizable normal extensions of K4:3 are decidable.
1.12 Quasi-normal modal logics All logics we have considered so far were normal, i.e., closed under the rule of necessitation '='. McKinsey and Tarski [1948] noticed, however, that by adding to S4 the McKinsey axiom ma = p ! p and taking
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0
ÆÆ
6 1 6
Æ1
. 2
@@ I F Æ0
..
G ! Figure 7.
the closure under modus ponens and substitution we obtain a logic|let us denote it by S4:10 |which is not normal in that sense. To understand why this is so, consider the frame F shown in Fig. 7. One can easily construct a model on F such that 0 6j= ma (0 sees a nal proper cluster). On the other hand, ma and all its substitution instances are true at 0 (0 sees a nal simple cluster), from which S4:10 f' : 0 j= 'g and so ma 62 S4:10 . A set of modal formulas containing K and closed under modus ponens and substitution was called by Segerberg [1971] a quasi-normal logic. The minimal quasi-normal extension of a logic L with formulas 'i , i 2 I , will be denoted by L + f'i : i 2 I g (i.e., the operation + presupposes taking the closure under modus ponens and substitution only). ExtL is the class of all quasi-normal logics above L. It is easy to see that a quasi-normal logic is normal i it is closed under the congruence rule p $ q=p $ q. Quasi-normal logics, introduced originally as some abstract (though natural) generalization of normal ones, attracted modal logicians' attention after Solovay [1976] constructed his provability logics GL and S. The former one treats as \it is provable in Peano Arithmetic" and describes those properties of Godel's provability predicate that are provable in PA; it is normal. The latter characterizes the properties of the provability predicate that are true in the standard arithmetic model, and in view of Godel's Incompleteness Theorem it cannot be normal. (For a detailed discussion of provability logic consult Modal Logic and Self-reference.) Solovay showed in fact that S = GL + p ! p: At rst sight S may appear to be inconsistent: Lob's axiom requires frames to be irre exive, while p ! p is refuted in them. And indeed, no Kripke frame validates both these axioms (in particular no consistent extension of S is normal). Having the algebraic semantics for normal modal logics, it is fairly easy to construct an adequate algebraic semantics for a consistent L 2 ExtK. Let M be a normal logic contained in L (for instance the greatest one, which is called the kernel of L) and AM its Tarski{Lindenbaum algebra (in Section 11 of Basic Modal Logic it was called the canonical modal algebra for M ).
140 The set
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
r = f[']M : ' 2 Lg
is clearly a lter in AM . By the well known properties of the Tarski{ Lindenbaum algebras, we then obtain the following completeness result: ' 2 L i under every valuation in AM the value of ' belongs to r. Structures of the form hA; ri, where A is a modal algebra and r a lter in A, are known as modal matrices. Thus, every quasi-normal logic is characterized by a suitable class of modal matrices. It is not hard to see that L is normal i it is characterized by a class of modal matrices with unit lters. Now, going over to the dual (Stone{Jonsson{Tarski representation) A+ of A in a modal matrix hA; ri and taking r+ to be the set of ultra lters in A containing r, we arrive at the general frame A+ with the set of distinguished points (or actual worlds) r+ . A formula ' is regarded to be valid in hA+ ; r+ i i under any valuation in A+ , ' is true at all points in r+ . Taking into account the Generation Theorem, we can conclude that every quasi-normal modal logic is characterized by a suitable class of rooted general frames in which the root is regarded to be the only actual world. It follows in particular that, as was rst observed by McKinsey and Tarski [1948], K4 + f'i : i 2 I g = K4 f'i : i 2 I g: However, one cannot replace here K4 by K or T. Note also that as was shown by Segerberg [1971], K, T and some other standard normal logics are not nitely axiomatizable with modus ponens and substitution as the only postulated inference rules. Duality theory between modal matrices and frames with distinguished points can be developed along with duality theory for normal logics (for details see [Chagrov and Zakharyaschev 1997]). Kripke frames with distinguished points were used for studying quasi-normal logics by Segerberg [1971]. Modal matrices were considered by Blok and Kohler [1983] (under the name of ltered algebras), Chagrov [1985b], and Shum [1985]. EXAMPLE 75. Consider the (transitive) frame G = hV; S; Qi whose underlying Kripke frame is shown in Fig. 7 and Q consists of ;, V , all nite sets of natural numbers and the complements to them in the space V (so ! 2 X 2 Q i there is n < ! such that m 2 X for all m n). Since G is irre exive and Noetherian, it validates GL. Moreover, we have hG; !i j= p ! p; for if under some valuation ! j= p then p must be true at every point. It follows that G with actual world ! validates S. (The reader can check that by making ! re exive we again obtain a frame for S.) By inserting the \tail" G as in Fig. 7 into nite rooted frames for GL below their roots and using the fact that GL has FMP, one can readily
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show that, for every formula ',
' 2 S i
^
(
2Sub'
!
) ! ' 2 GL:
It follows in particular that S is decidable. This example shows that the concepts of Kripke completeness and FMP do not play so important role in the quasi-normal case: even simple logics require in nite general frames. One possible way to cope with them at least in the transitive case is to extend the frame-theoretic language of the canonical formulas to the class ExtK4. Notice rst that the canonical formulas, introduced in Section 1.6, cannot axiomatize all logics in ExtK4. Indeed, hG; wi 6j= (F; D; ?) i there is a co nal subreduction f of G to F satisfying (CDC) for D and the following actual world condition as well: (AWC) f (w) is the root of F.
Now, consider the frame hG; !i constructed in Example 75. Since each set X 2 Q containing ! is in nite and has a dead end, it is impossible to reduce X to Æ or , and so hG; !i validates all normal canonical formulas. On the other hand, we clearly have hG; !i 6j= Bn for every n 1. So the logics K4BDn cannot be axiomatized by normal canonical formulas without the postulated necessitation. To get over this obstacle we have to modify the de nition of subreduction so that such sets as X above may be \reduced" at least to irre exive roots of frames. Given a frame G = hV; S; Qi with an irre exive root u and a frame F = hW; R; P i, we say a partial map f from W onto V is a quasisubreduction of F to G if it satis es (R1) for all x; y 2 domf such that f (x) 6= u or f (y) 6= u, (R2) and (R3).11 Thus, we may map all points in the frame G in Fig. 7 to , and this map will be a quasi-reduction of G to satisfying (AWC). Actually, every frame is quasi-reducible to . Now, given a nite frame F with an irre exive root a0 and a set D of antichains in F, we de ne the quasi-normal canonical formula (F; D; ?) as the result of deleting p0 from '0 in (F; D; ?) (which says that a0 is not self-accessible); the quasi-normal negation free canonical formula (F; D) is de ned in exactly the same way, starting from (F; D). It is not hard to see that (F; D; ?) (or (F; D)) is refuted in a frame hG; wi i there is a co nal (respectively, plain) quasi-subreduction of G to F satisfying (CDC) for D and (AWC). The following result is obtained by an obvious generalization of the proof of Theorem 44 to frames with distinguished points (for details see [Zakharyaschev 1992]). 11 Another possibility is to allow \reductions" of X to re exive points by relaxing (R2); cf. Section 2.6.
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THEOREM 76. There is an algorithm which, given a modal (negation free) formula ', constructs a nite set of normal and quasi-normal (negation free) canonical formulas such that K4 + ' = K4 + . For example, S = K4 + (Æ) + (). Since frames for S4 are re exive, we have COROLLARY 77. There is an algorithm which, given a modal formula ', constructs a nite set of normal canonical formulas built on re exive frames such that S4 + ' = S4 + . As a consequence we obtain THEOREM 78 (Segerberg 1975). ExtS4:3 = NExtS4:3.
Proof. We must show that every logic L 2 ExtS4:3 is normal, i.e., ' 2 L only if ' 2 L, for every '. Suppose otherwise. Then by Corollary 77, there exists (F; D; ?) 2 L such that (F; D; ?) 62 L. Let hG; wi be a frame validating L and refuting (F; D; ?). Since G j= S4:3, G is a chain of non-degenerate clusters. And since it refutes (F; D; ?) there is a co nal subreduction f of G to F. It follows, in particular, that F is also a chain of non-degenerate clusters and so D = ;. Let a be the root of F. De ne a map g by taking g(x) =
8 < :
f (x) if x 2 domf a if x 2 f 1(a)# domf unde ned otherwise.
It should be clear that g co nally subreduces G to F and g(w) = a. Consequently, hG; wi 6j= (F; ?), which is a contradiction. Let us now brie y consider quasi-normal analogues of subframe and co nal subframe logics in NExtK4. Those logics that can be represented in the form (K4 f (Fi ) : i 2 I g) + f (Fj ) : j 2 J g + f (Fk ) : k 2 K g are called (quasi-normal) subframe logics and those of the form (K4 f (Fi ; ?) : i 2 I g) + f (Fj ; ?) : j 2 J g + f (Fk ; ?) : k 2 K g are called (quasi-normal) co nal subframe logics. The classes of quasinormal subframe and co nal subframe logics are denoted by QSF and QCSF , respectively. The example of S shows that Theorem 52 cannot be extended to QSF and QCSF . Yet one can show that all nitely axiomatizable logics in QSF and QCSF are decidable. We omit almost all proofs and con ne ourselves mainly to formulations of relevant results. For details the reader is referred to [Zakharyaschev 1996].
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AA
A
F Au
AA A Fr1 AÆ0
143
AA A Fir A0 61 .. .
6
AA A Fir(!+1)A0 61 2 1
.. .
!
Figure 8. We use the following notation. For a frame F = hW; Ri with irre exive r root u and 0 < < !, Fir and F denote the frames obtained from F by replacing u with the descending chains 1 of irre exive D 0; : : : ; E and ir re exive points, respectively; Fir = W ; R ; P (!+1) (!+1) (!+1) (!+1) is the frame that results from F by replacing u with the in nite descending chain 0; 1; : : : of irre exive points and then adding irre exive root !, with P(!+1) containing all subsets of W fug, all nite subsets of natural numbers f0; 1; : : : g, all ( nite) unions of these sets and all complements to them in the space W(!+1) (see Fig. 8). Note that F is a quasi-reduct of every frame r ir of the form Fir , F or F(!+1) . The following theorem characterizes the canonical formulas belonging to logics in QSF and QCSF . THEOREM 79. Suppose L is a subframe or co nal subframe quasi-normal logic. Then rm (i) for every nite frame F with root u, (F; D; ?) 2 L i hF; ui 6j= L; rm (ii) for every nite frame F with irre exive root u, (F; D; ?) 2 L i D
E
hF; ui 6j= L, hFr1 ; 0i 6j= L and Fir(!+1) ; ! 6j= L.
Proof. We prove only (() of (ii). Let G = hV; S; Qi refute (F; D; ?) at its root w and show that hG; wi 6j= L. We have a co nal quasi-subreduction f of G to F such that f (w) = u. Consider the set U = f 1(u) 2 Q. Without loss of generality we may assume that U = U #. There are three possible cases. Case 1. The point w is irre exive and fwg 2 Q. Then the restriction of f to domf (U fwg) is a co nal subreduction of G to F satisfying (AWC) and so hG; wi 6j= L. Case 2. There is X U such that w 2 X 2 Q and, for every x 2 X , there exists y 2 X \ x". Then the restriction of f to domf (U X ) is a co nal subreduction of G to Fr1 satisfying (AWC) and so again hG; wi 6j= L.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Case 3. If neither of the preceding cases holds then, for every X U such that w 2 X 2 Q, the set DX = X X # of dead ends in X is a cover for X , i.e., X DX #, and w 2 X DX 2 Q. Put
X0 = DU ; : : : ; Xn+1 = DU (X0 [[Xn ) ; : : : ; X! = U
[
X :
Each of these sets, save possibly X! , is an S antichain of irre exive points and belongs to Q. Besides, X Xn# = n<! X for every n < !. Therefore, the map g de ned by
g(x) =
f (x) if x 2 V U if x 2 X ; 0 !
is a co nal quasi-subreduction of G to Fir satisfying (AWC). D E(!+1) Now using the fact that Fir (!+1) ; ! 6j= L and that the composition of (co nal) (quasi-) subreductions is again a (co nal) (quasi-) subreduction, it is not hard to see that hG; wi 6j= L. COROLLARY 80. All subframe and co nal subframe quasi-normal logics above S4 have FMP. EXAMPLE 81. As an illustration let us use Theorem 79 to characterize those normal and quasi-normal canonical formulas that belong to S. Clearly, either (Æ) or () is refuted at the root of every rooted Kripke frame. So all normal canonical formulas are in S. Every quasi-normal formula (F; D; ?) associated with F containing a re exive point is also in S, since (Æ) is refuted at the roots of F, Fr1 and Fir (!+1) . But no quasi-normal formula (F; D; ?) built on irre exive F belongs to S, because Fir (!+1) j= (Æ) and D E ir F(!+1) ; ! j= (), since f!g 62 P(!+1) . Notice that incidentally we have proved the following completeness theorem for S. THEOREM 82. S is characterized by the class D
E
f Fir(!+1) ; ! : F is a nite rooted irre exive frameg: Theorem 79 reduces the decision problem for a logic L in QSF or QCSF to the problem of verifying, given a nite frame F with root u, whether D E r ir hF; ui, hF1 ; 0i and F(!+1) ; ! refute an axiom of L. The two former frames present no diÆculties: they are D E nite. As to the D latter, it Eis not hard ir to see that, for instance, F(!+1) ; ! 6j= (G; ?) i Fir ; 1 , for some jGj, is co nally quasi-subreducible to G. Thus we obtain THEOREM 83. All nitely axiomatizable subframe and co nal subframe quasi-normal logics are decidable.
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One can also give a frame-theoretic characterization of the classes QSF and QCSF similar to Theorem 53. Let us say that a frame F with actual world u is a (co nal) subframe of a frame G with actual world w if F is a (co nal) subframe of G and u = w. THEOREM 84. L is a (co nal) subframe quasi-normal logic i L is characterized by a class of frames with actual worlds that is closed under (co nal) subframes.
1.13 Tabular logics Every logic L having the nite model property can be represented as the in-
tersection of some tabular logics, that is logics characterized by nite frames (or models, algebras, matrices, etc.):
L=
\
fLogF : F is a nite frame for Lg:
(It follows in particular that every fragment of L containing only those formulas whose length does not exceed some xed n < ! is determined by a nite frame; for that reason logics with FMP are also called nitely approximable.) In many respects tabular logics are very easy to deal with. For instance, the key problem of recognizing whether a formula ' belongs to a tabular L is trivially decided by the direct inspection of all possible valuations of ''s variables in the nite frame characterizing L. That is why the question \is it tabular?" is one of the rst items in the standard \questionnaire" for every new logical system. First results concerning the tabularity of modal logics were obtained by Godel [1932] and Dugundji [1940] who showed that intuitionistic propositional logic and all Lewis' modal systems S1{S5 are not tabular. (Note that using the same method Drabbe [1967] proved that the three non-normal Lewis' systems S1{S3 cannot be characterized by a matrix with a nite number of distinguished elements). For arbitrary logics in ExtK one can easily prove the following syntactical criterion of tabularity, which uses the formulas n = :('1 ^ ('2 ^ ('3 ^ ^ 'n ) : : : ));
n =
n^1
m=0
:m ('1 ^ ^ 'n );
tabn = n ^ n ; where 'i = p1 ^ ^ pi 1 ^ :pi ^ pi+1 ^ ^ pn . THEOREM 85. L 2 ExtK is tabular i tabn 2 L, for some n < !. Proof. A frame F = hW; Ri refutes n at a point x1 i a chain of length n starts from x1 , and F refutes n at x1 i there is a chain x1 Rx2 R : : : Rxm
146
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
of length m < n such that xm is of branching n, i.e., xm Ry1 ; : : : ; xm Ryn for some distinct y1 ; : : : ; yn . It follows that every rooted generated (by an actual world) subframe of the canonical frame for L containing tabn has at most 1 + (n 1) + + (n 1)n 2 points. As a consequence we immediately obtain COROLLARY 86. Every tabular modal logic has nitely many extensions and all of them are also tabular. The next theorem follows from general algebraic results of [Blok and Kohler 1983]; equally easy it can be proved using the characterization above. THEOREM 87. Every tabular logic L 2 ExtK is nitely axiomatizable.
Proof. According to Theorem 85, L is an extension of K + tabn , for some n < !. By Corollary 86, we have a chain K + tabn = L1 L2 Lk 1 Lk = L of quasi-normal logics such that fL0 2 ExtK : Li L0 Li+1 g = ;, for every i = 1; : : : ; k 1. It remains to notice that if L0 is nitely axiomatizable, L0 L00 and there is no logic located properly between L0 and L00 then L00 is also nitely axiomatizable (e.g. L00 = L0 + ', for any ' 2 L00 L0 ). Theorem 12 provides us in fact with an algorithm to decide, given a tabular logic L 2 NExtK4 and an arbitrary formula ', whether K4' = L. Indeed, notice rst that we have THEOREM 88. Each nitely axiomatizable logic L 2 NExtK4 of nite depth is a nite union-splitting, i.e., can be represented in the form
L = K4 f ] (Fi ; ?) : i 2 I g with nite I .
Proof. Let L = K4 ' be a logic of depth n and let m be the number of variables in '. We show that L coincides with the logic L0 = K4 f ] (G; ?) : jGj
nX +1 i=1
2mcm (i); G 6j= 'g
(cm (i) was de ned in Section 1.2). The inclusion L L0 is obvious. Suppose ' 62 L0. Then there is a rooted re ned m-generated frame F for L0 refuting '. Clearly, F is of depth n, since otherwise ] (G; ?) is an axiom of L0 for every rooted generated subframe G of F of depth n + 1 and so F 6j= L0, which is a contradiction. But then ] (F; ?) is an axiom of L0 , contrary to our assumption.
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Thus, all tabular logics in NExtK4 are nite union-splittings and so, by Theorem 12, we obtain the following THEOREM 89. Let L be a tabular logic in NExtK4. (i) (Blok 1980c) L has nitely many immediate predecessors and they are also tabular. (ii) The axiomatizability problem for L above K4 is decidable. For logics in NExtK this is not the case, witness Theorems 36 and 205. The tabularity criterion of Theorem 85 is not e ective. Moreover, as we shall see in Section 4.4, no e ective tabularity criterion exists in general. However, if we restrict attention to suÆciently strong logics, e.g. to the class NExtS4, the tabularity problem turns out to be decidable. The key idea, proposed by Kuznetsov [1971], is to consider the so called pretabular logics. A logic L 2 (N)ExtL0 is said to be pretabular in the lattice (N)ExtL0 , if L is not tabular but every proper extension of L in (N)ExtL0 is tabular. In other words, a pretabular logic in (N)ExtL0 is a maximal non-tabular logic in (N)ExtL0 . THEOREM 90. In the lattices ExtK and NExtK every non-tabular logic is contained in a pretabular one.
Proof. By Theorem 85, a logic is non-tabular i it does not contain the formula tabn , for any n < !. It follows that the union of an ascending chain of non-tabular logics is a non-tabular logic as well. The standard use of Zorn's Lemma completes the proof. If there is a simple description of all pretabular logics in a lattice, we obtain an e ective (modulo the description) tabularity criterion for the lattice. Indeed, take for de niteness the lattice NExtK4. How to determine, given a formula ', whether K4 ' is tabular? We may launch two parallel processes: one of them generates all derivations in K4 ' and stops after nding a derivation of tabn , for some n < !; another process checks if ' belongs to a pretabular logic in NExtK4 and stops if this is the case. The termination of the rst process means that K4 ' is tabular, while that of the second one shows that it is not tabular. Unfortunately, it is impossible to describe in an e ective way all pretabular logics in (N)ExtK and even (N)ExtK4: Blok [1980c] and Chagrov [1989] constructed a continuum of them. However, for smaller lattices like NExtS4 or NExtGL such descriptions were found by Maksimova [1975b], Esakia and Meskhi [1977] and Blok [1980c]. The ve pretabular logics in NExtS4 were presented in Section 17 of Basic Modal Logic. In NExtGL the picture is much more complicated.
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6 6 . ..
G!
1 6 . 2 .. m @ 6IHYPiPH@PHPHPP n . .. 3 1 6 G!m;n
a0
6 6@I@HH a2 b1 b2 6 a3 6 a4 a1 H YH
G!2;2
Figure 9. THEOREM 91 (Blok 1980c, Chagrov 1989). The set of pretabular logics in NExtGL is denumerable. It consists of the logics GL:3 = LogG! and LogG!m;n, for m 0, n 1, where G! and G!m;n are the frames depicted in Fig. 9. If hm; ni 6= hk; li then LogG!m;n 6= LogG!k;l . Using this semantic description of pretabular logics in NExtGL, it is not hard to nd nite sets of formulas axiomatizing them. Moreover, all of them turn out to be decidable. For we have THEOREM 92. Every non-tabular logic L 2 NExtK4 has a non-tabular extension with FMP, and so every pretabular logic in NExtK4 has FMP.
Proof. Since L is non-tabular and characterized by the class of its rooted nitely generated re ned frames, we have either a sequence Fi , i = 1; 2; : : : , of rooted nite frames for L of depth i, or a sequence Fi of rooted nite frames for L of width i. In both cases the logic LogfFi : i < !g L is non-tabular and has FMP. So we obtain the following result on the decidability of tabularity. THEOREM 93. The property of tabularity is decidable in NExtS4, ExtS4, NExtGL, ExtGL. Since a logic in ExtK4 is locally tabular i it is determined by a frame of nite depth, the property of local tabularity is decidable in the lattices mentioned in Theorem 93 as well. However, this is not the case for ExtK4 itself.
1.14 Interpolation One of the fundamental properties of logics is their capability to provide explicit de nitions of implicitly de nable terms, which is known as the Beth property (Beth [1953] proved it for classical logic). In the modal case we
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say a logic L has the Beth property if, for any formula '(p1 ; : : : ; pn ; pn+1 ) and variables p and q di erent from p1 ; : : : ; pn ,
'(p1 ; : : : ; pn ; p) ^ '(p1 ; : : : pn ; q) ! (p $ q) 2 L only if there is a formula (p1 ; : : : pn) such that
'(p1 ; : : : ; pn; p) ! (p $ (p1 ; : : : pn )) 2 L: The Beth property turns out to be closely related to the interpolation property which was introduced by Craig [1957] for classical logic. Namely, we say that a logic L has the interpolation property if, for every implication ! 2 L, there exists a formula , called an interpolant for ! in L, such that ! 2 L, ! 2 L and every variable in , if any, occurs in both and . While in abstract model theory interpolation is weaker than Beth de nability, for modal logics we have THEOREM 94 (Maksimova 1992). A normal modal logic has interpolation i it has the Beth property. Say also that a normal modal logic L has the interpolation property for the consequence relation `L, ` -interpolation for short, if every time when `L , there is a formula such that `L , `L and Var Var \ Var . (Here Var' is the set of all variables in '.) It should be clear that interpolation implies `-interpolation. By the end of the 1970s interpolation had been established for a good many standard modal systems. The semantical proofs, sometimes rather sophisticated, resemble the Henkin construction of the canonical models. Here are two examples of such proofs (which are due to Maksimova [1982b] and Smorynski [1978]). THEOREM 95 (Gabbay 1972). The logics K, K4, T, S4 have the interpolation property.
Proof. We consider only S4; for the other logics the proofs are similar. Suppose ! 62 S4 and ! 62 S4 for any whose variables occur in both and , and show that in this case ! 62 S4. Let t = ( ; ) be a pair of sets of formulas such that Var' Var if ' 2 and Var' Var if ' 2 . Say that t is inseparable if there are no and with Var Var \ Var such that Vn formulas 'i 2 , j 2W m ' !
2 S4 ,
! i i=1 i=1 i 2 S4. The pair t is called complete if for every ' and with Var' Var and Var Var , one of the formulas ' and :' is in and one of and : is in . LEMMA 96. Every inseparable pair t0 = ( 0 ; 0 ) can be extended to a complete inseparable pair.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Proof. Let '1 ; '2 ; : : : and 1 ; 2 ; : : : be enumerations of all formulas whose variables occur in and , respectively. De ne pairs t0n = ( 0n ; 0n ) and tn+1 = ( n+1 ; n+1 ) inductively by taking t0n =
( n [ f'n g; n ) if this pair is inseparable ( n [ f:'n g; n ) otherwise,
( 0n ; 0n [ f n g) if this pair is inseparable ( 0n ; 0n [ f: n g) otherwise S S and put t = ( ; ), where = n
tn+1 =
one of the pairs ( one of the pairs (
[ f'g; ) or ( [ f:'g; ) is inseparable and ; [ f g) or ( ; [ f: g) is inseparable.
We prove only the former claim. Suppose, on the contrary, that both pairs are separable, i.e., there are formulas 1 , 2 in variables occurring in both and such that, for some '1 ; : : : ; 'n 2 , 1 ; : : : ; m 2 , we have
'1 ^ ^ 'n ^ ' ! 1 2 S4; 1 !
1 _ _ m
2 S4; '1 ^ ^ 'n ^ :' ! 2 2 S4; 2 ! 1 _ _ m 2 S4: Then we obtain ('1 ^ ^ 'n ^ ') _ ('1 ^ ^ 'n ^ :') ! 1 _ 2 2 S4,
1 _ 2 ! 1 _ _ m 2 S4, from which '1 ^ ^ 'n ! 1 _ 2 2 S4; 1 _ 2 ! 1 _ _ m 2 S4; contrary to t being inseparable. Now we de ne a frame F = hW; Ri by taking W to be the set of all
complete and inseparable pairs and, for t1 = ( 1 ; 1 ), t2 = ( 2 ; 2 ) in W , t1 Rt2 i ' 2 1 implies ' 2 2 . Using the axioms p ! p and p ! p of S4, one can readily check that R is a quasi-order on W , i.e., F j= S4. De ne a valuation V in F by taking for every variable p 2 Var( ! ), V(p) = f( ; ) 2 W : either p 2 or p 2 Var and p 62 g. Put M = hF; Vi. By induction on the construction of formulas ' and with Var' Var , Var Var one can show that for every t = ( ; ) in F (M; t) j= ' i ' 2 ; (M; t) 6j=
i
2 :
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Indeed, the basis of induction follows from the de nition of V and the completeness and inseparability of t. The cases of the Boolean connectives present no diÆculty. So suppose ' = '1 . If t j= '1 then, for every t0 = ( 0 ; 0 ) 2 t", we have t0 j= '1 and so '1 2 0 . Suppose '1 62 . Then :'1 2 . Consider the pair t0 = ( 0 ; 0 ), where 0
= f:'1 g [ f :
2 g;
0 = f: :
: 2 g;
and show that it is inseparable. Assume otherwise. Then there is with Var Var \ Var such that, for some formulas 1 ; : : : ; n 2 , : n+1 ; : : : ; : m 2 ,
:'1 ^ 1 ^ ^ n ! 2 S4; ! : n+1 _ _ : m 2 S4: It follows that
:'1 ^ 1 ^ ^ n ! 2 S4; ! : n+1 _ _ : m 2 S4;
contrary to t being inseparable. Let t0 = ( 0 ; 0 ) be a complete inseparable extension of t0 . By the de nition of t0 , we have tRt0 and so '1 2 0 , contrary to :'1 2 0 0 and t0 being inseparable. Suppose now that '1 2 . Then for every t0 = ( 0 ; 0 ) such that tRt0 , we have '1 2 and so t0 j= '1 . Consequently, t j= '1 . The formula is treated in the dual way. To complete the proof it remains to observe that M 6j= ! . This proof does not always go through for di erent kinds of logics. However, sometimes suitable modi cations are possible. THEOREM 97. GL has the interpolation property.
Proof. Suppose ! has no interpolant in GL. Our goal is to construct a nite irre exive transitive frame refuting ! . This time we consider nite pairs t = ( ; ) such that all formulas in and are constructed from variables and their negations using ^, _, , . Without loss of generality we will assume and to be formulas of that sort. Say that a formula with Var Var \Var V t is separable if there is W such that ! 2 GL and ! 2 GL. It should be clear that if t = ( ; ) is a nite inseparable pair then in the same way as in the proof of Theorem 95 but taking only subformulas of and we can obtain a nite inseparable pair t? = ( ? ; ? ) satisfying the conditions: for every ' 2 Sub and 2 Sub , one of the formulas ' and :' (an equivalent formula of the form under consideration, to be more precise) is in ? and one of and : is in ? .
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Now we construct by induction a nite rooted model for GL refuting ! . As its root we take (f g? ; f g? ). If we have already put in our model a pair t = ( ; ) and it has not been considered yet, then for every ' 2 and every 2 , we add to the model the pairs
t1 = (f ; ; :'; ' : 2 t2 = f ; : 2
g? ; f ; : 2 g? );
g?; f ; ; : ;
: 2 g? ):
One can readily show that if t is inseparable then t1 and t2 are also inseparable. Put tR0t1 and tR0 t2 . The process of adding new pairs must eventually terminate, since each step reduces the number of formulas of the form ' and in the left and right parts of pairs. Let W be the set of all pairs constructed in this way and R the transitive closure of R0 . Clearly, the resulting frame F = hW; Ri validates GL. De ne a valuation V in F by taking, for each variable p,
V(p) = f( ; ) 2 W : p 2 g: As in the proof of Theorem 95, it is easily shown that ! is refuted in F under V. To clarify the algebraic meaning of interpolation we require the following well known proposition. PROPOSITION 98. If r is a normal lter12 in a modal algebra A then the relation r , de ned by a r b i a $ b 2 r, is a congruence relation. The map r 7! r is an isomorphism from the lattice of normal lters in A onto the lattice of congruences in A. Denote by A=r the quotient algebra A= r and let kakr = fb : a r bg. Say that a class C of algebras is amalgamable if for all algebras A0 , A1, A2 in C such that A0 is embedded in A1 and A2 by isomorphisms f1 and f2 , respectively, there exist A 2 C and isomorphisms g1 and g2 of A1 and A2 into A with g1(f1 (x)) = g2(f2 (x)), for any x in A0. If in addition we have
gi (x) gj (y) implies 9z 2 A0 (x i fi (z ) and fj (z ) j y) for all x 2 Ai , y 2 Aj such that fi; j g = f1; 2g, then C is called superamalgamable. Here Ai is the universe of Ai and i its lattice order. THEOREM 99 (Maksimova 1979). L has the interpolation property i the variety AlgL of modal algebras for L is superamalgamable. L has the ` interpolation property i AlgL is amalgamable. 12 A lter r is normal (or open, as in Section 10 of Basic Modal Logic) if a 2 r whenever a 2 r.
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Proof. We prove only the former claim. ()) Suppose L has the interpolation property and A0 , A1 , A2 are modal algebras for L such that A0 is a subalgebra of both A1 and A2 . With each element a 2 Ai , i = 0; 1; 2, we associate a variable pia in such a way that, for a 2 A0 , p0a = p1a = p2a . Denote by Li the language with the variables pia , for a 2 Ai , i = 0; 1; 2, and let L = L1 [ L2 . We will assume that L is the language of L. Fix the valuation Vi of Li in Ai , de ned by Vi (pia ) = a, and put i = f' 2 ForLi : Vi (') = >g:
Let be the closure of 1 [ 2 [ L under modus ponens. We show that, for every ' 2 ForLi , 2 ForLj such that fi; j g = f1; 2g, (13) ' !
2 i 9 2 ForL0 (' ! 2 i and ! 2 j ): Suppose ' ! 2 . Then there exist nite sets i i and j j such that
^
^
) 2 L: Since L has interpolation, there is a formula 2 ForL0 such that ^
i^'!(
i ^ ' ! 2 L;
^
j
!
j
! ( !
) 2 L;
from which ' ! 2 i and ! 2 j . The converse implication is obvious. Now construct an algebra A by taking the set fk'k : ' 2 g as its universe, where k'k = f : ' $ 2 g, k'k ^ k k = k' ^ k and k'k = k 'k, for 2 f:; g. One can readily prove that A 2 AlgL. De ne maps gi from Ai into A by taking gi (a) = kpia k. It is not diÆcult to show that gi is an embedding of Ai in A. And for a 2 A0 , we have
g1 (a) = kp0ak = g2 (a):
It remains to check the condition for superamalgamability: Suppose a 2 Ai , b 2 Aj , fi; j g = f1; 2g, and gi (a) gj (b). Then gi (a) ! gj (b) = > and so kpia ! pjb k = >, i.e., pia ! pjb 2 . By (13), we have 2 ForL0 with V( ) = c such that a i c j b. (() Assuming AlgL to be superamalgamable, we show that L has the interpolation property. To this end we require LEMMA 100. Suppose A0 is a subalgebra of modal algebras A1 and A2 , a 2 A1 , b 2 A2 and there is no c 2 A0 such that a 1 c 2 b. Then there are ultra lters r1 in A1 and r2 in A2 such that a 2 r1 , b 62 r2 and r1 \ A0 = r2 \ A0 . Suppose '(p1 ; : : : ; pm; q1 ; : : : ; qn ) and (q1 ; : : : ; qn ; r1 ; : : : ; rl ) are formulas for which there is no (q1 ; : : : ; qn ) such that ' ! 2 L and ! 2 L. We show that in this case there exists an algebra A 2 VarL refuting ' ! .
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Let A00 , A01 and A02 be the free algebras in AlgL generated by the sets fc1 ; : : : ; cn g, fa1 ; : : : ; am ; c1 ; : : : ; cn g and fc1 ; : : : ; cn ; b1 ; : : : ; bl g, respectively. According to this de nition, A00 is a subalgebra of both A01 and A02 . By Lemma 100, there are ultra lters r1 in A01 and r2 in A02 such that we have '(a1 ; : : : ; am ; c1 ; : : : ; cn ) 2 r1 and (c1 ; : : : ; cn ; b1 ; : : : ; bl ) 62 r2 . De ne normal lters ri = fa 2 A0i : 8m < ! ma 2 ri g and put A1 = A01 =r1 , A2 = A02 =r2 . Construct an algebra A0 by taking A0 = fkakr1 : a 2 A00 g. By the de nition, A0 is a subalgebra of A1, i.e., is embedded in A1 by the map f1 (x) = x. One can show that A0 is embedded in A2 by the map f2 (kxkr1 ) = kxkr2 . Then there are an algebra A for L and isomorphisms g1 and g2 of A1 and A2 into A satisfying the conditions of superamalgamability. De ne a valuation V in A by taking V(pi ) = g1 (kai kr1 ), V(qj ) = g1 (kcj kr1 ) = g2(kcj kr2 ) and V(rk ) = g2 (kbk kr2 ). Then V(') 6 V( ) because otherwise there would exist fi; j g = f1; 2g and z 2 A0 such that V(') i fi (z ) and fj (z ) j V( ). Thus, A 6j= ' ! and so ' ! 62 L. Using this theorem Maksimova [1979] discovered a surprising fact: there are only nitely many logics in NExtS4 with the interpolation property (not more than 38, to be more exact) and all of them turned out to be union-splittings. By Theorem 12, we obtain then THEOREM 101 (Maksimova 1979). There is an algorithm which, given a modal formula ', decides whether S4 ' has interpolation. We illustrate this result by considering a much simpler class of logics. THEOREM 102. Only four logics in NExtS5 have the interpolation property: S5 itself, the logic of the two-point cluster, Triv and For.
Proof. We have already demonstrated how to prove that a logic has interpolation. So now we show only that no logic L in NExtS5 di erent from those mentioned in the formulation has the interpolation property. Suppose on the contrary that L has interpolation. We use the amalgamability of the variety of modal algebras for L to show that an arbitrary big nite cluster is a frame for L, from which it will follow that L = S5. Figure 10 demonstrates two ways of reducing the three-point cluster to the two-point one. By the amalgamation property, there must exist a cluster reducible to the two depicted copies of the two-point cluster, with the reductions satisfying the amalgamation condition. It should be clear from Fig. 10 that such a cluster contains at least four points. By the same scheme one can prove now that every n-point cluster validates L. It would be naive to expect that such a simple picture can be extended to classes like NExtK4 or NExtK. Even in NExtGL the situation is quite
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ÆH YH ÆH Y H H H Æ H H HÆ @I HÆ Æ Y H H @ Æ H H Æ I HÆ @ @ @ @ Æ @ Æ @ Æ Figure 10. di erent from that in NExtS4: Maksimova [1989] discovered that there is a continuum of logics in NExtGL having the interpolation property. This result is based upon the following observation. For L 2 NExtK4, we call a formula (p) conservative in NExtL if + ( (?) ^ (p) ^ (q)) ! (p ! q) ^ (p) 2 L: For example, in NExtS4 conservative are p ! p, p $ p, and p $ p. THEOREM 103 (Maksimova 1987). If L 2 NExtK4 has the interpolation property and formulas i , for i 2 I , are conservative in NExtL, then the logic L f i : i 2 I g also has the interpolation property. Proof. Suppose ' ! 2 L f i : i 2 I g. Then there is a nite J I , say J = f1; : : : ; lg, such that ' ! 2 L f i : i 2 J g and so, as follows from the de nition of conservative formulas and the Deduction Theorem for K4,
+
l ^ j =1
( j (?) ^ j (p1 ) ^ ^ j (pn )) ! (' ! ) 2 L;
where p1 ; : : : ; pm; pm+1 ; : : : ; pk and pm+1 ; : : : ; pk ; pk+1 ; : : : ; pn are all the variables in ' and , respectively. Consequently
+
l ^
j =1
(+
( j (?) ^ j (p1 ) ^ ^ j (pk )) ^ ' !
l ^
j =1
( j (pm+1 ) ^ ^ j (pn )) ! ) 2 L:
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Since L has the interpolation property, there is (pm+1 ; : : : ; pk ) such that
+
l ^ j =1
+
( j (?) ^ j (p1 ) ^ ^ j (pk )) ^ ' ! 2 L;
l ^ j =1
( j (pm+1 ) ^ ^ j (pn )) ! ( ! ) 2 L:
Then we obtain ' ! 2 L f i : i 2 I g and ! i.e., is an interpolant for ' ! in L f i : i 2 I g.
2 L f i : i 2 I g,
Using the formulas
i = + (i+1 > ^ i+2 ? ! i+1 p _ i+1 :p) which are conservative in NExtGL, one can readily construct a continuum of logics in this class with the interpolation property. The set of logics in NExtGL without interpolation is also continual. In general, an interpolant for an implication ! 2 L depends on both and . Say that a logic L has uniform interpolation if, for any nite set of variables and any formula , there exists a formula such that Var and ! 2 L, ! 2 L whenever Var \ Var and ! 2 L. In this case is called a post-interpolant for and . Roughly speaking, a logic has uniform interpolation if we can choose an interpolant for ! 2 L independly from the actual shape of . Uniform interpolation was rst investigated by Pitts [1992] who proved that intuitionistic logic enjoys it. It is fairly easy to nd multiple examples of modal logics with uniform interpolation by observing that any locally tabular logic with interpolation has uniform interpolation as well. Indeed, for every formula and every set of variables , we can de ne a postinterpolant as the conjunction of a maximal set of pairwise non-equivalent in L formulas 0 such that Var 0 and ! 0 2 L (which is nite in view of the local tabularity of L). It follows, for instance, that S5 has uniform interpolation. In general, however, interpolation does not imply uniform interpolation: [Ghilardi and Zawadowski 1995] showed that S4 does not enjoy the latter, witness the following formula without a post-interpolant for frg in S4
p ^ (p ! q) ^ (q ! p) ^ (p ! r) ^ (q ! :r): Only a few positive results on the uniform interpolation of modal logics are known: Shavrukov [1993] proved it for GL, Ghilardi [1995] for K, and Visser [1996] for Grz. A property closely related to interpolation is so called Hallden completeness. A logic L is said to be Hallden complete if ' _ 2 L and
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Var' \ Var = ; imply ' 2 L or 2 L. Since every variable free formula is equivalent in D either to > or to ?, L 2 ExtD is Hallden complete whenever it has interpolation. K, K4, GL are examples of Hallden incomplete logics with interpolation: each of them contains > _ :> but not > and :>. On the other hand, S4:3 is a Hallden complete logic (see [van Benthem and Humberstone 1983]) without interpolation (see [Maksimova 1982a]). Actually, there is a continuum of Hallden complete logics in NExtS4 (see [Chagrov and Zakharyaschev 1993]). Hallden completeness has an interesting lattice-theoretic characterization. THEOREM 104 (Lemmon 1966c). A logic L 2 ExtK is Hallden complete T i it is -irreducible in ExtL. Since the lattice ExtS5 is linearly ordered by inclusion, all logics above S5 are Hallden complete. There are various semantic criteria for Hallden completeness (see e.g. [Maksimova 1995]). Here we note only the following generalization of the result of [van Benthem and Humberstone 1983]. THEOREM 105. Suppose a logic L 2 ExtK is characterized by a class C of descriptive rooted frames with distinguished roots. Then L is Hallden complete i , for all frames hF1 ; d1 i and hF2 ; d2 i in C , there is a frame hF; di for L reducible13 to both hF1 ; d1 i and hF2 ; d2 i. For more results and references on Hallden completeness consult [Chagrov and Zakharyaschev 1991]. 2 POLYMODAL LOGICS So far we have con ned ourselves to considering modal logics with only one necessity operator. From a theoretical point of view this restriction is not such a great loss as it may seem at rst sight. In fact, really important concepts of modal logic do not depend on the number of boxes and can be introduced and investigated on the basis of just one. We shall give a precise meaning to this claim in Section 2.3 below where it is shown that polymodal logic is reduced in a natural way to unimodal logic. However, there are at least two reasons for a detailed discussion of polymodal logic in this chapter. First, a number of interesting phenomena are easily missed in unimodal logic and actually appear in a representative form only in the polymodal case. For example, with the exception of NExtK4.3 and QCSF all known general decidability results in unimodal logic have been obtained by proving the nite model property. In fact, nearly all natural classes of logics in NExtK turned out to be describable by their nite frames. The situation 13
By reductions that map d to di .
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
drastically changes with the addition of just one more box. Even in the case of linear tense logics or bimodal provability logics one has to start with a thorough investigation of their in nite frames: FMP becomes a rather rare guest. While the result on NExtK4.3 indicated the need for general methods of establishing decidability without FMP, this need becomes of vital importance only in the context of polymodal logic. The second reason is that various applications of modal logic require polymodal languages. For example, in tense logic we have two necessitylike operators 1 and 2. One of them, say the former, is interpreted as \it will always be true" and the other as \it was always true". Kripke frames for tense logics are structures hW; R1 ; R2 i with two binary relations R1 and R2 such that R2 coincides with the converse R1 1 of R1 (which re ects the fact that a moment x is earlier than y i y is later than x). The characteristic axioms connecting the two tense operators are
p ! 1 2 p and p ! 2 1 p: For more information about tense systems consult Basic Tense Logic. Another example is basic temporal logic in which we have two necessitylike operators: one of them|usually called Next|is interpreted by the successor relation in ! and the other by its transitive and re exive closure. Details can be found in [Segerberg 1989]. Propositional dynamic logic PDL and its extensions, like deterministic PDL, can also be regarded as polymodal logics (see Dynamic Logic). A number of provability logics use two or more modal operators; see e.g. Boolos [1993]. In GLB, for instance, we have one operator 1 understood as provability in PA and another operator 2 interpreted as !-provability in PA. The unimodal fragments of GLB coincide with GL. The axioms connecting 1 and 2 are
1 p ! 2 p and 1 p ! 2 1 p: In epistemic logics we need an operator i for each agent i; i ' is interpreted as \agent i believes (or knows) '". One possible way to axiomatize the logic of knowledge with m agents is to take the axioms of S5 for each agent without any principles connecting N Nmdi erent i and j . We denote the resultant logic by m i=1 S5. Often i=1 S5 is extended by the common knowledge operator C with the intended meaning C' = E' ^ E2 ' ^ ^ En ' ^ : : : ;
V where E' = m i=1 i '
(see e.g. [Halpern and Moses 1992] and [Meyer and van der Hoek 1995]). The reader will nd more items for this list in other chapters of the Handbook.
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From the semantical point of view, many standard polymodal logics can be obtained by applying Boolean or various natural closure operators to the accessibility relations of Kripke frames. For instance, in frames hW; R1 ; : : : ; Rn i for epistemic logic the common knowledge operator is interpreted by the transitive closure of R1 [ [ Rn . Tense frames result from usual hW; Ri by adding the converse of R. Humberstone [1983] and Goranko [1990a] study the bimodal logic of inaccessible worlds determined by frames of the form W; R; W 2 R . This list of examples can be continued; for a general approach and related topics consult [Goranko 1990b; Gargov et al. 1987; Gargov and Passy 1990]. Let us see now how polymodal logics in general t into the theory developed so far. We begin by demonstrating how the concepts introduced in the unimodal case transfer to polymodal logic and showing that a few general results|like Sahlqvist's and Blok's Theorems|have natural analogues in polymodal logic. We hope to convince the reader that up to this point no new diÆculties arise when one switches from the unimodal language to the polymodal one. After that, in Section 2.2, we start considering subtler features of polymodal logics.
2.1 From unimodal to polymodal Let LI be the propositional language with a nite number of necessity operators i , i 2 I . A normal polymodal logic in LI is a set of LI -formulas containing all classical tautologies, the axioms i (p ! q) ! (i p ! i q) for all i 2 I , and closed under substitution, modus ponens and the rule of necessitation '=i ' for every i 2 I . If the language is clear from the context, we call these logics just (normal) modal logics and denote by NExtL the family of all normal extensions of L (in the language LI ). The smallest normal modal logic with n necessity operators is denoted by Kn (K = K1 , of course). Given a logic L0 in LI and a set of LI -formulas , we again denote by L0 the smallest normal logic (in LI ) containing L0 [ . A number of other notions and results also transfer in a rather straightforward way, e.g. Theorems 4 and 6, Proposition 5 and all concepts involved in their formulations. More care has to be taken to generalize Theorems 1, 2 and 3. Denote by M I the set of non-empty strings (words) over fi : i 2 I g which do not contain any i twice and put ^
^
I ' = fM ' : M 2 M I g; I m' = fnI ' : n mg: In the language LI the operator I serves as a sort of surrogate for in K. For example, the following polymodal version of Theorem 1 holds. THEOREM 106 (Deduction). For every modal logic L in LI , every set of
160
LI -formulas
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV , and all LI -formulas ' and , ; ` ' i 9m 0 ` m L
L
I
! ':
Theorems 2 and 3 can be reformulated analogously by replacing with
I (a logic L in LI is n-transitive if it contains I n p ! nI +1 p).
Basic semantic concepts are lifted to the polymodal case in a straightforward manner. The algebraic counterpart of L 2 NExtKn is the variety of Boolean algebras with n unary operators validating L. A structure F = hW; hRi : i 2 I i; P i is called a (general polymodal) frame whenever every hW; Ri ; P i, for i 2 I , is a unimodal frame. We then put
i X = fx 2 W : 8y (xRi y ! y 2 X )g: Di erentiated, re ned and descriptive frames and the truth-preserving operations can also be de ned in the same component-wise way. For instance, a frame F = hW; hRi : i 2 I i; P i is di erentiated if all the unimodal frames hW; Ri ; P i, for i 2 I , are di erentiated. F = hW; hRi : i 2 I i; P i is a (generated) subframe of G = hV; hSi : i 2 I i; Qi if all hW; Ri ; P i are (generated) subframes of hV; Si ; Qi, and f is a reduction of F to G if f is a reduction of hW; Ri ; P i to hV; Si ; Qi, for every i 2 I . There are some exceptions to thisSrule. A point r is called a root of F if it is a root of the unimodal frame hW; i2I Ri i. This does not mean that r is a root of all unimodal reducts of F. Another important exception: as before, a polymodal frame is {-generated if the algebra F+ is {-generated; however, this does not mean that the unimodal reducts of F are {-generated.
Splittings and the degree of Kripke incompleteness The semantic criterion of splittings by nite frames given in Theorem 15 transfers to polymodal logics by replacing with I . Again, all nite rooted frames split NExtL0 , if L0 is an n-transitive logic in LI . Notice, however, that n-transitivity is a rather strong condition in the polymodal case. For example, it is easily checked that the fusion S5 S5 as well as the minimal tense logic K4:t containing K4 are not n-transitive, for any n < ! (see Sections 2.2 and 2.4 for precise de nitions). In fact, only Æ splits the lattice NExt(S5 S5) and only splits NExtK4:t (see [Wolter 1993] and [Kracht 1992], respectively). S Call a frame hW; hRi : i 2 I ii cycle free if the unimodal frame hW; i2I Ri i is cycle free. Kracht [1990] showed that precisely the nite cycle free frames split NExtKn . It is not diÆcult now to extend Blok's result on the degree of Kripke incompleteness to the polymodal case. Note, however, that the degree of incompleteness of For in NExtKn is 2@0 whenever n 2. So, we do not have a polymodal analog of Makinson's Theorem. (An example of an incomplete
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maximal consistent logic in NExtK2 is the logic determined by the tense frame C(0; Æ) introduced in Section 2.5). THEOREM 107. Let n > 1. If L is a union-splitting of NExtKn , then L is strictly Kripke complete. Otherwise L has degree of Kripke incompleteness 2@0 in NExtKn .
Sahlqvist's Theorem and persistence The proof of the following polymodal version of Sahlqvist's Theorem is a straightforward extension of the proof in the unimodal case. Say that ' is a Sahlqvist formula (in LI ) if the result of replacing all i and i , i 2 I , in ' with and , respectively, is a unimodal Sahlqvist formula. THEOREM 108. Suppose that ' is equivalent in NExtKn to a Sahlqvist formula. Then Kn ' is D-persistent, and one can e ectively construct a rst order formula (x) in R1 ; : : : ; Rn and = such that, for every descriptive or Kripke frame F and every point a in F, (F; a) j= ' i F j= (x)[a]. Bellissima's result on the DF -persistence N of all logics in NExtAltn has a polymodal analog as well. Denote by i2I Altn the smallest polymodal logic in LI containingNAltn in all its unimodal fragments. It is easy to see that every L 2 NExt i2I Altn is DF -persistent and so Kripke complete. However, in contrast to the lattice NExtAlt1 |which is countable and all logics in which have FMP (see [Segerberg 1986] and [Bellissima 1988])| the lattice NExt(Alt1 Alt1 ) is rather complex: as was shown by Grefe [1994], it contains logics without FMP (even without nite frames at all) and uncountably many maximal consistent logics. Some FMP results Fine's Theorem on uniform logics can be extended to a suitable class of polymodal logics in LI , namely those logics that contain i >, for all i 2 I , and are axiomatizable by formulas ' in which all maximal sequences of nested modal operators coincide with respect to the distribution of the indices i of i and i , i 2 I . Now consider a result of Lewis [1974] which we have not proved in its unimodal formulation. Call a normal polymodal logic non-iterative if it is axiomatizable by formulas without nested modalities. Examples of noniterative logics are T = K p ! p, Altm Altn and K2 2 p ! 1p. THEOREM 109 (Lewis 1974). All non-iterative normal logics have FMP. Proof. Suppose the axioms of L = Kn have no nested modal operators and ' 62 L. By a '-description we mean any set of subformulas of ' together with the negations of the remaining formulas in Sub'. For each L-consistent '-description select a maximal L-consistent set containing . Denote by W the ( nite) set of the selected and de ne
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
F = hW; hRi : i 2 I ii and M = hF; Vi by taking Ri i i
^
2
and V(p) = f 2 W : p 2 g. It is easily proved that (M; ) j= i 2 , for all subformulas of ' and 2 W . Hence F 6j= '. It is also easy to see that for all truth-functional compounds of subformulas in ', (14) (M; ) j= i i i 2 : Consider now a model M0 = hF; V0 i and 2 . For each variable p put p
=
_ n^
: 2 V(p)
o
and denote by 0 the result of substituting p for p, for each p in . Then M0 j= i M j= 0 . In view of (14), we have M j= 0 because 0 has no nested modalities. Therefore, F j= and so F j= L.
Tabular Logics Needless to say that all polymodal tabular logics are nitely axiomatizable and have only nitely many extensions. (The proof is the same as in the unimodal case.) A more interesting observation concerns the complexity of polymodal logics whose unimodal fragments are tabular or pretabular. In fact, it is not diÆcult to construct two tabular unimodal logics L1 and L2 such that their fusion L1 L2 has uncountably many normal extensions (see e.g. [Grefe 1994]). However, those logics are DF persistent and so Kripke complete. Wolter [1994b] showed that the lattice
Æ
NExtT can be embedded into the lattice NExt(Log Æ6 S5) in such a way that properties like FMP, decidability and Kripke completeness are re ected under this embedding. It follows that almost all \negative" phenomena of modal logic are exhibited by bimodal logics one unimodal fragment of which is tabular and the other pretabular.
2.2 Fusions The simplest way of constructing polymodal logics from unimodal ones is to form the fusions (alias independent joins) of them. Namely, given two unimodal logics L1 and L2 in languages with the same set of variables and distinct modal operators 1 and 2 , respectively, the fusion L1 L2 of L1 and L2 is the smallest bimodal logic to contain L1 [ L2. If 1 and 2 axiomatize L1 and L2 , then L1 L2 is axiomatized by 1 [ 2 , i.e., L1 L2 = K2 1 2 . So the fusions are precisely those bimodal logics that are axiomatizable by sets of formulas each of which contains only one of 1 , 2 . From the model-theoretic point of view this means that a frame hW; R1 ; R2 ; P i validates L1 L2 i hW; Ri ; P i j= Li for i = 1; 2.
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PROPOSITION 110 (Thomason 1980). If logics L1 and L2 are consistent, then L1 L2 is a conservative extension of both L1 and L2 .
Proof. Suppose for de niteness that ' 62 L1 , for some formula ' in the language of L1 , and consider the Tarski{Lindenbaum algebras
AL (!) = A; ^A ; :A ; 1 and AL (!) = B; ^B ; :B ; 2 : 1
2
The Boolean reducts of them are countably in nite atomless Boolean algebras which are known to be isomorphic (see e.g. [Koppelberg 1988]). So we may assume A = B , ^A = ^B , :A = :B . Since the algebra AL1 (!)
that A refutes ', A; ^ ; :A ; 1; 2 is then an algebra for L1 L2 refuting '. Having constructed the fusion of logics, it is natural to ask which of their properties it inherits. For example, the rst order theory of a single equivalence relation has the nite model property and is decidable, but the theory of two equivalence relations is undecidable and so does not have the nite model property (see [Janiczak 1953]). So neither decidability nor the nite model property is preserved under joins of rst order theories. On the other hand, as was shown by Pigozzi [1974], decidability is preserved under fusions of equational theories in languages with mutually disjoint sets of operation symbols. For modal logics we have: THEOREM 111. Suppose L1 and L2 are normal unimodal consistent logics and P is one of the following properties: FMP, (strong) Kripke completeness, decidability, Hallden completeness, interpolation, uniform interpolation. Then L = L1 L2 has P i both L1 and L2 have P .
Proof. We outline proofs of some claims in this theorem; the reader can consult [Fine and Schurz 1996], [Kracht and Wolter 1991], and [Wolter 1997b] for more details. The implication ()) presents no diÆculties. So let us concentrate on ((). With each formula ' of the form i we associate a new variable q' which will be called the surrogate of '. For a formula ' containing no surrogate variables, denote by '1 the formula that results from ' by replacing all occurrences of formulas 2 , which are not within the scope of another 2, with their surrogate variables q2 . So '1 is a unimodal formula containing only 1 . Denote by 1(') the set of variables in ' together with all subformulas of 2 2 Sub'. The formula '2 and the set 2(') are de ned symmetrically. Suppose now that both L1 and L2 are Kripke complete and ' 62 L. To prove the completeness of L we construct a Kripke frame for L refuting '. Since we know only how to build refutation frames for the unimodal fragments of L, the frame is constructed by steps alternating between 1 and 2 . First, since L1 is complete, there is a unimodal model M based
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
on a Kripke frame for L1 and refuting '1 at its root r. Our aim now is to ensure that the formulas of the form 2 have the same truth-values as their surrogates q2 . To do this, with each point x in M we can associate the formula
'x =
^
^
f 2 1(') : (M; x) j= 1 g ^ f:
:
2 1('); (M; x) 6j= 1 g;
construct a model Mx based on a frame for L2 and satisfying '2x at its root y, and then hook Mx to M by identifying x and y. After that we can switch to 1 and in the same manner ensure that formulas 1 have the same truth-values as q1 at all points in every Mx . And so forth. However, to realize this quite obvious scheme we must be sure that 'x is really satis able in a frame for L2 , which may impose some restrictions on the models we choose. First, one can show that in the construction above it is enough to deal with points x accessible from r by at most m = md(') steps. Let X be the set of all such points. Now, a suÆcient and necessary condition for 'x to be L- (and so L2 -) consistent can be formulated as follows. Call a 1 (')-description the conjunction of formulas in any maximal L-consistent subset of 1 (') [ f: : 2 1(')g. It should be clear that 'x is L-consistent i it is a 1(')-description. Denote by 1 (') the set of all 1 (')-descriptions. It follows that all 'x , for x 2 X , are W L-consistent i (M; r) j= 1 m ( 1 ('))W1 . In other words, we should start m 1 with a model M satisfying '1 ^ 1 ( 1 (')) at its root r.WOf course, m 2 the subsequent models Mx, for x 2 X , must satisfy '2x ^ 2 ( 2 ('x )) , where 2 ('x ) is the set of all 2 ('x )-descriptions, etc. In this way we can prove that Kripke completeness is preserved under fusions. The preservation of strong completeness and FMP can be established in a similar manner. The following lemma plays the key role in the proof of the preservation of the four remaining properties. LEMMA 112. The following conditions are equivalent for every ': (i) ' 2 L1 L2 ; W (ii) m ( 1 ('))1 ! '1 2 L1 , where m = md('); 1
(iii)
2 m (W 2 ('))2 ! '2 2 L2 .
For Kripke complete L1 and L2 this lemma was rst proved by Fine and Schurz [1996] and Kracht and Wolter [1991]; actually, it is an immediate consequence of the consideration above. The proof for the arbitrary case is also based upon a similar construction combined with the algebraic proof of Proposition 110; for details see [Wolter 1997b]. Now we show how one can use this lemma to prove the preservation of the remaining properties. De ne a1 (') to be the length of the longest
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sequence 2 ; 1 ; 2 ; : : : of boxes starting with 2 such that a subformula of the form 2 (: : : 1(: : : 2 (: : : : : : ))) occurs in '. The function a2 (') is de ned analogously by exchanging 1 and 2 , and a(') = a1 (') + a2 ('). It is easy to see that _
a(') > a(
_
1 (')) or a(') > a(
2 (')):
The preservation of decidability, Hallden completeness, interpolation, and uniform interpolation can be proved by induction on a(') with the help of Lemma 112. We illustrate the method only for Hallden completeness. Notice rst that, modulo the Boolean equivalence, we have _
1 (' _ ) =
_
1 (') ^
_
1 ( ) ^
^
('; );
where ('; ) = f 1 ! : 2 : 1 2 1 ('); 2 2 1 ( ); 1 ! : 2 2 Lg: Suppose both L1 and L2 are Hallden complete. By induction on n = a('_ ) we prove that ' _ 2 L implies ' 2 L or 2 L whenever ' and have no common variables. The basis of induction is trivial. So suppose W a(' _ ) = n > 0 and ' _ 2 L. We may also assume that a(' _ ) > a( 1 (' _ )): By the induction hypothesis, it follows W W that ( W'; ) = ;. Hence, up to the Boolean equivalence, 1 (' _ ) = 1 (') ^ 1 ( ) and, by Lemma 112, _ _ m( 1 ('))1 ^ m( 1 ( ))1 ! (' _ )1 2 L1 ; 1
1
for m = md(' _ ). Then m _ m _ 1 1 1 ( 1 ( 1 (')) ! ' ) _ (1 ( 1 ( )) !
1)
2 L1
and, by the Hallden completeness of L1, one of the disjuncts in this formula belongs to L1 . By Lemma 112, this means that ' 2 L or 2 L. REMARK. This theorem can be generalized to fusions of polymodal logics with polyadic modalities. Note that in languages with nitely many variables both GL:3 and K are strongly complete but GL:3 K is not strongly complete even in the language with one variable (see [Kracht and Wolter 1991]). It is natural now to ask whether there exist interesting axioms ' containing both 1 and 2 and such that (L1 L2 ) ' inherits basic properties of L1; L2 2 NExtK. Let us start with the observation that even such a simple axiom as 1 p $ 2 p destroys almost all \good" properties because (i) we can identify the logic (L1 L2) 1 p $ 2 p with the sum of the translation of L1 and L2 into a common unimodal language and (ii) such properties as
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FMP, decidability, and Kripke completeness are not preserved under sums of unimodal logics (see Example 64 and [Chagrov and Zakharyaschev 1997]). Even for the simpler formula 2p ! 1 p no general results are available. To demonstrate this we consider the following way of constructing a bimodal logic Lu for a given L 2 NExtK:
Lu = (L S5) 2 p ! 1 p: The modal operator 2 in Lu is called the universal modality. Its meaning is explained by the following lemma: LEMMA 113 (Goranko and Passy 1992). For every normal unimodal logic L and all unimodal formulas ' and , ' `L i `Lu 2 ' ! :
Proof. Follows immediately from Theorem 19 (ii), since hW; R; P i j= L i hW; R; W W; P i j= Lu ; for every frame hW; R; P i and every unimodal logic L. The universal modality is used to express those properties of frames F = hW; R; W W i that cannot be expressed in the unimodal language. For example, F validates 2 (p ! 1 p) ! :p i it contains no in nite R-chains. Recall that there is no corresponding unimodal axiom, since K is determined by the class of frames without in nite R-chains. We refer the reader to [Goranko and Passy 1992] for more information on this matter. THEOREM 114 (Goranko and Passy 1992). For any L 2 NExtK, (i) L is globally Kripke complete i Lu is Kripke complete; (ii) L has global FMP i Lu has FMP. Proof. We prove only (i). Suppose that Lu is Kripke complete and ' 6`L . Then by Lemma 113, 2 ' ! 62 Lu and so 2 ' ! is refuted in a Kripke frame F = hW; R1 ; R2 i for Lu . We may assume that R2 = W W . But then ' `L is refuted in hW; R1 i. Conversely, suppose that L is globally Kripke complete and ' 62 Lu , for a (possibly bimodal) formula '. Using the properties of S5 it is readily checked that ' is (e ectively) equivalent in Ku to a formula '0 which is a conjunction of formulas of the form = 0 _ 2 1 _ 2 2 _ 2 3 _ _ 2 n such that 0 ; : : : ; n are unimodal formulas in the language with 1 . Let be a conjunct of '0 such that 62 Lu . Then : 1 6`L i , for every i 2 f0; 2; 3; : : : ; ng. Since L is globally complete, we have Kripke frames hWi ; Ri i for L refuting : 1 `L i , for i 2 f0; 2; : : : ; ng. Denote by hW; Ri the disjoint union of those frames. Then hW; R; W W i is a Kripke frame for Lu refuting '.
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We have seen in Section 1.5 that there are Kripke complete logics (logics with FMP) which do not enjoy the corresponding global property. In view of Theorem 114, we conclude that neither FMP nor Kripke completeness is preserved under the map L 7! Lu . Another interesting way of adding to fusions new axioms mixing the necessity operators is to use the so called inductive (or Segerberg's) axioms. First, we extend the language LI with m necessity operators by introducing the operators E and C and then let
ind = fEp $
^
i2I
i p; Cp ! ECp; C(p ! Ep) ! (p ! Cp)g:
Now, given L 2 NExtKm , we put
LECm = (L KE S4C ) ind; where KE and S4C are just K and S4 in the languages with E and C, respectively. The following proposition explains the meaning of the inductive axioms. PROPOSITION 115. A frame hW; R1 ; : : : ; Rm ; RE ; RC i validates LECm i hW; R1 ; : : : ; Rm i j= L, RE = R1 [ [ Rm and RC is the transitive re exive closure of RE . EXAMPLE 116. The logic (Alt1 D)EC1 is determined by the frame h!; S; i in which S is the successor relation in !. (Here we omit writing RE because RE = S .) For details consult [Segerberg 1989].14 No general results are known about the preservation properties of the map L 7! LECm . In fact, it is easy to extend the counter-examples for the map L 7! Lu to the present case (see [Hemaspaandra 1996]). However, at least in some cases|especially those that are of importance for epistemic logic|the logic LECm enjoys a number of desirable properties. THEOREM 117 (Halpern and Moses N 1992). For every m 1, the logics N Nm m S5)EC have FMP. ( m K ) EC , ( S4 ) EC and ( m m m i=1 i=1 i=1 N
Proof. We consider only L = ( m i=1 S5)ECm . The proof is by ltration and so the main diÆculty is to nd a suitable \ lter". Suppose that ' 62 L and let M = hhW; R1 ; : : : ; Rm ; RE ; RC i ; Ui be the canonical model for L. Denote by : the closure of a set of formulas under negations and de ne a lter = :1 [ :2 [ :3 , where 1 = Sub', 2 = fi : E 2 :1 g and 3 = fEC ; i C : C 2 :1 g. Certainly, is nite and closed under subformulas. Now, we lter M through , i.e., put W = f[x] : x 2 W g, 14 Krister Segerberg kindly informed us that this result was independently obtained by D. Scott, H. Kamp, K. Fine and himself.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
R 6R F ? 1
1
2
I@AKA@ A 6 A-?Fs
Figure 11. where [x] consists of all points that validate the same formulas in as x, and [x]Ri [y] i 8i 2 ((M; x) j= i ! (M; y) j= i ); ; RE = R1 [ [ Rm
and RC is the transitive and re exive closure of RE . A rather tedious ; R ; R i refutes ' under the inductive proof shows that hW ; R1; : : : ; Rm E C valuation U (p) = f[x] : x j= pg, p a variable in '. For details we refer the reader to [Halpern and Moses 1992] and [Meyer and van der Hoek 1995].
It would be of interest to look for big classes of logics L for which LECm inherits basic properties of L.
2.3 Simulation In the preceding section we saw how results concerning logics in NExtK can be extended to a certain class of polymodal logics. More generally, we may ask whether|at least theoretically|polymodal logics are reducible to unimodal ones. The rst to attack this problem was Thomason [1974b, 1975c] who proved that each polymodal logic L can be embedded into a unimodal logic Ls in such a way that L inherits almost all interesting properties of Ls . Using this result one can construct unimodal logics with various \negative" properties by presenting rst polymodal logics with the corresponding properties, which is often much easier. It was in this way that Thomason [1975c] constructed Kripke incomplete and undecidable unimodal calculi. Kracht [1996] strengthened Thomason's result by showing that his embedding not only re ects but also (i) preserves almost all important properties and (ii) induces an isomorphism from the lattice NExtK2 onto the interval [Sim; K ?], for some normal unimodal logic Sim. Thus indeed, in many respects polymodal logics turn out to be reducible to unimodal ones. Below we outline Thomason's construction following [Kracht 1999] and [Kracht and Wolter 1999]. To de ne the unimodal \simulation" Ls of a bimodal logic L, let us rst transform each bimodal frame into a unimodal one.
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So suppose F = hW; R1 ; R2 ; P i is a bimodal frame. Construct a unimodal frame Fs = hW s ; Rs ; P s i|the simulation of F|by taking
W s = W f1; 2g [ f1g; Rs = fhhx; 1i ; hx; 2ii : x 2 W g [ fhhx; 2i ; hx; 1ii : x 2 W g [ fhhx; 1i ; 1i : x 2 W g [ fhhx; 1i ; hy; 1ii : x; y 2 W; xR1 yg [ fhhx; 2i ; hy; 2ii : x; y 2 W; xR2 yg; s P = f(X f2g) [ (Y f1g) [ Z : X; Y 2 P; Z f1gg: This construction is illustrated by Fig. 11. One can easily prove that Fs is a Kripke (di erentiated, re ned, descriptive) frame whenever F is so. Notice also that if W = ; then Fs = . Now, given a bimodal logic L, de ne the simulation Ls of L to be the unimodal logic LogfFs : F j= Lg:
To formulate the translation which embeds L into Ls we require the following formulas and notations:
= ? ' = ( ! ') = ? ' = ( ! ') = : ^ : ' = ( ! '):
, and are de ned dually. Observe that the formula is true in Fs only at 1, is true precisely at the points in the set fhx; 1i : x 2 W g, and is true at the points fhx; 2i : x 2 W g and only at them. Put ps = p; (:')s = ^ :'s ; s (' ^ ) = 's ^ s ; (1 ')s = 's ; (2 ')s = 's : By an easy induction on the construction of ' one can prove LEMMA 118. Let M = hF; Vi be a bimodal model, X = fx : x j= g and let Ms = hFs ; Vs i be a model such that Vs (p) \ X = V(p) f1g, for all variables p. Then for every bimodal formula ', (M; x) j= ' i (Ms ; hx; 1i) j= 's ; M j= ' i Ms j= ! 's ; F j= ' i Fs j= ! 's :
Using this lemma, both consequence relations `L and `L can be reduced to the corresponding consequence relations for Ls .
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
PROPOSITION 119. Let L be a bimodal logic, a set of bimodal formulas and ' a bimodal formula. Then `L ' i ! s `Ls ! 's ; `L ' i ! s `Ls ! 's ; where ! s = f ! Æ : Æ 2 s g. To axiomatize Ls , given an axiomatization of L, we require the following formulas: (a) ! ( p $ p); ^ p ! p; (b) ! ( p $ p); (c) ! ( p $ p); (d) ^ p ! p; ^ p ! p; (e) ^ p ! p:
Let Sim = K f(a); : : : ; (e)g. Obviously, Fs is a frame for Sim whenever F is a bimodal frame. Consider now a di erentiated frame F = hW; R; P i for Sim which contains only one point where is true. (Actually, every rooted di erentiated frame for Sim satis es this condition.) Construct a bimodal frame Fs = hV; R1 ; R2 ; Qi, called the unsimulation of F, in the following way. Put V = fx 2 W : x j= g, V = fx 2 W : x j= g and U = fx 2 W : x j= g. Since _ _ 2 K, we have W = V [ V [ U . It is not hard to verify using (b) and (c) (and the di erentiatedness of F) that for every x 2 V there exists a unique x 2 V such that xRx , and for every y 2 V there exists yÆ 2 V such that yRyÆ. By (d), x = xÆ . Finally, we put R1 = R \ V 2 , R2 = fhx; yi 2 V 2 : x Ry g and Q = fX \ V : X 2 P g. It is easily proved that Fs is a bimodal frame. The name unsimulation is justi ed by the following lemma. LEMMA 120. For every di erentiated bimodal frame F, (Fs )s = F. Now we have: THEOREM 121. For every bimodal logic L = K2 ,
Ls = Sim ! s : Proof. Clearly, Sim ! s Ls . Assume that the converse inclusion does not hold. Then there exists a rooted di erentiated F such that F 6j= Ls but F j= Sim ! s . By Lemma 120, (Fs )s 6j= Ls . By the de nition of Ls , we then conclude that Fs 6j= L. And by Proposition 119, we have (Fs )s 6j= ! s , from which F 6j= ! s . Given L 2 [Sim; K ?], the logic Ls = f' : ! 's 2 Lg is called the unsimulation of L.
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LEMMA 122. If L is determined by a class C of frames in which is true only at one point then Ls = LogfFs : F 2 Cg. We are in a position now to formulate the main result of this section. THEOREM 123 (Kracht 1999). The map L 7! Ls is an isomorphism from the lattice NExtK2 onto the interval [Sim; K1 ?]. The inverse map is L 7! Ls . Both these maps preserve tabularity, (global) FMP, (global) Kripke completeness, decidability, interpolation, strong completeness, Rand D-persistence, elementarity.
Proof. To prove the rst claim it suÆces to show that (Ls )s = L for every L 2 [Sim; K ?]. That L (Ls )s is clear. Consider the set C of all di erentiated frames Fs such that F j= L and is true only at one point in F. By Lemma 122, C characterizes Ls . It is not diÆcult to show now that the class fF+s : F 2 Cg is closed under subalgebras, homomorphic images and direct products; so it is a variety. Consequently, C is (up to isomorphic copies) the class of all di erentiated frames for Ls . Take a di erentiated frame F for (Ls )s . Then Fs j= Ls . So there exists Gs 2 C which is isomorphic to Fs . Hence (Fs )s = (Gs )s and F j= L, since s G j= L. It follows that L is determined by fFs : F 2 Cg whenever L is determined by C . The preservation of tabularity, (global) FMP, (global) Kripke completeness, and strong completeness under both maps is proved with the help of Lemma 122 and the observation above. It is also clear that L is decidable whenever Ls is decidable. For the remaining (rather technical) part of the proof the reader is referred to [Kracht 1999] and [Kracht and Wolter 1999].
Besides its theoretical signi cance, this theorem can be used to transfer rather subtle counter-examples from polymodal logic to unimodal logic. For instance, Kracht [1996] constructs a polymodal logic which has FMP and is globally Kripke incomplete. By Theorem 123, we obtain a unimodal logic with the same properties.
2.4 Minimal tense extensions Now let us turn to tense logics which may be regarded as normal bimodal logics containing the axioms p ! 1 2 p and p ! 21 p. Usually studies in Tense Logic concern some special systems representing various models of time, like cyclic time, discrete or dense linear time, branching time, relativistic time, etc. Such systems are discussed in Basic Tense Logic, volume 6 of this Handbook (see also [Gabbay et al. 1994], [Goldblatt 1987] and [van Benthem 1991]). However, as before our concern is general methods which make it possible to obtain results not only for this or that particular system but for wide classes of logics. This direction of studies in Tense Logic is quite
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new and actually not so many general results are available. In this and the next section we consider two natural families of tense logics|the minimal tense extensions of unimodal logics and tense logics of linear frames. Our aim is to nd out to what extent the theory developed for unimodal logics in NExtK and especially NExtK4 can be \lifted" to these families. The smallest tense logic K:t is determined by the class of bimodal Kripke frames hW; R; R 1 i in which R is the accessibility relation for 1 and R 1 for 2 . Frames of this type are known as tense Kripke frames; general frames of the form hW; R; R 1 ; P i will be called just tense frames. Notice that not all unimodal general frames hW; R; P i can be converted into tense frames hW; R; R 1 ; P i because P is not necessarily closed under the operation
2 X = fx 2 W : 9y 2 X xR 1 yg: For instance, in the frame F of Example 7 we have 2 f! + 1g = f!g 62 P . Each normal unimodal logic L = K in the language with 1 gives rise to its minimal tense extension L:t = K:t . From the semantical point of view L:t is the logic determined by the class of tense frames hW; R; R 1; P i such that hW; R; P i j= L. The formation of the minimal tense extensions
is the simplest way of constructing tense logics from unimodal ones. Of \natural" tense logics, minimal tense extensions are, for instance, the logics of (converse) transitive trees, (converse) well-founded frames, (converse) transitive directed frames, etc. The main aim of this section is to describe conditions under which various properties of L are inherited by L:t. Notice rst that unlike fusions, L:t is not in general a conservative extension of L, witness L = LogF where F is again the frame constructed in Example 7: one can easily check that K4:t L:t. However, if L is Kripke complete then L:t is a conservative extension of L and so L0 :t = L:t implies L0 L. This example may appear to be accidental (as the rst examples of Kripke incomplete logics in NExtK). However, we can repeat (with a slight modi cation) Blok's construction of Theorem 35 and prove the following THEOREM 124. If L is a union-splitting of NExtK or L = For, then L0 :t = L:t implies L0 = L. Otherwise there is a continuum of logics in NExtK having the same minimal tense extension as L. It is not known whether there exists L 2 NExtK4 such that L:t is not a conservative extension of L. Theorem 124 leaves us little hope to obtain general positive results for the whole family of minimal tense extensions. As in the case of unimodal logics we can try our luck by considering logics with transitive frames. So in the rest of this section it is assumed that the unimodal and tense logics we deal with contain K4 and K4:t, respectively, and that frames are transitive. But even in this case we do not have general preservation results: Wolter [1996b] constructed a logic L 2 NExtK4 having FMP and such that L:t is not Kripke complete. However, the situation turns out to be not so hopeless
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if we restrict attention to the well-behaved classes of logics in NExtK4, namely logics of nite width, nite depth and co nal subframe logics. First, we have the following results of [Wolter 1997a]. THEOREM 125. If L 2 NExtK4 is a logic of nite depth then L:t has FMP. If L 2 NExtK4 is a logic of nite width then L:t is Kripke complete. It is to be noted that tense logics of nite depth are much more complex than their unimodal counterparts. For example, there exists an undecidable nitely axiomatizable logic containing K4:t1 1 ? (for details see [Kracht and Wolter 1999]). The minimal tense extensions of co nal subframe logics were investigated in [Wolter 1995, 1997a]. THEOREM 126. If L 2 NExtK4 is a co nal subframe logic then (i) L:t is Kripke complete; (ii) L:t has FMP i L is canonical; (iii) L:t is decidable whenever L is nitely axiomatizable. Before outlining the idea of the proof we note some immediate consequences for a few standard tense logics. EXAMPLE 127. (i) The logic of the converse well-founded tense frames is GL:t; it does not have FMP but is decidable. (ii) The logic of the converse transitive trees is K4:3:t; it has FMP and is decidable. (iii) The logic of the converse well-founded directed tense frames is GL:t K4:2:t; it does not have FMP and is decidable.
Proof. The proof of the negative part, i.e., that L:t does not have FMP if L is not canonical, is rather technical; it is based on the characterization of the canonical co nal subframe logics of [Zakharyaschev 1996]. The reader can get some intuition from the following example: neither Grz:t nor GL:t has FMP. Indeed, the Grzegorczyk axiom
2 (2 (p ! 2 p) ! p) ! p is refuted in h!; ; i and so does not belong to Grz:t; however, it is valid in all nite partial orders. The argument for GL:t is similar: take the Lob axiom in 2 and the frame h!; >;
h!; >;
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determined by a set of frames of the form Frp such that F is of nite depth. Indeed, suppose ' 62 L:t and consider a countermodel M = hF; Vi for ' based on a descriptive nitely generated tense frame F = hW; R; R 1; P i for L:t. Say that a point x 2 W is non-eliminable (relative to ') if there are a subformula of ' and S 2 fR; R 1g such that x 2 maxS fy 2 W : y j= g or x 2 maxS fy 2 W : y j= : g. Denote by We the set of non-eliminable points in W and construct a new model Me on the frame Fe = hWe ; R We ; R 1 We i by taking Ve (p) = V(p) \ We for all variables p in '. Clearly, the Kripke frame Fe is of nite depth (d(Fe ) 2l('), to be more precise). Besides, using Theorem 23 one can easily show that (Me ; y) j= i (M; y) j= , for all 2 Sub' and y 2 We . (Note that Theorem 23 is applicable in this case, since hW; R; P i is descriptive whenever
W; R; R 1; P is descriptive.) Moreover, the R-reduct hWe ; R We i of Fe is a co nal subframe of the R-reduct hW; Ri of the underlying Kripke frame of F. So Fe is a frame for L:t whenever L is canonical (= D-persistent). However, this is not so if L is not canonical. EXAMPLE 128. Consider the frame F = hW; R; R 1 ; P i, where hW; Ri is the re exive point 1 followed by the chain h!; >i and P consists of all co nite sets containing 1 and their complements. Then F j= GL:t but (for an arbitrary ') Fe contains 1 and so Fe 6j= GL:t. A rather tedious proof (see [Wolter 1997a]) shows, however, that there exists a replacement function rp for Fe such that Frp e validates L:t and all points in clusters from domrp are eliminable relative to R in F. (In the example above we put rpf1g = h!; >;
Vrp (p) \ rpC = fmj + i : j < !; ai 2 V(p)g: Using the fact that domrp contains only R-eliminable points, one can show by induction that, for every 2 Sub', (Me ; y) j= i (Mrp e ; y ) j= , if C (y) does not belong to domrp, and
fn 2 rpC : (Mrp e ; n) j= g = fmj + i : j < !; (Me ; ai ) j= g; if a cluster C = fa0; : : : ; am 1 g is in domrp. Thus Frp e refutes ',
which proves that L:t is Kripke complete. To show that all canonical logics L:t do have FMP we reduce Frp e once again. De ne an equivalence relation on We by induction on the R-depth dR (x) of a point x in Fe . Suppose that dR (x) = dR (y) and is already de ned for all points of R-depth < dR (x) and put x y if the following
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conditions are satis ed: (a) x j= i y j= , for all 2 Sub' (x ' y, for short), (b) if z is an R-successor of y and C (z ) 6= C (y) then there exists an R-successor z 0 of x with C (z 0 ) 6= C (x) such that z z 0 and vice versa, (c) the cluster C (x) is degenerate i C (y) is degenerate, (d) rpC (x) = rpC (y), (e) for each z 2 C (x) there exists z 0 2 C (y) such that z ' z 0 and vice versa. Let [x] denote the equivalence class generated by x. De ne a frame G = hV; S; S 1 i by taking V = f[x] : x 2 We g, and [x]S [y] i there are x0 2 [x] and y0 2 [y] such that x0 Ry0 . Since Fe is of nite depth, V is nite. Moreover, the map x 7! [x] is a reduction of the unimodal frame hWe ; R We i to hV; S i. It follows that G is a frame for L:t whenever L is canonical. De ne a valuation in G by putting [x] j= p i x j= p, for all x 2 We and all variables p in '. Then one can show that [x] j= i x j= , for all 2 Sub'. So G 6j= ', as required, which means that L:t has FMP. To prove the decidability of a nitely axiomatizable L:t we rst show its completeness with respect to a rather simple class of frames. De ne a replacement function rf for G as follows. For each cluster C in Fe the set [C ] = f[x] : x 2 C g is a cluster in G, and moreover, every cluster in G can be presented in this way. So we put rf [C ] = rpC , for all clusters [C ] in G. Notice that by (d), rf is well-de ned. It is easily shown now that rf rf the R-reduct of Frp e is reducible to the R-reduct of G and that G refutes '. Thus we obtain LEMMA 129. For each co nal subframe logic L,
L:t = LogfGrp : Grp j= L:t; G nite, rp a replacement function for Gg: So, to establish the decidability of a nitely axiomatizable L:t it is enough now to present an algorithm which is capable of deciding, given an rp for a nite G and ', whether Grp j= '. To this end we require the notion of a cluster assignment t = ht1 ; t2 i in a tense frame G, which is any function from the set of clusters in G into the set fm; jgfm; jg such that tC = (m; m) if C is degenerate (here m and j are just two symbols; m stands for \maximal" and j for \joker"). A valuation V in G is called '-good for (G; t) if the following conditions hold: if t1 C = j then C \ maxR (V( )) = ;, for all 2 Sub';
if t2 C = j then C \ maxR (V(
)) = ;, for all 2 Sub' . EXAMPLE 130. Let F be the frame constructed in Example 128 and suppose that tf1g = (j; m). Then each valuation V in F is '-good for (G; t) no matter what ' is, because 1 is eliminable relative to R. The point 1 is not R 1-eliminable, since 1 2 maxR 1 (>). Given a formula ', a nite frame F and a replacement function rp for F, we construct a nite frame G = hV; S; S 1 i with a cluster assignment 1
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t as follows. Let k be the number of variables in '. Then G is obtained from Frp by replacing every rpC = h!; >;
Lin ' = Lin (F1 ; t1 ) (Fn ; tn ):
Proof. Let (Fi ; ti ), 1 i n, be the collection of all nite frames with type assignments such that, for each i, (a) there is a countermodel Mi = hFi ; Vi i for ' in which Vi is '-good for (Fi ; ti ), (b) the depth of Fi does not exceed 4l(') + 1, and (c) no cluster in Fi contains more than 2v(') points, where v(') is the number of variables in '. Let F refute (Gi ; ti ) under a valuation U. By the de nition of (Fi ; ti ), the model Mi refutes '. De ne a valuation U0 in F by taking, for all variables p in ', [ U0 (p) = fU(px) : x 2 Vi (p)g: S It is not hard to show by induction that U0 ( ) = fU(px) : x 2 Vi ( )g for all 2 Sub', and so F refutes ' under U0 . Thus F j= ' implies F j= (Fi ; ti ) for every i. The converse direction is rather technical; we refer the reader to [Wolter 1996c]. \Canonical" axiomatizations of some standard linear tense logics are shown in Table 3, where we use the following abbreviations. Given a nite frame F = C1 Cn , we write ((C1 ; tC1 ) (Cn ; tCn )) instead of (F; t) and ( ; (C1 ; tC1 ) (Cn ; tCn )) instead of ((C1 ; tC1 ) (Cn ; tCn )) ((Æ; (j; j)) (C1 ; tC1 ) (Cn ; tCn )): ((C1 ; tC1 ) (Cn ; tCn ); ) is de ned analogously. T Now we exploit the formulas (F; t) to characterize the -irreducible logics in NExtLin. Recall that every logic L 2 NExtL0 is represented as \ \ L = fL0 L : L0 is -irreducibleg: So such a characterization can open the door T to a better understanding of the structure of the lattice NExtLin. The -irreducible logics will be described semantically as the logics determined by certain descriptive frames. DEFINITION 133. (1) Denote by k the non-degenerate cluster with k > 0 points. (2) Let !<(0) be the strictly ascending chain h!; <; >i of natural numbers, !<(1) the chain h!; ; i, !< (2) the ascending chain of natural numbers in which precisely the even points are re exive, !<(3) the chain in which precisely the multiples of 3 are re exive, and so on; !> (n) is the mirror image of !< (n).
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Table 3. Axiomatizations of standard tense logics
Ordt = Logfh; <; >i : an ordinalg = Lin ( ; (Æ; (j; m))) Et = Lin 1 > 2 > = Lin ( ; (; (m; m))) ((; (m; m)); ) On = Logh!n; <; >i = Ordt ((Æ; (m; j)) (Æ; (m; j))) ( ; (; (m; m))) {z
|
RD LD Zt Dsn
n+1
= LogfG : 8x(:xRx ! 9y(xRy ^ fz : xRzRyg = ;))g = Lin ( ; (; (m; m))) ( ; (; (m; m)) (Æ; (m; j))) = the mirror image of RD = LoghZ; <; >i = RD LD ((Æ; (j; j)) (Æ; (j; m))) ((Æ; (m; j)) (Æ; (j; j))) = Lin n1 +1 p ! n1 p = Lin ( ; (; (m; m) (; (m; m)); ) |
Qt Rt Rdt
}
{z
n+1
= LoghQ ; <; >i = Ds1 Et = LoghR; <; >i = Qt ((Æ; (m; j)) (Æ; (j; m))) = Logfh; ; i : an ordinalg = Lin ( ; ( 2 ; (j; m)))
}
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(3) C(0; 1 ) is the mirror image of the frame introduced in Example 128, i.e., C(0; 1 ) = h! < (0) 1 ; P i, where P consists of all co nite sets containing 1 and their complements. We generalize this construction to chains !< (n) and clusters k . Namely, for n < ! , k > 1 and
k = fa0 ; : : : ; ak 1 g, we put
C(n; k ) = h!< (n) k ; P i; where P is the set of possible values generated by fXi : 0 i k 1g, for Xi = fai g [ fkj + i : j 2 !g, 0 i k 1. C( k ; n) denotes the mirror image of C(n; k ). (4) C(0; 1 ; 0) = h! < (0) 1 ! > (0); P i, where P consists of all co nite sets containing 1 and their complements. It is easy to check that the frames de ned in (3) and (4) are descriptive and a singleton fxg is in P i x 62 k. For a class of frames C , we denote by C the class of nite sequences of frames from C and let [C ] = f[F] : F 2 C g. The class of nite clusters and the frames of the form (3) in De nition 133 is denoted by B0 ; put also B = fC(0; 1 ; 0)g [ B0 . THEOREM 134. Each logic L 2 NExtLin is determined by a set C [B]. If L is nitely axiomatizable then L = LogC for some set C [B0 ].
Proof. We explain the idea of the proof of the rst claim. Suppose that M = hF; Vi is a countermodel for = ((C1 ; tC1 ) (Cn ; tCn )) based on a descriptive frame F = hW; R; R 1; P i. We must show that there exists G 2 [B] refuting and such that LogG LogF. Consider the sets Wi = fy 2 W : (M; y) j=
_
fpx : x 2 Ci gg:
One can easily show that Wi are intervals in F and F = F1 Fn , for the subframes Fi of F induced by Wi . Moreover, G = [G] is as required if G = hG1 ; : : : ; Gn i is a sequence in B such that LogGi LogFi , and Gi 6j= (Ci ; tCi ), for 1 i n. Frames Gi with those properties are constructed in [Wolter96d]. EXAMPLE 135. The logic Qt is determined by the frames F 2 [B ] which contain no pair of adjacent irre exive points, and Rt is determined by the frames F 2 [B] which contain neither a pair of adjacent irre exive points nor a pair of adjacent non-degenerate clusters. It is notTdiÆcult to show now that the logics LogF, for F 2 [B], coincide with the -irreducible logics in NExtLin. Our rst aim is achieved, and
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in the remaining part of this section we shall draw consequences of this result. Using the same sort of arguments as in the proof of Theorem 126 and Kruskal's [1960] Tree Theorem one can prove COROLLARY 136. (i) All nitely axiomatizable logics in NExtLin are decidable. (ii) A logic L is nitely axiomatizable whenever there exists n < ! such that L 2 NExtDsn . It follows in particular that all logics in NExtQt and all logics of re exive frames are nitely axiomatizable and decidable. Now we formulate two corollaries concerning the Kripke completeness of linear tense logics. First, it is not hard to see that every logic in NExtLin characterized by an in nite frame in [B ] is Kripke incomplete. Using this observation one can prove COROLLARY 137. Suppose L 2 NExtLin and there is a Kripke frame of in nite depth for L. Then there exists a Kripke incomplete logic in NExtL. This result means in particular that in Tense Logic we do not have analogues of the unimodal completeness results of Bull [1966b] and Fine [1974c]. However, if a logic is complete then it is determined by a simple class of frames. Let K be the class frames containing nite clusters and frames of the form (2) in De nition 133. THEOREM 138. Each Kripke complete logic in NExtLin is determined by a subset of [K ]. One of the main types of logics considered in conventional Tense Logic are logics determined by strict linear orders, known also as time-lines. We call them t-line logics. All logics in Table 3, save Rdt , are t-line logics. T-line logics were de ned semantically, and now we are going to determine a necessary syntactic condition for a linear tense logic to be a t-line logic. Given a frame F, we denote by FÆ the frame that results from F by replacing its proper clusters with re exive points. Call L 2 NExtLin a t-axiom logic if L is axiomatizable by a set of formulas of the form (F; t) in which F contains no proper clusters. PROPOSITION 139. The following conditions are equivalent for all logics L 2 NExtLin: (i) L is a t-axiom logic; (ii) FÆ j= L implies F j= L, for every F 2 [B]. (iii) (G; t) 2 L implies (GÆ ; t) 2 L,15 for every nite G. 15
We assume that tC = tÆ whenever Æ replaces C in G.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Proof. The implications (i) ) (ii) and (iii) ) (i) are clear. To prove that (ii) ) (iii), suppose (GÆ ; t) 62 L. Then there exists a frame F 2 [B ] for L refuting (GÆ ; t). Without loss of generality we may assume that F contains no proper clusters. By enlarging some clusters in F we can construct a frame H 2 [B] such that HÆ = F and H 6j= (G; t). In view of (ii), H j= L and so (G; t) 62 L. It follows that the t-axiom logics form a complete sublattice of the lattice NExtLin. THEOREM 140. (i) All nitely axiomatizable t-axiom logics are Kripke complete. (ii) All t-line logics are t-axiom logics.
Proof. (i) Suppose that L = Lin f (GÆi ; ti ) : i 2 I g, for some nite set I . By Theorem 134, L is determined by a subset of [B0]. For F 2 [B0], let kF be the Kripke frame that results from F by replacing all C(n; k ) and C( k ; n) with !< (n) and !>(n), respectively. Then we clearly have LogkF LogF, and F j= (GÆ ; t) i kF j= (GÆ ; t). It follows that L is Kripke complete. (ii) Suppose that L is a t-line logic. By Proposition 139 (3), it suÆces to observe that F j= (GÆ ; t) i F j= (G; t), for all time-lines F and all nite G. So the fact that in Table 3 all t-line logics are axiomatized by canonical formulas of the form (GÆ ; t) is no accident. Finding and verifying axiomatizations of t-line logics becomes almost trivial now. EXAMPLE 141. Let us check the axiomatization of Zt in Table 3. Put
L = RD LD ((Æ; (j; j)) (Æ; (j; m))) ((Æ; (m; j)) (Æ; (j; j))): By Theorem 140, L is complete. By Theorem 138, L is then determined by a subset of [K ]. Clearly this set contains hZ; <; >i, possibly k for k > 0, and nothing else. But the logic of k contains Zt , for all k > 0. We conclude this section by discussing the decidability of properties of logics in NExtLin. In Section 4.4 it will be shown that almost all interesting properties of calculi are undecidable in NExtK and even in NExtS4. In NExtLin the situation is di erent, as was proved in [Wolter 1996c, 1997c]. THEOREM 142. (i) There are algorithms which, given a formula ', decide whether Lin ' has FMP, interpolation, whether it is Kripke complete, strongly complete, canonical, R-persistent. (ii) A linear tense logic is canonical i it is D-persistent i it is complete and its frames are rst order de nable.
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(iii) If a logic in NExtLin has a frame of in nite depth then it does not have interpolation. So NExtLin provides an interesting example of a rather complex lattice of modal logics for which almost all important properties of calculi are decidable. We shall not go into details of the proof here but discuss quite natural criteria for canonicity and strong completeness of logics in NExtLin required to prove this theorem. Denote by B+ the class of frames containing B together with frames C(n1 ; k ; n2 ) de ned as follows. Suppose k > 1, n1 ; n2 < ! are such that n1 + n2 > 0 and k = fa0 ; : : : ; ak 1 g. Then
C(n1 ; k ; n2 ) = h!< (n1 ) k !> (n2 ); P i; where P is the set of possible values generated by fXi : 0 i k 1g, for Xi = fai g [ fkj + i : j 2 !g [ fkj + i : j 2 !g and f0; 1 ; : : : ; n ; : : : g being the points in !> (n2 ). Let F be the class of frames of the form
hf0; : : : ; n1 g; <; >i 1 hf0; : : : ; n2 g; <; >i or hf0; : : : ; ng; <; >i : THEOREM 143. (i) A logic L 2 NExtLin is canonical i the underlying Kripke frame of each frame F 2 [B+ ] for L validates L as well. (ii) A logic L 2 NExtLin is strongly complete i for each frame F 2 [B+ ] validating L, there exists a Kripke frame G for L which results from F by replacing
every C(n; k ) with ! < (n) or ! < (n) H k , for some H 2 F , and every C( k ; n) with ! > (n) or k H ! > (n), for some H 2 F , and every C(n1 ; k ; n2 ) with ! < (n1 ) H ! > (n2 ), for some H 2 F .
EXAMPLE 144. The logic Rt is not canonical because C(2; 2 ) j= Rt but !<(2) 2 6j= Rt . However, Rt is strongly complete, since F j= Rt whenever G 2 [B+ ] validates Rt and F is obtained from G as in the formulation of Theorem 143 with H = 2 F . One can also use Theorem 143 to construct two strongly complete logics L1; L2 2 NExtLin whose sum L1 L2 is not strongly complete (see [Wolter 1996b]).
2.6 Bimodal provability logics Bimodal provability logics emerge when combinations of two di erent provability predicates are investigated, for example, if 1 is understood as \it
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is provable in PA" and 2 as \it is provable in ZF". In contrast to the situation in unimodal provability logic, where almost all provability predicates behave like the necessity operator in GL, there exist quite a lot of di erent types of bimodal provability logics. Various completeness results extending Solovay's completeness theorem for GL to the bimodal case were established by Smorynski [1985], Montagna [1987], Beklemishev [1994, 1996] and Visser [1995]. Here we will not deal with the interpretation of modal operators as provability predicates but sketch some results on modal logics containing the bimodal provability logic
CSM0 = (GL GL) 1 p ! 2 p 2p ! 1 2 p (named so by Visser [1995] after Carlson, Smorynski and Montagna). A number of provability logics is included in this class, witness the list below. (As in unimodal provability logic we have quasi-normal logics among them, i.e., sets of formulas containing K2 and closed under modus ponens and substitutions (but not necessarily under '=i '). Recall that we denote by L + the smallest quasi-normal logic containing L and .)
CSM1
= CSM0 2 (1 p ! p). (This is PRLZF in [Smorynski 1985] and F in [Montagna 1987].)
NB1 = CSM0 (:1 p ^ 2 p) ! 2 (1 q ! q). CSM2 = CSM1 + 1 p ! p. (This is PRLZF [Smorynski 1985] and F1 in [Montagna 1987].)
CSM3
= CSM2 + 2 p [Smorynski 1985].)
! p.
+ Re ection1 in
(This is PRLZF + Re ection2 in
NB2 = NB1 + 2 p ! p + 2 p ! 1 p. A remarkable feature of CSM0 is that|like in GL|we have uniquely determined de nable xed points. THEOREM 145 (Smorynski 1985). Let '(p) be a formula in which every occurrence of p lies within the scope of some 1 or some 2 . Then (i) there exists a formula containing only the propositional variables of '(p) di erent from p such that $ '( ) 2 CSM0 ; (ii) 1 ((p $ '(p)) ^ (q $ '(q))) ! (p $ q) 2 CSM0 . In the remaining part of this section we are concerned with subframe logics containing CSM0 , the main result stating that those of them that are nitely axiomatizable are decidable. All the provability logics introduced above turn out to be subframe logics, so we obtain a uniform proof of their decidability. An interesting trait of subframe logics in ExtCSM0 is that (as a rule) they are Kripke incomplete; in the list above such are CSMi ,
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i = 1; 2; 3, and NBi , i = 1; 2. The proof extends the techniques introduced by Visser [1995]; for details we refer the reader to [Wolter 1998]. First we develop|as was done for NExtK4 and NExtLin|a frame theoretic language for axiomatizing subframe logics in the lattice ExtCSM0 . A nite frame G = hW; R1 ; R2 i validates CSM0 i both R1 and R2 are transitive, irre exive, R2 R1 and
8x; y; z (xR1 y ^ yR2 z ! xR2 z ):
In this section all (not only nite) frames are assumed to satisfy these conditions, save irre exivity. A nite frame F is called a surrogate frame if it has precisely one root r and all points di erent from r are R2 -irre exive. Surrogate frames will provide the language to axiomatize subframe logics in ExtCSM0 . A normal surrogate frame hW; R1 ; R2 i is a surrogate frame in which the root r is R1 -irre exive. We write xRip y i xRi y and :yRi x. Given a frame G = hV; S1 ; S2 ; Qi for CSM0 and a surrogate frame F = hW; R1 ; R2 i, a map h from V onto W is called a weak reduction of G to F if for i 2 f1; 2g and all x; y 2 V , xSi y implies f (x)Ri f (y),
f (x)Rip f (y) implies 9z 2 V (xSi z ^ f (z ) = f (y)), f 1 (X ) 2 Q for all X W .
(The standard de nition of reduction is relaxed here in the second condition.) Each weak reduction to a CSM0 -frames is a usual reduction, since in this case Rip = Ri . A frame G is said to be weakly subreducible to a surrogate frame F if a subframe of G is weakly reducible to F. To describe weak subreducibility syntactically, with each surrogate frame F = hW; R1 ; R2 i we associate the formula
(F) = Æ(F) ^ 1 Æ(F) ! :pr ; where r is the root of F and ^ Æ(F) = fpx ! 1 py : xR1p y; x; y 2 W g ^ ^
fpx ! 2 py : xR2p y; x; y 2 W g ^ ^ fpx ! :py : x 6= y; x; y 2 W g ^ ^ fpx ! :1 py : :(xR1 y); x; y 2 W g ^ ^ fpx ! :2 py : :(xR2 y); x; y 2 W g:
LEMMA 146. For every surrogate frame F and every CSM0 -frame G, we have G 6j= (F) i G is weakly subreducible to F.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV Table 4. Axiomatizations of provability logics
CSM1 CSM0 + 1 p ! p
= CSM0 ( 6) = CSM0 + ()
CSM0 + 2 p ! p
= CSM0 + ( Æ1 )
Æ1
1 CSM0 + 2 p ! 1 p = CSM0 + ( Æ6 )
1- @ I @I 1 1 @ CSM0 ( ) ( @ ) 1- - I @ @I 1 1 ( @ ) ( @ )
NB1
=
It follows immediately that CSM0 (F) and CSM0 + (F) are subframe logics. Conversely, we have the following completeness result. THEOREM 147. (i) There is an algorithm which, given a formula ' such that CSM0 + ' is a subframe logic, returns surrogate frames F1 ; : : : ; Fn for which
CSM0 + ' = CSM0 + (F1 ) + + (Fn ): (ii) There is an algorithm which, given a formula ' such that CSM0 ' is a subframe logic, returns normal surrogate frames F1 ; : : : ; Fn such that CSM0 ' = CSM0 (F1 ) (Fn ): Table 4 shows axiomatizations of the logics introduced above by means of formulas of the form (F). In this section we adopt the convention that in gures we place the number 1 nearby an arrow from x to y if xR1 y and :xR2 y. An arrow without a number means that xR2 y (and therefore xR1 y as well). The proof of decidability is based on the completeness of subframe logics in ExtCSM0 with respect to rather simple descriptive frames. With every surrogate frame F we associate a nite set of frames E(F) = fFA : A 2 SeqFg:
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Loosely, it is de ned as follows. Let us rst assume that the root r of F is R2 -irre exive. Then the frames in E(F) are the results of inserting an in nite strictly descending R1 -chain, denoted by C (!), between each nondegenerate R1 -cluster C and its R1 -successors. This de nes R1 uniquely. However, R2 may be de ned in di erent ways, since a point R2 -seeing a point in C need not (but may) R2 -see certain points in the chain C (!). To be more precise, the set SeqF consists of all sequences A of the form
A = hAx : xR1 x; x 2 W i.
where Ax is a subset of fy 2 W C : yR2 xg such that for all y and z , y 2 Ax and zR1 y imply z 2 Ax . For each non-degenerate R1 -cluster C , denote by C (!) the set f(n; C ) : n 2 !g. Finally, given A 2 SeqF, we construct FA = hV; S0 ; S1 i as the frame satisfying the following conditions: V = W [ SfC (!) : C a non-degenerate R1 -cluster in Fg;
Ri = Si \ (W W ), for i 2 f1; 2g; S1 is de ned so that C (!) becomes an in nite descending chain be
tween C and its immediate successors;
for every non-degenerate R1 -cluster C ,
{ ((C (!) [ C ) (C (!) [ C )) \ S2 = ;, { for all y 2 W C and x 2 C (!), xS2 y i CR2 y, { for all y 2 W C , C = fj : 0 j m 1g and x 2 C (!), yS2 x i 9i 2 !9j m 1 (x = (im + j; C ) ^ y 2 Aj ), { for all x 2 C (!) and y 2 V C , xS2 y i CS2 y.
We illustrate this technical de nition by a simple example. EXAMPLE 148. Construct E(F) for the frame F in Fig. 12 (a). In this case we have two R1 -re exive points, namely c and d. So, SeqF consists of pairs hAc ; Ad i. There are four di erent pairs and so we have four frames in E(F): the frame in Fig. 12 (b) is Fh;;;i and that in (c) is Fhfag;fbgi . Fh;;fbgi is obtained from Fhfag;fbgi by omitting the R2 -arrows starting from a, save the arrow to c, and Fhfag;;i is obtained from Fhfag;fbgi by omitting the R2 -arrows starting from b, save the arrow to d. Suppose now that the root r of F = hW; R1 ; R2 i is R2 -re exive. We de ne FA as in the previous case, but this time we also insert an in nite strictly descending R2 -chain C (!) between r and its R1 -successors. We have de ned the relational component of our frames and now turn to their sets of possible values. Given FA = hV; S1 ; S2 i and a non-degenerate R1 -cluster C = fj : 0 j m 1g in F, let
PC = ffj g [ f(im + j; C ) : i 2 !g : j = 0; : : : ; m 1g
188
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
6 6 6 .
c 1-d Æ Æ
6 6
a
(a)
b
.. Æ1-Æ
6 6
(b)
- 6 -6 6 .
.. Æ1-Æ
6 6
(c)
Figure 12. and denote by P the closure of
ffxg : x 2 V; :xS1 xg [ fPC : C is a non-degenerate R1 -cluster in Fg under intersections and complements in V . The resultant general frame is denoted by G(FA ) = hV; S1 ; S2 ; P i. One can check that it is a descriptive frame for CSM0 . The following completeness result is proved similarly to that in Section 2.4. THEOREM 149. (i) Each subframe logic in NExtCSM0 is determined by a set of frames of the form G(FA ), in which F is a normal surrogate frame and A 2 SeqF. (ii) Each subframe logic in ExtCSM0 is determined by a set of frames with distinguished worlds of the form hG(FA ); ri in which F is a surrogate frame with root r and A 2 SeqF. As a consequence of Theorem 149 and the fact that, for each surrogate frame F with root r and each A 2 SeqF, both the logics of G(FA ) and hG(FA ); ri are decidable, we obtain THEOREM 150. All nitely axiomatizable subframe logics in ExtCSM0 are decidable. We conjecture that the method above can be extended to logics without the GL-axioms, i.e., all nitely axiomatizable subframe logics containing (K4 K4) 1p ! 2 p 2 p ! 1 2 p are decidable.
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2.7 Cartesian products of modal logics Polymodal logics can be used for talking about multi-dimensional relational structures such as Cartesian products of Kripke frames. The formation of products is probably the most natural way of introducing a concept of dimension in modal logic in order to re ect interactions between modal operators representing time, space, knowledge, actions, etc. Products of modal logics (i.e., the sets of polymodal formulas that are valid in Cartesian products of Kripke frames for those logics) have been studied in both pure modal logic (see e.g. [Segerberg 1973], [Shehtman 1978a], [Marx and Venema 1997], [Gabbay and Shehtman 1998], [Kurucz 2000], [Wolter 2000]) and applications in computer science and arti cial intelligence (see e.g. [Reif and Sistla 1985], [Fagin et al. 1995], [Baader and Ohlbach 1995], [Finger and Reynolds 1999], [Wolter and Zakharyaschev 1998, 1999b, 1999c, 2000]) since the 1970s. (Products of modal logics are also relevant to nite variable fragments of modal and intermediate predicate logics; see [Gabbay and Shehtman 1993].) The (Cartesian) product of two frames F1 = hW1 ; R1 i and F2 = hW2 ; R2 i is the bimodal frame of the form
F1 F2 = hW1 W2 ; Rh ; Rv i
in which, for all u1 ; u2 2 W1 and v1 ; v2 2 W2 ,
hu1 ; v1 i Rh hu2 ; v2 i i u1 R1 u2 and v1 = v2 ; hu1 ; v1 i Rv hu2 ; v2 i i u1 = u2 and v1 R2 v2 :
The subscripts h and v appeal to the geometrical intuition of considering Rh as the \horizontal" accessibility relation in F1 F2 and Rv as the \vertical" one. Let L2 be the bimodal language with boxes and (and their duals and ). Frames for this language will be denoted by F = hW; Rh ; Rv i, so that and are interpreted by the relations Rh and Rv , respectively. Products are just special frames of this form. Every product F = F1 F2 satis es the following two important properties: Rv Æ Rh = Rh Æ Rv and Rv 1 Æ Rh Rh Æ Rv 1, known as commutativity and the Church{Rosser property, respectively. F is commutative i it validates the formula com = p $ p; and F is Church{Rosser i it validates
chr = p ! p:
It is to be noted, however, that these properties are not characteristic for products: there are bimodal commutative and Church{Rosser frames that are not (isomorphic to) products of any two frames.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Given two classes of Kripke frames C1 and C2 , we de ne their (Cartesian) product C1 C2 by taking
C1 C2 = fF1 F2 : F1 2 C1 ; F2 2 C2 g: Let C1 and C2 be the classes of all Kripke frames for complete unimodal logics L1 and L2 . Their (Cartesian) product L1 L2 is de ned as the bimodal logic Log(C1 C2 ) in the language L2 . It is easy to see that L1 L2 is a conservative extension of both L1 and L2 , and that
L1 L2 (L1 L2 ) com chr: In some important cases the converse inclusion also holds: THEOREM 151 (Gabbay and Shehtman 1998). Suppose both L1 and L2 are axiomatized by variable free formulas and formulas of the form
n p ! m p: Then
L1 L2 = (L1 L2 ) com chr:
This theorem yields, for instance, axiomatizations for products of K, D, K4, T, S4, S5. However there are many products of standard logics which cannot be axiomatized in this canonical way, for instance, Grz Grz (see [Gabbay and Shehtman 1998]). Moreover, products of logics of linear frames may be even not recursively enumerable, e.g. GL:3 GL:3 (see [Reynolds and Zakharyaschev 2000]). (On the other hand, as was observed in [Gabbay and Shehtman 1998], if classes C1 ; : : : ; Cn are elementary and recursive then Log(C1 Cn ) is recursively axiomatizable.) In contrast to the unimodal case, usually it is rather diÆcult to prove positive results about products of even simple standard logics. Here we illustrate one of the methods of establishing FMP and decidability developed in [Wolter and Zakharyaschev 1998, 1999b] by applying it to S5 S5. Other techniques| ltration, nite depth method, and mosaic|can be found in [Gabbay and Shehtman 1998] and [Marx and Venema 1997]. S5 S5 is clearly determined by the class of products of universal frames which will be called S5-rectangles. Suppose we are given a formula ' and want to nd out whether it is satis able in some S5-rectangle. Let us call a type for ' any subset t of Sub' such that ^ 2 t i 2 t and 2 t; for every ^ 2 Sub',
: 2 t i 2= t;
for every :
2 Sub'.
A type-cluster for ' is a set T of distinct types for ' such that 8t 2 T 8 2 Sub' ( 2 t $ 9t0 2 T 2 t0 ):
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Let Q be a non-empty set elements a in which are labelled by type-clusters T for '. In other words, we can think of Q as consisting of pairs of the form T a with pairwise distinct a. A run through Q (or a Q-run, for short) is a function r from Q to the set of types for ' such that r(T a ) 2 T for every T a 2 Q, and
8T a 2 Q8 2 Sub' ( 2 r(T a ) $ 9scb 2 Q 2 r(scb )). Say that Q is a quasimodel for ' if, for every T a 2 Q and every t 2 T , there
is a Q-run r coming through t, i.e., r(T a ) = t. Q satis es ' if ' belongs to a type occurring in a type-cluster in Q. One can readily show that a formula ' is satis ed in an S5-rectangle i ' is satis ed in some quasimodel for '. We prove now that every satis able formula ' is satis ed in a quasimodel of some bounded size. Let Q be a quasimodel satisfying '. Construct a subquasimodel Q0 of Q in the following way. To begin with, we put in Q0 an element a0 from Q labelled by a type-cluster T containing a type with '. Then, for every t 2 T we take a run r coming through t, and for each 2 t select r(a) containing and put in Q0 the element a together with its copy a0 (labelled by the same type-cluster as a). Thus the resulting Q0 contains at most 2jSub'j 2 jSub'j elements. It is now easy to see that Q0 is a quasimodel satisfying '. For suppose we have an element a 2 Q0 labelled by T and a t 2 T . If a = a0 then, by the construction, we have a Q0 -run coming through t. Assume now that a 6= a0 . We know that there is a Q-run r0 through t. Let r0 (a0 ) = t0 . By the construction we have a Q0 -run r00 through t0 . But then the function r de ned by ( r0 (b) if b = a r(b) = r00 (b) otherwise, for b 2 Q0 , is a run in Q0 coming through t. Note that this gives us 22jSub'j 2 jSub'j runs in Q0 coming through all its types. As a consequence we obtain: THEOREM 152. Every formula satis able in an S5-rectangle is satis ed in an S5-rectangle containing at most 23jSub'j 4 jSub'j2 points. Thus S5 S5 has FMP and is decidable. Unfortunately, there is no general transfer theorem that could guarantee the preservation of such properties of logics as decidability, axiomatizability, or interpolation under the formation of products. If we consider only products of standard modal logics, then the results obtained so far can roughly be described as follows (for more details consult [Spaan 1993], [Marx and Venema 1997], [Gabbay and Shehtman 1998], [Marx and Areces 1998], [Marx 1999], [Wolter 2000], [Reynolds and Zakharyaschev 2000]):
192
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
logics of the form KL and S5L are usually decidable (in particular, for L 2 fK; T; K4; K4:3; S4; S5g);
products of logics determined by in nite linear orders are undecidable, the computational complexity of a decidable product is usually substantially higher than the complexity of its components; for example, the satis ability problem for S5 S5 is NEXP T IME -complete.
The decidability and FMP of logics like K4 K4, S4 S4, S4 S4:3 remain challenging open problems of the eld. In higher dimensions|for n 3|the rst results related to products of modal logics were obtained in the framework of algebraic logic: as follows from [Maddux 1980] and [Johnson 1969], S5n is undecidable and not nitely axiomatizable. However, as we mentioned above, products like S5n and Kn are recursively enumerable. It is worth noting that although Kn have FMP for all n < ! [Gabbay and Shehtman 1998], this could imply the decidabiliy of Kn only if the class of nite frames for Kn were recursive. We could have such a test if Kn were nitely axiomatizable; however this is not the case [Kurucz 2000]. To prove the decidability of Kn , it would also be enough to show that it has the product FMP, i.e., it is characterized by the class of products of n-many nite frames. But this approach does not work either: as has been shown by Hirsch et al. [2000], all logics L such that Kn L S5n are undecidable, non nitely axiomatizable, do not have the product FMP, and it is undecidable whether a nite frame is a frame for L. (The only known example of a decidable higher dimensional product of non-tabular logics is Altn [Gabbay and Shehtman 1998].) 3 SUPERINTUITIONISTIC LOGICS Although C.I. Lewis constructed his rst modal calculus S3 in 1918, it was Godel's [1933] two page note that attracted serious attention of mathematical logicians to modal systems. While Lewis [1918] used an abstract necessity operator to avoid paradoxes of material implication, Godel [1933] and earlier Orlov [1928]16 treated as \it is provable" to give a classical interpretation of intuitionistic propositional logic Int by means of embedding it into a modal \provability" system which turned out to be equivalent to Lewis' S4. Approximately at the same time Godel [1932] observed that there are in nitely many logics located between Int and classical logic Cl, which| together with the creation of constructive (proper) extensions of Int by Kleene [1945] and Rose [1953] (realizability logic), Medvedev [1962] (logic 16 Orlov's paper remained unnoticed till the end of the 1980s. It is remarkable also for constructing the rst system of relevant logic.
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193
of nite problems), Kreisel and Putnam [1957]|gave an impetus to studying the class of logics intermediate between Int and Cl, started by Umezawa [1955, 1959]. Godel's embedding of Int into S4, presented in an algebraic form by McKinsey and Tarski [1948] and extended to all intermediate logics by Dummett and Lemmon [1959], made it possible to develop the theories of modal and intermediate logics in parallel ways. And the structural results of Blok [1976] and Esakia [1979a,b], establishing an isomorphism between the lattices ExtInt and NExtGrz, along with preservation results of Maksimova and Rybakov [1974] and Zakharyaschev [1991], transferring various properties from modal to intermediate logics and back, showed that in many respects the theory of intermediate logics is reducible to the theory of logics in NExtS4. To demonstrate this as well as some features of intermediate logics is the main aim of this part. We will use the same system of notations as in the modal case. In particular, ExtInt is the lattice of all logics of the form Int + (where is an arbitrary set of formulas in the language of Int and + as before means taking the closure under modus ponens and substitution); we call them superintuitionistic logics or si-logics for short. Basic facts about the syntax and semantics of Int and relevant references can be found in Intuitionistic Logic, see volume 7 of this Handbook. A list of some \standard" si-logics is given in Table 5.
3.1 Intuitionistic frames As in the case of modal logics, the adequate relational semantics for si-logics can be constructed on the base of the Stone representation of the algebraic \models" for Int, known as Heyting (or pseudo-Boolean) algebras. It is hard to trace now who was the rst to introduce intuitionistic general frames|the earliest references we know are [Esakia 1974] and [Rautenberg 1979]|but in any case, having at hand [Jonsson and Tarski 1951] and [Goldblatt 1976a], the construction must have been clear. An intuitionistic (general) frame is a triple F = hW; R; P i in which R is a partial order on W 6= ; and P , the set of possible values in F, is a collection of upward closed subsets (cones) in W containing ; and closed under the Boolean \, [, and the operation (for !) de ned by
X Y = fx 2 W : 8y 2 x" (y 2 X ! y 2 Y )g: If P contains all upward closed subsets in W then we call F a Kripke frame and denote it by F = hW; Ri. An important feature of intuitionistic models M = hF; Vi (V, a valuation in F, maps propositional variables to sets in P ) is that V('), the truth-value of a formula ', is always upward closed. Every intuitionistic frame F = hW; R; P i gives rise to the Heyting algebra F+ = hP; \; [; ; ;i called the dual of F. Conversely, given a Heyting algebra
194
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV Table 5. A list of standard superintuitionistic logics
For Cl SmL KC LC SL KP BDn
= = = = = = = =
BWn BTWn Tn Bn NLn
= = = = =
Int + p Int + p _ :p Int + (:q ! p) ! (((p ! q) ! p) ! p) Int + :p _ ::p Int + (p ! q) _ (q ! p) Int + ((::p ! p) ! :p _ p) ! :p _ ::p Int + (:p ! q _ r) ! (:p ! q) _ (:p ! r) Int + bdn , where bd1 = p1 _ :p1 ; bdn+1 = pn+1 _ (pn+1 ! bdn ) W W Int + ni=0 (pi ! j=6 i pj ) V W W Int + 0i<jn :(:pi ^ :pj ) ! ni=0 (:pi ! j6=i :pj ) V W W W Int + ni=0 ((pi ! i=6 j pj ) ! i=6 j pj ) ! ni=0 pi Vn W Wn Int + i=0 (:pi $ i=6 j pj ) ! i=0 pi Int + nf n , where nf 0 = ?, nf 1 = p, nf 2 = :p, nf ! = > nf 2m+3 = nf 2m+1 _ nf 2m+2 , nf 2m+4 = nf 2m+3 ! nf 2m+1
A = hA; ^; _; !; ?i, we construct its relational representation A+ = hW; Ri by taking W to be the set of all prime lters in A (a lter r is prime if it
is proper and a _ b 2 r implies a 2 r or b 2 r), R to be the set-theoretic inclusion and P = ffr 2 W : a 2 rg : a 2 Ag: It is readily checked that A+ , the dual of A, is an intuitionistic frame, A = (A+ )+ and A+ is di erentiated, tight in the sense that
xRy i 8X 2 P (x 2 X ! y 2 X ); and compact, i.e., for any families X \
P
(X [ Y ) = fx 2 W : 8X 2 X8Y
and Y fW
X : X 2 P g,
2 Y (x 2 X ^ x 2 Y )g 6= ;
T whenever (X 0 [ Y 0 ) 6= ; for every nite subfamilies X 0 X , Y 0 Y . Frames with these three properties (actually di erentiatedness follows from tightness) are called descriptive. In the same way as in the modal case one can prove that F is descriptive i F = (F+ )+ . Duality between the
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195
Æ>
:::
nf 10Æ
Æ nf 9
p
6@I@*Æ61 4 Æ @ 6@I@@*Æ63 @ 6 Æ @ 6@I@@*Æ65 @Æ 7 8 Æ @ * 6@I@@ 6 10Æ @ @Æ 9 2Æ
@@ @ I I@ @ @ nf 7 Æ @ I@ Æ nf 8 @ nf 5 nf 6 Æ I@ Æ@ @ I@ @ @ nf 3 Æ @ I@ Æ nf 4 @ nf 2 Æ I@ Æ nf 1 @ @Æ ? AInt (1)
F<1 (1) Int
(a)
(b)
Figure 13. basic truth-preserving operations on algebras and descriptive frames (the de nitions of generated subframes, reductions and disjoint unions do not change) is also established by the same technique. Since every consistent si-logic L is characterized by its Tarski{Lindenbaum algebra AL , we conclude that L is characterized also by a class of intuitionistic frames, say by the dual of AL. Re ned nitely generated frames for Int look similarly to those for K4: the only di erence is that now all clusters are simple and the truth-sets must be upward closed. Fig. 13 showing (a) the free 1-generated Heyting algebra AInt(1) and (b) its dual FInt (1) will help the reader to restore the details. AInt(1) was rst constructed by Rieger [1949] and Nishimura [1960]; it is called the Rieger{Nishimura lattice. The formulas nf n de ned in Table 5 and used for the construction are known as Nishimura formulas (see also Section 3 of Intuitionistic Logic), in volume 7 of this Handbook. At the algebraic level the connection between Int and S4 discovered by Godel is re ected by the fact, established in [Mckinsey and Tarski 1946], that the algebra of open elements (i.e., elements a such that a = a) of every modal algebra for S4 (known as a topological Boolean algebra; see [Rasiowa and Sikorski 1963]) is a Heyting algebra and conversely, every Heyting algebra is isomorphic to the algebra of open elements of a suitable algebra for S4. We explain this result in the frame-theoretic language. Given a frame F = hW; R; P i for S4 (which means that R is a quasiorder on W ), we denote by W the set of clusters in F|more generally,
196
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
X = fC (x) : x 2 X g|and put C (x)C (y) i xRy, P = fX : X 2 P ^ X = X g = fX : X 2 P ^ X = X"g: It is readily checked that the structure F = hW; R; P i is an intuitionistic frame (for instance, (X ) (Y ) = (( X [ Y ))); we call it the skeleton of F. The skeleton of a model M = hF; Vi for S4 is the intuitionistic model M = hF; Vi, where V(p) = V(p). Denote by T the Godel translation pre xing to all subformulas of a given intuitionistic formula.17 By induction on the construction of ' one can easily prove the following LEMMA 153 (Skeleton). For every model M for S4, every intuitionistic formula ' and every point x in M, (M; C (x)) j= ' i (M; x) j= T ('): It follows that ' 2 Int implies T (') 2 S4. To prove the converse we should be able to convert intuitionistic frames F into modal ones with the skeleton (isomorphic to) F. This is trivial if F is a Kripke frame|we can just regard it to be a frame for S4, which in view of the Kripke completeness of both Int and S4, shows that T really embeds the former into the latter, i.e., ' 2 Int i T (') 2 S4: In general, the most obvious way of constructing a modal frame from an intuitionistic frame F = hW; R; P i is to take the closure P of P under the Boolean operations \, [ and !. It is well known in the theory of Boolean algebras (see [Rasiowa and Sikorski 1963]) that for every X W , X is in P i X = ( X1 [ Y1 ) \ \ ( Xn [ Yn ) for some X1 ; Y1 ; : : : ; Xn ; Yn 2 P and n 1. It follows that if X 2 P then
X = (X1 Y1 ) \ \ (Xn Yn ) 2 P P; and so P is closed under in hW; Ri and P coincides with the set of upward closed sets in P . Thus, hW; R; P i is a partially ordered modal frame; we shall denote it by F. Moreover, we clearly have F = F. If M = hF; Vi is an intuitionistic model then M = hF; Vi is a modal model having M as its skeleton. So by the Skeleton Lemma, (M; x) j= ' i (M; x) j= T (');
The translation de ned in [Godel 1933] does not pre x to conjunctions and disjunctions. However this di erence is of no importance as far as embeddings into logics in NExtS4 are concerned. 17
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for every intuitionistic formula ' and every point x in F. It is worth noting that if F = hW; Ri is a nite intuitionistic Kripke frame then F is also a Kripke frame. However, for an in nite F, F is not in general a Kripke frame, witness h!; i. The operator is not the only one which, given an intuitionistic frame F, returns a modal frame whose skeleton is isomorphic to F. As an example, we de ne now an in nite class of such operators. For Kripke frames F = hW; Ri and G = hV; S i, denote by F G the direct product of F and G, i.e., the frame hW V; R S i in which the relation R S is de ned component-wise:
hx1 ; y1 i (R S ) hx2 ; y2i
i x1 Rx2 and y1 Sy2:
Let 0 < k !. We will regard k to be the set f0; : : : ; k 1g if k < ! and f0; 1; : : : g if k = !. Denote by k an operator which, given an intuitionistic frame F = hW; R; P i, returns a modal frame k F = hkW; kR; kP i such that (i) hkW; kRi is the direct product of the k-point cluster k; k2 and hW; Ri (in other words, hkW; kRi is obtained from hW; Ri by replacing its every point with a k-point cluster); (ii) k F = F; (iii) I X 2 kP , for every I k and X 2 P . For instance, we can take kP to be the Boolean closure of the set
fI X : I k; X 2 P g: For a Kripke frame
F = hW; R; UpW i we can, of course, take kP and then k F =
kW; kR; 2kW
.
= 2kW
3.2 Canonical formulas The language of canonical formulas, axiomatizing all si-logics and characterizing the structure of their frames, can be easily developed following the scheme of constructing the canonical formulas for K4 outlined in Section 1.6 and using the connection between modal and intuitionistic frames established above. We con ne ourselves here only to pointing out the differences from the modal case and some interesting peculiarities; details can be found in [Zakharyaschev 1983, 1989] and [Chagrov and Zakharyaschev 1997]. Actually, there are two important di erences. First, in the de nition of subreduction of F = hW; R; P i to G the condition (R3) does not correspond to the fact that all sets in P are upward closed. We replace it by the following condition (R30 ) 8X 2 Q f 1 (X )# 2 P , where Q = fV X : X 2 Qg and P = fW X : X 2 P g. For a completely de ned f satisfying (R1) and (R2) the condition (R30 ) is clearly
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
:p 1Æ :q q @ I G
@
:p Æ2 :r r 6
@@
3
Æ :p
p
@
:q :p ! :q _ :r @Æ ::pp ! ! :r 0 Figure 14. equivalent to (R3) and so every reduction is also a subreduction. If G is a nite Kripke frame then (R30 ) is equivalent to 8z 2 V f 1 (z )# 2 P . G is a subframe of F if G is a subframe of F and the identity map on V is a subreduction of F to G. It is of interest to note that in the intuitionistic case (co nal) subreductions are dual to IC(N)-subalgebras of Heyting algebras which preserve only implication, conjunction (and negation or ?) but do not necessarily preserve disjunction. Second, we have to change the de nition of open domains. Now we say an antichain a (of at least two points) is an open domain in an intuitionistic model N relative to V a formula W ' if there ia a pair ta = ( a ; a ) such that a [ a = Sub', a ! a 62 Int and
2
a i a j=
for all a 2 a.
It is worth noting that in any intuitionistic model every antichain a is open relative to every disjunction free formula '. Indeed, let a be de ned by condition above and a = Sub' a . It should be clear that ^ 2 a i 2 a and 2 a . And if ! 2 a, 2 a but 2 a then a j= for every a 2 a and b 6j= forVsome b W2 a, whence b 6j= ! , which is a contradiction. It follows that a ! a 62 Int. EXAMPLE 154. Let us try to characterize the class of intuitionistic refutation frames for the Weak Kreisel{Putnam Formula
wkp = (:p ! :q _ :r) ! (:p ! :q) _ (:p ! :r): First we construct its simplest countermodel; it is depicted in Fig. 14, where by putting a formula to the left (right) of a point we mean that it is true (not true) at the point. Then we observe that every frame F refuting wkp is co nally subreducible to the frame G underlying this countermodel by
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the map f de ned as follows:
f (x) =
8 > > > > < > > > > :
0 1 2 3 unde ned
if x j= :p ! :q _ :r, x 6j= (:p ! :q) _ (:p ! :r) if x j= :p ! :q _ :r, x j= :p and x j= q if x j= :p ! :q _ :r, x j= :p and x j= r if x j= p or x j= :p ^ :q ^ :r otherwise.
However, the co nal subreducibility to G is only a necessary condition for F 6j= wkp, witness the frame having the form of the three-dimensional Boolean cube with the top point deleted. The reason for this is that the antichain f1; 2g is a closed domain in N: it is impossible to insert a point a between 0 and f1; 2g and extend to it consistently the truth-sets for the depicted formulas. Indeed, otherwise we would have a j= :p ! :q _ :r, a 6j= :q _ :r and so a 6j= :p, i.e., there must be a point x 2 a" such that x j= p, but such a point does not exist. In fact, F 6j= wkp i there is a co nal subreduction of F to G satisfying (CDC) for ff1; 2gg. Now, as in the modal case, with every nite rooted intuitionistic frame F = hW; Ri and a set D of antichains in it we can associate two formulas (F; D; ?) and (F; D), called the canonical and negation free canonical formulas, respectively, so that G 6j= (F; D; ?) (G 6j= (F; D)) i there is a (co nal) subreduction of G to F satisfying (CDC) for D. For instance, if a0 ; : : : ; an are all points in F and a0 is its root, then one can take
(F; D; ?) = where
^
ai Raj
ij
^
^
d2D
d ^ ? ! p0 ;
^
= ( pk ! pj ) ! pi ; :aj Rak ^ ^ _ ( pk ! pi ) ! pj ; d = aj 2d ai 2W d" :ai Rak
ij
? =
n ^
(
^
i=0 :ai Rak
pk ! pi ) ! ?:
(F; D) is obtained from (F; D; ?) by deleting the conjunct ? . THEOREM 155. There is an algorithm which, given an intuitionistic ', returns canonical formulas (F1 ; D1 ; ?); : : : ; (Fn ; Dn ; ?) such that Int + ' = Int + (F1 ; D1 ; ?) + + (Fn ; Dn ; ?):
So the set of intuitionistic canonical formulas is complete for ExtInt. If ' is negation free then one can use only negation free canonical formulas. And if ' is disjunction free then all Di are empty.
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Table 6 and Theorem 156 show canonical axiomatizations of the si-logics in Table 5. Using this \geometrical" representation it is not hard to see, for instance, that SmL, known as the Smetanich logic, is the greatest consistent extension of Int di erent from Cl; it is the logic of the two-point rooted frame. KC, the logic of the Weak Law of the Excluded Middle, is characterized by the class of directed frames. It is the greatest si-logic containing the same negation free formulas as Int (see [Jankov 1968a]). LC, the Dummett or chain logic, is characterized by the class of linear frames (see [Dummett 1959]). BDn and BWn are the minimal logics of depth n and width n, respectively (see [Hosoi 1967] and [Smorynski 1973]). Finite frames for BTWn contain n top points [Smorynski 1973] and nite frames for Tn are of branching n, i.e., no point has more than n immediate successors. THEOREM 156 (Nishimura 1960, Anderson 1972). Every extension L of Int by formulas in one variable can be represented either as
L = Int + nf 2n = Int + ] (Hn ; ?) or as
L = Int + nf 2n 1 = Int + ] (Hn+1 ; ?) + ] (Hn+2 ; ?); where Hn , Hn+1 , Hn+2 are the subframes of the frame in Fig. 13 generated by the points n, n +1 and n +2, respectively, and ] (F; ?) is an abbreviation for (F; D] ; ?), D] the set of all antichains in F. Jankov [1969] proved in fact that logics of the form Int + ] (F; ?) and only them are splittings of ExtInt. However, not every si-logic is a unionsplitting of ExtInt which means that this class has no axiomatic basis.
3.3 Modal companions and preservation theorems The fact that the Godel translation T embeds Int into S4 and the relationship between intuitionistic and modal frames established in Section 3.1 can be used to reduce various problems concerning Int (e.g. proving completeness or FMP) to those for S4 and vice versa. Moreover, it turns out that each logic in ExtInt is embedded by T into some logics in NExtS4, and for each logic in NExtS4 there is one in ExtInt embeddable in it. We say a modal logic M 2 NExtS4 is a modal companion of a si-logic L if L is embedded in M by T , i.e., if for every intuitionistic formula ',
' 2 L i T (') 2 M: If M is a modal companion of L then L is called the si-fragment of M and denoted by M . The reason for denoting the operator \modal logic 7! its si-fragment" by the same symbol we used for the skeleton operator is explained by the following
ADVANCED MODAL LOGIC Table 6. Canonical axioms of standard superintuitionistic logics
For
= Int + (Æ)
Æ
Cl
= Int + ( Æ6)
SmL
=
KC
=
LC
=
SL
=
KP
=
BDn
=
Æ Æ Æ Æ6 K A A Int + ( Æ ) + ( Æ6) Æ Æ AK Int + ( AÆ ; ?) Æ Æ AK Int + ( AÆ ) Æ 6 Æ Æ AK Int + ] ( AÆ ; ?)
Æ AK Æ1 Æ2 Æ Æ1 Æ2 Æ I 6 @ @I 6 Int + ( @Æ ; ff1; 2gg; ?) + ( @Æ ; ff1; 2gg; ?) Æ. n ..6 Æ1 Int + ( Æ60 ) n+1
z }| {
BWn
=
Æ Æ I @ Int + ( @Æ ) n+1
z }| {
BTWn =
Æ Æ I @ Int + ( @Æ ; ?) n+1
z }| {
Tn
=
Æ Æ I @ Int + ] ( @Æ ) n+1
z }| {
Bn
=
Æ Æ I@Æ ; ?) @
Int + ] (
201
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
THEOREM 157. For every M 2 NExtS4, M = f' : T (') 2 M g. Moreover, if M is characterized by a class C of modal frames then M is characterized by the class C = fF : F 2 Cg of intuitionistic frames.
Proof. It suÆces to show that f' : T (') 2 M g = LogC . Suppose that T (') 2 M . Then F j= T (') and so, by the Skeleton Lemma, F j= ' for every F 2 C , i.e., ' 2 LogC . Conversely, if F j= ' for all F 2 C then, by the same lemma, T (') is valid in all frames in C and so T (') 2 M . Thus, maps NExtS4 into ExtInt. The following simple observation shows that actually is a surjection. Given a logic L 2 ExtInt, we put
L = S4 fT (') : ' 2 Lg: THEOREM 158 (Dummett and Lemmon 1959). For every si-logic L, L is a modal companion of L.
Proof. Clearly, L L. To prove the converse inclusion, suppose ' 62 L, i.e., there is a frame F for L refuting '. Since F = F, by the Skeleton Lemma we have F j= L and F 6j= T ('). Therefore, T (') 62 L and so ' 62 L. Now we use the language of canonical formulas to obtain a general characterization of all modal companions of a given si-logic L. Our presentation follows [Zakharyaschev 1989, 1991]. Notice rst that for every modal frame G and every intuitionistic canonical formula (F; D; ?), G j= (F; D; ?) i G j= (F; D; ?) and so S4 T ( (F; D; ?)) = S4 (F; D; ?). The same concern, of course, the negation free canonical formulas. THEOREM 159. A logic M 2 NExtS4 is a modal companion of a si-logic L = Int + f (Fi ; Di ; ?) : i 2 I g i M can be represented in the form
M = S4 f (Fi ; Di ; ?) : i 2 I g f (Fj ; Dj ; ?) : j 2 J g; where every frame Fj , for j 2 J , contains a proper cluster.
Proof. (() We must show that for every intuitionistic formula ', ' 2 L i T (') 2 M . Suppose that ' 62 L and F = hW; R; P i is a frame separating ' from L. We prove that F separates T (') from M . As was observed above, F 6j= T (') and F j= (Fi ; Di ; ?) for any i 2 I . So it remains to show that F j= (Fj ; Dj ; ?) for every j 2 J . Suppose otherwise. Then, for some j 2 J , we have a subreduction f of F to Fj . Let a1 and a2 be distinct points belonging to the same proper cluster in Fj . By the de nition of subreduction, f 1 (a1 ) f 1(a2 )# and f 1 (a2 ) f 1(a1 ) #, and so there is an in nite chain x1 Ry1Rx2 Ry2R : : :
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in F such that fx1 ; x2 ; : : : g f 1(a1 ) and fy1; y2 ; : : : g f 1(a2 ). And since R is a partial order, all the points xi and yi are distinct. Since f 1 (a1 ) 2 P , there are Xi ; Yi 2 P such that
f 1 (a1 ) = ( X1 [ Y1 ) \ \ ( Xn [ Yn ): And since f 1 (a1 ) \ f 1 (a2 ) = ;, for every point yi there is some number ni such that yi 2 Xni and yi 62 Yni . But then, for some distinct l and m, the numbers nl and nm must coincide, and so if, say, yl Rym then xm 62 Ynm and xm 2 Xnl (for yl Rxm Rym, Xi = Xi ", Yi = Yi "). Therefore, xm 62 f 1(a1 ), which is a contradiction. The rest of the proof presents no diÆculties. This proof does not touch upon the co nality condition. So along with canonical formulas in Theorem 159 we can use negation free canonical formulas. Thus, we have:
S4 = S4:1 = Dum = Grz = Int; S4:2 = (S4:2 Grz) = KC; S4:3 = (S4:3 Grz) = LC; S5 = (S5 Grz) = Cl: COROLLARY 160. The set of modal companions of every consistent silogic L forms the interval
)] = fM 2 NExtS4 : L M L Grzg 1 (L) = [ L; L (ÆÆ and contains an in nite descending chain of logics.
Proof. Notice rst that (F; D; ?) and (F; D ) are in Grz i F contains ÆÆ )]. On the other hand, the a proper cluster. So 1 (L) [ L, L ( si-fragments of all logicsinthe interval are the same, namely L. Therefore, 1 (L) = [ L; L (ÆÆ )]. Now, if L is consistent then (Æ) 62 L and so we have
L L (Cn ) L (C2 ) L (C1 ) = For; where Ci is the non-degenerate cluster with i points.
This result is due to Maksimova and Rybakov [1974], Blok [1976] and Esakia [1979b]. Thus, all modal companions of every si-logic L are contained ÆÆ between the least companion L and the greatest one, viz., L ( ), which will be denoted by L. Using Theorems 159 and 44, we obtain
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
COROLLARY 161. There is an algorithm which, given a modal formula ', returns an intuitionistic formula such that (S4 ') = Int + . The following theorem, which is also a consequence of Theorem 159, describes lattice-theoretic properties of the maps , and . Items (i), (ii) and (iv) in it were rst proved by Maksimova and Rybakov [1974], and (iii) is due to Blok [1976] and Esakia [1979b] and known as the Blok{Esakia Theorem. THEOREM 162. (i) The map is a homomorphism of the lattice NExtS4 onto the lattice ExtInt. (ii) The map is an isomorphism of ExtInt into NExtS4. (iii) The map is an isomorphism of ExtInt onto NExtGrz. (iv) All these maps preserve in nite sums and intersections of logics. Now we give frame-theoretic characterizations of the operators and . Note rst that the following evident relations between frames for si-logics and their modal companions hold:
F j= M i F j= M; F j= L i F j= L; F j= L i F j= L; F j= L i k F j= L: THEOREM 163 (Maksimova and Rybakov 1974). A si-logic L is characterized by a class C of intuitionistic frames i L is characterized by the class C = fF : F 2 Cg.
Proof. ()) It suÆces to show that any canonical formula (F; D; ?) 62 L is refuted by some frame in C . Since F is partially ordered, (F; D; ?) 62 L, i.e., there is F 2 C refuting (F; D; ?) and so F 6j= (F; D; ?). (() is straightforward. To characterize we require LEMMA 164. For any canonical formula (F; D; ?) built on a quasi-ordered frame F, (F; D; ?) 2 S4 (F; D; ?), where D = fd : d 2 Dg and d = fC (x) : x 2 dg.
Proof. Let G be a quasi-ordered frame refuting (F; D; ?). Then there is a co nal subreduction f of G to F satisfying (CDC) for D. The map h from F onto F de ned by h(x) = C (x), for every x in F, is clearly a reduction of F to F. So the composition hf is a co nal subreduction of G to F, and it is easy to verify that it satis es (CDC) for D.
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THEOREM 165. A si-logic LS is characterized by a class C of frames i L is characterized by the class 0
Proof. ()) As was noted above, if F is a frame for L then k F is a frame for L. So suppose that a formula (F; D; ?), built on a quasiordered frame F = hW; RSi, does not belong to L and show that it is refuted by some frame in 0
(ii) (F; D; ?) 2 L i either F is partially ordered and (F; D; ?) 2 L or F contains a proper cluster.
Proof. (i) The implication ()) was actually established in the proof of Theorem 165, and the converse one follows from Lemma 164. (ii) Suppose (F; D; ?) 2 L. Then either F is partially ordered, and so (F; D; ?) 2 L, or F contains a proper cluster. The converse implication follows from (i) and the fact that (F; D; ?) 2 Grz for every frame F with a proper cluster. The results obtained in this section not only establish some structural correspondences between logics in ExtInt and NExtS4 and their frames, but may be also used for transferring various properties of modal logics to their si-fragments and back. A few results of that sort are collected in
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV Table 7. Preservation Theorem Property of logics
Preserved under
Decidability Kripke completeness Strong completeness Finite model property Tabularity Pretabularity D-persistence Local tabularity Disjunction property Hallden completeness Interpolation property Elementarity Independent axiomatizability
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No
Yes Yes Yes Yes No No Yes No Yes No No Yes Yes
Yes No No Yes Yes Yes No No Yes No No No Yes
Table 7; we shall cite them as the Preservation Theorem. The preservation of decidability follows from the de nition of and Theorem 167. That preserves Kripke completeness, FMP and tabularity is a consequence of Theorem 157. The map preserves Kripke completeness and FMP, since we can de ne k in Theorem 165 so that k hW; Ri = hkW; kRi; however, does not in general preserve the tabularity, because Cl = S5 is not tabular. The preservation of FMP and tabularity under follows from Theorem 163. On the other hand, Shehtman [1980] proved that does not preserve Kripke completeness (since preserves it and Grz is complete, this means in particular that Kripke completeness is not preserved under sums of logics in NExtS4). Some other preservation results in Table 7 will be discussed later. For references see [Chagrov and Zakharyaschev 1992, 1997].
3.4 Completeness In this section we brie y discuss the most important results concerning completeness of si-logics with respect to various classes of Kripke frames.
Kripke completeness That not all si-logics are complete with respect to Kripke frames was discovered by Shehtman [1977], who found a way
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207
to adjust Fine's [1974b] idea to the intuitionistic case (which was not so easy because intuitionistic formulas do not \feel" in nite ascending chains essential in Fine's construction; see Section 20 of Basic Modal Logic). Note however that Kuznetsov's [1975] question whether all si-logics are complete with respect to the topological semantics (see Intuitionistic Logic, volume 7 of this Handbook) is still open. As to general positive results, notice rst that the Preservation Theorem yields the following translation of Fine's [1974c] Theorem on nite width logics (si-logics of nite width were studied by Sobolev [1977a]). THEOREM 168. Every si-logic of width n (i.e., a logic in ExtBWn ; see Table 5) is characterized by a class of Noetherian Kripke frames of width n. The translation of Sahlqvist's Theorem gives nothing interesting for silogics. A sort of intuitionistic analog of this theorem has been recently proved by Ghilardi and Meloni [1997]. Here is a somewhat simpli ed variant of their result in which p, q, r, s denote tuples of propositional variables and , tuples of formulas of the same length as r and s, respectively. THEOREM 169 (Ghilardi and Meloni 1997). Suppose '(p; q; r; s) is an intuitionistic formula in which the variables r occur positively and the variables s occur negatively, and which does not contain any !, except for negations and double negations of atoms, in the premise of a subformula of the form '0 ! '00 . Assume also that (p; q) and (p; q) are formulas such that p occur positively in and negatively in , while q occur negatively in and positively in . Then the logic
Int + '(p; q; (p; q); (p; q)) is canonical. The preservation of D-persistence under (see [Zakharyaschev 1996]) and the fact (discovered by Chagrova [1990]) that L is characterized by an elementary class of Kripke frames whenever L is determined by such a class provide us with an intuitionistic variant of the Fine{van Benthem Theorem. THEOREM 170. If a si-logic is characterized by an elementary class of Kripke frames then it is D-persistent. As in the modal case, it is unknown whether the converse of this theorem holds. All known non-elementary si-logics, for instance the Scott logic SL and the logics Tn of nite n-ary trees (see [Rodenburg 1986]) are not canonical and even strongly complete either, as was shown by Shimura [1995]. (Actually he proved that no logic in the intervals [SL; SL + bd3 ] and [Int; T2 ], save of course Int, is strongly complete.) As far as we know, there are no examples of si-logics separating canonicity, D-persistence and strong completeness. (Ghilardi, Meloni and Miglioli have
208
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
recently showed that SL in any language with nitely many variables is canonical). Theorem 40 which holds in the intuitionistic case as well gives an algebraic counterpart of strong Kripke completeness.
The nite model property. The rst example of an in nitely axiomatizable si-logic without FMP was constructed by Jankov [1968b]|that was in fact the starting point of a long series of \negative" results in modal logic. A nitely axiomatizable logic without FMP appeared two years later in [Kuznetsov and Gerchiu 1970]. The reader can get some impression about this and other examples of that sort by proving (it is really not hard) that
Æ Æ 6 12 6 ÆÆÆÆ ÆÆÆÆ @IBM Æ I @ M B Æ ' = ( @Æ ) 2= L = Int + bw4 + ( @Æ ; ff1; 2gg) 1 2
but no nite frame can separate ' from L. (Notice by the way that L is axiomatizable by Sahlqvist formulas; see [Chagrov and Zakharyaschev 1995b].) FMP of a good many si-logics was proved using various forms of ltration; see e.g. [Gabbay 1970], [Ono 1972], [Smorynski 1973], [Ferrari and Miglioli 1993]. As an illustration of a rather sophisticated selective ltration we present here the following THEOREM 171 (Gabbay and de Jongh 1974). The logic Tn (see Table 5) is characterized by the class of nite n-ary trees.
Proof. First we prove that Tn is characterized by the class of nite frames of branching n. Suppose ' 62 Tn and M = hF; Vi is a model for Tn refuting '. Without loss of generality we may assume that F = hW; Ri is a tree. Let = Sub' and x = f 2 : x j= g, for every point x in F. Given x in F, put rg(x) = f[y] : y 2 x"g and say that x is of minimal range if rg(x) = rg(y) for every y 2 [x] \ x". Since there are only nitely many distinct -equivalence classes in M, every y 2 [x] sees a point z 2 [x] of minimal range. Now we extract from M a nite refutation frame G = hV; S i for ' of branching n. To begin with, we select some point x of minimal range at which ' is refuted and put V0 = fxg. Suppose Vk has already been de ned. If jrg(x)j = 1 for every x 2 Vk , then Sk we put G = hV; S i, where V = i=0 Vk and S is the restriction of R to V . Otherwise, for each x 2 Vk with jrg(x)j > 1 and each [y] 2 rg(x) di erent from [x] and such that z y for no [z ] 2 rg(x) f[x]g, we select a point u 2 [y] \ x" S of minimal range. Let Ux be the set of all selected points for x and Vk+1 = x Ux. It should be clear that x u (and rg(x) rg(u)), for every u 2 Ux , and so the inductive process must terminate. Consequently G 6j= '.
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It remains to establish that G j= Tn , i.e., G is of branching n. Suppose otherwise. Then there is a point x in G with m n +1 immediate successors x0 ; : : : ; xm , which are evidently in Ux because F is a tree. We are going to construct a substitution instance of Tn 's axiom bbn which is refuted at x in M. Denote by Æi the conjunction of the formulas in xi . Since all of them are true at xi in M, we have xi j= Æi ; and since i j for no distinct i and j , we have xj 6j= i if i 6= j . Put i = Æi , for 0 i < n, n = Æn _ _ Æm and consider the truth-value of the formula = bbn f 0 =p0; : : : ; n =png at x in M. W Since : ; m, we have x 6j= ni=0 i . Suppose that VnxRxi for every W i = 0; : : W W x 6j= W i=0 (( i ! i=6 j j ) ! i=6 j j ). Then y j= i ! i=6 j j and y 6j= i=6 j j , for some yW2 x" and some i 2 f0; : : : ; ng, and hence y 6j= i . Since xi j= i and xi 6j= i=6 j j , y sees no point in [xi ] and so y 6 x (for otherwise x would not be of minimal range). Therefore, xj y for some j 2 f0; : : : ; mg, and then y j= j if j < n and y j= n if j n, which is a contradiction. V W W It follows that x j= ni=0 (( i ! i=6 j j ) ! i=6 j j ), from which x 6j= , contrary to M being a model for bbn . It remains to notice that every nite frame of branching n is a reduct of a nite n-ary tree, which clearly validates Tn . Another way of obtaining general results on FMP of si-logics is to translate the corresponding results in modal logic with the help of the Preservation Theorem. THEOREM 172. Every si-logic of nite depth (i.e., every logic in ExtBDn , for n < !) is locally tabular. Note, however, that unlike NExtK4, the converse does not hold: the Dummett logic LC, characterized by the class of nite chains (or by the in nite ascending chain), is locally tabular. As we saw in Section 1.7, every non-locally tabular in NExtS4 logic is contained in Grz.3, the only prelocally tabular logic in NExtS4. But in ExtInt this way of determining local tabularity does not work: THEOREM 173 (Mardaev 1984). There is a continuum of pre-locally tabular logics in ExtInt. Besides, it is not clear whether every locally tabular logic in ExtInt (or NExtK4) is contained in a pre-locally tabular one. An intuitionistic formula is said to be essentially negative if every occurrence of a variable in it is in the scope of some :. If ' is essentially negative then T (') is a -formula, which yields
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THEOREM 174 (McKay 1971, Rybakov 1978). If a si-logic L is decidable (or has FMP) and ' is an essentially negative formula then L+' is decidable (has FMP). Originally this result was proved with the help of Glivenko's Theorem (see Section 7 in Intuitionistic Logic). Say that an occurrence of a variable in a formula is essential if it is not in the scope of any :. A formula ' is mild if every two essential occurrences of the same variable in ' are either both positive or both negative. Kuznetsov [1972] claimed (we have not seen the proof) that all si-logics whose extra axioms do not contain negative occurrences of essential variables have FMP. And Wronski [1989] announced that if L is a decidable si-logic and ' a mild formula then L + ' is also decidable. Subframe and co nal subframe si-logics|that is logics axiomatizable by canonical formulas of the form (F) and (F; ?), respectively|can be characterized both syntactically and semantically (see [Zakharyaschev 1996]). THEOREM 175. The following conditions are equivalent for every si-logic L: (i) L is a (co nal) subframe logic; (ii) L is axiomatizable by implicative (respectively, disjunction free) formulas; (iii) L is characterized by a class of nite frames closed under the formation of (co nal) subframes. That all si-logics with disjunction free axioms have FMP was rst proved by McKay [1968] with the help of Diego's [1966] Theorem according to which there are only nitely many pairwise non-equivalent in Int disjunction free formulas in variables p1 ; : : : ; pn (see also [Urquhart 1974]). Since frames for Int contain no clusters, Theorem 58 and its analog for co nal subframe logics reduce in the intuitionistic case to the following result which is due to Chagrova [1986], Rodenburg [1986], Shimura [1993] and Zakharyaschev [1996]. THEOREM 176. All si-logics with disjunction free axioms are elementary (de nable by 89-sentences) and D-persistent. Theorem 68 is translated into the intuitionistic case simply by replacing K4 with Int, with + and with . As a consequence we obtain, for instance, that Ono's [1972] Bn and all other logics whose canonical axioms are built on trees have FMP. Moreover, we also have THEOREM 177 (Sobolev 1977b, Nishimura 1960). All si-logics with extra axioms in one variable have FMP and are decidable.
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In fact Sobolev [1977b] proved a more general (but rather complicated) syntactical suÆcient condition of FMP and constructed a formula in two variables axiomatizing a si-logic without FMP (Shehtman's [1977] incomplete si-logic has also axioms in two variables).
Tabularity By the Blok{Esakia and Preservation Theorems, the situation with tabular logics in ExtInt is the same as in NExtGrz. In particular, L 2 ExtInt is tabular i BDn + BWn L for some n < ! i L is not a sublogic of one of the three pretabular logics in ExtInt, namely LC, BD2 and KC + bd3 . (The pretabular si-logics were described by Maksimova [1972].) The tabularity problem is decidable in ExtInt.
3.5 Disjunction property One of the aims of studying extensions of Int, which may be of interest for applications in computer science, is to describe the class of constructive silogics. At the propositional level a consistent logic L 2 ExtInt is regarded to be constructive if it has the disjunction property (DP, for short) which means that for all formulas ' and ,
'_
2 L implies ' 2 L or 2 L.
That intuitionistic logic itself is constructive in this sense was proved in a syntactic way by Gentzen [1934{1935]. However, Lukasiewicz (1952) conjectured that no proper consistent extension of Int has DP. A similar property was introduced for modal logics (see e.g. [Lemmon and Scott 1977]): L 2 NExtK has the (modal) disjunction property if, for every n 1 and all formulas '1 ; : : : ; 'n ,
'1 _ _ 'n 2 L implies 'i 2 L, for some i 2 f1; : : : ; ng:
The following theorem (in a somewhat di erent form it was proved in [Hughes and Cresswell 1984] and [Maksimova 1986]) provides a semantic criterion of DP. THEOREM 178. Suppose a modal or si-logic L is characterized by a class C of descriptive rooted frames closed under the formation of rooted generated subframes. Then L has DP i , for every n 1 and all F1 ; : : : ; Fn 2 C with roots x1 ; : : : ; xn , there is a frame F for L with root x such that the disjoint union F1 + + Fn is a generated subframe of F with fx1 ; : : : ; xn g x".
Proof. We consider only the modal case. ()) Let FL = hWL ; RL ; PL i be a universal frame for L, big enough to contain F1 + + Fn as its generated subframe. Assuming that FL is associated with a suitable canonical model for L, we show that there is a point x in FL such that x" = WL . The set 0 = f:' : 9y 2 WL y 6j= 'g
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is L-consistent (for otherwise '1 _ _'n 2 L for some '1 ; : : : ; 'n 62 L). Let be a maximal L-consistent extension of 0 and x the point in FL where is true. Then xRL y, for every y 2 WL . (() Suppose otherwise. Then there are formulas '1 ; : : : ; 'n 62 L such that '1 _ _ 'n 2 L. Take frames F1 ; : : : ; Fn 2 C refuting '1 ; : : : ; 'n at their roots, respectively, and let F be a rooted frame for L containing F1 + + Fn as a generated subframe and such that its root x sees the roots of F1 ; : : : ; Fn . Then all the formulas '1 ; : : : ; 'n are refuted at x and so '1 _ _ 'n 62 L, which is a contradiction. It should be clear that if we use only the suÆcient condition of Theorem 178, the requirement that frames in C are descriptive is redundant. Furthermore, it is easy to see that for L 2 NExtK4 we may assume n 2. And clearly a logic L 2 NExtS4 has DP i , for all ' and , ' _ 2 L implies ' 2 L or 2 L. As a direct consequence of the proof above we obtain COROLLARY 179. A modal or si-logic L has DP i the canonical frame FL = hWL ; RL i contains a point x such that x" = WL . Using the semantic criterion above it is not hard to show that DP is preserved under , and . It is also a good tool for proving and disproving DP of logics with transparent semantics. EXAMPLE 180. (i) Let F1 ; : : : ; Fn be serial rooted Kripke frames. Then the frame obtained by adding a root to F1 + + Fn is also serial. Therefore, D has DP. In the same way one can show that K, K4, T, S4, Grz, GL and many other modal logics have DP. (ii) Since no rooted symmetrical frame can contain a proper generated subframe, no consistent logic in NExtKB has DP. The rst proper extensions of Int with DP were constructed by Kreisel and Putnam [1957]: these were KP (now called the Kreisel{Putnam logic) and SL (known as the Scott logic). We present here Gabbay's [1970] proof that KP has DP. THEOREM 181 (Kreisel and Putnam 1957). KP has DP.
Proof. Using ltration one can show that KP is characterized by the class of nite rooted frames F = hW; Ri satisfying the condition (15)
8x; y; z (xRy ^ xRz ^ :yRz ^ :zRy ! 9u (xRu ^ uRy ^ uRz ^ 8v (uRv ! 9w (vRw ^ (yRw _ zRw))))):
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If F is such a frame then for each non-empty X W 1 , the generated subframe of F based on the set W (W 1 X )# is rooted; we denote its root by r(X ). Let F1 = hW1 ; R1 i and F2 = hW2 ; R2 i be nite rooted frames satisfying (15). We construct from them a frame F = hW; Ri by taking
W = W1 [ W2 [ U; where U = fX1 [ X2 : X1 W11 ; X2 W21 ; X1 ; X2 6= ;g, and xRy i (x; y 2 Wi ^ xRi y) _ (x; y 2 U ^ x y) _ (x = X1 [ X2 2 U ^ y 2 Wi ^ r(Xi )Ri y):
It follows from the given de nition that F1 + F2 is a generated subframe of F, W1 [ W2 is a cover for F and W11 [ W21 is its root. So our theorem will be proved if we show that (15) holds. Suppose x; y; z 2 W satisfy the premise of (15). Since (15) holds for F1 and F2 , we can assume that x = X1 [ X2 2 U . Let Y1 [ Y2 and Z1 [ Z2 be the sets of nal points in y" and z", respectively, with Yi ; Zi Wi . By the de nition of R, we have Yi ; Zi Xi . Consider u = (Y1 [ Z1 ) [ (Y2 [ Z2 ). Clearly, xRu, uRy and uRz . Suppose now that v 2 u". Let w be any nal point in v ". Then v 2 (Y1 [ Z1 ) [ (Y2 [ Z2 ) and so either yRw or zRw.
Other examples of constructive si-logics were constructed by Ono [1972] and Gabbay and de Jongh [1974], namely, Bn and Tn . Anderson [1972] proved that among the consistent si-logics with extra axioms in one variable only those of the form Int + nf 2n+2 , for n 5, have DP (for n = 6 the proof was found by Wronski [1974]; see also [Sasaki 1992]). Finally, Wronski [1973] showed that there is a continuum of si-logics with DP. The additional axioms of logics in all these examples contained occurrences of _; on the other hand, known examples of si-logics with disjunction free extra axioms, say LC, KC, Cl, BWn or BDn , were not constructive. This observation led Hosoi and Ono [1973] to the conjecture that the disjunction free fragment of every consistent si-logic with DP coincides with that of Int. We present a proof of this conjecture following [Zakharyaschev 1987]. First we describe the co nal subframe logics in NExtS4 with DP, assuming that every such logic L is represented by its independent canonical axiomatization (16) L = S4 f (Fi ; ?) : i 2 I g:
All frames in the rest of this section are assumed to be quasi-ordered. Say that a nite rooted frame F with 2 points is simple if its root cluster and at least one of the nal clusters are simple. Suppose F = hW; Ri is a
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simple frame, a0 ; a1 ; : : : ; am ; am+1 ; : : : ; an are all its points, with a0 being the root, C (a1 ); : : : ; C (am ) all the distinct immediate cluster-successors of a0 , and an a nal point with simple C (an ). For every k = 1; : : : ; n, de ne a formula k by taking k=
^
ai Raj ;i6=0
'ij ^
n ^ i=1
'i ^ '0? ! pk
V where 'ij , 'i were de ned in Section 3.2 and '0? = ( ni=1 pi ! ?). Now we associate with F the formula (F) = p0 _ 1 if m = 1, and the formula (F) = 1 _ _ m if m > 1. LEMMA 182. For every simple frame F, (F) 2 S4 (F; ?).
Proof. It is enough to show that G 6j= (F) implies G 6j= (F; ?), for any nite G. So suppose (F) is refuted in a nite frame G under some valuation. De ne a partial map f from G onto F by taking f (x) =
8 < :
a0 if x 6j= (F) ai if x 6j= i , 1 i n unde ned otherwise.
One can readily check that f is a subreduction of G to F. However it is not necessarily co nal. So we extend f by putting f (x) = an , for every x of depth 1 in G such that f (x#) = fa0 g. Clearly, the improved map is still a subreduction of G to F, and '0? ensures its co nality. Using the semantical properties of the canonical formulas it is a matter of routine to prove the following LEMMA 183. Suppose i 2 f1; : : : ; mg and G is the subframe of F generated by ai . Then (G; ?) 2 S4 i . We are in a position now to prove a criterion of DP for the co nal subframe logics in NExtS4. THEOREM 184. A consistent co nal subframe logic L 2 NExtS4 has the disjunction property i no frame Fi in its independent axiomatization (16) is simple, for i 2 I .
Proof. ()) Suppose, on the contrary, that Fi is simple, for some i 2 I . Since the axiomatization (16) is independent, every proper generated subframe of Fi validates L. By Lemma 182, (Fi ) 2 L and so either p0 2 L or j 2 L. However, both alternatives are impossible: the former means that L is inconsistent, while the latter, by Lemma 183, implies (G; ?) 2 L, where G is the subframe of Fi generated by an immediate successor of Fi 's root.
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AA G1 AA G2 A A AÆ Æ y AÆ I@ 6 @ @ Æ
x
Figure 15. (() Given two nite rooted frames G1 and G2 for L, we construct the frame F as shown in Fig. 15 and prove that F j= L. Suppose otherwise, i.e., there exists a co nal subreduction f of F to Fi , for some i 2 I . Let xi be the root of Fi . Since G1 and G2 are not co nally subreducible to Fi and since L is consistent, f 1 (xi ) = fxg. By the co nality condition, it follows in particular that y 2 domf . But then Fi is simple, which is a contradiction. Thus, by Theorem 178, L has DP. Note that in fact the proof of ()) shows that if L 2 NExtS4, F is a simple frame, (F; ?) 2 L and (G; ?) 62 L for any proper generated subframe G of F then L does not have DP. Transferring this observation to the intuitionistic case, we obtain THEOREM 185 (Minari 1986, Zakharyaschev 1987). If a si-logic is consistent and has DP then the disjunction free fragments of L and Int are the same. SuÆcient conditions of DP in terms of canonical formulas can be found in [Chagrov and Zakharyaschev 1993, 1997]. Since classical logic is not constructive, it is of interest to nd maximal consistent si-logics with DP. That they exist follows from Zorn's Lemma. Here is a concrete example of such a logic. Trying to formalize the proof interpretation of intuitionistic logic, Medvedev [1962] proposed to treat intuitionistic formulas as nite problems. Formally, a nite problem is a pair hX; Y i of nite sets such that Y X and X 6= ;; elements in X are called possible solutions and elements in Y solutions to the problem. The operations on nite problems, corresponding to the logical connectives, are de ned as follows:
hX1 ; Y1 i ^ hX2 ; Y2 i = hX1 X2 ; Y1 Y2 i ; hX1 ; Y1 i _ hX2 ; Y2 i = hX1 t X2 ; Y1 t Y2 i ; D E hX1 ; Y1 i ! hX2 ; Y2 i = X2X ; ff 2 X2X : f (Y1 ) Y2 g ; 1
1
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
Æ
Æ 1Æ6@I1Æ@I1Æ6 I @ 6 @ @@ @@ @ Æ Æ Æ Æ Æ Æ Æ Æ Æ 6@I@ @I@ 6 6@I@@I@61@I@161 @ @ @ @ @ Æ I@ Æ ÆI@@ Æ6 Æ Æ@I@ Æ6 Æ 1Æ @ @Æ @Æ @Æ Figure 16.
? = hX; ;i : Here X t Y = (X f1g) [ (Y f2g) and X Y is the set of all functions from X into Y . Note that in the de nition of ? the set X is xed, but arbitrary; for de niteness one can take X = f;g. Now we can interpret formulas by nite problems. Namely, given a formula ', we replace its variables by arbitrary nite problems and perform the operations corresponding to the connectives in '. If the result is a problem with a non-empty set of solutions no matter what nite problems are substituted for the variables in ', then ' is called nitely valid. One can show that the set of all nitely valid formulas is a si-logic; it is called Medvedev's logic and denoted by ML. In fact, ML can be de ned semantically. Medvedev [1966] showed that ML coincides with the set of formulas that are valid in all frames Bn having the form of the n-ary Boolean cubes with the topmost point deleted; for n = 1; 2; 3; 4, the Medvedev frames are shown in Fig. 16. Since Bn + Bm is a generated subframe of Bn+m , ML has DP. Moreover, Levin [1969] proved that it has no proper consistent extension with DP. The following proof of this result is due to Maksimova [1986]. THEOREM 186 (Levin 1969). ML is a maximal si-logic with DP.
Proof. Suppose, on the contrary, that there exists a proper consistent extension L of ML having DP. Then we have a formula ' 2 L ML. We show rst that there is an essentially negative substitution instance ' of ' such that ' 62 ML. Since '(p1 ; : : : ; pn ) 62 ML, there is a Medvedev frame Bm refuting ' under some valuation V. With every point x in Bm we associate a new variable qx and extend V to these variables by taking V(qx ) to be the set of nal points in Bm that are not accessible from x. By the construction of Bm , we have y j= :qx i y 2 x", from which
V(
_
x2V(pi )
:qx ) = V(pi ):
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217
W W Let ' = '( x2V(p1 ) :qx ; : : : ; x2V(pn) :qx ). It follows that V(' ) = V(') and so ' 62 ML. Thus, we may assume that ' is an essentially negative formula. Since KP ML, ML contains the formulas
ndk = (:p ! :q1 _ _ :qk ) ! (:p ! :q1 ) _ _ (:p ! :qk ) which, as is easy to see, belong to KP. Let us consider the logic
ND = Int + fndk : k 1g: Using the fact that the outermost ! in ndk can be replaced with $ and that (:p ! :q) $ :(:p ^ q) 2 Int, one can readily show that every essentially negative formula is equivalent in ND to the conjunction of formulas of the form : 1 _ _: l . So L ML contains a formula of the form : 1 _ _: l . Since L has DP, : i 2 L for some i. But then, by Glivenko's Theorem, : i 2 ML, which is a contradiction. REMARK. ML is not nitely axiomatizable, as was shown by Maksimova et al. [1979]. Nobody knows whether it is decidable. It turns out, however, that ML is not the unique maximal logic with DP in ExtInt. Kirk [1982] noted that there is no greatest consistent si-logic with DP. Maksimova [1984] showed that there are in nitely many maximal constructive si-logics, and Chagrov [1992a] proved that in fact there are a continuum of them; see also Ferrari and Miglioli [1993, 1995a, 1995b]. Galanter [1990] claims that each si-logic characterized by the class of frames of the form
hfW : W f1; : : : ; ng; W 6= ;; jW j 62 N g; i ; where n = 1; 2; : : : and N is some xed in nite set of natural numbers, is a maximal si-logic with DP.
3.6 Intuitionistic Modal Logics All modal logics we have dealt with so far were constructed on the classical non-modal basis. It can be replaced by logics of other types. For instance, one can consider modal logics based on relevant logic (see e.g. [Fuhrmann 1989]) or many-valued logics (see e.g. [Segerberg 1967], [Morikawa 1989], [Ostermann 1988]), and many others. In this section we brie y discuss modal logics with the intuitionistic basis. Unlike the classical case, the intuitionistic and are not supposed to be dual, which provides more possibilities for de ning intuitionistic modal logics. For a non-empty set M of modal operators, let LM be the standard propositional language augmented by the connectives in M. By an
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
intuitionistic modal logic in the language LM we understand any subset of LM containing Int and closed under modus ponens, substitution and the regularity rule ' ! = ' ! , for every 2 M. There are three ways of de ning intuitionistic analogues of (classical) normal modal logics. First, one can take the family of logics extending the basic system IntK in the language L which is axiomatized by adding to Int the standard axioms of K
(p ^ q) $ p ^ q and >: An example of a logic in this family is Kuznetsov's [1985] intuitionistic provability logic I4 (Kuznetsov used 4 instead of ), the intuitionistic analog of the provability logic GL. It can be obtained by adding to IntK (and even to Int) the axioms
p ! p; (p ! p) ! p; ((p ! q) ! p) ! (q ! p): A model theory for logics in NExtIntK was developed by Ono [1977], Bozic and Dosen [1984], Dosen [1985a], Sotirov [1984] and Wolter and Zakharyaschev [1997, 1999a]; we discuss it below. Font [1984, 1986] considered these logics from the algebraic point of view, and Luppi [1996] investigated their interpolation property by proving, in particular, that the superamalgamability of the corresponding varieties of algebras is equivalent to interpolation. A possibility operator in logics of this sort can be de ned in the classical way by taking ' = ::'. Note, however, that in general this does not distribute over disjunction and that the connection via negation between and is too strong from the intuitionistic standpoint (actually, the situation here is similar to that in intuitionistic predicate logic where 9 and 8 are not dual.) Another family of \normal" intuitionistic modal logics can be de ned in the language L by taking as the basic system the smallest logic in L to contain the axioms
(p _ q) $ p _ q and :?; it will be denoted by IntK . Logics in NExtIntK were studied by Bozic and Dosen [1984], Dosen [1985a], Sotirov [1984] and Wolter [1997e]. Finally, we can de ne intuitionistic modal logics with independent and . These are extensions of IntK, the smallest logic in the language L containing both IntK and IntK . Fischer Servi [1980, 1984] constructed a logic in NExtIntK by imposing a weak connection between the necessity and possibility operators:
FS = IntK (p ! q) ! (p ! q) (p ! q) ! (p ! q):
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219
A remarkable feature of FS is that the standard translation ST of modal formulas into rst order ones (see Correspondence Theory) not only embeds K into classical predicate logic but also FS into intuitionistic rst order logic: ' belongs to the former i ST (') is a theorem of the latter. According to Simpson [1994], this result was proved by C. Stirling; see also Grefe [1997]. Various extensions of FS were studied by Bull [1966a], Ono [1977], Fischer Servi [1977, 1980, 1984], Amati and Pirri [1994], Ewald [1986], Wolter and Zakharyaschev [1997], Wolter [1997e]. The best known one is probably the logic MIPC = FS p ! p p ! p p ! p p ! p p ! p p ! p introduced by Prior [1957]. Bull [1966a] noticed that the translation de ned by (pi ) = Pi (x), ? = ?, ( ) = , for 2 f^; _; !g, ( ) = 8x , ( ) = 9x is an embedding of MIPC into the monadic fragment of intuitionistic predicate logic. Ono [1977], Ono and Suzuki [1988], Suzuki [1990], and Bezhanishvili [1998] investigated the relations between logics in NExtMIPC and superintuitionistic predicate logics induced by that translation. In what follows we restrict attention only to the classes of intuitionistic modal logics introduced above. An interesting example of a system not covered here was constructed by Wijesekera [1990]. A general model theory for such logics is developed by Sotirov [1984] and Wolter and Zakharyaschev [1997]. Let us consider rst the algebraic and relational semantics for the logics introduced above. All the semantical concepts to be de ned below turn out to be natural combinations of the corresponding notions developed for classical modal and si-logics. For details and proofs we refer the reader to Wolter and Zakharyaschev [1997, 1999a]. From the algebraic point of view, every logic L 2 NExtIntKM , for M f; g, corresponds to the variety of Heyting algebras with one or two operators validating L. The variety of algebras for IntKM will be called the variety of M-algebras. To construct the relational representations of M-algebras, we de ne a frame to be a structure of the form hW; R; R ; P i in which hW; R; P i is an intuitionistic frame, R a binary relation on W such that R Æ R Æ R = R and P is closed under the operation X = fx 2 W : 8y 2 W (xR y ! y 2 X )g:
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
A -frame has the form hW; R; R ; P i, where hW; R; P i is again an intuitionistic frame, R a binary relation on W satisfying the condition
R
1ÆR
ÆR
1
= R
and P is closed under
X = fx 2 W : 9y 2 X xR yg: Finally, a -frame is a structure hW; R; R ; R ; P i the unimodal reducts hW; R; R ; P i and hW; R; R ; P i of which are - and -frames, respectively. (To see why the intuitionistic and modal accessibility relations are connected by the conditions above the reader can construct in the standard way the canonical models for the logics under consideration. The important point here is that we take the Leibnizean de nition of the truth-relation for the modal operators. Other de nitions may impose di erent connecting conditions; see below.) Given a -frame F = hW; R; R ; R; P i, it is easy to check that its dual
F+ = hP; \; [; !; ;; ; i is a -algebra. Conversely, for each -algebra A = hA; ^; _; !; ?; ; i we can de ne the dual frame
A+ = hW; R; R ; R ; P i by taking hW; R; P i to be the dual of the Heyting algebra hA; ^; _; !; ?i and putting r1 R r2 i 8a 2 A (a 2 r1 ! a 2 r2 );
r1 R r2 i 8a 2 A (a 2 r2 ! a 2 r1 ): A+ is a -frame and, moreover, A = (A+ )+ . Using the standard technique
of the model theory for classical modal and si-logics, one can show that a -frame F is isomorphic to its bidual (F+ )+ i F = hW; R; R; R ; P i is descriptive, i.e., hW; R; P i is a descriptive intuitionistic frame and, for all x; y 2 W , xR y i 8X 2 P (x 2 X ! y 2 X );
xR y i 8X 2 P (y 2 X ! x 2 X ):
Thus we get the following completeness theorem. THEOREM 187. Every logic L 2 NExtIntK is characterized by a suitable class of (descriptive) -frames, e.g. by the class fA+ : A j= Lg. Similar results hold for logics in NExtIntK and NExtIntK.
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As usual, by a Kripke frame we understand a frame hW; R; R ; R ; P i in which P consists of all R-cones; in this case we omit P . An intuitionistic modal logic L is D-persistent if the underlying Kripke frame of each descriptive frame for L validates L. For example, FS as well as the logics
L(k; l; m; n) = IntK k l p ! mn p; for k; l; m; n 0 are D-persistent and so Kripke complete (see Wolter and Zakharyaschev [1997]). Descriptive frames validating FS satisfy the conditions
! 9z (yRz ^ xR z ^ xR z ); xR y ! 9z (xRz ^ zR y ^ zRy); xR y
and those for L(k; l; m; n) satisfy
xRk y ^ xRm y ! 9u (yRl u ^ zRn u): It follows, in particular, that MIPC is D-persistent; its Kripke frames have the properties: R is a quasi-order, R = R1 and R = R Æ (R \ R ). On the contrary, I4 is not D-persistent, although it is complete with respect to the class of Kripke frames hW; R; R i such that hW; R i is a frame for GL and R the re exive closure of R . The next step in constructing duality theory of M-algebras and M-frames is to nd relational counterparts of the algebraic operations of forming homomorphisms, subalgebras and direct products. Let F = hW; R; R ; R ; P i be a -frame and V a non-empty subset of W such that
8x 2 V 8y 2 W (xR y _ xRy ! y 2 V ); 8x 2 V 8y 2 W (xR y ! 9z 2 V (xR z ^ yRz )): Then G = hV; R V; R V; R V; fX \ V : X 2 P gi is also a -frame
which is called the subframe of F generated by V . The former of the two conditions above is standard: it requires V to be upward closed with respect to both R and R . However, the latter one does not imply that V is upward closed with respect to R : the frame G in Fig. 17 is a generated subframe of F, although the set fx; z g is not an R -cone in F. This is one di erence from the standard (classical modal or intuitionistic) case. Another one arises when we de ne the relational analog of subalgebras. Given -frames F = hW; R; R ; R ; P i and G = hV; S; S ; S ; Qi, we say a map f from W onto V is a reduction of F to G if f 1(X ) 2 P for every X 2 Q and, for all x; y 2 W and u 2 V , xRy implies f (x)Sf (y), xR y implies f (x)S f (y), for 2 f; g, f (x)Su implies 9z 2 f 1 (u) xRz ,
222
M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV y R -z Æ Æ
KA RA R F AÆx
z
Æ
6R
G Æx
Figure 17. 1R 4
Æ0
Æ
6R
Æ
2
F
01S 4
Æ
Æ
Æ S 6 I @ 6 S @ S SS@Æ Æ
KA RA R AÆ
G 2
3
3
Figure 18.
f (x)S u implies 9z 2 f 1 (u) xR z , f (x)S u implies 9z 2 W (xR z ^ uSf (z )). Again, the last condition di ers from the standard one: given f (x)S f (y), in general we do not have a point z such that xR z and f (y) = f (z ), witness the map gluing 0 and 1 in the frame F in Fig. 18 and reducing it to G. Note that both these concepts coincide with the standard ones in classical modal frames, where R and S are the diagonals. The relational counterpart of direct products|disjoint unions of frames|is de ned as usual. THEOREM 188. (i) If G is the subframe of a -frame F generated by V then the map h de ned by h(X ) = X \V , for X an element in F+ , is a homomorphism from F+ onto G+ .
(ii) If h is a homomorphism from a -algebra A onto a -algebra B then the map h+ de ned by h+ (r) = h 1 (r), r a prime lter in B, is an isomorphism from B+ onto a generated subframe of A+.
(iii) If f is a reduction of a -frame F to a -frame G then the map f + de ned by f + (X ) = f 1 (X ), X an element in G+ , is an embedding of G+ into F+ .
(iv) If B is a subalgebra of a -algebra A then the map f de ned by f (r) = r \ B , r a prime lter in A and B the universe of B, is a reduction of A+ to B+ .
This duality can be used for proving various results on modal de nability. For instance, a class C of -frames is of the form C = fF : F j= g, for
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223
some set of L -formulas, i C is closed under the formation of generated subframes, reducts, disjoint unions, and both C and its complement are closed under the operation F 7! (F+ )+ (see Wolter and Zakharyaschev [1997]). Moreover, one can extend Fine's Theorem connecting the rst order de nability and D-persistence of classical modal logics to the intuitionistic modal case: THEOREM 189. If a logic L 2 NExtIntK is characterized by an elementary class of Kripke frames then L is D-persistent. These results may be regarded as a justi cation for the relational semantics introduced in this section. However, it is not the only possible one. For example, Bozic and Dosen [1984] impose a weaker condition on the connection between R and R in -frames. Fisher Servi [1980] interprets FS in birelational Kripke frames of the form hW; R; S i in which R is a partial order, R Æ S S Æ R, and
xRy ^ xSz ! 9u (ySu ^ zRu): The intuitionistic connectives are interpreted by R and the truth-conditions for and are de ned as follows
X = fx 2 W : 8y; z (xRySz ! z 2 X g; X = fx 2 W : 9y 2 X xSyg:
In birelational frames for MIPC S is an equivalence relation and
xSyRz ! 9u xRuSz: These frames were independently introduced by L. Esakia who also established duality between them and \monadic Heyting algebras". There are two ways of investigating various properties of intuitionistic modal logics. One is to continue extending the classical methods to logics in NExtIntKM . Another one uses those methods indirectly via embeddings of intuitionistic modal logics into classical ones. That such embeddings are possible was noticed by Shehtman [1979], Fischer Servi [1980, 1984], and Sotirov [1984]. Our exposition here follows Wolter and Zakharyaschev [1997, 1999a]. For simplicity we con ne ourselves only to considering the class NExtIntK and refer the reader to the cited papers for information about more general embeddings. Let T be the translation of L into LI pre xing I to every subformula of a given L -formula. Thus, we are trying to embed intuitionistic modal logics in NExtIntK into classical bimodal logics with the necessity operators I (of S4) and . Say that T embeds L 2 NExtIntK into M 2 NExt(S4 K) (S4 in LI and K in L ) if, for every ' 2 L , ' 2 L i T (') 2 M:
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
In this case M is called a bimodal (or BM-) companion of L. For every logic M 2 NExt(S4 K) put
M = f' 2 L : T (') 2 M g; and let be the map from NExtIntK into NExt(S4 K) de ned by
(IntK ) = (Grz K) mix T ( ); where L and mix = I I p $ p. (The axiom mix re ects the condition R Æ R Æ R = R of -frames.) Then we have the following extension of the embedding results of Maksimova and Rybakov [1974], Blok [1976] and Esakia [1979a,b]: THEOREM 190. (i) The map is a lattice homomorphism from the lattice NExt(S4 K) onto NExtIntK preserving decidability, Kripke completeness, tabularity and the nite model property. (ii) Each logic IntK is embedded by T into any logic M in the interval (S4 K) T ( ) M (Grz K) mix T ( ): (iii) The map is an isomorphism from the lattice NExtIntK onto the lattice NExt(Grz K) mix preserving FMP and tabularity. Note that Fischer Servi [1980] used another generalization of the Godel translation. She de ned T (') = T ('); T (') = I T (') and showed that this translation embeds FS into the logic (S4 K) I p ! I p I p ! I p: It is not clear, however, whether all extensions of FS can be embedded into classical bimodal logics via this translation. Let us turn now to completeness theory of intuitionistic modal logics. As to the standard systems I4 , FS, and MIPC, their FMP can be proved by using (sometimes rather involved) ltration arguments; see Muravitskij [1981], Simpson [1994] and Grefe [1997], and Ono [1977], respectively. Further results based on the ltration method were obtained by Sotirov [1984] and Ono [1977]. However, in contrast to classical modal logic, only a few general completeness results covering interesting classes of intuitionistic modal logics are known. The proofs of the following two theorems are based on the translation into classical bimodal logics discussed above.
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THEOREM 191. Suppose that a si-logic Int + has one of the properties: decidability, Kripke completeness, FMP. Then the logics IntK and IntK p ! p also have the same property.
Proof. It suÆces to show that there is a BM-companion of each of these systems satisfying the corresponding property. Notice that
((S4 T ( )) K) = IntK ; ((S4 T ( )) (K p ! p)) = IntK p ! p: So it remains to use the fact that if Int + has one of the properties under consideration then its smallest modal companion S4 T ( ) has this property as well (Table 7), and if L1, L2 are unimodal logics having one of those properties then the fusion L1 L2 also enjoys the same property
(Theorem 111).
Such a simple reduction to known results in classical modal logic is not available for logics containing IntK4 = IntK p ! p. However, by extending Fine's [1974] method of maximal points to bimodal companions of extensions of IntK4 Wolter and Zakharyaschev [1999a] proved the following: THEOREM 192. Suppose L IntK4 has a D-persistent BM-companion M (S4 K4) mix whose Kripke frames are closed under the formation of substructures. Then (i) for every set of intuitionistic negation and disjunction free formulas, L has FMP; (ii) for every set n 1,
of intuitionistic disjunction free formulas and every
L
n _ i=0
(pi !
_
j 6=i
pj )
has the nite model property.
One can use this result to show that the following (and many other) intuitionistic modal logics enjoy FMP: (1) IntK4 ;
(2) IntS4 = IntK4 p ! p (R is re exive);
(3) IntS4:3 = IntS4 (p ! q) _ (q ! p) (R is re exive and connected); (4) IntK4 p _ :p (R is symmetrical);
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
(5) IntK4 p _ :p (R is Euclidean); (6) IntK4 p _ :p (xRy ^ xR z ! yR z ). We conclude this section with some remarks on lattices of intuitionistic modal logics. Wolter [1997e] uses duality theory to study splittings of lattices of intuitionistic modal logics. For example, he showed that each nite rooted frame splits NExt(L n p ! n+1p), for L = IntK and L = FS, and each R -cycle free nite rooted frame splits the lattices of extensions of IntK and FS. No positive results are known, however, for the lattice NExtIntK . In fact, the behavior of -frames is quite di erent from that of frames for FS. For instance, in classical modal logic we have RGF = GRF , for each class of frames (or even -frames) F , where G and R are the operations of forming generated subframes and reducts, respectively. But this does not hold for -frames. More precisely, there exists a nite -frame G such that RGfGg 6 GRfGg. In other terms, the variety of modal algebras for K has the congruence extension property (i.e., each congruence of a subalgebra of a modal algebra can be extended to a congruence of the algebra itself) but this is not the case for the variety of -algebras. Vakarelov [1981, 1985] and Wolter [1997e] investigate how logics having Int as their non-modal fragment are located in the lattices of intuitionistic modal logics. It turns out, for instance, that in NExtIntK the inconsistent logic has a continuum of immediate predecessors all of which have Int as their non-modal fragment, but no such logic exists in the lattice of extensions of IntK . For a recent methodological approach to combining logics, see [Gabbay, 1988]. 4 ALGORITHMIC PROBLEMS All algorithmic results considered in the previous sections were positive: we presented concrete procedures for deciding whether an arbitrary given formula belongs to a given logic in some class or whether it axiomatizes a logic with a certain property. What is the complexity of those decision algorithms? Do there exist undecidable calculi18 and properties? These are the main questions we address in this chapter.
4.1 Undecidable calculi The rst undecidable modal and si-calculi were constructed by Thomason [1975c] (polymodal and unimodal), Isard [1977] (unimodal) and Shehtman 18 By a calculus we mean a logic with nitely many axioms (inference rules in our case are xed).
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227
[1978c] (superintuitionistic). However, we begin with the very simple example of [Shehtman 1982] which is a modal reformulation of the undecidable associative calculus T of [Tseitin 1958]. The axioms of T are
ac = ca; ad = da; bc = cb; bd = db; edb = be; eca = ae; abac = abacc: The reader will notice immediately an analogy between them and the axioms of the following modal calculus with ve necessity operators:
1 3 p $ 3 1 p 1 4 p $ 4 1 p 2 3 p $ 3 2 p 2 4 p $ 4 2 p 5 4 2p $ 2 5 p 5 3 1 p $ 15 p 1 2 13 p $ 1 2 1 33 p: Moreover, it is not hard to see that words x, y in the alphabet fa; b; c; d; eg are equivalent in T 19 i f (x)p $ f (y)p 2 K5 , where f is the natural L = K5
one-to-one correspondence between such words and modalities in language f1; : : : ; 5 g under which, for instance, f (cadedb) = 3 1 4 5 42 . It follows immediately that L is undecidable. Using the undecidable associative calculus of Matiyasevich [1967], one can construct in the same way an undecidable bimodal calculus having three reductions of modalities as its axioms. It is unknown whether there is an undecidable unimodal calculus axiomatizable by reductions of modalities. Another simple way of proving undecidability, known as the domino or tiling technique, was suggested by Harel [1983]. It is particularly useful in the case of multi-dimensional modal logics, say Cartesian products. Tiles can be thought of as 4-tuples of colours
t = hleft(t); right(t); up(t); down(t)i : A nite set T of tiles is said to tile N N if there is a map : N N such that for all i; j 2 N ,
7! T
up( (i; j )) = down( (i; j + 1)) and right( (i; j )) = left( (i + 1; j )). If we think of a tile as a physical 1 1-square with colours along its four edges, then a tiling of N N is just a way of placing an in nite number of 19 I.e., they can be obtained from each other by a nite number of transformations of the form w1 ww2 ! w1 vw2 , where w = v or v = w is an axiom of T .
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
tiles, each of a type from T , together to cover the rst quarter of the in nite plane, with no rotation of the tiles allowed and the colours on adjacent edges of adjacent tiles matching. The tiling problem for N N is formulated as follows: \given a nite set T of tiles, does T tile N N ?" Robinson [1971] proved that this problem is undecidable (in fact, co-r.e.-complete). We will demonstrate the use of tiling to show the undecidability of the logic (K K)u , i.e., the square of K (with boxes and ) extended with the universal modality (see Section 2.2); this result is due to Spaan [1993]. Given a nite set T of tiles, construct a formula 'T as the conjunction of the following formulas: W pt ; Vt2T t=6 t0 :(pt ^ pt0 ); Vt2T (pt ! Wup(t)=down(t0 ) pt0 ); Vt2T (pt ! Wright(t)=left(t0 ) pt0 ); (> ^ >): It is easily seen (see e.g. [Spaan 1993] or [Marx 1999]) that 'T is satis able in the product of two frames i T tiles N N . It follows that (K K)u is undecidable. Thomason's simulation and the undecidable polymodal calculi mentioned above provide us with examples of undecidable calculi in NExtK. However, to nd axioms of undecidable unimodal calculi with transitive frames, as well as undecidable si-calculi, a more sophisticated construction is required. Instead of associative calculi, let us use now Minsky machines with two tapes (or register machines with two registers). A Minsky machine is a nite set (program) of instructions for transforming triples hs; m; ni of natural numbers, called con gurations. The intended meaning of the current con guration hs; m; ni is as follows: s is the number (label) of the current machine state and m, n represent the current state of information. Each instruction has one of the four possible forms:
s ! ht; 1; 0i ; s ! ht; 0; 1i ; s ! ht; 1; 0i (ht0 ; 0; 0i); s ! ht; 0; 1i (ht0 ; 0; 0i): The last of them, for instance, means: transform hs; m; ni into ht; m; n 1i if n > 0 and into ht0 ; m; ni if n = 0. For a Minsky machine P , we shall write P : hs; m; ni ! ht; k; li if starting with hs; m; ni and applying the instructions in P , in nitely many steps (possibly, in 0 steps) we can reach ht; k; li. We shall use the well known fact (see e.g. [Mal'cev 1970]) that the following con guration problem is undecidable: given a program P and con gurations hs; m; ni, ht; k; li, determine whether P : hs; m; ni ! ht; k; li.
ADVANCED MODAL LOGIC
X d 6yXXXXyXXXd1 ÆX X yX y g yXXXd2 a XXX 1 X X I@X0 XyXXXg2 g@ I a0 @ 6 @a10 @I@2 6 a0 0 a1 a11 6a21 6 6 . a02 . a12 .6a22 .. .. .. a0t 1 a1k 1 a2l 1 .6a0t .6a1k *.6a2l
229
b
..J ]J
..
: : : : :J: e(t; k; l)
..
Figure 19. With every program P and con guration hs; m; ni we associate the transitive frame F depicted in Fig. 19. Its points e(t; k; l) represent con gurations ht; k; li such that P : hs; m; ni ! ht; k; li; e(t; k; l) sees the points a0t , a1k , a2l representing the components of ht; k; li. The following variable free formulas characterize points in F in the sense that each of these formulas, denoted by Greek letters with subscripts and/or superscripts, is true in F only at the point denoted by the corresponding Roman letter with the same subscript and/or superscript:
= > ^ >; = ?; = ^ ^ :2 ;
Æ = : ^ ^ :2 ; Æ1 = Æ ^ :2 Æ; Æ2 = Æ1 ^ :2 Æ1 ;
1 = ^ :2 ^ :Æ; 2 = 1 ^ :2 1 ^ :Æ; 00 = ^ Æ ^ :2 ^ :2 Æ; 10 = 1 ^ Æ1 ^ :2 1 ^ :2 Æ1 ; 20 = 2 ^ Æ2 ^ :2 2 ^ :2 Æ2 ; ^ ij+1 = ij ^ :2 ij ^ : k0 ; i= 6 k where i 2 f0; 1; 2g, j 0. The formulas characterizing e(t; k; l) are denoted by (t; 1k ; 2l ), where (t; '; ) =
t ^ i=0
0i ^ : 0t+1 ^ ' ^ :2 ' ^ ^ :2 :
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
We require also formulas characterizing not only xed but arbitrary con gurations: 1 = ( 10 _ 10 ) ^ : 00 ^ : 20 ^ p1 ^ :p1 ; 2 = 10 ^ : 00 ^ : 20 ^ p1 ^ :2 p1 ; 1 = ( 20 _ 20 ) ^ : 00 ^ : 10 ^ p2 ^ :p2 ; 2 = 20 ^ : 00 ^ : 10 ^ p2 ^ :2 p2 : Now we are fully equipped to simulate the behavior of Minsky machines by means of modal formulas. Let us consider for simplicity only tense logics and observe that F satis es the condition
8x8y9z (xRzR 1 y _ xR 1 zRy _ xRy _ xR 1 y _ x = y): So, for every valuation in F, a formula ' is true at some point in F i the formula
' = 1 ' _ 1 ' _ ' _ 1 ' _ ' is true at all points in F, i.e., the modal operator can be understood as \omniscience". Let be a formula which is refuted in F and does not contain p1 and p2 . With each instruction I in P we associate a formula AxI by taking: AxI = : ^ (t; 1 ; 1 ) ! : ^ (t0 ; 2 ; 1 ) if I has the form t ! ht0 ; 1; 0i,
AxI = : ^ (t; 1 ; 1 ) ! : ^ (t0 ; 1 ; 2 )
if I is t ! ht0 ; 0; 1i, AxI = (: ^ (t; 2 ; 1 ) ! : ^ (t0 ; 1 ; 1 )) ^ (: ^ (t; 10 ; 1 ) ! : ^ (t00 ; 10 ; 1 )) if I is t ! ht0 ; 1; 0i (ht00 ; 0; 0i), AxI = (: ^ (t; 1 ; 2 ) ! : ^ (t0 ; 1 ; 1 )) ^ (: ^ (t; 1 ; 20 ) ! : ^ (t00 ; 1 ; 20 )) if I is t ! ht0 ; 0; 1i (ht00 ; 0; 0i). The formula simulating P as a whole is
AxP =
^
I 2P
AxI:
Now, by induction on the length of computations and using the frame F in Fig. 19 one can show that for every program P and con gurations hs; m; ni, ht; k; li, we have P : hs; m; ni ! ht; k; li i
: ^ (s; 1m ; 2n ) ! : ^ (t; 1k ; 2l ) 2 K4:t AxP:
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231
Thus, if the con guration problem is undecidable for P then the tense calculus K4:t AxP is undecidable too. In the same manner (but using somewhat more complicated frames and formulas) one can construct undecidable calculi in NExtK4 and even ExtInt; for details consult [Chagrova, 1991] and [Chagrov and Zakharyaschev, 1997]. The following table presents some "quantitative characteristics" of known undecidable calculi in various classes of logics. Its rst line, for instance, means that there is an undecidable si-calculus with axioms in 4 variables and the derivability problem in it is undecidable in the class of formulas in 2 variables; = means that the number of variables is optimal, and indicates that the optimal number is still unknown. The number of variables in Class of logics undecidable calculi separated formulas ExtInt 4; 2 =2 NExtS4 3; 2 =1 ExtS4 3 =1 NExtGL =1 =1 ExtGL =1 =1 ExtS =1 =1 NExtK4 =1 =0 ExtK4 =1 =0 These observations follow from [Anderson, 1972; Chagrov, 1994; Sobolev, 1977a] and [Zakharyaschev, 1997a]. Say that a formula is undecidable in (N)ExtL if no algorithm can determine for an arbitrary given ' whether 2 L + ' (respectively, 2 L '). For example, formulas in one variable, the axioms of BWn and BDn are decidable in ExtInt. On the other hand, there are purely implicative undecidable formulas in ExtInt, and
:(p ^ q) _ :(:p ^ q) _ :(p ^ :q) _ :(:p ^ :q) is the shortest known undecidable formula in this class. Here are some modal examples: the formula (2 ? ! p _ :p) is undecidable in NExtGL, +:+ p _ + :+ :+ p in ExtS, ? in ExtK4 and NExtK4:t; in NExtK and NExtK4:t undecidable is the conjunction of axioms of any consistent tabular logic in these classes. However, no non-trivial criteria are known for a formula to be decidable; it is unclear also whether one can e ectively recognize the decidability of formulas in the classes ExtInt, (N)ExtS4, (N)ExtGL, ExtS, (N)ExtK4.
4.2 Admissibility and derivability of inference rules Another interesting algorithmic problem for a logic L is to determine whether an arbitrary given inference rule '1 ; : : : ; 'n =' is derivable in L, i.e., ' is
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
derivable in L from the assumptions '1 ; : : : ; 'n , and whether it is admissible in L, i.e., for every substitution s, 's 2 L whenever '1 s; : : : ; 'n s 2 L. (Note that derivability depends on the postulated inference rules in L, while admissibility depends only on the set of formulas in L.) Admissible and derivable rules are used for simplifying the construction of derivations. Derivable rules, like the well known rule of syllogism
'! ; ! ; '! may replace some fragments of xed length in derivations, thereby shortening them linearly. Admissible rules in principle may reduce derivations more drastically. Since ' 2 L i the rule >=' is derivable (or admissible) in L, the derivability and admissibility problems for inference rules may be regarded as generalizations of the decidability problem. If the only postulated rules in L are substitution and modus ponens, the Deduction Theorem reduces the derivability problem for inference rules in L to its decidability: '1 ; : : : ; 'n is derivable in L i '1 ^ ^ 'n ! 2 L: However, if the rule of necessitation '=' is also postulated in L, we have only '1 ; : : : ; 'n is derivable in L i '1 ; : : : ; 'n `L : For n-transitive L this is equivalent to n ('1 ^ ^ 'n ) ! 2 L, and so the derivability problem for inference rules in n-transitive logics is decidable i the logics themselves are decidable. In general, in view of the existential quanti er in Theorem 1, the situation is much more complicated. Notice rst that similarly to Harrop's Theorem, a suÆcient condition for the derivability problem to be decidable in a calculus is its global FMP (see Section 1.5). Thus we have THEOREM 193. The derivability problem for inference rules in K, T, D, KB is decidable. Moreover, sometimes we can obtain an upper bound for the parameter m in the Deduction Theorem, which also ensures the decidability of the derivability problem for inference rules. One can prove, for instance, that for K it is enough to take m = 2jSub'[Sub j . In general, however, the derivability problem for inference rules in a logic L turns out to be more complex than the decidability problem for L. (Recall, by the way, that there are logics with FMP but not global FMP.) THEOREM 194 (Spaan 1993). There is a decidable calculus in NExtK the derivability problem for inference rules in which is undecidable.
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233
Spaan proves this result by simulating in `L , L the decidable logic de ned below, the tiling problem for N N . The logic L is surprisingly simple:
L = Alt2
^
1i4
pi !
_
1i<j 4
(pi ^ pj ):
It is a subframe logic, so it is D-persistent and has FMP (because Alt2 L; see Theorem 22 and Proposition 59). Note also that the bimodal logic Lu (see Section 2.2) is a complete and elementary subframe logic which is undecidable because `L is undecidable. Using this observation one can construct a unimodal subframe logic in NExtK with the same properties. Let us turn now to the admissibility problem. It is not hard to see that the rules (::p ! p) ! p _ :p :p ! q _ r and :p _ ::p (:p ! q) _ (:p ! r) are admissible but not derivable in Int and p ^ :p=? is admissible but not derivable in any extension of S4.3 save those containing p ! p, in which it is derivable. (Recall that a logic L is said to be structurally complete if every admissible inference rule in L is derivable in L. We have just seen that Int as well as S4.3 are not structurally complete. For more information on structural completeness see e.g. [Tsytkin 1978, 1987] and [Rybakov 1995].) The following result strengthens Fine's [1971] Theorem according to which all logics in ExtS4.3 are decidable. THEOREM 195 (Rybakov 1984a). The admissibility problem for inference rules is decidable in every logic containing S4.3. An impetus for investigations of admissible inference rules in various logics was given by Friedman's [1975] problem 40 asking whether one can e ectively recognize admissible rules in Int. This problem turned out to be closely connected to the admissibility problem in suitable modal logics. We demonstrate this below for the logic GL following [Rybakov 1987, 1989]. First we show that dealing with logics in NExtK, it is suÆcient to consider inference rules of a rather special form. Let '(q1 ; : : : ; q2n+2 ) be a formula containing no and and represented in the full disjunctive normal form. Say that an inference rule is reduced if it has the form
'(p0 ; : : : ; pn; p0 ; : : : ; pn )=p0 : THEOREM 196. For every rule '= one can e ectively construct a reduced rule '0 = 0 such that '= is admissible in a logic L 2 NExtK i '0 = 0 is admissible in L.
Proof. Observe rst that if ' and do not contain p then '= is admissible in L i ' ^ ( $ p)=p is admissible in L. So we can consider only rules of
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
the form '=p0 . Besides, without loss of generality we may assume that ' does not contain . With every non-atomic subformula of ' we associate the new variable p . For convenience we also put p = pi if = pi and p = ? if = ?. We show now that the rule
p' ^
^
fp $ p p : = 1 2 2 Sub'; 2 f^; _; !gg ^ ^ fp $ p : = 1 2 Sub'g=p0 1
2
1
is admissible in L i '=p0 is admissible in L. For brevity we denote the antecedent of that rule by '00 . ()) Since every substitution instance of '00 =p0 is admissible in L, the V rule ' ^ 2 Sub' ( $ )=p0 and so '=p0 are also admissible in L. (() Suppose '=p0 is admissible in L and '00 s is in L, for some substitution s = f =p : 2 Sub'g. By induction on the construction of one can readily show that $ s 2 L. Therefore, ' $ 's 2 L. Since '00 s 2 L, we must have p's = ' 2 L, from which 's 2 L and so p0 s 2 L. Thus '00 =p0 is admissible in L. The rule '00 =p0 is not reduced, but it is easy to make it so simply by representing '00 in its full disjunctive normal form '0 , treating subformulas pi as variables. From now on we will deal with only reducedW rules di erent from ?=p0 (which is clearly admissible in any logic). Let j 'j =p0 be a reduced rule in which every disjunct 'j is the conjunction of the form
:0 p0 ^ ^ :m pm ^ :0 p0 ^ ^ :m pm ; where each :i and :j is either blank or :. We will identify such conjunc(17)
tions with the sets of their conjuncts. Now, given a non-empty set W of conjunctions of the form (17), we de ne a frame F = hW; Ri and a model M = hF; Vi by taking
'i R'j i
8k 2 f0; : : : ; mg(:pk 2 'i ! :pk 2 'j ^ :pk 2 'j ) ^ 9k 2 f0; : : : ; mg(:pk 2 'j ^ pk 2 'i ); V(pk ) = f'i 2 W : pk 2 'i g:
It should be clear that F is nite, transitive and irre exive. W THEOREM 197. A reduced rule j 'j =p0 is not admissible in GL i there is a model M = hF; Vi de ned as above on a set W of conjunctions of the form (17) and such that (i) :p0 2 'i for some 'i 2 W ; (ii) 'i j= 'i for every 'i 2 W ;
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235
(iii) for every antichain a in F there is 'j 2 W such that, for every k f0; : : : ; mg, 'j j= pk i 'i j= + pk for some 'i 2 a.
2
Proof. ()) We are givenW that there are formulas 0 ; : : : ; m in variables q1 ; : : : ; qn such that j 'j 2 GL and p0 62 GL, where Wby we denote f 0 =p0; : : : ; m =pmg. This is equivalent to MGL (n) j=W j 'j and MGL (n) 6j= p0 . De ne W to be the set of those disjuncts 'j in j 'j whose substitution instances 'j are satis ed in MGL (n). Clearly W 6= ;. Let us check (i) { (iii). W (i) Take a point x in MGL (n) at which p0 is false. As MGL (n) j= j 'j , we must have x j= 'i for some i. One of the formulas p0 or :p0 is a conjunct of 'i . Clearly it is not p0 . Therefore, :p0 2 'i . (ii) It suÆces to show that, for all 'i 2 W and k 2 f0; : : : ; mg, 'i j= pk i pk 2 'i . Suppose 'i j= pk . Then there is 'j 2 W such that 'i R'j and 'j j= pk . By the de nition of V and R, this means that pk 2 'j and pk 2 'i . Conversely, suppose pk 2 'i . Then x j= 'i and in particular x j= pk for some x in MGL (n). Let y be a nal point in the set fz 2 x ": z j= pk g. Since MGL (n) is irre exive, we have y j= pk , y 6j= pk and y j= 'j for some 'j 2 W . It follows that 'i R'j and 'j j= pk , from which 'i j= pk .
(iii) Let a be an antichain in F. For every 'i 2 a, let xi be a nal point in the set fy 2 WGL (n) : y j= 'i g. It should be clear that the points fxi : 'i 2 ag form an antichain b in FGL (n) and so, by the construction of FGL (n), there is a point y in FGL(n) such that y" = b". Then the formula 'j 2 W we are looking for is any one satisfying the condition y j= 'j , as can be easily checked by a straightforward inspection. (() The proof in this direction is rather technical; we con ne ourselves to just W a few remarks. Let M be a model satisfying (i){(iii). To prove that j 'j =p0 is not admissible in GL we require once again the n-universal model MGL (n), but this time we take n to be the number of symbols in the rule. By induction on the depth of points in M one can show that M is a generated submodel of MGL (n). W Our aim is to nd formulas 0 ; : : : ; m such that MGL (n) j= j 'j and MGL (n) 6j= p0 (here again = f 0 =p0; : : : ; m =pmg). Loosely, we need to extend the properties of M to the whole model MGL (n). To this end we can take the sets f'i g in FGL (n) and augment them inductively in such a way that we could embrace all points in FGL (n). At the induction step we use the condition (iii), and the required 0 ; : : : ; m are constructed with the help of (i) and (ii); roughly, they describe in MGL (n) the analogues of the truth-sets in M of the variables in our rule.
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A remarkable feature of this criterion is that it can be e ectively checked. Thus we have THEOREM 198. There is an algorithm which, given an inference rule, can decide whether it is admissible in GL. In a similar way one can prove THEOREM 199 (Rybakov 1987). The admissibility problem in Grz is decidable. We show now that the admissibility problem in Int can be reduced to the same problem in Grz and so is also decidable. To this end we require the following THEOREM 200 (Rybakov 1984b). A rule '= is admissible in Int i the rule T (')=T ( ) is admissible in Grz. As a consequence of Theorems 199 and 200 we obtain THEOREM 201 (Rybakov 1984b). The admissibility problem in Int is decidable. Although there are many other examples of logics in which the admissibility problem is decidable and the scheme of establishing decidability is quite similar to the argument presented above,20 proofs are rather diÆcult and only in few cases they work for big families of logics as in [Rybakov 1994]. Besides, all these results hold only for extensions of K4 and Int. For logics with non-transitive frames, even for K, the admissibility problem is still waiting for a solution. The same concerns polymodal, in particular tense logics. Chagrov [1992b] constructed a decidable in nitely axiomatizable logic in NExtK4 for which the admissibility problem is undecidable. It would be of interest to nd modal and si-calculi of that sort. A close algorithmic problem for a logic L is to determine, given an arbitrary formula '(p1 ; : : : ; pn ), whether there exist formulas 1 , : : : , n such that '( 1 ; : : : ; n ) 2 L. Note that an \equation" '(p1 ; : : : ; pn) has a solution in L i the rule '(p1 ; : : : ; pn)=? is not admissible in L. This observation and Theorem 195 provide us with examples of logics in which the substitution problem is decidable (see e.g. [Rybakov 1993]). We do not know, however, if there is a logic such that the substitution problem in it is decidable, while the admissibility one is not. The inference rules we have dealt with so far were structural in the sense that they were \closed" under substitution. An interesting example of a 20 Quite recently S. Ghilardi [1999a,b] has found another way of recognizing admissibility of inference rules. He showed that certain si- and modal logics L (in particular, Int, K4, S4, GL, Grz) have the following property. Given an L-consistent formula ', one can e ectively compute substitutions 1 ; : : : ; n such that i ' 2 L for every i = 1; : : : ; n, and if ' 2 L for some substitution , then is, up to provable equivalence, an instantiation of some of the i . A rule '= is then admissible in L i i 2 L for all i = 1; : : : ; n.
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nonstructural rule was considered by Gabbay [1981a]: ' _ (p ! p); where p 62 Sub' : ' It is readily seen that this rule holds in a frame F (in the sense that for every formula ' and every variable p not occurring in ', ' is valid in F whenever (p ! p) _ ' is valid in F) i F is irre exive and that K is closed under it (since K is characterized by the class of irre exive frames). We refer the reader to [Venema 1991] and [Marx and Venema 1997] for more information about rules of this type.
4.3 Properties of recursively axiomatizable logics Dealing with in nite classes of logics, we can regard questions like \Is a logic L decidable?", \Does L have FMP?", etc., as mass algorithmic problems. But to formulate such problems properly we should decide rst how to represent the input data of algorithms recognizing properties of logics. One can, for instance, consider the class of recursively axiomatizable logics (which, by Craig's [1953] Theorem, coincides with that of recursively enumerable ones) and represent them as programs generating their axioms. However, this approach turns out to be too general because the following analog of the Rice{Uspenskij Theorem holds. THEOREM 202 (Kuznetsov). No nontrivial property of recursively axiomatizable si-logics is decidable. Of course, nothing will change if we take some other family of logics, say NExtK4. The proof of this theorem (Kuznetsov left it unpublished) is very simple; we give it even in a more general form than required. PROPOSITION 203. Suppose L1 and L2 are logics in some family L, L1 is recursively axiomatizable, L1 L2 , L2 is nitely axiomatizable (say, by a formula ), and a property P holds for only one of L1, L2. Then no algorithm can recognize P , given a program enumerating axioms of a logic in L. Proof. Let 0 ; 1 ; : : : be a recursive sequence of axioms for L1 . Given an arbitrary (Turing, Minsky, Pascal, etc.) program P having natural numbers as its input, we de ne the following recursive sequence of formulas (where (n)1 and (n)2 are the rst and second components of the pair of natural numbers with code n under some xed e ective encoding): n if P does not come to a stop on input (n)1 in (n)2 steps n =
otherwise. This sequence axiomatizes L1 if P does not come to a stop on any input and L2 otherwise. It is well known in recursion theory that the halting problem is undecidable, and so the property P is undecidable in L as well.
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The reader must have already noticed that this proof has nothing to do with modal and si-logics; it is rather about e ective computations. To avoid this unpleasant situation let us con ne ourselves to the smaller class of nitely axiomatizable modal and si-logics and try to nd algorithms recognizing properties of the corresponding calculi. However, even in this case we should be very careful. If arbitrary nite axiomatizations are allowed then we come across the following THEOREM 204 (Kuznetsov 1963). For every nitely axiomatizable si-logic L (in particular, Int, Cl, inconsistent logic), there is no algorithm which, given an arbitrary nite list of formulas, can determine whether its closure under substitution and modus ponens coincides with L. Needless to say that the same holds for (normal) modal logics as well. Fortunately, the situation is not so hopeless if we consider nite axiomatizations over some basic logics. For instance, by Makinson's Theorem, one can e ectively recognize, given a formula ', whether the logic K ' is consistent. Other examples of decidable properties in various lattices of modal logics were presented in Theorems 89, 93, 101, and 142. In the next section we consider those properties that turn out to be undecidable in various classes of modal and si-calculi.
4.4 Undecidable properties of calculi The rst \negative" algorithmic results concerning properties of modal calculi were obtained by Thomason [1982] who showed that FMP and Kripke completeness are undecidable in NExtK, and consistency is undecidable in NExtK:t. Later Thomason's discovery has been extended to other properties and narrower classes of logics. In fact, a good many standard properties of modal and si-calculi (in reasonably big classes) proved to be undecidable; decidable ones are rather exceptional. In this section we present three known schemes of proving such kind of undecidability results. Each of them has its advantages (as well as disadvantages) and can be adjusted for various applications. The rst one is due to Thomason [1982]. Let L(n) be a recursive sequence of normal bimodal calculi such that no algorithm can decide, given n, whether L(n) is consistent. Such sequences, as we shall see a bit later, exist even in NExtK4:t. Suppose also that L is a normal unimodal calculus which does not have some property, say, FMP, decidability or Kripke completeness. Consider now the recursive sequence of logics L(n) L with three necessity operators. If L(n) is inconsistent then the fusion L(n) L is inconsistent too and so has the properties mentioned above. And if L(n) is consistent then, in accordance with Proposition 110, L(n) L is a conservative extension of both L(n) and L , which means that it is Kripke incomplete, undecidable and does not have FMP whenever
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L is so. Consequently, the three properties under consideration cannot be decidable in the class NExtK3 , for otherwise the consistency of L(n) would be decidable. By Theorem 123, these properties are undecidable in NExtK as well. Note however that, since Thomason's simulation embeds polymodal logics only into \non-transitive" unimodal ones, this very simple scheme does not work if we want to investigate algorithmic aspects of properties of calculi in NExtK4 and ExtInt. To illustrate the second scheme let us recall the construction of the undecidable calculus in NExtK4:t discussed in Section 4.1. First, we choose a Minsky program P and a con guration a = hs; m; ni so that no algorithm can decide, given a con guration b, whether P : a ! b. (That they exist is shown in [Chagrov 1990b].) Then we put = ? and add to K4:t AxP one more axiom (: ^ (s; 1m ; 2n ) ! : ^ (t; 1k ; 2l )) ! ; where c = ht; k; li is an arbitrary xed con guration. The resulting calculus is denoted by L(c). Suppose that P : a 6! c. Then one can readily check that the new axiom is valid in the frame F shown in Fig. 19 and prove that P : hs; m; ni ! ht0 ; k0 ; l0i i : ^ (s; 1 ; 2 ) ! : ^ (t0 ; 10 ; 20 ) 2 L(c): m
n
k
l
Therefore, L(c) is undecidable, consistent and does not have FMP. And if P : a ! c then L(c) is clearly inconsistent. It follows by the choice of P and a that consistency, decidability and FMP are undecidable in NExtK4:t. In fact, the argument will change very little if we take as the axiom of some tabular logic in NExtK4:t. So we obtain THEOREM 205. The properties of tabularity and coincidence with an arbitrary xed tabular logic (in particular, inconsistent) are undecidable in NExtK4:t Moreover, these results (except the consistency problem, of course) can be transferred to logics in NExtK. We demonstrate this by an example; complete proofs can be found in [Chagrov 1996]. We require the frame which results from that in Fig. 19 by adding to it a re exive point c0 and an irre exive one c1 so that c1 sees all other points save a and b and is seen itself only from a and b. As before, we denote the frame by F. PROPOSITION 206. Let be a formula refutable at some point in F different from c0 and > 2 K . Then the problem of deciding, for an arbitrary formula ', whether K ' = K is undecidable.
Proof. It should be clear that contains at least one variable, say r, and there are points in F at which r has distinct truth-values (under the
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
valuation refuting ); c0 and c1 are then the only points in F where the formulas 0 = 3r _ 3 :r and
1 = 0 ^ (r _ r _ 2 r) ^ (:r _ :r _ 2 :r) are true, respectively. Observe that from every point in F save c0 we can reach all points in F by 3 steps. So we can take = 3 . The formulas and should be replaced with = 1 ^ 2 1 , = 1 ^ :2 1 which (under the valuation refuting ) are true only at a and b, respectively. Now consider the logic
L(c) = K AxP (: ^ (s; 1m ; 2n ) ! : ^ (t; 1k ; 2l )) ! : If P : a ! c then L(c) = K . And if P : a 6! c then, using the fact that the set of points in F where is refutable coincides with the set of points from which every point of the form e(x; y; z ) is accessible by three steps, one can show that F j= L(c) and so L(c) 6= K . Putting, for instance, = p $ p, we obtain then that the problem of coincidence with LogÆ is undecidable in NExtK. Likewise one can prove the following THEOREM 207. (i) If a consistent nitely axiomatizable logic L is not a union-splitting of NExtK then the axiomatization problem for L above K is undecidable. (ii) The properties of tabularity and coincidence with an arbitrary xed consistent tabular logic are undecidable in NExtK. (iii) The problem of coincidence with an arbitrary xed consistent calculus in NExtD4 or in NExtGL is undecidable in NExtK. (iv) The properties of tabularity and coincidence with an arbitrary xed tabular (in particular, inconsistent) logic are undecidable in ExtK4. Of the algorithmic problems concerning tabularity that remain open the most intriguing are undoubtedly the tabularity and local tabularity problems in NExtK4. Note that a positive solution to the former implies a positive solution to the latter. Now we present the second scheme in a more general form used in [Chagrov 1990b] and [Chagrov and Zakharyaschev 1993]. Assume again that the second con guration problem is undecidable for P and a, and let be a formula such that L0 has some property P , where L0 is the minimal logic in the class under consideration. Associate with P , a and a con guration b formulas AxP and (a; b) such that (a; b) 2 L0 AxP i P : a ! b.
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Besides, and AxP are chosen so that AxP 2 L0 . Now consider the calculus L(b) = L0 AxP (a; b) ! ; where is some formula such that 2 L0 . If P : a ! b then we clearly have L(b) = L0 and so L(b) has P ; but if P : a 6! b then the fact that L(b) does not have P must be ensured by an appropriate choice of . (In the considerations above we did not need , i.e., it was suÆcient to put
= >). With the help of this scheme one can prove the following THEOREM 208. (i) The properties of decidability, Kripke completeness as well as FMP are undecidable in the classes ExtInt, (N)ExtGrz, (N)ExtGL. (ii) The interpolation property is undecidable in (N)ExtGL. (iii) Hallden completeness is undecidable in ExtInt, (N)ExtGrz, ExtS. These and some other results of that sort can be found in [Chagrov 1990b,c, 1994, 1996], [Chagrova 1991], [Chagrov and Zakharyaschev 1993, 1995b]. The third scheme was developed in [Chagrova 1989, 1991] and [Chagrov and Chagrova 1995] for establishing the undecidability of certain rst order properties of modal calculi (or formulas). The di erence of this scheme from the previous one is that now we use calculi of the form
L(b) = L0 AxP (a; b) _ ; where AxP satis es one more condition besides those mentioned above: it must be rst order de nable on Kripke frames for L0 . If P : a ! b then the formula AxP ^ ( (a; b) _ ) is equivalent to AxP in the class of Kripke frames for L0 and so is rst order de nable on that class or its any subclass. And if P : a 6! b then by choosing an appropriate one can show that AxP ^ ( (a; b) _ ) is not rst order de nable on, say, countable Kripke frames for L0 , as in [Chagrova 1989], or on nite frames for L0 , as in [Chagrov and Chagrova 1995]. In this way the following theorem is proved: THEOREM 209. (i) No algorithm is able to recognize the rst order de nability of modal formulas on the class of Kripke frames for S4 and even the rst order de nability on countable ( nite) Kripke frames for S4. The properties of rst order de nability and de nability on countable ( nite) Kripke frames of intuitionistic formulas are undecidable as well. (ii) The set of modal or intuitionistic formulas that are rst order de nable on countable ( nite) frames but are not rst order de nable on the
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV class of all (respectively, countable) Kripke frames mentioned in (i) is undecidable.
We conclude this section with two remarks. First, all undecidability results above can be formulated in the stronger form of recursive inseparability. For instance, the set of inconsistent calculi in NExtK4:t and the set of calculi without FMP are recursively inseparable. And second, some properties are not only undecidable but the families of calculi having them are not recursively enumerable; for example, the set of consistent calculi in NExtK4:t is not enumerable. However, for the majority of other properties the problem of enumerability of the corresponding calculi is open.
4.5 Semantical consequence So far we have dealt with only syntactical formalizations of logical entailment. However, sometimes a semantical approach is preferable. Say that a formula ' is a semantical consequence of a formula in a class of frames C if ' is valid in all frames in C validating . (One can consider also the local, i.e., point-wise variant of this relation.) Note that ' is a consequence of in the class of, say, Kripke frames for S4 i ' is a consequence of (p ! 2 p) ^ (p ! p) ^ in the class of all Kripke frames. But the consequence relation on nite frames is not expressible by modal formulas (as was shown in [Chagrov 1995], if (p ! 2 p) ^ ' is valid in arbitrarily large nite rooted frames then it is valid in some in nite rooted frame as well). In parallel with constructing and proving the undecidability of modal and si-calculi we can obtain the following THEOREM 210. The semantical consequence relation in the class of all (K4-, S4-, Int-) Kripke frames is undecidable. Moreover, if j= denotes one of these relations then there is a formula (a formula ') such that the set f' : j= 'g is undecidable. In a sense, formulas and ', for which f' : j= 'g is undecidable are analogous to undecidable calculi and formulas, respectively. However, this analogy is far from being perfect: for every formula , the sets f' : ` 'g and f' : ` 'g are recursively enumerable, which contrasts with THEOREM 211 (Thomason 1975a). There exists a formula such that f' : j= 'g is a 11 -complete set. Unfortunately, Thomason's [1974b, 1975b, 1975c] results have not been transferred so far to transitive frames, although this does not seem to be absolutely impossible. Chagrov [1990a] (see also [Chagrov and Chagrova 1995]) developed a technique for proving the analog of Theorem 210 for the consequence relation
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on all (K4-, S4-, GL-, Int-) nite frames. Moreover, since this relation is clearly enumerable, instead of \undecidable" one can use \not enumerable".
4.6 Complexity problems Having proved that a given logic is decidable, we are facing the problem of nding an optimal (in one sense or another) decision algorithm for it. The complexity of decision algorithms for many standard modal and si-logics is determined by the size of minimal frames separating formulas from those logics. For instance, as was shown by Jaskowski (1936) and McKinsey (1941), for every ' 62 S4 (or ' 62 Int) there is a frame F j= S4 with 2jSub'j points such that F 6j= '. The same upper bound is usually obtained by the standard ltration. Is it possible to reduce the exponential upper bound to the polynomial one? This question was raised by Kuznetsov [1975] for Int. It turned out, however, that it concerns not only Int. First, Kuznetsov observed (for the proof see [Kuznetsov 1979]) that if the answer to his question is positive, i.e., Int has polynomial FMP, then the problem \Are Int and Cl polynomially equivalent?" has a positive solution as well. (Logics L1 and L2 are polynomially equivalent if there are polynomial time transformations f and g of formulas such that ' 2 L1 i f (') 2 L2 and ' 2 L2 i g(') 2 L1 .) Then Statman [1979] showed that the problem \' 2 Int?" is P SP ACE -complete and so Kuznetsov's problem is equivalent to one of the \hopeless" complexity problems, namely \NP = P SP ACE ?". Complexity function
For a logic L with FMP, we introduce the complexity function
fL (n) = lmax min jFj ; (')n Fj=L '62L Fj6 =' where l('), the length of ', is the number of subformulas in ' and jFj the number of points in F. If there is a constant c such that fL(n) 2cn (or fL(n) nc or fL (n) c n); L is said to have the exponential (respectively, polynomial or linear) nite model property. The following result shows that Int does not have polynomial FMP. THEOREM 212 (Zakharyaschev and Popov 1979). log2 fInt(n) n. Proof. The exponential upper bound is well known and to establish the lower one it is suÆcient to use the formulas n =
n^1 i=1
((:pi+1 ! qi+1 ) _ (pi+1 ! qi+1 ) ! qi ) ! (:p1 ! q1 ) _ (p1 ! q1 ):
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It is not hard to see that n 2= Int and every refutation frame for n contains the full binary tree of depth n as a subframe. Likewise the same result can be proved for many other standard superintuitionistic and modal logics whose FMP is established by the usual ltration and whose frames contain full binary trees of arbitrary nite depth. Such are, for instance, KC, SL, K4, S4, GL. In the case of K the length of formulas that play the role ofp n is not a linear but a square function of n, which means that fK (n) 2 cn , for some constant c > 0, and so K does not have polynomial FMP either. As was shown in [Zakharyaschev 1996], all co nal subframe modal and si-logics have exponential FMP. It seems plausible that log2 fL (n) n for every consistent si-logic L di erent from Cl and axiomatizable by formulas in one variable. The construction of Theorem 212 does not work for logics whose frames do not contain arbitrarily large full binary trees. Such are, for instance, logics of nite width or of nite depth, and the following was proved in [Chagrov 1983]. THEOREM 213. (i) The minimal logics of width n < ! in the classes NExtK4, NExtS4, NExtGrz, NExtGL, ExtInt have polynomial FMP. (ii) Lin and all logics containing S4.3 have linear FMP. (iii) The minimal logics of depth n in NExtGrz, NExtGL, ExtInt have polynomial FMP, with the power of the corresponding polynomial n 1. (iv) The minimal logics of depth n in NExtK4, NExtS4 have polynomial FMP, with the power of the corresponding polynomial n.
Proof. (i) is proved by two ltrations. First, with the help of the standard ltration one constructs a nite frame separating a formula ' from the given logic L and then, using the selective ltration, extracts from it a polynomial separation frame: it suÆces to take a point refuting ' and all maximal points at which is false, for some 2 Sub' (in the intuitionistic case ! 2 Sub' should be considered). (ii) is proved analogously. To illustrate the proof of (iii) and (iv), we consider the minimal logic L of depth 3 in NExtGL. Suppose ' 2= L. Then there is a transitive irre exive model M of depth 3 refuting ' at its root r. Let i , for 1 i m, be all \boxed" subformulas of '. For every i 2 f1; : : : ; mg, we choose a point refuting i , if it exists. And then we do the same in the set x", for every chosen point x. Let M0 be the submodel formed by the selected points and r. Clearly, it contains at most 1 + m + m2 points. And by induction on the
ADVANCED MODAL LOGIC a1
-a2 -a3
an
245
- Æ -b1 -b2
bf (n)
Figure 20. construction of formulas in Sub' one can easily show that M0 refutes ' at r. To prove the lower bound one can use the formulas
n =
n ^
n ^
i=1
i=1
:( (pi+1 ! pi ) ^ n ^
i=1
(qi+1 ! qi ) ^
(> ^ + (:p
i+1 ^ pi )) ^ (? !
n ^ i=1
(:qi+1 ^ qi )))
which are not in L and every separation frame for which contains the full n-ary tree of depth 3, i.e., at least 1 + n + n2 points. However, even if frames for a logic with FMP do not contain full nite binary trees its complexity function can grow very fast, witness the following result of [Chagrov 1985a]. THEOREM 214. For every arithmetic function f (n), there are logics L of width 1 in NExtK4 and of width 2 in ExtInt, NExtGrz, NExtGL having FMP and such that fL(n) f (n).
Proof. We construct a logic L 2 NExtK4:3 whose complexity function grows faster than a given increasing arithmetic function f (n). De ne L to be the logic of all frames of the form shown in Fig. 20. To see that L satis es the property we need, consider the sequence of formulas 1 = p1 _ (p1 ! ((p ! p) ! p)); i+1 = pi+1 _ (pi+1 ! i ): Since these formulas are refuted at points of the form aj in suÆciently large frames depicted in Fig. 20, they are not in L. And since L contains the formulas : n ! (f (n) 1 > ^ f (n) ?); n cannot be separated from L by a frame with f (n) points. For logics of nite depth this theorem does not hold, since according to the description of nitely generated universal frames in Section 1.2, for every L 2 NExtK4BDk (k 3), we have fL(n) 2
2c n
2
k 2
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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV
for some constant c > 0. And as was shown in [Chagrov 1985a], one cannot in general reduce this upper bound. THEOREM 215. For every k 3, there are logics L of depth k in NExtGrz, NExtGL, ExtInt such that 2
fL(n) 2
2n
k 2
:
Proof. We illustrate the proof for k = 3 in NExtGL. Let L be the logic characterized by the class of rooted frames Fm for GL of depth 3 de ned as follows. Fm contains m dead ends, every non-empty set of them has a focus, i.e., a point that sees precisely the dead ends in this set, and besides the root there are no other points in Fm. It should be clear that L does not contain the formulas
m =
n ^ i=1
(pi+1 ! pi ) !
n ^ i=1
(pi ! pi+1 ):
On the other hand n is not refutable in a frame for L with < 2m points because the following formulas are in L:
: m !
^
X f1;:::;mg;X 6=;
^
(
i2X
Æi ^
^
i62X;1im
where Æi = p1 ^ ^ pi ^ :pi+1 ^ ^ :pm+1 .
:Æi );
Note, however, that the logics constructed in the proofs of the last two theorems are not nitely axiomatizable. We know of only one \very complex" calculus with FMP. THEOREM 216. log2 log2 fKP (n) n. For the proof see [Chagrov and Zakharyaschev 1997], where the reader can nd also some other results in this direction. Relation to complexity classes Let us return to the original problem of optimizing decision algorithms for the logics under consideration. First of all, it is to be noted that there is a natural lower bound for decision algorithms which cannot be reduced| we mean the complexity of decision procedures for Cl. This is clear for (consistent) modal logics on the classical base; and by Glivenko's Theorem, every si-logic \contains" Cl in the form of the negated formulas. Thus, if we manage to construct an e ective decision procedure for some of our logics then Cl can be decided by an equally e ective algorithm. (We remind the reader that all existing decision algorithms for Cl require exponential
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time (of the number of variables in the tested formulas). On the other hand, only polynomial time algorithms are regarded to be acceptable in complexity theory.) So, when analyzing the complexity of decision algorithms for modal and si-logics, it is reasonable to compare them with decision algorithms for Cl. For example, if a logic L is polynomially equivalent to Cl then we can regard these two logics to be of the same complexity. Moreover, provided that somebody nds a polynomial time decision procedure for Cl, a polynomial time decision algorithm can be constructed for L as well. The following theorem lists results obtained by [Ladner 1977], [Ono and Nakamura 1980], [Chagrov 1983], and [Spaan 1993]. THEOREM 217. All logics mentioned in the formulation of Theorem 213 are polynomially equivalent to Cl.
Proof. We illustrate the proof only for the minimal logic L of depth 3 in NExtGL using the method of [Kuznetsov 1979]. Suppose ' is a formula of length n. By Theorem 213, the condition ' 62 L means that M 6j= ', for some model M = hF; Vi based on a frame F for GL of depth 3 and cardinality c n2 . We describe this observation by means of classical formulas, understanding their variables as follows. Let x, y, z be names (numbers) of points in F, for 1 x; y; z c n2 . With every pair hx; yi of points in F we associate a variable pxy whose meaning is \x sees y". And with every 2 Sub' and every x we associate a variable qx which means \ is true at x". Denote by the conjunction q1' ^ q2' ^ ^ qc'n2 :
It means that ' is true in M. And let be the conjunction of the following formulas under all possible values of their subscripts: :pxx; pxy ^ pyz ! pxz ; q: $ :q ; x
qx ^ $ qx
^ qx ;
qx _ $ qx
_ qx ;
q x
x
$
c^ n2 y=1
(pxy ! qy ):
(The rst two formulas say that R is irre exive and transitive and the rest simulate the truth-relation in M.) Finally, we de ne a formula saying that our frame is of depth 3:
=
^
1x;y;z;ucn2
:(pxy ^ pyz ^ pzu ):
The formula ^ ^: is of length 50(cn2)5 and can be clearly constructed by an algorithm working at most polynomial time in the length of '. It is readily seen that ' 62 L i ^ ^ : is satis able in Cl. Thus we have
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polynomially reduced the derivability problem in L to that in Cl. Since the converse reduction is trivial, L and Cl are polynomially equivalent. The reader must have noticed that Theorem 217 lists almost all logics known to have polynomial FMP. Kuznetsov [1975] conjectured that every calculus having polynomial FMP is polynomially equivalent to Cl. This conjecture is closely related to some problems in the complexity theory of algorithms. We remind the reader that NP is the class of problems that can be solved by polynomial time algorithms on nondeterministic (Turing) machines. An NP -complete problem is a problem in NP to which all other problems in NP are polynomially reducible. (For more detailed de nitions consult [Garey and Johnson 1979].) The most popular NP -complete problem is the satis ability problem for Boolean formulas, i.e., the nonderivability problem for Cl. So the nonderivability problem for all logics listed Theorem 217 is NP -complete and Kuznetsov's conjecture is equivalent to a positive solution to the problem whether the nonderivability problem for every calculus with polynomial FMP is NP -complete. Note that if coNP = NP (for the de nition of the class coNP see [Garey and Johnson 1979]; we just mention that the derivability problem in Cl is coNP -complete) then Kuznetsov's conjecture does hold. But since \coNP = NP ?" belongs to the list of \unsolvable" problems under the current state of knowledge, it may be of interest to nd out whether Kuznetsov's conjecture implies coNP = NP . Another complexity class we consider here is the class P SP ACE of problems that can be solved by polynomial space algorithms. A typical example of a P SP ACE -complete problem is the truth problem for quanti ed Boolean formulas. The following theorem (which summarizes results obtained by Ladner [1977], Statman [1979], Chagrov [1985a], Halpern and Moses [1992] and Spaan [1993]) lists some P SP ACE -complete logics. THEOREM 218. The nonderivability problem (and so the derivability problem) in the following logics is P SP ACE -complete: Int, KC, K, K K, S4, S4 S4, S5 S5, GL, Grz, K:t and K4:t. It follows in particular that complexity is not preserved under the formation of fusions of logics (under the assumption NP 6= P SP ACE ), since nonderivability in S5 is NP -complete. For more information on the preservation of complexity under fusions consult [Spaan 1993]. Finally we note that the nonderivability problem in logics with the universal modality or common knowledge operator is mostly even EXP T IME complete, witness Ku [Spaan 1993] and S4EC2 [Halpern and Moses 1992]. The complexity of the nonderivabilty problem for Cartesian products of many standard modal logics is NEXP T IME -hard; S5 S5 and K S5 are examples of NEXP T IME -complete logics (see [Marx 1999]). (Note, by the way, that the known upper bound for K K is non-elementary.)
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5 APPENDIX We conclude this chapter with a (by no means complete) list of references for those directions of research in modal logic that were not considered above:
Congruential logics. These are modal logics that do not necessarily contain the distribution axiom (p ! q) ! (p ! q) but are closed under modus ponens and the congruence rule p $ q=p $ q. Segerberg [1971] and Chellas [1980] de ne a semantics for these logics; Lewis [1974] proves FMP of all congruential non-iterative logics and Surendonk [1996] shows that they are canonical. Dosen [1988] considers duality between algebras and neighbourhood frames and Kracht and Wolter [1999] study embeddings into normal bimodal logics.
Modal logics with graded modalities. The truth-relation for their possibility operators n is de ned as follows: x j= n p i there exist at least n points accessible from x at which p holds. An early reference is [Fine 1972]; more recent are [van der Hoek 1992] (applications to epistemic logic) and [Cerrato 1994] (FMP and decidability).
Modal logics with the di erence operator or with nominals (or names). The semantics of nominals is similar to that of propositional variables; the di erence is that a nominal is true at exactly one point in a frame. For the di erence operator [6=], we have x j= [6=]p i p is true everywhere except x. De Rijke [1993], Blackburn [1993] and Goranko and Gargov [1993] study the completeness and expressive power of systems of that sort. Closely related to the di erence operator is the modal operator [i] for inaccessible worlds: x j= [i]p i p is true in all worlds which are not accessible from x, see [Humberstone 1983] and [Goranko 1990a].
Modal logics with dyadic or even polyadic operators. For duality theory in this case see [Goldblatt 1989]. An extensive study of Sahlqvisttype theorems with applications to polyadic logics is [Venema 1991]. For connections with the theory of relational algebras see [Mikulas 1995] and [Marx 1995]. In those dissertations the reader can nd also recent results on arrow logic, i.e., a certain type of polyadic logic which is interpreted in Kripke frames built from arrows. An embedding of polyadic logics into polymodal logics is discussed in [Kracht and Wolter 1997].
Bisimulations. Bisimulations were introduced in modal logic by van Benthem [1983] to characterize its expressive power; see also [de Rijke 1996]. Visser [1996] used bisimulations to prove uniform interpolation. Recently, bisimulations have attracted attention because they form a
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common tool in modal logic and process theory. We refer the reader to collection [Ponse et al. 1996] for information on this subject. Modal logics with xed point operators, i.e., modal logics enriched by operators forming the least and greatest xed points of monotone formulas. These systems are also called modal -calculi. Under this name they were introduced and studied by Kozen [1983, 1988]; see also [Walukiewicz 1993, 1996] and [Bosangue and Kwiatkowska 1996]. Proof theory. Early references to studies of sequent calculi and natural deduction systems for a few modal logics can be found in Basic Modal Logic. More recently, (non-standard) sequent calculi for modal logics have been considered by Dosen [1985b], Masini [1992] and Avron [1996]; see also collection [Wansing 1996] and the chapter Sequent systems for modal logics later in this Handbook. For natural deduction systems see Borghuis [1993]; tableau systems for modal and tense logics were constructed in [Fitting 1983], [Rautenberg 1983], [Gore 1994] and [Kashima 1994]. Orlowska [1996] develops relational proof systems. Display calculi for modal logics were introduced by Belnap [1982]; see also [Wansing 1994] and collection [Wansing 1996]. Description logic, a formalism closely related to modal logic, was designed in arti cial intelligence by Brachman and Schmolze [1985] as a means for knowledge representation and reasoning (for a survey see [Donini et al. 1996]). Schild [1991] was the rst to observe that the basic description logic ALC is just a terminological variant of the polymodal K. Recently, in order to represent dynamic and intensional knowledge, combinations of description and modal logics have been introduced, see e.g. Baader and Ohlbach [1995], Baader and Laux [1995], and Wolter and Zakharyaschev [1998, 1999b,c]. ACKNOWLEDGMENTS
First of all, we are indebted to our friend and colleague Marcus Kracht who not only helped us with numerous advices but also supplied us with some material for this chapter. We are grateful to Hiroakira Ono and the members of his Logic Group in Japan Advanced Institute of Science and Technology for the creative and stimulating atmosphere that surrounded the rst two authors during their stay in JAIST in 1996{97, where the bulk of this chapter was written. Thanks are also due to Johan van Benthem, Wim Blok, Dov Gabbay, Silvio Ghilardi, Agnes Kurucz, Krister Segerberg, Valentin Shehtman, Dimiter Vakarelov, and Heinrich Wansing for their helpful comments and stimulating discussions. And certainly our work would be impossible without constant support and love of our wives: Olga, Imke and Lilia.
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The work of the rst author was partly nanced by the Alexander von Humboldt Foundation. A. Chagrov Tver State University, Russia F. Wolter Institute of Information Science, Leipzig University, Germany M. Zakharyaschev King's College London, UK BIBLIOGRAPHY
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JAMES W. GARSON
QUANTIFICATION IN MODAL LOGIC 0 INTRODUCTION
0.1 An Outline of this Chapter The novice may wonder why quanti ed modal logic (QML) is considered diÆcult. QML would seem to be easy: simply add the principles of rstorder logic to propositional modal logic. Unfortunately, this choice does not correspond to an intuitively satisfying semantics. From the semantical point of view, we are confronted with a number of decisions concerning the quanti ers, and these in turn prompt new questions about the semantics of identity, terms, and predicates. Since most of the choices can be made independently, the number of interesting quanti ed modal logics seems bewilderingly large. The main purpose of this chapter is to try to make sense of this seemingly chaotic terrain. Section 1 provides a review of the major systems. Section 2 explains the diÆculties in completeness proofs for QMLs, and presents strategies for overcoming them. Section 3 shows that some systems of QML behave like second-order logics; they have strong expressive powers and so are incomplete. The Appendix lists rules, systems, and semantical conditions covered in this chapter. Free logic serves, in one way or another, as the foundation for most of the systems we will study. We will argue in Section 1.2.1.2 that allegiance to rst-order logic is a source of ad hoc stipulations in semantics for QML. However, when the principles of free logic are adopted, complications can be avoided. Since free logic is such a crucial foundation for QML, we will give a brief description of it here. The reader who knows about free logic, or who wants to read Bencivena's chapter (in Volume 7 of this Handbook) on the topic, may skip section 0.2. Since free logics are usually formulated using = in QML in any case, we will brie y discuss identity in intensional logics in Section 0.3.
0.2 A Short Review of Free Logic One oddity of rst-order logic with identity is that it seems to provide an argument for the existence of God. From the provable identity g = g we may derive, 9xx = g by Existential Generalisation. If g abbreviates `God', then 9xx = g reads `God exists'. This anomaly is connected with the basic assumption made in the semantics for quanti cational logic that every constant (such as g) refers to an object in the domain of quanti cation.
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Objectual Domain
1:3
1:4
World Fixed Relative Domain Domains
Standard Predicates Intentional (except E is Predicates intentional)
1:2 Rigid Terms
Non{rigid Terms
1:2:1
1:2:2
World Fixed Relative Domain Domains 1.2.1.1 Q1 (Kripke)
1.3.1 QC
Global Local Terms Terms
1:2:1:2
Substantial Domain (some of the individual concepts)
1.3.2 QC (Thomason)
1.4.1 QS (Garson)
1.4.2 B1 (Parks)
1.2.2.2 Q3L (Bowen)
1.2.2.1 Q3 Free Classical (Thomason) Logic Logic 1:2:1:2:3
1.2.1.2.2 Q1R
Eliminate Truth value Terms gaps
1.2.1.2.3.1 QK (Kripke)
1:2:1:2:3:2 Nested No Restrictions Domains on Domains
1.2.1.2.3.2.2 QPL (Hughes & Creswell)
1.2.1.2.3.2.1 GK (Gabbay)
Figure 1. Roadmap Explanation of the quanti ed Modal Logic Roadmap This tree represents the structure of the discussion of quanti ed modal logic in this chapter. Each node contains a number indicating the section of this chapter where a topic is discussed. Branches from each node are labelled with the main options which one can choose at that point. The `leaves' of the tree are labelled with the name used in this chapter of the system which results from choosing the options on all branches leading to it. Beneath the name of each system is the name of an author associated with the system. The references in the bibliography associated with his name contain a description of the system in question.
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There are a number of ways for a believer in the principles of rst-order logic to handle this problem. One popular tactic is to count `God' as a de nite description IxGx, where Gx is interpreted to be true only of God. Then `God exists' translates to 9yy = IxGx. By Russell's theory of descriptions, this amounts to 9z (9yy = z ^ Gz ^ 8x(Gx ! x = z )), which is not a theorem. However, this reply depends on a debatable assumption, namely that for every name which may fail to refer, we can nd a predicate (or open sentence) which picks out that referent uniquely. Kripke [1972] presents strong evidence that we cannot nd such uniquely identifying predicates. Even if we could solve this problem, the use of Russell's theory causes another problem. We want to be able to say that `Pegasus has wings' is true, but that `Pegasus is a hippopotamus' is false. If we translated `Pegasus' away in these two sentences according to Russell's theory of descriptions, we obtain sentences of the shapes W (IxP x) and H (IxP x), which are both false since Pegasus does not exist. We do no better translating these sentences by 8x(P x ! W x) and 8x(P x ! Hx), because in this case both are vacuously true, since nothing satis es the predicate P . Free logic avoids these diÆculties by dropping the assumption that every name must refer to an object in the domain of quanti cation. As a result, the principles for the quanti ers are somewhat weaker. Let us assume that we have a primitive predicate E , whose extension is the domain of quanti cation. The revised axiom of Existential Generalisation becomes: (FEG) (P t ^ Et) ! 9xP x: The proof we gave for 9xx = g in rst-order logic is now blocked. Using (FEG), we may obtain 9xx = g from g = g only if we have already proven Eg, and Eg expresses what we are trying to prove. A complete system MFL of minimal free logic with identity can be constructed by de ning 9x and :8x: and adding the following rules to propositional logic plus identity theory: 8xP x for any term t (FUI) Et ! P t (FUG)
` A ! (Et ! P t) t is a term that does not appear in A ! 8xP x. ` A ! 8xP x
In these rules, and throughout this chapter, A and P x are w s, x is any variable, and P t is the result of substituting the term t properly for all occurrences of x in P x. It is an easy exercise to show that Et is equivalent in MFL to 9xx = t (where x is not t). So we could have de ned Et as 9xx = t, and avoided the introduction of a special predicate letter E . However, in some intensional logics, there is no way to de ne Et in terms of the rest of the primitive vocabulary, and so we have prepared for this by assuming that E is primitive.
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0.3 Identity in Intensional Logics The failure of the substitution of identical terms is a familiar criterion for identifying intensional expressions. For example, the invalidity of the famous argument: Scott is the author of Waverley King George wonders whether Scott is Scott King George wonders whether Scott is the author of Waverley serves as evidence that `King George wonders whether' is intensional. It should not surprise us, then, if we need to limit the rule of substitution of identities in intensional logics. One simple way to enforce the desired restriction is to allow substitution in atomic sentences only, as in the following system ID for identity: t = t0 where P t is an atom. (= In) t = t (= Out) P t ! P t0 Although the restriction to atomic sentences may seem strong, it has no e ect whatsoever in rst-order logic, because (= Out) insures the substitution of identities in all extensional sentences. However, in intensional logics, it does not guarantee substitution of identical terms which lie in the scope of intensional operators. Some may object to the view that the substitution of identicals fails. Russell, for example, gave an explanation of the invalidity of the argument about the author of Waverley which did not require any restrictions on the rule of substitution. Russell claimed that the description `the author of Waverley', does not count as a term. When the description is eliminated according to his theory, the rst premise of the argument no longer has the form of an identity. This tactic does not work, however, for arguments such as the following where there are no descriptions to eliminate: Cicero is Tully. King George knows that Cicero is Cicero. King George knows that Cicero is Tully. One reaction to this sort of example is to argue that the failure of the rule of substitution is a sign that the expression being substituted is not really a term. So the invalidity of the last argument shows that `Cicero' and `Tully' are not terms, and must be translated using corresponding descriptions: IxCx and IxT x. When this is done, the rst premise of the argument no longer has the form of an identity, and so does not count as a case of substitution. Notice, however, that adherence to the principle of unrestricted substitution leads us to a position similar to the one which resulted from adherence
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to the classical rules for quanti ers, we conclude that many of the expressions which we would ordinarily count as terms, must be treated instead as descriptions. We were forced before to deny the termhood of expressions which might fail to denote, and now we are compelled to deny it of expressions which might have synonyms. Since we have little guarantee that a given expression avoids either defect, we feel pressure, as Quine did, to claim that no expression of English should be rendered as a constant in rst-order logic. Given the simplicity of the alternative rules, the insistence on the classical rules for quanti ers and the unrestricted substitution of identities is, in our opinion, a prejudice, and one which blocks a natural exposition of an adequate foundation for quanti ed modal logics. 1 A TAXONOMY OF QUANTIFIED INTENSIONAL LOGIC One of the most signi cant points of di erence between semantical treatments of QML concerns the domain of quanti cation. Some systems quantify over objects, while others quanti er over what Carnap [1947] called individual concepts. The second approach is more general, but it is also more abstract, and more diÆcult to motivate. So we will open this account of QML with systems that use the objectual interpretation.
1.1 Some Semantical Preliminaries Before we begin, it will be helpful to de ne a few semantical ideas which we will use throughout this chapter. We assume that a quanti ed modal language is constructed from predicate letters, the primitive predicate constant E , terms (which include in nitely many variables) the logical constants :; !; ; =, and a quanti er 8x for each of the variables x. The predicate letters come equipped with integers indicating their arity. The propositional variables are taken to be 0-ary predicate letters, and well-formed formulas are de ned in the usual way. Given a set D, the extensions of terms and predicate letters are de ned just as they are in rst-order logic. The extension of a term is some member of D, and the extension of an i-ary predicate letter is a set of i-length sequences of members of D. Given a set W of indices (typically, possible worlds), the intension of an expression is simply a function which takes each member of W into an appropriate extension for that expression. Carnap's individual concepts are simply term intensions, that is, functions from the set of possible worlds into the domain of objects. Throughout this chapter, a Q-model hW; R; D; Q; ai will contain a set W of possible worlds, a binary relation R on W , a nonempty set D of possible objects, some item Q which determines the domain of quanti cation, and an assignment function a, which interprets the terms (including variables)
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and predicate letters by assigning them the corresponding kind of intensions with respect to W and D. If the quanti er rules of a system are based on free logic, then there will be a predicate letter E in the language. To ensure that E receives the proper interpretation as picking out the quanti er domain, we will assume that a Q-model for a language that contains E always meets the condition that a(E ) is Q. In some semantics, the terms are rigid designators, that is, their extensions are the same in all possible worlds. Usually such terms are assigned no intensions, but given extensions directly. However, in order to keep the description of a model as consistent as possible, we will assume that terms always have intensions, and that terms which are rigid designators simply meet the added condition that their intensions are constant functions. The symbol = will always be interpreted as contingent identity. This means that t = t0 is ruled true in a world just in case t and t0 have the same extension in that world. The truth value of a sentence A on a model hW; R; D; Q; ai at world w of W (written a(A)(w)) will be de ned by induction on the shape of A using the standard clauses for atomic sentences, :; ! and . When we present a given approach to the quanti ers, we usually will need only to say what Q is like, and to give the truth clause for the quanti er. The quanti ed modal logics we are going to discuss are all extensions of propositional modal logics which are adequate with respect to some class of Kripke frames. For example, we will consider extensions of S4, which are adequate (semantically consistent and complete) with respect to the class R(S4) of Kripke frames hW; Ri that are re exive and transitive. Usually we will not care which propositional modal logic is chosen as the foundation for our quanti ed logic. We will assume that some propositional modal logic has already been chosen, and that the frame of any Q-model is in R(S ). When we need to be explicit, we will talk of S -models, and mean models whose Kripke frames are in the set R(S ). The notions of Q-satis ability and Q-validity are determined by the concept of a Q-model exactly as in propositional modal logic.
1.2 The Objectual Interpretation 1.2.1 Rigid Terms. Kripke's historic paper [1963] serves as an excellent starting point for a discussion of logics with the objectual interpretation. One reason is that he made the important simplifying assumption that all terms of the language are rigid designators. Systems that allow nonrigid terms are, as we shall see, rather complicated, and so we will begin, as Kripke did, by assuming that the intension of every term is a constant function. This assumption validates the following two rules which we refer
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to together as (RT) (for rigid terms). t = t0 :t = t0 (RT) t = t0 :t = t0 The rigidity condition re ects the view that proper names have extensions, but no intensions. Since (RT) guarantees the substitution of identity in all contexts, it sits well with those who object to restrictions on substitution of identities. Kripke's paper also lays out two important options concerning the quanti er domains. The simplest of the two, the xed domain approach, assumes a single domain of quanti cation which contains, presumably, all the possible objects. The world-relative interpretation, on the other hand, assumes that the domain of quanti cation contains only the objects that exist in a given world, and so the domain varies from one world to another. 1.2.1.1 Fixed Domains: The System Q1. Although the xed domain approach is less general, it is attractive from the semantical point of view because we need only add the familiar machinery for 8x to the semantics of a modal logic in the following way. A xed domain objectual model with rigid terms (or Q1-model) is a sequence hW; R; D; Q1; ai, where the domain of quanti cation Q1 is D, the set of possible objects, and where a meets the condition (aRT), which guarantees that the term intensions are constant functions. (aRT) a(t)(w) is a(t)(w0 ) for all w; w0 in W: The truth value of a sentence on a model is then de ned using the following clause for the quanti er: (Q1) a(8xA)(w) is T i for all d in Q1; a(d=x)(A)(w) is T: (Here a(d=x) is the assignment like a save that a(x) = d.) For each propositional modal logic S , let the formal system Q1-S consist of the principles of S , rules for rst-order logic (ID), (RT), and the Barcan formula (BF): (BF) 8xA ! 8xA: One satisfying feature of the xed domain account is that most propositional modal logics S for which we can show completeness with respect to a set R(S ) of Kripke frames, have the feature that the system Q1-S is semantically consistent and complete with respect to Q1-S -validity. There are exceptions, however. For example, Cresswell [1995] explains that when R(S ) is convergent, completeness of Q1-S may fail.
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1.2.1.2 World-Relative Domains. 1.2.1.2.1 The Motivation for World-relative Domains. The xed domain interpretation is satisfying from the formal point of view, but it is not an accurate account of the semantics of quanti er expressions of natural language. We do not think that `There is a man who signed the Declaration of Independence' is true, at least not if we read `there is' in the present tense. Nevertheless, this sentence was true in 1777, which shows that the domains of the present tense quanti ers changes to re ect which objects exist at di erent times. The domain varies along other dimensions as well. For example, when I announce to my class that everyone did well on the midterm, it is understood that I am not praising the whole human race. Time, place, speaker, and even topic of discussion play a role in determining the domain in ordinary communication. There are also strong reasons for rejecting xed domains in modal languages. On the xed domain interpretation, the sentence 8x9y(y = x) (which reads `everything exists necessarily') is valid, but we would not ordinarily count this as a logical truth because we assume that di erent things exist in the di erent possible worlds. The defender of the xed domain interpretation can respond to these objections by insisting that the domain of 8x contains merely possible objects. Expressions whose domain depends on the context, can then be de ned using 8x and predicate letters. For example, the present tense quanti er can be de ned using 8x and a predicate letter that reads `presently exists'. One diÆculty with this proposal is that it requires the invention of predicates for all the di erent subdomains which we may ever intend for quanti er expressions, and it forces us to represent simple expressions of natural language di erently in di erent contexts of their use. It would be more satisfying if we could specify semantics for intensional logic which admits the context dependence of the domain. 1.2.1.2.2 World-Relative Models: Q1R- Semantics. Let us de ne a worldrelative objectual model with rigid terms (or Q1R-model) as a sequence hW; R; D; Q1R; ai, where Q1R is a function that assigns a subset D(w) to D to each possible world w, and where a meets condition (aRT). The truth clause for the quanti er reads as follows: (Q1R) a(8xA)(w) is T i for every d in D(w); a(d=x)(A)(w) is T: An adequate logic Q1R for Q1R-validity can generally be formulated by adding the principles MFL of free logic, rules ID for (intensional) identity, and (RT) to the underling modal logic. 1.2.1.2.3 Methods for Preserving Classical Quanti er Rules. The worldrelative interpretation of the quanti ers virtually demands the adoption of free logic. I say `virtually' because there are systems which use rstorder rules with the world-relative interpretation; however, they have serious
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limitations. To appreciate the diÆculties in trying to maintain the standard rules, notice rst that the sentence 9x(x = t) is true at a world on a model just in case the extension of t is in the domain of that world. However, 9x(x = t) is a theorem of rst-order logic, and so it follows that every term t of the language must refer to an object that exists in every possible world. This leads to two diÆculties. First, there may not be any one object that exists in all the worlds. Second, the whole motivation for the world relative approach was to re ect the idea that objects in one world may not exist in another; but if standard rules are used, no terms may refer to such objects. 1.2.1.2.3.1 Eliminate terms: the system QK. Kripke [1963] gives an example of a system for the world-relative interpretation which keeps the classical rules. The system QK has no terms other than variables. On a semantics where variables are given extensions in the domain, the validity of 9xx = y would demand that the extension of y be a member of every possible world. Kripke avoids this diÆculty by giving sentences with free variables the closure interpretation. So 9xx = y has the semantical e ect of 8y9xx = y, which is valid in free logic. From the semantical point of view, then, Kripke's system, has no terms at all, because the variables are really disguised universal quanti ers. Although Kripke has shown that modal extensions of rst-order logic with the world-relative interpretation are possible, his system underscores a theme which we have been developing throughout this chapter, namely that adoption of the classical rules forces us into an inadequate account of terms. Another oddity of Kripke's system is that he must weaken the necessitation rule: `if A is a theorem, then so is A'. Otherwise we would be able to derive 9xx = y which, since it is given the closure interpretation, says that any object of one domain exists in all the others. The rule is repaired by restricting it to closed sentences. 1.2.1.2.3.2 Nested domains and truth value gaps. There is a second problem with using classical logic with the world- relative interpretation which has exerted pressure on the way semantics for quanti ed modal logics is formulated. The principles of classical logic, along with the (unrestricted) rule of necessitation entail (CBF), the converse of the Barcan Formula. (CBF)
8xA ! 8xA:
It is not diÆcult to show that every world-relative model of (CBF) must meet condition (ND) (for `nested domains'). (ND) If wRw0 then D(w) is a subset of D(w0 ): To see this, notice that 8x9yy = x is Q1R-relative valid, and entails 8x9yy = x by (CBF). Our desire to avoid 8x9yy = x was one of the things which prompted the world-relative interpretation, for 8x9yy = x claims that any object which exists in the real world must also exist in all
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worlds which are possible relative to ours. Certainly, we want to allow that there are possible worlds where at least one of the things of our world fails to exist. If R is symmetric, then it follows from (ND) that all worlds accessible from ours have exactly the same domains. This result is re ected in the fact that the Barcan Formula (BF) is provable in systems as strong as B which use the standard quanti er rules. In models of S5 where all worlds are accessible from each other, (ND) demands that all domains be the same, in direct con ict with our intention to distinguish the domains. Despite these diÆculties in using classical principles with an unrestricted necessitation rule, several authors have de ned systems which preserve the classical rules. Typically, their systems simply adopt (ND). Yet other adjustments must be made, however, to preserve classical logic. The sentence 8xP x ! P t, for example, is not valid on a model where the extension of t at a world w is outside D(w), and the extension of P at w is D(w). One simple way to restore validity to the rule of Universal Instantiation is to stipulate that the terms are local, that is, the extension of a term at a world must be in the domain D(w) of that world. However, there are serious problems with this. According to this view, `Pegasus' and possibly `God' cannot count as terms since their extensions are not in the real world. As we have argued in Section 0.2, there are good reasons for wanting to count these as terms. Furthermore, we have been assuming that terms are rigid, so terms must have the same referent in all worlds. So the demand that terms be local entails that any term must have an extension which exists in all the worlds. In fact, the only objects at which the domains might vary are ones which are never named in any world. This undercuts the whole point of introducing world-relative domains, namely to accommodate terms that refer to things that may not exist in other possible worlds. The consequences of having terms that are both local and rigid are disastrous. There is another related idea, however, that looks as though it might work. If we assume that predicate letters are local, i.e. that their extensions at a world must contain only objects that exist at that world, then we will ensure that the classical sentence F t ! 9xF x (hence 8xP x ! P t) is valid. The reason is that from the truth of F t, it follows that t refers to an existing object, and from this it follows that 9xF x is true. Nevertheless, local predicates set up other anomalies, and they do not lead to the validation of the classical rules. To see why, consider :F t ! 9x:F t. From the truth of :F t, it does not follow that the extension of t is an existing object, and so it does not follow that 9x:F t is true. Not only do we fail to validate the rule of Existential Generalisation, but the valid principles cannot be expressed as axiom schemata. (We cannot write P t ! 9xP x for arbitrary sentences P t, because some of these instances are valid, and others are not.) In case we are using axioms and a rule of substitution of formulas for atoms, the problem re-emerges in the failure of the rule of substitution. Either way,
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the use of local predicates leads to serious formal diÆculties. There is a somewhat more plausible way to ensure the classical principles. A Strawsonian treatment would rule that a sentence has no truth value when it contains a term that does not refer to an existing object. Following this idea, we allow terms to refer to objects outside of the domain of a given world, but rule that sentences which contain such terms lack truth values. Valid sentences are then de ned as ones which are never false. As a result, 8xP x ! P t is valid, since any assignment that gives t an extension outside the domain for a world leaves the whole conditional without a value, and assignments that give t an extension inside the domain will make P t true if 8xP x is true. 1.2.1.2.3.2.1 The systems GKc and GKs. When truth value gaps are introduced, we are faced with a number of options concerning the truth clause for . On at least one of these options we may drop the nesting condition (ND) if we like and still obtain the classical rules. However, there are pressures that make us want to keep it. Suppose we are evaluating F t at w and the referent of t is in the domain D(w) of w. Then we expect to give F t a truth value on the basis of the values F t has in the worlds accessible from w. Unless we adopt (ND), there is no guarantee that t refers to an existing object in all accessible worlds, and so F t may be unde ned in some of them. Adopting the nesting condition ensures that we will always determine a value for P t at w on the basis of the values which F t is bound to have in all accessible worlds. If we drop (ND), however, there are two ways to determine the value of F t at w depending on whether the failure of F t to be de ned in an accessible world should make F t false or not. On the rst option, Gabbay's GKc [Gabbay, 1976, pp. 75 .], the necessitation rule must be restricted so that we can no longer derive (CBF). On the second option, GKs, (CBF) is derivable, but the truth of (CBF) in a model no longer entails (ND). Either way, the rules of the underlying modal logic must be changed. 1.2.1.2.3.2.2 The system QPL. For these reasons, the more popular choice [Hughes and Cresswell, 1968] has been to assume (ND) and to de ne satis ability as follows. A QPL-satis able set is one where none of its sentences is false in any world on some Q1R-model that meets (ND), and where any sentence which contains a term t with extension a(t)(w) 62 D(w) has no truth value at w. QPL-semantics is attractive from a purely formal point of view because we have relatively simple completeness proofs for systems that result from adding the principles of (classical) predicate logic to certain propositional modal logics, provided, that is, that the language omits =. Proofs are available, for example, for M and S4. In case the modality is as strong as B, the domains become rigid, and the completeness proof is carried out using methods developed for systems that validate the Barcan Formula.
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1.2.1.2.4 Conclusion: We Should Adopt Free Logic. The appeal of simple completeness proofs should not blind us to the fact that the stipulations required in order to preserve the classical principles do not always sit well with our intuitions. Our conclusion, then, is that there is little reason to attempt to preserve the classical rules in formulating systems with the objectual interpretation and world-relative domains. The principles of free logic are much better suited to the task. As we will see in Section 2, results for systems based on free logic are actually not that diÆcult, especially when identity is not present. 1.2.2 Non-rigid terms and world-relative domains 1.2.2.1 The System Q3. There are two important reasons why the assumption that all terms are rigid designators should be rejected. First, expressions like `the tallest man' clearly refer to di erent objects in di erent worlds. If we want to count descriptions among our terms, as we do on a Strawsonian account, we cannot accept the rigidity condition. Second, David Lewis [1968] contends that it makes no sense to talk of identity of objects across possible worlds. Objects from two di erent worlds are never identical, although it may make sense to talk of the counterpart of an object in another world. On counterpart theory, then, it is impossible for the intension of any term to be a constant function. Since it is important that a logical theory not rule out reasonable positions, we would like to relax the restriction that terms are rigid. Let us de ne a Q3-model, then, as a Q1R-model which (possibly) fails to meet condition (aRT). Something unexpected happens when we relax the assumption that terms are rigid. The rule (FUI) of instantiation for free logic is no longer Q3-valid. In order to see why, notice that the sentence (t = t ^ Et) ! 9xx = t is a consequence of (FUI). Since t = t is also provable there, we obtain (E ). (E ) Et ! 9xx = t:
If t reads `the author of \Counterpart Theory" ', then (E ) says that if the author of `Counterpart Theory' exists, then there is someone who is necessarily the author of `Counterpart Theory'. Intuitively, (E ) is unacceptable, and it is not diÆcult to back up this insight with a formal counter-example. Let us imagine a model with two worlds, r (real) and u (unreal) whose domains both contain two objects, namely David Lewis and Saul Kripke. Assume that both worlds are accessible from themselves and each other. Imagine that the extension of t at the real world r is Lewis, but that it is Kripke in the unreal world u. On this model, 9xx = t is false in r because neither Lewis nor Kripke is the extension of t in both worlds. Nevertheless, Et is true in r since the extension of t in the real world, namely David Lewis, is in the domain of r. This counterexample helps us appreciate the subtle reason why (FUI) has broken down. There is no question that David Lewis exists, and there is no
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question that the author of `Counterpart Theory' is identical to the author of `Counterpart Theory' in any world we choose. However, the claim that any one person counts as the author of `Counterpart Theory' in all worlds seems false. One way to help diagnose this situation is to reformulate Q3 semantics in an equivalent, but more complex way. Replace each object with the constant function which takes any world to that object. Seen this way, the items in our domain(s) are all intensions of rigid terms. The rule of instantiation is no longer valid because the domain of quanti cation includes only constant term intensions, whereas terms may have nonconstant intensions. The rules of free logic would be Q3-valid if we were to interpret the primitive predicate E so that Et is true in world w i the extension a(t)(w) of t 2 D(w) and a(t) is a constant function. Notice, however, that the extension of E must contain term intensions, and not objects, if it is to do this job. As a result, E is an intensional predicate, which means that substitution of identity does not hold for its term position. Substitution fails because E `David Lewis' is presumably true, while E `the author of \Counterpart Theory" ' is not, even though `David Lewis' and `the author of \Counterpart Theory" ' refer to the same thing in the real world. Aldo Bressan [1973] has championed the view that even scienti c language requires intensional predicates. His more general semantics de nes the extension of a one-place predicate at a possible world as a set of individual concepts (i.e. term intensions) not a set of objects. As a result, he has no diÆculty accommodating a primitive predicate which expresses rigidity. Hintikka [1970] chose more modest methods. He showed how to formulate a correct rule of instantiation for Q3 that does not require an intensional existence predicate. Notice that the sentence 9xx = t is true in a model at world w i the intension of t has the same value in all worlds accessible from w. Similarly, 9xx = t is true at w just in case the intension of t is constant in all worlds accessible from those worlds. While there is no one sentence that expresses that a term is rigid, a sentence of the shape 9x i x = t, where i is a string of i boxes, guarantees that the intension of t is constant across enough worlds so that i F t follows from 8x i F x when F t is atomic. This idea is generalised in Hintikka's formulation (HUI) of a valid rule of universal instantiation for nonrigid terms. (HUI)
8xP x (9x i x = t ^ : : : ^ 9x k x = t) ! P t
where i; : : : ; k is a list of integers which records for each occurrence of x in P x, the number of boxes whose scope includes that occurrence. In modal logic as strong as S4, this rule can be simpli ed considerably because there 9x i x = t is equivalent to 9xx = t. Thomason [1970] demonstrates the adequacy of Q3{S4, using (TUI) as the instantiation rule.
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8xP x 9xx = t ! P t
Completeness proofs for the weaker modalities have never been published as far as I know. Perhaps researchers have been daunted by the complexity of Hintikka's rule. It is interesting to note that even in the context of S4, Thomason was forced to adopt other complex rules for identity and the quanti er. Parsons [1975] has given a weak completeness result for a system that uses more standard rules, but he also shows that, in general, Thomason's rules cannot be simpli ed in the obvious way. 1.2.2.2 A Classical Logic with Local Terms: The System Q3L. There is a simple way to avoid the complicated instantiation rule needed in Q3. If we add the assumption that terms are local, that is, that the extension of a term at a world w is always in that world's domain, then we restore the classical quanti er rules. A Q3 model with local terms (Q3L-model) is a Q3-model which meets condition (L)
a(t)(w) 2 D(w) for all w in W , and all terms t. This condition could not be sensibly imposed for systems with rigid terms because then, any object referred to by a term would have to exist in all the domains. However, when terms are nonrigid, the domains can change as long as the extension of the terms change in corresponding ways. There is an important application of Q3L which Cocchiarella discusses in his chapter in Volume 3.4. If is to capture logical necessity, then we may think of possible worlds w as predicate logic models hDw; awi, each equipped with its own domain Dw, and assignment function aw. We expect an assignment function aw of a model hDw; awi to give extensions to the terms (and predicate letters) in the corresponding domain Dw. So it is only natural in this case to adopt nonrigid terms, world-relative domains, the objectual interpretation, and local terms. If we interpret A to mean that A is true in all models, then Q3Lsemantics cannot be axiomatised. However, if we give A the generalised interpretation where A is true i it is true on all models in an arbitrarily selected set of models, then Q3L is axiomatised by adding the principles of predicate logic to S5. A more general account stipulates that A is true on a model U just in case A is true in all models U 0 suitably related to U . In this case the underlying modality depends on the conditions we adopt on the accessibility relation between models. If we take this option, however, and the accessibility relation is not symmetric, then we are forced to assume nested domains (ND), in order to preserve the classical quanti er rules. Bowen [1979] investigates systems of this kind. Even if we are willing to give up the nesting condition, problems arise. Suppose we are evaluating 8xF x in a world w where object o exists, and w0 is an accessible world where o (L)
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does not exist. To determine the value of 8xF x, we need to nd the value of F x when x refers to o. This requires that we nd the value of F x in world w0 where o does not exist. At this point we are faced with the same options we described in Section 1.2.1.2.3.2. We may use truth value gaps, or we may rule that F x in this case is false. As we pointed out, both choices have disadvantages. Despite its application to certain notions of logical necessity, the local term condition (L) is not usually acceptable. In ordinary reasoning, we would nd the assumption that anything that exists in the real world exists in all worlds possible relative to our is quite implausible. For this reason, we are still interested in Q3 without local terms, even though the rules may be diÆcult.
1.3 The Conceptual Interpretation The systems we have discussed so far are not especially satisfying. We have good reasons for wanting to allow nonrigid terms in our language, and yet the rules we need for Q3 are quite complex, unless we move to a language with a primitive intensional predicate that expresses rigidity. On the other had, systems with local variables, like Q3L, have limited applications. One account of our diÆculties, as we explained earlier, is that our terms can be assigned any intension, while the domain(s) of quanti cation contain only constant intensions. Perhaps allowing nonrigid intensions in our domain might result in a better match between the quanti ers and the terms, and so yield simpler rules. Though it may seem philosophically dangerous to quantify over individual concepts, there are intuitions concerning tense and modality that support this choice. For example, imagine that our possible worlds are now states of the universe at a given time. The extension of a term at a given time will turn out to be a temporal slice of some thing, `frozen' as it is at that instant. Notice that things, since they change, cannot be identi ed with term extensions. Instead, things are world-lines, or functions from times into time slices, and so they correspond to term intensions or individual concepts. Since our ontology takes things, not their slices as ontologically basic, it is only natural to quantify over term intensions in temporal logic. Our reluctance to quantify over individual concepts may be an accident of nomenclature. The so called `objects' of a temporal semantics are not the familiar things of our world, while the formal entities that do correspond to things are misleadingly called `individual concepts'. 1.3.1. Fixed Domains: The System QC. Let us now formulate what we will call the conceptual interpretation of the quanti er. A conceptual model (or QC-model) is a sequence hW; R; D; QC; ai
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where QC is the set of functions from W into D. The truth clause for the quanti er reads as follows: (QC) a(8xA)(w) is T
i
for every f in QC; a(f=x)(A)(w) is T .
Here `a(f=x)' represents the assignment function identical to a except that the intension of x on a(f=x) is function f . Although the conceptual interpretation is designed to satisfy reasonable intuitions, there are a number of problems with it. One formal diÆculty is that no (consistent) system is complete for this semantics. Whenever we interpret the domain of any quanti er as a set of all functions, we run the risk that the language will have the expressive power of second-order arithmetic, with the result that Godel's Theorem applies. As we will show in Section 3, that is exactly what happens with QC. There are also intuitive diÆculties. First, notice that 9xx = t is QCvalid, and yet we have given an intuitive counterexample to it in Section 1.2.2.1. We do now want to say that there is something which is necessarily the author of `Counterpart Theory', because no one thing is the author of that paper in all possible worlds. However, on the conceptual interpretation, 9xx = t is true as long as we can nd some term intension which matches that of t in all possible worlds, and the term intension of t so quali es. This shows that the conceptual interpretation di ers from our ordinary reading of the quanti er. Another QC-valid sentence which may tantalise some readers is 9x9yy = x, which claims that there is something (God?) which necessarily exists. However the QC-validity of this sentence will do little to satisfy those who still search for an ontological argument for the existence of God. Any term intension will do to satisfy 9yy = x, simply because any term intension has the property that there is a term intension (namely itself) which agrees with it in accessible worlds. 1.3.2. World-relative Domains: The System Q2. The reader may think that we can repair these problems by introducing world-relative domains. Let us investigate the situation, then, when a Q2model is a sequence hW; R; D; Q2; ai, where Q2 is a function that assigns a domain D(w) to each world w. The quanti er truth clause now reads as follows. (Q2) a(8xA)(w) is T i for every function f : W ! D, if f (w) 2 D(w); a(f=x)(A)(w) is T . Unfortunately, the problems we mentioned still remain. First, the incompleteness result still applies to the new semantics. Second, although both 9xx = t and 9x9yy = x are no longer valid, they still do not receive their intuitive interpretations. For example, 9x9yy = x will turn out to be true on every model where the domains of the worlds all contain at least
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one object. In that case, any function that picks a member of D(w) for each world w will satisfy 9yy = x, and so verify 9x9yy = x.
1.4 The Substantial Interpretation As we showed in the last section, the conceptual interpretation of the quanti ers does not match the interpretation which we give to quanti er expressions in ordinary language. The sentence 9x9yy = x, which we interpret as making the very strong claim that some thing must exist in every possible world, is valid on the conceptual interpretation as long as no possible world has an empty domain. The di erence between our intuitive understanding of 9x9yy = x, and the conceptual interpretation is that the existence of a term intension that (say) picks out David Lewis in this world, a rock in another, a blade of grass in another, and so on, counts to verify 9x9yy = x. On the other hand, our intuitions demand that any term intension that veri es 9x9yy = x must be coherent in some sense; our concept of a thing brings with it some notion of what it would be like in other worlds. Only certain collections of objects, (and certainly not a collection consisting of David Lewis, a rock, a blade of grass, etc.) could count as the manifestations of a thing, and so only these collections should count to verify 9x9yy = x. In order to do justice to these intuitions, we must restrict the domain of quanti cation to the term intensions that re ect `the way things are' across possible worlds. Thomason [1969] suggests that the domain should contain only constant functions. The idea is that for 9x9yy = x to be true there must be one thing, identical across possible worlds, which exists in each one. This proposal is simply Q3, the objectual interpretation with non-rigid terms. We have already discussed some of the formal diÆculties with this option in Section 1.2.2. There are also intuitive objections similar to the ones which we used in arguing against systems with rigid terms. First, Thomason's account of substances is incompatible with counterpart theory, for on that view, the domains of the possible worlds are disjoint, and so there cannot be any constant term intensions to ll the domain of the quanti er. Second, in temporal logic, where objects are time slices, we do not want a thing to consist of the same time slice across di erent times. The slices of a thing picked out at di erent times may be quite di erent, but the world line composed of the slices still represents one uni ed thing. 1.4.1. The System QS. If we are to accommodate a variety of conceptions about what things are like, we should not assume that they are the constant term intensions (Q3), nor that they are all the term intensions (Q2). To be completely general, we introduce a set of term intensions for each world, to serve as its domain
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of quanti cation, and we will make no stipulations about what these sets contain. Let us now give a formal account of this approach. A world-relative substantial model (or QS-model) is a sequence hW; R; D; QS; ai, where QS is a function that assigns to each world w a set S (w) of functions from W into D. (We call S (w) the set of substances for world w.) The truth clause for the quanti er reads as follows: (QS) a(8xA)(w) is T i for every member f of S (w), a(f=x)(A)(w) is T . It is not diÆcult to see that 9x9yy = x is not valid on this semantics, for it would only be true in world w if there were a substance f in S (w0 ) in every world w0 accessible from w. Complete systems for QS can be constructed as long as we are willing to introduce the intensional predicate constant E to represent which functions count s substances in each possible world. An adequate system for this semantics very often results from adding the rules of MFL, and the rules ID for (intensional) identity to the underlying modal logic. As we will explain in Section 2.2.4, more general quanti er rules may be needed for weaker modal logics. We should note an important restriction on the rule of substitution of identities in QS. The constant E is an intensional predicate, and this means that substitution of term identities does not hold in its term position. When we formulate the rule of substitution for identities, we must make it clear that we do not consider Et to be an atomic sentence, for otherwise we would be able to deduce Et0 from t = t0 and Et. 1.4.2. Fully Intensional Predicates: The System B1. During our discussion of Q3, we pointed out that one way to simplify the instantiation rule is to introduce an intensional predicate E to the language. A predicate is intensional when its extension at a world w contains term intensions, and not objects as we ordinarily expect. To be more careful, the extension of an n-ary intensional predicate letter at a world is a set of n-length sequences of term intensions. Bressan [1973] presents a beautifully general modal logic, with descriptions and quanti ers for all types, which assumes that predicate letters are intensional in this sense. Clearly, such a strong language cannot be axiomatised. However, Parks [1976] has axiomatised the rst-order fragment B1 of Bressan's system, using the substantial interpretation of the quanti er. B1 uses S5 as its modal foundation, and a xed domain of substances. For this reason B1 validates classical quanti er rules and the Barcan Formula. However, more general languages with weaker modalities and world-relative domains of substances can be constructed using Bressan's more general treatment of predicates. In fact, we can add such predicate letters to QS without causing any major complications. All we need to do is adjust the
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rule of substitution of identities for those predicate letters so that substitution of one term for another is not allowed unless we already have a sentence which informs us that their intensions (not just their extensions at a given world) are the same. In weaker modal logics, this requires that we introduce a symbol for strong identity, interpreted so that a strong identity is true just in case the anking terms have the same intensions. Once this symbol is available, we simply adopt a rule of substitution of strong identities for term positions of the intensional predicate letters. 2 COMPLETENESS IN QUANTIFIED INTENSIONAL LOGIC
2.1 Why Completeness is Hard to Prove in Quanti ed Modal Logic Completeness proofs in QML are quite a bit harder than completeness proofs for propositional modal logic or rst-order logic. One reason that proofs are diÆcult is that sometimes there are none to nd, as is the case of the conceptual interpretation Q2. Even when a system is complete, the proof may be elusive, and diÆcult to formulate in a simple way. Another problem is lack of generality: a proof strategy may only work when the underlying modal logic is fairly strong (for example, as strong as S4), or when ad hoc conditions are placed on the models. One of the best ways to understand the methods used in completeness proofs for QML is to locate the main diÆculty which arises if we simply try to `paste together' proofs for quanti cational logic and propositional modal logic. In order to uncover the problem, let us review the crucial steps in the completeness proofs in each kind of logic. 2.1.1. Completeness Proofs for Propositional Modal Logics The most powerful method for proving completeness of a propositional modal logic S is to use maximally consistent sets. Completeness follows if we can show that any S -consistent set is S -satis able. (A set is S -consistent i there is no proof of a contradiction from the sentences in that set.) We begin by extending a given S -consistent set H to a maximally consistent set r (for real world) by Lindenbaum's Lemma. Then we build what we will call the standard model hW; R; ai for S . The set W of possible world of the model is taken to be the set of all maximally consistent sets of S , (on occasion, W contains just some of the maximally consistent sets related in some way to r). The relation R (of accessibility) is usually de ned so that wRw0 i if A 2 w, then A 2 w0 . Finally, the assignment function a is de ned for propositional variables p so that a(p)(w) is T i p 2 w. The central lemma (TL) (for Truth Lemma) in the proof shows that membership in w and truth in w on the standard model amount to the same thing.
286 (TL) a(A)(w) is T
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A 2 w.
Once (TL) is shown, it follows that all members of H are true at r on the standard model. We can also prove that hW; Ri 2 R(S ) (the set of Kripke frames that corresponds to S ), and so the standard model S -satis es H . The proof of (TL) is an induction on the construction of A, and the only really interesting case is when A has the shape B . (The case for propositional variables is trivial given the de nition of the standard model, and cases for : and ! simply depend on corresponding properties of maximally consistent sets w : :B 2 w i B 62 w, and B ! C 2 w i either B 62 w or C 2 w.) The proof of the case for takes the following form. a(A)(w) is T i if wRw0 then a(A)(w0 ) is T (1) i if wRw0 then A 2 w0 (2) i A 2 w. The only diÆcult part is to show the equivalence of (1) and (2). The inference from (2) to (1) is a simple consequence of the way we de ned R. In order to show that (1) implies (2), we show (:) instead. (:) if B 62 w, then there is a maximally consistent set w0 such that wRw0 and B 62 w0 . The proof of (:) makes a second use of the Lindenbaum Lemma. Given thatS B 62 w, we show the consistency of the set w = fA : A 2 wg f:B g. Then we use the Lindenbaum Lemma to extend w to a maximally consistent set w0 . The set w0 is such that wRw0 because for each sentence A in w, A 2 w0 ; it does not contain B since it is consistent and contains :B . 2.1.2. Completeness of First-order Logic In this section we will give a quick review of a completeness proof for PL, rst-order logic with identity. Again we show that any PL- consistent set is PL-satis able by rst extending H to a maximally consistent set r, written in language L. We then construct a model hD; ai from r as follows. The assignment function a is de ned so that the extension a(t) of t is ft0 : t = t0 2 rg, the equivalence class of terms ruled identical in r. The domain D contains a(t) for each term t. The assignment function a is de ned for i-ary predicate letters F so that hd1 ; : : : ; di i is a member of a(F ) just in case F t1 ; : : : ; ti 2 r and a(tj ) is dj for each of the tj of t1 ; : : : ; ti . Given the presence of principles of identity, it is not diÆcult to show that (TL) holds for atomic sentences on this model. In order to establish (TL) for all sentences, we must be sure that the set r meets one further condition concerning the quanti er, namely (8x). (8x) a(8xP x) is T i 8xP x 2 r. The proof of (8x) will be ensured if we can show that r is omega-complete (OC).
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(OC) If r ` P t, for every term t of L, then r ` 8xP x,for any variable x. (Here we write `r ` A' for `A is provable from the set of hypotheses r'.) Notice that (OC) is equivalent to (OC0 ). S (OC0 ) If r f:8xP xg is consistent, then S for some term t of L; r f:P tg is consistent. There are maximally consistent sets that are not omega-complete, so when we extend H to r using the Lindenbaum procedure, we must take special steps to guarantee (OC). Remember that the Lindenbaum method for extending a consistent set to a maximally consistent one begins by ordering the w s. A series of sets M0 = H; M1 ; : : : ; is then formed by letting Mi+1 be the result of adding the i + 1th w to Mi , i doing so would leave Mi+1 consistent. (Otherwise Mi+1 is Mi .) The maximally consistent set desired is the union of all the Mi . To ensure a set is omega-complete during this construction, we do the following. If Mi is the ith set formed in that construction, and :8xP x is the i + 1th sentence in our ordering of all the well-formed formulas, and if adding :8xP x to Mi would yield a consistent set, then we form Mi+1 from Mi by adding both :8xP x, and a sentence of the form :P t, where t is a term that is new to :8xP x and Mi . It is not too hard to see that adding this second sentence to Mi+1 cannot cause Mi+1 to become inconsistent, as long as Mi plus :8xP x was already consistent S as we have assumed. (The Sreason is that if Mi+1 = Mi f:8xP x; :P tg were inconsistent, then Mi f:8xP xg ` P t. Since t is foreign to both Mi Sand :8xP x, it follows by the rule of Universal Generalisation that S Mi f:8xP xg ` 8xP x, which entails that Mi f:8xP xg is inconsistent, contrary to our assumption.) We can also see from the second formulation (OC0 ) of omega-completeness that the result of the construction is omegacomplete, and so a saturated set. (A saturated set is a maximally consistent set that is omega-complete.) Now suppose we use this construction to produce a saturated extension r of H . As a result, we can show that (8x) holds in the model constructed from r by the following reasoning. a(8xP x) is T i for all d in D; a(d=x)(P x) is T (1) i for all terms t; a(a(t)=x)(P x) is T (2) i for all terms t; a(P t) is T (3) i for all terms t; P t 2 w (4) i 8xP x 2 w. The equivalence between (1) and (2) is proven by a straightforward induction on the length of P x. The equivalence of (2) and (3) is the result of the hypothesis of the induction; (3) entails (4) because r is omega-complete; and (4) entails (3) because of the rule of Universal Instantiation. Now that we have nished the proof of the case for 8x, we have a proof of (TL). It follows that the PL-model we have de ned satis es all the sentences
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of r and, hence, all sentences of our original set H . We conclude that any PL-consistent set is PL-satis able. 2.1.3. The DiÆculties in Quanti ed Modal Logics Notice that the method we described for constructing a saturated set for rst-order logic requires that we have an in nite set of terms of L which are foreign to H . Since we may have in nitely many sentences :8xP x to add, we need in nitely many `instances' :P t where t is new to the construction. As a result, the set w which we constructed using this method, contains an in nite set of terms of L which did not appear in H . Now let us imagine that we hope to prove completeness of a modal logic Q, which adds principles of rst-order logic to the propositional modal logic S . We begin with an Q-consistent set H which we hope to show is Q-satis able by extending H to a saturated set r written in language L. We then hope to construct the standard model, which will make all sentences of H true at r. DiÆculties arise when we try to prove (TL), for there is a con ict between what we need to ensure (8x) and () together. Condition (8x) demands that the set W of possible worlds be the set of saturated sets in language L, for the terms of L (actually their equivalence classes) determine the domain of the quanti cation of our model. On the other hand, the proof of condition () requires the following. From a given possible world w which contains :B , we must be able to construct a saturated set in language L which is an extension of w = fA : A 2 wg [ f:B g. The problem is that in order to extend w to a saturated set in L, we must nd an in nite set of terms of L that do not appear in w . However, the world w contains (P t ! P t) for each term t of L, with the result that all formulas P t ! P t appear in w . So there are no terms of L foreign to w . If we attempt to remedy the problem at this point by constructing a world w0 from W in a larger language L0 , then we nd ourselves in a vicious circle. Now we must prove property (8x) for L0 instead of L. This forces us to de ne W as the set of all saturated sets in language L0 , so that when we want to extend w to a saturated set, we must nd in nitely many terms of L0 foreign to w . However, w is now a saturated set in language L0 , and contains (P t ! P t) for all terms t of L0 . Again, we have no guarantee that there are any terms of L0 which do not appear in w .
2.2 Strategies for Quanti ed Modal Logic Completeness Proofs In this section, we will illustrate four di erent strategies for obtaining completeness proofs in QML. Each of them has its strengths and weaknesses. Ideally, we would like to nd a completely general completeness proof. The proof would demonstrate completeness of the most general semantics we have considered, namely QS. The proofs for all less general systems would
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then fall out of the general proof just as proofs for the stronger propositional modal logics result from the completeness proof for K. This would help clarify and unify quanti ed modal logic. The strategy we present in Section 2.2.4 comes closest to providing such a general proof. However any such method will face some limitations for the reasons discussed at the end of Section 2.2.1. 2.2.1. Strategy 1: Extend w to a saturated set without using any new terms (completeness of Q1) The completeness proof for Q1 given by Thomason [1970] is worth reviewing because it illustrates an important strategy for overcoming the problem which we outlined in Section 2.1.3. Remember our diÆculty was that we needed a way to extend a consistent set w to a saturated one, but we did not have an in nite set of terms missing from w in order to carry out the construction. The system Q1 uses xed domains, the objectual interpretation, and rigid terms. It veri es classical quanti er principles and the Barcan Formula. When these are present, it turns out that w is already omega-complete in the case of most modal logics. Since any consistent omega-complete set can be extended to a saturated set in the same language [Henkin, 1949], we can extend w to a saturated set without needing any extra terms. The details of this reasoning are given in the following lemmas. LEMMA 1. If w is omega-complete, then so is w [ f , provided f is nite.
Proof. Suppose that w is omega-complete. To show that w [ f is also omega-complete, let us assume that w [ f ` P t for all terms t. It follows that w ` ^f ! P t for all terms t, where ^f is the conjunction of the members of f . Since w is omega-complete, it follows that w ` 8x(^f ! P x) for any choice of variable x we like. If we choose a variable x foreign to ^f , it follows that w ` ^f ! 8xP x, and so w [ f ` 8xP x. By principles of quanti cational logic, we can replace the variable x of 8xP x for any other variable. It follows, then, that whenever w [ f ` P t for all terms t, then w [ f ` 8xP x, and so w [ f is omega-complete. LEMMA 2. Any consistent omega-complete set w can be extended to a saturated set written in the same language.
Proof. We construct a saturated extension of w using a variant of the method described in Section 2.1.2. Suppose that the set Mi plus :8xP x is consistent, so that we are to form Mi+1 by adding :8xP x and an instance :P t to Mi . Ordinarily, we would choose a term t foreign to both Mi and :8xP x in order to ensure that adding :P t will not cause Mi+1 to become
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inconsistent. In this case, however, we must use a term t which may already appear in w. When w is omega-complete as we have assumed, it follows by Lemma 1 that Mi is omega-complete as well.S (Mi is formed by adding only nitely many sentences to w.) Since Mi f:8xP xg is consistent, it follows by formulation (OC0 ) of omega- completeness that Mi [ f:P tg is consistent for some term t of L. So :P t can be consistently added to Mi for this choice of t, and since :P t entails :8xP x, the result of adding both these sentences to Mi remains consistent. Once we ensure that instances :P t are consistently added in this way, it is a simple matter to verify that the union of the Mi is a saturated extension of w. LEMMA S 3. If w is a saturated set which contains :B , then w = fA : A 2 wg f:B g is consistent and omega-complete.
Proof. We can show that w is consistent just as we do in propositional modal logic. By Lemma 1, w is omega-complete if fA : A 2 wg is. Assume now that fA : A 2 wg ` P t for every term t. By principles of the modal logic K; w ` P t for each term t, and since w is omega-complete, it follows that w ` 8xP x. By the Barcan Formula, it follows that w ` 8xP x. Since w is maximal, 8xP x 2 w, and so 8xP x 2 fA : A 2 wg. It follows that fA : A 2 wg ` 8xP x. LEMMA 4. If w is a saturated set that contains :B then w = fA : A 2 S wg f:B g can be extended to a saturated set written in the same language.
Proof. By Lemma 3, w is consistent and omega-complete. By Lemma 2, it can be extended to a saturated set in the same language. Now let us assume that the system Q1 results from adding rules of classical logic, rules (ID) for identity, and (RT) for rigid terms to propositional modal logic S . To show completeness, we prove, as usual, that every Q1consistent set is Q1-satis able. Given a consistent set, we extend it to a saturated set r written in language L in the usual way. We then construct the standard Q1-model hW; R; D; Q1; ai as follows. W is the set of all saturated sets that contain t = t0 just in case t = t0 2 r. R is de ned in the usual way. The extension a(t)(w) of term t is ft0 : t = t0 2 rg. D is the set of all term extensions. Sequence hd1 ; : : : ; di i 2 a(F )(w) i F t1 ; : : : ; ti 2 w and a(tj )(w) is dj for the dj of d1 ; : : : ; di . For most modal logics, we may show that hW; Ri 2 R(S ) just as we did in the completeness proof for S , and so once we prove the truth lemma (TL), we will know that the sentences of H are all true at r on this model. It will follow that H is Q1-satis able. The interesting cases in the proof of (TL), concern and 8x. The proof of (8x) can be carried out along the lines we speci ed in Section 2.1.2. To establish (), it is crucial to show (:).
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(:) if :B 2 w, then there is a member w0 of W such that wRw0 and :B 2 w0 . By Lemma 4, we extension w0 of S know that we can construct a saturated 0 fA : A 2 wg f:B g. We can show that this w is a member of W if we can show that t = t0 2 w0 i t = t0 2 r. Since w is a member of W , we already know that t = t0 2 w i t = t0 2 r. Notice that if t = t0 2 w, then by (RT), (t = t0 ) 2 w, and t = t0 2 w0 . If t = t0 62 w, then :t = t0 is, and so by (RT) :t = t0 2 w, and :t = t0 2 w0 . It follows that w0 contains exactly the identities of r and so is a member of W . Since fA : A 2 wg is a subset of w0 , we know that wRw0 , and so we have completed the proof of (:). Strategy 1 has important limitations. First, the method depends on using rst-order logic and the Barcan Formulas, so it is not applicable to systems that give a more general account of the quanti ers. Second, the completeness result is blocked for certain underlying modal logics S . We illustrate the problem with modal logics where R is convergent. In proving that the standard model is convergent for propositional modal logics, one assumes wRw0 and wRw00 , establishes the consistency of fA : A 2 w0 g [ fA : A 2 w00 g, and then employs the Lindenbaum Lemma to extend this set to a maximally consistent set w000 such that w0 Rw000 and w00 Rw000 . In the case of a quanti ed modal logic, we must know that fA : A 2 w0 g [ fA : A 2 w00 g is omega-complete as well as consistent before Lemma 2 can be used to extend it to a saturated set. However, there is no guarantee that fA : A 2 w0 g [ fA : A 2 w00 g will be omega-complete. It will not be, for example, if fA : A 2 w0 g contains each of P t1 ; P t3 ; : : :, and fA : A 2 w00 g contains 8xP x; P t2 ; P t4 ; : : :, and t1 ; t2 ; : : : is a list of all terms of L. Under these circumstances fA : A 2 w0 g [ fA : A 2 w00 g contains f 8xP x; P t1 ; P t2 ; P t3 ; : : :g and so is not omega- complete. DiÆculties of this kind can be expected whenever the proof that hW; Ri 2 R(S ) for the propositional modal logic S rests on proving the existence of a consistent set, and then extending it to a maximally consistent set by the Lindenbaum Lemma. (Convergence and density are two conditions where this technique is typically used.) In this kind of case, the proof that hW; Ri 2 (S ) may fail for the quanti cational logic when a consistent set formed fails to be omega-complete. The problem does not arise for most modal logics. Strategy 1 works to show completeness, for example, for systems whose corresponding conditions on R are preserved under subsets. (Conditions are preserved under subsets i when the conditions hold for hW; Ri they also hold for hW 0 ; R0 i, where W 0 is a subset of W and R0 is R restricted to W 0 .) Conditions preserved under subsets include the universal conditions, i.e. conditions on R that can be expressed with universal quanti ers alone. However, for systems whose conditions are not preserved under subsets, strategy 1 does not necessarily
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yield a completeness result. This failure is directly related to the fact that system Q1-S is not complete for a semantics with convergent R [Cresswell, 1995]. 2.2.2. Strategy 2: Build the set of possible worlds all in one construction (completeness for Q1{S5) Gallin [1975, p. 25 .] o ers another strategy for proving completeness of S5 systems that contain classical principles and the Barcan Formula. It is a clever technique which has applications to systems with weaker rules. Gallin avoids the complication we encountered in extending w to a saturated set by de ning the set of worlds of his standard model so that all the worlds w are saturated and already satisfying condition (:S5). (:S5) If :A 2 w, then there is a world w0 such that :A 2 w0 .
In S5, this condition is suÆcient for demonstrating the case of (TL) for formulas that begin with . Gallin shows how to build a whole collection W of saturated sets from a consistent set H , using a variation of the Lindenbaum construction. The sets in W are the possible worlds of the standard model. In order to coordinate the construction properly, let W be a sequence w0 ; w1 ; w2 ; : : : of possible worlds. W is constructed from a consistent set H , using a series W0 ; W1 ; W2 ; : : :. Each of the Wi contains a sequence w0 ; w1 ; w2 ; : : : of consistent sets, each of which is on its way to becoming saturated as we move to larger Wj . The Wi are also arranged so that eventually, (:S5) is met for each formula A. To de ne the Wi , we need a generalisation of the notion of consistency. We say that a sequence W of sets is consistent just in case no nite subset f of any of the sets w in W is such that ^ f ` p ^ :p. A formula A can be consistently added to world w of sequence W just in case doing so would leave the sequence W consistent. This de nition of consistency ensures not only that adding A to a world w leaves w consistent, but that adding A is also consistent with all the facts about all the other worlds. Now we are ready to de ne the series W0 ; W1 ; W2 ; : : :. We let W0 be the sequence such that its rst world w0 is H , and all the other worlds w1 ; w2 ; : : : are empty. We then order the pairs hi; Ai consisting of integers i and formulas A, and for each pair hi; Ai, we pick a term t(i; A), which is foreign to H , and all sentences of previous pairs in the ordering. For each Wj , we de ne Wj+1 as follows. We consider the j + 1th pair hi; Ai in the ordering and we add A to world wi of Wj i H can be consistently added to wi of Wj . (Otherwise we set Wj+1 equal to Wj .) In case A has the shape :8xP x, we also add :P t, where t is t(i; A). In case A has the shape :A, we also nd the rst empty set in the sequence Wj+1 , and we add :A to it. There is such an empty set in Wj+1 , because we have only added
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nitely many formulas to this point, and W0 contained an in nite sequence of empty sets. It is also clear that adding this formula could not cause Wj+1 to become inconsistent. Once the Wj have been de ned this way, we let W be the sequence we get by letting the ith world of Wj be the union over all the sets w0 ; w1 ; w2 ; : : : ; which were the ith worlds of W0 ; W1 ; W2 ; : : :. It is not diÆcult to prove that each of the worlds of W is a saturated set that meets property (:S5). Notice, however, that because of the special de nition Gallin uses for consistency, the demonstration that these sets are saturated requires the Barcan Formula and classical principles for the quanti ers. Gallin claims that this proof is signi cantly easier than the method we presented as strategy 1. We do not agree with Gallin's' taste in simplicity. However, this strategy is quite interesting, and it can be modi ed for use with weaker rules as [Menzel, 1991] shows. 2.2.3. Strategy 3: Allow the language to vary across possible worlds
The second strategy we are going to discuss is illustrated by a completeness proof [Garson, 1978] for QS, the most general semantics we have described. The same idea will be used to sketch the proof of the completeness for QPL along the lines of Hughes and Cresswell [1968, p. 147 .] and Gabbay [1976, p. 46 .]. 2.2.3.1 Completeness of QS. In systems with world-relative domains, the Barcan Formula is not valid, and so we no longer know that fA : A 2 wg is omega-complete. Notice, however, that since the domain of quanti cation varies from one possible world to the next, we are free to select a di erent language for each of the saturated sets which are in W in the standard model. When it comes time to construct a saturated set from w , we simply build a saturated set in a language larger than the one in which w is written. Since QS is based on free logic, we have to readjust our de nition of omega-completeness and, hence, our de nition of saturation. An omegacomplete set for free logic in language L is any set that meets condition (FOC). (FOC) If w ` Et variable x.
! P t for every term t of L, then w ` 8xP x for any
A free logic saturated set for L is simply any maximally consistent set w for which (FOC) holds. It is easy to prove that a consistent set written in language L can be extended to a set which is free logic saturated for a language with in nitely many more terms than L. To provide the proof simply replace `(Et ! P t)' for `P t' in the corresponding proof for rst-order logic (see Section 2.1.2).
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Let QS be the logic that results from adding the principles of MFL and ID to certain propositional modal logics S . We will explain more about which logics these are later. We will demonstrate the completeness of QS with respect to the set of all QS-models (world relative substantial models for S ). See Section 1.4.1 for the de nition of a QS-model. As usual we assume that set H is consistent in QS, and we extend H to a free logic saturated set r written in a language L. At this point, however, we consider a larger language L+ , which contains in nitely many terms which are not in L. We then de ne the set W of possible worlds for our standard QS-model hW; R; D; S; ai as the set of all free logic saturated sets written in some language L0 such that there are in nitely many terms of L+ that do not appear in L0 . The idea behind this is to guarantee that whenever wS 2 W , there will be in nitely many terms foreign to w = fA : A 2 wg f:B g so that w can be extended to a saturated set in language L+ . The other parts of the de nition of the standard QS-model are straightforward. R is de ned in the usual way: wRw0 i if A 2 w, then A 2 w0 . The intension a(t) of a term t given by a is de ned so that a(t)(w) is ft : t = t0 2 wg, the equivalence class of terms ruled identical in w. S is de ned so that s 2 S (w) i s is a(t) for some term t such that Et 2 w. The domain of possible objects D is simply the set of all term extensions in all the possible worlds. The intension a(F ) of an i-ary predicate letter F is given as one would expect: hd1 ; : : : ; di i 2 a(F )(w) i F t1 ; : : : ; ti 2 w and each of the a(tj )(w) is dj . The intension a(E ) is S . Because the members of w are free logic saturated sets written in di erent languages, we cannot prove the Truth Lemma (TL) for this standard model. If t does not appear in Lw, the language in which the saturated set w is written, then a(:F t)(w) is T , but :F t 62 w. However, there is a weaker formulation (wTL) which will still serve our purposes. (wTL) If A is a sentence of Lw, then a(A)(w) is T i A 2 w. The proof of (W TL) for cases other than and 8x is straightforward. The crucial step in the case for is to demonstrate (:). (:) If B is a sentence of Lw, then if :B 2 w then there is a w0 in W such that wRw0 and :B 2 w0 . We begin the proof by assuming that B isSa sentence of Lw, and that :B 2 w. We construct w = fA : A 2 wg f:B g which we show to be consistent in the usual way. Since w is a member of W , there must be an in nite set N of terms of L+ that do not appear in w. By the de nition of w , it is clear that none of these terms appear in w either. We could construct a free logic saturated set w0 from w using these terms. However, if w0 is to be a member of W , there must be an in nite set of terms of L+ foreign to w0 . In order to ensure that we do not `use up' all the terms in our
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construction of w0 , we divide N into two in nite sets N1 and N2 . We use N1 to extend w to a free logic saturated set w0 , and we leave N2 in reserve to ensure that w0 2 W . When w0 is constructed in this way, we can easily prove that wRw0 , and that :B 2 w0 , and so we have nished the proof of (:). We would have skipped the case for 8x if it were not for one ticklish point. Along the way, we need to show (ES). (ES) a(t0 ) 2 S (w) i Et0 2 w. ((ES) is also needed to show the case of formulas with the shape Et.) The proof of (ES) would seem to be trivial given our de nition of S (w), but it is not. The trouble comes in showing (ES) from left to right. Suppose that a(t0 ) 2 S (w). Then by the de nition of S (w), there is a term t such that a(t0 ) is at a(t) and Et 2 w. For ordinary predicates, this would be enough to ensure that Et0 2 w, for when a(t)(w) is a(t0 )(w), we have that t = t0 2 w, and so can substitute t0 for t. Remember, however, that E is an intensional predicate for which the rule of substitution of identities does not hold, so this reasoning will not work. We must nd some other way to ensure that Et0 2 w. Things look bad when we realise that t0 may not even be in the language Lw, in which case Et0 62 w. Luckily, our de nition of the standard model ensures that whenever a(t) is a(t0 ) then t and t0 are the same term. The reason is that when t 62 Lw, it follows that a(t)(w) = ft0 : t = t0 2 wg is empty. For any pair of distinct terms t; t0 we choose, we can always nd a language Lw such that t is in Lw and t0 is not. It follows that the only way that a(t) and a(t0 ) can be identical is if t is identical to t0 . We have that Et 2 w, so we conclude that Et0 2 w and our proof of (ES) is nished. Once Lemma (wTL) is established in this way, the completeness of QS is shown fairly easily. We have already extended the QS-consistent set H to a free logic saturated set r, and since there were in nitely many terms foreign to r in L+ , it turns out that r 2 W . By (wTL), it follows that all members of r (and so all members of H ) are true at r on the standard model, and so H is QS-satis able. Although this proof is satisfying because it shows completeness for a system with a very general treatment of the quanti ers, it does not count as the general sort of completeness proof which we desire. The reason is that the strategy does not work to establish completeness of systems that use less general treatments of the quanti ers. For example, we might hope to show the completeness of the objectual interpretation with world relative domains and rigid terms by considering the system which results from adding (RT) to QS. We would hope that (RT) would ensure that terms are rigid on our standard model, with the result that all members of S (w) are constant functions. However, these hopes cannot be realised using the present de nition of the standard model. In order to ensure that (wTL) holds for sentences
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t = t0 , we are virtually forced into de ning a(t)(w) as ft : t = t0 2 wg. If any term t is rigid on this model, it would follow that a(t)(w) is a(t)(w0 ), and so that w and w0 share exactly the same identities. Because every saturated set for L contains t = t for every term t of L, it follows that w and w0 must be written in languages with the same terms. However, the strategy of this completeness proof depends on allowing our languages to shift from one saturated set to the next. Using similar reasoning, we can see that it is pointless to hope for a completeness proof for systems with xed domains using the standard model of this section. There is another respect in which the variable language strategy lacks generality. The method does not work for all propositional modal logics S . (Garson's [1978] claim to the contrary is an error.) The reason is that when possible worlds are written in di erent languages, we lose an important property () which is needed in showing that hW; Ri on the standard model is in R(S ). () If wRw0 and A 2 w0 , then A 2 w.
This property fails if term t is in the language of w0 , but not the language of w, and A is (say) F t. The sentence F t cannot be in w because it is not in the language of w. For many modal logics (for example, D, M, and S4), we do not need () in order to show that hW; Ri 2 R(S ). However, for systems like B, the property seems indispensable. There are tricks one can use to overcome the diÆculty for individual systems, but the changing language strategy does not provide a proof that is general with respect to the underlying modal logic. 2.2.3.2 Completeness of QPL without identity. When = is absent from our language, the problems we described in extending the completeness proof of QS to systems that use the objectual interpretation can be overcome, at least for some of the propositional modal logics. We will illustrate this by sketching the proof for QPL with respect to a QPL-semantics, where we use the objectual interpretation, world-relative domains, the nesting condition (ND), and truth value gaps. (See Section 1.2.1.2.3.2). We will be assuming that the underlying modal logic S does not require property () for its completeness proof. Remember that the system QPL simply results from adding the rules of rst-order logic to S . Since we are using classical principles, we de ne the standard model using the ordinary de nition of saturation. Since identity is absent, we may simply let the extension of a term (at any world) be itself. This ensures the rigidity of the terms, and so the objectual interpretation for the domains. It is easy to arrange that domains are nested in the standard model by de ning R so that wRw0 i w0 contains the terms of w, and if A 2 w, then A 2 w0 . This calls for no changes in the proof of the case for .
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It is particularly convenient that we are allowed truth value gaps in this semantics, since we may consider each world w as de ning the class of sentences de ned at w. The formal neatness of truth value gaps at this point suggests that their introduction was not designed to meet philosophical intuitions, but rather to avoid formal complications in the completeness proof. 2.2.4. Strategy 4: Rede ne Saturation
Thomason's [1970] proof of the completeness of Q3 is the inspiration for the next strategy we are going to present. At the risk of repetition, we will give a second completeness proof for QS. Once we have presented the details, we will show how to modify the proof to obtain completeness results for Q3, and several other systems. Strategy 4 follows the outlines of strategy 1; however, the concept of omega-completeness is adjusted to re ect the fact that the Barcan Formula and classical principles of quanti cation are no longer available. As we have already pointed out, w is not omega-complete in logics that lack the Barcan Formula. However, w has a weaker property which ensures that w can be extended to a set that has a correspondingly weaker form of saturation, a form which nevertheless ensures a proof of the quanti er case of the Truth Lemma. Although this strategy turns out to be quite powerful, it has the disadvantage that we must reformulate the quanti cational principles in a more general, and more complex way. In order to help simplify our presentation, we will adopt a few abbreviations. We use ` 3 ' for strict implication, so that `A 3 B ' abbreviates `(A ! B )'. We will be working constantly with formulas that have the shape (GF), where parentheses are to be restored from right to left. (GF)
A1 ! A2
3
:::
3 Ai 3 B .
(For example, A ! B 3 C 3 D amounts to A ! (B 3 (C 3 D)), or A ! (B ! (C ! D)).) We will use `G(B )' to represent any sentence with
shape (GF), and G(C ) will be the sentence that results from replacing C for B in G(B ). Using this notation, we may now present two general rules for the quanti ers. (GUI)
G(8xP x) G(Et ! P t)
(GUG)
` G(Et ! P t) where t does not appear in G(8xP x): ` G(8xP x)
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We should make clear that G(A) may represent a sentence where any of the arrows (whether ! or 3 ) is missing in the pattern (GF). So all of the following, for example, are instances of the rule (GUI).
8xP x A ! 8xP x A 3 8xP x A 3 B 3 8xP x : Et ! P t A ! Et ! P t A 3 Et ! P t A 3 B 3 Et ! P t
The reader can verify that (GUI) and (GUG) are QS-valid. The system (GS) consists of (GUI), (GUG), (=In), (=Out), and principles for propositional modal logics S . The quanti er rules (GUI) and (GUG) appear to be very odd and cumbersome. However, GS has a simple and natural reformulation in natural deduction format. The propositional modal logic K may be formulated by introducing boxed subproofs:
Together with introduction and elimination rules for : (In)
(Elim)
A
.. . A A
.. .
.. . A (See [Konyndyk, 1986, p. 34 ].) When natural deduction rules are employed, GS may be reformulated using the standard free logic rules (FUI) and (FUG), with the understanding that these apply within any subproof. It is a straightforward matter to show that this natural deduction formulation is equivalent to GS. Another feature of GS is evidence for its naturalness. One would hope to construct a quanti ed modal logic with xed domains by adding Et as an axiom, thus ensuring that the free logic rules collapse to their classical counterparts. In QS, the addition of Et entails (CBF), but (BF) is independent, and must be added as a separate axiom. However, when Et is added to GS, both the Barcan Formula (BF) and its converse (CBF) are provable. It is pleasing that the generalised rules are symmetrical with respect to the adoption of the Barcan Formula and its converse.
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The concept of omega-completeness which corresponds to the rules of GS is (GOC) (for general omega-completeness). (GOC) If w ` G(Et ! P t) for every term t of L, then w ` G(8xP x), for any variable x. A GOC set is just a set with property (GOC), and a set is generally saturated (for language L) just in case it is a maximally consistent GOC set. Our next task is to state and prove analogues of Lemmas 1{4 of Section 2.2.1 for general omega-completeness and general saturation. S
LEMMA G1. If w is GOC, then so is w f , provided that f is nite. S
Proof. Suppose that w is GOC, and assume that for all terms t; w f ` G(P t). It follows that w ` ^f ! G(P t). By propositional logic, this sentence is equivalent to one with Sthe shape (GF), so we know that w ` ^f ! G(8xP x), and hence that w f ` G(8xP x). LEMMA G2. Any consistent set w with property (GOC) can be extended to a generally saturated set written in the same language.
Proof. If :G(8xP x) is the candidate for addition to Mi in the LindenS baum construction, and if Mi f:G(8xP x)g is consistent, then we add both :G(8xP x) and :G(Et ! P t) to Mi to form Mi+1 , for some term t which leaves Mi+1 consistent. There is such a term because w is GOC and so, by Lemma G1, Mi+1 is GOC. This construction preserves consistency, and results in a GOC set, and so it yields a generally saturated set. LEMMA G3. If w Sis a generally saturated set that contains :B , then w = fA : A 2 wg f:B g is consistent and GOC. Proof. The consistency of w is proven in the standard way. To show that w is GOC, assume that w ` G(Et ! P t) for any term t of L. It follows that fA : A 2 wg ` :B ! G(Et ! P t). By principles of propositional modal logic K, w ` (:B ! G(Et ! P t)), and so w ` :B 3 G(Et ! P t) for every term t of L. Since w is GOC, w ` :B 3 G(8xP x), and since w is maximal, :B 3 G(8xP x) 2 w. As a result, :B ! G(8xP x) 2 fA : A 2 wg, and so w ` G(8xP x). LEMMA G4. S If w is generally saturated and contains :B , then w = fA : A 2 wg f:B g can be extended to a generally saturated set written in the same language.
Proof. By Lemmas G2 and G3.
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2.2.4.1 Completeness of GS. Now that we have proven Lemmas G1{G4, only a few details need to be mentioned to nish a completeness proof for GS. We begin with a GS-consistent set, and we extend it to a generally saturated set r written in language L. (To do so, we merely generalise the standard construction so that when :G(8xP x) is added, then so is :G(Et ! P t), where t is new to the construction.) We de ne the standard GS-model so that W is the set of all generally saturated sets for L. Items R; D; S and a are de ned in exactly the way as they were in Section 2.2.1. We may also prove the stronger truth lemma (TL) in a straightforward way. The case for requires that we show that if :A 2 w, then there is a w0 in W such that wRw0 and :A 2 w0 , but this is easily established using Lemma G4. To prove the case for 8x we notice rst that all generally saturated sets are free logic saturated, because free logic omega-completeness (FOC) is a special case of (GOC) when G(Et ! P t) is Et ! P t. So we will have no diÆculty proving that a(8xP x)(w) is T i 8xP x 2 w as long as we can show (ES). (ES) a(t) 2 S (w) i Et 2 w. In order to show (ES) in Section 2.2.3.1, we proved that if t and t0 are distinct, then so are their intensions a(t) and a(t0 ). We can show this is true of the standard GS-model as follows. In all the systems we are considering, the sentence :t = t0 is consistent if t and t0 are distinct. So there is a generally saturated set in W that contains :t = t0 , and the extensions of t and t0 di er there. This method of proving completeness has a number of advantages. Since all our sets are generally saturated in the same language, we no longer face the diÆculties noted in Section 2.2.3 in showing that hW; Ri 2 R(S ). Property () now holds, and so the proof proceeds exactly the way it does in propositional modal logics. However, there are still modal logics for which the method does not apply. The proof is still blocked, for example, when R is convergent for reasons similar to the ones we explained at the end of Section 2.2.1. Sets we can show to be consistent which we would hope to extend to a generally saturated set by Lemma G2 need not be GOC. Although strategy 4 does not solve the completeness problem for all underlying propositional modal logics, it can be generalised in another way. Once a completeness proof is available for GS, the method may be modi ed to obtain completeness results for extensions of GS that correspond to less general treatments of the terms and the quanti ers. A number of variations on this theme will be explored in the next sections. Despite its generality, there is another problem with this method. The systems we have proven complete use the generalised quanti er rules (GUI) and (GUG). We would like to be able to show completeness for logics which use the more modest principles (FUI) and (FUG) of free logic. However, this is not always possible. Parsons [1975] has shown that (GUI) is independent
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from the free logic rules in Q3. One reason for the sporadic nature of published completeness results is that certain systems are complete only when the generalised quanti er rules are chosen. Determining the conditions under which the generalised rules are necessary is an interesting topic for future research. 2.2.4.2 Rigid Terms: Completeness of GQ1R. One advantage of strategy 4 is that it can be used to obtain completeness proofs for a variety of logics that use the objectual interpretation, even if they contain identity. A simple formulation of a system GQ1R which is complete for the objectual interpretation results from adding the rules (RT) and (=E) to GS to ensure that all the terms are rigid. t = t0 (=E) Et ! Et0 Remember that E is an intensional predicate in GS, and so the rule of substitution does not apply to it. However, once the terms are rigid, substitution of identicals is valid in all contexts, and so (=E) is valid. It is not diÆcult to show the completeness of GQ1R for the objectual interpretation with rigid terms and world relative domains. Only one change in the de nition of the standard model is required, along with a simple adjustment to the proof of (TL). We begin with a consistent set H , which we extend to r, a generally saturated set in L. We then de ne the standard model as before, except we ensure the rigidity of all the terms by restricting W to sets that contain exactly the identities of r. We must adjust the proof of the case for because we will need to know that w can be extended to a set that contains the same identities as r. However, this can be shown using virtually the same argument we gave in Section 2.2.1, using the fact that (RT) is provable in GQ1R. Because our terms are rigid, the proof of (ES) is simpli ed. Since substitution now holds in the term slot of E , the proof that Et 2 w i a(t) 2 S (w) no longer requires a demonstration that the intensions of t and t0 are identical only if t and t0 are identical. Since all term intensions are rigid on this standard model, and since our domains contain only term intension, we can modify the model by replacing each constant term intension in a domain D(w) with its value. The result is a Q1R-model which satis es r and hence, H . 2.2.4.3 Fixed Domains: Completeness of GQ1. It is a simple matter to verify that adding (CBF) (the converse of the Barcan Formula) to GS ensures that the standard model meets the nesting condition (ND). (CBF)
8xP x ! 8xP x.
(ND)
If wRw0 then D(w) is a subset of D(w0 ).
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The relationship between (CBF) and (ND) can be appreciated better when it is pointed out that (BF) is equivalent (in free logic plus modal logic K) to (E ). (E )
8xEx. Objection to (E ) prompted our interest in logics with world relative domains. It is not hard to see that any model that satis es (E ) meets the nesting condition. Presence of the Barcan Formula (BF) forces the `converse' condition (CND) on the standard model. (BF)
8xP x ! 8xP x
(CND) If wRw0 then D(w0 ) is a subset of D(w). Let us restrict the domain W of the standard model so that it contains only worlds such that rRi w, where Ri is the result of composing R with itself i times, and R0 is the identity relation. It follows from the presence of both (BF) and (CBF) that the domains of the standard model are all identical, and so can be collapsed into one. So we may use strategy 4 to give a completeness proof for a semantics with a xed domain of the quanti er, but with a possibly wider domain for the terms. In order to prove completeness for GQ1, we need only ensure that the terms are all given extensions in the domain of quanti cation. The standard model meets this condition when (E) is added to GS, and so we have an easy completeness proof of GQ1 = GQ1R + (E). (E)
Et
It is interesting to note that both (BF) and (CBF) are derivable as soon as (E) is added to GS. In free logic, the addition of Et would restore the classical quanti er rules, and so allow us to prove (CBF); but (BF) is still independent. It is pleasing that the generalised rules are symmetrical with respect to the adoption of the Barcan Formula and its converse. 2.2.4.4 Nonrigid Terms: Completeness of Q3. Something like strategy 4 was invented by Thomason to prove completeness of Q3{S4. The system he showed complete is necessarily based on the generalised quanti er rules. We will use strategy 4 here to prove completeness of several kinds of Q3 logics. In our discussion of systems with the objectual interpretation and non-rigid terms (Section 1.2.2), we pointed out that quanti er rules are quite complicated unless we introduce a primitive predicate that expresses that a term intension is a constant function. We have been presuming all along that there is a primitive predicate E in our language which is interpreted so that a(E ) is S , the set of `real' substances. So we will begin with proofs for systems with arbitrarily strong modal logic and a primitive
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existence predicate. Later we will show how to modify the proof for systems as strong as S4, so that the inclusion of a primitive predicate is not needed. There is a problem which arises when we allow non-rigid terms with the objectual interpretation which draws our attention to a step in the proof of (TL) which we have so far ignored. Let us look at the reasoning we will need to carry out the proof of the case for the quanti er. a(8xP x)(w) is T i for all d in D(w); a(d=x)(P x)(w) is T (1) i for all t, if a(t)(w) 2 D(w), then a(a(t)(w)=x)(P x)(w) is T (2) i for all t, if a(t)(w) 2 D(w), then a(P t)(w) is T i for all t; (Et ! P t) 2 w i 8xP x 2 w. The proof that (1) and (2) are equivalent requires the proof of (SL) (for Substitution Lemma). (SL) a(a(t)(w)=x)(P x)(w) is a(P t)(w). Unfortunately, (SL) is not always true if t is non-rigid. It is false, for example, for P t = F t on the following model. The set of worlds W contains (the real) world r, and (an unreal) world u, and they are both accessible from themselves and each other. The domain D contains two objects d, for (David Lewis) and s (for Saul Kripke). The term t (read `the author of \Counterpart Theory" ') has d as its extension in the real world, and s as its extension in the unreal world u. The extension of F (read `is author of \Counterpart Theory" ') contains d in r, and s in u. Notice now that a(a(t)(u)=x)(F x)(u) is a(s=x)(F x)(u), which is false, since s is not in the extension of F in both worlds. However a(F t)(u) is true because the extension of t is in the extension of F in each world. We see that (SL) fails for reasons closely related to the fact that substitution of identities fails for non-rigid terms. We did not face this problem for systems with rigid terms, because (SL) is true when a(t) is a constant function. The problem did not arise with the substantial interpretation because there the lemma we need (SSL) concerns substitution of intensions and is readily proven. (SSL) a(a(t)=x)(P x)(w) is a(P t)(w). Thomason tackles the problem posed by the failure of (SL) in a direct way. He stipulates that variables are rigid designators and uses variables, not terms, to x the domains of his standard model. The extension a(t)(w) is set to fx : x = t 2 wg, and the domain D(w) contains the extensions of all terms t such that Et 2 w. By adding the rules (RV), to the system, he can ensure that the standard model has rigid variables, using the methods we outlined in Section 2.2.4.2.
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x=y :x = y x=y x = y :x = y Ex ! Ey However, the use of rigid variables leads to further complications. In order to establish the case for identity in (TL), we need to know that if a(t)(w) is a(t0 )(w) then t = t0 2 w. The identity of a(t)(w) only establishes that x = t 2 w i x = t0 2 w, for all variables x. To show that t = t0 2 w, we need to know that there is some variable y such that y = t 2 w. This requires us to restrict the set W of possible worlds of our model to those that meet condition (V). (RV)
For all w in W , and all terms t of L, there is a y such that y = t 2 w. In order to meet condition (V) when it comes time to extend w to a set in W , Thomason added the following rule to this system. (V)
(G=)
` G(:y = t) ` G(p ^ :p)
The rule (G=) ensures that we can consistently add a sentence of the form y = t for each of the terms t during the construction of a saturated set, and to do so without extending the language. The system Q3 which we can show to be complete using this method is composed of GS, (RV), and (G=). The system Thomason [1970] showed to be complete lacked the primitive existence predicate E , and was built on S4. In S4, the sentence 9xx = t is true in the standard model just in case the intension of t is rigid. Also, the replacement of Et with 9xx = t in the rules of free logic results in valid quanti er rules. It follows that if S is S4 or stronger, we can formulate a complete system for Q3- S without a primitive existence predicate by replacing Et with 9xx = t in the rules of Q3-S. 3 UNAXIOMATISABILITY OF SOME QUANTIFIED INTENSIONAL LOGICS
3.1 Introduction Certain quanti ed modal languages are capable of expressing statements of arithmetic. These systems cannot be axiomatised, for if they were, they would be adequate for arithmetic, which is impossible by Godel's Theorem. In this section we will give examples of three quanti ed modal logics which are incomplete for this reason. First, we review Scott's result (reported in [Kamp, 1977]) that predicate tense logic is incomplete if time is described by the reals. Next we will discuss unaxiomatisability results [Fine, 1970] for propositional modal logics with quanti ers over propositional variables.
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Finally, we will show that Q2 cannot be formalised, at least not if the underlying modal logic is S4.3 or weaker. (This is Kripke's result reported in [Kamp, 1977].) The rest of this section contains preliminary material which we need later. A reader with a background in mathematical logic will probably want to skip to Section 3.2. 3.1.1 Languages that express arithmetic
The language PA (for Peano Arithmetic) contains quanti ers, =, a constant 0, and function symbols 0 , +, . A model hD; ai of PA consists of a nonempty domain D (of quanti cation), and an assignment function a that assigns to 0 a unary function a(0 ) from D to D, and to both + and , binary functions a(+) and a() from D D to D. A model is the standard model of arithmetic i D is the set of integers 0; 1; 2; : : : ; a(0 ) is the function that takes each integer into its successor, a(+) is the addition function, and a() is multiplication. Now suppose we have a language L which includes the symbols of PA and which contains a sentence SMA which is true on a model just in case it is the standard model of arithmetic. It follows that the valid sentences of L cannot be formalised. The reason is that the sentence A of arithmetic is true on the standard model just in case S MA ! A is a valid sentence of L. So any axiomatisation of L would provide a way to formalise the true sentences of arithmetic, and this, Godel showed, cannot be done. There is no need for SMA to pick out the standard model exactly. (In fact, it cannot.) It is easy to see that the same sentences are true on any pair of isomorphic models. So L will be unaxiomatisable as long as it contains a sentence SMA which is true only on models of PA that are isomorphic to the standard model. (To avoid talking all the time of isomorphic models, we will mean by a `standard model' any model isomorphic to the standard one.) We do not need 0;0 ; + and in the language in order to obtain this kind of incompleteness result. It is well known that constants and function symbols are eliminable in favour of corresponding predicate letters. For example, we may introduce the predicate Z for zero, and the sentence 9!xZx which ensures that the extension of Z is a singleton. (We use 9!xP x to abbreviate 9x(P x ^ 8y(P y ! x = y)), where y is chosen new to P x.) We may then conjoin 9!xZx to SMA, and replace each sentence P 0 of SMA involving 0, with 8x(Zx ! P x), which says the same thing. To eliminate 0 , we introduce a binary predicate letter N , and we add 9!yNxy to ensure that the extension of N is a unary function. We then replace axioms P x0 involving 0 , with 8y(Nxy ! P y). By introducing ternary predicates, for + and , and performing the same manoeuvre, we can complete the elimination of function symbols. It follows that any language which contains rst-order logic with identity and contains a sentence SMA which is true only on a
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standard model is incomplete, (if it is consistent). (In the case of a language that uses predicate letters, Z; N; T; P for arithmetic, hD; ai counts as a standard model i a is a function over these predicate letters which assigns them extensions, and hD; ai is isomorphic to another model of the same kind whose domain is the integers and which gives Z; N; T; P the extension zero, the successor function, plus, and times.)
3.2 Incompleteness of Predicate Tense Logic with Real Time It is crucial in physics that we represent moments of time using numbers. If time is atomic, and there is a rst moment, then the set of times looks like the integers of the standard model of arithmetic. We are more likely to think of time as dense, and so represent it using the rationals, or the reals. Scott showed that if time is mathematical in any of these senses, then predicate tense logic is incomplete. (The result is reported in [Kamp, 1977].) When we assume that the Kripke frame hW; Ri of any tense logic model hW; R; D; ai is such that W is the set of integers, and R the relation `less than', then we can nd a sentence SMA which is true only on standard models. Even when we consider frames hW; Ri where W is the set of rationals or reals, the same argument can be constructed. 3.2.1 Syntax and Semantics of Predicate Tense Logic Let us de ne T1 (tense predicate logic like Q1) in the following way. The syntax of T1 involves an alphabet which includes symbols of rst-order logic, and two sentential operators G and H (read `it will always be that' and `it was always the case that'). The more familiar operators F and P (read `it will be that' and `it was the case that') are de ned by F =df :G:, and P =df :H :. To formulate the semantics of T1 let us de ne a T1-model as a sequence hW; R; D; ai, where hW; Ri is like the integers in that sense that W is the set consisting of 0; 1; 2; : : :, and R is `less than'. The quanti er of T1 is interpreted with a xed domain D, so its truth clause is (Q1).
(Q1) a(8xP x)(w) is T i for all d in D; a(d=x)(P x)(w) is T . The truth clauses for G and H read as follows. (G) a(GA)(w) is T i if wRw0 , then a(A)(w0 ) is T . (H ) a(HA)(w) is T i if w0 Rw, then a(A)(w0 ) is T . For the moment, we will assume that terms are all rigid designators, so a(t)(w) is a(t)(w0 ) for all w; w0 in W . This restriction can be relaxed without changing the essentials of the incompleteness proof. Notice, then, that semantics for T1 is exactly like Q1, except that in T1 we have two intensional operators.
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3.2.2 The Expressive Capabilities of T1
If we had quanti ers and predicate letters in T1 whose domain were the set W of times, then the unaxiomatisability of T1 would be easy to show. In that case, sentences valid in T1 would be those that are valid on all frames hW; Ri where W is the integers. We could then construct the sentence Q consisting of the axioms of ( rst-order) arithmetic using predicate letters Z; N; P; T . (See [Boolos and Je rey, 1989, p. 161] for these axioms.) Sentence Q would serve as the sentence which expresses that a model is standard. Our problem is, however, that W is not the domain of quanti cation in T1. The quanti ers range instead, over the domain D of objects. Nevertheless, it is possible to nd a sentence of T1 that sets up a correspondence between members of W and members of D so that sentences that express properties of the domain D re ect corresponding properties in the set of worlds W . In order to show how this correspondence is brought about, let us rst give a few de nitions and facts concerning the things that T1 can express. First, we will de ne two operators A, and S (read `it is always the case that' and `it is sometimes the case that') as follows. AA = A ^ GA ^ HA;
SA = A _ F A _ P A:
Since W in every model of T1 is the set of integers, it is easy to verify the following facts about all models of T1. FACT 1. AA is true at w i A is true at every time w0 in W . FACT 2. SA is true at w i A is true at some time w0 in W . Now let us introduce the predicate letter E (read `exists'). We will use the following two sentences to ensure that every member of D is in the extension of E at some time, and that the extension of E is always either a singleton or empty. (F1)
8xS (Ex ^ H :Ex ^ G:Ex)
(F2)
A8x8y ((Ex ^ Ey ) ! x = y )
(Everything exists at exactly one time.) (No two things exist at the same time.)
Any model that makes both of these sentences true sets up a function from D into W , because for each member d of D, we know there is exactly one integer td of W at which d exists. Now let us introduce the following abbreviation. (<)
x < y =df S (Ex ^ F Ey).
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The sentence S (Ex ^ F Ey) is true at t for a just in case there is some time where a(x) exists, and a later time where a(y) exists. Since (F1) guarantees that an object exists at only one time, it follows that the pair d; d0 satis es the extension of < at any time just in case the integer td where d exists is less than the integer td0 where d0 exists. So < sets up an ordering on D that corresponds to the relation `less than' on the integers. Actually, < does not express all the facts about `less than' on the integers, because (F1) and (F2) do not guarantee that something exists at every time. The extension of < corresponds to `less than' restricted to WD the set of those times when objects exist. We could set up a one-one correspondence between W and D by adding the sentence A9xEx, but this will block the proof for the case of the rationals and the reals, as we will see. 3.2.3 Unaxiomatisability of T1
Now let us introduce an equivalence that xes the extension of predicate N as the successor function, and guarantees that every object in the ordering set up by < has a successor. (F3)
xNy $ x < y ^ 8z ((:z = y ^ x < z ) ! y < z ),
(F4)
9y(Ey ^ xNy).
If (F3) and (F4) are both true at any time t of W , then the pair hd; d0 i is in the extension of N at t just in case the corresponding times td ; td0 are such that td0 is the successor of td in WD (the set of times where objects exist). We may also de ne Z (read `is zero'), and guarantee that zero exists as follows. (F5)
Zx $ 8y:y < x
(F6)
9x(Ex ^ Zx).
These two sentences ensure that there is a least member t0 in the set WD of times at which objects exist. Let SMA be the conjunction of (F1){(F6) and Q , the result of eliminating 0;0 ; + and in favour of predicate letters Z; N; P and T in the axioms Q of rst-order arithmetic. We claim that SMA expresses that a model is standard in the following sense. LEMMA 5. For any model hW; R; D; ai of T1, and any w in W , a(SMA)(w) is T only if hD; a(w)i is standard. Here a(w) is the function that gives to each predicate letter F , the extension a(F )(w), that F receives on a at world w.
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Proof. Let us imagine a model hW; R; D; ai that satis es SMA at w. As we said, the truth of (F1) and (F2) sets up a correspondence between objects of D and a subset WD of W . (F3) ensures that a pair of objects hd; d0 i is in the extension of < just in case the corresponding numbers td; td0 bear the relation `less than'. (F4) and (F5) ensure that there is a successor for any number ti of the series, and (F6) ensures that there is a least member t0 in the series. It follows from this that the objects of D correspond to a sequence w0 ; w1 ; w2 ; w3 ; : : : of numbers of W ordered by `less than', with a rst member w0 . Furthermore, the extension of N picks out the successor function on this ordering. Given that the sentence Q is satis ed, we also know the extensions of P and T at w must be plus and times. It follows then that hD; a(w)i is a standard model of arithmetic. Let us suppose that A is any sentence of arithmetic, and that A is the result of eliminating 0;0 ; +; in favour of predicate letters Z; N; P; T in the usual way. We may now show that T1 is incomplete on the basis of the following theorem. THEOREM 1. SMA ! A is T1-valid i A is true on the standard model of arithmetic.
Proof. (left to right) Let hW; R; D; ai be a T1-model such that W is the integers (rationals, reals), R is the ordering `less than' on W; D is the integers, and d 2 a(E )(w) i d = w, and a(E )(w) is the empty set if w is not an integer. On this model a(SMA(w) is T , for any w in W . By the T1-validity of SMA ! A , it follows that a(A )(w) is T . Notice that A contains no intensional operators, and so its value is determined by the extensions of Z; N; P; T exactly as it would be on the extensional model hD; a(w)i. Since this is a standard model, A must be true on the standard model of arithmetic. (right to left) Suppose that A is true on the standard model for arithmetic and suppose that a(SMA)(w) is T at any w in W , on a model of T1. By Lemma 5, hD; a(w)i is standard, and so a(A )(w) is T . We conclude that SMA ! A is T1-valid. Notice that Theorem 1 can be proved as long as we begin with any frame
hW; Ri which contains a substructure which is isomorphic to the integers
ordered by `less than'. It follows that T1 is incomplete with respect to the integers, the rationals, the reals, and virtually any other conceivable numerical account of time. It is also clear that the same argument works for logics that have world-relative domains. In this case, we are already supplied with a primitive predicate E which picks out things that exist at a given time, and the same argument can be carried out for this E .
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3.3 Incompleteness of Second-Order Propositional Modal Logics Fine [1970] shows that second-order propositional modal logic (SOPML) is incomplete when the modality is S4.2 or less. SOPML is ordinary propositional modal logic, except we introduce quanti ers which bind the propositional variables. Here, we will give an incompleteness result for a somewhat di erent system called SOMA (for Second-Order Modal Arithmetic), assuming that the modality is S4.3 or less. SOMA is SOPML supplemented with a propositional constant 0, and connectives 0 ; +, and , of arities 1, 2, 2 respectively. The unaxiomatisability of SOMA is easier to prove than it is for SOPML, because SOMA already contains the notation for arithmetic. In the case of SOPML, we need to show how to get the e ect of the binary function symbols + and . The proof of SOMA allows us to display the main strategy used in the proof for SOPML, without having to cover this less central detail. Another reason for concentrating on SOMA is that its relatively easy incompleteness result provides a quick proof of the incompleteness of Q2, which we give in Section 3.4. 3.3.1 The Intuitions behind the Proof
We know that a system is incomplete as long as it contains a sentence that expresses that its models contain a standard model of arithmetic. It is well known that any model of both the axioms of rst-order arithmetic Q, and (MI) (the second-order axiom of mathematical induction) is a standard model. (See, for example [Boolos and Je rey, 1989, Ch. 18].) (MI) 8P ((P 0 ^ 8x(P x ! P x0 )) ! 8xP x).
Q can be expressed in rst-order logic; however, (MI) requires quanti cation over a monadic predicate letter. So any extension of rst-order logic that can achieve the e ect of quanti cation over monadic predicates will express arithmetic, and so be incomplete. The idea behind the incompleteness proof for SOMA, then, is to show that quanti cation over propositional variables in modal logic can be used to get the e ect of both quanti cation over worlds and quanti cation over predicates of worlds. To see how this is done, think of the intensions of propositional variables as truth sets. (The truth set of a propositional variable is just the set of worlds where it is true.) So quanti cation over propositional variables amounts to quanti cation over truth sets, that is, over properties of worlds. We also need to be able to quantify over objects of W if were are to express the axioms of arithmetic. This is done in SOMA by nding a way to say that an intension contains a single world. Then quanti cation over all singleton sets of worlds amounts to quanti cation over the individual worlds themselves.
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3.3.2 The Expressive Resources of SOMA
We turn now to the details involved in showing that SOMA can express arithmetic. To x our later discussion, we will give the semantics for SOMA here. A model hW; R; ai of SOMA assigns to each propositional variable p a subset a(p) of W , called the truth set of p. It also assigns to 0 ; +, and , functions that take us from subsets of W into new subsets of W in the case of 0 and from subsets of W W to W in the case of + and . Since we are proving incompleteness for SOMA-S4.3, we will assume that hD; Ri is re exive, transitive and connected. The clauses for sentences with shapes A0 ; A + B , and A B are as follows a(A0 ) = a(0 )(a(A)) a(A + B ) = a(+)(a(A); a(B )) a(A B ) = a()(a(A); a(B )) and the clauses for :; !, and are given in the usual way. For the quanti er we have the following. (8p)
w 2 a(8pP p) i for every subset s of W , w 2 a(s=p)(P p)
Here a(s=p) is the assignment just like a save that a(s=p)(p) is s. Most of the properties we can express in SOMA only apply to a portion of the model hW; R; ai. It will turn out, however, that this is enough for our purposes. We de ne the future of model hW; R; ai at w, as the model hW f; Rf; af i, where Wf is fw0 : wRw0 g and Rf and af are R and a restricted to Wf. The future at w, then, contains just those worlds accessible from w, and the portions of R and a which concern these worlds. For convenience, we will also call hW f; af i the future of hW; R; ai at w, where in this case, af (0 ); af (+) and af () are restricted to singleton sets of Wf. We are going to show how to construct a sentence which is true at w just in case the future hW f; af i at w is a standard model of arithmetic. In order to get the e ect of quanti cation over objects of W , let us present a sentence Ip of SOMA whose truth at w ensures that the intension of p is a singleton in the future at w. (I)
Ip =df p ^ 8q((p ! q) _ (p ! :q)).
There is an interesting intuition behind this de nition. Ip says that the sentences entailed by p form a maximally consistent set, for the rst conjunct says that p is consistent, and the second, that p entails any sentence or its negation. Given that worlds are maximally consistent sets, it follows that p could only be true at one world. Let us show now that Ip actually has this intended e ect. The rst conjunct ensures that there is a world accessible from w where p is true, so we know that a f (p) contains at least one member of Wf. Notice next that
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(p ! q) ensures that every world accessible from w 2 a f (p) only if it is
in a f (q), and so that a f (p) is a subset of a f (q). The e ect of the second conjunct of Ip, then, is to ensure that for any subset of Wf we choose, a f (p) will be a subset of either it or its complement. It follows that a f (p) must be a singleton, for suppose there were two distinct members w; w0 of a f (p). Then the subset s such that w 2 s and w0 62 s would have to include all the members of a f (p), or its complement would. In either case, one of w; w0 would have to be missing from a f (p). Since we have a way to express uniqueness of an intension, we can quantify both over worlds and their properties. In order to enforce the structure of the standard model on the future of w, we will need to express properties about the relation R. The abbreviation () shows how to do this. ()
p q =df (p ! q)
It is a simple matter to verify that whenever the intensions of p and q are singletons in the future of w, then p q is true at w just in case wp Rwq , where wp and wq are the worlds at which p and q are true in the future of w. Now let us de ne identity in SOMA. (=)
p=q =df (p $ q).
It is easy to see that (p $ q) is true at w, just in case a(p) and a(q) agree on members in Wf. So p = q is true at w i a f (p) is a f (q). We may de ne `<' from `' in the usual way. (<)
p < q =df p q ^ :p = q.
In order to write (MI) in SOMA, we need a way to express that a world has a property. We know that (p ! q) is true at w just in case the truth set of p is a subset of the truth set of q in the future of w. So in case the intension of p is a singleton containing wp ; (p ! q) says that the world wp is in the intension of q. Therefore, we will adopt the following abbreviation. (is)
p is q =df (p ! q).
3.3.3 Incompleteness of SOMA
Now we are ready to formulate arithmetic in SOMA. Let SMA be the conjunction of the following sentences. 1. I0 ^ 0 2. 8pIp0
3. 8p8q((Ip ^ Iq) ! (p0 = q0 ! p = q))
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4. 8q(Iq ! :q < 0)
5. 8p8q((Ip ^ Iq) ! (p0 = q ! (p < q ^ 8r((p < r ^ :r = q) ! q < r))) 6. 8p8qI(p + q) 7. 8p8qI(p q)
8. 8p(Ip ! p + 0 = p)
9. 8p8q((Ip ^ Iq) ! p + q0 = (p + q)0 )
10. 8p(Ip ! p 0 = p)
11. 8p8q((Ip ^ Iq) ! p q0 = (p q) + p) (MII) 8p((0 is p ^ 8q(Iq ! (q is p ! q0 is p)) ! 8r(Ir ! r is p)). Sentences (3){(5) establish the proper relationship between zero, the successor function and the relation Rf which is expressed by . Sentences (8){(11) are the axioms of Q, with propositional quanti ers restricted to I. (MII) is our formulation of the second-order axiom of mathematical induction. Our next task is to convince you that SMA is true at w just in case the future at w contains a standard model of arithmetic, in the sense of the following lemma. LEMMA 6. a(SMA)(w) is T on hW; D; ai i the future hWf, Rf, afi of hW; D; ai at w is such that hWf, afi is a standard model of arithmetic and af (0) is a singleton containing w, and Rf is `less than or equal to' on Wf.
Proof. (left to right) Let a numeral be a sentence 0i composed of the propositional constant 0, followed by i primes (0 ). Sentences (1) and (2) of SMA guarantee that the intension of any numeral is a singleton set in the future of w. The second conjunct of (1) establishes that w is in a(0), and so a f (0) is a singleton containing w. Now let wi be the singleton which is in the extension of numeral 0i . Sentence (3) guarantees that wi is not wj for i not equal to j , and so wi is not wi+1 . (MII) ensures that every member of Wf is wi for some numeral 0i , and so there is a one-one mapping between numerals and Wf. By sentences (4) and (5), we know that the wi form a sequence such that w0 Rfw1 ; w1 Rfw2 : : :, with w0 as the least member. By the re exivity and transitivity of Rf and (5) we may show further that i is less than or equal to j i wi Rfwj . So Rf is `less than or equal to'. Sentences (6) and (7) ensure that the intensions of (p + q) and (p q) are functions that range over singletons only. The remaining sentences (8){(11) ensure that a f (+) and af () correspond to addition and multiplication. We conclude that hW f; af i is indeed a standard model of arithmetic, and af (0) is indeed a singleton containing zero.
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(right to left) Suppose that hW f; Rf; a f i is the future of hW; R; ai at w, and suppose that hW f; a f i is a standard model. Suppose that a f (0) is a singleton containing w, and that Rf is `less than or equal to'. Since af is standard, we know that a f (0) = fwg contains the representative of zero on this model. It follows that wRfw0 for every w0 in W . Therefore, the future of hW; R; ai at w just is hW; R; ai. Now the reader can verify that sentences (1){(11) and (MII) are all true at w on hW; R; ai, and so a(SMA)(w) is T . This completes the proof of Lemma 6. Now we must check that sentences of arithmetic are true just in case their translations into SOMA are true in the future of a given w. We de ne A for sentences A of arithmetic as the result of replacing variables and quanti ers of A with prepositional quanti ers restricted to I , and replacing identity in A with the corresponding sentence of SOMA according to de nition (=). LEMMA 7. If hWf, afi is the future of hW; R; ai at w, and hWf, afi is a standard model of arithmetic, then A is true on the standard model i w 2 a(A ) on hW; R; ai.
Proof. The proof is by induction on the structure of A. The non-trivial cases occur when A has the shapes 8xP x, and t = s. The case for the quanti er runs as follows.
8xP x is true in arithmetic.
i P n is true on the standard model for every numeral n i w 2 a(P n) in the future of w, for every numeral n i w 2 a(P q) in the future of w for all q such that af (q) is a singleton i w 2 a(8p(Ip ! P p)). The case for identity requires rst that we show that the extension of any term t in the standard model of arithmetic corresponds to a f (t) in hW f; a f i. This is easily shown by induction on the structure of t. The rest of this case proceeds as follows. t = s is true in arithmetic i a0 (t) is a0 (s), for the a0 of the standard model i af (t) is af (s) in the future of w i a(t)(w0 ) is a(s)(w0 ) for w0 in W f i w0 2 a(t $ s) for w0 in W f i w 2 a(t $ s) This completes the proof of Lemma 7. We are now ready to prove the incompleteness of SOMA, using the following theorem.
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THEOREM 2. A is true on the standard model of arithmetic i SMA ! A is S.3-valid in SOMA.
Proof. (left to right) Suppose that A is true on the standard model of arithmetic. Let hW; R; ai by any S4.3-model of SOMA. Let w be any world in W , and suppose that w 2 a(SMA). It follows by Lemma 6 that hW f; a f i is a standard model of arithmetic, where hW f; Rf; a f i is the future of hW; R; ai at w. It follows by Lemma 7 that w is in a(A ). We conclude that (SMA ! A ) is S4.3-valid in SOMA. (right to left) Suppose that (SMA ! A ) is S4.3-valid in SOMA. Let hW; R; ai be the SOMA model such that W is the integers, R is the relation `equal or less than', the intension of 0 is the singleton containing zero, and the intensions of 0 ; +, and are the successor function, plus, and times, de ned on singleton sets of members of W . hW; Ri is clearly re exive, transitive and connected, so hW; R; ai is and S4.3-model. By Lemma 6, 0 2 a(SMA), and so by the validity of (SMA ! A ); 0 2 a(A ). However, the future of hW; R; ai at zero is a standard model of arithmetic, and so by Lemma 7, A is a true sentence of arithmetic. Theorem 2 establishes that SOMA with modality of strength S4.3 is incomplete. It follows also that SOMA is incomplete for all weaker logics (down to K). The reason is that when the following sentences of SOMA are true at w in a model of SOMA, then Rf in the future of w must be re exive, transitive, and connected.
8p(p ! p) 8p(p ! p) 8p8q((p ^ q) ! ((p ^ q) _ (p ^ q) _ q ^ p))) So by adding these sentences to SMA, we may carry out the proof of Theorem 2 for any system weaker than S4.3.
3.4 Incompleteness of
Q2
The proof of the incompleteness of SOMA can be used to show that Q2 cannot be axiomatised as long as the propositional modal logic is S4.3 or weaker. This is done by showing that there is a transformation that takes us from sentences of SOMA to sentences of Q2, so that A is valid in SOMA just in case A is valid in Q2. It follows that since SOMA can express arithmetic for modalities S4.3 or less, then so can Q2. The idea behind the transformation is to mimic propositional variables p of SOMA, whose intensions amount to functions from W into (T; F ), using corresponding individual variables xp , whose intensions take us from W to D. We will arbitrarily select a term t whose extension at a world w picks out the object of D which plays the role of the truth value T for that world.
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Then we will represent that p is true at world w, using the Q2 sentence xp = t. To simplify the proof, we will assume rst that Q2 is Q2 with function symbols 0;0 ; +; . These symbols can be eliminated later in favour of predicate letters of Q2. We de ne A for any sentence A of SOMA, as the Q2-sentence that results from replacing each variable p of A with a corresponding variable xp of Q2, and replacing each propositional variable p with the sentence xp = t. (Of course, we must be sure that the xp and t are distinct variables.) Let us prove the following lemma about this transformation. LEMMA 8. If hW; R; ai is a model of SOMA, and hW; R; D; Q2; a0 i is a model of Q2, and w0 2 a(p) i a0 (xp )(w0 ) = a0 (t)(w0 ), for all w0 in W , and D(w0 ) contains two members for all w0 such that wRw0 , then w 2 a(A) i a(A )(w) is T .
Proof. The proof of Lemma 8 is straightforward induction on the structure of A. Now we may show how to set up a correspondence between sentences of SOMA and Q2, according to the following theorem. THEOREM 3. A is valid in SOMA i 9x9y:x = y ! A is Q2 -valid.
Proof. (left to right) Assume that A is valid in SOMA, and consider a Q2 -model hW; R; D; Q2; a0 i such that 9x9y:x = y is true at any w in W . Then D(w0 ) contains two objects for every w0 such that wRw0 . Build a SOMA model hW; R; ai such that w0 is in a0 (p) i a0 (xp )(w0 ) is a(t)(w0 ). We have met the conditions for Lemma 8, and so by the validity of A we conclude that a0 (A )(w) is T . (right to left) Now assume that 9x9y:x = y ! A is Q2-valid. Let hW; R; ai be any SOMA-model, and w any member of W . Now de ne a Q2- model hW; R; D; Q2; a0 i as follows. Let D contain the objects T; F , and let D(w0 ) be D for each w0 in W . Let a0 (t)(w0 ) be T for all w0 in W , and let a0 (xp )(w0 ) be T if w0 is in a(p), and F otherwise. The value of a(9x9y:x = y)(w) is T , and so, by the validity of 9x9y:x = y ! A ; a(A )(w) is T . Since the conditions for Lemma 8 are met, we conclude that w is in a(A).
Theorem 3 establishes that Q2 is incomplete for all modalities S4.3 or weaker. It follows that Q2 is also incomplete, because function symbols of sentences of Q2 can be eliminated for corresponding predicate letters of Q2. The same sort of argument can be used to show the unaxiomatisability of QC, where we have a single domain of quanti cation. In fact, the proof is easier, since we need only the sentence 9x9y:x = y to ensure that the domain contains two objects.
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317
S5
It is interesting to ask whether these results apply to Q2 and SOMA for modalities as strong as S5. The answer is that they do not. Kripke has shown (see [Kamp, 1977]) that Q2S5 is axiomatisable, and Fine [1970] even shows that SOPML-S5 is decidable. The reason that the proof strategy that we have used does not work for S5 in that our method depends on our ability to build the structure of the standard model of arithmetic within the kinds of Kripke frames with which we were supplied. In the case of S5, however, the frames are equivalence classes, and there is no way to develop the ordering we need to construct the standard model.
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JAMES W. GARSON APPENDIX
List of Rules Page 269 (FUI)
8xP x for any term t Et ! P t ` A ! (Et ! P t) where t is any term not in A ! 8xP x ` A ! 8xP x
269
(FUG)
270
(=In)
270
(=Out)
273
(RT)
273
(BF)
8xA ! 8xA
275
(CBF)
278
(HUI)
8xA ! 8xA 8xP x (9x i x = t ^ : : : ^ 9x k x = t) ! P t
t=t t = t0 where P t is an atom P t ! P t0 :t = t0 t = t0 t = t0 :t = t0
where i; : : : ; k is a list of integers which records for each occurrence of x in P x, the number of boxes whose scope includes that occurrence. 280
(TUI)
297
(GUI)
297
(GUG)
8xP x 9xx = t ! P t G(8xP x) G(Et ! P t) ` G(Et ! P t) ` G(8xP x)
where t does not appear in G(8xP x). Here G(B ) is any sentence with the shape A1 ! A2 3 : : : 3 Ai 3 B and G(C ) is the result of replacing C for B in G(B ).
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(=E)
t = t0 Et ! Et0
302
(E)
Et
303
(RV)
x=y x = y
304
(G=)
` G(:y = t) ` G(p ^ :p)
:x = y x=y :x = y Ex ! Ey
319
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List of Systems PL = Predicate Logic, First-order Logic (without identity). S = an arbitrarily selected propositional modal logic as strong or stronger than K. 269
MFL = (FUI) + (FUG) Minimal Free Logic
270
ID = (=In) + (=Out) Intensional Identity Theory
273
Q1 = S + PL + ID + (RT) + (BF).
274
Q1R = S + MFL + ID + (RT).
275
QK = S + PL with no terms. The necessitation rule is restricted to apply only to closed sentences.
277
QPL = S + PL.
282
Q2 cannot be axiomatised, unless modality is as strong as S5.
284
QS = S + MFL + ID.
284
B1{S5 = S + PL + (BF) + ID + axiom of substitution for strong identities.
298
GS = S + (GUI) + (GUG) + ID.
301
GQ1R = GS + (RT) + (=E).
302
GQ1 = GQ1R + (E).
304
Q3 = S + GS + (RV) +(G=).
304
Q3{S4 = S4 + GS with 9xx = t for Et + (RV) + (G=). (Thomason's Q3 [1970] also contains rules for descriptions, and the axiom 9x9yx = y to guarantee that every domain contains at least one individual.)
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List of Conditions on Models A model has the form hW; R; D; Q; ai hW; Ri is the Kripke frame, D is the domain of possible objects, a is the assignment function, that gives terms and predicate letters their intensions over W and D, Q is an item that details the nature of the quanti er domain(s). For free logic with primitive predicate E we have a(E ) is Q. The truth value a(A)(w) of formula A at world w is de ned recursively by the following clauses. (:) a(:A)(w) is T i a(A)(w) is not T , (!) a(A ! B )(w) is T i a(A)(w) is F or a(A)(w) is T , () a(A)(w) is T i if wRw0 then a(A)(w0 ) is T and another clause for the quanti er which di ers in di erent semantics. To describe a semantics for quanti er modal logic, we give a description of Q, list any other conditions on the model and then give the truth clause for the quanti er. 273 Q1 A Q1-model has Q = Q1 = D, and meets (aRT). (aRT) a(t)(w) is a(t)(w0 ) for all w; w0 in W , (Q1) a(8xA)(w) is T i for all d in Q1, a(d=x)(A)(w) is T . 274 Q1R A Q1R-model has Q = Q1R a function that assigns subsets D(w) to the worlds w of W , and it meets (aRT). (Q1R) a(8xA)(w) is T i for every d in D(w); a(d=x)(A)(w) is T . 277 QPL A QPL-model is Q1R-model that meets (ND). Truth values are calculated using (TG) (truth value gaps). (ND) If wRw0 , then D(w) is a subset of D(w0 ). (TG) If a(t)(w) 62 D(w), then any sentence P t containing t has no truth value.
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277 GK A GK-model is a Q1R-model. Truth values are calculated using (TG). For GKc, use clause (c) for . For GKs, use (s). (c) a(A)(w) is T i if wRw0 , then A has a value at w and a(A)(w0 ) is T . (s) a(A)(w) is T i if wRw0 and A has a value at w, then a(A)(w0 ) is T . 278 Q3 A Q3-model has Q = Q1R = a function that assigns a domain D(w) to each possible world. It need not meet condition (aRT). The quanti er clause is (Q1R). 280 Q3L A Q3L-model is Q3-model that meets condition (L) (local terms). (L) a(t)(w) 2 D(w) for all w in W , and all terms t. 281 QC A QC-model has Q = QC = the set of all functions from W into D. (QC) a(8xA)(w) is T i for every f in QC, a(f=x)(A(w) is T . 282 Q2 A Q2-model has Q = Q2 = Q1R = a function that assigns a domain D(w) for each of the possible worlds w. (Q2) a(8xA)(w) is T i for every function f from W into D; a(f=x)(A)(w) is T . 284 QS A QS-model has Q = QS = a function that assigns to each world w a subset S (w) of the set QC of all functions from W into D. (QS) a(8xA)(w) is T i for every member f of S (w), a(f=x)(A)(w) is T . Department of Philosophy, University of Houston, USA.
BIBLIOGRAPHY
[Boolos and Je rey, 1989] G. Boolos and R. Je rey. Computability and Logic, 3rd edition. Cambridge University Press, (1st edition 1974), 1989. [Bowen, 1979] K. Bowen. Model Theory for Modal Logic. Reidel, Dordrecht, 1979. [Bressan, 1973] A. Bressan. A General Interpreted Modal Calculus. Yale University Press, 1973. [Carnap, 1947] R. Carnap. Meaning and Necessity. University of Chicago Press, 1947. [Cresswell, 1995] M. J. Cresswell. Incompleteness and the Barcan formula. Journal of Philosophical Logic, 24:379{403, 1995.
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[Fine, 1970] K. Fine. Propositional quanti ers in modal logic. Theoria, 36:336{346, 1970. [Gabbay, 1976] D. M. Gabbay. Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics. Reidel, Dordrecht, 1976. [Gallin, 1975] D. Gallin. Intensional and Higher-Order Modal Logic. North-Holland, Amsterdam, 1975. [Garson, 1978] J. Garson. Completeness of some quanti ed modal logics. Logique et Analyse, 21:153{164, 1978. [Henkin, 1949] L. Henkin. The completeness of the rst-order functional calculus. Journal of Symbolic Logic, 14:159{166, 1949. [Hintikka, 1970] J. Hintikka. Existential and uniqueness presuppositions. In Philosophical Problems in Logic. D. Reidel, Dordrecht, 1970. [Hughes and Cresswell, 1968] G. Hughes and H. Cresswell. An Introduction to Modal Logic. Methuen, London, 1968. [Kamp, 1977] H. Kamp. Two related theorems by D. Scott and S. Kripke, 1977. Xerox. [Konyndyk, 1986] K. Konyndyk. Introductory Modal Logic. University of Notre Dame Press, Notre Dame, Indiana, 1986. [Kripke, 1963] S. Kripke. Semantical considerations in modal logic. Acta Philosophica Fennica, 16:83{94, 1963. [Kripke, 1972] S. Kripke. Naming and necessity. In D. Davidson and G. Harman, editors, Semantics of Natural Language. Reidel, Dordrecht, 1972. [Lewis, 1968] D. Lewis. Counterpart theory and quanti ed modal logic. Journal of Philosophy, 65:113{126, 1968. [Menzel, 1991] C. Menzel. The true modal logic. Journal of Philosophical Logic, 20:331{ 374, 1991. [Parks, 1976] Z. Parks. Investigations into quanti ed modal logic. Studia Logica, 35:109{ 125, 1976. [Parsons, 1975] C. Parsons. On modal quanti er theory with contingent domains (abstract). Journal of Symbolic Logic, 40:302, 1975. [Thomason, 1969] R. Thomason. Modal logic and metaphysics. In K. Lambert, editor, The Logical Way of Doing Things. Yale University Press, 1969. [Thomason, 1970] R. Thomason. Some completeness results for modal predicate calculi. In K. Lambert, editor, Philosophical Problems in Logic. D. Reidel, Doredrecht, 1970. EDITOR'S NOTE The following recent book is of interest: BIBLIOGRAPHY
[Fitting and Mendelsohn, 1999] M. Fitting and R. L. Mendelsohn. First-order Modal Logic. Kluwer Academic Publishers, 1999.
JOHAN VAN BENTHEM
CORRESPONDENCE THEORY 1 INTRODUCTION TO THE SUBJECT
Correspondences When possible worlds semantics arrived around 1960, one of its most charming features was the discovery of simple connections between existing intensional axioms and ordinary properties of the alternative relation among worlds. Decades of syntactic labour had produced a jungle of intensional axiomatic theories, for which a perspicuous semantic setting now became available. For instance, typical completeness theorems appeared such as the following: A modal formula is a theorem of S4 if and only if it is true in all re exive, transitive Kripke frames. Indeed, S4 may also be shown to be the modal logic of the partial orders; which matches the most famous modal logic with perhaps the most basic type of classical relational structure. Such matchings extend to logics higher up in the S4-spectrum. For instance, S4.2 with its additional axiom
p ! p is complete with respect to those frames which are re exive, transitive and directed, or con uent:
8xyz ((Rxy ^ Rxz ) ! 9u(Ryu ^ Rzu)) Again, the latter condition is a `diamond property' of classical fame. Completeness results such as these have inspired a ourishing area of intensional Completeness Theory, witness the classic [Segerberg, 1971]. It took modal logicians some time, however, to realise that there are also direct semantic equivalences involved here, having nothing to do with deduction in modal logics. Indeed, the whole present Correspondence Theory arose out of simple observations such as the following, made in the early seventies. EXAMPLE 1. The T -axiom p ! p is true in a Kripke frame hW; Ri if and only if R is re exive. Here, `true in a frame' means true in all worlds, under all assignments to the proposition letters.
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Proof. `)': Consider any w 2 W . If p ! p is true in hW; Ri, then, in particular, it is true at w under the assignment V with V (p) = fv 2 W j Rwvg: Thus, p will be at w true by de nition | and, hence, also p: i.e. Rww. `(': By re exivity, truth at all R-alternatives implies actual truth. EXAMPLE 2. The S4-axiom p ! p is equivalent to transitivity.
Proof. By an analogous argument.
EXAMPLE 3. The S4.2-axiom p ! p de nes directedness.
Proof. `)': Consider arbitrary w; v; u 2 W such that Rwv; Rwu. Let the assignment V have V (p) = fs 2 W j Rvsg: Immediately, this gives truth of p at v. Therefore, p is true at w, whence p must hold as well. It follows that p is true at u; i.e. u has some R-successor in V (p) | whence v; u share a common R-successor. `(': If p is true at W , say because of some v with Rwv verifying p, then p will be true at all R-successors of w. For, all of these share at least one successor with v, by directedness. Not all correspondences are equally simple. For instance, S4.2 has a companion logic S4.1 obtained by enriching S4 with the `McKinsey Axiom' p ! p. This converse of the S4.2 axiom turns out to be much more complex. A well-known completeness theorem says that S4.1 axiomatises the modal theory of those Kripke frames which are re exive, transitive as well as atomic: 8x9y(Rxy ^ 8z (Ryz ! z = y)): (Notice that we need identity here, in addition to the predicate constant R.) We shall see later in Section 2.2 that the S4.1 axioms together (just) manage to de ne the above threefold relational condition, but that the McKinsey Axiom does not de ne atomicity on its own (it is weaker). Indeed, this simple modal principle does not possess a rst-order relational equivalent at all | a discovery made independently by several people around 1975.
Modal Formulas as Conditions on the Alternative Relation The general picture emerging here is that of modal axioms expressing certain `classical' constraints on the alternative relation in frames where they are valid. With hindsight, this observation is hardly surprising. After all, given
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some valuation, the clauses of the basic Kripke truth de nition amount to a translation from modal formulas into classical ones involving R. Thus, e.g.,
p ! p p ! p
8y(Rxy ! P y) ! P x 8y(Rxy ! P y) ! ! 8y(Rxy ! 8z (Ryz ! P z )); while the McKinsey Axiom p ! p becomes 8y(Rxy ! 9z (Ryz ^ P z )) ! 9y(Rxy ^ 8z (Ryz ! P z )): becomes becomes
Here the parameter `x' refers to the current world of evaluation, while unary predicate constants P (Q; : : :) denote the sets of worlds where the corresponding proposition letter p (q; : : :) holds. Let us pause, to realise how, by this simple observation alone, many established results about classical predicate logic can be transferred straightaway to modal logic. For instance, for Kripke frames plus a xed assignment (the modal `models' of Section 2.1), Compactness and Lowenheim{Skolem results are immediate. If, e.g. a set of modal formulas is nitely satis able in Kripke models (given suitable assignments), then its classical transcription will be nitely satis ed too. Hence, by ordinary compactness, the latter set is simultaneously satis ed in some structure hW; R; P; Q; : : :i: which forms a Kripke frame cum assignment verifying the original set. But, this perspective is not quite the one we need. In the evaluation of modal formulas according to the above truth definition, two factors are intermingled: the relational pattern of the worlds and the particular `facts', i.e. the assignment. But the latter | the particular denotations of constants P; Q; : : : | is not relevant to the role of modal formulas as relational constraints. Indeed, these may even obscure the issue. When, e.g. V (p) equals W; p ! p holds in all worlds | but this observation is completely uninformative about the true content of this axiom (viz. re exivity). In order to arrive at the proper perspective, one simply abstracts from the e ects of particular assignments, by means of a universal quanti cation over the unary predicates in the preceding translation. Thus, for instance,
(p _ q) ! (p _ q) now becomes
8P 8Q (8y(Rxy ! (P y _ Qy)) ! (8y(Rxy ! P y)_ _8y(Rxy ! Qy))): Notice that modal formulas now get second-order transcriptions, as opposed to the earlier rst-order ones.
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The parameter `x' has remained: the present relational conditions are still `local' in some actual world. A `global' condition is obtained by performing one more universal quanti cation, this time with respect to this world parameter. The distinction is not without importance. The local version is more suitable for the original Kripke structures hW; R; w0 i, in which some `actual world' w0 gured prominently, as well as for `non- normal' modal semantics, in which certain worlds are distinguished from others. The global reading is the more common one, however, which will be predominant in the sequel. Again, the very point of view embodied in the above translation is signi cant | even though some of the earlier transfer phenomena are lost. What is lost, for instance, are most useful forms of compactness, as well as the Lowenheim{Skolem property. There is no automatic guarantee through second-order logic that, if a modal formula is true in some uncountable Kripke frame (i.e. under all valuations) it will be true in its countable elementary subframes (again, under all valuations). Still, this very phenomenon will be used to drive a wedge between `essentially rst-order' and `essentially second-order' modal axioms in Section 2.2. Moreover, not all is lost. The above transcriptions are very simple second-order formulas, viz. so-called 11 -sentences, with all second-order quanti ers occurring in a universal pre x in front of a rst-order matrix. From classical logic, we still now a few things about 11 -sentences, that will turn out useful. (Cf. the chapters on Higher Order Logic and Algorithms in Volume 1 of this Handbook for background.) One such thing is involved in the following obvious question. In the light of earlier examples of correspondence, the present second-order transcriptions are exceedingly cumbersome. Compare, e.g. for the T -axiom p ! p,
8xRxx with 8x8P (8y(Rxy ! P y) ! P x): Yet it was the discovery of the former simple rst-order equivalents that motivated the above investigation in the rst place. Now for some modal formulas, the second-order complexity may be unavoidable | witness the example of McKinsey's Axiom. But at least, there arises an obvious basic Which modal formulas de ne rst- order relational conditions | and how do they manage it? Query:
By the above perspective, classical sources provide one immediate answer. A 11 -sentence is rst-order de nable if and only if it is preserved under the formation of ultraproducts, a fundamental construction in classical model theory. Through the above transcription, the same criterion applies to modal formulas. (The technical ins and outs of this point, as well as of related ones in this introduction, are postponed until the relevant sections: Sections 2.1 and 2.2 in this case.)
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Modal Correspondence Theory The preceding query has been the starting point for a systematic study of classical de nability of modal formulas, when viewed as relational principles. Now the mentioned ultraproduct characterisation is a very abstract, global one, rather removed from the actual business of nding correspondences. Also historically, it is a rather late development | and we shall therefore turn to more concrete themes, as they evolved. At rst sight, proving rst-order de nability seems a simple matter: just nd an equivalent, and show that it works. Still, there is the question how much system there is to this activity. For instance, Examples 1{3 exhibited regularities in their proofs. And indeed, closer inspection reveals that re exivity, transitivity and directedness may be obtained from the second-order transcriptions of the S4.2-axioms through certain substitutions of `minimal' de nable assignments. The heuristics behind this method is simply this. If, e.g. p ! p is true at x, then the most `parsimonious' way of verifying the antecedent (i.e. by having V (p) = fy j Rxyg) carries maximal information about the whole implication. This essentially, is why the substitution of Rxu for P u in
8x8P (8y(Rxy ! P y) ! P x) yields the equivalent formula
8x(8y(Rxy ! Rxy) ! Rxx): By the universal validity of the antecedent, the latter may be simpli ed to the usual statement of re exivity. A completely analogous line of thought produces transitivity from the transcription of p ! p. Some complications arise with antecedents as in p ! p; but the general idea remains the same. In this way, one discovers a large recursive class of modal formulas with e ectively obtainable rst-order equivalents. Nevertheless, this method of substitutions also has de nite limits. Notably, it does not work for all rst-order de nable modal formulas | as will be proved in Section 2.2 for the case of S4.1. In connection with this matter, the exact combinatorial complexity of the set of rst-order de nable modal formulas is still unknown | but there are reasons for fearing that it is not even arithmetically de nable (let alone, recursive or recursively enumerable). Disproving rst-order de nability is a more diÆcult matter. Indeed, how should one go about this at all? The common pattern in all examples in the literature comes to this: nd some semantic preservation property of rstorder sentences, which is lacked by the modal formula under consideration. Thus, e.g. the earliest published contribution by the present author was an example showing how the McKinsey Axiom sins against the Lowenheim{
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Skolem theorem. It holds in a certain uncountable Kripke frame (to be presented in Section 2.2.) without holding in any of a certain group of its countable elementary subframes. A classical example of this phenomenon occurs when Dedekind Continuity (itself a 11 -property) is added to the rst-order ordering theory of the rationals. The resulting 11 -sentence has uncountable models (notably, the reals); but, it even lacks countable models altogether. The modal examples of `essentially second-order' axioms to be found in Section 2.2 will serve to delimit the range of the above method of substitutions. As so often, the McKinsey Axiom again provides an illuminating example. The above heuristics of `minimal veri cation' typically fails for antecedents such as p, expressing some dependency | and rst-order failure is immediate. Besides the modal half of the story, so to speak, there also exists the opposite direction, looking from classical formulas to modal ones. Again, this inspires a basic Query. Which rst-order relational conditions are modally de nable? The `positive' side of this matter again concerns the establishing of valid equivalences. Thus, for instance, how does one nd a modal de nition for such a classical favourite as connectedness
8xyz ((Rxy ^ Rxz ) ! (Ryz _ Rzy)))? This time, the heuristics consists in imagining a situation where the property fails, together with a way of `maximally exploiting' this failure through modal formulas. In the above particular case, supposing that Rxy; Rxz; :Ryz; :Rzy, one sets p true at y (with p false at z ) and q true at z (with q false at y). This has the e ect of verifying the following formula at x: (p ^ :q) ^ (q ^ :p): Now, the original property itself will correspond to the negation of this modal `failure description', i.e.
:((p ^ :q) ^ (q ^ :p)): By some familiar equivalence transformations, this becomes
(p ! q) _ (q ! p); a principle known from the literature as Geach's Axiom. It remains to be shown, of course, that conversely, failure of this axiom implies failure of connectedness; but this is immediate. In order to crosscheck, one might also apply the earlier method of substitutions to (some
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suitable transform of) the Geach Axiom: and indeed, connectedness will ensue. The `negative' side again consists of disproofs. Here as well, these turn out to possess a particular interest | as we are forced to contemplate `typical behaviour' of modal formulas. A standard example is the following. Although re exivity was modally de nable, irre exivity turns out intractable: 8x:Rxx. But, failed attempts are no de nite refutations. What we need is some semantic property of modal formulas, as relational conditions on Kripke frames, which is not shared by this particular rst-order sentence. At this point, the modal model theory of Section 2.1 comes in. There, one nds that the following mappings play a fundamental role in the transmission of modal truth between Kripke frames: a p-morphism is a function f from a frame hW; R1 i to a frame hW2 ; R2 i which 1. preserves R1 , and 2. `almost' preserves R2 , in the following sense: `If R2 f (w)v, then there exists some u 2 W1 such that (a) R1 wu and (b) f (u) = v'. Under di erent names, this notion has had a career in standard logic already, e.g. the `Mostowski collapse' in set theory is of this kind. For the purposes of the present example, it need only be recorded that subjective p-morphisms preserve truth of modal formulas on Kripke frames. But then, irre exivity may be dismissed: it holds in the frame of the natural numbers with the usual order, but it fails in its p-morphic image (!) arising from the contraction to one single re exive point. This example will have given a taste of the actual eld-work in this area of Correspondence Theory. There also arises the more general question, of course, whether some combination of modally valid preservation requirements manages to characterise all and only the modally de nable rst-order sentences. This is indeed the case, and an elegant result to this e ect | involving p-morphisms as well as other basic constructions, will be proved in Section 2.4. The preceding survey by no means exhausts the range of questions that can be investigated in Correspondence Theory | but it does convey the spirit.
Correspondence and Completeness Three pillars of wisdom support the edi ce of Modal Logic. There is the ubiquitous Completeness Theory, the present Correspondence, or, more generally, De nability Theory | and nally, the Duality Theory between Kripke frames and `modal algebras' (cf. Section 2.3 below) has become an area of its own. Connections between the latter two will become apparent as Section
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2 unfolds | in particular, the above-mentioned characterisation of modally de nable rst-order sentences will be obtained as a consequence of the classic Birkho Theorem of Universal Algebra, applied to modal algebra. The relation between correspondence and completeness is less vital to subsequent developments. Moreover, it turns out to be rather complex | and indeed, only partially understood. Nevertheless, for those readers who are familiar with the basic notions of Completeness Theory, the following sketch of issues may serve to bring questions of correspondence closer to traditional concerns. The early completeness theorems in modal logic were brought under one heading in [Segerberg, 1971]: `modal logic L is determined by a class R of Kripke frames', i.e. L axiomatises the modal theory of R (on the basis of the minimal logic K). As before, two perspectives emerge here. First, one may start with a given class R, asking for a recursive axiomatisation L of its modal theory. In general, there is no guarantee for success here; but there is one helpful observation involving rst-order de nability. FACT 4. If R is elementary (i.e. de ned by a single rst-order sentence), then its modal theory is recursively axiomatisable.
Proof. Let = (R; =) de ne R. A modal formula ' belongs to the theory of R if and only if it holds in all frames in R. This may be restated as follows: 8x8P1 : : : 8Pn ('); where (') is the earlier rst-order translation of ', while p1 ; : : : ; pn are the proposition letters occurring in the latter formula. Now, the predicate variables P1 ; : : : ; Pn do not occur in the rst-order sentence , and, therefore the above implication is equivalent to 8x ('). But this is an ordinary rst-order implication. So, since the latter notion is recursively axiomatisable, the same must be true for membership of the modal theory of R. Axiomatisable, yes, but axiomatisable on the basis of the minimal modal logic K? Even this is true, choosing a suitable recursive set of axioms as in the proof of Craig's Theorem in classical logic and noticing that K contains modus ponens (which is all that is needed). Thus, in retrospect, the earlier completeness theorems for re exive, transitive orders (and other elementary classes) were quite predictable. The direction from classes of frames to logics is not the current one in modal logic; being more appropriate to areas such as tense logic, where temporal structures often precede temporal theories. Usually, one already possesses a certain logic L, asking for a class R of Kripke frames with respect to which it is complete. (Notice that, if any class R suÆces, then the whole class of Kripke frames validating L will.)
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Nowadays, we know that not all modal logics are in fact complete in the above sense, contrary to earlier expectations. This is the content of the celebrated `modal incompleteness theorems' in [Fine, 1974; Thomason, 1974]. But it has been hoped that, at least, all rst-order de nable axiom sets are complete. (Indeed, a defective proof to this e ect has circulated.) Even this more modest expectation was frustrated in [van Benthem, 1978]: FACT 5. The modal logic L with characteristic axioms p ! p p ! p (p ^ (p ! p)) ! p is rst-order de nable: its frames are just those satisfying the condition
8xy(Rxy $ x = y): But the characteristic axiom of the modal theory of the latter class of frames, viz. p $ p, is not minimally derivable from L. The relevant correspondence will be proved in Section 2.2. For the moment, it may be noticed that the third axiom de nes a notion of `safe return': from any R-successor of a world x, one can always return to x by following some nite R-chain of R-successors of x. The relevant argument is highly nontrivial, far outside the range of our earlier method of substitutions. Nevertheless, even the latter has its relevance for completeness theory, as we shall see presently. What the modal incompleteness theorems show is that the minimal modal logic K is to weak to produce all modally valid inferences. But of course, there may be stronger reasonable `base logics'. One particular example arises from the method of substitutions. For instance, in proving the equivalence of substitution instances with more current rst-order conditions, one uses an extremely natural second-order logic K2 with the following deductive apparatus: Some rst-order base complete with respect to modus ponens, similar axioms for the second-order quanti ers; with the following form of ` rst-order instantiation' allowed for rst-order formulas 8x'(X ) ! '( ): Through the earlier second-order transcription, K2 may be used as a modal base logic. Here is an example of some fame. In the metamathematics of arithmetical provability (cf. [Boolos, 1979] or Smorynski's in a later volume of this Handbook), the following two modal axioms are basic:
p ! p; (p ! p) ! p
(`Lob's Axiom'):
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The semantic import of the latter will be established in Section 2.2: it holds in those Kripke frames whose alternative relation is transitive, while possessing a well-founded converse. Moreover, transitivity is K2 -derivable from Lob's Axiom, by the substitution of
Rxu ^ 8y(Ruy ! Rxy)
for P u:
(The antecedent becomes universally valid, while the consequent expresses transitivity.) An advantage of K2 over K? No, around 1975, Dick de Jongh and Giovanni Sambin found a K -deduction for the rst axiom from the second after all. The two deductions are related, but systematic connections between K-deductions and K2 -deductions have not been explored up to date. Nevertheless, K2 is non-conservative over K in the modal realm. In [van Benthem, 1979b] we nd the following incompleteness theorem. FACT 6. The modal axiom
? _ ((p ! p) ! p); with ? the falsum, de nes the same class of Kripke frames as ? _ ?. But, the latter formula is not K-derivable from the former | even though it is K2 -derivable. Again, there is a correspondence involved here. But the idea is illustrated by a simple K2 -deduction at the back of this result: 1. 8P (8y(Rxy ! (8z (Ryz ! P z ) ! P y)) ! P x) (0 (p ! p) ! p0 ), 2. 8y(Rxy ! (8z (Ryz ! z 6= x) ! y 6= x)) ! x 6= x (x 6= u for P u), 3. 4. 5. 6.
:8y(Rxy ! (8z (Ryz ! z 6= x) ! y 6= x)), 9y(Rxy ^ 8z (Ryz ! z 6= x) ^ y = x), Rxx ^ 8z (Rxz ! z = 6 x) x= 6 x: a contradiction (?).
That K2 , in its turn, must be modally incomplete (as is any proposed recursively axiomatised base logic) follows from the general incompleteness results in [Thomason, 1975]. First-order de nability does not imply completeness. But, when a modal logic is both rst-order de nable and complete, it enjoys a very pleasant form of the latter property | viz. with respect to the underlying frame of its own Henkin model. (`First-order de nability plus completeness imply canonicity': cf. [Fine, 1975; van Benthem, 1980].) Such canonical modal logics will be characterised semantically in Section 2.4: notice that many
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of the familiar text book examples are of this kind. In fact, a canonical completeness proof, such as that for S4, often proceeds by means of rstorder conditions on the Henkin model, induced by the corresponding axioms. The relation between these familiar `Henkin arguments' and the above method of substitutions is at present still rather mysterious. Sahlqvist [1975] contains many examples of parallels; but Fine [1975] presents a problem. The modal formula (p _ q) ! (p _ q) axiomatises a canonical modal logic, without being rst-order de nable. Thus, we are still far from complete clarity in the area between completeness and correspondence.
Variations and Generalisations Logical model theory may be viewed as a marriage between ontology and language (or `mathematics' and `linguistics'). Accordingly, the semantics of propositional modal logic, our paradigm example up till now, exhibits the familiar triangle language
interpretation
structures
Or, from the above translational point of view, the components are prima facie language
translation
representation language
All these `degrees of freedom' may be varied in intensional logic | and thus there appears a whole family of `correspondence theories'. We shall explore some examples of recognised importance in Section 3. Here, let us just think about the various possibilities and their implications. Even within the domain of propositional modal logic, alternatives have been proposed for Kripke-type relational semantics. Jennings, Johnstone and Schotch [1980] contains the proposal to work with ternary alternative relations, employing the following notion of necessity:
' is true at x if 8yz (Rxyz ! '(y) _ '(z )): Their motivation was, amongst others, to create room for `non-cumulation' of necessities: the `Aggregation Axiom'
p ^ q ! (p ^ q) will no longer be valid. What happens to earlier correspondences in this new light? Old boundaries start shifting; e.g. p ! p remains rst-order de nable, but p ! p becomes essentially second-order on this semantics.
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This is compensated for by the phenomenon of formerly unexciting principles, such as the Aggregation Axiom (which was trivially valid before) springing into unexpected bloom: EXAMPLE 7. p ^ q ! (p ^ q) de nes
8xyz (Rxyz ! (y = z _ Rxyy _ Rxzz )): Proof. `)': Suppose the condition fails at x; y; z . Setting V (p) = W = fz g; V (q) = W
fyg;
will then verify p; q at x, while (p ^ q) is falsi ed (by Rxyz ). `(': Suppose that p; q hold at x, and consider Rxyz . Either y = z , whence y veri es both p and q (by Rxyy and the truth de nition), or Rxyy, implying the same conclusion, or Rxzz , in which case z veri es both p and q. So, (p ^ q) holds at x. As for the general theorems, forming the backbone of the subject, nothing essential changes in this ternary semantics. This example changed both the structures and the form of the truth de nition. What may not be generally realised is the variety o ered even when xing the two parameters of `language' and `structures'. Therefore, a short digression is undertaken here. The Kripke truth de nition is not sacrosanct | other clauses would have been quite imaginable. Thus, for instance, we may make the following OBSERVATION 8. The truth de nition `' is true at x if 8y((Rxy _ Ryx) ! '(y))' yields as a modal base logic KB; i.e. the minimal logic K plus the Brouwer Axiom p ! p.
Proof. The Brouwer Axiom de nes symmetry of the alternative relation; as may be seen by substituting u = x for P u. And indeed KB is complete with respect to the class of symmetric Kripke frames. Hence, any non-theorem ' of KB is falsi ed on some symmetric frame hW; Ri. But, on symmetric frames R coincides with the relation xy. (Rxy _ Ryx) (i.e. R united with its converse R ); whence ' also fails by the new evaluation. Conversely, if ' has a counter-example hW; Ri under the new truth de nition, then it has hW; R [ R i for an ordinary symmetric counter-example; whence it is outside of KB. Thus, there is a possible trade-o between truth de nition and requirements on the alternative relation. The exact extent of this phenomenon remains to be investigated. Notice for example how KB is equally well generated by the following truth de nition:
' is true at x if 8y((Rxy ^ Ryx) ! '(y)):
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The general principle behind such examples is this. FACT 9. If C (R) is any condition on R, and (x; y) some formula in R; = such that 1. If C (R) is satis ed, then R and xy: (x; y) coincide, 2. xy: (x; y) satis es C , then the modal logic determined by (the Kripke frames obeying) C may also be generated without conditions through the truth de nition
' is true at x if 8y( (x; y) ! '(y)): This rather subversive shift in perspective will not be investigated in this contribution. At this point, it merely serves to remind us that not a single aspect of the semantic enterprise is immune to revision. Leaving the realm of modal logic, of the many intensional candidates for a correspondence perspective, only a few have been explored up to date. In Section 3, some important examples are reviewed brie y, viz. tense logic, conditional logic and intuitionistic logic. These illustrate, in ascending order, certain diÆculties which tend to make Correspondence Theory rather more diÆcult (often also: more exciting) in many cases. These diÆculties have to do with `pre-conditions' on the alternative relation (not very serious), and the phenomenon of `admissible assignments' (rather more serious), to be explained in due course. Nevertheless, for instance, Intuitionistic Correspondence Theory will turn out to possess also some elegant features lacked by its modal predecessor. A few examples, even without proof, will render the above remarks more concrete. In tense logic, the correspondence runs between temporal axioms and properties of the temporal order (`before', `earlier than'). EXAMPLE 10 (`Hamblin's Axiom'). (p ^ Hp) ! F Hp de nes discreteness of Time: 8x9y >x8z
8xyz (y = z _ Cxyz _ Cxzy): Finally, in intuitionistic logic, (`intermediate') axioms impose constraints upon the possible growth patterns of stages of knowledge.
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EXAMPLE 12 (`Weak Excluded Middle'). gence' of growing stages, i.e. directedness:
:p _ ::p de nes `local conver-
8xyz ((x y ^ x z ) ! 9u(y u ^ z u)): Proofs, and further explorations are postponed until the relevant sections. At this stage, the experienced reader may predict that two nuts will be especially diÆcult to crack for any Correspondence Theory. The rst of these concerns the earlier tacit restriction to propositional logic: what happens in the predicate case? In Section 2.5 we shall see that no essential problems seem to arise | although the eld remains largely unexplored. A more formidable problem arises when the truth de nition for the intensional operators itself becomes of higher-order complexity. In that case, e.g. a search for possible rst-order equivalents of intensional axioms seems rather pointless. This eventuality arises when disjunction is evaluated barwise in Beth semantics for intuitionistic logic (i.e. ' _ is true at x if the '-worlds and -worlds together form a barrier intersecting each branch passing through x). The last word has not been said here, however. Philosophically, it seems a rather unsatisfactory division of semantic labour to let the truth de nition absorb structural complexity (in this case: the second-order behaviour of branches). The latter should be located where it belongs, viz. in the structures themselves. And indeed, the Beth semantics admits of a two-sorted rst-order reformulation in terms of nodes and paths, which generates a Correspondence Theory of the usual kind. All this is not to say that there are no limits to the useful application of a correspondence perspective. But, these are to be found in philosophical relevance, rather than technical impossibility. One should study correspondences only as long as they serve the purpose of semantic enlightenment | which is the shedding of light upon one conceptual framework by relating it systematically to another. 2 MODALITY In this chapter, modal correspondence theory will be surveyed against the background of modal model theory and modal algebra, whose basics are explained. (Cf. the chapter by Bull and Segerberg in this volume for the necessary background.)
2.1 Modal Model Theory The basic structures of modal semantics are introduced: frames, models and general frames. These may be studied either purely classically, or
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with a speci cally modal purpose. In both cases, the emphasis is not upon such structures in isolation, but upon their `categorial context': what are their relations with other structures, and which of these relations are truthpreserving? Thus, we will introduce the modal preservation operations of generated subframe, disjoint union, p-morphic image and ultra lter extension. Moreover, the fundamental classical formation of ultraproducts will be used as well. All these notions will appear again and again in later sections. Semantic structures. The structures used in the Kripke truth de nition are models M , i.e. triples hW; R; V i, where W is a nonempty set of worlds, R is a binary alternative relation on W , and V is a valuation assigning sets of worlds V (p) to proposition letters p. The notion explicated then becomes
M '[w]
: `' is true in M at w':
In our correspondence theory we also want to see the bare bones: a frame F is a couple hW; Ri as above, but without a valuation. There is nothing intrinsically `modal' about all this, of course. Frames are just the `directed graphs' of Graph Theory. In Sections 2.3 and 2.4, a third notion of modal structure will be required as well | intermediate, in a sense between models and frames. A general frame F is a couple hF; Wi, or alternatively, a triple hW; R; Wi such that F = hW; Ri is a frame, and W is a set of subsets of W , closed under the formation of complements, unions and modal projections. Formally, if X 2 W; then W X 2 W if X; Y 2 W; then X [ Y 2 W if X 2 W; then (X ) =def fw 2 W
j 9v 2 X : Rwvg 2 W:
The following example illustrates the e ect of restricted sets W. Consider the frame hN; i, where N is the set of natural numbers. Its modal theory contains such principles as p ! p; p ! p and Geach's Axiom: together forming the logic S4.3. Typically left out is the McKinsey Axiom p ! p; as it may be falsi ed in some in nite alternation of p; :p: say by V (p) = f2n j n 2 N g. But now, consider the structure hN; ; Wi, where W consists of all nite and all co nite subsets of N . It is easily checked that all three closure conditions obtain for W. Thus, we have a general frame here. Its logic contains the earlier one (`a fortiori'); but it also adds principles. Notably, the McKinsey Axiom can no longer be falsi ed, as the above `tell-tale' valuation is no longer admissible. Thus, S4.1 holds in this general frame, although it does not in the underlying `full frame'. And further increases in the modal theory are possible, by restricting W even more; e.g. there is even a most austere choice, viz. W = f;; N g, which yields a general frame validating the `classical logic' with axiom p $ p | which was still invalid in the previous general frame. Thus, one single underlying frame may still generate a hierarchy of modal logics.
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The original algebraic motivation for this notion (due to Thomason [1972]) will be given in Section 2.3. But here already, a direct logical reason may be given. Kripke frames are so-called `standard models' for modal formulas, considered as second-order 11 -sentences: the universal predicate quanti ers range over all sets of possible worlds. An intermediate possibility would have been to allow also `general models' in the sense of Henkin [1950]: in which this second-order range may be restricted, say to some set W. Usually, such ranges are to be closed under certain mild conditions of de nability | in order to verify reasonable forms of the universal instantiation (or `comprehension') axiom. This, of course, is precisely what happened in the above. The uses of this notion lie partly in modal Completeness Theory, partly in modal algebra. For the moment, it will not be a major concern. Semantic questions. Given a formal language, interpreted in certain structures, a plethora of questions arises concerning the interplay between more `linguistic' and more `structural' (or `mathematical') notions. We mention only a few fundamental ones. Arguably the ` rst question' of any model theory is that concerning the relation between linguistic indistinguishability (equality of modal theories) and structural indistinguishability (isomorphism) of semantic structures. How far do the webs of language and ontology diverge? In classical logic, we know that ( rst-order) elementary equivalence coincides with isomorphism on the nite structures, but no higher up: isomorphism then becomes by far the ner sieve. Now, the modal language on models behaves like the rst-order language of the rst translation in the introduction: nothing spectacular results. But the second-order notion seems more interesting in this respect. (Equality of second-order theories is quit`e strong: modulo the Axiom of Constructibility, it even implies isomorphism in all countable frames; cf. [Ajtai, 1979]). From Van Benthem [1985], which treats the analogous question for tense logic in Chapter 2.2.1, we extract THEOREM 13. Finite Kripke frames that are generated by a single point (cf. below) are isomorphic if and only if they possess the same modal theory. But, the countable Kripke frames Z Z (the integers, with each point replaced by a copy of the integers) and Q Z (the rationals, treated likewise) possess the same modal theory, without being isomorphic. In tense logic, the latter result means that the formal language cannot distinguish between locally discrete/globally discrete and locally discrete/globally dense Time. (The latter may well be that of our World.) In the context of modal logic, no such appealing interpretation is possible, whence we forego further discussion of the above result. From now on, we will con ne attention to a single theme, which again, is characteristic for much of what goes on in Model Theory.
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Truth-preserving operations. In evaluating the truth of a modal formula ' at a world w we only have to consider w itself, (possibly) its R-successors, (possibly) their R-successors, etcetera. Thus, only that part of the frame is involved which is `R-generated' by w, so to speak. In general, one never has to look beyond R-closed environments of w: an observation summed up in the following notion and result. DEFINITION 14. M1 (= hW1 ; R1 ; V1 i) is a generated submodel of M2 (= M ) if hW2 ; R2 ; V2 i) (notation: M1 ! 2
1. W1 W2
2. R1 = R2 restricted to W1 ,
3. V1 (p) = V2 (p) \ W1 , for all proposition letters p; i.e. M1 is an ordinary submodel of M2 , which has the additional feature that 4. W1 is closed under passing to R2 -successors. The next result is the famous `Generation Theorem' of Segerberg [1971]. M , then for all worlds w 2 W and all modal THEOREM 15. If M1 ! 2 1 formulas ', M1 '[w] i M2 '[w]. This is what happens inside a single model. When comparisons are desired between evaluation in distinct models, a more external connection is required. DEFINITION 16. A relation C is a zigzag connection between two models M1; M2 if 1. domain (C ) = W1 , range (C ) = W2 , (a) if Cwv and w0 2 W1 with R1 ww0 , then Cw0 v0 for some v0 2 W2 with R2 vv0 (`forth choice') 0 0 0 (b) If Cwv and v 2 W2 with R2 vv , then Cww for some w0 2 W1 with R1 ww0 (`back choice') 2. if Cwv, then w; v verify the same proposition letters. Starting from the basic case (3), the back-and-forth clauses ensure that evaluation of successive modalities in modal formulas yield the same results on either side: THEOREM 17. If M1 is zigzag-connected to M2 by C , then, for all worlds w 2 W1 ; v 2 W2 with Cwv, and all modal formulas ',
M1 '[w] i M2 '[w]:
Notation. M1 ! M2 for zigzag-connected models (by some C ).
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By a result in Van Benthem [1976], the Generation Theorem and the preceding `Zigzag Theorem' combined are characteristic for modal formulas as rst-order formulas in the sense of the introduction: THEOREM 18. A rst-order formula '(x) in the language with R; P; Q; : : : is logically equivalent to some modal transcription if and only if it is invariant for generated submodels and zigzag connections (in the above sense). For the case of pure frames, the above notions and results lead to the following three preservation results. F ) if DEFINITION 19. F1 is a generated subframe of F2 (F1 ! 2 1. W1 W2 ,
2. R1 = R2 restricted to W1 , 3. W1 is R2 -closed in W2 . In general logic, this type of situation is often described by saying that the `converse frame' hW2 ; R 2 i is an end extension of hW1 ; R 1 i: the added worlds all come `at the end'. From Theorem 15 we derive preservation under generated subframes: F , then F ' implies F ', for all modal COROLLARY 20. If F1 ! 2 2 1 formulas '. Here `F '' means `' is true in F ', in the global second-order sense of the introduction: at all worlds, under all valuations. But Theorem 15 also has an `upward' directed moral. DEFINITION 21. The disjoint union fFi ji 2 I g of a family of frames Fi = hWi ; Ri i is the disjoint union of the domains Wi , with the obvious coordinate relations Ri . Another direct application is preservation under disjoint unions: COROLLARY 22. If Fi ' (all i 2 I ), then fFi ji 2 I g ', for all modal formulas '. Next, turning to Theorem 17, one now needs a connection between frames which can be turned into a suitable zigzag relation between models over them. DEFINITION 23. A zigzag morphism from F1 to F2 is a function: W1 ! W2 satisfying 1. R1 ww0 implies R2 f (w)f (w0 ), i.e. f is an ordinary R-homomorphism; which has the additional backward property that 2. if F2 f (w)v, then there exists u 2 W1 with R1 wu and f (u) = v.
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This notion was mentioned under its current, but rather uninformative name of `p-morphism' in the introduction. Here is one more example: the map from nodes to levels (counting from the top) is a zigzag morphism from the in nite binary tree (with the descendant relation) onto the natural numbers (with the usual ordering). Notice also that injective (1-1) zigzag morphisms are even just isomorphisms. Theorem 17 now implies the `p-morphism' theorem of Segerberg [1971]. COROLLARY 24. If f is a zigzag morphism from F1 onto F2 , then, for all modal formulas '; F1 ' implies F1 '. For more `local' versions of these results, the reader is referred to [van Benthem, 1983]. More examples, and applications of Corollaries 20, 22, and 24 will be found in Section 2.4. A quick impression may be gained from the following sample observation (D. C. Makinson). The modal theory of any Kripke frame is either contained in the classical modal logic (characteristic axiom p $ p) or the `absurd' modal logic (characteristic axiom (p ^ :p)). For, any frame F either contains end points without R-successors, or it is serial (8x9yRxy). In the former case, such an end point by itself forms a generated subframe, and by Corollary 20, the logic of the frame is contained in that of the subframe | which is the absurd one. In the latter case, contraction to one single re exive point is a zigzag morphism, and by Corollary 24, the logic of the frame is contained in that of the re exive point | which is the classical one. We conclude by noting that these three notions are easily adapted to general frames, taking due precautions concerning the various sets W1 ; W2 . Here are the three necessary additions: In 19: add `W1 = fX \ W1 j X 2 W2 g'. In 21: add `the new W2 remains essentially the old W1 ' (but for the disjointness procedure used). In 23: add the following `continuity requirement', reminiscent of topology: `for all X 2 W2 ; f 1[X ] 2 W1 '. These will be needed in the duality theory of Section 2.3. Propositions and possible worlds. Another characteristic feature of modal semantics is the analogy between propositions and sets of possible worlds; as well as (moving up one stage in set-theoretic abstraction) that between possible worlds and maximal sets of propositions. Indeed, many philosophers would deny that there exist any di erences here. Let us investigate. The ideal setting here are general frames hW; R; Wi: the range is clearly identi able with a collection of `propositions' over W .
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Now, if worlds are to be considered as sets of propositions, then some obvious desiderata govern the connection between a world w and propositions X; Y associated with w: 1. X 2 w or Y 2.
2 w if and only if X [ Y 2 w X 62 w if and only if W X 2 w
(`analysis') (`decisiveness').
Accordingly, one considers only subsets w of W satisfying these two conditions. These are precisely the so-called ultra lters on W. What about the alternative relation to be imposed? Again, a common idea is that a world v is R-accessible to w if it `satis es all w's modal prejudices', i.e. whenever ' is true at w, ' should be true at v. The same idea may be expressed as follows: whenever ' is true at v; ' should be true at w. In the present context, this becomes the following stipulation: Rwv if for all X 2 v; (X ) 2 w: In this process, no new propositions have been created, whence the former propositions X now reappear as sets X = fw j X 2 wg. These considerations motivate DEFINITION 25. The ultra lter extension ue(G) of a general frame G = hW; R; Wi is the general frame hue(W; W); ue(R; W); ue(W)i, with 1. ue(W; W) is the set of all ultra lters on W, 2. ue(R; W)wv, if for each X 2 W such that X 2 v; (X ) 2 w, 3. ue(W) is fX j X 2 Wg. What this construction has done is to re-create G one level higher up in the set-theoretic air, so to speak, and some calculation will prove THEOREM 26. G and ue(G) verify the same modal formulas. Still, not everything need have remained the same: the world pattern of hW; Ri may di er from that of hue(W; W); ue(R; W)i. First, each old world w 2 W generates an ultra lter fX 2 W j w 2 X g and, hence, a corresponding new world in ue(W; W). But, unless W satis es certain separation principles for worlds, di erent old worlds may be identi ed to a single new one. (In the earlier example of hN; ; f;; N gi, only a single new world remains, where there used to be in nitely many!) On the other hand, the construction may also introduce worlds that were not there before. For instance, on the earlier general frame hN; , (co-) nite setsi, the co- nite sets form an ultra lter which induces a `point at in nity' in the resulting ultra lter extension. Indeed, it is easily seen that the latter consists of hN; i followed by just that in nite point.
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In Section 2.3, necessary and suÆcient conditions will be formulated guaranteeing that a general frame is `stable' under the construction of ultra lter extensions. In any case, it turns out that the process stabilises after one step at the most. Now, these considerations also apply to `full' Kripke frames. DEFINITION 27. The ultra lter extension ue(F ) of a frame F = hW; Ri is the frame hue(W ); ue(R)i, with 1. ue(W ) is the set of all ultra lters on W , 2. ue(R)wv if for each X W such that X 2 v; (X ) 2 w. This time, Theorem 26 does not hold, however. For, it only says that the modal theory of the general frame hW; R; power set of W i coincides with that of the induced general frame according to De nition 25. Now, the latter is, in general, a restriction of the full frame hue(W ); ue(R)i. Hence, we can only conclude to anti-preservation under ultra lter extensions: COROLLARY 28. If ue(F ) ', then F ', for all modal formulas '. Still, this structural notion can be made a little more familiar by connecting it with previous model-theoretic operations. First, the above-mentioned connection between old worlds and new worlds is 1-1 this time, and indeed isomorphic (consider suitable singleton sets): THEOREM 29. F lies isomorphically embedded in ue(F ). In general, this cannot be strengthened to `embedded as a generated subframe'. But, another connection with the earlier preservation notions may be drawn from [van Benthem, 1979a]. THEOREM 30. ue(F ) is a zigzag-morphic image of some frame F 0 which is elementarily equivalent to F .
Proof. One expands F to (F; X )X W , and then passes on to a suitably saturated elementary extension, by ordinary model theory. From the latter, a canonical function from worlds to ultra lters on F exists, which turns out to be a zigzag morphism. Ultraproducts and de nability. New, modally inspired notions concerning frames have been forged in the above. But old classical constructions may be considered as well. Of the various possibilities, only one is selected here, viz. the formation of ultraproducts. (For many other examples, cf. [van Benthem, 1985, Chapter I.2.1].) Its use has been indicated in the introduction already. The basic theory (and heuristics) of the notion of `ultraproduct' has been given in the Higher Order Logic chapter in volume 1 of this Handbook. (Cf. also [Chang and Keisler, 1973, Chapters 4.1 and 6.1].) We recall some of its outstanding features and uses.
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DEFINITION 31. For any family of Kripke frames fFi j i 2 I g with an ultra lter U on I , the ultraproduct U Fi is the frame hW; Ri with 1. W is the set of classes f , for all functions f 2 fWi j i 2 I g, where f is the equivalence class of f in the relation f g , fi 2 I j f (i) = g(i)g 2 U , 2. R is the set of couples hf ; g i for which fi 2 I j Ri f (i)g(i)g 2 U . This de nitional equivalence is lifted by induction to THEOREM 32 (`Los Equivalence'). For all ultraproducts, and all rst-order formulas '(x1 ; : : : ; xn ), U Fi '[f1 ; : : : ; fn ] i fi 2 I j Fi '[f 1 (i); : : : ; f n (i)]g 2 U: Thus, in particular, all rst-order sentences ' are preserved under ultraproducts in the following sense: if Fi '(all i 2 I ); then u Fi ': Conversely, `Keisler's Theorem' tells us that this is also enough. THEOREM 33. A class of Kripke frames is elementary if and only if both that class and its complement are closed under the formation of ultraproducts and isomorphic images.
Proof. Cf. [Chang and Keisler, 1973, Chapter 6.2].
A somewhat more liberal notion of de nability, viz. by means of arbitrary sets of rst-order formulas, yields so-called -elementary classes. Here the relevant characterisation employs a special case of ultraproducts. DEFINITION 34. An ultrapower U F is an ultraproduct with in each coordinate i the same frame F . Notice that by the Los Equivalence, U F is elementarily equivalent to F , i.e. both frames possess the same rst-order theory. THEOREM 35. A class of Kripke frames is -elementary if and only if it is closed under the formation of ultraproducts and isomorphic images, while its complement is closed under the formation of ultrapowers. All these notions will be used in the modal correspondence theory of the next section. In this connection, it should be observed that, as for the other kinds of modal semantic structure, ultraproducts of models and of general frames are easily de ned using the above heuristics. These will not be used in the sequel however. (Cf. [van Benthem, 1983].)
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The above de nability question for classical model theory leads to a clear modal task: `to characterise the modally de nable classes of Kripke frames'. In section 2.4 this matter will be investigated. We have arrived at the interplay between classical and modal model theory, which lies at the heart of modal correspondence theory.
2.2 Correspondence I: From Modal to Classical Logic Through the translation given in the Introduction, modal formulas may be viewed as de ning constraints on the alternative relation in Kripke frames. Some of these constraints are rst-order de nable, others are not. Examples are presented of both, after which the former class is explored. A mathematical characterisation is given for it, in terms of ultrapowers, and methods are developed for (dis-)proving membership of the class. The limits of these methods are established as well. First-order de nability. The class of modal formulas to be studied here is de ned as follows. DEFINITION 36. M1 consists of all modal formulas ' for which a rstorder sentence (in R; =) exists such that
F
' i F ;
for all Kripke frames F:
Various examples of formulas in M1 have occurred in the Introduction. For purposes of illustration, see Table 1 below. As these are all rather easy to establish, some readers may desire a more complex example. Here it is, straight from the incompleteness Example 5 in the Introduction. THEOREM 37. The conjunction of the formulas p ! p; p ! p and (p ^ (p ! p)) ! p is in M1. Proof. We shall show that this conjunction de nes the same class as the classical axiom p $ p, i.e. 8xy(Rxy $ x = y). The argument requires several stages. 1. 2.
p ! p imposes re exivity, p ^ (p ! p) ! p says the following: 8xy(Rxy ! 9n9z1 ; : : : ; zn (Rxz1 ^ : : : ^ Rxzn ^ ^Ryz1 ^ : : : ^ Rzn x)).
In other words, from any R-successor y of x, one may return to x by way of some nite chain of R-successors of x. In case the chain is empty, this reduces to just: Ryx. This (second-order!) equivalence is proved as follows (I. L. Humberstone): `)': Consider any y with Rxy. Let the good points be those R-successors z
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Table 1. Modal formula Condition p ! p 8xRxx p ! p 8xy(Rxy ! 8z (Ryz ! Rxz )) p ! p 8xy(Rxy ! 8z (Rxz ! 9u(Ryu ^ Rzu))) (p _ q) ! p _ q 8xy(Rxy ! 8z (Rxz ! z = y)) (p ! q) _ (q ! p) 8xy(Rxy ! 8z (Rxz ! (Ryz _ Rzy))) p ! p 8xy(Rxy ! y = x) ? 8x:9yRxy p ! p 8xy(Rxy ! Ryx) of x which can be reached from y through some nite chain (possibly empty) of R-successors of x. Then, set V (p) equal to the set of all R-successors of good points. This assignment produces the following e ects. 1. p is true at y (y being a successor of y, by re exivity), and, hence, p is true at x. 2. Any R-successor of x verifying p is itself a good point, whence all its R-successors belong to V (p). It follows that (p ! p) is true at x. Therefore, p itself must be true at x: i.e. x is R-successor of some good point, which was precisely to be proved. `(': Truth of p in x is discovered by merely following the relevant chain. 3. Now, having secured re exivity and `safe return', we can nd out what the McKinsey Axiom says in the present context. First, notice that all R-successors of any point x may be divided up into concentric shells Sn (x), where Sn (x) consists of those Rsuccessors y of x which return to x by n R- arrows (between Rsuccessors of x) but no less. For instance, S0 (x) only consists of x itself, S1 (x) contains immediate R- predecessors. Notice also that, if y 2 Sn+1 (x), then it must have some R-successor in Sn (x). The McKinsey Axiom makes this whole hierarchy collapse. Set V (p) = [fS2n (x) j n = 0; 1; 2; : : :g. Then p will be true at x, as follows from the above picture. For, if Rxy, and y 2 Sn (x), then either n is even | whence p holds at y (by de nition) and so p (by re exivity), or n is odd | whence y has an R-successor in Sn 1 (x) verifying p: which again veri es p at y. It follows that p must be true at x. So, p holds at some Rsuccessor of x. Which one? In the present situation, this can only be
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x itself. But then again, this means that there can be no shells Sn (x) with n odd. Thus, there is only S0 (w) : 8y(Rxy ! y = x). 4. Combining (1) and (3), the required conclusion follows: the three axioms together imply 8xy(Rxy $ y = x), and are obviously implied by it. The very unexpectedness of this argument will have made it clear that there is a creative side to establishing correspondences. Global and local de nability. Originally, Kripke introduced frames hW; R; w0 i, with a designated `actual world' w0 . From that point of view, the study of `local' equivalence becomes natural: F '[w] i [w]; where the rst-order formula has one free variable now. The reader may have noticed already that previous correspondence arguments often provide local versions as well. For instance, we had F p ! p[w] i F Rxx[w] F p ! p[w] i F 8y(Rxy ! 8z (Ryz ! Rxz ))[w]: The local notion is the more informative one, in that local correspondence of ' with (x) implies global correspondence of ' with 8x (x); but not conversely. Indeed, [van Benthem, 1976] contains an example of a formula in M1 which has no local rst-order equivalent at all! On the other hand, there are also circumstances under which the distinction collapses | e.g. on the transitive Kripke frames (W. Dziobiak; cf. [van Benthem, 1981a]). Finally, a word of warning. Local validity of, e.g. p ! p means `local transitivity', no more. The frame hN; fh0; ni j n 2 N g[fhn; n +1i j n 2 N gi is locally transitive in 0, without being transitive. First-order unde nability. There is a threshold of complexity below which second-order phenomena do not occur. THEOREM 38. All modal formulas without nestings of modal operators are in M1. Proof. Cf. [van Benthem, 1978]: a combinatorial classi cation suÆces. EXAMPLE 39. Lob's Axiom (p ! p) ! p is outside of M1. Proof. It suÆces to establish the following Claim: Lob's Axiom de nes transitivity plus well-foundedness of the converse of the alternative relation (i.e. there are no ascending sequences xRx1 Rx2 Rx3 ; : : :). For, by a wellknown classical compactness argument, the latter combination cannot be rst-order de nable (e.g. notice that it holds in hN; >i, but not in its nonisomorphic ultrapowers). First, assume that Lob's Axiom fails in F ; i.e. for some V and w,
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1. hF; V i (p ! p)[w], but 2. hF; V i 6 p[w] Also, assume transitivity of R: we will refute the well-foundedness of R , by constructing an endless ascending sequence of worlds wRw1 Rw2 : : :. Step 1: Chose any w1 with Rww1 where p fails (by (2)). By (1), p ! p is true at w1 , whence p fails again. Step 2: chose any w2 with Rw1 w2 where p fails. By (1) and transitivity, p ! p is true at w2 , etcetera: an endless sequence is on its way. Next, failure of either of the two relational conditions results in failure of Lob's Axiom. If transitivity fails, say Rwv; Rvu; :Rwu, then V (p) = W fv; ug veri es (p ! p) at w, while falsifying p. If well-foundedness fails, say wRw1 Rw2 ; : : :, then V (p) = W fw; w1 ; w2 ; : : :g produces the same e ect. More complex unde nability arguments will be discussed later on. First-order de nability and ultraproducts. Modal formulas could be regarded as 11 -sentences, witness the Introduction. Now, for the latter sentences, ultraproducts provide the touchstone for rst-order de nability: THEOREM 40. A 11 -sentence in R; = is rst-order de nable if and only if it is preserved under ultraproducts.
Proof. `)': This follows from the Los Equivalence (cf. Section 2.1). `(': Consider a typical such sentence:
8P1 : : : 8Pn '(P1 ; : : : ; Pn ; R; =) (' rst-order): Clearly it is preserved under isomorphisms (and so is its negation). Moreover, its negation (a `11 -sentence') is preserved under ultraproducts (cf. [Chang and Keisler, 1973, Chapter 4.1], for the easy argument). So, given the assumption on the sentence itself, Keisler's Theorem (33) applies. COROLLARY 41. A modal formula is in M1 if and only if it is preserved under ultraproducts. A second application says that no generalisation of our topic is obtained by allowing arbitrary sets of de ning rst-order conditions. COROLLARY 42. If a modal formula has a -elementary de nition, it has an elementary de nition.
Proof. -elementary classes are closed under the formation of ultraproducts.
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This characterisation of M1 is rather aspeci c, as it holds for all 11 sentences. Later on, we will exploit the speci cally modal character of our formulas to do better. Moreover, the characterisation is rather abstract, as ultraproducts are hard to visualise. Therefore, we now turn to more concrete methods for separating formulas inside M1 from those outside. Formulas beyond M1: Compactness and Lowenheim{Skolem arguments. In practice, non rst-order de nability often shows up in failure of the Compactness and Lowenheim{Skolem theorems. The rst was involved in the example of Lob's Axiom, the second will be presented now. EXAMPLE 43 (McKinsey's Axiom). p ! p is outside of M1.
Proof. Consider the following uncountably in nite Kripke frame F = hW; Ri: cf
b1n
bn
0
bn
a W = R =
fag [ fbn; b0n ; b1n j n 2 N g [ fcf j f : N ! f0; 1gg fha; bni; hbn ; b0n i; hbn ; b1ni; hb0n ; b0ni; hb1n ; b1ni j n 2 N g[ fha; cf i j f : N ! f0; 1gg [ fhcf ; bfn(n)i j n 2 N; f : N ! f0; 1gg:
We observe two things. 1. F
p ! p:
Thanks to the presence of the re exive endpoints b0n ; b1n , the validity of the McKinsey Axiom is obvious everywhere, except for a. So, suppose that, under some valuation V; p is true at a. By assumption, p is true at each bn , and hence p is true at b0n or b1n . Now, pick any function f : N ! f0; 1g such that bfn(n) is a p-world (each n 2 N ). Then p holds at cf , and hence p at a. By the downward Lowenheim{Skolem theorem, F possesses a countable elementary substructure F 0 whose domain contains (at least) a; bn ; b0n; b1n (all n 2 N ). As F is uncountable, many worlds (cf ) must be missing in
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W 0 . Fix any one of these, say cf0 . Notice, for a start, that c1 f0 cannot be in W 0 either. (For, the existence of `complementary' c-worlds is rstorder expressible; and F 0 veri es the same rst-order formulas at each of its worlds as F .) Now, setting V (p) = fbfn0 (n) j n 2 N g will verify p at a, while falsifying p. Thus, we have shown 2. F 0 2 p ! p. We may conclude that the McKinsey Axiom is not rst-order de nable | not being preserved under elementary subframes. In practice, failure of Lowenheim{Skolem or compactness properties is an infallible mark of being outside of M1. The reader may also think this to be the case in theory, by the famous Lindstrom Theorem. (Cf. Volume 1, chapters by Hodges or van Benthem and Doets.) But there is a little-realised problem: the Lindstrom Theorem does not work for languages with a xed nite vocabulary (cf. [van Benthem, 1976]). In our case of R; =, there do exist proper extensions of predicate logic satisfying both the Lowenheim and compactness properties. These are not modal examples, however | and it may well be the case, for all we know, that a modal formula ' belongs to M1 if and only if the logic obtained by adding ' to the rst-order predicate logic in R; = as a propositional constant has the Lowenheim and compactness properties. Indeed, up till now, all unde nability arguments (including the above) have always been found reducible to compactness arguments alone. The nal characterisation of M1. Corollary 41 may be improved by noting the following fact about Kripke frames, connecting the modal and classical notions of Section 2.1. fF j i 2 I g. LEMMA 44. U Fi ! U i Thus, ultraproducts are generated subframes of suitable ultrapowers. A second idea comes from the preceding section: outside of M1, we encountered non preservation under elementary equivalence, a notion tied up with ultrapowers by the Keisler{Shelah Theorem (cf. [Chang and Keisler, 1973, Chapter 6.1]). We arrive at the main result of [van Benthem, 1976]. THEOREM 45. (i) A modal formula is in M1 if and only if (ii) it is preserved under ultrapowers if and only if (iii) it is preserved under elementary equivalence.
Proof. (i) ) (iii) ) (ii) are immediate. (ii) ) (i): If ' is preserved under ultrapowers, then, by Lemma 44, it is also preserved under ultraproducts | because disjoint unions preserve modal truth (Corollary 22). Now apply Corollary 41.
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Again, this insight saves us some spurious generalisations. Besides `elementary', there are two more levels in the de nability hierarchy elementary -elementary
-elementary - elementary
higher-order A -elementary class is de ned by an in nite disjunction of rst-order sentences (-elementary classes by in nite conjunctions). The prime example of this phenomenon is niteness. -elementary classes arise from in nite disjunctions of in nite conjunctions, or vice versa: both cases (and all purported `higher' ones) collapse | and the hierarchy stops here, even in classical logic. The reason lies in the simple observation that a class of frames is -elementary if and only if it is closed under elementary equivalence. But the preceding result has a COROLLARY 46. Modal formulas are either elementary, or essentially higher-order. Unfortunately, even this better characterisation does not yield much e ective information concerning the members of M1. For, there are no syntactic criteria for preservation under ultrapowers. From [van Benthem, 1983], we will cite the catalogue of what little we know. DIGRESSION 47. 1. 11 -sentences in R; = of the purely universal form 8P1 : : : 8Pm8x1 : : : 8xn' (' quanti er-free) are preserved under ultraproducts. This tells us that p ! p, i.e. 8P 8x(P x ! 8y(Rxy ! P y)) must be in M1: but that was clear without such heavy artillery. 2. 11 -sentences in R; = of the universal-existential form 8P1 : : : 8Pm9x1 : : : 9xn ' (' quanti er-free) are preserved under ultrapowers. This is of no help whatsoever, as modal formulas have at least one universal rst-order quanti er (8x). 3. Further presents will not be forthcoming: any 11 -sentence in R; = is logically equivalent to one of the form 8P1 : : : 8Pm 8x1 : : : 8xn 9y1 : : : 9yn ' (' quanti er-free) So, all complexity occurs at this level already.
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Thus, other ways are to be developed for describing M1 e ectively. The method of substitutions. There is a common syntactic pattern to many examples of rst- order de nable modal formulas: certain antecedents, in combination with certain consequents enable one to `read o ' equivalents. Starting from the earlier examples p ! p; p ! p, one may notice successively that conjunctions and disjunctions are admissible as well; as long as one avoids or (: : : _ : : :) combinations to the left. A typical instance is the following result from [Sahlqvist, 1975]: THEOREM 48. Modal formulas ' ! are in M1, provided that 1. ' is constructed from the forms p; p; p; : : : ; ?; >, using only ^; _ and , while 2. ' is constructed from proposition letters, ?; >, using ^; _; and .
This theorem accounts for cases such as
(p ^ q) ! (p _ p _ q) which de nes
8xy(Rxy ! 8z (Rxz ! (z = y _ Rzy _ Ryz ))): Proof. The heuristics of the Introduction works: for each `minimal veri cation' of the antecedent, the consequent must hold. For further technical information (e.g. the monotonicity of the consequent is vital too), cf. [van Benthem, 1976], which also contains generalisations of the theorem. That is fatal, is shown by the McKinsey Axiom. The Fine Axiom (p _ q) ! (p _ q) does the same for (: : : _ : : :). Finally, the Lob Axiom (in the equivalent form p ! (p ^ :p)) demonstrates the danger of `negative' parts in the consequent. Thus, in a sense, we have a `best result' here. Notice that the class described is rather typical for modal axioms, which often assume this implicational form. Indeed, the most characteristic modal axioms are even simply reduction principles of the form (modal operators) p ! (modal operators) p. THEOREM 49. A modal reduction principle is in M1 if and only if it is of one of the following four types: ~ ! : : : : : : p, 1. Mp 2. 3.
~ , : : : : : : p ! Mp ~ ! N~ Mp ~ : : : (i times) : : : Mp
(where length (N~ ) = i),
CORRESPONDENCE THEORY ~ ! : : : (i times) : : : Mp ~ 4. N~ Mp
355
(where length (N~ ) = i).
Proof. Cf. [van Benthem, 1976] for the rather laborious argument.
Thus at least, important parts of M1 have been classi ed. This particular theorem nishes a project begun in [Fitch, 1973]. A general method of proof for Theorem 48 consists of the method of substitutions, introduced in the introduction. Here we shall merely illustrate how it works: a justi cation may be found in [van Benthem, 1983]. EXAMPLE 50. Write p ! p as
8P 8x(9y(Rxy ^ 8z (Ryz ! P z )) ! 8u(Rxu ! 9v(Ruv ^ P v))): Rewrite this to the equivalent
8xy(Rxy ! 8P (8z (Ryz ! P z ) ! 8u(Rxu ! 9v(Ruv ^ P v)))): Substitute for P : z:Ryz , to obtain
8xy(Rxy ! (8z (Ryz ! Ryz ) ! 8u(Rxu ! 9v(Ruv ^ Ryv)))): This is equivalent to
8xy(Rxy ! 8u(Rxu ! 9v(Ruv ^ Ryv))); i.e. directedness (con uence). Write (p ^ q) ! (p _ p _ q) as
8xy(Rxy ! 8P ((P y ^ 8z (Ryz ! Qz )) ! 8u(Rxu ! (P u_ _9v(Ruv ^ P v) _ Qu)))): Substitute for P : zy = z , and for Q : z:Ryz , to obtain (an equivalent of) the earlier connectedness. Write (p ^ p) ! p as
8xy(Rxy ! 8P ((P y ^ 8z (Ryz ! P z )) ! P x)): Substitute for P : z y = z _ Ryz , to obtain (an equivalent of) 8xy(Rxy ! (Ryx _ y = x)): Write p ! p as 8x8P (8y(Rxy ! 8z (Ryz ! P z )) ! 8u(Rxu ! P u)):
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Substitute for P : z R2 xz ; i.e. z 9v(Rxv ^ Rvz ), to obtain (modulo logical equivalence)
8x8u(Rxu ! 9v(Rxv ^ Rvu)); i.e., density of the alternative relation. In general, substitutions will be disjunctions of forms Rn yz (n = 0; 1; 2; : : :); the cases 0, 1 standing for =; R, respectively. Despite these advances, the range of the method of substitutions has it limits. To see this, here is an example of a formula in M1 with a quite di erent spirit. EXAMPLE 51. The conjunction of the K4.1 axioms, i.e. p ! p, p ! p is in M1.
Proof. p ! p de ned transitivity and, therefore, it suÆces to prove the following Claim. On the transitive Kripke frames, McKinsey's Axiom de nes atomicity: 8x9y(Rxy ^ 8z (Ryz ! z = y)): From right to left, the implication is clear. From left to right, however, the argument runs deeper. Assume that F is a transitive frame, containing a world w 2 W such that
8y(Rwy ! 9z (Ryz ^ z 6= y)): Using some suitable form of the Axiom of Choice (it is as serious as this . . . ), nd a subset X of w's R-successors such that 1. 8y 2 W (Rwy ! 9z 2 XRyz ) 2. 8y 2 W (Rwy ! 9z 2 (W
X )Ryz ). Setting V (p) = X then falsi es the McKinsey Axiom at w.
This complexity is unavoidable. We can, for example, prove THEOREM 52. (p ! p) ^ (p ! p) is not equivalent to any conjunction of its rst-order substitution instances.
Proof. Here is where the earlier general frame hN; , nite and co nite setsi comes in. First, an ordinary model-theoretic Observation. The nite and co nite sets of natural numbers are precisely those rst-order de nable in hN; i, possibly using parameters. Now, it was noticed already in Section 2.1 that the above formula holds in this general frame | and hence so do all its rst-order substitution
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instances. But the latter also hold in the full frame hN; i. So, if our formula were de ned by them, it would also hold in the full frame: which it does not. So, although he method of substitutions carves out a large, and important part of M1, it does not fully describe the latter class. The complexity of M1. The method of substitutions describes a part of M1 which may even be shown to be recursively enumerable (cf. [van Benthem, 1983]). But M1 over owed its boundaries. Indeed, there are reasons to believe that M1 is not recursively enumerable | probably not even arithmetically de nable. For, in the general case of 11 -sentences, we know THEOREM 53. First-order de nability of 11 -sentences is not an arithmetical notion.
Proof. (Cf. [van Benthem, 1983] or the Higher Order Logic Chapter in Volume 1 of this Handbook.) Other topics. Various other questions had to be omitted here. At least, one example should be mentioned, viz. that of relative correspondences. On several occasions, a restriction to transitive Kripke frames produced interesting shifts: global and local rst-order de nability collapse, the McKinsey Axiom becomes elementary, etc. A sample result is in [van Benthem, 1976]. THEOREM 54. On the transitive Kripke frames, all modal reduction principles are rst-order de nable. Thus, `pre-conditions' on the alternative relation are worth considering. In areas such as tense logic, our temporal intuitions even require them.
2.3 Modal Algebra An alternative to Kripke semantic structures is o ered by so-called `modal algebras', in which the modal language may be interpreted as well. The realm of modal algebras has its own mathematical structure, with subalgebras, direct products and homomorphic images as key notions. Now, backand-forth connections may be established between these two realms, through the Stone Representation. A categorial parallel emerges between the above triad of notions and the basic triad of Section 2.1: zigzag-morphic images, disjoint unions and generated subframes, respectively. Moreover, the earlier `possible worlds construction' for ultra lter extensions will be seen to arise naturally from the Stone Representation. The algebraic perspective. As in other areas of logic, the modal propositional language may also be interpreted in algebraic structures. These assume the
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form of a Boolean Algebra (needed to interpret the propositional base) enriched with a unary operation, in order to capture the modal operator. DEFINITION 55. A modal algebra is a tuple A = hA; 0; 1; +;0 ; i; where hA; 0; 1; +;0 i is a Boolean Algebra and is a unary operator satisfying the equations 1. (x + y) = x + y 2. 0 = 0.
Notice that corresponds to possiblity (): the necessity choice would have yielded equations 10 . (x y) = x y 20 . 1 = 1. This algebraic perspective at once yields a completeness result. THEOREM 56. A modal formula is derivable in the minimal modal logic K if and only if it receives value 1 in all modal algebras under all assignments. The concept of evaluation at the back of this goes as follows. Let V assign A-values to proposition letters. Then, V may be lifted to all formulas through the recursive clauses V (:') = V (')0 V (' _ ) = V (') + V ( ) V (') = V (') ; etc. Thus, a modal formula is read as a `polynomial' in 0 ; +; . The proof of the completeness Theorem 56 comes cheap. First, one shows by induction on the length of proofs that all K-theorems are `polynomials identical to 1'. Conversely, one considers the so-called Lindenbaum Algebra of the modal language, whose elements are equivalence classes of Kprovably equivalent modal formulas, with operations de ned in the obvious way through the connectives. The value 1 in this algebra is awarded to all and only the K-theorems: hence non- theorems are disquali ed as polynomials identical to 1. Such uses of modal algebra are a joy to some (cf. [Rasiowa and Sikorski, 1970]); to others they show that the algebraic approach is merely `syntax in disguise'. After all, the above result may be viewed as a re-axiomatisation of K, no more. For instance, notice that the hard work in the usual (Henkin type) model-theoretic completeness theorems consists in showing that nontheorems can be refuted in set-theoretic (Kripke)-models. To put this into a slogan, which will become fully comprehensible at the end of this chapter:
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HENKIN = LINDENBAUM + STONE. Nevertheless, the algebraic perspective has further uses, which are being discovered only gradually. First, notice that it o ers a more general framework than Kripke semantics. For the above Lindenbaum construction to work, one only needs the principle of Replacement of Equivalents; i.e. modally, closure under the rule if
`'$ ;
then
` ' $ :
(Algebraically, this just amounts to an identity axiom.) The above additional equations represent optional further choices. But even in the realm of the above modal algebra, there exists a whole discipline of universal algebraic notions and results, which turn out to be applicable to modal logic in surprising ways. Two instructive references are [Goldblatt, 1979] and [Blok, 1976]. Here we shall only skim the surface, taking what is needed for the modal de nability results of Section 2.4. Thus, we shall need the following three fundamental algebraic notions. DEFINITION 57. A1 is a modal subalgebra of A2 if A1 A2 , and the operations of A2 coincide with those of A1 on A1 . DEFINITION 58. The direct product fAi j i 2 I g of a family of modal algebras fAi j i 2 I g consists of all functions in the Cartesian product fAi j i 2 I g, with operations de ned component-wise: f + g = (f (i) +i g(i))i ; f = (f (i) )i ; etc. i
DEFINITION 59. A function f is a homomorphism from A1 to A2 if it respects all operations: f (a +1 b) = f (a) +2 f (b); f (a1 ) = f (a)2 ; etc. These three operations are fundamental in algebra because they characterise algebraic equational de nability. This is the content of `Birkho 's Theorem': A class of algebras is de ned by the validity of a certain set of algebraic equations (under all assignments) if and only if that class is closed under the formation of subalgebras, direct products and homomorphic images. (For a proof, cf. [Gratzer, 1968].) There is much more to Universal Algebra, of course, but this is what we shall need in the sequel. Kripke frames induce modal algebras. In order to tap the above resources, a systematic connection is needed between the earlier semantic structures and modal algebras. To begin with, each Kripke frame F = hW; Ri gives rise to the following modal algebra A(F ) = hP (W ); ?; W; [; ; i
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where is the modal projection of 2.1:
(X ) = fw 2 W
j 9v 2 XRwvg (X W ):
As for truth of modal formulas, it is immediate that a modal formula ' is true in F if and only if its corresponding modal equation a(') is identical to 1 in the algebra A(F ). For instance, truth of
(p _ q) ! (p _ q); or equivalently
:::(p _ q) _ (::p _ ::q)
is equivalent to the validity of the identity 0 0 0 0 0 0 0 (x + y) + (x + y ) = 1:
Thus, A maps single Kripke frames to modal algebras. But what happens to the characteristic modal connections between frames, as in Section 2.1? We shall take them one by one. First, if F1 is a generated subframe of F2 , then the obvious restriction map sending X W2 to X \ W1 is a modal homomorphism from A(F2 ) onto A(F1 ). (The key observation is that R2 -closure of W1 guarantees homomorphic respect for the projection operator .) Next, the algebra induced by a disjoint union fFi j i 2 I g is isomorphic, in a natural way, to the direct product fA(Fi ) j i 2 I g. One simply associates a set X of worlds in the former with the function (X \ Wi )i2I . Finally, and this happy ending will be predictable by now, if F2 is a zigzag-morphic image of F1 through f , then the stipulation
A(f )(X ) =def f 1 [X ] de nes an isomorphism between A(F2 ) and a subalgebra of A(F1 ). (This time, the two relational clauses in the de nition of `zigzag morphism' ensure that A(f ) respects projections.) Notice the reversal in direction in the latter case: this is a common phenomenon in these `categorial connections'. Modal algebras induce Kripke structures. There is a road back. Conversely, modal algebras may be `represented' as if they had come from an underlying base frame. The idea of this so-called Stone Representation is as follows. (It is due to Jonsson and Tarski around 1950.) Worlds w are to be created such that an element a in the algebra may be thought of as the set of w `in a'. But then, the desired correspondence between algebraic and set-theoretic operations becomes: no set w is in 0, all sets w are in 1; w is in a + b i w is in a or w is in b; w is in a0 i w is not in a:
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Thus, as w searches through A `where it belongs', it picks out a set X such that 0 62 X; 1 2 X; a + b 2 X i a 2 X or b 2 X; a0 2 X i a 62 X: Such sets X are called ultra lters on A. Thus, let
W (A) = all ultra lters on A: A suitable alternative relation may be found through the same motivation as in Section 2.1. hw; vi 2 R(A) i for each a 2 A; if a 2 v; then a 2 w: So, each modal algebra A induces a Kripke frame
F (A) = hW (A); R(A)i: This time, truth in A and truth in F (A) need not correspond, however. For, F (A) may harbour many more sets of worlds than just those corresponding to the elements a of the algebra | and hence it contains additional potential falsi ers. Thus, the implication goes only one way. The equation t1 = t2 is valid in A, where the polynomials t1 ; t2 correspond to the modal formulas '1 ; '2 , when '1 $ '2 is true in F (A). A complete equivalence is only restored by changing F (A) to the general frame
F (A) = hW (A); R(A); W(A)i; where W(A) consists of all sets of the form
fw 2 W (A) j a 2 wg (a 2 A): So, what we now get is a two-way connection between modal algebras and general frames | and here lies the genesis of the latter notion. Two ways; for, it is easily seen that all previous insights about the mapping A apply equally well to general frames, instead of merely `full' frames. Again, the interest of the present connection may be gauged by seeing what happens to the three fundamental algebraic operations when translated through F into Kripke-semantic terms. First, if A1 is a modal subalgebra of A2, then the obvious restriction map sending ultra lters w on A2 to ultra lters w \ A1 on A1 is a zigzag morphism from F (A2 ) onto F (A1 ). Next, the direct product of a family fAi j i 2 I g has an F -image containing the disjoint union fF (Ai ) j i 2 I g. No isomorphism need obtain, however: a slight aw in our correspondence.
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But nally, if A2 is a homomorphic image of A1 through f , then the map F (f ), de ned by setting
F (f )(w) =def f 1 [w]; sends A2 -ultra lters to A1 -ultra lters, in such a way that it embeds F (A2 ) isomorphically as a generated subframe of F (A1 ). Back and forth. So far, so good. Modal algebras induce general frames, and these, in their turn, induce modal algebras. But, what happens on a return-trip? One case is simple, by construction: THEOREM 60. A(F (A)) is isomorphic to A. The converse direction is more diÆcult. (F (A(G)) need not be isomorphic to F , for general frames G. This is precisely what we noted in connection with `possible world constructions' in Section 2.1. But, as was announced there, it can be ascertained which conditions on general frames G do guarantee such an isomorphism. DEFINITION 61. A general frame G = hW; R; Wi is descriptive if it satis es Leibniz' Principle for identity:
1. 8xy 2 W (x = y $ 8Z 2 W(x 2 Z $ y 2 Z )) as well as Leibniz' Principle for alternatives: 2. 8xy 2 W (Rxy $ 8Z 2 W(y 2 Z ! x 2 (Z ))): Moreover, it should satisfy Saturation: 3. each subset of W with the nite intersection property has a non-empty total intersection. The following basic result is in [Goldblatt, 1979]. THEOREM 62. F (A(G)) is isomorphic to G if and only if G is descriptive. The standard examples of descriptive frames are the general frames derived from Henkin models in modal completeness proofs, by taking for W the range of modally de nable sets of worlds. It may also be noticed that general frames G which are themselves of the form F (A) are always descriptive. Thus, for certain theoretical purposes, the `proper' bijective correspondence may be said to be that between modal algebras and descriptive frames, which are `stable' under the possible worlds construction described in Section 2.1. The categorial connection. The above connections between modal algebras and Kripke structures run deeper than might appear at rst sight. The
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general picture is that of two mathematical worlds, or `categories', which turn out to be quite similar in structure:
hModal algebras, homomorphisms intoi hGeneral frames, zigzag morphisms intoi: The earlier considerations may be summed up in the following two schemata:
G1 A(G1 )
f A(f )
G2
A1
A(G2 )
F (A1 )
f F (f )
A2 F (A2 )
So, A; F are what a category theorist would call `contravariant' functors. Therefore, information concerning the one category may sometimes be transferred to the other. Thus, a `categorial transfer' arises, of which we mention a few phenomena. (The following passage can be skipped by readers unfamiliar with Category Theory or Universal Algebra). The category of modal algebras has among its internal limit constructions the formation of terminals (viz. the degenerate single point algebras) and pull-backs. Hence, it is closed under nite limits in general. Through A; F , we may derive that the category of general frames is closed under nite co-limits, speci cally under initials (allowing the empty frame) and push-outs. (In this connection, the `adjointness' behaviour of A; F may be investigated.) The preservation behaviour of modal formulas under such limit constructions remains to be studied. An algebraically well-motivated notion is that of a free algebra. What corresponds to these in the realm of general frames? A surprising connection with modal completeness theory appears. The Stone representations of free algebras are essentially Henkin general frames (proposition letters correspond to free generators of the algebra). The latter structures were characterised semantically in [Fine, 1975], in terms of certain `universal embedding' properties with respect to zigzag morphisms. This turns out to follow directly, as the dual of the `homomorphic extension' de nition of free algebras. Our nal example concerns another algebraic classic, the notion of a subdirectly irreducible modal algebra (used with great versatility in [Blok, 1976]). These turn out to correspond almost (not quite) to rooted general frames whose domain consists of one root world together with its Rsuccessors, their R-successors, etcetera. The famous Birkho Theorem stating that Every (modal) algebra is a subdirect product of subdirectly irreducibles,
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may then be compared with the simple Kripke-semantic observation that Every general frame is a zigzag-morphic image of the disjoint union of its rooted generated subframes. These examples will have made it clear how the categorial connection between modal algebra and possible worlds semantics can be a very rewarding perspective.
2.4 From Classical to Modal Logic Reversing the direction of the earlier correspondence study (Section 2.2), there arises DEFINITION 63. P1 is the set of all rst-order sentences in R; = for which a modal formula exists de ning the same class of Kripke frames. All earlier examples of formulas in M1 also provide examples for P1, of course. Therefore, here are some more general results straightaway. Some methods exist for proving the existence of modal de nitions. THEOREM 64. Each rst-order sentence of the form 8xU', where U is a (possibly empty) sequence of restricted universal quanti ers, of the form
8u(Rvu !
(with u; v distinct)
followed by a matrix ' of atomic formulas u = v; Ruv combined through ^; _, belongs to P1.
Proof. The relevant combinatorial argument is based on the heuristics explained in the introduction. Cf. [van Benthem, 1976]. Some examples of formulas of this type are re exivity: 8xRxx; transitivity: 8x8y(Rxy ! 8z (Ryz ! Rxz )) and
connectedness: 8x8y(Rxy ! 8z (Rxz ! (Rzy _ Ryz ))):
Disproving de nability proceeds through counter-examples to preservation behaviour. EXAMPLE 65.
1. 9xRxx is outside of P1. It holds in hf0; 1g; fh1; 1igi; but not in its generated subframe hf0g; ?i. 2. 8x8yRxy is outside of P1.
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It is preserved under generated subframes, but not under disjoint unions. On hf0g; fh0; 0igi and hf1g; fh1; 1igi, the relation is universal; but not on hf0; 1g; fh0; 0i; h1; 1igi. 3. 8x:Rxx is outside of P1. It is preserved under generated subframes and disjoint unions; but not under zigzag-morphic images, witness the Introduction. 4. 8x9y(Rxy ^ Ryy) is outside of P1. It is preserved under all three operations mentioned up till now, but not inversely under the formation of ultra lter extensions. It can be shown to hold in ue(hN;
Proof. This argument is given in heuristic outline here, as it is one of the most elegant applications of algebraic results in modal semantics. Evidently, modally de nable classes of Kripke frames exhibit all the listed closure phenomena: the surprising direction leads from `closure' to `de nability'. First, notice that one closure condition can be added for free, by an earlier result. Theorem 30 implies that our class R of frames is itself closed under the formation of ultra lter extensions: if F 2 R, then the relevant elementary equivalent F 0 2 R (R being elementary), and hence so is its zigzag-morphic image ue(F ). Now the obvious strategy is to show that R equals MOD(Thmod(R)), i.e. the class of Kripke frames verifying each modal formula which is valid throughout R. The nontrivial inclusion here requires us to show that if F Thmod (R); then F 2 R; for every Kripke frame F : And here is where an excursion into the realm of modal algebra will help. F veri es Thmod (R), and hence A(F ) veri es the equational theory of the class fA(G) j G 2 Rg. (Recall the earlier correspondence between modal formulas and polynomials.) By Birkho 's Theorem, in a suitable version, this implies that A(F ) must be constructible as a homomorphic image of some subalgebra of some direct product fA(Gi ) j i 2 I g, with Gi 2 R. In a picture,
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A(F )
surjective homomorphism
A fA(Gi ) j i 2 I g:
Now the latter algebra is isomorphic to A(fGi j i 2 I g), by the earlier duality. Moreover, the latter disjoint union belongs to R | by the given closure conditions. So, the picture becomes, for some G 2 R:
A(F )
surjective homomorphism
A A(G):
Now, the transformation F turns this into the corresponding row
F A(F )
embedding as generated subframe
F (A)
surjective zigzag morphism
F A(G):
But then, nally, the following walk through the diagrams suÆces. G 2
R ) F A(G) = ue(G) 2 R (by the above observation) ) F (A) 2 R (closure under zigzag images) ) F A(F ) 2 R (closure under generated subframes) ) F 2 R (`anti-closure' under ultra lter extensions). Actually, this result does not yet characterise P1, as it talks about modal de nability by any set, nite or in nite. The additional strengthenings needed for zeroing in on P1 are hardly enlightening, however. The result also says a little bit more. Adding closure under ultra lter extensions, while removing the condition of elementary de nability, yields a characterisation of those classes of Kripke frames de nable by means of a canonical modal logic in the sense of the Introduction (i.e. one which is complete with respect to its Henkin frames). Moreover, the above proof heuristics may also be used to formulate a general closure condition on classes of Kripke frames necessary and suÆcient for de nability by means of just any set of modal formulas (`SA-constructions'; cf. [Goldblatt and Thomason, 1974]). As with the earlier ultrapower characterisation of M1, the above characterisation gives no e ective information concerning the formulas in P1. What is needed are `preservation theorems' giving the syntactic cash value of the given four closure conditions. Several of these have been given in [van Benthem, 1976], extending earlier results, e.g. of Feferman and Kreisel. Here is an idea. Preservation under generated subframes allows only formulas constructed from atomic formulas and their negations, using
8; ^; _ as well as restricted existential quanti ers 9v(Ruv^ (u; v distinct).
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Preservation under disjoint unions admits only one single universal quanti er in front: all others are to be restricted to the form 8v(Ruv !). Finally, preservation under zigzag images forbids the negations, and we are left with THEOREM 67. A rst-order sentence is preserved under the formation of generated subframes, disjoint unions and zigzag-morphic images if and only if it is equivalent to one of the form 8x (x), where (x) has been constructed from atomic formulas using only conjunction, disjunction and restricted quanti ers.
Proof. By elementary chain constructions, as in [Chang and Keisler, 1973, Chapter 3.1]. For preservation under ultra lter extensions, only some partial results have been found. (After all, the class of sentences preserved under such a complex operation need not even be e ectively enumerable.) As for the total complexity of P1, this may well be considerable | as was the case with M1. Are the two classes perhaps recursive in each other?
2.5 Modal Predicate Logic As in much technical work in this area, modal propositional logic has been studied up till now. Modal predicate logic, however important in philosophical applications, is much less understood. (Cf. Chapter 2.5 in this Handbook.) Nevertheless, in the case of Correspondence Theory, an excuse for the neglect may be found in Theorem 69 below. The un nished state of the art shows already in the fact that no commonly accepted notion of semantic structure, or truth de nition exists. Hence, we x one particular, reasonably motivated choice as a basis for the following sketch of a predicate-logical variant of the earlier theory. The language is the ordinary one of predicate logic, with added modal operators. Structures are tuples
M = hW; R; D; V i; where the skeleton hW; R; Di is a Kripke frame with a domain function D assigning sets of individuals Dw to each world w 2 W . The valuation V supplies the interpretation of the nonlogical vocabulary at each world. The truth de nition explicates the notion `'(x) is true in M at w for d'; where the sequence d assigned to the free individual variables x comes from Dw . Its key options are embodied in the clauses for the individual quanti ers: these are to range over Dw , plus that for the modal operator:
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'(x) is true at w for d if, for each R-alternative v for w such that d is in Dv ; '(x) is true at v for d.
Thus, necessity means `truth in all alternatives, where de ned'. As before, truth in a skeleton (at some world, for some sequence of individuals) means truth under all possible valuations. Again, in this way modal axioms start expressing properties of R; D | and their interplay. The relevant matching `working language' on the classical side will now be a two-sorted one: one sort for worlds, another for individuals. Its basic predicates are the two sortal identities, R between worlds, as well as the sort-crossing Exw : `x is in the domain of w', or `x exists at w'. EXAMPLE 68. The Barcan Formula 8xAx ! 8xAx de nes
8wv(Rwv ! 8x(Exv ! Exw)): Proof. `(': Assume 8xAx at w, and consider any R-alternative v. For all d 2 Dv ; d 2 Dw (by the given condition), whence Ad holds at w | and, hence, Ad holds at v. `)': The Barcan Formula will hold under the following particular assignment: Vu (A; d) = 1 if Rwu and d 2 Dw . This V veri es the antecedent, and hence the consequent. The relational condition follows. Thus, the Barcan Formula expresses an interaction between R and D. This is not accidental. For pure R-principles, we have the following conservation result. THEOREM 69. There exists an e ective translation from sentences ' of modal predicate logic to formulas p(') of modal propositional logic such that, if ' is equivalent to some pure R; =-sentence , then p(') already de nes in the sense of Section 2.2.
Proof. p merely crosses out quanti ers in some suitable way. For full details (here and elsewhere) cf. [van Benthem, 1983]. Besides the Barcan Formula, there are three further fundamental `de re/de dicto interchanges'. One of these provides a new example of non- rstorder de nability. EXAMPLE 70. 1. 2.
8xAx ! 8xAx is universally valid, 9xAx ! 9xAx de nes 8wv(Rwv ! 8x(Exw ! Exv)),
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9xAx ! 9xAx de nes an essentially higher-order condition on R; = ; E.
Despite the super cial resemblance to the McKinsey Axiom of section 2.2., the proof for the latter result is quite di erent from that of Example 43. Interested readers may notice that the above principle holds in worlds with a nite chain of overlapping two-element successors: f1; 2g; f2; 3g; f3; 4g; : : :; fn 1; ng; fn; n + 1g: But, it may fail in the presence of in nite such chains, and then compactness lurks. Further systematic re ection on the above `positive' result yields a method of substitutions again, with an outcome like that of Theorem 48: THEOREM 71. Formulas of the form ' ! , with ' constructed from atomic formulas pre xed by a (possibly empty) sequence of 8; , using only ^; _; 9 and , and constructed from atomic formulas using ^; _; 9; as well as 8; , are all uniformly rst-order de nable. The global mathematical characterisation of rst-order de nability remains essentially the same in this area, whence it is omitted here. Something which does not generalise easily, however, is the algebraic approach of Section 2.3. This is an endemic problem in classical (and intuitionistic) logic already: elegant algebraization stops at the gates of predicate logic. There could be an area of `modal cylindric algebra' of course (cf. [Henkin et al., 1971]), but none exists yet. (For an interesting related area, cf. the extension of modal propositional algebra to the modal program algebra of dynamic logicians (cf. [Kozen, 1979] or the Dynamic Logic chapter in volume 5 of this Handbook).) As a consequence, we still lack an elegant characterisation of the modally de nable fragment of the present two-sorted rst-order language. What we do have, however, is such a characterisation for that same language with parametrised predicate constants A(w; ) for the predicate constants A( ) of the modal predicate logic. Thus, this is the appropriate language for the rst-order transcription of the above truth de nition. The Barcan Formula, for example, becomes 8x(Exw ! 8v((Ewv ^ Exv) ! Avx)) ! ! 8v(Rwv ! 8x(Exv ! Avx)): As in Theorem 18, two characteristic modal relations suÆce for characterising the modal transcriptions among the class of all formulas of this language. In order to end on an optimistic note, here is the relevant result. First, modal predicate logic knows generated submodels, just as in Section 2.1. Moreover, the earlier zigzag relations may be enriched so as to incorporate individual back-and-forth choices, as in the Ehrenfeucht{Frasse approach to rst-order de nability.
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DEFINITION 72. A zigzag connection C between two models M1 ; M2 relates nite sequences (w; x) of equal length (w a world, x a sequence of individuals in the domain of w) in such a way that 1. all such sequences occur: those from M1 in the domain, those from M2 in the range of C 2. if C (w; x)(v; y) and w0 2 W1 , with R1 ww0 ; x 2 Dw , then C (w0 ; x)(v0 ; y) for some v0 2 W2 with R2 vv0 ; y 2 Dv0 , and analogously in the opposite direction (`world zigzag') 3. if C (w; x)(v; y) and d 2 Dw , then C (w; x d)(v; y e) for some e 2 Dv , and vice versa. 4. if C (w; x)(v; y), then the map (x)i between hDw ; Vw i and hDv ; Vv i.
! (y)i
(`individual zigzag') is a partial isomorphism
Now, transcriptions of modal formulas are invariant for generated submodels and zigzag connections, in the obvious sense. E.g. the latter have been made precisely in such a way that for modal ',
' is true at w for x
i
' is true at v for y,
when C (w; x)(v; y):
THEOREM 73. A formula ' = '(w; x) of the two-sorted world/individual language is (equivalent to the transcription of) a modal formula if and only if it is invariant for generated submodels and zigzag connections.
Proof. This follows from the main proof in [van Benthem, 1981b].
On the whole, exciting technical results are yet scarce in modal predicate logic | and Correspondence Theory is no exception.
2.6 Higher-Order Correspondence Modal formulas de ne second-order (11 ) conditions on the alternative relation in all cases, and rst-order conditions in some. In the perspective of abstract model theory, two possible generalisations arise here. Instead of the rst-order target language, one may consider suitable extensions. For instance, in Theorem 37, the relevant relational condition was de nable in L!1 ;! : rst-order logic with countable conjunctions and disjunctions. Not all modal formulas become de nable here, however. E.g. Lob's Axiom de ned a form of well-foundedness, which is known to be beyond L!1 ! , or indeed any language of the L1! -family. On the other hand, this time for instance, the de ning condition is already in `weak secondorder logic' L2 , allowing quanti cation over nite sets of individuals. Thus,
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various wider classes of de nability could be considered for modal formulas, short of 11 . And, in fact, even the latter case itself is interesting. Which 11 -sentences, for example, admit of modal de nitions? Given the general lack of semantic characterisations for such higher logics, such characterisations for their modal fragments are also diÆcult to obtain. One observation might be that both L!1 ! and L2 have the property of invariance for partial isomorphism (cf. van Dalen's chapter in Volume 1 of this Handbook). It will be of interest to study this preservation condition on modal formulas. In fact, no counter-examples have been discovered yet; but these do exist in tense logic. (The rationals hQ;
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JOHAN VAN BENTHEM
. Does the second-order prenex hierarchy induce an ascending corresponding hierarchy of modally de nable rst-order principles about the alternative relation? This possibly ascending hierarchy cannot exhaust all rst-order principles, as higher-order modal formulas do retain one basic preservation property: their local truth is invariant under passing to generated subframes. (The Generation Theorem 15 yields this consequence all the way up, not just for the original modal 11 -formulas.) But then, we know what this semantic constraint means in syntactic terms for rst-order formulas (cf. [van Benthem, 1976, Chapter 6]). These will be the `almost-restricted' ones consisting of one universal quanti er followed by a compound of atomic formulas with negation, conjunction and restricted quanti ers 9y(Rxy^). The other preservation properties of Section 2.1 are lost, however. As was observed earlier, irre exivity (8x:Rxx) becomes de nable and, hence, preservation under zigzag morphisms fails. Anti-preservation under ultra lter extensions fails, because the earlier example 8x9y(Rxy ^ Ryy) becomes de nable as well. (A straightforward de nition uses a propositional quanti er within a modal scope: 8p(p ! p). But there is a nonembedded substitute in the form of 9p(p ^ 8q(p ! (q ! q))).) Thus, we arrive at the following Question. Can every almost-restricted rst-order formula 8x'(x) be de ned at some level in the modal propositional quanti er hierarchy? Using `simulation' of restricted rst-order quanti cation by propositional quanti ers, one may indeed handle most obvious cases. Here is one illustration of the procedure Example. Let '(x) be 9y (Rxy ^ 8z (Ryz ! (Rzz _ (Rzy ^ Rzx)))). The idea is to de ne fxg; fyg; fz g, in a sense, as far as necessary (i.e. on the set consisting of x, its R-, R2- and R3 -successors) | and then to express all desired relations between these by means of modal formulas: Query
9px (px ^ 8qx(((px ^ qx) _ (px ^ qx ) _ (px ^ qx ) _ (px ^ qx )) ! ((px ! qx ) ^ (px ! qx) ^ (px ! qx ))) [this makes px unique to the extent indicated] ^ 9py (py ^ [same uniqueness statement] ^ 8pz (((py ^ pz )^ [same uniqueness statement]) ! (8qz (pz ! (qz ! qz ))[i.e. `Rzz '] _ (pz ^ px ^ py )[i.e. `Rzy ^ Rzx'])))). Accordingly, our conjecture is that the above question has a positive answer. We conclude with one further Question. Does the addition of propositional quanti ers within modal scopes add any power of expression?
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3 OTHER INTENSIONAL NOTIONS Modal logic is only one branch, be it a paradigmatic one, of intensional logic in general. But also in other intensional areas, a Correspondence Theory is possible. In some cases, the generalisation runs smoothly: existing notions and results may be applied at once, or after only minor modi cation. A case in point is tense logic, to be treated in Section 3.1. More challenging generalisations arise when the relevant intensional semantics exhibit strong peculiarities, diverging from the earlier modal case. Sometimes, these assume the form of pre-conditions on the alternative relation; but maybe the most important hurdle is when a restriction is proclaimed on `admissible assignments'. Both phenomena occur in conditional logic, the topic of Section 3.2. That, even under such circumstances, an interesting Correspondence Theory may remain, is shown by the example of intuitionistic logic in Section 3.3. These two new features do not exhaust the possible semantic variation. One may also move to the interplay of di erent kinds of intensional operators, for instance, using correspondence to connect di erent alternative relations. Example. In dynamic logic, two modal operators ; gure, which may be provided with two alternative relations R; R . (Recall that a means `after every successful computation of a', while the intuitive meaning of a is to be: `after any nite number of runs of a'.) Now, from a correspondence point of view, the well-known Segerberg Axioms p ! p p ! p (p ! p) ! (p ! p) de ne precisely the condition that R coincides with the transitive closure of R. The very exoticness of this example to many readers may help to show that Correspondence Theory is omnipresent. No systematic developments will be given in the following sections. Their purpose is to convey an impression of notions and themes, through mainly illustrative examples. Indeed, here is where the reader may wish to carry on the torch herself.
3.1 Tense Logic Traditionally, tense-logical structures have been taken to be temporal orders hT;
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JOHAN VAN BENTHEM
been that of Prior, adding operators G (`it is always going to be'), H (`it has always been') to some propositional base. We add F (`future'), P (`past') as derived notions. (Cf. the chapter on Basic Tense logic in volume 6 for the necessary background in tense logic.) Of the amazing diversity of `ontological' and `linguistic' questions concerning this temporal semantics, only a few themes will be mentioned here. (Cf. [van Benthem, 1985] for a varied exploration.) Explaining philosophical dicta. In his famous paper `The Unreality of Time', the philosopher McTaggart enunciated several temporal principles. One of these reads [McTaggart, 1908]: \If one of the determinations past, present and future can ever be applied to (an event), then one of them has always been and always will be applicable, though of course not always the same one." When translated into Priorean axioms, this becomes a list: 1. P q ! H (F q _ q _ P q) 2. P q ! GP q 3. q ! HF q 4. q ! GP q 5. F q ! HF q 6. F q ! G(F q _ q _ P q). What do these principles mean? The answer may be obtained through the method of substitutions ( tted to the temporal case | but such generalisations will be presupposed tacitly henceforth). EXAMPLE 74. 1. de nes left-connectedness: 8x8y < x8z < x(y < z _ z < y _ y = z ); 2. de nes transitivity: 8x8y < x8z > x y < z , 3. de nes >, 4. de nes >. If G; H had been interpreted through di erent relations x8z < x z < y,
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6. de nes right-connectedness: 8x8y > x8z > x(y < z _ z < y _ y = z ). Thus, the McTaggart temporal picture is one of linear ow. An incompleteness theorem. Simple transfer of earlier modal results establishes the seminal incompleteness result of [Thomason, 1972], in a very simple version. THEOREM 75. The tense logic axiomatised by
H (Hp ! p) ! Hp (Lob's Axiom) GF p ! F Gp (McKinsey Axiom) is incomplete.
Proof. Speci cally, this logic holds in no frame | and yet it is not inconsistent. First, as to the former statement, recall from Section 2.2 that 1. Lob's Axiom de nes transitivity of > and well-foundedness of <. By the former, < is transitive as well (transitivity is `independent of the temporal direction', or isotropic (cf. [van Benthem, 1985])). Thus, in this special case, Example 51 applies, and we have 2. McKinsey's Axiom de nes atomicity: 8x9y > x8z > y z = y. A consequence of the latter property is 8x9y > x y < y (cf. Example 65(4)). So, the temporal order must contain instantaneous loops : : : < y < y < y < : : :, which contradicts well-foundedness. Therefore, our logic holds in no frame. Nevertheless, it does hold in a general frame, viz. an earlier example from Section 2.1: hN; <; Wi, with
W = fX N j X is nite or N X is niteg: The reason was that refutations for the McKinsey Axiom are no longer `admissible', as these involve in nite alterations. (Thomason gives a speculation at this point concerning the Second Law of Thermodynamics: `event patterns stabilise'.) But then, the logic cannot be inconsistent: its Ktheorems hold in all general frames where it is valid. Tense-logical axioms for the temporal order. In [van Benthem, 1985], the following fundamental axioms are derived for any temporal order induced by a comparative (in the linguistic sense) `earlier than'.
1. irre exivity: 8x :x < x
(`no vortices in Time')
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2. transitivity: 8x8y > x8z > y z > x 3. almost-connectedness: 8x8y > x8z (x < z _ z < y)
(` ow')
(`arrows are comparative yard sticks')
A version of the latter principle may also be found as the key axiom in Leibniz' relational theory of Space-Time (cf. [Winnie, 1977]). Which tense-logical axioms correspond? From Section 2.4, we know that (1) is unde nable, (2) yields Gp ! GGp, while (3) just fails to fall under Theorem 67. What the latter result does give is a correspondence between
8x8y > x8z > y8u > x(y < u _ u < z ) and
(F (p ^ F q) ^ F r) ! (F (p ^ F r) _ F (r ^ F q)): Another example concerns particular temporal orders. One can never hope to fully de ne such frames categorically by their tense-logical theories. For, by the Generation Theorem, tense-logical formulas cannot distinguish between one single, or several parallel ows of Time | which latter picture is so familiar from contemporary science ction. Still, if disjoint unions of frames are disregarded, we have THEOREM 76. hN;
H (Hp ! p) ! Hp P p ! H (F p _ p _ P p) F p ! G(F p _ p _ P p) FT G(Gp ! p) ! (F Gp ! Gp) The proof is omitted here. But, e.g. the integers hZ;
Gp ! GGp; F p ! G(F p _ p _ P p); P T (D4.3) P q ! P q (`irrevocable past'): This logic is claimed to be appropriate for an analysis of the famous Diodorean `Master Argument', identifying possibility with actual or future truth | a version of what was later to become known as the principle of Plenitude: all metaphysical possibilities are eventually realised in this World.
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Our analysis of this claim runs as follows. Gp ! GGp de nes transitivity for <, the McTaggart Axiom de nes right-connectedness; while P T de nes left-succession: 8x9y y < x. The additional `mixing postulate' de nes
8xy(Rxy ! 8z (z < x ! z < y)): Claim
(1). 8xy(Rxy ! (y < x _ y = x _ x < y)).
Proof. Assume Rxy. Let z < x (by left-succession). Then z < y (`mix'). The conclusion follows by right-connectedness. Claim
(2). 8xy(Rxy ! (x < y _ x = y)).
Proof. If Rxy and y < x, then y < y (`mix'): contra irre exivity.
The outcome is this: without ever using transitivity, but with irre exivity (which is presupposed in White's whole set-up), a relational condition follows which is indeed de ned by the Diodorean challenge:
p ! (F p _ p): This is only one of the many possible semantics for temporal modalities, of course. The correspondence aspect of, e.g. the Occamist `branching time' of [Burgess, 1979] remains to be explored. Alternative temporal ontologies. Recently `interval structures' have been proposed as an alternative for the above traditional point ontology. From the manifesto of [Humberstone, 1979], a picture emerges of triples
hI; ; x8z x y > z ,
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a property known as left monotonicity, 2. de nes 8x8y >x8z y z > z , its dual property of right monotonicity. Finally, 3. de nes 8x8y x8z > y (9u z : u x _ 9u z : u > x), a form of a principle known as convexity. (`Stretches of time should be uninterrupted'.) Starting from the other side, one may impose basic postulates on ; <, asking for de nitions in this `interval tense logic'. For <, these might be the earlier-mentioned ones, for , a minimum seems to be the requirement of partial order, while monotonicity (and convexity) take care of minimal connections between <; . This would add only two axioms to the preceding ones, viz. S4 for inclusion. The further condition of anti-symmetry is not de nable | as may be seen by noting that the map n 7! n (modulo 2) is a -zigzag morphism sending the anti-symmetric frame hZ; i to the non-antisymmetric one hf0; 1g; fh0; 0i; h0; 1i; h1; 0i; h1; 1igi. Many more examples of further correspondences on top of this foundation may be found in Chapter II.3.2 of [van Benthem, 1985].
3.2 Conditionals From among the teeming multitude of `conditional logics', three specimens have been included here. As no work of the present kind has been done in this area at all, the following considerations are still very much rst steps. (Cf. the Conditional Logic chapter in volume 5 for a discussion of conditional logics.) Constructive implication Perhaps the single most e ective argument in favour of constructive, as opposed to classical implication is the natural deduction analysis. The natural rules for !-introduction and !-elimination give us only a fragment of all classical pure !-tautologies; axiomatised by
(A1) ' ! ( (A2)
! ') (' ! ( ! )) ! ((' !
) ! (' ! ))
plus the rule of modus ponens. A principle notably outside of this class is Peirce's Law ((' ! ) ! ') ! ': But really, the same elegance shows up in the Henkin completeness proof. In the usual proof, one starts from a given consistent set | and then has
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to extend this arbitrarily to just any maximally consistent one, in order to `break down' implications according to the classical truth table. A canonical model construction rather uses a unique natural model, viz. that consistent set together with all its consistent extensions, exploiting the evident decomposition rule ` ' ! if and only if 80 : if 0 ` ', then 0 ` : A perfect match arises with the following semantics. Structures are general frames F = hW; R; Wi, where R corresponds to the above inclusion relation, and W consists of all R-hereditary sets of worlds. (Propositions represent R-cumulative knowledge on this view.) A direct study of the above logic on these frames would yield rather clumsy conditions. One case will be exhibited, as it illustrates a variant concept of correspondence, viz. correspondence for rules rather than axioms. EXAMPLE 77. Modus Ponens de nes the condition `every world belongs to some nite R-loop'.
Proof. `(': Suppose that xRx1 R : : : Rxn Rx. Let V (p); V (q) be R-hereditary subsets of W , such that p; p ! q hold at x. Then, successively, p; q hold at x1 ; : : : ; xn , and nally at x. `)': Suppose that x belongs to no nite R-loop. Set V (p) := the smallest R-hereditary set containing x; V (q) = the R-hereditary closure of fy j Rxyg. This veri es p; p ! q at x; without verifying q. What will be done instead is to postulate the partial order behaviour of re exivity, transitivity and antisymmetry. Finer peculiarities of (A1), (A2) remain undetectable below this threshold. Further restrictions on R may now be imposed by stronger axioms; e.g. we can see why Peirce's Law is characteristic for classical logic. EXAMPLE 78. Peirce's Law de nes the restriction to single points:
:
8xy(Rxy ! y = x): Proof. `(': A simple calculation suÆces. `)': Suppose that Rxy; x 6= y. Set V (q) = ?; V (p) = fz j Rxz ^ x 6= z g. This makes (p ! q) ! p true at x (notice that p ! q is false at x itself), while falsifying p. (By the way, that V is admissible, i.e. that V (p) is Rhereditary, follows from the above general assumption.) But `intermediate' implication axioms exist as well. EXAMPLE 79. The following principle ((p ! q) ! p) ! (((q ! r) ! q) ! p)
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de nes a maximal length 3 for R-chains:
8xy(Rxy ! 8z (Ryz ! (x = y _ y = z _ 8u(Rzu ! z = u)))): Proof. Here is the relevant counter-example for the argument in the `)'direction. Assume that xRyRzRu, while x 6= y; y 6= z; z 6= u. Set V (r) = ;; V (q) = fv j Ruv ^ u 6= vg [ fv j Ryv ^ :Rvz g; V (p) = fv j Ryv ^ y 6= vg. The principle will be falsi ed at y. It has not been possible to nd other types of intermediate example. Hence, we conclude with a Conjecture. All principles of pure constructive implication de ne rstorder constraints on R; viz. restrictions to some nite chain length. Relevant implication
Of the various proposed semantics for relevance logic, here is a perspicuous example from [Gabbay, 1976, Chapter 15]. Structures are now tuples hW; R; V; 0i, where 0 is a special world providing a vantage point from which to compare other worlds through the ternary relation R. Intuitively, Ra bc is to mean that b is `included' in c, at least from the perspective of a. (One might think of, for example, `a-local inclusion': a \ b a \ c.) No prior conditions are imposed on this relation. This is not to say that these are not to be found at all. For instance, it may be shown that the mentioned local inclusion relation is characterised by two betweenness axioms: 1. Ra bc $ Rb ac
(interchanging boundaries)
2. (Ra bc ^ Rdae ^ Rd be) ! Rdce (I.e. if c 2 [a; b]; a 2 [d; e]; b 2 [d; e], then c 2 [d; e]: a form of convexity.) The explication of implication reads as follows:
' ! is true at a i , for all b; c such that Ra bc, if ' is true at b, then is true at c. As it stands, this de nition makes no implication laws universally valid. To obtain at least some indubitable principle, one therefore imposes a restriction on valuations. The most urgent case is that of p ! p. On the above bare semantics, it would correspond to 8xyz (Rxyz ! y = z ), collapsing the ternary relation. To avoid this, one again requires `cumulation': valuations V are only to assign subsets X of W subject to the constraint that 8xy 2 W (R0 xy ! (x 2 X ! y 2 X )).
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If this constraint is to extend automatically to sets X de ned by complex implicational formulas, then a mild form of transitivity is to be imposed on the ternary relation after all:
8xyzu((R0xy ^ Ry zu) ! Rxzu): Notice how this relates perspectives from di erent vantage points. But then, if reasonable forms of transitivity have become respectable, we also add ()8xyzu((R0xy ^ R0 yz ) ! R0 xz ). Now, at last, some genuine correspondences arise | of a `local' sort (cf. Section 2.2). EXAMPLE 80. 1. Modus Ponens de nes R0 00, 2. Axiom A1 de nes a curious form of `transitivity': 8xyzu((R0xy ^ Ry zu) ! R0 xu).
Proof. (Case (1) only) `(': This direction is immediate. `)': Let V (p) = f0g [ fx j R0 0xg; V (q) = fx j R0 0xg. By the above principle (), both assignments are admissible. Clearly, both p and p ! q are true at 0, whence also q: i.e. R0 00. Obviously, the second principle is not very plausible | but then, neither is (A1) for a relevance logician. A more interesting phenomenon in relevance logic, from the present point of view, is the treatment of negation. This formerly inconspicuous notion is now interpreted using a `reversal operation' + on worlds:
:' is true at a
i
' is true at a+ .
In this light, new combined correspondences appear, such as that between Contraposition and the reversal law
8xy(R0 xy ! R0 y+x+ ): Correspondence Theory may be applied to any kind of semantic entity. Counterfactual implication
Ramsey told us to evaluate conditionals as follows. Make the minimal adjustment of your stock of beliefs needed to accommodate the antecedent: then see if the consequent follows. Various syntactic and semantic implementations of this view exist, of which that of [Lewis, 1973] has deservedly won the greatest favour. A counterfactual ' ! is true in a world, on his
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account, if is true in all worlds most similar to that world given that ' holds in them. As the preceding account has some diÆculties in the in nite case, let us consider nite models hW; C; V i, where C is a ternary relation of comparative similarity: Cx yz for: `y is closer to x than z is'. Lewis gives three basic conditions on the relation `no closer': 1. transitivity: 8xyzu((:Cxyz ^ :Cx zu) ! :Cx yu), 2. connectedness: 8xyz (:Cxyz _ :Cx zy), 3. egocentrism: 8xy(:Cx xy ! x = y). Rewriting these for `closer', one nds to one's surprise that (2) is rather weak, being merely 20 . asymmetry: 8xyz (Cxyz ! :Cx zy). On the other hand, (1) becomes a strong principle 10 . 8xyu(Cxyu ! 8z (Cxyz _ Cx zu)), which we knew as almost-connectedness back in Section 3.1. From asymmetry and almost-connectedness, one may derive ordinary transitivity and irre exivity, whence the three `comparative' axioms of Section 3.1 emerge. These principles justify the appealing picture of `similarity spheres' around the reference world x. The tendency has been since 1973 to retain only transitivity and irre exivity as fundamental pre-conditions on C , leaving various forms of connectedness as optional extras. Thus, one nds an axiomatisation of this austere minimal conditional logic in [Burgess, 1981]. The truth de nition in this case may be taken to be the following:
'
!
is true at w if w holds in all '-worlds C -closest to w:
Indeed, this clause veri es the following list of principles without further ado: p ! p; p ! q; p ! r ` p ! q ^ r; p ^ q ! p; p ! r; q ! r ` p _ q ! r: It is only the last one which requires transitivity:
p
! q ^ r ` p ^ q ! r:
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383
Egocentrism is restored by adding the principle of Modus Ponens:
p
! q; p ` q
But, the original Lewis logic contained even further principles, such as the formidable ((p _ q) ! p) _ :((p _ q) ! r) _ q ! r: What does it express? As it happens, it restores almost-connectedness.
Proof. First, the axiom is valid under this additional assumption | by the above discussion. Next, suppose almost-connectedness fails; i.e. for some xyzu we have: Cx yz; :Cxyu; :Cx uz . By transitivity, it follows that :Cx zu. Now, set V (p) = fyg; V (q) = fz; ug; V (r) = fy; ug. Then z is q-closest among the worlds falsifying r. The two p _ q-closest worlds y; u both verify r. Finally, p fails in the p _ q-closest world u. Thus, Lewis' axiom has been refuted. Finally, to mention an example outside of Lewis' original logic, there is the Stalnaker principle of `Conditional Excluded Middle':
p
! q ^ p ! :q:
As was stated in the Introduction, this axiom even requires the similarity order to be a linear one. In the present nite case, this means that the above truth de nition reduces to:
'
!
is true at w if
holds in the closest '-alternative to w:
And that was the original Stalnaker explication of conditionals. The previous examples were all conditional axioms without nestings of !. This is typical for most current logics in this area. Relational conditions matching these have invariably been found to be rst-order ones. Hence, in view of Theorem 38, here is our Conjecture. All counterfactual axioms without nestings of conditionals are rst-order de nable. The reason for this restriction lies in the motivation for the present area. Entailment conditionals such as constructive implication, or modal entailment have often been proposed out of dissatisfaction with classical `nested principles', such as, say, p ! (q ! p) or Peirce's Law. The non-nested classical fragment was not called into question. Counterfactual conditionals, however, typically disobey classical implicational logic at the level of nonnested inferences, such as the monotonicity rule from p ! q to p ^ r ! q. Nevertheless, there are intrinsic reasons to be found inside the above semantics for considering nested axioms after all. For, one obvious omission
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JOHAN VAN BENTHEM
in the above list of semantic conditions was the lack of index principles relating the perspectives of di erent worlds. For instance, when we read C for a moment as relative proximity in Euclidean space, we nd the following Triangle Inequality
8xyz ((Cxyz ^ Cz xy) ! Cy xz ): And there are other elegant principles of this kind. Now, it is easily seen that such index principles are just what is involved when nested counterfactuals are evaluated: the perspective starts shifting. Thus, it will be rewarding to have correspondences here as well. One, not too exciting example is the following. The Absorption Law
p
! (q ! r) ` (p ^ q) ! r
de nes the index principle
8xyz (Cxyz ! 8u:Cy uz ): Better examples are still to be found. Indeed, e.g. the counterfactual logic of Euclidean space, the most natural geometric representation of our similarity pictures, is still a mystery.
3.3 Intuitionistic Logic Constructive conditional logic is only a part of the full intuitionistic logic, whose Kripke semantics extends the earlier constructive models. In this section, a sketch will be given of an Intuitionistic Correspondence Theory. (For details on intuitionistic logic, cf. van Dalen's chapter in volume 7 of this Handbook.) Kripke semantics, intermediate axioms and correspondence.
DEFINITION 81. An intuitionistic Kripke model M is a tuple hW; ; V i, where is a partial order (`possible growth') on W (`stages of knowledge'). The valuation V assigns -closed subsets of W to proposition letters (`cumulation of knowledge'). The truth de nition has the following familiar pattern,
M M M M
2 ?[w] ' ! [w] ' ^ [w] ' _ [w]
for all w 2 W; if M [v] for all v w such that M if M '[w] and M [w]; if M '[w] or M [w]:
Negation is de ned as usual (:' becoming ' ! ?).
'[v];
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385
The pre-condition of partial order was motivated earlier on. But, other choices may be defended as well. As is well-known, the above semantics was derived from the modal one, through the Godel translation g:
g(p) = p g(' ! ) = (g(') ! g( )) g(' ^ ) = g(') ^ g( ) g(' _ ) = g(') _ g( ) g(?) = ?: Now, there is a whole range of modal logics whose `intuitionistic fragment' (through g ) coincides with intuitionistic propositional logics. Amongst others, we have the THEOREM 82. Let X be any modal logic in the range from S4 to S4.Grz = S4 plus the Grzegorczyk Axiom
((p ! p) ! p) ! p: Then, for all intuitionistic formulas '; ' is intuitionistically provable in Heyting's logic if and only if g(') is a theorem of X . The earlier modal correspondences yield a corresponding semantic range, between `pre-orders' (re exive and transitive) and `trees': EXAMPLE 83. Grzegorczyk's Axiom de nes the combination of (i) re exivity, (ii) transitivity, and (iii) well-foundedness in the following sense: `from no w is there an ascending chain w = w1 w2 : : : with wi 6= wi+1 (i = 1; 2; : : :)'.
Proof. This goes more or less like the closely related Axiom of Lob. By the way, notice that (iii) implies anti-symmetry. Note also that, semantically, Grzegorczyk's axiom alone implies the S4-laws: syntactic derivations to match were found around 1979 by W. J. Blok and E. Pledger. Thus, a case may also be made for the Tree of Knowledge as a basis for intuitionistic semantics. Nevertheless, we shall stick to partial orders for a start. Above S4Grz, modal logics start producing greater g-fragments | the so-called intermediate logics, ascending to full classical logic. Intermediate axioms impose various restrictions on the pattern of growth for knowledge, classical logic forcing the existence of single (`complete') nodes. EXAMPLE 84. (i) Excluded Middle p _ :p de nes 8x8y(x y ! x = y).
Proof.`(' is immediate. `)': Suppose x y; x 6= y. (By anti-symmetry then y V (p) = fz j y z g. This falsi es both p and :p at x.
6 x.)
Set
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JOHAN VAN BENTHEM
(ii) Weak Excluded Middle :p _ ::p de nes directedness.
Proof. `(': Suppose that :p fails at x; say p holds at y x. Then consider any z x. As it shares a common successor with y, and V (p) is -hereditary, it has a successor verifying p, whence :p fails at z . So ::p holds at x. `)': Suppose that x y; z , where y; z share no common successors. Set V (p) = fu j z ug. (Like above, this is a -closed set.) Notice that x; y 62 V (p). It follows that :p fails at x (consider z ), but ::p fails as well (consider y). (iii) Conditional Choice (p ! q) _ (q ! p) de nes connectedness.
Proof. `(': Suppose that p ! q fails at x; i.e. some y x has p true, but q false. Now consider any z x such that q holds. Either z x, but then, by -heredity, q is true at y (quod non), or y z , and so, again by -heredity, p is true at z , i.e. q ! p is true at x. `)': Let x y; z with y 6 z; z 6 y. Set V (p) = fu j y ug; V (q) = fu j z ug. Then p ! q fails at x (watch y), and q ! p fails as well (watch z ).
Much more forbidding principles than these have been proposed as intermediate axioms. But surprisingly, these usually turned out to be rst-order de nable: EXAMPLE 85. (i) The Stability Principle (::p ! p) ! (p _ :p) de nes
8x:9yz (x y ^ x z ^ :9u(y u ^ z u) ^ ^ 8u(8s(u s ! 9t(s t ^ z t)) ! :9v(u v ^ y v))): (ii) The Kreisel-Putnam Axiom (:p ! (q _ r)) ! ((:p ! q) _ (:p ! r)) de nes
8x:9yz (x y ^ x z ^ :y z ^ :z y ^ ^ 8u((x u ^ u y ^ u z ) ! 9v(u v ^ :y v ^ :z v))): No matter how complex such axioms may seem at rst sight, proofs of the above assertions are quite simple exercises in `imagining what a counterexample would look like'. This recurrent experience led to the following conjecture in [van Benthem, 1976]: All intermediate axioms express rst-order constraints on growth of knowledge. Two conjectures refuted. The earlier hope was all but given up in the rst version of this chapter; as `Scott's Rule' turned out to be an essentially
CORRESPONDENCE THEORY
387
higher-order intermediate inference. The relevant argument was sharpened somewhat by P. Rodenburg: THEOREM 86. Scott's Axiom ((::p ! p) ! (p _ :p)) ! (:p _ ::p) de nes no rst-order condition on partial orders.
Proof. An elaborate Lowenheim{Skolem argument works, in the spirit of Example 43. As an illustration of the non-triviality of our present subject matter, it follows here. Step 1: Consider the following Kripke frame hW; i:
dcX
A c AA
AA A X
W consists of the in nite binary tree T , together with, for each node c in T and each -hereditary, co nal set X in Tc (i.e. the subtree with root c), some point dcX . is the usual order on T , together with
c dcX x, for all x 2 X dcX dcX 0 , if X 0 X .
. Scott's Axiom is true in hW; i. Proof. First, let c 2 T be a putative refutation. I.e., for some valuation V, Claim
1. (::p ! p) ! p _ :p is true at c, 2. :p _ ::p is false at c.
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JOHAN VAN BENTHEM
Then consider the node dcX , where X is the co nal hereditary set
Tc \ (V (p) [ V (:p)): One veri es successively that ::p ! p is true at dcX , whereas both p; :p are false. (E.g. if p were true at dcX , then p is true throughout X , whence ::p is true at c | whereas (2) says the opposite.) Thus, we have a contradiction with (1). A similar argument works for the case where c is of the form dcX itself. Step 2: A matter of cardinality: Claim. The above Kripke frame is uncountable. @ c Proof. In particular, there are 2 0 nodes of the form dX . For, each subset Y of N may be coded as follows, using (distinct) hereditary co nal subsets Y + of the in nite binary tree. Let Y = fy1; y2 ; y3 ; : : :g.
Y2
Y1
Y+
Y+
etc. going down the extreme right branch using the extreme left branches to code y1 ; y2 ; y3; : : :. For all nodes not arrived at in this way, one makes Y + co nal by means of the following stipulation:
y1
y3 , etc.
Y +
Y+
Y+
y2
Step 3: Take any countable elementary substructure F of hW; i containing the original binary tree.
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389
. Scott's Axiom may be falsi ed in F . Proof. Consider T as a double tree c0 Claim
c1
c2
and again Tc2 a countable sequence of `trees on a string': 2 c
T1
T3
T3 Let DX1 ; DX2 ; : : : be an enumeration of the points dCX0 remaining in F . Notice that, for each i 2 N , 1. nite intersections Ti \ X1 \ : : : \ Xn are still hereditary co nal in Ti , 2. the total intersection Ti \ fXj j j = 1; 2; : : :g is empty.
As for the latter observation, it suÆces to see that the assertion
8x9dCX
0
with dC0
6 x;
which holds in hW; i, can be expressed in rst-order terms in hW; i; and, hence, it has remained valid in the elementary substructure F . Now, de ne X1 = X1 Xn+1 = X1 \ : : : Xk for the smallest k such that Tn+1 \ X1 \ : : : \ Xk =6 Tn+1 \ Xn : Scott's Axiom may now be falsi ed at c0 , by setting X = [ fTi \ X j i = 1; 2; : : :g; V (p) = fy j 9x y x 2 X g: i
to see this, notice, that successively, 1. each point dXi has a successor (in Ti ) outside of V (p), 2. (::p ! p) ! p _ :p holds at c0 ,
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JOHAN VAN BENTHEM
3. :p _ ::p fails at c0 . We conclude that Scott's Axiom is not rst-order de nable | not being preserved under elementary subframes. This complex behaviour disappears on better-behaved structures. OBSERVATION 87. On trees, Scott's Axiom de nes the rst-order condition
8x:9yzu (x y ^ x z ^ z u ^ z 6= u ^ :9v(y v ^ z v)): This, and other experiences of its kind, led to a revised guess in the rst version of this chapter: On trees, all intermediate axioms express rst-order constraints on descendance. A proof sketch was added, involving semantic tableaux as `patterns of falsi cation', to be realised in trees. This conjecture was `almost' refuted in [Rodenburg, 1982]. The semantic tableau method runs into problems with disjunctions, and indeed we have the following counter-example. EXAMPLE 88. Consider the formula
= ((:p ^ :q ^ :r) ! (p ^ q ^ r)) ! (:p ^ :q ^ :r) with the simultaneous substitution of: p&q for p, p&:q for q, and :p&q for r. This is not rst-order de nable on partial orders. On suitably tree-like structures, it expresses the lack of `3-forks' of immediate successors as well as the absence of in nite comb-like structures. On trees, this negative example probably still works | but there is an instructive diÆculty here. The class of trees itself has a higher-order de nition; 11 , to be precise. Therefore, current model-theoretic arguments for disproving rst-order de nability (compactness, Lowenheim{Skolem) run the risk of employing constructions leading outside of this class. Higherorder preconditions are a problem for our Correspondence Theory. To illustrate this from a purely classical angle, the reader may consider a related problem, showing how soon the familiar methods of model theory fail us. Finiteness is rst-order unde nable on partial orders, even on trees. It is thus de nable on linear trees, however, viz. by `every non-initial node has an immediate predecessor'. What about the (at most) binary trees? This intermediate case seems to be open. The state of the subject. The progress of science is sometimes startling. Where the rst version of this chapter (1981) had some tentative examples, enlightenment reigns in the report [Rodenburg, 1982]. Of its many topics, only a few will be mentioned here. First, there are several semantic options | as indicated above, ranging from partial orders via `downward linear orders' to trees. But moreover,
CORRESPONDENCE THEORY
391
there is a legitimate choice of language. Despite appearances, it is the disjunction clause which is now strongly constructive in intuitionistic Kripke semantics. (`Choose now!' Classical logic would have a more humane clause in this setting: (' _ ), i.e. `' or eventually'.) Thus, it is of interest to consider both the full language and its _-free fragment. The semantic tableau method mentioned above, in combination with the above counter-examples, has led to the results in the following scheme: All formulas Partial Downward Trees rst-order de nable orders linear orders without _ with _
YES NO
YES NO
YES ?
But there are also matters of ` ne structure'. For instance, Scott's Axiom had only one proposition letter | and for such intuitionistic formulas we have the beautiful Rieger{Nishimura lattice. Now, Scott's Axiom merely seemed a t candidate for a counter-example among the intermediate axioms existing in the literature. Rodenburg has proved that it is also minimal in the Rieger-Nishimura lattice with respect to non rst-order de nability. (More precisely, an intuitionistic formula with one proposition letter is rstorder de nable on the partial orders if and only if it is equivalent to one of A1 ; : : : ; A9 in the lattice.) In the counter-examples needed for the latter result, a uniform method may be seen at work: compactness, in the form that sets of formulas which are nitely satis able in nite models are also simultaneously satis able (in some in nite model). Now, indeed, intuitionistic truth has a close connection with truth in nite submodels (cf. [Smorynski, 1973]). Our question is whether this may lead to the following improvement in the mathematical characterisation of rst-order de nability as given in Section 2.2. . An intuitionistic formula ' is rst-order de nable if and only if ' is preserved under ultraproducts of nite frames. Conjecture
Intuitionistic de nability. As with the direction `from intensional to classical', the case `from classical to intuitionistic de nability' shows many resemblances with our earlier modal study. For instance, a Goldblatt{Thomason type characterisation was proved in [van Benthem, 1983] (cf. our earlier Theorem 66):
A rst-order constraint on the growth pattern is intuitionistically de nable if and only if it is preserved under the formation of generated subframes, disjoint unions, zigzag-morphic images, lter extensions and ` lter inversions'.
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JOHAN VAN BENTHEM
Merely in order to illustrate this topic, which has a wider semantic signi cance, here is a sketch of the representation theory in the background. On the algebraic side, the intuitionistic language may be interpreted in Heyting Algebras hA; 0; 1; +; ; )i satisfying suitable postulates. Now, each Kripke (general) frame in the above sense induces such a Heyting Algebra, through its -hereditary sets, provided with suitable, obvious operations. But also conversely, a lter representation now takes Heyting Algebras to Kripke general frames. Indeed, the earlier categorial duality (cf. Section 2.3) is again forthcoming. The more general semantic interest of the construction is this. Despite the super cial similarity with structures consisting of the `complete' possible worlds, intuitionistic Kripke models should be regarded as patterns of stages of partial information. This comes out quite nicely in the above representation, where `worlds' are no longer complete ultra lters, but merely lters (in the _-free case) or `splitting' lters (for the full language). Filters F merely satisfy the closure condition that
a; b 2 F i a b 2 F; a minimal requirement on partial information. Also quite suggestively, the `modal' alternative relation collapses into inclusion (`growth'): 8a ) b 2 F 8a 2 F 0 : b 2 F 0 i F F 0 : The present-day supporters of `partial models' and `information semantics' would do well to study intuitionistic logic. Predicate logic. Again, correspondence phenomena do not stop at the frontier of predicate logic. This will be illustrated by means of some intuitionistic examples. Kripke models M = hW; ; D; V i will now be of the usual variety; in particular satisfying 1. 8xy(x y ! Dw Dv ) (monotonicity) 2. 8xy(x y ! 8d~ 2 Dx(Vx (P; d~) = 1 ! Vy (P; d~) = 1) (heredity). But other varieties, say with maps between the domains (cf. [Goldblatt, 1979]) would be suitable as well. The `de re/de dicto' interchange principles of Section 2.5 now have their obvious counterparts in the following quartetto: 1. 2. 3.
:9xAx ! 8x:Ax, 8x:Ax ! :9xAx, 9x:Ax ! :8xAx,
CORRESPONDENCE THEORY 4.
393
:8xAx ! 9x:Ax.
The rst three of these are universally valid on the present semantics. That they already hide quite some complexity is shown by the Godel translation of (3): (9x:Ax ! :8xAx); or
(9x:Ax ! 9x:Ax):
No wonder that (3), e.g. does not de ne precisely the above monotonicity constraint on domains | even though its modal cousin 9xAx ! 9xAx did. The rst really complex principle in Section 2.5 was the converse implication 9xAx ! 9xAx. We shall now investigate its intuitionistic cousin (4) | a rejected classical law. EXAMPLE 89. 1.
:8xAx ! 9x:Ax implies that all domains are equal: 8xy(x y ! Dx = Dy )
2. On frames with constant nite domain, the rst-order condition that
:8xAx ! 9x:Ax expresses
8x (9!d d 2 Dx _ 8y(x y ! 8z (x z ! 9u(y u ^ z u)))): Proof. Ad 1. Suppose that x y, but Dx 6= Dy . Make A true at y for all d 2 Dx , and similarly at all y0 y. This stipulation de nes an admissible assignment verifying :8xAx at x, while falsifying 9x:Ax. Ad 2. First, if jDxj = 1, then trivially, :8xAx ! 9x:Ax holds at x. (Recall that all domains are equal.) Next, if jDxj > 1, then one may argue as follows. If is directed above x in the above sense, then the assumption that 9x:Ax fails at x can be exploited to show that :8xAx must fail as well. For, let Dx = fd1 ; : : : ; dk g. By the assumption, Adi will be true at some xi x (1 i k). Then, by successive applications of directedness, there will be found a common successor y x1 ; : : : ; y xk , where 8xAx is true (by heredity). This falsi es :8xAx at x. If on the other hand, for some x; jDx j > 1 while is not directed above x, then, say, there exist x1 x; x2 x without common successors. Then pick any object d 2 Dx, making A true at x1 and all its - successors for all objects except d; while making A true at x2 and all its -successors for d only. This assignment veri es :8xAx at x, while falsifying 9x:Ax.
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JOHAN VAN BENTHEM
Thus, a classical quanti er axiom may express an interesting purely relational constraint on . Now, intuitionists are fond of saying that (4) is valid for nite domains: as we have seen, however, it does impose constraints even then. They go on to say that an extrapolation to the in nite case would be illegitimate. At least, our principle becomes much more complex then. THEOREM 90. :8xAx ! 9x:Ax is not rst-order de nable in general.
Proof. Consider the following structure, in which all worlds have a common domain N .
:::
:::
0 1 2
1 0 +1
! ( < !1)
i.e. hW; i has the relational pattern of
hN (!1 Z); i: .
Claim
:8xAx ! 9x:Ax is true in this frame.
Starting from any world x, assume that 9x:Ax fails. Then, for each n 2 N , An must hold at some ( n ; kn ) > x. As the co nality of !1 exceeds !, there exists some < !1 such that ( ; 0) > ( n ; kn )(n 2 N ). Now, by heredity, 8xAx must hold at ( ; 0) | whence :8xAx is false at x. Next, by the Lowenheim{Skolem theorem (as ever), this frame has countable elementary subframes. (Indeed, hIN; i itself is one.) But in these, our principle may be falsi ed using some countable co nal sequence x0 ; x1 ; : : : making A0 true from x0 upward, A1 from x1 upward, etcetera. As in earlier arguments, the conclusion of the theorem follows. Proof.
To nish this list of examples, it may be noted that a famous weaker variant of the above axiom does indeed de ne a rst-order constraint. EXAMPLE 91. Markov's Principle
8x(Ax _ :Ax) ^ ::9xAx ! 9xAx de nes the relational condition
8x9y x 8z y 8d(Edz ! Edx): Correspondence Theory remains surprising.
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Post-Script: quantum logic. Correspondences have not proved uniformly successful in intensional contexts. It seems only fair to nish with a more problematic example. A possible worlds semantics for quantum logic was proposed in [Goldblatt, 1974]. Kripke frames are now regarded as sets of `states' of some physical system, provided with a relation of `orthogonality' (?). From its physical motivation, two pre-conditions follow for ?, viz. irre exivity and symmetry. But in addition, there is also a restriction to `admissible ranges' for propositions, in the sense that these sets X W are to be orthogonally closed: 8x 2 (W X )9y 2 (W X )(:x?y ^ 8z 2 X y?z ): The key truth clauses are those for conjunction (interpreted as usual), and negation, interpreted as follows:
:' is true at x
if x is orthogonal to all '-worlds.
This semantics validates the usual principles for quantum logic, when _ is de ned in terms of :; ^ by the De Morgan law. But, one key principle remains invalid, viz. the ortho-modularity axiom
p $ (p ^ q) _ (p ^ :(p ^ q)): This axiom has a natural motivation in the Hilbert Space semantics for quantum logic | being the key stone in the representation of ortho-modular lattices as subspace algebras of suitable vector spaces. Thus, a minimal expectation would be that an enlightening correspondence is forthcoming with some constraint on the orthogonality relation ?. In reality, no such thing has happened. Quantum logicians pass onto general frames, into whose very de nition validity of ortho-modularity has been built in. Despite this cover-up, the fact remains that the relational possible worlds perspective fails to do its correspondence duties here. A set-back, or an indication that facile over-applicability of Kripke semantics need not be feared for? 4 CONCLUSION At a purely technical level, Correspondence Theory is an applied subject. Classical tools have been borrowed from model theory and universal algebra. In return to these mother disciplines, the subject o ers a good range of (counter-)examples, as well as prospects for generalisability to other suitably chosen fragments of higher-order logic. (Cf. [van Benthem, 1983].) From a more philosophical point of view, the whole enterprise may be described as nding out what possible worlds semantics really does for us.
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It is one thing to make conceptual proposals, and another to really probe their depths. The systematic study of connections between intensional and classical perspectives upon possible world structures is an exploration of the bene ts gained by the semantics. This chapter started with the observation that `complex' modal axioms turned out to express `simple' classical requirements (i.e. rst-order ones). We have investigated the range and limits of this, and related phenomena. Especially these limits have become quite clear | and, with them, the limits of fruitful application of Kripke semantics. This philosophical conclusion holds for all semantics, of course. But we have earned the moral right to say it, through honest toil. ACKNOWLEDGEMENTS The classical introduction to a systematic modal model theory remains [Segerberg, 1971]. Some rst applications of more sophisticated tools from classical model theory may be found in [Fine, 1975]. The algebraic connection was developed beyond the elementary level by L. Esakia, S. K. Thomason, R. I. Goldblatt and W. J. Blok. Two good surveys are [Blok, 1976] and [Goldblatt, 1979]. The proper perspective upon modal logic as a fragment of second-order logic was given in [Thomason, 1975]. An early appearance of correspondence theory proper is made in [Sahlqvist, 1975], full surveys are found in [van Benthem, 1983] for the case of modal logic and [van Benthem, 1985] for the case of tense logic. Other case studies are still in a preliminary state, with the exception of the intuitionistic treatise [Rodenburg, 1982]. University of Amsterdam
APPENDIX (1997) This chapter rst appeared in 1984. In the meantime, Modal Logic has evolved, but the basic structure of our original presentation remains valid. Therefore, we have left the old text unchanged, and merely added a short chronicle of further developments, including some answers to open questions. Generally speaking, correspondence methods have become a useful technical tool in pure and applied Modal Logic, without forming a major research area in their own right. A more principled motivation is given in van Benthem [1996a], where correspondence analysis is viewed as a central part in the philosophical quest for logical `core theories' of semantic phenomena in language and computation. In particular, correspondences suggest the introduction of new many-sorted models, inducing decidable geometries of `states' and `paths' in the study of time and computation.
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Extensions to Other Branches of Intensional Logic The rst signi cant extension of correspondence theory concerns Intuitionistic Logic. This involves the new feature that all valuations must be restricted to hereditary ones, leading only to formulas whose truth is preserved upward in the relational ordering. Rodenburg [1986] investigates this area in detail. In particular, he shows that the implication-conjunction fragment is totally rst-order, whereas disjunctions can lead to non- rst-orderness. Moreover, he introduces semantic tableau methods for explicit description of rstorder correspondents. A nal interesting feature is Rodenburg's analysis of intuitionistic Beth models which employ a second-order truth condition: a disjunction is true when its disjuncts `bar' all future paths. These also turn out to be amenable to correspondence analysis, over two-sorted frames with both points and paths. Restricted valuations also occur with the ternary relational models of Relevant Logic. A full correspondence analysis is given in Kurtonina [1995], which analyses the special e ects of working with features like distinguished points (actual worlds), non-standard connectives (including a new product conjunction), as well as the much poorer nonBoolean fragments found in categorial logics for grammatical analysis (cf. [van Benthem, 1991; Moortgat, 1996]). Further extensions have been made to Epistemic Logic [van der Hoek, 1992] and Partial Logics [Thijsse, 1992; Jaspars, 1994; Huertas, 1994]. Correspondence with restricted valuations for `convex' propositions has also been proposed in standard Temporal Logic (cf. van Benthem [1983; 1986; 1995b]). But also, most axioms for richer intervalbased versions have rst-order `Sahlqvist forms' [Venema, 1991]. Zanardo [1994] gives correspondences for modal-temporal models of branching spacetime. Finally, correspondence methods have turned out very useful in Algebraic Logic. Venema [1991], Marx and Venema [1996] present a systematic study of relational algebra and cylindric algebra along these lines, pointing out the Sahlqvist form of most familiar algebraic axioms, and calculating their frame constraints on algebraic `atom structures'. This establishes a much wider bridge between algebraic logic and modal logic than our earlier duality.
Restricted Frame Classes Correspondence behaviour may change on special frame classes. In this chapter, we have looked at some e ects of a restriction to transitive frames. But one can also investigate non- rst-order frame classes. Van Benthem [1989a] considers nite frames, where, amongst others, the McKinsey axiom still de nes a non- rst-order condition. In this area, standard compactnessbased model-theoretic techniques no longer work, and they must be replaced by a more careful combinatorial analysis with Ehrenfeucht-Frasse games of model comparison. (More generally, the nite model theory of modal logic
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is still undeveloped. Rosen [1995] proves some interesting transfer results, showing better nite model-theoretic behaviour than for rst-order logic in general.) Doets [1987] takes up modal Ehrenfeucht games in great depth, investigating, amongst others, correspondence over countable and over wellfounded frames. (For instance, the so-called Fine Axiom turns out to be rst-order over countable frames.)
Complexity This chapter contains some results on the (high) complexity of de nability problems for monadic 11 formulas. It turns out much harder to deal with the modal fragment of these. A lower bound for the complexity of rst-orderness of modal formulas has been found in Chagrova [1991]: M1 is undecidable. It seems likely that her methods (involving reductions of Minsky machine computation to correspondence statements) can also be made to yield non-arithmetical complexity. Conversely, undecidability of modal de nability for rst-order statements has been proved by Wolter [1993]: that is, P1 is undecidable, too. A more general investigation of time and space complexity for modal logics, and the `jumps' that may occur with di erent operator vocabularies, may be found in Spaan [1993]. It has improved decidability results for the so-called `subframe logics' de ned in Fine [1985], as well as `transfer' of complexity bounds from components to compounds in poly-modal logics (cf. [Kracht and Wolter, 1991]).
Correspondence and Completeness The main business of modal logic has been the search for completeness theorems over various frame classes. Correspondence theory bypasses this deductive information, focussing on direct semantic de nability. Nevertheless, Kracht [1993] shows how the two enterprises can be merged, by a suitably generalized form of modal de nability. Perhaps the most powerful result of this kind is the generalized Sahlqvist Theorem in Venema [1991], which shows that over suitably rich modal languages (possessing matched versions for each modality accessing all directions of its alternative relation), and allowing natural additional rules of inference beyond the minimal modal logic, the correspondence and the completeness version of the Sahlqvist Theorem converge in their proofs. The essential observation in the argument is as follows. In standard Henkin models for these richer systems, unlike in the standard case, all de nable subsets employed in the correspondence proof (such as singletons or successor sets) are modally de nable. Direct frame correspondences for modal rules of inference may be found in van Benthem [1985]. Over frames, the latter correspond to non-11 second-order formulas, but except for a few scattered observations in the literature, correspondence theory for modal rules of inference remains underexplored.
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Duality with Algebraic Logic Algebraic methods have been invaluable in nding key results on correspondence, such as the Goldblatt-Thomason characterization of the modally de nable rst- order formulas. Nevertheless, a purely model-theoretic reanalysis has been given in van Benthem [1993b], revolving around saturated models instead of descriptive frames. There is no de nite preference here, as it is precisely the interplay between algebraic and model-theoretic viewpoints that remains fruitful. For new uses of correspondence methods in algebraic logic, as well as new set-theoretic representations for Boolean algebras with additional modal operators, see Marx [1995], Mikulas [1995]. For instance, Marx has an in-depth study of the duality between algebraic amalgamation and logical interpolation. The latter methods no longer employ simple binary relations as in the Jonsson-Tarski Stone representation, but more complex set-theoretic constructs. (Modal correspondences over nitary relations occur in van Benthem [1992], with a nite neighbourhood semantics for logic programs.) Developing a systematic correspondence theory over such generalized relational structures then becomes the next challenge.
Extended Modal Logics Perhaps the most striking development in modal logic over the past ten years has been the systematic use of more powerful formalisms, with stronger modal operators over relational frames. A straightforward step is `polymodal logic', which gives the same expressive power over frames with more alternative relations. Examples of the latter trend are the indexed modalities < i > of propositional dynamic logic (cf. [Harel, 1984; Goldblatt, 1987; Harel et al., 1998]), or n-ary modalities accessing (n + 1)-ary alternative relations, as happens in relevant or categorial logics (cf. [Dunn, 2001; Kurtonina, 1995]). The correspondence theory of such extensions is straightforward, whereas there are interesting issues of `transfer' for axiomatic completeness, nite model property, or computational complexity: cf. [Spaan, 1993; Fine and Schurz, 1996]. Transfer may depend very much on the connections between the various modalities. A case in point is modal predicate logic, whose theory has rapidly expanded over the past decade. Van Benthem [1993a] surveys some striking contributions by Ghilardi and Shehtman. More interesting, from a correspondence point of view, is an increase in expressive power over the original binary relational frames. For temporal logic, the latter research line was initiated by Kamp's Theorem on functional completeness of the fSince, Untilg language over continuous linear orders. In modal logic, the rst systematic work emanated from the `So a School': cf., e.g., [Gargov and Passy, 1990; Goranko, 1990], Vakarelov [1991; 1996]. These papers study addition of various new operators, such as a universal
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modality ranging over all worlds (relationally accessible or not), or various operations on poly-modalities, such as `program intersection'. New frame constructions were invented to deal with these, such as `duplication'. De Rijke [1992] investigates the `di erence modality' (\in at least one di erent world"), which has turned out to be useful and yet tractable. A more general program for extending modal logic (viewed as a general `theory of information') occurs in van Benthem [1990] but the technical perspective is also clear in the pioneering paper Gabbay [1981]. Finally, de Rijke [1993] is an extensive model-theoretic investigation of de nability and correspondence for extended modal languages, producing generalized versions for many results in this chapter (such as frame preservation theorems or effective correspondence algorithms). Still another angle on all this will follow below.
Alternatives: Direct Frame Theory One may also analyze the frame content of modal logics more directly in terms of mathematical properties of graphs. Fine [1985] is a pioneer of this trend, emphasizing the good behaviour of `subframe logics' which are complete for frame classes that are closed under taking subframes. (Such logics make no `existential commitments'.) First-orderness is not a prominent consideration here: e.g., Lob's Axiom de nes a simple subframe logic. Zakharyashev [1992; 1995] is a sophisticated study of modal logic from this viewpoint. Nevertheless, his direct classi cation of modal logics into three stages of frame preservation behaviour may again be re ected in secondorder syntax and hence result in a form of correspondence theory at that higher level. A forthcoming monograph by Chagrov and Zakharyashev provides much more background, inluding references to earlier Russian sources (going back to Jankov in the sixties). Another excellent source, for many of the topics listed here, is the survey chapter [Chagrov et al., 1996].
Models, Bisimulation and Invariance Another noticeable shift of emphasis in the current literature leads away from frames to models as the primary objects of semantic interest. This move makes all of basic modal logic rst-order, via our standard translation. The main questions then address what makes modal logics special as subspecies of rst-order logic. In particular, what is the basic semantic invariance for basic modal logic, which should play a role like Ehrenfeucht games or `partial isomorphism' in rst-order model theory? A key result here is the semantic characterization of the modal fragment of rst-order logic (modulo logical equivalence) as precisely those formulas in one free variable which are invariant for generated submodels and our `zigzag relations' [van Benthem, 1976]. In modern jargon, this says that these for-
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mulas are precisely the ones invariant for bisimulation. The latter link was also developed in Hennessy & Milner [1985], which matches modal formalisms in di erent strengths with coarser or ner process equivalences. For up-to-date expositions of the resulting analogies between modal logics and computational process theories, cf. [van Benthem and Bergstra, 1995; van Benthem et al., 1994], as well as various contributions in the volume [Ponse et al., 1995]. This development has led to a new look at connections between modal formalisms and rst-order logic. For instance, there are striking analogies between the meta-theories of both logics, whose precise extent and explanation is explored in de Rijke [1993], and Andreka, van Benthem & Nemeti [1998]. In particular, the latter paper investigates the hierarchy of nite-variable fragments for rst-order logic as a candidate for a general account of modal logic (cf. [Gabbay, 1981; van Benthem, 1991] for this view). Typically, modal formulas need only two variables over worlds in their standard translation, temporal formulas only three, and so on. Finite-variable fragments are natural, and may be considered as functionally complete modal formalisms (cf. the insightful game-based analysis of Kamp's Theorem in Immerman & Kozen [1987]). Nevertheless, Andreka, van Benthem & Nemeti [1998] also turn up an array of negative properties, and eventually propose another classi cation for modal languages in terms of restricting atoms for bounded quanti ers. The resulting `guarded fragments' can be analyzed much like the basic modal language, including analogous bisimulation techniques. In particular, these bisimulations now relate nite sequences of objects instead of single worlds, as in manydimensional modal logics (cf. [Marx and Venema, 1996] for the theory of such formalisms). Their correspondence theory, taken with respect to natural generalized frame conditions for arbitrary rst-order relations, still remains to be understood. [van Benthem, 1996b] is a general study of dynamic logics for computation and cognition, pursued via these techniques. One of its central concerns is expressive completeness of modal process logics vis-a-vis process equivalences like bisimulation.
Connections with Higher-Order Logic and Set Theory From rst-order correspondence, forays can be made into higher-order de nability. Sometimes, this move is suggested by the modal language itself. E.g., in propositional dynamic logic, program iteration naturally translates into a countable disjunction of nite repetitions. Thus, translation into the in nitary standard language L!1 ! seems the evident route. In nitary frame correspondences were brie y considered in van Benthem [1983], and their modal model theory is explored in [de Rijke, 1993; van Benthem and Bergstra, 1995]. Of course, one may restore a balance here, and consider an in nitary modal counterpart of L! , allowing arbitrary set conjunctions and disjunctions, which would be the most natural formalism invariant for
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bisimulation. Barwise and Moss [1995] take this line, linking up truth on models and correspondence on frames. (Another perspective on in nitary modal logic is given in [Barwise and van Benthem, 1996].) Among a number of original results, they prove that a modal formula has all its in nitary substitution instances true in a model M i it is true (in the usual second-order sense) on the frame collapse of that model taken with respect to the maximal bisimulation over M . As a direct consequence, frame correspondences for modal formulas imply model correspondences in in nitary modal logic. (The issue of good converses is still open). The original motivation for this type of investigation was that it relates modal logics to (non-well-founded) set theories. Linkages of this kind are further explored in d'Agostino [1995] which also raises the issue of more complex correspondences for modal axioms. For instance, she shows that the second-order Lob Axiom holds in a frame i that frame is transitive while its collapse with respect to the maximal bisimulation is irre exive. More generally, then, the interesting point about many correspondences is not that they must always reduce modal axioms to rst-order ones, but rather the fact that they reformulate modal principles to any more perspicuous classical formalism. Another natural candidate of the latter kind is second-order monadic 11 logic (cf. [Doets and van Benthem, 2001]). In particular, Doets [1989] shows how modal completeness theorems can sometimes be extended to cover this whole language. Moreover, many e ective translation methods (see below) turn out to work for this broader language anyway. Finally, van Benthem [1989b] points out how rst-order correspondence theory, suitably restated for second-order 11 formulas, is a natural generalization which handles so-called computable forms of Circumscription in the AI literature (which involves reasoning from a second-order `predicate-minimal' closure for rst-order axioms; cf. [Lifshitz, 1985]).
Translations Correspondence has become a conspicuous theme in the computational literature on theorem proving with intensional logics. A number of algorithms have been proposed, some of them rediscoveries of the Substitution Method and its ilk (cf. [Simmons, 1994]) and even much older results in secondorder logic [Doherty, Lukasiewicz and Szalas, 1994], others working with new `functional` translations better geared towards complete standard Skolemization and Resolution (cf. Ohlbach [1991; 1993]). One interesting feature of some of these algorithms is that they also produce useful equivalents for second-order modal principles. For instance, the typically non- rst-order McKinsey Axiom gets a natural equivalent quantifying over both individual worlds and Skolem functions witnessing its (non-Sahlqvist) antecedent. Finally, we mention the use of set-theoretic interpretations of the standard translation in d'Agostino, van Benthem, Montanari & Policriti [1995], which
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read the universal modality as describing a power set. This translation also works with an explicit axiom system for general frames plus one axiom stating that the relational successors of any point in a frame form a set. This shift in perspective reduces theorem proving in modal logics to deduction in weak computational set theories. Many of these translations can also be formulated so as to deal with extended modal formalisms or larger fragments of second-order logic.
Designing New Logics Finally, correspondence techniques have been used in `deconstructing' standard logics and designing new ones. For instance, one can interpret rstorder predicate logic over possible worlds models (`labelled transition systems') with assignments replaced by abstract states connected by abstract relations Rx modelling variable shifts. Then, standard predicate-logical validities turn out to express interesting frame properties, constraining possible computations, e.g., by Church-Rosser con uence properties (which match the rst-order axiom 9y8x ! 8x9y ). Moreover, one may want to impose certain restrictions on admissible valuations, such as `heredity constraints' for axioms Py ! 8xPy or Py ! [y=x]Px (van Benthem [1997; 1996b] have details). These abstract models re ect certain dependencies between admissible object values that may exist for individual variables. This theme is investigated more explicitly in [Alechina and van Benthem, 1993; Alechina, 1995], which design new generalized quanti er logics over `dependence models', rst proposed by Michiel van Lambalgen | where again the force of possible axioms is measured at least initially in terms of (Sahlqvist) frame correspondences. Related modal approaches to rst-order logic are found in [Venema, 1991; Marx, 1995]. ADDED IN PRINT (1999) Handbooks appear according to their own rhythms. Two years have elapsed since the updates were written for this Appendix. Here are a few further items of interest. D'Agostino [1998] contains new material on de nability in in nitary modal logics, a topic also pursued further by Barwise and Moss. Meyer Viol [1995] has examples of correspondence for intuitionistic predicate logic showing how intermediate axioms can be quite surprising in their content. Hollenberg [1998] is an extensive study of de nability, invariance and safety in modal process languages. Gerbrandy [1998] has interesting theorems on modal de nability and bisimulation invariance in a setting of non-well-founded set theory, with applications to dynamic logic of epistemic updates. Gradel [1999] is an excellent survey of progress made on the program of decidable guarded rst-order languages extending modal logic,
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including also xed-point operators. Van Benthem [1998] is an up-to-date survey of the de nability/correspondence paradigm, and the corresponding `tandem approach' to modal and classical logics. Finally, two modern texts on modal logic that take correspondence seriously are Blackburn, de Rijke and Venema [1999] and van Benthem [1999]. BIBLIOGRAPHY
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INDEX 23-formula, 125
1 L1 -sentences, 328, 350 L{prime logic, 89 -irreducible logic, 89 {-complex logic, 115 {-generated frame, 93, 160 n-transitive logic, 87, 160
actual world, 140 actual world condition, 141 aggregation, 336 alternative relation, 339 amalgamability, 152 anti-preservation, 345 atom, 94, 99 atomic, 326 axiomatic basis, 89 axiomatization nite, 89 independent, 89 problem, 97 recursive, 89 Barcan Formula, 368 Beth Property, 149 bimodal companion, 224 Birkho 's Theorem, 359 bisimulation, 249, 400 canonical, 334 canonical formula, 119 intuitionistic, 199 quasi-normal, 141 canonicity, 100 categorial connection, 362 CDC, 118 closed domain, 118 closed domain condition, 118
cluster assignment, 175 co nal subframe formula, 125 co nal subframe logic, 125 quasi-normal, 142 compact frame, 92 compactness, 112, 327 comparative, 375 complete set of formulas, 89 completeness, 325 completeness theorems, 332 complex variety, 114 complexity, 367 complexity function, 243 conditional logic, 373 con guration problem, 228 con uent, 325 congruential logic, 249 conservative formula, 155 correspondence, 325 counterfactual, 381 cover, 94 cycle free frame, 98, 160 d-cyclic set, 94 deduction theorem, 87 deductively equivalent formulas, 86 degree of incompleteness, 108 depth of a frame, 94 descriptive, 362 descriptive frame, 92, 160 di erence operator, 249 di erentiated frame, 92, 160 direct products, 357 directed, 325 disjoint union, 339 disjunction property, 211 modal, 211 distinguished point, 140
410 downward directness, 105 duality, 331 Dummett logic, 200 dynamic logic, 373 elementary equivalence, 340 elementary logic, 107 essentially negative formula, 209 lter representation, 392 nite embedding property, 128 nite model property exponential, 243 global, 113 polynomial, 243 rst-order de nable, 347 rst-order equivalent, 328 rst-order unde nability, 349 xed point operator, 250 focus, 132 frame formula, 120 fusion, 162 Godel translation, 196 general frame, 338, 361 generated subframe, 339 generation theorem, 341 global de nability, 349 global derivability, 87 global Kripke completeness, 113 graded modality, 249 Hallden completeness, 156 Henkin model, 359 Heyting algebra, 193, 392 higher-order correspondence, 370 homomorphic images, 357 inaccessible world, 159 incompleteness, 375 independent set of formulas, 89 inference rule admissible, 232 derivable, 231 intermediate axioms, 390
interpolant, 149 post-, 156 interpolation property, 149 for a consequence relation, 149 intersection of logics, 88 interval, 377 intuitionistic frame, 193 intuitionistic logic, 373 intuitionistic modal frame, 219 intuitionistic modal logic, 218 invariance, 371 isomorphism, 340 Jankov formula, 120 Kreisel{Putnam logic, 212 Kripke frame, 92, 325 Lob's Axiom, 116 Lob's Axiom, 333 Lowenheim{Skolem, 327 Lindenbaum Algebra, 358 Lindstrom Theorem, 352 linear tense logic, 176 local de nability, 349 local tabularity, 123 logic of a class of frames, 92 Los Equivalence, 350 McKinsey Axiom, 326 Medvedev's logic, 216 minimal modal logic, 333 minimal tense extension, 172 Minsky machine, 228 modal algebra, 331, 357 modal companion, 200 modal degree, 101 modal incompletness, 333 modal matrix, 140 modal predicate logic, 367 modal projection, 339 modal reduction principle, 354 models, 327 negation, 381
411 negative formula, 102 Nishimura formula, 195 Noetherian frame, 116 nominal, 249 non-eliminability, 101 non-iterative logic, 161 normal lter, 152 normal form, 123 open domain, 118, 198 p-morphims, 331 p-morphism, 92 partial order, 325 persistence, 100 polymodal frame, 160 polymodal logic, 159 polynomial, 358 polynomially equivalent logics, 243 positive formula, 102 possible worlds, 343 preservation, 342 pretabularity, 147 prime lter, 194 prime formula, 89 propositional quanti ers, 371 propositions, 343 pseudo-Boolean algebra, 193 quasi-normal logic, 139 reduced frame, 101 reduction, 92, 160 weak, 185 re ned frame, 92 re ned re ned, 160 re exive, 325 relevance logic, 380 replacement function, 173 restricted quanti ers, 364 Rieger{Nishimura lattice, 195 root, 92, 160 rooted frame, 92 Sahlqvist formula, 107, 161
Sahlqvist Theorem, 354 saturation, 362 Scott logic, 212 second-order equivalent, 327 semantical consequence, 242 si-fragment, 200 si-logic, 193 similarity, 382 simulation of a frame, 169 simulation of a logic, 169 skeleton, 196 skeleton lemma, 196 Smetanich logic, 200 splitting, 96 union-, 97 splitting pair, 90 standard translation, 135 Stone representation, 357 strict Kripke completeness, 97 strict sf-completeness, 129 strong global completeness, 113 strong Kripke completeness, 112 strongly positive formula, 103 structural completeness, 233 subalgebras, 357 subdirectly irreducible, 363 subframe, 116, 145, 160, 198 co nal, 126, 145 generated, 92, 160 subframe formula, 125 subframe logic, 125, 127 quasi-normal, 142 subreduction, 116 co nal, 116 quasi-, 141 weak, 185 substitutions, 354 sum of logics, 88 superamalgamability, 152 superintuitionistic logic, 193 surrogate, 163 surrogate frame, 185 symmetry, 336
412 t-line logic, 181 tabularity, 145 Tarski's criterion, 89 temporal modalities, 377 temporal order, 375 tense frame, 172 tense logic, 171, 373 tight frame, 92 time-line, 181 topological Boolean algebra, 195 transfer, 399 transitive, 325 translation, 327 ultra lter extension, 339 ultra lters, 344 ultrapower, 346 ultraproduct, 328 undecidable formula, 231 uniform formula, 124 uniform interpolation, 156 universal frame of rank n, 93 universal modality, 166 untied formula, 105 upward closed set, 92 valuation, 339 weak Kreisel{Putnam formula, 198 zigzag connection, 341, 370 zigzag morphism, 342 Zigzag Theorem, 342
