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 eective 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 eective 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 dierent 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 dierentiation 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 eect 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 dierences 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 dierence between (1) and (2). One possibility would be to hold that while both (1) and (2) are of the same form, viz. (i) they dier 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 dierence between (1) and (2), he feels, is a dierence 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 dierent 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 dierence 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 dierence 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 eort 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 eect 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 eorts 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 eort 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 dierent 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 dierent. 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 dierent 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

12

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 eorts of the latter had been syntactic, but Prior, whose interests lay in temporal notions, gave an algebraic avoured analysis which in eect 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;

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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 oered. 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 dierence 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, diering from that in [Segerberg, 1971]), and usually one would require much more, for example, closure under congruence (cgr), monotonicity (mon), or necessitation (nec):

16

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 oer these slightly more general de nitions as well. In this paper|and here we oer 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

18

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 dierent 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 oer little of interest beyond what normal logics oer, 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:

BASIC MODAL LOGIC

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 eort 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 oer. 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 dierent 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 [Jerey, 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 eective 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 dierent 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 eective) 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 eect, 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 eective, 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 eort 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 eect

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, dierent 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 dierent 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 dierence 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 eect 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 dierentiated, 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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35

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 eect). 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 dierentiated 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 dierentiated; 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 dierentiated. 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 dierentiated 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.

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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 dierence 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 dierent point. It is now immediate that hWLr ; RLr ; VLr i veri es L, but since this dierentiated 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 dierentiated 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 dierentiated 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);

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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

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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 dierence, 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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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 dierence 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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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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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-aairs', i.e. maximal consistent sets of `propositions'. The natural de nition of S on these `states-of-aairs' 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. [Jerey, 1990] R. C. Jerey. 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 dierent 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 dierence 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 diers 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 dierent 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 dierentiated 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 dierentiated 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 dierentiated, 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 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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95

clusters of depth d + 1 follows from the dierentiatedness 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 dierent 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

To eliminate the variable X ranging over P , we can use two simple observations. The rst one is purely set-theoretic:

ADVANCED MODAL LOGIC (3)

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 eectively 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 eectively 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

\

fi (: : : ; 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, eectively 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 dierentiated, 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 [email protected] 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

[email protected]

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 = [email protected] .

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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 dierent from union-splittings has [email protected] 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 [email protected] (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].

ADVANCED MODAL LOGIC

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

[ 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 ;

=

^

fi ! 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

ADVANCED MODAL LOGIC

115

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

116

M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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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117

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.

118

M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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,

ADVANCED MODAL LOGIC

119

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 eectively 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 eective class of nite frames (or algebras, matrices, models, etc.) and that there is an eective (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 eectively 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 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 dierent 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 eectively 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 Subequivalent 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

[email protected] @I Æ61 @ 2Æ @ [email protected] 2 Æ@I Æ62 3Æ @ 3 Æ @Æ 3 [email protected] @I 6 4Æ @ 4 Æ @Æ 4 (c)

133 1Æ

[email protected]@Æ61 3 Æ @Æ 3 [email protected]@ 6 5 Æ @Æ 5 [email protected]@ 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 dierent 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, dierent 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

= 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 dierent 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 (fg? ; 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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153

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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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 dierent 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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155

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. dierent 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 fi : i 2 I g also has the interpolation property. Proof. Suppose ' ! 2 L fi : i 2 I g. Then there is a nite J I , say J = f1; : : : ; lg, such that ' ! 2 L fi : 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 fi : i 2 I g and ! i.e., is an interpolant for ' ! in L fi : i 2 I g.

2 L fi : 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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157

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 .

158

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 Nmdierent 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: Dierentiated, 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 dierentiated if all the unimodal frames hW; Ri ; P i, for i 2 I , are dierentiated. 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 [email protected] 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 [email protected] 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 eectively 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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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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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 ('; ) = f1 ! :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 (eectively) 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

[email protected]@ 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 (dierentiated, 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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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 dierentiated frame F = hW; R; P i for Sim which contains only one point where is true. (Actually, every rooted dierentiated 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 dierentiatedness 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 dierentiated 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 dierentiated 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 dierentiated 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 dierentiated frames for Ls . Take a dierentiated 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!; >; ; ; ). 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!; >; (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 dierent, 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 ! > > > < > > > > :

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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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 dierent 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 [email protected]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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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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205

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

206

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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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

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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) dierent 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 f0 =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 dierent 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 [email protected] 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 [email protected]@ @[email protected] 6 [email protected]@@[email protected]61@I@161 @ @ @ @ @ Æ [email protected] Æ Æ[email protected]@ Æ6 Æ Æ@[email protected] 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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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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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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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 dierent 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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221

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 dierence 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 diers 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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225

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 dierent 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 [email protected]X0 XyXXXg2 [email protected] I a0 @ 6 @a10 @[email protected]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 eectively 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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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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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 eectively 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 eectively 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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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 dierent 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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(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 eectively 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 eectively 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 eective 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 eective 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 eectively 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 dierence 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 dierent 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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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 eective decision procedure for some of our logics then Cl can be decided by an equally eective 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 dierence operator or with nominals (or names). The semantics of nominals is similar to that of propositional variables; the dierence is that a nominal is true at exactly one point in a frame. For the dierence 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 dierence 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.

268

JAMES W. GARSON QML 1 Conceptual Domain (All individual concepts)

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 ws, 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 eect 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 dierence 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 dierent 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 dierent things exist in the dierent 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 dierent subdomains which we may ever intend for quanti er expressions, and it forces us to represent simple expressions of natural language dierently in dierent 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 eect 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 dierent objects in dierent 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 dierent 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 diers 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 dierence 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 dierent times. The slices of a thing picked out at dierent times may be quite dierent, 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

JAMES W. GARSON i

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 ws. 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 dierent 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 .] oers 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 dierent 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 dierent 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 dierent 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 dier 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 Jerey, 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. (, 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),

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(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 dierent 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 oered 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 oers 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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JOHAN VAN BENTHEM

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

CORRESPONDENCE THEORY

363

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,

364

JOHAN VAN BENTHEM

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.

CORRESPONDENCE THEORY

365

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;

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 eective 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 eective 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 dierent 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 dierentiation 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 eect 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 dierences 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 dierence between (1) and (2). One possibility would be to hold that while both (1) and (2) are of the same form, viz. (i) they dier 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 dierence between (1) and (2), he feels, is a dierence 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 dierent 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 dierence 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 dierence 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 eort 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 eect 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 eorts 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 eort 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 dierent 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 dierent. 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 dierent 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

12

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 eorts of the latter had been syntactic, but Prior, whose interests lay in temporal notions, gave an algebraic avoured analysis which in eect 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 oered. 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 dierence 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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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, diering 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 oer these slightly more general de nitions as well. In this paper|and here we oer 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

18

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 dierent 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 oer little of interest beyond what normal logics oer, 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:

BASIC MODAL LOGIC

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 eort 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 oer. 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 dierent 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 [Jerey, 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 eective 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 dierent 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 eective) 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 eect, 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 eective, 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 eort 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 eect

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, dierent 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 dierent 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 dierence 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.

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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 eect 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 dierentiated, 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):

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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

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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

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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

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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

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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

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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

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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.

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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;

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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 eect). 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

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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 :

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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

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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

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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

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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 dierentiated 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 dierentiated; 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 dierentiated. 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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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 dierentiated 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.

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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.

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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

56

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 dierence 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 dierent point. It is now immediate that hWLr ; RLr ; VLr i veri es L, but since this dierentiated 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 dierentiated 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 dierentiated 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

57

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:

58

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.

BASIC MODAL LOGIC

59

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

60

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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61

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,

62

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

63

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

64

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 dierence, 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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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 dierence 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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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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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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ROBERT BULL AND KRISTER SEGERBERG

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-aairs', i.e. maximal consistent sets of `propositions'. The natural de nition of S on these `states-of-aairs' 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

[Ackerman, 1956] W. Ackerman. Begrundung einer strengen Implikation. Journal of Symbolic Logic, 21, 113{128, 1956. [Alban, 1943] M. J. Alban. Independence of the primitive symbols of Lewis' calculi of propositions. Journal of Symbolic Logic, 8, 24{26, 1943. [Anderson and Belnap, 1975] A. R. Anderson and N. D. Belnap. Entailment: The Logic of Relevance and Necessity, Vol. 1, Princeton University Press, Princeton, 1975. [Anderson, 1980] C. A. Anderson. Some axioms for the logic of sense and denotation: alternative (0). Nous, 14, 217{234, 1980. [Bayart, 1959] A. Bayart. Quasi-adequation de la logique modale du second ordre S5 et atequation de la logique du premier ordre S5. Logique et analyse, 2, 99{121, 1959. [Becker, 1930] O. Becker. Zur Logik der Modalitaten. Jarbuch fur Philosophie und phanonenologische Forschung, 11, 496{548, 1930. [Belnap, 1981] N. D. Belnap. Modal and relevance logics: 1977. In Modern Logic|A Survey, E. Agazzi, ed. pp. 131{151. Reidel,Dordrecht, 1981. [Beth, 1959] E. W. Beth. The Foundations of Mathematics: A Study in the Philosophy of Science. North-Holland, Amsterdam, 1959. [Blok, 1980] W. J. Blok. The lattice of modal algebras: an algebraic investigation. Journal of Symbolic Logic, 45, 221{236, 1980. [Block, 1980a] W. J. Blok. Pretabular varieties of modal algebras. Studia Logica, 39, 101{124, 1980. [Boolos, 1979] G. Boolos. The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press, Cambridge, 1979. [Bowen, 1978] K. A. Bowen. Model Theory for Modal Logic. Reidel, Dordrecht, 1978. [Bull, 1965] R.A. Bull. An algebraic study of Diodoreanmodal systems. Journal of Symbolic Logic, 30, 58{64, 1965. [Bull, 1965a] R. A. Bull. A modal extension of intuitionistic logic. Notre Dame Journal of Formal Logic, 6, 142{146, 1965. [Bull, 1966] R. A. Bull. That all normal extensions of S4.3 have the nite model property. Zeit Math. Logik Grund., 12, 341{344, 1966. [Bull, 1966a] R. A. Bull. MIPC as the formalisation of an intuitionist concept of modality. Journal of Symbolic Logic, 31, 609{616, 1966. [Bull, 1967] R. A. Bull. On the extension of S4 with CLMpMLp. Notre Dame Journal of Formal Logic, 8, 325{329, 1967. [Bull, 1969] R. A. Bull. On modal logic with propositional quanti ers. Journal of Symbolic Logic, 34, 257{263, 1969. [Bull, 1982] R. A. Bull. Review. Journal of Symbolic Logic, 47, 440{445, 1982. [Bull, 1983] R. A. Bull. Review. Journal of Symbolic Logic, 48, 488{495, 1983. [Carnap, 1942] R. Carnap. Introduction to Semantics. Harvard University Press, Cambridge, MA, 1942. [Carnap, 1947] R. Carnap. Meaning and Necessity: A Study in Semantics and Modal Logic. The University of Chicago Press, Chicago, 1947.

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[Chellas, 1980] B. F. Chellas. Modal Logic: An Introduction. Cambridge University Press, Cambridge, 1980. [Church, 1946] A. Church. A formulation of the logic of sense and denotation. Abstract. Journal of Symbolic Logic, 11, 31, 1946. [Church, 1951] A. Church. A formulation of the logic of sense and denotation. In Structure, Method and Meaning. Essays in Honor of Henry M. Scheer, P. Henle et al., eds. pp. 3{24. the Liberal Arts Press, NY, 1951. [Church, 1951a] A. Church. The weak theory of implication. In Kontrolliertes Denken: Untesuchungen zum Logikkalkul under der Einzelwissenschaften, Menne et al., eds. pp. 22{37. Kommissions-Verlag Karl Alber, Munich, 1951. [Church, 1973{74] A. Church. Outline of a revised formulation of the logic of sense and denotation. Nous, 7, 24{33; 8, 135{156, 1973{74. [Cresswell, 1967] M. Cresswell. A Henkin completeness theorem for T . Notre Dame Journal of Formal Logic, 8, 186{190, 1967. [Curley, 1975] E. M. Curley. The development of Lewis' theory of strict implication. Notre Dame Journal of Formal Logic, 16, 517{527, 1975. [Curry, 1950] H. B. Curry. A Theory of Formal Deducibility. University of Notre Dame Press, Notre Dame, IN, 1950. [Dugundj, 1940] J. Dugundj. Note on a property of matrices for Lewis and Langford's calculi of propositions. Journal of Symbolic Logic, 5, 150{151, 1940. [Dummett and Lemmon, 1959] M. A. E. Dummett and E. J. Lemmon. Modal logics between S4 and S5. Zeit Math. Logik. Grund., 3, 250{264, 1959. [Esaia and Meskhi, 1977] L. Esakia and V. Meskhi. Five critical modal systems. Theoria, 43, 52{60, 1977. [Feyes, 1965] R. Feys. Modal Logics. Edited with some complements by Joseph Dopp, E. Nauwelaerts, Louvainand Gauthier-Vallars, Paris, 1965. [Fine, 1970] K. Fine. Propositional quantifers in modal logic. Theoria, 36, 336{346, 1970. [Fine, 1971] K. Fine. The logics containing S4.3. Zeit. Math. Logik. Grund., 17, 371{ 376, 1971. [Fine, 1972] K. Fine. Logics containing S4 without the nite model property. In Conference in Mathematical Logic, London 1970, W. Hodges, ed. pp. 88{102. Vol. 255 Lecture Notes in Mathematics, Springer- Verlag, Berlin, 1972. [Fine, 1974] K. Fine. An incomplete logic containing S4. Theoria, 40, 23{29, 1974. [Fine, 1974a] K. Fine. An ascending chain of S4 logics. Theoria, 40, 110{116, 1974. [Fine, 1974b] K. Fine. Logics containing K4, Part I. Journal of Symbolic Logic, 39, 31{42, 1974. [Fine, 1975] K. Fine. Some connections between elementary and modal logic. In Proceedings of the Third Scandinavian Logic Symposium, S. Kanger, ed. pp. 15{31. NorthHollnd, Amsterdam, 1975. [Fine, 1975a] K. Fine. Normal formas in modal logic. Notre Dame Journal of Formal Logic, 16, 229{234, 1975. [Fine, 1977] K. Fine. Prior on the construction of possible worlds and instants. In Worlds, Times and Selves, A. N. Prior and K. Fine, eds. pp. 116{ 161. Duckworth, London, 1977. [Fine, 1977a] K. Fine. Properties, propositions and sets. Journal of Philosophical Logic, 6, 135{191, 1977. [Fine, 1978] K. Fine. Model theory for modal logic. Journal of Philosophical Logic, 7, 125{156, 1978/81. [Fine, 1980] K. Fine. First order modal theories, I: Sets. Nous, 15, 177{205; II: Propositions. Studia Logica, 34, 159{202; III: Facts. Synthese, 53, 43{122, 1980, 1981, 1982. [Fischer-Servi, 1977] G. Fischer-Servi. On modal logic with an intuitionist base. Studia Logica, 36, 141{149, 1977. [Fischer-Servi, 1981] G. Fischer-Servi. Semantics for a class of intuitionist modal calculi. In Italian Studies in the Philosophy of Science, M. L. Dalla Chiara, ed. pp. 59{72. Reidel, Dordrecht, 1981. [Fitch, 1937] F. B. Fitch. Modal functions in two-valued logic. Journal of Symbolic Logic, 2, 125{128, 1937.

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[Fitch, 1939] F. B. Fitch. Note on modal functions. Journal of Symbolic Logic, 4, 115{ 116, 1939. [Fitch, 1948] F. B. Fitch. Intuitionistic modal logic with quanti ers. Portugaliae Mathematica, 7, 113{118, 1948. [Fitch, 1952] F. B.Fitch. Symbolic Logic: An Introduction. Ronald Press, NY, 1952. [Follesdal, 1989] D. Follesdal. Von Wright's modal logic. In The Philosophy of Georg Henrik Von Wright, P. A. Schilpp, ed. 1989. [Follesdal and Hilpinen, 1971] D. Follesdal and R. Hilpinen. Denontic logic: An introduction. In Deontic Logic: Introductory and Systematic Reasings, R. Hilpinen, ed. pp. 1{35. Reidel, Drodrecht, 1971. [Friedman, 1975] H. Friedman. One hundred and two problems in mathematical logic. Journal of Symbolic Logic, 40, 113{129, 1975. [Gabbay, 1976] D. M. Gabbay. Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics. Reidel, Dordrecht, 1976. [Gabbay, 1981] D. M. Gabbay. Semantical Investigations in Heyting's Intuitionistic Logic. Reidel, Dordrecht, 1981. [Gerson, 1975] M. Gerson. The inadequacy of the neighbourhood semantics for modal logic. Journal of Symbolic Logic, 40, 141{148, 1975. [Gerson, 1975a] M. Gerson. An extension of S4 complete for the neighbourhood semantics but incomplete for the relational semantics. Studia Logica, 34, 333{342, 1975. [Gerson, 1976] M. Gerson. A neighbourhood frame for T with no equivalent relational frame. Zeit. Math. Logik. Grund., 22, 29{34, 1976. [Godel, 1933] K. Godel. Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergenisse eines mathematisches Kolloquiums, 4, 39{40, 1933. [Goldblatt, 1975] R. I. Goldblatt. First-order de nability in modal logic. Journal of Symbolic Logic, 40, 35{40, 1975. [Goldblatt, 1976] R. I. Goldblatt. Methamathematics of modal logic. Reports on Mathematical Logic, 6, 41{78; 7, 21{52, 1976. [Goldblatt and Thomason, 1975] R. I. Goldblatt and S. K. Thomason. Axiomatic classes in propositional modal logic. In Algebra and Logic, J. N. Crossley, ed. pp. 163{173. Vo. 450 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1975. [Grzegoczyk, 1981] A. Grzegoczyk. Individualistic formal approach to deontic logic. Studia Logica, 40, 99{102, 1981. [Guillaume, 1958] M. Guillaume. Rapports entre calculs propositionnels modaux et topologie impliques par certaines extensions de la methode e tableaux semantiques. Comptes renus hebdomaires des seances de l'Academie des Sciences, 246, 1140{1142, 2207{2210; 247, 1281{1283, Gauthiers- Villars, Paris, 1958. [Hallden, 1949] S. Hallden. Results concerning the decision problem of Lewis's calculi S3 and S4. Journal of Symbolic Logic, 15, 230{236, 1949. [Hansson and Gardenfors, 1973] B. Hansson and P. Gardenfors. A guide to intensional semantics. In Modality, Morality and other Problems of Sense and Nonsense. Essays Dedicated to Soren Hallden, pp. 151{167. Gleerup, Lund, 1973. [Hilpinen, 1971] R. Hilpinen. Deontic Logic: Introductory and Systematic Readings. Reidel, Dordrecht, 1971. [Hintikka, 1955] J. Hintikka. Form and content in quanti cation theory. Acta Philosophica Fennica, 8, 11{55, 1955. [Hintikka, 1957] J. Hintikka. Quanti ers in Deontic Logic. Societas Scientarum Fennica, Commentationes humanarum litterarum 23:4, Helsingfors, 1957. [Hintikka, 1961] J. Hintikka. Modality and quanti cation. Theoria, 27, 119{128, 1961. Revised veesion reprinted in Hinktikka [1969]. [Hintikka, 1962] J. Hintikka. Knowledge and Belief: An Introduction to the Logic of the Two Notions. Cornell University Press, Ithaca, NY, 1962. [Hintikka, 1963] J. Hintikka. The modes of modality. Acta Philosophica Fennica, 16, 65{82, 1963. Reprinted in Hintikka [1969]. [Hintikka, 1969] J. Hinktikka. Models for Modalities: Selected Essays. Reidel, Dordrecht, 1969. [Hintikka, 1969a] J. Hinktikka. Review. Journal of Symbolic Logic, 34, 305{306, 1969.

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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 dierent 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 dierence 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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87

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 diers 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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89

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 dierent 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 dierentiated 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 dierentiated 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 dierentiated, 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 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.

ADVANCED MODAL LOGIC

95

clusters of depth d + 1 follows from the dierentiatedness 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 dierent 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

To eliminate the variable X ranging over P , we can use two simple observations. The rst one is purely set-theoretic:

ADVANCED MODAL LOGIC (3)

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 eectively 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 eectively 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).

ADVANCED MODAL LOGIC

105

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

\

fi (: : : ; 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, eectively 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 dierentiated, 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 [email protected] 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

[email protected]

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 = [email protected] .

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111

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 dierent from union-splittings has [email protected] 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 [email protected] (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

[ 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 ;

=

^

fi ! 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

ADVANCED MODAL LOGIC

115

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

116

M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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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117

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 eectively 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 eective class of nite frames (or algebras, matrices, models, etc.) and that there is an eective (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 eectively 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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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 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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125

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 dierent 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 eectively 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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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127

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

128

M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 Subequivalent successors, Mn+1 6j= . And since we keep in Fn+1 points which

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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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

[email protected] @I Æ61 @ 2Æ @ [email protected] 2 Æ@I Æ62 3Æ @ 3 Æ @Æ 3 [email protected] @I 6 4Æ @ 4 Æ @Æ 4 (c)

133 1Æ

[email protected]@Æ61 3 Æ @Æ 3 [email protected]@ 6 5 Æ @Æ 5 [email protected]@ 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 dierent 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, dierent 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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

= 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 dierent 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

Now we construct by induction a nite rooted model for GL refuting ! . As its root we take (fg? ; 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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153

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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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 dierent 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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155

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. dierent 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 fi : i 2 I g also has the interpolation property. Proof. Suppose ' ! 2 L fi : i 2 I g. Then there is a nite J I , say J = f1; : : : ; lg, such that ' ! 2 L fi : 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 fi : i 2 I g and ! i.e., is an interpolant for ' ! in L fi : i 2 I g.

2 L fi : 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 Nmdierent 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: Dierentiated, 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 dierentiated if all the unimodal frames hW; Ri ; P i, for i 2 I , are dierentiated. 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 [email protected] 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 [email protected] 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 eectively 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 ('; ) = f1 ! :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 (eectively) 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

[email protected]@ 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 (dierentiated, 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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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 dierentiated frame F = hW; R; P i for Sim which contains only one point where is true. (Actually, every rooted dierentiated 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 dierentiatedness 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 dierentiated 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 dierentiated 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 dierentiated 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 dierentiated frames for Ls . Take a dierentiated 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!; >; ; ; ). 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!; >; (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 dierent, 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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183

(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 ! > > > < > > > > :

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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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 dierent 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 [email protected]Æ ; ?) @

Int + ] (

201

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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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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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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

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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) dierent 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 f0 =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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 dierent 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 [email protected] 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 [email protected]@ @[email protected] 6 [email protected]@@[email protected]61@I@161 @ @ @ @ @ Æ [email protected] Æ Æ[email protected]@ Æ6 Æ Æ@[email protected] 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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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 dierent 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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221

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 dierence 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 diers 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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225

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 dierent 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 [email protected]X0 XyXXXg2 [email protected] I a0 @ 6 @a10 @[email protected]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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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 eectively 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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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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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 eectively 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 eectively 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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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 dierent 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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(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 eectively 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 eectively 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 eective 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 eective 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 eectively 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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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 dierence 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 dierent 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

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-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 eective decision procedure for some of our logics then Cl can be decided by an equally eective 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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M. ZAKHARYASCHEV, F. WOLTER, AND A. CHAGROV

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 dierence operator or with nominals (or names). The semantics of nominals is similar to that of propositional variables; the dierence is that a nominal is true at exactly one point in a frame. For the dierence 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 dierence 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.

268

JAMES W. GARSON QML 1 Conceptual Domain (All individual concepts)

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 ws, 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 eect 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 dierence 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 dierent 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 dierent things exist in the dierent 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 dierent subdomains which we may ever intend for quanti er expressions, and it forces us to represent simple expressions of natural language dierently in dierent 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 eect 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 dierent objects in dierent 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 dierent 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 diers 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 dierence 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 dierent times. The slices of a thing picked out at dierent times may be quite dierent, 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

JAMES W. GARSON i

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 ws. 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 dierent 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 .] oers 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 dierent 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 dierent 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 dierent 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 dier 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 Jerey, 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. (, 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),

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(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 dierent 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 oered 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 oers 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;

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