Handbook of the History of Logic Volume 7 Logic and the Modalities in the Twentieth Century Handbook of the History of Logic Volume 7 Logic and the Modalities in the Twentieth Century ELSEVIER NORTH HOLLAND Handbook of the History of Logic Volume 7 Logic and the Modalities in the Twentieth Century Edited by Dov M. Gabbay Department of Computer Science King’s College London Strand, London, WC2R 2LS, UK and John Woods Philosophy Department University of British Columbia Vancouver, BC Canada, V6T 1Z1 and Department of Computer Science King’s College London Strand, London, WC2R 2LS, UK and Department of Philosophy University of Lethbridge Lethbridge, Canada ELSEVIER NORTH HOLLAND Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris San Diego - San Francisco - Singapore - Sydney - Tokyo Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2006 Copyright © 2006 Elsevier B.V. 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Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-51622-0 ISBN-10: 0-444-51622-0 ISBN-13: 978-0-444-51596-4 (set) ISBN-10: 0-444-51596-8 (set) For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1 CONTENTS vii Preface xi List of Contributors 1 Mathematical Modal Logic: A View of its Evolution Rob Goldblatt 99 Epistemic Logic Paul Gochet and Pascal Gribomont 197 Deontic Logic Paul McNamara 289 Relevant and Substructural Logics Greg Restall 399 A. N. Prior’s Logic Peter Øhrstrøm and Per F. V. Hasle 447 Modern Temporal Logic: The Philosophical Background Peter Øhrstrøm and Per F. V. Hasle 499 The Gamut of Dynamic Logics Jan van Eijck and Martin Stokhof 601 Situation Theory and Situation Semantics Keith Devlin 665 Dialogue Logic Erik Krabbe 705 Index This Page is Intentionally Left Blank PREFACE In its traditional sense, a modal logic is one whose logical vocabulary contains the modal expressions “possibly”, “necessarily” and “contingently”, construed as sentence operators. If the first half of the twentieth century can lay fair claim to having produced the deep and definitive accounts of classical logic, perhaps the dominant achievement in the next quarter century was the attainment of a firm semantic grip on a hefty plurality of modal systems, marked by the seminal works of Hintikka, Kanger and Kripke. The semanticizing of modal sentences — apart from the importance intrinsic to such an achievement — opened up an important tension between modal and classical logics. Perhaps the most significant difference is that, whereas classical systems are extensional, modal setups are intensional, a happenstance which various philosophers of logic have greeted with suspicion and — in some cases — incredulity. Some of the skeptics — Quine and Harman are two — raised doubts about whether a modal system could have the bona fides of a genuine logic. This notwithstanding, the great burst of energy in the modal research programmes since the 1950s has proved irresistible, and the central semantic idea of accessibility relations on alternative possible worlds has had a philosophical influence well beyond the confines of logic, especially in the philosophy of language. One of the byproducts of the modal groundswell is that there are a great many more interpreted systems of the modals “possibly” and “necessarily” than there are different meanings of these terms in ordinary English. It is easy to see that they are ambiguous in English, that “possibly” encompasses the quite different senses of logical, physical, causal, and practical possibility (ditto “necessarily”). But the sheer plurality of logical systems in which these terms are centrally at issue greatly exceeds this rather modest number of ordinary-language meanings. It is not wholly clear how to understand this proliferation. One possibility is that logic has a greater capacity to identify different concepts of possibility than do native speakers of languages such as English. It is also possible that the multiplication of heretofore unrecognized concepts of possibility is more a matter of the free creation of the theoretical logician. Whatever is to be said for these and other options, it is safe to say that, in having taken the modal turn, logic took on a more experimental character than was evident in the classical heyday. 1 Here, too, 1 Intimations of the experimental proclivities of modal logicians are evident in Aristotle’s modal logic, of which there are up to five distinct treatments of logical necessity. Then, too, the stream of systems produced by C.I. Lewis from 1912 into the 1930s encompasses vastly different axiomatizations. viii it is not entirely clear what to make of this. Of the golden age of classical logic it can be said with some confidence that logicians took themselves to be doing one of two things. Taking the implication relation as an example, either they were formalizing the pre-existing concept of implication or they were originating a concept designed to facilitate some larger purpose, such as the construction of logically precise languages adequate for science or capable of supporting the reductive burdens of logicism. Part of the answer may be that the model theoretic apparatus needed for the interpretation of modal systems is more complex, and admits of greater recombinations of its elements, than do the standard models of classical logic. Accordingly, it may be more natural for the modal theorist to reconfigure a possible worlds semantics and wait for the kind of, e.g., implication relation it embeds to “fall out”. We should take care not to over-press the contrast between analyzing pre- existing concepts and fashioning new ones. The distinction is present in Kant’s pre-critical writings, and persists in the works of his maturity. Kant saw analysis as making concepts clear , a job for philosophy. Synthesis was the business of making clear concepts , a task that falls to the mathematician. Since logic’s great classical interlude arose from the contributions of philosophers and mathematicians alike, we cannot say, especially in the aftermath of the paradox that dethroned intu- itive (or “analytic”) set theory, that classical logic is synthesis-free. Far from it. What is more, apart from the local disputes within classical logic itself, there were early rivals, such as intuitionism. Even so, classical logic has not been especially pluralistic, whereas modal logic is vigorously so. It is moreover a rather pacific and non-antagonistic pluralism, which attests further to its readiness to view logic as the exploration of mathematically interesting languages and model theoretic structures, with a focus that is a good deal less analytical or philosophical than most classical variations on classical semantics. It also bears on this issue that the semantics of some of the 20th century’s earliest axiom systems — Lewis’ S2 and S3 for example — must stretch themselves beyond ordinary recognition in order to keep up with the axioms. S2 and S3 cannot be semanticized in a normal worlds approach; so nonnormal worlds were postulated (more experimentation still), giving inadvertent anticipation of somewhat later paraconsistent developments. A further case in point is the attempt by relevance logicians to impose relevance constraints on the implication relation, so as to evade the classical theorem that everything whatever is implied by an inconsistency. In most relevant approaches, the disjunctive syllogism rule is demoted from a valid to a merely admissible rule. On its face a rather slight adjustment, this actually strips these logics of the truth-functional character of their classical predecessors. More intensionality still. An attraction of the traditional alethic modals is the ease with which they bear new interpretations in the breakthrough work of Hintikka, von Wright and others in epistemic and deontic logic. These were significant developments twice-over. On the one hand, the new logics of knowledge and belief, and of obligation and permission, were able to retain much of the syntactic and semantic machinery of Preface Preface ix their alethic predecessors, showing that all these logics are to a degree variations of each other. On the other hand, however, the logics of knowledge and obligation had taken yet another step away from classical logic. Not only are these newer logics intensional and more experimentally oriented than their classical vis-` a-vis, but there is now the looming presence of agents operating in time . We say “looming” rather than “overt” inasmuch as neither agents nor times are much developed, if at all, in the semantics of these particular systems. The ephemeral presence of agents and times is important in another respect. It indicates that logic was developing in ways that would satisfy a broader interpre- tation of the adjective “modal”. If one were to consult The New Oxford Dictionary of English , corrected reprint, 2001, it would be seen that in the entry for modal logic the first reference is not to a logic of possibility and necessity, but rather to a logic in which sentences are subject to “some qualification”. If, then, we were to accept the trichotomy of the basic modes of language introduced by the linguist C.W. Morris — the trichotomony between syntax, semantics and pragmatics — we would be reminded that a pragmatic approach to language is one that takes expressly into account the roles of language-users and the contexts in which they operate. By these lights, the developments in the 1950s and 1960s within the epistemic and deontic adaptations of alethic modal logic mark the transition of logic from a purely syntacto-semantic enterprise, to a Morrisean” enterprise in which agents, times and situations have a load-bearing role. This was the prag- matic turn in logic. 2 In the broad sense of “modal”, a pragmatic logic counts as modal, a happenstance that the Editors have allowed themselves to be guided by in organizing the present volume. In addition to chapters on the traditional modal logics, their epistemic and deontic variations and relevant logic, there are chapters on systems in which the times of utterance are taken note of, in which temporal change is tracked, in which an utterer’s situation is taken into account, in which interpersonal utterance is acknowledged, and in which agents compete with one another in the furtherance of their interests. In each case, the sentences of the logic are modified by these other factors — time, change, agents, situations, dialogue roles, and procedural strategies. Modal logics in the broad sense reflect another change in logic’s conception of itself. In the classical heyday, logicians were preoccupied with the analysis of properties (such as implication and logical truth) of abstractly linguistic constructions or of linguistic artifacts in relation to abstractly set theoretic structures. If such logicians gave any thought to the ins- and-outs of human reasoning in the here and now, it was much the received view that the classical laws were also norms of reasoning, albeit in a highly idealized form. Even so, the attention to reasoning was at best an afterthought. 3 With the rise of modern modal logic, the emphasis began to shift. Under press 2 The word “pragmatic” invites confusion. In its logico-epistemological sense, it is the Quinean doctrine that no principle of logic is immune from overthrow. In its linguistic meaning, it is a logic that takes express note of the role of linguistic agents. The second sense is intended here. 3 The pretensions of so-called natural deduction systems to be more “natural” than axiomatic systems reflected rather more a distrust of the epistemic privilege that logicians sought to extend to their axioms than to a burning interest in getting reasoning on the ground right. “ x of developments in computer science and argumentation theory (chiefly dialogue logic), logic started a shift toward a greater emphasis on reasoning. What we find in the chapters of this volume is an attempt, to the extent possible, to lodge pragmatic developments affecting agents and situations in the methodology and principal attainments of the classical analyses of implication and the like. No one thinks that modalizing the implication relation either narrowly or broadly will leave the classical analysis untouched. But, for the most part, there is a widespread desire on the part of modal logicians to retain as much of classical logic as comports with their modal ambitions. What we see in the proliferation of modal logics is not, therefore, a revolution in logic but a development. It is a development very much in progress as we write. But it is already wholly clear that it has broken the research programme in logic wide-open, and has given rise to questions and challenges that are not likely to be settled with any definiteness for some time to come. Once again the Editors are deeply and most gratefully in the debt of the vol- ume’s able authors. The Editors also warmly thank the following persons: Profes- sor Margaret Schabas, Head of the Philosophy Department, and Professor Nancy Gallini, Dean of the Faculty of Arts, at the University of British Columbia; Profes- sor Bryson Brown, Chair of the Philosophy Department and his successor Michael Stingl, and Professor Christopher Nicol, Dean of the Faculty of Arts and Sci- ence, at the University of Lethbridge; Professor Alan Gibbons, Head of the Com- puter Science Department, and his successor Andrew Jones, at King’s College London; Jane Spurr, Publications Administrator in London; Dawn Collins and Carol Woods, Production Associates in Lethbridge and Vancouver, respectively; and our colleagues at Elsevier, Senior Publisher, Arjen Sevenster, and Production Associate, Andy Deelen. Dov M. Gabbay King’s College London John Woods University of British Columbia and King’s College London and University of Lethbridge Preface CONTRIBUTORS Keith Devlin CSLI, Stanford University, 210 Panama Street, Stanford, CA 94305-4115, USA. devlin@csli.stanford.edu Jan van Eijck CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. jve@cwi.nl Dov M. Gabbay Department of Computer Science, King’s College London, Strand, London WC2R 2LS, UK dg@dcs.kcl.ac.uk Paul Gochet Universit ́ e de Li` ege, Institut Montefiore, B28, B-4000 Li` ege, Belgium. pgochet@ulg.ac.be Robert Goldblatt School of Mathematics, Statistics and Computer Science, Victoria University, P. O. Box 600, Wellington, New Zealand. Rob.Goldblatt@vuw.ac.nz Pascal Gribomont Universit ́ e de Li` ege, Institut Montefiore, B28, B-4000 Li` ege, Belgium. gribomont@montefiore.ulg.ac.be Per F. V. Hasle Department of Communication Aalborg University, Denmark. phasle@hum.aau.dk Erik C. W. Krabbe Department of Theoretical Philosophy, Faculty of Philosophy, Rijksuniversiteit Gronin- gen, Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands. E.C.W.Krabbe@rug.nl Paul McNamara Philosophy Department, Hamilton Smith Hall, University of New Hampshire, Durham, NH 03824, USA. paulm@cisunix.unh.edu Peter Øhrstrøm Department of Communication, Aalborg University, Denmark. poe@hum.aau.dk xii Greg Restall Department of Philosophy, University of Melbourne, Victoria 3010, Australia. restall@unimelb.edu.au Martin Stokhof ILLC / Department of Philosophy, Faculty of Humanities, Universiteit van Amsterdam, Nieuwe Doelenstraat 15, 1012 CP Amsterdam, The Netherlands. m.j.b.stokhof@uva.nl John Woods Philosophy Department, University of British Columbia, Vancouver, British Columbia Canada, V6T 1Z1 jhwoods@interchange.ubc.ca Contributors MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION Robert Goldblatt . . . there is no one fundamental logical no- tion of necessity , nor consequently of possi- bility . If this conclusion is valid, the subject of modality ought to be banished from logic, since propositions are simply true or false . . . [Russell, 1905] 1 INTRODUCTION Modal logic was originally conceived as the logic of necessary and possible truths. It is now viewed more broadly as the study of many linguistic constructions that qualify the truth conditions of statements, including statements concerning knowl- edge, belief, temporal discourse, and ethics. Most recently, modal symbolism and model theory have been put to use in computer science, to formalise reasoning about the way programs behave and to express dynamical properties of transi- tions between states. Over a period of three decades or so from the early 1930’s there evolved two kinds of mathematical semantics for modal logic. Algebraic semantics interprets modal connectives as operators on Boolean algebras. Relational semantics uses relational structures, often called Kripke models , whose elements are thought of variously as being possible worlds, moments of time, evidential situations, or states of a computer. The two approaches are intimately related: the subsets of a re- lational structure form a modal algebra (Boolean algebra with operators), while conversely any modal algebra can be embedded into an algebra of subsets of a relational structure via extensions of Stone’s Boolean representation theory. Tech- niques from both kinds of semantics have been used to explore the nature of modal logic and to clarify its relationship to other formalisms, particularly first and sec- ond order monadic predicate logic. The aim of this article is to review these developments in a way that provides some insight into how the present came to be as it is. The pervading theme is the mathematics underlying modal logic, and this has at least three dimensions. To begin with there are the new mathematical ideas: when and why they were Dov M. Gabbay and John Woods (Editors) c Handbook of the History of Logic. Volume 7 © 2006 Elsevier B V. All rights reserved. 2 Robert Goldblatt introduced, and how they interacted and evolved. Then there is the use of methods and results from other areas of mathematical logic, algebra and topology in the analysis of modal systems. Finally, there is the application of modal syntax and semantics to study notions of mathematical and computational interest. There has been some mild controversy about priorities in the origin of relational model theory, and space is devoted to this issue in section 4. An attempt is made to record in one place a sufficiently full account of what was said and done by early contributors to allow readers to make their own assessment (although the author does give his). Despite its length, the article does not purport to give an encyclopaedic coverage of the field. For instance, there is much about temporal logic (see [Gabbay et al. , 1994]) and logics of knowledge (see [Fagin et al. , 1995]) that is not reported here, while the surface of modal predicate logic is barely scratched, and proof theory is not discussed at all. I have not attempted to survey the work of the present younger generation of modal logicians (see [Chagrov and Zakharyaschev, 1997], [Kracht, 1999], and [Marx and Venema, 1997], for example). There has been little by way of historical review of work on intensional semantics over the last century, and no doubt there remains room for more. Several people have provided information, comments and corrections, both his- torical and editorial. For such assistance I am grateful to Wim Blok, Max Cress- well, John Dawson, Allen Emerson, Saul Kripke, Neil Leslie, Ed Mares, Robin Milner, Hiroakira Ono, Amir Pnueli, Lawrence Pedersen, Vaughan Pratt, Colin Stirling and Paul van Ulsen. This article originally appeared as [Goldblatt, 2003c]. As well as corrections and minor adjustments, there are two significant additions to this version. The last part of section 6.6 has been rewritten in the light of the discovery in 2003 of a solution of what was described in the first version as a “perplexing open question”. This was the question of whether a logic validated by its canonical frame must be characterised by a first-order definable class of frames. Also, a new section 7.7 has been added to describe recent work in theoretical computer science on modal logics for “coalgebras”. 2 BEGINNINGS 2.1 What is a Modality? Modal logic began with Aristotle’s analysis of statements containing the words “necessary” and “possible”. 1 These are but two of a wide range of modal connec- tives , or modalities that are abundant in natural and technical languages. Briefly, a modality is any word or phrase that can be applied to a given statement S to create a new statement that makes an assertion about the mode of truth of S : 1 For the early history of modal logic, including the work of Greek and medieval scholars, see [Bochenski, 1961] and [Kneale and Kneale, 1962]. The Historical Introduction to [Lemmon and Scott, 1966] gives a brief but informative sketch. Mathematical Modal Logic: A View of its Evolution 3 about when, where or how S is true, or about the circumstances under which S may be true. Here are some examples, grouped according to the subject they are naturally associated with tense logic: henceforth, eventually, hitherto, previously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being . . . deontic logic: it is obligatory/forbidden/permitted/unlawful that epistemic logic: it is known to X that, it is common knowledge that doxastic logic: it is believed that dynamic logic: after the program/computation/action finishes, the program enables, throughout the computation geometric logic: it is locally the case that metalogic: it is valid/satisfiable/provable/consistent that The key to understanding the relational modal semantics is that many modalities come in dual pairs, with one of the pair having an interpretation as a universal quantifier (“in all. . . ”) and the other as an existential quantifier (“in some. . . ”). This is illustrated by the following interpretations, the first being famously at- tributed to Leibniz (see section 4). necessarily in all possible worlds possibly in some possible world henceforth at all future times eventually at some future time it is valid that in all models it is satisfiable that in some model after the program finishes after all terminating executions the program enables there is a terminating execution such that It is now common to use the symbol for a modality of universal character, and for its existential dual. In systems based on classical truth-functional logic, is equivalent to ¬ ¬ , and to ¬ ¬ , where ¬ is the negation connective. Thus “necessarily” means “not possibly not”, “eventually” means “not henceforth not”, a statement is valid when its negation is not satisfiable, etc. Notation Rather than trying to accommodate all the notations used for truth-functional connectives by different authors over the years, we will fix on the symbols ∧ , ∨ , ¬ , → and ↔ for conjunction, disjunction, negation, (material) implication, and (material) equivalence. The symbol ⊤ is used for a constant true formula, equivalent to any tautology, while ⊥ is a constant false formula, equivalent to ¬⊤ We also use ⊤ and ⊥ as symbols for truth values. 4 Robert Goldblatt The standard syntax for propositional modal logic is based on a countably infinite list p 0 , p 1 , . . . of propositional variables , for which we typically use the letters p, q, r . Formulas are generated from these variables by means of the above connectives and the symbols and . There are of course a number of options about which of these to take as primitive symbols, and which to define in terms of primitives. When describing the work of different authors we will sometimes use their original symbols for modalities, such as M for possibly , L or N for necessarily , and other conventions for deontic and tense logics. The symbol n stands for a sequence · · · of n copies of , and likewise n for · · · ( n times). A systematic notation will also be employed for Boolean algebras: the symbols + , · , − denote the operations of sum (join), product (meet), and complement in a Boolean algebra, and 0 and 1 are the greatest and least elements under the ordering ≤ given by x ≤ y iff x · y = x The supremum (sum) and infimum (product) of a set X of elements will be denoted ∑ X and ∏ X (when they exist). 2.2 MacColl’s Iterated Modalities The first substantial algebraic analysis of modalised statements was carried out by Hugh MacColl, in a series of papers that appeared in Mind between 1880 and 1906 under the title Symbolical Reasoning , 2 as well as in other papers and his book of [1906]. MacColl symbolised the conjunction of two statements a and b by their concatenation ab , used a + b for their disjunction, and wrote a : b for the statement “ a implies b ”, which he said could be read “if a is true, then b must be true”, or “whenever a is true, b is also true”. The equation a = b was used for the assertion that a and b are equivalent, meaning that each implies the other. Thus a = b is itself equivalent to the “compound implication” ( a : b )( b : a ), an observation that was rendered symbolically by the equation ( a = b ) = ( a : b )( b : a ). MacColl wrote a ′ for the “denial” or “negative” of statement a , and stated that ( a ′ + b ) ′ is equivalent to ab ′ . However, while a ′ + b is a “necessary consequence” of a : b (written ( a : b ) : a ′ + b ), he argued that the two formulas are not equivalent because their denials are not equivalent, claiming that the denial of a : b “only asserts the possibility of the combination ab ′ ”, while the denial of a ′ + b “asserts the certainty of the same combination”. 3 Boole had written a = 1 and a = 0 for “ a is true” and “ a is false”, giving a tem- poral reading of these as always true and always false respectively [Boole, 1854, ch. XI]. MacColl invoked the letters ǫ and η to stand for certainty and impossibility, initially describing them as replacements for 1 and 0, and then introduced a third letter θ to denote a statement that was neither certain nor impossible, and hence 2 A listing of these papers is given in the Bibliography of [Lewis, 1918] and on p. 132 of Church’s bibliography in volume 1 of The Journal of Symbolic Logic A comprehensive bibliography of MacColl’s works is given in [Astroh and Kl ̈ uwer, 1998]. 3 This appears to conflict with his earlier claim that the denial of a ′ + b is equivalent to ab ′ “Actuality” may be a better word than “certainty” to express what he meant here (see [MacColl, 1880, p. 54]. Mathematical Modal Logic: A View of its Evolution 5 was “a variable (neither always true nor always false)”. He wrote the equations ( a = ǫ ), ( b = η ) and ( c = θ ) to express that a is a certainty, b is an impossibility, and c is a variable. Then he changed these to the symbols a ǫ , b η , c θ , and went on to write a τ for “ a is true” and a ι for “ a is false”, noting that a true statement is “not necessarily a certainty” and a false one is “not necessarily impossible”. In these terms he stated that a : b is equivalent both to ( a.b ′ ) η (“it is impossible that a and not b ”) and to ( a ′ + b ) ǫ (“it is certain that either not a or b ”). Once the step to this superscript notation had been taken, it was evident that it could be repeated, giving an easy notation for iterations of modalities. MacColl gave the example of A ηιǫǫ as “it is certain that it is certain that it is false that it is impossible that A ”, abbreviated this to “it is certain that a is certainly possible”, and observed that Probably no reader—at least no English reader, born and brought up in England—can go through the full unabbreviated translation of this symbolic statement A ηιǫǫ into ordinary speech without being forcibly reminded of a certain nursery composition, whose ever-increasing accumulation of thats affords such pleasure to the infantile mind; I allude, of course, to “The House that Jack Built”. But trivial matters in appearance often supply excellent illustrations of important general principles. 4 There has been a recent revival of interest in MacColl, with a special issue of the Nordic Journal of Philosophical Logic 5 devoted to studies of his work. In par- ticular the article [Read, 1998] analyses the principles of modal algebra proposed by MacColl and argues that together they correspond to the modal logic T, later developed by Feys and von Wright, that is described at the end of section 2.4 below. 2.3 The Lewis Systems MacColl’s papers are similar in style to earlier nineteenth century logicians. They give a descriptive account of the meanings and properties of logical operations but, in contrast to contemporary expectations, provide neither a formal definition of the class of formulas dealt with nor an axiomatisation of operations in the sense of a rigorous deduction of theorems from a given set of principles (axioms) by means of explicitly stated rules of inference. The first truly modern formal axiom systems for modal logic are due to C. I. Lewis, who defined five different ones, S1–S5, in Appendix II of the book Symbolic Logic [1932] that he wrote with C. H. Langford. Lewis had begun in [1912, p. 522] with a concern that the expositors of the algebra of logic have not always taken pains to indicate that there is a difference between the algebraic and ordinary meanings of implication. 4 Mind (New Series), vol. 9, 1900, p. 75. 5 Volume 3, number 1, December 1998, available at http://www.hf.uio.no/filosofi/njpl/vol3no1/index.html 6 Robert Goldblatt He observed that the algebraic meaning, as used in the Principia Mathematica of Russell and Whitehead, leads to the “startling theorems” that a false proposition implies any proposition, and a true proposition is implied by any proposition. These so-called paradoxes of material implication take the symbolic forms ¬ α → ( α → β ) α → ( β → α ) For Lewis the ordinary meaning of “ α implies β ” is that β can be validly inferred 6 from α , or is deducible 7 from α , an interpretation that he considered was not subject to these paradoxes. Taking “ α implies β ” as synonymous with “either not- α or β ”, he distinguished extensional and intensional meanings of disjunction, providing two meanings for “implies”. Extensional disjunction is the usual truth- functional “or”, which gives the material (algebraic) implication synonymous with “it is false that α is true and β is false”. Intensional disjunction is such that at least one of the disjoined propositions is “necessarily” true. 8 That reading gives Lewis’ “ordinary” implication, which he also dubbed “strict”, meaning that “it is impossible (or logically inconceivable 9 ) that α is true and β is false”. The system of Lewis’s book A Survey of Symbolic Logic [1918] used a primitive impossibility operator to define strict implication. This later became the system S3 of [Lewis and Langford, 1932], which introduced instead the symbol for possibility, but Lewis decided that he wished S2 to be regarded as the correct system for strict implication. The systems were defined with negation, conjunction, and possibility as their primitive connectives, but he made no use of a symbol for the dual combination ¬ ¬ 10 For strict implication the symbol 3 was used, with α 3 β being a definitional abbreviation for ¬ ( α ∧ ¬ β ). Strict equivalence ( α = β ) was defined as ( α 3 β ) ∧ ( β 3 α ). Here now are definitions of S1–S5 in Lewis’s style, presented both to facili- tate discussion of later developments and to convey some of the character of his 6 [Lewis, 1912, p. 527] 7 [Lewis and Langford, 1932, p. 122] 8 [Lewis, 1912, p. 523] 9 [Lewis and Langford, 1932, p. 161] 10 The dual symbol was later devised by F. B. Fitch and first appeared in print in 1946 in a paper of R. Barcan. See footnote 425 of [Hughes and Cresswell, 1968, fn. 425]. Mathematical Modal Logic: A View of its Evolution 7 approach. System S1 has the axioms 11 ( p ∧ q ) 3 ( q ∧ p ) ( p ∧ q ) 3 p p 3 ( p ∧ p ) (( p ∧ q ) ∧ r ) 3 ( p ∧ ( q ∧ r )) (( p 3 q ) ∧ ( q 3 r )) 3 ( p 3 r ) ( p ∧ ( p 3 q )) 3 q, where p, q, r are propositional variables, and the following rules of inference. • Uniform substitution of formulas for propositional variables. • Substitution of strict equivalents : from ( α = β ) and γ infer any formula obtained from γ by substituting β for some occurrence(s) of α • Adjunction : from α and β infer α ∧ β • Strict detachment : from α and α 3 β infer β 12 System S2 is obtained by adding the axiom ( p ∧ q ) 3 p to the basis for S1. S3 is S1 plus the axiom ( p 3 q ) 3 ( ¬ q 3 ¬ p ) S4 is S1 plus p 3 p , or equivalently p 3 p . S5 is S1 plus p 3 p The axioms for S4 and S5 were first proposed for consideration as further pos- tulates in a paper of Oskar Becker [1930]. His motivation was to find axioms that reduced the number of logically non-equivalent combinations that could be formed from the connectives “not” and “impossible”. He also considered the for- mula p 3 ¬ ¬ p , and called it the “Brouwersche axiom”. The connection with Brouwer is remote: if “not” is translated to “impossible” ( ¬ ), and “implies” to its strict version, then the intuitionistically acceptable principle p → ¬¬ p becomes the Brouwersche axiom. 2.4 G ̈ odel on Provability as a Modality G ̈ odel in [1931] reviewed Becker’s 1930 article. In reference to Becker’s discussion of connections between modal logic and intuitionistic logic he wrote It seems doubtful, however, that the steps here taken to deal with this prob- lem on a formal plane will lead to success. He subsequently took up this problem himself with great success, and at the same time simplified the way that modal logics are presented. The Lewis systems contain all truth-functional tautologies as theorems, but it requires an extensive analysis 11 Originally p 3 ¬¬ p was included as an axiom, but this was shown to be redundant by McKinsey in 1934. 12 Lewis used the name “Inference” for the rule of strict detachment. He also used “assert” rather than “infer” in these rules.