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• .. an Introduction ~- to and , Its Raymond Bradley / Norman Swartz

AN INTRODUCTION TO LOGIC AND ITS PHILOSOPHY

! I RAYMOND BRADLEY NORMAN SWARTZ Department of Philosophy Simon Fraser University

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PREFACE xv

TO THE TEACHER xvii

TO THE STUDENT xxi

POSSIBLE WORLDS 1 1. THIS AND OTHER POSSIBLE WORLDS 1 The realm of possibilities 1 What are the limits to the possible? 2 Possibility is not the same as conceivability 3 Possible worlds: actual and non-actual 4 distinguished from other kinds 6 The constituents of possible worlds 7 2. , , AND FALSITY 9 Truth and falsity defined 9 Truth in a 11 Truth in the actual world 12 The myth of degrees of truth 12 3. PROPERTIES OF PROPOSITIONS 13 Possibly true propositions 13 Possibly false propositions 13 Contingent propositions 14 Contradictories of propositions 14 Noncontingent propositions 15 Necessarily true propositions 16 Necessarily false propositions 17 More about contradictory propositions 18 Some main kinds of noncontingent propositions 19 Summary 24 Symbolization 25 4. RELATIONS BETWEEN PROPOSITIONS 28 Inconsistency 28 Consistency 30 Implication 31 Equivalence 35 Symbolization 41

vii CONTENTS

5. SETS OF PROPOSITIONS 42 Truth-values of -sets 42 Modal properties of proposition-sets 42 Modal relations between proposition-sets 44 Minding our "P's and "Q"s 47 6. MODAL PROPERTIES AND RELATIONS PICTURED ON WORLDS-DIAGRAMS 48 Worlds-diagrams for modal properties 49 Worlds-diagrams for modal relations 50 Interpretation of worlds-diagrams 50 A note on history and nomenclature 53 Capsule of modal relations 54 Appendix to section 6 57 7. IS A SINGLE THEORY OF TRUTH ADEQUATE FOR BOTH CONTINGENT AND NONCONTINGENT PROPOSITIONS? 58 8. THE "POSSIBLE WORLDS" IDIOM 62

PROPOSITIONS 65 1. INTRODUCTION 65 2. THE BEARERS OF TRUTH-VALUES 65 Thesis 1: Such things as beliefs, statements, assertions, remarks, hypotheses, and theories are the bearers of truth and falsity. 68 Thesis 2: Acts of believing (stating, asserting, etc.) are the bearers of truth-values. 68 Thesis 3: That which is believed, stated, etc., is what is true or false. 71 Thesis 4: Sentences are the bearers of truth-values. 71 Thesis 5: Sentence-tokens are the bearers of truth-values. 73 Thesis 6: Sentence-types are the bearers of truth-values. 74 Thesis 7: -free sentences are the bearers of truth-values. 75 Thesis 8: Context-free sentence-tokens are those things to which truth and falsity may be attributed. 76 Thesis 9: Context-free sentence-types are those things to which truth and falsity may be attributed. 76 Thesis 10: Propositions are those things to which truth and falsity may be attributed. 79 Thesis 11: Propositions are to be identified with the meanings of sentences. 80 Thesis 12: Propositions are to be identified with sets of possible worlds. 82 CONTENTS IX

Thesis 13: Propositions are abstract entities in their own right; that is, they are sui generis, they are not to be identified with any other kind of abstract entity. 84 Categorial differences between sentences and propositions 86 One final note 86 3. THE STRUCTURE OF PROPOSITIONS: A SPECULATIVE THEORY 87 Concepts 87 Attributes of concepts. 90 Identity conditions for concepts 92 Analysis of propositions 94 Identity conditions for propositions 96 4. ON REFERRING TO SENTENCES AND TO PROPOSITIONS 97 Techniques for referring to sentences 97 Basic techniques for referring to propositions 98 Advanced technique for referring to propositions: context- free 100 Untensed verbs in context-free references 103 5. THE OMNITEMPORALITY OF TRUTH 104 6. PROPOSITIONS, SENTENCES, AND POSSIBLE WORLDS 108 The uni-linguo proviso 110 The linguo-centric proviso 111 Securing to propositions 111 7. SENTENTIAL AND POSSIBLE-WORLDS TESTING 113 Sentential ambiguity 113 The method of possible-worlds testing 114 Janus-faced sentences 119 8. POSSIBLE-WORLDS PARABLES 121 Case Study 1: The thesis that persons (creatures) who lack a language cannot have reflective beliefs 122 Case Study 2: The thesis that persons (creatures) who lack a language cannot believe necessary 125 Case Study 3: The thesis that a justified belief in a true proposition constitutes knowledge 126

KNOWLEDGE 129 1. THE SUBJECT MATTER AND THE SCIENCE OF LOGIC 129 2. THE NATURE OF KNOWLEDGE 130 X CONTENTS

7. Is it a necessary condition of the truth of as knowing that P, that P should be true? 131 2. Is it a necessary condition of a's knowing that P, that a believe that P? 133 3. Is it a necessary condition of a's knowing that P, that a be justified in believing that P? 136 4. What might the missing fourth necessary condition for a's knowing that P be? 137 3. THE LIMITS OF HUMAN KNOWLEDGE 139 The known and the unknown 139 The knowable and the unknowable 140 4. EXPERIENTIAL AND RATIOCINATIVE KNOWLEDGE 142 Experiential knowledge 142 Ratiocinative knowledge 144 Appendix to section 4 149 5. EMPIRICAL AND A PRIORI KNOWLEDGE 149 of "empirical" and "a priori" 150 The non-exhaustiveness and non-exclusiveness of the experiential/ratiocinative distinction 151 The exhaustiveness and exclusiveness of the empirical/ a priori distinction 152 Is a priori knowledge certain? 155 6. EPISTEMIC AND MODAL STATUS CONSIDERED TOGETHER 156 7. Are there any contingent propositions which are know- able empirically? 157 2. Are there any contingent propositions which are know- able both experientially and ratiocinatively? 158 3. Are there any contingent propositions which are know- able ratiocinatively but which are not knowable experientially? 163 4. Are there any contingent propositions which are know- able by other than experiential or ratiocinative means? 164 5. Are there any contingent propositions which are unknowable? 167 6. Are there any noncontingent propositions knowable empirically? 168 7. Are there any noncontingent propositions which are knowable both experientially and ratiocinatively? 170 8. Are there any noncontingent propositions which are knowable ratiocinatively but which are not knowable experientially? 170 CONTENTS xi

9. Are there any noncontingent propositions which are knowable a priori but by means other than ratiocination? 171 10. Are there any noncontingent propositions which are unknowable? 172 Appendix to section 6: a complete classificatory scheme for the epistemic and modal distinctions 174 7. THE EPISTEMOLOGY OF LOGIC 175 THE SCIENCE OF LOGIC: AN OVERVIEW 179 1. INTRODUCTION 179 2. THE METHOD OF ANALYSIS 180 The objects of 180 Three levels of analysis 181 The idea of a complete analysis 183 The need for a further kind of analysis 184 Possible-worlds analysis 185 Degrees of analytical knowledge 187 3. THE OF ANALYSIS 189 Moore's problem 189 A Moorean solution 190 4. THE METHOD OF 192 The nature of inference 193 Valid and invalid propositional 195 Determining the of inferences: the problem of justification 196 Rules of inference 198 What kind of rule is a rule of inference? 200 Inference and the expansion of knowledge 201 5. INFERENCE WITHIN THE SCIENCE OF LOGIC 205 Inference within axiomatic systems: the example of S5 205 Inference within natural deduction systems 210 The theoretical warrant of the method of direct proof 215 6. A PHILOSOPHICAL PERSPECTIVE ON LOGIC AS A WHOLE 218 The indispensability of modal concepts within propositional 218 Problems about the reduction principles 220 Problems about the 224 logics 228 The move to logic 230 Traditional syllogistic 232 CONTENTS

Modern predicate logic 233 Modal notions in predicate logic 236 Modalities 237 Heterogeneous and homogeneous possible worlds 239 Is there really a logic of concepts? 240

TRUTH-FUNCTIONAL PROPOSITIONAL LOGIC 247 1. INTRODUCTION 247 2. TRUTH-FUNCTIONAL OPERATORS 247 The uses of "not" and "it is not the case that" 249 The uses of "and" 252 The uses of "or" 257 Interlude: compound sentences containing two or more sentential operators 261 The uses of "if... then ..." 263 The uses of "if and only if 269 Appendix: truth-tables for wffs containing three or more letters 272 3. EVALUATING COMPOUND SENTENCES 273 A note on two senses of "determined" 277 4. ELEMENTARY TRUTH-TABLE TECHNIQUES FOR REVEALING MODAL STATUS AND MODAL RELATIONS 279 Modal status 279 Modal relations 284 Deductive validity 290 5. ADVANCED TRUTH-TABLE TECHNIQUES 294 Corrected truth-tables 294 Reduced truth-tables 297 6. THE CONCEPT OF FORM 301 Sentences and sentential forms in a logic 301 The relationship between sentences and sentence-forms 302 7. EVALUATING SENTENCE-FORMS 306 The validity of sentence-forms 306 Modal relations 308 Implication 308 Equivalence 309 Inconsistency 309 Argument-forms and deductive validity 310 8. FORM IN A NATURAL LANGUAGE 311 CONTENTS xiii

9. WORLDS-DIAGRAMS AS A DECISION PROCEDURE FOR TRUTH-FUNCTIONAL PROPOSITIONAL LOGIC 313 10. A SHORTCUT FORMAL METHOD: REDUCTIO AD ABSURDUM TESTS 315 Summary 320

6 MODAL PROPOSITIONAL LOGIC 323 1. INTRODUCTION 323 2. MODAL OPERATORS 323 Non-truth-functionality 323 Modal and nonmodal propositions; modalized and non- modalized formulae 324 The interdefinability of the monadic and dyadic modal operators 327 3. SOME PROBLEMATIC USES OF MODAL EXPRESSIONS 329 "It is possible that" 329 Problems with the use of "it is necessary that"; the modal fallacy; absolute and relative necessity 330 4. THE MODAL STATUS OF MODAL PROPOSITIONS 333 5. THE OPERATOR "IT IS CONTINGENTLY TRUE THAT" 337 6. ESSENTIAL PROPERTIES OF RELATIONS 339 7. TWO CASE STUDIES IN MODAL RELATIONS: a light- hearted interlude 345 Case study 1: the of telling the truth 345 Case study 2: an invalid inference and an unwitting impossible 347 8. USING WORLDS-DIAGRAMS TO ASCERTAIN THE VALIDITY OF MODALIZED FORMULAE 350 Applications 351 The validity of the axioms of S5 356 The nonvalidity of the axiom set for S6 358 9. A SHORTCUT FORMAL METHOD FOR DETERMINING THE VALIDITY OF MODALIZED FORMULAE: modal reductios 359 10. THE NUMBER OF FORMALLY NON-EQUIVALENT SENTENCE-FORMS CONSTRUCTIBLE ON N SENTENCE- VARIABLES 365 xiv CONTENTS

11. LOOKING BEYOND TO INDUCTIVE LOGIC 370 The cardinality of a class and other concepts of class size 371 The concept of contingent content 372 Monadic modal functors 375 What are the prospects for a fully-developed inductive logic? 379 The concept of probabilification 381 A dyadic modal functor for the concept of probabilification 382

INDEX 385 Preface

Talk of possible worlds is now a commonplace within philosophy. It began, nearly three hundred years ago, within philosophical theology. Leibniz thought it reassuring to say that although our world contains much that is evil, it is nonetheless the best of all possible worlds. Few philosophers today find this very plausible. Nevertheless, talk of the set of all possible worlds — when stripped of the suggestion that the actual world is better than any others — is nowadays frequently invoked as a means of illuminating other areas of philosophy. Ethics, epistemology, philosophy of psychology, philos• ophy of language, and — most notably of all — logic, are all benefiting from the insights of what is called "possible worlds ". Unfortunately, most current talk of possible worlds is still regarded as the province of professionals; little of it has filtered down to those who are just beginning to learn their philosophy. Yet there is no good why this should be so. Although the higher reaches of possible-worlds semantics bristle with technical subtleties, its basic insights are really very simple. This book explains what those insights are and uses them to construct an integrated approach to both the and the science of logic itself. This approach, we believe, is especially suited to the needs of those who have difficulty with symbols. There are many persons who would like to learn something of philosophy and logic but who, because they are put off by the severely formal manner in which the logical part of philosophy is usually presented, are deterred from pursuing their intent. Their alienation is unfortunate and needless; we try to prevent it by taking more pains than usual to ensure that logical concepts are well understood before they are symbolized. Indeed, it is only in the last two chapters that the powers of symbolism are systematically exploited. Then, again, we would like to think that our approach will be helpful to those for whom symbolism holds no terrors but for whom the difficulty lies rather in seeing how there can be any real connection between the formal results of logic and the substantive inquiries undertaken in other parts of philosophy. Their intellectual schizophrenia reflects the that the rarefied results of formal logic resemble those of mathematics more closely than they do those of , epistemology, and the rest. But it neglects the fact, which we here emphasize, that the basic concepts of formal logic are hammered out on the same anvil of analytical inquiry as are those of other parts of philosophy. Grounding logic in talk of possible worlds, we believe, is one way of making logic seem more at home with its philosophical kin. Many of the arguments presented in this book are, and need to be, matters for philosophical debate. Yet seldom have we done more than hint at the parameters within which such debate arises. There are three main for this. First, we believe that the kind of questioning which we hope this book will generate is likely to be deeper if it is provoked by sustained argument for a single coherent point of view rather than if it stems from exposure to an eclectic display of divergent doctrines. Secondly, we are confident that serious students will, sooner or later, be treated — by their reading or their teachers — to arguments which will put ours into a broader perspective. Thirdly, we could not hope to do justice to competing points of view without making this book even longer than it is. Students in three countries — Australia, New Zealand, and Canada — have been the guinea pigs for the general approach and parts of the material in this book. We have benefited from the criticisms of many and also from the encouragement of the few who have gone on to become professional teachers of philosophy. We are indebted to scores of fine young minds. Specific acknowledgements go to two institutions and to a number of individuals. The Canada Council and the President's Research Grant Committee of Simon Fraser University have provided generous financial assistance for some of the research and editorial work on this book. Three research assistants, Michael Beebe, Jeffrey Skosnik, and Moira Gutteridge, have assisted in the preparation of

xv xvi Preface the manuscript: their help in compiling the references and index, in preparing some of the graphics, and in commenting on the manuscript, has been invaluable. Many professional colleagues have offered criticism and encouragement. We are especially indebted to Sidney Luckenbach, California State University at Northridge, Malcolm Rennie, Australian National University, and an anonymous reviewer for Basil Blackwell, our U.K. publisher. Each of these philosophers has offered extremely valuable comments on the manuscript. Even though we have not adopted all their suggestions, we hope that they will like the final version and will recognize their own contributions to it. Charles Hamblin, University of New South Wales, is the philosophical godfather of this book: his early unpublished classification of modal relations gave rise to the worlds-diagrams herein; his success in introducing students to logic through modal, rather than truth-functional concepts, has served as a model for our approach. Responsibility for the final shape and substance of the book lies squarely with the authors. The book contains many imperfections. We are aware of some of them but have not wished to fall prey to the perils of perfectionism by further delaying publication. Besides, we trust that errors of which we are not yet aware will be communicated to us by those who think the possible worlds approach worth promoting and who would like to see it brought closer to that state of perfection which only a non-actual possible world is likely ever to contain.

NOTE ON THE SECOND PRINTING

All typographical errors known to us have been corrected. Substantive changes occur on pages 78, 143, 146, 286, 295 and 296. To the Teacher

Approach Three main features determine the complexion of this book: (1) the way we characterize the subject matter of logic; (2) the way we characterize the methodology of logic; and (3) the fact that we present the science of logic in its philosophical rather than in its formal guise.

(1) The subject matter of logic, as we present it, is explicated mainly in terms of metaphysical talk about the set of all possible worlds and the ways in which concepts apply and propositions are true or false within those worlds. Philosophical reflection about the foundations of logic — and of mathematics, for that matter — tends to make metaphysical realists (Platonists, if you like) of us all. Both of us (the authors) would, if we could, happily live with the economies of nominalism. But, like so many others, from Plato to , , and David Lewis, we have felt compelled to posit a modestly rich of abstract entities. Our own catalog extends not only to numbers and sets, but also to concepts and propositions and — above all — to possible worlds. Those of you who think it possible to be more parsimonious will probably relish the task of showing how it can be done. At the very least, you should find our arguments grist for your own philosophical mills. One of the chief attractions of the Leibnizian metaphysic of possible worlds is that — as Kripke, Hintikka, and others demonstrated in the early 1960s — it enables us to give a semantical underpinning to much of the machinery of formal logic. And it enables us to do this in a way which flows naturally from the simple intuitions which most of us — laymen and philosophers alike — have about such concepts as consistency, inconsistency, implication, validity, and the like. We are all (if we have any logical insights at all) disposed to say such things as: that an argument is valid just when its imply its conclusion; that one proposition (or set of propositions) implies another just when it isn't possible for the former to be true in circumstances in which the latter is false; that one proposition is inconsistent with another just when it isn't possible for both to be true; and so on. It is only a short step from these modes of talking to those of Leibniz's possible worlds. And the step is worth taking. For once we take it, we have at our disposal an extremely powerful set-theoretic framework in terms of which to explain all these, and many other basic concepts of logic and philosophy. The set-theoretic framework which does so much of this explanatory work is depicted in this book by means of what we call worlds-diagrams. Our worlds-diagrams are grounded, in chapter 1, in simple and pictureable intuitions about the ways in which the truth-values of propositions may be distributed across the members of the set of all possible worlds. They are elaborated, in chapters 5 and 6, in such a way as to yield a new and effective decision-procedure for evaluating sentences and sentence-forms within prepositional logic, both truth-functional and modal. The logic whose formalism most adequately reflects our Leibnizian intuitions about implication, necessity, and the like is, of course, modal logic. More particularly, we believe and argue that it is that system of modal logic which C. I. Lewis called S5. Like William and Martha Kneale, we are persuaded that S5 is the system whose theses and rules "suffice for the reconstruction of the whole 6f logic as that is commonly understood".1 We give to modal logic in general, and to S5 in particular, both the philosophical and the pedagogical primacy which we believe is their due. This is why, for instance, we introduce the modal concept of logical implication (what Lewis called "strict implication") early in chapter 1 and postpone discussion of the truth-functional concept of material conditionality (often

1. The Development of Logic, Oxford, Clarendon Press, 1962, p. 583.

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misleadingly called "material implication") until chapter 5. It is, you will undoubtedly agree, a purely empirical question as to whether this order of presentation is pedagogically viable and desirable. We believe that it is — on the basis of experience. But we await the reports of others.

(2) The methodology of logic, as we present it, is explicated in terms of epistemological talk about the a priori methods of analysis and inference whereby knowledge of noncontingent propositions is possible. The epistemology of logic is an essential, though often neglected, part of the philosophy of logic. We try to treat it with more than usual care and thoroughness. Chapter 3 serves as an introduction to epistemology in general, with as much emphasis on noncontingent propositions as on contingent ones. To the question, What is the nature of human knowledge? we answer with a version of defeasibility theory. It would have been nice, perhaps, to have espoused one of the currently fashionable causal theories of knowledge. But causal theories, though plausible for cases of experiential knowledge, do not seem able — as at present formulated — to account for the kinds of a priori knowledge which we have in mathematics and logic. To the question, What are the limits of human knowledge? we answer by arguing for the unknowability in principle of at least some propositions: of some contingent ones by virtue of the falsity of ; of some noncontingent ones by virtue of the truth of Godel's Proof. To the question, What are the standard modes of knowledge-acquisition for humans? we offer two answers. One has to do with the natural history, as it were, of human knowledge: with its sources in experience and its sources in reason. The other has to do with the Kantian dichotomy between empirical and a priori knowledge: with whether or not it is possible to know a proposition by means other than experience. The two distinctions, we argue, are by no means equivalent. Thus it is, for example, that we are able to accommodate Kripke's claim that some necessary propositions of logic are knowable both experientially and a priori without in any way compromising Kant's belief in the exclusiveness and exhaustiveness of the empirical/ a priori distinction. The upshot of all this is that we offer ten different categories — rather than the usual four — under which to classify the epistemic status of various propositions; and further, that our knowledge of the truths of logic turns out to be possible under just two of them. Although we acknowledge, with Kripke, that parts of the subject matter of logic may on occasion be known experientially, it by no means follows that appeal to experience forms part of the distinctive methodology whereby that subject matter may be systematically explored. On the contrary, we argue, the methodology of logic involves two wholly a priori operations: the ratiocinative operations of analysis and of inference. In chapter 4 we first show how analysis — "the greater part of the business of reason", as Kant put it — and inference can yield knowledge of noncontingent propositions; and then go on to illustrate, by surveying the three main branches of logic — Prepositional Logic, Predicate Logic, and (what a growing number of philosophers call) Concept Logic — the sorts of knowledge which these methods can yield.

(3) The philosophy of logic which we present in this book is resolutely antilinguistic in several important respects. With respect to the bearers of truth-values, we argue against those who try to identify them with any form of linguistic entity, such as sentences, and quasilinguistic entities such as sentences taken together with their meanings. Propositions may be expressed by linguistic entities and apprehended by language-using creatures; but their existence, we hold, is not dependent upon the existence of either of these. With respect to the notion of necessary truth, we argue against those who suppose that necessary truth can somehow be explained in terms of rules of language, conventions, definitions, and the like. The truth of necessary propositions, we hold, does not require a different kind of explanation from the truth of contingent propositions. Rather it consists, as does the truth of contingent propositions, in "fitting the " — albeit the facts in all possible worlds, not just in some. To the Teacher xix

With respect to the notion of a priori knowledge, we argue against those who suppose that a priori knowledge is best explained in terms of our understanding of words and sentences and of the linguistic rules, conventions, or definitions which they obey. The linguistic theory of the a priori was never so persuasive as when it went tandem with the linguistic theory of necessary truth. But having rejected the latter we felt free, indeed obliged, to reject the former. Our alternative account of a priori knowledge relies, as already noted, on the twin notions of analysis and inference. It is, of course, propositions and their conceptual constituents which — on our view — are the proper objects of analysis. We try to show how what we call "constituent-analysis", when combined with possible- worlds analysis (roughly, truth-condition analysis), can yield knowledge both of the truth-value and of the modal status of logical propositions. And we try to show, further, how rules of inference may be justified by these same analytical methods. In short, we replace linguistic theories about truth-bearers and necessity with an ontological theory about propositions, possible worlds, and relations between them. And we replace the linguistic theory of a priori knowledge with an epistemological theory which gives due recognition to what philosophers have long called "the powers of reason". This is not to say that we ignore matters having to do with language altogether. On the contrary, we devote a lot of attention to such questions as how sentential ambiguity may be resolved, how ordinary language maps onto the conceptual notation of symbolic logic, how sentence-forms can be evaluated in order to yield logical knowledge, and so on. But we resist the view that the theory of logic is best viewed as a fragment of the theory of language. Just as the subject matter of logic needs to be distinguished from its epistemology, so both — on our view — need to be distinguished from theory of language. Many philosophical theses, other than those just canvassed, are developed in this book. We touch on, and in some cases discuss at length, the views of dozens of philosophers and a few mathematicians. This is seemly in a book whose primary emphasis is on the philosophy, rather than the formalism, of logic. And it serves to explain why we have found room in this book for little more than a brief sketch (in chapter 4) of the broad territory of logic, and why chapters 5 and 6, although lengthy, do no more than introduce the elements of prepositional logic (truth-functional and modal, respectively). Which brings us to:

Place in a Curriculum

The book is written so as to be intelligible, and hopefully even intriguing, to the general reader. Yet it is expressly designed for use as a textbook in first- or second-year courses at colleges and universities. It is not designed to replace handbooks in 'speed reasoning' or informal reasoning. Nor is it designed only for those students who plan to go on to more advanced work in philosophy and logic. As we see it, Possible Worlds could, either in part or as a whole, serve as an introduction to philosophy in general — as an introduction, that is, to the logical and analytical methods adopted by contemporary analytical philosophers. It could, either in part or as a whole, serve as an introduction to formal logic, in particular — as a philosophical introduction to the basic concepts with which formal logicians operate. So viewed, it should be used as a prolegomenon to the standard cumcular offerings in logic programs: to all those courses, that is, which develop natural deduction and axiomatic techniques for prepositional logic, quantification theory, and the like. For we touch on these and such-like matters only for purposes of illustration (if at all). Again, the book could, either in part or as a whole, serve as the text for a belated course in the philosophy of logic — as a way of opening up philosophical questions about logic to those whose introduction to logic has been of the more traditional formal kind. It has been our experience — and that of countless other teachers of logic — that the traditional approach, of plunging students straight into the formalism of logic, leaves so many lacunae in their understanding that additional XX To the Teacher instruction in the philosophy of logic is sooner or later seen as a necessity. Our own preference, of course, is to preempt these sorts of problems by introducing logic, from the outset, as an integral part of philosophy — to be more specific, as that part of philosophy which serves as a foundation for mathematics, but whose most intimate philosophical ties are with metaphysics on the one hand and with epistemology on the other. But those of you who do not share our pedagogical predilections on this matter may nevertheless find that Possible Worlds will help your students, later if not sooner, to understand why logic is as important to philosophers as it is to mathematicians. Ideally, the material in this book should be covered in a single course of something like 40 - 50 lecture hours. Alternatively, the material might be spread over two courses each of 20 - 25 lecture hours. Barring that, one will have to design shorter courses around particular selections from the book's six chapters. Here are two possiblilities: A short course on 'baby' formal logic, which keeps philosophical discussion at a minimum and maximizes formal techniques in propositional logic, could be structured around chapters 1, 5, and 6 (with chapter 4, perhaps, being treated in tutorial discussions). A short course on , which maximizes philosophical discussion and minimizes formal techniques, could be structured around chapters 1, 2, 3, and 4. Still again, the book may be used as an ancillary text for courses whose primary is importantly different from ours. For example, for courses in , teachers may wish their students to read chapter 2; for courses in epistemology, teachers may wish their students to read chapter 3; and for courses which — like so many general introductions to philosophy — include a brief survey of logic, teachers may wish their students to read chapter 1 and selections from chapter 4.

Some Practical Hints Many of the exercises — especially in chapters 1, 5, and 6 — since they require non-prose answers — may be more readily corrected (perhaps by exchanging papers in class) if the instructor prepares worksheets for students to complete. Exercises which require the completion (e.g., by addition of brackets) of worlds-diagrams are best handled by distributing photocopies of figure We hereby give our permission for the unlimited reproduction of this figure. You may find it useful to make the progressive construction of a glossary of terms a formal requirement in the course. Finally, much of the material in the book lends itself to testing by multiple-choice examinations. We would be happy to send sample copies to teachers whose requests are made on departmental letterheads. To the Student

Most of you already understand that logic is theoretically important as a foundation for mathematics and practically useful as a set of principles for rational thinking about any topic whatever. But what may not be clear to you is that logic also forms an integral part of philosophy. Since this book aspires to introduce you to logic in its philosophical setting, a few words of explanation are in order. What, for a start, is philosophy? More concretely: What is it to be a philosopher? Our answer — which will suffice for present purposes — is this:

To be a philosopher is to reflect upon the implications of our experience, of the beliefs we hold, and of the things we say, and to try to render these all consistent — or, as it were, to try to get them all into perspective.

Four of the terms we have just used are particularly significant: "reflect", "implications", "consistent", and "perspective". Philosophers are, first and foremost, reflective persons. We believe — as did Socrates — that an unexamined life is shallow, and that unexamined beliefs are often mischievous and sometimes dangerous. This is why we tend to be perplexed about, and to inspect more carefully, matters which less reflective persons either take for granted or brush aside as of no practical value. Any belief, dogma, or creed — social, political, moral, religious, or whatever — is subject to the philosopher's critical scrutiny. Nothing is sacrosanct; not even the beliefs of other philosophers. Do not be dismayed, then, to find in this book arguments which criticize other philosophical viewpoints; and do not be disturbed, either, if your teachers when dealing with this book find reason to criticize our own viewpoint. Out of such dialectic, philosophical insight may be born and progress towards truth may be made. Of the notions of implication and consistency we shall have much to say before long. For the moment, it will suffice to say that one way a philosopher, or anyone else for that matter, has of testing the credentials of any belief or theory is to ask such questions as these: "Is it implied by something we already know to be true?" (if so, it must itself be true); "Does it imply something we know to be false?" (if so, it must itself be false); and "is it consistent with other beliefs that we hold?" (if not, we are logically obliged, whether we like it or not, to give up at least one of them). Plainly, if you wish to be able to answer questions such as these you will need to know a good deal about the concepts of implication and consistency themselves. But these concepts are logical concepts. Little wonder, then, that logic is — as we said at the outset — an "integral part of philosophy". The trouble is, however, that when — with others in coffeehouses, pubs, or university classrooms, or, alone, in moments of reflective solitude — we ponder such deep philosophical questions as those about existence, freedom, responsibility, etc., most of us do not know how to handle, in the requisite disciplined way, the logical concepts on which such questions turn. Learning a little logic can help us to get our thinking straight in philosophy as well as elsewhere. As for trying to get everything into perspective: that, it is probably fair to say, is usually thought to be the most distinctive goal of philosophers. You will need to learn a good deal of logic, and a lot more philosophy, before achieving the kind of lofty vantage point reached by such great thinkers as Plato, Aristotle, Leibniz, Hume, Kant, Russell, and Wittgenstein. But one must begin somewhere. And perhaps one of the best places to start is by reflecting on the fact that the world of human experience is but one of many that could have been — that the actual world is, as we shall say, only one of many possible worlds. Thinking about other possible worlds is — as Leibniz recognized — a way of getting our own world into perspective. It is also — as we try to show — a way of providing both an introduction to, and secure theoretical foundations for, logic itself. Hence, the title of this book.

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PossiblePossible WorldsWorlds

1.1. THISTHIS ANDAND OTHEROTHER POSSIBLEPOSSIBLE WORLDSWORLDS

The Realm of Possibilities

TheThe yearyear isis A.D.AD. 4272.4272. LazarusLazarus LongLong isis 23602360 yearsyears old.old. AlthoughAlthough hehe hashas beenbeen nearnear deathdeath manymany times,times, hehe hasn'thasn't -— unlikeunlike hishis biblicalbiblical namesakenamesake -— requiredrequired thethe interventionintervention ofof a miraclemiracle toto recover.recover. HHee simplysimply checkschecks himselfhimself (or(or isis takentaken byby force)force) intointo a RejuvenationRejuvenation ClinicClinic fromfrom timetime toto time.time. WhenWhen wewe lastlast hearhear ofof himhim hehe isis undergoingundergoing rejuvenationrejuvenation again.again. TheThe yearyear isis nownow 42914291 andand LazarusLazarus isis beinbeingg treatedtreated inin hishis ownown portableportable clinicclinic aboardaboard thethe star-yachtstar-yacht "Dora""Dora" afterafter travelingtraveling backback inin timetime toto hihiss birthplacebirthplace inin KansasKansas andand beingbeing "mortally"mortally wounded"wounded" inin thethe trenchestrenches "somewhere"somewhere inin France"France".. AllAll this,this, andand muchmuch more,more, happenshappens toto thethe LazarusLazarus LongLong ofof RobertRobert A.A. Heinlein'sHeinlein's novelnovel Time Enough for Love. l1 InIn hishis novel,novel, HeinleinHeinlein startsstarts withwith a frameworkframework ofof personspersons whowho actuallyactually livedlived (e.g.,(e.g., WoodrowWoodrow WilsoWilson.andn and KaiseKaiserr WilhelWilhelmm II)II),, ooff placeplacess thathatt actuallactuallyy existedexisted (e.g.,(e.g., KansaKansass CitCityy anandd France)France),, anandd ofof eventsevents thathatt actuallactuallyy occurreoccurredd (e.g.,(e.g., U-boaU-boatt attackattackss anandd ththee entrentryy ooff ththee U.SU.S.. intintoo WorlWorldd WaWarr I)I),, anandd buildbuildss uupp a worlworldd ooff fictionalfictional persons, persons, places,places, andand events.events. HeHe carriescarries usus withwith him,him, inin make-believe,make-believe, ttoo anotheanotherr worlworldd differendifferentt frofromm ouourr actuaactuall oneone:: ttoo a merelmerelyy possiblpossiblee worldworld.. HoHoww mucmuchh ooff thithiss otheotherr possiblpossiblee worlworldd iiss believablebelievable?? HoHoww mucmuchh ooff iitt iiss really possiblepossible?? MucMuchh ooff iitt iiss crediblecredible.. FoForr instanceinstance,, thertheree could havhavee beebeenn a mamann namenamedd "Ira"Ira HowardHoward"" whwhoo diediedd iinn 18731873 anandd whoswhosee wilwilll instructeinstructedd hihiss trusteetrusteess ttoo setset uupp a foundatiofoundationn devotedevotedd ttoo ththee prolongatioprolongationn ooff humahumann lifelife.. FoForr alalll wwee knowknow,, IrIraa HowarHowardd mamayy bbee jusjustt aass historicahistoricall a personagpersonagee aass WoodroWoodroww WilsonWilson.. TToo bbee sure,sure, IrIraa HowarHowardd may bbee jusjustt aass mucmuchh a creaturcreaturee ooff Heinlein'Heinlein'ss imaginatioimaginationn aass iiss MinervMinervaa —- a computecomputerr becombecomee fleshflesh anandd bloobloodd anandd ononee ooff LazarusLazarus'' manmanyy mistressesmistresses.. BuButt thithiss mattermatterss notnot.. ForFor,, whatevewhateverr ththee historicahistoricall factfactss happehappenn ttoo bebe,, wwee cacann alwayalwayss supposesuppose -— counter)'actually,counterfactually, as aswe wesay say- —that thatthey theymight might havhavee beebeenn otherwiseotherwise.. WWee constantlconstantlyy makmakee sucsuchh suppositionsuppositionss iinn ththee worlworldd ooff reareall lifelife.. ThThee worlworldd ooff fiction needsneeds nono specialspecial indulgence.indulgence. WeWe easilyeasily can,can, andand dailydaily do,do, entertainentertain allall sortssorts ofof ununactualizeactualizedd possibilitiepossibilitiess abouaboutt pastpast,, presentpresent,, anandd futurefuture.. WWee thinthinkk abouaboutt thingthingss thathatt might havhavee happenedhappened,, mightmight bbee happeninhappeningg anandd might bbee abouaboutt ttoo happenhappen.. NoNott onlonlyy ddoo wwee ruefullruefullyy asaskk "Wha"Whatt iiff thingthingss had been thuthuss anandd thus?"thus?";; wwee alsalsoo wondewonderr "Wha"Whatt iiff thingthingss are ssoo anandd so?so?"" anandd "Wha"Whatt iiff thingthingss were to be suchsuch anandd such?such?"" CounterfactuaCounterfactuall suppositiosuppositionn iiss nonott mermeree idlidlee speculationspeculation.. NeitheNeitherr iiss iitt jusjustt a fancfancyy ooff ththee dreamedreamerr oorr a refugrefugee foforr ththee escapistescapist.. GiveGivenn thathatt wwee araree ssoo ofteoftenn ignoranignorantt ooff whawhatt is, wwee neeneedd a ricrichh

11.. NeNeww YorkYork,, G.PG.P.. Putnam'Putnam'ss SonsSons,, 1973.1973.

1 2 POSSIBLEPOSSIBLE WORLDSWORLDS

sensesense ofof whatwhat might be. InIn mattersmatters ofof practicepractice,, wewe needneed toto considerconsider alternativesalternatives wherewhere knowledgeknowledge iiss denieddenied us.us. InIn mattersmatters ofof theory,theory, wewe needneed toto considerconsider hypotheseshypotheses wherewhere factsfacts areare unknown.unknown. ActualityActuality is,is, asas itit were,were, surroundedsurrounded bbyy anan infiniteinfinite realmrealm ofof possibilitiespossibilities.. Or,Or, asas wewe mightmight otherwiseotherwise puputt it,it, ourour actualactual worldworld isis surroundedsurrounded bbyy anan infinityinfinity ofof otherother possiblpossiblee worlds.worlds. NNoo wonderwonder thenthen thathatt fictionfiction writerswriters likelike HeinleinHeinlein havehave littlelittle difficultydifficulty inin beguilinbeguilingg uuss withwith theirtheir storiesstories ofof possibilitiespossibilities,, mosmostt ofof whichwhich willwill nevernever bebe actualized.actualized. MightMight therethere notnot comecome a timetime whenwhen incestincest willwill bbee sociallysocially andand legallylegally acceptable,acceptable, whenwhen humanhuman lifelife willwill bbee prolongeprolongedd bbyy periodiperiodicc visitsvisits toto RejuvenationRejuvenation Clinics,Clinics, whewhenn computerscomputers willwill bbee embodiedembodied inin humanhuman flesh,flesh, oror whenwhen traveltravel fromfrom oneone galacticgalactic colonycolony toto anotheranother willwill bbee a commonplace?commonplace? MaybeMaybe ourour everydayeveryday counterfactualcounterfactual suppositionssuppositions areare muchmuch moremore mundane.mundane. ButBut whowho amongamong usus wouldwould rulerule thethe excitingexciting onesones entirelyentirely outout ofof order?order? TheThe factfact isis thatthat wewe can,can, andand do,do, conceiveconceive ofof socialsocial andand legal,legal, biologicabiologicall andand technological,technological, andand perhapperhapss eveneven ofof physicalphysical,, possibilitiepossibilitiess whicwhichh ththee worlworldd ofof factfact mamayy neveneverr encompass.encompass.

What are the limits to the possible?possible? ButBut areare therethere notnot limitslimits ofof somesome sortsort toto whatwhat wewe cancan conceiveconceive oror supposesuppose toto bbee possiblepossible?? DoesDoes jusjustt anythinganything go?go? HowHow aboutabout time-travel,time-travel, forfor instance?instance? InIn Heinlein'sHeinlein's novel,novel, LazarusLazarus recountsrecounts howhow hhee assumedassumed a biologicalbiological ageage ofof thirty-fivethirty-five soso asas toto traveltravel bacbackk inin timetime twenty-threetwenty-three hundredhundred yearsyears ttoo observeobserve howhow thingsthings werewere inin hishis childhoodchildhood andand (as(as itit turnsturns out)out) toto fallfall inin lovelove withwith hishis ownown mothermother..

SOMESOME EVENTSEVENTS ININ THETHE LIFELIFE OFOF LAZARUSLAZARUS LONG.LONG, SENIORSENIOR MEMBERMEMBER OFOF THETHE HOWHOWARARDD FAMILIESFAMILIES

FirstFirst runrun throughthrough timetimetI

is born 1------1stealssteals starshipstarship returnsreturns ttoo rejuvenated travels through In I------....-ltoto evacuateevacuate OldOld HomHomse at clinic space-tine on Kansas HowardHoward Terra,Terrat leadleadss on Secundus star-yacht, FamiliesFamilies GreatGreat DiasporDia.sporaa "Dora"

--- --'1912' '141 '16 1'17 1------1 )6 1- - u ------~--hu--142721------~ '1'1 .tS 21 SecondSecond runrun througthroughh timettime i lands wounded near in France) birthplace rescued by "Dora" and carried to 4291 FIGUREFIGURE (l.a)(ha)

CouldCould time-traveltime-travel occur?occur? CanCan youyou bbee suresure thatthat therethere isis notnot somesome otherother possiblpossiblee worldworld inin whichwhich itit occuroccurss eveneven ifif itit nevernever occurs,occurs, oror willwill occur,occur, inin ourour actualactual world?world? IsIs therethere anyany paradoparadoxx inin thethe ideaidea ofof LazaruLazaruss observingobserving himselfhimself asas a four-year-old?four-year-old? SeeSee exerciseexercise 4 onon pp.. 25.25.

ManyMany ofof usus maymay feelfeel thatthat thethe veryvery conceptconcept ofof time-traveltime-travel isis paradoxicalparadoxical.. IfIf time-traveltime-travel werewere possiblepossible thenthen shouldshould wewe notnot bbee ableable toto imagineimagine a persopersonn travelingtraveling bacbackk inin timetime andand fatheringfathering himself?himself? ButBut iiss thisthis reallyreally possiblepossible?? OurOur mindsminds bogglbogglee a bitbit.. ItIt isis supposedsupposed bbyy somesome thatthat herehere wewe havehave reachedreached ththee limitslimits ofof conceivability,conceivability, andand thatthat thethe worldworld whichwhich HeinleinHeinlein portrayportrayss throughthrough LazarusLazarus is,is, inin thisthis respectrespect atat least,least, notnot inin anyany sensesense a possiblpossiblee world.world. This,This, asas itit happens,happens, isis thethe viewview ofof anotheranother ofof thethe novel'novel'ss characters:characters: CarolynCarolyn Briggs,Briggs, chiefchief archivistarchivist ofof thethe HowardHoward Foundation.Foundation. AsAs sheshe puputt itit inin herher PrefacePreface ttoo thethe RevisedRevised EditionEdition ofof Lazarus'Lazarus' memoirs:memoirs: "An"An apocryphalapocryphal andand obviouslyobviously impossibleimpossible taletale ofof thethe laslastt eventsevents inin hishis lifelife hashas beebeenn includedincluded atat thethe insistenceinsistence ofof thethe editoreditor ofof thethe originaloriginal memoir,memoir, bubutt itit cannocannOtt bbee takentaken seriously."seriously." OnOn thethe otherother hand,hand, somesome ofof uuss maymay feelfeel thatthat time-travel,time-travel, ofof thethe kindkind thatthat LazaruLazaruss § 1 .' ThiThiss anandd OtheOtherr PossiblPossiblee WorldWorldss 3

iiss supposesupposedd ttoo havhavee undertakenundertaken,, iiss nonott whollwhollyy inconceivableinconceivable;; thathatt paradoparadoxx cacann bbee avoidedavoided;; anandd thathatt aa worlworldd iinn whicwhichh time-travetime-travell occuroccurss mamayy somedasomedayy bbee seeseenn ttoo bbee possiblpossiblee iiff anandd whewhenn ouourr conceptconceptss ooff space-timspace-timee becombecomee mormoree sophisticatedsophisticated.. WWee mamayy wanwantt ttoo agreagreee witwithh Carolyn'Carolyn'ss predecessopredecessorr —- JustinJustin FooteFoote,, chiechieff archivisarchivistt emerituemerituss —- whewhenn hhee appendappendss ththee followinfollowingg notnotee ttoo heherr PrefacePreface:: "M"Myy lovellovelyy anandd learnelearnedd successosuccessorr iinn officofficee doedoess nonott knoknoww whawhatt shshee iiss talkintalkingg aboutabout.. WitWithh ththee SenioSeniorr [i.e.[i.e.,, Lazarus]Lazarus],, ththee mosmostt fantastifantasticc iiss alwayalwayss ththee mosmostt probable."2probable."2 BuButt whatevewhateverr wwee sasayy abouaboutt time-travetime-travell —- whethewhetherr wwee thinthinkk iitt fallfallss withiwithinn oorr beyonbeyondd ththee limitlimitss ooff possibilitpossibilityy —- iitt iiss certaicertainn thathatt somsomee thingthingss araree not possiblpossiblee nnoo mattematterr hohoww sophisticatesophisticatedd ouourr conceptconceptss becomebecome.. ThThee suppositiosuppositionn thathatt time-travetime-travell botbothh will occuoccurr sometimsometimee iinn ththee futurfuturee an andd wilwilll not occuroccur anytimanytimee iinn ththee futurefuture,, iiss a cascasee ooff pointpoint.. IItt taketakess usus,, iinn a sensesense,, beyonbeyondd ththee boundboundss ooff conceivabilityconceivability.. IItt iiss nonott justjust paradoxical.paradoxical. ItIt is,is, asas wewe say,say, flatlyflatly self-contradictory. self-contradictory. AA supposedsupposed worldworld inin whichwhich somethingsomething lit­lit• erallerallyy botbothh iiss ththee cascasee anandd iiss nonott ththee cascasee iiss notnot,, iinn ananyy sensesense,, a possiblpossiblee worldworld.. IItt iiss aann impossiblimpossiblee oneone..

Possibility is not the same asas conceivabilityconceivability SSoo fafarr wwee havhavee spokespokenn aass iiff ththee boundarboundaryy betweebetweenn ththee possiblpossiblee anandd ththee impossiblimpossiblee werweree a functiofunctionn ofof humahumann psychologypsychology:: aass ifif,, thathatt isis,, iitt coincidecoincidedd witwithh ththee boundarboundaryy betweebetweenn thathatt whicwhichh wwee finfindd conceivablconceivablee anandd thathatt whicwhichh wwee findfind inconceivableinconceivable.. YetYet,, whilwhilee thertheree iiss a greagreatt amounamountt ooff overlaoverlapp betweebetweenn whawhatt iiss conceivablconceivablee anandd whawhatt iiss possiblepossible,, ththee twtwoo araree nonott ththee samesame.. IInn ththee firstfirst placeplace,, iitt shouldshould bbee cleaclearr oonn reflectioreflectionn thathatt ouourr inabilitinabilityy ttoo conceivconceivee ooff a certaicertainn statestate ooff affairaffairss doedoess nonott implimplyy ththee impossibilitimpossibilityy ooff thathatt statestate ooff affairsaffairs.. NotoriouslyNotoriously,, thertheree wawass a timtimee whewhenn ouourr ancestorancestorss thoughthoughtt iitt inconceivablinconceivablee thathatt ththee eartearthh shouldshould bbee roundround.. YeYett obviouslobviouslyy ththee possibilitpossibilityy ooff ththee earth'earth'ss beinbeingg rounroundd wawass iinn nnoo wawayy limitelimitedd bbyy theitheirr inabilitinabilityy ttoo conceivconceivee itit.. Secondly,Secondly, ouourr seemingseeming abilitabilityy ttoo conceivconceivee ooff a certaicertainn statestate ooff affairaffairss doedoess nonott implimplyy ththee possibilitpossibilityy ooff thathatt statestate ooff affairsaffairs.. FoForr manmanyy centuriescenturies,, mathematicianmathematicianss soughtsought a meanmeanss ooff squaringsquaring ththee circlcirclee (sought(sought a procedurproceduree wherebwherebyy ttoo construcconstructt foforr ananyy givegivenn circlcirclee a squaresquare ooff equaequall area)area).. PlainlyPlainly,, thetheyy woulwouldd nonott havhavee dondonee soso unlesunlesss thetheyy thoughthoughtt iitt conceivablconceivablee thathatt suchsuch a procedurproceduree existedexisted.. YeYett wwee nonoww knowknow,, anandd cacann proveprove,, thathatt ththee squaringsquaring ooff ththee circlcirclee iiss whollwhollyy beyonbeyondd ththee boundboundss ooff possibilitpossibilityy —­ thatthat thethe conceptconcept itselfitself isis self-contradictoryself-contradictory.. ButBut if,if, asas ourour firstfirst argument argument shows,shows, conceivabilityconceivability isis neitherneither aa necessarynecessary conditioncondition (conceivability(conceivability isis notnot needed forfor somethingsomething toto bebe possible),possible), nor,nor, asas ourour secondsecond argumentargument shows,shows, a sufficientsufficient conditioconditionn (conceivability(conceivability doesn'tdoesn't suffice toto establishestablish possibility),possibility), whatwhat thenthen are thethe conditionsconditions forfor something'ssomething's beingbeing possiblepossible?? OneOne mightmight bebe temptedtempted toto answeranswer thatthat itit isis conceivability without inconsistency oror coherentcoherent conceivability -— notnot jusjustt conceivabilityconceivability itselfitself -— whichwhich isis thethe measuremeasure ofof possibility.possibility. AfterAfter allall,, althoughalthough certaincertain ofof ourour ancestorsancestors apparentlyapparently diddid notnot havehave thethe psychologicalpsychological capacitycapacity toto conceiveconceive ooff thethe earthearth beingbeing round,round, thethe conceptconcept ofof thethe earthearth beingbeing roundround isis itselfitself a perfectlyperfectly coherentcoherent oorr self-consistentSelf-consistent conceptconcept andand hencehence isis oneone ofof whichwhich theythey couldcould -— inin thethe requisiterequisite sensesense -— havhavee conceivedconceived withoutwithout inconsistency.inconsistency. And,And, again,again, althoughalthough generationsgenerations ofof mathematiciansmathematicians thoughtthought thatthat ththee squaringsquaring ofof thethe circlecircle waswas a goalgoal whichwhich oneone couldcould intelligiblyintelligibly pursue,pursue, wewe nownow knowknow thatthat theythey coulcouldd notnot conceiveconceive ofof thatthat goalgoal withoutwithout inconsistency.inconsistency. Unfortunately,Unfortunately, thisthis answeranswer willwill notnot do;do; oror rather,rather, itit willwill notnot dodo fromfrom thethe standpointstandpoint ofof thosethose whwhoo wishwish -— asas wewe dodo inin thisthis bookbook -— toto explainexplain thethe principalprincipal conceptsconcepts ofof logiclogic inin termsterms ofof thethe notionnotion ofof a possiblepossible world.world. ForFor amongamong thosethose principalprincipal conceptsconcepts areare thethe conceptsconcepts ofof consistencyconsistency andand inconsistency.inconsistency. AndAnd ifif wewe thenthen invokeinvoke thethe conceptconcept ofof consistencyconsistency inin orderorder toto explainexplain whatwhat a possiblepossible worldworld isis andand howhow a possiblepossible worldworld differsdiffers fromfrom anan impossibleimpossible one,one, wewe exposeexpose ourselvesourselves toto aa chargecharge ofof circularity.circularity.

2.2. Time Enough For Love,Love, p.p.xvii. xvii. [It[It willwill bebeour ourpractice practice totogive givethe the completecompletebibliographical bibliographical referencereferenceonly only onon thethe firstfIrst occasionoccasion inineach eachchapter chapterof ofour ourciting citinga awork. work. ThereinafterThereinafter weweshall shall omitomitthe thedetails detailsof ofpublication.] publication.] 4 POSSIBLPOSSIBLEE WORLDWORLDSS

OurOur predicamentpredicament isis apparentlyapparently this.this. OnOn thethe oneone handhand,, wwee wiswishh ttoo avoiavoidd psychologism, viz.viz.,, anyany formform ofof theorytheory whichwhich makesmakes logiclogic a functionfunction ofof humanhuman psychologypsychology.. IInn ititss oldeolderr formform,, psychologismpsychologism heldheld thatthat thethe lawslaws ofof logiclogic werewere "the"the lawslaws ofof thought".thought". IItt treatetreatedd ththee sciencsciencee ooff logilogicc aass iiff iitt werweree a branchbranch ofof psychologypsychology concernedconcerned withwith thethe descriptiondescription ooff hohoww humahumann beingbeingss actuallactuallyy reason.reason. Psychologism,Psychologism, inin itsits traditionaltraditional form,form, isis nownow dead,dead, largelylargely aass a resulresultt ooff ththee effortsefforts,, earlearlyy iinn thithiss century,century, ofof Frege,Frege, Russell,Russell, andand Wittgenstein.Wittgenstein. WeWe wiswishh ttoo avoiavoidd introducinintroducingg iitt iinn a neneww formform.. YetYet thatthat isis whatwhat wewe wouldwould bebe doingdoing ifif wewe werewere toto explainexplain ththee principaprincipall conceptconceptss ooff logilogicc iinn termtermss ooff possiblepossible worldsworlds andand thenthen gogo onon toto explainexplain possiblepossible worldworldss themselvethemselvess iinn termtermss ooff ththee purelpurelyy psychologicalpsychological conceptconcept ofof conceivability.conceivability. OnOn thethe otherother handhand,, wwee wiswishh ttoo avoiavoidd ththee circularity iinn whicwhichh wewe wouldwould bebe involvedinvolved ifif wewe triedtried toto definedefine possiblpossiblee worldworldss iinn termtermss ooff ththee logicalogicall notionotionn ooff consistency,consistency, andand thenthen laterlater defineddefined consistencyconsistency inin termsterms ooff possiblpossiblee worlds.worlds. Fortunately,Fortunately, therethere isis a wayway outout ofof thisthis predicament.predicament. WWee cacann avoiavoidd circularitcircularityy anandd stilstilll nonott bbee trappedtrapped byby psychologism.psychologism. Consider,Consider, forfor a moment,moment, hohoww ththee chargchargee ooff circularitcircularityy iinn definitiondefinitionss mamayy iinn otherother casescases bebe avoided.avoided. WeWe looklook inin a dictionary,dictionary, forfor example,example, ttoo findfind ououtt whawhatt iitt iiss foforr somethinsomethingg ttoo bebe complexcomplex andand findfind thatthat thethe conceptconcept ofof complexitycomplexity isis opposedopposed ttoo thathatt ooff simplicitysimplicity;; wwee trtryy ttoo findfind outout whatwhat itit isis forfor somethingsomething toto bebe simplesimple andand findfind thatthat ththee concepconceptt ooff simplicitsimplicityy iiss opposeopposedd ttoo thathatt ooff complexity.complexity. YetYet suchsuch a circlecircle ofof definitionsdefinitions cancan bebe,, andand standardlystandardly isis,, brokenbroken.. IItt iiss brokebrokenn bbyy citingciting examples: sometimessometimes byby ostensionostension (pointing);(pointing); sometimessometimes bbyy namingnaming;; anandd sometimesometimess bbyy descriptiondescription.. Thus,Thus, inin thethe presentpresent casecase wewe cancan avoidavoid thethe traptrap ofof circularitycircularity —- anandd aatt ththee samsamee timetime,, ththee seductionseduction ofof psychologismpsychologism -— byby citingciting clear-cutclear-cut examplesexamples (they(they araree ofteoftenn callecalledd "paradig"paradigmm examples"examples")) ooff possiblepossible worlds,worlds, andand equallyequally clear-cutclear-cut examplesexamples ofof impossibleimpossible worldsworlds.. IntuitivelyIntuitively,, wwee woulwouldd wanwantt ttoo include,include, amongamong thethe possiblepossible worlds,worlds, worldsworlds inin whichwhich thertheree araree mormoree objectobjectss thathann iinn ththee actuaactuall worlworldd (e.g.,(e.g., inin whichwhich thethe earthearth hashas twotwo moons);moons); worldsworlds inin whicwhichh thertheree araree fewefewerr objectobjectss thathann iinn ththee actuaactuall worldworld (e.g.,(e.g., inin whichwhich thethe earthearth hashas nono moonmoon atat all);all); worldworldss iinn whicwhichh ththee samsamee objectobjectss exisexistt aass iinn thethe actualactual worldworld butbut havehave differentdifferent propertiesproperties (e.g.,(e.g., inin whicwhichh ththee long-supposelong-supposedd "canals"canals"" oonn MarMarss turturnn out,out, afterafter all,all, toto bebe relicsrelics ofof somesome pastpast civilization);civilization); andand soso onon.. AnAndd intuitivelyintuitively,, tootoo,, wwee woulwouldd wanwantt ttoo include,include, amongamong thethe impossibleimpossible worlds,worlds, worldsworlds inin whichwhich circlescircles cacann bbee squaredsquared,, worldworldss iinn whicwhichh therethere isis anan eveneven squaresquare rootroot ofof nine,nine, worldsworlds inin whichwhich time-traveltime-travel botbothh doedoess anandd doedoess nonott occuroccur,, anandd ssoo onon.. ToTo bebe sure,sure, descriptionsdescriptions cancan bebe givengiven ofof worldsworlds aboutabout whoswhosee possibilitpossibilityy oorr impossibilitimpossibilityy wwee havhavee nono clearclear intuitions.intuitions. ButBut forfor nearlynearly anyany distinctiondistinction thatthat wwee carcaree ttoo thinthinkk ooff thertheree wilwilll bbee problematiproblematicc casescases lyinglying nearnear thethe borderlineborderline ofof thatthat distinction.distinction. TheThe casecase ooff worldworldss iinn whicwhichh time-travetime-travell occuroccurss iiss justjust suchsuch a case;case; andand so,so, untiluntil recently,recently, waswas thethe casecase ofof worldworldss iinn whicwhichh mormoree thathann foufourr colorcolorss areare neededneeded toto demarcatedemarcate betweenbetween countriescountries onon a planeplane mapmap.. BuButt ththee difficultdifficultyy ooff decidingdeciding,, foforr certaicertainn cases,cases, onon whichwhich sideside ofof thethe distinctiondistinction theythey areare toto bbee locatedlocated shoulshouldd nonott bbee alloweallowedd ttoo discredidiscreditt thethe distinctiondistinction itself.itself. TheThe distinctiondistinction betweenbetween possiblepossible andand impossibleimpossible worldworldss iiss groundegroundedd iinn aann appeaappeall ttoo paradigmparadigm cases.cases. WeWe dodo notnot havehave toto bebe ableable toto settlesettle allall boundarboundaryy disputedisputess i inn ordeorderr ttoo havhavee a securesecure enoughenough footingfooting onon whichwhich toto proceedproceed inin ourour attemptsattempts ttoo explainexplain ththee workingworkingss ooff logiclogic..

Possible worlds: actual and non-actualnon-actual LetLet usus pausepause atat thisthis pointpoint andand reflect,reflect, inin philosophicalphilosophical fashion,fashion, oonn some ooff ththee thingthingss wwee havhavee beenbeen saying.saying. WeWe havehave beenbeen workingworking withwith a threefoldthreefold distinction,distinction, whicwhichh iiss impliciimplicitt iinn mucmuchh ooff ouourr thinking,thinking, betweenbetween thethe actualactual world,world, worldsworlds whichwhich areare non-actuanon-actuall bubutt possiblepossible,, anandd worldworldss whicwhichh areare neither.neither. ButBut whatwhat preciselyprecisely dodo wewe meanmean byby "possible"possible world"world"?? MorMoree basicallbasicallyy stillstill:: WhaWhatt preciselpreciselyy ddoo wewe meanmean byby "the"the actualactual world"?world"? WhenWhen wewe speakspeak ofof "the"the actualactual world"world" wewe dodo notnot meameann jusjustt ththee planeplanett oonn whicwhichh wwee livelive.. NoNorr ddoo wewe meanmean ourour solarsolar system,system, oror eveneven ourour galaxy.galaxy. WeWe havhavee spokespokenn ooff ththee actuaactuall worlworldd iinn aann allall­- encompassingencompassing wayway soso asas toto embraceembrace allall thatthat reallyreally existsexists -— ththee universuniversee aass a whole.whole. Again,Again, whenwhen wewe speakspeak ofof "the"the actualactual world"world" wewe dodo nonott meameann jusjustt ththee universuniversee aass iitt iiss nownow,, iinn ththee present.present. WhenWhen wewe identifyidentify itit -— asas aboveabove -— withwith allall thathatt reallreallyy existsexists,, wwee araree usinusingg "exists"exists"" iinn a §§ 11 ThisThis andand OtherOther PossiblePossible WorldsWorlds 55 timelesstimeless sense,sense, soso asas toto encompassencompass notnot onlyonly whatwhat existsexists nownow butbut alsoalso whatwhat onceonce existedexisted inin thethe pastpast andand whatwhat willwill comecome toto existexist inin thethe future.future. TheThe actualactual worldworld embracesembraces allall thatthat was,was, is,is, oror willwill be.be. NowNow itit isis clearclear thatthat thethe actualactual worldworld isis aa possiblepossible world.world. IfIf somethingsomething actuallyactually existsexists thenthen itit isis obviouslyobviously possiblepossible thatthat itit exists.exists. OnOn thethe otherother hand,hand, notnot everythingeverything thatthat possiblypossibly existsexists doesdoes soso actually.actually. NotNot allall possiblepossible worldsworlds areare actual.actual. ItIt follows,follows, therefore,therefore, thatthat thethe actualactual worldworld isis onlyonly oneone amongamong manymany possiblepossible worlds:worlds: thatthat therethere areare possiblepossible worldsworlds otherother thanthan ours.ours. Moreover,Moreover, givengiven thatthat bbyy "the"the actualactual world"world" wewe meanmean -— asas wewe agreedagreed aa momentmoment agoago -— everythingeverything thatthat was,was, is,is, oror willwill bebe thethe case,case, itit followsfollows thatthat byby "another"another possiblepossible world"world" wewe dodo notnot meanmean somesome planet,planet, starstar oror whatnotwhatnot thatthat actuallyactually existsexists andand whichwhich isis locatedlocated somewheresomewhere "out"out there"there" inin physicalphysical space.space. WhateverWhatever actuallactuallyy exists,exists, itit mustmust bebe remembered,remembered, belongsbelongs toto thethe actualactual worldworld eveneven ifif itit isis light-yearslight-years away.away. Other,Other, non-actual,non-actual, possiblepossible worlds,worlds, areare notnot locatedlocated anywhereanywhere inin physicalphysical space.space. TheyThey areare located,located, asas itit were,were, inin conceptualconceptual space;space; oror rather,rather, asas wewe maymay preferprefer toto say,say, inin logicallogical space.space.

AllAll PossiblePossible WorldsWorlds A TheThe Non-actualNon-actual actualactual worldsworlds worldworld —

I I

FIGURFIGUREE (1.b)(l.b)

OuOurr worlworldd (everythin(everythingg thathatt actuallactuallyy waswas,, isis,, oorr wilwilll bebe)) iiss onlonlyy ononee ooff aann infinitinfinitee numbenumberr ofof possiblpossiblee worldsworlds.. ItIt iiss ththee actuaactuall worldworld.. ThThee otherotherss araree non-actual.non-actual.

NotNotee thathatt iinn thithiss anandd otheotherr similasimilarr diagramdiagramss wwee represenrepresentt aann infinitinfinitee numbenumberr ooff possiblpossiblee worldworldss bbyy a rectanglrectanglee ooff finitfinitee sizesize.. ThThee rectanglrectanglee mamayy bbee thoughthoughtt ooff aass containincontainingg aann infinitinfinitee numbenumberr ooff pointpointss eaceachh ooff whicwhichh representrepresentss a differendifferentt possiblpossiblee worldworld.. IItt iiss onlonlyy foforr ththee saksakee ooff diagrammatidiagrammaticc conveniencconveniencee thathatt wwee represenrepresentt ththee actuaactuall worlworldd oonn thithiss anandd a fefeww subsequensubsequentt figures byby aa segmentsegment ofof thethe rectanglerectangle ratherrather thanthan byby aa singlesingle point.point. FromFrom aa logicallogical pointpoint ofof vievieww ththee actuaactuall worlworldd hahass littllittlee (i(iff anyany)) claiclaimm ttoo privilegeprivilegedd statusstatus.. IndeedIndeed,, iinn mosmostt ooff ththee worlds-diagramworlds-diagramss featurefeaturedd latelaterr wwee shalshalll havhavee nnoo neeneedd ttoo makmakee mentiomentionn ooff ththee actuaactuall worldworld,, lelett alonalonee ttoo featurfeaturee iitt prominently.prominently.

NoteNote,, furtherfurther,, thathatt ththee brackebrackett foforr non-actuanon-actuall worldworldss iiss lefleftt opeopenn aatt ththee left-hanleft-handd sidsidee ooff thithiss diagradiagramm anandd alalll diagramdiagramss oonn whicwhichh ththee actuaactuall worlworldd iiss featuredfeatured.. ThiThiss iiss ttoo signifsignifyy thathatt thertheree araree mormoree non-actuanon-actuall worldworldss thathann wwee havhavee herheree depicteddepicted.. ThThee clasclasss ooff non-actuanon-actuall worldworldss containcontainss alalll possiblpossiblee worldworldss otheotherr thathann ththee actuaactuall worlworldd anandd containcontainss aass welwelll alalll impossiblimpossiblee worldsworlds.. EverEveryy impossiblimpossiblee worlworldd iiss a non-actuanon-actuall worldworld.. 6 POSSIBLPOSSIBLEE WORLDSWORLDS

Logical possibility distinguished from other kinds ByBy "a"a possiblepossible world",world", itit shouldshould bebe emphasized,emphasized, wwee dodo not meameann onlonlyy a physically possiblpossiblee worldworld.. CountlessCountless worldsworlds whichwhich areare physicallyphysically impossibleimpossible areare numberenumberedd amonamongg ththee possiblpossiblee worldworldss wwee areare talkingtalking about.about. PhysicallyPhysically possiblepossible worldsworlds formform a propeproperr subsetsubset ooff alalll possiblpossiblee worldsworlds;; oror,, ttoo makmakee thethe contrastcontrast somewhatsomewhat sharper,sharper, wewe mightmight saysay thatthat thethe setset ofof physicallphysicallyy possiblpossiblee worldworldss formformss a propeproperr subsetsubset ofof allall logically possiblepossible worlds.worlds. A physicallyphysically possiblepossible worldworld isis anyany possiblepossible worlworldd whicwhichh hahass ththee samsamee naturanaturall lawlawss aass doedoess ththee 3 actualactual world.3 ThusThus thethe logicallylogically possiblepossible worldworld depicteddepicted iinn CharleCharless DickensDickens'' novenovell David Copperfield isis a physicallyphysically possiblepossible one:one: nono eventevent inin thatthat novelnovel violateviolatess ananyy naturanaturall lawlaw.. OOnn ththee otheotherr handhand,, WashingtonWashington Irving'sIrving's shortshort storystory Rip Van Winkle describesdescribes a physicallphysicallyy impossiblimpossiblee worldworld:: a worlworldd iinn whichwhich a personperson sleepssleeps withoutwithout nourishmentnourishment forfor twentytwenty yearsyears.. ThThee lattelatterr circumstancecircumstance,, a person'person'ss livinlivingg forfor twentytwenty yearsyears withoutwithout nourishmentnourishment (energy(energy intake),intake), violateviolatess certaicertainn lawlawss ooff thermodynamicsthermodynamics.. Nonetheless,Nonetheless, althoughalthough suchsuch a situationsituation isis thusthus physicallyphysically impossibleimpossible,, iitt iiss nonott logicalllogicallyy impossibleimpossible.. InIn

AllAll PossiblePossible WorldsWorlds r K Non-ActualNon-Actual WorldsWorlds

PhysicallyPhysically PhysicallPhysicallyy ImpossibleImpossible PossiblPossiblee ------..J'''------_"Y~-----~"'------"\A ThThee ActualActual WorlWorldd ..

FIGUREFIGURE (1.c)(l.c)

3.3. NoteNote that,that, onon thisthis account,account, thethe notionnotion ofof a physicallyphysically possiblpossiblee worlworldd iiss parasitiparasiticc upouponn (need(needss ttoo bbee defineddefined inin termsterms of)of) thethe broaderbroader notionnotion ofof a logicallylogically possiblepossible world.world. Similarly,Similarly, a statstatee ooff affairaffairss iiss physicallphysicallyy possiblepossible,, itit isis usuallyusually said,said, ifif itsits descriptiondescription isis consistent with withthe thenatural naturallaws laws(of (ofthe theactual actualworld). world).Yet Yetthe therelation relationof of consistencyconsistency -— asas wewe showshow inin sectionsection 4 -— isis itselfitself toto bbee defineddefined iinn termtermss ooff possiblpossiblee worldsworlds,, i.e.i.e.,, ooff logicallylogically possiblepossible worlds.worlds. § 1 This and Other Possible Worlds 7

some non-actual, physically impossible, but logically possible world in which the natural laws are different from those in the actual world, Rip Van Winkle does sleep (without nourishment) for twenty years. Of course, not every physically will be logically possible. The physically impossible world in which Rip Van Winkle sleeps for exactly twenty years without nourishment and does not sleep during those years without nourishment is a logically im­ possible one as well. The class of logically possible worlds is the most inclusive class of possible worlds. It includes every other kind of possible world as well: all those that are physically possible and many, but not all, that are physically impossible; and it includes all worlds which are technologically possible, i.e., physically possible worlds having the same physical resources and industrial capacity as the actual world, and many, but not all, which are technologically impossible. Very many different subsets of the set of all logically possible worlds may be distinguished. Many of these are of philosophical interest, e.g., physical, technological, moral, legal, etc. But we shall not much concern ourselves with them. For the most part, in this book our interest lies with the largest class of possible worlds, the class of logically possible worlds, and on a few occasions with the class of physically possible worlds. It is only in more advanced studies in logic that the special properties of various of these other subsets are examined. Hereinafter, whenever we use the expression "all possible worlds" without any further qualification, we are to be understood to mean "all logically possible worlds".

The constituents ofpossible worlds How are possible worlds constituted? It may help if we start with the possible world that we know best: the actual one. Following Wittgenstein, we shall say that the actual world is "the totality of [actually] existing states of affairs", where by "a state of affairs" we shall mean roughly what he meant, viz., an arrangement of objects, individuals, or things having various properties and standing in various relations to one another. It will help, however, if we adopt a slightly different terminology. Instead of the terms "objects", "things", and "individuals" we shall adopt the more neutral term "items".4 And in addition to the terms "properties" and "relations" we shall adopt the more generic term "attributes".5 An item (object or thing) is whatever exists in at least one possible world: physical objects like Model T Fords and starships; persons such as Woodrow Wilson and Lazarus Long; pla~es such as Old Home Terra (the earth) and Secunda; events such as World War I and the Great Diaspora of the Human Race; abstract objects such as numbers and sets; and so on. An attribute (property or relation) is whatever is exemplified or instanced (has instances) by an item or by items in a world; properties such as being red, being old, being distant, or being frightening; andrelations such as being faster than, being a lover of, being more distant than, or being earlier than. Typically, items are the sorts of things we would have to mention in giving a description of a possible world; they are things we can refer to. Attributes are the sorts of things that characterize items; they are the sorts of things which we ascribe to the objects of reference. Now it is clear, from the examples just given of items and attributes, that items exist in possible worlds other than the actual one, and that attributes are instanced in possible worlds other than the

4. Other terms which play roughly the same role in philosophical literature are "particulars" and "logical subjects".

5. The term "attribute" with th~ we have given it was introduced into recent philosophical literature by Carnap. As he put it: "In ordinary language there is no word which comprehends both properties and relations. Since such a word would serve a useful purpose, let us agree in what follows that the word 'attribute' shall have this sense. Thus a one-place attribute is a property, and a two-place (or a many-place) attribute is a relation." Introduction to Symbolic Logic and Its Applications, New York, Dover, 1958, p. 5. 8 POSSIBLPOSSIBLEE WORLDSWORLDS

actualactual one.one. How,How, then,then, dodo non-actualnon-actual possiblepossible worldsworlds differdiffer frofromm ththee actuaactuall oneone?? TheTheyy mamayy diffedifferr iinn threethree basicbasic ways.ways. OtherOther possiblepossible worldsworlds maymay containcontain ththee ververyy samsamee itemitemss aass ththee actuaactuall worlworldd bubutt differdiffer fromfrom thethe actualactual worldworld inin respectrespect ofof thethe attributesattributes whicwhichh thosthosee itemitemss instanceinstance;; e.g.e.g.,, diffedifferr iinn respectrespect ofof thethe EiffelEiffel TowerTower beingbeing purple,purple, thethe TajTaj MahalMahal beinbeingg greengreen,, anandd ssoo onon.. OOrr thetheyy mamayy contaicontainn atat leastleast somesome itemsitems whichwhich dodo notnot existexist inin thethe actualactual worlworldd (an(andd ipsipsoo factfactoo diffedifferr iinn respecrespectt ooff somsomee attributes);attributes); e.g.,e.g., theythey maymay containcontain LazarusLazarus LongLong oror SherlockSherlock HolmesHolmes.. OOrr stilstilll againagain,, thetheyy mamayy laclackk certaincertain itemsitems whichwhich existexist inin thethe actualactual worldworld (and(and ipsoipso factfactoo diffedifferr iinn respecrespectt ooff somsomee attributes)attributes);; e.g.e.g.,, theythey maymay lacklack StalinStalin oror Shakespeare.Shakespeare. SomeSome philosophersphilosophers havehave thoughtthought thatthat otherother possiblepossible worldworldss cacann diffedifferr frofromm ththee actuaactuall ononee onlonlyy iinn thethe firstfirst ofof thesethese threethree ways.ways. OtherOther philosophersphilosophers insistinsist thathatt thetheyy cacann diffedifferr frofromm ththee actuaactuall ononee iinn th thee otherother twotwo waysways asas well.well. OurOur ownown thinkingthinking inin thethe mattematterr iiss evidentevident:: iinn choosinchoosingg ththee worlworldd ooff TimeTime Enough for Love asas ourour example,example, wewe areare allowingallowing ththee existenceexistence ooff non-actuanon-actuall possiblpossiblee worldworldss somsomee ofof whosewhose itemsitems dodo notnot existexist inin thethe actualactual world.world. InIn chapterchapter 44,, sectiosectionn 66,, wwee wilwilll explorexploree brieflbrieflyy somsomee ooff thethe consequencesconsequences ofof beingbeing lessless generous.generous.

EXERCISES

InIn thisthis andand subsequentsubsequent exercises,exercises, itit isis essentialessential thatthat oneone assumassumee thathatt alalll ththee termtermss beinbeingg useusedd havhavee theirtheir standardstandard meanings,meanings, e.g.,e.g., thatthat "square""square" refersrefers ttoo a planeplane,, closedclosed,, four-sidefour-sidedd figurfiguree havinhavingg equaequall interiorinterior anglesangles andand equalequal sides.sides. ToTo bebe sure,sure, forfor anyany worwordd oorr sentencsentencee wwee carcaree ttoo thinthinkk ofof,, thertheree wilwilll bbee possiblepossible worldsworlds inin whichwhich thatthat wordword oror sentencesentence willwill meameann somethingsomething altogethealtogetherr differendifferentt frofromm whawhatt iitt meansmeans inin thethe actualactual world.world. ThisThis fact,fact, however,however, inin nnoo wawayy telltellss againsagainstt ththee facfactt thathatt whawhatt we refereferr ttoo byby thethe wordword "square""square" hashas fourfour sidessides inin allall possiblepossible worldsworlds.. ThThee exerciseexercisess poseposedd asaskk whethewhetherr ththee claimsclaims we areare herehere consideringconsidering areare truetrue inin anyany possiblpossiblee worldworld;; anandd thithiss iiss a whollwhollyy distincdistinctt mattematterr fromfrom whetherwhether ourour wordswords mightmight bebe usedused byby thethe inhabitantsinhabitants ooff otheotherr possiblpossiblee worldworldss ttoo makmakee differendifferentt claims. 66

Part A

For each ofofthe thefollowing, following, saysaywhether whetherthere thereis isa alogically logicallypossible possibleworld worldin inwhich which ititis istrue. true.

,.1. Frederick, of Gilbert and Sullivan's "Pirates of Penzance", has reached the age of 21 years afterafter only 5 birthdays. 2. There isis a square house allallof ofwhose whosewalls wallsface facesouth. south. 3. Epimenides,Epimemdes, thethe Cretan, spoke in truth when he said that everything that Cretans say isisfalse. false. 4. Mt. Everest isis lower than Mt. Cook. 5. Mt. Everest isis lower than Mt. Cook, Mt. Cook is lower than Mt. Whistler, and Mt. Whistler isis lowerlower thanthan Mt. Everest.Everest. 6. 2 + 2 i=2*4.4. 7. The PoPe.Pope believes thatthat2 2 ++ 2 i=2*4.4. 8. The Pope knows that 2 + 2 i=2*4.4.

6. ForFor moremore onon thisthis point,point, seesee ourour discussiondiscussion inin chapterchapter 22,, pppp.. 1l1Off,1 Off, ofof thethe uni-linguouni-linguo proviso.proviso. § 2 Propositions, Truth, and Falsity 9

9. No objects are subject to the law ofgravitation. 70. There is a mountain which is higher than every mountain.

Part B The following quotations are for reflection and discussion. For each of them try to decide whether the circumstance described could occur (a) in a logically possible world, and (b) in a physically possible world. 7. ., 'All right,' said the Cat; and this time it vanished qutte slowly, beginning with the end ofthe tail, and ending with the grin, which remained some time after the rest of it had gone." (Lewis Carroll, Alice's Adventures in Wonderland & Through the Looking Glass, New York, Signet Classics, 7960, p. 65.) 2. "As Gregor Samsa awoke one morning from uneasy dreams he found himseJj transformed in his bed into a gigantic insect. He was lying on his hard, as it were armor-plated, back and when he lifted his head a little he c:ould see his dome-like brown belly divided into stiffarched segments on top of which the bed quilt could hardly keep in position and was about to slide off completely. His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes." (Franz Kafka, The Metamorphosis, 7972, translated by Willa and Edwin Muir, New York, Schocken Books, 7948, p. 7.) 3. "NAT BARTLETT is very tall, gaunt, and loose-framed. His right arm has been amputated at the shoulder, and the sleeve on that side of the heavy mackinaw he wears hangs flabbily or flaps against his body as he moves. ... He closes the door and tiptoes carefully to the companionway. He ascends it a few steps and remains for a moment listening for some sound from above. Then he goes over to the table, turning the lantern very low, and sits down, resting his elbows, his chin on his hands, staring somberly before him." (Eugene O'Neill, Where the Cross is Made, copyright 7979 by Boni and Liveright. Reprinted in Twelve One-Act Plays for Study and Production, ed. S. MarlOn Tucker, Boston, Ginn and Co., 7929, pp. 202, 208.)

2. PROPOSITIONS, TRUTH, AND FALSITY

Truth andfalsity defined Items (i.e., objects, things), we have said, may exist in possible worlds other than the actual one. Likewise, attributes (i.e., properties and relations) may be instanced in possible worlds other than the actual one. Consider, now, any arbitrarily selected item and any arbitrarily selected attribute; and let us them, respectively, a and F. Then we can define "truth" and "falsity" as follows:

(a) it is true that a has F if, and only if, a has F; and (b) it is false that a has F if, and only if, it is not the case that a has F.

These definitions tell us, for instance, that where "a" stands for Krakatoa Island and "F" stands for the property of being annihilated in a volcanic eruption, then

It is true that Krakatoa Island was annihilated by a -volcanic eruption if and only if Krakatoa Island was annihilated by a volcanic eruption 1100 POSSIBLPOSSIBLEE WORLDSWORLDS

anandd againagain,, thathatt

IItt iiss falsfalsee thathatt KrakatoKrakatoaa IslanIslandd wawass annihilateannihilatedd bbyy a volcanivolcanicc eruptioeruptionn iiff anandd onlonlyy iiff iitt iiss nonott ththee cascasee thathatt KrakatoKrakatoaa IslanIslandd wawass annihilateannihilatedd bbyy a volcanivolcanicc eruptioneruption..

ThesThesee definitiondefinitionss accoraccordd witwithh ththee insighinsightt whicwhichh AristotleAristotle,, mormoree thathann .twotwo thousanthousandd yearyearss agoago,, expresseexpressedd thus:thus: TToo sasayy ooff whawhatt iiss thathatt iitt iiss nonott oorr ooff whawhatt iiss nonott thathatt iitt isis,, iiss falsefalse,, whilwhilee ttoo sasayy ooff whawhatt iiss thathatt iitt isis,, anandd ooff whawhatt iiss nonott thathatt iitt iiss notnot,, iiss true.7true.7

SeveraSeverall pointpointss abouaboutt thesthesee definitiondefinitionss araree wortworthh notingnoting::

(i(i)) AlthougAlthoughh Aristotle'Aristotle'ss accounaccountt suggestsuggestss thathatt iitt iiss personspersons'' sayingsayingss whicwhichh araree trutruee oorr falsfalsee (i.e.(i.e.,, whicwhichh areare,, aass wwee shalshalll puputt itit,, ththee bearerbearerss ooff truth-values)truth-values),, ouourr accounaccountt iinn (a(a)) anandd (b(b)) leaveleavess opeopenn thethe questioquestionn aass ttoo whawhatt iitt iiss whicwhichh iiss trutruee oorr falsefalse.. TherTheree iiss nnoo doubdoubtt thathatt sayingssayings,, alonalongg witwithh believingsbelievings,, supposingssupposings,, etc.etc.,, araree among ththee thingthingss whic whichh cacann bbee trutruee oorr falsefalse.. BuButt manmanyy contemporarycontemporary philosopherphilosopherss prefepreferr ttoo saysay thathatt iitt iiss primarilprimarilyy propositionspropositions whichwhich havehave thethe propertiesproperties ofof truthtruth oror falsityfalsity (i.e.(i.e.,, thathatt iitt iiss primarilprimarilyy propositionpropositionss whicwhichh araree bearerbearerss ooff truth-values), anandd woulwouldd holholdd thathatt thethe thingthingss personpersonss say,say, believebelieve,, suppose,suppose, etc.etc.,, araree trutruee oorr falsfalsee jusjustt whewhenn ththee propositionpropositionss whicwhichh thetheyy utterutter,, believebelieve,, oorr entertainentertain,, etc.etc.,, araree trutruee oorr falsefalse.. FoForr ththee reasonreasonss givegivenn aatt lengtlengthh iinn chaptechapterr 22,, w wee adopadoptt thethe 8 F, lattelatterr wawayy ooff talking.8 IfIf wwee lelett ththee letteletterr "P""P" standstand foforr ththee propositiopropositionn thathatt a hahass F, wwee cacann restatrestatee (a)(a) anandd (b)(b) asas

(a)*(a) * ThThee propositiopropositionn P (tha(thatt a hahass FF)) iiss trutruee iiff anandd onlonlyy iiff a hahass FF;; anandd (b)*(b) * TheThe propositionproposition P (that(that a hashas F)F) isis falsefalse ifif andand onlyonly ifif itit isis notnot thethe cascasee thatthat a hashas FF..

(ii)(ii) OurOur account,account, inin (b)(b) andand (b)*,(b)*, ofof thethe conditionsconditions inin whichwhich thethe propositionproposition thatthat a hashas F isis false,false, allowsallows forfor twotwo suchsuch conditions:conditions: thethe possiblepossible statestate ofof affairsaffairs inin whichwhich thethe itemitem a existsexists butbut failsfails toto havehave thethe attributeattribute F;F; andand thethe possiblepossible statestate ofof affairsaffairs inin whichwhich thethe itemitem a doesdoes not exist.exist. SinceSince anan attributeattribute cancan bebe instancedinstanced byby anan itemitem inin a possiblepossible worldworld onlyonly ifif thatthat itemitem existsexists inin thatthat possiblepossible world,world, thethe failurefailure ofof anan itemitem toto existexist inin a givengiven possiblepossible worldworld precludesprecludes itit fromfrom havinghaving anyany attributesattributes whateverwhatever inin thatthat world.world.

(iii)(iii) StrictlyStrictly speaking,speaking, ourour accountaccount -— andand Aristotle'sAristotle's -— ofof whatwhat itit isis forfor a propositionproposition toto bebe truetrue oror falsefalse appliesapplies onlyonly toto propositionspropositions whichwhich ascribeascribe properties toto items.items. ButBut itit isis easilyeasily enoughenough extendedextended toto dealdeal withwith thosethose propositionspropositions whichwhich ascribeascribe relations toto twotwo oror moremore items.items. WeWe cancan dealdeal withwith so-calledso-called "two-place""two-place" attributesattributes (i.e.,(i.e., relationsrelations holdingholding betweenbetween justjust twotwo items)items) asas follows:follows: wherewhere P isis R a proposition,proposition, a andand b areare items,items, andand R isis thethe two-placetwo-place attributeattribute (relation)(relation) whichwhich P assertsasserts toto holdhold R, betweenbetween a andand b, thenthen P isis truetrue ifif andand onlyonly ifif a andand b standstand toto eacheach otherother inin thethe relationrelation R, whilewhile P isis

7.7. Metaphysics,r,Metaphysics, T, 7(10117(101 lbb 2626 - 27),27), The Basic Works of Aristotle, ed.ed. R.R. McKeon,McKeon, NewNew York,York, RandomRandom House,House, 1941.1941.

8.8. A proposition,proposition, wewe shallshall argue,argue, isis toto bebe distinguisheddistinguished fromfrom thethe variousvarious sentencessentences whichwhich language-speakerlanguage-speakerss maymay useuse toto expressexpress it,it, inin muchmuch thethe samesame wayway asas a numbernumber isis toto bebe distinguisheddistinguished fromfrom thethe variousvarious numeralnumeralss whichwhich maymay bebe usedused toto expressexpress it.it. § 2 Propositions,Propositions, Truth,Truth, andand FalsityFalsity 1111

falsefalse ifif andand onlyonly ifif itit isis notnot thethe casecase thatthat a andand b standstand inin thethe relationrelation R.R. AndAnd accountsaccounts ofof truth-valuestruth-values forfor otherother relationalrelational propositionpropositionss involvinginvolving threethree oror moremore itemsitems maymay bbee constructedconstructed alongalong similarsimilar lineslines.. Alternatively,Alternatively, wewe cancan dealdeal withwith relationalrelational propositionpropositionss bbyy pointinpointingg outout thatthat wheneverwhenever anan itemitem a standstandss inin a relationrelation R toto oneone oror moremore otherother items,items, b, c, etc.,etc., a cancan bbee saidsaid toto havehave thethe relational property ooff standingstanding inin thatthat relationrelation toto b, c, etc.etc. AndAnd sincesince relationalrelational propertiepropertiess areare propertiesproperties,, thethe straightforwardstraightforward accountaccount givengiven inin (a)(a) andand (b)(b) suffices.suffices.

(iv) TheThe accountaccount ofof truthtruth whichwhich wewe areare herehere espousingespousing hashas beebeenn describeddescribed variouslyvariously asas "the"the CorrespondenceCorrespondence Theory",Theory", "the"the RealistRealist Theory",Theory", oror eveneven "the"the SimpleSimple Theory"Theory" ofof truth.truth. InIn effect,effect, i itt sayssays thatthat a propositionproposition,, P,P, isis truetrue ifif andand onlyonly if.if the'(possible)the (possible) statestate ofof affairs,affairs, e.g.,e.g., ofof a's havinghaving F,F, isis asas P assertsasserts itit toto be.be. ItIt definesdefines "truth""truth" asas a propertyproperty whichwhich propositionspropositions havehave jusjustt whenwhen theythey "correspond""correspond" toto thethe (possible)(possible) statesstates ofof affairsaffairs whosewhose existenceexistence theythey assert.assert. ItIt isis a "realist""realist" theorytheory ooff truthtruth insofarinsofar asas itit makesmakes truthtruth a realreal oror objectiveobjective propertpropertyy ofof propositionspropositions,, i.e.,i.e., notnot somethingsomething subjectivesubjective butbut a functionfunction ofof whatwhat statesstates ofof affairsaffairs reallyreally existexist inin thisthis oror thatthat possiblpossiblee world.world. AndAnd itit isis a "simple""simple" theorytheory ofof truthtruth insofarinsofar asas itit accordsaccords withwith thethe simplesimple intuitionsintuitions whichwhich mostmost ofof usus -— beforbeforee wewe trytry toto getget tootoo sophisticatedsophisticated aboutabout suchsuch mattersmatters -— havehave aboutabout thethe conditionsconditions forfor sayingsaying thatthat somethingsomething isis truetrue oror false.false.

(v)(v) IfIf itit seemsseems toto somesome thatthat thisthis theorytheory isis overlyoverly simplesimple andand notnot profounprofoundd enough,enough, thisthis isis probablprobablyy becausbecausee itit isis deceptivelydeceptively easyeasy toto confuseconfuse thethe questionquestion "What"What areare thethe conditionsconditions ofof truthtruth andand falsity?falsity?"" withwith thethe questionquestion "What"What areare thethe conditionsconditions forfor ourour knowledge ofof truthtruth andand falsity?"falsity?" ItIt isis notnot surprising,surprising, asas a consequence,consequence, thatthat thethe mostmost commonlycommonly favoredfavored allegedalleged rivalsrivals toto thethe CorresppndenceCorrespondence TheoryTheory areare thethe so-calledso-called CoherenceCoherence andand PragmatistPragmatist TheoriesTheories ofof Truth.Truth. ProponentsProponents ofof thesethese theoriestheories tendtend eithereither toto denydeny oror toto ignoreignore thethe distinctiondistinction betweebetweenn a proposition'proposition'ss being truetrue andand a proposition'sproposition's being known toto bbee true.true. WeWe maymay acceptaccept thethe coherencecoherence theorist'stheorist's claimclaim thatthat oneone wayway ofof gettinggetting toto knoknoww whatwhat propositionpropositionss areare truetrue isis toto determinedetermine whichwhich ofof themthem coherecohere withwith thethe restrest ofof thethe beliefbeliefss thatthat wewe holdhold toto bbee true.true. AndAnd wewe maymay alsoalso acceptaccept thethe pragmatist'pragmatist'ss claimclaim thatthat oneone wayway ofof gettinggetting toto knowknow whatwhat propositionpropositionss areare truetrue isis toto determinedetermine whichwhich ofof themthem itit proveprovess usefuluseful oror practicapracticall toto believebelieve.. ButBut atat thethe samesame timetime wewe wouldwould wish,wish, toto poinpointt outout thatthat neitherneither claimclaim warrantswarrants identifying truthtruth withwith coherencecoherence oror truthtruth withwith practicapracticall usefulnessusefulness.. AndAnd furtherfurther wewe wouldwould poinpointt outout thatthat botbothh theoriestheories areare parasitiparasiticc uponupon thethe "simple""simple" theorytheory ofof truth.truth. TheThe pragmatispragmatistt statesstates thatthat certaincertain hypotheseshypotheses are useful;useful; thethe coherencecoherence theoristtheorist statesstates thatthat certaincertain setssets ofof beliefbeliefss are coherent.coherent. YetYet each,each, inin soso doing,doing, isis implicitlyimplicitly makingmaking a claimclaim toto thethe simple truthtruth ofof these propositionspropositions.. TheThe conceptconcept ofof simplesimple truth,truth, itit seems,seems, isis notnot easileasilyy dispenseddispensed with.with. EvenEven thosethose whowho wouldwould likelike mostmost toto avoidavoid itit cannotcannot dodo soso..

Truth in a possible world ItIt shouldshould bbee evidentevident fromfrom ourour definitionsdefinitions (i)(i) thatthat a propositionproposition,, P,P, maymay assertassert thatthat anan itemitem a hashas attributeattribute F eveneven ifif a existsexists onlyonly inin a non-actualnon-actual possiblpossiblee world,world, andand (ii)(ii) thatthat P willwill bebe truetrue inin thathatt non-actualnon-actual worldworld providedprovided thatthat inin thatthat worldworld a hashas F.F. LetLet uuss ususee thethe letterletter "w""W" (with(with oror withoutwithout numericalnumerical indices)indices) toto referrefer toto anyany possiblpossiblee worldworld ofof ourour choosingchoosing (e.g.,(e.g., thethe worldworld ofof Heinlein'sHeinlein's Time Enough for Love). ThenThen ourour definitionsdefinitions allowallow us,us, forfor instance,instance, toto assertassert thethe truthtruth inin somesome specifiespecifiedd possiblpossiblee world,world, WI'Wj, ofof thethe propositiopropositionn thatthat LazarusLazarus LongLong hashas thethe relationalrelational propertpropertyy ofof beinbeingg a loveloverr ofof MinervaMinerva eveneven thoughthough inin somesome otherother specifiedspecified possiblpossiblee world,world, W 22 (e.g.,(e.g., thethe actualactual world),world), hehe doedoess notnot existexist andand a fortiorifortiori doesdoes notnot havehave thatthat propertyproperty.. Likewise,Likewise, theythey allowallow usus toto assertassert thethe falsityfalsity iinn somesome specifiedspecified possiblpossiblee world,world, W 2,2, ofof thethe propositiopropositionn thatthat LazarusLazarus hadhad MinervaMinerva asas a loverlover eveneven thoughthough inin somesome otherother specifiedspecified possiblpossiblee world,world, e.g.,e.g., WI'Wj, thatthat propositiopropositionn isis true.true. NeedlesNeedlesss toto say,say, theythey alsoalso allowallow usus toto assertassert thatthat a propositiopropositionn isis truetrue (or(or false)false) inin somesome unspecifiedimspecified world,world, W;W; or,or, asas wewe shallshall laterlater puputt it,it, toto assertassert thatthat a propositiopropositionn isis possiblpossiblyy truetrue (or(or false,false, asas thethe casecase maymay be).be). 1212 POSSIBLEPOSSIBLE WORLDSWORLDS

Truth in the actual world ItIt followsfollows fromfrom whatwhat wewe havehave jusjustt saidsaid thatthat -— contrarycontrary toto whatwhat isis oftenoften supposedsupposed -— thethe expressionsexpressions "is"is true"true" andand "is"is false"false" dodo not meanmean thethe samesame asas "is"is actuallyactually true"true" andand "is"is actuallyactually false".false". ToTo saysay thathatt a proposItionproposition isis actuallyactually truetrue (or(or false)false) isis toto saysay thatthat amongamong thethe possiblepossible worldsworlds inin whichwhich itit isis truetrue (or(or false)false) therethere isis thethe actualactual world.world. Thus,Thus, forfor instance,instance, thethe propositiopropositionn thatthat CanadaCanada isis northnorth ofof MexicoMexico iiss actuallyactually truetrue becausbecausee inin thethe actual worldworld CanadaCanada standsstands inin thethe relationrelation ofof beinbeingg north-ofnorth-of toto Mexico,Mexico, i.e.,i.e., hashas thethe relationalrelational propertpropertyy ofof beinbeingg northnorth ofof Mexico.Mexico. AndAnd equally,equally, thethe propositiopropositionn thatthat LazarusLazarus hashas MinervaMinerva asas a loverlover isis actuallyactually falsefalse becausbecausee inin thethe actual worldworld LazarusLazarus failsfails toto exist.exist. TheThe factfact thatthat "true""true" doesdoes notnot meanmean "actuaJly"actually true"true" shouldshould bbee evidentevident fromfrom twotwo considerations.considerations. First,First, ifif itit did,did, thenthen thethe occurrenceoccurrence ofof thethe qualifierqualifier "actually""actually" wouldwould bbee whollywholly redundantredundant (pleonastic).(pleonastic). Yet,Yet, asas wewe havehave alreadyalready seen,seen, itit isis not.not. Secondly,Secondly, ifif itit diddid meanmean thethe same,same, thenthen personpersonss inin otherother possiblpossiblee worldsworlds wouldwould notnot bbee ableable toto invokeinvoke tht~ee conceptconcept ofof truthtruth withoutwithout therebythereby makingmaking claimsclaims aboutabout this world,world, thethe actualactual oneone inin whichwhich youyou andand wewe findfind ourselves.ourselves. Consider,Consider, forfor illustrativeillustrative purposes,purposes, Lazarus'Lazarus' claimclaim (on(on thethe firstfirst pagepage ofof thethe narrativenarrative ofof Time Enough for Love): "It's"It's truetrue I'mI'm notnot handyhandy withwith thethe jabbejabberr theythey speakspeak here."here." TheThe yearyear isis A.D.AD. 42724272 andand thethe placplacee isis thethe fictionalfictional HowarHowardd RejuvenationRejuvenation ClinicClinic onon thethe fictionalfictional planeplanett ofof Secundus.Secundus. LazarusLazarus assertsasserts thatthat a certaincertain propositionproposition,, viz.,viz., thatthat hehe doesn'tdoesn't havehave facilityfacility withwith thethe locallocal language,language, isis true.true. WeWe understandunderstand himhim toto bbee sayinsayingg somethingsomething whichwhich isis truetrue ofof thethe fictionalfictional worldworld inin whichwhich hehe exists.exists. YetYet ifif "true""true" meant "actually"actually true"true" wewe shouldshould bbee obligedobliged toto understandunderstand himhim asas sayingsaying somethingsomething veryvery differentdifferent -— asas sayingsaying somethingsomething aboutabout hishis lacklack ofof facilityfacility withwith a languagelanguage thatthat existsexists inin thethe actualactual worldworld whichwhich youyou andand wewe inhabit.inhabit. AlthoughAlthough "true""true" doesdoes notnot mean "actually"actually true",true", itit cancan bbee -— andand oftenoften isis -— usedused toto refer toto actualactual truth;truth; whatwhat mattersmatters isis who usesuses it.it. WhenWhen personpersonss inin thethe actualactual worldworld claimclaim thatthat a propositiopropositionn isis truetrue theythey areare claimingclaiming thatthat itit isis truetrue inin theirtheir ownown world;world; andand sincesince theirtheir worldworld isis thethe actualactual world,world, itit turnsturns outout thatthat theythey areare attributingattributing actualactual truthtruth toto thethe propositionproposition.. However,However, whenwhen a persopersonn inin a non-actualnon-actual worldworld attributesattributes truthtruth toto a propositionproposition,, hehe isis attributingattributing toto thatthat propositiopropositionn truthtruth inin hishis own world;world; andand sincesince hishis worldworld isis not thethe actualactual world,world, itit turnsturns outout thatthat hehe isis attributingattributing non-actualnon-actual (but(but possiblepossible)) truthtruth toto thatthat propositionproposition..

The myth of degrees of oftruth truth ItIt isis worthworth pointinpointingg outout thatthat itit isis implicitimplicit inin botbothh Aristotle'sAristotle's accountaccount andand thethe oneone inin (a)*(a)* andand (b)*(b)* thathatt truthtruth andand falsityfalsity dodo notnot admitadmit ofof degrees.degrees. AlthoughAlthough therethere isis a commoncommon mannermanner ofof speechspeech whichwhich fostersfosters thisthis beliebelieff -— forfor example,example, "Jones'"Jones' reportreport waswas moremore truetrue thanthan Roberts'"Roberts'" -— strictlystrictly speakingspeaking thisthis man­man• nerner ofof speechspeech isis logicallylogically untenableuntenable.. A propositiopropositionn isis eithereither whollywholly truetrue oror whollywholly false.false. ThereThere isis nono suchsuch thingthing asas partiapartiall truth.truth. ConsiderConsider thethe simplestsimplest sortsort ofof case,case, thatthat inin whichwhich a propositiopropositionn P ascribesascribes anan attributeattribute F toto anan itemitem a: P isis truetrue ifif a hashas attributeattribute F,F, otherwiseotherwise P isis false.false. ThereThere isis nono provisionprovision,, nnoo room,room, inin thisthis explicationexplication ofof truth,truth, forfor P toto bbee anythinganything otherother thanthan whollywholly truetrue oror whollywholly false;false; forfor eithereither a hashas thethe attributeattribute F,F, oror itit isis notnot thethe casecase thatthat a hashas thethe attributeattribute FF.. ThereThere is,is, however,however, a wayway to.to reconcilereconcile thethe commoncommon mannermanner ofof speechspeech jusjustt reportedreported withwith thethe stringenciesstringencies ofof ourour definitiondefinition ofof truth;truth; andand thatthat isis toto regardregard suchsuch utterancesutterances asas "Jones'"Jones' reportreport waswas moremore truetrue thanthan Roberts'"Roberts'" asas ellipticalelliptical forfor somethingsomething ofof thisthis sort:sort: "A"A greatergreater numbernumber ofof thethe detailsdetails reportedreported bbyy JonesJones werewere truetrue thanthan thosethose reportedreported bbyy Roberts."Roberts." Similarly,Similarly, anan articlearticle inin Time magazine99 whichwhich borboree thethe titletitle "How"How TrueTrue isis thethe Bible?"Bible?" couldcould insteadinstead havehave beebeenn bettebetterr presentepresentedd underunder thethe titltitlee "How"How MuchMuch ofof thethe BibleBible isis True?"True?"

9.9. Dec.Dec. 30,1974.30, 1974. § 3 PropertiesProperties ofof PropositionPropositionss 1313

3.3. PROPERPROPERTIETIESS OFOF PROPOSITIONSPROPOSITIONS

AmongAmong thethe itemsitems whichwhich existexist inin variousvarious possiblpossiblee worldsworlds mustmust bebe includedincluded propositions;propositions; andand amongamong ththee attributesattributes whichwhich propositionspropositions instanceinstance inin variousvarious possiblepossible worldsworlds areare thethe propertiepropertiess ofof truthtruth anandd falsity.falsity. InIn everyevery possiblpossiblee worldworld eacheach propositiopropositionn hashas oneone oror otherother ofof thesethese propertiepropertiess ofof truthtruth anandd falsity.falsity. ButBut truthtruth andand falsityfalsity areare notnot thethe onlyonly propertiepropertiess whichwhich propositionpropositionss cancan have.have. ByBy virtuevirtue ofof ththee factfact thatthat inin anyany givengiven possiblpossiblee world,world, a propositiopropositionn hashas eithereither thethe propertpropertyy ofof truthtruth oror thethe propertyproperty ofof falsity,falsity, wewe cancan distinguishdistinguish otherother propertiesproperties whichwhich a propositionproposition cancan have,have, viz.,viz., possiblepossible truth,truth, possiblepossible falsity,falsity, ,contingency, nonnoncontingencycontingency,, necessarynecessary truth,truth, andand necessarynecessary falsity.falsity. TheseThese areare propertiepropertiess —­ modal properties wewe shallshall callcall themthem -— whichwhich propositionpropositionss havehave accordingaccording toto thethe wayway inin whichwhich theitheirr truth-valuestruth-values areare distributeddistributed acrossacross thethe setset ofof all possiblpossiblee worlds;worlds; according,according, thatthat is,is, toto whetherwhether theythey araree truetrue (or(or false)false) inin jusjustt some,some, inin all,all, oror inin nonenone ofof thethe totalitytotality ofof possiblpossiblee worlds.Iworlds.010 InIn distinguishindistinguishingg themthem wewe begibeginn toto approachapproach thethe heartheart ofof logiclogic itselfitself..

Possibly true propositions Consider,Consider, forfor a start,start, thosethose propositionpropositionss whichwhich areare truetrue inin atat leastleast oneone possiblpossiblee world.world. TheThe propositionproposition thatthat a particulaparticularr item,item, Lazarus,Lazarus, hadhad a particulaparticularr attribute,attribute, thatthat ofof havinghaving lotslots ofof timetime forfor lovelove inin hihiss twenty-threetwenty-three hundredhundred oddodd years,years, isis a casecase inin pointpoint.. ItIt isis a propositiopropositionn whichwhich isis truetrue inin thatthat possiblepossible 'though"though (so(so farfar asas wewe know)know) non-actualnon-actual worldworld ofof Heinlein'sHeinlein's novel.novel. WeWe shallshall saysay thatthat itit isis possibly true, oror againagain thatthat itit isis a possible truth. AnotherAnother exampleexample ofof a propositiopropositionn whicwhichh isis truetrue inin atat leastleast ononee possiblpossiblee worldworld isis thethe propositiopropositionn thatthat WoodrowWoodrow WilsonWilson waswas presidenpresidentt ofof thethe U.S.U.S. inin 1917.1917. InIn thithiss case,case, ofof course,course, oneone ofof thethe possiblpossiblee worldsworlds inin whichwhich thethe propositiopropositionn isis truetrue isis alsoalso thethe actualactual oneone.. NeverthelesNeverthelesss itit makesmakes sensesense toto saysay thatthat thisthis propositionproposition isis a possiblpossiblee truthtruth eveneven if,if, inin sayingsaying so,so, wewe araree notnot sayingsaying allall thatthat wewe areare entitledentitled toto say,say, viz.,viz., thatthat itit isis alsoalso anan actualactual truth.truth. ForFor actualactual truthstruths formform a subclasssubclass ofof possiblpossiblee truths.truths. A propositiopropositionn isis a possiblpossiblee truth,truth, then,then, ifif itit isis truetrue inin atat leastleast oneone possiblepossible worldworld -— actualactual oror non-actual.non-actual. A propositiopropositionn isis actually true ifif amongamong thethe possiblpossiblee worldsworlds inin whichwhich iitt isis truetrue therethere occursoccurs thethe actualactual worldworld.. InIn sayingsaying ofof a propositiopropositionn thatthat itit isis truetrue inin atat leastleast oneone possiblepossible worldworld itit shouldshould notnot bbee thoughtthought thathatt itit isis alsoalso beinbeingg claimedclaimed thatthat thatthat propositiopropositionn isis falsefalse inin somesome otherother possiblpossiblee world.world. WhenWhen wewe saysay thatthat aa propositiopropositionn isis truetrue inin somesome possiblpossiblee worldsworlds wewe leaveleave itit anan openopen questionquestion asas toto whetherwhether thatthat propositiopropositionn isis truetrue inin allall otherother possiblpossiblee worldsworlds asas well,well, oror isis falsefalse inin somesome ofof thosethose otherother possiblpossiblee worldsworlds..

Possibly falsefalse propositions HowHow aboutabout propositionpropositionss whichwhich areare falsefalse inin atat leastleast oneone possiblpossiblee world?world? SuchSuch propositionspropositions,, wewe shalshalll wantwant toto say,say, areare possibly false,false, oror areare possible falsities.falsities. NowNow somesome (but,(but, asas wewe shallshall seesee later,later, notnot all)all) propositionpropositionss whichwhich areare possiblpossiblyy truetrue areare alsoalso possiblpossiblyy false.false. TheThe propositiopropositionn thatthat LazarusLazarus hadhad lotslots ooff timetime forfor lovelove isis notnot onlyonly possiblpossiblyy true,true, becausbecausee therethere isis a possiblpossiblee worldworld inin whichwhich itit isis true;true; itit isis alsoalso possiblpossiblyy false,false, becausbecausee therethere areare otherother possiblpossiblee worldsworlds -— ourour ownown actualactual oneone amongamong themthem -— inin whicwhichh (so(so farfar asas wewe know)know) itit isis false,false, i.e.,i.e., actuallyactually false.false. Similarily,Similarily, thethe propositiopropositionn thatthat WoodrowWoodrow WilsoWilsonn waswas presideiltpresident ofof thethe UnitedUnited StatesStates inin 19171917 isis notnot onlyonly possiblypossibly (as(as wellwell asas actually)actually) true,true, becausebecause thertheree isis a possiblepossible (as(as itit happenshappens thethe actual)actual) worldworld inin whichwhich itit isis true;true; itit isis alsoalso possiblpossiblyy false,false, becausbecausee thertheree areare otherother possiblpossiblee worldsworlds -— ourour ownown actualactual oneone not amongamong themthem -— inin whichwhich itit isis false.false.

10.10. NotNotee thatthat althoughalthough attributionsattributions ofof noncontingentnoncontingent truthtruth andand noncontingentnoncontingent falsityfalsity will countcount asas attributionattributionss ofof modalmodal status,status, thethe correspondingcorresponding attributionsattributions ofof contingentcontingent truthtruth andand contingentcontingent falsityfalsity willwill not. TheThe reasonreason foforr allowingallowing anan ascriptionascription ofof contingencycontingency toto countcount asas anan ascriptionascription ofof modalmodal statusstatus bubutt notnot allowingallowing ascriptionsascriptions ooff contingentcontingent truthtruth oror contingentcontingent falsehoodfalsehood isis givengiven inin chapterchapter 6,6, sectionsection 55.. 1414 POSSIBLEPOSSIBLE WORLDSWORLDS

Contingent propositions AnyAny propositiopropositionn whichwhich notnot onlyonly isis truetrue inin somesome possiblpossiblee worldsworlds butbut alsoalso isis falsefalse inin somesome possiblpossiblee worldsworlds isis saidsaid toto bbee a contingent proposition. A contingentcontingent propositionproposition,, thatthat is,is, isis botbothh possiblpossiblyy trutruee andand possiblpossiblyy false.false. TheThe propositiopropositionn aboutabout WoodrowWoodrow WilsonWilson isis contingentcontingent andand happenshappens toto bbee truetrue iinn thethe actualactual world,world, whilewhile thethe propositiopropositionn aboutabout LazarusLazarus isis contingentcontingent andand happenshappens toto bbee falsefalse inin ththee actualactual world.world. TheThe formerformer isis a possiblpossiblee truthtruth whichwhich could havehave beebeenn false,false, eveneven thoughthough asas a mattermatter ooff actualactual factfact itit isis not;not; thethe latterlatter isis a possiblpossiblee falsityfalsity whichwhich could havehave beebeenn true,true, eveneven thoughthough asas a mattermatter ofof actualactual factfact itit isis notnot..

Contradictories of propositions SupposeSuppose wewe havehave a propositiopropositionn whichwhich isis contingentcontingent andand truetrue inin thethe actualactual world,world, suchsuch asas ththee propositiopropositionn thatthat thethe U.SU.S.. enteredentered WorldWorld WarWar I inin 1917:1917: thenthen itit isis truetrue inin somesome possiblpossiblee worldsworlds,, includingincluding thethe actualactual one,one, bubutt falsefalse inin allall thosethose possiblpossiblee worldsworlds inin whichwhich itit isis notnot true.true. WhatWhat mighmightt suchsuch a possiblpossiblee worldworld bbee inin whichwhich itit isis falsefalse thatthat thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917?1917? ItIt couldcould be,be, forfor example,example, a possiblepossible worldworld inin whichwhich thethe U.S.U.S. enteredentered WorldWorld WarWar I,I, notnot inin 1917,1917, bubutt inin 1918;1918; oror,, itit mightmight bbee a possiblpossiblee worldworld inin whichwhich WorldWorld WarWar I nevernever tooktook placplacee atat all,all, perhapperhapss becausbecausee universaluniversal peacpeacee hadhad beebeenn establishedestablished inin thatthat worldworld somesome yearsyears earlier,earlier, oror becausbecausee mankindmankind hadhad managedmanaged ttoo destroydestroy itselfitself quitequite accidentlyaccidently inin 1916;1916; oror againagain itit mightmight bbee a possiblpossiblee worldworld inin whichwhich thethe NortNorthh AmericanAmerican continent,continent, andand hencehence thethe U.S.,U.S., simplysimply diddid notnot exist.exist. InIn eacheach andand everyoneevery one ofof thesethese lattelatterr possiblpossiblee worlds,worlds, andand inin countlesscountless othersothers besidesbesides,, itit isis falsefalse thatthat thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917,1917, oror inin otherother words,words, itit isis truetrue thatthat itit isis notnot thethe casecase thatthat thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917.1917. LetLet usus callcall anyany propositiopropositionn whichwhich isis truetrue inin allall thosethose possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich a givegivenn

AllAll PossiblePossible WorldsWorlds A

P isis falsefalse P isis truetrue

v A. v J A contradictorcontradictoryy ooff ContingentContingent contingentcontingent propositiopropositionn PP propositiopropositionn PP

FIGUREFIGURE (1.d)(l.d)

A contingentcontingent proposition,proposition, P,P, isis representedrepresented asas beingbeing truetrue inin somesome possiblepossible worldworldss andand falsefalse inin allall thethe others.others. AnyAny contradictorycontradictory ofof propositiopropositionn P willwill thenthen bbee falsefalse iinn allall thosethose possiblpossiblee worldsworlds inin whichwhich P isis truetrue andand truetrue inin allall thosethose possibl possiblee worldsworlds inin whichwhich P isis false.false. § 3 PropertiePropertiess ooff PropositionPropositionss 1155 propositiopropositionn iiss falsefalse,, anandd whicwhichh iiss falsfalsee iinn alalll thosthosee possiblpossiblee worldsworlds,, iiff anyany,, iinn whicwhichh a givegivenn propositiopropositionn iiss truetrue,, a contradictory ooff thathatt proposition.proposition.!!11 TheThenn ththee propositiopropositionn thathatt iitt iiss nonott ththee cascasee thathatt ththee U.SU.S.. entereenteredd WorlWorldd WaWarr I iinn 19119177 iiss a contradictorcontradictoryy ooff ththee propositiopropositionn thathatt ththee U.SU.S.. dididd enteenterr WorlWorldd WaWarr I aatt thathatt timetime,, anandd vicvicee versaversa.. ThThee propositiopropositionn thathatt iitt iiss nonott ththee cascasee thathatt iitt diddid,, sincsincee iitt iiss a contradictorcontradictoryy ooff a trutruee contingencontingentt proposition proposition,, wilwilll bbee contingencontingentt anandd falsefalse.. ItIt wilwilll bbee falsefalse,, thathatt isis,, iinn alalll thosthose.possiblee possible worldsworlds,, includinincludingg ththee actuaactuall oneone,, iinn whicwhichh ththee U.SU.S.. entereenteredd WorlWorldd WaWarr I iinn 1917.1917. AgainAgain,, suppossupposee wwee havhavee a propositiopropositionn whic whichh iiss contingencontingentt anandd falsefalse,, sucsuchh aass ththee propositiopropositionn thathatt nnoo fightingoccurred occurredin inFrance Franceduring duringWorld World WarWarI. I. Then,Then, althoughalthoughthis this contingentcontingent propositionpropositionwill will bebe falsfalsee iinn somsomee possiblpossiblee worldworldss includinincludingg ththee actuaactuall oneone,, thertheree wilwilll bbee otheotherr possiblpossiblee worldworldss iinn whicwhichh iitt iiss truetrue.. ThesThesee otheotherr possiblpossiblee worldworldss wilwilll bbee preciselpreciselyy thosthosee iinn whicwhichh a contradictorcontradictoryy ooff thathatt propositionproposition,, e.g.e.g.,, thathatt fightinfightingg dididd occuoccurr iinn FrancFrancee aatt thathatt timetime,, wilwilll bbee false.false.

Noncontingent propositionspropositIOns WWee havhavee jusjustt beebeenn talkintalkingg abouaboutt thosthosee propositionpropositionss whicwhichh araree botbothh possiblpossiblyy trutruee and possiblpossiblyy false.false. TheTheyy areare,, wwee saidsaid,, ththee contingencontingentt propositionspropositions.. ArAree thertheree ananyy propositionpropositionss whicwhichh araree not botbothh possiblpossiblyy trutruee anandd possiblpossiblyy falsefalse?? AnAnyy propositionpropositionss whicwhichh araree jusjustt ththee ononee anandd nonott ththee otherother?? AnAnyy propositionspropositions,, forfor instanceinstance,, whicwhichh coulcouldd nonott possiblpossiblyy bbee falsefalse,, i.e.i.e.,, whicwhichh musmustt bbee truetrue?? OOrr againagain,, ananyy propositionpropositionss

AlAlll PossiblPossiblee WorldsWorlds ------'""'l /_------A .... ThThee Non-actuaNon-actuall actualactual worldworldss worlworldd A v 1 • "\ \

P is P isis truetrue I ~r~:true I "------. "„ 1 NecessarilyNecessarily truetrue propositionproposition PP

FIGUREFIGURE (1.e)(J.e)

IfIf P isis a necessarilynecessarily truetrue (noncontingently(noncontingently true)true) proposition,proposition, thenthen P isis truetrue inin allall possiblepossible worldworldss -— thethe non-actualnon-actual onesones asas wellwell asas thethe actualactual one.one.

11.11. ItIt isis commonlycommonly saidsaid thatthat a propositionproposition hashas oneone andand onlyonly oneone contradictory.contradictory. IfIf thisthis claimclaim werewere true,true, wwee ought,ought, accordingly,accordingly, toto speakspeak ofof the contradictorycontradictory ofof a propositionproposition ratherrather than,than, asas wewe havehave here,here, ofof a contradictory.contradictory. However,However, subsequentlysubsequently inin thisthis chapterchapter -— afterafter wewe havehave mademade a distinctiondistinction betweebetweenn propositional-identitypropositional-identity andand propositional-equivalencepropositional-equivalence -— wewe willwill showshow thatthat everyevery propositionproposition hashas anan infiniteinfinite numbernumber ofof non-identicalnon-identical equivalentsequivalents andand hencehence thatthat everyevery propositionproposition hashas anan infiniteinfinite numbernumber ofof non-identicalnon-identical,, butbut equivalent,equivalent, contradictoriescontradictories.. 1616 POSSIBLEPOSSIBLE WORLDSWORLDS

whichwhich couldcould notnot possiblpossiblyy bbee true,true, i.e.,i.e., whichwhich mustmust bbee false?false? SuchSuch propositionspropositions,, ifif therethere areare any,any, woulwouldd notnot bebe contingent.contingent. TheyThey wouldwould bbee whatwhat wewe shouldshould wantwant toto callcall noncontingent propositionspropositions.. Consider,Consider, first,first, whatwhat itit wouldwould bbee likelike forfor therethere toto bebe propositionpropositionss whichwhich must be true. SucSuchh propositionpropositionss wewe shouldshould callcall necessarily true propositionspropositions.. A necessarilynecessarily truetrue propositiopropositionn would,would, ooff course,course, bbee a possiblpossiblyy truetrue propositionproposition.. Indeed,Indeed, sincesince itit wouldwould notnot onlyonly bebe truetrue inin atat leastleast oneone possiblepossible worldworld butbut truetrue inin all possiblpossiblee worlds,worlds, itit wouldwould alsoalso bbee truetrue inin thethe actualactual world.world. SeeSee figurefigure (I.e).(l.e). A necessarilynecessarily truetrue proposition,proposition, inin short,short, wouldwould bebe bothboth possiblypossibly truetrue andand actuallyactually true.true. ButBut itit woulwouldd notnot bbee contingentcontingent andand true.true. ForFor a truetrue contingentcontingent propositionproposition,, remember,remember, isis falsefalse inin somesome possiblepossible world,world, whereaswhereas a necessarilynecessarily truetrue propositiopropositionn -— sincesince itit wouldwould bbee truetrue inin everyevery possiblpossiblee worldworld —­ couldcould notnot bebe falsefalse inin any.any. A necessarilynecessarily truetrue propositionproposition,, then,then, asas wellwell asas beingbeing bothboth actuallyactually anandd possiblypossibly true,true, wouldwould bebe noncontingently true.true. CanCan wewe givegive anan exampleexample ofof suchsuch a propositionproposition?? ExamplesExamples abound,abound, andand wewe shallshall examineexamine manymany iinn thethe coursecourse ofof thisthis bookbook.. ButBut forfor presenpresentt purposepurposess wewe shallshall satisfysatisfy ourselvesourselves withwith a particularlparticularlyy straightforwardstraightforward example.example. ItIt isis a fairlyfairly obviousobvious factfact thatthat notnot onlyonly cancan wewe ascribeascribe truthtruth andand falsityfalsity (and(and otherother attributes)attributes) ttoo individualindividual propositionspropositions,, bubutt alsoalso wewe cancan ascribeascribe variousvarious attributesattributes toto pairs ofof propositionspropositions.. FoForr example,example, wewe cancan assertassert ofof a paipairr ofof propositionpropositionss (1)(1) thatthat neitherneither isis true,true, (2)(2) thatthat onlyonly oneone isis true,true, (3)(3) thatthat atat leastleast oneone isis true,true, oror (4)(4) thatthat botbothh areare true,true, etc.etc. InIn assertingasserting somethingsomething ofof a pair ofof propositionspropositions oneone is,is, ofof course,course, expressingexpressing a propositiopropositionn whichwhich isis itselfitself eithereither truetrue oror false.false. ForFor example,example, ifif oneone werweree toto assertassert ofof thethe twotwo falsefalse contingentcontingent propositionspropositions

(1.1)(1-1) BenjaminBenjamin FranklinFranklin waswas a presidenpresidentt ofof SpainSpain,,

(1.2) CanadaCanada isis southsouth ofof MexicoMexico,,

thatthat oneone oror thethe otherother ofof themthem isis true,true, thenthen thethe propositiopropositionn whichwhich oneone hashas expressedexpressed isis itself,itself, obviouslobviouslyy enough,enough, contingentcontingent andand false.false. OurOur abilityability toto ascribeascribe truthtruth toto neitherneither of,of, oror toto oneone of,of, oror toto botbothh of,of, etc.,etc., a paipairr ofof propositionspropositions,, provideprovidess a meansmeans toto constructconstruct anan exampleexample ofof a necessarilynecessarily truetrue propositionproposition..

Necessarily true propositionspropositions ConsiderConsider thethe casecase inin whichwhich wewe ascribeascribe truthtruth toto oneone oror thethe otherother ofof a paipairr ofof propositionspropositions.. WeWe can,can, ooff course,course, dodo thisthis forfor anyany arbitrarilyarbitrarily selectedselected paipairr ofof propositionpropositionss whatever.whatever, ButBut let'slet's seesee whatwhat happenshappens whenwhen wewe dodo soso inin thethe casecase ofof ascribingascribing truthtruth toto oneone oror thethe otherother ofof a paipairr ofof contradictorycontradictory propositionspropositions.. Take,Take, forfor instance,instance, thethe contradictorycontradictory paipairr

(1.3) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917;1917;

(1.4) ItIt isis notnot thethe casecase thatthat thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917.1917.

AsAs wewe saw,saw, (1.3) isis truetrue inin allall thosethose possiblpossiblee worldsworlds inin whichwhich (1.4) isis false,false, andand (1.4) isis truetrue inin alalll thosethose possiblpossiblee worldsworlds inin whichwhich (1.3) isis false.false. SoSo inin everyevery possiblpossiblee worldworld oneone oror thethe otherother isis true.true. IIff wewe thenthen assertassert ofof suchsuch a paipairr thatthat oneone oror thethe otherother isis true,true, thethe propositiopropositionn wewe expressexpress willwill bbee truetrue iinn allall possiblpossiblee worlds.worlds. ThisThis propositiopropositionn willwill bbee

(1.5) EitherEither thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 oror itit isis notnot thethe casecase thatthat thethe U.SU.S.. enteredentered WorldWorld WarWar I inin 1917.1917. § 33 PropertiesProperties ofof PropositionsPropositions 1177

TheThe samesame holdsholds forfor any-any pairpair ofof contradictoriescontradictories whatsoever.whatsoever. ThinThinkk ooff ananyy propositiopropositIOnn yoyouu likelike:: foforr example,example, thethe contingentcontingent propositionproposition thatthat youyou (the(the reader)reader) araree nonoww (a(att ththee momenmomentt ooff readinreadingg thithiss sentence)sentence) wearingwearing a pairpair ofof blueblue jeans.jeans. ItIt doesn'tdoesn't mattermatter whethewhetherr thithiss propositiopropositionn iiss actuallactuallyy trutruee oorr false.false. BeingBeing contingent,contingent, itit isis truetrue inin atat leastleast somesome possiblepossible worldworldss —- eveevenn iiff nonott ththee actuaactuall ononee —- andand falsefalse inin allall thosethose inin whichwhich itit isis notnot true.true. NowNow thinkthink ofof a contradictorycontradictory ooff thathatt propositionproposition,, e.g.e.g.,, thathatt iitt isis notnot thethe casecase thatthat youyou areare wearingwearing a pairpair ofof blueblue jeansjeans.. WhetheWhetherr thithiss lattelatterr propositiopropositionn iiss actuallactuallyy truetrue oror false,false, itit willwill bebe truetrue inin allall thosethose possiblepossible worldsworlds inin whicwhichh ththee propositiopropositionn thathatt yoyouu araree wearinwearingg blueblue jeansjeans isis false,false, andand falsefalse inin allall thosethose possiblepossible worldsworlds iinn whicwhichh thithiss propositiopropositionn iiss truetrue.. FinallyFinally,, then,then, thinkthink ofof thethe propositionproposition thatthat either youyou areare wearingwearing a paipairr ooff blu bluee jeanjeanss or iitt iiss nonott ththee cascasee thathatt youyou areare wearingwearing a pairpair ofof blueblue jeans.jeans. ThisThis latterlatter proposition,proposition, jusjustt liklikee ththee propositiopropositionn (1.5)(1.5),, iiss trutruee iinn allall possiblepossible worlds,worlds, andand hence,hence, likelike (1.5),(1.5), isis necessarilynecessarily truetrue.. AnyAny propositionproposition whichwhich assertsasserts ofof twotwo simplersimpler onesones thatthat eithereither ononee oorr ththee otheotherr iiss truetrue,, i.e.i.e.,, thathatt aatt leastleast oneone isis true,true, isis calledcalled a disjunctive proposition:proposition: itit disjoins twtwoo simplesimplerr propositionpropositionss eaceachh ononee ooff whichwhich isis a disjunct. TheThe disjunctsdisjuncts inin a disjunctivedisjunctive propositiopropositionn neeneedd nonott bbee contradictoriecontradictoriess ooff ononee another,another, asas theythey areare inin thethe casecase ofof (1.5). ForFor instance,instance, ththee propositiopropositionn thathatt eitheeitherr yoyouu araree wearinwearingg aa pairpair ofof blueblue jeansjeans oror youyou areare dresseddressed forfor a formalformal occasionoccasion iiss a disjunctiodisjunctionn whoswhosee disjunctdisjunctss araree notnot contradictoriescontradictories ofof oneone another.another. ButBut inin thethe casecase whenwhen disjunctsdisjuncts araree contradictoriescontradictories,, iitt followsfollows,, aass wwee havehave seen,seen, thatthat thethe disjunctiondisjunction itselfitself isis necessarilynecessarily truetrue.. . ItIt isis importantimportant notnot toto thinkthink thatthat thethe onlyonly necessarilynecessarily trutruee propositionpropositionss araree thosthosee whicwhichh asserassertt ooff aa pairpair ofof contradictoriescontradictories thatthat oneone oror thethe otherother isis true.true. LaterLater wwee shalshalll citcitee exampleexampless ooff necessarilnecessarilyy trutruee propositionspropositions whichwhich areare notnot ofof thisthis sortsort..

Necessarily falsefalse propositions Consider,Consider, now,now, whatwhat itit wouldwould bebe likelike toto havehave a noncontingentnoncontingent propositiopropositionn whicwhichh must be jalse.Jalse. SuchSuch aa propositionproposition wewe shouldshould callcall a necessarily falsefalse proposition.proposition. AA necessarilynecessarily falsefalse propositionproposition would,would, ofof course,course, bebe a possiblypossibly falsefalse proposition.proposition. Indeed,Indeed, sincesince itit woulwouldd nonott onlonlyy bbee falsfalsee iinn aatt leasleastt somsomee possiblepossible worldsworlds butbut falsefalse inin all possiblepossible worlds,worlds, itit woulwouldd alsalsoo bbee falsfalsee iinn ththee actuaactuall worldworld.. AA necessarilynecessarily falsefalse proposition,proposition, inin short,short, wouldwould bebe bothboth possiblpossiblyy falsfalsee anandd actuallactuallyy falsefalse.. BuButt iitt woulwouldd nonott bebe contingentcontingent andand false.false. ForFor a falsefalse contingentcontingent proposition,proposition, rememberremember,, iiss trutruee iinn somsomee possiblpossiblee worlworldd whereaswhereas a necessarilynecessarily falsefalse propositionproposition -— sincesince itit isis falsefalse inin evereveryy possiblpossiblee worlworldd —- iiss nonott trutruee iinn anyany.. A necessarilynecessarily falsefalse proposition,proposition, then,then, asas wellwell asas beingbeing botbothh actuallyactually anandd possiblpossiblyy falsfalsee woulwouldd bbee noncontingently false. WhatWhat wouldwould bebe anan exampleexample ofof suchsuch a proposition?proposition? AgainAgain,, ouourr abilitabilityy ttoo sasayy somethinsomethingg abouaboutt ththee wayway truthtruth andand falsityfalsity areare distributeddistributed betweenbetween thethe membersmembers ofof a pair ofof propositionspropositions givesgives usus thethe meansmeans ofof constructingconstructing anan exampleexample ofof a necessarilynecessarily falsefalse propositionproposition.. JustJust asas wewe cancan saysay ofof anyany pairpair ofof propositionspropositions thatthat oneone oorr ththee otheotherr ooff thethemm iiss truetrue,, wwee cacann alsalsoo assertassert ofof anyany pairpair thatthat both arearetrue. true. ButButconsider consider whatwhat happenshappens whenwhen wewe saysay ofof aa pairpair ofof propositionspropositions thatthat theythey bothboth areare truetrue inin thethe casecase inin whichwhich thethe twotwo propositionpropositionss happehappenn ttoo bbee contradictoriescontradictories.. Consider,Consider, again,again, thethe contradictorycontradictory pairpair ofof propositionspropositions (1.3) anandd (1.4). SupposSupposee wwee werweree ttoo asserassertt ooff themthem thatthat theythey bothboth areare true.true. TheThe propositionproposition toto thisthis effecteffect woulwouldd bbee

(1.6) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 anandd iitt iISs nonott ththee cascasee thathatt ththee U.SU.S.. enteredentered WorldWorld WarWar I inin 1917.1917.

ItIt isis easyeasy toto seesee thatthat therethere isis nono possiblepossible worldworld inin whichwhich thithiss lattelatterr propositiopropositionn iiss truetrue.. ForFor,, aass wwee sasaww earlierearlier whenwhen discussingdiscussing thethe limitslimits ofof possibilitypossibility andand conceivability,conceivability, a supposesupposedd worlworldd iinn whicwhichh somethingsomething literallyliterally bothboth isis thethe casecase andand isis notnot thethe casecase isis notnot,, iinn ananyy sensesense,, a possiblpossiblee worldworld;; iitt iiss aann 1818 POSSIBLEPOSSIBLE WORLDSWORLDS

AllAll PossiblePossible WorldsWorlds .A-A. , TheThe Non-actualNon-actual ActualActual WorldsWorlds WorldWorld *- A " N..

P iiss P isis falsefalse falsefalse

•—— ^ NecessarilyNecessarily falsefalse propositionproposition P FIGUREFIGURE (1j)(1.f)

IfIf P isis a necessarilynecessarily falsefalse (i.e.(i.e. noncontingentlynoncontingently false)false) propositionpropOSltlon,, thenthen P isis falsefalse inin alalll possiblpossiblee worldsworlds -— thethe hon-actualnon-actual onesones asas wellwell asas thethe actualactual one.one. SuchSuch propositionpropositionss areare alsoalso saidsaid toto bbee self-contradictory, self-inconsistent, oror logically impossible.

impossibleimpossible one.one. TheThe casecase wewe consideredconsidered earlierearlier waswas thatthat ofof thethe suppositionsupposition thatthat timetime traveltravel bothboth will occuroccur sometimesometime inin thethe futurefuture andand will not occuroccur atat anyany timetime inin thethe future.future. WeWe cancan seesee thatthat it,it, likelike thethe casecase wewe areare presently considering,considering, involvesinvolves ascribingascribing truthtruth toto twotwo propositionpropositionss oneone ofof whichwhich isis a cbntradictorycontradictory ofof thethe other.other. AndAnd perhapperhapss wewe cancan alsoalso nownow seesee whywhy wewe earlierearlier saidsaid thatthat thatthat propositiopropositionn waswas self-contradictory, andand whywhy wewe wouldwould nownow wantwant toto saysay thatthat thethe presenpresentt casecase isis alsoalso self­self- contradictory.contradictory. ForFor ifif anyany propositiopropositionn ascribesascribes truthtruth toto botbothh membersmembers ofof a paipairr ofof contradictories,contradictories, thenthen thatthat propositiopropositionn isis oneone whichwhich hashas a contradictioncontradiction withinwithin itself.itself. AnyAny propositionproposition whichwhich assertsasserts ofof twotwo simplersimpler onesones thatthat bothboth ofof themthem areare true,true, isis calledcalled a conjunctive propositionproposition:: itit conjoins twotwo simplersimpler propositionpropositionss eacheach ofof whichwhich isis calledcalled a conjunct. TheThe conjunctsconjuncts inin a conjunctiveconjunctive propositiopropositionn needneed notnot bbee contradictoriescontradictories ofof oneone another,another, asas theythey areare inin thethe casecase ofof (7.6).(1.6). ForFor instance,instance, thethe propositiopropositionn thatthat youyou areare wearingwearing a paipairr ofof blubluee jeanjeanss andand youryour friendfriend isis wearingwearing a paipairr ofof tweedstweeds isis a conjunctionconjunction whosewhose conjunctsconjuncts areare not contradictories.contradictories. ButBut inin thethe cascasee wherwheree conjunctsconjuncts areare contradictories,contradictories, itit follows,follows, asas wwee havhavee seen,seen, thathatt ththee conjunctionconjunction isis necessarilynecessarily false.false.

More about contradictorycontradictorypropositions propositions A simplesimple wayway ofof describingdescribing thethe relationrelation whichwhich holdsholds betweenbetween twotwo propositIOnspropositions whichwhich areare contradictoriescontradictories ofof oneone anotheranother is,is toto saysay thatthat inin eacheach possiblpossiblee worldworld oneone oror otherother ofof thosethose propositionpropositionss iiss truetrue andand thethe otherother isis false.false. ThisThis descriptiondescription hashas twotwo immediateimmediate consequencesconsequences whichwhich wewe wouldwould dodo wellwell toto note.note. First,First, notenote thatthat anyany contradictorycontradictory ofof a contingentcontingent propositiopropositionn isis itselfitself a contingentcontingent propositionproposition.. FoForr ifif a propositiopropositionn isis contingent,contingent, i.e.,i.e., truetrue inin somesome possiblpossiblee worldsworlds andand falsefalse inin allall thethe others,others, thenthen sincesince anyany propositiopropositionn whichwhich isis itsits contradictorycontradictory mustmust bbee falsefalse inin allall possiblpossiblee worldsworlds inin whichwhich thethe formerformer iiss § 3 PropertiesProperties ofof PropositionsPropositions 1919 true,true, andand truetrue inin allall possiblpossiblee worldsworlds inin whichwhich thethe formerformer isis false,false, itit followsfollows immediatelyimmediately thatthat thisthis secondsecond propositiopropositionn mustmust bbee falsefalse inin somesome possiblpossiblee worldsworlds andand truetrue inin somesome others.others. ButBut thisthis isis jusjustt ttoo saysay thatthat thethe latterlatter propositiopropositionn isis contingent.contingent. Second,Second, notenote thatthat anyany contradictorycontradictory ofof a noncontingentnoncontingent propositiopropositionn isis itselfitself a noncontingentnoncontingent propositionproposition.. ForFor ifif a propositiopropositionn isis noncontingentlynoncontingently true,true, i.e.,i.e., truetrue inin allall possiblpossiblee worldsworlds andand falsefalse iinn none,none, thenthen anyany propositiopropositionn whichwhich isis itsits contradictorycontradictory mustmust bbee falsefalse inin allall possiblpossiblee worldsworlds andand truetrue iinn none,none, whichwhich isis jusjustt toto saysay thatthat itit itselfitself mustmust bbee nonnoncontingentcontingent,, andand moremore especially,especially, noncontingentlynoncontingently false.false. Similarly,Similarly, ifif a propositiopropositionn isis noncontingentlynoncontingently false,false, i.e.,i.e., falsefalse inin allall possiblpossiblee worldsworlds andand truetrue iinn none,none, thenthen anyany propositiopropositionn whichwhich isis itsits contradictorycontradictory mustmust bbee truetrue inin allall possiblpossiblee worldsworlds andand falsefalse iinn none,none, whichwhich isis jusjustt toto saysay thatthat itit itselfitself mustmust bbee nonnoncontingentcontingent,, andand moremore especially,especially, noncontingentlynoncontingently true.true. InIn short,short, ifif oneone membermember ofof a contradictorycontradictory paipairr ofof propositionpropositionss isis necessarilynecessarily truetrue oror necessarilynecessarily false,false, thenthen thethe otherother membermember ofof thatthat paipairr mustmust bbee necessarilynecessarily falsefalse oror necessarilynecessarily truetrue respectively.respectively. TToo citecite anan exampleexample ofof suchsuch a paipairr wewe needneed onlyonly returnreturn toto propositionpropositionss (1.5) andand (1.6):(1-6): (1.5)(1-5) isis truetrue iinn allall possiblpossiblee worldsworlds andand (1.6) isis falsefalse inin allall possiblpossiblee worlds;worlds; theythey are,are, inin short,short, contradictoriescontradictories ofof ononee another.another. BetweenBetween them,them, then,then, a propositiopropositionn andand anyany contradictorycontradictory ofof thatthat propositionproposition dividedivide thethe setset ofof alalll possiblepossible worldsworlds intointo twotwo subsetssubsets whichwhich areare mutually exclusive andand jointly exhaustive. OneOne ofof thesthesee subsetssubsets willwill comprisecomprise allall ofof thethe possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich thethe formerformer propositiopropositionn isis true,true, anandd thethe otherother subsetsubset willwill comprisecomprise allall ofof thethe possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich thethe latterlatter propositiopropositionn isis truetrue.. EachEach possiblpossiblee worldworld willwill belonbelongg toto oneone oror thethe other,other, bubutt notnot toto bothboth,, ofof thesethese subsets.subsets.

Some main kinds of noncontingent propositions SoSo far,far, thethe onlyonly exampleexample citedcited ofof a necessarilynecessarily truetrue propositioproposItIonn waswas thethe disjunctiondisjunction ofof a paipairr ooff contradictories,contradictories, viz.,viz., (1.5),.(1.5); andand thethe onlyonly exampleexample ofof a necessarilynecessarily falsefalse propositiopropositionn waswas thethe conjunctioconjunctionn ofof a paipairr ofof contradictories,contradictories, viz.,viz., (1.6). YetYet manymany noncontingentnoncontingent propositionpropositionss areare ofof kindskinds ververyy differentdifferent fromfrom these.these. InIn whatwhat follows,follows, wewe citecite somesome otherother kindskinds ofof examplesexamples (in(in nono specialspecial order),order), alalll ofof necessarilnecessarilyy truetrue propositionspropositions.. ItIt shouldshould bbee evident,evident, fromfrom whatwhat wewe saidsaid inin thethe previoupreviouss subsection,subsection, thatthat thethe contradictoriescontradictories ofof eacheach ofof thethe necessarilynecessarily truetrue propositionpropositionss citedcited willwill bbee necessarilynecessarily false.false.

1.7. OneOne mainmain kindkind ofof necessarilynecessarily truetrue propositiopropositionn isis exemplifiedexemplified bbyy

(1.7) IfIf somethingsomething isis redred thenthen itit isis coloredcolored..

NotNotee thatthat thisthis propositiopropositionn isis truetrue eveneven inin thosethose possiblpossiblee worldsworlds inin whichwhich therethere areare nono redred things.things. TToo assertassert (1.7) isis simplysimply toto assertassert thatthat if anythinganything isis redred thenthen itit isis colored;colored; andand thisthis propositiopropositionn isis trutruee botbothh inin worldsworlds inin whichwhich therethere areare redred thingsthings andand inin worldsworlds inin whichwhich therethere areare not. 1212 PhilosopherPhilosopherss havehave spentspent timetime analyzinganalyzing thethe reasons forforthe the necessarynecessary truthtruthof of propositionspropositions likelike (1.7).(1.7). Properties,Properties,in in general,general, tendtend toto comecome inin ranges: rangingrangingfrom from thethe moremoreor orless lessspecific specific toto thethe moremoregeneral. general. ThusThus thethe propertpropertyy ofof beinbeingg locatedlocated inin RayRay Bradley'sBradley's drawerdrawer inin hishis deskdesk atat SimonSimon FraserFraser UniversityUniversity isis a highlhighlyy specificspecific propertyproperty.. Or,Or, asas wewe shallshall prefepreferr toto say,say, itit isis a highlyhighly determinate propertyproperty.. TheThe propertpropertyy ooff beinbeingg locatedlocated inin BritishBritish ColumbiaColumbia isis lessless determinate;determinate; thatthat ofof beinbeingg locatedlocated inin CanadaCanada eveneven lessless soso;; thatthat ofof beinbeingg locatedlocated inin thethe northernnorthern hemispherehemisphere eveneven lessless soso again;again; andand soso on.on. AsAs wewe proceeproceedd alonalongg thethe scalescale ofof increasingincreasing generality,generality, i.e.,i.e., decreasingdecreasing determinateness,determinateness, wewe comecome finallyfinally toto thethe leasleastt determinatedeterminate propertpropertyy inin thethe range.range. ThisThis isis thethe propertpropertyy which,which, intuitively,intuitively, fallsfalls jusjustt shortshort ofof thethe mosmostt generalgeneral ofof allall propertiepropertiess -— thatthat ofof beinbeingg a thingthing -— viz.,viz., inin thethe presenpresentt instanceinstance thethe propertpropertyy ofof beinbeingg

12.12. ItIt hashas becombecomee standardstandard practicepractice,, inin thethe paspastt hundredhundred yearsyears oror so,so, toto construeconstrue sentencessentences ofof thethe forformm "All"All . . . are...... " asas ifif theythey saidsaid thethe samesame asas sentencessentences ofof thethe formform "For"For anythinganything whatever,whatever, ifif itit isis ...... thethenn itit is. . . ." OnOn thisthis account,account, thethe sentencesentence "All"All redred thingsthings areare colored"colored" expressesexpresses thethe necessarynecessary truthtruth (1.7).(7.7). 2020 POSSIBLEPOSSIBLE WORLDSWORLDS locatedlocated somewheresomewhere oror other.other. SuchSuch propertiepropertiess wewe shallshall callcall determinable propertiesproperties.. DeterminablDeterminablee propertiepropertiess areare thosethose undeunderr whichwhich allall thethe moremore oror lessless determinatedeterminate propertiesproperties ofof thethe rangerange areare subsumed.subsumed. TheyThey areare thosethose whichwhich wewe wouldwould citecite ifif wewe wishedwished toto givegive thethe leastleast determinatedeterminate possiblpossiblee answeranswer toto suchsuch generalgeneral questionsquestions aboutabout a thingthing as:as: "Where"Where isis it?"it?" (location);(location); "How"How manymany areare there?"there?" (number);(number); "How"How heavyheavy isis it?"it?" (weight);(weight); "What"What colorcolor isis it?"it?" (color);(color); andand soso on.on. Plainly,Plainly, onon thithiss analysis,analysis, thethe propertpropertyy ofof beinbeingg coloredcolored -— ofof havinghaving somesome colorcolor oror otherother -— isis a determinabledeterminabte propertpropertyy undeunderr whichwhich thethe relativelyrelatively determinatedeterminate propertpropertyy ofof beinbeingg redred isis subsumed.subsumed. ToTo bbee redred is,is, therefore,therefore, ipsipsoo factofacto toto bbee colored;colored; andand nothingnothing couldcould possiblpossiblyy bbee redred withoutwithout beinbeingg coloredcolored..

2.2. InIn thethe lightlight ofof thisthis understandinunderstandingg ofof determinabledeterminable propertiesproperties,, wewe areare ableable toto givegive a characterizationcharacterization ofof a secondsecond mainmain kindkind ofof necessarynecessary truth;truth; thatthat ofof whichwhich

(7.8)(1.8) AnyAny eventevent whichwhich occurs,occurs, occursoccurs atat somesome timetime oror otherother isis anan example.example. ThisThis kindkind ofof propositiopropositionn isis sometimessometimes calledcalled a category propositionpropositIOn (or(or eveneven a categoriap3categorial13 proposition):proposition): itit isis thatthat kindkind ofof propositiopropositionn inin whichwhich a determinabledeterminable propertpropertyy isis trultrulyy ascribedascribed toto anan itemitem ofof a certaincertain sort,sort, oror (as(as wewe shallshall say)say) ofof a certaincertain category. JustJust asas determinatedeterminate propertiepropertiess areare subsumablesubsumable underunder highlyhighly generalgeneral determinabledeterminable propertiesproperties,, soso areare particulaparticularr itemsitems anandd classesclasses ofof itemsitems subsumablesubsumable underunder highlyhighly generalgeneral categoriescategories ofof items,items, e.g.,e.g., thethe categoriescategories ofof materialmaterial objects,objects, ofof events,events, ofof persons,persons, ofof mentalmental processesprocesses,, ofof sounds,sounds, andand soso on.on. NoNoww whatwhat distinguishesdistinguishes itemsitems belonginbelongingg toto oneone categorycategory fromfrom thosethose belonginbelongingg toto anotheranother isis jusjustt thethe setset ofof determinabledeterminaWe (as(as distincdistinctt fromfrom determinate)determinate) propertiepropertiess whichwhich areare essentialessential toto suchsuch items. 1414 MaterialMaterial objectsobjects andand shadows,shadows, foforr instance,instance, shareshare certaincertain determinabledeterminate propertiesproperties,, e.g.,e.g., havinghaving somesome number,number, somesome shape,shape, andand somesome location.location. AndAnd thesethese determinabledeterminable propertiepropertiess areare essential toto themthem inin thethe sensesense thatthat nothingnothing coulcouldd possiblpossiblyy bbee a materialmaterial objectobject oror a shadowshadow ifif itit lackedlacked anyany ofof thesethese determinabledeterminable propertiesproperties.. ByBy wayway ooff ,contrast, thethe determinatedeterminate propertiesproperties,, whichwhich particulaparticularr itemsitems inin a givengiven categorycategory happenhappen toto have,have, areare not allall essential.essential. TheThe determinatedeterminate propertpropertyy ofof beinbeingg inin RayRay Bradley'sBradley's drawer,drawer, forfor instance,instance, isis nonott essentialessential toto hishis copycopy ofof Anna Karenina althoughalthough thethe determinabledeterminable propertpropertyy ofof beinbeingg locatedlocated somewheresomewhere oror otherother is.is. WhatWhat makesmakes materialmaterial objectsobjects andand shadowsshadows comprisecomprise differentdifferent categoriescategories isis thethe factfact that,that, thoughthough sharingsharing certaincertain determinabledeterminable propertiesproperties,, theythey dodo notnot shareshare others.others. TheyThey dodo notnot share,share, forfor instance,instance, thethe determinabledeterminable propertpropertyy ofof havinghaving somesome massmass oror other.other. AndAnd neitherneither ofof themthem hashas thethe determinabledeterminable propertpropertyy whichwhich seemsseems distinctivedistinctive ofof propositionspropositions,, viz.,viz., beinbeingg truetrue oror false.false. TheThe questioquestionn asas toto whatwhat thethe essentialessential determinabledeterminable propertiepropertiess ofof itemsitems inin a givengiven categorycategory turnturn outout toto bebe,, isis ononee whichwhich wewe cannotcannot pursupursuee here.here. SufficeSuffice itit toto saysay thatthat anyany truetrue propositiopropositionn which,which, likelike (7.8),(1.8), ascribesascribes toto itemsitems inin a certaincertain categorycategory a determinabledeterminable propertpropertyy whichwhich isis essentialessential toto thosethose items,items, willwill bbee necessarilynecessarily true.true. AndAnd so,so, forfor thatthat matter,matter, willwill bbee anyany truetrue propositionpropositionss which,which, likelike

(7.9)(1.9) PropositionsPropositions havehave nono colorcolor denydeny ofof itemsitems inin a certaincertain categorycategory thatthat theythey havehave a certaincertain determinabledeterminable propertyproperty..

13.13. NotNotee thethe spellingspelling ofof thisthis term.term. DoDo notnot confuseconfuse itit withwith "categorical"."categorical". TheThe twotwo areare notnot synonymssynonyms..

14.14. A relatedrelated kindkind ofof necessarynecessary truth,truth, thatthat whichwhich trulytruly ascribesascribes essentialessential propertiepropertiess toto so-calledso-called natural kindskinds -— as,as, forfor example,example, thethe propositionproposition thatthat goldgold isis aa metalmetal -— hashas recentlyrecently beenbeen thethe subjectsubject ofof muchmuch philosophicaphilosdphicall discussion.discussion. See,See, inin particularparticular,, SaulSaul Kripke,Kripke, "Naming"Naming andand NecessityNecessity"" inin Semantics of Natural Language,Language, ed.ed. HarmanHarman andand Davidson,Davidson, Dodrecht,Dodrecht, D.D. Reidel,Reidel, 1972.1972. Kripke,Kripke, somewhatsomewhat controversially,controversially, maintainsmaintains thatthat havinhavingg atomicatomic numbenumberr 7979 isis anan essentialessential propertpropertyy ofof gold.gold. § 3 PropertiesProperties ofof PropositionsPropositions 2121

3. StillStill anotheranother mainmain kindkind ofof necessarynecessary truthtruth isis exemplifiedexemplified bbyy thethe relational proposition

(7.10)(1.10) IfIf MollyMolly isis tallertaller thanthan JudiJudi thenthen itit isis notnot thethe casecase thatthat JudiJudi isis tallertaller thathann Molly.Molly.

TheThe necessarynecessary truthtruth ofof thisthis propositionproposition,, itit shouldshould bbee noted,noted, isis notnot inin anyany wayway a functionfunction ofof whicwhichh particulaparticularr itemsitems itit assertsasserts asas standingstanding inin thethe relationrelation ofof beinbeingg tallertaller than.than. Rather,Rather, itit isis a functionfunction ofof ththee essentialessential naturenature ofof thethe relationrelation ofof beinbeingg tallertaller than;than; ofof thethe fact,fact, thatthat is,is, thatthat nono mattermatter whatwhat possiblepossible worldsworlds oror whatwhat possiblposs,iblee itemsitems areare involved,involved, ifif thethe relationrelation ofof beinbeingg tallertaller thanthan holdsholds betweebetweenn twotwo items,items, x andand y,y, inin thatthat order,order, thenthen itit doesdoes notnot alsoalso holdhold betweebetweenn x andand y inin thethe reversereverse order.order. ThisThis facfactt aboutabout thethe essentialessential naturenature ofof thethe relationrelation ofof beinbeingg tallertaller thanthan isis sometimessometimes referredreferred toto bbyy sayingsaying thathatt thethe relationrelation ofof beinbeingg tallertaller thanthan isis anan "asymmetrical""asymmetrical" relation.relation. ItIt isis a factfact aboutabout thethe relationrelation ofof beinbeingg tallertaller thanthan whichwhich servesserves toto explainexplain whywhy notnot onlyonly (1.10)(1-10) bubutt countlesscountless otherother propositionspropositions,, viz.,viz., any propositionpropositionss ofof thethe forformm

(1.11)(1-11) IfIf x isis tallertaller thanthan y thenthen itit isis notnot thethe casecase thatthat y isis tallertaller thanthan x

areare necessarilynecessarily true.true. MoreMore generally,generally, thethe propertpropertyy ofof beinbeingg asymmetricalasymmetrical isis essentialessential toto countlesscountless otheotherr relationsrelations asas well,well, e.g.,e.g., beinbeingg thethe mothermother of,of, beingbeing olderolder than,than, beingbeing thethe successorsuccessor of,of, andand soso on.on. AndAnd foforr eacheach ofof thesethese relationsrelations therethere willwill bbee countlesscountless necessarilynecessarily truetrue propositionpropositionss analogousanalogous toto (1.10). IInn chapterchapter 6,6, sectionsection 6,6, wewe willwill investigateinvestigate otherother ofof thethe essentialessential propertiepropertiess whichwhich relationsrelations maymay have:have: (a)(a) symmetrysymmetry oror nonnonsymmetrsymmetryy (as(as alternativesalternatives toto asymmetry);asymmetry); (b)(b) transitivity,transitivity, intrarisitivity,intrarisitivity, oorr , nontransitivity;nontransitivity; andand (c)(c) reflexivity,reflexivity, irreflexivity,irreflexivity, oror nonreflexivity.nonreflexivity. WeWe willwill seesee thatthat everyevery relationrelation hahass atat leastleast oneone essentialessential propertpropertyy drawndrawn fromfrom eacheach ofof (a),(a), (b),(b), andand (c),(c), i.e.,i.e., hashas atat leastleast threethree essentialessential propertiepropertiess in toto. ThatThatthis thisis isso soprovides providesa arich richsource sourceof ofnecessary necessary truths.truths.For Forall allthose thosepropositions propositions whosewhose truthtruth cancan bbee ascertainedascertained bbyy anan appealappeal toto a factfact aboutabout anan essentialessential propertpropertyy ofof a relationrelation wilwilll themselvesthemselves bbee necessarynecessary truthstruths..

4. InIn sayingsaying thatthat propositionpropositionss ofof thethe severalseveral mainmain kindskinds discusseddiscussed soso farfar areare logicallylogically necessarynecessary truthstruths wewe are,are, itit shouldshould bbee noted,noted, usingusing thethe termterm "logically"logically necessary"necessary" inin a fairlyfairly broadbroad sense.sense. WeWe araree notnot sayingsaying thatthat suchsuch propositionpropositionss areare currentlycurrently recognizedrecognized asas truthstruths withinwithin anyany ofof thethe systemssystems ooff formalformal logiclogic whichwhich soso farfar havehave beebeenn developed.developed. Rather,Rather, wewe areare sayingsaying thatthat theythey areare truetrue inin alalll logicallylogically possiblpossiblee worlds.worlds. NowNow,, inin thisthis broabroadd sensesense ofof thethe term,term, thethe truths ojof mathematics mustmust alsalsoo countcount asas anotheranother mainmain kindkind ofof logicallylogically necessarynecessary truth.truth. BertrandBertrand RussellRussell andand AA.. NN.. Whitehead,Whitehead, itit iiss worthworth mentioning,mentioning, triedtried soonsoon afterafter thethe turnturn ofof thisthis centurycentury toto showshow thatthat thethe truthstruths ofof mathematicsmathematics werewere logicallylogically necessarynecessary inin thethe ratherrather stricterstricter sensesense ofof beinbeingg derivablederivable inin accordanceaccordance withwith thethe rulesrules ooff andand fromfrom thethe axiomsaxioms ofof then-knownthen-known systemssystems ofof formalformal logic.logic. ButBut whethewhetherr oror nonott theythey werweree successfulsuccessful -— anandd thithiss iiss stillstill a topitopicc forfor debatedebate -— thertheree cancan bbee littllittlee doubtdoubt thathatt mathematicamathematicall truthtruthss areare logicalllogicallyy necessarynecessary inin thethe broadebroaderr sense.sense. Indeed,Indeed, itit seemsseems thatthat earlyearly GreekGreek philosopherphilosopherss recognizerecognizedd themthem aass suchsuch welwelll beforbeforee AristotlAristotlee laidlaid thethe foundationfoundation ofof formalformal logiclogic inin thethe thirdthird centurycentury B.C.B.C. TheThe trutruee propositionpropositionss ofof mathematicsmathematics seemedseemed toto themthem toto bbee necessarnecessaryy inin a sensesense inin whicwhichh thosethose ofof historyhistory,, geography,geography, andand thethe likelike areare not.not. Admittedly,Admittedly, theythey doubtlessdoubtless wouldwould notnot thenthen havehave offeredoffered a "true"true inin alalll (logically)(logically) possiblpossiblee worlds"worlds" analysisanalysis ofof thethe sensesense inin whichwhich mathematicalmathematical truthstruths seemedseemed toto themthem toto bbee necessary.necessary. ButBut suchsuch anan analysisanalysis seemsseems toto fit wellwell withwith theirtheir judgmentjudgmentss inin thethe mattermatter.. ItIt maymay seemseem toto somesome thatthat therethere areare twotwo importantimportant classesclasses ofof exceptionsexceptions toto thethe generalgeneral accountaccount wwee areare givinggiving ofof mathematicalmathematical truths.truths. TheThe firstfirst hashas toto dodo withwith propositionpropositionss ofof whatwhat wewe oftenoften callcall applied mathematics.mathematics. IfIf bbyy "propositions"propositions ofof appliedapplied mathematics"mathematics" isis meantmeant certaincertain truetrue propositionpropositionss ooff physicsphysics,, engineering,engineering, andand thethe likelike whichwhich areare inferredinferred bbyy mathematicalmathematical reasoningreasoning fromfrom otheotherr propositionpropositionss ofof physicsphysics,, engineering,engineering, etc.,etc., thenthen ofof coursecourse suchsuch propositionpropositionss willwill notnot countcount asas necessarynecessary 2222 POSSIBLEPOSSIBLE WORLDSWORLDS truthstruths bubutt onlyonly asas contingentcontingent ones.ones. ButBut thenthen itit isis misleadingmisleading eveneven toto saysay thatthat theythey are,are, inin anyany sensesense,, propositionpropositionss ofof mathematics.mathematics. If,If, onon thethe otherother hand,hand, wewe meanmean bbyy "propositions"propositions ofof appliedapplied mathematics"mathematics" certaincertain propositionspropositions which,which, liklikee

(1.12) IfIf therethere areare 2020 applesapples inin thethe baskebaskett thenthen therethere areare atat leastleast 1919 applesapples inin ththee baskebaskett areare instances (i.e.,(i.e., applications)applications) ofof propositionpropositionss ofof purpuree mathematics,mathematics, thenthen ourour accountaccount isis sustained.sustained. ForFor therethere areare nono possiblpossiblee worldsworlds -— eveneven worldsworlds inin whichwhich apples,apples, baskets,baskets, andand whatnotwhatnot failfail toto existexist ­— inin whichwhich a propositiopropositionn likelike (1.12)(1-12) isis false.false. ThisThis sortsort ofof mathematicalmathematical propositiopropositionn shouldshould bbee distinguisheddistinguished fromfrom propositionpropositionss aboutabout thethe physicaphysicall resultsresults ofof puttinputtingg thingsthings together,together, e.g.,e.g.,

(1.13) OneOne literliter ofof alcoholalcohol addedadded toto oneone literliter ofof waterwater makesmakes 2 litersliters ofof liquiliquidd whichwhich notnot onlyonly isis notnot necessarilynecessarily truetrue bubutt isis contingentcontingent andand false.false. (See(See chapterchapter 3,3, pp.. 170).170). A secondsecond supposedsupposed exceptionexception hashas toto dodo withwith thethe propositionspropositions ofof geometry.geometry. Thus,Thus, itit maymay bebe saidsaid thathatt althoughalthough therethere waswas a timetime whenwhen propositionpropositionss likelike Euclid'Euclid'ss

(1.14) TheThe sumsum ofof thethe interiorinterior anglesangles ofof a triangletriangle isis equalequal toto twotwo rightright angleangless werewere universallyuniversally thoughtthought toto bbee necessarilynecessarily true,true, wewe nownow knowknow muchmuch betterbetter:: thethe development,development, inin ththee nineteenthnineteenth century,century, ofof non-Euclideannon-Euclidean (e.g.,(e.g., RiemannianRiemannian andand Lobachevskian)Lobachevskian) geometriesgeometries inin whichwhich ththee sumsum ofof thethe interiorinterior anglesangles ofof trianglestriangles areare assertedasserted toto bebe moremore thanthan oror lessless thanthan (respectively)(respectively) twotwo righrightt angles,angles, showsshows thatthat (1.14) isis notnot necessarilynecessarily true.true. InIn effect,effect, thethe objectionobjection isis thatthat withwith thethe adventadvent ooff non-Euclideannon-Euclidean geometriesgeometries inin thethe nineteenthnineteenth century,century, wewe havehave comecome toto seesee thatthat thethe answeranswer toto ththee questionquestion asas toto whatwhat isis thethe sumsum ofof thethe interiorinterior anglesangles ofof a triangle,triangle, isis a contingentcontingent oneone andand willwill varvaryy fromfrom possiblpossiblee worldworld toto possiblpossiblee worldworld accordingaccording toto thethe naturalnatural lawslaws inin thosethose worldsworlds.. ButBut thisthis objectionobjection failsfails toto makemake anan importantimportant distinction.distinction. ItIt isis correctcorrect soso longlong asas bbyy "triangles""triangles" iitt refersrefers toto physicaphysicall objects,objects, e.g.,e.g., trianglestriangles whichwhich surveyorssurveyors mightmight layoutlay out onon a field,field, oror paperpaper trianglestriangles whichwhich oneone mightmight cutcut outout withwith scissors.scissors. ButBut therethere areare otherother triangles,triangles, thosethose ofof purpuree geometry,geometry, whoswhosee propertiepropertiess areare not subjectsubject toto thethe physicaphysicall peculiaritiepeculiaritiess ofof anyany world.world. TheseThese idealized,idealized, abstract entitiesentities havehave invariantinvariant propertiesproperties.. WhatWhat thethe adventadvent ofof non-Euclideannon-Euclidean geometriesgeometries hashas shownshown usus aboutabout thesethese trianglestriangles isis thatthat wewe mustmust distinguishdistinguish variousvarious kinds:kinds: a EuclideanEuclidean triangletriangle willwill havehave interiorinterior angleangless whosewhose sumsum isis equalequal toto twotwo rightright angles;angles; a RiemannianRiemannian triangle,triangle, anglesangles whosewhose sumsum isis greatergreater thanthan twotwo rightright angles;angles; andand a LobachevskianLobachevskian triangle,triangle, anglesangles whosewhose sumsum isis lessless thanthan twotwo rightright angles.angles. SpecifySpecify whichwhich kindkind ofof abstractabstract triangletriangle youyou areare referingrefering to,to, andand itit isis a necessarynecessary truthtruth (or(or falsehood)falsehood) thatthat itsits anglesangles sumsum toto twotwo rightright angles,angles, lessless thanthan two,two, oror greatergreater thanthan twotwo..

5. AmongAmong thethe mostmost importantimportant kindskinds ofof necessarilynecessarily truetrue propositionspropositions areare thosethose truetrue propositionspropositions whichwhich ascribeascribe modal propertiepropertiess -— necessarynecessary truth,truth, necessarynecessary falsity,falsity, contingency,contingency, etc.etc. -— toto otheotherr propositionspropositions.. Consider,Consider, forfor example,example, thethe propositiopropositionn

(1.15) ItIt isis necessarilynecessarily truetrue thatthat twotwo plupluss twotwo equalsequals fourfour..

(1.15) assertsasserts ofof thethe simplersimpler propositiopropositionn -— viz.,viz., thatthat twotwo plupluss twotwo equalsequals fourfour -— thatthat itit isis necessarilynecessarily true.true. NoNoww thethe propositiopropositionn thatthat twotwo plupluss twotwo equalsequals four,four, sincesince itit isis a truetrue propositiopropositionn ofof mathematics,mathematics, isis necessarilynecessarily true.true. ThusThus inin ascribingascribing toto thisthis propositiopropositionn a propertpropertyy whichwhich itit doesdoes have,have, thethe propositiopropositionn (1.15) isis true. ButBut isis (1.15) necessarilynecessarily true,true, i.e.,i.e., truetrue inin allall possiblpossiblee worlds?worlds? TheThe answeranswer wewe shalshalll § 3 PropertiePropertiess ooff PropositionPropositionss 2233

wanwantt ttoo givgivee —- anandd foforr whicwhichh wwee shalshalll arguarguee iinn chaptechapterr 44,, pppp.. 185-88185-88,, anandd agaiagainn iinn chaptechapterr 66,, pp.. 333355 —- iiss thathatt trutruee propositionpropositionss whicwhichh ascribascribee necessarnecessaryy truttruthh ttoo otheotherr propositionpropositionss araree themselvethemselvess necessarilnecessarilyy truetrue,, i.e.i.e.,, logicalllogicallyy necessarynecessary.. AAss HintikkHintikkaa hahass puputt itit::

IItt seemseemss ttoo mmee obviouobviouss thathatt whatevewhateverr iiss logicalllogicallyy necessarnecessaryy herheree anandd nonoww musmustt alsalsoo bbee logicalllogicallyy necessarnecessaryy iinn alalll logicalllogicallyy possiblpossiblee statestatess ooff affairaffairss thathatt coulcouldd havhavee beebeenn realizerealizedd insteainsteadd ooff ththee actuaactuall one.lone.155

SimilaSimilarr considerationconsiderationss lealeadd uuss ttoo sasayy furthefurtherr thathatt ananyy trutruee propositiopropositionn whicwhichh ascribeascribess necessarnecessaryy falsitfalsityy oorr contingenccontingencyy ttoo anotheanotherr propositiopropositionn iiss itselitselff trutruee iinn alalll possiblpossiblee worldsworlds,, i.e.i.e.,, iiss necessarilnecessarilyy truetrue.. AndAnd ththee samsamee holdholdss foforr trutruee propositionpropositionss whichwhich,, liklikee

(1.16) ThThee propositiopropositionn thathatt thertheree araree 2200 appleappless iinn ththee baskebaskett iiss consistent witwithh ththee propositiopropositionn thathatt ththee baskebaskett iiss greegreenn asserassertt ooff twtwoo simplesimplerr propositionpropositionss thathatt thetheyy araree logicalllogicallyy relaterelatedd bbyy ononee ooff ththee modal relationsrelations,, implicationimplication,, equivalenceequivalence,, consistencyconsistency,, oorr inconsistencinconsistencyy (discussed(discussed iinn ththee nexnextt section).section).

6. ThThee precedinprecedingg fivefive kindkindss ooff necessarnecessaryy truttruthh araree nonott whollwhollyy exclusivexclusivee ooff ononee anotheanotherr anandd araree certainlcertainlyy nonott exhaustivexhaustivee ooff ththee wholwholee clasclasss ooff necessarnecessaryy truthstruths.. TherTheree araree otheotherr propositionpropositionss whicwhichh araree trutruee iinn alalll possiblpossiblee worldworldss bubutt foforr whicwhichh iitt iiss difficuldifficultt ttoo find ananyy singlesingle apaptt descriptiodescriptionn excepexceptt ttoo sasayy thathatt thetheyy araree propositionpropositionss whoswhosee truttruthh cacann bbee ascertaineascertainedd bbyy conceptual analysis. WheWhenn PlatoPlato,, foforr instanceinstance,, iinn hihiss dialoguedialogue,, Theatetus, analyzeanalyzedd ththee concepconceptt ooff knowledgknowledgee anandd camcamee ttoo ththee conclusioconclusionn thathatt knowinknowingg implieimpliess (among(among otheotherr thingsthings)) believingbelieving,, hhee provideprovidedd groundgroundss forfor uuss ttoo ascertainascertain,, aass beinbeingg necessarnecessaryy truthstruths,, a hoshostt ooff propositionspropositions,, suchsuch aass

(7.77)(1.17) IfIf ththee PopPopee knowknowss thathatt thertheree araree ninninee planetplanetss thethenn ththee PopPopee believebelievess thathatt thertheree areare ninenine planets.planets.

ThatThat hostshosts ofof suchsuch necessarynecessary truthstruths existexist shouldshould bebe evidentevident fromfrom thethe factfact thatthat thethe analyticanalytic truthtruth whichwhich PlatoPlato discovereddiscovered isis quitequite general,general, admittingadmitting countlesscountless instancesinstances rangingranging fromfrom thethe Pope'sPope's knowledgeknowledge (and(and consequentconsequent belief)belief) aboutabout thethe numbernumber ofof thethe planetsplanets toto thethe knowledgeknowledge (and(and consequentconsequent belief)belief) thathatt mostmost ofof usus havehave a!?outabout thethe earth'searth's beingbeing round,round, andand soso on.on. Likewise,Likewise, whenwhen a moralmoral philosophephilosopherr analyzesanalyzes thethe conceptconcept ofof moralmoral obligationobligation andand concludesconcludes thatthat beingbeing morallymorally obligedobliged toto dodo somethingsomething impliesimplies beingbeing ableable toto dodo it,it, thatthat philosopherphilosopher establishesestablishes thatthat a hosthost ofof propositions,propositions, ofof whicwhichh

(1.78)(1-18) IfIf thethe foreignforeign ministerminister isis morallymorally obligedobliged toto resignresign thenthen hehe isis ableable toto dodo ssoo isis onlyonly oneone instance,instance, areare necessarynecessary truths.truths. Characteristically,Characteristically, philosophersphilosophers -— inin theirtheir analysesanalyses ofof thesthesee andand otherother conceptsconcepts whichwhich figurefigure centrallycentrally inin ourour thinkingthinking -— areare notnot tryingtrying toto competecompete withwith scientistscientistss inin discoveringdiscovering contingentcontingent truthstruths aboutabout howhow thethe actualactual worldworld happenshappens toto bebe constituted.constituted. Rather,Rather, theythey areare tryingtrying toto discoverdiscover whatwhat isis implied byby manymany ofof thethe conceptsconcepts whichwhich scientistsscientists andand thethe restrest ofof usus taketake forfor granted.granted. AndAnd thethe propositionspropositions inin whichwhich theythey reportreport theirtheir analyticalanalytical discoveries,discoveries, ifif true,true, araree necessarilynecessarily truetrue..

15.15. JaakkoJaakko Hintikka,Hintikka, "The"The ModesModes ofof Modality"Modality" inin Acta Philosophica Fennica, vol.vol. 1616 (1963),(1963), pp.pp. 65-81.65-81. 2244 POSSIBLPOSSIBLEE WORLDSWORLDS

Summary MucMuchh ooff whawhatt wwee havhavee saisaidd ssoo fafarr concerninconcerningg worldworldss —- actuaactuall anandd non-actuanon-actuall —- anandd concerninconcerningg propositionpropositionss —- trutruee anandd falsefalse,, contingencontingentt anandd noncontingennoncontingentt —- abouaboutt itemitemss iinn thesthesee worldworldss cacann bbee summarizesummarizedd iinn ththee followinfollowingg diagramdiagram::

AlAlll PossiblPossible.e WorldWorldss NonNon­- ThThee ActuaActuall actual actual actuaactuall TrutTruthh ModaModall worldworldas worlworldd StatuStatuss StatuStatuss _____--''-- --y-_...A----,. ,,.._--"-...... ~------__-____,

··TRUE•TRUE Iinn alalll possiblpossiblee worlds···worlds- . Necessarily True • Non-contingent

··TRlJE•TRUE Iinn ththee actuaactuall worlworldd anandd FALSFALSEE Iinn aatt leasleastt ononee otheotherr Possibly possible world . PropPropo­o possible world True sitions sitions Contingent ..•FALSFALSEE iinn ththee actuaactuall worlworldd anandd TRUTRUEE iinn aatt leasleastt ononee otherother Possibly possiblpossiblee worlworldd .. False False . ·FALSEFALSE iinn alalll possiblpossiblee worldsworlds . Necessarily False

FIGURFIGUREE (1.g)(7.g) NotNotee thathatt onlonlyy thosthosee propositionpropositionss araree contingencontingentt whicwhichh araree both possiblpossiblyy trutruee and possiblpossiblyy falsefalse,, whereawhereass thosthosee propositionpropositionss araree noncontingennoncontingentt whicwhichh araree either nonott possiblpossiblyy falsfalsee (th(thee necessarilynecessarily truetrue ones)ones) or notnot possiblypossibly truetrue (the(the necessarilynecessarily falsefalse ones)ones)..

OtherOther importantimportant relationshipsrelationships betweenbetween classesclasses ofof propositionspropositions maymay bebe readread offoff thethe diagramdiagram iinn accordanceaccordance withwith thethe followingfollowing rule:rule: ifif thethe bracketbracket representingrepresenting oneone classclass ofof propositionspropositions (e.g.,(e.g., ththee necessarilynecessarily truetrue ones)ones) isis containedcontained withinwithin thethe bracketbracket representingrepresenting anotheranother classclass ofof propositionspropositions (e.g.,(e.g., thethe actuallyactually truetrue ones),ones), thenthen anyany propositionproposition belongingbelonging toto thethe formerformer classclass isis a propositiopropositionn belongingbelonging toto thethe latterlatter class.class. ForFor instance,instance, allall necessarilynecessarily truetrue propositionspropositions areare actuallyactually truetrue,, althoughalthough notnot allall actuallyactually truetrue propositionspropositions areare necessarilynecessarily truetrue..

EXERCISES

7.1. For each ofofthe the followingfollowing propositions,propositions, saysay (7)(1) whetherwhetherit itis iscontingent contingentor ornoncontingent, noncontingent,and and (2)(2) if noncontingent, then whether it is true or false.

a. All aunts are arefemales. females. b. On April 73,13, 7945,1945, all femalesare areaunts. aunts. c. If a bird is entirely black, black,then thenit is itnot is alsonot alsowhite. white. d. All black birds birdsare areblack. black. § 3 PropertiesProperties ofof PropositionsPropositions 2525

e. All blackbirds are black.black. ff. Some squares havehavesix six sides.sides. g. No mushrooms arearepoisonous. poisonous. h. It is illegal to jaywalk jaywalk inin Moscow.Moscow.

I.i. Whenever itit isis summersummer inin NewNew Zealand,Zealand, itit isis winterwinter inin Canada.Canada, j. What will be, will be.be.

2. Briefly explain why each ofofthe thefollowing following propositionspropositionsis isfalse. false.

k. If a proposition is actually true, then that propositionproposition isis alsoalsocontingent. contingent. l.I. If it is possiblepossiblefor fora aproposition proposition totobe betrue, true,then thenit it isispossible possiblefor forthat thatsame same proposition to be false. false. m. Even though aaproposition proposition isisactually actuallyfalse, false, ititneed neednot notbe. be. n. If a proposition is noncontingent, then it is actually true.true. o. If a proposition is possiblypossibly true,true, thenthenit it isis contingent.contingent.

3. Write a short story such that a person reading that story could tell fromfrom subtlesubtle cluesclues thatthat therethere isis no possiblepossibleworld world inin whichwhichall allthe theevents eventsrelated relatedoccur. occur.

4. Does the concept of time-travel really generate paradoxes?paradoxes? HereHere isis whatwhat LazarusLazarus sayssays aboutabout thethe matter when discussing it with two of his descendants beforebeforemaking makinghis hisown owntime-trip: time-trip:"That "Thatold old cliche about shooting your grandfather before he sires your father, then going fuff!fuff! like a soapsoap bubble -— and all descendants, too,too,meaning meaningboth bothof ofyou youamong amongothers others- — isisnonsense. nonsense.The Thefact fact that I'm here and you're here means that I didn'tdidn't do it -— or won't do it; the tenses of ofgrammar grammar aren't built forfor time-traveltime-travel -— butbut itit doesdoesnot notmean meanthat thatI Inever neverwent wentback backand andpoked pokedaround. around.I I haven't any yen to look at myselfmyself when I was a snot-nose; it'sit'sthe theera erathat thatinterests interestsme. me.If If II ranran across myselfmyselfas asa ayoung youngkid, kid,he he- —I /- —wouldn't wouldn'trecognize recognizeme; me;I Iwould wouldbe bea strangera strangerto tothat thatbrat. brat. He wouldn't give me a passing glance. I know, I waswas he." Discuss. (Robert A. Heinlein, TimTimee Enou~hEnough ForFor Love,Love, p.p. 358.)358.)

****** * * * * Symbolization MuchMuch ofof thethe successsuccess andand promispromisee ofof modernmodern logic,logic, likelike thatthat ofof mathematics,mathematics, arisesarises fromfrom thethe powerfupowerfull andand suggestivesuggestive symbolizationsymbolization ofof itsits concepts.concepts. FromFrom timetime toto time,time, asas wewe introduceintroduce andand discussdiscuss variouvariouss ofof thethe mostmost fundamentalfundamental conceptsconcepts ofof logic,logic, wewe willwill alsoalso introduceintroduce symbolssymbols whichwhich areare widelywidely acceptedaccepted asas standingstanding forfor thosethose concepts.concepts. AlreadyAlready wwee havhavee casuallycasually introducedintroduced somesome ofof thesethese symbolssymbols asas whenwhen,, forfor example,example, wewe 'used'used ththee capitalcapital letterletter "P""P" ofof thethe EnglishEnglish alphabetalphabet toto representrepresent anyany arbitrarilyarbitrarily chosenchosen propositionproposition;; whenwhen wewe usedused "a" toto representrepresent anyany arbitrarilyarbitrarily chosenchosen item;item; andand whenwhen wewe usedused "F""F" toto representrepresent anan arbitrarilyarbitrarily chosenchosen attributeattribute.. LetLet usus extendextend ourour catalogcatalog ofof symbolssymbols eveneven more.more. WeWe havehave jusjustt introducedintroduced severalseveral ofof thethe mosmostt importantimportant conceptsconcepts ofof modernmodern logic.logic. TheyThey areare standardlystandardly representedrepresented inin symbolssymbols asas follows:follows: 2626 POSSIBLEPOSSIBLE WORLDSWORLDS

1616 (1)(1) TheThe conceptconcept ofof falsity:falsity: " '"V"~ " [called[called tilde ]] (2)(2) TheThe conceptconcept ofof possibilitypossibility:: "0""0" [called[called diamond]diamond] (3)(3) TheThe conceptconcept ofof necessarynecessary truth:truth: "0""•" [called[called box] (4)(4) TheThe conceptconcept ofof contingency:contingency: "'V""V" [called[called nabla]nabla] (5)(5) TheThe conceptconcept ofof noncontingency:noncontingency: "l;""A" [called[called delta]delta)

EachEach ofof thesethese symbolssymbols maymay bebe prefixedprefixed toto a symbolsymbol (such(such asas "P","P", "Q","Q", "R")"R") whichwhich standsstands forfor a propositionproposition,, toto yieldyield a furtherfurther propositionapropositionall symbolsymbol (e.g.,(e.g., "'"V"~P"p",, "OR")."OR"). AnyAny sequencesequence ofof oneone oorr moremore propositionapropositionall symbolssymbols isis calledcalled a "formula"."formula".J717 EachEach ofof thesethese symbolssymbols maymay bebe defineddefined contextuallycontextually asas follows:follows:

18 " '"V^ PP"" =df"df "The"The propositiopropositionn P isis false" 18 lor[or "It"It isis falsefalse thatthat P"]P"] "The proposition P is possible" [or "It is possible that P"] "OP""OP" =df= df "The proposition P is possible" [or "It is possible that P"] "•P" "The proposition P is necessarily true" [or "It is "oP" = dfdf "The proposition P is necessarily true" [or "It is necessarily true that P"] necessarily true that P"] "'VP""VP" =df "The proposition P is contingent" [or "It is contingent that P"] df "The proposition P is contingent" [or "It is contingent that P"] "AP" "l;P" =df= df "The proposition P is noncontingent" [or "It is "The proposition P is noncontingentnoncontingent" [or "It is that P"] noncontingent that P"]

TheseThese symbolssymbols maymay bbee concatenated,concatenated, thatthat is,is, linkedlinked togethertogether inin a series,series, asas forfor example,example, "'"V" ~ '"V~ P".P".

ThisThis givesgives uuss thethe meansmeans toto expressexpress suchsuch propositionpropositionss asas thatthat P isis possiblpossiblyy falsefalse ("0("0 '"V~ P"),P"), necessarilynecessarily

falsefalse ("0("• '"V P"),P"), impossibleimpossible ("(" '"V~ 0OP")P"),, etcetc.. CertainCertain ofof thesethese combinationscombinations ofof symbolssymbols areare sometimessometimes unwittinglyunwittingly translatedtranslated intointo proseprose iinn incorrectincorrect ways,ways, andand oneone shouldshould bewarebeware ofof thethe pitfalls.pitfalls. InIn particularparticular,, combinationscombinations beginninbeginningg withwith "V"'V"" andand "l;""A" provprovee troublesome.troublesome. Consider,Consider, forfor example,example, ·"'i7P"."VP". ThereThere isis a temptationtemptation toto translatetranslate thisthis aass "it"it isis contingentlycontingently truetrue thatthat P,"P," oror equivalentlyequivalently asas "P"P isis contingentlycontingently true".true". NeitheNeitherr ofof thesethese proposedproposed translationstranslations willwill do.do. TheThe correctcorrect translationtranslation isis "It"It isis contingentcontingent thatthat P isis true."true." Why?Why? WhatWhat exactlyexactly iiss thethe differencedifference betweebetweenn sayingsaying onon thethe oneone handhand thatthat P isis contingentlycontingently true,true, andand onon thethe other,other, thatthat itit iiss contingentcontingent thatthat P isis true?true? JustJust this:this: toto saysay thatthat P isis contingentlycontingently truetrue isis toto saysay thatthat P isis bothboth contingentcontingent andand true;true; toto saysay thatthat itit isis contingentcontingent thatthat P isis true,true, isis toto saysay onlyonly thatthat P'sP's beinbeingg truetrue iiss contingent,contingent, i.e.,i.e., thatthat P isis truetrue inin somesome possiblpossiblee worldsworlds andand falsefalse inin somesome others.others. Clearly,Clearly, thenthen,,

16.16. RhymesRhymes withwith "Hilda"."Hilda".

17.17. InIn laterlater chapterschapters wewe willwill distinguishdistinguish betweenbetween formulaeformulae whichwhich areare constructedconstructed soso asas toto makemake sensesense (i.e.,(i.e., whichwhich areare well-formed)well-formed) andand thosethose whichwhich areare not.not. TheThe exactexact rulesrules forfor constructingconstructing well-formedwell-formed formulaeformulae withinwithin certaincertain systemssystems ofof symbolssymbols areare givengiven inin chapterschapters 5 andand 66..

18.18. TheThe "=dr"-symbol"=df"-symbol maymay bbee readread asas "has"has thethe samesame meaningmeaning as"as" oror alternativelyalternatively asas "equals"equals bbyy "definition"..

TheThe expressionexpression onon thethe leftleft sideside ofof thethe "=df"-symbol"=df"-symbol isis thethe expressionexpression beinbeingg introduced;introduced; itit isis knownknown bbyy ththee

technicaltechnical namename definiendum. TheThe expressionexpression onon thethe rightright handhand sideside ofof thethe "=dfde""-symbo-symbolI isis thethe oneone whose.whose meaningmeaning isis presumepresumedd alreadyalready understoounderstoodd andand isis beinbeingg assignedassigned toto thethe definiendum.definiendum. TheThe rightright handhand expressionexpression iiss calledcalled thethe definiens.definiens. § 3 PropertiesProperties ofof PropositionsPropositions 2727

toto saysay thatthat a propositionproposition isis contingentlycontingently truetrue isis toto saysay more thanthan merelymerely thatthat itsits truthtruth isis contingent.contingent. InIn otherother words,words, thethe expressionexpression "P"P isis contingentlycontingently true"true" isis toto bbee renderedrendered inin ourour symbolicsymbolic notationnotation asas "P"P andand 'VVP.P."" ItIt isis thusthus incorrectincorrect toto translatetranslate "'V"vPP"" alonealone asas "P"P isis contingentlycontingently true."true." Similarly,Similarly, "'V" V 'V~ P"P" isis toto bebe translatedtranslated asas "it"it isis contingentcontingent thatthat itit isis falsefalse thatthat P";P"; andand notnot asas "P"P isis contingentlycontingently false."false." AnAn easyeasy rulerule toto beabearr inin mindmind soso asas toto avoidavoid mistakesmistakes inin translationstranslations isis this:this: NeveNeverr translatetranslate "'V""V" oror "f,""A" adverbially,adverbially, i.e.,i.e., withwith anan "ly""ly" ending;ending; alwaysalways translatetranslate themthem asas "it"it isis contingentcontingent that"that" aDdand "it"it i~i« nonnoncontingencontingentt that"that" respectively.respectively. AdverbialAdverbial transli'tionstranslations inin thethe otherother cases,cases, i.e.,i.e., forfor "0""•" andand "0""0" areare freelyfreely permittedpermitted:: "it"it isis necessarilynecessarily truetrue that"that" andand "it"it isis possiblpossiblyy truetrue that"that" respectively.respectively.

EXERCISES

1.,. Let "A" stand for the proposition that Canada is north of Mexico. Translate each of the followingfollowing expressions into English prose.prose.

(a)(a) ,A (f)(0 'VO'VA~ <> ^A (k)(k) 'VAVA (p)(P) A^Af,'VA

(b)(b) 'VA^A (g)(g) oA• A (I)0) 'V'VAV^A (q)(q) 'V^ f,A A

(c)(c) OA (h)(h) o'VA• ^A (m)(m) 'V^VA'VA (r)(r) 'Vf,'VA^ A^A

(d)(d) O'VAO^A (i)(i) 'V~ 0• AA (n)(n) 'V^v^A'V 'VA

(e)(e) 'VOA^OA (j)(J) 'Vo'VA^O^A (0)(o) Af,A A

2. For each of the cases (c)(c) through (r)(r) above say whether the proposition expressed is true or false.false.

3. Letting "A" now stand for the proposition that all squares have four sides, say for each of thethe expressions (a)(a) - (r)(r) in question "1, whether the proposition is true or false.false.

4. Explain why "f,"A P" is notnot to be translated as "P is noncontingently true" but as "it is noncontingent that P is true." Find a proposition of which it is true that it is nonnoncontingentcontingent that it is true, but of which it is false that it is noncontingently true.true.

5. Say of each of the following which is true and which is false. (Note: it is actually true that some cows are infertile.)infertile.)

(s)(s) It is contingent that some cows are infertile.infertile.

(t)(t) It is contingent that it is not the case that some cows are infertile.infertile. (u)(u) It is contingently true that some cows are infertile.infertile. (v)(v) It is contingently false that some cows are infertile.infertile.

6. Say for each of the following whether it is true or false.false. (w)(w) It is noncontingent that 2 + 2 =— 4.4. (x)(x) It is noncontingently true that 2 + 2 =— 4.4. (y)(y) It is noncontingent that it is false that 2 + 2 = 4.4. 2288 POSSIBLPOSSIBLEE WORLDSWORLDS

(z(z)) It is noncontingently falsefalse that 2 + 2 —= 4. 7. Explain the difference in meaning in the two phrases "noncontingently true" and "not"not contingently true".true".

44.. RELATIONRELATIONSS BETWEEBETWEENN PROPOSITIONSPROPOSITIONS

JusJustt aass ththee modamodall propertiepropertiess ooff propositionpropositionss araree a functiofunctionn ooff ththee waywayss iinn whicwhichh ththee truth-valuetruth-valuess ooff thosthosee propositionpropositionss singlsinglyy araree distributedistributedd acrosacrosss ththee sesett ooff alalll possiblpossiblee worldsworlds,, ssoo ththee modal relationsrelations betweebetweenn propositionpropositionss araree a functiofunctionn ooff ththee waywayss iinn whicwhichh ththee truth-valuetruth-valuess ooff ththee membermemberss ooff pairs ooff propositionpropositionss araree distributedistributedd acrosacrosss ththee sesett ooff alalll possiblpossiblee worldsworlds.. WWee singlsinglee ououtt foufourr modamodall relationrelationss foforr immediatimmediatee attentionattention,, viz.viz.,,

inconsistency; consistency;consistency;implication; implication;and andequivalence. equivalence.

IInn termtermss ooff thesthesee wwee wilwilll finfindd iitt possiblpossiblee ttoo explaiexplainn mosmostt ooff ththee centracentrall logicalogicall conceptconceptss discussediscussedd iinn thithiss bookbook.. YeYett thesthesee modamodall relationsrelations,, wwee shallshall nonoww see,see, cacann themselvethemselvess bbee explainedexplained -— iinn mucmuchh ththee samesame wawayy asas ththee aboveabove discusseddiscussed modamodall propertiepropertiess -— iinn termtermss ofof possiblpossiblee worldsworlds..

Inconsistency TwTwoo propositionpropositionss araree inconsistent witwithh ononee anotheranother,, wwee ordinarilordinarilyy say,say, jusjustt whewhenn iitt iiss necessarnecessaryy thathatt iiff ononee iiss trutruee ththee otheotherr iiss falsefalse,, i.e.i.e.,, jusjustt whewhenn thetheyy cannocannott botbothh bbee truetrue.. TranslatinTranslatingg thithiss ordinarordinaryy taltalkk intintoo taltalkk ooff possiblpossiblee worldworldss wwee mamayy saysay thathatt twtwoo propositionpropositionss araree inconsisteninconsistentt jusjustt whewhenn iinn ananyy possiblpossiblee worldworld,, iiff anyany,, iinn whicwhichh ononee iiss trutruee ththee otheotherr iiss falsefalse,, i.e.i.e.,, jusjustt whewhenn thertheree iiss nnoo possiblpossiblee worlworldd iinn whicwhichh botbothh araree truetrue.. InconsistencyInconsistency iiss a generigenericc modamodall relationrelation.. ItIt hahass twotwo,, anandd onlonlyy twtwoo species:species: contradictiocontradictionn anandd contrariety.contrariety. isis thatthat speciesspecies ofof inconsistencyinconsistency whichwhich holdsholds betweenbetween twotwo propositionspropositions whenwhen theythey nonott onlyonly cannotcannot bothboth bebe truetrue butbut alsoalso cannotcannot bothboth bebe false.false. AsAs wewe sawsaw inin sectionsection 2,2, itit isis a relationrelation whichwhich holdsholds between,between, e.g.,e.g., thethe contingentcontingent propositionpropositionss

(1.3) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 andand (1.4) It is not the case that the U.S. entered World War I in 1917; (1.4) It is not the case that the U.S. entered World War I in 1917; and also holds between, e.g., the noncontingent propositions and also holds between, e.g., the noncontingent propositions (1.5) EitherEither thethe U.S.U.S. enteredentered WorldWorld WarWar IIin in 19171917 ororit it isis notnot thethe casecasethat that thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 andand (1.6) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 andand itit isis notnot thethe casecase thatthat thethe U.SU.S.. enteredentered WorldWorld WarWar I inin 1917.1917.

InIn anyany possiblepossible world,world, oneone oror otherother ofof thethe propositionspropositions inin a contradictorycontradictory pairpair isis truetrue andand thethe remainingremaining propositionproposition isis false.false. WhereWhere twotwo propositionspropositions areare contradictoriescontradictories ofof oneone anotheranother therethere isis nnoo possiblepossible worldworld inin whichwhich bothboth propositionspropositions areare truetrue andand nono possiblepossible worldworld inin whichwhich bothboth propositionpropositionss areare false.false. ToTo repeat:repeat: contradictoriescontradictories dividedivide thethe setset ofof allall possiblepossible worldsworlds intointo twotwo mutuallymutually exclusiveexclusive andand jointlyjointly exhaustiveexhaustive subsetssubsets.. § 4 RelationsRelations betweenbetween PropositionPropositionss 2929

Contrariety isis thatthat speciesspecies ofof inconsistencyinconsistency whichwhich holdsholds betweebetweenn twotwo propositionpropositionss whenwhen althougalthoughh theythey cannQtcannot botbothh bbee true,true, theythey neverthelesneverthelesss cancan botbothh bbee false.false. Consider,Consider, forfor instance,instance, thethe relatiorelationn betweebetweenn thethe contingentcontingent propositionpropositionss

(1.3) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 andand (1.19) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 1914.1914.

Plainly,Plainly, ifif oneone membermember fromfrom thisthis contrarycontrary paipairr isis true,true, thethe otherother isis false.false. TheThe truthtruth ofof oneone excludesexcludes thethe truthtruth ofof thethe other.other. SoSo ifif twotwo propositionpropositionss areare contrariescontraries itit mustmust bebe thatthat atat leastleast oneone isis false.false. MoreoverMoreover botbothh maymay bbee false.false. AfterAfter all,all, wewe cancan conceiveconceive ofof itsits havinghaving beebeenn thethe casecase thatthat thethe U.Su.s.. enteredentered WorldWorld WarWar I neitherneither inin 19141914 nornor inin 19171917 bubutt rather,rather, letlet usus suppose,suppose, inin 1916.1916. ThusThus whilewhile propositionpropositionss (1.3) andand (1.19) cannotcannot botbothh bbee true,true, theythey can botbothh bbee false.false. ThereThere isis somesome possiblepossible worldworld inin whichwhich (1.3) andand (1.19) areare false.false. BetweenBetween them,them, then,then, thethe propositionpropositionss ofof a contrarycontrary pairpair dodo notnot exhaustexhaust allall thethe possibilitiespossibilities.. InIn short,short, contrariescontraries dividedivide thethe setset ofof allall possiblpossiblee worldsworlds intointo twotwo mutuallymutually exclusiveexclusive subsetssubsets whichwhich areare not jointljointlyy exhaustive.exhaustive. BothBoth membersmembers ofof thethe contrarycontrary paipairr jusjustt consideredconsidered areare contingentcontingent propositionspropositions.. CanCan noncontingentnoncontingent propositionpropositionss alsoalso bbee contraries?contraries? CanCan a nonnoncontingencontingentt propositiopropositionn bbee a contrarycontrary ofof a contingentcontingent propositionproposition?? AsAs wewe havehave herehere defineddefined "contrariety","contrariety", thethe answeranswer isis "Yes""Yes" toto botbothh questions.questions.I919 Consider,Consider, first,first, twotwo propositionpropositionss whichwhich areare necessarilynecessarily false.false. SinceSince therethere areare nono possiblpossiblee worldsworlds iinn whichwhich eithereither isis true,true, therethere isis nono possiblpossiblee worldworld inin whichwhich bothboth areare true.true. ThatThat isis toto say,say, sincesince botbothh araree necessarilynecessarily false,false, theythey cannotcannot bothboth bebe true.true. ButBut equally,equally, sincesince bothboth areare necessarilynecessarily false,false, theythey areare botbothh falsefalse inin allall possiblpossiblee worlds,worlds, andand hencehence therethere isis a possiblpossiblee worldworld inin whichwhich botbothh areare false.false. Hence,Hence, sincesince twotwo necessarilynecessarily falsefalse propositionpropositionss cannotcannot botbothh bbee true,true, bubutt cancan botbothh bbee false,false, theythey areare contrariescontraries.. Consider,Consider, secondly,secondly, a paipairr ofof propositionpropositionss oneone ofof whichwhich isis necessarilynecessarily falsefalse andand thethe otherother ofof whichwhich isis contingent.contingent. SinceSince therethere areare nono possiblpossiblee worldsworlds inin whichwhich thethe necessarilynecessarily falsefalse propositiopropositionn isis true,true, therethere cancan bbee nono possiblpossiblee worldsworlds inin whichwhich bothboth itit andand thethe contingentcontingent propositiopropositionn areare true.true. ToTo bbee sure;sure; therethere willwill bbee somesome possiblpossiblee worldsworlds inin whichwhich thethe contingentcontingent propositiopropositionn isis true.true. ButBut inin allall thosethose possiblepossible worldsworlds thethe necessarilynecessarily falsefalse propositiopropositionn willwill bbee false.false. Hence,Hence, eveneven inin thosethose possiblpossiblee worldsworlds itit willwill nonott bbee thethe casecase thatthat both areare true.true. Moreover,Moreover, botbothh propositionpropositionss maymay bbee false.false. TheyThey willwill botbothh bbee falsefalse iinn allall thosethose possiblpossiblee worldsworlds inin whichwhich thethe contingentcontingent propositionproposition isis false.false. Hence,Hence, sincesince twotwo propositionpropositionss oneone ofof whichwhich isis necessarilynecessarily falsefalse andand thethe otherother ofof whichwhich isis contingentcontingent cannotcannot botbothh bbee true,true, bubutt cacann botbothh bbee false,false, theythey areare contrariescontraries.. NecessarilNecessarilyy falsefalse propositionspropositions,, itit isis clear,clear, areare profligatprofligatee sourcessources ofof inconsistency.inconsistency. EveryEvery necessarilynecessarily falsefalse propositiopropositionn isis a contradictorycontradictory of,of, andand hencehence inconsistentinconsistent with,with, everyevery necessarilynecessarily truetrue propositionproposition.. EveryEvery necessarilynecessarily falsefalse propositiopropositionn isis a contrarycontrary ofof anyany andand everyevery contingentcontingent proposition.proposition. AndAnd evereveryy necessarilynecessarily falsefalse propositionproposition isis a contrarycontrary ofof everyevery otherother necessarilynecessarily falsefalse propositionproposition.. Indeed.Indeed, wewe needneed onlyonly addadd thatthat thethe termterm "self-inconsistent""self-inconsistent" isis a synonymsynonym forfor thethe termterm "necessarily"necessarily false",false", iinn orderorder toto concludeconclude thatthat a necessarilynecessarily falsefalse propositiopropositionn isis inconsistentinconsistent withwith everyevery propositiopropositionn whatever,whatever, includingincluding itself,itself, i.e.,i.e., thatthat a necessarilynecessarily falsefalse propositiopropositionn isis self-inconsistent.^//'-inconsistent. FromFrom thethe factfact thatthat twotwo propositionpropositionss areare inconsistentinconsistent itit followsfollows thatthat atat leastleast oneone isis actually false.false. ForFor since,since, bbyy thethe definitiondefinition ofof "inconsistency","inconsistency", therethere isis nono possiblpossiblee worldworld inin whichwhich botbothh membersmembers ooff anan inconsistentinconsistent paipairr ofof propositionpropositionss areare true,true, everyevery possiblpossiblee worldworld -— includingincluding thethe actualactual worldworld -— isis a worldworld inin whichwhich atat leastleast oneone ofof themthem isis false.false. Inconsistency,Inconsistency, wewe maymay say,say, providesprovides a guaranteeguarantee ooff falsity.falsity. ButBut thethe converseconverse doesdoes notnot hold.hold. FromFrom thethe factfact thatthat oneone oror botbothh ofof a paipairr ofof propositionpropositionss isis

19.19. HistoricallyHistorically somesome logicianslogicians havehave usedused thethe termsterms "contradiction""contradiction" andand "contrariety""contrariety" asas ifif theythey appliedapplied onlonlyy inin casescases inin whichwhich botbothh propositionpropositionss areare contingent.contingent. ForFor moremore onon this,this, seesee thethe subsectionsubsection entitledentitled "A Note onon History cmdand Nomenclature", pppp.. 53-5453-54.. 3030 POSSIBLEPOSSIBLE WORLDSWORLDS actuallyactually false,false, itit doesdoes notnot followfollow thatthat theythey areare inconsistent,inconsistent, i.e.,i.e., thatthat inin everyevery possiblpossiblee worldworld oneone oorr botbothh ofof themthem isis false.false. ThusThus (1.4) andand (1.19) areare falsefalse inin thethe actualactual world.world. YetYet theythey areare nonott inconsistent.inconsistent. ThisThis isis fairlyfairly easyeasy toto show.show. ConsiderConsider thethe factfact thatthat (1.19) isis contingentcontingent andand hencehence isis trutruee inin somesome possiblpossiblee worlds,worlds, inin particularparticular,, inin allall thosethose possiblpossiblee worldsworlds inin whichwhich thethe U.S.U.S. entereenteredd WorldWorld WarWar I inin 1914.1914. ButBut inin eacheach ofof thesethese possiblpossiblee worlds,worlds, itit turnsturns outout thatthat (1.4) isis alsoalso true:true: anyany possiblpossiblee worldworld inin whichwhich thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 19141914 isis alsoalso a worldworld inin whichwhich itit isis notnot ththee casecase thatthat thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917.1917. AndAnd thisthis isis jusjustt toto saysay thatthat inin allall thosethose possiblpossiblee worldsworlds inin whichwhich (1.19) isis true,true, (1.4) isis alsoalso true.true. HenceHence therethere isis a possiblpossiblee worldworld inin whichwhich thesthesee actuallyactually falsefalse propositionpropositionss areare truetrue together.together. InIn short,short, fromfrom thethe factfact thatthat theythey areare falsefalse inin factfact itit doedoess notnot followfollow thatthat theythey areare inconsistent.inconsistent. BeingBeing falsefalse inin thethe actualactual world,world, itit turnsturns out,out, provideprovidess nnoo guaranteeguarantee ofof inconsistencyinconsistency..

EXERCISE

Can two propositions be contraries as well as contradictories oj one another? Explain your answer. Can two propositions be contraries as well as contradictories of one another? Explain your answer. * ** ** Consistency WhatConsistencydoes it mean to say that two propositiOns are consistent with one another? Given that we already know what it means to say that two propositions are inconsistent with one another, the What does it mean to say that two propositions are consistent with one another? Given that we answer comes easily: two propositions are consistent with one another if and only if it is not the case already know what it means to say that two propositions are inconsistent with one another, the that they are inconsistent. It follows that two propositions are consistent if and only if it is not the answer comes easily: two propositions are consistent with one another if and only if it is not the case case that there is no possible world in which both are true. But this means that they are consistent if that they are inconsistent. It follows that two propositions are consistent if and only if it is not the and only if there is a possible world in which both are true. case that there is no possible world in which both are true. But this means that they are consistent if As an example of the modal relation of consistency, consider the relation between the contingent and only if there is a possible world in which both are true. propositionsAs an example of the modal relation of consistency, consider the relation between the contingent propositions (7.3)(1.3) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 andand (7.20) Lazarus Long was born in Kansas in 1912. (1.20) Lazarus Long was born in Kansas in 1912. WhateverWhatever otherother relationrelation maymay holdhold betweebetweenn them,them, plainlplainlyy thethe relationrelation ofof consistencyconsistency does:does: itit needneed nonott bbee thethe casecase thatthat ifif oneone isis truetrue thethe otherother isis false;false; botbothh cancan bbee true.true. NNoo mattermatter whatwhat thethe factsfacts happehappenn toto bbee aboutabout thethe actualactual worldworld (no(no matter,matter, thatthat is,is, whetherwhether eithereither (1.3) oror (1.20) isis actuallyactually true),true), itit iiss possiblpossiblee thatthat botbothh ofof themthem shouldshould bbee truetrue -— whichwhich isis jusjustt toto saysay thatthat therethere isis atat leastleast oneone possiblpossiblee worldworld inin whichwhich botbothh areare true.true. IfIf twotwo contingentcontingent propositionpropositionss happenhappen botbothh toto bbee truetrue inin thethe actualactual world,world, thenthen sincesince thethe actuaactuall worldworld isis alsoalso a possiblpossiblee world,world, therethere isis a possiblpossiblee worldworld inin whichwhich botbothh areare true,true, andand hencehence theythey araree consistent.consistent. ActualActual truth,truth, thatthat isis toto say,say, provideprovidess a guaranteeguarantee ofof consistency.consistency. ButBut thethe converseconverse doesdoes nonott hold.hold. FromFrom thethe factfact thatthat twotwo propositionpropositionss areare consistentconsistent itit doesdoes notnot followfollow thatthat theythey areare bothboth truetrue iinn thethe actualactual world.world. WhatWhat doesdoes.folloJolloww isis thatthat therethere isis some possiblpossiblee worldworld inin whichwhich botbothh areare true;true; yeyett thatthat possiblpossiblee worldworld maymay bbee non-actual.non-actual. ThusThus (1.3) andand (1.20) areare consistent.consistent. YetYet theythey areare notnot botbothh truetrue inin thethe actualactual world.world. (1.3) isis truetrue inin thethe actualactual worldworld andand falsefalse inin somesome non-actualnon-actual worlds,worlds, whilwhilee (7.20)(1.20) isis falsefalse inin thethe actualactual worldworld andand truetrue inin somesome non-actualnon-actual worlds.worlds. HenceHence propositionpropositionss cancan bbee consistentconsistent eveneven ifif oneone oror botbothh isis false.false. InIn short,short, consistencyconsistency doesdoes notnot providprovidee a guaranteeguarantee ofof truthtruth.. SinceSince a necessarilynecessarily truetrue propositiopropositionn isis truetrue inin allall possiblpossiblee worlds,worlds, a necessarilynecessarily truetrue propositiopropositionn § 4 RelationRelationss betweebetweenn PropositionPropositionss 3311 wilwilll bbee consistenconsistentt witwithh ananyy propositiopropositIOnn whicwhichh iiss trutruee iinn aatt leasleastt ononee possiblpossiblee worldworld.. IItt wilwilll bbee consistentconsistent,, thathatt isis,, witwithh ananyy contingencontingentt propositiopropositionn anandd witwithh ananyy necessarilnecessarilyy trutruee propositionproposition.. IItt wilwilll bbee inconsisteninconsistentt onlonlyy witwithh thosthosee propositionpropositionss whicwhichh araree nonott trutruee iinn ananyy possiblpossiblee worldsworlds,, i.e.i.e.,, witwithh necessarilnecessarilyy falsfalsee onesones..

Implication OOff ththee foufourr modamodall relationrelationss wwee araree currentlcurrentlyy consideringconsidering,, iitt iiss probablprobablyy implicatioimplicationn whicwhichh iiss mosmostt closelcloselyy identifiedidentified,, iinn mosmostt personspersons'' mindsminds,, witwithh ththee concernconcernss ooff philosophphilosophyy iinn generagenerall anandd ooff logilogicc iinn particularparticular.. PhilosophyPhilosophy,, abovabovee allall,, iiss concerneconcernedd witwithh ththee pursuipursuitt ooff truthtruth;; anandd logilogicc —- ititss handmaidehandmaidenn —- witwithh discoverindiscoveringg neneww truthtruthss onconcee establisheestablishedd oneoness araree withiwithinn ouourr graspgrasp.. TToo bbee suresure,, philosopherphilosopherss anandd logicianlogicianss alikalikee araree concerneconcernedd ttoo avoid inconsistency (sinc(sincee ththee inconsistencinconsistencyy ooff twtwoo propositionpropositionss iiss a sufficientsufficient conditioncondition ofof thethe falsity ofof atat leastleast oneone ofof them)them) andand thusthus toto preserve consistency (since(sincethe the consistencconsistencyy ooff twtwoo propositionpropositionss iiss a necessarnecessaryy —- bubutt nonott sufficiensufficientt —- conditioconditionn ooff ththee truth ooff both)both).. BuButt iitt iiss iinn tracing implications thathatt thetheyy mosmostt obviouslobviouslyy advancadvancee theitheirr commocommonn concerconcernn witwithh ththee discoverdiscoveryy ooff neneww truthtruthss oonn ththee basibasiss ooff oneoness alreadalreadyy establishedestablished.. FoForr implicatioimplicationn iiss ththee relatiorelationn whicwhichh holdholdss betweebetweenn aann ordereorderedd paipairr ooff propositionpropositionss whewhenn ththee firsfirstt cannocannott bbee trutruee withouwithoutt ththee seconsecondd alsalsoo beinbeingg truetrue,, i.e.i.e.,, whewhenn ththee truttruthh ooff ththee firstfirst iiss a sufficientsufficient conditioconditionn ooff ththee truttruthh ooff ththee secondsecond.. LikLikee ththee relationrelationss ooff inconsistencinconsistencyy andand consistency,consistency, ththee relatiorelationn ofof implicatioimplicationn cancan bbee definedefinedd iinn termtermss ooff ouourr taltalkk ooff possiblpossiblee worldsworlds.. HerHeree araree threthreee equivalenequivalentt waywayss ooff soso definindefiningg itit::

(a(a)) a propositiopropositionn P implieimpliess a propositiopropositionn Q iiff anandd onlonlyy iiff Q iiss trutruee iinn alalll thosthosee possiblpossiblee worldsworlds,, iiff anyany,, iinn whicwhichh P iiss truetrue;;

(b(b)) a propositiopropositionn P implieimpliess a propositiopropositionn Q iiff anandd onlonlyy iiff thertheree iiss nnoo possiblpossiblee worlworldd iinn whicwhichh P iiss trutruee anandd Q false;2ofalse;20

(c(c)) a propositiopropositionn P implieimpliess a propositiopropositionn Q iiff anandd onlonlyy iiff iinn eaceachh ooff allall possiblepossible worldsworlds ifif P isis truetrue thenthen Q isis alsoalso truetrue..

AsAs anan exampleexample ofof thethe relationrelation ofof implicationimplication considerconsider thethe relationrelation whichwhich thethe propositiopropositionn

(7.3)(1.3) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 hashas toto thethe propositiopropositionn

(7.27)(1.21) TheThe U.S.U.S. enteredentered WorldWorld WarWar I beforebefore 1920.1920.

WhateverWhatever otherother relationsrelations maymay holdhold betweenbetween (7.3)(1.3) andand (7.27),(1.21), plainlyplainly thethe relationrelation ofof implicatioimplicationn does.does. AllAll threethree ofof thethe aboveabove definitionsdefinitions areare satisfiedsatisfied inin thisthis case.case. Thus:Thus: [definition[definition (a)](a)] (7.27)(1.21) isis trutruee inin allall thosethose possiblepossible worldsworlds inin whichwhich (7.3)(1.3) isis true;true; [definition[definition (b)](b)] therethere isis nono possiblepossible worldworld inin whicwhichh (7.3)(1.3) isis truetrue andand (7.27)(1-21) isis false;false; andand [definition[definition (c)](c)] inin eacheach ofof allall possiblepossible worlds,worlds, ifif (7.3)(1.3) isis trutruee thenthen (7.2/)(1.21) isis truetrue.. ToTo saysay thatthat a propositionproposition Q follows from a propositionproposition P isis justjust toto saysay thatthat P implies Q.Q. HencHencee thethe relationrelation ofof followingfollowing from,from, likelike itsits converse,converse, cancan bebe explainedexplained inin termsterms ofof possiblepossible worlds.worlds. Indeed,Indeed, thethe explanationexplanation cancan bebe givengiven byby thethe simplesimple expedientexpedient ofof substitutingsubstituting thethe wordswords "a"a propositionproposition Q

20.20. DefinitionDefinition (b),(b), itit shouldshould bebe noted,noted, amountsamounts toto sayingsaying thatthat P impliesimplies Q ifif andand onlyonly ifif thethe truthtruth ofof P iiss inconsistent withwith thethe falsityfalsity ofof Q.Q. Implication,Implication,in inshort, short, isis definabledefinable ininterms terms ofof inconsistency.inconsistency. 3232 POSSIBLEPOSSIBLE WORLDSWORLDS followsfollows fromfrom a propositiopropositionn P"P" forfor thethe wordswords "a"a propositiopropositionn P impliesimplies a propositiopropositionn Q"Q" inin eacheach ofof thethe definitionsdefinitions (a),(a), (b),(b), andand (c)(c) above.above. InIn termsterms ofof thethe relationrelation ofof implicationimplication (and(and hencehence inin termsterms ofof ourour talktalk ofof possiblpossiblee worlds)worlds) wewe cancan alsoalso throwthrow lightlight onon anotheranother importantimportant logicallogical concept:concept: thatthat ofof thethe deductive validity ofof anan inferenceinference oror argument.argument. True,True, wewe havehave notnot hithertohitherto hadhad occasionoccasion toto useuse thethe wordswords "deductively"deductively valid".valid". YetYet itit willwill bbee evidentevident toto anyoneanyone whowho hashas eveneven a superficialsuperficial understandinunderstandingg ofof thethe meaningsmeanings ofof thesethese wordswords thathatt much ofof ourour discussiondiscussion hashas consistedconsisted inin marshallingmarshalling deductivelydeductively validvalid argumentsarguments andand drawingdrawing deductivelydeductively validvalid inferences.2121 TimeTime andand againagain wewe havehave signaledsignaled thethe presencpresencee ofof argumentsarguments andand inferencesinferences bbyy meansmeans ofof suchsuch wordswords asas "hence","hence", "consequently","consequently", "therefore","therefore", andand "it"it followsfollows that";that"; andand implicitlyimplicitly wewe havehave beebeenn claimingclaiming thatthat thesethese argumentsarguments andand inferencesinferences areare deductivelydeductively valid.valid. ButBut whatwhat doesdoes itit meanmean toto saysay thatthat anan argumentargument oror inferenceinference isis deductivelydeductively valid?valid? AsAs a preliminarpreliminaryy itit maymay helphelp ifif wewe remindremind ourselvesourselves ofof somesome familiarfamiliar facts:facts: thatthat itit isis propositionpropositionss fromfrom whichwhich andand toto whicwhichh inferencesinferences areare drawn.drawn. Consider,Consider, then,then, thethe simplestsimplest sortsort ofof argumentargument (or(or correspondingcorresponding inferenceinference)) whichwhich featuresfeatures jusjustt oneone propositiopropositionn asas itsits premispremisee andand jusjustt oneone propositiopropositionn asas itsits conclusion;conclusion; andand letlet uuss designatedesignate thethe premispremisee "P""P" andand thethe conclusionconclusion "Q"."Q". ThenThen wewe cancan reformulatereformulate ourour questionquestion bbyy asking:asking: WhatWhat doesdoes itit meanmean toto saysay thatthat anan argumentargument oror inferenceinference fromfrom P toto Q isis deductivelydeductively valid?valid? ToTo saysay thatthat anan argumentargument oror inferenceinference fromfrom a propositiopropositionn P toto a propositiopropositionn Q isis deductivelydeductively validvalid iiss jusjustt toto saysay thatthat P impliesimplies Q,Q, oror (conversely)(conversely) thatthat Q followsfollows fromfrom P.22P.22 DeductiveDeductive validity,validity, then,then, whicwhichh isis a propertpropertyy ofof argumentsarguments oror inferences,inferences, cancan bebe explainedexplained inin termsterms ofof thethe modalmodal relationsrelations ofof implicationimplication andand followingfollowing from.from. And,And, likelike them,them, itit cancan bebe explainedexplained inin termsterms ofof possiblepossible worlds.worlds. InIn thisthis case,case, wewe needneed onlyonly adoptadopt thethe expedientexpedient ofof substitutingsubstituting thethe wordswords "an"an argumentargument oror inferenceinference fromfrom P toto Q isis deductivelydeductively valid"valid" forfor thethe wordswords "a"a propositionproposition P impliesimplies a propositionproposition Q"Q" inin eacheach ofof thethe definitionsdefinitions (a),(a), (b),(b), andand (c)(c) above.above. A casualcasual readingreading ofof ourour threethree definitionsdefinitions ofof "implication""implication" maymay suggestsuggest toto SOluesome thatthat onlyonly truetrue propositionpropositionss cancan havehave implications.implications. AfterAfter all,all, wewe defineddefined "implication","implication", inin (c)(c) forfor instance,instance, asas thethe relationrelation whichwhich holdsholds betweebetweenn a propositiopropositionn P andand a propositiopropositionn Q whenwhen inin allall possiblpossiblee worldsworlds ifif P iiss true thenthen Q isis alsoalso true.true. AndAnd wewe illustratedillustrated thethe relationrelation ofof implicationimplication bbyy citingciting a casecase wherewhere a trutruee propositionproposition,, viz.,viz., (1.3),(1-3), stoodstood inin thatthat relationrelation toto anotheranother truetrue propositionproposition,, viz.,viz., (1.21). DoesDoes thisthis meanmean thatthat falsefalse propositionpropositionss cannotcannot havhavee implications?implications? DoesDoes itit mean,mean, toto puputt thethe questionquestion inin otheotherr words,words, thatthat nothingnothing followsfollows fromfrom falsefalse propositionspropositions,, oror thatthat deductivelydeductively valivalidd argumentsarguments cannotcannot havhavee falsefalse premisespremises?? NoNott atat alLall. OnOn a moremore carefulcareful readingreading ofof thesethese definitionsdefinitions itit willwill bbee seenseen thatthat theythey saysay nothingnothing whateverwhatever aboutabout thethe actualactual truth-valuestruth-values ofof P oror Q;Q; i.e.,i.e., thatthat theythey saysay nothingnothing atat allall aboutabout whetherwhether P oror Q areare truetrue inin thethe actualactual world.world. HenceHence theythey dodo notnot rulerule outout thethe possibilitpossibilityy ofof a propositiopropositionn P implyingimplying a propositiopropositionn Q whenwhen P isis notnot truetrue bubutt false.false. InIn (c),(c), forfor instance,instance, wewe merelymerely saidsaid thathatt wherewhere P impliesimplies Q,Q, inin allall possiblpossiblee worldsworlds Q willwill bbee truetrue if P isis true.true. WeWe havehave notnot asserted thatthat P iiss truetrue inin thethe actualactual worldworld bubutt merelymerely entertainedentertained thethe supposition thatthat P isis truetrue inin somesome worldworld oror otherother;; andand thatthat isis somethingsomething wewe cancan dodo eveneven inin thethe casecase wherewhere P isis falsefalse inin thethe actualactual worldworld oror eveneven wherwheree P isis falsefalse inin allall possiblepossible worlds.worlds.

21.21. WhenWhen personpersonss drawdraw a conclusionconclusion outout ofof a propositiopropositionn oror a setset ofof propositionpropositionss theythey cancan bbee correctlycorrectly saidsaid toto bbee "inferring"inferring a proposition"proposition".. InferringInferring isis somethingsomething personpersonss do; itit isis not a logicallogical relationrelation betweebetweenn propositionspropositions.. ItIt isis notnot onlyonly grammaticallygrammatically incorrectincorrect toto speakspeak ofof oneone propositiopropositionn inferringinferring another,another, itit isis logicallylogically confusedconfused asas well.well. SeeSee H.W.H.W. Fowler,Fowler, A Dictionary of Modern English Usage, 2nd Edition, revisedrevised bbyy SirSir ErnestErnest Gowers,Gowers, Oxford,Oxford, ClarendonClarendon Press,Press, 1965,1965, pp.. 282.282.

22.22. NotNotee thatthat herehere wewe areare definingdefining "deductive"deductive validity",validity", notnot "validity""validity" per se. Later,Later, inin chapterchapter 4,4, wewe shallshall definedefine thethe widerwider conceptconcept ofof validityvalidity.. § 4 RelationsRelations betweenbetween PropositionPropositionss 3333

Moreover,Moreover, wewe mightmight jusjustt asas easilyeasily havehave chosenchosen toto illustrateillustrate thethe relationrelation ofof implicationimplication bbyy citingciting aa casecase wherewhere a falsefalse propositiopropositionn impliesimplies anotheranother propositionproposition.. ConsiderConsider suchsuch a case.case. TheThe propositionproposition

(7.79)(1.19) TheThe u.s.U.S. enteredentered WorldWorld WarWar I inin 19141914

isis contingentcontingent andand happenshappens toto bbee false;false; itit isis falsefalse inin thethe actualactual worldworld eveneven thoughthough itit isis truetrue inin atat leasleastt somesome non-actualnon-actual worlds.worlds. DoesDoes thisthis (actually)(actually) falsefalse propositiopropositionn havehave anyany implications?implications? (Equivalently(Equivalently:: DoDo anyany otherother propositionpropositionss followfollow fromfrom (1.19)? CanCan anyany otherother propositiopropositionn bbee inferredinferred withwith deductivdeductivee validityvalidity fromfrom (1.79)?)(1.19)?) ObviouslyObviously enough,enough, thethe answeranswer is:is: Yes.Yes. A falsefalse propositionproposition,, likelike a truetrue oneone,, willwill havehave countlesscountless implications.implications. ForFor instance,instance, thethe falsefalse propositiopropositionn (i.(1.19)19) impliesimplies allall thethe countlesscountless propositionpropositionss thatthat wewe couldcould expressexpress bbyy utteringuttering a sentencesentence ofof thethe formform "The"The U.S.U.S. entereenteredd WorldWorld WarWar I beforbeforee ..."" andand filling inin thethe blanblankk withwith thethe specificationspecification ofof anyany datedate whateverwhatever latelaterr thanthan 1914,1914, e.g.,e.g., 1915,1915, 1916,1916, 1917,1917, etc.,etc., etc.etc. TheThe crucialcrucial differencedifference betweebetweenn thethe implicationsimplications ofof a falsefalse propositiopropositionn andand thethe implicationsimplications ofof a truetrue propositiopropositionn lieslies inin thethe factfact thatthat onon thethe oneone hand,hand, a falsefalse propositiopropositionn hashas implicationsimplications somesome ofof whichwhich areare falsefalse -— asas isis thethe propositiopropositionn thatthat thethe U.S.U.S. entereenteredd WorldWorld WarWar I beforbeforee 19151915 -— andand somesome ofof whichwhich areare truetrue -— asas isis thethe propositiopropositionn (1.21) thatthat ththee U.S.U.S. enteredentered WorldWorld WarWar I beforbeforee 19201920 -— while,while, onon thethe otherother hand,hand, a truetrue propositiopropositionn hahass implicationsimplications allall ofof whichwhich areare truetrue.. Here,Here, then,then, areare twotwo importantimportant logicallogical factsfacts aboutabout thethe relationrelation ofof implication.implication, (i)(i) AllAll ththee implicationsimplications ofof a truetrue propositionproposition havehave thethe samesame truth-valuetruth-value asas thatthat proposition,proposition, i.e.,i.e., theythey 'preserve'preserve'' itsits truth.truth. ForFor thisthis reasonreason implicationimplication isis saidsaid toto bbee a truth-preserving relation.relation. InIn tracingtracing outout ththee implicationsimplications ofof a truetrue propositiopropositionn wewe cancan bbee ledled onlyonly toto furtherfurther truetrue propositionspropositions,, nevernever toto falsefalse onesones.. Or,Or, inin otherother words,words, thethe onlyonly propositionpropositionss thatthat followfollow fromfrom oror cancan bbee inferredinferred withwith deductivedeductive validitvalidityy fromfrom propositionpropositionss whichwhich areare truetrue areare propositionpropositionss whichwhich areare alsoalso true.true, (ii)(ii) TheThe implicationsimplications ofof a falsefalse propositiopropositionn needneed notnot havehave thethe samesame truth-valuetruth-value asas thatthat propositionproposition.. SomeSome ofof thethe implicationsimplications ofof a falsefalse propositiopropositionn areare themselvesthemselves false;false; bubutt othersothers areare true.true. Implication,Implication, wewe maymay say,say, isis notnot falsity-preserving.falsity-preserving. AmongAmong thethe propositionpropositionss whichwhich followfollow fromfrom oror cancan bebe inferredinferred withwith deductivdeductivee validityvalidity fromfrom propositionspropositions whichwhich areare false,false, therethere areare somesome truetrue propositionspropositions asas wellwell asas somesome falsefalse ones.ones. TheseThese simplesimple logicallogical factsfacts havehave importantimportant practicalpractical andand methodologicalmethodological applicationsapplications whenwhen itit comecomess toto thethe pursuipursuitt ofof truth.truth. ByBy virtuevirtue ofof (i),(i), itit followsfollows thatthat oneone ofof thethe waysways inin whichwhich wewe cancan advanceadvance ththee frontiersfrontiers ofof humanhuman knowledgeknowledge isis simplysimply toto reflectreflect uponupon,, oror reasonreason out,out, thethe implicationsimplications ooff propositionpropositionss wewe alreadyalready knowknow toto bbee true.true. ThisThis is,is, paradigmaticallyparadigmatically,, thethe wayway inin whichwhich advancesadvances araree mademade inin mathematicsmathematics andand logic.logic. ButBut itit isis alsoalso thethe wayway inin whichwhich unrecognizeunrecognizedd truthstruths cancan bbee discovereddiscovered inin otherother areasartas asas well.well. ManyMany ofof thethe advancesadvances mademade inin technologytechnology andand thethe appliedapplied sciencessciences,, forfor instance,instance, occuroccur becausbecausee someonesomeone hashas reasonedreasoned outout forfor particulaparticularr circumstancescircumstances thethe implicationsimplications ooff universauniversall propositionpropositionss alreadyalready acceptedaccepted asas truetrue inin thethe purpuree sciences.sciences. ByBy virtuevirtue ofof (ii),(ii), itit followsfollows thathatt wewe cancan alsoalso advanceadvance thethe frontiersfrontiers ofof humanhuman knowledge,knowledge, negativelynegatively asas itit were,were, bbyy testingtesting ththee implicationsimplications ofof hypotheseshypotheses ,whosewhose truth-valuestruth-values areare asas yetyet unknown,unknown, weedingweeding outout thethe falsefalse hypotheses,hypotheses, andand thusthus narrowingnarrowing downdown thethe rangerange ofof alternativesalternatives withinwithin whichwhich truthtruth maymay yetyet bebe found.found. AAnn exploratoryexploratory hypothesishypothesis isis puputt forwardforward andand thenthen testedtested bbyy seeingseeing whetherwhether itsits implicationsimplications 'hold'hold up'up' (as(as wewe say)say) IIIin thethe lightlight ofof experience.experience. OfOf course,course, ifif a hypothesishypothesis hashas implicationsimplications whichwhich experienceexperience showsshows toto bbee true,true, thisthis doesdoes notnot entitleentitle uuss toto concludeconclude thatthat thethe hypothesishypothesis itselfitself isis true.true. ForFor asas wewe havhavee seen,seen, therethere alwaysalways areare somesome implicationsimplications ofof a propositiopropositionn whichwhich areare truetrue eveneven whenwhen thethe propprop­• ositionosition itselfitself isis false.false. ButBut if,if, onon thethe otherother hand,hand, a hypothesishypothesis hashas anyany implicationsimplications whichwhich experienceexperience showsshows toto bbee false,false, thisthis doesdoes entitleentitle uuss toto concludeconclude thatthat thethe hypothesihypothesiss itselfitself isis false.false. ForFor aass wewe havehave seen,seen, therethere cancan bbee nono falsefalse implicationsimplications ofof a propositiopropositionn inin thethe casecase wherewhere thatthat propositiopropositionn itselfitself isis true.true. HenceHence ifif anyany ofof thethe implicationsimplications ofof a hypothesishypothesis turnturn outout toto bbee false,false, wewe maymay validlvalidlyy inferinfer thatthat thatthat hypothesishypothesis isis false.false. 3344 POSSIBLPOSSIBLEE WORLDSWORLDS

ThesThesee twtwoo factfactss abouaboutt ththee relatiorelationn ooff implicatioimplicationn araree reflectereflectedd iinn ththee standarstandardd methodologmethodologyy (o(orr 'logic'logic'' aass iitt iiss ofteoftenn calledcalled)) ooff scientifiscientificc enquiryenquiry.. ThThee coscostt ooff ignorinignoringg themthem,, whewhenn ononee iiss conductinconductingg scientifiscientificc researcresearchh oorr whewhenn ononee iiss pursuinpursuingg knowledgknowledgee iinn ananyy field whateverwhatever,, iiss thathatt ththee discoverdiscoveryy ooff truttruthh thethenn becomebecomess a completelcompletelyy haphazarhaphazardd mattermatter.. TwTwoo furthefurtherr importanimportantt logicalogicall factfactss abouaboutt ththee relatiorelationn ooff implicatioimplicationn deservdeservee noticnoticee anandd discussiondiscussion.. IItt followfollowss frofromm ththee definitiondefinitionss givegivenn ooff implicatioimplicationn thatthat:: (iii(iii)) a necessarilnecessarilyy falsfalsee propositiopropositionn implieimpliess ananyy an andd evereveryy propositionproposition;; anandd (iv(iv)) a necessarilnecessarilyy trutruee propositiopropositionn iiss implieimpliedd by ananyy anandd evereveryy propositiopropositionn whateverwhatever.. ConclusioConclusionn (iii(iii)) followfollowss frofromm ththee facfactt thatthat,, iiff a propositiopropositionn P iiss necessarilynecessarily falsfalsee thethenn thertheree iiss nnoo possiblpossiblee worlworldd iinn whicwhichh P iiss trutruee anandd a fortiorifortiori nnoo possiblpossiblee worlworldd sucsuchh thathatt iinn iitt botbothh P iiss trutruee anandd somsomee otheotherr propositiopropositionn Q iiss falsefalse;; ssoo thathatt [b[byy definitiodefinitionn (b)(b)]] P musmustt bebe saisaidd ttoo implimplyy QQ.. ConclusioConclusionn (iv(iv)) followfollowss frofromm ththee facfactt thathatt iiff a propositiopropositionn Q iiss necessarilnecessarilyy trutruee thethenn thertheree iiss nono possiblpossiblee worlworldd iinn whicwhichh Q iiss falsfalsee anandd a fortiorifortiori nnoo possiblpossiblee worlworldd sucsuchh thathatt iinn iitt botbothh Q iiss falsfalsee andand somsomee propositiopropositionn P iiss truetrue;; ssoo thathatt [agai[againn bbyy definitiodefinitionn (b)(b)]] Q musmustt bbee saisaidd ttoo bbee implieimpliedd bbyy PP.. ThesThesee conclusionsconclusions,, howeverhowever,, strikstrikee manmanyy personpersonss aass counterintuitivecounterintuitive.. SurelySurely,, iitt woulwouldd bbee saidsaid,, thethe necessarilnecessarilyy falsfalsee propositiopropositionn

(1.6) ThThee U.SU.S.. entereenteredd WorlWorldd WaWarr I iinn 19171917 anandd iitt iiss nonott ththee cascasee thathatt ththee U.SU.S.. entered World War I in 1917 entered World War I in 1917 does not imply the proposition does not imply the proposition (1.2) Canada is south of Mexico. (1.2) Canada is south of Mexico. And surely, it again would be said, the necessarily true proposition And surely, it again would be said, the necessarily true proposition (1.7) If some thing is red then it is colored (1.7) If some thing is red then it is colored is not implied by the proposition is not implied by the proposition (1.20) Lazarus Long was born in Kansas in 1912. (1.20) Lazarus Long was born in Kansas in 1912. ForFor thethe propositionspropositions inin thethe first pairpair havehave 'nothing'nothing toto dodo with'with' eacheach other;other; theythey areare notnot inin anyany senssensee aboutabout thethe samesame things;things; on~one isis 'irrelevant''irrelevant' toto thethe other.other. AndAnd thethe samesame wouldwould bebe saidsaid forfor thethe propositionspropositions inin thethe secondsecond pair.pair. TheseThese admittedlyadmittedly counterintuitivecounterintuitive results areare onesones toto whichwhich wewe devotedevote a goodgood dealdeal ofof discussiondiscussion iinn chapterchapter 4,4, sectionsection 6,6, pp.pp. 224-30.224-30. ForFor thethe present,present, jusjustt threethree briefbrief observationsobservations mustmust suffice.suffice. InIn thethe first place,place, (iii)(iii) andand (iv) oughtought notnot toto bebe viewedviewed solelysolely asas consequencesconsequences ofof somesome recentlyrecently developeddeveloped artificialartificial definitionsdefinitions ofof implication.implication. TheyThey areare consequences,consequences, rather,rather, ofof definitionsdefinitions whicwhichh philosophersphilosophers havehave longlong beenbeen disposeddisposed toto give;give; indeed,indeed, comparablecomparable definitionsdefinitions werewere adoptedadopted by,by, andand thethe consequencesconsequences recognizedrecognized by,by, manymany logicianslogicians inin medievalmedieval times.times. Moreover,Moreover, theythey areare immediateimmediate (even(even ifif notnot immediatelyimmediately obvious)obvious) consequencesconsequences ofof thethe definitionsdefinitions whichwhich mostmost ofof usus wouldwould naturallynaturally bebe inclinedinclined toto give:give: asas whenwhen wewe saysay thatthat oneone propositionproposition impliesimplies anotheranother ifif thethe latterlatter can'tcan't possibly bebe falsefalse ifif thethe formerformer isis true;true; oror again,again, asas whenwhen wewe saysay thatthat oneone propositionproposition impliesimplies anotheranother justjust whenwhen iiff thethe formerformer isis truetrue thenthen necessarily thethe latterlatter isis true.2323 OnceOnce wewe recognizerecognize thisthis wewe maymay bebe moremore readyready

23.23. ForFor discussiondiscussion ofof anan ambiguity,ambiguity, andand a possiblepossible philosophicalphilosophical confusion,confusion, lurkinglurking inin thesethese naturalnatural waysways ofof speaking,speaking, seesee chapterchapter 6,6, sectionsection 3.3. § 4 RelationsRelations betweenbetween PropositionPropositionss 3535 toto educateeducate ourour intuitionsintuitions toto thethe poinpointt ofof recognizingrecognizing (iii)(iii) andand (iv)(iv) asas thethe importantimportant logicallogical truthtruthss whichwhich theythey areare.. Secondly,Secondly, itit isis notnot hardhard toto understanunderstandd whywhy ourour uneducateduneducated intuitionsintuitions tendtend toto rebelrebel atat acceptinacceptingg (iii)(iii) andand (iv).(iv). ForFor thethe plaiplainn factfact ofof thethe mattermatter isis thatthat mostmost ofof thethe instancesinstances ofof implicationimplication whichwhich wwee areare likelylikely toto thinkthink ofof inin connectionconnection withwith thethe inferencesinferences wewe perforperformm inin dailydaily life,life, oror inin scientificscientific enquiry,enquiry, areare instancesinstances inin whichwhich thethe relationrelation ofof implicationimplication holdsholds betweebetweenn contingent propositionspropositions;; andand oneone contingentcontingent propositionproposition,, asas itit happens,happens, impliesimplies anotheranother onlyonly ifif therethere is a certaincertain measuremeasure ooff 'relevance''relevance' toto bbee foundfound betweebetweenn themthem -— onlyonly ifif theythey are,are, inin somesome sense,sense, 'about''about' thethe samesame things.things. NoNott surprisingly,surprisingly, then,then, wewe areare inclinedinclined toto indulgeindulge ourour all-too-commonall-too-common dispositiondisposition toto generalize -— ttoo suppose,suppose, thatthat is,is, thatthat allall casescases ofof implicationimplication mustmust bbee likelike thethe onesones withwith whichwhich wewe areare mostmost familiar.familiar. HadHad we,we, fromfrom thethe beginningbeginning,, attendedattended botbothh toto thethe consequencesconsequences ofof ourour definitionsdefinitions andand toto thethe factfact thathatt theythey allowallow ofof applicationapplication toto noncontingentnoncontingent propositionpropositionss asas wellwell asas contingentcontingent ones,ones, wewe mightmight neveneverr havehave comecome toto expectexpect thatthat allall casescases ofof implicationimplication wouldwould satisfysatisfy thethe allegedalleged relevancerelevance requiremenrequirementt when,when, inin thethe naturenature ofof thethe case,case, onlyonly somesome dodo.. Thirdly,Thirdly, inin chapterchapter 4,4, sectionsection 6,6, pppp.. 224-30,224-30, wewe prespresss thethe casecase furtherfurther forfor acceptanceacceptance ofof (iii)(iii) andand (iv)(iv) bbyy showing,showing, amongamong otherother things,things, thatthat thosethose whowho areare disposeddisposed toto rejectreject themthem areare likelylikely toto havehave otheotherr competing andand eveneven moremore compellingcompelling intuitionsintuitions onon thethe basibasiss ofof whichwhich theythey willwill bbee stronglystrongly disposeddisposed,, asas wellwell asas logicallylogically obliged,obliged, toto acceptaccept (iii)(iii) andand (iv).(iv). ButBut thethe detaileddetailed argumentargument onon thatthat cancan waitwait..

EXERCISES

1. Explain the difference between assertingasserting (1)(1) thatthatQ Qis isa afalse false implicationimplication ofofP, P, andandasserting asserting (2)(2) that it is false that P implies Q. 2. Give an example of two propositionsproplJsitions such that the latter is a false implication of the former. 3. Give an example of two propositions such that it is false that the former implies the latter.latter.

****** * *** Equivalence OnceOnce wewe havehave thethe conceptconcept ofof implicationimplication inin handhand itit isis easyeasy toto givegive anan accountaccount ofof thethe modalmodal relationrelation ooff equivalence.equivalence. ToTo saysay thatthat a propositiopropositionn P isis equivalentequivalent toto a propositiopropositionn Q isis jusjustt toto saysay thatthat thetheyy implyimply oneone another,another, i.e.,i.e., thatthat notnot onlyonly doesdoes PimplyP imply Q bubutt alsoalso Q impliesimplies P,P, i.e.,i.e., thatthat thethe relationrelation ooff mutual implicationimplication holdsholds betweebetweenn P andand Q.Q. NoNoww thethe relationrelation ofof implication,implication, asas wewe havehave alreadyalready seen,seen, cancan itselfitself bbee defineddefined inin termsterms ofof possiblepossible worlds.worlds. ItIt followsfollows thatthat thethe relationrelation ofof equivalenceequivalence isis likewiselikewise definabledefinable.. ConsiderConsider onceonce moremore howhow wewe defineddefined "implication"."implication". AnyAny ofof thethe definitions,definitions, (a),(a), (b),(b), oror (c),(c), willwill dodo.. LetLet uuss chousechoose (a).(a). ThereThere wewe saidsaid thatthat a propositiopropositionn P impliesimplies a propositiopropositionn Q ifif andand onlyonly ifif Q iiss truetrue inin allall thosethose possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich P isis true.true. Suppose,Suppose, now,now, thatthat P andand Q araree equivalent,equivalent, i.e.,i.e., thatthat notnot onlyonly doesdoes PimplyP imply Q bubutt alsoalso Q impliesimplies P.P. ThenThen notnot onlyonly willwill Q bbee truetrue iinn allall thosethose possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich P isis true,true, bubutt alsoalso thethe converseconverse willwill hold,hold, i.e.,i.e., P willwill bbee truetrue inin allall thosethose possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich Q isis true.true. ItIt followsfollows thatthat wherewhere twotwo propositionpropositionss areare equivalent,equivalent, ifif therethere areare anyany possiblpossiblee worldsworlds inin whichwhich onl(one^ ofof themthem isis true,true, thenthen inin exactlyexactly ththee samesame worldsworlds thethe otherother isis alsoalso true.true. MoreMore brieflybriefly,, twotwo propositionpropositionss areare equivalentequivalent ifif andand onlyonly ifif thetheyy havehave thethe samesame truth-valuetruth-value inin preciselpreciselyy thethe samesame setssets ofof possiblpossiblee worlds,worlds, i.e.,i.e., therethere areare nono possiblepossible worldsworlds inin whichwhich theythey differdiffer inin truth-value.truth-value. 3366 POSSIBLPOSSIBLEE WORLDSWORLDS

AAss aann examplexamplee ooff ththee relatiorelationn ooff equivalencequivalencee consideconsiderr ththee relatiorelationn whicwhichh ththee contingencontingentt propositiopropositionn

(1.2)(7.2) CanadCanadaa iiss soutsouthh ooff MexicMexicoo hahass ttoo ththee contingencontingentt propositiopropositionn

(1.22) MexicMexicoo iiss nortnorthh ooff CanadaCanada..

EveEvenn iiff wwee werweree merelmerelyy ttoo relrelyy oonn ouourr untutoreuntutoredd intuitionintuitionss mosmostt ooff uuss woulwouldd find iitt naturanaturall ttoo sasayy thathatt thesthesee twtwoo propositionpropositionss araree equivalentequivalent.. BuButt nonoww wwee cacann explaiexplainn whywhy.. WWee cacann poinpointt ououtt thathatt nonott onlonlyy doedoess (1.2)(7.2) implimplyy (1.22),(7.22), bubutt alsalsoo (1.22)(7.22) implieimpliess (1.2).(7.2). OrOr,, gettingettingg a littllittlee mormoree sophisticated,sophisticated, wwee cacann poinpointt ououtt thathatt iinn ananyy possiblpossiblee worlworldd iinn whicwhichh ononee iiss trutruee ththee otheotherr iiss alsalsoo trutruee anandd thathatt iinn ananyy possiblpossiblee worlworldd iinn whicwhichh ononee iiss falsfalsee ththee otheotherr iiss falsefalse.. IItt mattermatterss nonott aatt alalll thathatt botbothh propositionpropositionss happehappenn ttoo bbee falsfalsee iinn ththee actuaactuall worldworld.. AAss wwee havhavee alreadalreadyy seenseen,, falsfalsee propositionpropositionss aass welwelll aass trutruee oneoness cacann (an(andd dodo)) havhavee implicationsimplications.. AnAndd aass wwee cacann nonoww seesee,, falsfalsee propositionpropositionss aass welwelll aass trutruee oneoness cacann bbee equivalenequivalentt ttoo ononee another.another. NoncontingenNoncontingentt propositionpropositionss aass welwelll aass contingencontingentt oneoness cacann standstand iinn relationrelationss ooff equivalencequivalencee ttoo oneone anotheranother.. IndeedIndeed,, iiff wwee attenattendd carefullcarefullyy ttoo ththee definitiodefinitionn wwee havhavee givegivenn forfor equivalenceequivalence iitt iiss easeasyy toto see:see: (i(i)) thathatt alalll noncontingentlnoncontingentlyy trutruee propositionpropositionss forformm whawhatt iiss callecalledd aann equivalence-classequivalence-class,, i.e.i.e.,, a classclass alalll ooff whoswhosee membermemberss araree equivalenequivalentt ttoo ononee another;another;2424 anandd (ii(ii)) thathatt alalll noncontingentlnoncontingentlyy falsefalse propositionpropositionss likewislikewisee forformm aann equivalence-classequivalence-class.. ConclusioConclusionn (i(i)) followfollowss frofromm ththee facfactt thathatt iiff a propositiopropositionn iiss necessarilnecessarilyy trutruee iitt iiss trutruee iinn alalll possiblpossiblee worldworldss anandd henchencee iiss trutruee iinn preciselpreciselyy ththee samsamee setset ooff possiblpossiblee worldworldss aass ananyy otheotherr necessarilnecessarilyy trutruee propositionproposition.. ConclusioConclusionn (ii(ii)) followfollowss frofromm ththee factfact thathatt iiff a propositiopropositionn iiss necessarilnecessarilyy falsfalsee iitt iiss falsfalsee iinn alalll possiblpossiblee worldworldss anandd henchencee iiss falsfalsee iinn preciselpreciselyy ththee samesame setset ooff possiblpossiblee worldworldss aass ananyy otheotherr necessarilnecessarilyy falsfalsee propositionproposition.. ThesThesee twtwoo conclusionconclusionss strikestrike manmanyy personpersonss aass counterintuitivecounterintuitive.. Surely,Surely, iitt woulwouldd bbee said,said, thertheree iiss a difference betweebetweenn ththee necessarilnecessarilyy trutruee propositiopropositionn

(7.5)(1.5) EitherEither thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 oror itit isis notnot thethe casecase thatthat thethe U.SU.S.. enteredentered WorldWorld WarWar I inin 19171917 andand thethe necessarilynecessarily truetrue propositiopropositionn

(1.23) EitherEither CanadaCanada isis southsouth ofof MexicoMexico oror itit isis notnot thethe casecase thatthat CanadaCanada isis southsouth ooff Mexico.Mexico.

AfterAfter all,all, thethe conceptsconcepts involvedinvolved areare notnot thethe same.same. OneOne ofof thesethese propositionspropositions makesmakes referencereference toto anan itemitem calledcalled "the"the U.S."U.S." andand toto anan eventevent thatthat occurredoccurred atat a specificspecific momentmoment inin time.time. TheThe otherother makesmakes referencereference toto twotwo veryvery differentdifferent itemsitems calledcalled "Canada""Canada" andand "Mexico""Mexico" andand toto thethe geographicalgeographical locatiolocationn ofof oneone withwith respectrespect toto thethe other.other. How,How, then,then, cancan thethe twotwo propositionspropositions bebe equivalent?equivalent? Likewise,Likewise, itit wouldwould bebe said,said, therethere isis a differencedifference -— a conceptualconceptual difference,difference, oneone mightmight saysay -— betweenbetween thethe necessarilynecessarily falsefalse propositiopropositionn

24.24. InIn thisthis bookbook wewe areare usingusing thethe termterm "equivalence-class""equivalence-class" asas a synonymsynonym forfor "a"a classclass ofof equivalentequivalent propositions".propositions". OnOn thisthis reading,reading, itit isis possiblepossible forfor a propositionproposition toto belongbelong toto severalseveral equivalence-classes.equivalence-classes. MoreMore standardly,standardly, however,however, thethe termterm "equivalence-class""equivalence-class" isis usedused inin suchsuch a wayway thatthat a propositionproposition cancan bebe a membermember ooff onlyonly oneone equivalence-class.equivalence-class. ThereThere shouldshould bebe littlelittle causecause forfor confusion.confusion. TheThe moremore usualusual conceptionconception ooff equivalence-classequivalence-class cancan simplysimply bebe regardedregarded asas thethe logicallogical unionunion ofof allall thethe equivalence-classesequivalence-classes (as(as herehere defined)defined) ooff a propositionproposition.. § 4 RelationsRelations betweenbetween PropositionsPropositions 3737

(1.6) TheThe U.S.U.S. enteredentered WorldWorld WarWar I IIIin 19171917 andand itit isis notnot thethe casecase thatthat thethe U.SU.S.. enteredentered WorldWorld WarWar I inin 19171917

andand thethe necessarilynecessarily falsefalse propositiopropositionn

(1.24) CanadaCanada isis southsouth ofof MexicoMexico andand itit ISis notnot thethe casecase thatthat CanadaCanada ISis southsouth ooff MexicoMexico

whichwhich preventpreventss us,us, inin anyany ordinaryordinary sensesense ofof thethe word,word, fromfrom sayingsaying thatthat theythey areare "equivalent"."equivalent". TheThe samesame probleproblemm arisesarises inin connectionconnection withwith contingentcontingent propositionspropositions.. LetLet usus seesee how.how. Consider,Consider, forfor a start,start, thethe factfact thatthat anyany propositiopropositionn whichwhich assertsasserts ofof twotwo otherother propositionpropositionss thatthat botbothh areare truetrue willwill bbee truetrue inin allall andand onlyonly thosethose possiblpossiblee worldsworlds inin whichwhich both areare true.true. AfterAfter all,all, iinn anyany possiblpossiblee world,world, ifif any,any, inin whichwhich oneone werewere truetrue andand thethe otherother false,false, thethe claimclaim thatthat botbothh ofof thethemm areare trutruee woulwouldd bbee false.false. Suppose,Suppose, nownow,, thathatt wwee wantwant toto assertassert ofof a contingentcontingent propositiopropositionn thatthat botbothh iitt andand a noncontingentlynoncontingently truetrue propositiopropositionn areare true.true. ThenThen thethe propositiopropositionn inin whichwhich wewe assertassert theirtheir jointjoint truthtruth willwill bbee truetrue inin allall andand onlyonly thosethose possiblpossiblee worldsworlds inin whichwhich botbothh areare truetrue together.together. ButBut theythey willwill bbee truetrue togethertogether onlyonly inin thosethose possiblpossiblee worldsworlds inin whichwhich thethe contingentcontingent propositiopropositionn isis true.true. HenceHence thethe propositiopropositionn whichwhich assertsasserts thethe joinjointt truthtruth ofof twotwo propositionspropositions,, oneone ofof whichwhich isis contingentcontingent andand thethe otherother ofof whichwhich isis necessarilynecessarily true,true, willwill bbee truetrue onlyonly inin thosethose possiblpossiblee worldsworlds inin whichwhich thethe contingentcontingent propositiopropositionn isis true.true. ButBut thisthis meansmeans thatthat anyany propositiopropositionn whichwhich assertsasserts thethe joinjointt truthtruth ofof twotwo propositionpropositionss oneone ofof whichwhich isis contingentcontingent andand thethe otherother ofof whichwhich isis necessarilynecessarily truetrue willwill itselfitself bbee contingentcontingent andand equivalentequivalent toto thethe contingentcontingent propositionproposition.. ForFor example,example, supposesuppose thatthat wewe havehave a propositiopropositionn whichwhich assertsasserts botbothh thatthat a contingentcontingent propositionproposition,, letlet uuss saysay

(1.2) CanadaCanada isis southsouth ofof MexicMexicoo

andand thatthat a necessarilynecessarily truetrue propositionproposition,, letlet uuss saysay

(1.5) EitherEither thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 oror itit isis notnot thethe casecase thatthat thethe U.SU.S.. enteredentered WorldWorld WarWar I inin 19171917

areare true.true. ThisThis willwill bbee thethe propositiopropositionn

(1.25) CanadaCanada isis southsouth ofof MexicoMexico andand eithereither thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 19171917 oror itit isis notnot thethe casecase thatthat thethe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917.1917.

ThenThen itit followsfollows fromfrom whatwhat wewe havehave saidsaid thatthat (1.25) isis truetrue inin allall andand onlyonly thosethose possiblpossiblee worldsworlds iinn whichwhich itit isis truetrue thatthat CanadaCanada isis southsouth ofof Mexico,Mexico, i.e.,i.e., inin whichwhich (1.2) isis true.true. ButBut thisthis meansmeans nonott onlyonly thatthat (1.25) isis contingentcontingent bubutt alsoalso thatthat itit isis truetrue inin preciselpreciselyy thethe samesame setset ofof possiblpossiblee worldsworlds asas (1.2), andand hencehence thatthat (1.2) andand (1.25)(1-25) formform anan equivalence-class,equivalence-class, i.e.,i.e., areare equivalent.equivalent. ButBut areare (1.2) andand (1.25) identical? OurOur intuitionsintuitions areare likelylikely toto rebelrebel atat thethe veryvery suggestion.suggestion. AnAndd thisthis isis forfor thethe veryvery samesame sortssorts ofof reasonsreasons whichwhich wouldwould leadlead themthem toto reberebell atat thethe suggestionsuggestion thatthat thethe necessarilnecessarilyy truetrue propositionpropositionss (1.5) andand (1.23) areare identicalidentical.. PerhapsPerhaps thethe first poinpointt thatthat needneedss toto bbee mademade inin replyreply toto thesethese objectionsobjections isis thatthat thethe sensesense inin whicwhichh wewe areare sayingsaying thatthat twotwo contingentcontingent propositionpropositionss may bbee equivalent,equivalent, andand thatthat anyany twotwo necessarilynecessarily trutruee propositionpropositionss are equivalent,equivalent, andand thatthat anyany twotwo necessarilynecessarily falsefalse propositionpropositionss are equivalentequivalent -— isis simplysimply thatthat whichwhich isis conveyedconveyed inin ourour definition,definition, viz.,viz., thatthat membersmembers ofof eacheach ofof thesethese setssets ooff propositionpropositionss havehave thethe samesame truth-valuetruth-value inin thethe samesame setset ofof possiblepossible worlds.worlds. WeWe areare not claimingclaiming thatthat equivalentequivalent propositionpropositionss areare identicalidentical withwith oneone another.another. ToTo bbee sure,sure, therethere areare usesuses ofof thethe tertermm 3838 POSSIBLEPOSSIBLE WORLDSWORLDS

"equivalent""equivalent" inin ordinaryordinary discoursediscourse whichwhich fosterfoster thethe ideaidea thatthat "equivalent""equivalent" isis a precisprecisee synonymsynonym forfor "identical"."identical". ForFor instance,instance, someonesomeone whowho sayssays thatthat a temperaturetemperature ofof zerozero degreesdegrees CelsiusCelsius isis equivalentequivalent toto a temperaturetemperature ofof thirty-twothirty-two degreesdegrees FahrenheitFahrenheit mightmight jusjustt asas wellwell claimclaim thatthat thethe temperaturetemperature asas measuredmeasured onon oneone scalescale isis thethe samesame as,as, oror isis identicalidentical with,with, thethe temperaturetemperature asas measuredmeasured onon thethe otherother scale.scale. ButBut thethe claimclaim thatthat twotwo propositionpropositionss areare equivalentequivalent isis notnot toto bbee construedconstrued inin thisthis way.way. TwTwoo propositionpropositionss cancan havehave identicalidentical truth-valuestruth-values inin identicalidentical setssets ofof possiblpossiblee worldsworlds withoutwithout themselvesthemselves beinbeingg identical.identical. TheyThey cancan bbee identicalidentical inin these respectsrespects withoutwithout beingbeing identicalidenticalin inall all respects.respects. ThatThat isis toto say,say, theythey cancan bbee equivalentequivalent withoutwithout beinbeingg oneone andand thethe samesame propositionproposition.. DrawDraw a distinctiodistinctionn betweebetweenn equivalenceequivalence andand identity,identity, andand conclusionsconclusions (i)(i) andand (ii)(ii) nono longerlonger willwill seemseem counterintuitivecounterintuitive.. Similarly,Similarly, wewe shallshall thenthen bbee able,able, withwith consistency,consistency, toto saysay thatthat thethe twotwo equivalentequivalent contingentcontingent propositionpropositionss (1.2) andand (1.25)(1-25) areare likewiselikewise non-identical.non-identical. WhatWhat would bbee counterintuitivecounterintuitive wouldwould bebe thethe claimsclaims thatthat allall necessarilynecessarily truetrue propositionpropositionss areare identicalidentical withwith oneone another,another, thatthat allall necessarilynecessarily falsefalse propositionpropositionss areare identicalidentical withwith oneone another,another, andand thatthat allall equivalentequivalent contingentcontingent propositionpropositionss areare identicalidentical withwith oneone another.another. ForFor thenthen wewe shouldshould havehave toto concludeconclude thatthat therethere areare onlyonly twotwo nonnoncontingencontingentt propositionpropositionss -— a singlesingle necessarilynecessarily truetrue oneone andand a singlesingle necessarilynecessarily falsefalse one;one; andand thatthat allall equivalentequivalent contingentcontingent propositionspropositions areare identicalidentical.. ButBut preciselyprecisely howhow isis thethe distinctiondistinction betweebetweenn propositionapropositionall equivalenceequivalence andand propositionapropositionall identityidentity toto bbee drawn?drawn? MoreMore particularlyparticularly,, sincesince wewe havehave alreadyalready saidsaid whatwhat itit isis forfor twotwo propositionpropositionss toto bebe equivalent,equivalent, cancan wewe givegive anan accountaccount ofof propositionapropositionall identityidentity whichwhich willwill enableenable usus toto saysay thathatt propositionpropositionss maymay bbee equivalentequivalent bubutt non-identicalnon-identical?? ItIt isis worthworth noting,noting, forfor a start,start, thatthat discussionsdiscussions ofof identityidentity -— whetherwhether ofof thethe identityidentity ofof prClpositionspropositions oror ofof peoplepeople,, ofof shipsships oror ofof sealingsealing waxwax -— areare allall tootoo oftenoften bedevilebedeviledd bbyy difficultiesdifficulties eveneven inin posinposingg thethe probleproblemm coherently.coherently. WeWe cancan say,say, withoutwithout anyany sensesense ofof strain,strain, thatthat twotwo propositionpropositionss areare equivalent.equivalent. ButBut whatwhat wouldwould itit meanmean toto saysay thatthat twotwo propositionpropositionss areare identical?identical? IfIf theythey areare identicaidenticall howhow cancan theythey bbee two? Indeed,Indeed, howhow cancan wewe sensiblysensibly eveneven ususee thethe pluraplurall pronoupronounn "they""they" toto referrefer toto thatthat whichwhich wewe wantwant toto saysay isis one? One'sOne's headhead spins,spins, andand wewe seemseem toto bbee hedgedhedged inin betweenbetween inconsistencyinconsistency andand futility.futility. OneOne ofof thethe greatestgreatest philosopherphilosopherss ofof thethe twentiethtwentieth century,century, LudwigLudwig Wittgenstein,Wittgenstein, puputt itit aphoristicallyaphoristically::

RoughlyRoughly speaking,speaking, toto saysay ofof two thingsthings thatthat theythey areare identicalidentical isis nonsense,nonsense, whilewhile toto saysay ofof one thingthing thatthat itit isis identicalidentical withwith itselfitself isis toto saysay nothingnothing atat all.all.2525

OneOne wayway outout ofof thisthis incipientincipient dilemmadilemma lieslies inin thethe recognitionrecognition thatthat onon most,most, ifif notnot all,all, ofof thethe occasionsoccasions whenwhen wewe areare temptedtempted toto saysay thatthat twotwo thingsthings areare identical,identical, wewe couldcould equallyequally wellwell -— andand a lotlot moremore perspicuouslperspicuouslyy -— saysay thatthat twotwo linguisticlinguistic itemsitems symbolizesymbolize (refer(refer to,to, mean,mean, oror express)express) oneone andand thethe samesame thing.thing. InsteadInstead ofof sayingsaying -— withwith allall itsits attendantattendant awkwardnessawkwardness -— thatthat twotwo people,people, letlet usus saysay TullyTully andand Cicero,Cicero, areare identical,identical, wewe cancan saysay thatthat thethe namesnames "Tully""Tully" andand "Cicero""Cicero" referrefer toto oneone andand thethe samesame personperson.. InsteadInstead ofof sayingsaying thatthat twotwo propositionspropositions,, letlet uuss saysay thatthat VancouverVancouver isis northnorth ooff SeattleSeattle andand thatthat SeattleSeattle isis southsouth ofof Vancouver,Vancouver, areare identical,identical, wewe cancan saysay thatthat thethe sentencesentencess "Vancouver"Vancouver isis northnorth ofof Seattle"Seattle" andand "Seattle"Seattle isis southsouth ofof Vancouver"Vancouver" expressexpress oneone andand thethe samesame proposiproposi­• tion.2626 ThisThis essentially,essentially, isis thethe solutionsolution onceonce offered,offered, bubutt subsequentlysubsequently rejected,rejected, bbyy thethe greatgreat GermanGerman philosophephilosopherr andand mathematician,mathematician, GottlobGottlob Frege.Frege. AsAs hehe puputt it:it:

25.25. LudwigLudwig Wittgenstein,Wittgenstein, Tractatus Logico-Philosophicus, trans.trans. D.F.D.F. PearsPears andand B.F.B.F. McGuinness,McGuinness, London,London, RoutledgeRoutledge & KeganKegan Paul,Paul, 19611961 (original(original editionedition publishepublishedd inin GermanGerman underunder thethe titletitle Logisch-PhilosophicheLogisch-Philosophiche Abhandlung, 1921),1921), propositiopropositionn 5.53035.5303..

26.26. FurtherFurther reasonsreasons forfor adoptingadopting thethe distinctiondistinction betweebetweenn sentencessentences andand propositionpropositionss willwill bbee developeddeveloped atat lengthlength inin chapterchapter 2.2. § 44 RelationsRelations betweenbetween PropositionsPropositions 3399

WhatWhat isis intendedintended toto bebe saidsaid byby a = 6b seemsseems toto bbee thathatt ththee signsignss oorr namenamess 'a' anandd 'b''6' designatedesignate thethe samesame thing. 2727

AlthoughAlthough Frege'sFrege's suggestionsuggestion worksworks wellwell enoughenough whenwhen wwee wanwantt ttoo makmakee specifispecificc identity-claimsidentity-claims,, iitt doesdoes notnot enableenable usus toto avoidavoid thethe dilemmadilemma whenwhen wewe trytry ttoo formulatformulatee ththee conditionconditionss ooff identitidentityy foforr thingsthings quitequite generally,generally, thingsthings forfor whichwhich therethere areare nnoo linguisticlinguistic symbolsymbolss aass welwelll aass foforr thingthingss foforr whichwhich therethere are.are. AsAs itit stands,stands, Frege'sFrege's claimclaim suggestssuggests thatthat wwee shoulshouldd bbee ablablee ttoo sasayy somethinsomethingg alonalongg thesethese lines:lines: "Two"Two signssigns oror namesnames 'a''a' andand 'b''b' designatedesignate thethe samesame thinthingg iiff anandd onlonlyy ifif...... " (wher(wheree ththee blankblank isis toto bebe filled inin byby thethe specificationspecification ofof appropriateappropriate conditionsconditions ooff identity)identity).. BuButt iitt iiss jusjustt plaiplainn falsefalse thatthat toto makemake anan identity-claimidentity-claim is,is, inin general,general, toto assertassert thathatt twtwoo expressionexpressionss havhavee ththee samesame reference.reference. Frege'sFrege's reformulationreformulation worksworks wellwell enoughenough inin thethe casecase ooff itemitemss whicwhichh happehappenn ttoo havhavee beebeenn namednamed oror referredreferred toto byby someonesomeone oror otherother inin somesome languagelanguage oorr otherother.. BuButt araree thertheree nonott aatt leasleastt somsomee unnamedunnamed items,items, forfor whichwhich linguisticlinguistic symbolssymbols havehave notnot yeyett andand perhapperhapss neveneverr wilwilll bebe,, deviseddevised?? SurelSurelyy therethere mustmust bebe identity-conditionsidentity-conditions forfor thesethese itemsitems asas well.well. YeYett iiff thithiss iiss soso,, hohoww cacann wwee eveevenn beginbegin?? AAss wewe havehave alreadyalready seen,seen, wewe cancan hardlyhardly startstart offoff byby sayingsaying "Two"Two thingthingss araree identicaidenticall iiff anandd onlonlyy if..if .... " InIn orderorder toto givegive quitequite generalgeneral identity-conditionsidentity-conditions forfor anyany itemitemss whateverwhatever,, wwee woulwouldd ddoo welwelll toto,, aass itit were,were, 'turn'turn thethe problemproblem around'around' andand askask forfor thethe conditionsconditions ooff nonnon-identity.-identity. WWee woulwouldd ddoo wellwell,, thathatt is,is, toto ask,ask, "Under"Under whatwhat conditionsconditions shouldshould wewe feelfeel compelledcompelled ttoo sasayy thathatt thertheree araree two itemitemss ratheratherr thanthan justjust one?"one?" NotNot onlyonly is.is thisthis wayway ofof puttingputting thethe question.question-paradox-freeparadox-free,, bubutt iitt alsalsoo avoidavoidss ththee limitationslimitations implicitimplicit inin Frege'sFrege's formulationformulation.. TheThe answeranswer whichwhich commendscommends itselfitself toto mostmost thinkingthinking peoplepeople,, philosopherphilosopherss anandd nonphilosophernonphilosopherss alike,alike, isis essentiallyessentially thatthat whichwhich hashas comecome toto bebe knownknown asas Leibniz'Leibniz'ss PrinciplePrinciple.. LeibniLeibnizz puputt iitt thithiss wayway::

ThereThere areare nevernever inin naturenature twotwo beingsbeings whichwhich areare exactlyexactly alikalikee iinn whicwhichh iitt iiss nonott possiblpossiblee ttoo finfindd anan internalinternal difference...... 2288

InIn effect,effect, LeibnizLeibniz claimedclaimed thatthat itit isis impossibleimpossible forfor two itemsitems ttoo havhavee all theitheirr attributeattributess —- includinincludingg relationalrelational onesones -— inin common.common. PuttingPutting thethe pointpoint inin stillstill anotheranother wayway:. thertheree araree twtwoo itemitemss ratheratherr thathann oneone ifif andand onlyonly ifif oneone itemitem hashas atat leastleast oneone attributeattribute whichwhich ththee otheotherr doedoess notnot.. ArmedArmed withwith thisthis accountaccount ofof identity,identity, letlet usus returnreturn ttoo ththee tastaskk ooff distinguishindistinguishingg betweebetweenn propositionalpropositional equivalenceequivalence andand propositionalpropositional identity.identity. CanCan wwee givgivee aann accounaccountt ooff ththee conditionconditionss ooff propositionalpropositional identityidentity whichwhich willwill enableenable usus toto preservepreserve ourour intuitionintuitionss thathatt propositionpropositionss mamayy bbee equivalentequivalent andand yetyet non-identical?non-identical? ItIt seemsseems clearclear thatthat whatwhat guidesguides ourour intuitionsintuitions whenwhen wwee insisinsistt thathatt propositionspropositions,, contingencontingentt anandd noncontingentnoncontingent alike,alike, needneed notnot bebe identicalidentical eveneven whewhenn thetheyy araree membermemberss ooff ththee samesame equivalence-classesequivalence-classes isis somethingsomething likelike Leibniz'sLeibniz's Principle.Principle. WWee notnotee thathatt iinn eaceachh ooff ththee problematiproblematicc casescases preceding,preceding, oneone propositionproposition hashas atat leastleast oneone attributeattribute whicwhichh ththee otheotherr lackslacks,, anandd ssoo concludconcludee —­ inin accordanceaccordance withwith Leibniz'sLeibniz's PrinciplePrinciple -— thatthat thethe two,two, thougthoughh equivalentequivalent,, araree nonott identicalidentical.. LetLet usus callcall anyany attributeattribute whichwhich servesserves toto sortsort outout andand differentiatedifferentiate betweebetweenn twtwoo oorr mormoree itemitemss aa differentiating attribute. ThenThen wewe maymay saysay thatthat whatwhat guidesguides ouourr intuitionintuitionss aass ttoo ththee non-identitnon-identityy ooff

27.27. GottlobGottlob Frege,Frege, "On"On SenseSense andand Reference",Reference", inin Translations fromfrom the Philosophical Writings of Gottlob Frege, ed.ed. P.P. GeachGeach andand M.M. Black,Black, Oxford,Oxford, BasilBasil Blackwell,Blackwell, 1952,1952, pp.. 5656..

28.28. G.W.F.G.W.F. Leibniz,Leibniz, Monadology, trans.trans. R.R. Latta,Latta, London,London, OxfordOxford UniversitUniversityy PressPress,, 19651965,, SectioSectionn 99,, pp.. 222222.. ThisThis principleprinciple isis widelywidely knownknown asas thethe PrinciplePrinciple ofof IdentityIdentity ofof Indiscernibles.Indiscernibles. MorMoree aptlyaptly,, iitt mighmightt bbee callecalledd ththee PrinciplePrinciple ofof Non-IdentityNon-Identity ofof DiscerniblesDiscernibles.. 4040 POSSIBLEPOSSIBLE WORLDSWORLDS

thethe equivalentequivalent contingentcontingent propOSItIOnspropositions (7.2)(1.2) andand (7.25)(1.25) isis thethe factfact thatthat therethere isis atat leastleast oneone differentiatingdifferentiating attributeattribute whichwhich makesmakes themthem non-identical.non-identical. IndeedIndeed therethere areare several.several. TheThe attributeattribute ofof makingmaking referencereference toto thethe U.S.U.S. isis oneone ofof them:them: itit isis anan attributeattribute whichwhich (7.25)(1.25) hashas butbut (7.2)(1.2) doesdoes not.not. AndAnd therethere areare stillstill furtherfurther differentiatingdifferentiating attributesattributes which,which, inin thethe casecase ofof thesethese twotwo propositions,propositions, serveserve toto differentiatedifferentiate oneone fronifrom thethe other:other: (1.25)(1.25) makesmakes referencereference toto anan eventevent whichwhich (7.2)(1.2) doesdoes not;not; (7.25)(1.25) makesmakes referencereference toto a datedate whichwhich (7.2)(1.2) doesdoes not;not; andand soso on.on. InIn short,short, thethe itemsitems andand attributesattributes toto whichwhich oneone propositionproposition makesmakes referencereference areare notnot entirelyentirely thethe samesame asas thethe itemsitems andand attributesattributes toto whichwhich thethe otherother makesmakes reference.reference. HenceHence thethe propositionspropositions themselvesthemselves areare notnot thethe samesame butbut different.different. Similarly,Similarly, byby invokinginvoking Leibniz'sLeibniz's PrinciplePrinciple wewe maymay distinguishdistinguish thethe twotwo equivalentequivalent noncontingentnoncontingent propositionspropositions (1.5) andand (7.23):(1.23): (7.5)(1.5) refersrefers toto thethe U.S.,U.S., (1.23) doesdoes not;not; (7.23)(1.23) refersrefers toto CanadaCanada,, (7.5)(1.5) doesdoes not;not; etc.etc. OnceOnce again,again, wewe maymay concludeconclude thatthat twotwo equivalentequivalent propositionspropositions areare notnot identical.identical. ToTo sumsum up,up, equivalentequivalent propositionspropositions cannotcannot differdiffer fromfrom oneone anotheranother inin respectrespect ofof thethe attributeattribute ofof havinghaving thethe samesame truth-valuetruth-value inin thethe samesame setssets ofof possiblepossible worlds.worlds. ButBut theythey cancan differdiffer fromfrom oneone anotheranother inin respectrespect ofof otherother attributes.attributes. IdenticalIdentical propositions,propositions, byby wayway ofof comparison,comparison, cannotcannot differdiffer fromfrom oneone anotheranother inin respectrespect ofof thisthis oror anyany otherother attribute.attribute. TheyThey havehave allall ofof theirtheir attributesattributes inin common.common. InIn chapterchapter 2,2, sectionsection 2,2, wewe returnreturn toto thethe problemproblem ofof drawingdrawing a lineline betweenbetween propositionapropositionall equivalenceequivalence andand propositionalpropositional identityidentity andand comecome upup withwith a moremore preciseprecise statementstatement (in(in sectionsection 3)3) ofof thethe conditionsconditions forfor propositionalpropositional identityidentity..

EXERCISES

7.1. Which propositions (a. - e.) are inconsistentinconsistent with which propositionspropos1twns (i. - v.)? WhichWhich propositions (a. - e.) are consistentconsistent with which propositions (i. - v.)? Which propositionspropositions (a. - e.) implyimply which propositions (i. - v.)? And which propositions (a. - e.) are equivalenequivalentt to which propositions (i. - v.)?v.) ?

a. There are 8,098,789,243 stars. i1. All triangles have three sides.

b. All squares have fourfour sides. ll.ii There are fewerfewer than 77,567,224,38917,561,224,389 stars. c. Some squares have six sides. lll. There are more than 8,098,789,242 stars d. There are 8,098,789,243 stars or it is not m and fewerfewer than 8,098,789,244 stars.stars. the case that there are 8,098,789,243 stars.stars. lV. There are 124,759,332,511724,759,332,511 stars.stars. e. The Us.U.S. entered World War I in 7977.1917. W vv. The US.U.S. entered World War I after 1912.7972.

(Partial answer: a. is inconsistent with iv.iv. c. is inconsistent with i., ii., Hi.,iii., iv., and v. a. is consistent with i., ii., Hi.,iii., and v.)v.)

2. a. Is proposition A, defined below, consistent or inconsistent with proposition B?B?

"A" = Bill is exactly 6' tall.tall. "B" = Bill is exactly &6' 2/12"tall.tatl. § 4 RelationsRelations betweenbetween PropositionsPropositions 4141

b. Is proposition C, defined below, consistentconsistentor orinconsistent inconsistentwith withproposition propositionD? D?

"C" = Someone isisexactly exactly6' 6'tall. tall. "D" =— Someone isisexactly exactly6' 6'2" 2"tall. tall.

c. Is proposition E, defined below, self-consistentself-consistentor orself-inconsistent? self-inconsistent?

"E" =— Someone isisexactly exactly6' 6'tall talland and6' &2" 2"tall. tall.

3. Explain why it is misleadingmisleading to say such things as:as:

"In the actual world, Canada's being north ojof Mexico ISis inconsistent with Mexico's being north ojof Canada";Canada"; or "In the world ojof TimeTime EnoughEnough forfor Love,Love, the proposition that LazarusLazarus jallsfalls ininlove lovewith with two of his daughters implies the proposition that LazarusLazarus isisa ajather. father.""

*****$ % $ $ %

Symbolization

OurOur repertoirerepertoire ofof symbolssymbols cancan nownow bebe extendedextended toto encompassencompass hotnot onlyonly thethe modalmodal propertiesproperties representedrepresented bbyy "0", "~","0", "\7","V", andand "/:;.","A", bubutt alsoalso thethe modalmodal relationsrelations ofof consistency,consistency, inconsistency,inconsistency, implication,implication, andand equivalence.equivalence. TheyThey areare standardlystandardly representedrepresented inin symbolssymbols asas follows:follows:

(1)(1) TheThe conceptconcept ofof consistency:consistency: "o"0""

(2)(2) TheThe conceptconcept ofof inconsistency:inconsistency: "4>".p""

(3)(3) TheThe conceptconcept ofof implication:implication: "_""—»" [called[called arrow]

(4)(4) TheThe conceptconcept ofof equivalence:equivalence: ''."<—•...... ''" [called[called double-arrowJ29double-arrow]29

EachEach ofof thesethese symbolssymbols maymay bebe writtenwritten betweenbetween symbolssymbols whichwhich standstand forfor propositionpropositionss toto yieldyield furtherfurther propositionapropositionall symbolssymbols (e.g.,(e.g., "PoQ","PoQ", "P_Q")."P—»Q"). EachEach ofof thesethese symbolssymbols maymay bebe defineddefined contextuallycontextually asas follows:follows:

0 "P"P o Q"Q" =dfdf "P"P isis consistentconsistent withwith QQ"" "P .p Q" "P"P isis inconsistentinconsistent withwith QQ"" "P * Q" =dfdf

"P_Q""P-Q" =ddff "P"P impliesimplies QQ"" "P~Q" df "P"P isis equivalentequivalent toto Q"Q"

29.29. TheseThese latterlatter twotwo symbolssymbols areare notnot toto bbee confusedconfused withwith thethe twotwo symbolssymbols H:::>""o" andand He"" = " withwith whichwhich somesome readersreaders may alreadyalready bbee familiar.familiar. TheThe twotwo symbolssymbols H:;>""D" (called(called hook or orhorseshoe) horseshoe)and anHe"d " =(called " (calletriple-bar)d triple-bar) willwill bbee introducedintroduced laterlater inin thisthis boobookk andand willwill therethere bbee usedused toto standstand forfor thethe relationsrelations ofof materialmaterial conditionalitconditionalityy andand materialmaterial biconditionalitbiconditionalityy respectively.respectively. AtAt thatthat timetime wewe shallshall taketake somesome painpainss toto argueargue thatthat thethe relationsrelations ooff materialmaterial conditionalityconditionality andand materialmaterial biconditionalitbiconditionalityy areare distinctlydistinctly differentdifferent fromfrom anyany relationsrelations whichwhich havehave beenbeen introducedintroduced inin thisthis firstfirst chapter;chapter; inin particularparticular,, thatthat theythey areare distinctdistinct fromfrom implicationimplication andand equivalence.equivalence. 4242 POSSIBLEPOSSIBLE WORLDSWORLDS

EXERCISE

Refer to question 1 in the preceding exercise. Let the letters "A" - "E" stand for propositions a. - e. and the letters ''I''"J" - "N" for the propositions i.i, - v. Re-do question 1 expressing all youryour answers in the symbolism justjust introduced.introduced,

(Partial answer: A $~ M C ~], /, CC~ K, K, CC~ L,C ~C M4> MandCand C ~ N N

A o0], /, A o0 K, A o0 L, A o0 N)

5.5. SETSSETS OFOF PROPOSITIONSPROPOSITIONS

TheThe truth-values,truth-values, modalmodal propertiesproperties,, andand thethe modalmodal relationsrelations whichwhich maymay bebe ascribedascribed toto individuaindividuall propositionpropositionss andand toto pairpairss ofof propositionspropositions,, may,may, withwith equalequal proprietypropriety,, bbee ascribedascribed toto sets ooff propositionpropositionss andand toto pairpairss ofof setssets ofof propositionspropositions..

Truth-values of proposition-sets A setset ofof propositionspropositions willwill bebe saidsaid toto bebe true ifif everyevery membermember ofof thatthat setset isis true.true. AndAnd a setset ooff propositionpropositionss willwill bbee saidsaid toto bbee false ifif nonott everyevery membememberr ofof thathatt setset isis true,true, i.e.,i.e., ifif atat leastleast oneone membermember ofof thatthat setset isis false.false. NotNotee carefully:carefully: a setset ofof propositionpropositionss maymay bbee falsefalse eveneven thoughthough notnot evereveryy membermember ofof thatthat setset isis false.false. A singlesingle falsefalse membermember inin a setset ofof propositionpropositionss isis sufficientsufficient toto renderrender ththee setset false.false. AndAnd ofof coursecourse itit followsfollows fromfrom thisthis thatthat ifif a setset isis false,false, wewe areare notnot entitledentitled toto inferinfer ofof ananyy particulaparticularr membermember ofof thatthat setset thatthat itit isis false;false; wewe areare entitledentitled toto inferinfer onlyonly thatthat atat leastleast oneone membermember iiss false.false.

Example 1: A true set of propositions (1.26) {Snow{Snow isis white,white, TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 1917}.3o1917}.3°

Example 2: A false set of propositions (1.27) {Snow{Snow isis white,white, TheThe U.S.U.S. enteredentered WorldWorld WarWar II inin 1914}.1914}.

ItIt shouldshould bbee clearclear thatthat thethe twotwo expressionsexpressions (1)(1) "a"a setset ofof falsefalse propositionspropositions"" andand (2)(2) "a"a falsefalse setset ooff propositions"propositions",, dodo not meanmean thethe samesame thing.thing. A setset ofof falsefalse propositionpropositionss isis a setset all ofof whosewhose membermemberss areare false;false; a falsefalse setset ofof propositionpropositionss isis a setset at least oneoneof ofwhose whosemembers membersis isfalse. false.

Modal properties of proposition-sets A setset ofof propositionspropositions willwill bebe saidsaid toto bebe possibly true oror self-consistent ifif andand onlyonly ifif therethere existsexists aa possiblpossiblee worlworldd inin whicwhichh everyevery membermember ofof thatthat setset isis truetrue.. Self-consistencySelf-consistency isis a 'fragile''fragile' propertyproperty.. ItIt isis easilyeasily andand oftenoften unwittinglyunwittingly lostlost (see(see chapterchapter 6,6, sectionsection 7).7). ConsiderConsider thethe followingfollowing example:example:

30.30. HereHere andand subsequentlysubsequently inin thisthis boobookk wewe useuse a paipairr ofof bracebracess (i.e.,(i.e., "{""{" andand "}")"}") asas a meansmeans toto designatedesignate a set.set. § 5 SetsSets ofof PropositionPropositionss 4343

{April{April isis tallertaller thanthan Betty,Betty, BettyBetty isis tallertaller thanthan Carol,Carol, CarolCarol isis tallertaller thanthan Dawn,Dawn, DawnDawn isis tallertaller thanthan Edith,Edith, EdithEdith isis tallertaller thanthan Frances,Frances, FrancesFrances isis tallertaller thanthan April}April}..

NoticNoticee howhow everyevery subsetsubset consistingconsisting ofof any five ofof thesethese propositionpropositionss isis self-consistent.self-consistent. RemoveRemove thethe first,first, or thethe second,second, or thethe third,third, etc.,etc., andand thethe remaining setset isis self-consistentself-consistent (which(which ofof coursecourse isis not toto saysay thatthat itit isis true).true). ButBut inin reintroducingreintroducing thethe removedremoved propositiopropositionn andand consequentlyconsequently enlargingenlarging thethe setset toto whatwhat itit was,was, self-consistencyself-consistency isis lost.lost. AndAnd onceonce self-consistencyself-consistency isis lost,lost, inin thisthis setset asas inin anyany other,other, itit cancan nevernever bbee regainedregained bbyy adding moremore propositionspropositions.. SomeSome personpersonss thinkthink thatthat self-consistencyself-consistency cancan bebe restoredrestored bbyy insertinginserting a propositiopropositionn ofof thethe followingfollowing kindkind intointo a self-inconsistentself-inconsistent set:set: TheThe immediatelimmediatelyy precedinprecedingg propositiopropositionn isis false.false. ButBut thisthis devicedevice cancan never restorerestore self-consistency.self-consistency. (See(See thethe exercisesexercises onon pp.44.). 44.) A sesett ooff propositionpropositionss iiss possibly falsefalse iiff anandd onlonlyy iiff thertheree existexistss a possiblpossiblee worlworldd iinn whicwhichh aatt leastleast oneone membermember ofof thatthat setset isis false.false. A sesett ooff propositionpropositionss iiss necessarily true iiff anandd onlonlyy iiff evereveryy membememberr ooff thathatt sesett iiss necessarilnecessarilyy truetrue,, i.e.,i.e., ifif inin everyevery possiblpossiblee worldworld everyevery membermember ofof thatthat setset isis true.true. A sesett ooff propositionpropositionss iiss necessarily falsefalse oorr self-inconsistent iiff anandd onlonlyy iiff thertheree doedoess nonott exisexistt anyany possiblpossiblee worldworld inin whichwhich thatthat setset isis true,true, i.e.,i.e., ifif inin everyevery possiblpossiblee worldworld atat leastleast oneone propositiopropositionn oror anotheranother inin thatthat setset isis false.false. AndAnd finally,finally, a setset ofof propositionpropositionss isis contingent ifif andand onlyonly ifif thatthat setset isis neitherneither necessarilynecessarily truetrue nornor necessarilynecessarily false,false, i.e.,i.e., ifif andand onlyonly ifif therethere existsexists somesome possiblpossiblee worldworld inin whichwhich everyevery membermember ofof thatthat setset isis truetrue andand therethere existsexists somesome possiblpossiblee worldworld inin whichwhich atat leastleast oneone membermember ofof thatthat setset isis false.false.

EXERCISES

Part A

For each set below tell whether that set is (1) possibly true and/or possibly false; and (2) necessarilynecessarily true, necessarily false or contingent.contingent.

z.i. {Canada is north of Mexico, Hawaii is in the Pacific Ocean, Copper conducts electricity}electricity]

ii.zz. {Snow is white, Pine is a softwood, Coal is red}red]

lll.Hi. {There were exactly twelve tribes of Israel, There were exactly fourteen tribes of Israel}Israel]

w.iv. {All sisters are female, All triangles have three sides, All squares have four sides}sides]

v. {Some coffee cups are blue, Some coffee cups are green, Some coffee cups are yellow}yellow]

VI.vi. {Some triangular hats are blue, All triangles have three sides, Some squares have fivefive sides}sides]

vzz.vii. {All triangles have three sides, Some triangular hats are blue}blue]

VlZZ.viii. {Someone believes that today is Monday, Someone believes that today is Wednesday}Wednesday]

IX.ix. {Grass is green, Someone believes that grass is greengreen} ]

x. {All sisters are females, All females are sisters}sisters] 4444 POSSIBLEPOSSIBLE WORLDSWORLDS

Part B

1. Explain why self-consistency can never be restored to a self-inconsistent set of propositions by thethe device of inserting into that set a proposition of the sort: The immediately preceding proposition is false. 2. The example used above which reports the relative heights of April and Betty, etc., can be made self-consistent by the removal of the last proposition. What argument is to be used against thethe claim that it is the last proposition in the above set which 'induces' the self-inconsistency and hence is false?false? 3. A self-inconsistent set of three propositions of which every proper non-empty subset is self-consistent is called an antilogism.antilogism. An example would be:be:

{Lorna has three brothers,brothers, Sylvia has two brothers,brothers, Sylvia has twice as many brothers as Lorna}. Sylvia has twice as many brothers as Lorna ]. Find three examples ofantilogisms. Find three examples of antilogisms. 4. Find a set of three contingentcontingent propositions such that each pair of propositions drawn from that setset constitutes a self-inconsistent set. Example:Example: {Norman is shorter than Paul,Paul, Norman is the same height as Paul,Paul, Norman is taller than Paul}.Paul}. 5. Explain why one should notnot adopt the following definition of "necessary falsehood" for a set of propositions: A set of propositions is necessarily false if and only if every member of that set is necessarily false.false. ****** * ** * Modal relations between proposition-setsproposition-sets

TwoTwo setssets ofof propositionpropositionss willwill bbee saidsaid toto standstand inin thethe relationrelation ofof consistency ifif andand onlyonly ifif therethere existsexists somesome possiblpossiblee worldworld inin whichwhich allall thethe propositionpropositionss inin botbothh setssets areare jointljointlyy truetrue.. TwoTwo setssets ofof propositionpropositionss standstand inin thethe relationrelation ofof inconsistency ifif andand onlyonly ifif therethere doesdoes notnot existexist a possiblpossiblee worldworld inin whichwhich allall thethe propositionspropositions ofof bothboth setssets areare jointljointlyy truetrue.. OneOne setset ofof propositionpropositionss standsstands inin thethe relationrelation ofof implication toto anotheranother setset ofof propositionpropositionss ifif anandd onlyonly ifif allall thethe propositionpropositionss ofof thethe latterlatter setset areare truetrue inin everyevery possiblpossiblee world,world, ifif any,any, inin whichwhich allall thethe propositionpropositionss ofof thethe formerformer setset areare truetrue.. AndAnd twotwo setssets ofof propositionpropositionss standstand inin thethe relationrelation ofof equivalence ifif andand onlyonly ifif allall thethe propositionspropositions inin oneone setset areare truetrue inin allall andand jusjustt thosethose possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich allall thethe propositionpropositionss ofof thethe otherother setset areare truetrue.. ToTo illustrateillustrate thesethese definitions,definitions, wewe citecite thethe followingfollowing examples.examples.

Example 1:

TheThe setset ofof propositionspropositions

(1.28) {Ottawa{Ottawa isis thethe capitalcapital ofof Canada,Canada, AllAll menmen areare mortalmortal}}

isis consistent withwith thethe setset ofof propositionspropositions

(1.29) {Snow{Snow isis white,white, TodayToday isis Tuesday,Tuesday, SomeSome dogsdogs meow}meow}.. § 5 SetsSets ofof PropositionsPropositions 4545

Example 2:2:

TheThe setset ofof propositionpropositionss

(1.30) {April{April isis olderolder thanthan Betty,Betty, BettyBetty isis olderolder thanthan Carol,Carol, CarolCarol isis oldeolderr thanthan Diane}Diane}

isis inconsistent withwith thethe setset ofof propositionspropositions

(1.37)(1.31) {Diane{Diane isis olderolder thanthan Edith,Edith, EdithEdith isis olderolder thanthan April}April}..

Example 3:3:

TheThe setset ofof propositionpropositionss

(1.32) {Mary{Mary invitedinvited BrettBrett toto actact inin thethe playplay,, GreshamGresham invitedinvited SylviaSylvia toto actact iinn thethe playplay}}

implies thethe setset ofof propositionpropositionss

(1.33) {Sylvia{Sylvia waswas invitedinvited toto actact inin thethe play,play, MaryMary invitedinvited someonesomeone totoact act inin thethe play}play}..

Example 4:4:

TheThe setset ofof propositionspropositions

(1.34) {Today{Today isis WednesdayWednesday}}

isis equivalent toto thethe setset ofof propositionspropositions

(7.35)(1.35) {It{It isis laterlater inin thethe weekweek thanthan Tuesday,Tuesday, ItIt isis earlierearlier inin thethe weekweek thathann Thursday}.Thursday}.

AsAs wewe cancan see,see, some,some,although although notnotall, all, ofofthese thesesets setsof ofpropositions propositions containcontain moremore thanthan oneonemember, member, i.e.,i.e., moremore thanthan oneone propositionproposition.. DoesDoes thisthis meanmean thatthat thethe relationsrelations ofof consistency,consistency, inconsistency,inconsistency, etc.,etc., areare notnot alwaysalways dyadic,dyadic, oror two-placedtwo-placed relations?relations? NoNott atat all.all. ForFor a dyadicdyadic relationrelation isis a relationrelation whicwhichh holdsholds betweebetweenn twotwo items andand eacheach ofof thethe aboveabove setssets maymay bbee countedcounted asas a singlesingle itemitem eveneven ifif somesome ofof themthem havehave twotwo oror moremore members.members. HenceHence a relationrelation whichwhich holdsholds betweebetweenn twotwo sets ofof propositionpropositionss iiss stillstill a dyadicdyadic relationrelation eveneven ifif therethere isis moremore thanthan oneone propositiopropositionn inin eithereither oror botbothh sets.3131 InsofarInsofar asas modalmodal relationsrelations cancan obtainobtain betweebetweenn setssets ofof propositionpropositionss asas welwelll asas betweebetweenn singlesingle,, individualindividual propositionspropositions,, certaincertain consequencesconsequences followfollow whichwhich wewe wouldwould dodo wellwell toto exploreexplore..

31.31. NotNotee thatthat althoughalthough (7.35)(1.35) isis equivalent toto (7.34),(1.34), thethe setset (7.35)(1.35) doesdoes notnot itselfitself constituteconstitute aann equivalence-class,equivalence-class, i.e.,i.e., notnot allall itsits membersmembers areare equivalentequivalent toto oneone another.another. OneOne shouldshould bbee carefulcareful notnot toto supposesuppose thatthat thethe relationrelation ofof equivalenceequivalence cancan holdhold onlyonly betweenbetween equivalence-classes.equivalence-classes. 4646 POSSIBLPOSSIBLEE WORLDSWORLDS

ItIt needsneeds toto bebe pointedpointed outout thatthat wheneverwhenever a modalmodal relatiorelationn R holdholdss betweebetweenn twtwoo individualindividual propositions,propositions, P andand Q,32Q,32 therethere willwill alwaysalways bebe anan infinityinfinity ooff non-identicanon-identicall propositionpropositionss belonginbelongingg ttoo thethe samesame equivalence-classequivalence-class asas P,P, andand anan infinityinfinity ofof non-identicanon-identicall propositionpropositionss belonginbelongingg ttoo ththee samsamee equivalence-classequivalence-class asas Q,Q, andand thatthat any propositionproposition belonginbelongingg ttoo ththee formeformerr clasclasss wilwilll stanstandd iinn ththee relationrelation R toto any propositionproposition belongingbelonging toto thethe latterlatter class.class. ThisThis isiseasy easy toto prove.prove. Remember,Remember, first,first, thatthat forfor anyany propositionproposition P,P, whetherwhether contingencontingentt oorr noncontingentnoncontingent,, thertheree iiss a sesett ooff propositionspropositions eacheach ofof whichwhich isis truetrue inin preciselyprecisely thethe samesame setset ooff possiblpossiblee worldworldss aass PP;; thathatt iiss ttoo saysay,, anyany propositionproposition P,P, ofof whateverwhatever modalmodal status,status, isis a membermember ofof anan equivalence-class.equivalence-class. Secondly,Secondly, thethe equivalence-classequivalence-class toto whichwhich anyany givengiven propositiopropositionn P belongbelongss iiss a sesett ooff propositionspropositions withwith anan infiniteinfinite numbernumber ofof members.members. HowHow maymay wewe establishestablish thithiss lattelatterr claimclaim?? FoForr a startstart,, wwee mamayy notenote thatthat thethe setset ofof naturalnatural numbersnumbers isis a setset withwith anan infiniteinfinite numbenumberr ooff membersmembers.. NoNoww foforr eaceachh ooff thesethese naturalnatural numbersnumbers therethere existsexists a propositionproposition whichwhich assertsasserts thathatt ththee numbenumberr hahass a successorsuccessor.. HenceHence thethe numbernumber ofof suchsuch propositionspropositions isis itselfitself infinite.infinite. Moreover,Moreover, eaceachh ooff thesthesee propositionpropositionss iiss nonott onlonlyy true,true, butbut necessarilynecessarily true.true. ItIt followsfollows thatthat therethere isis anan infiniteinfinite numbenumberr ooff necessarnecessaryy truthstruths.. NowNow,, aass wwee sawsaw before,before, sincesince everyevery necessarynecessary truthtruth isis truetrue inin preciselyprecisely ththee samesame sesett ooff possiblpossiblee worldworldss aass evereveryy otherother necessarynecessary truth,truth, thethe setset ofof necessarynecessary truthstruths formsforms anan equivalence-class.equivalence-class. AndAnd,, aass wwee havhavee jusjustt seenseen,, thisthis equivalence-classequivalence-class mustmust havehave anan infiniteinfinite numbernumber ofof membersmembers.. IInn shortshort,, evereveryy necessarilnecessarilyy trutruee propositionproposition belongsbelongs toto anan equivalence-classequivalence-class whichwhich hashas anan infinitinfinitee numbenumberr ooff membersmembers.. BuButt iiff thithiss iiss soso,, thenthen thethe samesame mustmust bebe truetrue alsoalso ofof everyevery necessarilynecessarily falsefalse propositionproposition.. FoForr iitt iiss obviouobviouss thathatt thertheree mustmust bebe anan infiniteinfinite numbernumber ofof necessarilynecessarily falsefalse propositions:propositions: ttoo eacheach naturanaturall numbenumberr thertheree cacann bbee pairepairedd ofofff a necessarilynecessarily falsefalse proposition,proposition, e.g.,e.g., thethe propositionproposition thatthat thatthat numbernumber hahass nnoo successorsuccessor,, anandd iitt iiss equallyequally obviousobvious thatthat thisthis infiniteinfinite setset ofof propositionspropositions constitutesconstitutes aann equivalence-clasequivalence-classs witwithh alalll otherother propositionspropositions whichwhich areare necessarilynecessarily false.false. HenceHence everyevery necessarilnecessarilyy falsfalsee propositiopropositionn belongbelongss ttoo aann equivalence-classequivalence-class whichwhich hashas anan infiniteinfinite numbernumber ofof members.members. HowHow aboutabout contingentcontingent propositions?propositions? TheThe samesame resulresultt holdholdss foforr thethemm tootoo. . AAss wwee sasaww beforbeforee (p(p.. 37),37), forfor anyany contingentcontingent propositionproposition whatever,whatever, therethere existsexists anotheanotherr non-identicanon-identicall bubutt equivalentequivalent propositionproposition whichwhich assertsasserts thethe jointjoint truthtruth ofof botbothh thathatt propositiopropositionn anandd somsomee necessarilnecessarilyy trutruee proposition.proposition. ButBut therethere isis anan infiniteinfinite numbernumber ofof necessarilynecessarily trutruee propositionpropositionss ananyoney one ooff whicwhichh mamayy bbee assertedasserted toto bebe truetrue conjointlyconjointly withwith a givengiven contingentcontingent propositionproposition.. HencHencee foforr ananyy contingentcontingent propositionproposition whateverwhatever therethere existsexists anan infiniteinfinite numbernumber ofof non-identicanon-identicall bubutt equivalenequivalentt propositionpropositionss eacheach ofof whichwhich assertsasserts thethe jointjoint truthtruth ofof thatthat propositionproposition andand somesome necessarilynecessarily truetrue proposition.proposition. HenceHence everyevery contingentcontingent propositionproposition belongsbelongs toto anan equivalence-classequivalence-class whicwhichh hahass aann infinitinfinitee numbenumberr ooff members.members. Consider,Consider, inin thethe lightlight ofof allall this,this, twotwo individualindividual propositionspropositions,, A anandd BB,, whicwhichh stanstandd iinn somsomee modamodall relationrelation R.R. ForFor instance,instance, letlet usus supposesuppose thatthat A isis thethe contingentcontingent propositiopropositionn

(1.3) TheThe U.S.U.S. enteredentered WorldWorld WarWar I inin 19141914 andand B isis thethe contingentcontingent propositionproposition

(1.21) TheThe U.S.U.S. enteredentered WorldWorld WarWar I beforbeforee 1920.1920.

ThereThere isis anan infiniteinfinite numbernumber ofof non-identicalnon-identical propositionspropositions whicwhichh araree equivalenequivalentt ttoo AA,, anandd aann infinitinfinitee numbernumber ofof non-identicalnon-identical propositionspropositions whichwhich areare equivalentequivalent ttoo BB.. ThuThuss iitt followfollowss frofromm ththee fact thathatt (1.3)(1.3) (i.e.,(i.e., A)A) impliesimplies (1.21) (i.e.,(i.e., B)B) thatthat therethere isis anan infinitelinfinitelyy larglargee numbenumberr ooff propositionspropositions equivalentequivalent toto A eacheach ofof whichwhich impliesimplies anan infinitelyinfinitely largelarge numbenumberr ooff propositionpropositionss equivalenequivalentt ttoo B.B.

32.32. Or,Or, wewe mightmight equallyequally say,say, "between"between twotwo unitunit setssets {P}{P} andand {Q} ...... " § 5 SetsSets ofof PropositionsPropositions 4747

ParallelParallel conclusionsconclusions followfollow forfor eacheach ofof thethe otherother modalmodal relationsrelations ofof consistency,consistency, inconsistency,inconsistency, anandd equivalence.equivalence. InIn sum,sum, thethe poinpointt maymay bbee puputt thisthis way:way: WheneverWhenever twotwo propositionspropositions,, P andand Q,Q, standstand inin anyany modamodall relationrelation R,R, allall thosethose propositionpropositionss whichwhich areare equivalentequivalent toto P,P, ofof whichwhich therethere isis necessarilynecessarily anan infinitinfinitee number,number, willwill similarlysimilarly standstand inin thethe modalmodal relationrelation R toto eacheach ofof thethe infiniteinfinite numbernumber ofof propositionspropositions whichwhich areare equivalentequivalent toto QQ.. AnAn interesting,interesting, neglectedneglected corollarycorollary maymay bbee drawndrawn fromfrom thisthis principleprinciple.. InIn sectionsection 4 wewe arguedargued thathatt therethere areare onlyonly twotwo speciesspecies ofof thethe modalmodal relationrelation ofof inconsistency:inconsistency: eithereither twotwo inconsistentinconsistent propositionspropositions (and(and nownow wewe wouldwould addadd "proposition-sets")"proposition-sets") areare contrariescontraries oror theythey areare contradictories.contradictories. NoNoww whilewhile iitt hashas longlong beebeenn acknowledged,acknowledged, indeedindeed insistedinsisted upon,upon, thatthat nono propositiopropositionn hashas a uniqueunique (i.e.,(i.e., oneone andand onlonlyy one)one) contrary,contrary, itit hashas oftenoften beenbeen asas strenuouslystrenuously insistedinsisted thatthat everyevery propositiopropositionn doesdoes havehave a uniquuniquee contradictory,contradictory, i.e.,i.e., thatthat therethere isis oneone andand onlyonly oneone propositiopropositionn whichwhich standsstands inin thethe relationrelation ofof contradictioncontradiction toto a givengiven propositionproposition.. ButBut inin lightlight ofof thethe distinctiondistinction betweebetweenn propositional-identitpropositional-identityy anandd propositional-equivalencpropositional-equivalencee andand inin lightlight ofof thethe factfact thatthat modalmodal relationsrelations holdhold equallyequally wellwell betweebetweenn setsetss ofof propositionpropositionss asas betweebetweenn propositionpropositionss themselves,themselves, thesethese claimsclaims needneed toto bbee re-examined.re-examined. LetLet usus begibeginn withwith somesome examples.examples. ConsiderConsider thethe propositiopropositionn

(7.36)(1.36) TodayToday isis Wednesday.Wednesday.

AmongAmong itsits contrariescontraries areare

(7.37)(1.37) TodayToday isis MondayMonday;;

(7.38)(1.38) TodayToday isis Saturday.Saturday.

NoNoww let'slet's looklook atat somesome ofof itsits contradictories.contradictories. TheseThese willwill includincludee

(7.39)(1.39) TodayToday isis notnot Wednesday;Wednesday;

(7.40)(1.40) TodayToday isis notnot thethe dayday afterafter TuesdayTuesday;;

(7.47)(1.41) TodayToday isis notnot thethe dayday beforbeforee Thursday,Thursday, etc.etc.

WhatWhat difference cancan wewe detectdetect betweebetweenn thethe contrariescontraries ofof thethe propositiopropositionn (7.36)(1.36) andand thethe contradictoriescontradictories ofof thatthat samesame propositionproposition?? JustJust this:this: thethe contradictoriescontradictories ofof a givengiven propositiopropositionn formform anan equivalence-classequivalence-class (e.g.,(e.g., (7.39),(1.39), (7.40),(1.40), andand (7.47)(1.41) areare allall equivalentequivalent toto oneone another),another), whilewhile thethe contrariescontraries ofof a givengiven propositiopropositionn areare notnot allall equivalentequivalent toto oneone another.another. ThusThus whilewhile thethe claimclaim thathatt everyevery propositiopropositionn hashas a uniqueunique contradictorycontradictory cannotcannot bbee supported,supported, itit cancan bbee supersededsuperseded bbyy thethe trutruee claimclaim thatthat allall thethe contradictoriescontradictories ofof a propositiopropositionn areare logicallylogically equivalentequivalent toto oneone another,another, i.e.,i.e., thatthat thethe setset ofof contradictoriescontradictories ofof a propositiopropositionn isis itselfitself anan equivalence-class.equivalence-class. NNoo suchsuch claimclaim cancan bbee mademade forfor thethe contrariescontraries ofof a propositionproposition.. TheThe setset consistingconsisting ofof allall thethe contrariescontraries ofof a givengiven propositiopropositionn isis not a setset ooff equivalentequivalent propositionspropositions..

Minding our "P"s and "Q"s"Q"s

InsofarInsofar asas thethe kindskinds ofof propertiepropertiess andand relationsrelations wewe areare concernedconcerned toto ascribeascribe toto singlesingle propositionpropositionss may,may, asas wewe havehave jusjustt seen,seen, bbee ascribedascribed toto setssets ofof propositionspropositions,, wewe wouldwould dodo wellwell toto poinpointt outout thatthat botbothh propositionpropositionss andand setssets ofof propositionpropositionss maymay equallyequally wellwell bbee representedrepresented bbyy thethe samesame sortssorts ofof symbolssymbols iinn thethe conceptualconceptual notationnotation wewe useuse.. MoreMore specifically,specifically, whenwhen wewe writewrite suchsuch thingsthings asas "P"P standsstands inin thethe 4848 POSSIBLEPOSSIBLE WORLDSWORLDS

relationrelation R toto Q ifif andand onlyonly ifif...... ",", etc.,etc., wewe shouldshould bebe understoodunderstood toto bebe referring,referring, indiscriminately,indiscriminately, bbyy ourour useuse ofof "P""P" andand "Q","Q", botbothh toto singlesingle propositionpropositionss andand toto setssets ofof propositionspropositions.. II)In thethe nextnext sectionsection wewe shallshall introduceintroduce whatwhat wewe callcall "worlds-diagrams""worlds-diagrams" andand willwill labellabel partpartss ofof thethemm withwith "P"s"P"s andand "Q"s."Q"s. ForFor convenienceconvenience andand brevitbrevityy wewe oftenoften treattreat thesethese symbolssymbols asas ifif theythey referredreferred ttoo singlesingle propositionspropositions.. InIn factfact theythey oughtought toto bbee thoughtthought toto referrefer eithereither toto singlesingle propositionpropositionss oror ttoo proposition-setsproposition-sets..

EXERCISES

1. Which proposition-sets (a. - e.) are inconsistent with which proposition-sets (i. - v.)? WhichWhich proposition-sets (a. - e.) are consistent with which proposition-sets (i. - v.)? WhichWhich proposition-sets (a. - e.) imply which proposition-sets (i. - v.)? And which proposition-setsproposition-sets (a. - e.) are equivalent to which proposition-setspropositi?n-sets (i. - v.)?v.)?

a. {Today{ Today is Tuesday, Bill has missedmissed 1.i. {Bill has missed the bus}bus] the bus, Bill is late for work}work}

b. {Someone returned the wallet,wallet, 11.ii. {Someone who lost his keys Someone lost his keys}keys] returned the wallet}wallet}

c. {The Prime Minister is 6' tall,tall, uz.lii. {Mushroom omelets are notnot The Prime lvlinisterMinister is exactlyexactly poisonous, No mushroommushroom 5' 2/12" tall}tall] omelet is poisonous}poisonous}

d. {Some mushrooms are poisonous,poisonous, w.iv. {john{John is 15 years old, JohnJohn Some mushrooms are not poisonous}poisonous} is 5' 3/13" tall}tall}

e. {john{John is 15 years old and is 5' 3/13" tall}tall) v. {Although Bill has missed the bus, he is not late for work}work]

2. Which one of the ten sets of propositions in exercise 1 is a set of equivalent propositions?propositions?

3. Construct an equivalence-class of three propositions one of which is the proposition that Sylvia is Diane's mother.mother.

4. Construct an equivalence-class of three propositions one of which is the proposition that two plusplus two equals four.four.

6.6. MODALMODAL PROPERTIESPROPERTIES ANDAND RELATIONSRELATIONS PICTUREDPICTURED ONON WORLDS-DIAGRAMWORLDS-DIAGRAMSS

Worlds-diagramsWorlds-diagrams havehave alreadyalready beebeenn used:used: figure (l.b) introducedintroduced ourour basibasicc conventionsconventions forfor representingrepresenting anan infiniteinfinite numbernumber ofof possiblpossiblee worlds,worlds, actualactual andand non-actual;non-actual; andand figures (l.d), (l.e)(he),, andand (1(1.f)j) gavegave graphicgraphic significancesignificance toto ourour talktalk ofof thethe differentdifferent sortssorts ofof modalmodal statusstatus thatthat propositionpropositionss havehave accordingaccording toto whetherwhether theythey areare contingent,contingent, necessarilynecessarily truetrue oror necessarilynecessarily false,false, respectively.respectively. SoSo farfar wewe havehave givengiven thesethese diagramsdiagrams merelymerely anan illustrative role:role: ourour talk ofof possiblpossiblee worldsworlds couldcould havehave sufficedsufficed bbyy itself.itself. However,However, thesethese diagramsdiagrams cancan alsoalso bbee givengiven anan importantimportant heuristic role:role: theythey cacann facilitatefacilitate ourour discoverydiscovery andand prooprooff ofof logicallogical truthstruths whichwhich mightmight otherwiseotherwise eludeelude usus.. § 6 ModalModal PropertiesProperties andand RelationsRelations PicturedPictured onon Worlds-DiagramWorlds-Diagramss 4949

InIn orderorder thatthat wewe maymay betterbetter bebe ableable toto useuse themthem heuristicallyheuristically wewe adoptadopt thethe followingfollowing twotwo simplifyingsimplifying conventions:conventions:

(a)(a) WeWe usuallyusually omitomit fromfrom ourour diagramsdiagrams anyany representationrepresentation ofof thethe distinctiondistinction betweebetweenn thethe actuaactuall worldworld andand otherother (non-actual)(non-actual) possiblpossiblee worlds.worlds. WhenWhen thethe needneed arisesarises toto investigateinvestigate thethe consequencesconsequences ooff supposingsupposing somesome propositiopropositionn toto bbee actuallyactually truetrue oror actuallyactually false,false, thatthat distinctiondistinction can,can, ofof course,course, bbee reintroducedreintroduced inin thethe mannermanner displayed inin figuresfigures (1.d), (J.e)(l.e), , andand (1(1J),j), oror asas wewe shallshall seesee soon,soon, mormoree perspicuouslyperspicuously,, simplysimply bbyy placinplacingg anan "x""X" onon thethe diagramdiagram toto markmark thethe locationlocation ofof thethe actualactual worlworldd amongamong thethe setset ofof allall possiblpossiblee worlds.worlds. But,But, forfor thethe mostmost partpart,, wewe shallshall bbee concernedconcerned primarilprimarilyy withwith investigatinginvestigating thethe relationshipsrelationships betweebetweenn propositionpropositionss independentlyindependently ofof theirtheir truth-statustruth-status inin thethe actuaactuall world,world, andand soso shallshall havehave infrequentinfrequent needneed toto invokeinvoke thethe distinctiondistinction betweebetweenn actualactual andand non-actuanon-actuall worlds.worlds.

(b)(b) WeWe omitomit fromfrom ourour diagramsdiagrams anyany bracketinbracketingg spanningspanning thosethose possiblpossiblee worlds,worlds, ifif any,any, inin whichwhich a givengiven propositiopropositionn oror proposition-set3333 isis false.false. ThisThis meansmeans thatthat everyevery brackebrackett thatthat wewe ususee isis toto bbee interpretedinterpreted asas spanningspanning thosethose possiblpossiblee worldsworlds onlyonly inin whichwhich a givengiven propositiopropositionn (or(or proposition-setproposition-set)) isis true.true. InIn thethe eventevent thatthat a givengiven propositiopropositionn isis notnot truetrue inin anyany possiblpossiblee world,world, i.e.,i.e., isis falsefalse inin allall possiblepossible worlds,worlds, wewe 'locate''locate' thatthat propositiopropositionn bbyy meansmeans ofof a poinpointt placeplacedd outsideoutside (and(and toto thethe rightright of)of) thethe rectanglerectangle representingrepresenting thethe setset ofof allall possiblpossiblee worlds.worlds. InIn effecteffect wewe thusthus 'locate''locate' anyany necessarilynecessarily falsefalse propositiopropositionn amongamong thethe impossibleimpossible worldsworlds..

InIn lightlight ofof thesethese simplifyingsimplifying conventions,conventions, letlet usus firstfirst reconstructreconstruct thethe threethree basibasicc worlds-diagramsworlds-diagrams depictingdepicting thethe modalmodal propertiepropertiess ofof contingency,contingency, necessarynecessary truth,truth, andand necessarynecessary falsityfalsity (figures(figures (1.d)(1 .d),, (1.e)(l.e), , andand (1(1./J),j), andand thenthen considerconsider howhow theythey mightmight bbee supplementedsupplemented inin orderorder toto depictdepict modamodall relations.relations.

Worlds-diagrams for modal propertiesproperties

A singlesingle propositionproposition (or(or proposition-set)proposition-set) P,P, maymay bebe truetrue inin allall possiblepossible worlds,worlds, jusjustt some,some, oror none.none. ThereThere areare nono otherother possibilitiespossibilities.. If,If, then,then, wewe depictdepict thethe setset ofof allall possiblpossiblee worldsworlds bbyy a singlesingle boxbox,, iitt followsfollows thatthat wewe havehave needneed ofof threethree andand onlyonly threethree basibasicc worlds-diagramsworlds-diagrams forfor thethe modalmodal propertiepropertiess ofof a propositiopropositionn (or(or proposition-setproposition-set)) P.P. TheyThey areare::

p p " p 1 2 3 •

FIGUREFIGURE (1.k)(1.h)

33.33. SeeSee thethe subsectionsubsection "Minding"Minding ourour 'P's'P's andand 'Q's",'Q's", pppp.. 47-48.47-48. 5050 POSSIBLEPOSSIBLE WORLDSWORLDS

DiagramDiagram 1 inin figurefigure (7.h),(l.h), pp.. 49,49, depictsdepicts thethe contingencycontingency ofof a singlesingle propositiopropositionn (or(or proposition-set)proposition-set) P.P. TheThe propositiopropositionn P isis contingentcontingent becausbecausee itit isis truetrue inin somesome possiblpossiblee worldsworlds bubutt falsefalse inin allall ththee others.others. InIn effect,effect, diagramdiagram 1 isis a reconstructionreconstruction ofof figurefigure (1.d) mademade inin accordanceaccordance withwith thethe twotwo simplifyingsimplifying conventionsconventions specifiedspecified above.above. OurOur diagramdiagram givesgives thethe modalmodal statusstatus ofof P bubutt sayssays nothinnothingg aboutabout itsits actualactual truth-statustruth-status (i.e.,(i.e., truth-valuetruth-value inin thethe actualactual world)world).. DiagramDiagram 2 depictsdepicts thethe necessarynecessary truthtruth ofof a singlesingle propositiopropositionn P.P. TheThe propositiopropositionn P isis necessarilynecessarily true,true, sincesince itit isis truetrue inin allall possiblpossiblee worlds.worlds. InIn effect,effect, diagramdiagram 2 isis a reconstructionreconstruction ofof figurefigure (1.e) madmadee inin accordanceaccordance withwith ourour simplifyingsimplifying conventions.conventions. DiagramDiagram 3 depictsdepicts thethe necessarynecessary falsityfalsity ofof a singlesingle propositionproposition P.P. HereHere P isis necessarilynecessarily falsefalse sincesince i itt isis falsefalse inin allall possiblpossiblee worlds.worlds. InIn effect,effect, diagramdiagram 3 isis a simplificationsimplification ofof figurefigure (1(1.f).f) .

Worlds-diagrams forjor modal relationsrelations InIn orderorder toto depictdepict modalmodal relationsrelations betweebetweenn twotwo propositionpropositionss (or(or twotwo proposition-setsproposition-sets)) P andand Q,Q, wewe neeneedd exactlyexactly fifteenfifteen worlds-diagrams.worlds-diagrams. InIn thesethese worlds-diagramsworlds-diagrams (see(see figurefigure (1.i)(hi) onon pp.. 51),51), nono significancesignificance iiss toto bbee attachedattached toto thethe relativerelative sizes ofof thethe variousvarious segments.segments. ForFor ourour presenpresentt purposepurposess allall wewe needneed attenattendd toto isis thethe relativerelative placement ofof thethe segments,segments, oror asas mathematiciansmathematicians mightmight say,say, toto theirtheir topology.topology. ForFor thethe purposepurposess ofof thethe presenpresentt discussiondiscussion ourour diagramsdiagrams needneed onlyonly bbee qualitative,qualitative, notnot quantitative.3434

EXERCISE

Reproduce figurefigure (1.i)(l.i) and add brackets forjor "rvP""^P" and forjor "rv"^Q"Q" to each ojof the fifteenfijteen worlds-diagrams. ** ** *

Interpretation oJof worlds-diagrams DiagramsDiagrams 1 toto 4 depictdepict casescases wherewhere botbothh propositionpropositionss areare noncontingent.noncontingent. DiagramsDiagrams 5 toto 8 depictdepict casecasess wherewhere oneone propositiopropositionn isis noncontingentnoncontingent andand thethe otherother isis contingent.contingent. TheThe final sevenseven diagramsdiagrams (9(9 ttoo 15)15) depictdepict casescases wherewhere botbothh propositionpropositionss areare contingent.contingent. NoNoww eacheach ofof thesethese fifteenfifteen diagrams diagrams locateslocates twotwo propositions,propositions, PP andand Q,Q, withwith respectrespect toto thethe setset ofof allall possiblpossiblee worlds,worlds, andand thencethence withwith respectrespect toto oneone another,another, inin suchsuch a wayway thatthat wewe cancan determinedetermine whawhatt modalmodal relationsrelations oneone propositiopropositionn hashas toto thethe other.other. HowHow cancan wewe dodo thisthis?? TheThe modalmodal relationsrelations wewe havehave singledsingled outout forfor considerationconsideration soso farfar areare thosethose ofof inconsistency,inconsistency, consistency,consistency, implication,implication, andand equivalence.equivalence. Recall,Recall, then,then, howhow eacheach ofof thesethese fourfour relationsrelations waswas defineddefined:: P isis inconsistent withwith Q_ ifif andand onlyonly ifif therethere isis nono possiblepossible worldworld inin whichwhich bothboth areare true;true; P isis consistent withwith Q ifif andand onlyonly ifif therethere isis a possiblpossiblee worldworld inin whichwhich botbothh areare true;true; P implies Q ifif anandd onlyonly ifif [definition[definition (b)](b)] therethere isis nono possiblpossiblee worldworld inin whichwhich P isis truetrue andand Q isis false;false; andand P isis equivalent toto Q ifif andand onlyonly ifif inin eacheach ofof allall possiblpossiblee worldsworlds P hashas thethe samesame truth-valuetruth-value asas Q.Q. Recall,Recall, further,further, thathatt ourour devicedevice forfor depictingdepicting a propositiopropositionn asas truetrue inin a possiblpossiblee worldworld isis toto spanspan thatthat worldworld bbyy meansmeans ofof aa bracketbracket labeledlabeled withwith a symbolsymbol signifyingsignifying thatthat propositionproposition..

34.34. Later,Later, whenwhen wewe comecome toto discussdiscuss thethe conceptconcept ofof "the"the contingentcontingent content"content" ofof a propositionproposition,, wewe shallshall suggessuggestt howhow oneone mightmight wantwant toto reinterpretreinterpret thesethese worlds-diagramsworlds-diagrams soso thatthat thethe sizessizes ofof thethe segmentssegments dodo taketake oonn significance.significance. (See(See chapterchapter 6,6, sectionsection 11.)11.) §§6 6 ModaModall PropertiePropertiess anandd RelationRelationss PicturePicturedd oonn WWorlds-Diagramorlds-Diagramss 51

P P P A , . r

1 6 1111

...... I -v Q Q Q Q P P P .. . A

Q 2 7 Q 1122

~ Q P..

P P 3 8 1313

.... '---v--' .. -J Q Q Q

..P P...

P,Q 4 P/Q 9 14 .. , Q Q

P P P . A

5 10 1515

'---v----' .--J .,.....J Q Q Q

FIGUREFIGURE (l.i)(hi) 5252 POSSIBLPOSSIBLEE WORLDSWORLDS

TheThe requisiterequisite rulesrules forfor thethe interpretationinterpretation ofof ourour worlds-diagramsworlds-diagrams follofolloww immediatelyimmediately:: RuleRule 1:1: P isis inconsistentinconsistent withwith Q ifif andand onlyonly ifif therethere doesdoes nonott exisexistt ananyy sesett ooff possiblpossiblee worldworldss whicwhichh isis spannedspanned bothboth byby a bracketbracket forfor P andand bbyy a brackebrackett foforr Q;Q; RuleRule 2:2: P isis consistent withwith Q ifif andand onlyonly ifif therethere doesdoes exisexistt a sesett ooff possiblpossiblee worldworldss whicwhichh iiss spannedspanned bothboth byby a bracketbracket forfor P andand byby a brackebrackett foforr Q;Q; RuleRule 3:3: P impliesimplies Q ifif andand onlyonly ifif therethere doesdoes notnot existexist ananyy sesett ooff possiblpossiblee worldworldss whicwhichh iiss spannespannedd byby a bracketbracket forfor P andand whichwhich isis notnot spannedspanned bbyy a brackebrackett foforr Q (i.e.(i.e.,, iiff anandd onlonlyy iiff ananyy sesett ofof possiblepossible worldsworlds spannedspanned byby a bracketbracket forfor P isis alsalsoo spannespannedd bbyy a brackebrackett foforr Q);35Q);35 RuleRule 4:4: P isis equivalent toto Q ifif andand onlyonly ifif therethere doesdoes nonott exisexistt ananyy sesett ooff possiblpossiblee worldworldss whicwhichh iiss spannedspanned byby thethe bracketbracket forfor oneone andand whichwhich isis nonott spannedspanned bbyy ththee brackebrackett foforr ththee otheotherr (i.e.(i.e.,, thethe bracketsbrackets forfor P andand forfor Q spanspan preciselyprecisely ththee samesame sesett ooff worlds)worlds)..

ItIt isis thethe additionaddition ofof thesethese rulesrules ofof interpretationinterpretation thathatt givegivess ouourr worlds-diagraworlds-diagramm ththee heuristiheuristicc valuvaluee thatthat wewe earlierearlier claimedclaimed forfor them.them. ByBy applyingapplying themthem wwee cancan provprovee a larglargee numbenumberr ooff logicalogicall truthtruthss iinn aa simplesimple andand straightforwardstraightforward way.way. ConsiderConsider somesome examples:examples:

(i)(i) DiagramsDiagrams 2,2, 3,3, 4,4, 7,7, andand 8 comprisecomprise allall thethe casescases iinn whicwhichh ononee oorr botbothh ooff ththee propositionpropositionss P anandd Q isis necessarilynecessarily false.false. InIn nonenone ofof thesethese casescases isis therethere anyany setset ofof possiblepossible worldworldss spannespannedd botbothh bbyy a bracketbracket forfor P andand byby a bracketbracket forfor Q.Q. Hence,Hence, byby RuleRule 1,1, wewe maymay validlyvalidly inferinfer thathatt iinn alalll ooff thesethese casescases P isis inconsistentinconsistent withwith Q.Q. InIn short,short, if one or both of a pair of propositions isis necessarily false falsethen thenthose thosepropositions propositionsare areinconsistent inconsistentwith withone oneanother. another. (ii)(ii) DiagramsDiagrams 1,1, 2,2, 3,3, 5,5, andand 6 comprisecomprise allall thethe casescases iinn whicwhichh ononee oorr botbothh ooff ththee propositionpropositionss P andand Q isis necessarilynecessarily true.true. InIn eacheach ofof thesethese cases,cases, exceptexcept 2 andand 3,3, therethere isis a setset ooff possiblpossiblee worldworldss spannedspanned bothboth byby a bracketbracket forfor P andand byby a brackebrackett foforr QQ.. HenceHence,, bbyy RulRulee 22,, wwee mamayy validlvalidlyy inferinfer thatthat inin eacheach ofof thesethese cases,cases, exceptexcept 2 andand 3,3, P iiss consistenconsistentt witwithh QQ.. BuButt diagramdiagramss 2 anandd 3 areare casescases inin whichwhich oneone oror otherother ofof thethe twotwo propositionpropositionss iiss necessarilnecessarilyy falsefalse.. WWee mamayy concludeconclude,, therefore,therefore, thatthat a necessarily true proposition is consistent withwith anyany propositionproposition whateverwhateverexcept except a necessarilyfalse falseone. one. (iii)(iii) DiagramsDiagrams 3,3, 4,4, andand 8 comprisecomprise allall thethe casescases inin whicwhichh a propositiopropositionn P iiss necessarilnecessarilyy falsefalse.. IInn nonenone ofof thesethese casescases isis therethere a setset ofof possiblepossible worldsworlds spannedspanned bbyy a brackebrackett foforr PP.. HencHencee iinn nonnonee ofof thesethese casescases isis therethere a setset ofof possiblepossible worldsworlds whicwhichh iiss spannespannedd bbyy a brackebrackett foforr P anandd nonott spannedspanned byby a bracketbracket forfor Q.Q. Hence,Hence, byby RuleRule 3,3, wwee mamayy validlvalidlyy infeinferr thathatt iinn eaceachh cascasee iinn whicwhichh P isis necessarilynecessarily false,false, P impliesimplies Q nono mattermatter whetherwhether Q isis necessarilynecessarily trutruee (a(ass iinn 3)3),, necessarilynecessarily falsefalse (as(as inin 4),4), oror contingentcontingent (as(as inin 8).8). ByBy analogousanalogous reasoninreasoningg concerninconcerningg diagramdiagramss 22,, 44,, anandd 7 -— allall thethe casescases inin whichwhich a propositionproposition Q isis necessarilnecessarilyy falsfalsee —- wwee cacann shoshoww thathatt iinn eaceachh cascasee inin whichwhich Q isis necessarilynecessarily false,false, Q impliesimplies P nnoo mattematterr whethewhetherr P iiss necessarilnecessarilyy trutruee (a(ass iinn 2)2),, necessarilynecessarily falsefalse (as(as inin 4),4), oror contingentcontingent (as(as inin 7).7). IInn shortshort,, wwee mamayy concludconcludee thathatt a necessarily false proposition implies any and every proposition no matter what the modal status of that proposition. (iv)(iv) DiagramsDiagrams 1,1, 3,3, andand 6 comprisecomprise allall thethe casescases inin whicwhichh a propositiopropositionn Q iiss necessarilnecessarilyy truetrue.. IInn nonenone ofof thesethese casescases isis therethere a setset ofof possiblepossible worldworldss whicwhichh iiss not spannespannedd bbyy a brackebrackett foforr QQ.. Hence,Hence, byby RuleRule 3,3, wewe maymay validlyvalidly inferinfer thatthat inin eacheach cascasee iinn whicwhichh Q iiss necessarilnecessarilyy truetrue,, Q iiss impliedimplied byby a propositionproposition P nono mattermatter whetherwhether P iiss necessarilnecessarilyy trutruee (a(ass iinn 1)1),, necessarilnecessarilyy falsfalsee

35.35. ThisThis meansmeans thatthat P impliesimplies Q inin threethree cases:cases: (i)(i) wherewhere ththee brackebrackett foforr P spanspanss nnoo possiblpossiblee worldworldss aatt alalll (i.e.,(i.e., P isis necessarilynecessarily false);false); (ii)(ii) wherewhere thethe bracketbracket forfor P isis includedincluded withiwithinn ththee brackebrackett foforr QQ;; anandd (iii(iii)) wherwheree ththee bracketbracket forfor P isis coextensivecoextensive withwith thethe bracketbracket forfor Q.Q. § 6 ModalModal PropertiesProperties andand RelationsRelations PicturedPictured onon Worlds-DiagramsWorlds-Diagrams 5353

(as(as inin 3),3), oror contingentcontingent (as(as inin 6).6). ByBy analogousanalogous reasoningreasoning concerningconcerning diagramsdiagrams 1,2,1, 2, andand 5 -— allall thethe casescases inin whichwhich a propositiopropositionn P isis necessarilynecessarily truetrue -— wewe cancan showshow thatthat inin eacheach casecase inin whicwhichh P isis necessarilynecessarily true,true, P isis impliedimplied byby a propositionproposition Q nono mattermatter whetherwhether Q isis necessarilynecessarily trutruee (as(as inin 1),1), necessarilynecessarily falsefalse (as(as inin 2),2), oror contingentcontingent (as(as inin 5).5). InIn short,short, wewe maymay concludeconclude thatthat aa necessarily true proposition is implied by any and every proposition no matteTmatter what the modalmodal status ofofthat thatproposition. proposition.

TheThe specialspecial heuristicheuristic appealappeal ofof thesethese worlds-diagramsworlds-diagrams lieslies inin thethe factfact that,that, takentaken togethertogether witwithh certaincertain rulesrules forfor theirtheir interpretation,interpretation, wewe cancan literallyliterally see immediatelyimmediately thethe truthtruth ofof thesethese andand ofof manmanyy otherother propositionpropositionss aboutabout thethe modalmodal relationsrelations whichwhich propositionpropositionss havehave toto oneone another.another. TheThe additionaddition ooff stillstill furtherfurther definitionsdefinitions andand rulesrules ofof interpretationinterpretation laterlater inin thisthis boobookk willwill enableenable usus toto providprovidee mormoree perspicuouperspicuouss proofproofss ofof importantimportant logicallogical truthstruths -— includingincluding somesome whichwhich areare notnot asas well-knownwell-known aass thosethose soso farfar mentionedmentioned..

A note on history and nomenclaturenomenclature SoSo farfar wewe havehave givengiven namesnames toto onlyonly a fewfew ofof thethe modalmodal relationsrelations whichwhich cancan obtainobtain betweebetweenn twotwo propositionspropositions.. WeWe havehave spokenspoken ofof inconsistency (and(and itsits twotwo species,species, contradictioncontradiction andand contrariety),contrariety), ooff consistency, ofof implication, andand ofof equivalence. WithinWithin thethe philosophicaphilosophicall tradition,tradition, however,however, wewe findfind logicianslogicians talkingtalking alsoalso ofof modalmodal relationsrelations whichwhich theythey callcall "superimplication""superimplication" (sometimes(sometimes callecalledd "superalternation"),"superalternation"), "subimplication""subimplication" (sometimes(sometimes calledcalled "subalternation"),"subalternation"), "subcontrariety","subcontrariety", anandd "independence""independence" (sometimes(sometimes calledcalled "indifference")."indifference"). ByBy "superimplication""superimplication" andand "subimplication""subimplication" (or(or theirtheir terminologicalterminological alternates),alternates), traditionatraditionall logicianslogicians meantmeant simplysimply thethe relationsrelations ofof implication andand ofof following from respectively.respectively. ToTo saysay thatthat P standsstands inin thethe relationrelation ofof superimplication toto Q isis simplysimply toto saysay thatthat P implies Q,Q, whilewhile toto saysay thatthat P standsstands inin thethe relationrelation ofof subimplication toto Q isis simplysimply toto saysay thatthat P follows from Q (or(or thatthat Q implies P).P). InIn short,short, subimplicationsubimplication isis thethe converseconverse ofof superimplication,superimplication, i.e.,i.e., ofof implication.implication. ByBy "subcontrariety""subcontrariety" wewe meanmean thethe relationrelation whichwhich holdsholds betweebetweenn P andand Q whenwhen P andand Q can bothboth be true together but cannot both be false.false. ThatThat isis toto say,say, subcontrarietysubcontrariety isis thethe relationrelation whichwhich holdsholds betweebetweenn P andand Q whenwhen therethere isis atat leastleast oneone possiblpossiblee worldworld inin whichwhich botbothh areare truetrue (P(P andand Q araree 36 consistent)consistent) bubutt therethere isis nono possiblpossiblee worldworld inin whichwhich botbothh areare falsefalse (~'V P andand 'V~ Q areare inconsistent).36inconsistent). ByBy "independence""independence" wewe meanmean thethe relationrelation whichwhich holdsholds betweebetweenn P andand Q whenwhen nono relationrelation otheotherr thanthan consistencyconsistency holdsholds betweebetweenn thethe two.two. ThatThat isis toto say,say, independence isis thethe relationrelation whichwhich holdholdss betweebetweenn P andand Q whenwhen therethere isis atat leastleast oneone possiblpossiblee worldworld inin whichwhich botbothh areare truetrue (P(P andand Q araree consistent),consistent), therethere isis atat leastleast oneone possiblpossiblee worldworld inin whichwhich botbothh areare falsefalse ('V(~PP andand 'V Q areare alsalsoo consistent),consistent), therethere isis atat leastleast oneone possiblpossiblee worldworld inin whichwhich P isis truetrue andand Q isis false,false, andand therethere isis atat leasleastt oneone possiblpossiblee worldworld inin whichwhich P isis falsefalse andand Q isis truetrue.. OfOf thethe variousvarious modalmodal relationsrelations wewe havehave distinguished,distinguished, onlyonly oneone isis uniquelyuniquely depicteddepicted bbyy a singlesingle worlds-diagram.worlds-diagram. ThisThis isis thethe relationrelation ofof independence.independence. IndependenceIndependence isis depicteddepicted bbyy diagramdiagram 1515 andand bbyy thatthat diagramdiagram alone.alone. EachEach ofof thethe otherother modalmodal relationsrelations isis exemplifiedexemplified bbyy twotwo oror moremore ofof thethe fifteenfifteen worlds-diagrams.worlds-diagrams. Historically,Historically, thisthis factfact hashas notnot alwaysalways beenbeen recognized.recognized. WithinWithin traditionaltraditional logic,37logic,37

36.36. TheThe termterm "subcontrariety""subcontrariety" reflectsreflects thethe fact,fact, wellwell knownknown toto traditionaltraditional logicians,logicians, thatthat subcontrarsubcontraryy propositionpropositionss areare contradictoriescontradictories ofof propositionpropositionss whichwhich areare contrariescontraries andand standstand inin thethe relationrelation ofof sublmpllcatlonsubimplication toto thesethese contrarycontrary propositionspropositions.. ThusThus P isis a subcontrarysubcontrary ofof Q ifif andand onlyonly ifif (a)(a) '"-v P andand '"~ Q areare contraries,contraries, anandd (b)(b) thethe contrarycontrary propositionspropositions,, '"~ P and'"and ~ Q,Q, implyimply Q andand P respectively.respectively.

37.37. ByBy "traditional"traditional logic"logic" wewe meanmean thethe logiclogic whichwhich waswas foundedfounded bbyy AristotleAristotle (384-322(384-322 B.C.),B.C.), enrichedenriched bbyy ththee StoicsStoics andand Megarians,Megarians, andand effectivelyeffectively canonizedcanonized bbyy sixteenth-sixteenth- andand seventeenth-centuryseventeenth-century logicians.logicians. 5454 POSSIBLEPOSSIBLE WORLDSWORLDS

thethe termsterms "equivalence","equivalence", "contradiction","contradiction", "superimplication","superimplication", "contrariety","contrariety", "subimplication","subimplication", anandd "subcontrariety""subcontrariety" werewere oftenoften useusedd as if eacheach ofof them,them, too,too, werewere relationsrelations which,which, inin termsterms ofof ouourr worlds-diagrams,worlds-diagrams, wewe wouldwould uniqueluniquelyy depictdepict bbyy a singlesingle diagram.diagram. TraditionalTraditional logicianslogicians tendedtended toto thinthinkk ofof equivalenceequivalence asas thatthat relationrelation whichwhich wwee cancan reconstructreconstruct,, bbyy meansmeans ofof ourour worlds-diagrams,worlds-diagrams, asas holdinholdingg inin diagramdiagram 9 alone;alone; ofof contradictioncontradiction asas ifif itit couldcould bbee depicteddepicted inin diagramdiagram 1010 alone;alone; ofof superimplicatiosuperimplicationn (and(and hencehence ofof implication)implication) asas ifif itit couldcould bbee depicteddepicted inin diagramdiagram 1111 alone;alone; ofof contrarietycontrariety asas ifif itit coulcouldd bbee depicteddepicted inin diagramdiagram 1212 alone;alone; ofof subimplicationsubimplication asas ifif itit couldcould bbee depicteddepicted inin diagramdiagram 1313 alone;alone; anandd ofof subcontrarietysubcontrariety asas ifif itit couldcould bbee depicteddepicted inin diagramdiagram 1414 alone.alone. ThatThat isis toto say,say, theythey tendedtended toto thinkthink ooff thesethese relationsrelations asas ifif theythey couldcould holdhold onlyonly betweebetweenn propositionpropositionss botbothh ofof whichwhich areare contingent.contingent. (It(It isis easeasyy toto see,see, bbyy simplesimple inspection,inspection, thatthat diagramsdiagrams 9 throughthrough 1515 depictdepict thethe onlyonly casescases inin whichwhich botbothh propositionpropositionss areare contingent.)contingent.) OneOne consequenceconsequence ofof thethe traditionaltraditional preoccupatiopreoccupationn withwith relationsrelations betweebetweenn contingentcontingent propositionpropositionss was,was, andand sometimessometimes stillstill is,is, thatthat someonesomeone broughbroughtt upup inin thatthat tradition,tradition, ifif askeaskedd whatwhat logicallogical relationrelation holdsholds betweebetweenn a paipairr ofof contingent propositionpropositionss (any(any ofof thosethose depicteddepicted bbyy diagramsdiagrams 9 throughthrough 15),15), hadhad littlelittle difficultydifficulty inin invokinginvoking oneone ofof thethe standardstandard repertoirerepertoire ooff "equivalence","equivalence", "contradiction","contradiction", "superimplication","superimplication", "contrariety","contrariety", "subimplication","subimplication", "subcontrari"subcontrari­• ety",ety", oror "independence";"independence"; bubutt ifif askedasked whatwhat logicallogical relationrelation holdsholds betweenbetween a pairpair ofof propositionspropositions oneone oorr bothboth ofof whichwhich isis noncontingent (any(any ofof thosethose depicteddepicted byby diagramsdiagrams 1 throughthrough 8),8), simplysimply wouldwould nonott 38 knowknow whatwhat toto say.38say. A secondsecond andand moremore significantsignificant consequenceconsequence ofof thisthis preoccupationpreoccupation withwith relationsrelations betweenbetween contingentcontingent propositionspropositions was,was, andand stillstill is,is, thatthat thethe "intuitions""intuitions" ofof somesome (but(but not,not, ofof course,course, all)all) personpersonss underunder ththee influenceinfluence ofof thisthis traditiontradition areare notnot wellwell attunedattuned toto thethe analyses,analyses, inin termsterms ofof possiblpossiblee worlds,worlds, ofof relationsrelations likelike equivalenceequivalence andand implication;implication; theythey tendtend toto considerconsider thesethese analysesanalyses asas "counter-intuitive","counter-intuitive", anandd certaincertain consequenceconsequence ofof thesethese analysesanalyses asas "paradoxical"."paradoxical".3939

Capsule descriptions ojof modal relationsrelations ItIt maymay bbee helpfulhelpful toto gathergather togethertogether thethe descriptionsdescriptions wewe havehave givengiven inin variousvarious placeplacess ofof thosethose modamodall relationrelationss whichwhich areare ourour principlprinciplee concern.concern. P isis inconsistent withwith Q:Q: therethere isis nono possiblepossible worldworld inin whichwhich bothboth areare trutruee P isis a contradictory ofof Q:Q: therethere isis nono possiblepossible worldworld inin whichwhich bothboth areare trutruee andand nono possiblpossiblee worldworld inin whichwhich botbothh areare falsfalsee P isis a contrary ofof Q:Q: therethere isis nono possiblepossible worldworld inin whichwhich bothboth areare trutruee bubutt therethere isis a possiblpossiblee worldworld inin whichwhich botbothh araree falsefalse P isis consistent withwith Q:Q: therethere isis aapossible possible worldworldin inwhich which bothboth areare truetrue P implies (superimplies) Q:Q: therethere isis nono possiblepossible worldworld inin whichwhich P isis truetrue anandd Q isis falsfalsee P follows from (subimplies)(subimplies) Q:Q: therethere isis nono possiblepossible worldworld inin whichwhich Q isis truetrue anandd P isis falsfalsee

38.38. AnswersAnswers can,can, however,however, bbee given.given. Worlds-diagramWorlds-diagram 8,8, forfor example,example, depictsdepicts a relationrelation whichwhich satisfiessatisfies ththee descriptionsdescriptions givengiven ofof two modalmodal relations,relations, viz.,viz., implicationimplication andand contrariety.contrariety. ThatThat is,is, twotwo propositionspropositions,, P andand QQ,, whichwhich standstand inin thethe relationrelation depicteddepicted inin diagramdiagram 8,8, areare suchsuch thatthat (1)(1) P impliesimplies Q andand (2)(2) P andand Q araree contraries.contraries.

39.39. InIn chapterchapter 4 wewe developdevelop thisthis poinpointt furtherfurther andand suggestsuggest alsoalso thatthat somesome resistenceresistence toto thesethese analysesanalyses hashas itsits sourcesource inin a failurefailure toto distinguishdistinguish betweebetweenn validvalid inferenceinference andand demonstrability.demonstrability. § 6 Modal.PropertiesModal .Properties andand RelationsRelations PicturedPictured onon Worlds-DiagramsWorlds-Diagrams 5555

P isis equivalent toto Q:Q: inin eacheach possiblpossiblee world,world, P andand Q havehave matchinmatchingg truth-valuestruth-values

P isis a subcontrary ofof Q:Q: therethere isis a possiblpossiblee worldworld inin whichwhich botbothh areare trutruee bubutt therethere isis nono possiblpossiblee worldworld inin whichwhich botbothh areare falsefalse

P isis independentindependent ofof Q:Q: therethere isis a possiblpossiblee worldworld fnhi whichwhich botbothh areare truetrue,, a possiblepossible worldworld inin whichwhich bothboth areare false,false, a possiblpossiblee worlworldd inin whicwhichh P isis truetrue andand Q isis false,false, andand a possiblpossiblee worldworld inin whichwhich P isis falsefalse andand Q iiss truetrue

EXERCISES

Part A

1.,. Which worlds-diagramsworlds-diagrams, (in figurefigure (Li»){\.i)) represent cases in which P is necessarilynecessarily- true (i.e., in which oPOP obtains) ?

2. Which worlds-diagrams represent cases in which QQisis necessarily true? [OQ][•(?]

3. Which worlds-diagrams represent cases in which P is contingent? [VP[\7P]]

4. Which worlds-diagrams represent cases in which Q is contingent? [VQ[\7QJ]

5. Which three worlds-diagrams represent cases in which we may validly infer that P is actually truetrue (i.e., true in the actual world)?world) ?

6. Which seven worlds-diagrams represent cases in which P implies Q? [P---+Q][P—>Q]

7.7. Which seven worlds-diagrams represent cases in which Q implies P? [Q—[Q---+P]*P\

8. Which eight worlds-diagrams represent cases in which P is consistent with Q? [P 0o Q]

9. Which seven worlds-diagrams represent cases in which P is inconsistent with Q? [P Q]Q_]

10. Which three worlds-diagrams represent cases in which PandP and QQareare contradictories?contradictories?

11. Which four worlds-diagrams represent cases in which P and Q are contraries?contraries?

12. Which four worlds-diagrams represent cases in which P and QQareare subcontraries?subcontraries?

13. Which three worlds-diagrams represent cases in which PandP and QQareare equivalent? [P<—>Q][P...... Q]

14. Which worlds-diagrams represent cases in which P is possible and in which Q is possible? [VP[r~:p andand <>.0]0.0} 15. Which worlds-diagrams represent cases in which P is possible, Q is possible, and in which P is inconsistent with Q? [OP[OPandand OQOQandand (P Q)]Qj]

16. Which worlds-diagrams represent cases in which neither P implies Q nor Q implies P?P? [[^(P->Q)'V (P---+Q) andand^(Q->P)]'V (Q---+P)]

17. Which worlds-diagrams represent cases in which both P implies Q and P is consistent with Q?Q? [(P---+Q)[(P->Q) and (P(PoQ)}0 Q)] 5656 POSSIBLEPOSSIBLE WORLDWORLDSS

18. Which worlds-diagrams represent cases inin whichwhich bothbothP P impliesimplies QQand and PP isisinconsistent inconsistent with Q? [(P_Q){(P->Q) and (P(P*4> Q)]Q) ] 19. Which worlds-diagrams represent casescasesin inwhich whichboth bothP Pimplies impliesQ Qandand Q Qdoesdoes not notimply implyP? P? [(P_Q)[(P-+Q) and'"and^(Q->P)}(Q_P) ] 20. Which worlds-diagrams represent cases inin whichwhich PPand and QQ areareconsistent consistentwith withone oneanother another o but in which P does not notimply imply Q?Q?[(P [ (P0 Q) Q)and'" and ^(P_Q) (P—>Q)] ] 21. Which worlds-diagrams represent casescasesin inwhich whichP Pand andQ Qare areinconsistent inconsistentwith withone oneanother another and in which P does not notimply imply Q?Q?((P [ (P4> $Q) Q)and'" and ~(P-Q)] (P—>Q) ] 22. Which worlds-diagrams represent casescasesin inwhich whichP Pimplies impliesP? P?[P_PJ [P—>P] 23. Which worlds-diagrams represent cases in inwhich whichP Pis iscontingent, contingent,Q Qis isnecessarily necessarilytrue, true,and and in which P implies Q? [V'[VP,P, oQ,nQ and (P_Q)](P—>Q) ] 24. Which worlds-diagrams represent cases in inwhich whichP Pis isnecessarily necessarilytrue, true,Q Qis iscontingent, contingent,and and in which P implies Q? toP,[UP, V'Q,vQ, and (P-Q)](T—>QJ ] 25. Which worlds-diagrams represent cases inin whichwhich PP isisequivalent equivalent totoQ Qand and inin whichwhich PPand and Q are inconsistentinconsistent with one another? [(P[ (P<^->Q)...... Q) and and(P 4>(PQ)] QJ] 26. Which worlds-diagrams represent casescasesin inwhich whichP Pimplies'" implies ^P? P?[p- [P-*'" ~PJ P] 27. Which worlds-diagrams represent casescasesin inwhich which'" ^P Pimplies impliesP? P?[ '"[ P~ _P]P—tP] 28. Which worlds-diagrams represent casescasesin inwhich whichP Pis isconsistent consistentwith", with P?^ P?[Po'" [Po PJ~ P] 29. Which worlds-diagrams represent cases inin whichwhich bothbothP P impliesimplies '"^Q Q andand PP isisconsistent consistent with Q? [P-[/>->",QandPoQ] ^QandPoQ) 30. Which worlds-diagrams represent cases inin whichwhich PP isiscontingent contingentand and '"~P P isisnecessarily necessarily true? [V'[VPP and 0 d^P]'" P] 31 - 45. For each ofofthe thefifteen fifteenworlds-diagrams worlds-diagramsin infigure figure(1.i) (l.ifind) findany anytwo twopropositions propositionswhich whichstand stand in the relation depicted by that diagram.diagram.

Example: diagram 1414

Let "P" = There are fewer thanthan 30,00030,000galaxies. galaxies. Let "Q" = There are more thanthan 10,00010,000galaxies. galaxies.

PartBPart B

The expression "It is falsefalse thatthatP P impliesimplies Q"Q" isis ambiguousambiguous betweenbetweensaying saying"P "Pdoes doesnot notimply implyQ" Q"and and "P's being false implies Q" which may be symbolized unambiguously as "", (P_Q)"(P—>Q)" and "( '"~ P—*Q)P_Q)" "respectively. respectively. 46. Which worlds-diagrams represent casescasesin inwhich whichP Pdoes doesnot notimply implyQ? Q?['" [ (P-Q)~ (P^Q)}1 47. Which worlds-diagrams represent casescasesin inwhich whichP's P'sbeing beingfalse falseimplies impliesQ? Q?[ '"[ P-Ql^P—»Q] 48. Which worlds-diagrams represent cases inin whichwhich bothbothP's P's beingbeingtrue trueimplies implies QQ.and ar>-dP's P's being false impliesimplies Q? [(P_Q)[ (P->Q) andand ('" P_Q)(^P^QJ}1 § 6 ModalModal PropertiesProperties andand RelationsRelations PicturedPictured onon Worlds-DiagramsWorlds-Diagrams 5757

Part C

The expression "Q"Q is a false implication of P" means "Q is false and Q is an implication of P." In order to represent this relation on a set of worlds-diagrams we shall have to have aa devicedevicefor fordepicting depicting the actual world (i.e., for depicting that Q is false). Rather than persisting with the convention ofof figures (1.d)(l.d), , (1.e),(he), and (1(If),j), we shall hereinafter adopt the simpler convention of representing the location of the actual world among the set of possible worlds by "x"."X ". In worlds-diagrams 71 through 4, the "x""X" may be placed indiscriminately anywhere within the rectangle. But when we come toto diagrams 5 through 7515 we are faced with a number of alternatives. To show thesethesealternatives alternatives wewe willwill label each of the internal segments ofof thethe rectanglerectangle fromfrom leftleftto to rightright beginningbeginning withwith thethe letterletter "a"."a". Thus, forfor example,example, worlds-diagramworlds-diagram 7711 containingcontaining anan "x""X" inin thethe centralcentral segmentsegmentwould wouldbe bedesignated designated "77"11b".b". A worlds-diagram on which the location of the actual world is explicitly marked by an "x","X", we shall call a "-locating worlds-diagram".worlds-diagram". 49. Taking account of all the different ways the actual world may be depicted on a worlds-diagram, how many distinct reality-locating worlds-diagrams are there for two propositions? 50. Using the convention just discussed for describing a worlds-diagram on which the actual world is depicted, which worlds-diagrams represent cases inin whichwhich QQis isa afalse falseimplication implication ofofP?P? [(P---.QJ[(P-^Q) and 'V~QQ]] 57.51. Assume P to be the true proposition that there are exactly 730130 persons in some particular room, 2B. Let Q be the proposition that there are fewer than 740140 persons in room 2B. Draw a reality-locating worlds-diagram representing the modal relation between thesethesetwo two propositions. (For questions 52-55)52-55) AssumeAssumethat thatthere thereare areexactly exactly730 130persons personsin inroom room2B. 2B.Let LetP beP bethe theproposition proposition that there areare exactlyexactly 735135persons personsin inroom room2B, 2B,and andQ Qthe theproposition propositionthat thatthere thereare areat atleast least733 133persons persons in room 2B.2B. 52. Is Q implied by P? 53. Is Q true? 54. Is P true?true? 55. Draw a reality-locating worlds-diagram representing the modal relation between the propositions, P and Q.

Appendix to section 66 * ** ** ItIt isis anan interestinginteresting questionquestion toto ask,ask, "How"How manymany differentdifferent waysways maymay three arbitrarilyarbitrarily chosechosenn propositionpropositionss bbee arrangedarranged onon a worlds-diagram?"worlds-diagram?" TheThe answeranswer isis "255""255"..

ItIt isis possiblpossiblee toto givegive a generalgeneral formulaformula forfor thethe numbernumber (W(W'„n) ofof worldS-diagramsworlds-diagrams requiredrequired toto depicdepictt allall thethe possiblpossiblee waysways ofof arrangingarranging anyany arbitraryarbitrary numbernumber (n) ofof propositionspropositions.. ThatThat formulaformula isis4040

22n - Wnn == 22 " - 11

40.40. Alternatively,Alternatively, thethe formulaformula maymay bbee givengiven recursively,recursively, i.e.,i.e., WIWj = 3,3, andand

2 WW„n+r+ 7 =- (Wnn + 1)2-11) -1

UsingUsing thisthis latterlatter formulaformula itit isis easyeasy toto showshow thatthat thethe nextnext entryentry inin tabletable (7(1.j),J), i.e.,i.e., WW' ,would wouldbe be 3939 digitsdigitsin in 7 length.length. 5858 POSSIBLEPOSSIBLE WORLDSWORLDS

WeWe maymay calculatecalculate thethe valuevalue ofof Wn forfor thethe firstfirst fewfew valuesvalues ofof n.

n Wn

1 3

2 1515

3 255255

4 65,53565,535

5 4,294,967,2954,294,967,295

6 18,446,744,073,709,551,6118~446,744,073,709~551,6155

TABLETABLE (1.j)(l.j)

7.7. ISIS A SINGLESINGLE THEORYTHEORY OFOF TRUTHTRUTH ADEQUATEADEQUATE FORFOR BOTBOTHH CONTINGENTCONTINGENT ANDAND NONCONTINGENNONCONTINGENTT PROPOSITIONS?PROPOSITIONS?

InIn thisthis chapterchapter wewe havehave introducedintroduced oneone theorytheory ofof truth,truth, thethe so-calledso-called CorrespondenceCorrespondence TheoryTheory ofof TruthTruth whichwhich hashas itit thatthat a propositiopropositionn P,P, whichwhich ascribesascribes attributesattributes F toto anan itemitem a, isis truetrue ifif andand onlyonly ifif a hahass thethe attributesattributes F.F. SometimesSometimes thisthis theorytheory isis summedsummed upup epigrammaticallyepigrammatically bbyy sayingsaying thatthat a propositiopropositionn P isis truetrue ifif andand onlyonly ifif 'it'it fitsfits thethe facts'.facts'. NoNoww somesome philosopherphilosopherss whowho areare perfectlperfectlyy willingwilling toto acceptaccept thisthis theorytheory asas thethe correctcorrect accountaccount ofof ththee wayway thethe truth-valuestruth-values ofof contingentcontingent propositionpropositionss comecome about,about, havehave feltfelt thisthis samesame theorytheory toto bbee inadequateinadequate oror inappropriateinappropriate inin thethe casecase ofof noncontingennoncontingentt propositionspropositions.. WeWe woulwouldd dodo wellwell toto revierevieww thethe sortsort ofof thinkingthinking whichwhich mightmight leadlead oneone toto thisthis opinionopinion.. SupposeSuppose oneone werewere toto choosechoose asas one'sone's firstfirst exampleexample a contingentcontingent propositionproposition,, letlet usus saysay ththee propositiopropositionn

(1.42) CanadaCanada isis northnorth ofof MexicoMexico..

WhenWhen oneone asksasks whatwhat makesmakes (1.42)(1-42) actuallyactually true,true, thethe answeranswer isis obviousobvious enough:enough: thisthis particulaparticularr propositiopropositionn isis actuallyactually truetrue becausbecausee ofof certaincertain peculiapeculiarr geographicalgeographical featuresfeatures ofof thethe actualactual worldworld,, featuresfeatures whichwhich areare sharedshared bbyy somesome otherother possiblpossiblee worldsworlds bubutt definitelydefinitely notnot bbyy all.all. FollowingFollowing alongalong inin thisthis vein,vein, wewe cancan askask a similarsimilar questionquestion aboutabout noncontingentnoncontingent propositionspropositions.. LeLett usus taketake asas anan exampleexample t~ethe noncontingentnoncontingent propositiopropositionn

(1.43) EitherEither BoothBooth assassinatedassassinated AbrahamAbraham LincolnLincoln oror itit isis notnot thethe casecase thatthat BootBoothh assassinatedassassinated AbrahamAbraham LincolnLincoln.. § 77 IsIs aa SingleSingle TheoryTheory ofof TruthTruth AdequateAdequate forfor BothBoth ContingentContingent andand NoncontingentNoncontingent Propositions?Propositions? 5599

IfIf wewe nownow askask whatwhat makesmakes thisthis propositionproposition true,true, somesome personpersonss havhavee felfeltt thathatt ththee answeanswerr cannocannott bbee ooff thethe samesame sortsort asas thatthat justjust givengiven inin tnethe casecase ofof thethe exampleexample ooff a contingencontingentt propositionproposition.. FoForr thesthesee personspersons rightlyrightly notenote thatthat whereaswhereas thethe truth-valuetruth-value ofof (1.42)(1-42) varievariess frofromm possiblpossiblee worlworldd ttoo possiblpossiblee world,world, thethe truth-valuetruth-value ofof (1.43) doesdoes not.not. NoNo mattermatter hohoww anotheranother possiblpossiblee worlworldd mamayy diffedifferr frofromm ththee actualactual world,world, nono mattermatter howhow outlandishoutlandish andand farfetchedfarfetched thathatt worlworldd mighmightt seeseemm ttoo uuss iinn ththee actuaactuall world,world, thethe truth-valuetruth-value ofof (1.43) willwill bebe thethe samesame inin thatthat worlworldd aass iinn ththee actuaactuall worldworld.. IInn ththee actuaactuall world,world, BoothBooth diddid assassinateassassinate LincolnLincoln andand (1.43)(1-43) isis true.true. BuButt thertheree araree otheotherr possiblpossiblee worldworldss iinn whicwhichh LincolnLincoln diddid notnot gogo toto Ford'sFord's TheaterTheater onon AprilApril 14,14, 1865,1865, andand livelivedd ttoo bbee re-electere-electedd ttoo a thirthirdd termterm;; yetyet inin thosethose worlds,worlds, (1.43) isis true.true. Then,Then, too,too, therethere areare thosethose possiblpossiblee worldworldss iinn whicwhichh BootBoothh dididd shooshoott Lincoln,Lincoln, butbut thethe woundwound waswas a superficialsuperficial oneone andand LincolnLincoln recoveredrecovered;; yeyett iinn thosthosee worldsworlds,, tootoo,, (1.43)(1.43) isis truetrue.. InIn thethe eyeseyes ofof somesome philosophersphilosophers thesethese latterlatter sortssorts ofof factsfacts challengchallengee ththee correspondenccorrespondencee theortheoryy ooff truth.truth. InIn effecteffect thesethese philosophersphilosophers argueargue thatthat thethe truthtruth ofof noncontingennoncontingentt propositionpropositionss cannocannott bbee accountedaccounted forfor byby sayingsaying thatthat 'they'they fitfit thethe facts'facts' sincesince theythey remairemainn trutruee whatevewhateverr ththee factfactss happehappenn ttoo bebe.. HowHow couldcould thethe truthtruth ofof (1.43)(1-43) inin oneone possiblepossible worldworld bbee explainedexplained iinn termtermss ooff ononee sesett ooff factfactss anandd ititss truthtruth inin anotheranother possiblepossible worldworld inin termsterms ofof a differentdifferent setset ofof facts?facts? ThThee ververyy suggestiosuggestionn hahass seemeseemedd ttoo thesethese philosophersphilosophers toto constituteconstitute a fatalfatal weaknessweakness inin thethe correspondencecorrespondence theortheoryy ooff truthtruth.. IInn theitheirr vievieww therethere justjust doesn'tdoesn't seemseem toto bebe anyany properproper setset ofof factsfacts ttoo whicwhichh trutruee noncontingennoncontingentt propositionpropositionss 'correspond'.'correspond'. AndAnd so,so, theythey havehave beenbeen ledled toto proposepropose additional theorietheoriess ooff truthtruth,, theorietheoriess ttoo accounaccountt forfor thethe truth-valuestruth-values ofof noncontingentnoncontingent propositionspropositions.. OneOne ofof thethe oldestoldest supplementalsupplemental theoriestheories invokedinvoked toto accountaccount foforr ththee truttruthh ooff necessarnecessaryy truthtruthss holdholdss thatthat theirtheir truthtruth dependsdepends uponupon whatwhat werewere traditionallytraditionally calledcalled "The"The LawLawss ooff Thought"Thought",, oorr whawhatt mosmostt philosophersphilosophers nowadaysnowadays preferprefer toto callcall "The"The LawsLaws ofof Logic".Logic". PreciselPreciselyy whawhatt iiss encompasseencompassedd withiwithinn thesethese so-calledso-called lawslaws ofof thoughtthought hashas beenbeen anan issueissue ofof longlong disputedispute amonamongg philosophersphilosophers.. NeverthelessNevertheless,, iitt seemsseems fairlyfairly clearclear thatthat whateverwhatever elseelse isis toto bebe included,included, ththee lawlawss ooff thoughthoughtt ddoo includincludee ththee 'Laws'Laws'' ooff Identity,Identity, ofof TheThe ExcludedExcluded Middle,Middle, andand ofof Noncontradiction:Noncontradiction: roughlyroughly,, ththee 'laws'laws'' thathatt aann iteitemm iiss whawhatt iitt is,is, thatthat eithereither anan itemitem hashas a certaincertain attributeattribute oror itit doesdoes notnot,, anandd thathatt aann iteitemm cannocannott botbothh havhavee aa certaincertain attributeattribute andand failfail toto havehave itit.. ButBut thisthis supplementalsupplemental theorytheory turnsturns out,out, onon examination,examination, ttoo bbee whollwhollyy lackinlackingg iinn explanatorexplanatoryy powerpower.. ForFor ifif therethere isis controversycontroversy overover thethe mattermatter ofof whatwhat isis toto bbee includeincludedd amonamongg ththee lawlawss ooff thoughtthought,, thertheree hashas beenbeen eveneven greatergreater controversy,controversy, withwith moremore far-reachingfar-reaching implicationsimplications,, oveoverr ththee mattematterr ooff whawhatt sortsortss ofof thingsthings thesethese so-calledso-called "laws"laws ofof thought"thought" are. TheThe accountaccount whicwhichh seemseemss ttoo worworkk besbestt iinn explaininexplainingg whatwhat thethe lawslaws ofof thoughtthought are,are, isis thatthat whichwhich sayssays ofof themthem thathatt thetheyy araree themselvethemselvess nothinnothingg otheotherr thathann noncontingentlynoncontingently truetrue propositions.propositions. ButBut ifif thisthis accountaccount isis adoptedadopted —- anandd iitt iiss bbyy mosmostt contemporarcontemporaryy philosophersphilosophers -— thenthen itit isis veryvery hardhard toto seesee howhow thethe lawslaws ofof thoughthoughtt cacann servservee aass aann explanatioexplanationn ooff whatwhat itit isis thatthat thethe truth-valuestruth-values ofof noncontingentnoncontingent propositionspropositions dependependd onon.. ThThee troubltroublee iiss thathatt iiff aa certaincertain classclass ofof noncontingentlynoncontingently truetrue propositions,propositions, thethe honorehonoredd "laws"laws ooff thought"thought",, araree ttoo accounaccountt foforr thethe truthtruth (or(or falsity)falsity) ofof otherother (the(the run-of-the-mill)run-of-the-mill) noncontingennoncontingentt propositionspropositions,, thethenn somsomee furtherfurther,, presumablypresumably stillstill moremore honoredhonored propositionspropositions mustmust accountaccount forfor their truthtruth,, anandd ssoo oonn anandd oonn withouwithoutt end.end. InIn short,short, invokinginvoking somesome propositionspropositions toto accountaccount forfor ththee truttruthh oorr falsitfalsityy ooff otherotherss leadleadss ononee ttoo aann infiniteinfinite regressregress oror landslands oneone inin a circularity.circularity. A secondsecond supplementalsupplemental theory,theory, whichwhich wewe shallshall callcall "the"the linguisticlinguistic theortheoryy ooff necessarnecessaryy truth"truth",, holdholdss thatthat thethe truth-valuestruth-values ofof nonnoncontingencontingentt propositionspropositions comecome about,about, nonott througthroughh correspondenccorrespondencee witwithh ththee facts,facts, butbut ratherrather asas a resultresult ofof certaincertain 'rules'rules ofof language'.language'. CapitalizingCapitalizing oonn ththee ververyy worwordd "contingent""contingent",, thisthis theorytheory isis sometimessometimes expressedexpressed inin thethe followingfollowing way:way: TheThe truth-valuetruth-valuess ooff contingencontingentt propositionpropositionss areare contingent uponupon thethe facts,facts, whilewhile thethe truth-valuestruth-values ofof noncontingennoncontingentt propositionpropositionss araree not contingencontingentt uponupon thethe factsfacts butbut areare determineddetermined byby rulesrules ofof language.language. OrOr puputt stilstilll anotheanotherr wayway:: contingencontingentt truthtruthss (and(and falsehoods)falsehoods) areare factual;factual; noncontingentnoncontingent truthstruths (and(and falsehoods)falsehoods) araree linguisticlinguistic.. 6060 POSSIBLPOSSIBLEE WORLDSWORLDS

TheThe linguisticlinguistic theorytheory ofof necessarynecessary truthtruth doesdoes notnot belonbelongg onlonlyy ttoo ththee preservpreservee ooff philosophersphilosophers.. IItt iiss enshrinedenshrined alsoalso inin commonplacecommonplace talktalk ofof certaincertain propositionspropositions beinbeingg trutruee (o(orr falsefalse)) "b"byy definition"definition".. ThusThus,, itit mightmight bebe said,said, thethe propositionproposition

(1.44) AllAll rectanglesrectangles havehave fourfour sidessides

owesowes itsits truthtruth toto thethe factfact thatthat wewe humanhuman beingsbeings havehave resolveresolvedd ttoo define ththee tertermm "rectangle"rectangle"" aass "plane"plane closedclosed figurefigure withwith fourfour straightstraight sidessides allall ofof whosewhose anglesangles araree equal"equal".. ForFor,, givegivenn sucsuchh a stipulativstipulativee definition,definition, thethe truthtruth ofof (1.44) followsfollows immediatelyimmediately.. ThisThis talktalk ofof "definitional"definitional truths",truths", andand similarsimilar talktalk ofof "verbal"verbal truths"truths",, suggestsuggestss thathatt necessarnecessaryy truthtruthss areare notnot groundedgrounded -— asas areare contingentcontingent truthstruths -— inin correspondencecorrespondence betweebetweenn propositionpropositionss anandd statestatess ofof affairs,affairs, butbut areare groundedgrounded ratherrather inin somesome sortsort ofof correspondencecorrespondence betweebetweenn propositionpropositionss anandd ththee arbitraryarbitrary conventionsconventions oror rulesrules forfor thethe useuse ofof wordswords whichwhich wewe languagelanguage useruserss happehappenn ttoo adopt.adopt. Yet,Yet, despitedespite itsits initialinitial plausibility,plausibility, thisthis theorytheory seemsseems ttoo uuss defective.defective. ItsIts plausibility,plausibility, wewe suggest,suggest, derivesderives -— atat leastleast inin parpartt —- frofromm a confusionconfusion.. LeLett uuss concedconcedee thathatt whenwhen wewe introduceintroduce a newnew termterm intointo ourour languagelanguage bbyy definingdefining iitt (explicitl(explicitlyy oorr implicitlyimplicitly)) iinn termtermss ofof alreadyalready availableavailable expressions,expressions, wewe makemake availableavailable toto ourselvesourselves new ways of expressing truthstruths.. BuButt thithiss doesdoes notnot meanmean thatthat wewe therebythereby makemake availableavailable toto ourselvesourselves neneww truths.truths. OnceOnce wewe havehave introducedintroduced thethe termterm "rectangle""rectangle" bbyy definitiondefinition iinn ththee mannemannerr sketchesketchedd aboveabove,, ththee sentencesentence

(1.45) "All"All rectanglesrectangles havehave fourfour sides"sides" willwill expressexpress a necessarynecessary truthtruth inin a wayway inin whichwhich wwee coulcouldd nonott havhavee expresseexpressedd iitt earlierearlier.. BuButt ththee necessarynecessary truthtruth whichwhich itit expressesexpresses isis jusjustt thethe necessarynecessary truthtruth

(1.46) AllAll planeplane closedclosed figuresfigures withwith fourfour sidessides alalll ooff whosewhose anglesangles areare equalequal havehave fourfour sidessides andand thisthis isis a propositionproposition whosewhose truthtruth isis farfar lessless plausiblplausiblyy attributablattributablee ttoo definitiondefinition.. FoForr thithiss proposition,proposition, oneone isis moremore inclinedinclined toto say,say, wouldwould bbee trutruee eveneven iiff humahumann beingbeingss hahadd neveneverr existeexistedd anandd hadhad therethere nevernever beenbeen anyany languagelanguage whateverwhatever inin thethe worldworld.. TalkTalk aboutabout necessarynecessary truthstruths beingbeing "true"true byby definition"definition" mamayy alsalsoo derivderivee somsomee plausibilitplausibilityy frofromm ththee factfact thatthat ourour knowledge thatthat certaincertain propositionspropositions areare trutruee sometimessometimes stemsternss frofromm ouourr knowledge ooff ththee meanings,meanings, oror definitions,definitions, ofof thethe termsterms inin whichwhich theythey areare expressed.expressed. ThuThuss a persopersonn whwhoo comecomess ttoo knoknoww thatthat thethe expressionexpression "triac""triac" means the same as thethe expressionexpression "bidirectiona"bidirectionall triodtriodee thyristorthyristor"" mamayy therebythereby (even(even withoutwithout knowingknowing independentlyindependently whatwhat eithereither meansmeans)) knoknoww thathatt ththee sentencesentence

(7.47)(1.47) "All"All triacstriacs areare bidirectionalbidirectional triodtriodee thyristorsthyristors"" expressesexpresses a necessarilynecessarily truetrue proposition.proposition. ButBut thisthis doesdoes nonott meameann thathatt ththee necessarnecessaryy truttruthh ooff ththee propositionproposition soso expressedexpressed isis itselfitself somethingsomething thatthat sternsstems fromfrom ththee meaningmeaningss ooff thesthesee expressions.expressions. PerhapsPerhaps thethe gravestgravest defectdefect inin thethe linguisticlinguistic theorytheory ofof necessarnecessaryy truttruthh hahass ttoo ddoo witwithh ititss inabilitinabilityy ttoo explainexplain howhow itit isis thatthat necessarynecessary truths,truths, suchsuch asas thosethose ofof logilogicc anandd mathematicsmathematics,, cacann havhavee significansignificantt practicalpractical applicationsapplications inin thethe realreal world.world. InIn suchsuch appliedapplied sciencessciences aass aeronauticsaeronautics,, engineeringengineering,, anandd ththee like,like, arithmeticalarithmetical truthstruths maymay bebe appliedapplied inin waysways whichwhich yielyieldd importanimportantt neneww inferentiainferentiall truthtruthss aboutabout thethe worldworld aroundaround us:us: aboutabout thethe stressstress tolerancestolerances ofof bridgesbridges,, ththee efficiencefficiencyy ooff airplanairplanee propellerspropellers,, etcetc.. ButBut ifif thesethese necessarynecessary truthstruths areare merelymerely thethe resultresult ofof arbitraryarbitrary humahumann conventionconventionss foforr ththee ususee ofof § 7 IsIs a SingleSingle TheoryTheory ofof TruthTruth AdequateAdequate forfor BothBoth ContingentContingent andand NoncontingentNoncontingent Propositions?Propositions? 6161

mathematicalmathematical symbols,symbols, allall thisthis becomebecomess a seemingseeming miracle.miracle. WhyWhy shouldshould thethe worldworld conformconform soso felicitiouslyfelicitiously toto thethe consequencesconsequences ofof ourour linguisticlinguistic stipulations?stipulations? TheThe explanationexplanation ofof thethe successsuccess ofof logiclogic andand mathematicsmathematics inin theirtheir applicationsapplications toto thethe world,world, wewe suggest,suggest, lieslies elsewhere:elsewhere: inin thethe possible-worldpossible-worldss analysisanalysis ofof necessarynecessary truth.truth. NecessarNecessaryy truths,truths, suchsuch asas thosethose ofof mathematics,mathematics, applyapply toto thethe worldworld becausbecausee theythey areare truetrue inin allall possiblpossiblee worlds;worlds; andand sincesince thethe actualactual worldworld isis a possiblpossiblee world,world, itit follows thatthat theythey areare truetrue inin (i.e.,(i.e., applyapply to)to) thethe actualactual world.world. ThisThis accountaccount ofof thethe matter,matter, itit shouldshould bbee noted,noted, doesdoes notnot requirerequire a differentdifferent theorytheory ofof truthtruth fromfrom thatthat whicwhichh wewe havehave givengiven forfor contingentcontingent propositionspropositions.. ToTo saysay thatthat a propositiopropositionn isis truetrue inin thethe actualactual worldworld is,is, wewe havehave claimed,claimed, toto saysay thatthat itit fitsfits thethe factsfacts inin thethe actualactual world,world, i.e.,i.e., thatthat statesstates ofof affairsaffairs inin thethe actuaactuall worldworld areare asas thethe propositiopropositionn assertsasserts themthem toto bebe.. AndAnd likewise,likewise, toto saysay thatthat a propositiopropositionn isis truetrue inin somsomee otherother possiblpossiblee worldworld -— or,or, forfor thatthat matter,matter, inin allall possiblpossiblee worldsworlds -— isis jusjustt toto saysay thatthat inin thatthat otherother worldworld -— oror allall possiblpossiblee worldsworlds -— statesstates ofof affairsaffairs areare asas thethe propositiopropositionn assertsasserts themthem toto bebe.. OneOne anandd thethe samesame theorytheory ofof truthtruth sufficessuffices forfor allall cases:cases: thethe casecase whenwhen a propositiopropositionn isis truetrue inin allall possiblpossiblee worldworldss andand thethe casecase whenwhen itit isis truetrue inin jusjustt somesome (perhaps(perhaps thethe actualactual oneone included)included).. InIn orderorder toto seesee howhow thisthis singlesingle accountaccount ofof truthtruth sufficessuffices forfor allall cases,cases, letlet usus returnreturn toto thethe exampleexample ooff thethe noncontingentnoncontingent propositiopropositionn (1.43)(1-43) toto seesee howhow thisthis theorytheory mightmight bbee invokedinvoked toto explainexplain itsits truth.truth. ButBut first,first, letlet usus reflectreflect forfor a momentmoment onon anotheranother propositionproposition,, viz.,viz., thethe contingent propositiopropositionn

(1.48) EitherEither GeorgeGeorge WashingtonWashington waswas thethe first presidenpresidentt ofof thethe UnitedUnited StatesStates oorr BenjaminBenjamin FranklinFranklin waswas thethe first presidenpresidentt ofof thethe UnitedUnited StatesStates..

ThisThis propositionproposition,, beinbeingg contingent,contingent, isis truetrue inin somesome possiblpossiblee worldsworlds andand falsefalse inin allall thethe others.others. WhaWhatt featurefeature isis it,it, inin thosethose possiblpossiblee worldsworlds inin whichwhich (1.48) isis true,true, whichwhich accountsaccounts forfor itsits truth?truth? ThThee answeranswer isis obvious:obvious: (1.48)(1-48) isis truetrue inin allall andand onlyonly thosethose possiblpossiblee worldsworlds inin whichwhich eithereither WashingtonWashington oorr FranklinFranklin waswas thethe first presidenpresidentt ofof thethe UnitedUnited States.States. ButBut notenote -— asas inin thethe casecase ofof thethe noncontingentnoncontingent propositiopropositionn (1.43)(7.43) -•— howhow muchmuch variationvariation occursoccurs betweebetweenn thethe worldsworlds inin whichwhich thethe propositiopropositionn isis truetrue.. InIn thethe actualactual world,world, forfor example,example, Washington,Washington, notnot Franklin,Franklin, waswas thethe firstfirst presidenpresidentt ofof thethe UniteUnitedd States.States. ThusThus inin thethe actualactual world,world, (1.48)(1-48) isis true.true. ButBut inin somesome otherother possiblpossiblee worlds,worlds, Franklin,Franklin, notnot Washington,Washington, waswas thethe first presidenpresidentt ofof thethe UnitedUnited States.States. ButBut inin thatthat world,world, too,too, (1.48) fitsfits thethe facts,facts, i.e.,i.e., isis truetrue.. Here,Here, then,then, wewe havehave a paralleparallell toto thethe situationsituation wewe discovereddiscovered inin thethe casecase ofof thethe noncontingentnoncontingent propositiopropositionn (1.43),.(1.43); thatthat is,is, wewe havehave alreadyalready notednoted thatthat (1.43) isis truetrue inin somesome possiblpossiblee worldworld inin whicwhichh BoothBooth assassinatedassassinated LincolnLincoln andand isis truetrue inin somesome possiblpossiblee worldworld inin whichwhich hehe diddid not.not. TheThe relevantrelevant difference,difference, inin thethe presenpresentt context,context, betweebetweenn thesethese twotwo casescases -— thethe noncontingentnoncontingent propositiopropositionn (1.43)(1.43) andand thethe contingentcontingent propositiopropositionn (1.48)(1-48) -— lieslies inin thethe factfact thatthat thethe formerformer propositiopropositionn isis truetrue inin everyevery possiblpossiblee worldworld inin whichwhich BoothBooth diddid notnot assassinateassassinate Lincoln,Lincoln, whilewhile thethe latterlatter isis truetrue inin onlyonly somesome ofof thethe possiblpossiblee worldsworlds inin whichwhich WashingtonWashington waswas notnot thethe first presidentpresident.. ButBut thisthis differencedifference clearlyclearly hashas onlonlyy toto dodo withwith whetherwhether thethe setset ofof possiblpossiblee worldsworlds inin whichwhich eacheach propositiopropositionn fits thethe factsfacts comprisescomprises allall,, onlyonly some,some, oror nonenone ofof thethe totalitytotality ofof possiblpossiblee worlds.worlds. ItIt inin nono wayway challengeschallenges thethe claimclaim thatthat thethe truttruthh ofof botbothh (1.43) andand (1.48) isis toto bbee accountedaccounted forfor inin thethe samesame way,way, viz.,viz., inin theirtheir fitting thethe facts.facts. SummingSumming up,up, wewe maymay saysay thatthat therethere isis nono special probleproblemm aboutabout thethe truth-valuestruth-values ofof noncontingentnoncontingent propositionspropositions.. TheyThey areare truetrue oror falsefalse in exactly the same sort of way thatthat contingentcontingent propositionpropositionss araree truetrue oror false;false; i.e.,i.e., dependingdepending onon whetherwhether oror notnot theythey fit thethe facts.facts. TheThe suppositionsupposition thatthat therethere areare twotwo kindskinds ofof truth,truth, oror thatthat therethere isis a needneed forfor twotwo theoriestheories ofof truth,truth, isis misconceived.misconceived. Propositions areare trutruee oror false;false; alsoalso theythey areare contingentcontingent oror noncontingent.noncontingent. AndAnd thesethese variousvarious propertiepropertiess ofof propositionpropositionss (two(two forfor truth-statustruth-status andand twotwo forfor modal-status)modal-status) cancan combinecombine inin variousvarious waysways (four,(four, toto bbee exact).exact). ButBut iitt shouldshould notnot bebe thoughtthought thatthat inin sayingsaying ofof a propositionproposition thatthat itit is,is, forfor example,example, noncontingentlynoncontingently true,true, wewe areare sayingsaying thatthat itit exemplifiesexemplifies oneone ofof twotwo typestypes ofof truth. Rather,Rather, thisthis expressionexpression oughtought toto bbee construeconstruedd asas "noncontingent"noncontingent and true".true". ViewingViewing thethe mattermatter inin thisthis fashion,fashion, itit isis easyeasy toto seesee thatthat thethe questionquestion 6622 POSSIBLPOSSIBLEE WORLDSWORLDS

abouaboutt ththee speciaspeciall wawayy iinn whicwhichh ththee truttruthh (o(orr falsityfalsity)) ooff noncontingennoncontingentt propositionpropositionss iiss supposesupposedd ttoo comcomee abouaboutt does not even arise. LogiLogicc doedoess nonott requirrequiree mormoree thathann ononee theortheoryy ooff truth.truth.

88.. THTHEE "POSSIBLE-WORLDS"POSSIBLE-WORLDS"" IDIOMIDIOM

ThroughouThroughoutt thithiss boobookk wwee speaspeakk ofteoftenn ooff otheotherr possiblpossiblee worldsworlds.. WhWhyy havhavee wewe,, iinn commocommonn witwithh manmanyy otheotherr philosopherphilosopherss anandd logicianslogicians,, adopteadoptedd thithiss idiomidiom?? ThThee answeanswerr lieliess iinn ititss enormouenormouss heuristiheuristicc valuevalue.. ManManyy contemporarcontemporaryy philosopherphilosopherss believbelievee thathatt ththee possible-worldpossible-worldss idioidiomm provideprovidess a singlsinglee theoreticatheoreticall frameworframeworkk powerfupowerfull enougenoughh ttoo illuminatilluminatee anandd resolvresolvee manmanyy ooff ththee philosophicaphilosophicall problemsproblems surroundinsurroundingg sucsuchh mattermatterss aass

11.. ThThee logicalogicall notionnotionss ooff necessitynecessity,, contingencycontingency,, possibilitypossibility,, implicationimplication,, validityvalidity,, anandd ththee liklikee

22.. ThThee distinctiodistinctionn betweebetweenn logicalogicall necessitnecessityy anandd physicaphysicall necessitynecessity

33.. ThThee adequacadequacyy ooff a singlesingle theortheoryy ooff truthtruth

44.. ThThee identitidentityy conditionconditionss ooff propositionpropositionss anandd ooff conceptsconcepts

5.5. ThThee disambiguatiodisambiguationn ooff sentencessentences

66.. ThThee linlinkk betweebetweenn ththee meaninmeaningg ooff a sentencesentence anandd ththee truth-conditiontruth-conditionss ooff ththee proposition(sproposition(s)) iitt expressesexpresses

7.7. ThThee techniqutechniquee ooff refutatiorefutationn bbyy imaginarimaginaryy counterexamplescounterexamples

8.8. ThThee epistemiepistemicc concepconceptt ooff thathatt whicwhichh iiss humanlhumanlyy knowableknowable

99.. ThThee distinctiodistinctionn betweebetweenn accidentaaccidentall anandd essentiaessentiall propertiesproperties

10.10. TheThe conceptconcept ofof thethe contingentcontingent contentcontent ofof a propositiopropositionn

11.11. TheThe conceptconcept ofof probabilificatioprobabilificationn

EachEach ofof thesethese isis dealtdealt with,with, inin thethe orderorder given,given, inin thisthis book.book. ButBut thethelist list goesgoes beyondbeyond thetheconfines confines ofof thisthis book.book. RecentRecent workwork couchedcouched withinwithin thethe possible-worldspossible-worlds idiomidiom includesincludes workwork iinn

12.12. EthicsEthics

13.13. CounterfactualsCounterf actuals

14.14. EpistemicEpistemic logiclogic

15.15. PropositionalPropositional attitudesattitudes andand muchmuch moremore besides.besides. ToTo citecite somesome examples:examples: C.B.C.B. DanielsDaniels inin The Evaluation of Ethical Theories:Theories:

ForFor thethe purposespurposes ofof thisthis book,book, anan ethicalethical theorytheory isis simplysimply a determinationdetermination ofof a uniqueunique setset ofof idealideal worlds.worlds. AnAn idealideal world,world, relativerelative toto a theorytheory thatthat determinesdetermines itit asas ideal,ideal, isis a possiblepossible worldworld,, a fairyfairy worldworld perhaps,perhaps, inin whichwhich everythingeverything thethe theorytheory sayssays isis ideallyideally truetrue isis inin factfact true.true. AAnn idealideal worldworld relativerelative toto anan ethicalethical theorytheory isis a worldworld inin whichwhich everythingeverything thethe theorytheory sayssays oughtought ttoo 41 bebe thethe casecase isis thethe case.case.41

41.41. Halifax,Halifax, NovaNova Scotia,Scotia, DalhousieDalhousie UniversityUniversity Press,Press, 1975,1975, p.p. 11.. § 8 TheThe "Possible-Worlds""Possible-Worlds" IdiomIdiom 6363

DavidDavid Lewis,Lewis, inin Counterfactuals:Counter/actuals:

'IflIf kangaroos had no tails, they would topple over' seemsseems toto meme toto meanmean somethingsomething likelike this:this: iinn anyany possiblpossiblee statestate ofof affairsaffairs inin whichwhich kangarooskangaroos havehave nnoo tails,tails, andand whicwhichh resembleresembless ourour actuaactuall statestate ofof affairsaffairs asas much asas kangarooskangaroos havinghaving nono tailstails permitspermits itit to,to, thethe kangarooskangaroos toppletopple over.over. I shallshall givegive a generalgeneral analysisanalysis ofof counterfactualcounterfactual conditionsconditions alongalong thesethese lines.lines. MyMy methodsmethods areare thosethose ofof muchmuch recentrecent workwork inin possible-worlpossible-worldd semanticssemantics forfor intensionalintensional logic. 4242

AndAnd ]aakkoJaakko Hintikka,Hintikka, inin "On"On thethe LogicLogic ofof Perception"Perception"::

WhenWhen doesdoes a knowknow (believe,(believe, wish,wish, perceiveperceive)) moremore thanthan b? TheThe onlyonly reasonablereasonable generalgeneral answeranswer seemsseems toto bbee thatthat a knowsknows moremore thanthan b ifif andand onlyonly ifif thethe classclass ofof possiblpossiblee worldsworlds compatiblecompatible witwithh whatwhat hehe knowsknows isis smallersmaller thanthan thethe classclass ofof possiblpossiblee worldsworlds compatiblecompatible withwith whatwhat b knows.4343

TheThe extraordinaryextraordinary success,success, andand thethe veritableveritable explosionexplosion ofof researchresearch adoptingadopting thethe possible-worldspossible-worlds idiomidiom wawass nonott thethe motivatiomotivationn oror eveneven thethe expectationexpectation inin itsits adoption.adoption. AsAs a mattermatter ofof fact,fact, thethe idiomidiom isis notnot especiallyespecially new;new; onlyonly itsits widespreadwidespread adoptionadoption is.is. TalkTalk ofof possiblpossiblee worldsworlds cancan bbee foundfound throughoutthroughout thethe writingswritings ofof thethe greatgreat GermanGerman mathematicianmathematician andand philosopherphilosopher,, GottfriedGottfried LeibnizLeibniz (1646-1716).(1646-1716). LeibnizLeibniz waswas quitequite atat homehome inin thethe possible-worldpossible-worldss idiom.idiom. HeHe likedliked toto philosophizphilosophizee inin termsterms ofof thatthat idiom,idiom, askingasking suchsuch questionsquestions as,as, "Is"Is thisthis thethe besbestt ofof allall possiblpossiblee worlds?"worlds?" andand "Why"Why diddid GodGod createcreate thisthis particulaparticularr worldworld ratherrather thanthan somesome otherother possiblpossiblee world?"world?" AndAnd writingwriting ofof thosethose truthstruths whichwhich wewe (and(and hhee onon occasion)occasion) havehave calledcalled "necessary","necessary", hehe pennepennedd thesethese germinalgerminal lines:lines:

TheseThese areare thethe eternaleternal truths.truths. TheyThey diddid notnot obtainobtain onlyonly whilewhile thethe worldworld existed,existed, butbut theythey woulwouldd alsoalso obtainobtain ifif GODGOD hadhad createdcreated a worldworld withwith a differentdifferent planplan.. ButBut fromfrom these,these, existentialexistential oorr contingentcontingent truthstruths differdiffer entirely.44entirely.44

Leibniz'sLeibniz's stylestyle andand idiomidiom werewere inin advanceadvance ofof theirtheir time.time. OtherOther logicians,logicians, andand perhapperhapss LeibnizLeibniz too,too, sawsaw nono particulaparticularr advantage inin thisthis wayway ofof talking,talking, merelymerely anan alternative.alternative. ItIt waswas nOlnoi. untiuntill thethe earlyearly 19601960ss thatthat philosopherphilosopherss suchsuch asas KripkeKripke andand HintikkaHintikka returnedreturned toto Leibniz'sLeibniz's idiomidiom andand useusedd itit botbothh toto illuminateilluminate thethe philosophicaphilosophicall basebasess ofof logiclogic andand toto puspushh thethe frontiersfrontiers ofof logiclogic andand manymany otherother areasareas ooff philosophphilosophyy inin newnew directions.directions. NonethelessNonetheless,, itit isis importantimportant toto keepkeep thisthis talktalk ofof 'other'other possiblpossiblee worlds'worlds' inin itsits propeproperr philosophicaphilosophicall perspectiveperspective.. WeWe mustmust nevernever allowallow ourselvesourselves toto regardregard thisthis mannermanner ofof talktalk asas ifif itit werewere talktalk aboutabout actuallyactually existingexisting partsparts ofof thisthis world.world. WhoeverWhoever supposessupposes thatthat otherother possiblepossible worldsworlds areare basicbasic entitiesentities ooff thethe physicalphysical universeuniverse hashas failedfailed toto appreciateappreciate thethe pointpoint ofof ourour earlierearlier insistenceinsistence thatthat otherother possiblepossible worldsworlds areare notnot 'out'out there'there' inin physicaphysicall space.space. OtherOther possiblpossiblee worldsworlds areare abstract entitiesentities likelike numbersnumbers

42.42. Cambridge,Cambridge, Mass.,Mass., HarvardHarvard UniversityUniversity Press,Press, 1973,1973, pp.. 11..

43.43. Models for Modalities: Selected Essays, Dordrecht,Dordrecht, D.D. Reidel,Reidel, 1969,1969, pp.. 157.157.

44.44. "Necessary."Necessary andand ContingentContingent Truths"Truths" translatedtranslated fromfrom Opuscules et fragmentsfragments iniditsinedits de Leibniz, eded.. Courturat,Courturat, Paris,Paris, FelixFelix Alcan,Alcan, 1903,1903, pp pp.. 16-22,16-22, reprintedreprinted inin From Descartes to Locke ed.ed. T.V.T.V. SmithSmith anandd M.M. Grene,Grene, Chicago,Chicago, TheThe UniversityUniversity ofof ChicagoChicago Press,Press, 1957,1957, pppp.. 306-312.306-312. WeWe areare gratefulgrateful toto ourour colleague,colleague, DavidDavid Copp,Copp, forfor callingcalling thisthis passagpassagee toto ourour attention.attention. 6464 POSSIBLEPOSSIBLE WORLDSWORLDS

andand propositions.4545 WeWe feelfeel drivendriven toto posipositt themthem becausbecausee wewe seemseem unablunablee toto makemake ultimateultimate sensesense ooff logiclogic withoutwithout them.them. PositingPositing thethe existenceexistence ofof thesethese abstractabstract entitiesentities allowsallows usus toto unifyunify logic,logic, toto rigorizerigorize it,it, toto expandexpand it,it, andand mostmost importantlyimportantly inin ourour eyeseyes toto understand itit..

45.45. ThisThis distinctiondistinction betweebetweenn abstractabstract andand non-abstractnon-abstract (i.e.,(i.e., concrete)concrete) entitiesentities willwill bbee examinedexamined atat greatergreater lengthlength inin thethe nextnext chapter.chapter.