AN INTRODUCTION TO LOGIC AND ITS PHILOSOPHY

• .. an Introduction
~- to Logic and

, Its Philosophy

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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Copyriigghhtt©© 1979 byy Raymond Bradley anndd Norman Swartz

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Contents

P R E F A C E xv
T O T H E T E A C H E R xvii
T O T H E S T U D E N T xxi

POSSIBLE WORLDS 1

1. T H I S A N D O T H E R POSSIBLE W O R L D S 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
Logical possibility distinguished from other kinds 6

The constituents of possible worlds 7

2. PROPOSITIONS, T R U T H , A N D F A L S I T Y 9
Truth and falsity defined 9
Truth in a possible world 11
Truth in the actual world 12
The myth of degrees of truth 12

3. P R O P E R T I E S O F P R O P O S I T I O N S 13 19
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
Summary 24

Symbolization 25

4. R E L A T I O N S B E T W E E N P R O P O S I T I O N S 28
Inconsistency 28
Consistency 30
Implication 31
Equivalence 35
Symbolization 41

vii

CONTENTS

5. S E T S O F P R O P O S I T I O N S 42
Truth-values of proposition-sets 42
Modal properties of proposition-sets 42
Modal relations between proposition-sets 44
Minding our "P's and "Q"s 47

6. M O D A L P R O P E R T I E S A N D R E L A T I O N S P I C T U R E D O N
W O R L D S - D I A G R A M S 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 descriptions of modal relations 54
Appendix to section 6 57

7. IS A S I N G L E T H E O R Y O F T R U T H A D E Q U A T E F O R B O T H
C O N T I N G E N T A N D N O N C O N T I N G E N T PROPOSITIONS? 58

8. T H E " P O S S I B L E W O R L D S " I D I O M 62

PROPOSITIONS 65 75

1. I N T R O D U C T I O N 65

2. T H E B E A R E R S O F T R U T H - V A L U E S 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: Context-free sentences are the bearers of truth-values.
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 86
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
One final note 86

3. T H E S T R U C T U R E OF PROPOSITIONS: A SPECULATIVE
T H E O R Y 87
Concepts 87
Attributes of concepts. 90
Identity conditions for concepts 92
Analysis of propositions 94
Identity conditions for propositions 96

4. O N REFERRING T O SENTENCES A N D T O PROPOSITIONS 97
Techniques for referring to sentences 97
Basic techniques for referring to propositions 98
Advanced technique for referring to propositions: context-
free references 100
Untensed verbs in context-free references 103

5. T H E OMNITEMPORALITY OF T R U T H 104

6. PROPOSITIONS, SENTENCES, A N D POSSIBLE WORLDS 108
The uni-linguo proviso 110
The linguo-centric proviso 111
Securing reference to propositions 111

7. SENTENTIAL AMBIGUITY A N D 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 truths 125
Case Study 3: The thesis that a justified belief in a true
proposition constitutes knowledge 126

KNOWLEDGE 129 129

1. T H E SUBJECT M A T T E R A N D T H E SCIENCE OF LOGIC
2. T H E N A T U R E OF K N O W L E D G E 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. T H E LIMITS OF H U M A N K N O W L E D G E 139
The known and the unknown 139
The knowable and the unknowable 140

4. EXPERIENTIAL A N D RATIOCINATIVE K N O W L E D G E 142
Experiential knowledge 142
Ratiocinative knowledge 144
Appendix to section 4 149

5. EMPIRICAL A N D A PRIORI K N O W L E D G E 149
Definitions 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 A N D M O D A L STATUS CONSIDERED
T O G E T H E R 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 171
knowable a priori but by means other than ratiocination?

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. T H E EPISTEMOLOGY OF LOGIC 175

T H E SCIENCE OF LOGIC: AN OVERVIEW 179

1. I N T R O D U C T I O N 179

2. T H E M E T H O D OF ANALYSIS 180
The objects of philosophical analysis 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. T H E P A R A D O X OF ANALYSIS 189
Moore's problem 189
A Moorean solution 190

4. T H E M E T H O D OF I N F E R E N C E 192
The nature of inference 193
Valid and invalid propositional inferences 195
Determining the validity 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. I N F E R E N C E W I T H I N T H E 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 O N LOGIC AS
A W H O L E 218
The indispensability of modal concepts within propositional logics 218
Problems about the reduction principles 220
Problems about the paradoxes 224
Relevance logics 228
The move to predicate logic 230
Traditional syllogistic 232

CONTENTS

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

TRUTH-FUNCTIONAL PROPOSITIONAL LOGIC 247

1. INTRODUCTION 247

2. T R U T H - F U N C T I O N A L 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. E V A L U A T I N G C O M P O U N D SENTENCES 273 279
A note on two senses of "determined" 277

4. E L E M E N T A R Y T R U T H - T A B L E TECHNIQUES FOR

REVEALING MODAL STATUS AND MODAL RELATIONS
Modal status 279
Modal relations 284
Deductive validity 290

5. A D V A N C E D T R U T H - T A B L E TECHNIQUES 294
Corrected truth-tables 294
Reduced truth-tables 297

6. T H E CONCEPT OF F O R M 301
Sentences and sentential forms in a logic 301
The relationship between sentences and
sentence-forms 302

7. E V A L U A T I N G 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. F O R M IN A N A T U R A L L A N G U A G E 311

CONTENTS xiii

9. WORLDS-DIAGRAMS AS A DECISION PROCEDURE FOR
T R U T H - F U N C T I O N A L PROPOSITIONAL LOGIC 313

10. A S H O R T C U T F O R M A L M E T H O D : REDUCTIO AD
ABSURDUM TESTS 315
Summary 320

6 MODAL PROPOSITIONAL LOGIC 323

1. INTRODUCTION 323

2. M O D A L 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 M O D A L 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. T H E M O D A L STATUS OF M O D A L PROPOSITIONS 333

5. T H E OPERATOR "IT IS C O N T I N G E N T L Y T R U E T H A T " 337

6. ESSENTIAL PROPERTIES OF RELATIONS 339

7. T W O CASE STUDIES IN M O D A L RELATIONS: a light-
hearted interlude 345
Case study 1: the pragmatics of telling the truth 345
Case study 2: an invalid inference and an unwitting
impossible description 347

8. USING WORLDS-DIAGRAMS T O ASCERTAIN T H E
VALIDITY OF M O D A L I Z E D F O R M U L A E 350
Applications 351
The validity of the axioms of S5 356
The nonvalidity of the axiom set for S6 358

9. A S H O R T C U T F O R M A L M E T H O D FOR DETERMINING
T H E VALIDITY OF M O D A L I Z E D F O R M U L A E : modal
reductios 359

10. T H E NUMBER OF F O R M A L L Y NON-EQUIVALENT
SENTENCE-FORMS CONSTRUCTIBLE ON N SENTENCE-
VARIABLES 365

xiv CONTENTS

11. LOOKING BEYOND M O D A L LOGIC T O 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 statement 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 semantics".

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 reason 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 philosophy of logic 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 fact that the
rarefied results of formal logic resemble those of mathematics more closely than they do those of
metaphysics, 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 reasons 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
Gottlob Frege, Hilary Putnam, and David Lewis, we have felt compelled to posit a modestly rich
ontology 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
premises 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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xviii To the Teacher

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 verificationism; 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 facts" — albeit the facts in all possible worlds, not just in some.

T o the Teacher xix

W i t h respect to the notion of a priori knowledge, we argue against those who suppose that a priori
knowledge is best explained i n terms of our understanding of words and sentences and of the linguistic
rules, conventions, or definitions w h i c h they obey. T h e linguistic theory of the a priori was never so
persuasive as when it went tandem with the linguistic theory of necessary truth. B u t having rejected
the latter we felt free, indeed obliged, to reject the former. O u r 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. W e 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. A n d 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. A n d we replace the linguistic theory
of a priori knowledge w i t h an epistemological theory w h i c h gives due recognition to what philosophers
have long called "the powers of reason". T h i s is not to say that we ignore matters having to do with
language altogether. O n 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 i n 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.

M a n y philosophical theses, other than those just canvassed, are developed in this book. W e touch
on, and in some cases discuss at length, the views of dozens of philosophers and a few mathematicians.
T h i s is seemly in a book whose primary emphasis is on the philosophy, rather than the formalism, of
logic. A n d it serves to explain why we have found room i n 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

T h e 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 i n first- or second-year courses at colleges and universities.
It is not designed to replace handbooks in 'speed reasoning' or informal reasoning. N o r is it designed
only for those students who plan to go on to more advanced work in philosophy and logic.

A s we see it, Possible Worlds could, either i n 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 i n 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).

A g a i n , the book could, either i n 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 i n 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 philosophical logic, 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 focus is importantly
different from ours. For example, for courses in philosophy of language, 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

M o s t of you already understand that logic is theoretically important as a foundation for mathematics
and practically useful as a set of principles for rational t h i n k i n g 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.
W h a t , for a start, is philosophy? M o r e concretely: W h a t is it to be a philosopher? O u r answer —
which will suffice for present purposes — is this:

T o 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. W e believe — as did Socrates — that an
unexamined life is shallow, and that unexamined beliefs are often mischievous and sometimes
dangerous. T h i s 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. A n y belief, dogma,
or creed — social, political, moral, religious, or whatever — is subject to the philosopher's critical
scrutiny. N o t h i n g is sacrosanct; not even the beliefs of other philosophers. D o not be dismayed, then, to
find i n this book arguments which criticize other philosophical viewpoints; and do not be disturbed,
either, if your teachers when dealing w i t h 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.

O f the notions of implication and consistency we shall have much to say before long. F o r 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 t r u e ? " (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 w i t h other beliefs that we h o l d ? " (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 w i l l need to know a good deal about the concepts of
implication and consistency themselves. B u t these concepts are logical concepts. L i t t l e wonder, then,
that logic is — as we said at the outset — an "integral part of philosophy". T h e 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 w h i c h such questions turn. L e a r n i n g a little logic can help us to get our thinking straight
in philosophy as w e l l as elsewhere.

A s 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. Y o u w i l l need to learn a good deal of logic, and a lot more
philosophy, before achieving the k i n d of lofty vantage point reached by such great thinkers as Plato,
Aristotle, Leibniz, H u m e , Kant, Russell, and Wittgenstein. But one must begin somewhere. A n d
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. T h i n k i n g about other possible worlds is — as L e i b n i z recognized — a way of getting
our o w n w o r l d 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.

xxi



1

Possible Worlds

11. THIS AND OT HHEERR POSSIBLE WORLLDDS

Thhee Realm of Possibilities

The year is AA.DD. 44227722.. LLaazzaarruuss Long is 22336600 yearrs oldd.. Althhoouugghh hhe hhaas bbeen nnear ddeath mmaannyy ttimmees,
hhe hhaasn't -— uunnllikkee hhis bbiibblliiccaall nnaammesake -— rreeqquuiirreedd tthhe interrvvennttion of aa mmiirraaccllee tto rreeccooverr.. He
simpply chhecks hhimmself (or is taakken bbyy forrcce) into aa RRejjuuvvennaattiionn Clinic frroomm timme tto timmee. WWhheenn we
laast hhear of hhimm hhe is uunnddeerrggooiinng rreejjuuvvennaattion aaggaaiinn.. TThhee year is nnow 4291 aanndd LLaazzaarruuss is being
trreeaated inn hhis own pporrttaabblle clinniicc aabboaarrdd thhe starr-yacht ""DDora"" aafterr trraavveellinngg bbaackk inn timme to hhis
bbirrtthhppllaacce inn KKaannssaass aanndd bbeing ""mmoorrttaalllyy wounded"" inn thhe trreennchhes ""ssommewhheerree inn Frraannce"".

All thhiis, aanndd mmuucchh mmoorree,, hhaappppens tto thhe LLaazzaarruuss Long of RRobberrtt A. HHeeinnlein''s nnoovveel Timmee Enough
for Loovvee..l1 IInn hhis nnoovveel, HHeeiinnlleeinn starrts with aa frraammeeworrkk of pperrsons whho aacttuuaallly livedd (e.gg., WWooooddrrooww
Wilsonn.aanndd Kaiser WWiilhheellmm II), of places that aactuuaallly existed (e.gg., KKaannssaass City aanndd FFrraannccee)),, aanndd ooff
evennttss that aactuuaallly occuurrreedd (e.gg., U-boat aattaacks aanndd the entrry of the U.SS.. into World Waarr I), aannd
buuildds up aa worrldd offfiiccttioonnaal lpperrssonns, pplaaces, aanndd evennttss.. He carrrriiees uus with hhimm,, inn mmaakkee--bbeellieve, to
aannotherr worrldd ddiffeerreenntt frroomm our aactuuaal one: to aa mmerreely possible wwoorrld.

How mmuucchh of thhis otherr possible worrldd is believvaable?? HHow mmuucchh of it is really possible?? Much of itt
is crreeddiibbllee.. FForr instance, there could hhaave been aa mmaann nnammed ""IIrraa Howaarrd" who ddiedd in 18733 aannd
wwhhoossee will instruucted his ttrruusstteeeess to sett upp aa foundaattion ddevotedd to the prroolonngaattion of huummaann life. FFoorr
all we kknnooww, Irraa Howaarrd mmaay bbe justt aas hhistorriicaall aa perrsonagee aas WWoooddrrooww Wilson. TToo bbe surree, IIrraa
Howaarrd may bbe justt aas mmuucchh aa crreaturre of HHeinlein's imaaginaattion aas is Minerva —- aa coommppuutteerr
becomme fflleshh aanndd bloodd aanndd one of Lazarus' mmaanny mmiissttrreesssseess.. But thhis mmaattters nnott. FFoorr,, whateveerr the
historrical faacttss hhaappppen to be, we can aalways supposee -— counterf)a'acctutuaalllyl,y,asaswweesasayy- —ththaat tththeyeymmigighhtt
have beenn otherrwise.. WWe connsttaanntlyy maakke such supppositioonnss in the worrldd of rreaal life. Thhe worrldd ooff
ficcttiioonn nneeeeddss nnoo ssppeecciiaall inindduulglgeennccee.. WWeeeeaassiillyy ccaann, ,aannddddaailiylyddoo, , eenntteerrttaaiinn aalll ssoorrtstsooff uunnaaccttuuaalilizzeedd
possibilittieess about past, prresent, andd futurree. WWe think about things that might have happppennedd, mmiigghhtt
be happppenning andd might be about to happppeenn. Not only ddo we rruueefuullly ask ""WWhhaatt if things had bbeeeenn
tthhuuss aanndd tthhuuss??"";; wwee aallssoo wwoonnddeerr ""WWhhaatt iiff tthhiinnggss aarree ssoo aanndd ssoo??"" aanndd ""WWhhaatt iiff tthhiinnggss wweerree ttoo bbee ssuucchh
aanndd ssuucchh??"" CCoouunntteerrffaaccttuuaall ssuuppppoossiittiioonn iiss nnoott mmeerree iiddllee ssppeeccuullaattiioonn.. NNeeiitthheerr iiss iitt jjuusstt aa ffaannccyy ooff tthhee
ddrreeaammeerr oorr aa rreeffuuggee ffoorr tthhee eessccaappiisstt.. GGiivveenn tthhaatt wwee aarree ssoo oofftteenn iiggnnoorraanntt ooff wwhhaatt iiss,, wwee nneeeedd aa rriicchh

1. New Yoorrk, G.P. Putnam's Sons, 1973.

1

2 POSSIBLE WORLDDS

sennssee of what might bee.. IInn mmaattters of pprraaccttiiccee, we nneed to considderr aalternnaattives where kknnoowledge is

ddennied uuss. In mmaattters of thheorryy, we nneed to considderr hhyyppootthheesseess where faaccttss aarree uunnkknnoowwnn..

Actuaality is, aas it werre, surrrroouunnddeedd bbyy aann infinite rreeaalmm of ppoossibbilities. Or, aas we mmight ootthheerrwwiise

ppuut it, ourr aactuuaal worrlldd is surrrroouunnddeedd bbyy aann infinity of otherr ppossible worrllddss.. No wonderr thhen that

fiction wrriterrs like HHeeinnlleeinn hhaave little ddifficulty inn bbeguuiling uus with thheir sttorriieess of ppoossibbilities, mmost

off whhichh will nneverr bbe aacttuuaalliizzedd. Might thhere nnot come aa time when inncest will bbe socially aanndd legally
aacceptabble, when hhuummaann life will bbe pprroollonnged bbyy pperriiooddiicc visittss to RRejjuuvvennaattionn Clinnics, when
computers will bbe emmbboddied inn hhuummaann flesh, or when trraavveell frroomm one gaalactic colony to aannotherr will be
aa ccommmmoonnppllaacce?? MMaayybbee oouurr eeverryddaay ccounntterrffaactuuaal ssuuppppoossitions aarree mmuucchh mmoorree mmuunnddaannee.. BBuutt wwhho
aammoonng uus would rruullee thhe exciting oonneess enttirrely out of orrdderr?? TThhee fact is that we caann,, aanndd ddoo,, conncceive
ooff ssoocciiaall aanndd lleeggaall,, bbiioollooggiiccaall aanndd tteecchhnnoollooggiiccaall,, aanndd ppeerrhhaappss eevveenn ooff pphhyyssiiccaall,, ppoossssiibbiilliittiieess wwhhiicchh tthhee
wwoorrlldd ooff ffaacctt mmaayy nneevveerr eennccoommppaassss..

WWhhaatt are thee limits to thee ppoossssiibblele??

Buutt aarre thhere nnot limits of sommee sort to what we cann conceive orr supppose to bbe ppossible?? Does justt
aannyytthhinng go? HHow aabbout timme-trraavveell, forr instance?? In HHeeinnlein''s nnoovveell, LLaazzaarruuss rreecounts hhow he
aassuumedd aa bbiologicaal aage of thhirrttyy--ffive so aas to trraavveell bbaackk in time twenty--tthhrreeee hhuunnddrreedd yearrs to
observe hhow thhings wwerree in hhis chhildhhooodd aanndd (as it tuurrnns out) to fall inn love with hhis own motherr.

SOOMMEE EVV EENNTTSS IN THHEE LI FEE OFF LAA ZZAARRUUSS LONN GG,. SENNIOR MM EEMMBBEERR OFF THHEE HH OOWWAARRDD FFAAMMIILLIIEESS
FF iirrsstt rruun tthhrroouuggh ttiimmeetI

is born1I --- --------- ------. .-.-. -1l sstteeaallss ssttaarrsshhiipp rrettuurrnns tto rejuvenated travels through
In ttoo eevvaaccuuaattee OOlldd HHoommes at clinic space-tine on
TTeerrrraat, lleads on Secundus star-yacht,
Kansas HHoowwaarrd GGrreeaatt DDiiaas.sppoorra "Dora"

FFaammililiies

'1'1 .tS- - - --'1912' '16 1'17 1-- - - - - -1 21 )6 1- - u - - -- - -~--hu--142721- - -- - -- ---~
'141

Seconndd rruun tthhrroouugh lands wounded
ttiimme ti near in France)
rescued by
birthplace "Dora" and
carried to
4291 FIGUURREE ((lh.aa))

Could timme-trraavveel occurr?? Cann youu bbe surre that thhere is nnot sommee otherr ppossible worrlldd in whhichh it occcuurs
evenn if it nneverr occurrss, orr will occurr,, inn ourr aactuuaal worrldd?? IIs thherre aanny ppaarraaddoox inn thhe idea of Lazarus
obbserrving hhimmself aas aa fourr--yeaarr--ooldd?? See exerrcciise 4 onn pp.. 25.

Many of uus mmaay feel that thhe verry concepptt of timme-trraavveel is ppaarraaddooxxiiccaall.. IIf timme-trraavveel wwerree ppoosssible
thhen shouuld we nnott bbe aabble to imaagine aa pperrson trraavveellinng bbaackk inn time aanndd fathherrinng hhimmsself? Buutt is
thhis rreeaalllly ppossible?? OOuurr mminndds bbooggggllee aa bbitt.. IIt is supppposedd bbyy sommee that hherre we hhaave rreeaachhed the
limits of connceivabbiility, aanndd that thhe worrlldd whhich HHeeinnlleeinn pporrttrraayyss thhrroouugghh LLaazzaarruuss is, inn thhis rreesppect
aat leaasstt, nnot in aanny sennssee aa ppossible worrlldd.. Thhis, aas it hhaappppeenns, is thhe view of aannotherr of thhe novel's
chaarraacctteerrss:: Carroolyn Brriiggggss,, chief aarrcchhiivviist of thhe Howard FFouunnddaattiioonn.. Ass she ppuut it inn hherr PPrreefaace to
thhe RReevvised Eddition of Lazarruus' mmeemmooirrss:: ""AAnn aappoocrryypphhaall aanndd obbviouusly immppoossible ttaalle of thhe laastt
evennttss in hhis life hhaas bbeen includdedd aat thhe innsistennce of thhe editorr of thhe orriigginnaall mmeemmooirr,, bbuut it cannnnOot
bbe taakken serriously."" OOnn thhe otherr hhaanndd,, sommee of uus mmaay feel that timme-trraavveell, of thhe kind that LLaazzaarrus

§ 11 .' This aanndd OOtherr PPoossible Worrlds 3

is suuppposedd to haave unndderrttaakkeenn,, is not whholly inconceivabblle;; that parraaddooxx can be avoidedd;; andd thatt aa
worrld in which time-trraavveel occcuurs mmaay soommeddaay be seenn to be possibble if andd whenn ouur connceepptts ooff
space--ttiimmee becoomme mmorre sophisticaated. WWe mmaay waanntt to agree witth Carrolyn's prreddecesssoorr —- JJuussttiinn
Foote, chief arrcchhiivviistt emmerritus —- whenn he apppendds the folloowwing note to herr Prreeffaaccee:: "Myy lovely aanndd
leeaarrnedd succeessssoorr in offiiccee ddooees not knnow whhaatt shhe is talkkinngg abbouutt. Witthh the Seniorr [i.e., Lazarruus], tthhee
mmost fantaasttic is aalways the mmost pprroobbaabblele."."22

But whaateveerr we say aabbout timme-trraavveell —- whetthheerr we thhinkk it falls withhinn or beyondd the limits ooff
possibility —- it is certtaainn that somme thhings arre nott possible no mmaattterr how sophhisticcaattedd ourr conceppts
becomme. Thhee suppppoossition that timme-trraavveell bothh willll occuurr sommetimme in the futurre anndd will noott ooccccuurr
annytime in the futuurree,, is aa caase of point. IIt taakkees us, in aa sennssee, beyondd the bounndds of conceivabbiility. It is
nnot jjuusstt ppaarraaddooxxiiccaall.. IItt iiss,, aass wwee ssaayy,,ffllaattlylysseellff-connttrraaddiiccttoorryy..AAssuuppppoosseeddwwoorrllddiinnwwhhiicchhssoommeetthhiinngglliitt--
erraally bothh is the caase aanndd is not the caase is nott, in aanny sennssee, aa possible world. IIt is ann immppoosssibble one.

Possibilittyy iss nott thee sammee aassccoonncceeiivvaabbiilliittyy

So farr we haave spoken aas if the bounnddaarryy bbetwween the possible aanndd the immppoosssibble werre aa funnccttionn ooff
humman ppsychhoollooggyy:: aas if, that is, it coinciddedd withh thhe bbouunnddaarryy bbeettwweenn that whhiichh we find
connceivabble aanndd thhat whhiichh we findd innconncceeivaabbllee.. Yeett,, whhiile thherre is aa grreat aammoouunntt of oovveerrllaapp
bbeettwweenn what is connceivabble aanndd what is ppoossibble, thhe two aarree nnot thhe samme.

In thhe ffiirsstt pplaaccee, it shhouuldd bbe cleaar onn rreefflection that ourr innaabbiilliittyy to connceive of aa cerrttaaiinn staattee ooff
aaffairrs ddooeess nnot impply thhe immppoossssiibbiilliittyy of that staatte of aaffaairrss.. Nottoorriioouussllyy,, thherre waas aa timme whenn
our aannccestorrs thhought it inconncceivabblle that thhe earrtthh shhouuldd bbe rrouunndd. Yett obbvviouussly thhe ppoosssibbility ooff
thhe earrtthh''s bbeinng rround waas inn nnoo waay limmittedd bbyy thheirr innaabbiilliittyy to connceive it.

Seconddllyy, ourr seemming aabbiillittyy to connceive of aa cerrttaaiinn staattee of aaffairrs ddooeess nnot impply thhe ppossibbilitty
of that staatte off aaffaairrss.. FForr mmaannyy centuurriieess, mmaatthheemmaattiicciaanns soughht aa mmeaanns of squuaarriinngg thhe cciirrcle
(souught aa pprrooccedduurree whherrebby to construuct forr aanny given cirrccllee aa squaarre of eqquuaall aarreeaa)).. Plainnly, they
would nnot hhaavve ddonne so uunnlleessss thhey thhought it connceivabble that suuchh aa pprrooccedduurree existed. Yett we noww
kknnooww,, aanndd cann pprroovvee,, that thhe squuaarriinngg of thhe cirrccllee is whhoollly bbeyondd thhe bbouunnddss of ppoosssibbility —-
tthhaat tthhe concept itself is sseellff--ccoonnttrraadictorry.

But if, aass ourr ffiirrsst taarrgguummeenntt sshhoowwss,, ccoonncceeiivvaabbiilliittyy iiss nneeiitthheerr aa nneecceesssaarryy ccoonnddiittiioonn ((ccoonncceeiivvaabbiilliittyy iiss
nnoot neeeeddeedd forr sommethhinng ttoo bbee ppoossssibbllee), nnoorr,, aass ourr second aarrgguummeenntt shhoowws, aa suufficiieenntt ccoonndditionn
(connceivabbiility ddooesnn''t suuffffiiccee ttoo estaabblish ppoossssiibbiilliittyy)),, wwhhaat tthhenn arree tthhe connddiittiionnss forr ssommeetthhinng's
bbeeinng ppoossible?

OOnne mmiighhtt bbee ttemmppttedd ttoo aannswerr tthhaat it is conceivability wwiitthhoouutt incoonnssiisstteennccyy orr ccoohheerreennt t
conceivability -— nnoot juust conncceeivaabbiilliittyy itself -— wwhhiicchh is tthhe mmeeaasurre of ppoossssiibbiilliittyy.. Afterr aall,
aalthhouughh cerrttaaiinn of ouurr aannccestorrs aappppaarreennttllyy ddiidd nnoot hhaavve tthhe ppssyycchhoollooggiiccaall caappaacciittyy ttoo conncceeive off
tthhe earrtthh bbeeinngg rroouunndd, tthhe concept of tthhe eaarrtthh bbeeinngg rround is itself aa ppeerrffeeccttly coherent orr
sSeellff-consistteenntt concept aanndd hheennce is onne of wwhhiicchh tthheey couulldd -— inn tthhe rreeqquuiissiittee sseennssee -— haave
conncceeivedd wwithout innconnssistennccyy. And, aaggaaiinn,, aaltthhoouugghh generraattions of mmaatthheemmaattiicciiaannss tthhouught tthhaat the
sqquuaarriinng oof tthhee ccirrccllee wwaas aa ggooaall wwhhiicchh oonne ccoouulldd inntteelllliiggiibbllyy ppuurrssuuee,, wwe nnooww kknnooww tthhaatt tthheey ccoouuld
nnoott cconncceeive oof tthhaatt ggooaall wwithhouutt inncconnssisttennccyy.

UUnnforrttuunnaatteellyy, tthhiiss aannsswerr will nnoott ddoo;; oorr rraatthheerr,, itt will nnoott ddoo frromm tthhee ssttaannddppooiinntt oof tthhoossee whho
wwishh -— aass wwe ddoo inn tthhiiss bbooookk -— ttoo eexppllaaiinn tthhee pprriinnccipaal cconnceptts oof looggiicc inn tterrmmss oof tthhee nnoottiioonn oof a
ppoossssibble worrlldd.. FFoorr aammoonngg tthhoossee pprriinnccipaal cconnceptts aarree tthhe cconncepts oof cconnssistency aanndd innconnssisttennccyy.
And if wwe tthheenn innvvookkee tthhe cconncept oof cconnssistency inn oorrddeerr ttoo eexppllaaiinn wwhhaatt aa ppoossssibble worrldd is aanndd hhooww
aa ppoossssiibbllee wwoorrlldd ddiifffferrss ffrroomm aann iimmppoossssiibbllee oonnee,, wwee eexxppoossee oouurrsseellvveess ttoo aa cchhaarrggee oof cirrcuulaarriittyy..

22.. Time Enough FFoorr LLoovvee,, pp. .xxvvii.i. [[IIttwwilill bbeeoouurrpprraacctitciceetotoggiviveeththeeccoommppleletetebbibibliliooggrraapphhicicaall rreefeferreennceceonolnyly
oonn tthhee fIirrsstt ooccccaassiioonninineeaacchhcchhaapptteerrooffoouurrccitiitninggaawwoorrkk. .TThheerreeininaafftteerrwweesshhaalllloommititththeeddeetatailislsooffppuubblilcicaatitoionn.].]

4 PPOOSSSSIIBBLLEE WWOORRLLDD S

OO uur predicammeennt is apparently thiss.. OO n the onee hand, wwee wwiish to avooid ppssyycchhoololgoigsimsm, , vviizz., any
formm of theory which makeess logic a function off human psycchhoollooggyy.. In its oollddeerr ffoorm, psychologism
held that the laws of logic were "thee lawss off thoughht"".. It treatedd thee scciieennccee ooff llooggicc aass iiff iit wweerree a
branch of psychologyy concceerned wwiith the description off howw humman bbeeiinnggss acctuallyy reason.
Psychologism, inn its traaddiittiioonnaall formm,, is noww dead, largelyy as a resuullt ooff thee eeffffoorrttss,, eeaarllyy iin this
century, of Frege, Russellll,, and WWiitttgeenstein. WW ee wwiish to avoid introoduuccingg it in a neww ffoorm. YYeet
that is whaat we would be doingg if wwee weere to explain the princcippaall ccoonncceeppttss ooff llooggiicc iin teermmss ooff
possible woorrlds and then go on to exppllaain possssiibbllee wwoorrllddss tthemmsseellvveess iin tteerrmmss ooff tthhee ppuurrely
psyychological conncceepptt of conceivaabilityy. OO nn the otthheerr hhand, wwee wwiisshh tto aavooiidd tthhee cciirrccuulalarirtiyty iinn wwhhiicchh
wwee wwoouulldd bbee iinnvvoollvveedd iiff wwee ttrriieedd ttoo ddeeffiinnee ppoossssiibbllee wwoorrllddss iinn tteerrmmss ooff tthhee llooggiiccaall nnoottiioonn ooff
ccoonnssiisstteennccyy,, aanndd tthheenn llaatteerr ddeeffiinneedd ccoonnssiisstteennccyy iinn tteerrmmss ooff ppoossssiibbllee wwoorrllddss..

Fortunateellyy, there is a wayy out of this predicammeennt. WWee ccaan avooid cciircularityy and ssttiillll nnoott be
traappped by psychologism. CCoonnsider, for a moommeennt, how the cchhaarggee ooff cciircularityy in deeffiinniitions mmaayy iin
other cases be avoided. WW ee look in a dictionaryy, for exxaammppllee,, to finndd oouutt wwhhaatt it is ffoor ssoommeetthhiinngg to
be complex and find that the concceepptt of commplexxiity is oppoosseedd to thaatt ooff simplicittyy;; wwee try to finndd ouut
whaat it is for something to be simple and find that the ccoonncceepptt ooff simplicityy is ooppppoosseedd to thhaatt ooff
complexityy. YYeet such a circle of definitions can be, and standardlyy is, bbroken. Itt is bbrokkeenn bbyy cciitting
exampleess:: sometimeess by ostension (pointiinngg);; sommeetiimmeess by naming; and soommeettiimmeess byy ddeessccrriipption.
Thus, inn the preseenntt case we can avoid the trap off circularityy —- and att thee saammee time, thee seducctioon
of psychologism -— by citinngg clleeaarr--ccuutt exaammpplleess (ttheyy aarree oofftteenn ccaalllleedd ""ppaarraaddigmm eexxaammpplleess"")) ooff
ppoossssiibbllee wwoorrllddss,, aanndd eeqquuaallllyy cclleeaarr--ccuutt eexxaammpplleess ooff iimmppoossssiibbllee wwoorrllddss.. IInnttuuiittiivveellyy,, wwee wwoouulldd wwaanntt ttoo
iinncclluuddee,, aammoonngg tthhee ppoossssiibbllee wwoorrllddss,, wwoorrllddss iinn wwhhiicchh tthheerree aarree mmoorree oobbjjeeccttss tthhaann iinn tthhee aaccttuuaall wwoorrlldd
((ee..gg..,, iinn wwhhiicchh tthhee eeaarrtthh hhaass ttwwoo mmoooonnss));; wwoorrllddss iinn wwhhiicchh tthheerree aarree ffeewweerr oobbjjeeccttss tthhaann iinn tthhee aaccttuuaall
wwoorrlldd ((ee..gg..,, iinn wwhhiicchh tthhee eeaarrtthh hhaass nnoo mmoooonn aatt aallll));; wwoorrllddss iinn wwhhiicchh tthhee ssaammee oobbjjeeccttss eexxiisstt aass iinn tthhee
aaccttuuaall wwoorrlldd bbuutt hhaavvee ddiiffffeerreenntt pprrooppeerrttiieess ((ee..gg..,, iinn wwhhiicchh tthhee lloonngg--ssuuppppoosseedd ""ccaannaallss"" oonn MMaarrss ttuurrnn
oouutt,, aafftteerr aallll,, ttoo bbee rreelliiccss ooff ssoommee ppaasstt cciivviilliizzaattiioonn));; aanndd ssoo oonn.. AAnndd iinnttuuiittiivveellyy,, ttoooo,, wwee wwoouulldd wwaanntt ttoo
iinncclluuddee,, aammoonngg tthhee iimmppoossssiibbllee wwoorrllddss,, wwoorrllddss iinn wwhhiicchh cciirrcclleess ccaann bbee ssqquuaarreedd,, wwoorrllddss iinn wwhhiicchh tthheerree
iiss aann eevveenn ssqquuaarree rroooott ooff nniinnee,, wwoorrllddss iinn wwhhiicchh ttiimmee--ttrraavveell bbootthh ddooeess aanndd ddooeess nnoott ooccccuurr,, aanndd ssoo oonn..

To be sure, descriptions can be given off wwoorlds aboouutt wwhhoossee possibilityy oor immpossibilityy wwee have
noo clear intuuiittiioonnss.. But for nearly any distiinnccttiioonn that wwee ccaarree to think ooff theerree wwiillll bbee problemmaaticc
cases lyyinngg near the borderline of that distiinnccttiioonn.. TT hhee case off wwoorrlldds in wwhhiicch timee-travel ooccccuurrss is
just such a case;; and so, until recently, wwaass the case off wwoorrllds in wwhhiicch mmoorree than ffoouurr ccoolloorrss are
needed too demarccaattee betwweeeenn countries on a planee map. BBuutt the difficculty ooff decciding, ffoor cceerrtain
cases,, on whhich side of the distiinnccttiioonn they arree to bbee looccaatteedd sshould nnoott bbee aalllowweedd ttoo ddiissccrreeddiitt thee
distincction itsseellf. Thhee distiinnccttiioonn bettwweeeenn possssiibbllee and imppossssible wwoorrllddss iiss ggrroouunnddeedd iin aann aappppeeaall tto
paraaddiiggm cases.. WW ee do not have to be ablee to seettttllee all bboouunnddaarryy ddiissppuutteess iin oorrddeerr ttoo hhaavee aa secuurree
eennoouugghh ffoooottiinngg oonn wwhhiicchh ttoo pprroocceeeedd iinn oouurr aatttteemmppttss ttoo eexxppllaaiinn tthhee wwoorrkkiinnggss ooff llooggiicc..

Poossssiibbllee worldss:: actuuaall andd nononn-a-accttuual

Lett us pausee att this point and reflecct, in philosophiiccall ffaasshiion, on some ooff thee thingss wwee havvee been
saayyingg. We havve been woorkingg wwiith a threefold distinnccttiioonn,, wwhhicch is iimmpplliicciitt iin mmuucch ooff ouur
thhiinnkkiinngg, betweeeenn the actual wwoorrld,, wwoorrlldds wwhhiich arree nnonn--aaccttuuaall bbuutt ppoossssiibbllee,, aanndd wwoorrllddss wwhhiicchh arree
nneeiitthheerr.. BBuutt wwhhaatt pprreecciisseellyy ddoo wwee mmeeaann bbyy ""ppoossssiibbllee wwoorrlldd"" ?? MMoorree bbaassiiccaallllyy ssttiillll:: WWhhaatt pprreecciisseellyy ddoo
wwee mmeeaann bbyy ""tthhee aaccttuuaall wwoorrlldd""??

WW hhe n we speakk of "thee actual ww oorldd"" wwee do not meeaan juusstt the pllaanneett oon wwhhiicch wwee llive. NNoorr do
we mean our solar system, or eveen our galaxyy.. WW ee have spokkeenn ooff thee acctuall wwoorrld iin aann aallll-
encomppaassssiinng waay so as to embbrraaccee alll thaatt really exxiissttss -— tthee uunniivveerrssee aass aa whhole.

AAggaaiin, whheenn wee speaakk of "tthhee acttual ww oorrldd"" wwee do nnoott mmeeaann jjuusstt tthhee uunniivveerrssee aass iitt iiss nnooww,, iinn tthee
presenntt. WW hhen wee identify it -— as above -— wwiitth all tthhaatt rreally eexxiissttss,, wwee aarree uussiinng ""eexxiissttss"" iinn aa

§§11 TThhiiss aannddOOtthheerr PPoossssiibbllee WWoorrllddss 55

ttiimmeelleessss sseennssee,, ssoo aass ttoo eennccoommppaassss nnoott oonnllyy wwhhaatt eexxiissttss nnooww bbuutt aallssoo wwhhaatt oonnccee eexxiisttedd iinn tthhee ppaasstt

aand wwhhaatt wwiillll ccoommee ttoo eexxiisstt iinn tthhee ffuuttuurree.. TThhee aaccttuuaall wwoorrlldd eemmbbrraacceess aalll tthhaatt wwaass,, iiss,, oorr wwiillll bbee..
NNoww iitt iiss cclleeaarr tthhaatt tthhee aaccttuuaall wwoorrlldd iiss aa ppoossssiibbllee wwoorrlldd.. IIff ssoommeetthhiinngg aaccttuuaalllyy eexxiissttss tthheenn iitt iiss

oobbvviioouussllyy ppoossssiibbllee tthhaatt iitt eexxiissttss.. OOnn tthhee ootthheerr hhaanndd,, nnoott eevveerryytthhiinngg tthhaatt ppoossssiibbllyy eexxiissttss ddooeess ssoo
aaccttuuaalllyy.. NNoott aalll ppoossssiibbllee wwoorrllddss aarree aaccttuuaall.. IItt ffoolllloowwss,, tthheerreeffoorree,, tthhaatt tthhee aaccttuuaall wwoorrlldd iiss oonnllyy oonnee

aammoonngg mmaannyy ppoossssiibbllee wwoorrllddss:: tthhaatt tthheerree aarree ppoossssiibbllee wwoorrllddss ootthheerr tthhaann oouurrss.. MMoorreeoovveerr,, ggiivveenn tthhaatt by
""tthhee aaccttuuaall wworrldd"" wwee mmeeaann -— aass wwee aaggrreeeedd aa mmoommeenntt aaggoo -— eevveerryytthhiinngg tthhaatt wwaass,, iiss,, oorr wwiillll bbee tthhee
ccaassee,, iitt ffoolllloowwss tthhaatt bbyy ""aannootthheerr ppoossssiibbllee wworrldd"" wwee ddoo nnoott mmeeaann ssoommee ppllaanneett,, ssttaarr oorr wwhhaattnnoott tthhaatt
aaccttuuaallllyy eexxiissttss aanndd wwhhiicchh iiss lloocaatteedd ssoommeewwhherree ""oouutt tthheerree"" iinn pphhyyssiiccaall ssppaaccee.. WWhhaatteevveerr actuuaalllly
eexxiissttss,, iitt mmuusstt bbee rreemmeemmbbeerreedd,, bbeelloonnggss ttoo tthhee aaccttuuaall wwoorrlldd eevveenn iiff iitt iiss llighhtt--yyeaarrss aawwaayy.. OOtthheerr,,
nnoonn--aaccttuuaall,, ppoossssiibbllee wwoorrllddss,, aarree nnoott llocaatteedd aannyywwhheerree iinn pphhyyssiiccaall ssppaaccee.. TThheey aarree lloocaatteedd,, aass iitt wwerree,
in ccoonncceeppttuuaall ssppaaccee;; oorr rraatthheerr,, aass wwee mmaayy pprreeffeerr ttoo ssaayy,, iinn llooggiiccaall sppaaccee.

AA llll PPoossssiibbllee WWoorrlldss The

A aacttuual
wwoorrlld
NNoonn--aaccttuuaall
wwoorrllds + "«'---,

— <A* II

FIGURE ((1l..bb)

Ouurr world (everrythhinngg that aacttuuaallllyy waas, is, orr will bbe) is onnly one of aann infinite nnuummbbeerr ooff
ppoossible worrllddss.. IIt is thhe aacttuuaall worrldd. TThhee otherrs aarree nnoonn--aaccttuuaal.l.

Note that in thhis anndd otherr simmillaarr ddiagrraammss we rrepprreesent ann infinite nuummbbeerr of possibble
worrldds by a rreectaannggle of finite size. Thhee rreectaannggle mmaayy be thought of as contaainniinngg ann innfiinnitte
numberr of points each of whhichh rrepprreesents a ddiffeerrennt possible world. IIt is onnly forr the sake ooff
diagrrammmmaattic convenience that we rrepprreesent the actuuaall world on thhis anndd a few subseeqquueenntt
figguurreessbbyy aasseeggmmeenntt ooff ththee rreecctatanngglele rraaththeerrththaannbbyyaassiningglele ppooinint.t.FFrroomm aalologgicicaal lppooininttooff
vieww the actuuaal worrld has litttle (iff anny) claaimm to prriivviilegedd status. Inddeeedd, in mmost of tthhee
worrlds-ddiiaaggrrams feeaattuured laaterr we shaall haave no needd to mmaakke mmennttion of the actuuaal world, leett
aalloonnee ttoo ffeeaattuurree iitt pprroommiinneennttlyly..

Note, fuurther, that the brraacckkeett for non-aactuuaall worrldds is leefftt open at the leefftt--hhaannd sidde of tthis
diaaggrraamm and all diaagrrammss on which the actuaal woorrldd is feeaatturreedd.. This is to siggnniffyy thhaatt tthheerree
arree morre non-actuuaall worlds thhan we have herre deppictedd. The claasss off non-actuuaall wwoorrllddss
coonntains all possibblee worlds otthheerr thhan thhe actuaal woorrldd and coonntains as well all iimmppossiibbllee
worlds. Eveerry immppossible woorrldd is a non-actuuaal wwoorrldd.

6 POSSSIIBBLLEE WORLDS

LLooggicaall possibility distinguisheedd fromm othher kkiinds

By ""aa ppoossssiibbllee wwoorld"",, iitt sshhoouulldd bbee eemmpphhaassiizzeedd,, wwee ddoo not mmeean onnly aa physsiiccaalllyy possible wwoorldd.
CCoouunnttlleessss wwoorrlds wwhhiicchh aarree pphhyyssiiccaallly immppoossssiibbllee aarree nnuummbbeerred ammoonngg tthhee possible worllddss wwee aarree
ttaalkingg aabboouutt.. PPhhyyssiiccaalllly ppoossssiibble wwoorlds form aa pproper subbset ooff all possibble worlds; oorr,, too mmake tthhee
ccoonnttrast ssoommeewhat sshhaarrppeer, wwee mmiigghhtt ssaayy tthhaat tthhee sseett ooff pphhyyssiiccaally possibble worlddss foorrmmss aa prooppeerr
ssuubbsset ooff aallll logiccaally ppoossssiibblee wwoorrlddss.

A pphhyyssiiccaalllly ppoossssiibble wwoorld iiss aannyy ppoossssibble world whhich hhaass tthhee samme natural laaws aass does tthhee
aaccttuuaall wwoorrlldd..33 TThhuus tthhe loggiiccaalllly ppoossssiibble world ddeeppiictedd inn Chhaarrles Dickens' noovel Davviidd CCopperfiieelldd
iiss aa pphhyyssiiccaalllly ppoossssiibbllee oonnee:: nnoo eevenntt iinn tthhaat nnooveel violatess aannyy natural laaww.. OOnn thhee othheerr hhaanndd,
WWaasshhinnggttonn IIrrvinng's sshhoort ssttoorry RRiip VVaann WWiinkkle ddeessccribes aa pphhyyssiccaally immppoossssiibble world: aa worldd inn
wwhhiichh aa ppeerrssonn ssleeps wwithout nnoouurrishhment foorr ttwweenntty yeaarrss.. TThhee latter circcuumstannccee, aa perssoonn''ss liivviinngg
for ttwweennttyy yyeeaarrss wwithout nnoouurrishhmment (energy intakkee), violates certainn laaws oof tthheerrmmodynnammics.
NNoonneetthheelleessss,, aalthougghh ssuucchh aa ssituation iiss thus pphhyyssiiccaallly immppoossssiibbllee,, itt iss not logicaally imppoossssiibbllee.. IInn

AA llll PPoossssiibbllee WWoorrllddss

rK

NNoonn--AAccttuuaall
WWoorrllddss

PP hhyyssiiccaallll y PPhhyyssiiccaallllyy
IImmppoossssiibblle PPoossssiibbllee

--------..J'A ''--------_"Y~-----~"'-------"\

TThhe AAccttuuaall

WWoorrlld

..

FFIIGGUURREE (1l..cc)

33.. NNoottee tthhaatt,, oonn tthhiis aaccccoouunntt,, tthhee nnoottionn ooff aa pphhyyssiiccaally possible world iss parasiticc uponn (needdss too bbee ddeeffiinneedd
iinn tteerrmmss ooff)) tthhee bbrrooaaddeerr nnoottiioonn ooff aa llooggiiccaallllyy ppoossssiibbllee wwoorrlldd.. SSiimmiillaarrllyy,, aa ssttaattee ooff aaffffaaiirrss iiss pphhyyssiiccaallllyy ppoossssiibbllee,,
iitt iiss uussuuaallllyy ssaaiidd,, iiff iittss ddeessccrriippttiioonn iiss ccoonnssiisstetennt twwithiththtehenantautruarlallalwaws s(o(fofthteheacatcutaulalwworolrdl)d.).YYetetthtehererlealtaitoionnooff
ccoonnssiisstteennccyy -— aass wwee sshhooww iinn sseeccttiioonn 44 -— iiss iittsseellff ttoo bbee ddeeffiinneedd iinn tteerrmmss ooff ppoossssiibbllee wwoorrllddss,, ii..ee..,, ooff llooggiiccaalllyly
ppoossssiibbllee wwoorrllddss..

§ 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 impossible world 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 of possible 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~ meaning 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.

88 POSSIBLLEE WORLDS

aaccttuuaall oonnee.. HHooww,, tthheenn,, ddoo nnoonn--aaccttuuaall ppoossible worrldds ddifferr from the actuuaal one?? They mmaay difffeerr iinn
tthhrreeee bbaassiicc wwaayyss. OOtthher ppoosssible wworrllddss mmaayy contaain the verry same itemms as the actuaal world bbuutt
ddiiffer frroomm tthhe aaccttuuaall wwoorrlldd inn rreesppect of tthhe aattttrriibbuuttes whhichh tthhoosse itemms instance; e.g.,, difffeerr iinn
rreessppeect oof tthhe EEiiffel TToowwerr bbeeinng ppuurrppllee,, tthhe Taj Mahal bbeing grreenn, anndd so onn. Orr they mmaay ccoonnttaaiinn
aatt lleeaasstt ssoommee itemms wwhhiicchh ddoo nnoot eexxist inn tthhe aactuuaall worrlldd (and ipso facto diffeerr in rrespecctt of ssoommee
aattttrriibbuutteess));; ee.g., tthheey mmaayy connttaainn LLaazarruus Long orr Shherrllockk HHoolmmeess.. Orr still agaainn, they maay llaacckk
ccerrttaaiinn iittemms wwhhiicchh eexxiistt inn tthhe aaccttuuaall wworrlldd (andd ipso facto ddiffeerr in rrespectt of somme attrribbuuttees); e.g.,
tthheey mmaayy laacckk SSttaalliinn oorr Shhaakkeesppearree.

SSoommee pphhiillosopherrs hhaavve tthhouught tthhat otherr ppoossible worrldds cann ddiffeerr from the actuaal onne only iinn
tthhee firrsstt oof tthheessee tthhrreee wwaayyss. OOtthher pphhiilosoppherrs insistt that thhey can ddiffeerr from the actuaal onne in tthhee
ootthherr ttwwo wwaayyss aass wweellll.. OOuurr ownn tthhiinnkkiinngg inn thhe mmaattterr is eviddent: in choosinng the worrldd of TTiimme
Enough foorr Love aass ouurr exaammppllee, wwe aarree aallowing thhe existennce of nnonn-actuuaal possibblee worlds somme oof
wwhhoossee iittemms ddoo nnoott eexxiistt inn tthhe aaccttuuaall worrldd. In chapterr 4, sectioonn 6, we will expploorre brriieefly soomme ooff
tthhee cconnsequencess oof bbeeinngg lleessss ggennerroouuss.

EXXEERRCCISIESSES

IInn tthhiiss aanndd ssuubbseqquenntt exerrccises, it is essenntiaall that one aassuumme that all the terrms being usedd hhavee
tthheeirr ssttaannddaarrdd mmeeaanniinnggss, e.g., tthhaat ""ssqquuaarre"" rreeferrs to aa pplaannee, closed, fourr--sidedd figguurre haaving eeqquuaall
iinntterriorr aanngglleess aanndd eeqquuaall siddes. TToo bbee suurree, forr aanny worrdd orr sennttenncce we carre to think of, therre will bbee
ppoossssibble wwoorrllddss inn wwhhiicchh tthhaat wworrdd orr ssenntteennccee will mmean somethinng aaltoggeetthheerr difffeerreenntt frrom whhaatt iitt
mmeeaanns iinn tthhee aaccttuuaall worrlldd.. TThhis faacctt,, hhoowwever, inn nno way tteells aagainnstt the fact thatt whhaatt we rrefeerr ttoo
bbyy tthhee wwoorrdd ""ssqquuaarree"" hhaas fourr ssiddeess inn aallll ppoossible worrllddss.. Thhe exerrcciises posedd ask whetthheerr thee
cclaaimmss wwee aarree hheerre cconnsiderriinngg aarree ttrruuee inn aanny ppoossible world; aanndd thhis is a wholly distinnctt mmaatttteerr
ffrroomm wwhhetherr oouurr wworrddss mmiighht bbee uused bbyy tthhe inhaabbitanntts of otherr possibblee worrlds to mmaakke difffeerreenntt
ccllaaiimmss..66

Part A

For eeaacchhooffththeefofollolowwiningg,, sasayywwhheeththererththereereisisaalologgicicaallylyppoosssibiblelewwoorrldldininwwhhicichhititisistrtureu.e.

,1.. Frederick, of Gilbert and Sullivaann''ss "Pirates of Penzance",, haass reachheedd the agee of 211 yearrss aafftteerr
oonnllyy 5 birthddaayyss..

22.. TThheerree iiss aa sqquuaarree hhoouusseeaallllooffwwhhoosesewwalalsllfsafcaecesosuothu.th.

33.. EEppiimmeenmiddeess,, thhee Cretan, sppookkee inn truutthh whheenn hee said thaatt everything thaatt Cretans saayy iissfafalsles.e.

44.. MMtt.. EEvveerreesstt iss lowweerr thhaann MMt. Cookk..

55.. MMtt.. EEvveerreesstt iss lowweerr thhaann MMtt.. Cooookk,, MMt. Coookk iss lower thaann Mt. Whistler, and Mt. Whistler iiss
loowweerr thhaann MMtt.. EEveresstt..

66.. 2 ++ 2 i2=*44..
+77.. TThhee PPooPpee. bbeelliieevveessththaat t22 + 2 i=2*44. .
+88.. TThhee Pooppee kknnoowwss thhaatt 2 + 2 i2=*4..

66.. FFoorr mmoorree oonn tthhiis ppooiinntt,, see ourr ddiiscuussion inn chapterr 2, pppp.. 1l11OOfff,f, ooff tthhee uunni-i-lliinngguuoo pprroovviissoo..

§ 2 Propositions, Truth, and Falsity 9

9. No objects are subject to the law of gravitation.

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 of the 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 stiff arched 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 and falsity 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 name 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

10 PPOOSSSSIIBBLLEE WWORLDS

andd again, thhatt

Itt is false that Krakatoa Island was annihilated by a volcanic eruptioonn iiff
andd only if it is nott thee case that Krakatoa Island was annihilated by a
volcanic eruption.

Thesee definitions accord with thee insight which Aristotle, more thann .ttwo thoussaanndd yearss ago,
expresssseedd thus:

TToo say of whatt is that it is not or of whatt is not that it is, is false, wwhile
to say of whatt is that it is, and of whatt is not that it is not, is trruuee.7.7

Several points abouutt these definitions aree worth noting:

(i) Althougghh Aristotle'ss accounntt suggests that it is personnss'' sayings which aree true or false (i.e.,
which are, as we shall putt it, thee bearers of truth-valuueess)),, our accounntt in (a) and (b) leaves open the
question as to whatt it is which is true or false. Theree is no doubtt that sayings, along with bbeelliievings,
supposings, etc., are among the things which can be true or false. But many ccoontemporary
philosophers prefeerr to say that it is primariillyy pprrooppoossitions wwhhiicchh hhaavvee tthhee pprrooppeerties ooff ttrruuth oorr ffaallssiittyy
(i.e., that it is primariillyy propositions which are bearers of truth-valluueess)),, and would hold that the
things personnss say, believe, suppose, etc., are true or false justt when the propositions which they utter,
believe, or entertainn,, etc., are true or false. For the reasons given at length in chaptteerr 2, we adoptt the
latteerr way of talkkiinngg..88 If we let the letterr "P"" standd for the proposition that a has F,, we can restate (a)
and (b) as

(a)** The proposition P (that a has F)) is true if and only if a has FF;;
and

(b)** The proposition P (that a has F)) is false if and only if it is not the case
that a has F..

(ii) Our account, in (b) and (b)*, of the conditions in which the proposition that a has F is false,
allows for two such conditioonnss:: the possible state of affairs in which the item a exists but fails to have
the attribuuttee F;; and the possible state of affairs in which the item a does not exist. Since an aattttrriibute
cann be instanced by an item in a possible world only if that item exists in that possible world, the
failure of an item to exist in a given possible world precludes it from having any attribuutteess wwhhatever
in that world.

(iii) Strictly speaking, our account -— and Aristottllee''ss -— of what it is for a proposition to be true
orr false applies only to propositions which ascribe properttiieess to items. But it is easily enough eexxtended
to deal with those propositions which ascribe relations to two or more items. We can deal with
so-called "two-place" attributeess (i.e..,, relations holddiinngg between just two items) as follows: where P is
a prooppoossiittiioonn,, aa and b are items, and RR iss the two-place attribute (relation) which P aasssseerrttss to hold
between aa and b,, then P is true if and onlyy if aa and b stand to each other in the relation R,, whillee P is

7. Mettaapphhyysiscisc,s,Tr,, 7(1100111l bb 26 - 27),, The Basic WWoorrkkss of Aristotle, ed. RR.. MMcKKeeonn,, Neww Yorrkk,, RRaannddom
HHoouussee,, 11994411.

8. A pprrooppoossiittiioonn,, we shhaalll aarrgguuee,, is tto be ddiisttinnguuishhedd frrom tthe vaarriioouuss sseenntteenncceess wwhhiichh languuaagge-sppeaakerrs
mmay uuse tto eexxpprreessss it, in mmuucchh tthe ssaammee way aas aa nnuummbberr is tto bbe ddiisttinnguuishhedd frrom tthe vaarriioouuss nuummeerraalls
wwhhiichh mmay bbe uusseedd tto eexxpprreessss it.

§ 2 Prooppoossiittiioonnss,, Trutthh,, and FFalsity 1111

false if and only if it is not the case that a and b stand in the relation R.. Andd accounts of ttrruutthh--values
for otherr relational propositions involving three or more items may be constructedd along similar lines.
Alternatively, we can deal with relational propositions by pointing out that whenever an item a stands
in a relation RR to one or more otherr items, b, c, etc., a can be said to have the relational property ooff
standing in that relation to b, c, etc. Andd since relational propertieess are properties, the ssttraightforward
account given in (a) and (b) suffices.

(iv) The account of truth which we are here espousing has been described variouslyy as ""the
Correspondence Theory", "thee Realist Theory", or even "thee Simplee Theory" of truth. In effect, it
says that a propositionn,, P, is true if and only iff. the'((ppoossssible) state of affairs, e.g., of a's having F,, iissaass
P asserts it to be. It defines "truth" as a property which propositions have just when they
"correspond" to the (possible) states of affairs whose existence they assert. It is a "realist" theory ooff
truth insofar as it makes truth a real or objective property of propositions, i.e., not something
subjective but a function of what states of affairs really exist in this or that possible world. Andd it is a
"simple" theory of truth insofar as it accords with the simple intuitions which most of us -— beforee we
try to get too sophisticated aboutt such matters -— have aboutt the conditions for saying that something
is true or false.

(v) If it seems to some that this theory is overly simple and not profound enough, this is probably
because it is deceptively easy to confusee the question "What are the conditions of truth and falsity?"
with the question "What are the conditions for our knowledge of truth and falsittyy??"" It is not
surprising, as a consequencee,, that the most commonly favored alleged rivalss to the CCoorresppondence
Theory are the so-called Coherence and Pragmatistt Theories of Truth. Proponenttss of these tthheories
tend either to deny or to ignore the distinctioonn betweenn a proposition's being true and a proposition's
being known to be true. We may acceepptt the coherencee theorist''ss claim that one way of getting to know
what propositions are true is to determinnee which of them cohere with the rest of the beliefs that we
hold to be true. Andd we may also accept the pragmatiisstt''ss claim that one way of getting to know what
propositions are true is to determinnee which of them it proves useful or practical to believee.. But at the
same time we would wish, to point out that neitherr claim warrannttss identiffyyiinngg truth with coherencee or
truth with practical usefulnesss.. Andd furthheerr we would point out that both theories are parasitic upon
the "simple"" theory of truth. The pragmattiisstt states that certain hypotheesseess are useful; the ccoohherence
theorist states that certain sets of beliefs are coherent. Yet each, in so doingg,, is implicitly making a
ccllaaiimm ttoo tthhee ssiimmppllee ttrruutthh ooff tthheessee pprrooppoossiittiioonnss.. TThhee ccoonncceepptt ooff ssiimmppllee ttrruutthh,, iitt sseeeemmss,, iiss nnoott eeaassiillyy
ddiissppeennsseedd wwiitthh.. EEvveenn tthhoossee wwhhoo wwoouulldd lliikkee mmoosstt ttoo aavvooiidd iitt ccaannnnoott ddoo ssoo..

Truth in a possible world

It should be evident from our definitions (i) that a propositionn,, P, may assert that an item a has
attribuuttee F even if a exists only in a non-actual possible world, and (ii)) that P wwiill be true in that

non-actual world provided that in that world a has F.. Let us use the letter "wW"" (with or without

numerical indices) to referr to any possible world of our choosing (e.g., the world of Heinleinn''ss Time
Enoughh for Love). Then our definitions allow us, for instance, to assert the truth in some specified
possible world, WIj', of the proposition that Lazarus Long has the relational property of being a lover
off Minerva even though in some otherr specified possible world, W 22 (e.g., the actual worldd)),, he does
not exist and a fortiori does not have that property. Likewise, they allow us to assert the falsity in
ssoommee ssppeecciiffiieedd ppoossssiibbllee wwoorrlldd,, WW 22 ,, ooff tthhee pprrooppoossiittiioonn tthhaatt LLaazzaarruuss hhaadd MMiinneerrvvaa aass aa lloovveerr eevveenn
tthhoouugghh iinn ssoommee ootthheerr ssppeecciiffiieedd ppoossssiibbllee wwoorrlldd,, ee..gg..,, WWIj', tthhaatt pprrooppoossiittiioonn iiss ttrruuee.. NNeeeeddlleessss ttoo ssaayy,, tthheeyy
aallssoo aallllooww uuss ttoo aasssseerrtt tthhaatt aa pprrooppoossiittiioonn iiss ttrruuee ((oorr ffaallssee)) iinn ssoommee uimnssppeecciiffiieedd wwoorrlldd,, WW;; oorr,, aass wwee
sshhaallll llaatteerr ppuutt iitt,, ttoo aasssseerrtt tthhaatt aa pprrooppoossiittiioonn iiss ppoossssiibbllyy ttrruuee ((oorr ffaallssee,, aass tthhee ccaassee mmaayy bbee))..

12 POS S I B L EE WWOORRLLDDSS

Truth in the actual world

It follows from what we have justt said that -— contrary to what is often supposedd -— the eexxpressions
"is true" and "is false" do not mean the same as "is actually true" and "is actually false". To say that
a proposIition is actually true (or false) is to say that among the possible worlds inn which it is true (or
false) there is the actual world. Thus, for instance, the proposition that Canada is north of Mexico is
actually true because in the actual world Canada stands in the relation of being north-of to Mexico,
i.e., has the relational property of being north of Mexico. Andd equally, the proposition that Lazarus
has Minerva as a lover is actually false because in the actual world Lazarus fails to exist.

The fact that "true"" does not mean "actuaJlly true" should be evident from two considerations.
First, if it didd,, then the occurrence of the qualifier "actually" would be wholly redunnddaanntt (pleonastic).
Yet, as we have already seen, it is not. Secondllyy,, if it did mean the same, then personss in other
possible worlds would not be able to invoke tth~e concept of truth without therebyy making claims aabout
thiiss world, the actual one in which you and we find ourselves. Considerr,, for illustrativvee ppuurrpposes,
Lazarus' claim (on the first page of the narrative of Time Enoughh for Love): "It''ss true I'm not handy
with the jabbbeerr they speakk here." The year is AA.DD.. 4272 and the place is the fictional Howard
Rejuvenation Clinic on the fictional planett of Secundus. Lazarus asserts that a certain proposition,
viz., that he doesn''tt have facility with the local language, is true. We underssttaanndd him to be saying
ssoommeetthhiinngg wwhhiicchh iiss ttrruuee ooff tthhee ffiiccttiioonnaall wwoorrlldd iinn wwhhiicchh hhee eexxiissttss.. YYeett iiff ""ttrruuee"" mmeeaanntt ""aaccttuuaallllyy ttrruuee""
wwee sshhoouulldd bbee oobblliiggeedd ttoo uunnddeerrssttaanndd hhiimm aass ssaayyiinngg ssoommeetthhiinngg vveerryy ddiiffffeerreenntt -— aass ssaayyiinngg ssoommeetthhiinngg
aabboouutt hhiiss llaacckk ooff ffaacciilliittyy wwiitthh aa llaanngguuaaggee tthhaatt eexxiissttss iinn tthhee aaccttuuaall wwoorrlldd wwhhiicchh yyoouu aanndd wwee iinnhhaabbiitt..

Although "true"" does not meeaann "actually true", it can be -— and often is -— used to reffeerr to actual
truth; what matters is who uses it. When personss in the actual world claim that a proposition is true
they are claiming that it is true in their own world; and since their world is the actual world, it turns
out that they are attributing actual truth to the propositioonn.. However,, when a person in a non-actual
world attribuutteess truth to a propositionn,, he is attributing to that proposition truth in his own world;
and since his world is not the actual world, it turns out that he is attributing non-actual (but possible)
truth to that proposition.

The mmyytthh of deeggrreeeessoofftrtruuthth

It is worth pointing out that it is implicit in both Aristotllee''ss account and the one in (a)* and (b)* that
truth and falsity do not admit of degrees. Althouugghh there is a common manneerr of speech which ffoosstters
this belieff -— for example, "Jones' reportt was more true than Roberts'" -— strictly speaking this man-
ner of speech is logically untenable. A proposition is either wholly true or wholly false. There is no
such thing as partial truth. Consider the simplest sort of case, that in which a proposition P ascribes an
attribuuttee F to an item a:: P is true if a has attribuuttee F, otherwise P is false. There is no provissiioonn,, no
room, in this explicattiioonn of truth, for P to be anything otherr than wholly true or wholly false; for
either a has the attribuuttee F, or it is not the case that a has the attribuuttee F.

There is,, however, a way to. reconcile the common manneerr of speech justt reportedd with the
stringencies of our definition of truth; and that is to regard such utterances as "Jones'' report was more
true than Roberts'" as elliptical for something of this sort: "AA greater numberr of the details reported
by Jones were true than those reportedd by Roberts." Similarly, an article in Time magazinnee99 which
bore the title "Howw True is the Bible?" could instead have been better presenntteedd underr the title
"Howw Much of the Bible is True?"

9. Dec.. 30, 11997744.

§ 33 PPrrooppeerrttiieess oof PPrrooppoossiittionns 13

3. PROPERTIES OF PROPOOSITIIOONNS

Among thhe itemms whhichh exxist inn vaarriioouuss ppoossible worrldds mmuust bbe includdeedd pprrooppoossiittiioonnss;; aanndd aammoonngg the
aattttrriibbuuttees whhichh pprrooppoossiittionns instance inn vaarriioouuss ppoossible worrldds aarree thhe pprrooppeerrttiiees of trruutthh aanndd
faalsittyy. IInn every ppoossible worrlldd each pprrooppoossiittiionn hhaas one orr otherr of tthheessee pprrooppeerrttiiees of trruutthh aanndd
faalsittyy. Buutt trruutthh aanndd faalsity aarree nnot thhe onnly pprrooppeerrttiiees whhichh pprrooppoossiittionns cann hhaavvee. Byy virttuuee of the
fact that inn aannyy given ppoossible worrldd, aa pprrooppoossiittiionn hhaas eitthherr thhe pprrooppeerrttyy of trruutthh orr thhe pprrooppeerrttyy ooff
faalsityy, we cann ddistinguuish otherr pprrooppeerrttiiees whhichh aa pprrooppoossiittiionn cann hhaavvee, vizz..,, ppoossible trruutthh, ppoosssible
faalsittyy, conttinngenncy, nnoonnconnttinnggennccy, nnecesssaarry trruutthh, aanndd nnecesssaarry faalsittyy. TThheese aarre pprrooppeerrttiiees —-
moddaall properties we shhaalll call thhem -— whhichh pprrooppoossiittionns hhaavve aacccorrddiinngg to thhe waay inn whhichh theirr
trruutthh--vvaalluuees aarree ddistrribbuutteedd aacrrooss thhe sett of all ppoossible worrllddss;; aacccorrddiinngg,, that is, to whethherr thhey are
ttrruuee ((oorr ffaallssee)) iinn jjuusstt ssoommee,, iinn aallll,, oorr iinn nnoonnee ooff tthhee ttoottaalliittyy ooff ppoossssiibbllee wwoorrllddss.I.010 IInn ddiissttiinngguuiisshhiinngg
tthheemm wwee bbeeggiinn ttoo aapppprrooaacchh tthhee hheeaarrtt ooff llooggiicc iittsseellff..

Possibllyy truuee pprrooppoossiittiioonnss

CConnsiderr, forr aa starrtt, tthhoossee pprrooppoossiittionns whhichh aarree trruue inn aat leaast one ppoossible worrldd. TThhee pprrooppoossiittiioonn
that aa ppaarrttiiccuullaarr itemm,, Laazarruuss, hhaadd aa ppaarrttiiccuullaarr aattttrriibbuuttee,, that of hhaavvinngg lottss of timme forr love inn hhis
twenty-tthhrreee hhuunnddrreedd odddd yearrss, is aa caassee inn ppooinntt.. IIt is aa pprrooppoossiittiionn whhichh is trruuee inn that ppoossible
'"tthhoouugghh (so farr aas we kknnooww) nnonn-aactuuaall worrlldd of HHeeinnlein''ss nnoovveell.. WWee shhaalll say thhat it is possibly truee,,
or aagaainn that it is aa possiblee truutthh.. Annotthheerr exammpple of aa pprrooppoossiittiionn whhichh is trruue inn aat leaast one
ppoossible worrlldd is thhe pprrooppoossiittiionn that WWooooddrrooww WWilson was pprreesident of thhe U..SS.. inn 1917. IInn thhis
caassee,, of courrssee, one of thhe ppoossible worrldds inn whhichh thhe pprrooppoossiittiionn is trruuee is aalso thhe aactuuaall one.
Neverrthhelesss it mmaakkees ssennssee to say that thhis pprrooppoossiittiioonn is aa ppoossible trruutthh even if, inn saaying so, we are
nnot saaying aallll that we aarre ennttitled to saay, vizz..,, that it is aalso aann aacttuuaall trruutthh. FFoorr aacttuuaall trruutthhss form a
suubbclass of ppoossible trruutthhss.. A pprrooppoossiittiionn is aa ppoossible trruutthh, thheenn, if it is trruuee inn aat leaast one ppoossible
world -— aactuuaall orr nnoonn--aaccttuuaall.. A pprrooppoossiittiionn is actually truuee if aammoonngg thhe ppoossible worrlddss inn whhichh it
is trruue thherre occurrs thhe aactuuaall world.

In saaying of aa pprrooppoossiittiionn that it is trruue inn aat leaast one ppoossible worrlldd it shhouuldd nnot bbe thhought that
it is aalso bbeing claimmedd that thhat pprrooppoossiittiionn is faalse inn sommee otherr ppoossible worrldd. WWhhen we say that aa
pprrooppoossittion is trruue inn sommee ppoossible worrldds we leave it aann oppen qquueestion aas to whethherr thhat pproppositionn
is trruue inn aallll otherr ppoossible worrldds aas well, orr is false inn sommee of tthhoossee otherr ppoossible worldds.

Possibllyy faallssee pprrooppoossiittiioonnss

How aabbout pprrooppoossiittionns whhichh aarree false inn aat least one ppoossible worrldd?? Suucchh pprrooppoossiittiioonnss, we shhall
want to saay, aarree possiblyy faallssee,, oorr aarree ppoossssiibblleeffaallssiititeiess. . NNooww ssoommee((bbuutt,, aass wwee sshhaalllsseeee llaatteerr,, nnoott aalll))
pprrooppoossiittionns whhichh aarree ppoossibly trruue aarree aalso ppoossibly faalse. TThhee pprrooppoossiittiionn that LLaazarruus hhaadd lottss off
timme forr love is nnot onnly ppoossibly trruuee,, bbecause thherre is aa ppoossible worrlldd inn whhichh it is trruuee;; it is aalso
ppoossibly faalse, bbecause thherre aarree otherr ppoossible worrldds -— ourr own aactuuaall one aammoonngg thhemm -— inn whichh
(so farr aas we kknnooww) it is faalse, i.e., aacttuuaallly faalse. Similarriily, thhe pprrooppoossiittiionn thhat WWooooddrrooww WWilson
was pprreessiddenilt of thhe Unniitteedd SSttaatteess inn 19177 is nnot onnly ppoossibly (as well aas aaccttuuaallllyy) trruuee,, bbecause there
is aa ppoossible (as it hhaappppenns thhe aactuuaall) worrlldd inn whhichh it is trruuee;; it is aalso ppoossibly faalse, bbecause there
aarree otherr ppoossible worrldds -— ourr own aactuuaall one noott aammoonngg thhemm -— inn whhichh it is faallssee.

10. Note that although attrriibbuuttiioonns of noncontingennt trruutthh andd noncontingennt falsity will count as attrribuuttions
of moddaall status, the corrreessppoonnddiinngg attrriibbuuttiioonns of contingennt trruutthh andd contingennt falsity will noott.. Thhee rreaason forr
allowing ann ascrrippttiioonn of contingency to count as ann ascrrippttiioonn of moddaall stattuuss but not allowing ascrripttions ooff
contingennt trruutthh orr contingennt falsehood is given in chapter 6, section 5.

14 POSSIBLE WWOORRLLDDS

Contingent propositions

Anyy pprrooppoossiittiioonn whhiicchh nnot onnllyy is trruuee inn some ppoossibble worrllddss bbuutt aalso is faalse inn somme possible
worrlddss is saaidd to bbee aa contingenntt proposition.. A contingent pprrooppoossiittiioonn,, thhaat is, is bbotthh ppoosssibbly ttrrue
andd ppoosssibbly faalse. TThhee pprrooppoossiittiioonn aabbouutt WWooooddrrooww WWilson is contingent aanndd hhaappppeennss ttoo bbee trruuee iinn
thhe aacttuuaall worrld, whhiile thhe pprrooppoossiittiioonn aabbouutt Lazarruus is contingent aanndd hhaappppeennss ttoo bbee faalse inn the
aactuuaal worrldd. TThhee forrmmeerr is aa ppoossible trruuth whhiicchh could hhaavve bbeen faalse, even thhoouugghh aas aa mmaatttterr ooff
aactuuaal faact it is nnoott;; thhe latter is aa ppoossibble faalsity whhiichh could hhaavve bbeen trruuee,, even thhoouugghh aas aa mmaatttteerr
off aactuuaal fact it is not.

Contradictories off propositions

Suuppppoosse we hhaavve aa pprrooppoossiittiioonn whhiicchh is contingent aanndd trruuee inn thhe aacttuuaall worrld, suuchh aas the
pprrooppoossittion thhaat thhe U..SS.. enterred World WWaarr II inn 1917: thhenn it is trruuee inn some ppoossible wwoorrlds,
including thhe aacttuuaall onnee, bbuutt faalse inn aallll tthhoossee ppoossible worrllddss inn whhiicchh it is nnot trruuee.. What mmigghht
suchh aa ppoossibble world bbee inn whhiicchh it is faalse thhaat thhe U.SS.. enterred World Waarr II inn 1917?? IIt couulldd bbee,,
foorr exammppllee, aa ppoossibble world inn whhiichh thhe U..SS.. enterred World WWaarr II, nnot inn 1917, bbuutt inn 1918; oorr,
it mmiighhtt bbe aa ppoossible world inn whhiichh World Waarr II nneverr took pplaacce aat aallll,, pperrhhaappss bbecauuse uunniivveerrssaall
ppeaace hhaadd bbeen estabblished inn thhaat world some yearrss eaarrlliieerr,, orr bbecauuse maannkkind hhaadd mmaannaaggeedd to
ddestrrooy itself qquuiitte aaccciddennttllyy inn 1916; orr aaggaainn it mmiighhtt bbee aa ppoossibble world inn whhiicchh thhe Norrth
Amerrican conttinnenntt, aanndd hhennce thhe U..SS..,, simmppllyy ddiidd nnot exxistt.. IInn eaachh aanndd eveerryyoonne of tthheessee laatttteerr
ppoossibble worrllddss,, aanndd inn counnttless otherrs bbeesiddes, it is faalse thhaat thhe U.SS.. enterred World Waarr II inn 11917,
oorr iinn ootthheerr wwoorrddss,, iitt iiss ttrruuee tthhaatt iitt iiss nnoott tthhee ccaassee tthhaatt tthhee UU..SS.. eenntteerreedd WWoorrlldd WWaarr II iinn 11991177..

Lett uus call aannyy pprrooppoossiittiioonn whhiicchh is trruuee inn aallll tthhoossee ppoossibble worrllddss,, if aannyy,, inn whhiicchh aa givenn

AA llll PPoossssiible WWoorrllddss

A

P iiss ffaallssee P iiss ttrruuee

v vA . J

AA coonnttrraaddiiccttoorryy ooff CCoonnttiinggeenntt
coonnttiingeenntt pprrooppoossiittionn PP pprrooppoossiittionn PP

FIGUURREE ((1l.d))

A contingent pprrooppoossiittiioonn,, P,, is rreepprreesseennttedd aas bbeeinngg ttrruuee inn somme ppoossibble wworrlds
andd faalse inn aallll thhe othherrss.. Annyy contrraaddictorry of pprrooppoossittion PP will thhenn bbee faalse iinn
all tthhoossee ppoossible worrllddss inn whhiichh PP is trruuee aanndd trruuee inn aallll tthhoossee ppoossibble worrllddss iinn
whhich PP is faalse.

§§ 3 PPrrooppeerrttiies of Prrooppoossitions 15

prroppoositionn is false, and which is false in all thhoose possiibbllee worldds, if anny, in which a givveenn pprroposittioonn
is trruue, a contradictory of thaatt prrooppoossiittiioonn.!.!11 Thenn the prrooppoosition thaatt it is not the case thaatt the UU..SS.
enntteerreedd Worrld Warr I in 191177 is a conntraaddictory of the prrooppoosition thaatt the U.S.. didd enntteerr
Worrld Warr I at thaatt time, and vice versa. The prrooppoosition thaatt itt is not the case thaatt itt didd,, sinncee it iss
a contrraaddiictorryy of aa trruue continnggeenntt pprrooppoossiittiioonn,, will bbe continnggeenntt aanndd false. IIt will bbe false, that is,
in all thhoose.ppoossibblee worrldds, includdinngg the actuaal one, in which the U.SS.. enterredd World Warr I in 11917.
Again, supposee we have a prrooppoossition which is conntinnggeenntt and false, such as the prrooppoossition that nno
fighhttiinnggooccccuurrreeddininFFraranncecedduurirninggWWoorrldldWWaarrI.I.TThheenn, ,aaltlhthoouugghhththisisccoonntitninggeenntt pproroppoosistiitoionnwwililllbbee
false in somme possibblee worlds includdinngg the actuaal one, therre will be othheerr possibblee worlds in which itt
iiss ttrruuee.. TThheessee ootthheerr ppoossssiibbllee wwoorrllddss wwiillll bbee pprreecciisseellyy tthhoossee iinn wwhhiicchh aa ccoonnttrraaddiiccttoorryy ooff tthhaatt
pprrooppoossiittiioonn,, ee..gg..,, tthhaatt ffiigghhttiinngg ddiidd ooccccuurr iinn FFrraannccee aatt tthhaatt ttiimmee,, wwiillll bbee ffaallssee..

Noncontingent propossiittIiOons

Wee have justt beenn talkinng about thhoosse prrooppoositions which arre both possibly trruue and possibly faalse.
They arree, we saidd, the conntinnggeenntt prrooppoossittionns. Arree therre any prrooppoositions which arre nott both possibbly
trruue andd possibly false?? Annyy prrooppoositions which arre justt the one andd not the otherr? Annyy prrooppoossittions,
foorr instannce, whhich couldd nnot ppossibly bbe false, i.e., whhich mmuust bbe trruue?? Orr aagaainn, aanny pprrooppoossitions

/_ - - - - - - - - - -AA-l-lll-PP-oo-sssAsiibb. . -lle- -WWo-or-rlld-ss- - - - - - - - - - - - - - - - ' " " ' l

TThhee

NNoonn--aaccttuuaall acct uu aall

wwoorrlldss wwoorrlldd

v A "\ \

1•
I ~r~: IPis
P iiss ttrruuee true

"----------------------------- 1

. NNeecceessssaarriillyy ttrruue"„ pprrooppoossiittiioon PP

FIGURE ((J1..e)e)

Iff PP is aa nnecessaarily ttrruue (noncontingennttlyy ttrruuee) pprrooppoossiittiioonn,, tthhen PP is ttrruue inn aallll ppoossible worrlds
-— tthhe nnoonn-aactuuaall oonneess aas wwell aas tthhe aaccttuuaall onnee.

1111.. IItt iiss ccoommmmoonnllyy ssaaiidd tthhaatt aa pprrooppoossiittiioonn hhaass oonnee aanndd oonnllyy oonnee ccoonnttrraaddiiccttoorryy.. IIf tthhiiss cclaaiimm wweerree ttrruuee,, wwe
oouugghhtt,, aaccccoorrddiinnggllyy,, ttoo ssppeak oof thhee ccoonnttrraaddiiccttoorryy oof aa pprrooppoossiittiioonn rraatthheerr tthhaann,, aass wwee hhaavvee hheerree,, oof a
ccoonnttrraaddiiccttoorryy.. HHoowweevveerr,, ssuubbsseqquuennttly iinn tthhiiss cchhaapptterr -— aaffterr wwee hhaavvee mmaaddee aa ddiissttiinnccttiioonn bbetwweeenn
pprrooppoossiittiioonnaall--iddeennttiittyy aanndd pprrooppoossiittiioonnaall--eqquuiivaalence -— wwee wwiillll sshhooww tthhaatt eevverryy pprrooppoossiittiioonn hhaass aann innfinnite
nnuummbbeerr oof nnoonn--iiddeennttiiccaall eeqquuiivvaalents aanndd hheennccee tthhaatt eevverryy pprrooppoossiittiioonn hhaass aann iinnffinnite nnuummbbeerr oof nonn--iddeennttiiccaall,
bbuutt eeqquuiivvaalleenntt, contrraaddiiccttoorriieess.

1166 POSSIBLE WWOORRLLDDS

whhich couuldd nnot ppoossibly bbe trruuee,, i.e., whhich mmuust bbe false? Suuchh pprrooppoossiittiionns, if thherre aarre aannyy, would
nnot bbe contingent. TThheeyy wouuld bbe what we shouuld want to caallll nonconntinggeenntt prrooppoossiittionns.

CConnsiderr, firrsstt,, what it wouuld bbe like forr thhere to bbe pprrooppoossittions whhichh mmuusstt bee trruuee.. Suuchh
pprrooppoossittions we shouuld caallll neceessssaarriillyy truuee pprrooppoossiittiionnss. A nnecessaarilyy trruue pprrooppoossiittionn wouuldd,, off
courrsse, bbe aa ppoossibly trruue pprrooppoossiittiioonn.. IInnddeeedd, since it wouuldd nnot onnly bbe trruue inn aat leaast one ppoosssible
world bbuut trruue inn alll ppossible worrllddss, it wouuld aallssoo bbe trruue inn thhe aactuuaal worrlldd.. See figuurre ((Il..ee))..

A nnecessaarilyy ttrruue pprrooppoossiittiioonn,, inn shorrtt, wouuldd bbe bbothh ppoossibly trruue aanndd aacttuuaallly ttrruuee.. Buutt it would
nnot bbe continngeenntt aanndd trruuee.. FForr aa trruue continnggeenntt pprrooppoossiittiioonn,, rreemmeemmbbeerr,, is faallssee inn sommee ppoosssible
worrldd, wwhheerreeaass aa nnecessaarilyy trruue pprrooppoossiittionn -— since it wouuld bbe trruue inn every ppossible worrlldd —-
could nnot bbe faallse inn aannyy.. A nnecessaarilyy trruue pprrooppoossiittiioonn,, thhenn, aas well aas bbeing bbothh aacttuuaallly aannd
ppoossibly trruuee,, wouuld bbe noncontingently truee..

Can we give aann example of such aa pprrooppoossittion?? EExxaammpplleess aabboouunndd,, aanndd we shhaalll examinne mmaanny in
thhe courrse of thhis bboookk.. Buut forr pprreesenntt ppuurrppoosses we shhaalll satisffyy ouurrsseelvess with aa ppaarrticcuulaarrly
strraaightfoorrwwaarrd exammpplle.

IIt is aa faairrllyy obbvious fact that nnot onnly cann we aascrribbe trruutthh aanndd faalsity (and otherr aattttrriibbuuttes) to
individual pprrooppoossiittiionns, bbuut aallssoo we cann aascrribbe varriiouus aattttrribbuuttes to pairs of pprrooppoossiittiionnss. FFoorr
exammpple, we cann aassserrtt of aa pair of pprrooppoossitions (1) that nneitthherr is trruuee,, (2) that onnly one is trruuee,, (3)
that aat least one is trruuee,, orr (4) that bbothh aarre trruuee,, etc. In aasserrting somethinng of aa pairr of pprrooppoossittions
one is, of courrsse, exprressinngg aa pprrooppoossiittionn whhichh is itselff eitthheerr trruue or faalse. FForr exammpple, if one werre
to aassserrtt of thhe two faallse continngeenntt pprrooppoossittions

(1.-11)) Benjaammiinn Frraannkklin was aa pprreesidennt of SSppain,

(1.22)) Canada is south of MMexico,

that one or thhe otherr of thhem is trruuee,, thhen thhe pprrooppoossiittionn whhichh one hhaas exprresseedd is ittseelf, obviously
enouughh, continngeenntt aanndd faallssee.

Ourr aabbiillity to aascrribbe trruutthh to nneitthherr of, orr to one of, orr to bbothh of, etc., aa pair of prrooppoossiittionns,
pprroovviddees aa mmeans to consttrruct aann exampple of aa nnecessaarilyy trruue pprroppoosition.

Necessarilyy truuee pprrooppoossiittiioonnss

CConnsider thhe caassee inn whhich we aascrribe trruutthh to one orr thhe otherr of aa pair of pprrooppoossiittiionnss. WWee caann,, off
courrsse, ddoo thhis forr aanny arrbbitrraarriily selectedd pair of pprrooppoossitions whhaatteverr,. Buut let''ss see what hhaappppenns
when we ddoo so inn thhe caassee of aascrriibbiinngg trruutthh to one orr thhe otherr of aa pair of contrraaddiiccttorryy prrooppoossiittionns.
Taakke, forr instannce, thhe contrraaddiiccttorryy pairr

(1.3) TThhee U..SS.. enterredd World WWaarr II inn 119917;

(1.44)) IIt is nnoot thhe caassee that the U.S.. enterredd World WWaarr II inn 119917.

As we saw, (1.33)) is trruue inn aallll tthhoossee ppossible worrldds inn whhich (1.44)) is faalse, aanndd (1.44)) is trruue inn aall
tthhoossee ppossible worrldds inn whhichh (1.33)) is faalse. So inn every ppossible worrlldd one orr thhe otherr is trruuee.. Iff
we thhen aassserrtt of such aa pair that one or thhe otherr is trruuee,, thhe pprrooppoossiittionn we expprreessss will bbe trruue in
all ppossible worrllddss.. Thhis pprrooppoossiittionn will be

(1.55)) Eitherr thhe U..SS.. enterredd World WWaarr II inn 19177 orr it is nnot thhe caassee that thhe UU.S.
enterredd World WWaarr II inn 119917.

§ 33 PPrrooppeerrttiieess oof PPrrooppoossiittiioonns 17

TThhe ssaammee hhoollddss forr aannyy- ppaair of contrraaddiiccttoorriies whhaattsoeveerr.. Think of anny prrooppoossiittIiOon you like:: ffoorr

eexaammppllee, tthhe ccontingenntt pprrooppoossiittiionn tthhat youu (the rreeaadderr) aarre nnow (aatt the mmomenntt of rreadding this

sseenntteennccee)) wweaarriinngg aa ppaair of bbluue jeanns. IIt ddooesn''tt mmaattterr whetthheerr thhis prrooppoossition is actuaally true oorr

ffaallssee.. BBeeinngg cconnttingent, itt is ttrruuee inn aat leaastt sommee ppossible worrldds —- even if not the actual one —- aanndd

ffaallssee iinn aallll tthhoossee inn wwhhiichh it is nnot ttrruuee.. Now thhinkk of aa contrraaddiictorry of that prrooppoossiittionn, e.g..,, thaatt iitt

iiss nnoott tthhe ccaassee tthhaat youu aarree wwearrinng aa pair of bblue jeans. WWhhether thhis lattteerr prrooppoossition is aacttuallyy

ttrruuee oorr ffaallssee, itt will bbee ttrruuee inn aallll tthhoossee ppoossible worrldds inn whhich the prrooppoossition that you arre wweeaarrinngg

bblluuee jjeeaannss iiss faallsse, aanndd ffaallssee inn aallll tthhoossee ppoossible worrldds in whhich thhis prrooppoossition is trruuee. Finally,

tthheenn,, tthhiinnkk oof tthhe pprrooppoossiittiionn tthhaat eithheerr youu aarre wearring aa pairr of blue jeanns orr it is not the case thhaatt

yyoouu aarree wweaarriinngg aa ppaair of bbluue jeanns. TThhis latterr pprrooppoossiittiioonn,, justt like the prrooppoossition (1.5), is true iinn

aalll ppoosssibble wwoorrllddss,, aanndd hheenncce, likke (1.5), is nnecessaarilyy truue.

Anny pprrooppoossiittiioonn wwhhiichh aasssserrttss of ttwo simppler oonneess that eitthheerr one or the othheerr is trruuee, i.e.,, thaatt aatt

lleeaasstt oonne is ttrruuee,, iss caallledd aa disjunctive pprrooppoossiittiioonn:: it disjoins two simmplerr prrooppoositionns eachh onne ooff

wwhhiicchh iiss aa ddiissjjuunncctt.. TThhee ddiisjuunnctts inn aa ddisjuunnccttive pprrooppoossittion need not be conntraaddictorries of onnee

aannootthheerr,, aass tthheey aarree inn tthhe ccaassee of (1.5). FForr instannce, the pprrooppoossittion that eitthheerr you arre weaaring aa

ppaairr oof bblluuee jjeeaannss oorr youu aarree ddrreesssed forr aa formmaal occasionn is aa ddisjunncttion whhoose disjjuncts arre nnoott

cconnttrraaddiiccttoorriiees oof oonne aannootthheerr.. Buutt inn tthhe ccaassee when ddisjuncts aarre contrraaddiiccttoorriiees, it folloowws, as we

hhaavvee sseeeenn,, tthhaat tthhe ddiisjjuunnccttiioonn itself is nneecessaarilyy trruue. .

IIt iiss iimmppoorrttaanntt nnoott ttoo tthhinnkk tthhaat tthhe onnly nnecessaarilyy trruue pprrooppoossitions arre thhoosse which asseerrt of aa

ppaairr oof cconnttrraaddiiccttoorriiees tthhaat onne orr tthhe otherr is trruuee.. Later we shaall cite exaammpleess of necessssaarrillyy ttrrue

pprrooppoossiittiionns wwhhiicchh aarree nnoot of tthhiis sortt.

Neecceessssaarriillyy faallssee pprrooppoossiittiioonnss

CCoonnssiidderr,, nnooww,, wwhhat itt wwouuldd bbee likke tto hhaavve aa nnonncontinnggeenntt prrooppoossittion which muusstt beejJaalslsee. .SSuucchhaa
pprrooppoossiittiioonn wwe sshhoouuldd caall aa necceessssaarriillyy ffaalslsee pprrooppoossitiitoionn.. AA nneecceessssaarriillyy ffaallssee pprrooppoossiittiioonn wwoouulldd,, oof
ccoouurrssee,, bbee aa ppoosssibbly ffaallssee pprrooppoossiittiioonn.. IInnddeeedd, since it would not only be false in at leeaast ssoommee
ppoosssibble wwoorrllddss bbuutt ffaallssee inn allll ppoossible worrllddss, it would aallso be faalse in the actual world. AA
nneeccessarrily ffaallssee pprrooppoossiittiioonn,, inn shhorrtt, wouuldd bbe bbothh ppossibly faalse aanndd actuuaally false. But it would nnoott
bbee cconnttingenntt aanndd faallssee. FFoorr aa faallssee continngeenntt pprrooppoossiittiioonn,, rremmemmbberr, is trruue in somme possiibbllee wwoorrlldd
wwhheerreeaass aa nneeccessarrily ffaallssee pprrooppoossiittiionn -— since it is faallse inn everry possibblee worrldd —- is not true in aany.
A nneeccessarrily ffaallssee pprrooppoossiittiioonn,, tthheenn,, aas well aas bbeing bboth aactuuaallly aanndd possibly false would bbee
nnoonnccoonnttiinnggeennttly faallssee..

WWhhaatt wwoouulldd bbee aann exammpple of suuchh aa pprrooppoossittion?? Again, our abbiility to say sommetthhinngg abouutt thhe
wwaayy ttrruutthh aanndd faallsity aarree ddiistrribbuutteedd bbeettwweeenn thhe mmemmbberrs of aa ppair ooff pprrooppoossiittiioonnss ggiivveess uuss tthhe mmeeaanns
oof cconnstrruuccttiinngg aann eexammpple of aa nneecessaarily faallssee pprroppoosition.

JJuusstt aass wwe ccaann ssaay oof aannyy ppair of pprrooppoossittions that one or the otherr of themm is trruuee, we can aalssoo
aasssserrtt oof aannyy ppaairr tthhaat bbootthh aarreettrruuee. .BBuuttccoonnssiiddeerrwwhhaatt hhaappppeennsswwhheenn wwee ssaayy ooffaappaaiirrooff pprrooppoossiittiioonnss
tthhaatt tthheey bbootthh aarree ttrruuee inn tthhe ccaassee inn whhichh thhe two pprrooppoossitions happppen to be coonnttrraddictorries.
CCoonnssidderr,, aaggaaiinn,, tthhe connttrraaddiiccttoorryy ppair of pprrooppoossittions (1.3)) aanndd (1.4). Supppose we werre to asseerrt ooff
tthheemm tthhaatt tthheey bbootthh aarree ttrruuee.. TThhee pprrooppoossiittionn to thhis effectt would be

(11..66)) TThhee UU..SS.. entered World WWaarr I inn 19177 aanndd it iIsS not the caase thatt the UU..SS.
enntered Worrld WWaarr II inn 119917.

IIt iiss eeaassyy ttoo ssee tthhaat tthherre iss nnoo ppoossible worrlldd inn whhich thhis latterr prrooppoossition is trruuee. Forr,, as we saaww
eeaarrllierr wwhhen ddiisscuussing tthhe limmitts of ppoossibility aanndd conceivabbility, a supposeedd worrldd in wwhhicchh
ssommethhing literraallllyy bbootthh is tthhe ccaassee aanndd is nnot thhe caassee is nnot, inn aanny sennssee, a possibblee world; it is aann

18 PO SSSSIIBBLLEE WWOORRLLDDSS

AA llll PPoossssiibblle WWoorrlldss ,

.-AA . TThhee
AA ccttuuaall
NNoonn--aaccttuuaall
WWoorrlldss .. WWoorrlld
"*- A
N
P iiss ffaallssee
P iiss
f aa l s e

•—— ^

NNeecceessssaarriillyy ffaa llssee pprrooppoossiittiioonn PP

FF II GG UU RREE ((11.jf))

Iff P is a necessarily false (i.e. noncontingently false) prooppOoSsiltilon,, then P is false in aall
possible worlds -— the hnon-actual oonnes as well as the actual one. Such propositions are
also said to be self-contradictory, self-inconsistent, or logicaallllyy impossibbllee..

impossible one. The case we considered earlier was that of the supposition that time travel both will
occur sometime in the futurree and will not occur at any time in the future. We can see that it, like the
case we are presently consideringg,, involvveess ascribing truth to two propositions one of which is a
cbontradictory of the other. Andd perhaps we can also now see why we earlier said that that proposition
was self-contradictory, and why we would now want to say that the present case is also self-
contradictory. For if any proposition ascribes truth to both memberss of a pair of contradictories, then
that proposition is one which has a contradiction within iitself.

Anyy proposition which asserts of two simpler ones that both of them are true, is called a
conjuncttiivvee propositiioonn:: it conjoins two simpler propositions each of which is called a connjjuunncctt.. The
conjuncttss in a conjunctive proposition need not be contradictories of one anotherr,, as they are in the
case of (71.6). For instance, the proposition that you are wearing a pair of blue jeans and your friend
is wearing a pair of tweeds is a conjunction whose conjuncttss are not contradictories. But in the case
where conjuncttss are contradictories, it follows, as we have seen, that the conjunction is necessarily
false.

More abboouuttccoonntrtraaddicictotorryypprrooppoossitiitoionnss

AA simple way of describing the relation which holds betweenn two propoossiittIiOons which are
contradictories of one anothheerr is, to say that in each possible world one or otherr of those propositions is
true and the otherr is false. This description has two immediate consequueenncceess which we would do well
to note.

First, note that any contradictory of a contingenntt proposition is itsellff a contingentt propositiioonn.. FFor
iff a proposition is contingent, i.e., true in some possible worlds and false in alll the others, then since
any proposition which is its contradictory must be false in alll possible worlds in which the former is

§ 33 PPrrooppeerrttiieess oof PPrrooppoossiittionns 19

trruuee, aanndd trruue inn aallll ppossible worrldds inn whhichh thhe formmerr is faalse, it follows immmeeddiaattely that thhis
secoonndd pprrooppoossiittionn mmuust bbe faallse in sommee ppossible worrldds aanndd trruue in sommee othherrs. Buut thhis is just to
say that thhe latterr pprrooppoossiittionn is cconnttinngent.

Secondd, nnoottee that aanny contrraaddiiccttorryy of aa nnonncontinngeenntt pprrooppoossiittionn is itselff aa nnoonncconnttingennt
pprrooppoossiittiionn. FForr if aa pprrooppoossiittionn is nnonncontingentlyy trruuee,, i.e., trruue inn aallll ppossible worrldds aanndd faallssee in
nnonne, thhen aanny pprrooppoossiittionn whhichh is its contrraaddiicttorryy mmuust bbe faallsee inn aallll ppossible wwoorrlldds aanndd trruue in
nnonne, whhich is just to say that it itselff mmuust bbe nnonncontingennt, aanndd mmorree especially, nnoonncconnttinngently
faalse. Similarrly, if aa pprrooppoossiittionn is nnonncontingentlyy faalse, i.e., faallssee in aallll ppossible worrldds aanndd trruue in
nnonne, thhen aanny pprrooppoossiittionn whhichh is its contrraaddiiccttorryy mmuust bbe trruue inn aallll ppossible worrldds aanndd faallssee in
nnonne, whhichh is just to say that it itselff mmuust bbe nnonncontingennt, aanndd mmorree especially, nnoonncconnttinngently
trruuee.. In shorrtt, if one mmemmbber of aa contrraaddiiccttorryy pair of pprrooppoossitions is nnecessaarilyy trruue orr nneeccessarrily
faalse, thhen thhe otherr mmemmbber of that pair mmuust bbe nnecessaarilyy faallssee orr nnecessaarilyy trruue rreessppeeccttiively. To
cite aann example of such aa pair we nneed onnly rreetuurrnn to pprrooppoossitions (1.55)) aanndd (1.-66)):: (1.-55)) is trruue in
all ppossible worrldds aanndd (1.66)) is faallse in aallll ppossible worrllddss;; thhey aarree,, inn shorrtt, contrraaddiiccttorriies of one
aannothherr.

Betweeenn thhemm, thhenn, aa pprrooppoossiittionn aanndd aanny contrraaddiiccttorryy of that pprrooppoossiittionn ddividdee thhe sett of all
ppossible worrldds into two suubbsseettss whhichh aarre mutuuaallllyy exclusive aanndd joiinnttllyy exhaauussttiivvee.. OOnnee of tthese
suubbsseettss will compprriise aallll of the ppossible worrllddss, if aannyy, inn whhichh thhe formmerr pprrooppoossiittionn is trruuee,, aannd
thhe otherr suubbssett will compprriise aallll of the ppossible worrllddss, if aannyy, inn whhich thhe latterr pprrooppoossiittionn is trruue.
Each ppossible worrlldd will bbelong to one orr thhe otherr, bbuut nnot to bbotthh,, of tthheessee ssuubbsseettss..

Sommee mainn kinds of noncontingent propositions

So farr,, thhe onnly example cited of aa nnecessaarilyy trruue pprrooppoossIittIionn was thhe ddisjuunnccttionn of aa pair ooff
contrraaddiiccttoorriieess, viz..,, (1..55)),;. aanndd thhe onnly example of aa nnecessaarilyy faallsee pprrooppoossiittionn was thhe coonnjunctionn
of aa pair of contrraaddiiccttoorriieess, viz..,, (1.6). Yett mmaanny nnonncontinngeenntt pprrooppoossitions aarre of kkinnddss veerry
ddiffeerreenntt frrom tthheessee.. In what follows, we cite sommee otherr kkinnddss of examppless (in nno sppecial orrddeerr)),, aall
of nnecessaarilyy trruue pprrooppoossiittiionns. IIt shouuld bbe evident, frrom what we said inn thhe pprreevviouus suubbsseection,
that thhe contrraaddiiccttorriies of each of thhe nnecessaarilyy trruue pprrooppoossitions cited will bbe nnecessaarilyy faallsse.

17.. OOnnee mmaaiinn kind of nnecessaarilyy trruue pprrooppoossiittionn is exempplifieedd bby

(1.77)) If somethinng is rreedd thhen it is coolloorred.

Note that thhis pprrooppoossiittionn is trruue evenn inn tthhoossee ppossible worrldds inn whhichh thherre aarre nno rreedd thhinnggs. To

aassserrtt (1.77)) is simpply to aassserrtt that iiff aannyytthhinng is rreedd thhen it is colorredd;; aanndd thhis pprrooppoossiittionn is true

bbothh inn worrldds inn whhichh thherre aarre rreedd thhings aanndd in worrldds inn whhichh thhere aarre nnoott..1122 PPhhilosoopphers
hhaave sppennt time aannaallyyzzinng thhe reaassoonnssfoforrththeenneecceessssaarryy ttrruuththooffpproroppoosistiitoionnsslilkikee ((11.7.7)).. PProroppeertriteies,s,iinn
generraall, tend to come inn rannggeess:: rraannggininggffrroommththeemmoorreeoorrlelesssssppeeccififiicc totoththeemmoorreeggeenneerraal.l. TThhuusstthhee
pprroopperrtty of bbeing located inn RRaayy Brraaddlleeyy''ss ddrraawwerr inn hhis ddesk aat Simmonn Frraaser Unniverrssiittyy is aa highly
sppecific pprrooppeerrttyy.. OOrr,, aas we shhaalll pprreeferr to say, it is aa hhighhly deteerrmmiinnaattee pprrooppeerrttyy.. TThhee pprrooppeerrttyy off
bbeing locaated inn Brritish Columbia is leessss ddetermminaatte; that of bbeing located inn Canada evenn leessss so;
that of bbeing located inn the nnorrtthherrnn hhemmisppherre evenn leessss so aagaainn;; aanndd so onn.. Ass we pprroocceed along
thhe scaalle of incrreaasing generraallity, i.e., ddecrreaasing ddetermminaattenesss,, we come finaallly to thhe leeaastt
ddeterrmminate pprrooppeerrttyy inn thhe rraannggee. Thhis is thhe pprrooppeerrttyy whhiichh,, inttuuiittiivveelly, faalls just short of thhe mmost
generaal of aallll pprrooppeerrttiies -— that of bbeing aa thhinng -— viz..,, inn thhe pprreesenntt instance thhe pprrooppeerrttyy of being

12. IIt hhaas bbecomme standarrdd pprraaccttiiccee, inn the ppaast hhuunnddrreedd years orr so, to consttrrue senntteenncceess of the forrm
""AAll . . . aarree.. ...... "" aas if thhey said the sammee aas senntteenncceess of the form ""FFoorr aannyytthhinng whaatteverr, if it is ..... then
it is. . . ."" OOnn thhis aaccount, the sennttennccee ""AAll rreedd thhings aarre colorred"" expprreesssseess thhe nnecessssaarryy trruutthh ((17.7))..

200 POSSIBBLE WWORRLLDSS

located somewhere or other. Such properties we shall call deetteerrmmininaablbele properties. DDetterrmminable
properties are those under which alll the more or less determinatee properties of the range are
subsumed. They are those which we would cite if we wishedd to give the least determinatee ppossible
answer to such general questions about a thing as: "Whhere is it?" (location); "How many are tthheerree??"
(numbber); "How heavy is it?" (weight); "WWhhat color is it?" (color); and so onn.. Plainly, on this
analysis, the property of being colored -— of havinngg some color or other -— is a determinabtle property
uunderr whichh thhee relatively deterrmmiinnattee ppropperttyy off bbeiinng red is subbssuummeedd.. To bbe redd is, thereforree,, iippso
facttoo to bbe coolloredd;; annd nothinngg coouuldd ppossibbllyy bbe redd witthhouut bbeiinng ccoolloorreedd.

2. In the light of this unddeerstanddiinngg of determinable properties, we are able to give a
charactteerriizzaattiioonn of a second main kind of necessaarryy truthh; that of which

(17..88)) Annyy event which occuurs, occcuurs at some time or other

is an example. This kind of propposiittiioonn is sometimes called a caatteeggoorryy prrooppoossititiIoOnn (or even a
categoorriiaapl313 proppoossiittiioonn)):: it is that kind of propposiittiioonn inn which a determinable property is truly
ascribed to an item of a certaiinn sort, or (ass we shall say) of a certaiinn caatteeggoorryy.. Just as ddeetteerminate
properties are subsummabblle under highly general determinable properties, so are particular items and
classes of items subsuummabblle under highly general categgoorriieess of items, e.g., the categgoorriieess of material
objects, of events, of persons, of mental processes,, of sounddss,, and so onn.. Noww what distinnguuiisshheess items
belonging to one categgoorryy fromm those belonnggiinngg to anotherr is just the set of determinaWble (as distinct
from determinate)) properties which are essential to such itteemmss. .1144 Materiiaall objects and shadows, for
instance,, shhaarree ceerrttaaiinn deetteerrmmiinnaabtlee pproperties, e.g.,, havinngg sommee nummbbeerr,, sommee shape, annd some
locaattiion. Anndd these determmiinnable ppropperttiieess arree esssseennttiiaall to thhemm inn thhee seennssee thhatt nothinngg ccoouulldd
pposssiibbly bbe a materriiaall obbjecctt orr a shhaddoww if it lackked anny of these determmiinnable pproperties. By way ooff
ccoonnttrraasstt,, tthhee ddeetteerrmmiinnaattee pprrooppeerrttiieess,, wwhhiicchh ppaarrttiiccuullaarr iitteemmss iinn aa ggiivveenn ccaatteeggoorryy hhaappppeenn ttoo hhaavvee,, aarree
nnoott aallll eesssseennttiiaall.. TThhee ddeetteerrmmiinnaattee pprrooppeerrttyy ooff bbeeiinngg iinn RRaayy BBrraaddlleeyy''ss ddrraawweerr,, ffoorr iinnssttaannccee,, iiss nnoott
eesssseennttiiaall ttoo hhiiss ccooppyy ooff AAnnnnaa KKaarreennininaa aalltthhoouugghh tthhee ddeetteerrmmiinnaabbllee pprrooppeerrttyy ooff bbeeiinngg llooccaatteedd
ssoommeewwhheerree oorr ootthheerr iiss.. WWhhaatt mmaakkeess mmaatteerriiaall oobbjjeeccttss aanndd sshhaaddoowwss ccoommpprriissee ddiiffffeerreenntt ccaatteeggoorriieess iiss tthhee
ffaacctt tthhaatt,, tthhoouugghh sshhaarriinngg cceerrttaaiinn ddeetteerrmmiinnaabbllee pprrooppeerrttiieess,, tthheeyy ddoo nnoott sshhaarree ootthheerrss.. TThheeyy ddoo nnoott sshhaarree,,
ffoorr iinnssttaannccee,, tthhee ddeetteerrmmiinnaabbllee pprrooppeerrttyy ooff hhaavviinngg ssoommee mmaassss oorr ootthheerr.. AAnndd nneeiitthheerr ooff tthheemm hhaass tthhee
ddeetteerrmmiinnaabbllee pprrooppeerrttyy wwhhiicchh sseeeemmss ddiissttiinnccttiivvee ooff pprrooppoossiittiioonnss,, vviizz..,, bbeeiinngg ttrruuee oorr ffaallssee.. TThhee qquueessttiioonn
aass ttoo wwhhaatt tthhee eesssseennttiiaall ddeetteerrmmiinnaabbllee pprrooppeerrttiieess ooff iitteemmss iinn aa ggiivveenn ccaatteeggoorryy ttuurrnn oouutt ttoo bbee,, iiss oonnee
wwhhiicchh wwee ccaannnnoott ppuurrssuuee hheerree.. SSuuffffiiccee iitt ttoo ssaayy tthhaatt aannyy ttrruuee pprrooppoossiittiioonn wwhhiicchh,, lliikkee ((17..88)),, aassccrriibbeess ttoo
iitteemmss iinn aa cceerrttaaiinn ccaatteeggoorryy aa ddeetteerrmmiinnaabbllee pprrooppeerrttyy wwhhiicchh iiss eesssseennttiiaall ttoo tthhoossee iitteemmss,, wwiillll bbee
nneecceessssaarriillyy ttrruuee.. AAnndd ssoo,, ffoorr tthhaatt mmaatttteerr,, wwiillll bbee aannyy ttrruuee pprrooppoossiittiioonnss wwhhiicchh,, lliikkee

(17..99)) Propositions have no color

deny of items inn a certaiinn categgoorryy that they have a certaiinn determinable property.

13.. Note the spelling of this term. Do not confuse it with "ccaategorical"".. The two are not ssynonyms.

14.. A related kind of necessaarryy truth, that which truly ascribes essential properties to so-called naattuurraall kkiinndds
-— aass,, ffoorr eexxampllee,, tthhee pprrooppoossiittiioonn tthhaatt ggooldd iiss aa mmeettaall -— hhaass rreeccently bbeeeenn tthhee ssuubbjjeecct ooff mmuucchh pphhilosdophhiiccaall
discussion. See, in particular, Saul Krripke, "Naammiingg and Necessity" inn Seemmaanntitcicss ooff Naattuurraal l LaLnangguuaaggee,
ed. Harman and Davidson, Dodrecht, D.. Reidel, 1997722.. Krripke, somewhhatt controversially, maintains that hhaavving
atomic number 799 is an essential property of gold.

§ 33 PPrrooppeerrttiieess oof PPrrooppoossiittionns 2211

3.. Still aannotherr mmaaiinn kind of nnecessssaarryy trruutthh is exempplifieedd bbyy thhe relational proposition

(71.1100)) If Molly is taallerr thhann Judi thhen it is nnot thhe caassee that JJudi is taallerr than
Molly.

The nnecessssaarryy trruutthh of thhis pprrooppoossiittiioonn,, it shouuld bbe nnottedd, is nnot inn aanny way aa funncttionn of which
parrtticulaar itemmss it aassserrttss aas standding inn thhe rreelaattionn of bbeing taaller thhaann.. RRaatthheerr,, it is aa funnccttionn of the
esseennttiiaall nnaaturre of the rreelaattionn of bbeing taallerr thhaann;; of thhe factt, that is, that nno mmaattterr what ppoosssible
worrldds orr what ppoossss,iibble itemmss aarre innvvoolvedd,, if thhe rreelaattionn of bbeing taaller thhaann hholdds bbeettwweeenn two
items, x aanndd y, inn that orrddeerr,, thhen it ddooeess nnot aallssoo hhooldd bbeettwweenn x aanndd y inn thhe rreeverrse orrddeerr.. Thhis fact
aabbout thhe esseennttiiaall nnaaturre of thhe rreelaattionn of bbeing taaller thhann is sommeettimmeess rreeferrreedd to bbyy saying that
thhe rreelaattionn of bbeing taallerr thhann is aann ""aassyymmmmeettrriiccaal"" rreellaattiioonn.. IIt is aa fact aabbout thhe rreelaattionn of being
taaller thhann whhichh serrvveess to explain why nnot onnly (1.-1100)) bbuut couunnttlleessss otherr pprrooppoossiittiionns, viz..,, aany
pprrooppoossitions of thhe forrm

(1.-1111)) If x is taallerr thhaann y thhen it is nnot thhe caassee that y is taaller thhaann x

aarre nnecessaarilyy trruuee.. More generraallly, the pprrooppeerrttyy of bbeing aasymmmmeettrriiccaal is esseennttiaall to couunnttlleessss otherr
rreelaattions aas well, e.g., bbeing the mmotherr of, bbeing older thhaann,, bbeing thhe suucccessorr of, aanndd so onn.. And forr
each of tthheessee rreelaattions thhere will bbe couunnttlleessss nnecessaarilyy trruue pprrooppoossitions aannaallogouus to (1.10)).. In
chapterr 6, sectioonn 6, we will innvvestigatte otherr of thhe esseennttiaall pprrooppeerrttiies whhichh rreelaattions mmaayy hhaavvee:: (a)
symmetrry orr nnonnsymmmetry (as aalternnaattives to aasymmmeettrryy));; (b) trraannssiittiivviittyy,, inttrraarrisisiittiivviittyy,, orr
, nnonnttrraannssiittiivviittyy;; aanndd (c) rreefflexivity, irrrreefflleexxiivvittyy, or nnonnrreefflleexxivity. WWee will see that every rreelaattionn hhaas
aat least one esseennttiiaall pprrooppeerrttyy ddrraawwnn frrom each of (a), (bb), aanndd (c), i.e., hhaas aat leaast thhrree eessential
pprrooppeerrttiies inn toottoo. . TThhaattththisisisissosopprorovvidideessaarircichhsosouurcrceeooffnneecceessssaarryy trturuthths.s.FFoorraallllththoosesepproroppoosistiitoionnss
wwhhoossee trruutthh cann bbe aascerrtainedd bby aann aappppeeaal to aa fact aabbout aann esseennttiiaall pprrooppeerrttyy of aa rreelaattionn will
tthheemmsseellvveess bbe nnecessssaarryy trruutthhss.

4.. In saying that pprrooppoossitions of the severral mmaaiinn kkinnddss ddiscusseedd so farr aarre logically nneecessaarry
trruutthhs we aarree,, it shouuld bbe nnottedd, uusing the termm ""llooggiicaally nnecessssaarryy"" inn aa fairrly brroad sennssee.. WWee are
nnot saying that such pprrooppoossitions aarre currrreennttllyy rreeccognnized aas trruutthhs within aanny of thhe systemmss ooff
formaal logic whhichh so farr hhaave bbeen ddeveloped. RRaatthheerr,, we aarre saying that thhey aarre trruue inn aall
logically ppossible worrllddss.. Now, inn thhis brroad sennssee of thhe terrmm,, thhe truthss ofj matheemmaattiiccss mmuust also
count aas aannotherr mmaaiinn kind of logically nnecessssaarryy trruutthh. Berrtrand RRuusssseelll aanndd AA.. N.. WWhhiitteehheeaadd,, it is
worth mmeennttiioonninngg, trriiedd soon aafterr the turn of thhis centurry to shhoww that thhe trruutthhs of mmaatthheemmaattiics
wwerree logically nnecessssaarryy inn thhe rraatthherr strricterr sennssee of bbeing dderriivaabblle inn aaccorddaanncce with thhe rruullees off
aanndd ffrroomm tthhee aaxxiioommss ooff tthheenn--kknnoowwnn ssyysstteemmss ooff ffoorrmmaall llooggiicc.. BBuutt wwhheetthheerr oorr nnoott tthheeyy wweerree ssuucccceessssffuull
-— aanndd tthhiiss iiss ssttiillll aa ttooppiicc ffoorr ddeebbaattee -— tthheerree ccaann bbee lliittttllee ddoouubbtt tthhaatt mmaatthheemmaattiiccaall ttrruutthhss aarree llooggiiccaallllyy
nneecceessssaarryy iinn tthhee bbrrooaaddeerr sseennssee.. IInnddeeeedd,, iitt sseeeemmss tthhaatt eeaarrllyy GGrreeeekk pphhiilloossoopphheerrss rreeccooggnniizzeedd tthheemm aass
ssuucchh wweellll bbeeffoorree AArriissttoottllee llaaiidd tthhee ffoouunnddaattiioonn ooff ffoorrmmaall llooggiicc iinn tthhee tthhiirrdd cceennttuurryy BB..CC.. TThhee ttrruuee
pprrooppoossiittiioonnss ooff mmaatthheemmaattiiccss sseeeemmeedd ttoo tthheemm ttoo bbee nneecceessssaarryy iinn aa sseennssee iinn wwhhiicchh tthhoossee ooff hhiissttoorryy,,
ggeeooggrraapphhyy,, aanndd tthhee lliikkee aarree nnoott.. AAddmmiitttteeddllyy,, tthheeyy ddoouubbttlleessss wwoouulldd nnoott tthheenn hhaavvee ooffffeerreedd aa ""ttrruuee iinn aallll
((llooggiiccaallllyy)) ppoossssiibbllee wwoorrllddss"" aannaallyyssiiss ooff tthhee sseennssee iinn wwhhiicchh mmaatthheemmaattiiccaall ttrruutthhss sseeeemmeedd ttoo tthheemm ttoo bbee
nneecceessssaarryy.. BBuutt ssuucchh aann aannaallyyssiiss sseeeemmss ttoo ffiitt wweellll wwiitthh tthheeiirr jjuuddggmmeennttss iinn tthhee mmaatttteerr..

IIt mmaayy seemm to sommee that thhere aarre two imporrtaanntt claasssseess of exceppttionnss to thhe generaal aaccouunt we
aarre giving of mmaatthheemmaattiicaal trruutthhss.. TThhee ffirrsstt hhaas to ddoo with pprrooppoossitions of what we oftenn call appliedd
mmaatthheemmaattiiccs. If bbyy ""pprrooppoossiittions of aapppplliiedd mmaatthhemmaatticss"" is mmeant certaainn trruue pprrooppoossitions off
pphhyyssiics, engineering, aanndd the like whhich aarre inferrreedd bbyy mmaatthheemmaattiicaal rreeaasonning frrom otherr
pprrooppoossitions of pphhyyssiics, engineering, etc., thhen of courrse such pprrooppoossitions will nnot count aas nneecessaarry

22 POSSIBLE WWOORRLLDDS

trruutthhs bbuut onnly aas continngeenntt onneess.. Buut thhen it is mmisleading evenn ttoo say that thhey aarree,, inn aanny sense,
pprrooppoossitions of mmaatthheemmaattiiccs. IIf, onn thhe otherr hhaanndd,, we mmean bbyy ""pprrooppoossiittions of aapppplliiedd mmaatthheemmaattiics""
certaain pprrooppoossitions whhiichh,, like

(1.1122)) IIf thhere aarre 20 aapppplles inn thhe bbaaskett thhen thhere aarre aat least 19 aapppplles inn the
bbaasket

aarree inssttaanncceess (i.e., aapppplliicaattions) of pprrooppoossitions of ppuurree mmaatthheemmaattiics, thhen ourr aaccouunt is suustaained.
Forr thhere aarre nno ppossible worrldds -— evenn worrldds inn whhichh aappppllees, bbaaskkets, aanndd whatnott fail to exist -—
in whhichh aa pprrooppoossiittionn like (1.-1122)) is faalse. Thhis sorrt of mmaatthheemmaattiicaal pprrooppoossiittionn shouuld be
ddistinguishhedd from pprrooppoossitions aabbout thhe pphhyyssicaal rreesuults of ppuuttttinng thhings togetthherr, ee.g.,

(1.1133)) OOnnee liter of aalcohhool aaddddedd to one liter of waterr mmaakkees 2 liters of liquid

whhich nnot onnly is nnot nnecessaarilyy trruue bbuut is continngeenntt aanndd faalse. (See chapterr 3, pp.. 117700).
A seconnd supppposedd excepptioonn hhaas to ddoo with thhe pprrooppoossittions of geometry. Thhuus, it mmaayy bbe said that

aalthough thhere was aa time when pprrooppoossitions like Euclid's

(1.1144)) Thhee sum of thhe interriorr aanngglleess of aa trriiaannggle is equuaal to two rriighhtt annggles

wwerree uunniiverrssaallly thhoughht to bbe nnecessaarilyy trruuee, we nnow kknnow mmuucchh bbetter: thhe ddevelooppmmennt, inn the
nnineteeenntthh centurryy, of non-EEuuclidean (e.gg.., RRiemmaannnniiaann aanndd LLoobbaacchheevvsskkiiaann)) geommeettrriieess inn whhichh the
summ of the interriorr aanngglleess of trriiaannggles aarre aasserrtedd to bbe mmorre thhaann orr leessss thhaann (rreesppectivelyy)) two righht
aannggles, shhoowwss that (1.1144)) is nnot nnecessaarilyy trruuee.. In effect, thhe obbjjection is that with thhe aaddvent off
non-EEuuclidean geommeettrriieess inn the nnineteeenntthh centurryy, we hhaave come to see that thhe aannsweerr to the
qquueestionn aas to what is the sum of thhe interriorr aanngglleess of aa trriiaannggllee, is aa continngeenntt one aanndd will vvaarry
frrom ppossible worrlldd to ppossible worrlldd aaccorrddiinngg to the nnaattuurraall laawwss inn tthhoossee worldds.

Buutt thhis obbjjection faails to mmaakke aann imporrtaanntt ddistinnccttiionn.. IIt is corrreect so long aas bbyy ""ttrriiaannggles"" it
rreefers to pphhyyssicaal obbjjects, e.g., trriiaannggles whhich surrvveyorrs mmight laayyoouut onn aa fieldd,, orr ppaapper ttrriiaanngglles
whhich one mmight cut out with scissoorrs. But thhere aarre otherr trriiaannggllees, tthhoossee of ppuurree geommetry, whose
pprrooppeerrttiies aarre noott subjectt to the pphhyyssicaal ppecuuliarritties of aanny worrldd. TThheese ideaallizedd, abssttrraacctt eennttititieiess
hhaave invarriaanntt pprrooppeerrttiieess. WWhhaatt the aaddvent of nnonn--EEuucclliiddeeaann geommeettrriieess hhaas shown uus aabbout tthheese
trriiaannggles is that we mmuust ddistinguish varriiouus kkinnddss:: aa EEuucclliddeeaann trriiaannggle will hhaave interriorr annggles
wwhhoossee sum is equuaal to two rriighht aannggles; aa RRiemmaannnniiaann trriiaannggllee, aanngglleess wwhhoossee sum is grreaterr thhann two
rright aannggles; aanndd aa LLoobbaacchheevvsskkiiaann trriiaannggllee, aanngglleess wwhhoossee sum is lessss thhann two rriighht aannggles. Sppeecify
whhich kind of aabbstrraact trriiaannggle youu aarre rreeferrinng to, aanndd it is aa nnecessssaarryy trruutthh (orr falsehhooodd) that its
aanngglleess sum to two rriighht aannggles, leessss thhann two, or grreaterr thhaann two.

5.. Among thhe mmost imporrtaanntt kkinnddss of nnecessaarilyy trruue pprrooppoossitions aarree tthhoossee trruue pprrooppoossittions
whhich aascrribbe mmooddaall pprrooppeerrttiies -— nnecessssaarryy trruutthh, nnecessssaarryy faalsity, contingency, etc. -— to otherr
pprrooppoossiittiionnss. CCoonnssiiddeerr,, forr exammpple, thhe prroppoosition

(1.1155)) IIt is nnecessaarilyy trruue that two ppluus two equaals foouurr.

(1.1155)) aassserrttss of the simpler pprrooppoossiittionn -— viz.,, that two ppluus two equaals fourr -— that it is nneeccessarrily
trruuee. Now thhe pprrooppoossiittionn that two ppluus two equaals fourr,, since it is aa trruue pprrooppoossiittionn of mmaatthheemmaattiiccs,
is nnecessaarilyy trruuee.. Thuus inn aascrriibbiinngg to thhis pprrooppoossiittionn aa pprrooppeerrttyy whhichh it ddooeess hhaavvee, thhe prroppoosition
(1.1155)) is truuee.. Buutt is (1.1155)) nnecessaarilyy trruuee, i.e., trruue inn aallll ppossible worrldds?? TThhee aannsweerr we shall

§§ 3 PPrrooppeerrttiiees of Prrooppoositions 23

waanntt to give —- and forr which we shaall arrgue in chhaapterr 4, ppp. 185-88, and again in chhaapterr 6, p. 3355
—- is thhaatt true prrooppoositions which ascrribe neceessssaarryy truthh to otthheerr prrooppoositions arre tthhemmselvess
necessssaarrilyy trruuee, i.e., logically necessssaarry. As Hinntikkkkaa has put iitt:

It seemms to me obviouus thatt whatevveerr is logically neceessssaarryy herre and now must also be llooggiiccaalllyy
necessssaarryy in all logically possibble staattees of affairrs that couldd haave been rreeaallized instteeaadd of the
actual oonnee..l155

Simmillaarr considdeerrationns leaad us to say furthherr that anny trruue prrooppoossiittionn which ascribes necessssaarryy falsity
orr contingenncy to anotherr prrooppoossittionn is itsellff trruue in all possibblee worrlddss, i.e., is necesssaarrilyy trruuee. AAnndd
the same holds for trruue prrooppoossitions whhichh,, lliikke

(1.166)) Thhe prrooppoossittionn that therre arre 20 apppples in the baskett is conssiisstteenntt with the
prrooppoossition that the baskett is grreenn

asseerrtt of two simmpler prrooppoossitions that they arre logically rrelated by one of the moddaall relations,
impplication, equuivaalence, consistency, orr inconsistteenncyy (discuussseedd inn the nnext section).

6.. TThhee pprreecceddinng fivvee kkinnddss of nnecessssaarryy trruutthh aarre nnot whhoolly exclusivee of one aannotherr aanndd are
certaainnly nnot exhauustivee of the whhole claasss of nnecessssaarryy trruutthhss.. TThheerree aarre otherr pprrooppoossiittions whhichh are
trruue inn aallll ppossible worrldds bbuut forr whhichh it is ddifficult to findd aanny single aappt ddescrrippttionn except to say
that thhey aarre pprrooppoossittions wwhhoossee trruutthh cann bbe aascerrtained bbyy conceptuuaall analysis.. WWhhen PPlaattoo,, foorr
instannce, inn hhis ddiaaloguue, Thheeaatteettuuss,, aannaallyyzed the conceppt of kknnoowwledge aanndd camme to the conncluusioonn
that kknnoowwinngg immppllies (ammong otherr thhinngs) bbelievinng, hhe pprroovviiddeedd grrouunndds forr uus to aascerrttaaiinn, aas beinng
nnecessssaarryy trruutthhss,, aa hhoostt of pprrooppoossiittiioonnss, such as

(71.7177)) If the PPoppe kknnoows that there aarre nnine pplanets thhen the PPoppe bbeellievvees that there
aarree nninne pplaanneetts.

That hhoossttss of suuchh nnecesssaarry ttrruutthhss eexxist shhouuldd bbe eviddennt frrom tthhe faact tthhat tthhe aannaallyyttiicc ttrruutthh wwhhiicchh
PPlaatto ddiscoveredd is qquuiitte generraall, aaddmmiittttiinngg counnttless instances rraannggiinngg frrom tthhe PPoppe''s kknnoowwlleddge
(andd conseeqquueenntt bbeelief) aabbouut tthhe nnuummbbeerr of tthhe pplannets tto tthhe kknnoowwledge (anndd conseeqquueenntt bbeelief) that
mmoost of uus hhaavve aa!b?out tthhe earrthh''s bbeeinng rrouunndd, aanndd so onn.. LLiikkeewwiissee,, whhen aa mmorraal philoossoopphheerr
aannaallyyzes tthhe conceppt of mmorraal obbligattionn aanndd concludes tthhat bbeeinng mmoorraalllyy obbliged ttoo ddoo something
immpplliies bbeeinng aabblle tto ddoo itt, tthhat pphhiilosoppherr esttaabblliishheess tthhat aa hhoosstt of pprrooppoossiittiioonnss,, of which

(1.-7188)) IIf tthhe forreeignn mmiinnisterr is mmoorraalllyy obbliged tto rreessignn tthhen hhe is aabblle ttoo ddoo so

is oonnllyy oonne innstaannccee, aarree nneecesssaarry ttrruutthhss.. CChhaarraacctteerriissttiiccaallllyy, pphhiilosopherrs -— inn tthheeirr aannaallyyses oof thheese
aanndd oothher conncepts wwhhiicchh figurree cennttrraallllyy inn ouurr tthhiinnkkiinngg -— aarree nnoott ttrryyiinngg ttoo commppete wwithh scienttists
in ddiisscoverriinngg contingennt ttrruutthhss aabboouutt hhoow tthhe aaccttuuaall wwoorrlldd hhaappppeenns ttoo bbee connstituuttedd. RRaatthheerr,, tthhey
aarree ttrryyiinngg ttoo ddiiscoverr wwhhaat is implieedd bbyy mmaannyy oof tthhe conncepts wwhhiicchh sscciennttiissttss aanndd tthhe rreest oof uuss taakke
forr ggrraanntteedd.. AAnndd tthhe pprrooppoossiittiionnss inn wwhhiicchh tthheey rreeppoorrtt tthheeirr aannaallyyttiiccaall ddiisscoverriiees, if ttrruuee,, arre
nneeccessarrily trruuee.

1155. JJaaaakkkkoo Hintikkaa, ""TThhee MMooddeess of MModaality"" inn Acta Philosophhiiccaa Fennica, vooll.. 116 ((11996633)),, pppp.. 65-811.

24 POSSSIBBLE WORLDS

Summary

Muchh off whhaatt we have saidd so faarr connceerrning worlds —- actual and non-actual —- and ccoonncceerrnniinngg
prrooppoositioonns —- true and false, coonnttiinnggeenntt and noncoonnttiinnggeenntt —- abouutt ittemms in thheese worlds can bbee
suummmaarrizedd in the follloowwiinng ddiaaggrraamm:

.AAllll PPoossssiibbllee WWoorrllddss

NNoonn-- TThhee AAccttuuaall
aaccttuuaall
wwoorrllddsa aaccttuuaall TTrruutthh MMooddaall
_ _ _ _ _--''--
wwoorrlldd SSttaatt uu ss SSttaatt uu ss

--y-_...A----,. ,,.._--"-..........~------ __-____,

··•TTRRUUEE Iinn aallll ppoossssiibbllee wwoorrllddss-··· . Necessarily
True
• Non-contingent

PPrrooppoo- ··•TTRRlUJEE Iinn tthhee aaccttuuaall wwoorrlldd aanndd . Possibly
ss iittiioonn ss True
FFAALLSSEE Iinn aatt llee aasstt oonee oottheerr False
ppoossssiibbllee wwoorrlldd Possibly
False
..•FFAALLSSEE iinn tthhee aaccttuu aall wwoorrlldd aanndd Contingent

TTRUEE iinn aatt lleaasstt oonee oottheerr

ppoossssiibbllee wwoorrlldd ..

. ·FFAALLSSEE iinn aallll ppoossssiibbllee wwoorrllddss . Necessarily
False

FIGURE (17.g)

Note that onnly tthhoossee pprrooppoossitions aarre continnggeenntt whhich aarre bootthh ppossibly trruue and ppossibly false,
wwhheerreeaass tthhoossee pprrooppoossitions aarre nnonconnttiinnggeenntt whhich aarre eithheerr nnot ppossibly faalse (the nneeccessarrily
ttrruue oonneess)) orr nnot ppoossibly ttrruue (the nneecessaarilyy faallssee onneess).

OOtther immpporrttaanntt rreelaattionships bbeettwweeenn cclaasssseess of pprrooppoossiittions mmaayy bbee rreeaadd off tthhe ddiaagrraamm in
aaccorddaanncce wwith tthhe following rruullee:: if tthhe bbrraacckkeett rreepprreessenntting onne cclaassss of pprrooppoossiittions (ee..gg..,, thhe
nneecessaarilyy ttrruuee oonneess)) is contained wwithhinn tthhe bbrraacckkeett rreepprreessenntting aannotherr cclaassss of pprrooppoossiittions
(ee..gg..,, tthhe aaccttuuaallly ttrruuee oonneess)),, tthhen aanny pprrooppoossiittionn bbeelonging tto tthhe formmerr cclaassss is aa prrooppoosition
bbeelonging ttoo tthhe latterr claassss.. FForr instannce, aallll nneecessaarilyy ttrruuee pprrooppoossiittions aarree aaccttuuaallly trruuee,
aalthhough nnot aallll aaccttuuaallly ttrruue pprrooppoossiittions aarree nneecessaarilyy trruuee.

EXXEERRCCISIESSES

71.. Foorr eaacchhooffththeeffoollloowwiinngg pprrooppoossitiitoionnss,,ssaayy((71)) wwhheeththeerrititisisccoonntitninggeennt toorrnnoonnccoonntitninggeennt,t,aanndd((22))

iiff nonccoonnttiinnggeenntt,, thheenn whheetthheerr itt iss trruuee or fallssee..

aa.. AAllll aauunntstsaraerefefmemaleasl.es.
bb.. OOnn AApprriil 7133,, 71994455,, allll ffeemmaalelsesaraereauanutsn. ts.
cc.. If a bbiirrdd iss eennttirireelylyblbalcak,ckth,etnhietnis itnoist anlosot awlshoite.white.
dd.. AAlll bbllaacckkbibrdirsdasrearbelabckl.ack.

§ 33 PPrrooppeerrttiiees oof PPrrooppoossiittiionns 25

e.. All blackbirds aree bbllaacckk. .

ff. Sommee squuaarreesshhaavveessiixxssididese.s.

g. No mushhrroooommssaarreeppooisiosononuosu.s.

h.. It iss illegal toojjaayywwaalk ininMMoossccooww. .

Ii.. WWhheenneevveerr iitt iissssuummmmeerriinn NNeeww ZZeeaallaanndd, ,iitt iisswwiinntteerr iinn CCaannaaddaa.,

j.. WWhhaatt will bee,, will bbe..

2.. Brieflyy explain why eaacchh ooffththeefofolllolowwininggpproroppoosistitoionsnsisisfaflasles.e.

k. Iff a proposition iss actually truuee,, thheenn thhaattpprrooppoossiittiioonn iissaalslsooccoonntitnignegnetn. t.
lI.. If itt issppoossssiibblleeffoorraapprrooppoossiittiioonn totobbeetrtruuee, ,ththeennititisisppoosssisbilbelefofrorthtahtastasmaeme

proposition too bbeeffaalslsee. .

m. Even thoouugghh aapprrooppoossiittiioonn iissaaccttuuaalllyyffaalslsee, ,ititnneeeeddnnoottbbe.e.

n.. Iff a proposition iss noncontinnggeenntt,, thheenn it iss actually true..

oo.. If a proposition issppoossssiibbllyyttrruuee, ,tthheenniittiissccoonntitninggenetn.t.

3.. WWrriittee a shoorrtt stoorryy succhh thhaatt a persoonn reading thhaatt stoorryy could telll frroomm ssuubbttllee cclluueesstthhaatt tthheerree isis
nnooppoossssibiblelewwoorrllddininwwhhicichhaallltthheeeevveenntstsrerlealtaetdedococcucru. r.

4.. Dooeess thee conncceepptt of time-travel really geneerraattee ppaarraaddooxxeess?? HHeerree iiss wwhhaatt LLaazzaarruus ssaayyssaabboouut tththee
matter whenn discussing it with twoo of hiss desscceennddaanntstsbbefeofroeremmakaiknignghihsioswonwntimtiem-etr-itpr:ip":T"hTahtaotldold
cliche aboouutt shoooottiinngg your granddffaatthheerr beeffoorree he sirreess your fatherr,, thheenn going fuufff! likee a ssooap
bubbbllee -— and all desscceennddaanntsts, , toto,o,mmeeaannininggbbooththooffyyoouuaammoonnggooththeresrs-— isisnnoonnsesnesnes.e.TThheefafactct
thhaatt I'm herree and you'rree herree meaannss thhaatt I ddidn''t doo it -— orr won't doo it;; thee teennsseessooffggrraammmmaarr
aren't builtt foorr ttiimmee--ttrraavveell -— bbuutt iitt ddooeessnnototmmeaenanthtahtatI Inneveevrerwwenetntbabcakckaannddppookkeeddaaroruonudn.d.I I
haven't any yen too looookk at mmyyselff when I waass a snoott--nnoossee;;iti'ts'sththeeeeraraththaat tinintetreersetsstsmme.e.IIffIIrraann
aaccrroossssmmyysseellffaassaayoyouungngkikdi,d,hehe- —I /- —wwouoludlnd'nt 'trerceocgonginziezemme;e;I Iwwouoludldbebea satrsatrnagnegretro ttohatht abtrabtr. at.
HHee wwoouullddnn''tt ggiivvee mmee aa ppaassssiinngg ggllaannccee.. II kknnooww,, II wwaass hhee.."" DDiissccuussss.. ((RRoobbeerrtt AA.. HHeeiinnlleeiinn,, TTiimmee
EEnnoouug~hh FFoorr LLoovvee,, pp.. 335588..))

** ** * * * * * *

Symbolization

Much of the suucccceessss aanndd pprroommiise of mmodderrnn logic, like that of mmaatthheemmaattiics, aarriisseess frromm the powerful
aandd suuggggeessttiivvee symmbboolizationn of its concepts. Frrom time to timme, aas we intrrodduuce aanndd ddiisscuussss varrious
of the mmoosstt fundaammenntaal conncceppttss of logic, we will aallssoo intrrodduuce symbbols whhich aarre widely aacccepted
aas standding forr tthhoossee ccoonnccepptts.

Already we hhaave casuaally intrrodduucced sommee of tthheessee symbbols aas whhen, forr exampple, we ''uused the
cappital letterr ""PP"" of the EEnngglliisshh aalphabett to rrepprresenntt aanny aarrbbiittrraarriillyy chosenn pprrooppoossiittionn;; when we
uused "a" to rrepprresenntt aanny aarrbbiittrraarriillyy chosenn itemm;; aanndd when we uused "F"" to rrepprresenntt aann aarrbbiittrraarriliyly
chosenn aattrriibbuuttee.

Let uus exteenndd our cataalog of symbbols evenn mmoorree.. WWe hhaave justt intrrodduuccedd severral of thhe mmostt
important conncceppttss of mmodderrnn logic. TThheey aarre standaarrddlly rreepprreesennteedd inn symbbols aas ffoolllows::

26 POO SSSSI BB LLEE WWOO RR LLDDSS

(1) Thee coonncceeppt of falsiittyy:: "" '~"V"" [ccalled tildee1616 ]]
(2) Thee coonncceepptt off ppossssibiliittyy:: "" 0"" [ccalled ddiaiammoonndd]]
(3) Thee coonncceeppt of necceessssaarryy truth: "" 0• " [ccalled box]
(4) Thee coonncceepptt off coonnttingencyy:: "'V"" [ccalled nnaabbllaa]
(5) Thee coonncceepptt off nonnccoonnttiinngeenncyy:: "lA;" " [ccalled delttaa])

Each of these symbboollss maayy be prefixed too a symbol (such ass "" P"",, "QQ"",, "R"")) wwhich stands forr a
propoossiition, too yieldd a further propoossitional symbol (e.g., ""'"~V Pp"",, "OO RR "" ). Annyy sequence of one or
moorree propoossitional symbboollss iss called a ""fformmuullaa"".J.71 7

Each of these symboollss maayy be defined coonnttexxtually ass ffoollloowwss::

"" ^'"V P"" ="dd ff "" TThhee propoossiition P iss fallssee""1188 l[or "It iss false that P""]]
"" OOPP"" ==ddf
"" •oPP"" = ddf "" TThhee propoossiition P iss poossible" [or "It iss poosssibblee that P""]]

"" 'VVPP"" = ddff "" TThhee propoossiition P iss neceesssarily truuee"" [or "It is
"" Al;PP"" ==ddf nneecceessssaarriillyy ttrruuee tthhaatt PP""]]

"" TThhee pprrooppoossiittiioonn PP iiss ccoonnttiinnggeenntt"" [[oorr "" IItt iiss ccoonnttiinnggeenntt tthhaatt PP""]]
"" TThhee pprrooppoossiittiioonn PP iiss nnoonnccoonnttiinnggennennootntn""ccoo[[noontrrtiinn""ggIIeettnniittss tthhaatt PP""]]

Theessee symboollss maayy be coonnccaatteennaatteedd,, that iss,, linked toggeetthheerr inn a series, ass forr example, ""'"~V ~'"V P".

TT hhiiss gives uss thhee meeaannss too exxpprreessss such propoossiitioonns ass that P iss poossibly false (""00 ~'"V P"), neceessarily
faalse (""•0 '"V P"")),, imppoossssiiblle ("" ~'"V 0O P"), eettc.

CCeerrttainn of these coommbbiinnaattioonnss of symbboollss arree sommeettiimmeess unwitttiinnggllyy trannssllaatteedd intoo prossee iin

incoorrrecct wways, anndd one should beewwaarree of thhee pitfalls. Inn parttiiccuullaarr,, coommbbiinnaattioonnss begginning wwiitth ""'VV"
anndd "" lA;"" provvee trooublesommee.. Coonnsider, for example, ·""'Vi7PP".. Theere iss a temptatiioon too translattee this as
"" iitt iss coonnttiinngeennttly true thatt P,,"" orr equuivaalentllyy ass "PP iss coonnttiinngeennttly truuee"".. NNeeiitther off thesee pprrooppoosed
ttrraannssllaattiioonnss ww iillll ddoo.. TT hhee ccoorrrreecctt ttrraannssllaattiioonn iiss "" IItt iiss ccoonnttiinnggeenntt tthhaatt PP iiss ttrruuee.."" WW hhyy?? WW hhaatt eexxaaccttllyy iiss
tthhee ddiiffffeerreennccee bbeettwweeeenn ssaayyiinngg oonn tthhee oonnee hhaanndd tthhaatt PP iiss ccoonnttiinnggeennttllyy ttrruuee,, aanndd oonn tthhee ootthheerr,, tthhaatt iitt iiss
ccoonnttiinnggeenntt tthhaatt PP iiss ttrruuee?? JJuusstt tthhiiss:: ttoo ssaayy tthhaatt PP iiss ccoonnttiinnggeennttllyy ttrruuee iiss ttoo ssaayy tthhaatt PP iiss bbootthh
ccoonnttiinnggeenntt aanndd ttrruuee;; ttoo ssaayy tthhaatt iitt iiss ccoonnttiinnggeenntt tthhaatt PP iiss ttrruuee,, iiss ttoo ssaayy oonnllyy tthhaatt PP''ss bbeeiinngg ttrruuee iiss
ccoonnttiinnggeenntt,, ii..ee..,, tthhaatt PP iiss ttrruuee iinn ssoommee ppoossssiibbllee wwoorrllddss aanndd ffaallssee iinn ssoommee ootthheerrss.. CClleeaarrllyy,, tthheenn,,

16. RRhhymmees ww iitthh ""HH ilddaa"".

17. In later cchhaapptteers we ww iillll distiinngguuiisshh beetwweeeenn foormmuulae ww hhich arree ccoonnssttrucctedd so as too mmake sense ((i.e.,
wwhhiicch arree wweellll--ffoorrmmeedd)) anndd thosee ww hhich arree nott.. Thee exxaacctt rules for ccoonnssttructing wweelll-foormeed foormmuulae within
cceerrtain systemss of symbboollss arree given inn cchhaapptteers 5 anndd 6.

18. Thee "== ddf "r"- s-syymmbbool l mayy be read as "hass thhee samee mmeeaning ass"" or alteerrnnaattiivveellyy ass "eeqquuaallss by ddeeffiinnitiioonn"".
TT hhee exxppressionn on thhee lefft side of thhee "== ddf "f"- s-syymmbbool l iss thhee exxppressionn bbeeing intrroodduucceedd; itt iss known by tthe

technical namee definieenndduumm.. Thee exxppressionn on thhee right hand side off thhee "" ==d fd"e-"s-ysymmbbooIl iss thhee one whoossee.

mmeeaanning iss presumeedd already understooodd anndd iss bbeeing assigned too thhee definiendduumm. Thee right hand exxppressionn is
ccaalllleedd tthhee ddeeffiinniieennss..

§§ 3 Propertiiees of Propositions 2277

to say that a proposition is coontingently true is to say more than meerely that its truth is coontingent. In

otheerr woorrds, the expression "" PP is coontingently truuee"" is to be rendered in ouur symbboollic notation as "" P

and 'VV P .."" It is thus incorrect to translate "" 'vVPP"" alone as "" PP is coontingently truuee.."" Similarlyy,

"" 'VV '~V P"" is to be translated as "" iitt is coonntinggeenntt that it is false that P"; and not as "" PP is contingently

false.""

AA nn easyy rule to beeaarr in mi nd so as to avoid miissttakeess in translations is thiss:: NNeevveerr translate "" 'VV"" or

"" Af,"" adverbiaallllyy,, i.e., wiith an "" lly"" endingg;; alwayyss translate them as "" iitt is coonntinggeenntt that" aDnd "" iitt ii«~

nonccoonnttiinggeenntt that" respectiivvely. AAddvveerrbbiiaal trannsslia'tions in the otheerr cases,, i.e., foor "" •0"" and "" 00"" are

freely permitteedd:: "" iitt is necesssarily true that" and "" iitt is possibly true that" respectiivveellyy.

EEXXEERCRISCEISSES

1,.. Leett "AA"" staanndd foorr thee prrooppoosistiitoinon that Caannaaddaa is noorrtthh ooff MMeexxiiccoo. . Trannssllaattee each ooff thee fofollolowwiinngg
exxpprreessssiioonnss intoo Enngglilsihsh pprroossee.

(a)) ,A (f0) '~VO<>'V^AA (k)) 'VVAA ((Pp)) Af,^'VA A
(b)) '^VAA (gg)) o• AA (0I)) 'VV^'VAA ((qq)) '^V fA, A
(c)) OAA (h)) o•'V^AA (mm)) '^VV'AVA (r) '^Vf,'AV^AA

(d)) OO^'VAA (ii)) '~V 0• AA (n)) '^Vv^'VA'VA

(e)) '^VOOAA ((Jj)) '^VOo^A'VA (o0)) Af,AA

2. FFoorr each ooff thee caseess (c) throouugghh (rr) above say wheetthheerr thee prrooppoosistiitoinon expreesssseedd is true orr ffaallssee..

3. Leetttiningg "AA"" nooww staanndd foorr the prrooppoosistiitoinon that all squareess have foouurr sides, say foorr each ooff tthhee
expprreessssiioonnss (a) - (r) in queessttiioonn "1, whetthheerr thee prrooppoosistiitoinon is true orr ffaallssee..

4. Exxpplalainin whhyy "Af, PP"" is nott to bee traannssllaatteedd as "PP is noonnccoonntitnignegnetnlytly truee"" but as "it is

noonnccoonntitnignegnetnt that PP is truee.."" FFiinndd a prrooppoosistiitoinon ooff whiicchh it is true that it is nonccoonntitningegnetnt that
it is true, but ooff whiicchh it is falsee that it is noonnccoonntitnignegnetnlytly ttrruuee..

5. Say ooff each ooff thee foollloowwiningg whiicchh is true anndd whiicchh is faalssee.. (Notee:: it is accttuuaalllyy true that some
cows aree ininfeferrttiillee.)

(s) It is coonnttiinnggenetnt that some cows aree ininfeferrttiillee..

(t) It is coonnttiinnggenetnt that it is noott thee casee that some cows aree ininfeferrttiillee..

(u) It is coonnttininggenetnlytly true that some cows aree ininfeferrttiillee..

((vv)) It is coonnttininggenetnlytly falsee that some cows aree ininfeferrttiillee..

6. Say foorr each ooff thee foollloowwiningg wheetthheerr it is true orr ffaallssee..

+(w) It is noonnccoonntitnignegnetnt that 2 + 2 =— 4.
+(x) It is noonnccoonntitnignegnetnlytly true that 2 + 2 =— 44.
+(y) It is noonnccoonntitnignegnetnt that it is falsee that 2 + 2 == 44.

28 POSSIIBBLLEE WORLDS

+(z) It is noncontingently ffalse thatt 2 + 2 —= 4.

77.. Exxppllaainin the difference in meaninngg in the twoo pphrasseess "noncontingently true" and ""nnoott
contingently ttrrue".

4. RELATIONS BBEETTWWEEEENN PROPOSITIONS

JJuust as the mmooddaall prrooppeerrttiiees of prrooppoossiittionns arre a funncttionn of the waayys in whhichh the trruutthh--vvaalluuees ooff
thhoosse prrooppoossiittionns singly arre ddistrribbuutteedd acrrooss the sett of all possible worrllddss,, so the moddaall rreellaatitoionnss
bbetwween prrooppoossiittionns arre a funncttionn of the waayys in whhichh the trruutthh--vvaalluuees of the mmemmbberrs of pairss ooff
prrooppoossiittionns arre ddistrribbuutteedd acrrooss the sett of all possible worrllddss.. WWee single out fourr mmooddaall relations
forr immmeddiate attttennttionn, vviiz.,

inconsistteennccyy;; ccoonnssisistetenncyc;y;imimpplilciacatitoionn;;anadndeqequuiviavlaelnecnec.e.

In terrmms of tthheessee we will finnd it ppoossible to explainn mmost of the centrraall logicaal concepts ddiscussedd iinn
thhis bbooookk.. Yett tthheessee mmooddaall rreellaattiioonnss,, we shhaalll nnow see, cann tthheemmsseellvveess bbe explained -— inn mmuucchh the
samme waay aas thhe aabboovve ddiscussed mmooddaall pprrooppeerrttiieess -— inn terrmms of ppoossible wwoorrlds.

Inconssiisstteennccyy

Two pprrooppoossiittiionns aarree inconsisstteenntt with one aannootthheerr,, we orrddinarriily saay, just whhen it is nnecesssaarry thhaatt
if one is trruuee thhe otherr is faalse, i.e., just whhen thhey cannnot bbothh bbe trruuee.. TTrraannsslaatting thhis orrdinnaarry
talk into taalkk of ppoossible worrllddss we mmaayy say that two pprrooppoossiittiionns aarree inconsisstteenntt just whhen inn aany
ppoossible worrld, if aannyy,, inn whhichh one is trruue thhe otherr is faalse, i.e., just whhen thherre is nnoo ppossible
world inn whhichh bbothh aarree ttrruue.

IInnccoonnssiistencyy is aa generric mmooddaall rreellaattiioonn.. IIt hhaas twwo, aanndd onnly two sppeecciieess:: conttrraaddiiccttiioonn aanndd
contrraarriieettyy..

Contradiction is tthhaat ssppeecciieess of innconnsistency wwhhiicchh hhoolddss bbeettwweeenn ttwwo pprrooppoossiittiioonnss wwhhenn tthhey not
onnly cannnnot bbootthh bbee ttrruuee bbuutt aallso cannnnot bbootthh bbe faallssee. AAss wwe saw inn section 2, it is aa rreellaattiioonn wwhhiicchh
hhoolddss bbeettwween, e.g., tthhe contingennt prrooppoossitions

(11..33)) TThhee UU..SS.. ennterred World War II inn 11917
aannd

(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
(11..55)) EEither tthhe UU..SS.. eenntterredd WWorrld WWaarr IIiinn11991177 oorrititisisnnoottththeeccaasseetthhaattththeeUU.S.S. .

eenntterredd WWorrld War II inn 119917
aanndd

(11..66)) TThhee UU..SS.. eenntterredd WWorrld WWaarr II inn 11917 aanndd itt iss nnoott tthhe ccaassee tthhaatt tthhe UU.S.
eenntterredd WWorrld WWaarr II inn 11991177.

IInn aannyy ppoossssibble worrlldd,, oonne oorr ootthherr oof tthhe pprrooppoossiittiioonnss inn aa cconnttrraaddiiccttoorryy ppaair iss ttrruuee aanndd tthhe
rreemmaaiinniinngg pprrooppoossiittiioonn is faallssee.. WWhheerree ttwwo pprrooppoossiittiioonnss aarree cconnttrraaddiiccttoorriieess oof oonne aannootthherr tthheerree iss no
ppoossssibble worrldd inn wwhhiicchh bbootthh pprrooppoossiittiioonnss aarree ttrruuee aanndd nnoo ppoossssibble worrldd inn wwhhiicchh bbootthh prrooppoossittionns
aarree faallssee.. TToo rreeppeeaatt:: cconnttrraaddiiccttoorriieess ddiivviiddee tthhe sset oof aallll ppoossssibble wwoorrllddss innttoo ttwwo mmuuttuuaalllyy eexxccluussive
aanndd jjointtly eexhhaauussttive suubbsseettss.