Possible Worlds: An Introduction to Logic and Its Philosophy
Raymond Bradley and Norman Swartz
http://www.sfu.ca/~swartz/pw/text/pw_all.pdf
Copy made available for public use by authors
Table of Contents
PREFAC E xv
T O TH E TEACHE R xvii
T O TH E STUDEN T xxi
POSSIBLE WORLDS 1
1. THIS AN D OTHE R POSSIBLE WORLD 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, TRUTH , AN D FALSIT 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. PROPERTIES O F PROPOSITIONS 13
Possibly true propositions 13
Possibly false propositions 13
Contingent propositions 14
Contradictories of propositions 14
Noncontingent propositions 15
Necessarily true propositions 16
Necessarily false propositions 17
More about contradictory propositions 18
Some main kinds of noncontingent propositions 19
Summary 24
Symbolization 25
4. RELATION S BETWEE N PROPOSITIONS 28
Inconsistency 28
Consistency 30
Implication 31
Equivalence 35
Symbolization 41
vii CONTENTS
5. SETS O F PROPOSITIONS 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. MODA L PROPERTIES AN D RELATION S PICTURE D O N
WORLDS-DIAGRAMS 48
Worlds-diagrams for modal properties 49
Worlds-diagrams for modal relations 50
Interpretation of worlds-diagrams 50
A note on history and nomenclature 53
Capsule descriptions of modal relations 54
Appendix to section 6 57
7. IS A SINGLE THEOR Y O F TRUT H ADEQUAT E FO R BOT H
CONTINGEN T AN D NONCONTINGEN T PROPOSITIONS? 58
8. TH E "POSSIBLE WORLDS " IDIOM 62
2
PROPOSITIONS 65
1. INTRODUCTIO N 65
2. TH E BEARER S O F TRUTH-VALUE 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. 75
Thesis 8: Context-free sentence-tokens are those things to
which truth and falsity may be attributed. 76
Thesis 9: Context-free sentence-types are those things to
which truth and falsity may be attributed. 76
Thesis 10: Propositions are those things to which truth and
falsity may be attributed. 79
Thesis 11: Propositions are to be identified with the meanings
of sentences. 80
Thesis 12: Propositions are to be identified with sets of
possible worlds. 82
PROPOSITIONS 65
1. INTRODUCTIO N 65
2. TH E BEARER S O F TRUTH-VALUE 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. 75
Thesis 8: Context-free sentence-tokens are those things to
which truth and falsity may be attributed. 76
Thesis 9: Context-free sentence-types are those things to
which truth and falsity may be attributed. 76
Thesis 10: Propositions are those things to which truth and
falsity may be attributed. 79
Thesis 11: Propositions are to be identified with the meanings
of sentences. 80
Thesis 12: Propositions are to be identified with sets of
possible worlds. 82
3.
KNOWLEDGE 129
1. TH E SUBJECT MATTER AND TH E SCIENCE OF LOGIC 129
2. TH E NATURE OF KNOWLEDGE 130
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. TH E LIMITS OF HUMA N KNOWLEDGE 139
The known and the unknown 139
The knowable and the unknowable 140
4. EXPERIENTIAL AND RATIOCINATIVE KNOWLEDGE 142
Experiential knowledge 142
Ratiocinative knowledge 144
Appendix to section 4 149
5. EMPIRICAL AND A PRIORI KNOWLEDGE 149
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
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. TH E LIMITS OF HUMA N KNOWLEDGE 139
The known and the unknown 139
The knowable and the unknowable 140
4. EXPERIENTIAL AND RATIOCINATIVE KNOWLEDGE 142
Experiential knowledge 142
Ratiocinative knowledge 144
Appendix to section 4 149
5. EMPIRICAL AND A PRIORI KNOWLEDGE 149
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
9. Are there any noncontingent propositions which are
knowable a priori but by means other than ratiocination? 171
10. Are there any noncontingent propositions which are
unknowable? 172
Appendix to section 6: a complete classificatory scheme for
the epistemic and modal distinctions 174
7. TH E EPISTEMOLOGY OF LOGIC 175
4
THE SCIENCE OF LOGIC: AN OVERVIEW 179
1. INTRODUCTION 179
2. TH E METHO 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. TH E PARADOX OF ANALYSIS 189
Moore's problem 189
A Moorean solution 190
4. TH E METHO D OF INFERENCE 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. INFERENCE WITHIN TH 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 ON LOGIC AS
A WHOLE 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
Modern predicate logic 233
Modal notions in predicate logic 236
Modalities de dicto and de re 237
Heterogeneous and homogeneous possible worlds 239
Is there really a logic of concepts? 240
5
TRUTH-FUNCTIONAL PROPOSITIONAL LOGIC 247
1. INTRODUCTION 247
2. TRUTH-FUNCTIONAL OPERATORS 247
The uses of "not" and "it is not the case that" 249
The uses of "and" 252
The uses of "or" 257
Interlude: compound sentences containing two or more
sentential operators 261
The uses of "if... then ..." 263
The uses of "if and only if 269
Appendix: truth-tables for wffs containing three or more
letters 272
3. EVALUATING COMPOUND SENTENCES 273
A note on two senses of "determined" 277
4. ELEMENTARY TRUTH-TABLE TECHNIQUES FOR
REVEALING MODAL STATUS AND MODAL RELATIONS 279
Modal status 279
Modal relations 284
Deductive validity 290
5. ADVANCED TRUTH-TABLE TECHNIQUES 294
Corrected truth-tables 294
Reduced truth-tables 297
6. TH E CONCEPT OF FORM 301
Sentences and sentential forms in a logic 301
The relationship between sentences and
sentence-forms 302
7. EVALUATING SENTENCE-FORMS 306
The validity of sentence-forms 306
Modal relations 308
Implication 308
Equivalence 309
Inconsistency 309
Argument-forms and deductive validity 310
8. FORM IN A NATURAL LANGUAGE 311
9. WORLDS-DIAGRAMS AS A DECISION PROCEDURE FOR
TRUTH-FUNCTIONAL PROPOSITIONAL LOGIC 313
10. A SHORTCUT FORMAL METHOD: REDUCTIO AD
ABSURDUM TESTS 315
Summary 320
6
1 to 10 have to be included
11. LOOKING BEYOND MODAL LOGIC TO INDUCTIVE
LOGIC 370
The cardinality of a class and other concepts of class size 371
The concept of contingent content 372
Monadic modal functors 375
What are the prospects for a fully-developed inductive logic? 379
The concept of probabilification 381
A dyadic modal functor for the concept of probabilification 382
