Possible Worlds: An Introduction to Logic and Its Philosophy

Title ......Page 2
Copyright ......Page 4
Dedication ......Page 5
Contents ......Page 7
PREFACE ......Page 15
TO THE TEACHER ......Page 17
TO THE STUDENT ......Page 21
The realm of possibilities ......Page 23
What are the limits to the possible? ......Page 24
Possibility is not the same as conceivability ......Page 25
Possible worlds: actual and non-actual ......Page 26
Logical possibility distinguished from other kinds ......Page 28
The constituents of possible worlds ......Page 29
Truth and falsity defined ......Page 31
Truth in a possible world ......Page 33
The myth of degrees of truth ......Page 34
Possibly false propositions ......Page 35
Contradictories of propositions ......Page 36
Noncontingent propositions ......Page 37
Necessarily true propositions ......Page 38
Necessarily false propositions ......Page 39
More about contradictory propositions ......Page 40
Some main kinds of noncontingent propositions ......Page 41
Summary ......Page 46
Symbolization ......Page 47
Inconsistency ......Page 50
Consistency ......Page 52
Implication ......Page 53
Equivalence ......Page 57
Symbolization ......Page 63
Modal properties of proposition-sets ......Page 64
Modal relations between proposition-sets ......Page 66
Minding our "P's and "Q"s ......Page 69
6. MODAL PROPERTIES AND RELATIONS PICTURED ON WORLDS-DIAGRAMS ......Page 70
Worlds-diagrams for modal properties ......Page 71
Interpretation of worlds-diagrams ......Page 72
A note on history and nomenclature ......Page 75
Capsule descriptions of modal relations ......Page 76
Appendix to section 6 ......Page 79
7. IS A SINGLE THEORY OF TRUTH ADEQUATE FOR BOTH CONTINGENT AND NONCONTINGENT PROPOSITIONS? ......Page 80
8. THE "POSSIBLE WORLDS" IDIOM ......Page 84
2. THE BEARERS OF TRUTH-VALUES ......Page 87
Thesis 2: Acts of believing (stating, asserting, etc.) are the bearers of truth-values. ......Page 90
Thesis 4: Sentences are the bearers of truth-values. ......Page 93
Thesis 5: Sentence-tokens are the bearers of truth-values. ......Page 95
Thesis 6: Sentence-types are the bearers of truth-values. ......Page 96
Thesis 7: Context-free sentences are the bearers of truth-values. ......Page 97
Thesis 9: Context-free sentence-types are those things to which truth and falsity may be attributed. ......Page 98
Thesis 10: Propositions are those things to which truth and falsity may be attributed. ......Page 101
Thesis 1 1: Propositions are to be identified with the meanings of sentences. ......Page 102
Thesis 12: Propositions are to be identified with sets of possible worlds. ......Page 104
Thesis 13: Propositions are abstract entities in their own right; that is, they are sui generis, they are not to be identified with any other kind of abstract entity. ......Page 106
One final note ......Page 108
Concepts ......Page 109
Attributes of concepts. ......Page 112
Identity conditions for concepts ......Page 114
Analysis of propositions ......Page 116
Identity conditions for propositions ......Page 118
Techniques for referring to sentences ......Page 119
Basic techniques for referring to propositions ......Page 120
Advanced technique for referring to propositions: context-free references ......Page 122
Untensed verbs in context-free references ......Page 125
5. THE OMNITEMPORALITY OF TRUTH ......Page 126
6. PROPOSITIONS, SENTENCES, AND POSSIBLE WORLDS ......Page 130
The uni-linguo proviso ......Page 132
Securing reference to propositions ......Page 133
Sentential ambiguity ......Page 135
The method of possible-worlds testing ......Page 136
Janus-faced sentences ......Page 141
8. POSSIBLE-WORLDS PARABLES ......Page 143
Case Study 1: The thesis that persons (creatures) who lack a language cannot have reflective beliefs ......Page 144
Case Study 2: The thesis that persons (creatures) who lack a language cannot believe necessary truths ......Page 147
Case Study 3: The thesis that a justified belief in a true proposition constitutes knowledge ......Page 148
1. THE SUBJECT MATTER AND THE SCIENCE OF LOGIC ......Page 151
2. THE NATURE OF KNOWLEDGE ......Page 152
1. Is it a necessary condition of the truth of a's knowing that P, that P should be true? ......Page 153
2. Is it a necessary condition of a's knowing that P, that a believe that P? ......Page 155
3. Is it a necessary condition of a's knowing that P, that a be justified in believing that P? ......Page 158
4. What might the missing fourth necessary condition for a's knowing that P be? ......Page 159
The known and the unknown ......Page 161
The knowable and the unknowable ......Page 162
Experiential knowledge ......Page 164
Ratiocinative knowledge ......Page 166
5. EMPIRICAL AND A PRIORI KNOWLEDGE ......Page 171
Definitions of "empirical" and "a priori" ......Page 172
The non-exhaustiveness and non-exclusiveness of the experiential/ratiocinative distinction ......Page 173
The exhaustiveness and exclusiveness of the empirical / a priori distinction ......Page 174
Is a priori knowledge certain? ......Page 177
6. EPISTEMIG AND MODAL STATUS CONSIDERED TOGETHER ......Page 178
1. Are there any contingent propositions which are knowable empirically? ......Page 179
2. Are there any contingent propositions which are knowable both experientially and ratiocinatively? ......Page 180
3. Are there any contingent propositions which are knowable ratiocinatively but which are not knowable experientially? ......Page 185
4. Are there any contingent propositions which are knowable by other than experiential or ratiocinative means? ......Page 186
5. Are there any contingent propositions which are unknowable? ......Page 189
6. Are there any noncontingent propositions knowable empirically? ......Page 190
8. Are there any noncontingent propositions which are knowable ratiocinatively but which are not knowable experientially? ......Page 192
9. Are there any noncontingent propositions which are knowable a priori but by means other than ratiocination? ......Page 193
10. Are there any noncontingent propositions which are unknowable? ......Page 194
Appendix to section 6: a complete classificatory scheme for the epistemic and modal distinctions ......Page 196
7. THE EPISTEMOLOGY OF LOGIC ......Page 197
1. INTRODUCTION ......Page 201
The objects of philosophical analysis ......Page 202
Three levels of analysis ......Page 203
The idea of a complete analysis ......Page 205
The need for a further kind of analysis ......Page 206
Possible-worlds analysis ......Page 207
Degrees of analytical knowledge ......Page 209
Moore's problem ......Page 211
A Moorean solution ......Page 212
4. THE METHOD OF INFERENCE ......Page 214
The nature of inference ......Page 215
Valid and invalid propositional inferences ......Page 217
Determining the validity of inferences: the problem of justification ......Page 218
Rules of inference ......Page 220
What kind of rule is a rule of inference? ......Page 222
Inference and the expansion of knowledge ......Page 223
Inference within axiomatic systems: the example of S5 ......Page 227
Inference within natural deduction systems ......Page 232
The theoretical warrant of the method of direct proof ......Page 237
The indispensability of modal concepts within propositional logics ......Page 240
Problems about the reduction principles ......Page 242
Problems about the paradoxes ......Page 246
Relevance logics ......Page 250
The move to predicate logic ......Page 252
Traditional syllogistic ......Page 254
Modern predicate logic ......Page 255
Modal notions in predicate logic ......Page 258
Modalities de dicto and de re ......Page 259
Heterogeneous and homogeneous possible worlds ......Page 261
Is there really a logic of concepts? ......Page 262
2. TRUTH-FUNCTIONAL OPERATORS ......Page 269
The uses of "not" and "it is not the case that" ......Page 271
The uses of "and" ......Page 274
The uses of "or" ......Page 279
Interlude: compound sentences containing two or more sentential operators ......Page 283
The uses of "if... then ..." ......Page 285
The uses of "if and only if" ......Page 291
Appendix: truth-tables for wffs containing three or more letters ......Page 294
3. EVALUATING COMPOUND SENTENCES ......Page 295
A note on two senses of "determined" ......Page 299
Modal status ......Page 301
Modal relations ......Page 306
Deductive validity ......Page 312
Corrected truth-tables ......Page 316
Reduced truth-tables ......Page 319
Sentences and sentential forms in a logic ......Page 323
The relationship between sentences and sentence-forms ......Page 324
The validity of sentence-forms ......Page 328
Implication ......Page 330
Inconsistency ......Page 331
Argument-forms and deductive validity ......Page 332
8. FORM IN A NATURAL LANGUAGE ......Page 333
9. WORLDS-DIAGRAMS AS A DECISION PROCEDURE FOR TRUTH-FUNCTIONAL PROPOSITIONAL LOGIC ......Page 335
10. A SHORTCUT FORMAL METHOD: REDUCTIO AD ABSURDUM TESTS ......Page 337
Summary ......Page 342
Non-truth-functionality ......Page 345
Modal and nonmodal propositions; modalized and non-modalized formulae ......Page 347
The interdefinability of the monadic and dyadic modal operators ......Page 349
"It is possible that" ......Page 351
Problems with the use of "it is necessary that"; the modal fallacy; absolute and relative necessity ......Page 352
4. THE MODAL STATUS OF MODAL PROPOSITIONS ......Page 355
5. THE OPERATOR "IT IS CONTINGENTLY TRUE THAT" ......Page 359
6. ESSENTIAL PROPERTIES OF RELATIONS ......Page 361
Case study 1: the pragmatics of telling the truth ......Page 367
Case study 2: an invalid inference and an unwitting impossible description ......Page 369
8. USING WORLDS-DIAGRAMS TO ASCERTAIN THE VALIDITY OF MODALIZED FORMULAE ......Page 372
Applications ......Page 373
The validity of the axioms of S5 ......Page 378
The nonvalidity of the axiom set for S6 ......Page 380
9. A SHORTCUT FORMAL METHOD FOR DETERMINING THE VALIDITY OF MODALIZED FORMULAE: modal reductios ......Page 381
10. THE NUMBER OF FORMALLY NON-EQUIVALENT SENTENCE-FORMS CONSTRUCTIBLE ON N SENTENCE-VARIABLES ......Page 387
11. LOOKING BEYOND MODAL LOGIC TO INDUCTIVE LOGIC ......Page 392
The cardinality of a class and other concepts of class size ......Page 393
The concept of contingent content ......Page 394
Monadic modal functors ......Page 397
What are the prospects for a fully-developed inductive logic? ......Page 401
The concept of probabilification ......Page 403
A dyadic modal functor for the concept of probabilification ......Page 404
INDEX ......Page 407