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