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Sam Cumming is Assistant Professor of Philosophy at the University of California, Los Angeles.
Preface to Second Edition xiv
Preface to Revised Edition xv
Acknowledgments xvi
Introduction to Teacher 1
1 A Brief Introduction to Key Terms 5
1.1 Arguments 5
1.1.1 What is a Statement? 6
1.1.2 Premises and Conclusion 6
1.2 Putting Arguments into a Standard Format 7
1.3 Multiple Conclusions 9
1.4 Deductive Validity 10
1.5 Soundness 13
1.6 Missing Premises and Conclusions 13
2 Argument Forms and Propositional Logic 17
2.1 Formal Validity 17
2.2 Quotation Marks 19
2.3 Metalinguistic Variables 21
2.4 Non-formal Validity 23
2.5 The Need for Propositional Logic 24
2.5.1 Symbolic Notation 25
2.6 The Type/Token Distinction 26
3 Conjunction 31
3.1 Logical Conjunction 31
3.2 Distinguishing Deductive from Non-deductive Aspects of Conjunction 33
3.3 Phrasal Logical Conjunctions 34
3.4 Series Decompounding 34
3.5 Using ‘Respectively’ 35
3.6 Symbolizing Logical Conjunctions 35
4 Negation 42
4.1 Logical Negation 42
4.2 Some Other Negative Expressions 43
4.3 A Point about Methodology 45
4.4 A Point on Ambiguity 45
4.5 Symbolizing Logical Negations 45
4.6 Ambiguity and the Need for Groupers 46
4.7 Review of Symbols 47
4.8 Using ‘Without’ 48
4.9 Argument Forms Continued 48
4.10 Symbolizing Logical Negations Continued 51
5 Truth Tables 56
5.1 Well-formed Formulas 56
5.2 Scope 57
5.3 Main Connective 58
5.4 Truth Tables 59
5.4.1 Truth Table Analyses of Statements 61
5.4.2 Truth Table Analyses of Arguments 64
6 Disjunction 68
6.1 Logical Disjunction 68
6.2 Disjunction and Negation 69
6.3 Iterations and Groupers 71
6.4 Inclusive versus Exclusive ‘Or’ 73
6.5 Symbolizing Logical Disjunctions Continued 76
7 Conditionals 79
7.1 Conditionals with Constituent Statements 79
7.2 Conditionals without Constituent Statements 80
7.3 Logical Conditionals 80
7.4 Symbolizing Conditionals in PL 82
7.5 Necessary and Sufficient Conditions 82
7.6 Only If 84
7.7 Unless 86
7.8 Since, Because 88
7.9 Conditionals and Groupers 89
7.10 If and Only If 90
7.11 A Revised Grammar for Well-formedness in PL 91
7.12 Summarizing Truth Tables 99
7.12.1 Validity 99
7.12.2 Contradiction, Tautology, Contingency 102
7.12.3 Consistency 104
7.12.4 Logical Equivalence 105
8 Truth Trees 109
8.1 Reviewing Validity 109
8.2 Tree Trunks and Compound and Atomic Statements 110
8.3 Truth Tree Rules 111
8.3.1 Non-branching Rules 111
8.3.2 Branching Rules 112
8.4 Strategies 114
8.5 Truth Trees and Invalidity 117
8.6 Propositional Logic and Counter-examples (Counter-models) 121
8.7 Logical Properties and Relations Revisited 123
8.7.1 Consistency 123
8.7.2 Contradiction, Tautology, Contingency 124
8.7.3 Logical Equivalence 126
9 Property Predicate Logic 129
9.1 Limits of Propositional Logic 129
9.2 Singular Terms 130
9.3 Property Predicates 132
9.4 Quantifiers 134
9.4.1 Simple Existential Quantifier Statements 135
9.4.2 Symbolizing Simple Existential Statements 135
9.4.3 Simple Universal Quantifier Statements 137
9.4.4 Negations of Existentials 138
9.5 Complex Predicates 139
9.6 Well-formedness in PPL 142
9.7 Quantifiers Modifying General Terms 145
9.7.1 Existential Quantifiers and General Terms 145
9.7.2 Universal Quantifiers and General Terms 147
10 Evaluating Arguments in Property Predicate Logic 155
10.1 Quantifiers and Scope 156
10.2 The Truth Tree Method Extended 157
10.2.1 Quantifier Exchange Rule (QE) 157
10.2.2 Universal Quantifier Rule (UQ) 158
10.2.3 Existential Quantifier Rule (EQ) 161
10.3 Super Strategy 164
10.4 Property Predicate Logic and Counter-examples (Counter-models) 166
10.5 PPL Logical Equivalences and Non-equivalences 168
10.6 Other Logical Properties and Relations 170
10.6.1 Consistency 170
10.6.2 Logical Equivalence 170
10.6.3 Contradiction, Logical Truth, Contingency 171
11 Property Predicate Logic Refinements 172
11.1 Literal Meaning 172
11.2 ‘Any’ as an Existential 173
11.3 Restrictive Relative Clauses 175
11.4 Pronouns Revisited 176
11.4.1 Deixis and Anaphora 176
11.4.2 Quantification and Anaphora 177
11.5 Only 180
11.6 Restrictive Words in English 182
11.7 Evaluating Symbolizations of English in Logical Notation 185
12 Relational Predicate Logic 191
12.1 Limits of Property Predicate Logic 191
12.2 Convention 1: Number 193
12.3 Convention 2: Order 194
12.4 Convention 3: Active/Passive Voice 195
12.5 Convention 4: Single Quantifiers 197
12.6 Variables 199
12.6.1 Convention 5: Variables and Quantifiers 200
12.6.2 Convention 6: Variables and Property Predicates 200
12.6.3 General Comments about Variables 201
13 Relational Predicate Logic with Nested Quantifiers 207
13.1 Multiply General Statements 209
13.2 Universal Quantifier Procedure 212
13.3 Existential Quantifier Procedure 213
13.4 Double Binding Variables 213
13.4.1 Kicking Out 216
13.5 Systematic and Analytic Procedures 217
13.6 A Grammar for Well-formedness in RPL 218
13.7 Nested Quantifiers, Variables, and Scope 220
13.8 Order and Scope Refinements 221
13.8.1 The Order and Scope Procedure 224
13.9 Summary of the Overall Procedure for Symbolizing English Statements with Nested Quantifiers into RPL 226
14 Extending the Truth Tree Method to RPL 229
14.1 RPL Arguments without Quantifiers 229
14.2 RPL Arguments without Nested Quantifiers 230
14.3 RPL Arguments with Nested Quantifiers 232
14.4 Choosing Singular Terms to Instantiate 233
14.5 Infinite Truth Trees for RPL Arguments 234
14.6 Summary of Truth Tree Strategies 236
14.7 Relational Predicate Logic and Counter-examples (Counter-models) 239
15 Negation, Only, and Restrictive Relative Clauses 244
15.1 Negation 244
15.2 ‘Only’ as a Quantifier 246
15.3 Restrictive Relative Clauses 249
15.3.1 The Quantificational Restrictive Relative Clause Procedure 250
15.4 Quantifiers and Anaphora 252
15.4.1 Repair Algorithm 254
15.5 Anaphora and Restrictive Relative Clauses 257
15.6 Anaphora Across Sentences 262
15.7 Quantification in English 265
16 Relational Predicate Logic with Identity 268
16.1 Limits of Relational Predicate Logic 268
16.2 Extending the Truth Tree Method to RPL= 270
16.2.1 Identity-out Rule 270
16.2.2 Identity-in Rule 271
16.3 Sameness and Distinctness in English 273
16.3.1 ‘Only’ Again 273
16.3.2 Words of Distinction: Except, But, Other (than), Besides, Else 274
16.4 Numerical Adjectives 276
16.4.1 At Least n 276
16.4.2 At Most n (No More than n) 279
16.4.3 Exactly n 281
16.4.4 Counting Pairs 283
16.4.5 Combinatorics (optional) 283
16.5 Definite Descriptions 284
16.5.1 The Definite Description Quantifier Procedure 288
16.5.2 Definite Descriptions as Anaphors 289
16.5.3 Plural Definite Descriptions 289
17 Verbs and their Modifiers 294
17.1 Prepositional Phrases 294
17.2 The Event Approach 296
17.3 Indirect Support of the Event Approach 298
17.3.1 Fixing Referents and Binding Anaphoric Pronouns 298
17.3.2 Quantification over Events 299
17.3.3 Conversational Inferences and Events 300
17.3.4 Methodological Reflections 300
17.4 Adverbial Modification 301
17.5 Problems with the Event Approach 304
Appendix 308
A1 Conjunction 308
A1.1 Prepositional Phrases 308
A1.2 Conversational Inferences and Deductive Validity 309
A1.3 Relative Clauses 311
A2 Negation and Disjunction 314
A2.1 Modalities and Negation 314
A2.2 Disjunction and Conversational Inferences 315
A3 Conditionals 315
A3.1 Explication of the Material Conditional Truth Table 315
A3.1.1 Paradoxes of implication 318
A3.1.2 Conditionals and conversational inferences 318
A3.1.3 Paradoxes of implication revisited 320
A3.2 ‘If ’s and ‘Then’s without Conditionality 321
A4 Property Predicate Logic 321
A4.1 Only 321
A4.2 Conversational Inferences 322
A4.2.1 Existential import 322
A4.2.2 Scalar inferences 323
A4.3 More on Literal Meaning 324
A4.4 Adjectival Modification and Predication 325
A4.5 A Non-standard Quantifier – Most 329
A5 Relational Predicate Logic 330
A5.1 Passive Voice: Another Argument for Variables 330
A5.1.1 Passive voice for nested quantifier procedure 332
A5.2 Properties of Relations 333
A5.2.1 Symmetry, asymmetry, non-symmetry 333
A5.2.2 Transitivity, intransitivity, non-transitivity 334
A5.2.3 Total reflexivity, reflexivity, irreflexivity, and non-reflexivity 335
A6 Relational Predicate Logic with Identity 337
A6.1 ‘Only’ and Existential Import 337
A6.2 Descriptions and Anaphora 338
A6.3 Plural Anaphora 339
A6.3.1 Plural definite descriptions as anaphors 344
A6.3.2 Singular indefinite antecedents of plural pronouns 344
A6.3.3 Partitives 346
A6.4 Existence 347
A6.5 Intensionality 348
A6.6 Properties of the Identity Relationship 348
A6.7 The Superlative 349
A6.8 Identity and Predicative Adjectives 350
A7 Verbs and their Modifiers 350
A7.1 Infinitives and Gerunds 351
A7.2 Reference to Events 353
A7.3 The Logic of Perceptual Verbs 354
Answers for Selected Exercises 356
Chapter 1 356
Chapter 2 357
Chapter 3 358
Chapter 4 361
Chapter 5 363
Chapter 6 364
Chapter 7 366
Chapter 8 373
Chapter 9 378
Chapter 10 381
Chapter 11 392
Chapter 12 397
Chapter 13 398
Chapter 14 400
Chapter 15 413
Chapter 16 419
Chapter 17 426
Appendix 427
Logical Symbols 429
Index 430