Contents
Preface for Students xiii
Preface for Instructors xv Acknowledgements xix
PART I: ARGUMENTS
Chapter 1: Arguments
1.1 Introduction 3
1.2 Identifying Arguments 9
1.2.1 Inference Indicators 9
1.2.2 Explanations 10
1.2.3 Implicit Arguments 11
1.2.4 Enthymemes 13
1.3 Natural Arguments 18
1.3.1 Argument and Inference 18
1.3.2 TechniquesofDiagramming 19
Chapter 2: Validity
2.1 Validity 25
2.1.1 Defining Validity 25
2.1.2 Soundness 27
2.2 Argument Forms and Formal Validity 30
2.3 Evaluating Natural Arguments 34
PART II: STATEMENT LOGIC
Chapter 3: Statements and Conditionals
3.1 Statements and Compounds 41
3.1.1 Statements 41
3.1.2 Compounds 42
3.1.3 Statement Operators 44
3.2 Conditional Statements 51
3.3 Modus Ponens 55
3.3.1 Argument Form and Substitution Instance 55
3.3.2 Affirming the Consequent 58
Chapter 4: Negation
4.1 Symbolizing Negations 61
4.1.1 Negations 61
4.1.2 Contradictories 62
4.2 Modus Tollens 64
4.2.1 Modus Tollens and Double Negation 64
4.2.2 Denying the Antecedent 67
4.3 Inference and Implication 71
Chapter 5: Conjunction
5.1 Symbolizing Conjunctions 75
5.2 Rules of Inference for Conjunction 79
5.3 Evaluating Extended Arguments 84
Chapter 6: Disjunction
6.1 Symbolizing Disjunctions 93
6.2 Rules of Inference for Disjunctions 98
6.2.1 Disjunctive Syllogism 98
6.2.2 Disjunction 101
6.2.3 De Morgan’s Laws 102
Chapter 7: Conditional Proof
7.1 More on Symbolizing 109
7.1.1 Disjunctions in Conditionals 109
7.1.2 ‘Unless’ 110
7.1.3 ‘Otherwise,’ ‘Else’ 110
7.2 More Rules Involving Conditionals 112
7.2.1 Conditional Proofand Supposition 112
7.2.2 The Hypothetical Syllogism 115
7.3 Supposition in Natural Argument 119
Chapter 8: Biconditionals
8.1 Necessary and Sufficient Conditions 125
8.1.1 ζOnly If 125
8.1.2 Necessary and Sufficient Conditions 127
8.2 Biconditionals 129
8.2.1 Symbolizing 129
8.2.2 Conversational Implicature 130
8.2.3 Rules of Inference 131
Chapter 9: Dilemmas
9.1 Dilemmas 135
9.2 Natural Dilemmas 141
Chapter 10: Reductio Arguments
10.1 Reductio ad Absurdum 147
10.2 Natural Reductio Arguments 156
Chapter 11: Review and Consolidation
11.1 Rules of Inference 165
11.1.1 Rules of Inference and Equivalence Rules 165
11.1.2 Two Simplifying Modifications 167
11.1.3 Proof Strategies 169
11.2 Derived Rules 174
Chapter 12: SL as a Formal System
12.1 Rules of Formation 179
12.1.1 Symbols, Formulas, and Wffs 179
12.1.2 Consistency and Completeness 183
12.2 Sequents, Theorems, and Axioms 186
12.2.1 Sequents and Theorems 186
12.2.2 Axioms and the Propositional Calculus 188
Chapter 13: Truth Tables
13.1 Truth Tables and Statements 193
13.1.1 Truth Tables 193
13.1.2 Material Implication 196
13.1.3 Tautologies, Contradictions, and Contingent Statements 196
13.1.4 Logical Equivalence 198
13.2 Truth Tables and Validity 201
13.2.1 The Full Truth Table Method 201
13.2.2 Invalid Argument Forms 204
13.3 The Brief Truth Table Method 207
Chapter 14: Truth Trees for SL
14.1 Truth Trees 215
14.1.1 The Truth Tree Method 215
14.1.2 Decomposition Rules 219
14.2 Statements, Consistency, and Completeness 226
14.2.1 Tautologies, Contradictions, and Logical Equivalence 226
14.2.2 Consistency and Completeness 228
PART III: PREDICATE LOGIC
Chapter 15: Syllogistic Logic
15.1 C ategory Logic 233
15.1.1 Aristotlefs Logic 233
15.1.2 A-, E-, I-, and O-Statements 235
15.1.3 Ambiguous Statements 237
15.2 Carroll Diagrams 239
15.2.1 CarrolTs Diagrams 239
15.2.2 Existence and Non-Existence 242
15.2.3 Conversion 243
15.3 Evaluating Validity of Syllogisms 245
Chapter 16: Universal Quantification
16.1 Universal and Singular Statements 253
16.1.1 UniversalQuantification 253
16.1.2 Only" and "Nothing BuT 255
16.1.3 Singular Statements and Individual Names 256
16.2 Rules of Inference: UI and UG 260
Chapter 17: Existential Quantification
17.1 Particular S tatements 267
17.1.1 Existential Quantification 267
17.2 Rules of Inference 269
17.2.1 Existential Instantiation 269
17.2.2 Existential Generalization 273
17.2.3 ProofStrategy 276
Chapter 18: Advanced Class Logic
18.1 Arguments with More than 3 Predicates 281
18.1.1 Carroll Diagrams for 4 or 5 Categories 281
18.1.2 Sorites 286
18.2 Existential Import 290
18.2.1 On Giving Universal Statements Existential Import 290
18.2.2 Penevalid Arguments 293
18.2.3 Non-Emptiness of the UD 295
Chapter 19: Asyllogistic Arguments
19.1 More on Symbolizing 303
19.1.1 Non-Classical Statements 303
19.1.2 Anyf 304
19.2 Asyllogistic Proofs: QN 307
19.3 Predicate Logic as a Formal System 314
19.3.1 Symbols, Formulas, and Wffs 314
19.3.2 Propositional Functions and Quantifier Scope 318
Chapter 20: Relational Logic
20.1 The Logic of Relations 321
20.1.1 Relations 321
20.1.2 Symbolizing Relations 323
20.1.3 Nested Quantifiers 324
20.1.4 Relational Proofs 324
20.2 Properties of Binary Relations 327
20.2.1 Transitivity, Symmetry, and Reflexivity 327
20.2.2 Equivalence Relations 329
Chapter 21: Logic with Identity
21.1 Identity and Quantity 335
21.1.1 Symbolizing Identities and Quantities 335
21.1.2 Russell fs Theory of Definite Descriptions 339
21.2 Inferences Involving Identity 343
21.2.1 The Rule of Inference SI 343
21.2.2 Properties of Identity 345
21.3 Ordering Relations 347
Chapter 22: Relational Arguments
22.1 More on Symbolizing Relational Statements 351
22.1.1 A Methodfor Symbolizing 351
22.1.2 Prenex Forms 354
22.2 Relational Arguments 358
22.2.1 Arguments beyond the Scope of Traditional Logic 358
22.2.2 Ambiguities and the Quantifier Shift Fallacy 361
Chapter 23: Truth Trees for PL
23.1 Predicate Logic Truth Trees 367
23.1.1 Truth Tree Rulesfrom Statement Logic 367
23.1.2 Additional Truth Tree Rules for Quantifications 368
23.1.3 Negated Quantifier Decomposition Rules 370
23.1.4 Effective Completeness 372
23.2 Trees for Relational Logic and Identity 376
23.2.1 Truth Tree Rules in Relational Logic 376
23.2.2 Additional Truth Tree Rulesfor Identity and Diversity 379
Chapter 24: Other Logics
24.1 Second Order Logic 385
24.2 Modal Logic 386
24.3 Deontic Logic 387
24.4 Quantum Logic 388
24.5 Intuitionistic Logic 390
Appendix 1: The Paradoxes of Material Implication 395
Appendix 2: A Little History: Consequentiae 401
Appendix 3: Logic Diagrams 405
Glossary 413
Index 425