Understanding Symbolic Logic

Title ......Page 1
Copyright ......Page 2
Dedication ......Page 3
Contents ......Page 5
Preface ......Page 13
1 INTRODUCTION TO LOGIC ......Page 17
1. Why Study Logic? ......Page 18
2. What Logic Is All About ......Page 20
3. Induction and Deduction ......Page 22
4. Form and Validity ......Page 24
5. Truth and Validity ......Page 28
6. The Nature of Symbolic Logic ......Page 30
7. The Scope of Symbolic Logic ......Page 31
Study Questions ......Page 33
Exercises ......Page 34
2 THE STRUCTURE OF SENTENTIAL LOGIC ......Page 37
1. Simple and Compound Sentences ......Page 38
2. Sentential Operators ......Page 42
3. The Structure and Symbolism of Sentential Logic ......Page 44
Definitions ......Page 46
Exercises ......Page 47
3 COMPUTING TRUTH VALUES ......Page 49
1. Truth Tables for the Operators ......Page 50
2. Computing Truth Values ......Page 58
3. Truth-functional Operators ......Page 61
4. Non-truth-functional Operators ......Page 62
Definitions ......Page 64
Exercises ......Page 65
4 SYMBOLIZING ENGLISH SENTENCES ......Page 67
1. Simple Sentences ......Page 68
2. Truth-functional and Non-truth-functional Compounds ......Page 70
3. Symbolizing Truth-functional English Operators ......Page 71
4. Symbolizing Multiply Complex Sentences ......Page 81
Exercises ......Page 85
5 TRUTH TABLES FOR TESTING VALIDITY ......Page 90
1. Constructing Base Columns for Truth Tables ......Page 91
2. The Truth Table Test for Validity ......Page 95
3. Shortcut Validity Tests ......Page 100
4. Mechanical Decision Procedures ......Page 105
Study Questions ......Page 107
Exercises ......Page 108
6 FURTHER APPLICATIONS OF THE TRUTH TABLE METHOD ......Page 111
1. Tautologies, Contradictions, and Contingencies ......Page 112
2. Logical Implication and Logical Equivalence ......Page 116
3. Rules of Inference, Logical Implication, and Logical Equivalence ......Page 119
4. Consistency ......Page 120
5. Four Kinds of Truth Table Problems and the Relations Between Them ......Page 122
Definitions ......Page 123
Exercises ......Page 124
7 THE PROOF METHOD: EIGHT BASIC INFERENCE RULES ......Page 129
1. Form and Substitution Instance ......Page 131
2. The Proof Process ......Page 134
3. Eight Basic Inference Rules ......Page 136
4. Derivations and Proofs ......Page 144
5. Constructing Simple Proofs ......Page 146
6. Constructing More Complex Proofs ......Page 151
Definitions ......Page 154
Exercises ......Page 155
8 REPLACEMENT RULES ......Page 163
1. The Structure of Replacement Rules ......Page 164
2. The Ten Replacement Rules ......Page 165
3. Constructing Simple Proofs with Replacement Rules ......Page 174
4. Strategies for More Complex Proofs ......Page 178
Summary of Replacement Rules ......Page 183
Exercises ......Page 184
9 CONDITIONAL PROOF AND INDIRECT PROOF ......Page 191
1. Conditional Proof ......Page 192
2. Indirect Proof ......Page 196
3. Discharging Assumptions; Restrictions on C.P. and LP. ......Page 200
4. Using C.P. and LP. ......Page 201
5. Proofs of Theorems ......Page 205
6. Invalidity ......Page 207
7. Truth and Proof ......Page 208
Summary of Rules of Conditional Proof and Indirect Proof ......Page 209
Exercises ......Page 211
10 SINGULAR SENTENCES ......Page 217
1. Singular Sentences and Propositional Functions ......Page 219
2. Symbolizing Singular Sentences ......Page 222
Exercises ......Page 225
11 QUANTIFIERS ......Page 228
1. Universal and Existential Quantifiers ......Page 229
2. Free and Bound Variables; Scope of a Quantifier ......Page 233
3. Negated Quantifiers ......Page 234
Exercises ......Page 238
12 CATEGORICAL PROPOSITIONS ......Page 241
1. The Four Categorical Propositions ......Page 242
2. Individuals, Sets, and Properties ......Page 245
3. Venn Diagrams ......Page 246
4. Symbolizing Categorical Propositions ......Page 249
5. Negated Categorical Propositions ......Page 252
7. Symbolizing English Categorical Sentences ......Page 254
Summary of Categorical Propositions ......Page 260
Exercises ......Page 261
13 COMPLEX SUBJECTS AND PREDICATES ......Page 265
1. Complex Subjects and Predicates ......Page 266
2. Equivalent Symbolizations ......Page 270
Exercises ......Page 274
14 QUANTIFIER FORM AND TRUTH-FUNCTIONAL COMPOUNDS OF QUANTIFIER STATEMENTS ......Page 279
1. Quantifier Form ......Page 280
2. Truth-functional Compounds and Quantifier Form ......Page 281
3. Symbolizing Truth-functional Compounds ......Page 283
Definitions ......Page 284
Exercises ......Page 285
15 PROOFS IN PREDICATE LOGIC ......Page 289
1. Preliminary Statement of the Four Quantifier Rules ......Page 290
2. Instances of Quantified Formulas ......Page 292
3. The Rules of Universal Instantiation (U.I.) and Existential Generalization (E.G.) ......Page 293
4. The Rules of Existential Instantiation (E.I.) and Universal Generalization (U.G.); Flagging Restrictions ......Page 294
5. Constructing Proofs for \"Pure\" Quantifier Arguments ......Page 301
6. Constructing Proofs for Arguments Containing Truth-functional Compounds ......Page 306
7. Constructing Proofs of Quantifier Theorems ......Page 308
Statement of the Quantifier Rules, with All Necessary Restrictions ......Page 310
Exercises ......Page 311
16 INVALIDITY IN QUANTIFIER LOGIC ......Page 315
1. The Natural Interpretation Method ......Page 316
2. Truth Conditions for Quantifier Statements ......Page 319
3. The Model Universe Method ......Page 320
Exercises ......Page 327
17 SYMBOLIZATION IN RELATIONAL PREDICATE LOGIC ......Page 329
1. Relational Predicates and Singular Sentences ......Page 330
2. Multiple Quantifiers ......Page 333
3. Quantifier Negation ......Page 340
4. Categorical Relational Statements; Complex Subjects and Predicates ......Page 343
5. Symbolizing English Sentences ......Page 347
Exercises ......Page 351
18 PROOFS AND INVALIDITY FOR RELATIONAL PREDICATE LOGIC ......Page 357
1. Proofs in Relational Predicate Logic ......Page 358
2. Invalidity in Relational Predicate Logic ......Page 365
Exercises ......Page 369
19 IDENTITY AND DEFINITE DESCRIPTIONS ......Page 371
1. Identity Statements and Their Negations ......Page 372
2. Exceptives and \"Only\" Statements ......Page 373
3. Superlatives ......Page 376
4. Numerical Statements ......Page 377
5. Definite Descriptions ......Page 380
Exercises ......Page 382
1. Rules for Identity ......Page 387
2. Proofs Containing Identity Statements ......Page 391
Summary of Identity Rules ......Page 395
Exercises ......Page 396
21 WELL-FORMED FORMULAS FOR SENTENTIAL LOGIC ......Page 399
Exercises ......Page 401
22 PROOF TREES FOR SENTENTIAL LOGIC ......Page 403
Exercises ......Page 408
23 USING VENN DIAGRAMS TO PROVE VALIDITY ......Page 409
Exercises ......Page 413
24 PROOF TREES FOR PREDICATE LOGIC ......Page 414
Exercises ......Page 424
ANSWER TO STARRED EXERCISES ......Page 425
INDEX ......Page 475