First Course in Mathematical Logic

This introduction to rigorous mathematical logic is simple enough in both presentation and context for students of a wide range of ages and abilities. Starting with symbolizing sentences and sentential connectives, it proceeds to the rules of logical inference and sentential derivation, examines the concepts of truth and validity, and presents a series of truth tables. Subsequent topics include terms, predicates, and universal quantifiers; universal specification and laws of identity; axioms for addition; and universal generalization. Throughout the book, the authors emphasize the pervasive and important problem of translating English sentences into logical or mathematical symbolism. 1964 edition. Index.

Preface	p. v
Symbolizing Sentences	p. 1
Sentences	p. 1
Sentential Connectives	p. 2
The Form of Molecular Sentences	p. 5
Symbolizing Sentences	p. 9
The Sentential Connectives and Their Symbols	p. 12
Or	p. 14
Not	p. 16
If ... then ...	p. 19
Grouping and Parentheses	p. 21
The Negation of a Molecular Sentence	p. 29
Elimination of Some Parentheses	p. 33
Summary	p. 35
Logical Inference	p. 43
Introduction	p. 43
Rules of Inference and Proof	p. 44
Modus Ponendo Ponens	p. 44
Proofs	p. 48
Two-Step Proofs	p. 49
Double Negation	p. 52
Modus Tollendo Tollens	p. 54
More on Negation	p. 58
Adjunction and Simplification	p. 61
Disjunctions as Premises	p. 64
Modus Tollendo Ponens	p. 65
Sentential Derivation	p. 69
More About Parentheses	p. 77
Further Rules of Inference	p. 79
Law of Addition	p. 80
Law of Hypothetical Syllogism	p. 83
Law of Disjunctive Syllogism	p. 88
Law of Disjunctive Simplification	p. 92
Commutative Laws	p. 95
De Morgan's Laws	p. 98
Biconditional Sentences	p. 103
Summary of Rules of Inference	p. 107
Table of Rules of Inference	p. 108
Truth and Validity	p. 110
Introduction	p. 110
Truth Value and Truth-Functional Connectives	p. 111
Conjunction	p. 111
Negation	p. 112
Disjunction	p. 113
Conditional Sentences	p. 114
Equivalence: Biconditional Sentences	p. 117
Diagrams of Truth Value	p. 118
Invalid Conclusions	p. 122
Conditional Proof	p. 129
Consistency	p. 138
Indirect Proof	p. 146
Summary	p. 152
Truth Tables	p. 161
Truth Tables	p. 161
Tautologies	p. 169
Tautological Implication and Tautological Equivalence	p. 171
Summary	p. 176
Terms, Predicates, and Universal Quantifiers	p. 180
Introduction	p. 180
Terms	p. 183
Predicates	p. 185
Common Nouns as Predicates	p. 187
Atomic Formulas and Variables	p. 191
Universal Quantifiers	p. 197
Two Standard Forms	p. 205
Universal Specification and Laws of Identity	p. 211
One Quantifier	p. 211
Two or More Quantifiers	p. 223
Logic of Identity	p. 230
Truths of Logic	p. 236
A Simple Mathematical System: Axioms for Addition	p. 241
Commutative Axiom	p. 241
Associative Axiom	p. 245
Axiom for Zero	p. 255
Axiom for Negative Numbers	p. 258
Universal Generalization	p. 263
Theorems with Variables	p. 263
Theorems with Universal Quantifiers	p. 267
Index	p. 273
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