First-order logic

Front Cover......Page 1
Preface to the Dover Edition......Page 8
Preface to the First Edition......Page 9
Contents......Page 13
Part I. Propositional Logic from the Viewpoint of Analytic Tableaux......Page 15
§0. Foreword on Trees......Page 17
§1. Formulas of Propositional Logic......Page 18
§2. Boolean Valuations and Truth Sets......Page 23
§1. The Method of Tableaux......Page 29
§2. Consistency and Completeness of the System......Page 39
§1. Analytic Proofs of the Compactness Theorem......Page 44
§2. Maximal Consistency: Lindenbaum\'s Construction......Page 50
§3. An Analytic Modification of Lindenbaum\'s Proof......Page 52
§4. The Compactness Theorem for Deducibility......Page 54
Part II. First-Order Logic......Page 55
§1. Formulas of Quantification Theory......Page 57
§2. First-Order Valuations and Models......Page 60
§3. Boolean Valuations vs. First-Order Valuations......Page 65
§1. Extension of Our Unified Notation......Page 66
§2. Analytic Tableaux for Quantification Theory......Page 67
§3. The Completeness Theorem......Page 71
§4. The Skolem-Lowenheim and Compactness Theorems for First-Order Logic......Page 77
§1. Analytic Consistency......Page 79
§2. Further Discussion of Analytic Consistency......Page 81
§3. Analytic Consistency Properties for Finite Sets......Page 83
7 The Fundamental Theorem of Quantification Theory......Page 84
§1. Regular Sets......Page 85
§2. The Fundamental Theorem......Page 87
§3. Analytic Tableaux and Regular Sets......Page 89
§4. The Liberalized Rule D......Page 92
8 Axiom Systems for Quantification Theory......Page 93
§0. Foreword on Axiom Systems......Page 94
§1. The System Q_1......Page 95
§2. The Systems Q_2, Q_2*......Page 98
9 Magic Sets......Page 100
§1. Magic Sets......Page 101
§2. Applications of Magic Sets......Page 103
§1. Synthetic Consistency Properties......Page 105
§2. A More Direct Construction......Page 110
Part III. Further Topics in First-Order Logic......Page 113
§1. Gentzen Systems for Propositional Logic......Page 115
§2. Block Tableaux and Gentzen Systems for First-Order Logic......Page 123
§1. Gentzen\'s Hauptsatz......Page 124
§2. An Abstract Form of the Hauptsatz......Page 125
§3. Some Applications of the Hauptsatz......Page 129
§1. Prenex Formulas......Page 131
§2. Prenex Tableaux......Page 132
§1. Gentzen\'s Extended Hauptsatz......Page 135
§2. A New Form of the Extended Hauptsatz......Page 136
§3. Symmetric Gentzen Systems......Page 138
§1. Craig\'s Interpolation Lemma......Page 141
§2. Beth\'s Definability Theorem......Page 145
§1. Clashing Tableaux......Page 147
§3. A Symmetric Form of the Fundamental Theorem......Page 150
17 Systems of Linear Reasoning......Page 155
§1. Configurations......Page 156
§2. Linear Reasoning......Page 165
§3. Linear Reasoning for Prenex Formulas......Page 167
§4. A System Based on the Strong Symmetric Form of the Fundamental Theorem......Page 168
References......Page 170
Subject Index......Page 171
Back Cover......Page 180