**1 Introduction 1**

1.1 Outline of the book 1

1.2 Assumed knowledge 6

**2 Propositions and truth assignments 17**

2.1 Introduction 17

2.2 The construction of propositional formulas 19

2.3 The interpretation of propositional formulas 31

2.4 Logical equivalence 48

2.5 The expressive power of connectives 63

2.6 Logical consequence 74

**3 Formal propositional calculus 85**

3.1 Introduction 85

3.2 A formal system for propositional calculus 87

3.3 Soundness and completeness 100

3.4 Independence of axioms and alternative systems 119

**4 Predicates and models 133**

4.1 Introduction: basic ideas 133

4.2 First-order languages and their interpretation 140

4.3 Universally valid formulas and logical equivalence 163

4.4 Some axiom systems and their consequences 185

4.5 Substructures and Isomorphisms 208

**5 Formal predicate calculus 217**

5.1 Introduction 217

5.2 A formal system for predicate calculus 221

5.3 The soundness theorem 242

5.4 The equality axioms and non-normal structures 247

5.5 The completeness theorem 252

**6 Some uses of compactness 265**

6.1 Introduction: the compactness theorem 265

6.2 Finite axiomatizability 266

6.3 Some non-axiomatizable theories 272

6.4 The L¨owenheim–Skolem theorems 277

6.5 New models from old ones 289

6.6 Decidable theories 298

**Bibliography 309**

**Index 311**

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