TABLE OF CONTENTS: Preface ix Chapter 1. Algebra of Classes and Propositional Calculus 1 1.1 Boole 1 1.2 Jevons 10 1.3 Peirce 12 1.4 Schroder 17 Chapter 2. The Theory of Relatives 31 2.1 Introduction 31 2.2 Basic concepts of the theory of relatives 33 2.3 Basic postulates of the theory of relatives 40 2.4 Theory of relatives and model theory 51 2.5 First-order logic of relatives 66 Chapter 3. Changing the Order of Quantifiers 73 3.1 Schroder's proposal 73 3.2 Lowenheim's approach 81 3.3 The problem of expansions 87 3.4 Skolem functions 94 Chapter 4. The Lowenheim Normal Form 107 4.1 The Lowenheim normal form of an equation 107 4.2 Comments on Lowenheim's method 113 4.3 Conclusions 122 Chapter 5. Preliminaries to Lowenheim's Theorem 129 5.1 Indices and elements 129 5.2 Types of indices 132 5.3 Assignments 135 5.4 Types of equations 138 Chapter 6. Lowenheim's Theorem 143 6.1 The problem 143 6.2 An analysis of Lowenheim's proof 148 6.3 Reconstructing the proof 191 Appendix. First-Order Logic with Fleeing Indices 207 A.1 Introduction 207 A.2 Syntax 207 A.3 Semantics 211 A.4 The Lowenheim normal form 217 A.5 Lowenheim's theorem 220 References 227 Index 237