Contents
Introduction 1
1 Deductive Systems and Matrix Semantics 5
1.1 The Lattice of Theories 6
1.2 Matrix Semantics 8
1.3 Deductive Systems as Elementary Theories 9
1.4 The Elementary Leibniz Equivalence Relation 10
1.4.1 Protoalgebraic Logics 12
2 Equational Consequence and
Algebraic Semantics 13
2.1 Algebraic Semantics 14
2.2 Equivalent Algebraic Semantics 19
2.2.1 Uniqueness 22
2.2.2 Axiomatization 24
3 Th e Lattice of Theories 27
4 Tw o Intrinsic Characterizations 34
4.1 The Leibniz Operator 34
4.2 A Second Intrinsic Characterization 39
5 Matrix Semantics and Algebraizability 42
5.1 Matrix Semantics and Algebraic Semantics 42
5.2 Applications and Examples 46
5.2.1 Modal Logics 46
5.2.2 Entailment and Relevance Logics 48
5.2.3 Pure Implicational Logics 49
5.2.4 Two Logics with the Same Algebraization 54
5.2.5 Intuitionistic Propositional Logic without Implication . 56
5.2.6 Equivalential Logic 56
iv
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