# Applying Algebraic Logic; a General Methodology

Connections between Algebraic Logic and (ordinary) Logic. Algebraic counterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g., L admits a strongly complete Hilbert-style inference system iff Alg(L) is a finitely axiomatizable quasi-variety. Similarly, L is compact iff Alg(L) is closed under taking ultraproducts; L has the Craig interpolation property iff Alg(L) has the amalgamation property, etc.)

## Contents

1. Introduction . . . . 2
2. General framework for studying logics . . . . 4
1. Defining the framework . . . . . 4
2. Distinguished logics . . . . 11
3. Answers/solutions for important and hard exercises of section 2.2 . . . . 32
3. Bridge between the world of logics and the world of algebras . . . . 34
1. Fine-tuning the framework . . . . 34
2. Algebraic characterizations of completeness and compactness properties via Alg_m and Alg_|= (main theorems) . . . . 40
4. Generalizations. . . . . 59
5. Appendix A: New kinds of logics . . . . 63
6. Appendix B: Further equivalence results . . . . 67
7. References . . . . 71