Algebraic logic, where does it stand today?

Abstract

This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject.