Algebraizable logics - Blok, W. J., 1947- (Memoirs of the American Mathematical Society, 1989)
The main result of the paper is an intrinsic characterization of algebraizability in terms of the Leibniz operator [capital Greek]Omega, which associates with each theory [italic]T of a given deductive system [script]S a congruence relation [capital Greek]Omega[italic]T on the formula algebra. [capital Greek]Omega[italic]T identifies all formulas that cannot be distinguished from one another, on the basis of [italic]T, by any property expressible in the language of [script]S. The characterization theorem states that a deductive system [script]S is algebraizable if and only if [capital Greek]Omega is one-to-one and order-preserving on the lattice of [script]S-theories, and in addition preserves directed unions. Several other characteristics are given. The results and concepts are illustrated by a large number of examples from modal and intuitionistic logic, relevance logic, and classical predicate logic.