Filter distributive logics
A deductive system C (or sentential logic) on the algebra S of formulas is filter distributive if the lattice of C-filters of A is distributive for all algebras A similar to S. (F is a C-filter of A if (A,F) is a matrix validating C.) \(Th(C)=\{X\subseteq S /\) \(C(X)=X\}\). Theorem II.5 gives a syntactic characterization of finite logics C for which the lattice \(<Th(C),\subseteq >\) is distributive. (A logic C is called finite if the operator C is algebraic.) Moreover, for C finite and equivalential, this is shown to be equivalent to the filter distributivity of C. (Equivalential logics, a concept similar to but weaker than Block and Pigozzi's ''algebraizable'' logics, were studied by the Author in Stud. Logica 40, 227-236, 355-372 (1981; Zbl 0476.03032 and Zbl 0492.03008).) A C-filter F is C-prime if it is finitely meet irreducible and a matrix (A,D) for C is C-prime if D is a C-prime filter. The Author investigates the condition that the class of C-prime matrices for C be axiomatizable (in the usual first order sense): Theorem V.2 states that if this condition holds for a finite filter distributive logic C, then any axiomatic strengthening C' of C whose class of C'-prime matrices is still axiomatizable, is obtained from C by adding a finite set of axioms. This and other results are on a pair with Baker's theorem on the finite basis for congruence distributive equational classes.
syntactic characterization of finite logics
congruence distributive equational classes
scientific article; zbMATH DE number 3924755
scientific article