Matrices, primitive satisfaction and finitely based logics
\textit{K. A. Baker}'s primitive satisfaction theorem (stated for congruence-distributive equational classes of algebras [cf. Trans. Am. Math. Soc. 190, 125-150 (1974; Zbl 0291.08001)]) is modified and reformulated so as to apply to a class of logical matrices Matr(C) such that the lattice of C-filters is distributive. A matrix version of Jónsson's lemma is a corollary to this theorem. Then some properties of propositional logics with disjunction are studied. The main result states that a standard finitely equivalent logic C with disjunction can be axiomatized by a finite number of rule schemata if and only if the class of finitely subdirectly irreducible matrices in the class Matr(C) is finitely (first-order) axiomatizable.
scientific article; zbMATH DE number 3966040