A henkin-style proof of completeness for first-order algebraizable logics

This paper considers Henkin’s proof of completeness of classical first-order logic and extends its scope to the realm of algebraizable logics in the sense of Blok and Pigozzi. Given a propositional logic L (for which we only need to assume that it has an algebraic semantics L and a suitable disjunction) we axiomatize two natural first-order extensions L∀m and L∀ and prove that the former is complete with respect to all models over algebras from,L while the latter is complete with respect to all models over relatively finitely subdirectly irreducible algebras. While the first completeness result is relatively straightforward, the second requires non-trivial modifications of Henkin’s proof by making use of the disjunction connective. As a byproduct, we also obtain a form of Skolemization provided that the algebraic semantics admits regular completions. The relatively modest assumptions on the propositional side allow for a wide generalization of previous approaches by Rasiowa, Sikorski, Hájek, Horn, and others and help to illuminate the “essentially first-order” steps in the classical Henkin’s proof.