Hereditarily Structurally Complete Intermediate Logics: Citkin’s Theorem Via Duality

Abstract

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics.

Uniform Interpolation and Admissible Rules

“…Closely connected are hereditarily structurally complete logics that are logics for which each extension is structurally complete. Citkin (1978) provides a characterization of such intermediate logics and Bezhanishvili and Moraschini (2022) give an alternative proof of this result via duality theory. Rybakov (1995) gives a characterization of hereditarily structurally complete transitive modal logics.…”