Characteristic Formulas Over Intermediate Logics

Abstract

We expand the notion of characteristic formula to infinite finitely presented subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presented subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of countable algebras, while they are not axiomatizable by standard Yankov (Jankov) formulas. We also give the examples of intermediate logics that are not axiomatizable by characteristic formulas of infinite algebras. Further, using the Godel–McKinsey–Tarski translation, we extend these results to the varieties of interior algebras and normal extensions of S4. For this, using Maksimova’s Translation Lemma, we show that a finite presentation of a given Heyting algebra can be extended to its modal span. So, Maksimova’s Lemma allows us to extend the properties established for the finitely presented Heyting algebras to interior algebras.