Stable modal logics
Guram Bezhanishvili, Nick Bezhanishvili, Julia Ilin
Abstract:
We develop the theory of stable modal logics, a class of modal logics introduced by Bezhanishvili, Bezhanishvili & Iemhoff (to appear). We give several new characterizations of stable modal logics, and show that there are continuum many such. Since some basic modal systems such as K4 and S4 are not stable, for a modal logic L, we introduce the concept of an L-stable extension of L. We prove that there are continuum many S4-stable modal logics, and continuum many K4-stable modal logics between K4 and S4. We axiomatize K4-stable and S4-stable modal logics by means of stable canonical formulas of Bezhanishvili, Bezhanishvili & Iemhoff (to appear), and discuss the connection between S4-stable modal logics and stable superintuitionistic logics of Bezhanishvili & Bezhanishvili (to appear). We conclude the paper with examples of K4-stable modal logics, and compare K4-stable modal logics to subframe and splitting transitive modal logics.
Keywords: Modal logic, modal consequence relation, canonical formula, canonical rule, intuitionistic logic