Stable Canonical Rules for Intuitionistic Modal Logics
Cheng Liao
Abstract:
This thesis develops the theory of stable canonical formulas and rules for intuitionistic modal logics and Heyting-Lewis logics. We prove that every intuitionistic modal (or Heyting-Lewis) multi-conclusion consequence relation is axiomatizable by stable canonical rules. This allows us to assume without loss of generality that rules that we consider are stable canonical rules in many cases when we study intuitionistic modal logics and Heyting-Lewis logics, which turns out to be quite useful. In particular, our method gives an alternative proof of the Blok-Esakia theorem for intutionsitic modal logics, and helps us find an error in the proof of that theorem for Heyting-Lewis logics. Besides, using stable canonical rules, we also prove an analogue of the Dummett-Lemmon conjecture for intutionistic modal multi-conclusion consequence relations which states that an intuitionistic modal multi-conclusion consequence relation is Kripke complete if and only if its least modal companion is.