Stable Canonical Rules for Intuitionistic Modal Logics

TL;DR

This paper introduces stable canonical rules for intuitionistic modal logics, providing new axiomatization methods, alternative proofs for key theorems, and confirming the Dummett-Lemmon conjecture in this context.

Contribution

It establishes that all intuitionistic modal multi-conclusion consequence relations are axiomatizable by stable canonical rules and extends key theorems and conjectures in the field.

Findings

Proved axiomatizability of all intuitionistic modal multi-conclusion consequence relations.

Provided an alternative proof of the Blok-Esakia theorem for intuitionistic modal logics.

Confirmed the Dummett-Lemmon conjecture for intuitionistic modal multi-conclusion consequence relations.

Abstract

This paper develops stable canonical rules for intuitionistic modal logics, which were first introduced for superintuitionistic logics and transitive nor mal modal logics in [1] and [2] respectively. We first prove that every in tuitionistic modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. This allows us to assume, without loss of generality, that rules considered by us are stable canonical ones. The idea turns out to be useful. In particular, using stable canonical rules, we get an alterna tive proof of the Blok-Esakia theorem for intuitionistic modal logics which was first proved in [3] and generalize it to multi-conclusion consequence re lations. We also prove the Dummett-Lemmon conjecture for intuitionistic modal multi-conclusion consequence relations, which, as far as we know, is a new result.