Blok-Esakia Theorems via Stable Canonical Rules

TL;DR

The paper introduces a new uniform method using stable canonical rules to study modal companions of superintuitionistic rule systems, providing alternative proofs of key theorems and extending results to richer logical signatures.

Contribution

It develops a general approach with stable canonical rules that simplifies proofs of the Blok-Esakia theorem and related conjectures, extending them to bi-superintuitionistic and tense rule systems.

Findings

Alternative proofs of the Blok-Esakia theorem and Dummett-Lemmon conjecture.

Extension of the Blok-Esakia theorem to bi-superintuitionistic and tense systems.

Proof of the Kuznetsov-Muravitsky isomorphism for certain modal rule systems.

Abstract

We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic $\logic{KM}$ and modal rule systems extending the provability logic $\logic{GL}$. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases.