Pre-filtrations, Pre-stable Canonical Rules, and the Kuznetsov-Muravitsky Isomorphism

TL;DR

This work introduces pre-filtration and pre-stable canonical rules to study the Kuznetsov–Muravitsky system $KM$ and its connections to $GL$ and related modal logics. By exploiting dualities between modal Heyting algebras and order-topological spaces, the authors obtain new proofs of the Kuznetsov–Muravitsky isomorphism and of Esakia-type theorems, while also deriving preservation results (e.g., Kripke completeness and finite model property) via monomodal companions and translations. The key technical innovations are the pre-filtration construction, which overcomes failures of standard filtration for KM and GL, and the framework of pre-stable canonical rules, which encode truth-functional structure on the full algebra while localizing non-truth-functional content to finite domains. Together, these methods yield a unified, algebraic approach to proving isomorphisms and preservation results, with potential extensions to Magari algebras and other intuitionistic modal systems. The results deepen understanding of KM, its normal extensions, and their algebraic/relational semantics, offering a versatile toolkit for future explorations of pre-stable logics and monomodal companions.

Abstract

We introduce pre-filtration and pre-stable canonical rules for the Kuznetsov-Muravitsky system of intuitionistic modal logic and provide a new proof of the Kuznetsov-Muravitsky isomorphism, along with several preservation results. The proofs employ these rules and a duality between modal (Heyting) algebras and their corresponding order-topological spaces.

Figures (2)

• Figure 1: The $\mathtt{GL}$-space $\mathfrak{X}$
• Figure 2: The $\mathtt{KM}$-space $\mathfrak{Y}$

Theorems & Definitions (101)

• Proposition 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof