Stable Canonical Rules and Formulas for Pre-transitive Logics via Definable Filtration

TL;DR

This paper extends the theory of stable canonical rules using definable filtration, providing axiomatizations, algebraic characterizations, and new stable logics for pre-transitive modal systems.

Contribution

It generalizes stable canonical rules with definable filtration, applies this to pre-transitive logics, and introduces m-stable canonical formulas for enhanced axiomatizations.

Findings

Axiomatization of extensions via stable canonical rules

FMP of K4^{m+1}_1-stable logics

Existence of continuum many stable logics beyond K4 and subframe logics

Abstract

We generalize the theory of stable canonical rules by adopting definable filtration, a generalization of the method of filtration. We show that for a modal rule system or a modal logic that admits definable filtration, each extension is axiomatizable by stable canonical rules. Moreover, we provide an algebraic presentation of Gabbay's filtration and generalize stable canonical formulas and the axiomatization results via stable canonical formulas for $\mathsf{K4}$ to pre-transitive logics $\mathsf{K4^{m+1}_{1}} = \mathsf{K} + \Diamond^{m+1} p \to \Diamond p$ $(m \geq 1)$. As consequences, we obtain the fmp of $\mathsf{K4^{m+1}_{1}}$-stable logics and a characterization of splitting and union-splitting logics in the lattice $\mathsf{NExt}\mathsf{K4^{m+1}_{1}}$. There are continuum many $\mathsf{K4^{m+1}_{1}}$-stable logics that are neither $\mathsf{K4}$-stable logics nor subframe logics. Finally, we introduce $m$-stable canonical formulas, strengthening the axiomatization results for these logics.