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

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.