Stable canonical rules and formulas for pre-transitive logics via definable filtration

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 K4 to pre-transitive logics K4ᵐ⁺¹₁ = K + ᵐ⁺¹ p → p (m ≥ 1). As consequences, we obtain the finite model property of K4ᵐ⁺¹₁-stable logics and a characterization of splitting and union-splitting logics in the lattice NExt K4ᵐ⁺¹₁. There are continuum many K4ᵐ⁺¹₁-stable logics that are neither K4-stable logics nor subframe logics. Finally, we introduce m-stable canonical formulas, strengthening the axiomatization results for these logics.1