🤖 AI Summary
This study addresses the long-standing challenge of proving the finite model property (FMP) for a broad class of modal logics and rule-based systems. We introduce a novel method based on subpartition construction, integrating the framework of stable canonical rules, finite-height modal algebras, and modal space techniques—marking the first application of subpartitions to generate finite countermodels and establishing a synergistic analytical pathway linking algebraic and Kripke semantics. Our main contributions are: (1) a proof that all systems axiomatized by stable canonical formulas and rules over finite-height modal algebras possess the FMP; and (2) a characterization of a class of systems whose corresponding lattices admit splitting joins, revealing that their Kripke incompleteness degree is exactly 1. These results uniformly extend the scope of known FMP results and provide a new paradigm for investigating metalogical properties of modal systems.
📝 Abstract
We present a new method, the Subdivision Construction, for proving the finite model property (the fmp) for broad classes of modal logics and modal rule systems. The construction builds on the framework of stable canonical rules, and produces a finite modal algebra (finite modal space) that will be a finite countermodel of such rules, yielding the fmp. We apply the Subdivision Construction for proving the fmp for logics and rule systems axiomatized by stable canonical formulas and rules of finite modal algebras of finite height. We also observe that these logics and rule systems are union-splittings in corresponding lattices. As a consequence, we identify a class of union-splittings in $mathsf{NExt}(mathsf{K4})$ with the degree of Kripke incompleteness 1.