Chopping More Finely: Finite Countermodels in Modal Logic via the Subdivision Construction

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.