Finite-level modal logics

Abstract

We shall say that a logic ~£~ belongs to level ~ where O<~<~, if in ~ there holds i~formula ~ but not formula 6n_i ; M£~ o if ~0 o The logics of ~ ~ shall be called the finite-level logics; 6w-d~Uw6 ~ Since in S# the formulas ~---6n+ , are deducible for each ~ , each logic MeJ~ belongs to exactly one of the levels ~n ' g~w. This proposed classification of modal logics is closely related to be classification of hyperintuitionistic logics proposed by Hosoi [6]. As was shown in [2] there is a homomorphism ~ of the lattice J~ onto the lattice of hyperintuitionistic logics. For g-$4 .... a modal logic ME~ belongs to level ~ if and only if the hyperintuitionistic logic ff(~) is a logic of level g in Hosoi's sense [6]. A. V. Kuznetsov [i] proved that all finitelevel hyperintuitionistic logics are locally tabular (i.e., the sets of pseudo-Boolean algebras corresponding to these logics are locally finite). Our basic result in this paper is in some sense a generalization of Kuznetsov's result. A modal logic M is locally tabular if the set of topological Boolean algebras (TBAs) in which all formulas of M are true is locally finite, i.e., every finitely generated algebra in this set is finite. In other words, M is locally tabular if for every n the number of pairwise inequivalent (in M ) formulas in the variables ~,...,~ is finite. (Formulas ~ and # are considered equivalent in M if