The lattice of modal logics: an algebraic investigation

Abstract

Abstract Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jónsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → P or containing an axiom of the form □ m p ↔□ m +1 p for some natural number m is . Furthermore, we show that there exists an immediate predecessor of classical logic (axiomatized by p ↔□ p ) which is not characterized by any finite algebra. The existence of modal logics having immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra.