Modal logics over lattices

Abstract

Lattice theory has various close connections with modal logic. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logics over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the modal languages of tense logic and polyadic modal logic to talk about lattices via standard Kripke semantics. We first obtain a series of complete axiomatizations of tense logics over lattices, (un)bounded lattices over partial orders or strict orders. In particular, we solve an axiomatization problem left open by Burgess (1984) [8]. The second half of the paper gives a series of complete axiomatizations of polyadic modal logic with nominals over lattices, distributive lattices, and modular lattices, where the binary modalities of infimum and supremum can reveal more structures behind various lattices.