On Modal Logics of Subset Spaces

Studies in Logic, Vol. 18, No. 3 (2025): 1–24                        PII: 1674-3202(2025)-03-0001-24

Shengyang Zhong

Abstract. In modal logic, topological semantics is an intuitive and natural special case of neighbourhood semantics. This paper stems from the observation that the satisfaction relation of topological semantics applies to subset spaces which are more general than topological spaces. The minimal modal logic which is strongly sound and complete with respect to the class of subset spaces is found. Soundness and completeness results of some famous modal logics (e.g. S4, S5 and Tr) with respect to various important classes of subset spaces (e.g. intersection structures and complete fields of sets) are also proved. In the meantime, some known results, e.g. the soundness and completeness of Tr with respect to the class of discrete topological spaces, are proved directly using some modifications of the method of canonical model, without a detour via neighbourhood semantics or relational semantics.