Discrete and Topological Correspondence Theory for Modal Meet­Implication Logic and Modal Meet­Semilattice Logic in Filter Semantics

Studies in Logic, Vol. 18, No. 3 (2025): 25–66                        PII: 1674­3202(2025)­03­0025­42

Fei Liang, Zhiguang Zhao

Abstract. In the present paper, we give a systematic study of the discrete correspondence theory and topological correspondence theory of modal meet­implication logic and modal meetsemilattice logic, in the semantics provided in [21]. The special features of the present paper include the following three points: the first one is that the semantic structure used is based on a semilattice rather than an ordinary partial order; the second one is that the propositional variables are interpreted as filters rather than upsets, and the nominals, which are the “first­order” counterparts of propositional variables, are interpreted as principal filters rather than principal upsets; the third one is that in topological correspondence theory, the collection of admissible valuations is not closed under taking disjunction, which makes the proof of the topological Ackermann lemma different from existing settings.