Modal Logics over Bounded Lattices and Its Variety

Studies in Logic, Vol. 18, No. 6 (2025): 1–18                      PII: 1674­3202(2025)­06­0001­18

Xiaoyang Wang

Abstract. This article extends the foundational work of Wang and Wang on modal logic over lattices. Building upon their framework using polyadic modal logic with binary modalities ⟨sup⟩ and ⟨inf⟩ under standard Kripke semantics to axiomatize lattice structures, we focus on the modal characterization of bounded lattices and their extensions relevant to logical systems. By introducing nullary modalities 1 (maximum element) and 0 (minimum element), we first establish a modal axiomatic system for bounded lattices. Subsequently, we provide pure formula characterizations of complementation and orthocomplementation relations in lattices, along with corresponding completeness results. As key applications, we present modal characterizations of fundamental logical algebraic structures: Boolean algebras, orthomodular lattices, and Heyting algebras. The last section develops novel axiomatization results for atomic lattices and atomless lattices. Throughout this work, all axiomatic systems are shown to be strongly complete via pure­formula extensions, demonstrating how hybrid modal languages with nullary operators can uniformly capture boundary elements, complementation properties, and lattice­theoretic operations central to both classical and non­classical logics.