Modal Logics over Semi-lattices and Lattices with Alternative Axiomatization

Studies in Logic, Vol. 17, No. 3 (2024): 51–73                   PII: 1674­3202(2024)­03­0051­23

Xiaoyang Wang

Abstract. This paper builds on the previous work starting by X. Wang and Y. Wang (2022, 2023) on modal logics over lattices, exploring further the complex relationship between modal logic and lattice theory. In our initial research, we utilized polyadic hybrid logic with binary modalities ⟨sup⟩, ⟨inf⟩ to discuss lattices via standard semantics. This paper introduces a focused examination of meet semi-lattices, structures in which not every pair of elements necessarily has a supremum. To address meet semi­-lattices, it employs the language of polyadic hybrid logic with unary modality P and binary modality ⟨inf⟩. Subsequently, a complete axiomatization of polyadic hybrid logic over semi-­lattices is obtained. In our earlier work, the definition of lattices was primarily based on partial order relations. In the latter part of this paper, an alternative definition of lattices that aligns more with an algebraic perspective is proposed, and the corresponding axiomatic results are provided.