半格模态逻辑以及格的模态公理化新方法

Modal Logics over Semi-lattices and Lattices with Alternative Axiomatization

本文在《Modal Logic over Lattices》的基础工作之上,进一步探索了模态逻辑与格理论之间的关系。在之前的研究中,使用带二元模态词<sup>,<inf>的多元混合逻辑通过标准克里普克语义讨论格结构。本文将讨论如何使用模态逻辑刻画下半格结构。为了刻画下半格,本文使用了带有一元模态词P和二元模态词<inf>的多元混合逻辑语言并给出了半格上的多元混合逻辑的完整公理化。在已有的相关结果中,格的定义主要基于偏序关系。在本文的后半部分,提出了一种更符合代数视角的格的替代定义,并给出了相应的模态公理化结果。

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.