一阶逻辑中逻辑度量空间与理论相容性的拓扑性质  

Topological properties of compatibility theory and logical metric space in first-order logic

作　　者：王前 惠小静[1] 袁一丹 许倩 WANG Qian;HUI Xiaojing;YUAN Yidan;XU Qian(College of Mathematics and Computer Science,Yan’an University,Yan’an 716000,China)

机构地区：[1]延安大学数学与计算机科学学院,陕西延安716000

出　　处：《湖北大学学报(自然科学版)》2025年第2期173-179,共7页Journal of Hubei University:Natural Science

基　　金：国家自然科学基金(12261090、12301456);延安大学研究生教育创新计划项目;延安大学创新训练项目(YCX2023011)资助。

摘　　要：设Φ是全体不含函数符号的一阶闭逻辑公式之集。本研究基于一阶逻辑中的公理化真度理论对逻辑度量空间的结构进行分析,搭建逻辑度量空间与拓扑概念之间的联系,证明逻辑度量空间(Φ,ρ)中没有孤立点,并且度量空间(Φ,ρ)存在着既开又闭集组成的拓扑基。最后论证Φ中一个闭逻辑理论Γ是相容的充要条件。得到结论:Γ是相容的当且仅当Γ在逻辑度量空间中不含内点;Γ是相容的当且仅当Γ在逻辑度量空间中不含非空正则球面。LetΦbe the set consisting of all closed first-order formulas containing no function symbols.Based on the axiomatic truth degree theory in first-order logic,we analysed the structure of logical metric space,constructed the relationship between logical metric space and topological concept,and proved that there was no isolated point in logical metric space(Φ,ρ),and there was a topological basis composed of open and closed sets in metric space(Φ,ρ).Finally,we proved that a closed logic theoryΓinΦwas a necessary and sufficient condition for compatibility.It is concluded thatΓis consistent if and only ifΓhas no interior point in the logical metric space,Γis consistent if and only ifΓdoes not contain a non-empty regular sphere in a logical metric space.

关 键 词：一阶逻辑 逻辑度量空间 零维空间 相容性 

分 类 号：O141.1[理学—数学]