基于n值关系语义的命题模态逻辑系统研究  

Research on a Propositional Modal Logic System Based on n-Valued Relational Semantics

作　　者：周张泉 杨成彪 刘军 ZHOU Zhang-quan;YANG Cheng-biao;LIU Jun(School of Command and Control Engineering,Army Engineering University of PLA,Nanjing 210000,China;School of Computer Science and Engineering,Southeast University,Nanjing 210000,China;School of Information Science and Engineering,Nanjing Audit University Jinshen College,Nanjing 210000,China)

机构地区：[1]陆军工程大学指挥控制工程学院,江苏南京210000 [2]东南大学计算机科学与工程学院,江苏南京210000 [3]南京审计大学金审学院信息科学与工程学院,江苏南京210000

出　　处：《计算机技术与发展》2024年第2期71-77,共7页Computer Technology and Development

基　　金：江苏省高等学校基础科学(自然科学)研究面上项目A类(22KJB520003)。

摘　　要：传统的多值模态逻辑系统将关系语义中的状态及状态间的关系进行了多值化处理。然而,实际应用中状态间的关系往往是确定的,无需多值化。针对这种情况,基于?ukasiewicz代数系统提出了一种新的命题模态逻辑n值关系语义。在所提出的n值关系语义中,针对状态进行了多值化处理,同时保持了状态间关系的确定性。通过对逻辑公式的形式化定义以及可满足性和有效性的分析,证明了n值关系语义下经典命题模态逻辑系统K,T,S4和S5的正确性。进一步地,给出了极大一致集与典范模型在n值关系语义下的定义,并完成了上述经典命题模态逻辑系统的完备性证明。上述结论表明基于n值关系语义的命题模态逻辑系统能够涵盖并捕捉到经典逻辑系统中的所有有效命题。综上所述,所提出的基于?ukasiewicz代数系统的n值关系语义提供了一种在实际应用中处理多值状态及确定的状态间关系的方法。这种方法在扩展命题模态逻辑系统的形式化定义与关系语义是可行且有效的。Traditional multi-valued modal logic systems are established by mapping both of states and relations to multi-valued spaces in terms of their relational semantics.However,in practical applications,the relations between states are often deterministic and do not require multi-valuation.To address this issue,a new kind of n-valued relational semantics for propositional modal logic,based on ukasiewicz algebra system,has been proposed.In the proposed n-valued relational semantics,states are treated with multi-valuation while maintaining the determinism of relations between states.By formalizing the logical formulas and analyzing their satisfiability and validity,the correctness of the classical propositional modal logic systems K,T,S4,and S5 under the n-valued relational semantics has been demonstrated.Furthermore,the definitions of maximal uniform sets and canonical models have been provided under the n-valued relational semantics,along with the completeness proofs of the aforementioned classical propositional modal logic systems.This implies that the propositional modal logic system based on n-valued relational semantics can cover and capture all valid propositions in these classical systems.In conclusion,the n-valued relational semantics based on ukasiewicz algebra system offers a method to handle deterministic relations between multi-valued states in practical applications.This method has shown feasibility and effectiveness in expanding the formal definition and relational semantics of propositional modal logic systems.

关 键 词：模态逻辑 多值逻辑 关系语义 ?ukasiewicz系统 正确性和完备性 

分 类 号：TP301.2[自动化与计算机技术—计算机系统结构]