Research on a Propositional Modal Logic System Based on n-Valued Relational Semantics
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 onŁukasiewicz 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,then-valued relational semantics based on Łukasiewicz algebra system offers a method to handle deter-ministic 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.
modal logicmulti-valued logicrelational semanticsŁukasiewicz systemcorrectness and completeness
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