具有模态词□φ=□_(1φ)∨□_(2φ)且可靠与完备的公理系统

Sound and Complete Axiomatic System with a Modality □φ=□_(1φ)∨□_(2φ)

提出具有模态词□φ=□_(1φ)∨□_(2φ)的命题模态逻辑,给出其语言、语法与语义,其公理化系统是可靠与完备的,其中,□_1与□_2是给定的模态词.该逻辑的公理化系统具有与公理系统S5相似的语言,但具有不同的语法与语义.对于任意的公式φ,□φ=□_(1φ)∨□_(2φ);框架定义为三元组<W,R_1,R_2>,模型定义为四元组<W,R_1,R_2>;在完备性定理证明过程中,需要在由所有极大协调集所构成的集合上构造出两个等价关系,其典型模型的构建方法与经典典型模型的构建方法不同.如果□_1的可达关系R_1等于□_2的可达关系R_2,那么该逻辑的公理化系统变成S5.

This paper proposes a propositional modal logic with a modality □φ=□1V□2φ, and specifies the language, the syntax and the semantics for the logic. The axiomatic system for □ is sound and complete, where □1 and □2 are given in this paper. The axiomatic system for the logic has the similar language, but has the different syntax and semantics. For any formula φ, □φ=□1V□2φ; the frame for the axiomatic system is defined as an tripleW,R1,R2, and the model is defined as quadruple W,R1,R2,I. When the completeness theorem is proved, two equivalence relations are constructed on the set that is made up of all the maximal consistent sets. The construction method of a canonical model for the axiomatic system is different from the classical canonical model. If the accessibility relation R1 for □1 is the accessibility relation R2 for □2, then the axiomatic system for □ changes into S5.