提出具有模态词□φ=□₁V□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=□₁V□₂φ;框架定义为三元组W,R₁,R₂,模型定义为四元组W,R₁,R₂,I;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R₁等于□₂的可达关系R₂,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.

国家自然科学基金(61363047, 61173063, 60773059, 61035004); 江西省教育厅科技厅项目(GJJ14748); 江西省科技厅项目(20111BBE50008, 2011ZBBE50035, 20112BBE50052)