模态逻辑的集合论语义与互模拟不变性

含有命题变元的非良基集合能够被看作解释模态语言的模型。任给非良基集合a,一个命题变元p在a上真当且仅当p属于a。命题联结词的解释与古典命题逻辑相同。一个公式3A在a上真当且仅当存在集合b属于a,使得A在b上是真的。在一个集合中,属于关系被看作可及关系。在这种思想下,我们可以定义从模态语言到一阶集合论语言的标准翻译。对任意模态公式A和集合变元x,可以递归定义一阶集合论语言的公式ST(A,x)。在关系语义学下,van Benthem刻画定理是说,在带有唯一的二元关系符号R的一阶语言中,任何一阶公式等价于某个模态公式的标准翻译当且仅当这个一阶公式在互模拟下保持不变。因此,模态语言是该一阶关系语言的互模拟不变片段。同样,我们可以在集合上定义互模拟关系,证明van Benthem刻画定理对于集合论语义和集合上的互模拟不变片段成立,即模态语言是一阶集合论语言的集合互模拟不变片段。

Set-theoretic Semantics for Modal Logic and Invariance of Bisimulation

A non-well founded set with propositonal variables can be regarded as a model for interpreting modal languages.For any non-well founded set a,a propositonal variable p is true on a if and only if p belongs to a.The explanation of propositional conjunctions is the same as that of classical propositional logic.A formula 3 A is true on a if and only if there is a set b belonging to a,so that a is true on b.In a set,belonging relation is regarded as accessible relation.Under this kind of thinking,we can define the standard translation from modal language to first-order set theory language.For any modal formula A and set variable x,the formula ST(A,x)of the first-order set theory language can be recursively defined.In relational semantics,van Benthem’s characterization theorem states that in a first-order language with a unique binary relational symbol R,any first-order formula is equivalent to the standard translation of a modal formula if and only if this first-order formula remains unchanged under bisimulation.Therefore,modal language is an invariant segment of the first-order relational language.Similarly,we can define the bisimulation relationship on sets,and prove that van Benthem’s characterization theorem holds for the set theory semantics and the constant segments of bisimulation on sets,that is,the modal language is bisimulation invariant segments of the first-order set theory language.