论模态逻辑的集合论语义

On Set-theoretic Semantics for Modal Logic

引入非良基集合可以为模态逻辑提供一种新的语义学。这种语义是在集合上解释模态语言,使用集合中作为元素的集合之间的属于关系解释模态词,并在集合中采用命题变元作为本元,从而解释原子命题的真假。在这种新的语义下,从模型构造的角度看可以引入几种非标准的集合运算:不交并、生成子集合、p-态射、树展开等等,证明模态公式在这些运算下的保持或不变结果。利用这些结果还可以证明一些集合类不是模态可定义的。

Logic and set theory have very strong connections. In the study of logical seman- tics, set theoretic notions are often used. Conversely, sets can also be taken as semantic structures on which logical languages can be interpreted. In this paper, the aim is to analyze modal logic via taking sets as models which are used to calculate truth values of modal formulas. P. Aczel introduced non-well-founded sets which correspond to non- well- founded graphs. Since those graphs often occur in modal logic as frames, we can used non-well-founded sets to cope with them. Given a set a, the truth of a propositional variable p is defined as the membership between p and a. Thus we introduce propositional letters which are taken as urelements, i.e., elements that can form sets but not sets or classes themselves. The modal formula ◇φ is true at a set a if and only if there exists a set b which is a member of a and φ is true at b. In model theory, preservation and definability are fundamental problems. Then some non-standard set-theoretic operations are defined for this purpose. The key notion in these definitions is the concept set transitive closure of a given set. It works like a Kripke model but different. It has no underline frame which is the skeleton of a Kripke model. The main results of the paper are the preservation or invariance results of all modal formulas under constructions of sets. This shows that the basic modal language is a nice language to talk about sets. But the expressive power of modal logic over sets has lim- its. Some classes of sets cannot be defined in basic modal language under set-theoretic semantics. For instance, the class of all finite sets cannot be defined since it violates the disjoint union of set transitive closures. The irreflexivity which says that there are no sets which is a member of itself is also non-definable in basic modal language. This hints us that there are many problems here. We should find out more expressive languages to talk about classes of sets. We also need to compare the set-theoretic semantics and Kripke semantics more deeply so as to find out how we should represent the distinction between frame and model more precisely in set-theoretics semantics.