On a Set Theory With Uncertain Membership Relations

We logically model uncertainty by expanding language without changing logical reasoning rules. We expand the language of set theory by adding new predicate symbols, uncertain membership relations ∈+ and ∈−. We define the set theory ZF± as an extension of ZF with new symbols in classical logic. In this system we can represent uncertainty which is naturally represented in the model of 3-valued logic. We also show modal operator for formulae written in the language of set theory can be defined by using these new predicates and extended separation axiom.