On a set theory with uncertain membership relations Shunsuke Yatabe Yuzuru Kakuda Makoto Kikuchi Department of Computer and Systems Engineering, Kobe University, Rokko dai 1-1, Nada, Kobe 657-8501, Japan Abstract. 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. 1 Introduction Uncertainty appears on propositions’ truth values in logic. There are two standpoints for rep- resenting uncertainty of proposition P . One is that the uncertainty is represented by the truth values of P . On this standpoint, uncertain is a truth value other than true and false. It is concerned what logical reasoning rule is essential for uncertain propositions. The rules are analyzed without changing the language; many existing uncertainty studies are based on this standpoint. 3-valued logic and fuzzy logic are well-known. The other is that the uncertainty is represented by introducing a new symbol which insists that ”P is uncertain”. On this stand- point, it is concerned how we expand our language to represent such uncertainty without changing logical rules; it is analyzed that what kind of symbol is demanded. There are many non-logical studies of uncertainty on this standpoint, but few logical studies; it might be re- garded as just an application of classical logic. This paper is devoted to a logical study on uncertainty on this standpoint. We propose an extension of set theory in classical logic by in- troducing new predicate symbols ∈ + , ∈ - for uncertain membership relations. They represent a partial information about the membership relation ∈, and a formal language for membership relations on models in Kleene’s 3-valued logic can be defined by these symbols. So we can say ZF ± links two standpoints, uncertainty representation by changing logical rules to one by expanding the language. And we show modal operator for any formulae can be defined within ZF ± and an extended separation axiom. We can represent knowledge and information widely in set theory. Almost all the math- ematical arguments can be formulated with the notion of set formally. The notion of mem- bership relation ∈ between sets is essential as primitive predicate. However we can’t repre- sent uncertainty in set theory; traditionally mathematical truth has been regarded as certain property. Some ways of extending set theory have been thought to represent uncertainty by changing logical rules. One is a theory based on multi-valued logic, whose truth values can