Modern Set

In this paper, we intend to generalize the classical set theory as much as possible. we will do this by freeing sets from the regular properties of classical sets; e.g., the law of excluded middle, the law of non-contradiction, the distributive law, the commutative law,etc.... The fuzzy set theory succeeded in freeing sets from the law of excluded middle and the law of contradiction. However, in order to extend our language, it is more or less unreasonable to keep the commutative law and the distributive law. This modern idea of sets keeps the concept of membership functions but their value are not necessarily in [0,1]; nor do these modern sets form a lattice necessarily. Especially noteworthy is that modern sets are more general than generalized fuzzy set, (Please refer to Nakajima [6]). Here is the hierarchy of generality: Modern sets ≥ L-fuzzy set ≥ Generalized Fuzzy set ≥ fuzzy set ≥ classical set. These modern sets become the classical sets under the restriction to Ox, Ix for all x in X, and no further conditions are required The world of natural languages can not be ruled by a single classical logic because the real physical world consists of many aspects, each of which obeys a different logic. As is well-known, the distributive law does not always hold in every logic system. For example, in classical and intuitionistic logic it holds, but in quantum logic it fails. The Commutative law seems invalid when we talk about anything with temporal order in our daily conversation since every event in time is not reversible. ”He is a student and a male”. I can also say, ”He is a male and a student” but mean the same thing. The two significations commute of course. ”I have a health insurance and a car insurance”. In this case, the two significations commute. But how about this case ”He went to a supermarket and then a drug store”. In this case, the two significations do not commute anymore since there is temporal order. Furthermore, we have no expression for ”He went to a supermarket and then a drug store” in classical logic or fuzzy logic. In other words, the properties from classical logic, which we take for granted, do not express our thoughts that well.