Fuzzy logic and fuzzy set theory

It is known that for every complete Heyting algebra (cHa) f2, we can construct a O-valued universe V ~, which is a model of intuitionistic set theory. The intuitionistic set theory on V ~ is based on an intuitionistic logic, which is determined by the structure of the truth value set f2. The closed unit interval [0, 1] of real numbers is a cHa which is linearly ordered and dense with respect to the order of real numbers. In [6], we presented an intuitionistic logic and intuitionistic set theory on [0, 1 ]-valued universe V t~ 11, which are referred to as intuitionistic fuzzy logic and intuitionistic fuzzy set theory. The intuitionistic fuzzy logic is Gentzen's intuitionistic predicate logic with additional axioms and inference rules, which asserts that the truth value set is linearly ordered and dense cHa. [0, 1] has, not only the structure of linearly ordered dense cHa, but also arithmetical structure of real numbers. Thus, we can define the operations , -i-, and. on [0, 1] by