Toward complex fuzzy logic

Complex fuzzy logic is a postulated logic system that is isomorphic to the complex fuzzy sets recently described in a previous paper. This concept is analogous to the many-valued logics that are isomorphic to type-1 fuzzy sets, commonly known as fuzzy logic. As with fuzzy logics, a complex fuzzy logic would be defined by particular choices of the conjunction, disjunction and complement operators. In this paper, an important assertion from a previous paper, that only the modulus of a complex fuzzy membership should be considered in set theoretic (or logical) operations, is examined. A more general mathematical formulation (the property of rotational invariance) is proposed for this assertion, and the impact of this property on the form of complex fuzzy logic operations is examined. All complex fuzzy logics based on the modulus of a vector are shown to be rotationally invariant. The case of complex fuzzy logics that are not rotationally invariant is examined using the framework of vector logic. A candidate conjunction operator was identified, and the existence of a dual disjunction was proven. Finally, a discussion on the possible applications of complex fuzzy logic focuses on the phenomenon of regularity as a possible fuzzification of stationarity.