The theory and applications of generalized complex fuzzy propositional logic

The current definition of complex fuzzy logic has two limitations. First, the derivation uses complex fuzzy relations; hence, it assumes the existence of complex fuzzy sets. Second, current theory is based on a restricted polar representation of complex fuzzy proposition, where only one component of a complex fuzzy proposition carries fuzzy information. In this chapter we present a novel form of complex fuzzy logic. The new theory, referred to as generalized complex fuzzy logic, overcomes the limitations of the current theory and provides several advantages. First, the derivation of the new theory is based on axiomatic approach and does not assume the existence of complex fuzzy sets or complex fuzzy classes. Second, the new form supports Cartesian and polar representation of complex logical propositions with two components of fuzzy information. Hence, the new form significantly improves the expressive power and inference capability of complex fuzzy logic. Finally, the new form is compatible with (yet independent of) the definition of complex fuzzy classes; thereby providing further improvement in the expressive power and inference capability. The chapter surveys the current state of complex fuzzy sets, complex fuzzy classes, and complex fuzzy logic; and provides a new and generalized complex fuzzy propositional logic theory. The new theory has potential for usage in advanced complex fuzzy logic systems and latent for extension into multidimensional fuzzy propositional and predicate logic. Moreover, it can be used for inference with type 2 (or higher) fuzzy sets. Furthermore, the introduction of complex logic can be used for analysis of periodic temporal fuzzy processes where the period is fuzzy.