A New Look at Type-2 Fuzzy Sets and Type-2 Fuzzy Logic Systems

We propose the concept of a conditional fuzzy set and prove that a type-2 fuzzy set is equivalent to a conditional fuzzy set. We show that both the conditional fuzzy sets and the type-2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables, and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other, whereas in the type-2 fuzzy set framework, the secondary variable depends on the primary variable by definition. It is this dependence between the primary and secondary variables that makes the type-2 fuzzy sets a complex subject, while the conditional fuzzy sets do not have this built-in dependence and, thus, are much easier to analyze. With the fuzzy relation formulation, powerful tools in fuzzy set theory such as Zadeh's compositional rule of inference can be used to obtain the marginal fuzzy sets of the type-2 and conditional fuzzy sets, transforming the type-2 problems back to the conventional type-1 domain. With the help of the marginal fuzzy set concept, we show that a type-2 fuzzy logic system can be designed in the same way as designing a type-1 fuzzy logic system.