Generalized Fuzzy Logic Systems and Generalized Defuzzification Operators

The applications of fuzzy logic systems in control require defuzzified output. These applications can be seen as approximation problems. We revisit the problem of fuzzy logic systems in the framework of metric spaces and show that most results do not rely on specific properties of Rn or of the fuzzy logic. Definitions of the defuzzification, grounded exclusively in the properties in metric spaces, are introduced and applied to singletons. These definitions do not require measure functions on the space. Therefore, approximation foundations can be built for fuzzy systems on metric spaces, with singleton outputs, without the need of a measure function defined on the output metric space.