Fuzzy Logics Arising from Strict De Morgan Systems

In the application of fuzzy logic to expert systems, fuzzy control, and the like, it is not a single logic that is used, but a plethora of distinct logics. The choice used in a speci…c application is often ad hoc— decided on the basis of empirical factors or mere whim. There is a technical commonality to these logics in that they all arise in the same manner through the speci…cation of an algebra of truth values— that is, a set of truth values equipped with algebraic operations corresponding to each of the logical connectives in question. In our paper [9] we described in detail the technical setup common to these logics and, using tools from universal algebra, we showed how one can use algebra to answer many of the central questions about the logics thus obtained. This chapter is a continuation of the work started there. Using our results, it is our hope that the users of fuzzy logics will have the option of making their choices of fuzzy logic based on properties of the logics and that they will have tools available to use these logics more e¢ ciently. In this chapter, after reviewing some of the results and de…nitions from [9], we apply our methods to the study of fuzzy logics arising from truth value algebras on the unit interval that are strict De Morgan systems. This class of logics includes the ones in which the fuzzy conjunction is given by the usual multiplication of real numbers, a common choice in applications. In [9] we mainly considered applications to the original fuzzy logic introduced by Zadeh and interval-valued fuzzy logic. For these logics the conjunction is idempotent and we showed that in the idempotent setting, these two logics and classical Boolean logic are the only possibilities. As we will see here things are much more unwieldy once we drop idempotence. The main question we will address is when two choices of truth value algebras within the class of strict De Morgan systems yield the same logic or even comparable