Implications

Abstract

Fuzzy implications are ageneralization of the classical two-valued implication to the multi-valued setting. As one of the main operations in fuzzy logic, they play avital role both in the theory and applications, such as multivalued mathematical logic, fuzzy logic systems, fuzzy control, approximate reasoning, expert systems, image processing, and data analysis. Here, we focus on nilpotent logical systems and examine two different kinds of implications and the concept of a weak ordering property. Furthermore, we consider both R- and S-implications with respect to the three naturally derived negations from the previous chapter. The formulae and the basic properties of these implications are given which will come in handy when we implement fuzzy logic into neural architecture in Chap. 9.