The X-mu representation of fuzzy sets
Fuzzy sets are a good model of the flexible definitions used in human language, but do not always give results in accordance with human reasoning. One reason is that operations on fuzzy sets do not obey all identities from crisp set algebra, such as the law of the excluded middle. Additionally, when used to represent fuzzy numbers they lead to a generalised interval arithmetic rather than a fuzzy version of standard arithmetic. In this paper, we outline the \(X\hbox {-}\mu \) approach, a new method of representing, visualizing and calculating functions of fuzzy values. This fuzzy representation retains standard Boolean operations and leads to standard arithmetic when applied to numerical quantities. It includes the idea of strict fuzzy values, which do not incorporate set or interval-based uncertainty. Using simple examples, we illustrate the \(X\hbox {-}\mu \) approach and outline the notion of a strict fuzzy value.
fuzzy numbers strict fuzzy values \(x\hbox {-}mu\) representation fuzzy excluded middle gradual elements