The X-mu representation of fuzzy sets — Regaining the excluded middle

Fuzzy sets are a good model of the flexible definitions used in human language, but are not always in accordance with human reasoning because they do not satisfy the law of the excluded middle. In this paper, we outline the X-μ approach, a new method of representing, visualizing and calculating functions of fuzzy quantities. Using simple examples, we illustrate that the law of the excluded middle is satisfied with the X-μ approach, although it is not always possible to recover standard membership functions from the results of a calculation.