Distinguishability quantification of fuzzy sets

Distinguishability is a semantic property of fuzzy sets that has a great relevance in the design of interpretable fuzzy models. Distinguishability has been mathematically defined through different measures, which are addressed in this paper. Special emphasis is given to similarity, which exhibits sound theoretical properties but its calculation is usually computationally intensive, and possibility, whose calculation can be very efficient but it does not exhibit the same properties of similarity. It is shown that under mild conditions – usually met in interpretable fuzzy modeling – possibility can be used as a valid measure for assessing distinguishability, thus overcoming the computational inefficiencies of similarity measures. Moreover, procedures that minimize possibility also minimize similarity and, consequently, improve distinguishability. In this sense, the use of possibility is fully justified in interpretable fuzzy modeling.