Do exact shapes of fuzzy sets matter?

Abstract

An often raised objection to fuzzy logic can be articulated as follows. Fuzzy sets are intended to represent vague collections of objects. On the one hand, boundaries of such collections are inherently imprecise. On the other hand, fuzzy sets are defined by membership degrees which are precise. There is, therefore, an obvious discrepancy between the imprecision of a collection and the precision of a fuzzy set representing the collection. The objection leads to a question posed in the title of this paper: do exact shapes of fuzzy sets matter? That is, do the exact values of membership degrees of fuzzy sets matter? This is a fundamental question, particularly when fuzzy sets are supplied as an input to further processing. This paper presents a particular answer to this question. We focus on the case when the fuzzy sets in question are part of the input I from which the output O is produced in a way which can be described by logical formulas. This case covers several widely used fuzzy logical models including extension principle, products of fuzzy relations, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. We present formulas which say how similar is the original output O to a new output O′ when the original input I is replaced by a new input I′. The presented formulas provide us with an analytical tool which enables us to answer the question of whether (and to what extent) the exact shapes of fuzzy sets in a particular fuzzy logical model matter. We present application of our formulas to selected topics including the above-mentioned extension principle, fuzzy relational products, Zadeh's compositional rule of inference, fuzzy automata, and properties of fuzzy relations. Based on the analysis, we argue that in several applications of fuzzy logic, the exact shapes of fuzzy sets do not really matter.