Fuzzy Scales and Measurement

Abstract

I have already mentioned that a weakness in fuzzy set theory has been its lack of a reasonable basis in measurement. While mathematical contributors to the theory (e.g. Sugeno 1974) have attempted to ground fuzzy sets in axiomatics or some more general formalism (e.g. category theory), the exact relationship between fuzzy sets and various measurement problems remains incompletely specified. Dubois and Prade’s (1980) otherwise excellent monograph relegates the problem of empirically determining membership functions to a subsection titled “Where do ‘they’ come from?”, which consists of little more than a catalog of different devices researchers have used to assign membership values. Since 1980 there have been sporadic attempts to embed the construction of membership functions in various traditional psychometric approaches to measurement and scaling, and these will be critically discussed in section 3.1. Some attention has also been given this problem from cognitive anthropologists (cf. Kempton 1984), and of course the current work in prototype theory ultimately may contribute some insights.