A measure-theoretic axiomatization of fuzzy sets

Fuzzy Sets: History and Basic Notions

“…There is both a huge literature outside fuzzy sets in multiple criteria decision-making and a very big amount of tools for criteria aggregation in fuzzy set theory. However there is little work bridging the gap between fuzzy sets and conjoint measurement, while this has been for a long time a natural open question (partially addressed by researchers like Türksen (1991), andBollman-Sdorra et al (1993)). The situation seems to be better with relational methods in preference modeling where there is a very active field on fuzzy relations clearly positioned with respect to the non-fuzzy tradition (Fodor and Roubens, 1994;Slowinski, 1998); iv) Qualitative approaches to reasoning and decision.…”

“…Bollman- Sdorra et al (1993) propose a measurement-theoretic setting whose properties enable the basic fuzzy set system, that is, the membership function, the minimum and maximum operations for intersection and union connectives, to be meaningful. More on measurement issues appear in Chapter 3 of this volume, by Bilgic and Türksen. When fuzzy sets are modelled by complete preordering relations, they can be used as ordinal possibility distributions which are the basis of qualitative possibility theory (e.g.…”

Elicitation of membership functions: how far can theory take us?

“…In such a theory one can discuss the representation of a qualitative structure and the meaningfulness of such a representation (see Figure 2). This approach has been applied to fuzzy membership functions [19,13,14,18,6,2,3,4] which resulted in valuable insights as to what membership functions mean. Specifically, the ordinal nature of the membership function is agreed upon [19,18,6,3] and the min-max calculus is justified in this framework.…”

“…This approach has been applied to fuzzy membership functions [19,13,14,18,6,2,3,4] which resulted in valuable insights as to what membership functions mean. Specifically, the ordinal nature of the membership function is agreed upon [19,18,6,3] and the min-max calculus is justified in this framework. Also, the axioms required to move into "cardinal" (interval and ratio) scales with appropriate triangular norms and conorms are extracted and their suitability to fuzzy set theory (based on different interpretations) is discussed [2,3,4] (see Figure 3).…”

“…Measurement theory provides valuable insights to the meaning of membership functions and this path has been followed by many fuzzy set researchers [19,13,14,18,6,2]. However, measurement theory does not guide us as to how one should elicit membership functions.…”