Ordinal decomposability and fuzzy connectives

The benefit of computing with linguistic terms is now generally accepted. Fuzzy set theory provides us the conceptual tool for the interpretation and evaluation of linguistic concepts and expressions. It constitutes a quantification of the compatibility degree of objects with the associated linguistic concept through a membership function. When we make computation using fuzzy membership values such as in the evaluation of fuzzy rules confidence, the implicit assumptions are that the membership values have quantitative semantics (the extensive scale assumption) and that the numeric values are commensurate among the different fuzzy sets generated by the different concepts involved (the common scale assumption). In most situations these assumptions are difficult to justify and may lead to various anomalies. The membership values are more suitably interpreted only as ordinal scales where the numeric representations reflect compatibility orderings. In this paper, we examine the concept of fuzzy intersection and union from the perspective of decomposability and ordinal conjoint structure in measurement theory. We determine conditions under which a weak order, induced by a fuzzy set or otherwise, can be decomposed into other weak orders. We show particular cases of ordinal decomposability which correspond naturally to our concept of fuzzy intersection and union. This perspective of fuzzy connectives help us resolve some of the difficulties related to the above assumptions. (C) 2002 Elsevier Science B.V. All rights reserved.