Ordinal decomposability and fuzzy connectives

✍ Scribed by John W.T. Lee

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 255 KB
• Volume: 136
• Category: Article
• ISSN: 0165-0114
• DOI: 10.1016/s0165-0114(02)00270-1

✦ Synopsis

The beneÿt 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 quantiÿcation 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 conÿdence, the implicit assumptions are that the membership values have quantitative semantics (the extensive scale assumption) and that the numeric values are commensurate among the di erent fuzzy sets generated by the di erent concepts involved (the common scale assumption). In most situations these assumptions are di cult to justify and may lead to various anomalies. The membership values are more suitably interpreted only as ordinal scales where the numeric representations re ect 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 di culties related to the above assumptions.