This paper is just a first approach to the idea that the membership function μP of a fuzzy set labelled P is, basically, a measure on the set of linguistic expressions x is P for each x in the corresponding universe of discourse X. Estimating that the meaning of P (relatively to X) is nothing else than the use of P on X, these measures seem to be reached by generalizing to a preordered set the concept of Fuzzy Measure, introduced by M. Sugeno, when the preorder translates the primary use of the predicate P, that is a basic relationship like x is less P than y. The paper only deals with predicates whose use is made by means of numerical characteristics, but those cases on which the characteristics (if they exist) are not of a numerical nature are not considered. By generalizing DeLuca-Termini's sharpened order some typical membership functions are studied as measures.
@article{urn:eudml:doc:39173,
title = {A reflection on what is a membership function.},
journal = {Mathware and Soft Computing},
volume = {6},
year = {1999},
pages = {201-215},
zbl = {0968.28013},
mrnumber = {MR1774567},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39173}
}
Trillas, Enric; Alsina, Claudi. A reflection on what is a membership function.. Mathware and Soft Computing, Tome 6 (1999) pp. 201-215. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39173/