A Semantic Base for the Fuzzy Extension-Principle

Abstract

A distance/semantic is attached to fuzzy acceptabilities. Within the frame of this semantic, a modified version of the extension principle proves to be obligatory. It is derived without referring to fuzzy logic and coincides only  under certain assumptions with the usual version of the extension principle. Over that, it is shown under not very restricting assumptions that membership-functions which are compatible with the above semantic have to be automatically of that triangle- or trapezoid type which is so popular in fuzzy applications.