Vague partition, fuzzy sets and their axiomatical foundations

TL;DR

This paper develops an axiomatic foundation for fuzzy sets and vague phenomena, introducing new notions like vague partitions and redefining fuzzy sets through an axiomatic approach based on membership degrees.

Contribution

It establishes a formal axiomatic system for membership degrees and redefines fuzzy sets via vague partitions, providing a comprehensive mathematical model for vagueness.

Findings

Defines vague variables and partitions with useful properties

Redefines fuzzy sets based on the axiomatic system

Provides a mathematical framework for modeling vagueness

Abstract

Based on the in-depth analysis of the nature and features of vague phenomenon, this paper focuses on establishing the axiomatical foundation of the membership degree theory for vague phenomenon, presents an axiomatic system of governing membership degrees and their interconnections. Some important basic notions, such as vague variable, vague partition etc. are defined, their useful properties are characterized. Moreover, the notion of fuzzy set is also redefined by the notion of vague partition on the basis of the axiomatic system. Hence, this work can serve as a mathematical model of dealing with the phenomena of vagueness by axiomatical approach from the many-valued point of view, as well as the axiomatical foundation of fuzzy sets and its applications. The thesis defended in this paper is that the difference among vague attributes is the key point to recognize and model vague phenomena, membership degrees should be considered in a vague membership space. In other words, vagueness should be treated from a global and overall (instead of local) of view.