On the fundamentals of fuzzy set theory
type: Item
instance of: scholarly article
title: On the fundamentals of fuzzy set theory
zbMATH Open document ID: 0860.03038
author: Ulrich Höhle
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-04-21T00:00:00Z
review text: In the following considerations we restrict our interest to foundation-theoretically oriented aspects of fuzzy set theory (resp. fuzzy logic), especially to their mathematical semantics. We distinguish two mainstreams: \(\bullet\) Plausibility theory (including possibility and probability theory). \(\bullet\) Non-classical model theory. In both cases lattice-valued (resp. \([0,1]\)-valued) maps play a dominant role. The purpose of this paper is twofold: On the one hand, we give for the first time a coherent presentation of some of the most important results of these disciplines; on the other hand, we apply these methods to long-standing open problems of fuzzy set theory. A typical situation is fuzzy control theory. Starting from such fundamental phenomena as indistinguishability and local extent of existence we develop in terms of singletons and many-valued equalities the mathematical semantics of the so-called fuzzy control situation. In particular, we give a mathematically sound solution of the fuzzification and defuzzification problem [\textit{C. C. Lee}, IEEE Trans. Syst. Man Cybern. 20, 404-418, 419-435 (1990; Zbl 0707.93036, Zbl 0707.93037)].
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 94D05
Mathematics Subject Classification ID: 93C42
Mathematics Subject Classification ID: 03C90
Mathematics Subject Classification ID: 94A15
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 930223
zbMATH Keywords: possibility theory
zbMATH Keywords: monoidal logic
zbMATH Keywords: plausibility theory
zbMATH Keywords: non-classical model theory
zbMATH Keywords: fuzzy set
zbMATH Keywords: fuzzy logic
zbMATH Keywords: semantics
zbMATH Keywords: probability theory
zbMATH Keywords: indistinguishability
zbMATH Keywords: local extent of existence
zbMATH Keywords: singletons
zbMATH Keywords: many-valued equalities
zbMATH Keywords: fuzzy control
zbMATH Keywords: fuzzification
zbMATH Keywords: defuzzification
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0285
OpenAlex ID: W1988623045
DOI: 10.1006/JMAA.1996.0285
description: scientific article; zbMATH DE number 930223