An axiomatic approach to fuzzy set theory

doi:10.1016/0020-0255(90)90036-A

Information Sciences

Volume 52, Issue 1, October 1990, Pages 75-83

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Abstract

An axiomatic basis for the concept of “fuzzy set” is established, following the classical class theory developed by von Neumann and Bernays (VNB). The set of axioms we exhibit is the minimal set that enables a formal definition of fuzzy sets, without assuming a universe of discourse, and requires fewer axioms than any other published formal definition of fuzzy sets.

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There are more references available in the full text version of this article.

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Citation Excerpt :
Moreover in 1965, Zadeh (1975) has established the notion of fuzzy logic, which is a multilevel extension of classical logic where propositions can get truth-values in the interval [0, 1], and are not limited to one of the two values (Kandel, 1987). Almost five decades before, his pioneering work has shown a multitude of research work and applications related to signal processing (Mhalla, Jerbi, Dutilleul, Craye, & Benrejeb, 2010), knowledge representation (Cordova & Leyton, 2010), control theory (Hossu, Fagarasan, Hossu, & Iliescu, 2010), and reasoning (Ronen, Shabtai, & Guterman, 2002; Tamir & Kandel, 1990; Tamir & Kandel, 1995). In the educational context, Careaga and Avendaño (2009) proposed the Pedagogical Model of Knowledge Management, to address three processes: development of intellectual capital, capitalization of knowledge and development of social capital.
The aim of this quantitative, exploratory and descriptive research is to know if teachers that work in intercultural contexts develop ICT competences by using virtual platforms. The innovation of the teaching-learning processes is promoted through the virtual platform, keeping a connection by forming school networks among schools in indigenous contexts. 51 teachers from 10 schools from the Bío–Bío and Araucanía regions in Chile were part of the development of the networks. The main results show that the participant teachers increased their ICT competences in the different dimensions of the use of the ICT: pedagogical management, knowledge management, deepening of knowledge; and social, ethic and legal dimensions, by applying the knowledge and talent management models.
Axiomatic theory of intuitionistic fuzzy sets
2000, Fuzzy Sets and Systems
Fuzzy semantic analysis and formal specification of conceptual knowledge
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Conceptual knowledge can be specified using one of the methods of formal specification of the semantics of a computer program: axiomatic semantics, denotational semantics, or operational semantics. For example, axiomatic semantics can be used to specify the conceptual knowledge of a medical doctor in an expert system for medical diagnosis. The problem is, however, that the knowledge of the expert is not always crisp and well defined. In such cases, a mean for specifying fuzzy conceptual knowledge is required. This paper proposes a method for the specifications of fuzzy conceptual knowledge. To this end, the concepts of fuzzy axiomatic semantics and fuzzy denotational semantics are developed. Fuzzy semantics is a generalization of classical semantics.
A new approach to the similarity relations in the fuzzy set theory
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The subtle theory of “fuzzy” sets has long been proposed by Zadeh as an extension of the classical theory of “hard” sets. In this paper we show how the properties of reflexivity and transitivity of a fuzzy relation may be reconsidered if they are used in view of another intuitive support and new mathematical developments. As a consequence, we propose a new modified system of leading definitions for the fuzzy equivalence relationship that could be used profitably to support the concept of fuzzy partition. The role of an internal basis is also emphasized. Corollary properties are derived from this new approach.
On the axiomatic theory of fuzzy sets
1993, Fuzzy Sets and Systems
Complex intuitionistic fuzzy classes
2016, 2016 IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2016

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An axiomatic approach to fuzzy set theory

Abstract

Inform. and Control

Inform. Sci.

Internat. J. Man-Machine Stud.

Abstract Set Theory

Fuzzy Mathematical Techniques with Applications

Fuzzy statistics and policy analysis

Set valued set theory: Part one

Notre Dame J. Formal Logic

Categories of L-fuzzy sets

Bull. Amer. Math. Soc.