General Foundational Remarks on Fuzzy Set Theory

2011

It has been accepted that the fuzzy sets do not form a field. In this article, we are going to put forward an extension of the definition of fuzziness. With the help of this extension, we would be able to define the complement of a fuzzy set properly. This in turn would allow us to assert that fuzzy sets do form a field. In fact, the fuzzy membership value and the fuzzy membership function for the complement of a fuzzy set are two different things. This confusion has created a stumbling block towards accepting the theory of fuzzy sets as a generalization of the classical theory of sets.

Journal of Mathematical Analysis and Applications, 1972

Some new algebraic properties of the class _Lp(I) of the "fuzzy sets" are stressed; in particular it is pointed out that the class of the generalized characteristic functions furnished with the lattice operations proposed by Zadeh is a Brouwerian lattice. The possibility of inducing other different lattice operations to the whole class s(I) or to a suitable subclass of it is considered. The problem of the relationship between "fuzzy sets" and classical set theory is finally remarked. A qualitative comparison with similar situations appearing in the axiomatic formulation of quantum mechanics and in the classical theory of probability is made. * This paper is a slightly revised version of the report LC50 of the Laboratorio di Cibemetica de1 C.N.R.

10

Fuzzy Sets and Systems, 2002

We present a new and natural interpretation of fuzzy sets and fuzzy relations where the basic notions and operations have quite natural meanings. We interpret fuzzy sets and fuzzy relations in a cumulative Heyting valued model for intuitionistic set theory, and deÿne the basic notions and operations naturally in the model. As far as fuzzy sets and fuzzy relations are considered as extensions of crisp sets and relations, this interpretation seems to be most natural. In the interpretation the canonical embedding from the class of all sets into the model plays an important role. We distinguish generalized fuzzy sets, fuzzy subsets of crisp sets, and membership functions of fuzzy sets on crisp sets. Thus we present a foundation for developing a general theory of fuzzy sets where fuzzy subsets of di erent sets can be treated in a natural and uniform way.

Information Sciences, 1990

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.

Springer eBooks, 1996

International Journal of Information Engineering and Electronic Business, 2013

In this art icle, we would like to revisit and comment on the definit ion of co mplementation of fu zzy sets and also on some of the theories and formulas associated with this. Furthermore, the existing probability-possibility consistency principles are also revisited and related results are v iewed fro m the standpoint of the Randomness-Fuzziness consistency principles. It is found that the existing definition of complementation as well as the probability-possibility consistency principles is not well defined. Consequently the results obtained from these would be inappropriate fro m our standpoints. Hence we would like to suggest some new defin itions for so me of the terms often used in the theory of fuzzy sets whenever possible.

The aim of this talk is provide a motivation for Zadeh's Fuzzy Set Theory (ZFST), present elements of the same, present elements of Goguen's Fuzzy Set Theory (GFST) which generalizes ZFST and present elements of f-Set Theory which further generalizes GFST.

2009

Studies in Fuzziness and Soft Computing, Volume 244 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: kacprzyk@ibspan.waw.pl Further volumes of this series can be found on our homepage: springer.com Vol. 228. Adam Kasperski Discrete Optimization with Interval Data, 2008 ISBN 978-3-540-78483-8 Vol. 229. Sadaaki Miyamoto, Hidetomo Ichihashi, Katsuhiro Honda Algorithms for Fuzzy Clustering, 2008 ISBN 978-3-540-78736-5 Vol. 230.

Fuzzy Sets and Systems, 2007

In this paper we stress the relevance of a particular family of fuzzy sets, where each element can be viewed as the result of a classification problem. In particular, we assume that fuzzy sets are defined from a well-defined universe of objects into a valuation space where a particular graph is being defined, in such a way that each element of the considered universe has a degree of membership with respect to each state in the valuation space. The associated graph defines the structure of such a valuation space, where an ignorance state represents the beginning of a necessary learning procedure. Hence, every single state needs a positive definition, and possible queries are limited by such an associated graph. We then allocate this family of fuzzy sets with respect to other relevant families of fuzzy sets, and in particular with respect to Atanassov's intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassov's model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed.