A First View on the Alternatives of Fuzzy Set Theory

Abstract

During many centuries probability theory and error calculus have been the only models to treat imprecision and uncertainty. In the past three decennia however a lot of new models have been introduced for handling incomplete information. Undoubtly fuzzy set theory initiated by Zadeh [1] in 1965 plays the central role. Besides this widely applied theory, many other models pretending to be competitive with fuzzy set theory have been launched: rough set theory, flou set theory, L-flou set theory, intuitionistic set theory ... Allthough we are convinced that freedom and openness in research has be be respected in a degree as large as possible we will warn for increasing the number of alternatives to a degree that introduces confusion and hence diminishes the trust in our models, especially for new practitioners. As I already mentioned in a previous paper [2], time has come to clean up the existing material, to summarize the relevant theoretical models and indicate their relevance by means of convicting concrete examples, by which I do not mean the stipulation in a superficial way of some possible areas of application but indeed the description of real situations and the solution of real problems. In this talk we will focuss on several alternative theories and try to indicate their mutual relationships and their relationships to earlier existing mathematical models.

We will start with a short overview of the basic operations on the class of fuzzy sets and indicate the loss of structure (in this case a Kleene algebra) comparing to the Boolean algebra of (crisp) set theory. Secondly we will briefly outline Goguen’s [3] extension to L-fuzzy set theory where L denotes a complete lattice. Next we will introduce the structure of Gentilhomme’s flou set theory as well as its extension to n-flou sets and L-flou sets as given by Negoïta and Ralescu [5]. In the last model L denotes a complete lattice satisfying some supplementary condition (L): (L)(8A L)(8a 2 L)(a < supA) (9b 2 A)(a b)) For an extensive treatment of such lattices, the so-called kite-tail lattices, we refer to [6]. The sixth model that will presented is the theory of twofold fuzzy sets as introduced by Dubois and Prade [7] in 1987. Finally Pawlak’s [8] rough set theory will be briefly described.

In the second part of this paper we will outline some links between the models mentioned above. In particular it will be shown that:

• 1. Fuzzy set theory is an extension of flou set theory.

• 2. L-fuzzy set theory is equivalent to L-flou set theory for L a complete lattice satisfying the supplementary condition (L).

• 3. f0, lg-flou set theory is equivalent to set theory.

• 4. f0, ½, lg-flou set theory is equivalent to flou set theory.

• 5. [0, 1]-flou set theory is equivalent to fuzzy set theory.

• 6. Twofold fuzzy set theory extends flou set theory.

• 7. Twofold fuzzy set theory is equivalent to fuzzy set theory.

• 8. Flou set theory extends rough set theory.