Fuzzy Implications: Some Recently Solved Problems

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference

Choice procedures in Pairwise Comparison - Multiple-attribute Decision Making Methods

We consider extensions of some classical rational axioms introduced in conventional choice theory to valued preference relations. The concept of kernel is revisited using two ways: one proposes to determine kernels with a degree of qualification and the other presents a fuzzy kernel where every element of the support belongs to the rational choice set with a membership degree. Links between the two approaches is emphasized. We exploit these results in Multiple-attribute Decision Aid to determine the good and bad choices. All the results are valid if the valued preference relations are evaluated on a finite ordinal scale.

Hybrid probabilistic-possibilistic mixtures and utility functions

The paper was written during the stay of the second author as visiting professor at University Paul Sabatier, in Toulouse, in June, 1999.International audienceA basic building block in the standard mathematics of decision under uncertainty is the notion of probabilistic mixture. In order to generalize decision theory to non probabilistic uncertainty, one approach is to generalize mixture sets. In the recent past it has been proved that generalized mixtures can be non trivially defined, and they have been instrumental in the development of possibilistic utility theory. This paper characterizes the families of operations involved in generalized mixtures, due to a previous result on the characterization of the pairs of continuous t-norm and t-conorm such that the former is conditionally distributive over the latter. What is obtained is a family of mixtures that combine probabilistic and possibilistic mixtures via a threshold. It is based on a restricted family of t-conorm/ t-norm pairs which are very special ordinal sums. Any practically useful theory of pseudo-additive measures must use such special pairs of operations in order to extend the additivity property, and the notion of probabilistic independence

Putting Rough Sets and Fuzzy Sets Together

This paper draws from and continues a previous article by the authors, entitled “Rough fuzzy sets and fuzzy rough sets”, that appeared in the Int.J.of General Systems in 1990International audienceIn this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences