Fuzzy set theory is based on a `fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis...

This paper is devoted to the study of two kinds of implications on a finite chain $L$: $S$-implications and $R$-implications. A characterization of each kind of these operators is given and a lot of different implications on $L$ are obtained, not only from smooth t-norms but also from non smooth ones. Some additional properties on these implications are studied specially in the smooth case. Finally, a class of non smooth t-norms including the nilpotent minimum is characterized. Any t-norm in this class...

Statistical Inference deals with the drawing of conclusions about a random experiment on the basis of the information contained in a sample from it. A random experiment can be defined by means of the set of its possible outcomes (sample space) and the ability of observation of the experimenter. It is usually assumed that this ability allows the experimenter to describe the observable events as subsets of the sample space. In this paper, we will consider that the experimenter can only express the...

The aim of this paper is to present and study one important class of divergence measure between fuzzy subsets, and one important class of divergence measure between fuzzy partitions, each of them having some specific properties. In the first case, the divergence measure attempts to quantify the degree of difference between two fuzzy subsets ? and ? by comparing the fuzziness of both ? and ? with the fuzziness of the intermediate fuzzy subset. In the second case, we use this divergence between subsets...

In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice $L$ based on a priori given t-norm acting on $[a,1]$ and t-conorm acting on $[0,a]$ for an arbitrary element $a\in L\setminus \{0,1\}$. We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice.

We study here the behavior of the t-norms at the point (1/2, 1/2). We indicate why this point can be considered as significant in the specification of t-norms. Then, we suggest that the image of this point can be used to classify the t-norms. We consider some usual examples. We also study the case of parameterized t-norms. Finally using the results of this study, we propose a uniform method of computing the parameters. This method allows not only having the same parameter-scale for all the families,...

On caractérise toutes les entropies-floues qui sont des valuations des treillis P(X) des parties floues d'un ensemble fini X, on presente la construction de certaines entropies floues et on analyse leur caractère de valuation de treillis aiguisés Sh(g), g belonging to P(X).

The symmetric implicational method is revealed from a different perspective based upon the restriction theory, which results in a novel fuzzy inference scheme called the symmetric implicational restriction method. Initially, the SIR-principles are put forward, which constitute optimized versions of the triple I restriction inference mechanism. Next, the existential requirements of basic solutions are given. The supremum (or infimum) of its basic solutions is achieved from some properties of fuzzy...