Obtaining Contradiction Measures on Intuitionistic Fuzzy Sets From Fuzzy Connectives

Abstract

In a previous paper1, we proposed an axiomatic model for measuring self-contradiction in the framework of Atanassov fuzzy sets. This way, contradiction measures that are semicontinuous and completely semicontinuous, from both below and above, were defined. Although some examples were given, the problem of finding families of functions satisfying the different axioms remained open. The purpose of this paper is to construct some families of contradiction measures firstly using continuous t-norms and t-conorms, and secondly by means of strong negations. In both cases, we study the properties that they satisfy. These families are then classified according the different kinds of measures presented in the above paper.

The Notion of Weak-Contradiction: Definition and Measures

“…In addition, we have related the measure C m (A, B) with different frameworks by fixing a specific kind of density of contradiction. That way, we have characterized the 5 The results of the measures follow easily by using Proposition 12 and Corollary 6. two extremal cases of the measure C m (A, B) according to the density of contradiction considered.…”

“…We consider that the situation above, albeit extremely simplified, is enough to motivate, at least, the consideration of arbitrary negations in the definition of N -contradiction. 2) Mapping f as variables: Maybe the most important difference with respect to the approaches [5], [8] lies on the conceptual understanding of how to measure contradictions. In such approaches, a specific f -contradiction is fixed a priori.…”

“…Thus, the contradiction is determined by the information provided by the fuzzy sets themselves, without any a priori assumption. In other words, whereas in [5] and [8], the notion of contradiction is modeled by a specific N -contradiction, in our approach, the contradiction is represented by the whole set of different f -weak-contradictions. Note that this difference changes completely the conceptualization of the notion of contradiction; under this approach, "the notion of contradiction" is not just a relation between fuzzy sets but a set of relations.…”

“…The following example is related to approaches where a negation operator to determine the contradiction is fixed a priori [5], [8], [15], [18]. We can link our measure of contradiction to those approaches by assuming the following equivalence: Two fuzzy sets are contradictory if f A,B ≤ n, where n is the negation operator chosen to represent the contradiction (i.e., the n-weak-contradiction).…”