Abstract: For inference purposes in both classical and fuzzy logic, neither the information itself should be
contradictory, nor should any of the items of available information contradict each other. In order to avoid these
troubles in fuzzy logic, a study about contradiction was initiated by Trillas et al. in 5 and 6. They introduced the
concepts of both self-contradictory fuzzy set and contradiction between two fuzzy sets. Moreover, the need to
study not only contradiction but also the degree of such contradiction is pointed out in 1 and 2, suggesting
some measures for this purpose. Nevertheless, contradiction could have been measured in some other way. This
paper focuses on the study of contradiction between two fuzzy sets dealing with the problem from a geometrical
point of view that allow us to find out new ways to measure the contradiction degree. To do this, the two fuzzy
sets are interpreted as a subset of the unit square, and the so called contradiction region is determined. Specially
we tackle the case in which both sets represent a curve in 0,12. This new geometrical approach allows us to
obtain different functions to measure contradiction throughout distances. Moreover, some properties of these
contradiction measure functions are established and, in some particular case, the relations among these different
functions are obtained.
Keywords: fuzzy sets, t-norm, t-conorm, fuzzy strong negations, contradiction, measures of contradiction.
ACM Classification Keywords: F.4.1 Mathematical Logic and Formal Languages: Mathematical Logic (Model
theory, Set theory); I.2.3 Artificial Intelligence: Deduction and Theorem Proving (Uncertainty, “fuzzy” and
probabilistic reasoning); I.2.4 Artificial Intelligence: Knowledge Representation Formalisms and Methods
(Predicate logic, Representation languages).
A GEOMETRICAL INTERPRETATION TO DEFINE CONTRADICTION DEGREES BETWEEN TWO FUZZY SETS
Carmen Torres, Elena Castiñeira, Susana Cubillo, Victoria Zarzosa