Measuring contradiction in fuzzy logic

Abstract

Several methods have been proposed within fuzzy logic for inferring new knowledge from the original premises. However, there must be some guarantee that the results contradict neither each other nor the initial information. In 1999, Trillas et al. introduced the concepts of both contradictory set and contradiction between two sets. Moreover, we established the need to study not only contradiction but also the degree of such contradiction in E. Castiñeira et al., “Degrees of contradiction in fuzzy sets theory”, Proceedings IPMU'02, 2002a, pp. 171–176, Annecy (France), E. Castiñeira et al., “Contradicción entre dos conjuntos”, Actas ESTYLF'02, 2002b, pp. 379–383, León (Spain) (in Spanish) establishing some measures for this purpose. Nevertheless, contradiction could have been measured in some other way. Elena Castiñeira Holgado was born in Asturias (Spain). She received her Bachelor Degree in Mathematics from the Complutense University of Madrid in 1985, and her Ph.D. in Computer Sciences from the Technical University of Madrid in 1994, with research on fractals. She has been working on non-classical logics, and, especially, fuzzy logic since 1997. The main goal of this paper is to propose an axiomatic definition of measure of contradiction both for a set and between two sets. A requirement for such modelling is that some minimum conditions be met. We also examine how well some measures proposed throughout the fuzzy logic literature fit this definition. Finally, we obtain some results on the relationship between measures of contradiction and the measures of ambiguity introduced by Fishburn for classical logic and by Yager for imprecision.