An axiomatic model for measuring contradiction and image-contradiction between two AIFSs

The importance of dealing with contradictory information or of deriving contradictory consequences in inference processes justifies undertaking a theoretical study on the subject of contradiction. In [S. Cubillo, E. Castiñeira, Contradiction in intuitionistic fuzzy sets, in: Proceedings of the Conference IPMU’2004, Perugia, Italy, 2004, pp. 2180–2186] we defined contradictory and image-contradictory Atanassov intuitionistic sets, where we established that two sets A and B are image-contradictory, with respect to a given intuitionistic negation image, if A implies image, and are contradictory if they are image-contradictory for some negation image. The purpose of this article is to thoroughly examine the model for measuring contradiction between two Atanassov intuitionistic fuzzy sets irrespective of a fixed negation, proposed in [C. Torres-Blanc, E.E. Castiñeira, S. Cubillo, Measuring contradiction between two AIFS, in: Proceedings of the Eighth International FLINS Conference, Madrid, Spain, 2008, pp. 253–258], and also to introduce a mathematical model to measure image-contradiction between sets, where image is an intuitionistic negation. First, we justify and determine the minimum axioms that a function must satisfy to be able to be used as a measure of contradiction or a measure of image-contradiction. Also, we introduce some early examples of valid functions that conform to the model. Then, we establish the conditions for these measures to be continuous from below or continuous from above. Finally, we build families of contradiction and image-contradiction measures, establishing how they are relate to each other, and we look at how they behave with respect to continuity.