An axiomatic model for measuring contradiction and NN-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 NN-contradictory Atanassov intuitionistic sets, where we established that two sets A and B   are NN-contradictory, with respect to a given intuitionistic negation NN, if A   implies N(B)N(B), and are contradictory if they are NN-contradictory for some negation NN.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 NN-contradiction between sets, where NN 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 NN-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 NN-contradiction measures, establishing how they are relate to each other, and we look at how they behave with respect to continuity.