Measures of Self-Contradiction on Atanassov's Intuitionistic Fuzzy Sets: An Axiomatic Model

Trillas et al. (Soft Comput 1999;3(4):197-199 and In: Proc 18th Int Conf of the North American Fuzzy Information Processing Society (NAFIPS), New York;1999; pp 28-32) introduced the concepts of self-contradictory fuzzy set and contradictory fuzzy sets in an attempt to mark out when an inference process is not coherent. Later, contradiction was studied along the same lines in Cubillo and Castineira (In: Proc X Conf of Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2004), Perugia (Italy); (2004). 2180-2186) within the framework of Atanassov's intuitionistic fuzzy sets (AIFSs). The aim of this paper is to axiomatically model self-contradiction measures on AIFSs. After introducing some functions to measure negation-dependent or -independent degrees of self-contradiction of an AIFS in Castineira, Cubillo, and Torres (Mathware Soft-Comput 2006; 13:139-156), a preliminary axiomatic model for measuring the self-contradiction of AIFSs was presented in Castineira et al. (In: Proc XI Conf of Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006), Paris (France); 2006. pp 2391-2398). Being a very early model, it turned out to be incomplete. For this reason, this paper takes up the study started in Castineira et al. (2006) again with a view to filling up its gaps. Here, we present a more complete model that envisages the continuity of self-contradiction measures from a broader perspective. The concepts of semicontinuous and completely semicontinuous, from both below and above, are now introduced, and a classification result is shown. (C) 2009 Wiley Periodicals, Inc.