Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application

With the demand for knowledge-handling systems capable of dealing with and distinguishing between various facets of imprecision ever increasing, a clear and formal characterization of the mathematical models implementing such services is quintessential. In this paper, this task is undertaken simultaneously for the definition of implication within two settings: first, within intuitionistic fuzzy set theory and secondly, within interval-valued fuzzy set theory. By tracing these models back to the underlying lattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MV algebras), and on the other hand we are able to expose in a clear manner the two models formal equivalence. This equivalence, all too often neglected in literature, we exploit to construct operators extending the notions of classical and fuzzy implication on these structures; to initiate a meaningful classification framework for the resulting operators, based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine the intuititive ideas giving rise to both approaches as models of imprecision and apply them in a practical context. 2003 Elsevier Inc. All rights reserved. * Corresponding author. Tel.: +32-9-264-47-72; fax: +32-9-264-49-95. E-mail addresses: [email protected] (C. Cornelis), [email protected] (G. Deschrijver), [email protected] (E.E. Kerre). URL: http://fuzzy.UGent.be. 0888-613X/$ see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0888-613X(03)00072-0 www.elsevier.com/locate/ijar International Journal of Approximate Reasoning 35 (2004) 55–95