Fuzzy Sets and Systems 157 (2006) 622 – 627 www.elsevier.com/locate/fss Advances and challenges in interval-valued fuzzy logic C. Cornelis ∗ , G. Deschrijver, E.E. Kerre Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium Available online 28 October 2005 Abstract Among the various extensions to the common [0, 1]-valued truth degrees of “traditional” fuzzy set theory, closed intervals of [0, 1] stand out as a particularly appealing and promising choice for representing imperfect information, nicely accommodating and combining the facets of vagueness and uncertainty without paying too much in terms of computational complexity. From a logical point of view, due to the failure of the omnipresent prelinearity condition, the underlying algebraic structure L I falls outside the mainstream of the research on formal fuzzy logics (including MV-, BL- and MTL-algebras), and consequently so far has received only marginal attention. This comparative lack of interest for interval-valued fuzzy logic has been further strengthened, perhaps, by taking for granted that its algebraic operations amount to a twofold application of corresponding operations on the unit interval. Abandoning that simplifying assumption, however, we may find that L I reveals itself as a very rich and noteworthy structure allowing the construction of complex and surprisingly well-behaved logical systems. Reviewing the main advances on the algebraic characterization of logical operations on L I , and relating these results to the familiar completeness questions (which remain as major challenges) for the associated formal fuzzy logics, this paper paves the way for a systematic study of interval-valued fuzzy logic in the narrow sense. © 2005 Elsevier B.V. All rights reserved. Keywords: Interval-valued fuzzy logic; Logical connectives; Algebraic structures; Representability; Fuzzy logics 1. Introduction Interval-valued fuzzy set theory (apparently introduced independently in the mid-seventies by Grattan-Guinness [11], Jahn [14], Sambuc [19] and Zadeh [21]) is an increasingly popular extension of fuzzy set theory where traditional [0, 1]- valued membership degrees are replaced by intervals in [0, 1] that approximate the (partially unknown) exact degrees. Hence, not only vagueness (lack of sharp class boundaries), but also a feature of uncertainty (lack of information) can be addressed intuitively. Moreover, interval-valued fuzzy sets (IVFSs) are considerably easier to handle in practice than the similarly inspired type-2 fuzzy sets (of which IVFSs are in fact a special case, called “interval type-2 fuzzy sets” in that context, see e.g. [17]); as further evidence of their wide relevance, they also subsume the syntactically equivalent frameworks of Atanassov’s intuitionistic fuzzy sets [1] and of Gau and Buehrer’s vague sets [7]. ∗ Corresponding author. E-mail addresses: Chris.Cornelis@UGent.be (C. Cornelis), Glad.Deschrijver@UGent.be (G. Deschrijver), Etienne.Kerre@UGent.be (E.E. Kerre) URL: http://www.fuzzy.UGent.be (Fuzziness and Uncertainty Modelling). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.10.007