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In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter.

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Chicago
Van Gasse, Bart, Chris Cornelis, and Glad Deschrijver. 2011. “Interval-valued Algebras and Fuzzy Logics.” In 35 Years of Fuzzy Set Theory : Celebratory Volume Dedicated to the Retirement of Etienne E. Kerre, ed. Chris Cornelis, Glad Deschrijver, Mike Nachtegael, Steven Schockaert, and Yun Shi, 261:57–82. Berlin, Germany: Springer.
APA
Van Gasse, Bart, Cornelis, C., & Deschrijver, G. (2011). Interval-valued algebras and fuzzy logics. In Chris Cornelis, G. Deschrijver, M. Nachtegael, S. Schockaert, & Y. Shi (Eds.), 35 Years of fuzzy set theory : celebratory volume dedicated to the retirement of Etienne E. Kerre (Vol. 261, pp. 57–82). Berlin, Germany: Springer.
Vancouver
1.
Van Gasse B, Cornelis C, Deschrijver G. Interval-valued algebras and fuzzy logics. In: Cornelis C, Deschrijver G, Nachtegael M, Schockaert S, Shi Y, editors. 35 Years of fuzzy set theory : celebratory volume dedicated to the retirement of Etienne E. Kerre. Berlin, Germany: Springer; 2011. p. 57–82.
MLA
Van Gasse, Bart, Chris Cornelis, and Glad Deschrijver. “Interval-valued Algebras and Fuzzy Logics.” 35 Years of Fuzzy Set Theory : Celebratory Volume Dedicated to the Retirement of Etienne E. Kerre. Ed. Chris Cornelis et al. Vol. 261. Berlin, Germany: Springer, 2011. 57–82. Print.
@incollection{1224869,
  abstract     = {In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of {\textquoteleft}p implies q{\textquoteright} and {\textquoteleft}p and q{\textquoteright}, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter.},
  author       = {Van Gasse, Bart and Cornelis, Chris and Deschrijver, Glad},
  booktitle    = {35 Years of fuzzy set theory : celebratory volume dedicated to the retirement of Etienne E. Kerre},
  editor       = {Cornelis, Chris and Deschrijver, Glad and Nachtegael, Mike and Schockaert, Steven and Shi, Yun},
  isbn         = {9783642166280},
  language     = {eng},
  pages        = {57--82},
  publisher    = {Springer},
  series       = {Studies in Fuzziness and Soft Computing},
  title        = {Interval-valued algebras and fuzzy logics},
  url          = {http://dx.doi.org/10.1007/978-3-642-16629-7\_4},
  volume       = {261},
  year         = {2011},
}

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