Chapter

35 Years of Fuzzy Set Theory

Volume 261 of the series Studies in Fuzziness and Soft Computing pp 57-82

Interval-Valued Algebras and Fuzzy Logics

  • Bart Van GasseAffiliated withDepartment of Applied Mathematics and Computer Science, Ghent University
  • , Chris CornelisAffiliated withDepartment of Applied Mathematics and Computer Science, Ghent University
  • , Glad DeschrijverAffiliated withDepartment of Applied Mathematics and Computer Science, Ghent University

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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 ‘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.