Abstract
Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory.
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Deschrijver, G. (2013). Implication Functions in Interval-Valued Fuzzy Set Theory. In: Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds) Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol 300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35677-3_4
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DOI: https://doi.org/10.1007/978-3-642-35677-3_4
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