# Similarity in fuzzy reasoning.

• Volume: 2, Issue: 3, page 197-228
• ISSN: 1134-5632

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## Abstract

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Fuzzy set theory is based on a fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL-monoids that can be understood as a generalization of MV-algebras. Residuation is a basic concept in GL-monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval.

## How to cite

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Klawonn, Frank, and Castro, Juan Luis. "Similarity in fuzzy reasoning.." Mathware and Soft Computing 2.3 (1995): 197-228. <http://eudml.org/doc/39046>.

@article{Klawonn1995,
abstract = {Fuzzy set theory is based on a fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL-monoids that can be understood as a generalization of MV-algebras. Residuation is a basic concept in GL-monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval.},
author = {Klawonn, Frank, Castro, Juan Luis},
journal = {Mathware and Soft Computing},
keywords = {Lógica multivaluada; Algebras de Wajsberg; Conjuntos difusos; Algebras de Boole; Lógica matemática; Lógica difusa; residuation; indistinguishability; fuzzy equivalence relations; fuzzy sets; approximate reasoning; GL-monoids; generalization of MV-algebras},
language = {eng},
number = {3},
pages = {197-228},
title = {Similarity in fuzzy reasoning.},
url = {http://eudml.org/doc/39046},
volume = {2},
year = {1995},
}

TY - JOUR
AU - Klawonn, Frank
AU - Castro, Juan Luis
TI - Similarity in fuzzy reasoning.
JO - Mathware and Soft Computing
PY - 1995
VL - 2
IS - 3
SP - 197
EP - 228
AB - Fuzzy set theory is based on a fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL-monoids that can be understood as a generalization of MV-algebras. Residuation is a basic concept in GL-monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval.
LA - eng
KW - Lógica multivaluada; Algebras de Wajsberg; Conjuntos difusos; Algebras de Boole; Lógica matemática; Lógica difusa; residuation; indistinguishability; fuzzy equivalence relations; fuzzy sets; approximate reasoning; GL-monoids; generalization of MV-algebras
UR - http://eudml.org/doc/39046
ER -

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