Abstract
This paper introduces a new class of fuzzy closure operators called implicative closure operators, which generalize some notions of fuzzy closure operators already introduced by different authors. We show that implicative closure operators capture some usual consequence relations used in Approximate Reasoning, like Chakraborty’s graded consequence relation, Castro et al.’s fuzzy consequence relation, similarity-based consequence operators introduced by Dubois et al. and Gerla’s canonical extension of classical closure operators. We also study the relation of the implicative closure operators to other existing fuzzy inference operators as the Natural Inference Operators defined by Boixader and Jacas and the fuzzy operators defined by Biacino, Gerla and Ying.
Highlights
Many works have been devoted to extend the notions of closure operators, closure systems and consequence relations from two valued logic to many valued logic
Propositions of L will be denoted by lower case letters p; q; . . . , and fuzzy sets of propositions by upper case letters A, B, etc
Using the notation of closure operators and the notion of degree of inclusion, the relationship between graded consequence and fuzzy consequence relations become self-evident. The former is defined only over classical sets while the latter is defined over fuzzy sets, but both yield a fuzzy set of formulas as output
Summary
Many works have been devoted to extend the notions of closure operators, closure systems and consequence relations from two valued logic to many valued logic. To conclude this brief overview, let us point out the in the classical setting there are well-known relationships of interdefinability among closure operators, consequence relations and closure systems. Using the notation of closure operators and the notion of degree of inclusion, the relationship between graded consequence and fuzzy consequence relations become self-evident As already mentioned, the former is defined only over classical sets while the latter is defined over fuzzy sets, but both yield a fuzzy set of formulas as output. Infq2B Ce ðAÞðqÞ (recall that B is a classical set); (fc3) fuzzy cut: if B Ce ðAÞ Ce ðA [ BÞ Ce ðAÞ
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