On fuzzy implication in De Morgan algebras

Istituto di Matematica, Facoltà di Architettura, Università di Napoli, Via Monteoliveto 3, 80134 Napoli, Italy
Fuzzy Sets and Systems (Impact Factor: 1.99). 11/1989; 33(2):155-164. DOI: 10.1016/0165-0114(89)90238-8


The behaviour of some fuzzy implications with respect to measure of fuzziness and their deductive power is studied in a lattice framework. The point of view of Bandler and Kohout [1] and Willmott [7] is followed.

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    • "The fuzziness of elements x of the set of true values L (we use here L=[0; 1]) may be measured as follows [1] [8]: d(x)=x ∧ n(x). As it was shown in [3], the negation of elements changes the fuzziness of result such that d(n(x))¿d(x) if the negation is contracting and d(n(x))6d(x) if the negation is expanding. "
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    ABSTRACT: Common features in various fuzzy logical systems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.
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    ABSTRACT: Fourteen properties and their interdependencies contributed to the analysis of six classes implication operators–S-implications, R-implications, QL-implications, Force implication, f- generated implication operator and g- generated implication operator– are explored in this paper. It is found that all the properties can be inferred from three mutually independent properties. Then, the proven concerning which properties are true, false or satisfied for each of the six classes of implication operators is given. Based on which, the author get the property I(x(n(x)) = n (x) for all x ∈[0,1] that does not hold for force implication, f-generated implication operator and g-generated implication operator.
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