Advances in Fuzzy Implication Functions

Abstract

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An Algorithm for Producing Fuzzy Negations via Conical Sections

“…The last developments in the theory of fuzzy implications (if…X…then…Y…) indicate that fuzzy negation is enough to generate an algorithmic process of production for fuzzy implications, (c.f. [2,3]). For example, supposing that in the context of an application, the fuzzy implication Yager is selected, which is generated in the following way:…”

“…indicate that fuzzy negation is enough to generate an algorithmic process of production for fuzzy implications, (c.f. [2,3]). For example, supposing that in the context of an application, the fuzzy implication Yager is selected, which is generated in the following way: I(x, y) = f −1 (x• f (y)), where f is a decreasing function; if f is replaced by a fuzzy negation, then an algorithm for producing fuzzy implications is automatically generated.…”

“…It should be noticed that Proposition 3 in paragraph 3 Main Results and particularly Equation 14, constitute new results that might be used in the production of implications, merely from negations (c.f. [2,3]), and by extension in reaching conclusions. In general, the algorithmic approach of negations is virtually non-existent in the applications of fuzzy sets and implications, without any reasons explaining such a gap.…”

“…Fuzzy negations play a crucial role in the creation of De Morgan triples and also in the construction of fuzzy implications (c.f. [1][2][3]). As a general admission, the construction of fuzzy negations is the foundation stone of the triples De Morgan construction and fuzzy implications.…”

On the Use of Conjunctors With a Neutral Element in the Modus Ponens Inequality

“…The latter are approximate reasoning schemes that allow to infer a conclusion of the form "y is Q * " from two premises: a fuzzy proposition "x is P * " and a fuzzy conditional statement "If x is P, then y is Q". The inequality, which involves a fuzzy implication function [1,[6][7][8][9] I ∶ [0, 1] 2 → [0, 1] (used to model the fuzzy conditional) and a bivariate aggregation function [10][11][12][13] The aggregation of the premises in these inference processes has traditionally been performed by means of triangular norms [4,14,15], even though lately other functions such as overlap functions [16,17] and conjunctive uninorms [18,19] have also been investigated for this purpose. Another recent paper, Ref.…”