The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions

Abstract

Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications R O -implications, the strong implications (G, N )-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N . Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x, I(x, y)) ≤ y, for any fuzzy implication I and overlap function O, and, in particular, for R O -implications, (G, N )-implications, QL-implications and D-implications derived from tuples (O, G, N ), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T -conditionality and U -conditionality, for t-norms T and uninorms U , respectively. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ inference processes via the generalized Modus Ponens (GMP) and Modus Tollens (GMT), carried out via the Zadeh's Compositional Rule of Inference [5]. 1 The GMP, which is the focus of this paper, can be implemented by a scheme enabled by the functional inequality called the Law of T -Conditionality, which, for a t-norm T and a fuzzy implication I, is stated by:In fact, (TC) means x * T (x → y) ≤ y, which generalizes the Modus Ponens x ∧ T (x → y) y to the fuzzy context [7, 8].In the literature, the studies on the T -Conditionality have been done just for the three main families of fuzzy implications, namely, R-, (S, N )and QL-implications. 2 R-implications are generalizations to [0, 1] of Boolean implications defined by the identity given, for a universe set X, bywhere A, B ⊆ X, and the intersection is generalized by a t-norm. This class of implications is related to a residuation concept from the intuitionistic logic. On the other hand, (S, N )-implications are generalizations to [0, 1] of the Boolean material implication defined aswhere ∨ and ¬ are...

Fuzzy implication functions constructed from general overlap functions and fuzzy negations

“…In Figure 1, we illustrate the main results presented in this section. Note that the intersections between the families of (G, N ), QL, R O and D-implications had already been presented in [18,20,22,23,24].…”

General Grouping Functions

“…For all properties and related concepts on overlap functions and grouping functions, see [2,5,7,9,10,21,[23][24][25].…”

“…Overlap and grouping functions have been largely studied since they are richer than t-norms and t-conorms [18], respectively. Regarding, for instance, some properties like idempotency, homogeneity, and, mainly, the self-closeness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions [7,8,10,12]. For example, there is just one idempotent t-conorm (namely, the maximum t-conorm) and two homogeneous t-conorms (namely, the maximum and the probabilistic sum of t-conorms).…”

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

“…Recently, some authors [18,19] have also dealt with the use of conjunctive uninorms instead of t-norms, studying the case of RU-implications, and others (see Ref. [16]) have considered the use of overlap functions, dealing with the so-called O-conditionality. Since t-norms and conjunctive uninorms are special cases of conjunctors with a neutral element, in the current paper we will recover and generalize some of these results.…”

“…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.…”