A review of the relationships between implication, negation and aggregation functions from the point of view of material implication

Pradera, Ana; Beliakov, Gleb; Bustince, Humberto; Baets, Bernard De

doi:10.1016/j.ins.2015.09.033

Cited by 88 publications

(42 citation statements)

References 43 publications

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“…2 We remark that those studies were done under some restricted conditions, e.g., the continuity of the underlying operators is, in general, assumed [1,9]. 2 argument itself, as the work by Beliakov et al [15,16], who applied fuzzy implications and aggregation functions to construct more flexible consensus measures, avoiding the symmetry property of similarity and distance functions that could, in some circumstances, be too restrictive when considering weighted consensus models.The discussion above justifies the use of other kinds of aggregation functions that are able to model conjunctions, disjunctions and implications in some way in a more general context (see, e.g., the works by Ouyang [36], Pradera et al [37] and Pinheiro et al [38]). In this direction, Bustince et al [12,35] introduced overlap and grouping functions that are particular cases of continuous aggregation operators given by (not necessarily associative) increasing commutative functions, satisfying appropriate boundary conditions.…”

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confidence: 70%

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

Dimuro

Bedregal

Fernández

et al. 2019

International Journal of Approximate Reasoning

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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...

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confidence: 70%

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

Dimuro

Bedregal

Fernández

et al. 2019

International Journal of Approximate Reasoning

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“…Note that this look at fuzzy connectives allows to transform the results obtained in some particular subclass ℋ of extended Boolean function to the dual class ℋ obtained by the duality , ∈ {0, 1, … , 7}. So, for example, all results known for conjunctors can be transformed into the corresponding results for disjunctors (when the duality is considered) or into the corresponding results for implications (when the duality is considered, see also [10]). …”

Section: Discussionmentioning

confidence: 99%

Aggregation and Related Functions in Fuzzy Set Theory: Dualities and Monotonicity-Based Generalizations

Mesiar¹,

Kolesárová²

2017

JAMRIS

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“…Definition An implication operator (in the sense of Fodor and Roubens, see Baczyński and colleagues) is a mapping

I : {[0, 1]}^{2} \to [0, 1]

such that, for every

x, y, z \in [0, 1]

, if

x \leq z

then

I (x, y) \geq I (z, y)

; if

y \leq z

then

I (x, y) \leq I (x, z)

;

I (0, x) = 1

;

I (x, 1) = 1

;

I (1, 0) = 0

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Based on this fact, and taking into account the works by Kitainik, Sinha and Dougherty, Young, Fan et al, Bustince et al, and Zhang, the aim of this paper is to introduce a new axiomatization for fuzzy subsethood measures which allows us to: Construct subsethood measures using an appropriate aggregation of appropriate operators where the operators possess some properties which are usually demanded from implication operators; in particular, to find an aggregation function

MJX-tex-caligraphicscriptA : {[0, 1]}^{n} \to [0, 1]

(

n \geq 2

) such that, for all

A, B \in F (X)

, it holds

σ (A, B) = \underset{i = 1}{\overset{n}{MJX-tex-caligraphicscriptA}} (I (A (x_{i}), B (x_{i}))),

where

I : {[0, 1]}^{2} \to [0, 1]

is an operator satisfying some of the properties required from implication operators . These properties that we are going to require to the operators

I

, as for instance,

I (x, y) = 0 if and only if x = 1 and y = 0

have been widely studied in literature .…”

Section: Introductionmentioning

confidence: 99%

“…Construct subsethood measures using an appropriate aggregation of appropriate operators where the operators possess some properties which are usually demanded from implication operators 19,20 ; in particular, to find an aggregation function →  : [0, 1] [0, 1] n ( ≥ n 2) 21-23 such that, for all ∈ A B F X , ( ), it holds is an operator satisfying some of the properties required from implication operators. [24][25][26][27] These properties that we are going to require to the operators I , as for instance, 28…”

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Similarity measures, penalty functions, and fuzzy entropy from new fuzzy subsethood measures

et al. 2019

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In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.

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A review of the relationships between implication, negation and aggregation functions from the point of view of material implication

Cited by 88 publications

References 43 publications

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

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

Aggregation and Related Functions in Fuzzy Set Theory: Dualities and Monotonicity-Based Generalizations

Similarity measures, penalty functions, and fuzzy entropy from new fuzzy subsethood measures

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