A new class of fuzzy implications. Axioms of fuzzy implication revisited

In the recent years, several new construction methods of fuzzy implications have been proposed. However, these construction methods actually care about that the new implication could preserve more properties. In this paper, we introduce a new method for constructing fuzzy implications based on an aggregation function with F (1, 0) =1, a fuzzy implication I and a non-decreasing function φ, called FIφ-construction. Specifically, some logical properties of fuzzy implications preserved by this construction are studied. Moreover, it is studied how to use the FIφ-construction to produce a new implication satisfying a specific property. Furthermore, we produce two new subclasses of fuzzy implications such as UIφ-implications and GpIφ-implications by this method and discuss some additional properties. Finally, we provide a way to generate fuzzy subsethood measures by means of FIφ-implications.

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CONTRAPOSITIVE SYMMETRY OF DISTRIBUTIVE FUZZY IMPLICATIONS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488502001880 ◽

2002 ◽

Vol 10 (supp01) ◽

pp. 135-147 ◽

Cited By ~ 19

Author(s):

MICHAŁ BACZYŃSKI

Keyword(s):

Functional Equation ◽

Functional Equations ◽

Strong Negation ◽

Fuzzy Implications ◽

Fuzzy Implication

Recently, we have examined the solutions of the system of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z), where T : [0, 1]2 → [0, 1] is a strict t-norm and I : [0, 1]2 → [0, 1] is a non-continuous fuzzy implication. In this paper we continue these investigations for contrapositive implications, i.e. functions which satisfy the functional equation I(x, y) = I(N(y), N(x)), with a strong negation N : [0, 1] → [0, 1]. We show also the bounds for two classes of fuzzy implications which are connected with our investigations.

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On the Ordering Property and Law of Importation in Fuzzy Logic

International Journal of Artificial Life Research ◽

10.4018/jalr.2010070102 ◽

2010 ◽

Vol 1 (3) ◽

pp. 17-30

Author(s):

Huiwen Deng ◽

Huan Jiang

Keyword(s):

Fuzzy Logic ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Fuzzy Implications ◽

Fuzzy Implication ◽

The Law ◽

Necessary And Sufficient ◽

Triple I Method

In this paper, the authors investigate the ordering property (OP), , together with the general form of the law of importation(LI), i.e., , whereis a t-norm andis a fuzzy implication for the four main classes of fuzzy implications. The authors give necessary and sufficient conditions under which both (OP) and (LI) holds for S-, R-implications and some specific families of QL-, D-implications. Following this, the paper proposes the sufficient condition under which the equivalence between CRI and triple I method for FMP can be established. Moreover, this conclusion can be viewed as a unified triple I method, a generalized form of the known results proposed by Wang and Pei.

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An Algorithm for Producing Fuzzy Negations via Conical Sections

Algorithms ◽

10.3390/a12050089 ◽

2019 ◽

Vol 12 (5) ◽

pp. 89 ◽

Cited By ~ 3

Author(s):

Souliotis ◽

Papadopoulos

Keyword(s):

Artificial Intelligence ◽

Neural Networks ◽

Structural Element ◽

Specific Field ◽

New Class ◽

Fuzzy Implications ◽

Classical Negation ◽

Geometrical Concepts ◽

Computational Processes

In this paper we introduced a new class of strong negations, which were generated via conical sections. This paper focuses on the fact that simple mathematical and computational processes generate new strong fuzzy negations, through purely geometrical concepts such as the ellipse and the hyperbola. Well-known negations like the classical negation, Sugeno negation, etc., were produced via the suggested conical sections. The strong negations were a structural element in the production of fuzzy implications. Thus, we have a machine for producing fuzzy implications, which can be useful in many areas, as in artificial intelligence, neural networks, etc. Strong Fuzzy Negations refers to the discrepancy between the degree of difficulty of the effort and the significance of its results. Innovative results may, therefore, derive for use in literature in the specific field of mathematics. These data are, moreover, generated in an effortless, concise, as well as self-evident manner.

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Generating Fuzzy Implications by Ordinal Sums

Tatra Mountains Mathematical Publications ◽

10.1515/tmmp-2016-0018 ◽

2016 ◽

Vol 66 (1) ◽

pp. 39-50 ◽

Cited By ~ 3

Author(s):

Paweł Drygaś ◽

Anna Król

Keyword(s):

Fuzzy Implications ◽

Fuzzy Implication

Abstract This paper deals with ordinal sums of fuzzy implications. Some of the known constructions are recalled and new ways of generating fuzzy implications from given ones are proposed. Sufficient properties of fuzzy implications as summands for obtaining a fuzzy implication as a result are presented.

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(T,N) and Residual Fuzzy Co-Implication in Dual Heyting Algebra with Applications

10.20944/preprints201608.0027.v1 ◽

2016 ◽

Cited By ~ 1

Author(s):

Iqbal H. Jebril

Keyword(s):

Heyting Algebra ◽

Fuzzy Implications ◽

Fuzzy Implication ◽

New Concepts

Recently, many authors have been interested to introduce fuzzy implications over t-norms and t-conorms. In this paper, we introduce (S,N) and residuum fuzzy implication for Dubois t-norm and Hamacher's t-norm. Also, new concepts so-called (T,N) and residual fuzzy co-implication in dual Heyting Algebra are investigated. Some examples as well as application are discussed as well.

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Modifications of Łukasiewicz’s intuitionistic fuzzy implication

Notes on Intuitionistic Fuzzy Sets ◽

10.7546/nifs.2021.27.3.32-39 ◽

2021 ◽

Vol 27 (3) ◽

pp. 32-39

Author(s):

Alžbeta Michalíková ◽

◽

Eulalia Szmidt ◽

Peter Vassilev ◽

◽

...

Keyword(s):

Basic Properties ◽

Fuzzy Implications ◽

Intuitionistic Fuzzy ◽

Fuzzy Implication

In [6], G. Klir and B. Yuan named after J. Łukasiewicz the implication p \rightarrow q = min(1, p+q). In a series of papers, 198 different intuitionistic fuzzy implications have been introduced, and their basic properties have been studied. Here we introduce six new implications which are modifications of Łukasiewicz’s intuitionistic fuzzy implication, and we describe and prove some of their properties.

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