Fuzzy Implication Research Articles

On functional equations and inequalities related to some reasoning schemes that involve fuzzy implications

In this paper, we investigate some functional equations and inequalities which are associated with recognized reasoning schemes like hypothetical syllogism, modus ponens, modus tollens and reduction to absurdity. First, we recall facts and theorems that might be found in the literature. Next, we show some properties and relationships that take place between these equations and inequalities. Moreover, we present some of their solutions among well-known families of fuzzy implications – R-implications, (S,N)-implications and Yager's implications.

Modus Ponens property of T-power based implications

Among the great bunch of additional properties which fuzzy implications may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication when both the antecedent and the consequent are modified using the same quantifier modeled by using powers of t-norms. In order to find the fuzzy implications that satisfy this property, a new class of fuzzy implications named T-power based implications has been proposed. In this paper we study when this kind of fuzzy implication satisfies the Modus Ponens property with respect to a continuous t-norm T♯. Then a characterization theorem on this topic in terms of residual implication generated from a summand of T♯ is presented.

On the ordinal sum of fuzzy implications: New results and the distributivity over a class of overlap and grouping functions

Similar to the construction of ordinal sums of overlap functions, Baczyński et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL-implication constructed from tuples (O,G,N⊤) or a D-implication derived from grouping function G, and show some conclusions about its relations with (G,N)- and RO-implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions.

Fuzzy implications based on strong negations

Fuzzy implications based on strong negations

A Study of GD′- Implications, a New Hyper Class of Fuzzy Implications

In this paper, we introduce and study the GD′-operations, which are a hyper class of the known D′-operations. GD′-operations are in fact D′-operations, that are generated not only from the same fuzzy negation. Similar with D′-operations, they are not always fuzzy implications. Nevertheless, some sufficient, but not necessary conditions for a GD′-operation to be a fuzzy implication, will be proved. A study for the satisfaction, or the violation of the basic properties of fuzzy implications, such as the left neutrality property, the exchange principle, the identity principle and the ordering property will also be made. This study also completes the study of the basic properties of D′-implications. At the end, surprisingly an unexpected new result for the connection of the QL-operations and D-operations will be presented.

Pythagorean fuzzy full implication multiple I method and corresponding applications

The overwhelming majority of existing decision-making methods combined with the Pythagorean fuzzy set (PFS) are based on aggregation operators, and their logical foundation is imperfect. Therefore, we attempt to establish two decision-making methods based on the Pythagorean fuzzy multiple I method. This paper is devoted to the discussion of the full implication multiple I method based on the PFS. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO), Pythagorean fuzzy biresiduum, and the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum. In addition, the full implication multiple I method for Pythagorean fuzzy modus ponens (PFMP) is established, and the reversibility and continuity properties of the full implication multiple I method of PFMP are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in a decision-making problem. The advantages of the new method over existing methods are also explained. Overall, the proposed methods are based on logical reasoning, so they can more accurately and completely express decision information.

On the weak intuitionistic fuzzy implication based on △ operation

In this paper, a new weak intuitionistic fuzzy implication is introduced. Fulfillment of some axioms and properties, together with the Modus Ponens and Modus Tollens inference rules are investigated. Negation induced by the newly proposed implication is presented.

A general framework for the characterization of (S,N)-implications with a non-continuous negation based on completions of t-conorms

The characterization of (S,N)-implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [0,1]2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [0,1]2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of (S,N)-implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [0,1]2 or a strict t-conorm.

Study of Two Families of Generalized Yager’s Implications for Describing the Structure of Generalized (h,e)-Implications

In this study, we analyze the family of generalized (h,e)-implications. We determine when this family fulfills some of the main additional properties of fuzzy implication functions and we obtain a representation theorem that describes the structure of a generalized (h,e)-implication in terms of two families of fuzzy implication functions. These two families can be interpreted as particular cases of the (f,g) and (g,f)-implications, which are two families of fuzzy implication functions that generalize the well-known f and g-generated implications proposed by Yager through a generalization of the internal factors x and 1x, respectively. The behavior and additional properties of these two families are also studied in detail.

On the distributivity of fuzzy implications and the weighted S-implications

The distributivity properties of fuzzy implication functions play an important role in fuzzy research. Making use of the solution of the autodistributivity functional equations, we give the necessary and sufficient conditions of two types of the distributivity property of fuzzy implications. It is essentially based on the weighted disjunctive operator, and new weighted and mean (S,N)-implication functions are introduced. Then using the pliant operators – where all the operators have a common generator function – we give a parametric form of the implication, namely the neutral value-dependent implication, and we show that some weakened properties of the fuzzy weighted implications are valid such as the identity principle, the ordering principle, the T-conditionality, the strong negation principle, etc. With this new concept, we use the fixed point of the negation as a threshold. We give a specific form of the new implication.

A Study of (T, N)– and (N ′, T, N)– Implications

In this paper, we revisit and study the basic properties of two families of fuzzy implications, the so-called and implications. More specific, we study when these fuzzy implications satisfy, or not, the neutrality property , the exchange principle , the identity principle and the ordering property . Moreover, a study is presented for the law of importation with respect to a t- norm. Also, we study the relation of conjugation in implications.

N-vertical generated implications and their distributivities over t-norms and t-conorms

One of the methods to construct a new fuzzy implication from given ones is to transform the given implications and appropriately scale the variables. In this paper, we firstly construct a N-vertical generated implication by changing the linear transformations of two initial implications of the recent introduced vertical e-threshold generated implication. Then we give the relationship between these two classes of fuzzy implications, that is, they are different except for some special cases. Moreover, we study some basic properties of N-vertical generated implications. Finally, we focus on the distributivity of this class of implications over t-norms and t-conorms, and obtain the nice preservation of distributivity from given implications to the corresponding N-vertical generated implication.

A comprehensive study of upward fuzzy preference relation based fuzzy rough set models: Properties and applications in treatment of coronavirus disease.

In this paper, we first introduce a new type of rough sets called α‐upward fuzzified preference rodownward fuzzy preferenceugh sets using upward fuzy preference relation. Thereafter on the basis of α‐upward fuzzified preference rough sets, we propose approximate precision, rough degree, approximate quality and their mutual relationships. Furthermore, we presented the idea of new types of fuzzy upward β‐coverings, fuzzy upward β‐neighborhoods and fuzzy upward complement β‐neighborhoods and some relavent properties are discussed. Hereby, we formulate a new type of upward lower and upward upper approximations by applying an upward β‐neighborhoods. After employing the upward β‐neighborhoods based upward rough set approach to it any times, we can only get the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Subsequently, we presented the idea to combine the fuzzy implicator and t‐norm to introduce multigranulation (ℐ,T)‐fuzzy upward rough set applying fuzzy upward β‐covering and some relative properties are discussed. Finally we presented a new technique for the selection of medicine for treatment of coronavirus disease (COVID‐19) using multigranulation (ℐ,T)‐fuzzy upward rough sets.

Some notes on the category of fuzzy implications on bounded lattices

Some notes on the category of fuzzy implications on bounded lattices

Characterizations and Applications of Fuzzy Implications Generated by a Pair of Generators of T-Norms and the Usual Addition of Real Numbers

Motivated by several problems on constructions, algebraic properties, in particular the law of importation (LI) and the flexible ordering property (FOP), and applications in approximate reasoning, the present article defines a new class of fuzzy implications, called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications, by a pair of multiplicative and additive generators of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -norms and the usual addition instead of the multiplication or division usually used in the literature. The relations to other generator generated implications are clarified, the intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(S,N)$</tex-math></inline-formula> -implications are discussed in detail and the intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R$</tex-math></inline-formula> -implications are proved to be conjugates of the Łukasiewicz implication. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -norm solutions to the (LI) equation for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications in several cases are characterized in terms of generated t-norms, and two special subclasses of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications are characterized through (LI) and (FOP) on the other hand. A particularly interesting corollary shows that a binary operation on [0,1] enjoying (OP) satisfies (LI) with the Łukasiewicz <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -norm if and only if it is the Łukasiewicz implication. These results will partially solve or enrich the related open problems pending for fuzzy implications. As applications of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications, two fuzzy reasoning methods, called triple I method and parallel hierarchical triple I method, based on (LI) and (FOP) are proposed in order to set a logic foundation for Zadeh’s compositional rule of inference (CRI) and to remedy the dependency of Jayaram’s hierarchical CRI on the order of inputs, respectively.

On an Intuitionistic Fuzzy Form of the Goguen’s Implication

In the present paper we construct a new intuitionistic fuzzy implication, giving intuitionistic fuzzy form to Goguen’s implication. Some of its basic properties are studied and illustrated with examples. Geometrical interpretations of the different forms of the new implication are given. Other forms of the Goguen’s implication are discussed.

Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables

In this paper, we present a new Fuzzy Implication Generator via Fuzzy Negations which was generated via conical sections, in combination with the well-known Fuzzy Conjunction. The new Fuzzy Implication Generator takes its final forms after being configured by the fuzzy strong negations and combined with the most well-known fuzzy conjunctions TM, TP, TLK, TD, and TnM. The final implications that emerge, given that they are configured with the appropriate code, select the best value of the parameter and the best combination of the fuzzy conjunctions. This choice is made after comparing them with the Empiristic implication, which was created with the help of real temperature and humidity data from the Hellenic Meteorological Service. The use of the Empiristic implication is based on real data, and it also reduces the volume of the data without canceling them. Finally, the MATLAB code, which was used in the programming part of the paper, uses the new Fuzzy Implication Generator and approaches the Empiristic implication satisfactorily which is our final goal.

A Fuzzy Take on the Logical Issues of Statistical Hypothesis Testing

Statistical Hypothesis Testing (SHT) is a class of inference methods whereby one makes use of empirical data to test a hypothesis and often emit a judgment about whether to reject it or not. In this paper, we focus on the logical aspect of this strategy, which is largely independent of the adopted school of thought, at least within the various frequentist approaches. We identify SHT as taking the form of an unsound argument from Modus Tollens in classical logic, and, in order to rescue SHT from this difficulty, we propose that it can instead be grounded in t-norm based fuzzy logics. We reformulate the frequentists’ SHT logic by making use of a fuzzy extension of Modus Tollens to develop a model of truth valuation for its premises. Importantly, we show that it is possible to preserve the soundness of Modus Tollens by exploring the various conventions involved with constructing fuzzy negations and fuzzy implications (namely, the S and R conventions). We find that under the S convention, it is possible to conduct the Modus Tollens inference argument using Zadeh’s compositional extension and any possible t-norm. Under the R convention we find that this is not necessarily the case, but that by mixing R-implication with S-negation we can salvage the product t-norm, for example. In conclusion, we have shown that fuzzy logic is a legitimate framework to discuss and address the difficulties plaguing frequentist interpretations of SHT.

Multiobjective capacitated green vehicle routing problem with fuzzy time-distances and demands split into bags

Real-life challenges require proactive measures. Good transportation services and greener alternatives demand steadfast research in the area. Here, an attempt has been made to address such a problem. A particular case of VRP, with deliveries split into bags and triangular fuzzy travel times, has been modelled to minimise fuel emissions. The concepts of fuzzy rule-based implication for ranking and for comparing fuzzy numbers with numeric values, an expected value model, have been drawn upon. A discrete fuzzy-hybridised GA has been developed. Multiple experiments on data in existing works, parameter tuning, and comparative analysis have been performed, thereby corroborating the model's efficacy.

Rule extraction based on linguistic-valued intuitionistic fuzzy layered concept lattice

As one of the research tools for data processing and knowledge discovery, concept lattice can effectively extract information. In daily life, due to the ambiguity and uncertainty of the decision environment, different experts may provide different evaluation information according to individual needs which may be expressed with linguistic values. In order to handle these problems, we study the rule extraction method of linguistic-valued intuitionistic fuzzy layered concept lattice. First, we propose a linguistic-valued intuitionistic fuzzy formal decision context based on the intuitionistic fuzzy lattice implication algebra, which can simultaneously process the obtained comparable and incomparable linguistic information from both positive and negative aspects. Furthermore, by setting different linguistic-valued trust degrees, we put forward the linguistic-valued hierarchical concept construction operator. On this basis, a linguistic-valued intuitionistic fuzzy layered concept lattice can be constructed to meet the requirements of different experts at different levels. And then through the relationship between the conditional concept and decision concept in the obtained concept set, we get a rule set of the formal decision context. Finally, we present a rule extraction method to help people make more reasonable decisions, combining the confidence and support degree of linguistic-valued intuitionistic fuzzy decision rules. And a practical example involving individual financial investment decision-making is used to verify the efficiency and applicability of the proposed approach.