The law of contraposition is one of the most well-known tautologies in classical logic. In addition, there are two other notions of contrapositions: the law of right contraposition and the law of left contraposition. In fuzzy logic, these tautologies are modeled by a fuzzy implication and a fuzzy negation. The characterization of fuzzy implications satisfying the laws of right or left contraposition with respect to an arbitrary fuzzy negation is a problem that has gained attention recently. This article aims to propose characterizations of such implications, presenting necessary and sufficient conditions under which a fuzzy implication satisfies these laws with respect to an arbitrary fuzzy negation.

A compatibility-based characterization of fuzzy implications over 2-uninorms satisfying the law of importation

Migrativity properties of uninorms over fuzzy implications

In fuzzy set theory, the migrativity of aggregation functions stands out as a noteworthy subject from both theoretical and practical perspectives. The present paper undertakes a thorough study of the migrativity of fuzzy implications over uni-norms and null-norms. Firstly, leveraging the ordinal sum representations of t-norms and t-conorms, we investigate the migrativity of fuzzy implications over t-norms and t-conorms, which shows some positive results. Further, we characterize the migrative functional equation for fuzzy implications over uni-norms in the usual classes such as uni-norms locally internal on the boundary, uni-norms continuous on the boundary except at their neutral elements, representable uni-norms, idempotent uni-norms, uni-norms continuous in the open unit square, and uni-norms with continuous underlying operators. The results indicate that the structure of the solutions of the migrative functional equation is closely related to the natural negation of fuzzy implications. Finally, we explore the migrativity of fuzzy implications over null-norms.

Fuzzy implication operators are vital in the modeling of uncertain reasoning, particularly in approximate reasoning and fuzzy inference systems. The objective of this survey is to provide a structured and comprehensive overview of fuzzy implication, intuitionistic fuzzy implication, and hesitant fuzzy implication. We examine their properties, representations, and characterizations. We discuss a number of findings about fuzzy negations, fuzzy implications, intuitionistic fuzzy implication, and hesitant fuzzy implications, including their characterizations with respect to the identity principle and ordering property, which lead to fundamental results.

A study of fuzzy polynomial implications with respect to the existing generating methods

In this paper, we investigate the discretization of fuzzy implications using rounding functions. Our discretization method is a unified framework of the upper and lower discretization methods of Munar-Covas et al. Furthermore, we examine the extent to which the essential properties of these fuzzy implications are preserved in our discretization process.

Representations of the fuzzy implications satisfying the law of left or right contraposition

This research investigates the identification problem of fuzzy systems represented by fuzzy relational equations and TSK-type fuzzy models under uncertainty. The research object is the nonlinear dynamic model of a steam generator of a thermal power plant, for which accurate modeling is essential due to its complex behavior. The scientific problem addressed in the article is determining the optimal fuzzy implication and developing an identification algorithm that minimizes modeling error for nonlinear technological objects. An identification approach based on max–min composition is constructed using a fuzzy rule base to model input–output relationships. Structural and parametric identification procedures are formulated to select the criteria, parameters, and structural components of the fuzzy difference model. Several nonlinear control algorithms and multiple implication types are tested on the steam generator model. Experimental analysis shows that the ALI1 implication achieves the minimum mean square error among the evaluated implications, providing more accurate fuzzy relational mapping. The obtained results improve the quality of fuzzy system identification and enable the synthesis of an efficient fuzzy control strategy for nonlinear industrial processes. The developed method can be practically applied in real-time modeling, control, and optimization of thermal power plant units

The formula ¬A = (A → ((A → A) ∧ ¬(A → A))) is a tautology in the classical propositional logic. In this paper, we determine all intuitionistic fuzzy implications that satisfy this formula together with the classical intuitionistic fuzzy negation or with the negation generated by this implication.

This paper presents a unified framework for constructing two-branched fuzzy implications and families of copulas based on the same composition principles involving monotone and convex functions. The proposed methodology yields operators with a genuine dual structure, where each branch satisfies distinct boundary and monotonicity conditions while remaining consistent with the general axioms of copulas. By systematically combining monotone generators with convex transformations, new families of fuzzy implications and copulas are obtained, both exhibiting enhanced analytical properties such as strengthened two-increasing behavior, adjustable dependence strength, and flexible convexity with continuous transitions. Convexity ensures the two-increasing property, while continuity guarantees the completeness and mathematical soundness of the constructions. Remarkably, certain copulas produced under this framework display Archimedean-like features—symmetry and associativity—thus providing new theoretical instruments for the advancement of fuzzy logic and dependence modeling.

The Bandler-Kohout subproduct (BKS) method acts as one of the two representative fuzzy relational inference (FRI) strategies. Observing the BKS method using constraint modeling, two fuzzy implications, respectively, produce expression to the factors of inference mechanism and rule base. However, these two factors normally reflect different connotations from the perspectives of artificial intelligence applications and logical meaning. Enlightened by such idea, in this study, we propose and investigate the differently implicational BKS (DBKS) method. Initially, main properties of DBKS are validated. The reversibility and interpolativity of DBKS are proved under certain conditions. The equivalent relationship is verified between interpolativity and continuity for DBKS. The robustness of DBKS is confirmed from both the similarity and the extensional hull. Posteriorly, the computational performance of DBKS is analyzed. In DBKS, the preservation of the indistinguishability holds for input fuzzy sets, and it is proved that the first-aggregate-then-infer (FATI) reasoning strategy of DBKS is equivalent to the first-infer-then-aggregate (FITA) one. To improve the computational efficiency, the hierarchical DBKS method is presented. In addition, the fuzzy system is established on the strength of the DBKS method, the singleton fuzzifier and the centroid defuzzifier. Its response function is analyzed and a universal approximator is built by the fuzzy system via DBKS. At the end, we compare the results of DBKS with BKS by virtue of two examples in affective computing. It is discovered that DBKS can create superior forms of FRI in comparison to those produced by BKS.

Weak implications as ordinal sums of fuzzy implications and co-implications

Fuzzy Inference Systems (FIS) are used to help people to take decisions in complex situations or when a human expert is needed. Their particularity is that they can manage the imprecision and vagueness of knowledge by applying approximate reasoning. The main approach of approximate reasoning is the Compositional Rule of Inference (CRI), whose definition contains two operators as parameters: a \(t\)-norm and a fuzzy implication. However, since its creation, the fuzzy community considers only one combination of (\(t\)-norm, implication) in fuzzy applications, which is (min, min). For that, we are interested in studying the behavior of other combinations (\(t\)-norm, implication) and in checking their efficiency. In this paper, we combine the product \(t\)-norm with fifteen implications in the CRI. Then, for every combination, we check the satisfaction of the axiomatics of approximate reasoning. This axiomatics is a set of criteria that model human intuitions. This study allows us to identify the best combinations that coincide with human reasoning in order to guarantee an inference result close to the expert's opinion.

A feature selection method driven by fuzzy implication granularity

In this paper, we investigate intuitionistic fuzzy WSBG-ideals and intuitionistic fuzzy implicative WSBG-ideals within the framework of Sheffer stroke BG-algebras. We establish new algebraic structures that extend classical Boolean and BG-algebra frameworks by synthesizing intuitionistic fuzzy set theory, introduced by Atanassov, with the Sheffer stroke operation. We demonstrate a fundamental connection between intuitionistic fuzzy implicative WSBG-ideals and their level sets, showing that the level set of an intuitionistic fuzzy implicative WSBG-ideal corresponds to an implicative WSBG-ideal of the Sheffer stroke BG-algebra. Furthermore, we explore the properties of intuitionistic fuzzy WSBG-ideals, proving that every intuitionistic fuzzy implicative WSBG-ideal is also an intuitionistic fuzzy WSBG-ideal. However, the converse does not always hold. This work provides new insights into the algebraic properties of Sheffer stroke BG-algebras, enabling novel reasoning methods under uncertainty and paving the way for further applications in fuzzy logic and computational models.

It is well known that monotonicity has been an important defining criterion for fuzzy logic connectives, such as fuzzy negations, t-norms, t-conorms and fuzzy implications.Also, a stronger version of monotonicity, namely strict monotonicity, establishes some significant representation theorems of continuous fuzzy negations, continuous t-norms and continuous tconorms.In this work, we propose the strict monotonicity for fuzzy implications and investigate some necessary conditions on fuzzy implications to fulfill the same.Also, the relationship between the basic properties, functional equations of fuzzy implications and the strict monotonicity will be investigated.Further, we examine the strict monotonicity for fuzzy implications that do come from different families of fuzzy implications and show that the strict monotonicity is a necessary condition for fuzzy polynomial implications, fuzzy rational implications and some subclasses of (S, N ) and f -generated fuzzy implications.

Soft set theory was first introduced by D. A. Molodtsov [D. Molodtsov, Soft set theory first results, Computers & Mathematics with Applications, 37(4–5) (1999) 19–31] in 1999 as an effective approach to deal with uncertainty. Over time, fuzzy soft set theory has evolved significantly, integrating concepts from both fuzzy set theory and soft set theory to enhance decision-making processes. Acharjee and Medhi [S. Acharjee and S. Medhi, Logical connectives of fuzzy soft set theory, New Mathematics and Natural Computation, 21(1) (2025) 339–352] introduced several advanced concepts related to logical connectives in fuzzy soft set theory. Building on Molodtsov’s foundational work, this paper explores new developments in fuzzy soft logic, extending the findings of Acharjee and Medhi. Specifically, we introduce the concepts of [Formula: see text]-fuzzy soft implication and [Formula: see text]-fuzzy soft bi-implication, along with their related results within the framework of fuzzy soft logic.

Research on the construction of logical connectives using total (admissible) orders is a prolific area of study. Using such orders, a new method for constructing implication functions is defined on the set of discrete fuzzy numbers with support of a closed interval of a given finite chain and whose membership values belong to a finite set of fixed values. This method is based on the use of discrete implication functions defined on a finite chain. Furthermore, a bijective correspondence between the set of implication functions on the aforementioned subset of discrete fuzzy numbers and the set of discrete implication functions defined on the discrete chain is shown. Basic properties of these implication functions are thoroughly investigated, concluding that they are preserved under the proposed construction method. This result highlights the robustness and generality of the method, providing a systematic way to extend discrete implication functions to more complex structures while preserving their underlying properties.

Cross-migrative property of disjunctive uninorms over fuzzy implications