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Abstract In Massanet et al. (2017) [1] a new property of fuzzy implication functions, called the invariance property with respect to powers of a continuous t-norm, was introduced and its application in approximate reasoning was studied. In the same paper, the novel family of power based implications was presented as a family of fuzzy implication functions that satisfy such invariance property. Unfortunately, this fact is not entirely true since as we will prove in this paper, there exist some power based implications generated from specific ordinal sum t-norms which do not fulfil the invariance property. Thus, a characterization of which continuous t-norms can be used to generate power based implications satisfying the power based invariance property is presented. Additionally, as an alternative solution, we introduce in this paper a slight modification of this property in such a way that all power based implications satisfy it. [ABSTRACT FROM AUTHOR]

The modification (relaxation or intensification) of the antecedent or the consequent in a fuzzy “If, Then” conditional is an important asset for an expert in order to agree with it. The usual method to modify fuzzy propositions is the use of Zadeh's quantifiers based on powers of t-norms. However, the invariance of the truth value of the fuzzy conditional would be a desirable property when both the antecedent and the consequent are modified using the same quantifier. In this paper, a novel family of fuzzy implication functions based on powers of continuous t-norms which ensure the aforementioned property is presented. Other important additional properties are analyzed and from this study, it is proved that they do not intersect the most well-known classes of fuzzy implication functions. [ABSTRACT FROM AUTHOR]

In a group decision making problem, experts often express their opinions through the so-called preference relations. In recent years, several different definitions of preference relations depending on the framework and the nature of the problem have been introduced. These approaches vary from interval-valued fuzzy preference relations to incomplete fuzzy linguistic preference relations. In this paper, a novel definition of preference relation, the so-called subjective linguistic preference relation, is proposed. These preference relations are based on the concept of subjective evaluations, introduced in the linguistic computational model based on discrete fuzzy numbers. In this framework, the experts have more flexibility to express their opinions and the solid mathematical background of this model is a guarantee of no loss of information. Finally, an example of a multi-expert decision making problem with a hierarchical multi-granular linguistic context is analyzed to illustrate the potential of the proposed method and its advantages with respect to other methods. [ABSTRACT FROM AUTHOR]

In fuzzy logic and approximate reasoning the inference rule given by the Modus Tollens usually derives into an inequality involving three logical operators: a conjunction, an implication function and a negation. Until now, in this scenario the conjunction has been commonly modeled by a t-norm, but recently the possibility of using a more general conjunction has been pointed out. In this work, we want to generalize the Modus Tollens inequality by using a conjunctive uninorm instead of a t-norm, leading to the so-called U -Modus Tollens. First, we give a study of this new property for implication functions in general and then we specially focus on residual implications derived from uninorms. In all cases, we prove that there are a lot of solutions of the U -Modus Tollens and we give a characterization of all the solutions in some particular cases. [ABSTRACT FROM AUTHOR]

In the last decade, several new construction methods of fuzzy implication functions from one or more given ones have been proposed. Following this line of research, in our paper some construction methods based on polynomial functions of three variables are presented. Concretely, these methods provide a (possibly new) fuzzy implication function from a given one and a quadratic polynomial function. A complete characterization of those quadratic functions adequate to obtain a fuzzy implication function through this strategy is presented. Moreover, the invariant fuzzy implication functions with respect to this method are determined. Finally, some quadratic construction methods preserving some specific additional properties are analysed. [ABSTRACT FROM AUTHOR]