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The aim of this short paper is to give a simple look to the historical development of logical connectives for fuzzy sets and fuzzy logic. Concepts that have been (and still are) in the core of extensive theoretical research, like conjunction, disjunction, complement, subsethood and fuzzy conditionals, are considered in an informal way. An extensive list of essential references helps the interested reader to find sources for deeper study of the main subjects. [ABSTRACT FROM AUTHOR]

In this article discrete fuzzy numbers are used to model complete and incomplete qualitative information and some methods to aggregate this kind of information are proposed. When the support of discrete fuzzy numbers is a closed interval of the chain L n = { 0 , 1 , … , n } , they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms. On the other hand, when the support is not an interval of L n , the corresponding discrete fuzzy number can be interpreted as an incomplete linguistic expert valuation. In order to aggregate this kind of information, we propose two different methods. One of them deals with the construction of aggregation functions on the set of discrete fuzzy numbers from discrete aggregation functions defined on L n . The other one presents several procedures for estimating the missing information based on the so-called discrete associations . Finally, the proposed aggregation methods are used in a multi-expert decision making problem and a concrete example is given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a method to construct aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers from a couple of discrete aggregation functions. The interest on these discrete fuzzy numbers lies on the fact that they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms. Finally, a linguistic decision making model based on a pair of aggregation functions defined on discrete fuzzy numbers is given. [Copyright &y& Elsevier]

Abstract: In recent years, some new construction methods of fuzzy implications from other given ones have been proposed. One of them, the so-called threshold generation method, preserves important properties such as the exchange principle or the law of importation under some minimal conditions. This method is based on an adequate scaling on the second variable of the two initial fuzzy implications. In this paper, we propose a new method to generate fuzzy implications from two given ones in the same spirit of the threshold generation method but now through an adequate scaling on the first variable of the given fuzzy implications. The new implications, called vertical threshold generated implications, are deeply studied focusing on the preservation of the most common properties of fuzzy implications from the initial ones to the generated implication. Moreover, they are fully characterized by means of the -vertical section of the implication. [Copyright &y& Elsevier]

Abstract: In this paper, a new construction method of a fuzzy implication from two given ones, called threshold generation method, is introduced. It is a generalization of the way of construction of the recently introduced h-implications, which are fully characterized in this work. The threshold generation method allows to control, up to a certain level, the increasingness on the second variable of the fuzzy implication through an adequate scaling on that variable of the two given implications. The natural propagation of the most usual properties of fuzzy implications from the initial ones to the constructed implication is studied and the necessary and sufficient conditions in order to ensure this propagation are presented. In particular, the preservation of the contrapositive symmetry on threshold generated implications needs of another construction method of a fuzzy implication from a given one and a fuzzy negation. [Copyright &y& Elsevier]

Abstract: In this paper, the recently introduced construction method of a fuzzy implication from two given ones, called threshold generation method, is studied. This method preserves several of the most usual properties of fuzzy implications from the initial ones to the threshold generated implication. In particular, the preservation of the exchange principle, the law of importation and the distributivities with t-norms and t-conorms, stating the minimal necessary conditions to ensure this fact is studied. From these results, some properties of h-implications, as a particular case of threshold generated implications, are pointed out. [Copyright &y& Elsevier]

Abstract: In this paper aggregation functions defined on the set of all discrete fuzzy numbers whose support is a subset of consecutive natural numbers are introduced and the particular cases of uninorms and nullnorms are studied in detail. These aggregation functions are constructed from discrete aggregation functions (defined on a finite chain) and they are applied to the aggregation of subjective evaluations. [Copyright &y& Elsevier]

Abstract: Some open problems on fuzzy implications dealing with the so-called importation law are studied and totally or partially solved in this work. A weaker version of the law of importation, called the weak law of importation, is introduced. The relationships of these two properties and the exchange principle are studied. In particular, it is proved that the law of importation is stronger than the exchange principle. On the other hand, the three properties are equivalent for some kind of fuzzy implications, those that satisfy a boundary property. Along this study, new characterizations of (S,N)-implications, R-implications and their counterparts for uninorms based on the weak law of importation are showed. [Copyright &y& Elsevier]

Although the concept of reverse is given in (Publ. Math. Debrecen 12 (1973) 21–39) not only for a t-norm but also for any associative operator, the definition does not work in so general a case. In this paper, the concept of reverse is extended in an appropriate way to some kinds of associative operators like t-conorms, uninorms and t-operators. All continuous and reversible t-conorms are characterized and, after some results, an analogous characterization is also given for all semi-continuous reversible uninorms and all continuous reversible t-operators. In the last case, reversibility becomes a kind of transformation that converts a uninorm into a t-operator and viceversa. Finally, related to reversibility, uninorms and t-operators of moderate growth are studied. [Copyright &y& Elsevier]

• A new approach to Z-numbers, called mixed-discrete Z-numbers, is introduced. • The second component of mixed-discrete Z-numbers is managed as a discrete fuzzy number. • The computational complexity of the operations is greatly reduced with this approach. • An application to reach a decision from a group of experts is presented. One of the main goals of computing with words is the accurate modeling of natural language. In this direction, Z-numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers, (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as a measure of certainty or sureness of A. This structure allows to model many imprecise sentences of the natural language, but has the drawback of the complexity and hight computational cost of their operations, because the second component is usually considered from a probabilistic point of view. Since the computational problems are caused by the second component, we present in this paper a new approach called mixed-discrete Z-numbers. In this new approach the first component will be managed as a usual fuzzy number, and the second one as a discrete fuzzy number with support in a finite chain. That is, the second component B of a Z -number is modeled as a linguistic valuation based on a discrete fuzzy number and the operations on these second components are managed through aggregation functions on discrete fuzzy numbers. Understanding B as a measure of certainty and not as a measure of probability, greatly improves experts' flexibility, allows to model situations where no probability distribution is known, and reduces greatly the computational complexity of Z -numbers operations. After studying these new Z-numbers and their operations, an application to reach a decision from a group of experts is presented in order to show the potential of this approach. [ABSTRACT FROM AUTHOR]

Abstract Recently, the so-called family of T -power based implications was introduced. These operators involve the use of Zadeh's quantifiers modelled by powers of t-norms in its definition. Due to the fact that Zadeh's quantifiers constitute the usual method to modify fuzzy propositions, most of the members of this family of fuzzy implication functions satisfy an important property in approximate reasoning such as the invariance of the truth value of the fuzzy conditional when both the antecedent and the consequent are modified using the same quantifier. In this paper, an in-depth analysis of this property is performed by characterizing all binary functions satisfying it. From this general result, a fully characterization of the subfamily of T -power based implications which satisfy the invariance property is presented. Furthermore, a characterization of the whole family of T -power based implications is proved in which a novel property which slightly resembles the hypothetical syllogism plays a key role. [ABSTRACT FROM AUTHOR]

Abstract In any fuzzy rules based system the inference management is usually carried out by the so-called fuzzy implication functions. In this framework, the Modus Ponens property becomes essential to make forward inferences and it is well known that this inference rule is guaranteed when the conjunction and the implication function used in the process satisfy the corresponding functional inequality. Such inequality has been extensively studied for many kinds of implication functions when the conjunction is modelled by a t-norm. However, the use of conjunctive uninorms to model conjunctions is an increasingly widespread option in fuzzy systems and for this reason the study of the Modus Ponens with respect to a conjunctive uninorm U instead of a t-norm T , which we call here U -Modus Ponens or U -conditionality, becomes very important. In this paper this new property is deeply analyzed and it is shown that usual implications derived from t-norms and t-conorms do not satisfy it, but many solutions appear among those implications derived from uninorms. In particular, the U -Modus Ponens for the case of residual implications derived from uninorms, or RU -implications, is investigated in detail when the uninorm U lies in any of the four most usual classes of uninorms. [ABSTRACT FROM AUTHOR]

In this paper all uninorms locally internal in the region A ( e ) (given by the complement in [ 0 , 1 ] 2 of [ 0 , e ] 2 ∪ [ e , 1 ] 2 , where e is the neutral element of the uninorm) having continuous underlying operators are studied and characterized, by distinguishing some cases. When the underlying t-norm and t-conorm are not given by ordinal sums, it is proved that uninorms locally internal in A ( e ) are in fact all possible uninorms with these underlying operators (except when both the t-norm and the t-conorm are strict in which case there is also the class of representable uninorms), leading to a finite number of possibilities. When at least one of the continuous underlying operators is given by an ordinal sum, again there are other possible uninorms than those that are locally internal in A ( e ) , but all uninorms with this property are also characterized. In this case, infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm. [ABSTRACT FROM AUTHOR]

Abstract Probabilistic and survival implications are two kinds of fuzzy implication functions that combine the imprecision modelled by fuzzy concepts and the imprecision modelled by the probability theory. Both kinds of fuzzy implication functions are derived from copulas through two different construction methods, and since their introduction in 2011 and 2012 respectively, they have been deeply studied. In this paper an axiomatic characterization of both families is given and it is proved that both families coincide. The mentioned characterizations are obtained by reversing these construction methods in order to obtain copulas from fuzzy implication functions. As it is expected the so-called 2-increasingness property on fuzzy implication functions plays an important role in the characterization theorems. [ABSTRACT FROM AUTHOR]

Distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives. For this reason, the distributivity property has been extensively studied for several families of operations like triangular norms and conorms, some kinds of uninorms and nullnorms (also called t-operators) and even for some generalizations of them. In this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms, leading to many new solutions. [ABSTRACT FROM AUTHOR]

This paper deals with a more general way of defining conjectures than that presented in Trillas et al. Here conjectures are defined once an operator of consequences in the sense of Tarski is given, and like in Trillas et al. they result classified in consequences, hypotheses and speculations. With all that, it appears for the first time the actually non-surprising idea that a formalization of deduction seems to be required previously to formalize induction and abduction. [ABSTRACT FROM AUTHOR]

In this paper we characterize all idempotent uninorms defined on a finite ordinal scale. It is proved that any such discrete idempotent uninorm is uniquely determined by a decreasing function from the set of scale elements not greater than the neutral element to the set of scale elements not smaller than the neutral element, and vice versa. Based on this one-to-one correspondence, the total number of discrete idempotent uninorms on a finite ordinal scale of n + 1 elements is equal to 2n. [ABSTRACT FROM AUTHOR]