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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]

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]

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]

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]