Recently, two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications were introduced by Grzegorzewski [6, 7, 9]. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In this paper we generalize these two classes and propose a new kind of construction methods for fuzzy implications which are ba...

Recently, Grzegorzewski [9] introduced two new families of fuzzy implication functions called survival implications and survival S-implications. These two classes of multivalued implications are based on conditional copulas. In the same article the author gave a motivation to his ideas and he analyzed some properties of these classes of fuzzy implications. In this paper the laws of contrapositi...

In fuzzy set theory, comparison of fuzzy sets plays an important role. Among the several ways to compare fuzzy sets, we address the logical theoretic approach using fuzzy implications. We propose a general framework allowing to generate many measures of comparison: inclusion, similarity and distance, and study their properties. Since the literature on the use fuzzy implications for defining suc...

The characterization of S-implications generated from strong negations presented firstly by Trillas and Valverde in 1985 is wellknown in the literature. In this paper we show that some assumptions are needless and present two characterizations of S-implications with mutually independent requirements. We also present characterizations of (S,N)-implications obtained from continuous fuzzy negation...

It is proved that the single input and single output (SISO) fuzzy systems based upon genuine many-valued implications are universal approximators. It is shown theoretically that fuzzy control systems based upon genuine many-valued implications are equivalent to those based upon t-norm implications, the general approach to construct fuzzy systems is given. It is also shown that defuzzi8er based ...

Recently, Yager in the article “On some new classes of implication operators and their role in approximate reasoning” [12] has introduced two new classes of fuzzy implications called the f -generated and g-generated implications. Along similar lines, one of us has proposed another class of fuzzy implications called the h-generated implications. In this article we discuss in detail some properti...

In this work, by Zadeh’s extension principle, we extend representable uninorms and their fuzzy implications (coimplications) to type-2 fuzzy sets. Emphatically, we investigate in which algebras of fuzzy truth values the extended operations are type-2 uninorms and type-2 fuzzy implications (coimplications), respectively.

Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set I of all fuzzy implications. To this end, we propose a binary operation, denoted by ~, which makes (I,~) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup S of this monoid and using their represe...

In this paper we study a connection between quasicopulas and generator triples of fuzzy preference structures. We study also conditions under which fuzzy implications can be used to construct generator triples. As we show, an important property of fuzzy implications that enables us to construct a generator triple from a fuzzy implication, is the contrapositive symmetry.

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results ...