Since the seminal paper of fuzzy set theory by Zadeh in 1965, many extensions have been proposed to overcome the difficulty for assigning the membership degrees. In recent years, a new extension, the hesitant fuzzy sets, has attracted a lot of interest due to its usefulness to handle those problems in which it is difficult to provide accurately a single membership value; since for hesitant sets, membership values are given by a whole set of values. On the other hand, since fuzzy negations have an important role in applications as well as in the theoretical approach to of fuzzy logics, it is important to study an extension of the concept of fuzzy negation for hesitant fuzzy degrees (elements). In this paper, we propose such a definition and we study some of the main properties of this new concept.

The main contribution of this paper is to introduce an autonomous definition of the connective “fuzzy exclusive or” (fuzzy Xor, for short), which is independent from others connectives. Also, two canonical definitions of the connective Xor are obtained from the composition of fuzzy connectives, and based on the commutative and associative properties related to the notions of triangular norms, triangular conorms and fuzzy negations. We show that the main properties of the classical connective Xor are preserved by the connective fuzzy Xor, and, therefore, this new definition of the connective fuzzy Xor extends the related classical approach. The definitions of fuzzy Xor-implications and fuzzy E-implications, induced by the fuzzy Xor connective, are also studied, and their main properties are analyzed. The relationships between the fuzzy Xor-implications and the fuzzy E-implications with automorphisms are explored