Basic intuitionistic principles in fuzzy set theories and its extensions (A terminological debate on Atanassov IFS)

In this paper we contribute to the terminological debate about Atanassov's use of the term “Intuitionistic” in defining his structure based on ortho-pairs of fuzzy sets. In particular, we stress that it is defined as “intuitionistic” a negation which from one side does not satisfy a standard property of the intuitionistic Brouwer negation (contradiction law) and on the contrary asserts some principles rejected by intuitionism (strong double negation law and one of the de Morgan laws). An algebraic Brouwer negation is studied in the context of IFS showing that it can be induced from a Heyting implication. A similar situation occurs in the case of standard fuzzy sets (FS). Some conditions which allow one to distinguish from the algebraic point of view FS from IFS are treated. Finally, a particular subclass of IFS consisting of ortho-pairs of crisp sets (denoted by ICS) is studied, showing that shadowed sets can be algebraically identified with ICSs.