New Intuitionistic Fuzzy Operations

Abstract

When discussions for the name “IFS” started and I understood that the general critique is that over the IFSs we can define operation “negation” that is not classical, I started searching the implications and negations that have non-classical nature. My first step (naturally, not in the most suitable direction) was to introduce the operators, defined in Section 5.8. As it is mentioned there, one of them satisfies axiom A → ¬¬A, but does not satisfy axiom ¬¬A → A, and another satisfies axiom ¬¬A → A, but does not satisfy axiom A → ¬¬A, where operation ¬ is changed with one of these operators. On one hand, these operators are difficult to use, and on another – they are not close to the idea for the operation “negation”. So, I started searching for another form of implication and of negation-type of operators, and in a series of papers [51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 67, 81, 89, 92, 97, 109, 110, 543] more than 140 different operations for implication and more than 35 different operations for negation are defined. The research is in two aspects: logical and set-theoretical. Firstly, we give short remarks on the logical aspect. I hope that in future these results will be a basis for an independent book. After this, the basic results related to implication and negation operations over IFSs will be discussed and their basic properties will be described.

In this chapter we stop using the overline symbol for operation “negation” and instead we use symbol ¬. Since,a lot of different negations are introduce here, they are indexed ¬1, ¬2,... . Practically, the overline symbol coincides with the classical negation ¬1.