On Intuitionistic Fuzzy Sets Theory

The origin of my idea of intuitionistic fuzziness was a happenstance – as a mathematical game. I read the Russian translation of A. Kaufmann’s book [301] and decided to add to the definition, a second degree (degree of nonmembership) and studied the properties of a set with both degrees. Of course, I observed that the new set is an extension of the ordinary fuzzy set, but I did not immediately notice that it has essentially different properties. So the first research works of mine in this area followed, step-by-step, the existing results in fuzzy sets theory. Of course, some concepts are not so difficult to extend formally. It is interesting to show that the respective extension has specific properties, absent in the basic concept.

In this Section, operations and relations over IFSs extending the definitions of the relations and operations over fuzzy sets (see e.g. [218, 297, 301, 612]) are introduced. Conversely, the fuzzy sets relations and operations will turn out to be particular cases of these new definitions.

In this Chapter, several geometrical interpretations of the IFSs are discussed.

This Chapter includes several operators over IFSs which have no counterparts in the ordinary fuzzy set theory.

Initially, following [11, 39], we introduce two operators over IFSs that transform an IFS into a fuzzy set (i.e., a particular case of an IFS). They are similar to the operators “necessity” and “possibility” defined in some modal logics. Their properties resemble these of the modal logic (see e.g. [227]).

Following [18, 19, 39], we construct an operator which represents both operators □ from (4.1) and \(\diamondsuit\) from (4.2). It has no analogue in the ordinary modal logic, but the author hopes that the search for such an analogue in modal logic will be interesting.

Following the idea of a fuzzy set of level α (see, e.g. [301]), in [39] the definition of a set of (α,β)-level, generated by an IFS A, has been introduced, where α, β ∈ [0,1] are fixed numbers for which α + β ≤ 1.

First, let us emphasize that here we do not study the usual set-theoretic properties of the IFSs (i.e. properties which follow directly from the fact that IFSs are sets in the sense of the set theory – see Section 2.1). For example, given a metric space E, one can study the metric properties of the IFSs over E. This can be done directly by topological methods (see e.g. [421]) and the essential properties of the IFSs are not used. On the other hand, all IFSs (and hence, all fuzzy sets) over a fixed universe E generate a metric space (in the sense of [421]), but with a special metric (cf., e.g., [301]): one that is not related to the elements of E and to the values of the functions μ _A and ν _A defined for these elements.

First, we define six versions of another operation over IFSs – namely, Cartesian products of two IFSs. To introduce the concept of intuitionistic fuzzy relation, we use these operations.

Let E ₁ and E ₂ be two universes and let be two IFSs over E ₁ and over E ₂, respectively.

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.

An year after defining IFSs, in 1984, Stefka Stoeva and the author extended the concept of IFS to “Intuitionistic L-fuzzy set”, where L is some lattice, and they studied some of its basic properties [103]. All these results are included in [39] and hence, they not be discussed here.

Let me start with a recollection of my first acquaintance with Prof. Lotfi Zadeh. It was in 2001 in Villa Real, Portugal, where Prof. Pedro Melo-Pinto organized a school on fuzzy sets. Prof. Zadeh was invited for a 3-hour lecture, which he concluded with presentation of slides with articles by Samuel Kleene, Kurt Gödel and other luminaries of mathematical logic, who have written against the fuzzy sets. The fact that the sublime mathematician and logician Gödel had sometimes made slips in his judgments can be confirmed by the cosmologists, yet I was astonished by his opinion. Of course, nowadays, when we are aware of the enormous number of publications in the field of fuzzy sets, as well as of the various impressive applications of these, it is easy to say that Gödel had mistaken. However, I have been long tormented by the question why these mathematicians had opposed the fuzzy sets while they did not have anything against the three- and multi-valued logics of Jan Lukasiewicz. Thus I reached the conclusion that the reason for the then negative attitude towards fuzzy sets was hidden in the presence of the [0, 1] interval as the set of the fuzzy sets’ membership function (see, e.g, [301, 592, 593]).