Fuzzy Sets and Systems 156 (2005) 492 – 495
www.elsevier.com/locate/fss
Discussion
Some notes on (Atanassov’s) intuitionistic fuzzy sets
Przemysław Grzegorzewski∗ , Edyta Mrówka
Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
Available online 22 June 2005
Abstract
The paper contains few comments and positions the paper on (Atanasov’s) intuitionistic fuzzy sets presenting
points related to terminology, connections with other mathematical structures and possible interpretations.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Intuitionistic fuzzy sets; Interval-valued fuzzy sets
1. Introduction
In conventional fuzzy set, a membership function assigns to each element of the universe of discourse a
number from the unit interval to indicate the degree of belongingness to the set under consideration. Since
Zadeh introduced fuzzy sets in 1965 [19], many new approaches and theories treating imprecision and
uncertainty have been proposed. In 1986, Atanassov [1] introduced the concept of an intuitionistic fuzzy
set which is characterized by two functions expressing the degree of belongingness and the degree of
nonbelongingness, respectively. This idea, which is a natural generalization of a standard fuzzy set, seems
to be useful in modelling many real life situations, like negotiation processes, etc. Another well-known
generalization of a standard fuzzy set is the so-called interval-valued fuzzy set. Generally, the idea of
interval-valued fuzzy sets was attributed to Gorzałczany [5] and Türkşen [18], but actually they appear
earlier independently in the papers by Grattan-Guiness [6], Jahn [13] and Sambuc [17]. In fact, these
two approaches are mathematically equivalent; however, they have arisen on different grounds and have
different semantics. For more details we refer the reader, e.g., to [2].
∗ Corresponding author. Tel.: +48 22 3735 78; fax: +48 22 3727 72.
E-mail addresses: [email protected] (P. Grzegorzewski), [email protected] (E. Mrówka).
0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2005.06.002
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Since many controversies have arisen around intuitionistic fuzzy sets recently, Dubois et al. [3] have
undertaken an attempt to clarify all misunderstandings, clarify terminology etc. This short note is just our
voice in the matter.
2. Terminology
We agree with the opinion expressed in [3] and by many other researchers that the term “intuitionistic
fuzzy sets” is very unfortunate because the structure suggested by Atanassov has nothing in common with
intuitionistic mathematics and logic. We also agree that — although many theoretical and applied paper
devoted to intuitionistic fuzzy sets have appeared — this name should be changed since it is so misleading
and generates superfluous polemics. However, taking into account the numerous papers published under
this unsuitable name, we suggest that looking for the more appropriate name for Atanassov’s sets would
be as desirable as maintaining the acronym IFS. Thus, besides new names suggested in [3], we propose
the following names:
•
•
•
•
•
•
incomplete fuzzy sets,
inaccurate fuzzy sets,
imperfect fuzzy sets,
indefinite fuzzy sets,
indeterminate fuzzy sets,
indistinct fuzzy sets.
In our opinion, all these names not only maintain the acronym IFS but they reveal much better the
underlying idea of Atanassov’s sets than the word “intuitionistic”.
3. Interpretation of intuitionistic fuzzy sets
Some of the critics of Atanassov’s sets argue that because of the correspondence between intervalvalued fuzzy sets and intuitionistic fuzzy sets, all considerations devoted to the letter are redundant. We
have to disagree with this argumentation since the mathematical equivalence is one thing and particular
semantics is another thing. Moreover, this very particular semantics is often what matters for applications.
For example, it is obvious that each function is a relation and each relation is a subset of a Cartesian
product. Hence, each function is a set and it would be possible to develop the whole theory using the
language of set theory only. However, it seems more natural to describe the relationship between two
variables in a functional form. Thus, to sum up, we have nothing against using different terminology and
apparatus for similar or even equivalent objects, provided that we are aware of these relationships. And
we are grateful to the authors of [3] because of their efforts for “purifying the scene”.
But, going back to semantics and possible interpretations, in our opinion Atanassov’s sets give us a
very natural tool for modelling preferences. Sometimes it seems to be more natural to describe imprecise
and uncertain opinions not only by membership functions. It is due to the fact that in some situations
it is easier to describe our negative feelings than the positive attitude. Even more, quite often one can
easily specify objects or alternatives one dislikes, but simultaneously cannot specify clearly what he really
wants. Let us consider a situation observed in a real estate agency. Very often a customer looking for
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P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 156 (2005) 492 – 495
an apartment is not convinced completely on the location and considers several variants. It is obvious
that some districts are more preferable than others, when there are also districts that customer dislikes. It
seems that intuitionistic fuzzy sets are very useful for modelling situations like this. For example, it may
happen that a person asked about his favorite district in Warsaw cannot definitely choose whether it is
Ochota, Mokotów or Żoliborz, but he feels sure that he hates Wola. Thus we may apply an intuitionistic
fuzzy set for modelling the preferences of a customer, where the membership function shows the degree
that a given district is the most preferred one, while the nonmembership function indicates the degree
that a given district should not be taken into consideration (see, e.g., [10–12]).
We meet a similar situation when comparing preferences expressed by means of orderings which
admit uncertainty due to imprecision, vagueness and hesitance, i.e., where we can indicate objects which
are surely better than others, which are surely worse than others, but we also have objects which are
indifferent and incomparable. In this case intuitionistic fuzzy sets give us also perfect and very natural
tools for modelling such improper orderings (see, e.g. [7–9]).
It is worth noting that the problem of representation and handling of bipolar information in possibility
theory framework was also considered by Dubois, Kaci and Prade [4].
4. Intuitionistic fuzzy sets and other structures
The authors of [3] have noticed, interval-valued fuzzy sets were rediscovered several times under other
names. Often they appeared on different ground and had their own semantics. To the list of such objects
mentioned in [3], we want to add the so called shadow sets suggested by Pedrycz [14] and developed
later together with Vukovich (see [15,16]). From the mathematical point of view, a shadow set A is in fact
nothing more than a particular interval-valued fuzzy set where a lower fuzzy set of AL and a upper fuzzy
sets of AU take values not in the interval [0, 1] but AL , AU : X → {0, 1} (or, in other words, a shadow
set is an intuitionistic fuzzy set such that its membership function A and nonmembership function A
are of the type A , A : X → {0, 1}). However, shadow sets have quite nice and natural interpretation
and thus there is nothing wrong in that they exist under their own name.
5. Conclusions
We are very grateful to the authors of [3] for beginning this — in our opinion — fruitful discussion
which tries to find a constructive solution to a dilemma concerning Atanassov’s sets. We believe that such
discussion could clarify most misunderstandings and clean the ground for further development of that
interesting theory.
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