WHAT IS FUZZY SET THEORY? Re: fuzzy and probability

Ellen Hisdal (ellen@ifi.uio.no)
Wed, 24 Feb 1999 22:45:14 +0100 (MET)

Response to email letter to fuzzy group from Nan-Chieh Chiu
njchiu@eos.ncsu.edu
Operations Research
North Carolina State University
Sat, 30 Jan 1999
Subject:Fuzzy and probability.
Also Lotfi's email letter to the fuzzy group,
and the discussion following it at the end of 1998.
WHAT IS FUZZY SET THEORY?
> Can anyone recommend a book, (or an article) that has a
> good discussion on the difference between fuzzy set theory
> and probability theory.

Dear Nan-Chieh Chiu,

Your question is a very important one. It has to do with the existence
of a serious ambiguity as to what fuzzy set theory is. This ambiguity
was also illustrated by Lotfi Zadeh's letter to the fuzzy group,
and the discussion following it at the end of 1998.

Personally I consider fuzzy set theory and fuzzy logic to be a theory
which tries to explain how human beings operate with linguistic values
of variables such as `tall', `young', `very young' etc..
And how human beings can assign partial grades of membership
(mu element of the real interval [0,1]) to an object
or to a numerical attribute value u in a class. For example,
a grade of membership of 0.5 to a height u=170cm in the class `tall woman'.

I think that all Fuzzy Settians will agree with me that the above
subject is an important part of Fuzzy Set theory; Lotfi Zadeh deserves
much honor for introducing the concept of a partial grade of membership
of an object (or attribute value) in a class in order to explain how
humans think and reason with the aid of linguistic values of variables.
Furthermore he deserves honor for drawing our attention to the
importance of taking uncertainty into account in all cases (everyday
and technological ones) in which exact values are unavailable.

However, for many Fuzzy Settians, there exists another part of
fuzzy set theory. This is the claim that the concept of a partial
grade of membership of an object or an attribute value in a class
does not have a probabilistic interpretation. I consider this claim
to be a pure DOGMA which is not worthy of a serious scientific theory.
The reason for this is that a careful probabilistic interpretation
of grades of membership gives more generally reasonable results for labels
with connectives (such as the label `tall OR medium') than the max-min
operators which LZ (Lotfi Zadeh) suggests for the connectives
(such as the max operator for OR, and min for AND).

Somewhat superficially stated, LZ's max-min fuzzy set theory replaces
the + and x (times) operators of probability theory by max and min
respectively. These operations give reasonable results in some cases,
such as for `tall OR VERY tall', and unreasonable ones in others
(e.g. for `tall OR medium' (OR=inclusive OR)
whose `max' membership curve has a sharp dip around the boundary between
`tall' and `medium').

The unreasonable cases disappear in the TEE model for grades of membership
(see ref [1] below). Again very superficially stated,
the TEE model interprets mu (u=170cm)
tall (mu with subscript `tall')
as P(tall|u), the probability that a woman of height 170cm will be
assigned the label tall by a subject.

This is in contrast to the usual comparison performed by Fuzzy Settians
between the grade of membership for `tall' on the one hand,
and P(u|tall) on the other. P(u|tall woman) is the probability that a
woman labeled `tall' has a measured numerical height value u.

It IS true that P(u|tall) and the grade of membership of u in the
class `tall (woman)' have generally quite different numerical values.
So do P(u|tall) and P(tall|u) for the same height u.

(P(u|tall) must sum up to 1 over all intervals in a (quantized) U universe
(or integrate to 1 in a continuous U universe). This does not hold for
P(tall|u)=[mu(u) for `tall']. P(tall|u) must, for a given u, sum up to 1
over all height labels which a subject uses.)

Because of the unreasonable values which can occur in LZ's max-min theory
for the connectives, many other operators for the connectives have
been have been suggested instead of max and min in the course of time.
Except for the TEE model, all the other suggestions carefully avoid a
probabilistic interpretation because of THE DOGMA. This sad dogmatic
attitude should be unacceptable from a scientific point of view.

I hope that this summary of the situation may be of some help to you.

Let me mention that some researchers deal with the extra complication
of replacing numerical probability values by linguistic ones. For example,
they may replace an approximate probability value of 0.95
by the linguistic value `very big', and use a membership curve for `VERY big'.
However, this complicated question has no connection with the basic issues
mentioned in this letter.

Finally let me add that fuzzy set theory and fuzzy logic also try to
be a generalization of classical logic (propositional calculus).
Instead of operating only with truth values True (grade of
membership 1) and False (grade of membership 1), truth values between 0
and 1 are also accepted in fuzzy logic. Such intermediate truth values
have been used before in many valued-logics. The best known of these
is perhaps the one due to Lukasiewicz [2] (pages 87 and 131),
It has quite a bit of similarities to fuzzy logic.

A logic which completely carries through the analogy between truth
values and probabilities is described in reference [3].

Best greetings,
Ellen Hisdal

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Ellen Hisdal | Email: ellen@ifi.uio.no
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REFERENCES

[1]
@incollection{ruan,
author = {Hisdal, E.},
title = {Open-Mindedness and Probabilities versus Possibilities},
booktitle={Fuzzy Logic Foundations and Industrial Applications},
publisher = {Kluwer Academic Publishers, Boston},
year = {1996},
editor = {Da Ruan},
pages = {27-55} }

[2]
@book{borkowsky,
editor = {Borkowsky, L.},
title = {Jan Lukasiewicz, Selected Works},
publisher = {North Holland},
year = {1970} }

[3]
@book{hisdal,
author = {Hisdal, Ellen},
title = {Logical Structures for Representation of Knowledge
and Uncertainty},
publisher = {Physica Verlag, A Springer-Verlag Company},
year = {1998} }

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