Fuzziness vs. probability

Vilem Novak (novakv.prf1.osu@prf1.osu.cz)
Tue, 19 Aug 1997 12:26:00 +0200 (MET DST)

Dear Prof. Woodall,

I have received two papers of you (An Overview of Comparisons Between
Fuzzy and Statistical Methods; A probabilistic alternative to fuzzy
logic controllers, IIE Trans 1997, 459-467) which open again the
problem of probability vs. fuzziness. Though a lot of work on this
topic still should be done, I hoped that the basic question has
already been solved. To my surprize, I see that this is not true. Let
me very briefly touch these questions again - I do not claim to give
complete answer. However, the basic distinction seems to me clear, and
the work to be done should especially clarify the details.

Both these concepts are mathematical characterization of situations in
which we regard the phenomena surounding us; they are concerned with
the amount of information we have at disposal (or can have at
disposal), which is limited (mostly in principle). Thus, fuzziness
(fuzzy set, fuzzy approach, fuzzy logic) is a quantification (possible
mathematization) of the VAGUENESS phenomenon. This, in its basis,
concerns grouping of objects obtained using some property of objects,
say p(x). Formally the grouping is
X = {x| p(x)}.
Let me stress that in general, X is not a set. If p(x) is too complex
(and in the reality it mostly is the case) it is not possible to
characterize the grouping X precisely; its boundaries are unsharp.
Fuzzy set is then characterization of X which uses some scale. The use
of the interval [0, 1] is non-substantial.

Probability, on the other hand, is a quantified characterization
(possible mathematization) of the lack of information about OCCURRENCE
of some phenomenon. The word ``occurrence'' inherently contains time,
i.e. probability (or more generally, uncertainty phenomenon in narrower
sense) is always connected with the question whether the phenomenon in
question may be regarded during some time or not (i.e. whether it
occurs). In vague characterization of grouping, time plays no role. We
may think about what does it mean e.g. "small tree" without
considering whether we will actually see some small tree or not.

Using a paraphraze of my colleague, Dr. M. Mares, ``probability deals
with a question, whether something OCCURS, while (vagueness)
fuzziness deals with a question, WHAT has (not) occured''. Some
more detailed discussion including a lot of formalism can be found in
my paper

Novak, V.: Paradigm, Formal Properties and Limits of Fuzzy Logic. Int.
J. of General Systems 24(1996), 377--405.

Let me stress that reality (our regarding it) is in most cases both
vague as well as uncertain. This is also source of some discussions
concerning the non black/white situations describable by probability.
Yes, we need probability to describe them but fuzziness as well. You
demonstrate this also in your paper on probabilistic controller: the
technical methods you used are probabilistic (moments - why not;
reasonable technique) but your ``probability distributions'' are
basically fuzzy sets. Their probabilistic interpretation is forced.

Please, I solicit you, the probability theoretists, stop this
unfruitful endeavour to prove that ``fuzziness is some kind of hidden
probability''. This is nonsense. Let all of us better study both these
phenomena and try to find their interconnection which surely is very
deeply hidden somewhere. Maybe, we will find some other unifying and
deep theory satisfactory for all and worth of stydying. At any case,
fuzzy logic is a beautiful non-trivial mathematical theory. Try to read
some other papers or books than those you cite in your papers. So far
unpublished but very convincing is the book

Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer 1998 (to appear)

and a lot of other, mathematically deep papers.

Yours sincerely

Vilem Novak

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Vilem Novak, DSc., Associate Professor
University of Ostrava
IRAFM (Institute for Research and Applications of Fuzzy Modeling)
Brafova 7
701 03 Ostrava 1
Czech Republic

tel: +420-69-611 80 80
fax: +420-69-22 28 28
e-mail: novakv@osu.cz
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