Re: What do the measures mean?

savinov@usa.net
Sun, 25 Oct 1998 23:36:39 +0100 (MET)

In article <3627534C.B0B56753@student.utwente.nl>,
Stefan van den Oord <s.m.vandenoord@student.utwente.nl> wrote:
> Hello,
>
> For quite some time I've been thinking about different kinds of
> uncertainty and about the different theories and measures for it.
>
> I think there are three kinds of uncertainty: chance, possibility and
> certainty. Let me illustrate this with an example. I see a (Dutch)
> person who I do not know. Because of my (statistical) knowledge that of
> all Dutch people there is always a (not very large) part that is ill,
> there is a small chance that this person I see is ill. However, the
> possibility that he is ill is one; I don't have proof that he is not
> ill. The certainty that he is ill is zero, again because I have no proof
> of it. (If I do have proof, the possibility remains one, but the
> certainty becomes more than zero.)
>
> These three concepts are modeled by probability theory (chance) and
> possibility theory (possibility and certainty). Why do we need fuzzy set
> theory, and if we do, what is the meaning of the fuzzy membership
> degree? What do you think about this?

In my opinion, you are right that there are two main approaches for the
case of min/max operations:

- Negative where we deal with possibilities and where 1 means uncertainty
(no any information), 0 means impossibility, <= means logical
consequence |=. In particular, such a behavior can be modeled by means
of DNF.

- Positive where we deal with positive information and where 0 means
uncertainty (no any information), 1 means absolute certainty (the
corresponding event/state/object takes place), >= means logical
consequence |=. In particular, such a behavior can be modeled by means
of CNF.

Obviously these two cases are dual and in some sense (as all truly dual
things) incomparable, i.e., we are not able to deal with both these cases at
once -- in any approach we should choose what formalism to apply. Of course,
in many approaches especially applied ones these methods can be mixed by
means of some technical (out of logic, out of formalism) procedure and then
we obtain hybrid method, however, here we discuss pure case.

For example, databases store essentially positive information since if there
is a record about some person then we certainly know that this is the fact
(e.g., the person is ill) while if such a record is absent it means
absolutely nothing (absolute uncertainty). Each record can be represented as
a conjunction (of values) and all records are combined with logical OR and
represent DNF. A system with constraints (i.e., some negative, restrictive
information) can serve as an example of system with possibilistic (negative)
mode of information representation. Usually in such systems (especially in
knowledge based systems) we describe what is NOT possible instead of what is
really happened, i.e., we describe (in an intensional way, e.g., by means of
rules) what states/objects/events has never appeared (meaningless from the
logical point of view). In particular, one such piece of information can be
represented by means of one disjunction (or implication) and a number of such
disjunctions combined with logical AND represent CNF.

In probabilistic case we use other (more flexible) operations but the main
difference is that with this approach we are able to deal with both positive
and negative information.

In my opinion, when we use the term fuzzy its meaning is formally undefined,
i.e., it means only that we use only one modality (modality means negative or
positive) in contrast to probabilistic case. Usually the concrete modality and
concrete interpretation is defined indirectly from context of one or another
formalism. Thus the term fuzzy set theory (as it used now) is too general
and does not specify such "subtleties" as possibility, necessity etc.
In great extent it is because of "commercial" reasons -- if I call my theory
"ABC set theory" there will be less chances on success, so everybody uses this
notion as he wants and inserts into word "fuzzy" some additional semantics...

By the way, this issue is rather "dangerous" for discussion :-)
since it is rather general and everybody has his own frequently
opposite opinions.

Regards,

Sasha Savinov

--
Alexandr A. Savinov, PhD
Senior Scientific Collaborator, Laboratory of AI Systems
Inst. Math., Moldavian Acad. Sci.
str. Academiei 5,  MD-2028 Kishinev, Moldavia
Tel: +3732-73-81-30, Fax: +3732-73-80-27
mailto:savinov@math.md
http://www.geocities.com/ResearchTriangle/7220/

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