Re: Fuzzy Logic

To clarify things a bit: Dimitry is right concerning fuzzy sets, however, you have to take care when you want to compute with these membership grades. Typically, you will use fuzzy logic to do so. There are different possible interpretations of what a gradual membership value actually are. However, all interpretations have in common that these memberships correspond to gradual truth values (of a certain event!), whereas in probability/possibility theory you are dealing with the uncertainty of events. This has important consequences on how to compute with these values (conjunction/disjunction, etc.). While fuzzy logic is (as the name suggests) a logic, possibility theory for example is not even fully computational!

For a more detailed discussion, please see:
@article{DuboisPrade01,
author = {D. Dubois and H. Prade},
title = {Possibility theory, probability theory and multi-valued logic: A clarification},
journal = {Annals of Mathematics and Artificial Intelligence},
year = {2001},
volume = {32},
pages = {35--66}
}

Dmitry A. Kazakov schrieb:
On 4 Dec 2006 16:22:25 -0800, hmhallani@xxxxxxxxx wrote:

Someone asked me what was supposed to be an easy question: What does
the y-axis represents in the Membership Function? I know it is the
degree to which an input belongs to the appropriate fuzzy set. But the
person said, i need a more tangible answer.

So please can you give me an answer to this question.

It is a conditional set measure mA. The set being measured is the singleton
{x}. So y=mA({x}). Any meaning of m is beyond the scope of the fuzzy set
theory. In possibility theory m means the possibility that {x} belongs to
the set A:

y=pos(A|{x})

In probability theory it would the probability of.

In general it is the theory of uncertainty you take, which assigns a
"meaning" to the measure. However, any such theory would do this in some
opaque terms like "probability" or "possibility," which themselves cannot
be explained and taken for fundamental, intuitively defined. (Actually
there were many attempts to reduce probability to something else. None of
those was any successful. In my view it were just impossible to do.)

.