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From: spg@uhura.neoucom.edu (Shiva P. Gautam)
Subject: Re: Fuzzy logic compared to probability
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References: <312B60FB.41C67EA6@colorado.edu> <4gh66g$r8@elna.ethz.ch>
Date: Thu, 22 Feb 1996 18:18:23 GMT
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Xref: glinda.oz.cs.cmu.edu comp.ai.fuzzy:6778 sci.stat.math:9355

Andreas Poncet (poncet@isi.ee.ethz.ch) wrote:
: In article <312B60FB.41C67EA6@colorado.edu>, Robert Dodier <dodier@colorado.edu> writes:

: > I am interested in a certain question concerning fuzzy logic
: > and probability. I am trying to figure out whether there are
: > theoretical reasons to prefer one to the other in an application
: > concerning the computation of degrees of certainty.

: Well, usually, in the papers about fuzzy logic, it is claimed that fuzzy
: membership functions are not to be mixed up with probabilities.
: But what kind of probabilities?  The arguments used are typically that
: a "membershipness" can be given for any single observation, whereas no
: "probability" - in the sense of frequency - can be defined. Fine, but
: what if we consider the Bayesian subjective view of probability, namely
: as a degree of belief in (or a degree of relevance of) a given
: proposition? Then it seems to me that a "membershipness" is in fact
: a CONDITIONAL PROBABILITY.

I do not know much about fuzziness as I heard it a month ago for the first
time. If conditional probability which is the foundation of Bayesian
statistics in some sense is equivalent to membershipness, then Bayesian
statistics seems a component of fuzziness. However, it does not sound true
the other way around.

: As an example: instead of saying, e.g., about a specific man, that
: his height x has "tall"-membership, say, 0.8, 
: one could express this equivalently as P(A|x) = 0.8,
: with the proposition A defined as
:                   A = "the man is tall"

: > It would appear that if the fuzzy logic concept of `degree of
: > truth' is the same as the Bayesian `degree of belief,' then 
: > either fuzzy logic is the same as probability or else less
: > powerful (as it would have to be inconsistent).

: Fuzzy logic couldn't convinced me yet, for there seems to be much "ad
: hockery", as in the choice of the "tent" form, e.g., of membership
: functions. Why such a form? How can it incorporate prior information?
: (Probability theory decomposes the problem of specifying P(A|x) in two
: steps: the prior P(A) and the likelihood p(x|A).)
: Furthermore, how to consistently specify JOINT membership functions of DEPENDENT
: characteristics (such as "tall"-"heavy")? Again, using probability
: theory guarantees consistent answers.
 
Questions may sometimes be more important than answers. I can argue that
even in probabilty the answers are not consistent from mathematical
viewpoint. For the same problem one person can make one decision ( say for
a p-value) and another person may make another decision (because he was
not comfortable with that p-value etc.).
Fuzzy logic seems to accept the simultaneous existencce of opposites. IMHO
it is closer to human experince concerning decision making process.
Classical logic has yes or no. Fuzzy logic adds 'may be' to that. 
It (fuzzy logic) seems to deal with classical paradoxes. For example, every
set is subset of itself -we learn early in our courses. But consider a set
or collection of lakes. The collection itself is not lake so does not
belong to itself?  If the membership is defined according to area covered
by water, then the collection is a 0.8 lake (say).
I am begining to sound like I know but I don't know. As Andreas said there
seems lots of ad hock things. May be people will work on these problems.
It has been just thirty years of fuzzy logic, and it originated in
engineering. And people on the other side are used to loan but not borrow,
used to talk and not listen for thousands of years. 
Experience shows that Human decision making proc  : Opinions? Reactions?

Shiva


