Re: fuzzy inference

"Andrzej Pownuk" <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>...
> 
> If we assume that fuzzy sets theory is a part of probability theory then
> everything is correct.
> 
> However some authors (for example Earl Cox) say that fuzzy sets cannot be
> converted to probability.
> 
> If we accept that, then as you can see the situation is a little strange.
> 
Dr. Pownuk has gone a long way toward placing this problem into a
proper framework. I find this situation more than a little strange!

Probability was originally developed to deal with gambling, systems
that exist in discrete and mutually exclusive states (outcomes).
Fuzzy, on the other hand, usually deals with systems that can exist in
discrete states that are NOT mutually exclusive. As Earl Cox says:

"The difference between probability and fuzzy logic is clear when we
consider the underlying concept that each attempts to model.
Probability is concerned with the undecidability in the outcome of
clearly defined and randomly occurring events, while fuzzy logic is
concerned with the ambiguity or undecidability inherent in the
description of the event itself. Fuzziness is often expressed as
ambiguity rather than imprecision or uncertainty and remains a
characteristic of perception as well as concept."

What nobody seems to question is whether there is any difference in
the mathematics of these two approaches. I claim that there is no
difference; the only problem lies in our failure to extend the
original focus of probability on mutually exclusive states to states
that may not be mutually exclusive. We are not having a problem with
mathematics, but with words and concepts that we assume are handed
down by the Almighty. In some respects, fuzzy math and probability
have become religions.

Consider the following, where a is the truth value of A:

Fuzzy, min/max:                 Probability, max positive association:
A AND B = min(a, b)             p(A AND B) = min(p(A), p(B))
A OR B = max(a, b)              p(A OR B) = max(p(A), p(B))

Fuzzy, product/sum:             Probability, statistically
independent:
A AND B = ab                    p(A AND B) = p(A)p(B)
A OR B = a + b - ab             p(A OR B) = p(A) + p(B) - p(A)p(B)

Fuzzy, bounded sum/difference:  Probability, max negative association:
A AND B = max(0, 1 - (a + b))   p(A AND B) = max(0, 1 - (p(A) + p(B)))
A OR B = min(1, a + b)          p(A OR B) = min(1, p(A) + p(B))

Would anyone seriously propose that the fuzzy and probability formulas
have no relation to each other? Or that there is no relation between
the fuzzy truth value of A and the probability that A is true?

William Siler