From: mailbox@dmitry-kazakov.de   
      
   On 25 Sep 2003 06:24:40 -0700, wsiler@aol.com (William Siler) wrote:   
      
   >"Andrzej Pownuk"  wrote in message news:   
   3f7206eb$0$25930$3b214f66@usenet.univie.ac.at>...   
   >>   
   >> 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:   
      
   I do not think that mutual independence of states is a key issue. One   
   can always split original states into smaller ones which then will be   
   independent. It is no matter if such "virtual" outcomes realize in the   
   reality. Quarks also never appear unbound.   
      
   >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?   
      
   But similarity in formulae tells nothing about the reasons of this   
   similarity.   
      
   >Or that there is no relation between   
   >the fuzzy truth value of A and the probability that A is true?   
      
   No longer they are 0 or 1. But 0.5 probability has in my view nothing   
   to do with 0.5 of a fuzzy truth value. Or else, why should it have? As   
   I see it, probability and fuzzy truth values are different measures,   
   which have different properties and are useful *because* both are   
   different. Otherwise, there were absolutely no reason in having fuzzy   
   truth values. One can evolve random sets and be happy with them. Can   
   one? No, because, though one could get similar results with random   
   sets, their application in most cases would be ungrounded, because the   
   premises they rely upon cannot be proved in real-life cases. So the   
   whole system were built on sand.   
      
   ---   
   Regards,   
   Dmitry Kazakov   
   www.dmitry-kazakov.de   
      
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