From: mailbox@dmitry-kazakov.de   
      
   On Sun, 13 Feb 2005 13:19:01 +0100, Ernst Murnleitner wrote:   
      
   >>>>>Why not simple multiplying the (resulting) membership value(s) by the   
   >>>>>wight?   
   >>>>   
   >>>>Because it is infeasible. Values of the membership function are not   
   >>>>multiplicative in the classic fuzzy set theory.   
   >   
   > It is suggested in some fuzzy books. But maybe it is more logical not to   
   > use a WITH operator but use an additional condition like:   
   >   
   > if .... AND weight = full then ...   
   >   
   > If one sets the (one) set the way, that a weight of 0.5 would be a truth   
   >   value of 0.5 (e.g. a triangle beginning at 0 with a truth value of 0   
   > and have a maximum of 1 at 1), couldn't this be used instead of WIDTH?   
      
   I think it could, and it is fully legal in my view. Better it could be   
   just:   
      
      if (... AND weight) then (...)   
      
   Here weight is fuzzy logical. However I tend to apply changes to the right   
   side of "then". In your notation:   
      
      if (...) then (... AND weight)   
      
   (though I don't like the word "weight".) This happens if we have a training   
   set and the examples there cannot be attributed to the classes in a certain   
   way. I.e. when the teacher is not sure about the classification.   
      
   BTW, both seems to be quite equivalent if the measure is possibility:   
      
      Pos (A | B & Weight) = Pos (A & Weight | B)   
      
   Yet one cannot work with only the possibilities. And for the necessities   
   the above is not true. However if the result is propagated to other rules,   
   then its weight will re-appear on the left side again.   
      
   Anyway, it seems to me a good way to do deal with the issue, but it still   
   has some after-taste. Is that "weight" about the rule, or about the left   
   (right) side? Is it no matter? I cannot answer these questions...   
      
   >> PROD-operator is used empirically without any justification. It isn't   
   >> fuzzy. It might be called probabilistic under some, extremely rare,   
   >> circumstances. Though as normally used, it is plain shamanic.   
   >   
   > OK, you can find it in many books about fuzzy logic. But for me, this   
   > operator never seemed to be logic used as "AND". As you explain: A and A   
   > is not A?   
   >   
   > However, I used it for WITH. If you want to weight a rule by e.g. 0.5:   
   > Multiplication by 0.5 could be regarded as propability.   
      
   I must confess that I used to use multiplication too. It was sort of:   
      
   1. Nec(A) <= Pr(A) <= Pos(A)  -- borrowed from Dubois/Prade   
   2. Pr(A&B) = Pr(A)*Pr(B)  -- if A and B are independent   
   3. Nec(A)*Nec(B) <= Pr(A&B) <= Pos(A)*Pos(B)   
      
   Now it is very far-fetched to me, so I feel myself almost guilty for   
   posting something like that! (:-))   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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