From: mailbox@dmitry-kazakov.de   
      
   On Tue, 08 Feb 2005 22:51:09 +0100, Ernst Murnleitner wrote:   
      
   > Dmitry A. Kazakov 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.   
   >   
   > Why not? The PROD Activiation within the inference is also a   
   > multiplication of membership values (truth values).   
   >   
   > I think, a multiplikation (= PROD Operator, I call it PROD-WITH here) is   
   > suitable for WITH, as well as the MINIMUM Operator would be.   
      
   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.   
      
   Of course one could introduce some measure of truth (neither fuzzy, nor   
   probabilistic) where m(A and B)=m(A)*m(B). But that would be a pretty   
   counter intuitive one. Because for any A such that m(A)<1, m(A and A and A   
   ...)=0. So (A and A) is not A, or what? BTW, in Zadeh's fuzzy A and A = A.   
   Ergo,  PROD is *inherently* incompatible with fuzzy.   
      
   Further, returning to the Modus Ponens rule. Why you are so sure that in a   
   system m(A and B)=m(A)*m(B), the solution of (A, A=>B) would be   
   m(B)=m(A)*m(A=>B)? That requires a proof.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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