From: mailbox@dmitry-kazakov.de   
      
   On Mon, 07 Apr 2008 14:07:30 -0400, Kirk Zurell wrote:   
      
   > Dmitry A. Kazakov wrote:   
      
   >> 3. Logic is not geometry. Though it is true that the equation x^2 + y^2 =   
   >> r^2 implies under certain conditions certain statements about axes of a   
   >> circle defined in a certain way. But it would be silly to expect fuzzy   
   >> logic to capture that. Same is true for crisp logic too. Neither is   
   >> geometry.   
   >   
   > Admittedly I was approaching the problem from our control   
   > background, looking at a circle as just another output waveform.   
      
   Well, but waveform is not a fuzzy curve in the sense of a fuzzy subset over   
   R^2. That is when you draw a curve with a soft pencil and them rub the   
   drawing a bit. (OK, that would be rather a random set, because it   
   distributes a fixed amount of graphite [i.e. the set measure is additive],   
   but I think you get the idea.)   
      
   >>> If we can't describe   
   >>> its qualities accurately in human linguistic terms, we might not   
   >>> be able to encode it in fuzzy logic.   
   >>   
   >> You do not need to describe a circle accurately. You can do it   
   >> inaccurately, that's the whole idea.   
   >   
   > Meant to say "operably" or similar. In the model I was using and   
   > failing with, there is only that curve: no information about,   
   > say, the forces that bring it into being.   
      
   Hmm, that should be simply to do. For this you could describe its as, say,   
      
   x = W cos (t)   
   y = H sin (t)   
      
   Considering W and H fuzzy, will give us a fuzzy shape. We could also   
   continue fuzzifying it by adding fuzzy phase like   
      
   x = W cos (t+P)   
   y = H sin (t)   
      
   where P is a fuzzy number around 0, or e fuzzy-valued function of t. etc.   
      
   But I think you have a different model in mind. Like a *crisp* curve which   
   is "almost" a circle or looks like a circle. This is a different construct.   
   Here the curves are crisp. Fuzzy is a subset of all possible crisp curves.   
   For each crisp curve the truth value tells how it is possible for the curve   
   to appear a circle.   
      
   That brings us bug to the issue - the model first.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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