From: mailbox@dmitry-kazakov.de   
      
   William Siler wrote:   
      
   > "Dmitry A. Kazakov"  wrote in message   
   > news:...   
   >> Bartosz Bien wrote:   
   >>   
   >> > I'm building the world out of some crisp and fuzzy pieces (of   
   >> > information ;)), employing a Mamdani FLC for the decision making.   
   >> > Therefore, some of my rules utilize both a fuzzy granule and a crisp   
   >> > boolean value, e.g.:   
   >> >   
   >> > 1. if A is low then C is bad   
   >> > 2. if A is high and B is false then C is good   
   >> > 3. if A is high and B is true then C is very good   
   >> >   
   >> > In this example, physically having or not having B (boolean) influences   
   >> > the decision C when A is high. I am thinking of incorporating such   
   >> > cases in the double-input-single-output FLC, where input B has two   
   >> > singletons valued 0.0 - false, and 1.0 - true. Is this approach best or   
   >> > am I missing a point somewhere?   
   >>   
   >> I see nothing wrong with that. Actually it is how intuitionistic   
   >> propositions generalize boolean ones by replacing {0,1} (false, true)   
   >> with   
   >> [0,1]x[0,1]. So for an intuitionistic B one considers the pair (Pos(B),   
   >> Pos(notB)), or alternatively (Pos(B), Nec(B)=1-Pos(notB)). In the latter   
   >> representation:   
   >>   
   >> true = (1,1)   
   >> false = (0,0)   
   >> uncertain = (1,0)   
   >> contradictory = (0,1)   
   >>   
   >> The advantage is that you can handle crisp and fuzzy propositions in same   
   >> way and you can deal with uncertainty, when, for instance, B is reported   
   >> to be both true and false (=uncertain/unknown).   
   >   
   > I agree with most of what Dmitry has said. However, although his   
   > equation   
   >   
   > Nec(B) = 1 - Pos(notB)   
   >   
   > is certainly in the books (Klir and Yuan eq. 7.25, p. 189), it is   
   > incorrect. The correct formulation is   
   >   
   > Nec(B <= 1 - Pos(notB)   
      
   Correct, both! (:-))   
      
   When "not" is defined as 1-x, then Nec(B)=1-Pos(notB) follows from the de   
   Morgan rules. But that holds no longer B considered fuzzy. If B becomes   
   intuitionistic, then Pos(B) and Pos(notB) are at best just estimations of   
   some unknown, ideal, maybe non-existing B':   
      
   Nec(B) <= Nec(B') <= Pos(B') <= Pos(B)   
      
   And same for notB, notB':   
      
   Nec(notB) <= Nec(notB') <= Pos(notB') <= Pos(notB)   
      
   Provided these estimations we will immediately have:   
      
   Nec(B) <= Nec(B') = 1 - Pos(notB') <= 1 - Pos(notB)   
      
   What is behind the scenes is rather a philosophical question. One can   
   postulate that a fuzzy B' always exists, but cannot be sensed. Or even that   
   there is a crisp B'. Technically there is little difference between:   
      
   a. I know how to do "not", but B is inaccurate to get notB   
   b. I do not know how to do "not" at all   
      
   The result is same, one have to go for both B and notB.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   www.dmitry-kazakov.de   
      
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