From: mailbox@dmitry-kazakov.de   
      
   On Mon, 19 Apr 2004 08:47:50 GMT, "EarlCox"   
    wrote:   
      
   >Who ever said that "The sole idea behind fuzzy is to express uncertainty"?   
   >Fuzzy logic involves sets with elastic membership functions, but membership   
   >in a set is not uncertain.  Membership is a definite degree. It is just not   
   >one or zero. If I have the fuzzy set Tall with a linear increasing   
   >membership function between 4 and 7, and I have a height of 5ft 3in, it has   
   >a membership of, say, [.82]. This is not uncertain. I can use this degree of   
   >membership as a measure of evidence in future operations that correlate   
   >input with output -- that is, I can transform the morphology of an outcome   
   >fuzzy space through the aggregation of evidence to reflect the predicate   
   >(antecedent) evidence intrinsic in each fuzzy relation. In any case, the   
   >outcome of a fuzzy model can be very precise. There are thousands of example   
   >in the control world -- I would hate to think that the fuzzy ATO in the   
   >Sendai subway system, carrying hundreds of thousands of commuters every day,   
   >is guided by imprecise measures. The ATO fuzzy controller is more accurate   
   >than not only the human operator but the previous PID mathematical   
   >controllers. Space shuttle docking controllers use fuzzy logic. Industrial   
   >crane balancing and arm movement controller use fuzzy logic. A simple   
   >inverted pendulum balancing system, one of the earliest examples of fuzzy   
   >control, generates a very precise and very robust solution. Why would you   
   >think that the outcome of a fuzzy logic system is imprecise????   
      
   Any legal statement in Boolean logic is either true or false. This   
   includes the statements like if x is a member of X. If the outcome   
   appears to be 0.3, it is imprecise. Whether this imperfection is the   
   result of lacking knowledge (of any sort) is no matter.   
      
   >It is kind of naive assertions, lack of any background knowledge, and   
   >unfounded suppositions that causes sooooo much trouble in discussing the   
   >properties of fuzzy systems.   
   >   
   >I know what the extension principle means. My response was in reply to your   
   >statement: "Fuzzy logic is an extension of the Boolean logic. As such it   
   >cannot contradict to what it extends" well, OK, at the edges of the   
   >hypercube bounding the zero and one points in membership, Fuzzy Logic and   
   >Boolean Logic are equivalent. But these points account for a minute   
   >sub-universe of obervables in domain of fuzzy sets. As a larger theory of   
   >knowledge, fuzzy logic contradicts the underlying law of bivalence and its   
   >derived conditions (such as the excluded middle and non-contradiction.)   
   >Actually, fuzzy logic is a superset of Boolean Logic -- it doesn't extend   
   >Boolean Logic in the same sense that object oriented languages, like Java,   
   >extend a class definition. In any case, you can most definitely get   
   >different answers form the same data. You get different answers because the   
   >underlying logic used to generate the answers is different.   
      
   Sorry, but I do not see how that can happen. My point is the   
   following, let f(X1,X2,...,XN) is a Boolean predicate. Then its value   
   in fuzzy logic will not change so far all Xi are crisp. From this   
   directly follows that whatever system you build the answers it would   
   give on crisp data HAVE to include the answers of the corresponding   
   deterministic system. You can get less precise results, but you never   
   can get a contradiction.   
      
   --   
   Regards,   
   Dmitry Kazakov   
   www.dmitry-kazakov.de   
      
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