From: earlcox@earlcoxreports.com   
      
   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????   
      
   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.   
      
   Earl   
      
   "Dmitry A. Kazakov"  wrote in message   
   news:f107809haup7ugki4vpnhq8ve7j41sf57k@4ax.com...   
   > On Mon, 19 Apr 2004 01:35:39 GMT, "EarlCox"   
   >  wrote:   
   >   
   > >"Dmitry A. Kazakov"  wrote in message   
   > >news:c5una7$5ufrs$1@ID-77047.news.uni-berlin.de...> EarlCox wrote:   
   > >>   
   > >> > I have a difficult time understanding this hierarchy. Why should   
   fuzzy   
   > >be   
   > >> > a last resort? Fuzzy logic is a system of logic in the same way that   
   > >> > Boolean logic is a system of logic. It is essentially (but not   
   > >necessarily   
   > >> > always) a logic of continuous variables. It is possible to introduce   
   > >fuzzy   
   > >> > logic as the logic of choice into almost any modeling approach.   
   > >>   
   > >> Why one should need fuzzy logic, if the conventional logic would give   
   > >*same*   
   > >> result? An additional complexity is only then necessary when it gives   
   > >> something new. This is what I meant. When fuzzy logic starts to work,   
   it   
   > >> shows its strength in giving inexact results (similar to statistics).   
   This   
   > >> strength is also its weakness, we might get a fuzzy answer where an   
   exact   
   > >> one exists. As for stochastic problems, the situation is worse. Fuzzy   
   is   
   > >> not an extension of random, though it might be viewed as a very rough   
   > >> estimation of it. This is why I consider fuzzy as the last resort to be   
   > >> used when other approaches do not work. Fortunately or not, quite often   
   > >> they do not.   
   > >>   
   > >   
   > >Fuzzy logic does NOT give inexact results!   
   > >   
   > >See, this is the problem!!!! This is the result of inexperience and a   
   > >confusion of name with mechanism. Fuzzy logic is not a statistical   
   approach   
   > >nor does it have behaviors that are like statistical models. Fuzzy models   
   > >produce valid, concrete, precise, reliable and definite results.   
   >   
   > Of course not. The sole idea behind fuzzy is to express uncertainty   
   > and to deal with it. Uncertain is not precise.   
   >   
   > >The rest of all this commentary is equally confused. Of course Fuzzy   
   Logic   
   > >can contradict Boolean Logic. The intersection of A and Not-A is not an   
   > >empty set, as a trivial example, since set membership is not dichotomous.   
   >   
   > It is a silly example. This is not a contradiction. True that not   
   > every result of Boolean logic holds in fuzzy. But carefully observe,   
   > that it will, if all sets involved are crisp. This is what "extension"   
   > means mathematically. The important consequence of this is that you   
   > cannot get different answers on the same data.   
   >   
   > --   
   > Regards,   
   > Dmitry Kazakov   
   > www.dmitry-kazakov.de   
      
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