From: earlcox@earlcoxreports.com   
      
   Dmitry,   
      
     Ah, now I see the problem. You think that a fuzzy set is less precise than   
   a Boolean (crisp) set!! Many of us who work with fuzzy logic take just the   
   opposite view. Let the crisp set TALL be defined as x > 6ft, then I am   
   forced, at the boundary point of 6ft to suddenly switch between short (not   
   Tall) and Tall. As I approach this boundary of 6ft the Boolean set makes no   
   distinction between degrees of membership -- it does not provide any   
   information about the entropy compactness of the set.  It acts light a light   
   switch -- on or off. Hence, I am becoming less precise in the   
   epistemological sense.On the other hand, a fuzzy set version of TALL gives   
   us degrees or grades of membership in the set. (To continue the metaphor,   
   the fuzzy set acts like a dimmer switch, it gives us degrees of brightness).   
   These degrees are NOT the result of any uncertainty in the set -- on the   
   contrary, we are certain that the height is 5ft and the degree of membership   
   is [.75]. There is no "imperfection" in fuzzy logic. The membership value is   
   precise. It is just as precise, in its own theory of knowledge, as the   
   dichotomous 1 and 0 of Boolean Logic.   
      
   All of this stems, often subconsciously, from a sense that Boolean logic is   
   the only CORRECT form of logic and that all other logics are inferior. This   
   is, indeed, the gist of your argument. This feeling is often reinforced by   
   philosophers of logic (such as the late Willard Van Orman Quine) and   
   engineering professors (Kalman's name comes instantly to mind) who are   
   influenced by the name "fuzzy" and know almost nothing about the actual   
   nature of fuzzy logic.The fact is, of course, that fuzzy logic is not an   
   ad-hoc, heuristic, or inferior logic, nor is it a bastardized form of   
   Boolean logic. Fuzzy Logic is a powerful reasoning system for performing   
   logically concise and provable operations on sets that allow degrees of   
   membership. Boolean logic, on the other hand, has always been a way to   
   approximate such sets. When dealing with sets that have imprecise or elastic   
   boundaries -- note that this does not necessarily mean uncertain   
   boundaries -- logicians since the time of Aristotle have been forced to   
   approximate these sets through Boolean logic. Hence, it is the fuzzy set   
   that is more precise and the Boolean set that is less precise. Forcing us to   
   use crisp logic to represent such sets as Tall, large, small, rapidly   
   changing, dense, close, far, etc. forces us to give up information. We take   
   the entire set membership of people from four feet to five feet, eleven and   
   three quarters inches and assign them a membership of [0]. We throw away the   
   degree of membership which tells how representative they are of this concept   
   Tall. We eliminate this measure which can be used as supporting evidence in   
   reasoning systems.   
      
    Think about this Dmitry -- you will, I hope, see that fuzzy logic is not a   
   step-child of Boolean logic, nor is it a last resort for modeling complex   
   systems. It is a logic that exposes more knowledge about the structure of a   
   set and allows us to gain information-theoretic advantages over the   
   limitations of Boolean logic.   
      
   earl   
      
      
      
      
      
   "Dmitry A. Kazakov"  wrote in message   
   news:4sk780pav3ifk4df9jfks6pod11ba19kco@4ax.com...   
   > 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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