Re: Stupid question

From: S. F. Thomas (sf.thomas@verizon.net)
Date: Tue Nov 27 2001 - 12:54:22 MET

  • Next message: Hal Lin: "search software(about ellipsoidal fuzzy system)"

    > In the world of computer programme, binary nos. (1 and 0) are used but as we
    > all know that there is a lot of things between 1 and 0 in the world of fuzzy
    > algorithm. Then, contradition is here.

    There is no contradiction. We sometimes forget that fuzzy set theory
    is a branch of bivalent mathematics. The key idea of fuzzy set theory
    is to render terms in natural language which are fuzzy with respect
    to some underlying universe of discourse U, say, as a crisp object,
    subject to all the rules of bivalent mathematics, in the form of the
    membership function. Effectively, an object that is fuzzy with
    respect to U (e.g. a term such as "tall" with respect to U=height) is
    rendered as something crisp with respect to [0,1]^U, i.e
    mu[TALL]:U->[0,1]. There is nothing fuzzy about a membership
    function! (In the same way and in the same sense there is nothing
    random about a probability distribution function.) Therefore, there
    is no contradiction in elaborating a fuzzy model making use of binary
    computers. Contrary to what some writers would seem to want to
    maintain, it is not necessary to posit a non-classical, non-binary
    logic at the foundations either of mathematics, or of computation, as
    a requirement of modeling fuzzy systems. Bivalent logic, whether in
    mathematics or in computation, is a perfectly adequate foundation
    upon which to base a fuzzy-set theory. The notion of "fuzzy logic"
    obscures this fact.

    Hope this is helpful. See my "Fuzziness and Probability" (Wichita:
    ACG Press, 1995) for an exhaustive treatment of the relevant
    philosophical and mathematical foundation.

    Regards,
    S. F. Thomas

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