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