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Sampo Syreeni <decoy_at_iki.fi> wrote in news:Pine.SOL.4.62.0703020152510.3230
@kruuna.helsinki.fi:
> On 2007-03-01, V.J. Kumar wrote:
>
>>> By "real" I meant "real valued". Because of that, there's nothing in
>>> fuzzy set theory you couldn't handle with rather boring and classical
>>> measure theoretic tools.
>>
>> Are you saying that the membership function is just another name for
>> the classical measure?
>
> No, because most classical measures are additive. Yet, measure theory
> contains all of the tools which enable fuzzy measures to be handled.
The classical measure theory did not have any such tools until Michio Sugeno, a fuzzy logic theorist, developed them. Sugeno measure is not reducible to the classical additive measure naturally.
>Cf.
> e.g. the full body of theory on outer measures in various topological
> spaces.
>
>> How would you define a degree of truth in classical logic without
>> resorting to any additonal tools which would be cheating?
>
> The way it's done in fuzzy logic, of course.
You did not answer the question. "The way used it's done in fuzzy logic" is not reducible to classical logic.
>As I said, that's based on
> a classical foundation.
What classical foundation do you have in mind ? The notion of graded truth value is not classical by any stretch of imagination. And it is not reducible to 'true/false' without loss of information.
>
>> How would you define the membership function as measure?
>
> http://en.wikipedia.org/wiki/Fuzzy_measure_theory
Sugeno measure is not a membership function. See Dubois' "Fuzzy sets and probability" where he discusses similarities and differences between probability and possibility (fuzzy) measure and shows that neither is reducible to the other.
Whilst, as I said earlier, fuzzy logic/possibility theory significance is debatable, the mathematical structures are sufficiently different, despite profound similarities, for you to claim glibly that f.l. is trivially reducible to its classical counterpart. Received on Thu Mar 01 2007 - 20:55:33 CST