**Previous message:**Vladik Kreinovich: "conference in Istanbul in 2002"**Maybe in reply to:**Vilem Novak: "fuzzy proofs and law of excluded middle"**Next in thread:**Nico du Bois: "Re: fuzzy proofs and law of excluded middle"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

DieSpamDie wrote:

*>
*

*> Perhaps I'm missing something. If my membership value for
*

*> the set of tall people is .7, then I am at the same time
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*> 70% a member of the set and 30% not a member of the set.
*

*> That seems like a clear denial of the law of the excluded
*

*> middle to me.
*

You are missing something. Whether or not LEM applies, qua theory,

depends upon the precise rules sought to be applied for the

connectives AND and OR, also NOT. Qua empirical reality in the

application domain of natural language semantics, the existence of

LEM it would seem to me is not vitiated by fuzziness. As you point

out, however, the matter is not straightforward. To resolve the

difficulty, it is essential to keep clear the distinction between

meta-language and object language. Furthermore, it must be kept in

mind that in the meta-language -- the language used not only to talk

*about* fuzziness, but also to render precise the ways in which fuzzy

may be characterized -- there is logical bivalence. The meta-language

way of saying what you have said is to say, there are x (in the

universe of height values) such that both

mu[TALL](x) > 0

and

mu[NOT TALL](x) > 0.

This however is not a sufficient demonstration of the failure of LEM.

Again in the meta-language, the test is whether there exist x such

that

mu[TALL OR NOT TALL](x) < 1.

This is clearly the case if OR is modeled by the max rule and NOT by

the one-minus rule, and LEM fails, also its companion with respect to

conjunction, the law of (non-)contradiction, LC. If OR is modeled by

the bounded-sum (Lukasiewicz) rule, namely

a OR b = min(1, a+b)

where a and b represent membership functions in general, and

evaluation is point-wise, then

a OR NOT a = min(1, a+(1-a)) = 1

regardless of the point x at which the membership functions are

evaluated, and LEM holds. That is, the meta-language declares that

for any term A, fuzzy or otherwise,

mu[A OR NOT A](x) = 1, for all x.

Correspondingly,

a AND b = max(0, a+b-1)

is the Lukasiewicz or bounded-sum rule of conjunction, and LC holds,

since

a AND NOT a = max(0, a+1-a-1) = 0.

Hence, although there exist x such that both, for example,

mu[TALL](x) > 0

and

mu[NOT TALL](x) > 0,

there are no x for which

mu[TALL AND NOT TALL](x) is other than 0.

This is the meta-language way of affirming that in the (fuzzy) object

language, such terms as "tall and not tall" would violate the law of

non-contradiction. Which is why no witness in court would remain very

credible if she were to describe her attacker as "tall and not tall",

and the (admitted) fuzziness of the term "tall" would not be

sufficient to rescue her credibility. (Note that for her to say her

attacker was "not tall and not short" would be an entirely different

proposition, as would be her saying her attacker was of "medium

height". Any term and its negation, though, however fuzzy, would

remain the constant absurdity. And any term in disjunction with its

negation, eg. "tall or not tall", would yield the constant tautology,

in accordance with LEM.)

Hope this is helpful. Of course it raises immediately the issue of

when do which rules of conjunction and disjunction apply. It seems

clear to me that in the application domain of natural language

semantics, the min-max rules of the meta-language do not always

succeed in modeling the fuzzy object-language reality. I have

addressed these issues at some length, and starting rigorously from

first principles, in my _Fuzziness and Probability_ (ACG Press,

1995).

*> - Glenn
*

Regards,

S. F. Thomas

*> Vilem Novak wrote in message <200105281502.f4SF2ZU08654@cx.osu.cz>...
*

*> >The law of excluded middle holds in fuzzy logic, too; however, not
*

*> >with weak connectives \vee, \wedge but with other kinds of
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*> >connectves such as Lukasiewicz conjunction and disjunction.
*

*> >###################################################
*

*> >Prof. Vilem Novak, DSc.
*

*> >University of Ostrava
*

*> >IRAFM (Institute for Research and Applications of Fuzzy Modeling)
*

*> >30. dubna 22
*

*> >701 03 Ostrava 1
*

*> >Czech Republic
*

*> >
*

*> >tel: +420-69-6160 234
*

*> >fax: +420-69-6120 478
*

*> >mob: +420-602-576 477
*

*> >e-mail: Vilem.Novak@osu.cz
*

*> >WEB: http://ac030.osu.cz/irafm/
*

*> >###################################################
*

############################################################################

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