Re: fuzzy proofs and law of excluded middle

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Fri Jun 08 2001 - 11:32:53 MET DST

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    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
    > 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
    > >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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