Re: "Inevitable Illusions", Law of Excluded Middle etc.

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Wed May 23 2001 - 11:46:27 MET DST

  • Next message: afss2002@isical.ac.in: "AFSS 2002"

    Ellen Hisdal wrote:
    <snip>
    > It is shown there that when a subject takes the possibility of the
    > presence of fuzziness into account, then he realizes that an object
    > whose height is near the lower threshold for 'tall' may, in the presence
    > of fuzziness (e.g. errors of measurement; these are one example of fuzziness),
    > sometimes be labeled `tallī, and sometimes `NOT tall'.

    I would expect that to be true.

    > We therefore do not expect the Law of the Excluded Middle to hold
    > exactly in Fuzzy Set Theory.

    It can be. The problem of fuzziness and LEM is resolved in my opinion
    with rules for AND, OR and NOT which yield the following result:
    There exist x such that both

            mu[TALL](x) > 0

    and

            mu[NOT TALL] > 0,

    but for _all_ x,

            mu[TALL AND NOT TALL](x) = 0,

    and

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

    THis reflects the common-sense observation that the fuzziness in the
    term "tall", for example, would not rescue a witness from the
    derision of the court were she to say "the perpetrator was tall and
    not tall." In other words, LEM and law of contradiction may still
    hold under a fuzzy set theory, and without doing away with the
    fuzziness. This result is developed in some detail in my _Fuzziness
    and Probability_ (ACG Press, 1995). LEM and LC of course do not hold
    if AND and OR are modeled exclusively by the min and max operators
    respectively.

    Regards,
    S. F. Thomas

    PS. Good seeing you here again, Prof. Hisdal.

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