Re: fuzzy proofs and law of excluded middle

From: WSiler@aol.com
Date: Sat Jun 23 2001 - 07:39:33 MET DST

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    In a message dated 5/16/01 8:50:08 PM Central Daylight Time,
    ulrich.bodenhofer@scch.at writes:

    >>Don't believe anybody who calls that a crisis! Elkan's statement was simply
    a dull error. He, more or less, assumed that fuzzy logic obeys all laws of
    Boolean algebras (which is not true) and proved, under these FALSE
    assumptions, that it collapses into Boolean logic. Read the original Elkan
    paper
    (it is somewhere on the Web, for sure) and you will understand
    (since you seemingly have read Ruspini's reply already).

    The question arises whether something which is not a Boolean algebra
    may be considered as a concept of logic. Nowadays, the accepted opinion
    is yes! I would not like to go into detail, but to recommend the following
    books:
    >>

    By "accepted opinion", I presume you mean accepted by most fuzzy
    mathematicians. Certainly there are many others, including most AIers, who do
    not accept the opinion that Elkan has simply committed a "dull error".

    To restate the stituation in less euphemistic terms, Elkan showed (without
    explicitly so stating) that fuzzy logic fails to obey the laws of excluded
    middle and non-contradiction. This is not just a matter of "obeys all laws of
    Boolean algebras"; it is a matter of not obeying laws of logic which have
    been accepted for a couple of thousand years. This is not a "dull error", but
    an annoyiing statement of fact.

    Many fuzzy mathematicians assert that this failure is a virtue. After having
    been involved in creating fuzzy expert systems and a fuzzy expert system
    shell for over 15 years, I can not accept that this failure is a good thing.
    In some circumstances it produces highly counter-intuitive results. For
    example, if "~2" is a triangular fuzzy two, then the intersection of "~2 and
    NOT ~2 is bimodal, and the union "~2 OR NOT ~2" has two notches in it.

    It is, however, quite possible to define multivalued logics in a way which
    does obey both excluded middle and contradiction. Jim Buckley and I have a
    couple of fairly recent papers in Fuzzy Sets and Systems which defines such a
    family of logics. If the Zadehian max-min logic is taken as a default, then
    Elkan's proof crumbles.

    William Siler

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