RE: fuzzy proofs and law of excluded middle

From: Ulrich Bodenhofer (ulrich.bodenhofer@scch.at)
Date: Mon Jun 25 2001 - 19:34:38 MET DST

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    >
    > 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".
    >
    BTW, the way Elkan did this was logically dull! First assuming that the
    laws would hold in the fuzzy case and showing then that "fuzzy logic"
    collapses into the Boolean case is really weird!
    >
    > 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.
    >
    What is logic? The laws of (Boolean) logic as we use it today are not
    absolute, aren't they? What about Goedel, Post, Lukasiewicz, Wajsberg,
    etc.? They did not even know what "fuzzy" is, and I really think that
    what they did was logic!
    >
    > 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.
    >
    For sure not by definition. In fact, it very much depends on the underlying
    logic and which concept of approximate equality you assume.
    >
    > 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.
    >
    I will take a (very critical) look at them. Anyway, Lukasiewicz logic (which
    is
    known since the 30ies) obeys these two laws anyway, so I am quite curious
    what the papers may contain.

    Ulrich

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