Re: fuzzy proofs and law of excluded middle

From: Prof. Dr. Siegfried Gottwald (gottwald@rz.uni-leipzig.de)
Date: Wed Jun 27 2001 - 22:27:49 MET DST

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    I mean some additional comments may be helpful.

    > 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".
    >

    Sorry, but I think that Bodenhofer was right with his evaluation.
    However, the reason may be completely another one as given by him.

    > 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.

    Well, it is just this what is not true. What Elkan discussed was
    neither the law of excluded middle, nor the law of non-contradiction
    at all.

    Let me concentrate on the law of excluded middle, for the law of
    non-contradiction the situation is completely similar.

    The law of excluded middle is a metalogical statement, related to
    classical logic, saying that a proposition has to be true or has to
    be false. By the very approach toward fuzzy sets, as well as by the
    very approach toward many-valued logics, this law has to fail there.
    And this is completely trivial: it is just the heart of the matter
    in fuzzy and many-valued topics that the law of excluded middle has
    to fail there, because one accepts degrees in between "true" and
    "false". To make a paper out of this obvious fact is really
    astonishing. (And giving an award to such a paper even more.)

    However, there is something more complicated here. And this comes
    from the fact that the FORMULA $p \lor \neg p$ in some (weak) sense
    CODES (or: represents) this law of excluded middle inside CLASSICAL
    PROPOSITIONAL CALCULUS.

    That this representation is not a faithful one in any situation
    becomes clear just from the fact that this formula $p \lor \neg p$
    may become a logical truth in a suitable system of many-valued logic,
    if one reads the connectives $\lor$ and $\neg$ in a suitable way:
    e.g. if they are the Lukasiewicz arithmetical disjunction, and the
    Lukasiewicz negation.

    >
    > 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.

    But, let me repeat: it is the heart of the matter.

    > 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.
     
    Here, however, the OR is the max-disjunction. If one considers also
    here the Lukasiewicz arithmetic disjunction, together with the
    1-..-negation (i.e. Lukasiewicz negation), "~2 OR NOT ~2" becomes
    the universal fuzzy set (over the intended universe of discourse) -
    and this is not at all counterintuitive (for me). So the core problem
    seems to be how to choose the (generalized) connectives: particularly
    in cases in which there are more than only one candidate in the
    generalized (fuzzy or many-valued) setting.

    Siegfried Gottwald

    ------------------------
    Prof. Siegfried Gottwald
    Universitaet Leipzig
    Institut fuer Logik und Wissenschaftstheorie
    Burgstrasse 21

    D-04109 Leipzig/Germany

    email: gottwald@rz.uni-leipzig.de
    phone: (0341) 97 35770/71
    fax: (0341) 97 35798

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