Re: Fuzzy proofs.

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Thu May 31 2001 - 15:34:23 MET DST

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    Ulrich Bodenhofer wrote:
    >
    > Hm, in any case you have to be aware which kind of fuzzy logic you are
    > assuming.
    > There is NOT a single unique kind of fuzzy logic. There are infinitely many
    > ways
    > to define the three connectives /\, \/, and =>.

    While this is certainly true, one promise of the fuzzy set theory, as
    opposed to fuzzy logic per se, was that it provided a way to model
    certain aspects of natural language semantics. In this manifestly
    empirical domain, the question which arises is whether the
    tautological rules of inference known from the standard bivalent
    logic which provides the rules of logic and inference of the
    meta-language within which the theorems of fuzzy set theory, and
    indeed of fuzzy logic, are advanced, may be retained within a
    suitably constructed fuzzy set theory, and while retaining the
    essential fuzziness. As an empirical matter, I would argue that the
    tautologies familiar from the meta-language must have their
    counterparts in the object language where the fuzziness resides. In
    another thread, I have said that the fuzziness in the term "tall" for
    example, cannot rescue a witness from the derision of the court if
    she were to say "the perpetrator was tall and not tall." Similarly,
    the modus ponendo ponens as applied in natural language is no
    respecter of fuzziness. For example the syllogism:

            All rich men are happy - (major premise)
            John is rich - (minor premise)
            Therefore, John is happy- (Conclusion)

    carries through regardless of the fuzziness of the terms "rich" and
    "happy". And likewise for all of the well-known tautologies of the
    classical bivalent logic, which rely not at all on the meaning of the
    object-language propositions, rather only on their form, for example

            P & (P->Q) -> Q

    where the meanings of P and Q and P->Q are not so much at issue as
    the form of the compound proposition of which they are constituent
    parts. So now, the question for a fuzzy set theory is whether, given
    that fuzziness is no excuse for the failure of these tautologies, is
    how to make the fuzzy set theory reflective of these laws of
    semantics which continue to hold in the real-word, natural-language
    semantic domain. The "logic" of such a (reformulated if need be)
    fuzzy set theory of semantics should drop out a posteriori from the
    theory, rather than stand alone as a putative "logic" -- one of
    infinitely many -- in search of an application domain. At any rate,
    when the fuzzy set theory is anchored in the natural language domain
    which is its motivating point of departure, the issue is not so much
    whether or not there are infinitely many ways of "defining" the
    logical connectives of AND, OR and NOT, but when does which apply,
    and how can they be fused into a harmonious whole that obeys the
    observed, empirical, tautological rules of inference that remain a
    feature of natural language semantics even when fuzziness intrudes.
    The program to which these observations give rise is to show that,
    within a properly formulated fuzzy set theory of semantics, the rules
    of inference based on the well-known tautologies of *form*, may be
    preserved within a semantic theory in which the rules of inference
    are developed based on the preservation of semantic *content*, which
    latter is the essential contribution of the fuzzy theory...it is a
    way of modeling semantic *content*. I have given the beginnings of
    such an enterprise in my _Fuzziness and Probability_ (ACG Press,
    1995). LEM, LC, and modus ponendo ponens are all retained within the
    reformulated theory...and without the precisiation stratagem being
    adopted of simply getting rid of the fuzziness. And, btw, we would
    see fuzzy not so much as being logically prior to crisp as some have
    maintained, but as outgrowth, in the same way that binary computers
    based on a bivalent logic allow us to explore the higher reaches of
    fuzziness in computable fuzzy models of various problem domains. And
    within the outgrowth, there is a crisp subclass that behaves exactly
    likely the objects which populate the bivalent meta-language. It is a
    boot-strapping metaphor that is in play, so common in nature, of
    complexity emerging as an outgrowth of essential simplicity. As they
    say in another context, "as above, so below," but I digress.

    All of this is no help whatsoever to the original poster, I am aware.
    It also may be of no help whatsoever to committed abstract logicians.
    But it just might be of interest to those who were intrigued by the
    original promise of Zadeh's fuzzy set theory, which was to cope in
    precise metalanguage ways with the fuzziness of natural language,
    while being true to the empirically observable laws in this original
    domain of application. The rules in domains other than natural
    language semantics may be different I would agree, but even there, we
    always need a meta-language, don't we?

    > Best regards,
    > Ulrich
    >

    Regards,
    S. F. Thomas

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