Re: Fuzzy proofs.

From: Wise (weiz@spam-block.sympac.com.au)
Date: Thu May 31 2001 - 12:30:07 MET DST

  • Next message: Ulrich Bodenhofer: "Re: Fuzzy proofs."

    Sidney

    I have to say by the end of your comments
    I did not know if you were supporting or decrying fuzzy logic,
    but if you are asking can fuzzy logic accommodate the "syllogism":

    All rich men are happy
    John is rich
    Therefore, John is happy.

    I would argue yes.

    First the terms rich and happy are just names,
    labels we have assigned to sets. That these sets could
    also be named wealthy or joyful AND still represent the same
    universe and this is not really fuzzy mathematical. It's just
    a cause for a distraction about the underlying maths and logic.

    The proposition is

    All men that [to some extent] belong to the rich set are [to some degree]
    also members of the happy set.
    AND
    John is [to some degree] rich.

    I am therefore able to cope with the conclusion that

     John [being a rich man] belongs [to some extent] to the happy set.

    Yep we dismiss with the words in the square brackets when holding a
    discussion
    as we do with a lot of other superflious words when
    we speak. Natural conversation has a lot of implied understanding . For
    example I had to somehow
    know that John was a man and that a man was singular men etc etc..

    Now here is the sexy bit.

    While we could conduct this [fuzzy] conversation and have drawn the
    lingusitic conclusions,
    I think you call them tautological rules, If you now wanted to provide me
    just a few numeric
    details I could calculate just how dam happy john was.

    More if you wanted to keep john happy we could hook him up to a fuzzy
    control system that
    would meter out to John enough money to keep him at a given happy
    equilibrium.

    Even more... as John's happy levels changed, the fuzzy control system
    could be hooked into a fuzzy learning algorithm that could learn the new
    levels of rich for
    johns changing happyness and adjust the metering of wealth for john to
    sustain that new level.

    if you are right and Zadeh promised that he would deliver a system that
    coped with natural
    language AND one that would be true to the empirical rules of the domain. I
    am here to argue
    he has delivered. And more he has laid a foundation for developments that
    have significant
    implications for drawing out the intelligence hidden in natural language.

    All this is in our recent history and yet the bivalet model, far older,
    struggled to define with how tall was tall.

    No competition Sidney or Fuzzy speaking bivalet logic is has a small
    membership value in the competition for fuzzy worth set !

    Rob W

    Sidney Thomas wrote in message <3B0D5899.78F40859@verizon.net>...
    >
    >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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