Re: Fuzzy proofs.

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Wed May 30 2001 - 18:17:43 MET DST

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    Wise wrote:
    >
    > Sidney
    >
    > I have to say by the end of your comments
    > I did not know if you were supporting or decrying fuzzy logic,

    I am merely affirming the tautology of natural language semantics,
    and of bivalent logic, known as ponendo modus ponens, and asserting
    that fuzziness in the "labels" is not sufficient to overturn modus
    ponendo ponens, nor law of excluded middle, nor the law of
    contradiction, etc. I am further asserting that the original min-max
    Zadehian fuzzy set theory is wrong to the extent these laws fail...
    but wrong not in any absolute sense, because it is clearly right some
    of the time, rather wrong in these particulars. It is my opinion that
    the fuzzy set theory can be rescued in this regard, and without doing
    away with the fuzziness, by developing a fuller theory in which the
    rules for AND and OR are allowed to depart in certain well-defined
    ways from the min-max ones which got the ball rolling so to speak.
    Min-max emerges as just a particular case of a larger theory that
    also includes the product-sum rules as a particular case, and the
    bounded-sum rules as a particular case, and there is a sort of
    meta-rule internal to the logic of the whole system, rather than an
    arbitrary choice imposed from outside, that mediates among these
    extreme cases. What further emerges is that LEM and LC and modus
    ponendo ponens are affirmed as rules of form within the system which
    however are derived as a matter of fuzzy-mathematical necessity using
    preservation of semantic *content* as the meta-semantic basis for
    drawing the required conclusions. The matter may be stated very
    briefly as follows:

                     
                      { (1-t)ab + t min(a,b) , if t >= 0
            a AND b = {
                      { (1+t)ab - t max(0,a+b-1), if t < 0
     
    and t is a semantic consistency coefficient depending only on the
    membership *functions* a and b, determined essentially by the
    correlation coefficient between the two functions. The
    specializations of this rule are of course very familiar, and apply
    for the special cases when respectively t=1, t=0, and t= -1, giving

            a AND b = min(a,b), (t = 1),

            a AND b = ab, (t = 0), and

            a AND b = max(0,a+b-1), (t = -1).

    The corresponding rule of disjunction is

                     
                     { (1-t)(a+b-ab) + t max(a,b) , if t >= 0
            a OR b = {
                     { (1+t)(a+b-ab) - t min(1,a+b), if t < 0

    And the well-known special cases when t=1, t=0, and t= -1
    respectively are

            a OR b = max(a,b), (t = 1),

            a OR b = a+b-ab, (t = 0),

            a OR b = min(1,a+b) (t = -1).

    But I repeat, and emphasize, that the semantic consistency
    coefficient t is not an arbitrary externally imposed parameter to
    give us the answer that we want, rather is a function of the curves
    being connected. At any rate, with these generalized rules of
    conjunction and disjunction, and with the one-minus rule for
    negation, it may be shown that LEM, LC and modus ponens hold. THis
    last moreover holds whichever rule of implication -- (NOT a OR b) or
    (a AND b OR NOT a) -- is used.

    My point about the syllogism regarding rich and happy was precisely
    to make the point that the fuzziness of the terms involved are not
    sufficient to undo modus ponens. And likewise, the fuzziness of the
    term tall is not sufficient to undo either LEM or LC, which was my
    earlier point about the hypothetical witness in court asserting that
    the perpetrator was "tall and not tall", which remains, in the
    natural language with which I am familiar, the constant absurdity.
    The rules given above resolve this very easily. The semantic
    consistency coefficient binding "tall" and "NOT tall" is -1,
    corresponding to the correlation coefficient of -1 between any term
    and its negation under the one-minus rule. Hence using the general
    rules above, we get:

            a AND NOT a = max(0, a + (1-a) -1) = 0,

    as LC would require. And furthermore,

            a OR NOT a = min(1, a+(1-a) ) = 1

    as LEM would require.

    In this matter I neither support nor decry fuzzy logic. THe question
    for me, which Zadeh's seminal contribution has made possible, is how
    to reconcile the fuzziness in natural language semantics, with the
    tautologies of bivalent logic which have served us so well, and which
    to my mind remain empirical laws of natural language semantics, which
    fuzziness per se provides no basis for overturning, because LEM, LC,
    and so forth are rules of *form*, and apply irrespective of semantic
    content, and in particular irrespective of the fuzziness of the
    terms, eg. rich, happy, tall, that may be involved. A theory based on
    content can validate those rules of form, but if it appears to
    conflict with them, I would assert that the theory based on content
    needs to be reworked. That is one of the things that concerned me in
    my _Fuzziness and Probability_ (ACG Press, 1995).

    I hope this is helpful.

    Regards,
    S. F. Thomas

    > 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 math 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 superfluous 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
    > linguistic 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 happiness 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 bivalent model, far older,
    > struggled to define with how tall was tall.
    >
    > No competition Sidney or Fuzzy speaking bivalent 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 =>.

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