Re: Thomas' Fuzziness and Probability

From: S. F. Thomas (
Date: Sun Aug 19 2001 - 07:41:44 MET DST

  • Next message: Robert Dodier: "Re: Thomas' Fuzziness and Probability"

    I see your difficulty. You think that if A is a fuzzy term, and its
    membership function is denoted simply by a, let's say, then the
    one-minus rule of negation gives the membership function of NOT A as
    1-a. Hence the "middle" is included, so to speak, and LEM and LC
    should fail, as indeed it obviously does if the min-max rules are then
    applied. For we have A AND NOT A being modeled in the meta-language as
    min(a,1-a), which gives us the well-known middle with a peak at 0.5
    (assuming of course that a has its max at 1, its min at 0, and there
    is gradation in-between).

    Now let's try another rule of conjunction, in particular the
    Lukasiewicz bounded-sum rule, for which we have for two membership
    functions a and b, and their corresponding terms A and B,

            mu[A AND B] = a AND b = max(0, a+b-1).

    In the particular case where B is NOT A, and b=1-a, we have under this

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

    and in accordance with the law of contradiction, the term A AND NOT A
    is rendered as the comstant absurdity whose membership value is
    everywhere 0. LC is upheld.

    So again the fuzziness of any term A, does not require failure of LC
    (or LEM)... it depends on the rules used for combination of terms. And
    in the theory that I have developed, where the rules of combination
    are not defined at the outset, rather *derived* from other more basic
    semantic considerations, it is possible to see that the rules of
    combination may be self-selecting, depending upon the semantic
    relation between A and B, sufficiently captured in the corresponding
    membership functions a and b. So the question in a sense is not what
    rule to use, so much as when does which apply. Sometimes the
    self-selection yields min-max, sometimes bounded-sum, sometimes
    product and product-sum, and in general an infinity of linear
    combinations of these extreme cases. Since you have the book, I refer
    you to Section 3.4.1, p. 115. And when this self-selection takes
    place, the general rule of combination specializes to the Lukasiewicz
    rules for any term and its negation -- essentially because the
    correlation coefficient between a and 1-a is necessarily -1,
    corresponding to negative semantic consistency binding the two terms
    -- and LC and LEM are upheld. That certainly is what my intuition
    requires, and I have never been able to come up with a thought
    experiment in which, in the object language, LEM and LC are required
    to fail, *because* of fuzziness per se. If you allow something other
    than min-max in the meta-language, that can be modeled. So you can
    have points u in the domain say such that *both* a>0 and (1-a)>0
    (which is what fuzziness requires), yet no point such that a AND NOT a
    > 0, which Lukasiewicz in particular would give you. This corresponds
    to what we can have in the object language, where Jane might say her
    attacker was "tall", John might say he was "not tall", but neither may
    say the attacker was "tall and not tall", unless either one explicitly
    steps into a meta-language, for example by saying "some would say he
    is tall, others not", but the pure object-language construct "tall and
    not tall" remains the constant absurdity in the language with which I
    am familiar.

    Hope that helps.

    S. F. Thomas

    PS. I'll be offline for a couple of days as I am about to embark on a
    move across country. See you all when I get back online.

    Joe Pfeiffer <> wrote in message news:<>...
    > Ah, I think I've finally got it. Tell me if this rephrasing is what
    > you have in mind.
    > LC is only called into question if one accepts a fairly tortured
    > extension of LC into fuzzy logic: namely, ``if x has any membership
    > at all in A, x can have no membership in ~A.'' But it would be more
    > reasonable to define ~A as that set such that if x's membership in A
    > is m, x's membership in ~A is (1-m). Notice that this also gives us
    > LC for crisp sets as a special case.
    > In the example (and substituting what I believe to be more reasonable
    > phrasing -- the original phrasing got in the way of my understanding),
    > if the witness were to say that the assailant was ``tall but average''
    > it would be ridiculous, as that would be asserting membership of 1 in
    > both tall and (not tall)[see note below]. If the witness were to say
    > that the assailant was ``sort of tall and sort of average'' then she
    > would be asserting only that the assailant's membership in tall was in
    > (0,1), and his membership in (not tall) was also in (0,1). This would
    > be completely reasonable.
    > [note] As in my last post I assume ~tall == (short U average). If we
    > assume that any membership in tall implies no membership in short,
    > then any reasonable definition of union will give us that for anyone
    > with any membership at all in tall, ~tall = average.
    > PS: I'd like to comment, as the originator of this thread, that it
    > has done more to solidify my understanding of fuzzy logic than
    > everything I've read to date. Thanks to you all.

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