Elkan and the core problem of generalized connectives

From: S. F. Thomas (sfrthomas@yahoo.com)
Date: Sat Jun 30 2001 - 13:50:11 MET DST

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    Prof. Dr. Siegfried Gottwald (gottwald@rz.uni-leipzig.de) wrote:

    (( major cuts))
    > 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.

    This is a major burden of my _Fuzziness and Probability_ (1995, ACG
    Press), which addresses the question -- I believe successfully -- when
    does which set of rules apply. For most assuredly the min-max rules
    are sometimes correct, the bounded-sum rules sometimes correct, and
    the product-sum rules sometimes correct, but none of these are
    *always* correct. Moreover, these are extreme cases of an infinity of
    rules in general which however are in a sense linear combinations of
    these extreme cases. And the precise linear combination in each case
    is a function of the degree of correlation of the respective
    membership functions. The greater the degree of positive correlation
    >... which correlates further with some corresponding notion of
    positive semantic consistency, for example in the usage of the terms
    "TALL" and "VERY TALL" ... the more correct are the min-max rules.
    Conversely the greater the degree of negative correlation (and the
    greater the corresponding degree of negative semantic consistency, for
    example any term and its negation), the more correct the Lukasiewicz
    (bounded-sum) rules. And in between, the closer to zero correlation,
    or simple semantic independence, the more correct the product-sum
    rules. The result is a generalized set of connectives which, in terms
    of closeness to one or other of the extreme cases, are self-selecting
    in context.

    Some interesting results emerge. When terms belong to different
    universes of discourse, for example "TALL" and "HEAVY", there are no
    constraints of semantic consistency between them, and reflecting such
    independence, the product-sum rules are appropriate. This is relevant
    when we consider the material implication relation. For example, if we
    consider the material implication "TALL -> HEAVY", one approach from
    classical logic would be to consider this to mean "HEAVY OR NOT TALL".
    This disjunction would be modeled by the product-sum rule based on the
    semantic independence between the two different universes of discourse
    (weight and height in this case). If we consider the alternative
    rendition for the material implication of "(TALL AND HEAVY) OR NOT
    TALL", there is negative semantic consistency binding the two halves
    of the disjunction, because under the one-minus rule for negation,
    there is perfect negative correlation between "TALL" in the first part
    of the disjunction, and "NOT TALL" in the second; the bounded sum rule
    is therefore appropriate. Amazingly, these two versions of material
    implication may be shown to yield the same result when using the
    generalized connectives. This was left as conjecture in my _Fuzziness
    and Probability_, but based on the results in recent papers by Siler
    et al just cited on this thread by Siler, this result may now readily
    be proven.

    This solves the problem (if it is admitted to be such) posed by
    Turksen in 1979, namely that these two variants of material
    implication yield inconsistent results under the min-max calculus.
    Elkan's theorem, and especially its method of proof, appears to me a
    variant of Turksen's earlier finding, and the problem it poses (again
    if it is admitted to be such, rather
    than greeted with some or other stratagem of denial) equally is solved
    by this generalized rule for connectives.

    Btw, the failure of LEM and LC are of course also corrected in this
    development. No doubt Siler means to imply that failure of LEM and LC
    under the min-max calculus are of a piece with the particular failure
    to which Elkan drew attention, whether or not the latter explicitly
    addressed LEM and LC (which he did not). Further btw, I would agree
    that Elkan erred in his exposition, arguably when he simply assumed
    the "logical equivalence" of different renditions of the material
    implication relation familiar from the classical bivalent logic, which
    seems to me to be a form of begging the question. And it certainly
    gave fuzzicists in denial an excuse to dismiss the deeper underlying
    problem, rather than address the "core problem" which you properly
    identify above.

    > Siegfried Gottwald
    > ------------------------
    > Prof. Siegfried Gottwald

    S. F. Thomas

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