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From: WSiler@aol.com
Date: Sun Aug 19 2001 - 06:40:17 MET DST

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    In a message dated 8/16/01 4:00:19 AM Central Daylight Time,
    earldcox1@home.com writes:

    << > In article , Robert Dodier writes:
    > >
    >> Any such definition must ignore the relation between elements in a
    compound: if truth(B')=truth(B), then in any proposition containing A and B,
    I can swap in B' in place of B, and get exactly the same truth value for the
    compound; whether the elements are redundant, contradictory, or completely
    unrelated doesn't enter the calculation.
    >>

    This is only true if we have truth-functional logic. Truth-functionality and
    the failure to obey excluded middle and non-contradiction are sacred cows in
    the fuzzy-math literature. However, we pay a price for this. The hard fact is
    that in the real world, these sacred cows often make no sense.

    Consider, for example, Earl's fuzzy set in which Short had membership 0.15,
    Medium had membership 0.85, and Tall had membership 0.65. Common sense tells
    us that Short OR Medium OR Tall should be 1, instead of .85, and that Short
    AND Medum AND Tall should probably be zero instead of 0.15.

    This is easily achieved if we see that Short, Medium and Tall are all
    negatively associated. In my system, if A and B are not semantically
    inconsistent, we can use any multivalued logic we please including Zadeh's;
    but if A and B are semantically inconsistent, we MUST use A OR B = min(1, a +
    b), and A AND B = max(0, (a + b) - 1). Applying this to Earl's example, Short
    OR Medium OR Tall = 1, and Short AND Medium AND Tall = 0.

    This logic is not truth functional; we have to parse the (complex)
    proposition to see what logic we should use. But the results give us a
    multivalued logic which makes sense both mathematically and to the layman.

    In your example, suppose that the proposition we wish to evaluate is A AND B.
    You state that if truth(B') = truth(B), that A AND B' has the same truth
    value as A AND B. Sounds reasonable, and seems to be in accordance with the
    rule of substitution. We take the Zadeh logic as a default. Suppose that B
    and A are semantically unrelated, and that the truth value of A, B and B' are
    all 0.5. Then A AND B = A AND B'
    = 0.5.

    Now suppose that A is actually B, and that B' is actually NOT B. Let truth(A)
    be 0.5, truth(B) be 0.5; then (B') = 0.5 With your method truth(B AND NOT B)
    is 0.5, as is the truth(B OR NOT B). If, however, we parse these expressions
    and use the appropriate logic operators, we get truth(B AND NOT B) = 0, and
    truth(B OR NOT B) = 1, which makes perfect sense to a biologist like myself.

    If we parse Elkan's two expressions and use the appropriate logic above with
    any multivalued logic as the default, we find that the two expressions are
    perfectly equivalent, and Elkan's proof crumbles.

    Conclusion: we better be careful how we apply the rules by which we evaluate
    complex propositions and derive one logical proposition from another when
    using multivalued logics.

    William Siler

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