In a message dated 8/16/01 4:00:19 AM Central Daylight Time,
<< > In article <firstname.lastname@example.org>, Robert
>> 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. I know that
truth-functionality has been accepted as a given in the fuzzy-math
literature. However, we pay a terrible price for truth functionality. That
price is the loss of excluded middle and contradiction, which underlies
Elkan's paper of some years ago. The hard fact is that in the real world,
this loss often makes 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 (or something like
that). 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 made reasonable if we see that Short, Medium and Tall areall
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, 1 - (a + b). Applying this to Earl's example, Short
OR Medium OR Tall = 1, and Short AND Medium AND Tall = 0.
This logic may not be considered truth functional, I suppose; 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 proposiition 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. (let lower case letters be truth values.) 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'
Now suppose that A is actually B, and that B' is actually NOT B. Let truth(A)
=0.5, truth(B) is 0.5 and truth(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
Conclusion: we better be careful how we apply the classical logic rules by
which we derive one logical proposition from another when using multivalued
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