**Previous message:**WSiler@aol.com: "Re: Thomas' Fuzziness and Probability"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

In a message dated 8/16/01 4:00:19 AM Central Daylight Time,

earldcox1@home.com writes:

<< > In article , Robert Dodier writes:

*> >
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*>> 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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