# Re: does fuzzy bound probability?

Subject: Re: does fuzzy bound probability?
WSiler@aol.com
Date: Tue Nov 21 2000 - 15:55:30 MET

In a message dated 11/21/00 5:10:46 AM Central Standard Time, scott@ramas.com
writes:

<< Because (standard) fuzzy arithmetic makes no assumptions about the
> dependence or independence among quantities, it has been suggested
> that fuzzy arithmetic might be able to provide bounds on probability
> distributions in cases where the dependence among input variables
> cannot be specified empirically.
>>

I'm not sure what basis was offered for the statement that fuzzy logic makes
no assumptions about independence, but I'm afraid that the statement is not
true. Standard Zadehian min-max fuzzy logic assumes implicitly that the
operands of the logical operations AND and OR are positively associated as
much as possible, just as the probabilistic AND (a*b) and OR (a + b - a*b)
assumes independence (zero association) and the bounded sum and difference
logic AND (max(0, a + b - 1)) and OR (min(a +b - 1)) assume maximum negative
association. Jim Buckley and I have a paper or two in FS&S which discuss this
point and present a family of multivalued logic parameterized in terms of the
correlation coefficient between the operands, based either on neccessity (a
AND NOT a) or past history.

Consequently, any manipulations which employ fuzzy logical operations (such
as the extension principle) implicitly make an assumption about the
independence of the operands. That would seem to include fuzzy arithmetic.

William Siler

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