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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Defining fuzzy descriptors (was  NOT and DIFF)
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Date: Mon, 24 Mar 1997 10:59:34 GMT
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WSiler (wsiler@aol.com) wrote:
: The Buckley-Siler family of logics has one parameter which specifies which
: logic of the family to apply; r, the correlation coefficient between the
: truth values of the two operands, assuming that the operand values can
: span the full range of zero to one.

: Clearly, if the two operands have truth values of say 0.25 and 0.36 the
: correlation coefficient cannot be be one, since that would require all
: values of the operands be identical. Also, the correlation coefficient
: cannot be -1, since that would require that the operands be complements.
: Only a restricted range of r is possible for specified values of the
: operand truth values. That is the restriction imposed by our logic family.

Yes, but in the context of investigating associativity, you
are asking whether, a AND (b AND c) is the same as (a AND b) AND c.
Therefore, letting d = a AND b, and e = b AND c, you end up
asking whether when you take coefficients appropriate 
respectively to a AND e, and d AND c, the results are the same.
Your development makes it clear that these coefficients 
cannot be chosen independently, and moreover, that there is
in fact a *tighter* restriction than would be the case if
coefficients were independently chosen for combining a AND e,
and d AND c, respectively.  

In my development, I did not
find a way to identify or characterize such a tighter restriction, 
but it was clear, as you also indicate, that if such a tighter
restriction--also the dependency connection between them--
were not respected, then associativity need not hold.  And
similarly for distributivity.

But what I continue to maintain is that if it holds in your
development (with appropriate restrictions), it must also 
hold in my development (with analogous restrictions), arguing
purely from the algebraic parallel between my development and
yours.  Don't forget, in your development, you have a AND b
ranging from a low of max[a+b-1,0] when the parameter r is
lowest, to a high of min[a,b], when the parameter r is highest,
with ab in the middle corresponding to the result when the
parameter r is zero.  It is the same in my development, 
but with the algebraic difference that the 
(semantic consistency) coefficient I use ranges in
general from -1 to +1, instead of from your rl to ru.
There is therefore a one-to-one algebraic correspondence
between your development and mine, and hence there must exist
in my development, analogously with yours, appropriate
restrictions on the semantic consistency coefficient that
would permit associativity (also distributivity) to hold.
In the absence of such restrictions, which I failed to
identify or characterize, associativity and distributivity
would appear not to hold in general, as I discovered, and as 
you would obviously agree.

: Bill Siler

Regards,
S. F. Thomas
