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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: New Fuzzy Logic
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Date: Tue, 24 Sep 1996 20:53:44 GMT
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William Siler (williamsiler@delphi.com) wrote:
: Reposting article removed by rogue canceller.

There is some resistance to progress... obviously :)

(( cuts ))
: We first propose a source of fuzziness. We suppose that the
: truth value > 0 and < 1 of a fuzzy logical statement A is drawn
: from a number of underlying (probably implicit) correlated
: random variables whose values alpha[i] are binary, i.e. 0 or 1
: with a Bernoulli distribution, and that the truth value of A is
: a simple average of these binary values. (George Klir (1994)
: proposed a similar process where the random values are binary
: opinions of experts as to truth or falsehood of a statement.)

This is nothing new, btw.  Gaines proposed it in 1975, Thomas
in 1979 and again in 1995, Hisdal in 1985 and in a series of
subsequent papers.  (Email me for references.)  Fuzzy orthodoxy
has resisted the concept, but it cannot be on technical grounds,
nor ultimately on semantic or philosophical grounds.  But that
is a touchy subject -- for reasons I have not been able fully to
fathom -- and perhaps another thread.

(( cuts ))
:   13. A AND (B OR C) = (A AND B) OR (A AND C), any appropriate
: r.
:   14. A OR (B AND C) = (A OR B) AND (A OR C), any appropriate
: r.

I am pleased to see distributivity restored along with
law of excluded middle and law of contradiction.  I could not
establish such a result in Thomas (1995).  Although the development
there also makes use of the correlation coefficient, it is
defined not with respect to the statistical process outlined
by Siler and Buckley, but with respect to the membership functions
of the underlying fuzzy terms.  This in turn leads to a different
derivation of the rules of combination, for which distributivity
does not in general hold.  There are however, two functions in
my development -- the term determinant of semantic consistency,
phi, and the distance determinant of semantic consistency, psi 
(Thomas, 1995; p. 100) -- which are left unspecified except broadly 
as to shape and to anchoring values at extreme points.  The 
Siler/Buckley development can be regarded as a special case in 
which the functions phi and psi are fully specified, and for which
distributivity holds.  This is an important result,
which I for one heartily welcome.  It further strengthens a
fuzzy development which proceeds from objectivist foundations for
the notion of grade of membership, and which rejects as a clear
nonsense the sorts of "weird" membership functions obtained
when the original min/max rules are unquestioningly applied to such 
constructs as "tall and not tall".  For a long time, fuzzy logic
accepted these weird constructs as the price it seemed we had to
pay for the otherwise compelling semantics associated with the
concept of fuzzy; if not Thomas (1995), then Siler/Buckley
should convince us all that we can have our fuzzy cake and eat it
too.  I look forward to reading the fully elaborated paper, which
may attract further comment from me when I do.  In the meantime,
my hearty congratulations to Siler and Buckley.

: References:
:  
<snip>
: Thomas, SF (1994). Fuzzy Logic and Probability. ACG Press,
: Wichita, KS.

Er... that's not quite correct.  It should read as follows:
S. F. Thomas (1995).  Fuzziness and Probability.  ACG Press, etc.

Regards,
S. F. Thomas

