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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory? 
Message-ID: <1994Dec18.160403.980@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Sun, 18 Dec 1994 16:04:03 GMT
References: <1994Dec4.174744.27946@decan.gate.net> <1994Dec14.152840.22904@newsserver.rrzn.uni-hannover.de>
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mackw@bytex.com wrote:

: > : S. F. Thomas (sthomas@decan.gate.net) wrote:
: > 
: > : : I still insist that any utterance "A and not A" has no meaning 
: in 
: > : : natural language, at least not in the normal sense of the 
: conjunctive
: > : : "and".  

: I'll agree that in most common usage the English statement "A and not 
: A" is not meaningful, but there are lesser used cases where this 
: statement is used either for humor or emphasis where it does have 
: meaning.  Let me rewrite our burglar example using "6 foot" instead of 
: tall.

: WITNESS: The burglar was 6 foot tall.
: ATTORNEY: But my client is 6 foot one.
: WITNESS: Your client is the burglar.
: ATTORNEY: But 6 foot one is not 6 foot.  Are you saying the burglar
: 	was both 6 foot tall and not 6 foot tall?
: WITNESS: In this case yes, the burglar was 6 foot tall and not
: 	6 foot tall.

: Yes, the example sounds contrived, but the point is the statement 
: "6 foot tall and not 6 foot tall" is meaningful in this context.  We 
: would not consider the information useful and we probably accuse the 
: attorney of arguing a trivial point, but we can understand the 
: statement (likewise "A and not A" is meaningful but not terribly 
: useful in fuzzy logic).

: A fuzzy logic interpretation of the witness's final statement is that 
: the burglar has a high degree of membership in the fuzzy set "6 foot," 
: but being 6 foot one, he also has a low degree of nonmemebership in 
: this set.  This is explicitly stating some of the implicit meaning in 
: the witness's statement.  In this regard, fuzzy logic does a fairly 
: good job of translating the meaning of the english statement into a 
: form which a computer (or computer programmer) may be able to handle.

I see this as an issue of "combination of evidence".  What you have are
two _different_ assertions regarding the assumed same unknown of interest --
the height of the client/burglar, which let us call X.  Thus

	WITNESS:  X is 6 foot			(A)

	LAWYER :  X is 6 foot one               (B)

We are concerned with finding a restriction on X which is consistent 
with both A and B.  As these are two independent utterances, there is
no requirement of semantic consistency between them, and I find it
appropriate in that case to use a product rule of combination.  I
would agree that a min rule of combination might appear to "work" also,
and would not violate the law of contradiction, but my investigations
have revealed that the min/max rules are appropriate under
conditions of strong positive semantic consistency, which would not
apply to two different speakers.  The product rule is more appropriate
in this case, inferentially more correct, and as easily handled by computer.

You should also note that if it were the case that the client was
only five foot, so that we have

	LAWYER : X is 5 foot			(C)

and we sought to reconcile A with C, we would have a combined 
characterization for X which would yield no solution, whatever the 
rule of combination used, and the client would presumably be off the hook!

Now, the "admission" extracted by the lawyer from the witness when the 
latter is confronted with B, thus:

	WITNESS: X is NOT 6 foot		(D)

is really quite contrived, considering the fuzziness of the usage of the
term "6 foot" which is implicit in this context.  Had the witness had a
chance to use some sort of measuring device, we would have had a 
description such as "X is 6:0", in which case A and B could still be
reconciled through reference to intrinsic measurement error in the 
disparate measurement devices used by the witness and the lawyer.
But Aristotle's general semantic law as to the absurdity of any 
utterance "A and not A" remains intact. In my opinion.

: Wayne Mack

Cheers!
S.F.Thomas

P.S.  By the way, what kind of membership function would your computer
generate for the term "6 foot and not 6 foot"?  Choose any rule of
combination you like.


