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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Defining fuzzy descriptors (was  NOT and DIFF)
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References: <33116361.40EC@calvanet.calvacom.fr> <m2sp216qts.fsf@mailhost.neuroinformatik.ruhr-uni-bochum.de>
Date: Thu, 13 Mar 1997 11:40:04 GMT
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David Kastrup (dak@mailhost.neuroinformatik.ruhr-uni-bochum.de) wrote:
: Maurice Clerc <mcft10@calvanet.calvacom.fr> writes:

: > You are certainly right: my idea of what "consistency" means is a bit
: > ... fuzzy ;-)
: > 
: > In fact, it is a mixture of logical consistency and psychological
: > validity. Personnaly, I am more interested by the secont part. I don't
: > need a logic where "blue or blue" can be different from just "blue".

: Well, so you'll not find
: I'll do, and I'll do and I'll do
: (as said by one of the witches in MacBeth) more impressive than
: I'll do.

No, but it'd probably be a good first approximation. :)

: Using Min/Max Fuzzy logic will do fine for this criterion.

True.  But there're other glaring anomalies.

: > Also a logic where  "tall or not tall" can be not completely true seems
: > to be not "psychologically valid"  (although some tests indicate that
: > something like  "tall and not tall" can indeed have a meaning).

: Oh come on, you're locked in binary mode.  Fuzzy logic is logic
: concerned with truth values like "somewhat".

It is possible to have, for some x, both 

	mu[TALL](x) > 0 

and

	mu[NOT TALL](x) > 0

while insisting that, for all x,

	mu[TALL AND NOT TALL](x) = 0 

and

	mu[TALL OR NOT TALL](x) = 1.

: If you would answer that somebody is somewhat tall, then he'll be
: somewhat not tall as well probably.  

Yes (see above), but it would certainly be bizarre for someone 
to say that somebody is "somewhat tall and not somewhat tall".

: To answer the question "is he a
: tall person or is he a not tall person" then with "oh, absolutely"
: seems somewhat strange.

Yes, because, for all x,

	mu[TALL OR NOT TALL](x) = 1,

therefore the question has not been answered, in the sense of
providing a restriction over the space of height values that
would allow the listener/interrogator to infer something
meaningful about the height of the person in question.

: > Maybe could we define toghether what exactly could mean "the (NOT,
: > AND,OR) triad is logically inconsistent" ? Here are some ideas.
: > 
: > On the one hand, you a have a (fuzzy) logic with some rules/axioms like
: > A1 :=   NOT(A AND B) = NOT A OR NOT B
: > A2 :=   A OR (B AND C) = (A OR B) AND (A OR C)
: > A3 :=   NOT NOT A = A
: > A4 :=   A AND (B OR NOT B) = A
: > (please, don't stop here, because of A4)
: > etc.

: In short you are complaining that not all axioms sufficient for
: defining a strict two-valued logic do not hold in fuzzy logic.  Big
: deal.  If we take an axiom set which can only be fulfilled by binary
: logic, we need not wonder that it will not hold for continous
: memberships.

: Of course you have to abandon some axioms of binary logic.  Which ones
  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: and why, is a matter of preference and usefulness.  Several different
: fuzzy logics have different choices, with different flexibility and
: degrees of freedom.

Not so fast.  What I should expect is that more realistic axioms
must be adopted which reduce to the axioms of binary logic 
under restrictions appropriate to a binary context.  In other
words, a proper fuzzy logic should in some sense *contain* binary 
logic, not be flatly inconsistent with it.

: -- 
: David Kastrup                                     Phone: +49-234-700-5570
: Email: dak@neuroinformatik.ruhr-uni-bochum.de       Fax: +49-234-709-4209
: Institut fr Neuroinformatik, Universittsstr. 150, 44780 Bochum, Germany

Regards,
S. F. Thomas
