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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: [Recap] NOT and DIFF
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Date: Wed, 29 Jan 1997 00:58:47 GMT
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Maurice Clerc (mcft10@calvanet.calvacom.fr) wrote:
(( cuts ))
: Now, what about  the NOT operator ? As you know, there are some semantic
: difficulties to say in the same time something like
: i) All values are "covered" by fuzzy sets A, B, C, D (say bell curves)
: ii) NOT(A) is given by 1-mu(A)

: for it implies  Universe = A or B or C or D
: and                      NOT(A)#(B or C or D)
: as we expect  NOT(A) = Universe except A

I continue to think you are posing a non-problem.
I believe that with proper rules for the OR operator,
it is possible to have, in your motivating example,

	NOT(A) = B OR C OR D = Universe except A,

as your intuition demands.  I also note that, in
your example, if, for example, the max rule is used 
for the OR operator, you do *not* have

	Universe = A OR B OR C OR D,

as my ... and presumably your ... intuition also 
demand.  Yet you do not focus on finding appropriate
rules for OR and AND.  If you get those right, then
the complementation rule for NOT turns out to be
unexceptionable.  At least its questioning is no
longer motivated in just the way you attempt.  The
rules for OR and AND proposed in my _Fuzziness and
Probability_ (1995), pp. 115 ff., obviate the problem 
you have posed.
	
Regards,
S. F. Thomas

P.S. I leave tomorrow on a mid-winter vacation for
two and a half weeks, so I won't be able to follow 
this thread further until my return in mid-Feb.
