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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: [Recap] NOT and DIFF
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Date: Fri, 21 Feb 1997 12:16:05 GMT
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Maurice Clerc (mcft10@calvanet.calvacom.fr) wrote:
: S. F. Thomas wrote:
: > 
: > 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.   
: 
: That is why, among others reasons, the max rule seems to be not very
: good. Anyway, that was just the beginning of my "Recap." and, of course,
: we have already discussed about this particular point.
: 
: I would be _much_ _more_ interested by comments about the last formula I
: give, I mean 
:  
: DIFF(A_i) = OR contraction(A_j/A_i)    
: for  all j (even for j=i assuming that contraction(A_i/A_i) = empty set)
: 
: where contraction(A_j/A_i) is a concept/object/fuzzy
: set more specific than A_j, and depending from A_i (more precisely, from
: the intersection of A_j and A_i)
: 
: This model can be tested from a psychological point of view. 
: Example (simplified):
: You have 30 blue cards, from "almost white" to "almost black".
:  
: Step 1
: ------
: To put the cards along  a scale from "not at all dark blue" to
: "completely/really dark blue"
: 
: It gives a fuzzy set DB
: 
: Step 2 
: ------
: To put the cards along  a scale from "not at all NOT light blue" to
: "completely/really  NOT light blue"
: 
: It gives a fuzzy set NLB
: 
: Among others predictions, what the DIFF/contraction model says is 
: "DB is strictly included in NLB"

This result may be obtained without getting rid of the
complementation (one-minus) operator for NOT.

: Another point is that in this model, the opposite of a concept is
: depending from the context (the other fuzzy sets of the "world"), which
: is not the case, for example, with the (1-mu) model.

As pointed out before, I believe this to be incorrect.
It is true if the max rule for the OR operator is used,
but this is not always appropriate (Thomas, 1995, p. 117).
When rules more appropriate to the context you describe
are applied, your problem disappears.  See op. cit.,
pp. 126-129 for examples.  Note in particular that although
I speak of "rules more appropriate", their selection is
not ad hoc, rather based on *one* overarching formula which
specializes to a variety of familiar rules (max, bounded-sum,
product-sum) based on the value of a parameter which 
itself is not arbitrarily selected, rather follows from
the shape and location of the constituent fuzzy sets
sought to be combined.

: So, another prediction is that NLB is not exactly the same  if you say
: _before_ the test there are three classes (light_blue, normal_blue,
: dark_blue).  

NLB elaborated in the way you describe will presumably
not be exactly the same as 1-LB.  However, IMO, it will 
take more than minor experimental variation to dislodge the 
complementation rule for the NOT operator.  The latter
follows from the basic semantic consideration that a
listener must feel free to infer that the negative of
a proposition has been denied whenever its positive has
been asserted.  (At the level of object language; at the
level of meta-language, it is of course possible that a
referent point in a universe of discourse may, to varying 
degrees, be desribable both by a term and its negation,
which however is not the same thing.)

In any case, I still don't see that your DIFF/contraction
construct solves any problem that is not already solved
if appropriate modifications are made to the rules for
AND and OR operators, as earlier alluded.  The problem you 
address ultimately lies, IMO, with the inadequacy of the min/max 
rules for AND and OR, not with the inadequacy of the 
complementation rule for NOT.  All of this is not to
say that the abstract notions of DIFF and contraction may not 
be useful in some other context.

: > 
: > 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.
: 
: We are NOT(in_a_hurry) ;-)
: 
: Have a good time !

Thank you.  I had a truly wonderful time.

: Maurice Clerc

Regards,
S. F. Thomas
