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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Defining fuzzy descriptors (was  NOT and DIFF)
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Date: Wed, 12 Mar 1997 00:56:45 GMT
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Maurice Clerc (mcft10@calvanet.calvacom.fr) wrote:
: William Siler <wsiler@aol.com> wrote:

:  > S.F. Thomas wrote, relevant to the "DIFF" exchange:
: > 
: > >I continue to maintain that the anomaly that you have used to motivate
: > this enterprise stems from the inadequacy of the min/max rules for AND
: > and OR, rather than from the inadequacy of the complementation rule for
: > NOT.
: > 
: > I tend to agree with Thomas. Certainly the myriad formulae for AND and OR
: > indicate that there is a lot of room for variation here, although there is
: > no consensus as to when to use what. Perhaps Clerc is simply trying to
: > ensure that NOT does not escape the controversy.

: Right. More precisely, I think that in any logic the triplet
: (NOT,AND,OR) has to be consistent. In fuzzy logic, we already know
: (Elkan) that it is not the case for ("one minus", MIN, MAX). We also
: know the Elkan's contradiction doesn't exist any more for some other
: "triplets", yours, for example. But AFA I know there is no research
: about this very topic: in fuzzy logic, what are the constraints between
: (NOT, AND, OR) to be sure a given triplet is consistent ?

I come at it from a different angle.  Ellen Hisdal also...
though I do not presume to speak for her.
We start with minimal meta-semantic principles for AND, OR
and NOT (see my previous post) which form premises of
deductive argument.  The principles are "minimal" in the
sense that they say, minimally, what AND, OR, and NOT *mean*,
as context-free operators.  In the development that I then
advance, it turns out that there is an infinity of rules 
that may be derived for AND and OR, while there is only
one for NOT.  The rules for AND and OR come in pairs
though, and once the two operands are specified, the 
AND/OR rule-pair are uniquely determined.  Thus the notion
of "consistency" of which you speak is in this way 
automatically satisfied... at least my interpretation of
your notion of consistency.

More than that, the constraint that forces the rules for 
AND and OR to come in "consistent" pairs, may be traced
precisely to the complementation rule for NOT.  In other
words, if that is taken away, the rules for AND and OR
would no longer be tied together, or at least not in
the same way.  To see this notion explicitly developed,
see the proof of my Lemma 5, Part IV (Thomas, 1995, 
pp.112-113).  I think the notions there developed 
of semantic consistency, positive and negative, give a 
theoretical framework, in some sense, for the notion of 
consistency to which you allude.

: Now I think such a research can't be very fruitful if you say at the
: very beginning "Well, for NOT, no problem,  it is always "one minus" ". 

But if one starts with the minimal meta-semantic principles
of which I have spoken, the complementation rule is a
*derived* rule, not an assumed one.  The principle from
which it derives is very easily stated (see Postulate 4,
Thomas, 1995, p. 91):

   For any speaker S and proposition P the negation operator
   NOT satisfies 
   
   	X(S, NOT P) = 1  <==> X(S,P) = 0

where S is drawn from a population of speakers of the object
language in question, P is a generic (calibrational)
proposition of the object language, and X(S,P) = 1 if 
speaker S would affirm proposition P, and X(S,P) = 0
otherwise.  The language convention is fuzzy, hence 
two speakers S and S' need not agree with respect to their
affirmation/disaffirmation responses to any given calibrational
proposition P.  And, assuming lack of recall from one 
calibrational response to another, individual speakers need 
not be perfectly consistent in their usage, meaning that
a given speaker's response to a given calibrational proposition
may vary from trial to trial, in the same way it is assumed 
to vary from speaker to speaker for the same proposition.

If there is some other meta-semantic principle which
defines the meaning of NOT for the object language that
is in harmony with the meaning of its equivalent in the
metalanguage, I'd happily consider it, and examine its
implications, including for the construction of AND/OR
pairs and the consistency notions that we both seem to
think they ought to satisfy.  But I have difficulty
with the mere assertion of any number of rules for AND, 
OR and NOT.  That is a never-ending enterprise, 
because the population of candidate rules is infinite.

: Especially for some psychological tests seem to show in some cases
: a) the "one minus" rule doesn't work 
: b) the NOT operator has to be context dependant

This may be a misunderstanding of what is happening
semantically in context.  I have made this point in
an earlier post.

: > Clerc's viewpoint would be more comprehensible to me if his motivation for
: > proposing DIFF were clearer. Since he has come up with several ideas as to
: > how to formulate DIFF, it seems clear that his motivation is his driving
: > force. Maurice, could you explain once again just why it is that you focus
: > on the "one minus" rule for NOT rather than on the AND and OR, and what is
: > the motivation behind DIFF?

<snip>
: More seriously, I am studying reasoning systems under closed world
: assumption (CWA). For crisp logic in discrete closed world, you didn't
: need a special NOT operator. If the "universe" is {RED, GREEN, BLUE},
: then NOT(RED) is simply (GREEN OR BLUE).

: Another way to say that is that NOT is completely context dependent.

: Can we apply fuzzy logic in such a case ? First, we have to define the
: universe of discourse, say the wawe length. You then define three fuzzy
: sets centered respectively on 650nm, 530nm and 470nm.  
: Now you have to choose the triplet (NOT, AND, OR).

: I don't think it is a good idea to choose NOT := "one minus", for it is
: completely context free. That is why I suggested the DIFF operator. Not
: I am saying it is the "best" one, not even a good one. Just in order to
: prove it is indeed possible to have plausible context dependent NOT
: operators.

I don't think a context-free rule is a bad thing.  The fact that
we can speak of operators in the abstract (eg. P AND Q where P
and Q are arbitrary propositions; NOT P, where P is an arbitrary
proposition) suggests precisely that these operators have an 
essential meaning that is free of context.  

I agree, however, that in many real-world situations, 
there is an implicit context that limits meaning.  For 
example, if we are talking about men, to say that "John 
is not tall" contains a context restriction appropriate 
to heights of men.  Hence NOT TALL should not leave open 
the possibility that John has the height of a
gnat, and neither should NOT SHORT leave open the possibility
that John has the height of the Empire State Building.
But such a recognition of context is rather
easily resolved if the context itself (eg. of heights of men,
as distinct from, say, buildings) is conceived of as being an
implicit term which is ANDed to any compound expression which
otherwise might appear unreasonably unbounded with respect to
the underlying abstract universe of discourse.  For example,
NOT SHORT in the context of heights of men seemingly embracing 
King Kong, would become, implicitly or explicitly, 
(NOT SHORT) AND (CONTEXT of MEN'S HEIGHTS), which would
presumably rule out anything over nine feet or so.

Now consider another situation: We have a weight-scale
to measure people's weights, and our term set corresponding
to all the measurement reports of which the scale is
capable runs from "0" to "300" in 1 lb increments.  
Strictly speaking, each of these 301 measurement reports,
although numerical, is fuzzy, in the sense that each
refers to a continuum of abstract weight values, ideally
comprising an exact interval such as 178 +/- 0.5, for the
measurement report "178", but practically, considering 
measurement error, it must be acknowledged that the boundaries
are not the crisp ones that we would like, but rather are
fuzzy.  Now, John weighs himself, and declares "I'm not
200".  Under the complementation rule, we might have a
strange-looking fuzzy set which has membership equal to
1 everywhere, except for around 200, eg.:

1 ____________    _________
              \  /
               \/ 
0             200         300

Complementation with respect to terms as you suggest, eg.
NOT("200") = OR("0","1",...,"199", "201",...,"300") is also 
problematic, as we well know, because in that semantic 
context, what John means to allow is anything from 0 to 199, 
and "not 200" is meant to exclude everything from 200 on up.  More
likely still, what we can infer from what John says is that
he is somewhere in the range 195 to 199.  Maybe.  The point
is that the use of language is plastic enough that not
every use of the word "not" constitutes negation in the 
strict sense of the postulate earlier given.  What John
is up to in a weight-conscious world is a kind of defensive
denial, in which from his denial of "200" it may readily
be inferred by a human and humane listener that every other
term above 200 would similarly be denied.  I have
in a previous post given the "double negative" as another 
situation where what should be inferred is intensification,
not negation of the negation.  No doubt there are other
examples like these, but they do not, to my mind, detract
from the "pure" meaning for negation from which the 
one-minus rule derives.

I do not discourage your quest.  But "context 
dependence" does not per se motivate me to join in it.

: -----------------------------------------------------------
: Maurice CLERC
: http://calvaweb.calvacom.fr/mcft10/
: ------------------------------------------------------------

Regards,
S. F. Thomas

