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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Fuzziness & Society of Mind
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Date: Mon, 9 Dec 1996 01:26:14 GMT
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Maurice Clerc (mcft10@calvanet.calvacom.fr) wrote:

: I would like to discuss an idea given to me by the social psychologist
: Jim Kennedy <KennedyJ@po1.ocsp.bls.gov> (just the idea. The development
: below is mine, and he is not responsible for the mistakes you could
: find). It seems it could reconcile fuzzyness and probability, and also
: give a sound  (consistent) way to reformulate fuzzy operations (like
: AND, OR, NOT).

The idea that you sketch, in its essence--ie. identifying the
notion of grade of membership with the proportion of a 
population concurring with a suitably posed calibrational
proposition--goes back at least to Gaines (1975), as William 
Siler has already pointed out in his reply.  In addition to 
the works he has cited, you may also want to take a look at 
my _Fuzziness and Probability_ (1995), where the subject has 
been treated at book length, in both its philosophical and
technical aspects, with extensions to foundational questions 
of deductive and inductive (statistical) inference, and 
decision analysis.

The AND and OR operations that you propose are the product
and product-sum operations respectively, which emerge in my
treatment as merely special cases of an infinity of laws,
as follows:

The general law is of the form (Thomas, 1995; p.117):

            { (1-t).a.b + t.min[a,b],           1 >= t >= 0
 a AND b  = { 
            { (1+t).a.b - t.max[0,a+b-1],      -1 <= t < 0

where a and b represent the membership functions of two
fuzzy sets A and B ranging on the same universe of
discourse, and  -1 <= t <= 1 is a semantic consistency
coefficient that depends only on the membership functions
a and b.  The semantic consistency
coefficient t is postulated to be a function of the 
correlation coefficient between the two membership
functions.  

As Ellen Hisdal has pointed out on this group from
time to time, and also in various of her papers,
when the membership functions are identified with 
probabilities of word usage, it is possible to *derive* 
appropriate rules of combination, rather than to 
postulate them directly.  It is in this fashion that the
general law mentioned above is derived.  As will
be evident, there is in fact an infinity of laws,
depending upon the value of t, but three cases
are of interest, and yield results already quite familiar
from the literature:

 a AND b = a.b	        (t=0 -- semantic independence)

 a AND b = min[a,b] 	(t=1 -- positive semantic consistency)

 a AND b = max[0,a+b-1] (t=-1 -- negative semantic consistency)

There are corresponding rules for union, as follows:

 a OR b = a+b-a.b	(t=0)

 a OR b = max[a,b]	(t=1)

 a OR b = min[1,a+b]	(t=-1)

It should be noted that the min-max rules (t=1) and 
the bounded-sum rules (t=-1) are intertwined through the 
negation postulate and the correlation coefficient: they 
are flip sides of the same coin, so to speak, not
separate, arbitrary laws.

The novel aspect of what you propose lies in choosing not 
to identify a failure to concur with simple negation, which 
spares you from having to adopt the complementation (one-minus) 
rule for negation.  In this way, you seek to avoid what you
perceive to be a problem, namely that a particular agent
can answer YES to both "OLD" and "NOT-OLD".  I disagree 
with this notion, as I think it is based on a misconception
as to what constitutes negation.  Basically, the misconception
I think lies in thinking that an agent, having affirmed that
50 years is "OLD", has not necessarily disaffirmed that 
50 years is "NOT OLD".  I think that the one necessitates
the other, as being what negation ("NOT") means.  I do not
believe that it is meaningful to affirm one thing without
leaving a listener feeling free to infer that its negative
has been denied.  Within a fuzzy set theory, this last may
appear to be a naive statement, given the "shades of gray"
sought to be illuminated, but if one properly understands
the respective roles of object-language and meta-language,
the confusion disappears.  In any event, a fuzzy set theory
without the unary negation operator would be paltry indeed,
and would in fact preclude the result referred to earlier
in which the min-max and bounded-sum rules appear as
flip-sides of the same coin.  That is an insight I see no
good reason to give up.

: Thanks in advance for any comment,
: Maurice Clerc

I give further, specific comments below, interspersed
with quotes from your original...

: ======================================================================
: Fuzziness is seen here as a probabilistic distribution, result of an
: "introspective" survey.

This is a misconception I think... Even when the notion of
the grade of membership is identified with the probability
of an occurrence, namely that of an agent concurring with
one of your calibrational YES/nothing propositions, that
still does not mean that fuzziness may be identified with
a "probabilistic distribution".  It may, however, be identified
with an appropriate notion of "semantic likelihood", which
remains distinct from, though related to, the underlying 
concept of probability that has been invoked.

: In order to be clearer,  I will speak of "number of agents", but, of
: course,  I could  (an should, for the total agent number is usually
: unknown) speak of "ratios".

: Let's suppose we have a society of N (say N=100) elementary (binary) 
: intelligent agents. They are "intelligent", for they understand quite
: well what it is told. They are "binary" for they just can say "YES", or 
: nothing at all. 

Why not simply "YES" and "NO"?  

: We tell them there are just four possibilities for the variable "age of
: a man": YOUNG, MIDDLE_AGED, OLD, VERY_OLD, and we suppose these four
: possibilities are covering "well" the domain. That means each agent will
: answer "YES" at least one time if we ask it 
: "John is x years old. Is he YOUNG ? is he MIDDLE_AGED ? Is he OLD ? Is
: he VERY_OLD ?" for each domain value x.

: We ask them, for example
: 1) John is 50 years old. Is he MIDDLE_AGED ?   => 77  YES
: 2) Is he OLD ?       => 29 YES

: Now, can we guess what would be the  result if we ask "Is he OLD and
: MIDDLE_AGED ?"

: For 77+29 = 108 , we are sure at least 8 agents would answer YES., for
: they said YES to 1) and to 2). This the "min" point of view.

: For MIN(77, 29)=29, we are sure no more than 29 agents would answer YES.
: This is the "max" (optimistic) point of view.

: But, in fact, it is quite easy to compute the most probable number of
: distinct agents which answered YES to 1), and also YES to 2). It is just
: a classical probability problem. Here, the result is 23.

Not as a matter of necessity, as argued earlier.  You are
assuming semantic independence in the use of the terms 
OLD and MIDDLE-AGED, which may or may not be true as an
empirical matter.

: --------------
: In the same manner, we can find the most probable answer to the question
: "Is he OLD or MIDDLE_AGED ?"
: The "min" (pessimistic) point of view is given by MAX(77,29)=77, which
: means all the 29 agents are already in the 77. 
: But, in fact, the most probable ist that 7 of them are not in the 77. So
: the most probable answer is 77+7=84.

: Below, I give complete  fuzzy sets for MIDDLE_AGED, OLD, and fuzzy AND,
: fuzzy OR by using this method. If you draw the curves, you can see
: MIDDLE_AGED and OLD are bell curves 'defined), and so is "fuzzy AND"
: (computed). 

: age	MIDDLE_AGED	OLD	    fuzzy AND	       fuzzy OR
<table snipped>
: --------------
: 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

: A consistent way would be to say the NOT operator doesn't exist ! 

Drastic solution to a non-problem, as argued earlier.

: More
: precisely, for example, you could write:
: NOT(OLD) = JUNG or MIDDLE_AGED or VERY_OLD
: As you can guess the result is not as simple as something like
: 1-mu(OLD).

: This approach means you can't compute NOT(A) just like that: you have to
: know the context (the others fuzzy sets covering the domain, which are,
: in fact, the granular world representation). An interesting point is
: that the Elkan's contradiction doesn't exist any more (see Note).

: But it is then more difficult to compute fuzzy sets as  NOT(A AND NOT
: B).(This i another topic, but you have sometimes to take cores into
: account).

: In fact, this approach is independent of what we have seen above (the
: "agent society"), but, from this point of view, it just means that an
: agent can answer "YES, he is OLD", and also "YES, he is not OLD"
: (remember, an agent as a _very_ limited vocabulary !). 

But then you have the problem earlier mentioned, namely 
that one is never free to infer from a positive statement
that its negative has been denied.  I assert that that
does not conform with the empirical reality, fuzziness
notwithstanding.  Go to any courtroom if you need further 
convincing.

<rest snipped>

Regards,
S. F. Thomas
