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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Fuzzy logic compared to probability
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Date: Sat, 2 Mar 1996 05:01:43 GMT
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Tom Whalen (dscthw@gsusgi2.Gsu.EDU) wrote:
: On Tue, 27 Feb 1996, Andreas Poncet wrote:

: > As an example: instead of saying, e.g., about a specific man, that
: > his height x has "tall"-membership, say, 0.8, 
: > one could express this equivalently as P(A|x) = 0.8,
: > with the proposition A defined as
: >                   A = "the man is tall"

: This would seem to imply one of the following two scenarios:

< Scenario 1 snipped >

: (2)
: There exists a population of observers 80& of whom will say without 
: equivocation "the man is tall!" and 20% of whom will say without 
: equivocation "The man is NOT tall!"

: A fuzzy interpretation would be that the proposition "he is tall" conforms 
: to our overall knowledge better than any statement with truth value .79 
: but not so well as any statement with truth value .81.

IMO, there is less than first meets the eye, to the distinction sought 
to be made between Scenario 2 above, and the "fuzzy interpretation".
Wittgenstein, the philosopher, who had much to say about semantics,
said that "the meaning of a word is its use in the language."  That
simple insight illuminates much of what has been troublesome in the
foundations of fuzzy set theory.  It is easy to identify word usage
with Scenario 2.  Usage varies from person to person and from context 
to context, and for the same person, sometimes from one time
to the next.  But it is because usage is not entirely random, and
not entirely arbitrary, that it is possible for communication to
take place, and for listeners to be able to infer something meaningful
from statements such as a witness might testify to in court: 
"the perpetrator was tall".  And it is precisely because there is
some *probabilistic* uncertainty -- with respect to *usage*, yes or no,
that a competent speaker of the language would, eg., use the descriptor
tall to describe any given height value -- such as sketched in scenario 2,
that a word such as tall is *fuzzy* in the inferences it allows.
(I am not one of those who would deny fuzzy its separate existence
merely because of its intimate relation with probabilities -- of
word usage.  Fuzzy is different from probability, but related to
it, in the same way that likelihood is related to, but distinct
from probability.  In fact there is a sort of duality relationship
about which I have already expounded in an earlier post.)
The probabilistic uncertainty in word usage (eg. of the descriptor
"tall"), where a yes/no sample space applies, induces a (semantic) 
likelihood function over the space of hypotheses (height values 
consistent with the term in question).  At bottom, this is what
the membership function is.  And it is quite consistent with 
Prof. Whalen's "fuzzy interpretation", which itself requires
interpretation of the operational phrase "conforms to our overall
knowledge".  It is precisely scenario 2, seen also as an
operationalization of Wittgenstein's insight, that gives the
required further interpretation.  For the "overall knowledge"
which must be operationalized is nothing but the familiarity
we all have, as speakers of the language, of
*frequencies* of word usage (of a descriptor in question to describe 
a point in question).  

Parenthetically, the requirement of "without equivocation"
is really a red herring.  For utterances are always made or not
made, however profusely they may be hedged.  Descriptors are
used or not used, and once used, there is no question of 
equivocation.  But when we put ourselves into the metalanguage,
we know, eg.,  that there are x such that both mu[TALL](x) > 0 and
mu[NOT TALL](x) > 0, as matters of usage, and so we make the
mistake of thinking that a statement such as "x is tall" is
somehow equivocal.  It is not; it is merely fuzzy.  It is
the same mistake that has led many fuzzy theorists into the
trap of thinking that the laws of excluded middle and
the of contradiction should not apply to fuzzy.  But we 
well know that a statement such as "x is tall and not tall"
would be laughed out of court were a witness to a crime so to 
describe the perpetrator.  And the fuzziness of the term "tall"
would not be sufficient to rescue the witness from the court's
derision.  The resolution starts with
recognizing that we may have mu[TALL AND NOT TALL](x) = 0
for *all* x, to recognize, in the metalanguage, the absurdity 
of the object-language expression "tall and not tall", in accordance 
with the law of contradiction, even if, now in the
metalanguage, both mu[TALL](x) > 0 and mu[NOT TALL](x) > 0
for some x, to recognize the fuzziness of the object 
langugage terms "tall" and its complement "not tall". 

Once we give up the notion that, in the object language,
speakers make, or have in mind, statements of the form
"x is tall 0.8 (not tall 0.2)", rather they should
simply be taken at their word "x is tall" (thereby
simply disaffirming its negation, "x is not tall"),
we are free, as modellers in the metalanguage, to
ask the basic question of scenario 2:  At what rate
would speakers of the language use, yes or no, the 
descriptor in question to describe any given point 
in question.  It is the rate of such usage that then
allows us, in the metalanguage, to characterize the
fuzzy range of points to which any descriptor,
in actual use, could refer.  In this way we would have a
measurable interpretation of Prof. Whalen's "fuzzy 
interpretation", which otherwise would remain undefined, 
and therefore ultimately unsatisfying and inadequate 
as a basis for a theory purporting to describe
an observable phenomenon, that of language-use.  

This is not to say that one cannot proceed quite far on 
an intuitionistic basis, as is particularly
easy to do in such fields as controller design, where 
"word usage" may follow conventions laid down by designer fiat,
rather than by conventions of word usage in the general population.
But as applied to natural language semantics, there is an 
external reality out there that is subject to observation,
and measurement.  It is on such observable, measureable bedrock
that the fuzzy set theory (of semantics) -- with its associated
logic -- should try to ground itself.  Without that, fuzzy 
would remain vulnerable to the charge of being ultimately 
"ad hoc".  That would be a shame, because it does not
have to be.

: Tom Whalen     dscthw@gsusgi2.gsu.edu  voice (404)651-4080  fax (404)651-3498
: Professor of Decision Science

Regards,
S. F. Thomas
