Newsgroups: sci.stat.math,comp.ai.fuzzy
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!howland.reston.ans.net!news.sprintlink.net!tequesta.gate.net!decan!sthomas
From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory? 
Message-ID: <1994Dec3.021337.18334@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Sat, 3 Dec 1994 02:13:37 GMT
References: <hubey.786314318@pegasus.montclair.edu> <1994Dec2.190131.9720@newsserver.rrzn.uni-hannover.de>
X-Newsreader: TIN [version 1.2 PL2]
Followup-To: sci.stat.math,comp.ai.fuzzy
Lines: 102
Xref: glinda.oz.cs.cmu.edu sci.stat.math:3468 comp.ai.fuzzy:3535

mackw@bytex.com wrote:

: In article <hubey.786314318@pegasus.montclair.edu>, 
: <hubey@pegasus.montclair.edu> writes:
: > sthomas@decan.gate.net (S. F. Thomas) writes:
: > 
: > >Like Watanabe (1978), I also find it difficult to accept the
: > >result of the Zadehian min-max calculus that the fuzzy term "tall and
: > >not tall" should be anything less than the logical absurdity, as the 
: law of
: > >contradiction requires.  One would lose all credibility as a witness
: > >in court if one were to testify that the burglar was "tall, but not
: > >tall".

: When I was in college, one of the forwards on the basketball team was 
: 6'6" and was routinely described as not being tall.  Seeing him around 
: campus, I had an entirely different opinion.  He was both tall and not 
: tall or, more correctly, he was tall to a high degree, but was still not 
: tall to some degree.  He was not completely tall or completely not tall, 
: much less completely tall *and* completely not tall.

: As to a witness in court, I would not be surprised to hear a description 
: that "the burglar was tall but not too tall."  Again the interpretation 
                                     ^^^
"Tall" and "too tall" are of course different terms, with different
membership functions, and the conjunction of "tall" and "not too tall"
is quite meaningful to me...  and would be, I imagine, to any jury you 
care to choose.  The law of (non-)contradiction is not violated.

Similarly, if the explicit contexts are different, then the term "tall"
can obviously refer to quite different points in the universe of height
values.  Thus the term "tall for the general population" (A) is not the
same as the term "tall for a basketball player" (A').  So, again, the 
conjunction " A and not A' " does not violate the law of (non-)contradiction.  

I still insist that any utterance "A and not A" has no meaning in 
natural language, at least not in the normal sense of the conjunctive
"and".  Now, I do not disagree that your basketball player's
height could be at one and the same time both tall to some degree, and 
not tall, to some degree.  But this is at a higher level of language
which recognizes variation with respect to speaker, and variation with
respect to time, for a given speaker.  If the witness gets on the stand
and says "some would say the perpetrator is tall, others not", that is
perfectly understandable to a jury, which recognizes it (although they
may not call it such) as a meta-language kind of statement.  The witness
is describing not only the perpetrator in his best use of language at
the time, but also making a statement about the use of language for 
a larger population.  The examining lawyer may then well ask him to get 
back into object language (so to speak) and just speak for himself.
The other "out" has of course to do with the passage of time.  A
competent speaker of the language could of course at one time say
"the perpetrator is tall", and at another time say "the perpetrator is
not tall".  Nobody is perfectly consistent in his use of language.  A
jury would, I think, be understanding of such little inconsistencies
if the statements were taken at different times.

: of the statement from a fuzzy logic view is that tall is not an absolute 
: characteristic, but one of degrees.  Likewise "not too tall" can be 
: interpretted as "not tall to a low degree."

Well it depends on how you define your complementation operator.
That is not how I would define it based on my understanding of the 
use of natural language.  Nor is that how Zadeh defines it.

: Fuzzy logic basically redefines the meaning of "truth" to allow degrees 
: of truth.  

I have no problem here.

: Within the boundaries of its definitions, fuzzy logic appears 
: (at least to me) to be self consistent,  

Again no problem here.  As a mathematical formalism I have no quarrel
with defining the min-max operators the way they have been in the 
Zadehian development.  The difficulty I have is when it is asserted that
this formalism accurately represents the use of terms in a natural 
language.  

: and it also models, at least in 
:  some cases, descriptions used in the real world, which would imply that 
: fuzzy logic may also be useful.

Quite so.  In the redevelopment of fuzzy set theory found in
"Fuzziness and Probability", I do not argue in general with the min-max
rules.  But I do assert that they do not always apply.  And the theory
says precisely when which rule should apply, among (i) the min-max
rules (ii) the product and product-sum rules and (iii) the bounded-sum
rules.  Basically, one needs a notion of semantic consistency between
terms.  When terms are "consonant" under this notion of semantic 
consistency, then the min-max rules apply.  This applies, for example,
to the terms "tall" and "very tall".  When terms are "dissonant"
under this same notion, as for any term and its negation, the bounded-sum 
rules apply, which do not violate the law of (non-)contradiction.  The product
and product-sum rules apply for terms which are semantically independent,
eg. when different speakers are involved, or the same speaker at 
different times, or different universes of discourse.

: Wayne Mack

Cheers!
S.F.Thomas

