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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory?
Message-ID: <1994Dec4.170437.27737@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Sun, 4 Dec 1994 17:04:37 GMT
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H. M. Hubey (hubey@pegasus.montclair.edu) wrote:
: sthomas@decan.gate.net (S. F. Thomas) writes:

: >H. M. Hubey (hubey@pegasus.montclair.edu) wrote:

: >: So then if not(tall) is short, and not(short) is tall, then
: >: "tall and not(tall)" is equivalent to "not(short) and not(tall)".
: >: But this is not a contradiction is normal everyday conversation

: >I disagree.  In the English language as I understand it, there is a
: >big difference between "short" and "not tall".  "Tall and not tall"
: >remains a contradiction in my mind, while "not(short) and not(tall)"
: >is perfectly understandable to me as an English speaker approximating

: The good thing about fuzzy reasoning is that it always seems to work :-).
: But joking aside, from the view of classical bivalent logic, either words
: like tall/short, hot/cold, near/far are related or not. If we don't take
: tall=not(short) or short=not(tall), then I don't see how standard logic
: will work without producing contradictions. 

Which is why fuzzy set theory is an important contribution...

: We don't have any choice
: [from the standard logic view] except to accept them. 

Again I disagree.  Consider the following excerpt from 
"_Fuzziness_and_Probability_" (from Chap. III -- "Fuzzy Set
Theory of Semantics"):

   "We have developed a theory of semantics using ordinary
   bivalent logic and set theory in a meta-language describing
   terms in an object language, whose meanings are allowed to be
   fuzzy in a sense precisely [ie. non-fuzzily] characterized
   in the definition of a fuzzy subset.
   
   One could ask whether some of the [fuzzy] terms in our object language
   could be separated into a sub-class of "crisp" terms corresponding
   to the ordinary bivalent sets which populate the meta-language;
   and, moreover, whether the rules for intersection, union and
   complementation for this crisp sub-class of terms, is in harmony with
   the rules with which we are accustomed in the meta-language.
   Happily, this is so.

    .... [math stuff showing this claim deleted] ....

   That is to say, the fuzzy subsets corresponding to crisp terms
   behave exactly like ordinary sets.  This is as we should require:
   there is indeed some harmony between object language crisp terms
   and metalanguage ordinary sets.
   
   This is interesting as well from a philosophical standpoint.
   It is by no means obvious that vagueness and fuzziness in natural
   language should fall ultimately under the ambit of a two-valued
   logic.  Giles (1971, p. 322) for example has written that the
   notion of the fuzzy set is prior to that of set.  In the same vein,
   Goguen (1974, p. 514) writes: "Ideally we would like a
   foundation for fuzzy sets which is independent of ordinary
   set-theory ..."
   Goguen proceeds to axiomatize fuzzy set theory in the language
   of category theory, but as a theory of semantics it remains
   non-empirical.  Within the empirical framework which is proposed in
   this development, it would appear that what is fuzzy with respect
   to [a universe of discourse] U , e.g. "tall" [with respect to
   the space of height values], can be rendered as something crisp in 

                 U
            [0,1]

   e.g.  mu[TALL]: U -> [0,1].  Evidently we could define fuzzy sets 
   of type 2, as Zadeh has done, corresponding to fuzzy sets with fuzzy
   membership values, in which case what is fuzzy with respect to 

                 U
            [0,1]

   could be rendered as something crisp in

                      U
                 [0,1]
            [0,1]

   It would appear that there is no end to the process of
   'crispification': starting with the intuitively well-accepted
   canons of classical two-valued logic we may bootstrap ourselves
   through the higher reaches of fuzziness.  For philosophers,
   not to mention computer scientists, this would no doubt be
   considered a happy result: a non-classical logic at the
   foundations of mathematical reasoning or computation is a
   forbidding thought indeed."

So... the whole point of fuzzy set theory is to crispify,
crispify, crispify...  And that is done by raising the 
metalanguage as high as necessary to ensure bivalency 
at that level... not by attempting to do away with the
fuzziness at object-language level.  And not by attempting
to shake the foundations of mathematics, philosophy, or
anything else by suggesting anything as scary as an
infinitely multi-valued logic at the foundations of all
mathematical reasoning.

(( cuts ))

: I was comparing fuzzy logic to bivalent logic, as you can see. If we
: start out with already a whole range of values, then if this is not
: from the point of view of standard logic, then why do we need to save
: either the LEM or L of Contradiction at all?

Because in the object language, that is what ordinary common-sense
notions of the meanings of the connectives "and" and "or" require,
in accordance with Aristotelian dictum.  Now, it is entirely
acceptable for someone to propose a purely mathematical formalism
where there are connectives named /\ for the conjunctive
and \/ for the disjunctive, where /\ and \/ do not behave in
accordance with Aristotelian dictum.  My trouble is that I would
then not accept that these connectives, however unimpeachable
as internally consistent mathematical formalism, correctly
describe the _empirical_ reality of language-use as I see it.

: In any case, you can already see something like this in Chomsky's 
: phonemics, in which two distinctive features High and Low are used
: and it's possible to assign High&Low or Not(high)&Not(Low).

I am not familiar with Chomsky's phonemics, so I won't comment on
what he intended, or did not intend, when he did this.

: --
: 						-- Mark---

Cheers!
S.F.Thomas
