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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory?
Message-ID: <1994Dec2.034646.12369@decan.gate.net>
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Date: Fri, 2 Dec 1994 03:46:46 GMT
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H. M. Hubey (hubey@pegasus.montclair.edu) wrote:

(( cuts ))

: 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 term "medium".  Here is another excerpt from "Fuzziness and Probability":

   "The restoration of the law of contradiction and the law of
   excluded middle have ... quite empirical significance.
   The failure of these two laws has been a thorn in the side of the
   Zadehian development of fuzzy set theory.  Many authors have
   however grown insensitive to the thorn, indeed some, like Kochen
   (1975) have made a virtue out of the failure of these two laws:
   
      People do not have to dichotomize, as prescribed by Aristotle,
      that  x  is either large or not large; the above remark
      is less objectionable to those who see Aristotelian logic as
      a normative guide to cognition if we replaced 'not large' by 'small'.
   
   The quote is revealing, for it seems to imply that to require
   that  x  is either large or not large is the same as requiring that
    x  is either large or small, which is of course a quite different
   proposition ...  Moreover, I would go so far as to say that properly understood,
   the Aristotelian dictum is not merely a "normative guide to
   cognition" which presumably we may follow or not follow as we
   choose, but a principle of semantics with positive, empirical
   significance.  To the question "is  x  large?", if one answers "no"
   then is one as a listener supposed to infer that what is meant could be
   either or both " x  is large" _and_ " x  is not large"?  Granted, one is
   not restricted to a yes/no answer.  One may reply "more or less".
   But isn't one then affirming " x  is more or less large" and 
   should it not then be inferred that one is _dis_affirming
   " x  is _not_ more or less large".  People do not have to dichotomize 
   where choice of descriptors is concerned -- one is free to hedge one's 
   descriptions as profusely as one desires; but where utterances are concerned,
   it is very much in the nature of these that they are dichotomous
   -- either they are made or not made, and to infer any meaning
   from a positive utterance it seems to me a listener must feel
   free to infer that its negative has been denied.  Otherwise by excluding 
   nothing the utterance would have to be counted as meaningless.
   Aristotle was right.
   
   None of this denies the possibility (with which fuzzy set
   theory is intimately concerned) that a given  x  may legitimately,
   within the language convention, be called both "large" and "not
   large" by different people, or even by the same person at
   different times.  In this sense, (take note that this is at a
   higher level of meaning -- language turning around on itself,
   trying to "catch itself by the tail") it is reasonable to say that
    x  is large and not large, seemingly in violation of the law of
   contradiction.  But there is no violation.  In our _object_ language,
   "large /\ ~large" is still the meaningless term, and this is
   supposed to correspond with the "meaning" of the primitive
   utterance "large and not large".  But going a higher level to our
   _meta_language, we can say _there_ that there exist  x  such that
   mu[LARGE]( x ) > 0  and  mu[~LARGE]( x ) > 0 ,
   so that in this sense and at this level we
   could say, meaningfully, that for some  x ,  x  is large and not
   large.  But at this same level we may also say, for all  x ,
   mu[LARGE /\ ~LARGE]( x ) = 0 ,
   which is the meta-language way of
   saying "LARGE /\ ~LARGE" is the meaningless term, in accordance
   with Aristotle's dictum.
   
   The distinctions made here are important, for they have the
   effect of restoring the laws of excluded middle and of
   contradiction, without removing the essential fuzziness in terms such as
   "tall" and "large".  It is an achievement at which many may balk,
   as trying to have one's cake and eat it too.  At least one author
   (Watanabe 1978) has also insisted on the Aristotelian dictum and
   set about trying to "salvage" Zadehian fuzzy set theory on this
   score.  Unfortunately, in salvaging the beast, he "killed all its
   charms", reducing fuzzy sets to their so-called "core subsets" for
   which the Boolean lattice laws apply, and in the process did away
   with the fuzziness.  The present development should be of interest
   for retaining the beast's fuzzy charms while restoring
   Aristotelian respectability."

: Of course, the standard argument against LEM is the time
: contingency, going back to Aristotle's "Will there be a sea
: battle tomorrow?"

: The answer can't be either Yes or NO since in either case we'd
: be foretelling the future.

Recall the part of the excerpt above:

   "None of this denies the possibility (with which fuzzy set
   theory is intimately concerned) that a given  x  may legitimately,
   within the language convention, be called both "large" and "not
   large" by different people, or even by the same person at
   different times.  In this sense, (take note that this is at a"
   ^^^^^^^^^^^^^^^

: >appropriate, and it forces one to address the issue when does which
: >apply.  In so doing, the law of contradiction and the law of excluded
: >middle are upheld as a matter of necessity, a happy result if one is
: >disposed to accept these as having positive empirical significance as
: >semantic law.  The law of self-contradiction does not hold, as in the
: >Zadehian calculus, if one takes the containment relation between fuzzy


: This is probably due to the simple fact that not(x) is often written
: as 1-x, so that P+P'=P+(1-P)=1  but we don't get P(1-P)=0 as
: a happy coincidence(!) except at the boundaries.

Yes, but as stated in the previous post, the law of self-contradiction
is restored if the rule of implication is defined with reference to
possibility distributions rather than membership functions, and possibility
distributions are seen as "semantic likelihood" functions unique only
up to similarity transformations.

: --
: 						-- Mark---

Cheers!
S.F.Thomas
