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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory? 
Message-ID: <1994Dec4.174744.27946@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Sun, 4 Dec 1994 17:47:44 GMT
References: <D09E0u.2At@acsu.buffalo.edu>
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Sri_G (madhv-s@acsu.buffalo.edu) wrote:
: S. F. Thomas (sthomas@decan.gate.net) wrote:

: : 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".  

(( cuts relating to the ball which is blue and not blue ))

: The conjunctive "and" here is being used *not* to join contradictory
: propositions about the ball, but to join propositions about different
: parts of the ball. This, however, is *not* the sense of the conjunctive
: "and" in "tall and not tall".

Good point.  It's like the "and" in the phrase "the red, white and blue".

(( cuts ))

: I wonder how the dependence of the membership function on the speaker
: is handled in practice. Is there an alternative to having different
: membership functions tall1, tall2 ...., one for each speaker ?  (In
: that case, if speaker 1 says "the man was tall", and speaker 2 says
: "the man was not tall", this could be represented as "tall1(x) and not
: tall2(x)", which would not violate NC).

Quite.  And the rule of combination for this last "and" is the 
product rule, not the minimum rule.  As to the question is there
an alternative to having different membership functions tall1,
tall2 etc., the answer is yes... If you don't know tall1,
tall2, etc., you can substitute
a generic membership function "tall" which represents the use of this
term for the population as a whole.  If you know tall1, tall2 etc.
then of course you use those.  See the following quotation:

      "No, I have never seen Mr.
   Elton," she replied..."is he -- is he
   a tall man?"
      "Who shall answer that question?"
   cried Emma. "My father would say 'Yes',
   Mr. Bentley, 'No', and Miss Bates and I
   that he is just the happy medium".
   -- Jane Austen, _Emma_.

: : 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.

: Errr... if these two statements were to be made by the same witness on 
: the stand within the course of a trial, even a dim-witted lawyer would
: pounce upon it as an inconsistency in the witness's testimony ...

Ok, ok.  I was trying to be kind... :)  Small point, but I guess
the key here is whether the passage of time could be considered
sufficient to have dulled the memory.  If it has, then the witness
presumably has no business testifying in the first place, and the
lawyer would be quite right to "pounce", as you say.

: : 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.

: Couldn't agree more.

I am glad _someone_ does...

: - Sri_G

Cheers!
S.F.Thomas
