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From: hubey@pegasus.montclair.edu (H. M. Hubey)
Subject: Re: Fuzzy theory or probability theory?
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Date: Thu, 1 Dec 1994 20:38:38 GMT
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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".  The fuzziness of the term tall is not sufficient, in my view,
>to persuade a jury that such a contradiction could have positive
>meaning within the English language convention.  Similarly, the law of
>excluded middle requires that the disjunction "tall or not tall"
>should be the constant tautology.  Again, I am not persuaded that the
>fuzziness of the term tall is sufficient to justify the result of the
>min-max calculus under which not all elements of the universe have
>full membership, tautologically, in this disjunction.  Finally, the
>law of self-contradiction requires that any term which implies its own
>negation must be the logical absurdity.  Under the min-max calculus,
>any term, not necessarily the absurdity, whose membership function is
>everywhere less than half must imply its own negation.


I don't see any problems [of course, with a re-interpretation :-)]

Look at the problem from the point of view of bivalency.

Then not(tall) is just another word for "short".  If that's not the
case, [that is if we proceed by assuming that there's no connection
between tallness and shortness], then we could easily derive
contradictory results.

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
in which concepts such as tall-short, good-evil, hot-cold, early-late
etc, are used to describe a continuously varying quantity as a first
order approximation.  Then not(tall) and tall could easily mean
"of average height", so there's no reason in continuous valued
logic for the law of noncontradiction [LNC] to always hold. 

Similarly "tall or not(tall)" doesn't always have to be equal to
True if we take tall and others as intervals. 
AFter all, P*not(P) when negated gives P+not(P) so if 
P*not(P) =/= 0, then there's no reason why P+not(P)=1.

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.


>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.




--
						-- Mark---
....we must realize that the infinite in the sense of an infinite totality, 
where we still find it used in deductive methods, is an illusion. Hilbert,1925
