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From: hardy@phalen.stat.umn.edu (Michael Hardy)
Subject: Re: Fuzzy theory or probability theory?
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Date: Wed, 23 Nov 1994 15:05:45 GMT
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In article <3aqfdr$11b1@hearst.cac.psu.edu>,  <caj@jerry.psu.edu> wrote:


>Probability theory measures the chance of an object belonging (entirely,
>the standard way) to some set.

	[snip]

>The easiest way to recognize that fuzzy membership and chance membership
>are two different measures is to recognize that probability (chance) 
>*must* be measured on the 0,1 scale for it reflects a percentage of
>some whole (i.e. what percentage of these trials will be TRUE?).
>
>Fuzzy theory uses the 0,1 scale only because Boole (original Boolean logic)
>adopted this scale to make mathematical formulas of logic convenient
>and direct (i.e. AND (X,Y) = X*Y).  Fuzzy theory can be done on any 
>numeric interval (0,10 or 20,987, etc.) without loss of cohesion although
>the set and logic math will be more complex.


	Whence this strange assertion that probability theory _must_ be done
"on the 0,1 scale"?  What if I define the "odds" in favor of a proposition A
to be Odds(A) = P(A)/[1-P(A)]?  Then Odds(notA)=1/Odds(A), and
Odds(A&B)=Odds(A)*Odds(B|A)/[1+Odds(A)+Odds(B|A)], and
Odds(A|B)=Odds(A)*[Odds(B|A)/(1+Odds(B|A))]/[Odds(B|notA)/(1+Odds(B|notA))],
etc., etc.  Then the odds in favor of any proposition are in the interval from
zero to infinity rather than the interval from zero to one, "without loss of
cohesion although the . . . math will be more complex".  Other alternatives
are available as well.


	Mike Hardy


