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Re: Programming is the Engineering Discipline of the Science that is Mathematics

From: Keith H Duggar <duggar_at_alum.mit.edu>
Date: 9 Jun 2006 13:49:33 -0700
Message-ID: <1149886173.466244.173050@g10g2000cwb.googlegroups.com>


vc wrote:
> Erwin wrote:
> > > This is why PT is a /generalization/ of logic. It
> > > reduces to logic when applied to truth-valued
> > > statements. Just as gamma reduces to factorial for
> > > natural arguments. (Again no quibbles about offset by
> > > 1 etc).
> > >
> >
> > You mean like :
> >
> > AND (p1, p2) === p1*p2
>
> P(p1 and p2) is not equal P(p1)*P(p1) in general, so no
> such 'generalization' is possible.

Wow! vc is going off the VI deep end at the moment. "no such 'generalization' is possible"? Saying that PT is not a generalization is one thing; but, none possible??

Erwin, what vc was referring to is that

P(AB) = P(A|B)P(B) -or-
P(AB) = P(B|A)P(A)

where | means given and AB is short for "A and B". This is called the product rule. Something that vc seems not to know (given his questions in the other post) is that in the limit of true (0) and false (1) the conditional probability product rule reduces to the logical conjunction truth table. Here is the proof

g : P(A)  = 0
p : P(AB) = P(B|A)P(A)
u : P(AB) = 0

g : P(B)  = 0

p : P(AB) = P(A|B)P(B)
u : P(AB) = 0
g : P(A)  = 1
g : P(B)  = 1
s : P(~B) = 0
m : P(A)  = P(AB) + P(A~B)
p : P(A)  = P(AB) + P(A|~B)P(~B)
u : P(A)  = P(AB)
c : P(AB) = P(A)
u : P(AB) = 1

thus

P(A) : P(B) : P(AB)

0 : 0 : 0
0 : 1 : 0
1 : 0 : 0
1 : 1 : 1

descriptions

g : given
p : product rule
s : sum rule
m : marginalization (derived from sum rule)
u : substitution

Received on Fri Jun 09 2006 - 15:49:33 CDT

Original text of this message

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