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From: OsherD on 8 May 2010 20:30
From Osher Doctorow

The maximum Probable Causation/Influence (PI, also written P(A-->B)):

1) P(A-->B) = 1 + P(AB) - P(A)

occurs when P(AB) = P(A), in which case P(A-->B) is 1 on a scale of 0
to 1, that is to say with values in the interval [0, 1].

P(AB) = P(A) occurs if and only if:

2) P(AB) = P(A) iff AB = A with probability 1, which is to say either
A is a subset of B or P(A) = 0 (except possibly for a set of
probability 0).

However, if Probability is proportional to Lebesgue Measure, which for
Euclidean-like space means that the Probability of an object is
proportional to its volume, which is arguably intuitive under a wide
variety of conditions, then:

3) P(A) proportional to Lebesgue measure implies that P(A) = 0 for A a
0, 1, or 2-dimensional set embedded in 3-dimensional space or 3+1
dimensional spacetime.

Of course, (3) is essentially the Holographic Principle, since n-1 or
lower dimensional subsets of a n dimensional Euclidean-like
space(time) have Lebesgue measure 0, for n any positive integer.

Osher Doctorow
From: OsherD on 8 May 2010 20:34
From Osher Doctorow

It also turns out that P ' (A-->B) = 1 in the above condition where
P(A-->B) = 1 because in P ' (A-->B), we require P(B) < = P(A), and for
P(A) = 0 this means that P(B) = 0, so:

1) P ' (A-->B) = 1 + P(B) - P(A) for P(B) < = P(A) = 1 when P(A) = 0.

Osher Doctorow
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