Quantum Gravity 350.9: P{(A-->B)(B-->A) ' } = P(A ' B) > 1/2 iff P(B) > P(AB) + 1/2

[OsherD]

From Osher Doctorow

Theorem. P{(A-->B)(B-->A) ' } = P(A ' B) > 1/2 iff P(B) > P(AB) +
1/2.

Proof. P(B) - P(AB) = P(A ' B) = P{(A-->B)(B-->A) ' } (the last
equality proven previously) > 1/2 iff P(B) > 1/2 + P(AB). Q.E.D.

Notice that the Theorem implies that for the Memory case (P(A ' B) >
1/2) we have P(B) > 1/2 since P(AB) > = 0 always.

This raises a question as to whether P(C) > 1/2 for all sets C implies
that C are unbounded in an unbounded Universe. Since A and B above
are considered bounded and A ' , B ' unbounded, this is an exception
to the "rule".

More clarity on this is cast by the fact that:

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

which is direct from the definitions, from which:

2) P(A-->B) - P(B-->A) > 0 iff P(B) > P(A)

So we can have P(A) < 1/2 even if P(B) > 1/2. The indications are
that the Effect (B) can have P(B) > 1/2 but that P(A) < 1/2 or < = 1/2
for the Cause (A) bounded. Note that we want the left side of (2) to
hold in order for Probable Correlation not to hold at least in the
sense of the expression (A-->B)(B-->A) ' having relatively high
probability.

Osher Doctorow