Quantum Gravity 245.0: Causation in PI, Conditional Probability, QFT, GR

From Osher Doctorow

In Probable Causation/Influence (PI), "A causes B maximally" (but
probabilistically) is interpreted as either:

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

or:

2) P ' (A-->B) = 1 + P(B) - P(A) = 1 for P(B) < = P(A) (equivalent to
P(B) = P(A))

Equation (1) is equivalent to either (with probability 1) A is a
subset of B or P(A) is 0.

Equation (2) is equivalent to P(A) = P(B), whether or not A = B (with
probability 1) or whether or not both have probability 0.

Notice carefully that P and P ' are two variants of PI which coincide
when B is a subset of A (with probability 1).

What the above reduces to is the result that in PI (dropping
probabilistic qualifications for brevity here), "A causes B maximally"
is equivalent to either A being a subset of B, A and B having the same
probability, or A having probability 0.

Another way to say this is that in PI, subsets cause their supersets
maximally, "equivalent probability orbit sets" (P(A) = P(B)) cause
each other maximally or one causes the other maximally", and
probability 0 sets cause any set maximally (a variant of the
Holographic Principle in the latter case).

Then the degree to which sets approximate the above conditions
probabilistically is the degree to which A causes B.

The main difficulty with Conditional Probability, used extensively in
mathematical probability/statistics and physics, is that it excludes
probability 0 sets and small neighborhoods of them because of "blow-
up" nearby, doesn't discriminate between the two types P and P ' even
"analogously", and makes it more difficult to understand the above
relationships because of its use of quotient equations. I'll try to
discuss this later, although Readers can find my frequent past
discussion of Conditional Probability's defects in earlier postings.

Osher Doctorow
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