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From: Osher Doctorow on 2 Jul 2010 08:59
From Osher Doctorow

The two types of Probable Causation are:

1) P(A-->B) = 1 + P(AB) - P(A)
2) P ' (A-->B) = 1 + P(B) - P(A), where P(B) < = P(A)

Probable Correlation is defined as:

3) P(A<-->B) = P{(A-->B)(B-->A)} = P(AB) + P(A ' B ' )

Now suppose that B is a subset of A with probability 1, which is to
say except for subsets of probability 0. We can write this as B C A
with C representing the usual set inclusion operator, or B =
INTERNALITY(A) where we could use a subscript on INTERNALITY to
distinguish B from other subsets of A. Then P(AB) = P(B) since AB =
B iff B is a subset of A, and also P(A ' B ' ) = P(A ' ) = 1 - P(A)
because B C A iff A ' C B ', so using (2), (1) and (3) respectively
become:

4) P(A-->B) = 1 + P(B) - P(A) = P ' (A-->B), for B C A with
probability 1 (w.p. 1)
5) P(A<-->B) = P(B) + 1 - P(A) = 1 + P(B) - P(A) = P(A-->B) = P ' (A--
>B), for B C A w.p.1

Thus, the "internal world" of a set/event is (Probably) Correlated and
unifies P and P ' as Probable Causations with Probable Correlation.
This is not necessarily true of the "external world" or the combined
(partly) internal and external world of a set/event, and in fact it is
false for the former if there is some subset of the external world
with probability > 0. Let us write the first statement as a
Principle, although it is a proven Principle or Theorem:

6) Theorem (or Principle): The "internal world" of a set/event is
(Probably) Correlated and unifies P and P ' as Probable Causations
with Probable Correlation.

The Quantum Theory emphasis on Correlations and Expectations can be
regarded as merely a reflection of (6) rather than as a new version of
mathematics, but it then produces erroneous results in many cases
involving interactions or externalities of events.

Notice that if P(B C A) is not 1, then P (A-->B) = 1 + P(AB) - P(A)
does not equal P ' (A-->B) because P(AB) does not equal P(B) since B
has a subset of probability > 0 outside A.

Osher Doctorow

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