From Osher Doctorow
The equation for force transmission:
1) P(F1F2) = 1 (with F1, F2 fields)
is especially interesting because it implies that:
2) P(F1 U F2) = 1
since F1F2 as the intersection of F1 and F2 is a subset of F1 U F2 and
probability is monotone (increasing with increasing subset inclusion).
But (1) and (2) imply that:
3) P(F1F2 --> F1 U F2) = 1
since the left hand side is:
4) 1 + P(F1F2) - P(F1 U F2) = 1 + 1 - 1 = 1
noting that F1F2 intersected with F1 U F2 is F1F2.
However, we don't even require (1) to prove (3), since (3) is always
true. Recall that P(A-->B) = 1 iff A is a subset of B with
probability 1 (i.e., up to a set of probability 0) or if P(A) = 0
(which doesn't have to be considered up to sets of probability 0).
So what is the purpose of this particular exercise, that is to say
equations (1)-(4)? Well, from the previous parts of this thread, the
point is that when we are considering scenarios like P(F1F2) = 1 or
P(F1F2) > 0 as laws of force field transmission, we are automatically
in the Probable Influence (PI) machinery since P(F1F2 --> F1 U F2) = 1
is satisfied. We still have to examine the P(F1F2) or P(F1F2F3F4) and
so on expressions, but we are on "solid ground".
Osher Doctorow
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