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Science Forum Index » Physics Forum » Quantum Gravity 261.1: Causation is Composed of Not...
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| OsherD... |
 Posted: Mon May 26, 2008 6:52 pm |
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From Osher Doctorow
In comparing the two types of Causation-related Probability, Probable
Causation/Influence (PI) P(A-->B) and Conditional Probability (CP) P(B|
A), the respective equations are:
1) P(A-->B) = 1 + y - x, x = P(A), y = P(AB) where y "represents" the
Effect B
2) P(B|A) = y/x for x not 0, x = P(A), y = P(AB)
But P(AB) < = P(A) is a law of Probability, so both P(A-->B) = 1 + y -
x and P(B|A) = y/x are between 0 and y, which is to say that 0 < = y <
= x except that x isn't 0 in P(B|A).
This tells us that the Proximity of y to x, the Effect to the Cause,
increases (Probable) Causation, in both cases to the extent that if y
= x, then P(A-->B) = 1 + 0 = 1 and P(B|A) = P(A)/P(A) = 1. The
distance from y to x in typical metrics, for example in Euclidean
space, actually decreases Causation.
To see this more clearly, notice that:
2) P(A-->B) = 1 + y - x = 1 - (x - y)
So x - y as it increases (remember that x > = y) diminishes P(A-->B),
but as x - y decreases to 0, P(A-->B) increases to 1.
Similarly with P(B|A) except that x = 0 is not allowed (P(B|A) is
undefined for x = P(A) = 0).
So Proximity or "nearness" a type of partial inverse of distance/
metric "farness", increases with Causation, where the "nearness"
refers to the Cause and Effect variables x and y respectively. Those
variables if normalized into [0, 1] with y < = x can be velocity,
distance, acceleration, mass, or whatever. They are not exclusively
distance or displacement or length dimensionally, and this is where
exclusive reliance on the metric goes astray, although in some sense
Proximity could be regarded as having a length dimension L or more
likely an inverse length dimension L^(-1).
A second component of Causation is time, which is always part of the
Cause x although both x and y can be dimensionally time as in t2 - t1
or t1 - t2 for two different times or a delayed versus an advanced
time variable. That doesn't mean that x = t always. It would just be
extremely strange if x didn't involve t (time), that is to say if x
didn't increase with time. But it can also involve additional
variables involving length, mass, temperature, electrical charge, or
whatever.
Thus, when we look at the "kernel" of the derivative of normalized
function f:
1) [f(t + h) - f(t)]/h, h > 0
or its PI analog provided that normalization into [0, 1] has occurred:
2) 1 + [f(t + h) - f(t)] - h, h > = 0 (where h is a t or time type
variable)
it is not merely time (h) that is Causal but the proximity of f(t + h)
- f(t) to h! Since both (1) and expression (2) are < = 1 in
normalized form, the derivative has an upper bound of 1 in this
representation and when it or its kernel attains 1, then Causation is
maximum.
This may look to some Readers like a one-sided derivative since we
haven't discussed h < 0, but Readers can work on this question to
obtain a satisfactory answer if they choose to.
Osher Doctorow |
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| OsherD... |
| Posted: Mon May 26, 2008 6:55 pm |
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From Osher Doctorow
In the 9th line of typing in the previous post, I mean to say "between
0 and 1," not "between 0 and y."
Osher Doctorow |
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