Correlation is Evidence of Causation

A proof done with conditional probability.

Definition 1   correlation               :    c
Definition 2   causation                 :    a
Definition 3   not everything correlates :  P(c)   < 1
Definition 4   causation give correlation:  P(c|a) = 1

   P(a|c)                  : evidence for causation
 = P(c|a) P(a) / P(c)      : Bayesian inference
 =      1 P(a) / P(c)      : definition 4
 > P(a)                    : definition 3

Conclusion: P(a|c) > P(a)  : Correlation is evidence of causation. Q.e.d.

This proof is quite simple. You may notice that I did not include
proper definitions of correlation or causation. Correlation is easy,
since it is clearly defined as matrix math. Causation however, is
worse. I did not find any definition in my statistics books, and
Googling it gave meagre results. It found mainly claims that
correlation is not causation, without explaining what either is. Since
my proof is for people discussing like that, I decided to drop the
explanations too, since they are not necessary, and fewer would
understand them anyway. It is only necessary to know that causation
give correlation.

Same goes for Wikipedia, such as its Post_hoc_ergo_proper_hoc page. 

Its main subjects are again correlation and causation, but without
defining them usefully. Or one can accept all the similar articles as
dealing in boolean logic, and that explains their use of words quite
nicely, but boolean logic is not science, while Bayesian inference is,
because it deals in probabilities. This, and other logical
"fallacies", are just _logical_ fallacies. When translated into
probabilities, they are often true. So, what are they? Where do they
come from? Are they some kind of fallout after a war between
philosophers and scientists, or what?


(I originally made a proof with a definition of causation, but the more I
 studied, the more vague it became, until I ended up with evidence as a
 mathematical synonym for causation, and 4D edgy integrals of 3D
 simplex pieces of probability space, giving a proof that the
 correlation of A and B is evidence that A is evidence of B. But this
 went too far away from the original goal, so I dropped it.)

Quote from Daniel Dvorkin:
The correlation between ignorance of statistics 
and using "correlation is not causatison" as an argument 
is close to 1.