Random thing that I can't recall seeing on LW: Suppose A is evidence for B, i.e. P(B|A) > P(B). Then by Bayes, P(A|B) = P(A)P(B|A)/P(B) > P(A)P(B)/P(B) = P(A), i.e. B is evidence for A. In other words, the is-evidence-for relation is symmetric.

For instance, this means that the logical fallacy of affirming the consequent (A implies B, and B is true, therefore A) is actually probabilistically valid. "If Socrates is a man then he'll probably die; Socrates died, therefore it's more likely he's a man."