If correlation doesn’t imply causation, then what does?

This cool 2012 paper by Michael Nielsen has been posted, discussed, and often undeservedly ignored eight or nine times on HN. There is, if I remember correctly, a follow-on which includes some material contributed by Judea Pearl, which is also worth reading. Turing Award winner Judea Pearl's books and papers are the seminal sources; start with his Book of Why. Pearl's contribution was to show that there are circumstances where it it possible to separate events which are causal and events which are randomly correlated.