"is" in language can often mean "is a subset of", I was using the term that way. Whilst you are right that "=>" disclaims any philosophical relationship, the pr

"is" in language can often mean "is a subset of", I was using the term that way. Whilst you are right that "=>" disclaims any philosophical relationship, the proof covers all definitions of "cause" that one might reasonably come up with. So "you can't conclude A => B" implies (with probability 1) that "you can't conclude A causes B": The proof only defines "causation" as some event "a" for which "P(c|a) = 1". This is the same property that "=>" has in propositional logic, and there is no implication of philosophical causation here either. But the proof still works, as a consequence of its definitions. So in other words, the proof says: if causation causes correlation then P(a|c) > P(a) (i.e. correlation is evidence of causation) but we can't say causation is definitely true (P(a) = 1), however you want to define "causes" as long as it has the property that P(c|a) = 1.