What do people mean when they say (as when cautioning against the validity of a scientific study, for example) that "correlation does not imply causation"? Isn't causation just perfect correlation? And if so, doesn't that mean that the caveat in question does not concern causal claims per se, but inductive claims more generally?

Suppose there is a single type of event A that always causes two other events, B and C. Suppose, moreover, that, whenever B occurs, it is always caused by A, and similarly for C. Then B will be perfectly correlated with C, but by hypothesis is not caused by it.

This is typically what people worry about, though of course the correlation is rarely perfect, in fact. For example, people with AIDS almost always have HIV. There's a very strong correlation. We also know now that HIV does cause AIDS. But the correlation by itself does not show that HIV causes AIDS. Maybe having HIV is actually another effect of whatever it is that actually causes AIDS. Your immune system is depressed, and so you have this virus that other people do not have. Not actually true, but it might have been true.

But perhaps what is bothering you is a slightly different thought: This kind of strict correlation can't just be accidental, can it? Surely something must explain it! But, as I have said, having one cause the other is not the only possible explanation. There might be a common, third cause of both.

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