A critical mistake in the lead analysis is false assumption: where there is a causal relation between two variables, they will be correlated. This ignores that causes often cancel out. (Of course, not perfectly, but enough to make raw correlation a generally poor guide to causality.
I think you have a fundamentally mistaken epistemology, gwern: you don't see that correlations only support causality when they are predicted by a causal theory.
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What about the problem that if you admit that logical propositions are only probable, you must admit that the foundations of decision theory and Bayesian inference are only probable (and treat them accordingly)? Doesn't this leave you unable to complete a deduction because of a vicious regress?