Lillie Pittman

2022-07-21

Causation doesn't imply correlation and correlation does not imply
causation
Let P := |corr(X,Y) > .5| let Q := exits a relation F: $X\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Y$
Then the often stated line of correlation does not imply causation is simply! $Q\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}P$.
It is also true that causation does not imply correlation. So! $Q\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}P$
But $\left(P\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Q\right)\vee \left(Q\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}P\right)$ is a tautology.

tiltat9h

Your P and Q are not propositions; they are predicates. That is, the truth value of P varies depending on X and Y.
The correct translation of "Correlation does not imply causation" is not ! $!\left(P\to Q\right)$. Instead it is
$!\left(\left(\mathrm{\forall }X,Y\right)\left(P\left(X,Y\right)\to Q\left(X,Y\right)\right)$