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: $X\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y$

Then the often stated line of correlation does not imply causation is simply! $Q\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P$.

It is also true that causation does not imply correlation. So! $Q\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P$

But $(P\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Q)\vee (Q\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P)$ is a tautology.

causation

Let P := |corr(X,Y) > .5| let Q := exits a relation F: $X\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y$

Then the often stated line of correlation does not imply causation is simply! $Q\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P$.

It is also true that causation does not imply correlation. So! $Q\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P$

But $(P\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Q)\vee (Q\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}P)$ is a tautology.

tiltat9h

Beginner2022-07-22Added 14 answers

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 ! $!(P\to Q)$. Instead it is

$!((\mathrm{\forall}X,Y)(P(X,Y)\to Q(X,Y))$

Given this, you'll find that you can't formulate your paradox.

Note, incidentally, that the key issue here has nothing to do with correlation and causation. You could take statement "P" to be "Chris is a women" and "Q" to be "Chris is a parent". If we allow for the fact that there are many people named Chris in the world, these statements are predicates with truth values that depend on which Chris you're talking about.

Now it is not true that "All women are parents" and it is not true that "All parents are women", so if your logic were correct, you could apply it here equally well and get the same paradox without ever mentioning causation or correlation. The resolution remains the same.

The correct translation of "Correlation does not imply causation" is not ! $!(P\to Q)$. Instead it is

$!((\mathrm{\forall}X,Y)(P(X,Y)\to Q(X,Y))$

Given this, you'll find that you can't formulate your paradox.

Note, incidentally, that the key issue here has nothing to do with correlation and causation. You could take statement "P" to be "Chris is a women" and "Q" to be "Chris is a parent". If we allow for the fact that there are many people named Chris in the world, these statements are predicates with truth values that depend on which Chris you're talking about.

Now it is not true that "All women are parents" and it is not true that "All parents are women", so if your logic were correct, you could apply it here equally well and get the same paradox without ever mentioning causation or correlation. The resolution remains the same.

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