# Does correlation imply causation?

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In episode 55 of The Effort Report podcast, Roger Peng makes an interesting observation: correlation implies causation all the time! That’s how we know most of things in science, including that smoking causes lung cancer.

(around 00:37:30)

Roger is not wrong, but the informal language makes his statement ambiguous.

In formal logic, we almost never say “A implies B” \((A \Rightarrow B)\), except as a shorthand for something else.

Instead, we say A implies B under a set of assumptions Γ, or

\[\Gamma \vdash A \Rightarrow B.\]

Therefore, the statement “A implies B” can be valid under some Γ and invalid under others.

When we say “correlation does not imply causation”, we implicitly mean “without any additional assumptions”, that is,

\[\varnothing \nvdash \text{X correlates with Y}\Rightarrow\text{X causes Y}.\]

But under a suitable set of *causal assumptions* Γ, correlation may very well imply causation, which is one way to interpret Roger’s claim:

\[\Gamma \vdash \text{X correlates with Y}\Rightarrow\text{X causes Y}.\]

Elizabeth Matsui actually points to this difference when she says “correlation does not *equal* causation”.

To learn about causal assumptions, read the excellent overview of causal inference by Judea Pearl or one of his many other works.