Pearl evaluates the counterfactual “had X been x” by deleting arrows in a causal diagram, or equations in a structural formula. That is, the minimal alteration on a causal diagram to cause X to assume the value x. 

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Lewis was also intrigued by human’s ability to ideate counterfactuals. He described the ability as being able to come up with an alternative universe where our variable of concern is the only thing not held constant. 

For example, one world where Joe took the aspirin and another where he didn’t. At the point that Joe did/didn’t take the aspirin, what might be the effect on that universe?

Pearl notes that in order to judge the causal strength of a variable, we need to assess the “closeness” of the world resulting from a counterfactual to our world. 


Lewis’s Definition for Counterfactuals

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Structural Models with Counterfactuals

“The value Y would have taken for individual u, had X been assigned the value x”.

$Y_X=x^{(u)}$, or $Y_X(u)$. Note that these “potential outcomes” are defined at the individual level, not the population. 


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