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I often read that the counterfactual conditional "is standardly held to be closed under entailment", but I am not sure if I understand what it means: Say p counterfactually implies q, and q implies r; does p counterfactually imply r? And would this be the correct way to write it down?

((p ☐→ q) ∧ (q → r)) → (p ☐→ r)

I'm inclined to say yes, but I'd like to have a solid proof. I tried to look my question up in different handbooks and Lewis's "Counterfactuals", but was unable to answer the question (which is probably my fault).

I'd be happy for any input and recommendations for further reading!

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It is important not to confuse entailment with implication. Closure under entailment means that if p counterfactually implies q, and q entails r then p counterfactually implies r. Given that Lewis, whose notation you have adopted here, interprets counterfactuals in terms of a possible worlds semantics, it should be sufficient to strengthen the implication to a strict implication as follows:

((p ☐→ q) ∧ ☐(q → r)) → (p ☐→ r)

On Lewis' account we might read this as: if, in those relevant close possible worlds in which p is true q is also true, and all q worlds are r worlds, then in those relevant close PWs in which p is true r is also true.

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