The Theorem below is surely known. But the consequence about Molinism is interesting. It is related to arguments by Mike Almeida.
Definition. The claim A→B is a conditional providing A→B entails the material conditional "if A, then B".
Remark: This is of course a very lax definition of a conditional (B counts as a conditional, as does not-A), so the results below will be fairly general.
Definition. A→B is localized provided A&B entails A→B.
Remark: Lewisian and Molinist subjunctives are always localized.
Definition. Adams' Thesis holds for a conditional claim A→B providing P(A→B)=P(B|A).Definition. The claim B is (probabilistically) independent of A provided P(B|A)=P(B). (If P(A)>0, this is equivalent to P(A&B)=P(A)P(B).)
Theorem 1. Suppose A→B is a localized conditional. Then Adams' Thesis holds for A→B if and only if A→B is independent of A.
Proof. First note that if A→B is a localized conditional, then, necessarily, A&(A→B) holds if and only if A&B holds. Therefore P(A→B|A)=P(A&(A→B)|A)=P(A&B|A)=P(B|A). Now P(A→B|A)=P(A→B) if and only if A→B is independent of A. ■
Remark: It follows that Molinist conditionals do not satisfy Adams' Thesis. For in Molinist cases, God providentially decides what antecedents of conditionals to strongly actualize on the basis of what Molinist conditionals are true, and hence A is in general dependent on A→B (and thus A→B is in general dependent on A).[note 1]
