But even if van Inwagen's argument fails, there is a probabilistic variant that does not rely on (11). This argument is inspired by some remarks I got from Peter Forrest. Instead of (11), the argument uses the following claim:

Instead of concluding that the BCCF in fact does not have an explanation, the argument will conclude that some worlds have a BCCF that does not have an explanation. We proceed as follows. Making use of all the other premises of van Inwagen's argument, generalized to hold in all worlds, we get the claim that in every possible world there is an explanation of the BCCF, and the explanans is a necessary proposition. Now, if q is a necessary truth, then P(p|q) = P(p). Conditioning on necessary truth gets us no new probabilistic information beyond prior probabilities. Hence, in any world w, if the BCCF p of w is explained by a necessary truth, then P(p) > 1/2 by (28). Therefore, the BCCF of every possible world has probability greater than 1/2. But the BCCFs of different worlds are mutually exclusive, since any two worlds differ in the truth-value of some contingent proposition, and then the BCCF of one of the worlds will contain that proposition and that of the other will contain its denial. Hence, if p1 and p2 are the BCCFs of two distinct worlds, we have P(pj or p2) = P(pi) + P(p2) > 1/2 + 1/2 = 1. But no probability can be bigger than 1, and absurdity ensues again.

A defender of the PSR could, of course, deny (12), which this version of the argument presupposes, since otherwise we could have self-explanatory contingent explanations of the BCCF. A desperate, but not entirely unjustified, alternate measure would be to deny the assumption that if q is necessary, then P(p|q) = P(p), perhaps allowing that this is true if q is a tautology, and maybe even any narrowly logically necessary truth, but not if it is a substantive necessary truth, such as that horses are mammals, that water is H2O or that God values unity and happiness. It could, then, be the case that p1 has probability greater than 1/2 given one necessary truth q1, while a proposition p2 incompatible with p1 has probability greater than 1/2 given another necessary truth q2. For instance, perhaps that the universe consisting only of a single particle has high probability given that God values unity, and that the universe containing infinitely many happy persons has high probability given that God values happiness, even though it is a necessary truth that God values both unity and happiness.

The best way out for the PSR's defender, however, seems to be to oppose (28). First of all, statistical relevance theories of explanation deny (28), and, more broadly, (28) may be a manifestation of the mistaken conflation of explanation with prediction that plagued both the deductive-nomological (Hempel & Oppenheim 1948) and inductive-statistical (Hempel 1962) models of explanation. A standard counterexample to these models is the syphilis/paresis case (Scriven 1959; see also the discussion in Salmon 1990, sec. 2.3), which is also a counterexample to (28). We can explain why a person has paresis in terms of the earlier having of latent untreated syphilis, even though latent untreated syphilis leads to paresis only in a minority of cases.

Second, it is plausible that citing the relevant actual cause of an event explains the event. Indeed, to give causes is a paradigmatic way of explaining. But causation can filter through indeterministic events of probability less than 1/2. This is particularly clear in the case of forensic explanations. George murderously pushes Maurice off a very high cliff. Maurice falls and drowns. Unbeknownst to George, Maurice is actually a cliff diver, and had a 75 percent chance of survival for the fall. Nonetheless, George's murderous push killed Maurice, and George's having pushed Maurice explains why Maurice died. Granted, in this case it does not explain everything about why Maurice died. It does not explain, for instance, why in this case Maurice did not manage to swim out or why lack of oxygen kills earthly vertebrates. But it is still a fine explanation. In any case, it could well be that even after we answered all of these questions, it would be that the explanans made the explanandum less than 50 percent probable - there could be indeterministic quantum events in Maurice's brain behind Maurice's inability to swim out.

Third, if libertarianism holds, and if a plausible account of action requires one to say that free choices are explained by the agent's reasons, we have reason to deny (28). For it seems likely that libertarian-free agents can act on reasons that they had probability less than 1/2 of acting on.

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