Monday, February 4, 2013

Causal probability and counterfactuals

The causal probability of an event B on an event A is cPA(B)=∑KP(K)P(B|AK), where the Ks are a partition based on the relevant dependency hypotheses compatible with A. (Compare to P(B|A)=∑KP(K|A)P(B|AK).) A standard proposal in the literature is that

  1. the degree of the assertibility of an indicative "If A, then B" is equal to the conditional probability P(B|A) of B on A.
Consider the parallel thesis that
  1. the degree of assertibility of a subjunctive conditional or counterfactual AB is equal to the causal probability cPA(B) of B on A.
This thesis would unify the Stalnaker and Lewis (and Skyrms) approaches to causal decision theory as closely as possible. For according to the Stalnaker version, the causal expected value of an option A is:
  1. EV(A) = ∑BU(BA)P(AB),
where the sum is over a partition based on outcomes. On the Lewis/Skyrms approach, it will be:
  1. EV(A) = ∑BU(BA)cPA(B).
Now, if AB has truth value, then the degree of asssertibility of AB is equal to P(AB), and hence by (2) we have P(AB)=cPA(B). And so the two formulae are equivalent. If, on the other hand, AB has no truth value, then P(AB) in (3) makes no sense. But we can replace it with Assertibility(AB), which is basically the most natural replacement for P(AB) when AB has no truth value, and the revised (3) will come to the same thing as (4). So that's nice.

Notice, however, that this approach may not be compatibility with Molinism. For according to Molinism, God knows some conditionals of free will AB, where B is a free action and A is a maximally specific set of antecedents, for sure. If P is God's probabilities, then in such cases:

  1. 1=cPA(B)=∑KP(K)P(B|AK).
But because A is maximally specific, it will be compatible with only one relevant dependency hypothesis, say K0, describing how B depends on A. So 1=P(K0)P(B|AK0). It follows that P(B|AK0)=1 and P(K0)=1. But now we see that there is a dependency hypothesis K0 such that, together with A, it probabilistically necessitates B. But that can't be acceptable to a libertarian.

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