Toward a Probabilistic Model of Divine Providence

One of my long-term goals is to explore the consequences for divine providence on the assumption that open theism (OT) is correct. As I shall understand it here, OT is defined by the following commitments:

1. Monotheism: There exists one and only one God who is personal; necessarily existent; and essentially omnipotent, omniscient, and morally perfect. Every concrete reality metaphysically distinct from God is freely created by God ex nihilo and owes its continued existence to God's sustaining activity.
2. Future Contingency: There are some possible future states of affairs of which it is true that they both might and might not obtain.
3. Incompatibility of Future Contingency and Exhaustively Settled Divine Foreknowledge: Whatever both might and might not happen cannot be truly known as something that either will or will not happen.
4. Divine Epistemic Openness Regarding Future Contingents: Insofar as the future is contingent, God knows it as something that might and might not happen and does not know it as something that either will or will not happen. This follows from (2) and (3).
   
5. Divine Temporality: At least since creation, God stands in real temporal relations with his creation. This follows from (2) and (4).

Now, suppose that it is now T0, that God would like to bring about a certain state of affairs S at T2, and that S's obtaining at T2 is a future contingent. What can God do to affect whether S obtains at T2? It depends on S and on God's willingness to tolerate S's not obtaining at T2.

If S is an intrinsically indeterministic state of affairs, one that by its very nature precludes antecedent necessitation like Peter's freely (in a libertarian sense) denying Christ, then given (3) there is nothing God can do to guarantee that that state of affairs comes about. At most, God can act in ways that would influence whether that state of affairs comes about by directly altering the propensities. For example, if God at T1 were unilaterally to cause several people near Peter to accuse him of being a disciple and also to plant strongly in Peter's mind the idea that if he is found out as a disciple then he will likely meet with a painful death, then perhaps God could ratchet up the propensity of Peter's freely denying Christ somewhere close to 1. He would have to careful, though, not to overdo it. If God's influencing activities overwhelm Peter, then he ceases to be a libertarian free agent in the circumstances. And even if they don't quite overwhelm, but come very close to doing so, it becomes doubtful whether Peter could justly be held morally responsible for his actions.

If S is not an intrisically indeterministic state of affairs, however, then matters are much simpler, providentially speaking, since there is nothing preventing God from unilaterally bringing it about. He may not always want to do so, however. If God, say, were to unilaterally intervene at T1 so as to bring about Peter's verbally denying Christ at T2, then he compromises Peter's integrity as a moral agent and thereby compromises the quality of his relationship with Peter. Insofar as God values genuine, free relationships with his creatures he will refrain from unilaterally intervening to compel them to action. But perhaps on occasion the benefits outweigh the costs?

Anyway, for God of OT to make the right providential decisions, he needs to be able to assess the propensities of various possible states of affairs obtaining and to update these assessments as conditions change. Suppose, for example, that God wants to know at T0 what the propensity is for S to obtain at T2. In other words, suppose he wants to evaluate Prob(S at T2 | T0), i.e., the probability that S obtains at T2 given the state of the world at T0. How can he do this?

Well, presumably it would have to work something like this: At T0 God calculates the propensities for each possible states of the world at T1. To keep it simple, let's suppose that there are three such states (A, B, C) that are mutually independent and jointly exhaustive. Thus, God evaluates

Prob(A at T1 | T0) = a
Prob(B at T1 | T0) = b
Prob(C at T1 | T0) = c

Then, he can evaluate the conditional propensity of S on each of these possibilities. Thus,

Prob(S at T2 | A at T1) = d
Prob(S at T2 | B at T1) = e
Prob(S at T2 | C at T1) = f

Prob(S at T2 | T0) is then equal to

Prob(A at T1 | T0)*Prob(S at T2 | A at T1) +
Prob(B at T1 | T0)*Prob(S at T2 | B at T1) +
Prob(C at T1 | T0)*Prob(S at T2 | C at T1) = ad + be + cf

Once T1 arrives, and either A, B, or C occurs, then the other probabilities become irrelevant. If A occurs at T1, then Prob(S at T2 | T0) no longer matters. What's relevant now is Prob(S at T2 | A at T1).

Realistic cases, however, could be much more complicated. There might be no neat partition into a small finite number of mutually exclusive and jointly independent possibilities. Presumably the calculations wouldn't bother an infinitely intelligent being like God, but it would increase the number of providential variables, thereby requiring much more extensive intervention on God's part if he wants to achieve a specific outcome.