In my dissertation, I defended a causal power account of modality on which something is possible just in case either it’s actual or something can bring about a causal chain leading to its being actual. I noted at the time that unless there is a necessary first cause, this leads to an odd infinite branching view on which any possible world matches our world exactly once you get far enough back, but nonetheless every individual event is contingent, because if you go back far enough, you get a causal power to generate something else in its place. Rejecting this branching view yields a cosmological argument for a necessary being. To my surprise when I went around giving talks on the account, I found that some atheists were willing to embrace the branching view. And since then Graham Oppy has defended it, and Schmid and Malpass have cleverly used it to attack certain cosmological arguments.

I want to note a curious, and somewhat unappealing, probabilistic feature of the backwards-infinite branching view. While it is essential to the view that it be through-and-through contingentist, assuming classical probabilities can be applied to the setup, then the further back you go on a view like that, the closer it gets to fatalism.

For let *S*_{t} be a
proposition describing the total state of our world at time *t*. Let *Q*_{t} be the
conjunction of *S*_{u} for all *u* ≤ *t*: this is the total
present and past at *t*. Here is
what I mean by saying that the further back you go, the closer you get
to fatalism on the backwards-infinite branching view:

- lim
_{t→− ∞}*P*(*Q*_{t}) = 1.

I.e., the further back we go, the less randomness there is. In our time, there are many sources of randomness, and as a result the current state of the world is extremely unlikely—it is unlikely that I would be typing this in precisely this way at precisely this time, it is unlikely that the die throws in casinos right now come out as they do, and so on. But as we go back in time, the randomness fades away, and things are more and more likely.

This is not a completely absurd consequence (see Appendix). But it is also a surprising prediction about the past, one that we would not expect in a world with physics similar to ours.

**Proof of (1):** Let *t*_{n} be any
decreasing sequence of times going to − ∞. Let *Q* be the infinite disjunction *Q*_{t1} ∨ *Q*_{t2} ∨ ....
The backwards-infinite branching view tells us that *Q* is a necessary truth (because any
possible world has *Q*_{t} is true for
*t* sufficiently low). Thus,
*P*(*Q*) = 1. But now
observe that *Q*_{t1}
implies *Q*_{t2}
implies *Q*_{t3}
and so on. It follows from countable additivity that lim_{n→∞} *P*(*Q*_{tn}) = *P*(*Q*) = 1.

**Appendix:** Above, I said that the probabilistic
thesis is not absurd. Here is a specific model. Imagine a particle that
on day − *n* for *n* > 0 has probability 2^{−n} of moving one meter
to the left and probability 2^{n} of moving one meter to
the left, and otherwise it remains still. Suppose all these steps are
independent. Then with probability one, there is a time before which the
particle did not move (by the Borel-Cantelli lemma). We can coherently
suppose that necessarily the particle was at position 0 if you go far enough back, and then the
system models backwards-infinite branching. However, note an unappealing
aspect of this model: the movement probabilities are time-dependent. The
model does not seem to fit our laws of nature which are time-translation
symmetric (which is why we have energy conservation by Noether’s
theorem).