Here is the right way to see A-theory (this is particularly accurate as an account of Tom Crisp's presentism). There are infinitely many possible worlds, one of which, w0, is actual. Which world is actual changes with time: tomorrow, say, w17 will be actual (a world encodes everything that is objectively the case—but the A-theorist thinks there is an objective difference between how things are today and how they will be tomorrow). Of course, w0 and w17 are not unrelated: at w0 it is true that tomorrow* w17 will be actual, while at w17 it was true that yesterday* w17 was actual (here, "tomorrow*" and "yesterday*" are narrow-scope versions of our usually wide-scope terms).
Moreover, there are three crucial relations that can hold between worlds: S and E.
We say that S(w1,w2) if and only if w1 and w2 are simultaneous—i.e., it is the same time in both of them. When I say that it might have been the case that right now I am writing a post on growing-block theories, this implies there is a world w simultaneous with the actual world w0 such that at w, I am writing such a post. This relation is an equivalence relation—it is reflexive, symmetric and transitive.
The earlier-than relation E(w1,w2) holds if and only if at w2 it is true that w1 was actual. However, it is important to see that the S and E relations are not of a kind. S holds between the actual world and many, many worlds that have never been, are not, and never will be actual. E only holds (in some order) between the actual world and worlds that have been or will be actual. The relation E is transitive. Moreover, no two worlds related by S are related by E. In particular no world is earlier than itself..
Socrates was sitting if and only if there is a world w1 such that E(w1,w0) and at w1 Socrates is sitting.
The relations S and E probably have to be taken to be primitive.
If E(w1,w2) then I will say that w2 is in w1's future and w1 is in w2's past.
We can now precisely characterize closed-future and closed-past views. We have a closed future (past) if and only if the collection of worlds that the actual world is earlier (later) than the actual world is totally ordered by E. It is a consequence of this that no two worlds later (earlier) than the actual world are simultaneous. Everybody, I assume, believes in a closed past. Closed-futurist A-theory, then, is the view the collection of all worlds can be partitioned into disjoint subcollections, in one of which subcollections are all the worlds that aren't E-related to any world (these are the worlds that have no past or future), and each of the other subcollections is totally ordered by E.
Open futurists, on the other hand, think that there are two future worlds that are not E-related—in fact, they typically think there are two future worlds that are simultaneous.
Presentism then adds the further claim that at a world w, only those things are existent that are presently* existent. While A-theory is a theory of the structure of time, presentism is a theory of the ontology of each world. They neatly complement each other, because the presentist has an elegant answer to the question of what it is for an object or event to exist at a time:
- x existed at t in w if and only if there is a world w1 such that (a) it is t in w1, (b) E(w1,w) and (c) x exists in w1
- x exists at t in w (where w is such that it is t in it) if and only if x exists in w
- x will exist at t in w if and only if there is a world w1 such that (a) it is t in w1, (b) E(w,w1) and (c) x exists in w.
Our timeline (which is branching if open-futurism is true) is the collection of the actual world and all worlds that are E-related to the actual world.
This formulation shows the problem of how A-theory, divine immutability and omniscience could all be true (after all, doesn't it contradict immutability if God has to keep on updating his beliefs) is the same as the problem of how contingentism, strong aseity and omniscience could all be true, where contingentism is the doctrine that not everything is necessary and strong aseity is the doctrine that God has exactly the same intrinsic properties in all worlds. Moreover, this formulation also shows that an A-theorist who believes in divine immutability is someone who believes a restricted version of strong aseity—restricted to the worlds in our timeline. There is thus a good plausibilistic argument from A-theoretic immutability to strong aseity—why restrict to our timeline?
Where does the B-theorist stand in regard to all this? She insists that at every world it is true that in the past and the future, the very same world is actual. Of course, a different Lewisian "centered" world will be actual in the future. The above is really a matter of formalism, so it does not solve any really hard problems, so it does not solve the problem of how worlds and centered worlds differ.
The formalism does, however, highlight some problems for presentism. Consider the problem of induction for presentists. Normally, only what happens in the actual world is relevant for inductive inferences. We know that most worlds are different enough from the actual world that doing induction over them will mix us up completely. But to have any hope of induction being useful, the presentist has to insist on making use not just of data about what happens in the actual world, but also data about what happens in worlds earlier than the actual. I don't know that the presentist can answer this. Here's another problem. Finding out that I promise something in a world w1 gives me very little reason to do the promised deed in w0. But if I also find out that E(w1,w0), then I have reason to do this. This seems magical, and unless the A-theorist can give us a good substantive account of the E-relation, this will be unexplained.