Two presentist ways of seeing worlds

If presentism is true, then right now, call it t₂, the proposition B that Bucephalus exists is false, but it once was true, namely at t₁. Now, at every time a token of the following sentence expresses a truth:

0. For all p, a proposition p is true if and only if it is true at the actual world.

Now, let's imagine ourselves at t₁. Then Bucephalus exists. Thus, B is true. Moreover, (0) expresses a truth, and so B is true at the actual world. So at t₁ the sentence

1. B is true at the actual world

expresses a truth. But now let's return to our time. B is false. But (0) expresses a truth, and so the sentence

2. B is not true at the actual world

does expresses a truth. Thus, (1) expresses a truth when said at t₁ but expresses a falsehood when said at t₂. This shows that either:

3. "The actual world" refers to different worlds at different times

or

4. The proposition that p is true at w can change in truth value, even if "p" and "w" refer rigidly to a proposition and a world, respectively.

Thus, the presentist has two ways of understanding possible worlds. Either possible worlds are tensed, so that at every time we inhabit a different possible world (that's option (3)) or else the "true at" relation is tensed, so that we inhabit the same world at different times, or when we say at t that p is true at w, we say something true if and only if p is true at t at w.

I think there is a problem for (4). Let p be the proposition that horses do or do not exist. Let t be the actual present time. Then p is true at every world, since it's a necessary truth. Now consider a world w where the time sequence does not include t. There are several options for this. Maybe in w, time comes to an end in 2011. Maybe time is discrete in w while in our world it is continuous, and so w either includes no times from our world or else w "skips over" t. Or maybe for some other reason the time sequence in w is radically different from our world's time sequence. Then p is true at w. But on (4), when we say that p is true at w, that is true if and only if p is true at t at w. But nothing is true at t at w, since t isn't a time at w.

Here's a slightly different way to see the point. When p is true at w, it is true either because there are or because there are not horses at w (this is an uncontroversial case of disjunctive grounding). Suppose it's true because there are not horses at w. But at which time are the horses not there at w? After all, w could have horses at some but not other times. Presumably, the relevant time is the present time. On proposal (3), every world comes along with its own present time, and this is fine. But on proposal (4), a world's relevant present time is our present time, and w doesn't have our world's present time.

One could try to solve this with counterpart theory for times. But one can suppose w won't have a counterpart to our time.

Here's a bolder move to defend (4) against our argument: The accessibility relation between worlds differs between times. The proposition p isn't true at all worlds, but only at all accessible worlds (this may or may not involve a denial of S5—S5 does not say that all worlds are accessible, but only that accessibility is an equivalence relation). And a world is only accessible if it includes the present time (or a counterpart to it?). This has the implausible consequence that what is metaphysically possible changes with time. For instance, if in w the time sequence comes to an end with 2011, then the proposition that w is actual was possible in 2011, but is no longer possible. But it's implausible that what is metaphysically possible changes with time.

If this is right, then the presentist should embrace (3). But is (3) plausible? Do we really live in different worlds at different times?

The presentist's other move is simply to abandon talking about worlds, and instead talk about, say, abstract times (in the Crisp sense).