### Tense Logic and the End of Time

Over the past four months I've been working off-and-on on a paper on tense logic, in which I argue against the common assumption (common, that is, in philosophy of time circles) that the mere fact that some event happens at time t is sufficient for it to have always been the case prior to t, that that event was going to happen at t. More precisely, the thesis I want to reject is

(1) □(∀p)(∀t)(∀u: u<t)(IS(p,t) → WAS(WILL(p,t),u))I like to call this the IS implies WAS(WILL) principle, or IIWW for short.

i.e., necessarily, for all propositions p, and for all times t and u such that u is prior to t, p is the case at t, then at u it was the case that p will be the case at t.

The main defense of (1) runs as follows: For any state of affairs S and time t, either S obtains at t or S does not obtain at t. One or the either must be true. Now jump back to an earlier time, u, and consider the truth value at u of the propositions "S will obtain at t" and "S will not obtain at t", where the future-tense is to be understood in such a way that it carries no modal force. Thus, "S will obtain at t" is supposed to mean merely "At future time t, S obtains." Since, it must be the case at t either that S obtains or that S does not obtain, and since "S will obtain at t" means merely "At future time t, S obtains" and "S will not obtain at t" means merely "At future time t, S does not obtain", it seems that either "S will obtain at t" or "S will not obtain at t" must be true at u, prior to t, and that which is true at u is determined solely by what obtains at t.

If the foregoing argument is sound, then (1) is correct. But is it sound? I don't think so. And I think I can show this by pointing to the possibility of time coming to an end. First of all, is it possible that time come to an end? I don't see why not. There's no obvious contradiction, and any argument that purported to demonstrate a contradiction would have to show that the mere fact that one event comes after another requires that there be a third event after that one, and so on. In other words, to show that there could not possibly be an end to time one would have to show that "A occurs after B" implies "Some event, C, occurs after B". That inference is certainly not valid as it stands, and I don't see any way to make it both valid and sound.

So let's suppose that it is possible that time come to an end. What bearing does this have on (1)? A lot. Consider our pair of propositions "S will obtain at t" and "S will not obtain at t". Let us suppose that the time is now t-minus 10 and that at precisely t-minus 5 time stops for good. In that case, there never exists a time t. Hence, S neither obtains nor does not obtain at t. Hence, both "S will obtain at t" and "S will not obtain at t" are false. And what that means is that those two propositions, even when interpreted in the strictly non-modal way that I've indicated, are not contradictories, but contraries.

Let me restate things to make my point clear. My argument does not depend on the assumption that time will come to an end, but only on the logical possibility that it come to an end. Once that is granted, (1) must be false. That it is now true that S will obtain at t requires not only that S obtain at future time t, but also that there be a future time t. Hence, the mere fact that S obtains at t does not suffice to ground the prior truth of the proposition "S will obtain at t". What we would also need to know is that as of that prior time it is not possible that there not be a future time t.

Okay, but if "S will obtain at t" and "S will not obtain at t" are contraries and thus can both be false, then what is true when both are false? What, in other words, is the third possibility? Simply this, that there is no future time t.

The significance of this argument for my purposes concerns the issues of whether the 'Peircean' or the 'Ockhamist' system of tense logic is to be preferred. As I discussed in a previous post, there is a prima facie problem for the Peircean, for it seems hard to deny that there are such propositions as the Ockhamist's non-modal propositions about the future. Given that there are such propositions, by the argument I gave on behalf of (1) above, it seems that the only way for the Peircean to reject (1) is to deny bivalence. So the Peircean seems to face a serious dilemma: Either (a) take the implausible tack of denying that Ockhamist-style propositions about the future are really propositions, or (b) take the implausible tack of denying the principle of bivalence. What I have now argued by appealing to the possibility of ending time is that this is a false dilemma. The Peircean can concede the Ockhamist his propositions and retain bivalence by showing that even on the Ockhamist's own terms 'will' and 'will not' are not contradictories, but contraries.

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