Alex Pruss recently posted an interesting new objection to alethic openness at his blog. His argument is not that long, so for convenience I’ll reproduce it here.
According to open future views, the proposition that in 2015 a fair and indeterministic coin lands heads has some probability but is not true. However, that proposition is apt to become true in 2015. So the probability of the proposition isn’t the same as the probability of the proposition being true, since it’s certainly not true now, but might well become true in 2015.
So far so good (or bad). Suppose God promises you that from 2015 onward, every year, a fair and indeterministic coin will be tossed. Now let Q be the proposition that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads. Now note that on open future views Q can never possibly become true. For on any date, the proposition requires for its truth that there will be infinitely fair and indeterministic heads results still past that date, and on open future views a proposition that requires an undetermined future event won’t be true.
So, open future views have to say that it’s impossible for Q to ever be true. But a proposition such that it’s impossible for it ever to be true should get probability zero. But the probability that of the infinitely many coin tosses, infinitely many will be heads is 1 according to classical probability theory. So open future views should be rejected.
In response, the first point to make is that Alex’s talk of propositions having a probability needs some refinement, for propositions in and of themselves don’t have probabilities because propositions don’t in and of themselves define the reference class or possibility space in relation to which probabilities have to be defined. Thus, we can’t say what the probability of <2 + 2 = 4> is until we’re given more information. We could be talking about the probability of <2 + 2 = 4>’s existing, or of its coming to exist, of its being true, or of its becoming true, or of its being believed, etc. What Alex seems to mean by the probability of a proposition is the probability of its either being or becoming true.
The second point is that we should get clear about just what sorts or probabilities we’re talking about. Among other things, probabilities can be construed as relative frequencies, degrees of belief (credences), or as propensities (chances). In the case at hand, since we’re interested in whether a proposition is or could become true, not whether or to what extent it is believed, we can set credences aside. Furthermore, since we’re not interested in how often a proposition is or becomes true, but simply is the question of whether it is or becomes true, we can set frequencies aside. So the probabilities that Alex has in mind are chances, or single-case objective probabilities.
These clarifications in mind, let’s restate Alex’s argument:
Suppose God promises that every year, from 2015 onward without end, at least one fair and indeterministic coin will be tossed. Since it is, as Alex assumes and I will grant, metaphysically impossible that God renege on a promise once the promise is made, God’s promise is enough to make it true that <Every year, from 2015 onward without end, a fair and indeterministic coin will be tossed>. If God has promised this, then He’s going to make sure it happens, one way or another. But God’s promise obviously isn’t enough to make it true that <Every year, from 2015 onward without end, a fair and indeterministic coin is tossed and lands heads.> Given indeterminism, only a coin’s actually landing heads every year can make that true. But here’s the problem: Since the series is, by stipulation, never completed, it seems that necessarily there will never come a time when the proposition has become true. And if it necessarily cannot become true, then it seems that it must have zero chance of becoming true. But it also seems that it should have a non-zero chance of becoming true because by hypothesis it is now genuinely possible, however improbable, that some indeterministic coin or other lands heads in every year from 2015 on, without end. So we have an apparent contradiction: Is the chance of the proposition’s becoming true zero or non-zero? (Alex goes on to note that given infinitely many such coin tosses over infinitely many years there should be infinitely many heads, but this isn’t really germane to the question at hand except as support for the claim that the chance of the proposition’s becoming true is non-zero. There could be an infinite number of heads over an infinite number of years without it’s being true that at least one coin lands heads in every year.)
What should open futurists say about this? I think there are at least three options:
(1) Argue that the scenario is metaphysically impossible. There is literally zero chance that God would make such a promise.
(2) Argue that propositions that cannot become true (because they pertain to the resolutions of future contingents in the infinite future) need not have a zero chance of becoming true, but may rather have an infinitesimal (and thus non-zero) chance of becoming true.
(3) Argue that the proposition in question, viz., <Every year, from 2015 onward without end, a fair and indeterministic coin is tossed and lands heads> is ill-defined and so isn’t really a proposition or even a truth-bearer.
Of these, I think (1) and (2) are the most promising. As for (3), I see nothing obviously problematic with the proposition in question. It’s not self-referential. It’s not vague. While there is ambiguity in that “a fair and indeterministic coin” could refer to either a single fair coin that is tossed in each successive year or to some-fair-coin-or-other, I don’t see why the argument couldn’t work on either interpretation. In other words, the ambiguity isn’t debilitating. Is there some other type of semantic pathology that could get (3) off the ground? Perhaps, but we need to be told what that is and why it’s a problem. I don’t see any promising candidates.
(1) strikes me a very plausible line to take. The example is obviously very artificial and there’s no reason to think God would or even could have any interest in making such a trivial promise. The downside to this response is that it doesn’t readily generalize. Hence, Alex may be able to reinstate his objection by modifying the example slightly. In fact, he does just this in the same blog post by giving a second argument that runs along similar lines:
Here’s another argument in the same vein. Suppose I know I will have an eternal afterlife, and I promise you that I will freely pray for you every day, ad infinitum, starting November 1, 2014. On open future views, the object of my promise is a proposition that can never be true. But it’s clearly a bad thing to promise something that can never be true. Yet what I promised wasn’t a bad thing to promise. So open future views are false.
One might even have the direct intuition that one could keep the promise. That intuition is incompatible with open future views.
The example in this second argument involves a promise that one might conceivably have an interest in making, so the argument can’t be deflected as above. Of course, with the change in example come other possibilities for rejoinder. For example, one might argue that in a heavenly setting one’s free will on the matter is effectively settled upon making the initial promise. Hence, there need be no more indeterminism after that point. The truth of the proposition can be grounded in the promise in the same way that in the first example God’s promise is enough to make it true that <Every year, from 2015 onward without end, a fair and indeterministic coin will be tossed>.
Now it may be that any anti-alethic openness example that Alex or someone else might come with can be given a plausible ad hoc rebuttal of this sort. But that’s a tenuous line to take. It would be far better to have a more general response, one that can be applied with appropriate adjustments to a wide range of such examples. I think response (2) points the way toward a general solution.
Let Q stand for the proposition <Every year from 2015 on without end at least one fair coin is tossed in indeterministic circumstances and lands heads>.
Given that there is at least one causally possible future in which this occurs, and let’s stipulate that this is so, it follows that the chance of Q’s becoming true must be non-zero. (It would only be zero if there were no casually possible futures in which the sequence described in Q obtains, but by hypothesis that’s not the case.) This chance, however, while non-zero can’t have any finite value because the limit of the odds of such a sequence of heads obtaining is zero. So the chance must be infinitesimal, i.e., non-zero but smaller than any finite number. (For anyone who might be worried about infinitesimals, let me just say that they are now rigorously defined mathematical quantities. See here for details.)
So far so good, but we now have to explain in what sense it is necessary that Q not become true (because it always depends on unresolved future contingents). Here’s the problem: To get the contradiction that he wants, Alex has to conflate two different ways of thinking about the future. When we say that the chance of Q’s becoming true is non-zero, as explained in the preceding paragraph, we are thinking of the future as a completed totality, as a complete possible future. But when Alex argues that the chance of Q’s becoming true is zero, he’s thinking of the future as an incomplete totality, indeed, as a potentially infinite series. For any finite number of years, Y, after 2015 the chance of Q’s becoming true by 2015 + Y, is necessarily zero because by hypothesis there will always be future unresolved coin tosses that could falsify Q. But now the semblance of contradiction vanishes, for the following two claims are not inconsistent:
- The chance of Q’s becoming true is non-zero.
- The chance of Q’s becoming true by any finite number of years after 2015 is zero.
A similar distinction solves Alex’s second argument, for when the future is conceived of as a completed totality the object of my promise to freely pray for you everyday can become true—there is a causally possible future in which I do just that. Hence, like (1), the chance of that proposition’s becoming true is non-zero.