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.

Alan:

Thank you Alan for your careful attention. I am very sorry: I screwed up my original formulation. When I saw what you quoted, I at first thought that there was something wrong with the quotation, and then realized that the fault was mine. Here’s what I should have said (meant to have said?): “let Q be the proposition that there are infinitely many years starting with 2015 during each of which a fair and indeterministic coin lands heads”.

On this formulation, P(Q)=1, or at worst P(Q)=1-infinitesimal. So your solution (2) won’t work. Solution (2) worked against my original misformulation.

I think option (1) is unpromising (pun not intended). Consider this intuitive argument. Every week for eternity in heaven you will have a free choice whether to sing your (then) third-most-favorite hymn. The chance that you will sing that hymn in any given future week in heaven is pretty high, I expect, but conservatively let’s say that it’s at least 1/100, regardless of what happened in the preceding weeks (if you sung that hymn a lot, your probability will be low, I guess, but not lower than 1/100). Let Q be the proposition that you sing your third-most-favorite hymn on infinitely many weeks in heaven. As long as in each future week the chance of your singing your third-most-favorite is at least 1/100, regardless of what happened in the preceding weeks, it’s a theorem of classical probability that P(Q)=1. (And it’s pretty intuitive. In any sequence of N years, you’d be pretty confident that you sing that hymn at least around N/100 times. So in a sequence of infinitely many years, you probably sang it infinitely often.)

Thanks for the reply, Alex!

If you look closely at my blog post above, I think you’ll see that I anticipate your reformulation of Q. Here you propose that Q be the proposition that there are infinitely many years starting with 2015 during each of which a fair and indeterministic coin lands heads, and you claim that, so defined, P(Q)=1 based on the law of large numbers.

But this can’t be right. I grant that by the law of large numbers that if there’s an infinite number of fair coin tosses then there will (with a chance infinitesimally close to 1) be an infinite number of heads. But Q, as you stipulate it in your reply, doesn’t just imply an infinite number of heads. It also implies that those heads will satisfy a certain distribution requirement, namely, that there be heads *in every year*. But that doesn’t follow from the law of large numbers.

Moreover, given the distribution requirement, rather than P(Q)=1 it should intuitively be vanishingly close to zero, for no matter how many times a fair coin is tossed in a given year there is always a non-zero chance of no coin’s landing heads in that year. Given that successive tosses are independent, P(Q) is the product of an infinite number of quantities less than one, i.e., P(Q) = P(1 – no heads in 2015) * P(1 – no heads in 2016) * … etc., and so should have a limit of zero.

You go on, however, to suggest another, very different sort of reformulation, where Q now just says that some type of indeterministic event (e.g., my singing my third favorite hymn in heaven) happens an infinite number of times, with no stipulation that such an event-type follows any particular distribution requirement (e.g., happening every year). In this case, if that type of indeterministic event is *guaranteed* to happen infinitely often then of course P(Q) *virtually equals* 1. But the open futurist can happily agree with that. (I wouldn’t say that the chance strictly *equals* 1 since by hypothesis there are causally possible futures in which the event never occurs, or occurs only finitely many times, but those futures are infinitely unlikely.) That P(Q) virtually equals 1 depends only on whatever it is that *guarantees* that that event-type happens infinitely many times (plus the law of large numbers). So I don’t see any reason the open futurist should feel compelled to say that P(Q) must be significantly other than 1 in this case. Indeed, all that matters for the open futurist is that in nearly all causally possible futures the event-type occurs an infinite number of times.

Alan:

“But Q, as you stipulate it in your reply, doesn’t just imply an infinite number of heads. It also implies that those heads will satisfy a certain distribution requirement, namely, that there be heads *in every year*. But that doesn’t follow from the law of large numbers.”

I don’t see how my formulation implies that there will be heads in every year. Q says “that there are infinitely many years starting with 2015 during each of which a fair and indeterministic coin lands heads”. This does not mean that there is heads in every year. I did not say that the infinitely many years are consecutive, just that there is some set of infinitely many years. I guess what I said could be interpreted to mean that 2015 is a year with a heads toss. I didn’t mean that. All I meant was that the set of years starting with 2015 during which such a coin lands heads is an infinite set.

As for my second example, you write: “In this case, if that type of indeterministic event is *guaranteed* to happen infinitely often then of course P(Q) *virtually equals* 1.” But it’s not *guaranteed* to happen. You could just never sing that hymn. That’s within your freedom. It’s just very, very unlikely–zero or infinitesimal probability–that that’s what would happen.

Alex:Q says “that there are infinitely many years starting with 2015 during each of which a fair and indeterministic coin lands heads”. This does not mean that there is heads in every year.Alan:It sure seems that way to me, Alex. When you say “years … during each of which” you are directly implying that in each year at least one fair and indeterministic coin lands heads.Alex:I did not say that the infinitely many years are consecutive, just that there is some set of infinitely many years.Alan:I disagree. If in “each” year from 2015 on a coin lands head, then we have a consecutive sequence of years (2015, 2016, 2017, …) during each of which a coin lands heads.Alex:All I meant was that the set of years starting with 2015 during which such a coin lands heads is an infinite set.Alan:Alright, but this is quite different than what you said previously. Now you’re saying that there’s an infinite number of years sometime during which at least one indeterministic coin (“a coin”) lands heads. Let Q’ be “An indeterministic coin lands heads sometime after 2014.” The chance of Q` becoming true is equal to the ratio of causally possible futures in which it occurs to all causally possible futures. Given that coin flips are often indeterministic and that a great many coins are likely to be flipped within the next several years, P(Q`) should be very close to 1.Alex:As for my second example, … it’s not *guaranteed* to happen. You could just never sing that hymn. That’s within your freedom. It’s just very, very unlikely–zero or infinitesimal probability–that that’s what would happen.Alan:Fair enough. There are two different event-types at work in this example which I should have distinguished. First, there is the event-type E1, my freely singing *a* hymn during any given heavenly week. Second, there is the more specific event-type E2, my freely singing *my then third-favorite hymn* during any given heavenly week. Your example requires that there be an infinite number of occasions on which E2 occurs for, as you put it, Q is “the proposition that you sing your third-most-favorite hymn on infinitely many weeks in heaven.” Now, granted, nothing guarantees that Q ever becomes true. Indeed, nothing guarantees that I freely sing any hymns at all during any given heavenly week, much less my then third-favorite hymn. But as long as there are causally possible futures in which I do so, P(Q) is non-zero. Now the law of large numbers (LLN), you say, implies that P(Q) should be not just non-zero, but very close to 1. But that’s only true *if* there is a very high chance (I won’t say a “guarantee”) of E1 events occurring an infinite number of times. Let’s call that chance C1. The chance of E2 events occurring an infinite number of times, call it C2, can’t be higher than C1. Now let’s *stipulate* that C1 is very high. It then follows from LLN that C2 is also very high. My point is simply that if there are conditions in place that make C1 very high (and thus that, by LLN, make C2 very high), the open futurist can base a high chance for Q *on those conditions*. He need not base the chance on infinitely distant indeterministic events.In general, no matter what example you may give, the open futurist is going to approach it by saying that chances are equal to ratios of causally possible futures (CPFs). If a chance is very high, then the relevant event-type occurs in all or nearly all CPFs. If a chance is very low, then it fails to occur in all or nearly all CPFs. And so on. (Of course, some mathematical nuance is required when the ratio involves infinite quantities.) In every case, however, the chance is grounded in whatever causal or metaphysical factors determine which futures are causally possible and which aren’t.