The paradox of dry eternity

0. Introduction

Like many people, I am participating in ‘dry January’, meaning that I am not drinking any alcohol during the month of January. I’m also thinking about the Grim Reaper paradox, and have spent much of the year thinking about the infinite future debate between Morriston and Craig. Interestingly, all of these things have come together, in a paradox I shall now dub the ‘dry eternity’ paradox.

Part of the inspiration for this comes from a paper I read recently by Yishai Cohen, called ‘The Endless Future: a Persistent Thorn in the Kalam Cosmological Argument‘(2015). In that, Cohen agrees with Morriston that if a beginningless past is an actual infinite, then so is an endless future, and thus if a beginningless past is impossible for being an actual infinite, so is an endless future. Cohen also argues that if the grim reaper argument shows that the past has to be finite, then a parallel version shows that the future must have an end as well. His argument is critiqued by Jacobus Erasmus in a paper called Cohen on the Kalam Cosmological Argument (2016). Erasmus’ rebuttal is that Cohen’s version of the grim reaper argument presupposes that it is possible for God to actualise an ungrounded causal chain, which can be plausibly denied.

I think that the ‘dry eternity’ paradox escapes Erasmus’ reply.

1. 1. Two versions of the Grim Reaper paradox

Here is how Erasmus sets up the grim reaper paradox:

Suppose that the temporal series of past events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each past moment in time, a unique Reaper was assigned to issue a death warrant iff no previous Reaper had already issued a death warrant. (Cohen on the Kalam Cosmological Argument, p. 52)

This results in a contradictory state of affairs. Firstly, for all times t_n, there must have been a warrant issued prior to t_n. That’s because if no warrant had been issued by t_n-1, then the reaper at t_n-1 would issue their warrant (resulting in a warrant going out prior to t_n).

But, this same reasoning also applies to t_n-1 itself, giving us the contrary proposition. That’s because we can also say that the warrant won’t be issued at t_n-1, because if it had not been issued by t_n-2 it would have been issued by the t_n-2 reaper (i.e. before t_n-1).

Thus, we have both that the warrant must have been issued at some time prior to t_n, but also that there is no time prior to t_n at which it could be issued. Contradiction.

Cohen applies this to the endless future. All we need to do is change the relevant tenses in Erasmus’ quote from above to get the following:

Suppose that the temporal series of future events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each future moment in time, a unique Reaper is assigned to issue a death warrant iff no future Reaper will issue a death warrant.

But now we can derive a mirror image contradiction. Suppose that the reaper at t₀ checks to see if any future reapers will issue warrants or not. At this point there are two options:

i) no future reaper will issue a warrant (in which case the t₀ reaper issues theirs)

ii) some future reaper will issue a warrant (in which case the t₀ reaper does nothing)

Suppose that at t₀ no future reaper will issue their warrant, meaning that the reaper at t₀ can issue theirs. If it were the case that no future reaper issues a warrant at t₀, then, in particular, it is also true that at t₁ no future reaper will issue a warrant (consider: if it it true today that I will never drink again, then it will also be true tomorrow that I will never drink again). But if it is true at t₁ that no future reaper will issue a warrant, this would mean that the reaper at t₁ does issue their warrant! And, plainly enough, the reaper at t₁ is in the future of the reaper at t₀. So if, at t₀, no future reaper will issue a warrant, then some future reaper (such as the one at t₁) will issue a warrant! Contradiction.

Let’s take the other horn. Suppose at t₀ some future reaper will issue their warrant, meaning that the reaper at t₀ can stand down. Let’s suppose it is the reaper at t₁. Then we are right back to the beginning of the first horn again. For the reaper at t₁ will only issue their warrant if none of the future ones will. But if it is true at t₁ that none of the future ones will, then this is also true at t₂ as well, resulting in that reaper issuing their warrant which in turn brings about another contradiction exactly like the one from above.

Cohen discusses two objections that Koons posed to him in correspondence. The first of these is that in Koons’ version of the paradox, reapers are sensitive to what past reapers have done, but in Cohen’s version they have to be sensitive to what future reapers will do; yet it isn’t possible to have causal sensitivity to future events in the same way as to past ones. The reply Cohen makes to this is that an omniscient God could communicate the future to the reapers such that they know what the others will do, thus overcoming this causal asymmetry. Koons’ second point is that it isn’t possible for God to create beings who embody his omniscience. Even if that is true, the reapers themselves do not have to be omniscient (and can be quite ignorant of, say, how many coins I have in my pocket), just so long as God ensures that they know the behaviour of future reapers. In addition, Cohen points out that Koons’ reason for thinking that the reapers cannot embody omniscience has to do with avoiding causal loops, but it is not clear that there are any causal loops as such in this story (the behaviour of reaper n+1 does not depend on the behaviour of reaper n, etc). Thus it is far from clear that Koons has a successful reply here. One could avoid this by denying the possibility of an omniscient being that knows the future and can communicate it to reapers, of course, but a theist (in particular a Christian theist) will be unlikely to pick that option.

2. Erasmus’ objection

Erasmus’ objection comes at this from a different angle. He says that Cohen’s version of the grim reaper paradox (GRP) requires the following two principles:

K1. It is possible for God to predetermine an endless future

K2. It is possible for God to actualise an ungrounded causal chain

An ‘ungrounded causal chain’ “has a non-well founded relation (xRy, zRx, zRv, wRv, … ) because the chain lacks a first cause” (Erasmus, 2016, p 53). The behaviour of the reaper at t₀ is determined by (or grounded in) the behaviour of reapers that are in its future. But the behaviour of the reapers in its future, such as the one at at t₁, are themselves determined by (or grounded in) the behaviour of reapers in the future of them as well. Thus there is no ‘first cause’, or grounding, for the behaviour of the reapers. Let us suppose that this is an ungrounded causal chain, and that it is also (metaphysically?) impossible for God to actualise such a causal chain.

He then goes on to show that K2 is doing all the work in generating the paradox because it also applies to ungrounded causal chains that are purely spatial in character. Here is his spatial version of the example:

For example, suppose that time had a beginning and has an end. Accordingly, the predetermined series of future events is finite. Suppose further that space is inhabited only by an actually infinite row of successive Grim Reapers such that (1) there is a first Reaper but no last Reaper, (2) each Reaper is located at a unique spatial point, and (3) all the Reapers are facing the same direction. Now, suppose that God has predetermined that, at noon tomorrow, each Reaper will swing his scythe iff no Reaper in front of him swings his scythe. Accordingly, the same contradiction as above will result at noon tomorrow, namely, regardless of whether the first Reaper swings his scythe, it is both true and false that some Reaper in front of the first Reaper swings his scythe. The contradiction disappears, however, if (K2) is false. (Ibid)

Erasmus’ conclusion then is that denying K1 is not enough to block the contradiction, as it reappears in the spatial case. But denying K2 blocks both contradictions, and as such K1 is not the offending assumption. In effect, he is saying that Cohen’s GRP doesn’t show that the future must have an end. Rather, it just shows that God cannot actualise ungrounded causal sequences.

3. The Dry Eternity Paradox

Now is time to present my version of the paradox that does works even if K2 is false. It does not require that God actualises any ungrounded causal sequences. All that it requires in addition to K1, is one additional assumption:

K3. God can act based on his (presently available) knowledge of future events.

Suppose God has decided to undertake an infinite version of dry January. That is, he has decided to stop drinking (say) holy water forever. However, he enjoys a drop of holy water (who doesn’t?), and wants to to have one final sip. Accordingly, he determines to obey the following rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, he will celebrate with his final drink. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a drink all day).

Again, we are caught in a dilemma:

Firstly, suppose that, at t₀, God consults his comprehensive knowledge of the future, and discovers that he never again drinks after t₀. He immediately downs a shot to celebrate (who wouldn’t?). But in that case, when he does his check the next day, at t₁, he then will (again) discover that he will never have another drink, and immediately pour himself a drink to celebrate! So even though he rewarded himself yesterday for never having another drink, he is having another drink! Contradiction.

On the other horn, suppose that, at t₀, God consults his comprehensive knowledge of the future, and discovers that he does indeed have a drink at some day after t₀. Accordingly, he doesn’t celebrate by having a drink on t₀. But in that case, there must be some future day at which he has a drink. Suppose it is t₁. In that case, it must be that at t₁ God will check to see if he will have any subsequent drinks, and find that he will not, resulting in him pouring the last drink. But now we are back at the start of the first horn, because his check at at t₂ will also reveal a dry eternity ahead, at which point he will reward himself with another final drink! Contradiction again.

So we clearly have the exact same paradox again. This time however, it is not clear that God has actualised an ungrounded causal chain. After all, at each day God knows the future, and can merely consult his own (presently available) knowledge to see what happens in the coming days. We can imagine him writing it all down in a big book and every day he consults the book. Whatever causal story that happens each day that he consults the book, it is not clear that it is an ungrounded causal chain.

4. Replies?

One might deny that God can check his own knowledge to see what he knows about the future and act on it (K3). This would be weird. Why can’t God do that? Does he not know what he foreknows? Is he repressing it? Can he not act on what he knows? He seems to act on his foreknowledge on most versions of theism (specifically any where he has a plan, or reveals the future in prophecy, etc). Denying K3 leaves only the most austere versions of deism, it seems to me. Christianity seems hard to reconcile with its denial in any case.

Objecting to the possibility of the book doesn’t help unless it is really an objection to God’s omniscience, which a theist probably isn’t going to opt for (apart from Open Theists). Denying the possibility of an omniscient being would avoid the paradox of dry eternity though.

One could avoid the problem by denying the possibility of an endless future. The whole point of Morriston’s original reply to Craig was to say that if the past must have a beginning, then the future must have an end. This would vindicate Morriston’s challenge against Craig. It would show that either time has both a beginning and an end, or no beginning and no end, but that there is no third option.

The seemingly only other target we can find is the rule that God undertakes to obey. Perhaps it is not a proper rule, and that somehow it isn’t possible for God to undertake to obey it. Yet this seems rather strange. Consider this similar rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, will celebrate with a chocolate bar. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a chocolate bar).

Nothing paradoxical follows from this rule. Obeying this rule means that God drinks every day up to the day when he has his final drink, after which he eats chocolate bars every day. God can obviously follow that rule.

But what could stop him from undertaking to follow the rule obtained by merely swapping out the word ‘chocolate bar’ with ‘have his final drink’? Of course, it would lead to contradiction if he were to go this route, and that is a reason to think that it is (somehow) metaphysically impossible for him to swap those words around and undertake to follow the resulting rule. On the other hand, that is just to say that this is one of the things that could be denied to avoid the paradox. It doesn’t motivate thinking that it is impossible. We could ad hoc postulate anything is metaphysically impossible to avoid any paradox.

Something has to go to avoid the paradox of dry eternity.

The Grim Reaper Paradox, and the original solution, part 1

0. Introduction

Robert Koons has published a version of the ‘Grim Reaper paradox‘, also popularised by Alexander Pruss. One of the assumptions that sets up the paradox must be abandoned in order to avoid the contradiction. Koons and Pruss’ conclusion is that the assumption that an infinite past is possible should go.

It turns out that there is a potential solution to the paradox which was proposed by its originator Jose Benerdete, and more recently brought out in a paper by John Hawthorne (2000). I’m going to explain that solution here, and then (hopefully) develop it to the Koons/Pruss version in a part 2.

1. The Grim Reaper paradox

The Grim Reaper paradox was first put forwards by Jose Benardete in his 1964 book ‘Infinity‘ (p. 259). Although the developer of the paradox, Benardete was a passionate defender of the actual infinite, announcing in the blurb on the cover of his book that:

“This book is an attack on finitism in all its forms … A metaphysics of the actual infinite is offered as the solution to the contemporary crisis in the foundations of mathematics”

It is perhaps a consequence of the clarity with which he engaged with the topic was that his examples have since been able to be used to argue for precisely the opposite thesis; that is, for finitism. This is how the grim reaper paradox has been used in the contemporary philosophy of religion debate, primarily by Alexander Pruss and Robert Koons, who argue for a version of finitism (‘causal finitism’).

Here is how Benardete states the paradox:

“A man is shot through the heart during the last half minute by A. B shoots him through the preceding 1/4 minute, C during the 1/8 minute before that, &c. ad infinitum. Assuming that each shot kills instantly (if the man were alive), the man must already be dead before each shot. Thus he cannot be said to have died of a bullet wound” (p. 259)

Koons describes this story as follows:

“The story leads quickly to a contradiction, on the assumption that Fred [i.e. A] does not die unless one of the Reapers kills him. At least one Grim Reaper must act, since if all of the Reapers whose numbers are greater than 1 do nothing, then Reaper #1 will act. However, it is impossible for any Grim Reaper to act, since, for any n, Grim Reaper #n cannot do so unless Fred survives until its assigned deadline at 1/2ⁿ seconds after midnight. It is impossible for Fred to survive that long, since Fred’s surviving until Reaper #n’s deadline entails that no Grim Reaper with a number larger than (n+2) has acted, but, in that case, Reaper #(n+1) must have acted.” (Does the Universe Have a Cause?, p. 4)

The contradiction lies in the fact that Fred will surely die before the end of the minute, but that also there is no Grim Reaper who will kill him. What this paradox seems to show, at least according to the finitists, is that there could not be finite duration of time (such as a minute) that is actually divided into infinitely many sub-regions.

2. The original solution, part 1: Benerdete.

Benardete did not draw the same conclusions as Pruss and Koons from the grim reaper paradox, and remained committed to the possibility of actually infinitely divided durations and lengths. Shortly after stating the paradox, he makes the following comments about it:

As to the dead man, although he did not die of any single bullet wound, his death was certainly caused by the infinite fusillade of shots. Here, again, although he is already dead prior to each shot, he remains alive at any assigned instant which is prior to them all. Thus he cannot be said to have died at any moment of time whatever! (Infinity, p. 260)

These brief comments offer only a hint of how to understand the response. The outline can be made out however.

Firstly, Fred is alive at every moment prior to all the shots. Yet, for each shot he is already dead before that one is fired. It follows from this that “he cannot be said to have died at any moment of time whatever!” We can spell this out by saying that although there is no first moment in which it is true that Fred is dead, there is a last moment at which Fred is alive. And this is to be expected. The series of bullets (reapers) is an open infinite sequence; while there is a last bullet at 12:01, there is an endless sequence of bullets ever closer to 12:00, and in particular there is no first bullet. Given the continuity of time, we can think of the transition from 12:00 to later times as a Dedekind cut on the real numbers:

In the above diagram, the left line is time as it approaches 12:00, during which Fred is alive. This is a closed set (which is what the square join indicates), in that 12:00 is a member of that set, but it is the final member. So Fred is alive at every time up to and including 12:00. However, the line on the right is an open set (which is what the curved join indicates). So Fred is dead at every time strictly after 12:00. This is what Benardete means by “he cannot be said to have died at any moment of time whatever”. There is no first moment at which he is dead (rather, there is just a last moment at which he is alive).

This much of the solution seems fairly straightforward. It is addressing the time of death, and we have the following information. When is Fred alive? All the way up to and including 12. When is he dead? At all times later than 12. If time is continuous and actually infinitely divisible, this is what it is like to transition from one state to another (it is like a Dedekind cut on the real number line).

The second bit is where the difficulty lies. It involves the cause of death. Benardete has it that no bullet (reaper) is the individual that causes the death of Fred (“he did not die of any single bullet wound”). Yet, the totality of all of them does cause Fred to die (“his death was certainly caused by the infinite fusillade of shots”). But how can it be that the totality of the bullets causes him to die, when none of them individually causes him to die?

3. The original solution, part 2: John Hawthorne

This is where Hawthorne comes in. His paper Before-Effect and Zeno Causality develops Benerdete’s solution, spelling out the principles in greater detail.

Hawthorne first considers the case of a ball rolling towards an open-infinite Zeno-sequence of walls. 2 miles away there is a wall; 1 and 1/2 miles away is another wall; 1 and 1/4 miles away is another wall; 1 and 1/8 miles away there is another wall, etc. Thus, there is an infinite sequence of walls, ever closer to the point that is exactly one mile away. There is no wall which is the ‘closest’ to the one mile point (which makes it an open sequence). Suppose the walls are impenetrable and cannot be knocked over (etc). The ball is rolled towards the walls. What happens as it arrives at the one mile mark? Hawthorne’s answer is as follows:

“The ball does not proceed beyond a mile and it does not hit a wall.” (p. 625)

We are puzzled by this combination partly because we have fuzzy intuitions about what happens when there is ‘contact’ between objects and open series in this sort of setting (where space is continuous and actually infinitely divided). But we can spell it out by specifying what we mean by ‘contact’ in such a way that it makes sense. Hawthorne calls ‘open-closed contact’ what happens when “A closed surface contacts an open surface insofar as there is no unoccupied space in between the two surfaces.” When the ball arrives at the one mile point, it has achieved open-closed contact, in the sense that the closed surface of the ball has no unoccupied space between it and the infinite series of walls. Here is what Hawthorne says:

Consider the fusion of walls. Call it Gordon. On reflection it is clear that the sphere contacts Gordon. Gordon has an open surface. When the ball stops proceeding at the one mile mark, there is no unoccupied space between the sphere and Gordon. Contact occurs …  So the ball is stopped by contact: The ball hits something, though the thing that it hits is not one of the walls. (p. 626)

If we find this puzzling still, Hawthorne has the diagnosis at hand. It is because we are assuming what he calls the ‘contact principle’, namely:

If y is the fusion of x’s and z contacts y, then z contacts one of the x’s.

He makes the final following comments about this principle:

That principle holds for the finite case. But it is false if the x’s are infinite in number. Once we are clear about this, there is no residual puzzle, nor anything further to learn about the wall case. It is clear what happens in worlds that satisfy the original description: At a mile, the ball makes contact with the fusion of walls, which is rigid and impenetrable. As a result, it does not proceed further. The ball does not, however, make contact with any wall. (Ibid)

It is weird that the ball can be stopped by making contact with the totality (or ‘fusion’) of all the walls even though it does not contact any particular wall,  but that is partly because we are taking an intuition which is applicable to the finite cases only, and trying to apply it to the infinite case. In finite cases, there will be a first wall, and contact would be defined in relation to that wall. However, this is precisely what we do not have in this case, and relying on an intuition that presupposes that there is one will lead us into trouble.

 

      4. From walls to reapers

When it comes to the case of the infinite sequence of bullets (or reapers), Hawthorne basically applies the same analysis. Instead of the ‘contact principle’, he identifies the analogous principle and calls it the ‘change principle’:

“If x is the fusion of y’s and y’s are individually capable only of producing effect e by undergoing change, then x cannot … produce effect c without undergoing change”

This principle is also true in the finite case but false for the infinite case. Each of the bullets (reapers) are only capable of producing the effect of killing Fred by undergoing change (by being shot, or by swinging their scythes, etc). The assumption is that this applies to the totality of bullets (reapers). And it does, in finite cases. If there were only 10 bullets, then the totality would have to change, in the sense that one of the bullets would have to be fired at Fred, for it to bring about its effect. Yet, in the case of the infinite sequence of bullets (reapers), this is not the case. The totality can bring about the effect even though none of its elements (or the totality itself) changes in any way.

Going right back to Koons, we can apply this to his comments. In the quote I gave of his from above, he begins by saying:

“The story leads quickly to a contradiction, on the assumption that Fred does not die unless one of the Reapers kills him.” (Emphasis added)

What Hawthorne’s approach questions is this assumption. It is false that one of the reapers kills Fred, but that doesn’t mean that the effect is not brought by the totality of reapers. In the infinite case at hand, that is what happens. Once again, the weird behaviour of the infinite confounds our intuitions. Yet, giving up this intuition saves the situation from contradiction, and thus avoids the paradox.

 

5. Conclusion

Of course, some will argue that a ball cannot be stopped by a bunch of walls unless one of those walls makes contact with the ball, and the move from finite to the infinite context should make no difference to this.

Hawthorn is sensitive to the seemingly radical nature of the conclusion, admitting that it is a “big metaphysical surprise”. He goes on to finish by saying that:

“The Contact Principle, in full generality, could be given up fairly readily on
reflection. The Change Principle has a rather deeper hold on us. It seems to us scarcely thinkable that mundane causal powers—say that of killing with a machete—could combine so as to logically entail the causal power of producing some effect without the agent of the effect undergoing change. Nevertheless, surprising as this may be, the Change Principle should be rejected. The diagnosis is complete. The logic of each case is very much in order. And our puzzlement has been traced in each case to some faulty principle relating fusions to parts. Once we discard those principles, we will have no problem in accepting the required conclusions about what happens in each case” (p. 630-631)
I intend to write a part 2 to this, where I apply Hawthorne’s analysis to the version of the Grim Reaper paradox in which the target is not the actually infinitely divided duration, but the actually infinite past.