The Grim Reaper Paradox and the Kalām Argument
A Pruss' argument
Alexander Pruss and Robert Koons1 have established the Grim Reaper paradox as an argument in favor of Kalām’s argument. The paradox has various versions, but it runs like this:
There are infinite Grim Reapers that have a double function: first, to check the state’s victim’s life; second, if the victim was alive at Tn, a Grim Reaper kills him instantaneously. Tn is equaled to the noon, and Tn-1 is equal to Tn - 1/2 Tn (=11:30), Tn-2 is equal to Tn-1 - 1/2 Tn-1, and so on. Now, I will establish a setting. Fred, the victim, is alive at noon. Fred was alive at Tn, so a Grim Reaper (GR 1) kills him. However, Fred was also alive at 11:30 (Tn-1), so another Grim Reaper (GR 2) kills him before GR 1. But again, Fred was alive at 11:15 (Tn-2), and another Grim Reaper (GR 3) kills him instantaneously, but he was alive at 11:7 (Tn-3), therefore, another Grim Reaper (GR 4) has to kill him before. The same happens time and time again.
Here’s the paradox: Fred is alive or dead? He is alive and dead at the same time. Why? Certainly, Fred has to be dead at noon, because a Grim Reaper (Gr 1) killed Fred, however, at the same time GR 1 didn’t kill him. The same can be said of the prior (or posterior, depending on how you see) cases. The problem is that Fred did not die, but at the same time did actually die. If none of the Grim Reapers did kill him, Fred will be alive at 12:00. But that’s impossible, at least one of them has to kill him. But again, they did not kill him. So, at 12:00, Fred is alive and dead. If he is dead at 12:00, at least one of the Grim Reapers has to kill him.
So, another question arises: which of the Grim Reapers kills Fred? To kill Fred, GR 1 has to check if he is alive at that moment, he is alive, GR 1 kills him instantaneously. But, that does not occur, namely, the GR 1 does not kill him. And so on with other cases. Thus, no one of the Grim Reapers kills Fred. At all moments, Fred is alive and dead.
Fine, now, how does the GR Paradox relate to the Kalām argument? Pruss argues (in “From the Grim Reaper paradox to the Kalaam argument”2) in this way:
If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible.
If Hilbert's Hotel were possible, the GR Paradox could happen.
The GR Paradox cannot happen.
Therefore, there cannot be a backwards infinite sequence of events.
If we remember Hilbert’s Hotel, we recognize that the hotel is an actual infinite, the reason is that all rooms are infinite, at the same time and sense. So, how does relate a backward infinite sequence of events to Hilbert’s Hotel? First, it is impossible that a backward infinite sequence of events was an actual infinite: an actual infinite is already formed, it does not need to be formed. All the members of an actual infinite set exist right now. While a backward infinite sequence of events is forming, therefore, at least, it is a potential infinite: it has no beginning but has a final element, namely, the present moment, and it is formed over time.3 However, this objection doesn’t apply to Pruss’ argument. He doesn’t say that a backward infinite sequence of events entails an actual infinite.
That which he says is: if a backwards infinite sequence of events does exist, Hilbert’s Hotel would be possible. Also, he gives an argument for the first premise:
Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result.
So, in each of them (infinite sequence of events), says Pruss, a hotel room is created. As the number of rooms is infinite, and the rooms do not destroy, Hilbert’s Hotel is formed. Hilbert’s Hotel is an actual infinite. Thus, thanks to the infinite sequence of events, an actual infinite does exist. Someone with a good eye could say: “Ok, but, according to the prior objection, it is impossible that an actual infinite can be formed, and that argument speaks that an actual infinite is formed, the objection has to function with it as long as it entails an actual infinite.” Now, I will examine this objection, relating it to this argument for the first premise.
Is the Hilbert’s Hotel possible?
But first, I will review the objection, applying it to a prior argument that I reviewed in another post. That argument goes like this: a beginningless series of events in time entails an actual infinite because the infinity cannot be traversed, successive intervals of time (with an infinite length) cannot be traversed; in order to be traversed, a series needs two extremes, namely, a beginning and final. A beginningless series of events in time has no beginning, thus, from eternity exists a series of events in time infinite. The conclusion of the argument is that entails an actual infinite because the present will not be and time cannot traverse a backward series of infinite events in time.
Applying the objection to it, it says that an actual infinite cannot be formed, due to an actual infinite, so that to be a true infinity, needs to be already with its all members, an infinity with a first and final element cannot be an actual infinite. Because an actual infinite has an infinite number of members. In addition to this, a beginningless series of events in time occurs in time, and it tries to reach the final element. All infinite events in time, each of them, occurs in time: one follows after one. So, an actual infinite cannot be a beginningless series of events in time, its all members are not formed yet.
Now it’s time to apply the objection to Pruss’ argument. It says that if a backward infinite series of events in time could be, Hilbert’s Hotel would be possible. Given that the backward succession of events is infinite, the amount of time is also infinite. Hilbert’s Hotel has an infinity of rooms, they’re countless. With a specific amount of time, perfectly someone can create a hotel room.
So, with an infinite amount of time, perfectly an infinite number of persons, through time, can create an infinite number of hotel rooms. If it’s possible that infinite persons, through time or inside the same time frame, can create infinite numbers of hotel rooms, at some moment of the infinite time, Hilbert’s Hotel would be possible.
By “at some moment of the infinite time” I refer to one moment of the backward succession of infinite events. There is an infinite amount of time, I don’t see as an impossible action that infinite persons can create an infinite number of hotel rooms with that amount of time.
At a succession of infinite events (or infinite succession of events), as it has an infinite amount of time, and if a hotel room can be created with a specific amount of time, it logically follows that with infinite time, there could be infinite hotel rooms. Also, at an infinite succession of events in time, because of it does exist for eternity, it’s not inconceivable that such a hotel would exist for eternity. Perfectly, this hotel can exist for eternity, because an infinite time de facto exists (and infinitely many parts too), so, it’s not impossible such a hotel can exist for eternity. Thus, such a hotel would not be impossible, except if there is a backward finite succession of events in time.
The objection, again, does not apply to it. Since Hilbert’s Hotel does exist for eternity, it can be at whatever moment of the infinite time (and, therefore, with infinitely many parts). I don’t know what a moment can be, we are dealing with infinity. But if such a hotel is possible to exist thanks to a backward infinite succession of events, if it could be created, and if all that I have said is true, this succession of events does not (and couldn’t) exist.
Also, all this seems to me counter-intuitive, but, at the same time, it seems that it is not impossible if there could be a backward infinite succession of events. If Hilbert’s Hotel does exist for eternity, an actual infinite seems to exist too. Thus, a beginningless series of infinite events in time entails an actual infinite, that is, Hilbert’s Hotel.4
Therefore, the first premise is true and that argument too (besides the fact that breaks that objection). This premise is which holds the whole argument, the reason is that if it isn’t true, the second premise isn’t either, and so on with the other premises. Anyway, the third premise can be true independently of the prior premises,5 but, because of the argument is deductive, the first needs to be true for the argument to be true. Having said that, I will examine the second premise.
How do these paradoxes relate?
Pruss deploys an argument in favor of the second premise:
Argument for (2): If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes. And that would result in the GR Paradox.
This argument seems to be so specific or tight, nevertheless, even if it is, that doesn’t make it invalid. Instead of that, the important thing here is if that argument is incoherent, thus, the argument for (2) would not support the main argument, and for that, need to argue an objection against it or to show a contradiction.
So, if Hilbert’s Hotel would be possible, thus, the GR Paradox is also possible, says the argument. The reason for that, argues Pruss, is that each room of this hotel, it’s a factory of Grim Reapers. Pruss here shows that exists an entailment of Hilbert’s Hotel to GR Paradox. How is that? If we suppose that it’s possible that in each of the hotel rooms and the staff in each room could set a GR to go off at Tn, and so on, why would be impossible the GR Paradox? This is not controversial, because it’s a logical conclusion.
I would like to make a double distinction: 1) direct implication; 2) indirect implication. The first type would be Pruss’ argument, that is, x entails y (the HH entails the GR Paradox directly). The second type is basically an existing relation (or analogy) between two things that share a common element or various common elements, and because of that relation, if one would be possible, another one would be too. For the second type, I will deploy an argument for (2):
The Hilbert’s Hotel (HH) is an actual infinite (AI).
If it can exist at the backward infinite succession of events, an AI would exist.
If it’s possible that an AI can exist in those conditions, thus, an AI is not impossible that exist.
The GR Paradox is an AI.
4.1. Given that it’s an AI, if an AI would be possible in the same conditions that the HH, the GR Paradox would too.
So, it can exist in the same circumstances.
Now, I have a reason for (1), (2), and (3) premises, the possibility of HH at the backward infinite succession of events, that I gave above. If that reason is true, it’s possible the existence of an actual infinite. So, the only thing that lacks the argument is if the GR Paradox is an actual infinite.
I would like to appeal at the same reason that I used to HH: if it has an infinite number of members, and if they are existing, it is an actual infinite. What is the GR Paradox? It tells us that an infinite number of Grim Reapers are scheduled to go off for Fred, and then, they will try to kill him. It seems to me that this point is not problematic, and it’s not hard to see that the GR Paradox is an actual infinite. In itself considered, the “members” of the set (Grim Reapers) exist at the same time.
Is it justified the step from the fourth step to the fifth? I consider yes. If we take (3), the mere possibility of an AI in those circumstances, and if it’s true, all AI can exist. If it’s true that the GR Paradox is an actual infinite, whatever that it contains, they’re (the set’s members) infinite in number, and not merely a potential infinite. This is, if it’s possible in a backward infinite succession of events, it necessarily exists from eternity (to avoid the objection above mentioned). Therefore, given that it’s an AI, and for (3), it would exist, equal to the HH.
In this way, and if all this is true, we prove that Hilbert’s Hotel entails the GR Paradox through two arguments. Now, the following is purely going down. Given the nature of the GR Paradox, and the absurdities that entail, necessarily it’s impossible. (3) of the central argument is not problematic.
So, given that it’s a deductive argument, the conclusion necessarily follows. Because of the first premise of Pruss’s argument, that a backward infinite succession of events entails Hilbert’s Hotel, and that it also entails the GR Paradox, if this is not possible, the other also is not. If P is, also X; if X is, also Y; if Y is not, also X, and so on. It’s basically a Modus Tollendo Tollens.
Conclusion and further usage of the GR Paradox
In conclusion, this argument seems (to me, at least) not to be wrong. However, the point of view of Pruss has shifted (this he comments on in the same post, in a commentary), and now he says that
I am happy with the possibility of Hilbert's Hotel. The difficulty lies not with a HH in general, but a HH that lies as a whole in the causal history of an event (in this case, the end of the life of the person who is to be killed by the reapers). The problem isn't with an actual infinite as such, as with an actual infinity of causes that causally impinge on one thing [my italics].
He now understands the problem, not in the possibility of an actual infinite, but instead, in the context of causal infinitism. The paradox settles down on the possibility (or impossibility) of a causal history of an event, and in that sense, he uses the paradox. For instance, in Infinity, Causation, and Paradox, he addresses several paradoxes of infinity, and he argues those paradoxes can be resolved for the causal finitism.
But, for now, that’s all.
Koons, R. C. (2014). A New Kalam Argument: Revenge of the Grim Reaper. Noûs, 48(2), 256–267. https://doi.org/10.1111/j.1468-0068.2012.00858.x; Pruss. A. (October 2, 2009). From the Grim Reaper paradox to the Kalaam argument. Alexander Pruss’s Blog. https://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html; Pruss, A. Causal Finitism and the Kalam argument. Available at http://alexanderpruss.com/papers/kalaam.pdf
For now, I have an interest in that article, probably in the future, I will review another Pruss’s argument (and Koons’s argument) from the Grim Reaper Paradox.
This objection, however, only applies to Craig’s first philosophical argument (cf. 2008. Reasonable Faith: Christian Faith and Apologetics (3.a ed.). Crossway Books.), specifically, the second premise. In that premise, Craig says that a beginningles series of events in time entails an actual infinite. (Thanks to that objection, it seems to me that doesn’t entail). But it does not finish with that argument: there could be other contexts where the argument functions very well to demonstrate that a beginningless series of events in time is impossible. In addition to that, we will notice that Pruss’s argument reinforces this Craig’s argument, and avoids this objection.
This argument, on its own, works to support the Kalām argument, and, more specifically, the first philosophical argument for the impossible existence of an actual infinite.
The Grim Reaper Paradox can be used in other cases and not necessarily (and only exclusively) in this argument. That same thing Pruss has made with other uses (namely, arguments) of the paradox without appeal to Hilbert’s Hotel.