Pruss' grim reaper paradox
Philosopher and mathematician Alexander Pruss in 2008, and again in 2009, discussed on his blog the following paradox: Consider an infinite collection of grim reapers indexed by the positive integers n=1,2,..., where the nth grim reaper is scheduled to kill Fred at 11:00am + 1/n minutes, and where Fred's life is otherwise safe during that period. By hypothesis, he dies from the hand of a grim reaper, say the kth grim reaper. However the (k+1)th grim reaper visited Fred before the kth grim reaper, which means that Fred must already be dead by the time the kth grim reaper visits him. This is a contradiction, and we conclude that there is a logical error in the construction of this unusual and hypothetical situation.
Naturally we ask, what is the error? Well, there are at least two highly questionable assumptions at work. Most infamously, we have the notion that there can be infinitely many physical objects (such as grim reapers) in the universe. But we also must assume that it is possible to subdivide time into infinitely small increments. Moreover, we ought to proceed cautiously given the danger posed by the odd and unfamiliar character of the scenario that we may have unwittingly assumed other potentially controversial facts of which we are not presently aware.
It's an impressively clever paradox, and Pruss goes on to use it in quite a creative way: For as it turns out, one can reduce the strength of the afore-mentioned assumptions required by the paradox by additionally assuming the past-infinitude of time. Instead of directly assuming the possibility of an infinite collection of physical objects, we can posit merely infinite spacial and material resources. For in that case, it is possible that for each time t=-n, where n is a positive integer, a physical object is constructed. Then by the time t=0, an infinitude of objects, say the grim reapers and/or the rooms in Hilbert's Hotel, have been constructed. Furthermore, if we take on the assumption that time is arbitrarily divisible (as opposed to infinitely divisibile) then for each positive integer n we can posit an event at time t=-n causing an event (such as a grim reaper trying to kill Fred) to occur at t=1/n. With these lesser assumptions in hand, we can have a grim reaper sent out from every room in Hilbert's Hotel at the appropriate time to kill Fred as postulated by the GR paradox.
Pruss tries to advance this as an argument against the past-infinitude of time, but here I must disagree with him. Rather, it serves only as an argument against that certain combination of all three of the unverified assumptions it requires. Furthermore, I suggest that we may also need one further controversial assumption, that the duration of time it takes to kill Fred can be made arbitrarily short. For suppose it takes at least some fixed positive length e>0 of time (in minutes) to kill Fred. Then we can find an integer N such that n>N implies 1/n < e and hence we have infinitely many grim reapers killing Fred during the period between 11am and 11am + e minutes. In this case it is not obviously problematic to say that Fred died at the cumulative effect of all those grim reapers working to kill him between 11am and 11am + e minutes. Of course, this raises in turn the question of when precisely it becomes the case that Fred is dead. We would like to say that this at least is an instantaneous event, but can we be so sure? On the other hand, we don't necessarily need to say that Fred's death occurs instantaneously as long as we assume there is no fixed length of time e > 0 required to kill Fred. So perhaps it takes the nth reaper 1/n^2 minutes to kill Fred. In that case, there would be no overlap between possible killing events, i.e. the nth reaper would already be finished killing Fred by the time the (n-1)th reaper comes to visit.
At any rate, it seems clear that there are more assumptions at work in Pruss' argument than just the past-infinitude of time. I thank him very much for describing the paradox---if it is his own invention then he deserves some significant credit for that---and look forward to any further thoughts he has on the subject. Of course, I would be especially pleased if he happens to address some of the above concerns.
Naturally we ask, what is the error? Well, there are at least two highly questionable assumptions at work. Most infamously, we have the notion that there can be infinitely many physical objects (such as grim reapers) in the universe. But we also must assume that it is possible to subdivide time into infinitely small increments. Moreover, we ought to proceed cautiously given the danger posed by the odd and unfamiliar character of the scenario that we may have unwittingly assumed other potentially controversial facts of which we are not presently aware.
It's an impressively clever paradox, and Pruss goes on to use it in quite a creative way: For as it turns out, one can reduce the strength of the afore-mentioned assumptions required by the paradox by additionally assuming the past-infinitude of time. Instead of directly assuming the possibility of an infinite collection of physical objects, we can posit merely infinite spacial and material resources. For in that case, it is possible that for each time t=-n, where n is a positive integer, a physical object is constructed. Then by the time t=0, an infinitude of objects, say the grim reapers and/or the rooms in Hilbert's Hotel, have been constructed. Furthermore, if we take on the assumption that time is arbitrarily divisible (as opposed to infinitely divisibile) then for each positive integer n we can posit an event at time t=-n causing an event (such as a grim reaper trying to kill Fred) to occur at t=1/n. With these lesser assumptions in hand, we can have a grim reaper sent out from every room in Hilbert's Hotel at the appropriate time to kill Fred as postulated by the GR paradox.
Pruss tries to advance this as an argument against the past-infinitude of time, but here I must disagree with him. Rather, it serves only as an argument against that certain combination of all three of the unverified assumptions it requires. Furthermore, I suggest that we may also need one further controversial assumption, that the duration of time it takes to kill Fred can be made arbitrarily short. For suppose it takes at least some fixed positive length e>0 of time (in minutes) to kill Fred. Then we can find an integer N such that n>N implies 1/n < e and hence we have infinitely many grim reapers killing Fred during the period between 11am and 11am + e minutes. In this case it is not obviously problematic to say that Fred died at the cumulative effect of all those grim reapers working to kill him between 11am and 11am + e minutes. Of course, this raises in turn the question of when precisely it becomes the case that Fred is dead. We would like to say that this at least is an instantaneous event, but can we be so sure? On the other hand, we don't necessarily need to say that Fred's death occurs instantaneously as long as we assume there is no fixed length of time e > 0 required to kill Fred. So perhaps it takes the nth reaper 1/n^2 minutes to kill Fred. In that case, there would be no overlap between possible killing events, i.e. the nth reaper would already be finished killing Fred by the time the (n-1)th reaper comes to visit.
At any rate, it seems clear that there are more assumptions at work in Pruss' argument than just the past-infinitude of time. I thank him very much for describing the paradox---if it is his own invention then he deserves some significant credit for that---and look forward to any further thoughts he has on the subject. Of course, I would be especially pleased if he happens to address some of the above concerns.
Comments
"Most infamously, we have the notion that there can be infinitely many physical objects (such as grim reapers) in the universe." Pruss' argument surely doesn't depend on Grim Reapers being physical, though—for the sake of his paradox, couldn't they be immaterial, spiritual beings? (This is a thought experiment, after all.)
I like your paradox better than the GR paradox, yes, since it seems to avoid requiring us to divide time into arbitrarily small moments. In fact it might be better still to posit bell-ringers instead of blippers. That way the bell-ringers could be independently defined, whereas each blipper depends on infinitely many other blippers in its conceptual construction.
In the end though, it comes down I think to what models we're going to use. The bottom line---and I wish I would have made this point in the blog entry itself---is that an inconsistent model never forces us to regard any individual element of that model as itself inconsistent or incoherent. There doesn't have to be a "controversial assumption" which single-handedly causes the inconsistency. It could just be that the model is badly-constructed.
As for the immateriality of grim reapers, you are absolutely correct that Pruss' argument doesn't mention this or depend on it. However presumably he wants to reach an audience of primarily materialists. If he's going to assume the grim reapers are immaterial beings, then that's one whopper of a controversial assumption from the perspective of a materialist! Better to say that we just have an infinitude of material-spacial resources.
--Ben
It seems to me that the principle concerns at play here are theological and 'in house' so to speak. This is probably particularly the case because Catholics are necessarily committed to the B theory of time for doctrinal reasons (the commitment to the metaphysical simplicity of God, defined and proclaimed at the fourth lateran council and reaffirmed again at Vatican 1), which is precisely the doctrine with which Craig and other A theorists (all of whom are evangelical, which is no accident) take issue.
Moreover, even if one is some kind of materialist, I think it is hardly controversial to elucidate a paradox with a thought experiment which is logically possible. So long as the materialist can admit that an immaterial being is logically possible they can have no objection to the use of such beings in a thought experiment to demonstrate a paradox or contradiction. It is only if they think that materialism is modally/logically necessary that this argument will have no purchase on their views concerning infinite sequences of events. However, it is very difficult to sympathize with such a strict materialist, and one must wonder what kinds of arguments they might have for that position.
In any case, I think that the paradox's usefulness is not principally that it can be used to support a premise in an argument for the existence of God (which is perhaps a happy accident), but rather that it can do so without the radical consequences of Craig's arguments against all actual infinities.
Doesn't the argument depend on an assumption that if an infinite amount of universes exist, that they are causually connected?