Kalam, Part 3: The Grim Reaper

In Jose Benardete’s paradox, we are to suppose that there is an infinite number of Grim Reaper mechanisms, each of which is engineered to do two things: first, to check whether the victim, Fred, is still alive at the Grim Reaper’s appointed time, and, second, if he is still alive, to kill him instantaneously, and, if he is already dead at the appointed time, to do nothing. The last Grim Reaper (Reaper 1) performs this dual task at exactly one minute after noon. The next-to-last Reaper, Reaper 2, is appointed to perform the task at exactly one-half minute after noon. In general, each Reaper number n is assigned the moment 1/2^(n-1) minutes after noon. There is no first Reaper: for each Reaper n, there are infinitely many Reapers who are assigned moments of time earlier than Reaper n’s appointment.
 
It is certain that Fred does not survive the ordeal. In order to survive the whole ordeal, he must still be alive after one minute after twelve, but, we have stipulated that, if he survives until 12:01 p.m., then Reaper 1 will kill him. We can also prove that Fred will not survive until 12:01, since in order to do so, he must be alive at 30 seconds after 12, in which case Reaper 2 will have killed him.  In the same way, we can prove that Fred cannot survive until 1/2^(n-1) minutes after 12, for every n. Thus, no Grim Reaper can have the opportunity to kill Fred. Thus, it is impossible that Fred survive, and also impossible that any Reaper kill him! However, it seems also to be impossible for Fred to die with certainty and yet to do so without any cause.
 
The original Grim Reaper paradox requires some assumption about causality: that Fred cannot die unless someone or something kills him. I would like to eliminate that dependency. Consider the following variation: the Grim Placer. In place of asking whether a pre-existing victim Fred is dead or alive, we will focus instead on the question of whether or some Grim Placer has issued a death warrant. Let’s say that each Grim Placer #n can issue a death warrant by placing a particular kind of point-sized particle in a designated position, at exactly the distance of d/2^n meters from a plane P. Each Grim Placer #n checks to see if a particle is already at a distance of d/2^i  meters from plane P, for some i > n: that is, he checks to see if any earlier Placer has issued a “warrant”. If a particle has already been placed in one of the designated spots, then the Grim Placer #n does nothing, other than maintaining the status quo.
 
If there is no particle in an appropriate location, then the Grim Placer #n issues his warrant, placing a particle exactly d/2^n meters from P. We can now prove both that at 12:01 that some particle is located within d meters of the plane, and that no particle is located there. Suppose that there is no particle at any location d/2^i meters from plane P, for any i. This is impossible, since if there were no particle d/4 meters from P, then Grim Placer #1 would place a particle in the position d/2 meters from P.
 
Thus, there must at 12:01 pm be some particle in an appropriate position. Suppose that the particle is located at that time in position d/2^n meters from P, for some n. This means that every Grim Placer whose number is greater than n did nothing, contrary to our hypothesis. Thus, this option is also impossible.

​As Alexander Pruss has observed (Pruss 2009), the Grim Reaper paradox suggests not only that no finite time period can be divided into infinitely many sub-periods but also that it is impossible that there should exist infinitely many time periods, all of which are earlier than some event. It seems to provide grounds for thinking that time must be bounded at the beginning: that there must be a first period of time. If not, we could simply construct a new version of the Grim Placer paradox. As in the original version, we postulate the possibility of a Grim Placer, who creates a particle and places it at a designated spot, if and only if no particle is already located at a spot corresponding to any earlier Placer. In this version, Placer 1 is set to act at the first moment of 1 B.C., Placer 2 at the first moment of 2 B.C., and so on ad infinitum. Once again we can generate the contradiction: some particle must be placed within d meters of the plane, but there is no finite distance from the plane such that a particle could have been placed there.