May 17, 2015
Grim Reaper Paradox
Dear Dr. Craig,
I recently viewed your defense of the Kalam Cosmological Argument video at the Baylor University Alvin Plantinga conference, and I was intrigued by the new grim reaper argument against an infinite series of causal events. I've searched throughout the web and have found very little on this argument. I was wondering what exactly your thoughts were on this argument and if you will be adding it to your repertoire of arguments against an eternal series of causal events. Thanks for your tireless work, and I pray that the lord blesses your ministry as it has infinitely blessed my life.
It was a privilege as well as a good deal of fun to be involved in this conference honoring so great a thinker as Alvin Plantinga. There are a number of variations on the Grim Reaper Paradox, which you can read about in José Benardete’s entertaining book, Infinity: An Essay in Metaphysics (Oxford: Clarendon Press, 1964), pp. 259-61.
For those who are unfamiliar with the Pruss-Koons version1, we are invited to imagine that there are denumerably infinitely many Grim Reapers (whom we may identify as gods, so as to forestall any physicalistic objections). You are alive at midnight. Grim Reaper #1 will strike you dead at 1:00 a.m. if you are still alive at that time. Grim Reaper #2 will strike you dead at 12:30 a.m. if you are still alive then. Grim Reaper #3 will strike you dead at 12:15 a.m., and so on. Such a situation seems clearly conceivable—given the possibility of an actually infinite number of things—but leads to an impossibility: you cannot survive past midnight, and yet you cannot be killed by any Grim Reaper at any time!2
Graham Oppy, perhaps the world’s most formidable anti-theistic philosopher, responds to a similar paradox concerning infinitely many bells, each of which rings at its appointed time with a deafening peal. You cannot retain your hearing past midnight, and yet you cannot be rendered deaf by the pealing of any bell! So how does Oppy respond? He says that your deafness is not the result of any particular peal but is the collective effect of infinitely many peals.3 This response not only involves a most bizarre form of retro-causation, as Benardete points out4, but is in any case inapplicable to the Grim Reaper version, since once you are dead no further Grim Reaper will swing his scythe, so that collective action is out of the question.
Pruss and Koons show how to re-formulate the paradox so that the Grim Reapers are spread out over infinite time rather than over a single hour. For example, one can arrange things such that each Grim Reaper will swing his scythe on January 1 of each past year if you have managed to live that long. You cannot survive to the present and yet there is no Grim Reaper that kills you at any time.
I haven’t given a great deal of thought to the paradox, but it seems to me to be immediately directed against the continuity of time and as such is quite convincing. We should not think of time as composed of durationless instants, between any two of which there is an infinite number of instants. The paradox reinforces several other philosophical arguments against construing time on the model of a geometrical line composed of points. Fortunately, since the advent of quantum theory, philosophers, and physicists as well, have exhibited much greater openness to taking time and space to be discrete rather than dense. In fact, many think that the continuity of spacetime in General Relativity is what needs to go if we are to have a unified physical theory of the world.
The question will be whether this paradox can be successfully reformulated to rule out the infinity of the past. There is no difficulty in thinking of the past as a sequence of years, with a Grim Reaper stationed on January 1 of each year. The difficulty, rather, is that once you push the sequence of years back to infinity, you lose the analogue of midnight in the Grim Reaper story. There is no infinitely distant point at which you were alive and from which you try, unsuccessfully, to endure to the present. Since the series of past years has no beginning, not even an infinitely distant one, it makes no sense to say that though you were once alive, you cannot live to the present. On this scenario, with Grim Reapers stretching back infinitely into the past, you were never alive!
So the analogy to the Grim Reaper Paradox is not tight. We’ll have to make a counterfactual claim instead: something like “You would be alive at any time t, unless you were struck down first by a Grim Reaper.” That seems to be a coherent claim. You could be a being not susceptible to what Aristotle called generation and corruption and so naturally sempiternal (everlasting). You would perish only if destroyed by a Grim Reaper. But then the paradox arises: you cannot be alive today and yet you cannot be killed by any Grim Reaper (since, again, an earlier one would have already destroyed you before any given Grim Reaper swings his scythe).
It seems to me that this is a version of the kalam argument against the possibility of an infinite past formed by successive addition. I want to renew my invitation at the Baylor conference for theistic philosophers to explore this question more deeply.
1. Robert C. Koons, “A New Kalam Argument: Revenge of the Grim Reaper,” Noûs 48 (2014): 256-267; Alexander Pruss, “From the Grim Reaper paradox to the Kalam argument,” http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html, October 2, 2009; Alexander Pruss, “Probability on infinite sets and the Kalaam argument’, http://alexanderpruss.blogspot.com/2010/03/probability-on-infinite-sets-and-kalaam.html, March 16, 2010.
2. For those who don’t get it, the point is that before any Grim Reaper can cut you down, you will be already dead, since prior to any Grim Reaper’s action there is an actually infinite number of earlier Grim Reapers each ready to kill you.
3. Graham Oppy, Philosophical Perspectives on Infinity (Cambridge: Cambridge University Press, 2006), p. 83.
4. Benardete, Infinity, p. 259.