The Grim Reaper paradox was, a while back, used by both Alexander R. Pruss and Robert C. Koons to try to run a tenseless kalam cosmological argument for the existence of God. A little later, on Pruss’ blog, there were two developments: first, there was an argument to distinguish different kinds of actual infinities, so that the relevant kind which is absurd is “an actual infinity of causal influences on a single event” which, it seems to me, is entirely reasonable. I still think that it is absurd to postulate an actually infinite number of beings, but at least I don’t feel bound to worry about an actually infinite future or anything like that (about which I used to worry a great deal given arguments presented by Dr. William Lane Craig for his Kalam argument, and which can be made into an argument for the A-theory of time). Following this it was recognized by Pruss, again on his blog, that the Grim Reaper paradox doesn’t do the work it was supposed to do after all. The Grim Reaper paradox is supposed to show that there cannot be an infinite past, since at every moment of time tn, it is possible that a grim reaper come into existence which has the quality of checking at some time between two future times whether George is alive, and if so, kills George. However, between any two times ti and ty, assuming that ty≠ti, there are an infinite number of times.

- It is possible at any time tn that a Grim Reaper come into existence
- It is possible that there be an infinite series of such times in the past.
- There is a logically possible world in which at every past time tn a Grim Reaper came into existence set to go off between some time between two times ti and ty, where ti<ty, where George is alive at ti, and where there is no time between ti and ty at which a Grim Reaper is not set to go off.
- If 2, then George is dead by ty, and has been killed by a particular Grim Reaper
- If 2, then George cannot have been killed by a particular Grim Reaper (therefore has not been).

So, the problem is deeper, and it doesn’t prove that there cannot have been an infinity of past times. Note that if this Grim Reaper paradox could prove that the past could not be infinite, then it would also prove that the future cannot be actually infinite (which is theologically problematic on Catholicism).

However, we can say that all this just goes to show that the absurd kinds of infinities are 1) “an actual infinity of causal influences on a single event,” 2) an actually infinite number of beings and perhaps 3) an actually infinite number of events.

Let’s say that an event is not the same thing as a time. There are several ways we could define an event so as to distinguish our 3) from our 2). For instance, we could say that an event is a time at which some proposition becomes true or false (changes truth value). Now, suppose we revisit the argument with that in mind; we find that even though there is no particular point in time at which the event *G*: “A Grim Reaper set to go off between ti and ty” cannot begin to exist, we can say that there cannot be an actually infinite set of such events as *G*. Dr. Craig is always pretty careful to formulate his arguments against the infinitude of the past by stipulating that it is absurd to think that there be an actually infinite number of ‘events’ and he avoids the vocabulary of ‘times’ altogether. I think it’s a mistake, however, to reify ‘events’. I don’t think events really are things, instead I think an ‘event’ is a tool of language, and we might say that events and times must be coextensive. For instance, say that over two intervals of time not a single predicate is different, so that these two times are indiscernibly identical. Well, first, obviously that means that the times are identical (by the principle of the identity of indiscernibles), and second, there can be no such thing as ‘time’ as long as things are in a quiescent state (*pace* phenomenalism about time). Thus, any two times are not identical if and only if there is at least one property which obtains at one and does not obtain at the other. Interestingly, in order to get an actually infinite number of Grim Reapers it would have to be the case that at some time tn there is a finite number of Grim Reapers, and at another time tn+1 there is an infinite number of them. If at any time there is a finite number of Grim Reapers, then at no time after that time, however far removed, can it become true that there is an actually infinite number of them unless there is some time at which ‘an infinite number of Grim Reapers’ comes into existence. Talking about an actually infinite number of times might, therefore, be confused. Maybe times are indivisibly simple, and they just reduce to particular states of affairs.

Coming back to ‘events’, perhaps, then, Dr. Craig is right to say that an actually infinite number of events is not logically possible, but I don’t think that’s quite right, because that would mean that, assuming an actually infinite future, there would be an actually infinite number of events.

Not sure what I think about this yet. I’m still very happy to accept 1) and 2) above, but I’ll have to think more about whether events, defined as I have defined them, are identical with times.