Grim Toe-Cutters

Imagine that Fred has all ten toes at 10 am, and there are infinitely many grim reapers. When a grim reaper wakes up, it looks at Fred, and if he has all his ten toes, it cuts one off and destroys it; otherwise, it does nothing. There are no other toe-cutters around.

Suppose, further, that grim reaper wake-up times can be set by you to any times between 10 and noon, endpoints not included. If you set the activation times to be such that there is a first activation time after 10 am (e.g., the nth reaper wakes up 60/n minutes before noon), there is no paradox of any sort. But if you set the times such that they are all after 10 am, but before every activation time there is another activation time, then… well, then logic guarantees that Fred will get a toe cut off infinitely many times and will regrow a toe infinitely many times! For without toe-regrowing, we get a paradox.

This is, of course, logically and metaphysically possible. Toes can regrow, and it is metaphysically though perhaps not physically possible for them to do so quickly. But what is amazing is that just by setting wake-up times for grim toe-cutters, we can make this miracle happen.

Grim Reapers and logical impossibility

The main objection to the Grim Reaper paradox as an argument against infinite causal sequences is the Unsatisfiable Pair (UP) objection that notes that paradox sets up an impossible situation—and that’s why it’s impossible!

I’m exploring a response that distinguishes metaphysical and (narrowly) logical unsatisfiability. The Grim Reaper situation is not logically unsatisfiable. The UP objection (well, really, Unsatisfiable Quadruple) notes that the following cannot all be true:

1. For all n > 0, the nth reaper wakes up at 60/n minutes after 10 am and kills Fred if and only if Fred is alive.

2. Fred is alive at 10 am.

3. There are no possible causes of Fred’s death other than those described in (1).

4. There are no possible causes of Fred’s resurrection.

But all that’s needed to have these four claims hold is for each reaper to kill Fred and then have Fred causelessly come back to life before the next one kills him. And while I think causeless resurrections are metaphysically impossible, they are (narrowly) logically coherent.

In other words, for the UP objection to work, the unsatisfiability must be metaphysical, not merely narrowly logical. But this, I think, negatively affects the force of the UP objection. For instance, in my Infinity book I consider Grim Reapers with adjustable wake-up times, and I note that for some wake-up time settings (say, the nth reaper wakes up 60/n minutes before noon) there is no paradox, and I ask what metaphysical force prevents the wake-up time settings from being the paradoxical ones. Daniel Rubio in a review of the book responds (in the context of a parody) that “no metaphysical thesis is required to explain this impossibility; the fact that it would lead to a contradiction is enough.” But in fact a metaphysical thesis is required to explain the impossibility, since there is no contradiction (in the narrowly logical sense) in (1)–(4).

Perhaps this is not a big deal. After all the metaphysical thesis here, that causeless events are impossible, is one that I do accept. But nonetheless it is a metaphysical thesis, as such on par with causal finitism, and hence when we consider the explanation of the impossibility of the Grim Reaper story and the impossibility of various other of the causal paradoxes that I discuss, there is something appealing about seeing the case as nonetheless offering support for causal finitism, which explains all of them, while the thesis about causeless events being impossible does not.

Unreliable Grim Reapers

As usual, Fred is alive at 10 am, and there is an infinite sequence of Grim Reapers, where the nth has an alarm set for 60/n minutes after 10 am, and if the alarm goes off, it checks if Fred is dead, and swings its scythe at Fred if and only if Fred is alive. But here’s the twist. These Grim Reapers are unreliable killers. The probability that the nth Reaper’s swing would succeed in killing Fred is 1/n^p, where p is some positive real number, the same for each Reaper, and independently of all other relevant events.

Here’s the fun thing. It seems possible for Fred to survive the whole ordeal. All it takes is for every Grim Reaper to fail at killing Fred. Nothing absurd happens then. Moreover, it seems this isn’t the only way for absurdity to be avoided in this case. We could also suppose that the nth Reaper kills Fred, while Reapers n + 1, n + 2, … all fail.

Suppose we adopt what seems the best alternative to Causal Finitism, namely the Inconsistent Pair response to the original Grim Reaper paradox, which says that the reason the original paradox is impossible is simply because it embodies an Inconsistent set of propositions—some Reaper has to kill Fred and none can. If that’s what’s wrong with the original Grim Reaper paradox, then it seems we have to accept my Unreliable Reaper story as possible.

But things are a little bit more complicated. The only way to avoid paradox in the Unreliable Reaper story is if there is some n ≥ 0 such that all the Reapers starting with Reaper n + 1 fail. But now suppose that 0 < p ≤ 1. Then the event that all the Reapers starting with Reaper n + 1 fail is less than or equal to (1−1/(n+1)^p)(1−1/(n+2)^p)(1−1/(n+3)^p)... = 0 (this is because Σ_k 1/k^p = ∞ if p ≤ 1). Thus the probability that we have avoided paradox is 0. Hence, if we have to avoid paradox, a specific zero probability event—namely, the event of paradox-avoidance—has to happen (the probability of a countable disjunction of zero probability events is zero). But if it has to happen, it can’t be probability zero, but must be probability one!

Perhaps here we bring back the Inconsistent Pair response. We say that my Unreliable Reaper story is impossible if p ≤ 1, because if p ≤ 1, then a zero probability event has probability one, which is inconsistent. No such problem occurs if p > 1. Thus, on this version of the Inconsistent Pair response, my Unreliable Reaper story is impossible if the success probability of the nth Reaper is 1/n^p for p ≤ 1 but possible if p > 1. And that’s pretty counterintuitive.

The paradox of the Jolly Givers

Consider the Grim Reaper (GR) paradox. Fred’s alive at midnight. Only a GR can kill him. Each GR has an alarm with a wakeup time. When the alarm goes off, the GR looks to see if Fred’s alive, and if he is, the GR kills him. Otherwise, the GR does nothing. Suppose the alarm times of the GR’s are 12:30 am, 12:15 am, 12:07.5 am, …. Then Fred’s got to be dead, but no GR could have killed him. If, say, the 12:15 GR killed him, that means Fred was alive at 12:07.5, which means the 12:07.5 GR would have killed him.

A Hawthorne answer to the GR paradox is that the GRs together killed Fred, though no one of them did.

Here’s a simple variant that shows this can’t be true. You hang up a stocking at midnight. There is an infinite sequence of Jolly Givers, each with a different name, and each of which has exactly one orange. There are no other oranges in the world, nor anything that would make an orange. When a JG’s alarm goes off, it checks if there is anything in the stocking. If there is, it does nothing. If there is nothing in the stocking, it puts its orange in the stocking. The alarm times are the same as in the previous story.

The analogous Hawthorne answer would have to be that the JGs together put an orange in the stocking. But then one of the JGs would need to be missing his orange. But no one of the JGs is missing his orange, since no one of them took it out of his pocket. So, the orange would have had to come out of nowhere.

And, to paraphrase a very clever recent comment, if it came out of nowhere, why would it be an orange, rather than, say, a pear?

I think the JG paradox also suggests an interesting link between the principle that nothing comes from nothing and the rejection of supertasks.

You're not killed by the fusion of the Grim Reapers

A grim reaper (GR) is a device that activates at a pre-set time. It checks if Fred--the victim--is alive. If he is, it kills him. If he isn't alive, it does nothing. For the Grim Reaper Paradox, we're supposed to imagine one GR set for 12:30, another for 12:15, another for 12:07.5, and so on. Before each time for which a GR is set, there is an earlier one. But Fred is alive alive at 12:00. Paradox ensues when we notice that Fred must be dead at 12:30 (else that 12:30 GR would have killed him), but no GR could have killed him, since if he were alive at its activation time, he would have been alive when the previous one activated, and hence would have been killed then at least.

John Hawthorne has claimed that Fred is not killed by any one GR, but by them altogether. More precisely, Fred is killed by their mereological sum.

Here's a gruesome way to see the problem with this solution. We can number the GRs in reverse: the 12:30 GR is number 1, the 12:15 one is number 2, and so on. Then suppose that the odd-numbered GRs kill by decapitating and the even-numbered ones kill by stabbing in the heart. Given the setup, Fred is either decapitated or stabbed in the heart but not both. But which one?! If he were decapitated, he would have been first stabbed. If he were stabbed, he would have been decapitated before that.

Another argument against an infinite past?

I wonder if this very neat argument can't be used to provide another Grim Reaper style argument against an infinite past? The argument nicely fits with the intuition that Grim Reaper induces in me, namely that no event can have an infinite number of events in its causal history.

Koons on grim reapers and Kalaam arguments

I've been telling people about this paper by Rob Koons, defending a grim-reaper based Kalaam cosmological argument, but I didn't realize it was online.

The Grim Reaper Paradox

Here is a version of the Grim Reaper paradox. Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive, and if you're alive, it instantaneously kills you, and if you're not alive, it doesn't do anything.[note 1] Suppose there are countably infinitely many Grim Reapers, and before they go to bed for the night, each sets his alarm for a time (not necessarily the same time as the other Reapers) strictly between 8 and 9 am. Suppose, also, that no other kind of death is available for you, and that you're not going to be resurrected that day.

Then, you're going to be dead at 9 am, since as long as at least one Grim Reaper wakes up during that time period, you're guaranteed to be dead. Now whether there is a paradox here depends on how the Grim Reapers individually set their alarm clocks. Suppose now that they set them in such a way that the following proposition p is true:

(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.

Here's a useful Theorem: If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.

Now, if p is true, then no Grim Reaper kills you. For suppose that a Grim Reaper who wakes up at some time t₁, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t₁, say at t₀. But if so, then you're going to be dead right after t₀, and hence the Grim Reaper who woke up at t₁ is not going to do anything, since you're dead then. Hence, if p is true, no Grim Reaper kills you. On the other hand, I've shown that it is certain that a Grim Reaper kills you. Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

The above argument shows that some arrangements of Grim Reaper alarm clock times, namely the ones that make p be true, are impossible, because they result in your being dead and not dead at the same time. But no such objection can be made to other arrangements of Grim Reaper alarm clock times. For instance, if Grim Reaper 177 wakes up at 8:05 am, and all the other Grim Reapers happen to wake up later, there is no difficulty--Number 177 kills you, and you're dead at 9 am.

Now we have a trilemma. Either all mathematical combinations of Grim Reaper alarm clock times strictly between 8 and 9 am are possible in the above story, or some but not all, or none (in the last case, the story above is impossible whatever the times are). The hypothesis that some but not all are possible seems unlikely. Look: it's midnight, say, and we have all of these Grim Reapers setting their alarm clocks. It would be really, really odd if they were somehow compelled by the metaphysics of the situation to set their times in one of the privileged ways, unless it turns out that there are only finitely many moments of time between 8 and 9 am, so that p cannot be true. (Indeed, by the Theorem given above, these privileged ways of setting times are very unlikely if the Reapers are choosing independently, assuming that all real-numbered times between 8 and 9 am exist, which the Theorem assumes.) That leaves two hypotheses: That all the combinations are possible or none. If all the combinations are possible, so will be the ones that make p true (e.g., Reaper 1 waking up at 8:30:00, Reaper 2 at 8:15:30, Reaper 3 at 8:07:30, Reaper 4 at 8:03:45, and so on). And that's not possible.

So either there are only finitely moments of time between 8 and 9 am, or no combination of Grim Reaper alarm clock settings is possible. In the latter case, it basically follows that it's just impossible to have infinitely many Grim Reapers, whether their wakeup times are arranged so as to result in a paradox or not. So why can't there be infinitely many Grim Reapers? It seems that the only reason to suppose there can't be infinitely many Grim Reapers, even in cases where no paradox is generated, is if one thinks there can't be an actual infinity of objects in existence. And if there can't be an actual infinity of objects in existence, then there can't be an actual infinity of times in the past, since if there were an actual infinity of times, surely a new object could come into existence at each of those times.

So either there are only finitely moments of time between 8 and 9 am, or there are only finitely moments of time in the past. But if there are only finitely many moments of time in the past, there were only finitely many moments of time yesterday between 8 and 9 am, and today is no different. So in either case, a bounded interval of times contains only finitely many moments.

I am not fully convinced by this argument, but I don't have a very good response.

[This post is revised. I am grateful to Bill Craig for pointing out some sloppiness in the original.]