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.]
12 comments:
Reading this blog is like going to a mental amusement park. It must be really fun to be one of your students.
I suspect that time is, in fact, discrete. After all the physical universe in general seems to be quantum (that is, discrete), so why not time as well? IOW perhaps there are in fact no actual infinities (other than God).
Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.
I think what follows is that there is no Grim reaper kills you and you are such that some Grim reaper or other kills you. I'm not sure that's absurd to believe, since it involves an infinite disjunction. If you take the infinite disjunction of propositions of the form 'Reaper n killed a', that disjunction has to be true. But, since the disjunction is infinitely long, you know that no particular disjunct true. It would be nice to have a truth-operator in this context which took wide scope on the disjunction, and narrow scope on each disjunct. Similar expressions are unproblematic in other modal contexts. Ask whether some Reaper was a wrongdoer. W(A v B) (someone or other did wrong) and ~WA & ~WB (no one in particular was a wrongdoer) are perfecly consistent.
Mike:
"there is no Grim reaper kills you and you are such that some Grim reaper or other kills you"
Putting this in terms of quantifiers, I get:
(for all g)(GR(g)->g does not kill you) and you are such that (for some g)(GR(g) and g does kill you).
This seems an explicit contradiction.
(Here, -> is the material conditional, and GR is the predicate that one is a grim reaper.)
Alex,
There are models for infinite choices on which both (1) and (2) are true. No theorem that would generate contradiction is valid in those models.
1. ~Mp1 & ~Mp2 & . . .& ~Mpoo
2. N(p1 v p2 v . .v poo)
Here we have it is not possible that Reaper1 kills you and not possible that Reaper2 kills you and. . .and not possible that Reaper00 kills you. But necessarily Reaper1 or . .Reaper00 kills you.
Does ~Mp entail ~p does Np entail p on these models?
Sorry, a word dropped out: Does ~Mp entail ~p and does Np entail p on these models?
Does ~Mp entail ~p does Np entail p on these models?
Give 'N' the interpretation 'is obligatory that; and 'M' the interpretation 'is permissible that'. Let the worlds increase in value infinitely, w1, w2, w2, . . .w00. Let 'Np' be true just in case some world where p is true is better than any world where ~p is true. Finally, let p1 = God actualizes w1. In that case, both (1) and (2) of jan 25, 12:58, are true. I haven't taken the time to work it out, but I'll be those interpretations are also right for the gr puzzle, too. Let me try to show that.
John Hawthorne thinks the mereological sum of all the reapers is what kills you.
Graham Oppy says this, too.
But how can the mereological sum kill you if none of the reapers ever swings a scythe, as in my setting? Remember my setup: "if you're alive, it instantaneously kills you, and if you're not alive, it doesn't do anything". In the case where the reaper wakeup times have a limit point at 8 am, no reaper does anything. But if no individual reaper does anything, neither does the mereological sum do anything.
Assuming continuous (nondiscrete) time.
If there were 10 reapers spaced out across the hour, the reaper that arrived at 8am + 1hr/10, i.e. 8:06.
In general the reaper that arrives at 8:00 plus 1 hour/n kills. Where n is the number of reapers.
Since we have an infinite number of reapers here, n= infinity, hence the reaper that arrives at lim( n-> infinity) of 8 + 1/n kills.
Essentially the reaper that arrives at 8+1/infinity kills.
Effectively this would be 8, but not quite!
How is this fundamentally different to Zeno's Achilles-and-tortoise paradox?
- an infinite set of events (visits)
- of increasingly small duration
- contained within a finite, bounded time interval
- resulting, at the limit, in a binary event (killing Fred, or overtaking a tortoise).
Sure - the infinite series runs the other way ( in time ), and there is a cause-effect relationship between each Grim Reaper, but how does that negate the solution to Zeno's paradox?
The difference is that the Grim Reaper story sets up a contradiction: the victim is killed by a reaper but no reaper kills him.
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