Friday, January 25, 2008

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.]

15 comments:

zippy said...

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).

Mike Almeida said...

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.

Alexander R Pruss said...

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.)

Mike Almeida said...

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.

Alexander R Pruss said...

Does ~Mp entail ~p does Np entail p on these models?

Alexander R Pruss said...

Sorry, a word dropped out: Does ~Mp entail ~p and does Np entail p on these models?

Mike Almeida said...

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.

Jeremy Pierce said...

John Hawthorne thinks the mereological sum of all the reapers is what kills you.

Alexander R Pruss said...

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.

Dave said...

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!

Mk said...

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?

Alexander R Pruss said...

The difference is that the Grim Reaper story sets up a contradiction: the victim is killed by a reaper but no reaper kills him.

Unknown said...

Here's the situation in language easier for me to digest this (admittedly, much less fun, but maybe it will help clarify for others or expose where I may go wrong in this post):

(1) {a1, a2, ...} is a sequence in (0,1)
(1a) such that for each t>0, exists an a_n in (0,t)
(2) the smallest a_n does action X

We have

(1a) => smallest a_n does not exist. So we can't say the smallest one did something, so (1a) must be impossible for any sort of entity that undertakes an action. I think everyone would agree.

So, no particular grim reaper set his alarm earliest. Surely it's impossible, therefore, for the alarm settings (1a) to exist. What more this means, I'm much clear.

I don't understand the rejection of the idea that some but not all combinations could be admissible.

"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"

Why? By what measure is it odd? By our intuition on finite sets? How do we know that intuition holds up? This appears not to be a logical argument at all. Hence, I suggest it may be a false dichotomy.

Small further issue:

"(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.)"

The theorem shows that it is unlikely if the alarm times are chosen independently AND uniformly. I don't understand the why the latter assumption should necessarily be imposed and give us license to declare it very unlikely. (I don't understand why it should be excluded either, but I'm just pointing out it's an additional assumption of the theorem.)

Even if they were uniformly iid, probability 1 does not imply the event occurs. For a single clock, we have P{X =/= c} = 1 for any c in (8,9) -- so some event with probability zero must occur on every alarm setting.

Alexander R Pruss said...

"I don't understand the rejection of the idea that some but not all combinations could be admissible."

One thing I've since come to realize is that these Grim Reaper arguments are based on an Aristotelian metaphysics of localized powerful particulars. On such an Aristotelian metaphysics, the particulars have what powers they have, and should be freely recombinable in their different locations.

"Even if they were uniformly iid, probability 1 does not imply the event occurs."

Sure, but probability 1 implies that the event can occur. :-)

Frank Antoine said...

Hi, I've been searching online for the history of reapers and I find this very rare kind of arguement I am not sure what the calculations are for. What I really want to ask though Is that by any means are these feedback you all create really coherent at all?

I dont mean to be a problem but i just wanted to say i found this somewhat amusing.

(Getting back into character)

And also I'd like to know what Is so important about 8 and 9 nine o clock.

Why are we going off into calculations?

And why did the unknown guy come in and post also that everyone would agree?

Is this real or a joke? I really dont mean to disturb you but I saw this and was wondering: If You can comment me on google plus. I would like to know. There was a comment zippy had made that said you make it like a mental amusement park.

Yes. I agree. What is so important about 8 and 9 o clock?
Is this real information you exaggerate or just what you just put up to speak a code only a few can understand?

What is the point of this? (I dont mean to be the agressive stranger or anything but..) I also was wondering,, please guys, dont hate me immediately,,but:
Why is it that zippy is the only one who is coherent on this post? Zippy said exactly how it looks and feels, I just don't understand. Zippy came and got fascinated by your writing just like i did. And when i think about it. Well...*gulp* we all could use a bit more discretion in this crazy world..hmmmm

Please,,listen to a helpful stranger,,
So Ok. Next up:

Take No offense, please,,please,,listen,,

please, please,,,listen:

You DO know that the average reader just doing research wont understand this stuff right? And i say that with the expectation that i just came here to research history: but i kind need to maths it seems. Hold on. This is the most maths i know:

Jesus was a shimigami promoter of long life: Please no offense of my math attempt: (j ws + A prmt of long life) Actually i think your maths is better..Is your maths real maths? It looks real.

Can I atleast get to be the stranger who get to hear What inspired this? ,,,,And,,,,I'm also Not sure if I sound pleasant enough,,but i thought that if you play around with the word 'reaper' like that,,I would definately want to talk to you if you don't mind.

I am not exactly a collage grad, but I just saw your post. What inspired this study sir?

Did you also watch the grudge, being human and Bleach the animation?

Impossible people always annoy and harrass you to leave your space.

Overall, i did get a real smile from this Blog. Ahhh and i analyse that if this was a joke,, you must know ALL about chronic dry eye and legal killings. You might also know what circadian rhythmic antidote take away stress, fear or anxiety.

Bro. me and you are on the same quest....Bro hug?

Please please please, Follow me on facebook family. Stawker Wilson.

Good day to you now until if you see fit to reply.