tag:blogger.com,1999:blog-3891434218564545511.post9070923529978278749..comments2024-03-28T13:23:50.623-05:00Comments on Alexander Pruss's Blog: The Grim Reaper ParadoxAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger147125tag:blogger.com,1999:blog-3891434218564545511.post-30378503710044548902023-10-29T17:48:35.467-05:002023-10-29T17:48:35.467-05:00Suppose that Reaper #n has its alarm go off at 1/n...Suppose that Reaper #n has its alarm go off at 1/n hours past 8 a.m.<br /><br />Then, granted, no particular one of the reapers will kill you, because any killer would have been preceded by an earlier killer. And yet you must die, because any reaper who found you still alive would have killed you.<br /><br />One resolution to the paradox is that the <i>aggregate</i> of the infinitely many reapers who arise before 8 + 1/epsilon a.m. do, collectively, kill you. Yes, you will be already dead by the time any particular reaper checks in on you, so no one reaper is the killer. Nonetheless, the whole setup is such that the infinitely many reapers collectively kill you.<br /><br />While this seems implausible (how can an individual action happen if no one person commits the action?), perhaps the following consideration will make it more plausible: The premises of the problem entail that the compound action of <br /><br />(1) finding that you're alive and then <br /><br />(2) killing you<br /><br />can be completed in an arbitrarily short span of time. This means that this scenario is happening in a universe with nothing like a light-speed limit. Both<br /><br />(1) the reapers' information processing (checking whether you're alive, concluding that you are, and then initiating the action of killing you), and <br /><br />(2) the lethal physical effects of their processing (namely, your death, as a fait accompli that is detectable by the next reaper, no matter how quickly the next reaper arrives)<br /><br />happen arbitrarily quickly. In a universe like this (very different from our own), it's plausible that the compound action can be carried out in zero time. And if it can be carried out in zero time, then perhaps no <i>one</i> actually has to do it for it to happen.<br /><br />But then what <i>exactly</i> is it that kills you, if it's not the scythe of any one particular reaper? Here I would say that the "how" of your death is underspecified by the problem. The premises entail that you will die at 9 a.m. (because you must be dead at every point in time thereafter), but the premises underdetermine how precisely your death will happen.<br /><br />If a premise were added to the problem that said "You cannot die unless some particular reaper's scythe kills you", then I would say that the premises collectively are inconsistent.Tyrrell McAllisterhttps://www.blogger.com/profile/03742116091097551615noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-58285803104498265322023-10-29T17:39:33.213-05:002023-10-29T17:39:33.213-05:00This comment has been removed by the author.Tyrrell McAllisterhttps://www.blogger.com/profile/03742116091097551615noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-3963420558702530122022-10-24T14:15:08.452-05:002022-10-24T14:15:08.452-05:00Interesting post and discussion! I don't have ...Interesting post and discussion! I don't have anything meaningful to add, just a cosmetic complaint: "8:15:30" should be "8:15:00". :-)jcsahnwaldthttps://www.blogger.com/profile/05536598140736834255noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-78169639816798152772021-12-29T12:03:26.308-06:002021-12-29T12:03:26.308-06:00Yes, denying the infinite divisibility of time is ...Yes, denying the infinite divisibility of time is a way out of this paradox. But there are many other paradoxes in the book, and not all of them can be escaped like that.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-23866876216923034862021-12-28T16:28:09.168-06:002021-12-28T16:28:09.168-06:00Dr. Pruss, the argument assumes at first that time...Dr. Pruss, the argument assumes at first that time can be divided infinitely ("(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t"), I would like to know why an infinite amount of existing objects would imply that time can be divided infinitely. (sorry if the question is silly, I am just getting into these matters).Francohttps://www.blogger.com/profile/12215632250488178566noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-20110994375848371972021-11-09T09:51:43.315-06:002021-11-09T09:51:43.315-06:00I should note that this post preceeds my infinity ...I should note that this post preceeds my infinity book by about a decade. Some of the issues discussed in this thread are discussed in greater detail in the book.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-68390428581831388942021-11-09T09:50:24.933-06:002021-11-09T09:50:24.933-06:00A reasonable rearrangement principle makes it very...A reasonable rearrangement principle makes it very like that such a world is possible if infinite causal histories are possible. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-73714305536415088442021-11-09T09:30:26.644-06:002021-11-09T09:30:26.644-06:00Dr. Pruss,
Don't you need to prove that such ...Dr. Pruss,<br /><br />Don't you need to prove that such a world is even possible? <br /><br />And let's say that there is a possible world, and this reasoning shows there cannot be infinitely many GRs in that world. How does that help us in our world?<br /><br />Can't I just return this to you with "all we need to do is embed the story in a possible world where there are infinitely many such GRs and this would show that it's impossible for any of these GRs to kill Fred."JosiahThttps://www.blogger.com/profile/13321984874681720818noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-50261944255802339952021-11-01T20:49:32.640-05:002021-11-01T20:49:32.640-05:00Then time can't be infinitely divided in that ...Then time can't be infinitely divided in that possible world, but you still need an argument that it's true of this world. (I don't doubt that, if it's true of that possible world, it's true of all possible worlds, but that still needs a demonstration.)jqbhttps://www.blogger.com/profile/07510836914645398165noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-640914788139347142021-11-01T20:41:37.193-05:002021-11-01T20:41:37.193-05:00Josiah:
All we need to do is embed the story in a...Josiah:<br /><br />All we need to do is embed the story in a possible world that doesn't contain anything else with the causal power to kill Fred.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-85130895998212219372021-11-01T17:50:07.855-05:002021-11-01T17:50:07.855-05:00It's fascinating how some people will misuse a...It's fascinating how some people will misuse and abuse an argument to their own ends ... see, e.g., https://www.reddit.com/r/DebateAnAtheist/comments/qkge8y/an_older_version_of_the_kalam/jqbhttps://www.blogger.com/profile/07510836914645398165noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-92020286981636907752021-09-07T13:32:42.020-05:002021-09-07T13:32:42.020-05:00Dr. Pruss,
It's 100% clear to me that Fred is...Dr. Pruss,<br /><br />It's 100% clear to me that Fred is dead at 9am. There's no need to prove this anymore for my benefit.<br /><br />But you haven't proved the assertion that one of the GRs killed Fred. You've proved that Fred is dead (I agree) and then asserted that the only thing that can kill Fred is one of these countably many GRs. What's wrong with Fred being killed by Rudolph the Red Scythed Reaper, who did so at/before 8am because he wasn't allowed to join in (your) reaper games? Logic isn't requiring one of the countable GRs to kill Fred, it's you, and to me that assumption you're making is clearly what's causing the paradox.<br /><br />Now, since Fred is surely dead at 9am, something must have killed Fred. You COULD define what killed Fred to be a GR. BUT to "force" the paradox, you'd have to prove that this defined GR is one of your GRs. In particular, you have to prove that it's a GR that woke up AFTER 8am and therefore killed Fred after 8am. Given the other conditions, I think it's clearly impossible to prove that this GR satisfies that conditions and therefore is clearly not one of your GRs.<br /><br />Here is IMO a clear mathematical analogy to your "paradox."<br /> Let X be a subset of [0,1]. Define m=inf X to be an element of X such that m <= x for all x in X and if n <= x for all x in X, then m>= X.<br /><br />If X is a finite set, then everything is fine. So let X be (0,1], i.e. a cleverly defined INFINITE set. Then m is not an element of X. This is a paradox because m=inf X was defined to be an element of X. Therefore infinite sets cannot exist.<br /><br />The conclusion that infinite sets don't exist is clearly (I hope) not true, so there must be something wrong with the argument. Naturally the error in the argument is the exact thing that causes the contradiction, that is that I made the silly definition that required inf X to be an element of X. If I drop that requirement, then we have a perfectly good object and no paradox. <br /><br />Analogously, if you drop your assumption that one of YOUR GRs killed, Fred, the paradox doesn't occur.<br />JosiahThttps://www.blogger.com/profile/01592908118729462831noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-29448026213640989012021-09-07T10:45:41.161-05:002021-09-07T10:45:41.161-05:00Well, Fred would be alive all the time because for...Well, Fred would be alive all the time because for each GR, there is always a temporally prior GR, and by definition any GR with a temporally prior GR can't and won't act to kill Fred since the prior GR fulfills the role of killing Fred if Fred is alive. Kinda like if you had an infinite row of children beginning to your right and stretching endlessly to your left, and the first one had a ball in front of him, but the rule was that if there was another person next to them they can't kick the ball and have to move it to the other person.<br /><br />But this is a distributive property which applies to all members of the infinite row, so no kid will kick the ball because every kid has another one next to them. So the ball wouldn't be kicked.Wesley C.https://www.blogger.com/profile/05903323901343952714noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-43250478195384526042021-09-07T10:15:04.694-05:002021-09-07T10:15:04.694-05:00Why wouldn't they act? If Fred is alive when a...Why wouldn't they act? If Fred is alive when a GR wakes up, the GR kills him. So it can't be that Fred is alive all the time.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-31777838095844257772021-09-07T09:45:39.240-05:002021-09-07T09:45:39.240-05:00@Alex,
I think you said that it can be proven t...@Alex, <br /><br />I think you said that it can be proven that Fred IS killed or DOES die in the end, which I think is the most important part of the whole paradox. Because as I see it, if there are infinitely many reapers, and for each individual reaper if there is any one prior to them they won't act, this basically means Fred won't die as none of the reapers will act given that there are infinitely many reapers in the continuous time sequence.<br /><br />So, is there any definitive reason to think that Fred DOES die at the end, other than stipulation?Wesley C.https://www.blogger.com/profile/05903323901343952714noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-36417376080476934712021-09-07T09:36:41.174-05:002021-09-07T09:36:41.174-05:00One does need the assumption that nothing in the s...One does need the assumption that nothing in the situation other than a GR can kill Fred.<br /><br />Then one argues: Suppose for a reductio Fred is not dead at the end of the story. Then GR_7 didn't kill Fred. The only way that could have happened is if Fred was already dead at x_7. But then Fred is dead at the end of the story. So, Fred IS dead at the end of the story. But if you are dead, there is something that killed you. And only a GR can kill Fred.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-30946865416630310872021-09-07T02:45:48.069-05:002021-09-07T02:45:48.069-05:00Dr. Pruss,
I believe your argument is roughly:
...Dr. Pruss,<br /><br />I believe your argument is roughly: <br /> Let x=(x1,x2,...) in (0,60]^inf where x_i and x_j are distinct if i and j are distinct<br /> Defined an Event, E_x (=Ex for shorthand) by the following conditions:<br /> (1) Fred is alive at 8am<br /> (2) For each i = 1, 2, . . ., if Fred is alive at 8am + x_i min, then GR_i kills Fred.<br /> (3) Fred must be killed by GR_i for some i.<br /><br /> Then if inf{x_i} is not equal to x_j for some j, there is a contradiction as it's impossible for any GR_j to kill Fred. Since there is at least one x which would create this paradox (uncountably many x's, actually), we can analyze one such case and conclude that while it's possible to define any finite number of GR's, it is impossible to have infinitely many and supertasks are not possible.<br /><br />Please forgive me if I'm missing your argument, but if I'm not terribly off, my issue would be simply: When you created a paradox with Event Ex, you immediately concluded the issue must be with condition (2). <br />I think a perfectly reasonable possibility is that condition (3) isn't satisfied (and if inf{x_i}=0, then neither is (1)). It's obvious that no individual GR can be given the condition that it must kill Fred, as for the situation to be conceivable, we must allow a GR to do nothing in the event that Fred was already dead at that GR's "time to shine." So for each GR, it's a possibility that something else killed Fred, but for some reason, YOU are adding the condition that when we consider all GR's, one of them must have killed Fred.JosiahThttps://www.blogger.com/profile/01592908118729462831noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-65855775104728060622021-05-13T12:47:57.847-05:002021-05-13T12:47:57.847-05:00Hmm...it still seems that the infinite series canc...Hmm...it still seems that the infinite series cancels out the causal power of each reaper. Because for any reaper N, there is necessarily always one prior to it. And for any reaper N, it won't be the <b>actual cause</b> of the victim's death because there is a reaper prior to it that would cause the death instead. But this applies to all the reapers in the infinite series, so individually and as a whole no reaper can be the actual cause of the victim's death.<br /><br />So the victim doesn't die.Wesley C.https://www.blogger.com/profile/05903323901343952714noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-3855794128924440432021-05-13T12:18:56.888-05:002021-05-13T12:18:56.888-05:00Well, we can prove that the victim must die.
Choo...Well, we can prove that the victim must die.<br /><br />Choose any Grim Reaper N. Either the victim is already dead by the time N activates or not. If the victim is already dead, the victim dies. If the victim is not dead, then N activates and kills him, and the victim dies. Either way, the victim dies. QEDAlexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-31377921038055441832021-05-13T04:21:58.677-05:002021-05-13T04:21:58.677-05:00I think the GR set-up entails that the victim won&...I think the GR set-up entails that the victim won't die, due to each GR necessarily having an earlier GR, which means the later GR won't do anything as the earlier one would precede it. <br /><br />Just as if you had an infinite series of people starting on the right and extending to the left, where each person had a ball in front of them they would kick only on the condition that there is no person besides them on the left - and no ball would be kicked because every person has someone beside them - so too would no GR do anything as the earlier GR's negate the later ones from acting.<br /><br />So unless it can be proved that the victim <b>must</b> die, or that the GRs don't negate each other in the whole infinite set, it seems the paradox is solved because none of the GRs will be causally active due to a negating-rule-relation extending on the whole infinite set.Wesley C.https://www.blogger.com/profile/05903323901343952714noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-25078008565324816602021-04-26T09:45:57.999-05:002021-04-26T09:45:57.999-05:00AJ:
Well, the post only argues for the conclusion...AJ:<br /><br />Well, the post only argues for the conclusion that between any two times there are only finitely many times. Thus, between 9 and 10 am, there are only finitely many times, between 10 and 11 there are only finitely many times, etc. The argument in the post doesn't show that actual infinities can't exist (though that's one of the options considered). <br /><br />As far as this argument goes, it applies equally to the past and the future. <br /><br />I think actual infinities CAN exist. My current thinking is that the problem lies with causal chains that involve infinitely many causes behind an event. Such causal chains are impossible.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-84344673878586678752021-04-25T21:30:35.603-05:002021-04-25T21:30:35.603-05:00Though I'm 12 years late I have a question abo...Though I'm 12 years late I have a question about this paradox. Wouldnt this also prove that the future is finite too? If actual infinities can't exist then this paradox should apply to the future too right?<br /><br />This paradox seems like a double-edged sword.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-7373334853941364052021-02-06T06:29:26.536-06:002021-02-06T06:29:26.536-06:00Actually, it is the setting of the alarm clocks th...Actually, it is the setting of the alarm clocks that closes the interval. As soon as you have just one alarm clock set, you have an interval closed by a first time (which may be modified by the setting of another clock to an earlier time.)ARaybouldhttps://www.blogger.com/profile/16609714269978643038noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-53374775842718037222021-02-06T05:16:47.484-06:002021-02-06T05:16:47.484-06:00My apologies for posting so much, but I woke up th...My apologies for posting so much, but I woke up this morning with what feels like a minor epiphany: in any Grim Reaper scenario, the act of the assassin 'converts' the open interval into a (half-)closed one, by identifying / defining / picking out the first reaper.ARaybouldhttps://www.blogger.com/profile/16609714269978643038noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-27740760827463500552021-02-05T22:08:11.544-06:002021-02-05T22:08:11.544-06:00Alexander, thank you for taking the time to read w...Alexander, thank you for taking the time to read what we hoi polloi are writing at one another, let alone deigning to give us some pointers. I can imagine your eyes rolling as you look at the knots we tie ourselves up in!<br /><br />I must say, however, that you have left me in your dust with this post. Firstly, given the quote you start with, I tentatively figure that your two numbered arguments are to be taken as the two sides of the paradoxical contradiction that Grim Reaper paradoxes set up.<br /><br />Secondly, I can't imagine any way in which the execution (pun intended) of a Grim Reaper paradox involves backwards causal reasoning, whether infinite or finite. That leaves me guessing that you may be referring to the setup for a convergent-series Grim Reaper scenario, which seems to require infinite backwards-in-time reasoning to set the alarms. As far as I can tell, however, your version of the Grim Reaper scenario avoids this infinite-backwards-reasoning in its setup, which would (if my other assumptions about what you are saying here are correct) seem to save that form of Grim Reaper paradox from the no-infinite-reverse-causal-reasoning objection.<br /><br />---<br /><br />That aside, my response-du-jour to Grim Reaper paradoxes is that the paradox is purely mathematical, and has nothing to do with time. It goes like this:<br /><br />Firstly, so long as we avoid attempting to identify the assassin by its time of awakening, I think we can come up with a consistent story: you are killed by the first reaper to wake, acting alone. We can still do that, and also identify the assassin, if the wakeup times are uniformly distributed over a <i>closed</i> interval - the killer is the one who wakes on the lower bound. We can arguably also do it for all convergent-series Grim Reaper paradoxes, if we accept the limit of the series as the time the assassin wakes (which is a mathematically-respectable choice, I believe.)<br /><br />The problem of saying when the lone assassin woke comes with open intervals, because they don't have a least number. This does not have anything to do with time specifically, however, as we can have the same paradox without introducing time - for example, the sequence of rational numbers in the open interval (8,9) exists at 8.5 but not at 8, yet there is no end to be found anywhere, let alone between these numbers: we have a totally-ordered set with no end in either direction, yet it does not contain every number. Grim Reapers do not extend this paradox to time; they demonstrate it with a (hypothetical) example using time.<br /><br />If there is a circumstance in which this mathematical paradox would have an observable consequence in the physical world, I would agree that it is a paradox in that aspect of the physical world in which it is manifest, but Grim Reaper paradoxes do not achieve that for time, or any other physical observable.ARaybouldhttps://www.blogger.com/profile/16609714269978643038noreply@blogger.com