The Grim Reaper Paradox, and the original solution, part 1

0. Introduction

Robert Koons has published a version of the ‘Grim Reaper paradox‘, also popularised by Alexander Pruss. One of the assumptions that sets up the paradox must be abandoned in order to avoid the contradiction. Koons and Pruss’ conclusion is that the assumption that an infinite past is possible should go.

It turns out that there is a potential solution to the paradox which was proposed by its originator Jose Benerdete, and more recently brought out in a paper by John Hawthorne (2000). I’m going to explain that solution here, and then (hopefully) develop it to the Koons/Pruss version in a part 2.

  1. The Grim Reaper paradox

The Grim Reaper paradox was first put forwards by Jose Benardete in his 1964 book ‘Infinity‘ (p. 259). Although the developer of the paradox, Benardete was a passionate defender of the actual infinite, announcing in the blurb on the cover of his book that:

“This book is an attack on finitism in all its forms … A metaphysics of the actual infinite is offered as the solution to the contemporary crisis in the foundations of mathematics”

It is perhaps a consequence of the clarity with which he engaged with the topic was that his examples have since been able to be used to argue for precisely the opposite thesis; that is, for finitism. This is how the grim reaper paradox has been used in the contemporary philosophy of religion debate, primarily by Alexander Pruss and Robert Koons, who argue for a version of finitism (‘causal finitism’).

Here is how Benardete states the paradox:

“A man is shot through the heart during the last half minute by A. B shoots him through the preceding 1/4 minute, C during the 1/8 minute before that, &c. ad infinitum. Assuming that each shot kills instantly (if the man were alive), the man must already be dead before each shot. Thus he cannot be said to have died of a bullet wound” (p. 259)

Koons describes this story as follows:

“The story leads quickly to a contradiction, on the assumption that Fred [i.e. A] does not die unless one of the Reapers kills him. At least one Grim Reaper must act, since if all of the Reapers whose numbers are greater than 1 do nothing, then Reaper #1 will act. However, it is impossible for any Grim Reaper to act, since, for any n, Grim Reaper #n cannot do so unless Fred survives until its assigned deadline at 1/2n seconds after midnight. It is impossible for Fred to survive that long, since Fred’s surviving until Reaper #n’s deadline entails that no Grim Reaper with a number larger than (n+2) has acted, but, in that case, Reaper #(n+1) must have acted.” (Does the Universe Have a Cause?, p. 4)

The contradiction lies in the fact that Fred will surely die before the end of the minute, but that also there is no Grim Reaper who will kill him. What this paradox seems to show, at least according to the finitists, is that there could not be finite duration of time (such as a minute) that is actually divided into infinitely many sub-regions.

2. The original solution, part 1: Benerdete.

Benardete did not draw the same conclusions as Pruss and Koons from the grim reaper paradox, and remained committed to the possibility of actually infinitely divided durations and lengths. Shortly after stating the paradox, he makes the following comments about it:

As to the dead man, although he did not die of any single bullet wound, his death was certainly caused by the infinite fusillade of shots. Here, again, although he is already dead prior to each shot, he remains alive at any assigned instant which is prior to them all. Thus he cannot be said to have died at any moment of time whatever! (Infinity, p. 260)

These brief comments offer only a hint of how to understand the response. The outline can be made out however.

Firstly, Fred is alive at every moment prior to all the shots. Yet, for each shot he is already dead before that one is fired. It follows from this that “he cannot be said to have died at any moment of time whatever!” We can spell this out by saying that although there is no first moment in which it is true that Fred is dead, there is a last moment at which Fred is alive. And this is to be expected. The series of bullets (reapers) is an open infinite sequence; while there is a last bullet at 12:01, there is an endless sequence of bullets ever closer to 12:00, and in particular there is no first bullet. Given the continuity of time, we can think of the transition from 12:00 to later times as a Dedekind cut on the real numbers:

dedekind2.png

In the above diagram, the left line is time as it approaches 12:00, during which Fred is alive. This is a closed set (which is what the square join indicates), in that 12:00 is a member of that set, but it is the final member. So Fred is alive at every time up to and including 12:00. However, the line on the right is an open set (which is what the curved join indicates). So Fred is dead at every time strictly after 12:00. This is what Benardete means by “he cannot be said to have died at any moment of time whatever”. There is no first moment at which he is dead (rather, there is just a last moment at which he is alive).

This much of the solution seems fairly straightforward. It is addressing the time of death, and we have the following information. When is Fred alive? All the way up to and including 12. When is he dead? At all times later than 12. If time is continuous and actually infinitely divisible, this is what it is like to transition from one state to another (it is like a Dedekind cut on the real number line).

The second bit is where the difficulty lies. It involves the cause of death. Benardete has it that no bullet (reaper) is the individual that causes the death of Fred (“he did not die of any single bullet wound”). Yet, the totality of all of them does cause Fred to die (“his death was certainly caused by the infinite fusillade of shots”). But how can it be that the totality of the bullets causes him to die, when none of them individually causes him to die?

3. The original solution, part 2: John Hawthorne

This is where Hawthorne comes in. His paper Before-Effect and Zeno Causality develops Benerdete’s solution, spelling out the principles in greater detail.

Hawthorne first considers the case of a ball rolling towards an open-infinite Zeno-sequence of walls. 2 miles away there is a wall; 1 and 1/2 miles away is another wall; 1 and 1/4 miles away is another wall; 1 and 1/8 miles away there is another wall, etc. Thus, there is an infinite sequence of walls, ever closer to the point that is exactly one mile away. There is no wall which is the ‘closest’ to the one mile point (which makes it an open sequence). Suppose the walls are impenetrable and cannot be knocked over (etc). The ball is rolled towards the walls. What happens as it arrives at the one mile mark? Hawthorne’s answer is as follows:

The ball does not proceed beyond a mile and it does not hit a wall.” (p. 625)

We are puzzled by this combination partly because we have fuzzy intuitions about what happens when there is ‘contact’ between objects and open series in this sort of setting (where space is continuous and actually infinitely divided). But we can spell it out by specifying what we mean by ‘contact’ in such a way that it makes sense. Hawthorne calls ‘open-closed contact’ what happens when “A closed surface contacts an open surface insofar as there is no unoccupied space in between the two surfaces.” When the ball arrives at the one mile point, it has achieved open-closed contact, in the sense that the closed surface of the ball has no unoccupied space between it and the infinite series of walls. Here is what Hawthorne says:

Consider the fusion of walls. Call it Gordon. On reflection it is clear that the sphere contacts Gordon. Gordon has an open surface. When the ball stops proceeding at the one mile mark, there is no unoccupied space between the sphere and Gordon. Contact occurs …  So the ball is stopped by contact: The ball hits something, though the thing that it hits is not one of the walls. (p. 626)

If we find this puzzling still, Hawthorne has the diagnosis at hand. It is because we are assuming what he calls the ‘contact principle’, namely:

If y is the fusion of x’s and z contacts y, then z contacts one of the x’s.

He makes the final following comments about this principle:

That principle holds for the finite case. But it is false if the x’s are infinite in number. Once we are clear about this, there is no residual puzzle, nor anything further to learn about the wall case. It is clear what happens in worlds that satisfy the original description: At a mile, the ball makes contact with the fusion of walls, which is rigid and impenetrable. As a result, it does not proceed further. The ball does not, however, make contact with any wall. (Ibid)

It is weird that the ball can be stopped by making contact with the totality (or ‘fusion’) of all the walls even though it does not contact any particular wall,  but that is partly because we are taking an intuition which is applicable to the finite cases only, and trying to apply it to the infinite case. In finite cases, there will be a first wall, and contact would be defined in relation to that wall. However, this is precisely what we do not have in this case, and relying on an intuition that presupposes that there is one will lead us into trouble.

 

      4. From walls to reapers

When it comes to the case of the infinite sequence of bullets (or reapers), Hawthorne basically applies the same analysis. Instead of the ‘contact principle’, he identifies the analogous principle and calls it the ‘change principle’:

“If x is the fusion of y’s and y’s are individually capable only of producing effect e by undergoing change, then x cannot … produce effect c without undergoing change”

This principle is also true in the finite case but false for the infinite case. Each of the bullets (reapers) are only capable of producing the effect of killing Fred by undergoing change (by being shot, or by swinging their scythes, etc). The assumption is that this applies to the totality of bullets (reapers). And it does, in finite cases. If there were only 10 bullets, then the totality would have to change, in the sense that one of the bullets would have to be fired at Fred, for it to bring about its effect. Yet, in the case of the infinite sequence of bullets (reapers), this is not the case. The totality can bring about the effect even though none of its elements (or the totality itself) changes in any way.

Going right back to Koons, we can apply this to his comments. In the quote I gave of his from above, he begins by saying:

“The story leads quickly to a contradiction, on the assumption that Fred does not die unless one of the Reapers kills him.” (Emphasis added)


What Hawthorne’s approach questions is this assumption. It is false that one of the reapers kills Fred, but that doesn’t mean that the effect is not brought by the totality of reapers. In the infinite case at hand, that is what happens. Once again, the weird behaviour of the infinite confounds our intuitions. Yet, giving up this intuition saves the situation from contradiction, and thus avoids the paradox.

 

5. Conclusion

Of course, some will argue that a ball cannot be stopped by a bunch of walls unless one of those walls makes contact with the ball, and the move from finite to the infinite context should make no difference to this.

Hawthorn is sensitive to the seemingly radical nature of the conclusion, admitting that it is a “big metaphysical surprise”. He goes on to finish by saying that:

“The Contact Principle, in full generality, could be given up fairly readily on
reflection. The Change Principle has a rather deeper hold on us. It seems to us scarcely thinkable that mundane causal powers—say that of killing with a machete—could combine so as to logically entail the causal power of producing some effect without the agent of the effect undergoing change. Nevertheless, surprising as this may be, the Change Principle should be rejected. The diagnosis is complete. The logic of each case is very much in order. And our puzzlement has been traced in each case to some faulty principle relating fusions to parts. Once we discard those principles, we will have no problem in accepting the required conclusions about what happens in each case” (p. 630-631)
I intend to write a part 2 to this, where I apply Hawthorne’s analysis to the version of the Grim Reaper paradox in which the target is not the actually infinitely divided duration, but the actually infinite past.

20 thoughts on “The Grim Reaper Paradox, and the original solution, part 1”

  1. Alex — I’m broadly sympathetic to what you’ve argued here and I found the Hawthorne paper fascinating.

    Nonetheless, I’m unsure about Hawthorne’s solution to the grim reaper paradox.

    Let’s suppose that an infinitude of guns is pointed at Fred and an infinitude of guns is pointed at Sue. Let’s assume that if a gun fires at Fred, then Fred is killed, and if a gun fires at Sue, then Sue is killed. After one minute has elapsed, if no gun has yet fired at Fred, then gun 1 will fire at Fred. After half a minute has elapsed, if no gun has yet fired at Fred, then gun 2 will fire at Fred. And so on — after 1/n of a minute, if no gun has yet fired at Fred, then gun n will fire at Fred. No guns ever fire at Sue.

    After the time interval, Fred has been killed yet — if the Hawthorne solution is to be believed — no bullets struck Fred and no guns were ever fired. Fred and Sue are in the same situation because while an infinitude of guns were pointed at both of them, no gun was ever fired. What explains the difference between Fred and Sue?

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    1. The situation has a clear difference; in the Fred case gun n is triggered to kill him if he arrives at time n, but that’s not true for Sue. I suppose we can ask whether that is enough of a difference, but it marks out a plausibly relevant distinction between the two cases at least.

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      1. What does it amount to to say that the gun is triggered to shoot? In both of the cases in my version of the thought experiment, the actual actions that the guns perform are the same for both Fred and Sue. Is it what the guns would have done at a nearby possible world? Or what the shooters intentions are? Those seem like clear differences between Fred’s situation and Sue’s situation, but they’re not differences that are picked out by processes the guns actually undergo.

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      2. It could mean various unproblematic things I think. Imagine that each gun is a mechanism, with a computer programmed to check and kill if it registers Fred is alive. Or that there are reapers with dispositions such that they will kill if they see Fred.

        Koons is explicit that any story like this, which has a powers story of the modality would do.

        In your case, we could explain the difference between Fred and Sue’s cases by saying that the guns pointed at Fred have a computer programmed to shoot the guns at Fred if he is alive at time n, whereas the guns pointed at Sue (even though they would kill her if shot) are not connected to any mechanism that checks to see if she is there.

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      3. Suppose that every reaper ends up swinging their scythe acausally. That is, reaper 1 swings their scythe at 12:01 if Fred is still alive then, reaper 2 swings their scythe at 12:00:30, etc, but there is no state of affairs that makes it so they do that. In other words, for all moments up to 12:00 this world is identical to one in which no reaper will swing their scythe. In fact, even after 12:00, the only difference between those two worlds is that Fred is dead in one of them. At the very least, this is rather bizarre.

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  2. Interesting. I’m not sure I now have had time to read and understand this very well, but I would say that one likely problem with these kinds of arguments is that they are mathematical idealizations that likely are not physically possible in real life (e.g. the shot example, the speed of light, maybe Planck time, second law of thermodynamics, ect.).

    It is worth noting that maybe they could be resolved if we take into account the transfinite numbers, like the cardinals (aleph-null, aleph-one, ect.) And the hyperreals (omega, omega + 1, epsillon-null, ect.)

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    1. They are idealisations, but they are for assessing metaphysical possibilities in the broadest sense, not physically realistic possibilities. The actual world may be unlike this world. Nevertheless, if such a world is coherent on its own terms it could play the role of blocking an argument that requires that no such world is coherent.

      On the second point, we need just one type of infinity here, which is that of the continuum (or aleph-1). The denunerable infinite is not enough, but more than continuum many points is not required to set up the paradox.

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      1. Alex — I was surprised to see you say that denumerable infinity won’t work here. As far as I can tell, it will work. The series 1, 1/2, 1/3, … can obviously be placed in a one-to-one correspondence with the integers and so is denumerably infinite.

        Perhaps the thought is that the thought experiment requires time to be dense. But the rationals are both dense and denumerably infinite, so a set being dense does not show that the set is non-denumerably infinite.

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      2. Oh, yes you are right. Density is the crucial thing, and of course the rational numbers are dense. So either aleph-0 or aleph-1 would be enough. But not more is required.

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      3. The above point does have an interesting point, though using ordinal, rather than ‘hyperreals’.

        Namely, if the ordinal omega corresponds to the above series of infinite shooters, then say a firing-squad with the successor ordinal of omega (omega+1) people WOULD have a first shooter. It’s easy to see why.

        If omega is the order sequence,

        1 < 2 < 3 < 4 < 5 < …

        Then the firing sequence in seconds after some moment, being in an ordering, would be,

        …< 1/32 < 1/16 < 1/8 < 1/4 < 1/2 < 1,

        which is just the above sequence with a relabeling and put in the opposite ordering, so we could have just wrote this as the ordering, using the position in the line-up rather than time.

        … < 5 < 4 < 3 < 2 < 1,

        But if the firing squad above now has omega+1 people, then we have an order sequence,

        1 < 2 < 3 < 4 < 5 < … < 1*

        Then, since person 1* is the last person in the sequence, reversing the sequence,

        1* < … < 5 < 4 < 3 < 2 < 1,

        puts 1* in the first position, making them the first shooter.

        So in this case, there is a first moment of death.

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      4. I think this would mean including 12pm as the lower bound on the set of times during which Fred is dead. And I think that if we modify the example like this then there becomes a first moment of death, but no last moment of life. More importantly though, it means that the analysis would be that the first reaper kills Fred. So the general point is that when the reapers form a set with a closed boundary, the causal agent is relatively unproblematic to identity; it’s the lower bound. This doesn’t solve the problem though, because that concerns what to say about what happens in the transition across an open boundary. So while it presents a less weird scenario if we add in a first member as you do, I’m not sure how much light it sheds on the issue in question with Benerdete/Hawthorne’s examples. It’s sort of changing the subject subtly to something easier to understand.

        Quite possible I’m missing something subtle in what you are saying though.

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  3. Are we sure “Fred is alive at every time up to and including 12:00….So Fred is dead at every time strictly after 12:00. “?

    Here is an argument that Fred is dead at 12:00.

    1. Let’s call the reaper who checks to see if Fred is alive at 12:01 r0, and the reaper who checks at 12:00:30 r1, and so on.

    2. Consider the time that passes between each Reaper checking if it’s their job to kill Fred.

    30 seconds pass between r1 and r0
    15 seconds pass between r2 and r1
    7.5 seconds pass between r3 and r2

    30*(1/(2^N)) seconds pass between rN+1 and rN

    3. Consider the total time passed by the entire series of reapers:

    sum from N = 0 to infinity of: 30*(1/2^N)

    This is an infinite geometric series, which has a known closed form solution, in this case it is:

    30*(1/(1-0.5) = 60

    The entire series of reapers waits 60 seconds.

    4. The entire series of reapers waits 60 seconds. Wouldn’t this entail that Fred is dead 60 seconds before the last reaper performs their check at 12:01? namely at 12:00 on the nose?

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    1. well, no. the measure of (12:00,12:01] is still 60 seconds. In general, the measure of (a,b) is equal to the measure of [a,b]

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  4. Upon further reflection, it seems to me that the paradox can be altered slightly to escape this solution. Suppose instead of swinging their scythes iff Fred is still alive, the reapers decided to act this way:
    Reaper 1 will clap at 12:01:00 iff no other reaper has clapped previously.
    Reaper 2 will clap at 12:00:30 iff no other reaper has clapped previously.
    etc.
    What would happen in this case, given the absence of a shared effect all the reapers are aiming for? Surely, it couldn’t be that a clap is heard at 12:00, since that resolves nothing! it would still be the case that either no reaper clapped (in which case reaper 1 should have clapped) or it would be that reaper n clapped, in which case reaper n+1 should have clapped. And the contradiction persists.

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    1. Right. I’m not sure what Hawthorne would say here. It would be odd if a clap sounds even though no reaper claps their hand.

      On the other hand I’m also not sure if this really is any stranger than the example discussed in his paper, or just another way of saying the same thing.

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      1. The reason I think this hypothetical is more problematic than the one discussed in the paper is that it doesn’t seem to assume the “change principle”. In the original example, the contradiction emerges when we add in our premises the assumption that “Fred does not die unless one of the Reapers kills him.”. Rejecting this proposition allows us to say something along the lines of “Fred is killed by the totality of Reapers”. But we need not assume some analogous proposition in the clap example.

        Perhaps we should think of it this way: In the first case, after 12:00 (Fred’s last moment alive) for all n, Reaper n will kill Fred if he is still alive. But Fred is already dead at that point, and thus the material conditional is satisfied (that is, it is true). The reapers in the second example, however, will clap iff no *reaper* has clapped earlier. What happens at 12:00 is irrelevant here, because reaper n only cares about what the reapers n+1, n+2,… did. In this case, the contradiction can be derived rather straightforwardly.
        If no reaper claps, reaper 1 should have clapped. Contradiction.
        If reaper n is the last reaper that clapped, then reapers n+1,n+2,… didn’t clap, which implies that reaper n+1 should have clapped. Contradiction.

        I’m not sure where a “change principal” is assumed here.

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      2. Yes, so I think you are right about this. There are cases where Hawthorne’s proposed solution just doesn’t seem available. You seem to have picked out a way of specifying GRP setups that avoids his solution.

        I wonder if we are going to have to admit that space is finite as well because of this. We can rearrange our clapping reapers into a spatial line, such that all at one moment each will clap iff no reaper to the left of him claps. Or that the future is finite if the setup is such that each claps iff no future reaper does. Or that space is discrete by making a converging Zeno sequence out of them and saying that each claps iff none of the next ones in the sequence do.

        Somehow, even though I can’t quite see my way through it, I’m suspicious that one simple pattern of reasoning proves so much about metaphysics.

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      3. Alex — No, we don’t have to admit to the discreteness of space because of the grim reaper thought experiment. The thought experiment that you laid out — where we consider grim reapers who clap at one moment iff a grim reaper to their left claps assumes that there is such a thing as a spatially extended moment. That is, you’ve made the assumption that two points in space can objectively be at the same time. If that assumption fails, then nothing follows from your thought experiment about the nature of space.

        On the orthodox (Minkowskian) interpretation of relativity, there is no such thing as spatially extended moments. Two space-time points can objectively be at the same time iff they are numerically identical.

        Tim Maudlin goes a step further. Maudlin has offered both a reworking of topology and a new metaphysical interpretation of relativity that follows from his new topology. As it turns out, Maudlin’s topology is formally equivalent to standard (point set) topology in that every theorem that can be proved in one has a corresponding theorem that can be proved in the other. Maudlin’s topology makes clear that distinct points of space that do not bare before or after relations to each other (ie, points at space-like separation) do not bare sensible fundamental spatial relations to each other.

        One may object that this only shows a result that holds for possible worlds where relativity is true. In reply, the grim reaper argument for discrete space only shows what would obtain at worlds where relativity is false. We live at a world where relativity appears to be true, or at least cannot be ruled out by mere philosophical argumentation.

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      4. I’m not sure about it either. Foreknowledge often creates paradoxes either way, so I can’t really say if the problem in that case (or the problem in your paradox of dry eternity for that matter) is the infinitude of the future or the knowledge of the future.

        “I’m suspicious that one simple pattern of reasoning proves so much about metaphysics.”
        I kinda agree and kinda disagree with this sentiment. I personally think that *any* reasoning that tells us this much about something so foundational should be met with skepticism. If it was a polysyllogism with numerous premises, for instance, my immediate thought would have been “Great, this is obviously false. Let’s try finding out where it went wrong.”, since a subtle mistake can easily hide itself behind a lot of wordy premises and rules of inference. On the other hand, GRP is very straightforward and simple. In my opinion, the only reason it is worthy of serious consideration *is* the simplicity and the directness. It makes it seem like there barely is any room for an error to be hiding. Barely any place to poke holes at.

        The simplicity is its strongest point, as I see it. Still, I also hold an agnostic position for the time being. It certainly needs more thought.

        I’ll take the chance (since I seem to have your attention) to thank you for the work you’re putting out. I love thoughtology and really enjoy listening to the other conversations you have. You’ve really helped with the development of my interests and my thought in the last couple of years. Thank you, Alex.

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  5. I would like to expand on Keith Peterson’s response and your reply to it, as I suspect it is quite salient that the paradox goes away if you reformulate it over a closed interval (see, for example, Alexander Pruss’ formulation, in which the reapers activate according to this principle: “for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t” – all we need to do, to get a non-paradoxical scenario, is to change the interval over which the reapers are distributed to the closed one [8,9], and correspondingly, the interval before t to the half-closed one [8,t).)

    This strongly suggests to me that the paradox is not one of time, or of anything else physical, but is one of mathematics. After all, the origin of our measure of time is arbitrarily decided, so moving it infinitesimally should not make a difference, and that is even before we get into multiple observers that are moving relative to one another.

    I think all the arguments establishing the paradox are a form of “reaper n is not the assassin, because an earlier-acting reaper did the deed”, together with the fact that whichever reaper you pick always has predecessors – but the latter depends on the puzzle being defined with an open interval. Changing that to the closed interval gives us one assassin that we know is the first – and it is his scythe alone that is bloodied.

    To me, the use of “know is” in the previous sentence suggests that this is an epistemic paradox, not a metaphysical one. I recently came across this brief note on Paul Benacerraf’s response to the related Thompson’s Lamp paradox, saying that “Thomson’s experiment does not contain enough information to determine the state of the lamp at t = 1, rather like the way nothing can be found in Shakespeare’s play to determine whether Hamlet was right- or left-handed.” Maybe the difference between the open-interval and closed-interval cases is that the latter gives us one important extra piece of information?

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