[Cross-posted on my own blog.]

A Grim Reaper (GR) timed to go off at t0 is an entity which does the following at exactly t0. If Fred is not alive at t0, the GR does nothing at t0. If Fred is alive at t0, the GR instantaneously annihilates Fred. (If instantaneous action is not logically possible, one can complicate the situation by allowing shorter and shorter time intervals for these actions.) The GR Paradox then is this scenario. Fred is alive at 11:00 am today, and that he does not die today unless killed by a GR and he does not get resurrected today. There are infinitely many GRs, timed to go off in a staggered way at the respectively times t1,t2,… where tn is equal to 11:00 am + 1/n minutes. Well, by 11:02 am, Fred is certainly dead, since it is impossible that he survive a time at which a GR is timed to go off. But when was he killed? He wasn’t killed by the 11:00 am + 1 minute GR, because if he were alive just before 11:01 am, then he would have been alive at 11:00 am + 1/2 minute, when another GR went off, and he can’t survive a GR going off. It seems that none of the GRs could have killed him, because before each, there was another. So we have a contradiction: he both was and was not killed. Somebody has suggested that Fred is killed by the mereological sum of all the GRs, but that’s mistaken in the present setting because the GRs check if Fred is already dead before they do anything, so in the present setting, none of them actually do anything–and if they don’t do anything, how can they kill Fred?

The Kalaam argument needs the premise that there couldn’t be a backwards infinite sequence of events. Here is an argument for this:

- If there could be a backwards infinite sequence of events, Hilbert’s Hotel would be possible.
- If Hilbert’s Hotel were possible, the GR Paradox could happen.
- The GR Paradox cannot happen.
- Therefore, there cannot be a backwards infinite sequence of events.

Actually, one could make steps 1 and 2 into a single step, but this is more fun, and, if it works, establishes the interesting corollary that Hilbert’s Hotel couldn’t exist.

Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result.

Argument for (2): If Hilbert’s Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes. And that would result in the GR Paradox.

The argument for (3) was already given at the beginning of the post.

For about two years, I’ve smelled this argument coming, but I think my vanity has kept me from seeing it. I still have to confess that I have a really hard time accepting the corollary that Hilbert’s Hotel couldn’t exist–that corollary seems extremely counterintuitive to me. I wish I had some good way out.

On the other hand, establishing a major premise of an argument for the existence of God is a very happy outcome.

First it’s widgets *against* a finite past, now this! A couple comments:

First, although I didn’t see this at first, one neat thing about this argument is that it doesn’t seem to rule out an infinite *future* sequence of events even on an eternalist view of time (given that an infinity of future events could not cause a GR scenario that precedes those events in time). Your argument seems to rely on there being a causal order to time, but it is neutral with respect to presentism/eternalism. That’s good.

Second, the inference from the possibility of an infinity of past events to the possibility of a GR scenario seems to presuppose that temporal atomism is false. For if temporal atomism were true, then presumably it would be necessarily true. And if it were necessarily true, then GR would be impossible to construct even if an infinite hotel were possible to construct. (Similar reasoning applies to an analogous GR scenario expressed in terms of infinitely divisible spaceâ¦). But the presupposition that temporal atomism is false is equivalent to the proposition that there are infinitely many past moments. In other words, your argument against the possibility of infinitely many past events seems to presuppose that there are infinitely many past moments in time. There is a bit of a tension here: if there are infinitely many past moments, then one might think (via your widget arguments, say) that there could be a distinct event at each of those moments.

Clarification: by ‘GR’ or ‘GR scenario’, I meant the paradoxical scenario in which there are infinitely many GRs setup in the way you specify.

And another thought from my wife: every event, including Fred’s death, takes a non-zero amount of time. If so, then for any amount of time it might take Fred to die, an infinite number of grim reapers will have fired before Fred is dead. Let E be the event of those infinite grim reapers firing on Fred. E is the cause of Fred’s death.

Here’s a further thought (this one is mine). Consider an analogy. A block smashes a window. Assume the block is infinitely divisible (each top half has a top half, etc.) Ask: which *part* of the block caused the window to crash or was the *first* to hit the window? None, if there was no east-most part and the block was traveling east through the windowâ¦ That being so, we might say that the block *as a whole* has a left-most edge which was the first thing to go through the window and to smash it. Ok, now what Iâd like to propose is that events themselves can be combined to form sums or wholes. And I propose that there can be boundaries of the whole that no part overlaps (like in the block example). And finally, I propose that the event of *an infinite number of GRs shooting at Fred* has a temporal boundary that no individual GR shooting event overlaps. This boundary occurs at 11:00 sharp. That’s when Fred begins to die…

Still, in reply to the above comment, even if (necessarily) every causal transaction takes a finite amount of time, each GR can be more efficient (take less time to kill) than the one to act after it…

Actually, Alex, I’m not entirely sure you need the modal formulation to avoid contradiction. Specify truth conditions for any GR killing Fred. The simple idea is that a GR kills Fred at t iff there is no earlier GR that kills Fred at any earlier time t’. It then turns out that no particular GR kills Fred, since by the truth conditions there is no GR for which there exists no earlier GR that has killed Fred. But it is true for the disjunction of all GR’s that there is no earlier GR that has killed Fred at some earlier time. So the disjunction of GR’s can kill Fred, though no particular GR can do it.

Mike,

Does it make sense for their to be a *disjunction* of concrete objects? Or perhaps you have in mind a state of affairs consisting of the disjunction of all the GR events… Clever idea, but how is it that Fred should be dead even if there is no physical cause of death? I mean what caused the fatal wound? And what might that wound look like?

*Does it make sense for their to be a disjunction of concrete objects?*

It’s a little tricky, that’s for sure, since I want to attribute the property of killing Fred to a disjoined object. So, [Sue v Helen v Bob v Jane] killed Fred. But I think that has to be true, since Fred is dead, and they (the disjoined) are the sole suspect. I’m not sure there has to be no physical cause of death. It just has to be true that no particular GB inflicted it. It does raise interesting questions about how a group might kill someone without any member of the group doing so.

Josh:

The argument does, indeed, presuppose that temporal atomism is false. I am thinking it’s going to be hard to reconcile temporal atomism with relativity theory, and one’s going to have weird phenomenon like in Zeno’s Stadium paradox.

“the presupposition that temporal atomism is false is equivalent to the proposition that there are infinitely many past moments”

I don’t see it. Temporal atomism could be true and there could still be infinitely many past atomic moments. Conversely, it could be that there have to be only finitely many past moments, e.g., due to some subtle facts about the nature of regresses, but these moments would be Aristotelian rather than atomic. Aristotelian discretism about time says that in any finite interval, there are finitely many moments. However, these moments need not be equally spaced, and between any two moments there is potentially another.

“every event, including Fred’s death, takes a non-zero amount of time”

If so, you need a little trick, just as you need a little trick if it takes a non-zero amount of time for a GR to fire. The trick is this: suppose that GR #n, if it fires, kills Fred within 1/(4n) minutes. So, GR #1 is potentially operative from 11:01:00 to 11:01:15, GR #2 is potentially operative from 11:00:30 to 11:00:45, GR #3 is potentially operative from 11:00:15 to 11:00:22.5, and so on.

If relativity theory is correct, this is problematic due to the speed of light limit on action unless Fred and the GRs are point particles, with the GRs closer and closer to Fred. Moreover, it seems implausible that relativity theory should be *necessarily* correct, and all we need is some possible world in which to run the situation.

Mike:

Like Josh, I don’t know what a disjunction of objects, other than propositions or sentences, is.

I think you may simply be pointing out that the GR Paradox is an incoherent situation as I’ve described it. It is.

Mike:

How can a group kill without any member of the group doing anything other than *observing*? Recall how it works. GR #n observes to see if Fred is alive. If he’s not alive, the GR does nothing at all. Being observed does not kill, unless perhaps one is Schrodinger’s cat.

Note, by the way, that there are many sets of timings for GRs that do not give rise to a paradox. It is only with certain sets of timings that paradox arises. So there is no problem with GRs per se. It seems to me that the lesson to be learned is that time is discrete in the Aristotelian way, and the past is finite. I continue to find the necessity of this conclusion counterintuitive.

*How can a group kill without any member of the group doing anything other than observing? Recall how it works. GR #n observes to see if Fred is alive. If he’s not alive, the GR does nothing at all. Being observed does not kill, unless perhaps one is Schrodinger’s cat.*

You must be adding premises to the case. Everything that is said by way of description is consistent with the disjunction killing Fred. Are there other premises you want to add? If not, then exactly how they achieve this is a matter we can argue about–perhaps there are several competing ways it could be done–but it is not relevant to the fact that the case does not, as described, give rise to a contradiction. That was the conclusion I was after.

*Mike:Like Josh, I don’t know what a disjunction of objects, other than propositions or sentences, is.
I think you may simply be pointing out that the GR Paradox is an incoherent situation as I’ve described it. It is.*

I guess I don’t see the problem. Take the sentence, Smith and Wesson makes a good firearm, uttered at the the onset of their collaboration. Do I mean to say Smith makes a good firearm and Wesson makes a good firearm? Hardly. Is there a troubling metaphysics of conjoined objects? Again, hardly. So, when I say that Smith or Wesson make a good firearm, take me as saying that either, each, or both together do. In this case it’s the latter: both together do. And when I say that Sue or Helen or Bob killed Fred, take me as saying that either, each or all together did. The case rules out both ‘either and each’, so I conclude that all together did.

awww, why did I say “equivalent”? I only meant to say that if temporal atomism is false, then there are infinitely many past moments. Also, by ‘temporal atomism’ I meant that every time is ultimately built up out of finitely many temporal atoms (times that contain no times as proper parts), where ‘x ultimately built out of y’ means that the “mereological distance” from x to y is finite (that is to say, it is not the case that for every proper part p of of a time, there is a proper part of p not identical to an atomic time). Unless I’m mistaken, a GR scenario cannot be constructed given temporal atomism (so defined): a GR scenario entails that there is a time interval t, such that for every proper part p of t, there is a proper part of p not identical to an atomic time (or else that there are an infinite number of point-sized atomic times). And unless I’m mistaken, if temporal atomism (so defined) is false, then every time contains an infinite number of times (or else there are an infinite number point-sized atomic times). Am I right?

If so, then it appears that the inference from the possibility of an infinity of events to the possibility of a GR scenario presupposes the possibility of an infinite number of past times.

Mike:

Doing something collectively is not the same as its being done by a disjunction. But I have a plausible heuristic (it’s not exactly right–there are quantum cases–but it’s in the vicinity of something that surely is right): if no member of a collective does anything but observe, the collective does nothing but at most observe.

Josh:

I don’t think the argument presupposes the falsity of temporal atomism so defined. But I think the reasoning behind the argument does show the incompatibility of a certain kind of Aristotelian temporal discretism with an infinite past (which would have worried Aristotle, but by now he hopefully knows that the past is finite). Here’s why. According to Aristotelian temporal discretism–this is stipulative!–for any actual moments t0 and t1, and any rational number q between 0 and 1, it is possible for there to be an actual moment tq of time between t1 and t2, such that d(tq,t1) = q d(t2,t1), where d is temporal distance (if times are numbers, then d(t2,t1)=|t2-t1|. (E.g., if q = 1/3, then it is possible for there to be an actual moment of time that’s exactly one-third of the way from t0 to t1.) This, I take it, is one plausible way of making precise Aristotle’s claim that time is not infinitely subdivided but is infinitely subdividable.

Now, fix any pair of times t1 and t2. Let q1,q2,… be any enumeration of all rational numbers between 0 and 1. Let t-1,t-2,… be an infinite number of actual times, all of them prior to t1. It is possible, given Aristotelian temporal discretism, that for all n the following holds: at time t-n, somebody sets an alarm to go off between t1 and t2, such that the temporal distance of the alarm’s going off from t1 is equal to qn d(t2,t1), and nothing stops the alarm.

Why is this possible? Well, here is one reason to think this. Plainly, it would be possible that for every single past January 1, someone set an alarm to go off exactly a day later. Nothing in the Aristotelian discretism is threatened by this. But it would be really weird–and this intuition is at the heart of the GR paradox–if some mathematically coherent combinations of alarm times were logically possible and others were not. So it would be possible that for every n, on January 1 of the year 2009-n, the alarm was set for January 2 of the year 2009-n, but it would not be possible that for every n, on January 1 of the year 2009-n the alarm was set for January 1 of 2010 at 12:00 + qn minutes. (It is true that on Aristotelian discretism for some values of n, there is actually no moment at 12:00 + qn minutes. But that’s OK: this is what distinguishes Aristotelian discretism from what I called atomistic temporal discretism. Even though there is no moment at that time, it can be possible to do something that would make there be a moment then, just as even though George will never have a child, it can be possible for him to do something that would make him have a child.)

If one thinks that causal interactions between these alarms are going to lead to trouble, just suppose the alarms to happen in largely causally isolated parts of an infinite universe. (Here, use al Ghazali’s argument that an infinite past implies the possibility of an actual infinite: if an infinite past is possible, it is possible that God each year in the past created one immortal soul; then there would now be an infinite number of immortal souls.)

Alright, so grant me that the construction is possible. If so, then there is a possible world where the interval between t1 and t2 is densely populated by an infinity of moments. This is incompatible with Aristotelian discretism.

Hence, the Aristotelian discretist has to either deny the possibility of an infinite number of past events, or severely restrict which past events are and which are not allowed–in particular, arbitrary alarm-setting events will not be allowed.

So, yes, I think the argument does work given Aristotelian discretism, though its conclusion is incompatible with it. That’s because it’s a reductio. The crucial point is that for no actual or possible time t does Aristotelian discretism rule out the possibility of making a GR fire at t.

The argument does not work for discrete time theories on which there are no possible times beyond the actual ones.

It is very interesting to me that these arguments bear an uncanny similarity to some of the considerations in my widget-based arguments against Craig. But I think the present arguments are stronger, though I am not yet completely clear on what was wrong with those.

*Doing something collectively is not the same as its being done by a disjunction.*

Geeze, I said nothing of the kind. I said,

*And when I say that Sue or Helen or Bob killed Fred, take me as saying that either, each or all together did. The case rules out both ‘either and each’, so I conclude that all together did.*

So the ‘disjunction did it’ is true under the assumptions of the case. The disjunction plus the assumptions entails that the collective did it, and the collective doing it, plus the assumptions, entails the disjunction did it.

Alex,

Very interesting. Thanks for writing that out. I will think carefully about those remarks on discretism. I want to get clearer in my mind the relationship between the GR reasoning and the widget reasoning and understand exactly why one should win out over the other… You’ve opened up a fascinating (and I think important) topic of inquiry here.

Joshua:

Here is one reason for the GR reasoning to win out. The definition of each widget includes an infinite future. We might reasonably worry whether that infinite future is coherent. But each GR’s definition is coherent. The question is whether we can combine them. But, intuitively, some sort of principle of recombination, restricted appropriately, should be able to help there. However, I see difficulties there (e.g., it is very hard to get a principle of recombination that works well–for instance, there is no world where every innocent person suffers for eternity; maybe one can make an ad hominem move–the atheist should accept a principle of recombination).

Alex,

I see an outline of your paper. You set up your argument and then deal with objections to bring out a deeper understanding of the argument. Although I now see that I was mistaken in my âatomismâ objection, there is a related objection that I envision being the final objection of your paper. Itâs a âp-widgetâ objection. It goes something like this: If there *is* a smallest, non-zero length that a time can be, then the possibility of an infinity of past events does not imply the possibility of a GR scenario. On the other hand, if there *is not* a smallest, non-zero length that a time can be, then a p-widget is possible, and if a p-widget is possible, then so is an infinity of past events. Therefore, the inference from the possibility of an infinity of past events to the possibility of a GR scenario presupposes that there is not a smallest, non-zero length that a time can be and therefore that there can be a p-widget and therefore that there can be an infinity of past events.

In reply, we can argue that the inference from there being no smallest possible, non-zero length of time to the possibility of a p-widget is only plausible if there are no good reasons to think that p-widgets are impossible. But your argument provides a good reason to think that p-widgets are impossible. For by the reasoning of your argument, if a p-widget were possible, then so would a GR scenario. But a GR scenario is clearly not impossible. To use a Plantingian distinction, the temptation to think that a p-widget is possible is not based upon our actually âseeingâ that it is possible but is rather based upon an initial *failure to see* that it is impossible. But your GR-scenario reveals that a p-widget is indeed impossible.

Still, we might want to know what âmakesâ a p-widget impossible. Or, for that matter, what is to prevent God from counting though all the natural numbers before 1:00 this afternoon by counting â1â at 12:30, â2â at 12:45, â3â at 12.52.30, etc.?

Itâs really tempting to say that what makes a p-widget impossible is that there is a smallest possible, non-zero length of time (call this âatomismâ). Question: is it more evident to you that atomism is false than that *if* atomism is false, then a p-widget is possible? If the latter were more evident, then your GR argument combined with your p-widget argument would seem to suggest that atomism is true, not that there cannot be an infinity of past eventsâ¦ For me, the feeling that atomism cannot be true is maybe 1.2 times stronger than the feeling that if atomism isnât true, then a p-widget could be constructedâ¦ But then Iâm left wondering, âwhat makes a p-widget impossible? And why couldnât God keep counting faster and faster?â

Perhaps what you say about an infinity of future events being built into the definition of a p-widget is on the right track. The problem may not be with the coherence of a certain infinite future per say but with the infinite complexity that perhaps would have to be built into a p-widget (and/or its environment) for it to possibly replicate in the way you described. Perhaps such infinite complexity within concreta is impossibleâ¦ But honestly, this isnât an entirely satisfying response to me. What makes infinite complexity of a certain kind impossible? I’d like to keep thinking about what–other than atomism–might reasonably make a p-widget impossible.

I’m having trouble seeing that there is a genuine GR Paradox. Why doesn’t the argument show that there a contradiction only if there is an infinite series of GRs and Fred exists at t0? So long as Fred doesn’t exist at t0, there is no paradox of someone’s being killed and not killed. If the reply is that the world with Fred at t0 and the infinite series of GRs is possible, how do we know this? Maybe the argument should be taken as an impossibility proof of this world.

Josh:

Good thoughts.

Here is one approach. Times are spacelike hypersurfaces. Since such hypersurfaces can intersect one another in various complex ways, it is hard to see how there could be a minimum spacing between them. Working this out would have significant empirical consequences that differ from those of standard relativity theory. So, this is a more serious departure from relativity theory than that, say, of presentists who posit a preferred reference frame.

Moreover, if there is a minimum duration of time, then spatial motion is weird. Either objects jump from place to place without being at intervening places, or else there is also a minimum spatial distance. This makes space be discrete, and hence necessarily not isotropic. This, in turn, generates further departures from standard theories.

Of course, one can suppose that the discretization scale is so small that empirically none of this matters. But it’s still weird. For instance, it means that a cube cannot rotate without changing shape.

Alex:

Yes, I think those sorts of arguments against there being a minimum duration of time are powerful.

Still, if there is no minimum duration of time, then it is perplexing to me why a p-widget could not exist? What prevents God from making a p-widget?

A non-theist might give this answer: p-widgets would have to be infinitely complex or be in an infinitely complex environment, for in general new things cannot be produced ex nihilo (that is, without using previously existing materials). But there cannot be infinitely many concrete things.

But a non-theist who says that might also say this: There is an infinity of past events. Events are not concrete things (think Chisholm events + presentism). So although there is an infinity of past events, there is not, nor could there be, an infinity of concrete objects.

You say that with an infinity of past events a GR scenario could be constructed. But the non-theist replies that it could only be constructed if there are infinitely many resources (concrete things) to construct the infinitely many grim reapers. But there are not.

One replies that the GR scenario can be set up with just one grim reaper who decides ahead that for each of the infinitely many apointed times, he will shoot if Fred isn’t already dead. But the non-theist replies that that such a psychological state is impossible, perhaps because it could only be realized within an infinitely complex concrete system.

So this non-theist has an answer to why a p-widget is impossible. But that very answer combined with other reasonable background beliefs prevents her from accepting the inference from the possibility of an infinite past to the possibility of a GR scenario.

Perhaps a reply to all this is to modify the GR scenario so that it involves one GR who is such that as a matter of random, unplanned fact, he will shoot Fred if Fred is alive for each of the infinite times within the finite interval of time. If one accepts the possibility of an infinity of events, then it’s difficult to see how one might rule out the possibility of this modified GR scenario.

I’m still wondering, though, why God couldn’t create a p-widget. Maybe the answer is close to the one the non-theist gives above: there cannot be infinitely many concrete things, and p-widgets would have to be infinitely complex or be in an infinitely complex environment, for in general new things cannot be produced ex nihilo by the direct or indirect result of a p-widget…

**I was wrong about the Kalam argument (maybe I still am)**

I’ve always been somewhat skeptical of the Kalam argument. But recently I’ve had a change of sentiment: I now think the argument is defensible–at least to someone with my background beliefs about time and causation. Previously, there were three obstac…

I’m going to assume that a Grim Reaper scenario is impossible for just the reason Alex gives. I’m also going to assume that if a beginningless series of discrete past events is possible then a Hilbert’s Hotel is possible. My question concerns premise 2 of the original argument.

2. If Hilbert’s Hotel were possible, the GR Paradox could happen.

Here is Alex’s argument for 2:

If Hilbert’s Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes. And that would result in the GR Paradox.

I agree that if a Hilbert’s Hotel were possible, each room is such that *it could have* the following property: A GR appropriately correlated with the number of that room is set to go off. But it does not follow that it’s possible that *every* room in the hotel has this property. From

(x) it is possible that Fx

it does not follow that

It is possible that (x)Fx.

So, then, granted that the Grim Reaper scenario generates a contradiction and is impossible, it seems that Alex has shown only that no hotel – not even an infinite one – could “house” /all/ the GRs in his scenario. Why? For the completely innocuous reason that the scenario itself is impossible.

If this is right, then the argument fails to show either that a Hilbert’s Hotel is impossible or that a beginningless series of events is impossible.

Hi Wes!

This gap is supposed to be bridged by a plausible rearrangement principle. The easiest way to do this is with indistinguishable indeterministic Grim Reapers.

Thanks for the quick reply, and sorry for getting into this so late, Alex. At the risk of asking you to repeat something you’ve already said in one of these threads, would you mind answering the following questions?

1. What is this “plausible rearrangement principle?”

2. How does it work with the “indistinguishable indeterministic GRs?”

I did quickly look through the relevant threads, found just one reference to a “plausible rearrangement principle,” but didn’t really understand what I was reading and couldn’t go on from there. Sorry if I’m being a pain.

I thought your original argument was quite interesting, and I’d like to be sure I understand the reasoning behind premise 2.

The “plausible rearrangement principle” is in part a promissory note and in part a suggestion by Josh Rasmussen in this or the next thread. The idea is basically that if we’ve specified a “dispositional nature” for an object, we get to put in as many duplicates of the object as we like.

The indeterministic case is, I think, in a comment in the other thread, as well as this post.

I guess all this should get written up in a paper, but right now I am still unsure of how exactly to make the rearrangement principle go.

So we have a contradiction: he both was and was not killed. Somebody has suggested that Fred is killed by the mereological sum of all the GRs, but that’s mistaken in the present setting because the GRs check if Fred is already dead before they do anything, so in the present setting, none of them actually do anything–and if they don’t do anything, how can they kill Fred?Alex,

It’s Hawthorne that takes the mereological route. I don’t find it plausible either. I’ve tried to model these cases modally (though I’ve objections from, among others, Williamson and Cresswell, about the propriety of modal models). I think there is some sense in which none of the GR’s

cankill Fred. There is no world, for instance, in which the stipulations of the situation hold and any GR kill’s Fred. So, roughly, it is true that, S = N~K1 & N~K2 & .. .& N~Kn (where ‘K1′ is GR1 kills Fred, N is for necessary, I keep time indices suppressed). We’ll say MKn at t iff. (Vt’)(Vx)((t’ L t) & (x L n) only if ~Kx at t’) (M is possible, L is less than). But S fails to entail S’ = N~(K1 v K2 v . . .v Kn). If that’s true, we avoid contradiction. And I think that’s true if we reject the limit assumption for times t (there is no earliest time). That’s the rough idea in any case. Effectively, I don’t think Grim Reapers are impossible; I don’t think they generate a contradiction.