The Logical Form of the Grim Reaper Paradox

[Edit: it turns out that something quite similar to this is argued for in this paper by Nicholas Shackel]

0. Introduction

The Grim Reaper Paradox (GRP) comes in various different forms. Sometimes it is about the divisibility of time, and sometimes it is about whether the past (or future) is finite. Even when we fix on which of these issues it is aimed at, there are also lots of different ways it can be cashed out. It can be reapers swinging their scythes, or placing point particles on planes, etc. Much of the discussion can be on how these details are to be understood.

Here I want to highlight what the GRP is at the most abstract level. It might be that once we think about it from this rarefied perspective, without the complications about what exactly the details are supposed to be, that we can see the paradox more clearly.

1. The Schema

Any GRP has a logical form, which I shall refer to as the schema. Let’s just have as a toy example, the following:

The past has no beginning. There is an eternal machine such that each day at midnight, it checks to see if it has printed out anything yet from its printer. If it has, then it hibernates for the rest of the day. If it has not printed anything out yet, it immediately prints out the date and then hibernates for the rest of the day.

This is enough to generate our paradox. If it had not already printed anything out, this means that yesterday it would have run the same check and printed out the date. So it can’t be that the machine finds nothing printed out today. But that applies also to yesterday too, and every previous day. So although it can’t be that no date is printed out, no date could be printed on the paper.

The way to conceptualise this abstractly is as follows. There is a rule that characterises this example (and all the others). It is a universal condition that applies at all times. That condition says that some proposition p (which might be that a reaper kills Fred, or places a point particle on a plane, or that a machine prints out a date, etc) happens at a time if and only if p does not happen at any earlier time:

For all t (p at t iff for all t’ (if t'<t, then ~p at t’))

It says that p is true at t if and only if p is not true at any earlier time.

The schema on its own is not unsatisfiable. That is to say, if there are only finitely many times, then the schema can be true. In particular, the schema is true if there are only finitely many times and p is true at the first time. At that first time, p is true at t, and on the other side of the biconditional, the nested conditional (if t'<t, then ~p at t) has a false antecedent, and as it is in the scope of a universal quantifier it is vacuously true. So both sides of the schema are true. At all other times, p is not true at t (so the left side is false), and on the other side of the biconditional we have a condition that says that ~p is true at all earlier times, which is false because (as we just went through), p is true at the first time. So in all cases, the biconditional holds.

But if there is no first time, then we run into the familiar problem. If p is true at some time t, then the right side of the biconditional says that no earlier time, t’, could have p true at t’. But then take t’. It is also the case of that time that no earlier time, t”, has p true at it either. So given the left side of the biconditional, p is true at t’. Contradiction.

If p is not true at any time, then the left side of the condition is false for some arbitrary time t. But if p is not true at any time, then its not true at any time t’ earlier than t, which makes the right side of the biconditional true, which in turn implies p is true at t. Contradiction.

2. Conclusion

Now we have a purely logical version of the argument, freed from any distractions about reapers, or point particles, or eternal machines. The GRP really just says:

1. There is no first time t
2. For all t (p at t iff for all t’ (if t'<t, then ~p at t’))

As we have just seen, you can’t have both of these together. That is the GRP.

10 thoughts on “The Logical Form of the Grim Reaper Paradox”

1. Alexander Delaney says:

Alex, I think I need some help understanding things because the example of the eternal printing machine, as you laid it out, seems nonsensical to me. Okay, as I understand it you have a finite machine doing a simple daily task for eternity and there seems to be a paradox. (This could have serious implications because it sounds like the world we live in.) However, it seems that implicit in your description of the scenario, the eternal machine has an initial condition. (The printer started out with nothing printed.) How can an eternal machine have an initial condition?

As I see it, this problem can be avoided if the machine keeps track of what was printed for every day of its operation and uses only that information to make a printing decision. However, if such a machine is running eternally then to keep track of all days of operation it must have an infinite memory (and so an infinite number of discrete parts). It should be no surprise when paradoxes arise in a world with infinite things in it, but that doesn’t seem to be the world we live in so I can breath a sigh of relief.

If we restrict the machine to be of finite size, then it could only keep track of what was printed on each day up to some maximum number of past days, N. If the machine uses only that information for its decision making, then the machine would print out once every N days for eternity. That’s kind of boring, but I don’t see any paradoxes either. Am I missing something?

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1. Hi.

So I don’t know where you got the feeling that there was an implicit initial condition. It’s neither implicit nor explicit. To be explicit; there is no first state of the machine.

I don’t think the problem goes away if the machine remembers the past perfectly. Say that today, at t, it remembers that for all earlier times t’ it never printed anything out. Given that input it should print out today’s date. But if that’s right, then what did it do yesterday? If today it could remember every past state as involving not printing anything out, then that would also be true yesterday as well. So yesterday it would have printed out yesterday’s date. And that contradicts our assumption that today it remembered never printing out any date in the past.

On the other hand, if at t it remembers printing out the date at some past time t’, then at t’ it must have remembered never printing anything prior to that. But consider the day before t’; it would have made the same assessment then and printed that date (this is the same reasoning as above). So it’s contradictory on either route we take. So perfectly remembering the past doesn’t help, and it also involves the implication that it somehow requires infinite processing power which is weird enough as well.

In reality though, my example doesn’t require the machine to remember anything (other than its own basic program). Each day it wakes up and scans the tray of the printer to see if there is anything in it. If there is, it does nothing and goes back to sleep. If there isn’t, then it prints out today’s date and goes back to sleep. It doesn’t even need to have any memory at all.

I’m assuming that the paper stays in the same place forever and doesn’t decay. And that the ink doesn’t dry up, or the material the machine is made of doesn’t crumble to dust. Etc. So there are some assumptions here, but we could just dream up some conditions to explain them away. Maybe each day is a new exact replica of the machine made out of new stuff, etc. So the details don’t bother me too much. I’m not seeing any principled reason why such a machine wouldn’t be possible which comes from looking at these sorts of details.

Really, the point of the post was supposed to be that there is a logical form we can see which helps because then we can stop having conversations like this one! I think the problem is fundamentally logical, not physical.

Anyway, I hope that helps understand a bit more what I was trying to say.

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1. Alexander Delaney says:

Thanks for the reply, Alex! My background is in engineering, not philosophy, so I’m coming from a deep hole of ignorance and I really appreciate you taking the time to explain things.

I do agree with you that the fundamental problem is a logical one. The GR scenarios seem to be processes that perform a task only on their first cycle of operation. If the process is eternal, it makes no sense to ask about something that is dependent on its “first cycle.” If we’re just interested in why GR scenarios have paradoxes, I agree, it’s case closed. However, there are still interesting questions we can ask about things that do not depend on the first cycle, and these questions seem to depend on the physical realization of a particular scenario, especially on how the check for task completion is made.

For example, we can say that for your eternal printing machine, on any day that we look at the machine, there will definitely be a date printed out. It just makes no sense to ask what that date is. (Incidentally, in my previous reply I probably didn’t phrase my objection to your printer example very well. The problem is that the date printed out implicitly depends on the initial state of the machine [the memory of whether something was printed], but eternal machines have no initial state so it’s nonsensical to ask what date is printed.) Other GR scenarios might perform the check for task completion by first searching all past records. If the search for one past day takes a nonzero amount of time, then we can say that on any given day, the task will not be be completed.

Anyway, that’s just an engineer’s point of view of what is interesting.

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2. Alexander Delaney says:

Alex, I didn’t want to get sidetracked in my last reply, but there are a few things in your reply that bothered me. I don’t claim that my objections are valid, but if they are not I am hoping that you could provide an explanation to help clear up my thinking.

You wrote:
“In reality though, my example doesn’t require the machine to remember anything (other than its own basic program)…. It doesn’t even need to have any memory at all.”

How is that possible since the machine is not stateless? Either something has been printed or it has not. This requires only a single bit of memory, but it is memory nonetheless, and it is precisely this bit of memory that the machine uses to make its decision.

You wrote:
“Say that today, at t, it remembers that for all earlier times t’ it never printed anything out.”

Asking about the state “for all earlier times” entails asking about the machine’s “first state,” which makes no sense for an eternal machine. Any reasoning that follows along these lines will give meaningless results.

I do think it is valid to reason about the state of the machine on an actual given day based on logical possibilities for prior states in the finite past. For example, a machine like your printer follows a simple rule for its state change at every day: if (~bIsSomethingPrinted) then bIsSomethingPrinted = 1; From this we could reason that for any given day, bIsSomethingPrinted must be 1. However, asking questions that depend on what bIsSomethingPrinted was in its first state are nonsensical.

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Hi Alex!

This print machine thought experiment seems quite successful at showing that this particular causal chain is impossible; or rather per your logical form, that we cannot have the conjunction of 1 and 2.

I know the finitists will say that, this argument shows that infinite causal chains are impossible because they always lead to a contradiction. I know the eternalist rebuttal is “just because this *particular* causal chain (conjunction of 1 and 2) is impossible, it does not follow that all causal chains are impossible”.

However, I find this rebuttal to be rather unpersuasive, and can’t help but feel it would be unreasonable to even maintain my agnosticism on the causal finitude of our universe. I don’t have an argument, but if I am being perfectly honest, my intuition seems to have swung from agnosticism to an actual belief that the universe is causally finite.

Are you agnostic on the causal infinitude of our universe? I would ask you to explain why you remain agnostic, but asking someone to justify agnosticism seems silly. I am pretty lost as to what to think atm, and would love to hear your perspective on what your belief is with regards to the causal finitude of the universe.

Thanks so much for your content!

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1. Sorry for delay in replying.

The way I see it causal finitism is one explanation of why 1 and 2 can never be true together – because 1 can never be true at all.

But there is another view, which is more conservative. It’s just that 1 and 2 can never be true together, without also holding that 1 is impossible. At the end of the day, I think 1 and 2 are basically logically inconsistent, so that a story that tries to have both in there together is subtly inconsistent.

If that is right, then we don’t need a further explanation why we can’t have 1 and 2 true together; it’s just logically impossible.

Koons thinks we motivate the example with appeal to a ‘patchwork principle’ or ‘recombination principle’. This says that rearrangements of possible things are possible. If a dragon is possible and a unicorn is possible then both a dragon and a unicorn together are possible.

But this is a questionable guide to modality. It’s helpful in some respects, but ever since Lewis first proposed it, there has been a lot of discussion about how it should be restricted. All I’m suggesting is that if the combination of things is inconsistent, then that arrangement is not possible. That’s a very minimal constraint. Far less radical than causal finitism.

So maybe causal finitism is true, but should we jump to that conclusion based on the GRP? Not while there are other analyses that have the same explanatory power and are less ontologically costly.

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4. HeWhoYawns says:

Hi Alex,

Have you taken a look at Benacerraf’s treatment of the topic? I find his input rather intriguing, though I haven’t seen you talk about it. I don’t really think his proposal succeeds, but maybe you can try playing around with it and figuring something out.

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1. araybold says:

I had not seen your reply here when I mentioned Benacerraf’s views in my reply to the previous Grim Reaper post, but as you can see, I am following his lead, at least in one respect.

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5. araybold says:

Yesterday, I drafted a short reply, but did not post it. This morning, I woke up with this one almost fully-formed in my head…

Firstly, a note about the proposition p(t). It is not something about the state of the system, such as “at time t, the printer has a page in its out-tray” – that simply does not work (“the printer has a page in its out-tray at time t only if, at all times before t, it did not” is false.) It is, rather, the statement that, at time t, the relevant event (printing, in this case) happened.

I think it helps to rewrite Alex’s rule thus, using the identity P Q = (P and Q) or (~P and ~Q):

For all t ( for all t’ < t  (p(t) and p(t')) or (~p(t) and ~p(t')) )

I think I can say that Alex's rule is a general one for propositions that are true at most once, which is the case for Grim Reaper paradoxes. It is also the case in what Alex calls the familiar problem, where p is the proposition that t is the least number: given that there cannot be more than one least number, the first conjunction is false, but from the truth of this rule, the second is true, and therefore ~p(t) for any t – we have here the standard argument for there being no least number (itself being just the argument for there being no largest number, reversed), which amounts to a simple contradiction: whichever number you pick, there's always a lesser one.

Even as children we get this one (or at least the largest number variant) – but if it is not paradoxical that, whichever number you pick, there's always a lesser one, then what is paradoxical about "whichever time you pick, it is not the earliest time the event could occur"?

Well, there is the issue of the printer's state, of course. If, however, we were to find that the printer did not have something in its out-tray, then we would know that it is not working as specified – which is another way of saying, I think, that this situation would be inconsistent with the problem statement – it seems to require the printer to have something in its out-tray whenever we look.

But what has it printed? According to the problem statement, it would be a date. It cannot be a date in any calendar we could possibly know of, however: while proleptic dates can be used without a problem after the calendar has been created (the year of Julius Caesar's assassination,44 BCE in the Gregorian calendar, is an example), they cannot logically be used at the time they specify (a document date-stamped 44 BCE must be a forgery!) This, I think, suggests that the problem, as stated, is subtly inconsistent, or at least requiring backwards causality, which I won't consider further here. The printer can be displaying some printed output, or just a blank page, but nothing that would allow us to deduce a particular time at which the printing occurred.

So, I suspect there is a way for everything to be consistent here (excepting there being a date on the output), and no more paradoxical than infinity often is – it seems to be the same issue as in Hilbert's Hotel. it looks rather weird, but if one chooses to contemplate an outlandish and utterly infeasible thought experiment, one should not be surprised if the result is counter-intuitive.

The usual variants of this paradox all compress the possible action into an interval of finite duration, which tends to heighten the feeling of paradox (as the action is not pushed into the infinite past), but I agree with Alex that this is not relevant to the essential form (though it does depend on an additional fact about numbers that we don't learn so quickly: there are infinite rationals in any continuous finite interval.) As I mentioned in a reply to Alex’s previous post on the topic, they all depend on the interval being defined as an open one; use the corresponding closed one, and the problem goes away – there is a well-defined time for the event. The paradox is nothing more, I think, than that you can, at most, say either when was the last time before the event, or when the event occurred, but not both – which just follows from there not being a closest rational or real number to any given number – and the choice is made in the problem statement.

Two final points: Alex’s logical form of the paradox shows that it is about numbers, not time, so it seems unlikely that it could have an explanation that depends on, and thereby tells us about, the nature of time specifically (or anything else physical, for that matter.) Secondly, it is about what we can know (as in the last sentence of the previous paragraph), which, I think, shows it to be a puzzle of epistemology, not metaphysics.

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6. Ψ says: