The paradox of dry eternity

0. Introduction

Like many people, I am participating in ‘dry January’, meaning that I am not drinking any alcohol during the month of January. I’m also thinking about the Grim Reaper paradox, and have spent much of the year thinking about the infinite future debate between Morriston and Craig. Interestingly, all of these things have come together, in a paradox I shall now dub the ‘dry eternity’ paradox.

Part of the inspiration for this comes from a paper I read recently by Yishai Cohen, called ‘The Endless Future: a Persistent Thorn in the Kalam Cosmological Argument‘(2015). In that, Cohen agrees with Morriston that if a beginningless past is an actual infinite, then so is an endless future, and thus if a beginningless past is impossible for being an actual infinite, so is an endless future. Cohen also argues that if the grim reaper argument shows that the past has to be finite, then a parallel version shows that the future must have an end as well. His argument is critiqued by Jacobus Erasmus in a paper called Cohen on the Kalam Cosmological Argument (2016). Erasmus’ rebuttal is that Cohen’s version of the grim reaper argument presupposes that it is possible for God to actualise an ungrounded causal chain, which can be plausibly denied.

I think that the ‘dry eternity’ paradox escapes Erasmus’ reply.

    1. Two versions of the Grim Reaper paradox

Here is how Erasmus sets up the grim reaper paradox:

Suppose that the temporal series of past events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each past moment in time, a unique Reaper was assigned to issue a death warrant iff no previous Reaper had already issued a death warrant. (Cohen on the Kalam Cosmological Argument, p. 52)

This results in a contradictory state of affairs. Firstly, for all times tn, there must have been a warrant issued prior to tn. That’s because if no warrant had been issued by tn-1, then the reaper at tn-1 would issue their warrant (resulting in a warrant going out prior to tn).

But, this same reasoning also applies to tn-1 itself, giving us the contrary proposition. That’s because we can also say that the warrant won’t be issued at tn-1, because if it had not been issued by tn-2 it would have been issued by the tn-2 reaper (i.e. before tn-1).

Thus, we have both that the warrant must have been issued at some time prior to tnbut also that there is no time prior to tat which it could be issued. Contradiction.

Cohen applies this to the endless future. All we need to do is change the relevant tenses in Erasmus’ quote from above to get the following:

Suppose that the temporal series of future events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each future moment in time, a unique Reaper is assigned to issue a death warrant iff no future Reaper will issue a death warrant.

But now we can derive a mirror image contradiction. Suppose that the reaper at t0 checks to see if any future reapers will issue warrants or not. At this point there are two options:

i) no future reaper will issue a warrant (in which case the treaper issues theirs)

ii) some future reaper will issue a warrant (in which case the treaper does nothing)

Suppose that at tno future reaper will issue their warrant, meaning that the reaper at t0 can issue theirs. If it were the case that no future reaper issues a warrant at t0, then, in particular, it is also true that at tno future reaper will issue a warrant (consider: if it it true today that I will never drink again, then it will also be true tomorrow that I will never drink again). But if it is true at tthat no future reaper will issue a warrant, this would mean that the reaper at tdoes issue their warrant! And, plainly enough, the reaper at tis in the future of the reaper at t0. So if, at t0, no future reaper will issue a warrant, then some future reaper (such as the one at t1) will issue a warrant! Contradiction.

Let’s take the other horn. Suppose at tsome future reaper will issue their warrant, meaning that the reaper at t0 can stand down. Let’s suppose it is the reaper at t1. Then we are right back to the beginning of the first horn again. For the reaper at twill only issue their warrant if none of the future ones will. But if it is true at tthat none of the future ones will, then this is also true at tas well, resulting in that reaper issuing their warrant which in turn brings about another contradiction exactly like the one from above.

Cohen discusses two objections that Koons posed to him in correspondence. The first of these is that in Koons’ version of the paradox, reapers are sensitive to what past reapers have done, but in Cohen’s version they have to be sensitive to what future reapers will do; yet it isn’t possible to have causal sensitivity to future events in the same way as to past ones. The reply Cohen makes to this is that an omniscient God could communicate the future to the reapers such that they know what the others will do, thus overcoming this causal asymmetry. Koons’ second point is that it isn’t possible for God to create beings who embody his omniscience. Even if that is true, the reapers themselves do not have to be omniscient (and can be quite ignorant of, say, how many coins I have in my pocket), just so long as God ensures that they know the behaviour of future reapers. In addition, Cohen points out that Koons’ reason for thinking that the reapers cannot embody omniscience has to do with avoiding causal loops, but it is not clear that there are any causal loops as such in this story (the behaviour of reaper n+1 does not depend on the behaviour of reaper n, etc). Thus it is far from clear that Koons has a successful reply here. One could avoid this by denying the possibility of an omniscient being that knows the future and can communicate it to reapers, of course, but a theist (in particular a Christian theist) will be unlikely to pick that option.

2. Erasmus’ objection

Erasmus’ objection comes at this from a different angle. He says that Cohen’s version of the grim reaper paradox (GRP) requires the following two principles:

K1. It is possible for God to predetermine an endless future

K2. It is possible for God to actualise an ungrounded causal chain

An ‘ungrounded causal chain’ “has a non-well founded relation (xRy, zRx, zRv, wRv, … ) because the chain lacks a first cause” (Erasmus, 2016, p 53). The behaviour of the reaper at t0 is determined by (or grounded in) the behaviour of reapers that are in its future. But the behaviour of the reapers in its future, such as the one at at t1, are themselves determined by (or grounded in) the behaviour of reapers in the future of them as well. Thus there is no ‘first cause’, or grounding, for the behaviour of the reapers. Let us suppose that this is an ungrounded causal chain, and that it is also (metaphysically?) impossible for God to actualise such a causal chain.

He then goes on to show that K2 is doing all the work in generating the paradox because it also applies to ungrounded causal chains that are purely spatial in character. Here is his spatial version of the example:

For example, suppose that time had a beginning and has an end. Accordingly, the predetermined series of future events is finite. Suppose further that space is inhabited only by an actually infinite row of successive Grim Reapers such that (1) there is a first Reaper but no last Reaper, (2) each Reaper is located at a unique spatial point, and (3) all the Reapers are facing the same direction. Now, suppose that God has predetermined that, at noon tomorrow, each Reaper will swing his scythe iff no Reaper in front of him swings his scythe. Accordingly, the same contradiction as above will result at noon tomorrow, namely, regardless of whether the first Reaper swings his scythe, it is both true and false that some Reaper in front of the first Reaper swings his scythe. The contradiction disappears, however, if (K2) is false. (Ibid)

Erasmus’ conclusion then is that denying K1 is not enough to block the contradiction, as it reappears in the spatial case. But denying K2 blocks both contradictions, and as such K1 is not the offending assumption. In effect, he is saying that Cohen’s GRP doesn’t show that the future must have an end. Rather, it just shows that God cannot actualise ungrounded causal sequences.

3. The Dry Eternity Paradox

Now is time to present my version of the paradox that does works even if K2 is false. It does not require that God actualises any ungrounded causal sequences. All that it requires in addition to K1, is one additional assumption:

K3. God can act based on his (presently available) knowledge of future events.

Suppose God has decided to undertake an infinite version of dry January. That is, he has decided to stop drinking (say) holy water forever. However, he enjoys a drop of holy water (who doesn’t?), and wants to to have one final sip. Accordingly, he determines to obey the following rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, he will celebrate with his final drink. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a drink all day).

Again, we are caught in a dilemma:

Firstly, suppose that, at t0, God consults his comprehensive knowledge of the future, and discovers that he never again drinks after t0. He immediately downs a shot to celebrate (who wouldn’t?). But in that case, when he does his check the next day, at t1, he then will (again) discover that he will never have another drink, and immediately pour himself a drink to celebrate! So even though he rewarded himself yesterday for never having another drink, he is having another drink! Contradiction.

On the other horn, suppose that, at t0, God consults his comprehensive knowledge of the future, and discovers that he does indeed have a drink at some day after t0. Accordingly, he doesn’t celebrate by having a drink on t0. But in that case, there must be some future day at which he has a drink. Suppose it is t1. In that case, it must be that at t1 God will check to see if he will have any subsequent drinks, and find that he will not, resulting in him pouring the last drink. But now we are back at the start of the first horn, because his check at at twill also reveal a dry eternity ahead, at which point he will reward himself with another final drink! Contradiction again.

So we clearly have the exact same paradox again. This time however, it is not clear that God has actualised an ungrounded causal chain. After all, at each day God knows the future, and can merely consult his own (presently available) knowledge to see what happens in the coming days. We can imagine him writing it all down in a big book and every day he consults the book. Whatever causal story that happens each day that he consults the book, it is not clear that it is an ungrounded causal chain.

4. Replies?

One might deny that God can check his own knowledge to see what he knows about the future and act on it (K3). This would be weird. Why can’t God do that? Does he not know what he foreknows? Is he repressing it? Can he not act on what he knows? He seems to act on his foreknowledge on most versions of theism (specifically any where he has a plan, or reveals the future in prophecy, etc). Denying K3 leaves only the most austere versions of deism, it seems to me. Christianity seems hard to reconcile with its denial in any case.

Objecting to the possibility of the book doesn’t help unless it is really an objection to God’s omniscience, which a theist probably isn’t going to opt for (apart from Open Theists). Denying the possibility of an omniscient being would avoid the paradox of dry eternity though.

One could avoid the problem by denying the possibility of an endless future. The whole point of Morriston’s original reply to Craig was to say that if the past must have a beginning, then the future must have an end. This would vindicate Morriston’s challenge against Craig. It would show that either time has both a beginning and an end, or no beginning and no end, but that there is no third option.

The seemingly only other target we can find is the rule that God undertakes to obey. Perhaps it is not a proper rule, and that somehow it isn’t possible for God to undertake to obey it. Yet this seems rather strange. Consider this similar rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, will celebrate with a chocolate bar. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a chocolate bar).

Nothing paradoxical follows from this rule. Obeying this rule means that God drinks every day up to the day when he has his final drink, after which he eats chocolate bars every day. God can obviously follow that rule.

But what could stop him from undertaking to follow the rule obtained by merely swapping out the word ‘chocolate bar’ with ‘have his final drink’? Of course, it would lead to contradiction if he were to go this route, and that is a reason to think that it is (somehow) metaphysically impossible for him to swap those words around and undertake to follow the resulting rule. On the other hand, that is just to say that this is one of the things that could be denied to avoid the paradox. It doesn’t motivate thinking that it is impossible. We could ad hoc postulate anything is metaphysically impossible to avoid any paradox.

Something has to go to avoid the paradox of dry eternity.

17 thoughts on “The paradox of dry eternity”

  1. Here is my abbreviated take on the subject.

    1. Actual infinities are logically impossible.
    2. The past is actual.
    3. Therefore an infinite past is logically impossible.

    4. The future is not actual.
    5. An infinite future is not logically impossible on the grounds of being an actual infinity.

    7. God’s thoughts, plans, and knowledge are actual.
    8. God’s thoughts, plans and knowledge are not infinite.
    9. If the future is infinite, God has not thought about all of it, planned all of it and does not know all of it.

    10. The Bible teaches that the future is infinite (eternal life).
    11. If the Bible is true than at least some of the future is yet to be determined.
    12. If the Bible is true than at least some of the future is open.

    As I have grown as a Christian, I have had many reasons to reject determinism, but spent most of my early years believing in absolute divine foreknowledge. In recent years, I have come to a version of what has been called “Open Theism”. The same reasoning that led me to the cosmological argument, led me to Open Theism.

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    1. “1. Actual infinities are logically impossible.
      2. The past is actual.
      3. Therefore an infinite past is logically impossible.”

      There is a subtle but widespread misunderstanding here. The term ‘actual infinite’ doesn’t mean ‘infinite and actual’. It means something such that a proper part of it is equivalent to the whole it (like the way the even numbers are equinumerous with the natural numbers). We should probably use a different term, like ‘completed infinity’. So a completed infinity can be made up of things that are mere potentialities, and doesn’t have to just have the property of being made up of actual things. Imagine I start counting whole numbers and you ask me at some point how many I have still got left to count. Those future counting events have not happened yet (so they are not ‘actual’ yet) but we can think about how many of them there are, and it is clear that there is a completed infinity of them (we could line up proper parts of the numbers I haven’t yet counted with the whole of them and see that they are equivalent, like ‘every other number I have left to count’ and ‘every number I have left to count’ are equivalent to each other). The question is whether anything that concretely exists could have the property of being such that proper parts of it are equivalent to the whole.

      “4. The future is not actual.
      5. An infinite future is not logically impossible on the grounds of being an actual infinity.”

      Well, if the future has no end, then there are infinite future events still left to happen, and that is an actual infinite (a completed infinite), as the counting example above shows.

      “7. God’s thoughts, plans, and knowledge are actual.
      8. God’s thoughts, plans and knowledge are not infinite.
      9. If the future is infinite, God has not thought about all of it, planned all of it and does not know all of it.”

      You say you are an open theist. That means that future contingents have no truth value. It doesn’t mean that there are true propositions about the future that god doesn’t know. Open theism is compatible with god’s knowledge being infinite (he could know all mathematical truths), and with him knowing everything there is to know about the future. For me, the problem with open theism (or any view according to which future contingents have no truth value) is the semantic relativism that comes with it. I say ‘the coin will land heads’, which has no truth value because it is a future contingent. So what I said was neither true nor false. But then it lands (say) heads and I say about my previous sentence ‘what I just said was true after all’. So it wasn’t nether true nor false, it was true. The best account of this is from John McFarlane, and on that truth is relativised to two points in time and what I said changes value from neither to true as time passes. Much simpler to just say it was true but I didn’t know it at the time.

      “10. The Bible teaches that the future is infinite (eternal life).
      11. If the Bible is true than at least some of the future is yet to be determined.
      12. If the Bible is true than at least some of the future is open.”

      It depends on what you mean by ‘open’. If it means that there are some propositions, p, such that it is possible that it will be that p and it is possible that it will not be that p, then that is compatible with the future being fully known by God. But more importantly, if the bible teaches that there is eternal life, then it teaches that the future is a completed infinity, which is no more or less absurd than the infinite past.

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  2. In his article, Erasmus made a difference between future days being actual infinite and potential infinite. Your argument seems to work on actual infinite future days. Is there a version of your argument which also works in predetermined potential infinite future days? Because the view that potential infinite number of future days exist seems to me what defenders of the kalam advocate for. Sorry for my bad english.

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    1. Erasmus’ version of the Grim Reaper Argument presupposes that time is actually infinite:

      “Suppose that the temporal series of past events is actually infinite and that an actually infinite number of Grim Reapers exist.” (p. 51)

      On the question of whether a defender of the kalam is an advocate of the actually infinite future, or a merely potential infinite future, anyone (whether they advocate the kalam or not) who thinks that the future has no end to it is an advocate of the actually infinite future. As time passes, more and more days become present and then past, and the number of days that have gone through that is ever increasing. It is an always increasing but always finite accumulation. Imagine that starting today, each day that passed I put a new grain of sand onto a pile. This pile would get ever bigger, without limit, but always remaining finite. The pile would be potentially infinite. But it would correspond not to the future, but to the past. Each grain would stand for a day that is now past. The future days would be ahead of me still, and there would not be a potentially infinite amount of them , but an actually infinite amount of them. No matter how big our pile of sand is, there is always the same amount of time still to come (if time has no end). And that’s just like Hilbert’s hotel. If we escort guests out of the hotel one by one and make them stand in a line, that line will be ever increasing without limit, but the amount of guests still left in the hotel will always remain the same; there will always be an actually infinite amount of guests left. So unless you think that time has an end, you think that the future is actually infinite. And if my paradox shows that this is impossible, then time must have an end at some point. And if my paradox is faulty for some reason, so is the original grim reaper paradox. The endless future and beginningless past stand or fall together.

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    2. Why doesn’t it work for predetermined potential infinite future days? It seems to me the only way to escape the dry paradox is to say that the future will end. If the future will end, then the future isn’t potentially infinite.

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      1. Well, the difficulty is in expressing what we mean by the potential infinite carefully. When we do it doesn’t mean what Craig has convinced so many people it means, it seems to me.

        The confusion is because if time has a start, but no end, then we can think of the number of days that *has been* as a potential infinite. We could count them off each day, and the number we would have counted would always remain finite, but would be increasing without limit (with infinity as a limit). Craig thinks this means that the future is potentially infinite, but that’s confused. It is the past that is potentially infinite (it is the past that is growing as time passes, without limit while remaining finite). The future, in contrast, remains infinite here. No matter how many days have passed, if time has no end, then there is always the same amount of days left to come; like taking guests out of Hilbert’s hotel one by one, there is always the same amount left to escort out.

        So I think it is just confused to even talk about the future being endless but not actually infinite.

        Interestingly, if the future has an end (at t0), and the past doesn’t, then we could think about the future as potentially infinite. Here is a function, F(x), which takes t values as its input and returns the interval between that and t0. So if x = t-195, then F(x) = 195. This second value is how much future is left at a given moment of the infinite past. So at time -195, there are 195 moments left to come. As we decrease the value of x (which we can without limit) the value of F(x) increases (also without limit, or with infinity as a limit if you like). The amount of future each moment in the past has increases the further back in time we go, but is always finite. In this way we can cook up a version of the future being genuinely potentially infinite, but *only on the assumption that the past is beginningless and the future has an end*! 🙂

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      2. My thought is that their view on potential infinity can be granted, and I don’t think the paradox goes away, for on their own definition of potential infinity, an ending future isn’t potentially infinite. You can think of it like an internal critique. So I don’t understand doubtful’s thought that your dry paradox wouldn’t apply Erasmus’ view.

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  3. Ok then. If I shelve my worry about an endless future being an actual infinite, it doesn’t work merely because a potential future is still an endless future. Ok.

    I’m not satisfied with that because there is no analysis of what a potential future is, other than the one I gave above. So we are left without a clear idea of what the term is supposed to mean, apart from endless. But I guess that’s enough.

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  4. My point is that, in Craig and Loke’s view , as far as I understand, number of predetermined future days are actual infinite, but in Erasmus’ alternative view, number of predetermined future days are potential infinite. So when you ask, what is the number of the days that aren’t predetermined? The answer is zero because future days which aren’t predetermined dont exist. Or when you ask, what is the number of days that aren’t created? We can say the answer is equal to number of predetermined days and there are potential infinite number of them. Erasmus also doesnt support the infinity arguments, but he comes with an alternative undertanding of future. I guess he is thinking like if some theists want to defend the grim reaper paradox against the atheists who defend the view that past is actual infinite, those theists must give up the clasic undertanding of theism’s future view and accept his alternative view because he is aware that grim reaper paradox is also problematic for theism that holds future days as being actual infinite. So in his alternative view theists can champion grim reaper argument against atheists without any problem. In Craig and Loke’s view ,as far as I understand, future days exist as abstract objects so number of them are always infinite. But in Erasmus’ alternative view number of abstract objects are potential infinite because number of predetermined days are also potential infinite and they are equal to each other. One can ask whether non-predetermined future days are abstract objects? In his view, they are not. Non-predetermined future days simply do not exist. They are neither in concrete nor in abstract form. So we can ask what is the number of days that God will predetermine or create? The answer is potential infinite because number of created days always will be potential infinite. So it seems to me, in Erasmus’ alternative view, there is no question that has an answer which include actual infinite.
    But my question is that, is potential infinity really safe from paradoxes? When we look at literature there are many paradoxes for actual infinities but I don’t know whether if paradoxes for potential infinite exist. So I ask, is there a version of dry eternity paradox which is problematic for predetermined potential infinite future days, or is there a paradox which is problematic for potential infinities in general?

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    1. Craig’s position is that the endless future is merely potentially infinite (see: https://www.reasonablefaith.org/writings/scholarly-writings/the-existence-of-god/taking-tense-seriously-in-differentiating-past-and-future-a-response-to-wes/). I have a paper coming out soon with Wes Morriston where we show why that doesn’t make sense though.

      Loke thinks that future events are abstract objects (see: https://philpapers.org/rec/LOKOBP). But Craig is a nominalist and so doesn’t really believe in abstract objects at all.

      I’m not 100% sure, but from conversation with Erasmus he might think that the endless future is merely potentially infinite. I don’t think he has seen me and Wes’ paper yet though, so he might change his mind 🙂

      In any case, if the idea is that postulating a merely potentially future escapes the paradox, I want to see how that works. As Hugh suggests above, if it means the future has an end, then it would escape the paradox but at quite a high cost (also it’s not clear to me how a future with an end can be potentially infinite rather than just finite). But we need an analysis of what the words ‘potentially infinite” means in this context. Unlike the actual infinite, there is no canonical mathematical representation for the potential infinite. Until someone cooks something up (like I did in a comment above) we are just throwing words around with nothing behind them when we say ‘potentially infinite’ here. If you don’t like my analysis, what is a better one? I’m all ears for someone to present one.

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    1. (Specifically, his casual finitism thesis rules out these kinds of paradoxes, and plausibly rules out an infinite past, without ruling out an infinite future.)
      (Also, I suspect a more careful account of God’s foreknowledge and contingent will would put up some roadblocks to your dry eternity paradox as well.)

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      1. I have got Pruss’ book, although I need to study it more.

        But I’m not sure the dry eternity case involves causation at all. Are God’s actions caused by his foreknowledge? They are determined by it, but I’m not sure it’s causation necessarily.

        Here is another version of this paradox from Erasmus:

        At each time t, God says “everything I will say will be false”

        The relationship between the propositions that God expresses and the truth values they have is semantic, and not one of causation (the dog being on the floor doesn’t *cause* the proposition ‘the dog is on the mat’ to be true). So it seems to me that the connection to causation isn’t essential to these paradoxes.

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      2. Pruss considers, in one or two places in his book, the extension of his causal finitism to cover similar but non-causal explanatory relationships as well, with God’s foreknowledge specifically in view in some od the paradoxes.

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      3. Interesting. I’ll have to have a look at what he says. The second version I put above doesn’t even require God to have foreknowledge though, and only utilises semantic determination. It would be interesting to know how causal finitism covers that case.

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  5. Thanks for writing these accessible articles!
    Consider a contingent future event e such that, for some continuous interval I, every element of I corresponds to some possible outcome of e. Is someone (like W. L. Craig) who believes that it is metaphysically impossible for an actual infinite to exist committed to denying that e is metaphysically possible?
    The following simple argument is familiar:
    (1) Suppose that X is a continuous uniform random variable over I.
    (2) ∑P(X=x) = 1.
    (3) P(x=a) ≥ 0, for every a∈I.
    (4) If P(x=a) = 0, then ∑P(X=x) = 0.
    (5) If P(x=a) > 0, then ∑P(X=x) > 1.
    (6) Therefore, there can be no CURVs.
    Of course, the Kolmogorov axiomatisation gets around this problem and we hold that P(X=a) = 0 since we accept that ∫f(x)dx = 1. However, I feel that the opponent might reply that there is no parallel move to be made in the real-world case; P(X=a) just cannot be zero if ‘a’ corresponds to a possible outcome.
    I think there’s some kind of parallelism between, “If P(x=a) = 0, then ∑P(X=x) = 0” and, “If no grim reaper is responsible for killing Fred, then the (mereological) sum of grim reapers is not responsible for killing Fred.” Is there a relation?

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