Grim Reaper Paradoxes and Patchwork Principles | Ft. @Friction

Did you know that light bulbs could disprove the Kalam cosmological argument? Well, it's true, and in this video, we're going to explain.

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Okay, obviously, I'm being a bit factious. It's not light bulbs that will be our focus, but instead, objects very much like light bulbs. And they don't necessarily disprove the Kalam. Instead, we think the challenge they pose is plausible, philosophically interesting, and advances debates about the Kalam into fascinating new territory. But that doesn't grab your attention as well, does it?

So, I've been saying "we" and "are" because today, I'm co-presenting with Troy from Friction Philosophy. Our paper, "Grim Reaper Paradoxes and Patrick Principles: Severing the Case for Finitism," was recently accepted for publication at the Journal of Philosophy. And here, we'll be presenting the paper's arguments in a more popular form. You can find a link to the paper in the description, and you can also find Troy's channel linked in the description. I highly recommend checking it out. He hosts lots of excellent discussions with philosophers, and we'll even be making another video over on his channel discussing different aspects of our paper and replying to comments on this video. So, let's turn it over to Troy to outline today's video.

All right, so we'll start by explaining Benard paradoxes, of which the Grim Reaper Paradox is one variant. We'll then explain how these are used to support the Kalam cosmological argument, with a special emphasis on Rob Koons' Grim Reaper argument. We'll then raise two objections to the Grim Reaper argument. The first objection is a new kind of "companions in guilt" style argument based on a new paradox involving light bulbs of a special sort. The second objection appeals to a principle about exact duplication to show that one of the premises in the Grim Reaper argument is false.

So, Benard paradoxes are a family of paradoxes that trace back to Jose Benard and his book, *Infinity: An Essay in Metaphysics*. We'll explain this family of paradoxes by way of example. So, imagine there's this guy named Fred, and there are grim reapers bent on killing him. Each grim reaper has their own set time to kill Fred and will succeed in doing so if, of course, you know, Fred is alive at that time. Suppose, moreover, that there isn't just one grim reaper with this murderous intent, but infinitely many. Numbering the reapers, we suppose that Reaper 1 is set to kill Fred at 12:30 p.m., Reaper 2 at 12:15, Reaper 3 at 12:07 and 30 seconds, and so on for each of the infinite many reapers with times getting ever closer to noon, to 12 p.m. In general terms, Reaper n is set to kill Fred if Fred is alive at 1/2 to the power of n hours past noon. Alternatively, we could suppose that the reaper designated times are evenly spaced, one day apart, throughout an infinite past. That is to say, Reaper 1 is set to kill Fred yesterday, Reaper 2 is set to kill Fred two days ago, Reaper 3 three days ago, and so on, where Reaper n is set to kill Fred n days ago.

Now, both of these scenarios actually result in a paradox. Fred surely cannot survive today or to today, since if he was alive at Reaper 1's time, well, Reaper 1 would have killed him. So, some reaper must have killed Fred. But it also doesn't seem that any particular reaper can kill him at their designated time, since that would require that he was alive at the earlier reapers' designated times and so would have been killed by them instead. In other words, if we supposed that Reaper n killed Fred for some arbitrary n, then he must have been alive at least at Reaper n+1's time. But then Fred would have been killed by Reaper n+1. And if he's killed by Reaper n+1, then Reaper n finds a dead Fred and hence does not kill him. So, Reaper n does not kill Fred after all. Since the number n was arbitrary, you know, it could be true of any particular reaper in the sequence, it follows that none of the reapers kill Fred. Consequently, we can infer both that some reaper must have killed Fred, um, but also that none of the reapers killed him, and that's, of course, a contradiction.

Now, talking of grim reapers killing Fred is not essential to Benard paradoxes. You can get Benard paradoxes using all sorts of entities with all sorts of actions to be performed. The point of raising this specific Benard paradox, often called the Grim Reaper Paradox, is to simply illustrate one example of the general form of Benard paradoxes that can be concretely filled out in various ways. What's common to all Benard paradoxes is a kind of abstract structure, which can be characterized as satisfying two conditions. The first is the unbegun condition, or UC. According to this condition, there's a sequence of things with no first member. For example, there's a sequence of reapers with no first reaper. The second condition is the "at if and only if nowhere before" condition, or A&BC. According to this condition, each member in that sequence satisfies some condition if and only if no earlier member in the sequence satisfies that condition. For example, each reaper in the sequence kills Fred if and only if none of the earlier reapers kill Fred.

It's important to highlight that UC and A&BC cannot be true of the same sequence. It doesn't matter how we fill in the details regarding the members of the sequence, the conditions the members satisfy, and so on. This scenario will be inconsistent precisely because it has this inconsistent abstract structure. This is a point that's very clearly and convincingly made in Nicholas Shackel's 2005 article, "The Form of the Benard Dichotomy."

Now, you might, of course, question whether the Grim Reaper Paradox actually satisfies this abstract structure. Since, while it does satisfy UC, it may not be immediately obvious that it satisfies A&BC. After all, the task of any given reaper is to kill Fred if and only if he is alive at its designated time, not to kill Fred if and only if no earlier reaper killed him. However, if we add in the assumptions that Fred is killed only if he is killed by a reaper, and Fred is alive at some time if and only if he wasn't killed prior to that time, then we can straightforwardly deduce that each reaper kills Fred if and only if no reaper earlier than that reaper kills Fred. And also, if we don't add these or equivalent assumptions, then we actually don't get any contradictory paradox on our hands. For instance, something other than a reaper might have killed Fred before the reapers, in which case each reaper just does nothing and no contradiction arises. Or Fred might miraculously resurrect in between each reaper's designated time, in which case each reaper ends up killing Fred, and again, no contradiction arises. So, if we want to actually get a contradictory paradox, we need to add these assumptions so that both UC and A&BC, or their equivalents, are satisfied.

Now, at this point, you might be thinking, "Hey, these paradoxes are an interesting curiosity, but like, what on Earth do they have to do with the Kalam?" Well, the relevance is that several philosophers, including Rob Koons, Alex Pruss, and Jacobus Erasmus, have forwarded arguments from Benard paradoxes for various finitist theses. A finitist thesis is just a thesis that says some kind of infinity is impossible. One finitist thesis is causal finitism, which says that it's impossible for anything to have infinitely many causes. Another finitist thesis is temporal finitism, which says that infinite pasts are impossible. If we could use Benard paradoxes to argue for causal or temporal finitism, we could then use one of these finitist theses to motivate the claim that the universe began to exist, thereby motivating the second premise of the Kalam. The Kalam, of course, says that whatever begins to exist has a cause. The universe began to exist. Therefore, the universe has a cause. At least, that's the first stage of reasoning in the Kalam cosmological argument. And so, you can see how someone might attempt to wield these finitist theses on behalf of that second premise, that the universe began to exist.

Of course, technically, we need additional assumptions to deliver that conclusion, the conclusion that the universe began to exist. For instance, technically, we need to add that nothing physical exists timelessly, sans, or without the beginning of metric time. Also, that nothing physical existed in a non-metric time prior to the beginning of metric time. And also, that nothing physical exists timelessly, simpliciter. And maybe even that the A-theory of time is true, depending on our analysis of what it is to begin to exist. But we'll leave these additional claims aside for now. Just note that the beginning of the universe does not by itself follow from temporal or causal finitism, even though they may help in establishing it. So, that, in short, is why Benard paradoxes are relevant to the Kalam.

So, how would we argue for a finitist thesis from Benard paradoxes? The general form of such an argument would go like this: Premise 1: If a given finitist thesis is false, then Benard paradoxes are possible. Premise 2: Benard paradoxes are impossible. Premise 3: So, the given finitist thesis is true. For example, we might argue that if there could be the sort of beginningless temporal sequences required for the Grim Reaper paradox stated earlier, then that Grim Reaper paradox would be possible. We could then conclude that since that Grim Reaper paradox is not possible, that there cannot be, uh, unbegun temporal sequences of those sorts.

To us, the first premise appears quite implausible on its face. There doesn't appear to be any inconsistency or metaphysical absurdity in the denial of these finitist theses. There are, as far as we can tell, possibly unbegun temporal sequences, and there are not possibly any Benard scenarios, and so Benard scenarios are not possibly realized in any situation involving an unbegun temporal sequence. But surely, there can be other unbegun temporal sequences that involve no absurd scenarios like this. Perhaps there are other reasons to affirm certain finitist theses, but this just doesn't seem to cut it. Nevertheless, those who forward these sorts of arguments for finitist theses based on Benard paradoxes, um, we'll call these "B arguments" for short, offer more in the way of supporting the conditional premise, um, premise one from above.

In our paper, we focused on Rob Koons' presentation in his 2014 article, "A New Kalam Argument: Revenge of the Grim Reaper." Therein, he offers a Grim Reaper argument for temporal finitism and for the view that there cannot be beginningless sequences of ever smaller intervals of time scrunched into a finite period of time. Koons' Grim Reaper argument will be our focus going forward.

So, in the Grim Reaper argument, or GR for short, there are four premises, P1 through P4, that are argued to be jointly inconsistent with the assumption for reductio, labeled H1, that it's possible for there to be beginningless sequences of ever smaller intervals of time scrunched into a finite period of time. So, the general idea is that given premises P1 through P4, it follows that H1 is false. A second version of the GR uses premises P1 through P3 to argue for the falsity of a different assumption, H2, according to which an infinite past is possible. And so, again, the argument here would basically be: given premises P1 through P3, H2 is false. In other words, there can't be infinite pasts.

The two criticisms we level toward the GR target both versions because they only concern the first three premises, P1 through P3. So, for simplicity, we'll actually only be focusing on the second version of the GR going forward, which targets the possibility of infinite pasts. We can state the relevant premises of the GR as follows, leaving out the fourth premise about spacetime's arbitrary compressibility, since it's relevant to our ensuing discussion, and we can just grant Koons that assumption for the sake of argument.

The first premise states that an individual reaper is possible. And so, you can see on the slide, P1, which you are calling "Possible Reaper," says that some possible spacetime region contains a reaper with the power and disposition to place a particle at a certain location if and only if no particle has been placed in that location.

The second premise is the Patrick Principle. Note that in stating this principle, "quote-unquote spacetime region" refers both to the spacetime region itself and also the contents or occupants of that region. That is, the various things that exist within the spacetime region. So, then, here's that second premise, P2, called the Patrick Principle. This principle says that if there's a possible world with enough spacetime room to fit an arbitrary arrangement of individually possible spacetime regions, then there's a possible world in which exact intrinsic duplicates of those regions coexist together in that arrangement.

Now, I know that that sounds a little bit complicated, but let us break this down for you because it's easy to understand once you break it down. The idea here is that we can recombine and rearrange any collection of spacetime regions that are each individually possible in such a way that the regions can all jointly exist together within a given arrangement, assuming, of course, that some possible world's spacetime has enough kind of spatial and temporal room to be able to fit or accommodate that arrangement of regions. For instance, there's an individually possible spacetime region whose spatial boundaries are the boundaries of my water bottle and whose temporal duration is 24 hours. There's also an individually possible spacetime region whose spatial boundaries are the boundaries of a pool minus a spatial region exactly the size of my water bottle in the middle of it, and whose temporal duration is likewise 24 hours. Finally, there's a possible world whose spacetime is at least 24 hours long and whose spatial size is large enough to fit a pool containing a water bottle inside. Consequently, there is a possible world with enough spatial-temporal room to fit a particular arrangement of our two individually possible spacetime regions that you can see on the slide, namely, the arrangement with the pool and the water bottle filling the water bottle-shaped cavity inside the pool. By the Patrick Principle, it follows that there's a possible world in which duplicates of those regions exist in precisely that arrangement. In other words, it's possible for there to be a water bottle that is intrinsically exactly like mine, submerged in a pool. As we hope this example illustrates, the Patrick Principle is actually not that complicated. It just allows us to take individually possible spacetime regions, combine them together in various arrangements. The principle then allows us to infer that duplicates of those regions can actually coexist together in those arrangements, so long as those arrangements can actually fit within some possible spacetime.

Here are some other applications of the Patrick Principle. A room containing a talking donkey is possible. A room containing Donald Trump is possible. So, since there's a possible world with a spacetime large enough to fit two side-by-side rooms, it follows by the Patrick Principle that there's a possible world in which a duplicate talking donkey and a duplicate of Donald Trump exist in side-by-side rooms. So, that again is the Patrick Principle. Basically, it allows us to take individually possible regions, recombine them and rearrange them in various ways, and it allows us to infer that duplicates of those regions can all coexist together in those arrangements, provided, again, there's some possible spacetime with enough spatial and temporal room to be able to fit the relevant arrangement.

Now, consider the third and final premise, P3, which is what we're calling the intrinsicality of powers and dispositions. This premise says that the powers and dispositions ascribed to each reaper are properties intrinsic to that reaper in its corresponding region. Likewise, whether those powers and dispositions are realized or manifested or exercised is intrinsic to a reaper in its corresponding region. Here, a reaper's intrinsic properties in a given spacetime region are those which the reaper possesses independently of what happens in other regions. For instance, the reaper has a certain shape, and its having this shape does not depend on what happens in other regions elsewhere in the world. Whether the reaper has its shape is not a function of what goes on in other regions. By contrast, whether this reaper is the tallest reaper in the world, that does depend on what happens in other regions. It depends if there are other reapers in other regions which are taller than this particular reaper. And so, this feature would be extrinsic to the reaper in question. But returning to shape again, whether the reaper has its shape is not a function of what goes on in other regions. Like in the case of a reaper's being the tallest reaper that there is, according to P3, the same thing is true of the realized powers and dispositions of a reaper. Both its power and disposition, as well as whether that power and disposition are realized, is intrinsic to a reaper in its container region. Whether a reaper has or realizes its power and disposition does not depend on what happens elsewhere in the world, according to P3.

And finally, we can state the assumption for reductio as follows, which is that an infinite past is possible. By the way, a reductio is basically a form of argument where you assume a hypothesis to be true, then you go on to derive a contradiction, and then you infer that your original assumption is false. So, again, we can state the assumption for reductio as follows, which is that an infinite past is possible. This assumption for reductio says that some possible world contains infinitely many ever earlier temporal parts, or in simpler terms, an infinite past is possible.

So, here is how the GR proceeds. Assume that H2 is true. We will then derive a contradiction from this assumption, given our premises P1 through P3, from which it will follow that our assumption for reductio is false. In other words, H2 is false, and so the past cannot be infinite.

So, let's tease this out. So long as reapers are individually possible, as per P1, it's pretty clear that there will be infinitely many possible worlds, each of which has a region containing one reaper that realizes its power and disposition to place a particle at a certain location if and only if no particle has been placed in that location. But now consider the arrangement of spacetime regions consisting of infinitely many of these individually possible reaper-containing regions in a beginningless sequence stretching back into an infinite past. Now, by H2, some possible world has an infinite past, and hence there is a possible world with enough spatio-temporal room to be able to fit the aforementioned beginningless arrangement of reaper-containing regions. By the Patrick Principle, stated in P2, it follows that there's a possible world in which intrinsic duplicates of those regions coexist together in precisely that beginningless arrangement. But by P3, the realized powers and dispositions of reapers are intrinsic to them. And hence, the possible world that we've just inferred using the Patrick Principle, containing a beginningless sequence of intrinsic duplicates of reapers which realize their power and disposition, this world will be such that each of the reapers in the beginningless arrangement in the world all realize their power and disposition to place a particle at a certain location if and only if no particle has been placed at that location. So, at least assuming that only reapers create and place particles, it then follows that there's a possible world in which there's a beginningless sequence of reapers, each of which places a particle if and only if no earlier reaper places a particle. But the sequence of reapers in this possible world satisfies both UC and A&BC from earlier, right? It contains a sequence of reapers with no first member, thereby satisfying UC, and each reaper in the sequence places a particle if and only if no earlier reaper places a particle, thereby satisfying A&BC. So, there's a possible world in which a sequence satisfies UC and A&BC. But alas, there is no such possible world, since any such world contains a contradictory Benard paradox. This is what we saw from earlier. The conditions UC and A&BC are jointly inconsistent.

So, then, we've now reached a contradiction. They both is such a possible world. We infer that using the Patrick Principle and premises P1 through P3, together with the assumption for reductio, that there's a possible world with an infinite past, but also there can't be such a possible world because it instantiates a Benard paradox. It satisfies conditions UC and A&BC. So, we've reached a contradiction. There both is and is not a possible world containing this sequence of reapers. So, it then follows that at least given premises P1 through P3, it follows that our original assumption for reductio is false. In other words, H2 is false. Infinite pasts are impossible.

So, that is the Grim Reaper argument, or GR. It's an argument from premises P1, P2, and P3 to the conclusion that infinite pasts are impossible. And as we said earlier, another variant of the GR takes these three premises, adds another premise about the arbitrary compressibility of spacetime, and concludes that it's not possible for there to be a beginningless sequence of ever smaller intervals of time scrunched into a finite period of time.

Now, in what follows, we'll develop two problems for the GR. Note that we're setting aside lots of other criticisms of the GR that I myself have published. For more on those, you can see my *Synthese* article, my *Philosophical Studies* article, article with Alex Malpass, my single-authored *Mind* article, my *Mind* article with Alex Malpass, and my Kalam playlist more generally. All of those papers can be read for free and are linked in the description. So, definitely check out that description. And while you're there, smash that like button, of course. And in fact, I've put together a short blog post over at MajestyOfReason.wordpress.com, the objectively best site that has ever existed, containing links to everything I've done on the Grim Reaper Kalam argument, including both published articles and videos. So, yeah, here's just a brief glimpse into what that post contains, that published articles and whatnot. But anyway, let's now turn it over to Troy to introduce our first problem for the GR. Again, we got two problems for it, and now we're going to be going into the first problem.

All right, the first, the first problem is a new "companions in guilt" style argument against the conjunction of the premises of the Grim Reaper argument, GR. Uh, we developed three companion premises, P1 star, P2 star, and P3 star, and use these in our argument as follows: Premise 1: If P1 and P2 and P3 is true, so the conjunction of those three assumptions is true, then P1 star and P2 star and P3 star is true. That conjunction is true. Premise 2: P1 star and P2 star and P3 star is not true. By which we just mean that the entire conjunction is not true. That is, at least one of the conjuncts is false. And so, Conclusion 3: P1 and P2 and P3 is not true. The details of each premise, as well as the content of of P1 star through P3 star, will be elaborated next. We'll first introduce a finite Benard-like paradox, after which we will use this paradox to motivate premises one and two in turn.

So, our paradox involves light bulbs of a special sort, which we will call bulbs with a capital B to distinguish them from ordinary light bulbs. Each bulb can be in one of two mutually exclusive states: on or off. A bulb was able and disposed to be on if and only if there was no bulb to its left which is on, and otherwise to be off. One bulb is to the left of a second bulb just in case either one, the glass of the first bulb is touching the electrical contact of the second bulb, or two, there's a sequence of bulbs from the first bulb to the second bulb that touch each other in this way. Given these stipulations, a linear sequence of four bulbs would appear as follows. Given that a bulb is able and disposed to be on if and only if no leftward bulb is on, a bulb with no bulbs to its left, such as bulb one in this image, will be on. And since each of bulbs two, three, and four has a bulb to its left that is on, they each are off, as you can see here.

We can now provide P1 star as an analog to P1, which is "Possible Bulb." According to P1 star, some possible spacetime regions contain a bulb with the power and disposition to be on if and only if no bulb to its left is on. We will let P2 star be the same as P2, that is, the Patrick Principle. Finally, we provide P3 star as an analog to P3, and P3 star is the "Intrinsicality of Bulb's Powers and Dispositions." Um, according to which, the powers and dispositions described to each bulb are properties intrinsic to that bulb in its corresponding region. Likewise, whether those powers and dispositions are realized is intrinsic to a bulb in its corresponding region.

We'll now argue for the first premise in our "companions in guilt" argument, which says that if P1 and P2 and P3 is true, then so too is P1 star and P2 star and P3 star. To do this, we will argue for each of the following: one, if P1 is true, then P1 star is true. Two, if P2 is true, then P2 star is true. And three, if P3 is true, then P3 star is true.

First, why is P1 star true if P1 is true? Well, like an individual grim reaper, an individual bulb seems possible. It is both conceivable and imaginable, and these are widely taken to be evidence of possibility. Um, a bulb was relevantly similar to lots of actual mechanical systems whose states are sensitive to things in their environment. Its constitution is similar to actual light bulbs, which we know are possible, and so on. In short, the modal epistemological supports wielded on behalf of a reaper's individual possibility seem equally applicable to a bulb's individual possibility. Metaphysically speaking, moreover, a bulb, along with its specified power and disposition, is quite mundane. To bring this point out, imagine a world containing two such bulbs. Suppose that both bulbs are on. Now suppose that someone connects them such that one is to the immediate left of the other. In this case, the leftward bulb would remain on, while the rightward bulb would turn off. This is purely perfectly innocent. If anything, the bulbs are less strange than their reaper counterparts, which are capable of performing arbitrarily precise actions in arbitrarily small intervals of time, which is needed for the variant of the GR targeting the possibility of beginning sequences of of ever smaller temporal intervals as scrunched into a finite period of time. Bulbs seem quite tame in comparison. So, if we're granting the possibility of individual reapers, it seems that we should likewise grant the possibility of individual bulbs. And so, if P1 is true, then so too is P1 star.

Second, P2 star is true if P2 is true because P2 star is the same as P2. They're literally the same. And third, P3 star is true if if P3 is true because the support provided for for P3 applies equally well to P3 star. Each bulb has the power to be on or off under certain circumstances, and its having that power does not depend on anything else being arranged in a certain way. In the reaper case, Koons argues that each reaper has power to produce a particle of a certain kind under certain circumstances. Its having that power does not depend on anything else being arranged in a certain way. And it seems fair to say the same here. The state of a bulb simply varies with the circumstances according to the powers and dispositions we suppose it to have intrinsically. This is exactly parallel to what is assumed in the reaper case. The state of a reaper, say, creating a particle, varies with the environment, say, whether a previous reaper placed a particle, according to the powers and dispositions we suppose it to have intrinsically. Consequently, if P3 is true, then so is P3 star. The par or symmetry between them is evident.

In light of all the preceding, we conclude that premise one of our "companions in guilt" argument is true: if P1 and P2 and P3 is true, then so too is P1 star and P2 star and P3 star.

Let's now turn to the second premise of our "companions in guilt" argument, which says that P1 star and P2 star and P3 star is false. The whole conjunction is false. To establish this, let's first introduce the following assumption about possible spacetimes, H2 star: "Possibility of a Circular Spatial Arrangement." This assumption, H2 star, says that some possible world contains enough spatio-temporal room to accommodate a circular spatial arrangement of 16 bulb-sized regions, as depicted in the image on the slide. Now, H2 star is pretty obvious. For instance, our world's spacetime is big enough to be able to fit a circular arrangement of bulb-sized objects. I could literally go and buy, say, 16 light bulbs and arrange them in a spatial circle, just like that depicted in the image. So, I don't think there's any questioning H2 star.

So, in the course of establishing our second premise, that P1 star and P2 star and P3 star is false, we'll next show that the conjunction of P1 star and P2 star and P3 star and H2 star is false. Since bulbs are possible individually, per P1 star, clearly there are 16 possible worlds, each of which contains a region containing only one bulb, which realizes its power and disposition to be on if and only if no bulb to its left is on. Now, consider a circular spatial arrangement of those regions, such that each bulb is to the immediate left of the next bulb, forming a loop of bulbs, just as depicted in the image on the slide. By H2 star, some possible world has enough spatial-temporal room to be able to fit the aforementioned circular spatial arrangement of bulb-containing regions. By the Patrick Principle, stated in P2 star, it follows that there's a possible world in which intrinsic duplicates of these regions coexist together in precisely that circular arrangement. But by P3 star, the realized powers and dispositions of bulbs are intrinsic to them. And hence, the possible world containing a circular spatial arrangement of intrinsic duplicates of bulbs, which realize their power and disposition, will also be a world such that each of the bulbs in the circular arrangement in the world all realize their powers and dispositions to be on if and only if no bulb to their left is on.

So, it then follows that there's a possible world, as shown here at the bottom of the slide, in which there's a circular arrangement of bulbs, each of which is on if and only if none of the bulbs to its left is on. But alas, such a world is contradictory. Pick any arbitrary bulb n, where n ranges from 1 to 16, as shown in the image here. Suppose, for reductio, that bulb n is on. Then, since any given bulb is on if and only if none of the bulbs to its left are on, it follows that none of the bulbs to the left of bulb n are on. But because of the circular arrangement, bulb n is one of the bulbs to the left of bulb n. So, since none of the bulbs to the left of bulb n is on, and since bulb n is one of the bulbs to the left of bulb n, it follows that bulb n is not on. So, bulb n is on by our original assumption, and yet bulb n is not on, and that, of course, is a contradiction.

So, then, our original assumption for reductio is false. In other words, bulb n is not on. And again, since bulb n was totally arbitrary, it follows that none of the bulbs are on. And that brings us to the subconclusion 10 here: none of the bulbs are on. But if none of the bulbs are on, then none of the bulbs to the left of bulb one are on. So, none of the bulbs to the left of bulb one are on. Yet, remember that any given bulb is on if and only if none of the bulbs to its left are on. And since none of the bulbs to the left of bulb one are on, it follows that bulb one is on. So, some bulb is on. And we've now reached an explicit contradiction. We concluded earlier in step 10 that none of the bulbs are on, but we've just now concluded in step 15 that some bulb is on after all, and that, of course, is inconsistent.

So, notice what we've done. From P1 star and P2 star and P3 star and H2 star, we'd derive at the conclusion that there's a possible world in which there's a circular arrangement of bulbs, each of which is on if and only if none of the bulbs to its left is on. But such a world is literally inconsistent, as we just showed. And hence, there is no such possible world. From this, it follows that the conjunction of P1 star and P2 star and P3 star and H2 star is false. And since H2 star is obviously true, it then follows that the conjunction of P1 star and P2 star and P3 star is false. And that, of course, is just the second premise in our "companions in guilt" argument. Oh, and by the way, on this argument here on the slide, to see how five follows from three and four, notice that three is just saying saying that the conjuncts aren't all true together. So, if we know that one of the conjuncts is true, as four says, then it just follows that the other conjuncts aren't all true together after all. If those other conjuncts are all true together, well, then, since the remaining conjunct is also true per four, the whole conjunction would be true, and that, of course, contradicts three. And so, the conjunction of P1 star and P2 star and P3 star is false. It can't be the case that all of those conjuncts are true. At least one of those conjuncts must be false.

We've now argued that both premises in our "companions in guilt" argument are true, from which the conclusion follows that the conjunction of P1 and P2 and P3 is false. Since these are the premises of the Grim Reaper argument, or GR, it follows that one of the premises in the GR is false, and hence the GR is unsound. Accordingly, and that concludes our first objection to the GR.

In the paper, we respond in depth to five objections to our "companions in guilt" argument. For purposes of space, we won't cover those objections here, but we will talk about some of them in our video on Troy's channel. We also invite the curious listener to check out the paper itself to see the objections and our responses. You might even find that we address one of your own objections in there. Again, the paper is linked in the description, and so, of course, is my Patreon. Yeah, that's right. As a board-certified grifter, I need to beg for money in every single video. So, if you see value in the work that I do, please consider becoming a patron. Here is a fun fact: for many, many months, patrons have already had access to our paper, even when it was under review and not publicly available. And that's true of many of my other papers as well. This is just one of the exclusive goodies that patrons enjoy. Anyway, thanks for considering it, and let's turn it over to Troy to articulate our second objection to the GR.

So, as we said before, our second objection is going to concern a problem for the intrinsicality assumption. So, we'll begin with a general statement of a reaper's realized power and disposition, or RPD, as follows: X has RPD means that X has the realized power and disposition to create and place a particle at a certain location if and only if no particle has already been placed at that location. Notice that since RPD is the realized power and disposition, it follows that anything with RPD actually does create and place a particle at a location if and only if no particle has already been placed at that location. As we have seen, that GR requires that RPD is intrinsic to the reapers which have it. But this assumption is problematic, and I'll explain why.

Consider an ostensibly uncontroversial patchwork inference with just two individually possible regions, R1 and R2, and possible worlds W1 and W2 respectively. Suppose each region contains a reaper, GR1 in region R1 and GR2 in region R2, as diagrammed here. And each of these reapers has RPD intrinsically. Suppose also that both of these reapers, GR1 and GR2, are creating and placing a particle at their respective regions because no particle has been placed before their allotted time at their respective worlds. Since there's clearly a possible spacetime with enough room to accommodate duplicates of these two regions in a sequence, our world should suffice. We can use the Patrick Principle to infer that there is a possible world containing exact duplicates of those regions in sequence, R1 prime before R2 prime, each containing a counterpart reaper, GR1 prime and GR2 prime, where again, the contents of R1 prime exactly duplicate the contents of R1, and the contents of R2 prime exactly duplicate the contents of R2. In other words, by the patch principle, it's possible to have a sequence of two reaper-containing regions, one before the other, each of whose contents are an exact duplicate of the contents of the corresponding reaper-containing regions from W1 and W2. We can also ensure that the that we patch together before and after R1 prime and R2 prime, they're devoid of anything else that creates and places a particle, just as a side. All of this is to say something which should be pretty uncontroversial to proponents of the GR: you can have grim reaper sequences of two reapers where the original reapers were reapers that were creating and placing a particle.

But the problem is that this is already inconsistent. For we can derive inconsistent conclusions about what the reapers are doing at the inferred patch-together world, W prime. To see this, I'm going to give two separate inferences to these inconsistent conclusions. First, because the contents of R1 prime and R2 prime exactly duplicate the contents of regions R1 and R2 respectively, and because R1 and R2 contain reapers GR1 and GR2 creating and placing a particle, each creating and placing a particle, it follows that R1 prime and R2 prime likewise contain reapers GR1 prime and GR2 prime creating and placing a particle. Otherwise, the contents of the regions wouldn't be exact duplicates. The second inference to the opposing conclusion is that because both GR1 and GR2 have RPD intrinsically, and because GR1 prime and GR2 prime are intrinsic duplicates of GR1 and GR2 respectively, it follows that GR1 prime and GR2 prime will likewise have RPD. Now, GR1 prime is initial in a in a reaper sequence, and no particle is created and placed before GR1 prime's allotted time. So, GR1 prime proceeds to create and place a particle, since, recall, it has RPD and so realizes its power and disposition to create and place a particle if and only if no particle is created and placed before it's allotted time. So, again, GR1 prime creates and places a particle. But notice that GR2 prime then does not create and place a particle. That's because GR2 prime likewise has RPD and hence realizes its power and disposition to create and place a particle if and only if no particle is created and placed before it's allotted time. Since a particle is created and placed before its allotted time by GR1 prime, it follows that GR2 prime does not create and place a particle. This is depicted on the slide here. You can see GR1 prime, like, creating and placing a particle there, and GR2 prime is not. It's just sitting there. Anyway, notice the contradiction that we've derived here. Given the first inference, GR2 prime creates and places a particle. But given the second inference, GR2 prime does not create and place a particle. That clearly will not suffice. So, something must be wrong.

Now, we can summarize the assumptions which generate this inconsistency as follows: One, initial GRs grim reapers are possible. That is, it's possible for there to be a reaper which is initial in a sequence of reapers with nothing before it that creates and places a particle. This follows from the premise that individual reapers are possible, which is just P1 of the GR. Two, the Patrick Principle. This is just P2 of the GR. Three, the intrinsicality of a GR's realized power and disposition, which is just P3 of the GR. Four, some possible spatial-temporal framework can fit two GRs in sequence, which is obviously true. For instance, the actual world's spacetime has enough room to be able to fit two reapers in sequence, one after the other. Five, if region R contains a GR that is creating and placing a particle, and if the contents of region R prime exactly duplicate the contents of R, then a GR is creating and placing a particle in R prime. In other words, duplication of regions preserves the activities of the reapers at those regions. And then six, if region R contains a GR that has RPD intrinsically, and if the contents of region R prime exactly duplicate the contents of R, then R prime contains a GR that also has RPD. As we explained, the GR requires one, three, since they're just premises or follow from premises of the GR, and again, four, which just says that there's a a possible spacetime region to accommodate two reapers in sequence, is obviously true. Consequently, then, I, one of these to avoid the contradiction, is not an option for the proponent of the GR. The GR also requires six. So, recall, uh, according to six, if a region contains a grim reaper that has RPD intrinsically, an exact duplicate of that region will also contain a grim reaper that has RPD. After all, if if this is false, if six is false, then we actually cannot infer from P1 through P3 that the patch-together world has reapers with the realized power and disposition required to create a grim reaper paradox, which in turn is needed for the GR to succeed as a reductio of the possibility of an infinite past. Put differently, if six is false, then it could be the case that each of the infinitely many individually possible regions contains a GR that has RPD intrinsically, even though though there aren't any GRs in the patch-together world with RPD. In such a case, no Grim Reaper Paradox arises in the patch-together world, and that would be disastrous for the GR. Since one through three and six are required for the GR, well, four is clearly true, as as we said, the GR proponent must reject five.

Recall five says that exactly duplicating a region containing a grim reaper preserves the actions and activities of the reaper at that region. So, again, the GR proponent has to reject that assumption five. But to us, this rejection strikes us as plainly incorrect. Five is clearly true. At least for any ordinary understanding of exact duplication. If you were to examine two regions and notice that different things are happening inside those regions, for example, a GR in one region is creating and placing a particle, whereas no GR in the other region is creating and placing a particle, you would clearly be correct to conclude that the contents of those regions do not exactly duplicate each other. They're just different things going on at those regions. They're not exact duplicates. So, in short, the GR proponent is forced to reject five. But since five is clearly true, the GR must have gone wrong somewhere. Hence, the GR is unsound.

Now, in the paper, we actually use this objection to the GR to diagnose exactly where the GR goes wrong. Specifically, premise three, stating the intrinsicality of a reaper's realized powers and dispositions, is actually false. Even if their characteristic powers and dispositions are intrinsic to reapers, whether those powers and dispositions are realized or exercised or manifested depends on the contents of other regions in the world. In particular, whether a reaper realizes its power and disposition to create and place a particle if and only if no particle has been placed before depends on the actions of previous reapers in other regions. If a reaper, say Bob, is creating and placing a particle, then whether Bob realizes this power and disposition depends on whether some previous reaper in a separate region has created and placed a particle. Right? If a if a previous reaper has done so, well, then Bob, it actually turns out, has failed to realize that power and disposition, since Bob creates and places a particle despite a particle having already been placed. Supposing otherwise enabled us to infer absurdly that a non-initial reaper in a patch-together world could still realize its power and disposition despite creating and placing a particle, even though a particle had already been placed. So, it is simply false that a reaper's realized power and disposition is intrinsic. It's extrinsic, as it depends on what happens in regions elsewhere in the world. Consequently, the third premise of the GR, namely P3, is false, and thus the GR is unsound.

And by the way, the companion premise P3 star is also false for the exact same reason. Even if a bulb's power and disposition are intrinsic to it, whether that power and disposition are realized depends on the contents of other regions. In particular, whether a bulb B realizes its power and disposition to be on if and only if no leftward bulb is on partly depends on the states of leftward bulbs in other regions. If B is on, then whether B realizes its power and disposition depends on whether some leftward bulb in a separate region is on. After all, if a leftward bulb is on, then B, it turns out, has actually failed to realize its power and disposition, since again, B is on despite some leftward bulb being on as well. So, it is false that our bulb's realized power and disposition is intrinsic to a bulb. It's extrinsic, as it depends on what happens in other regions elsewhere in the world. Consequently, P3 star, just like P3, is false.

So, we've come full circle, if you'll pardon the pun. We first argued that the GR is unsound using our "companions in guilt" argument. We then developed a second objection to the GR based on the notion of exact duplication and then used this objection to pinpoint and explain exactly which premise of the GR is false. And for precisely the same reason, the companion premise P3 star is false. The central takeaway is that the GR fails. It is unsound.

So, there we go. That is the basic gist of our forthcoming article in the Journal of Philosophy. Stay tuned for the video we'll be doing on Troy's channel, Friction Philosophy, where we'll talk about other aspects of the paper not covered here and some of your own objections or questions. You can leave those in the comment section below, or you can just leave a comment because it helps the almighty algorithm. And speaking of the algorithm, as always, please smash that like button, turn on that little bell for notifications, consider supporting me on Patreon or through a one-time donation. Links to both of those are in the description. And of course, consider supporting Friction Philosophy on Patreon. Got to spread the love. Links to his channel are in the description. And again, the one-time donation link is down there as well. Think of a one-time donation as like a tip. And know, not that tip, get your mind out of the gutter. And hey, you tip waiters who bring you extra breadsticks, and waiters get paid for what they do. But I'm not getting paid, and I'm not bringing you mere breadsticks. I'm trying to help you better understand the fundamental nature of reality with caution, care, love, and philosophical rigor. So, if you tip waiters, you should also tip me. And that concludes my case for tipping me through a one-time donation. That was my triple collateral and free-for-all. Anyway, what better way to end is there than I'm Joe Mid. This is The Majesty of Reason. And peace out.

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