This is a follow-up to a previous post, in which I presented an argument against the conceivability of infinite regresses. Some critics (Schmid and Malpass, forthcoming) have argued that the Patchwork Principle too strong. Perhaps we should add an exception:
Revised Patchwork Principle. If (i) a certain causal structure S is conceivable, (ii) a finite process P is conceivable, and (iii) scenario A consists of inserting one or more copies of P into slots in S, each of which is large enough to accommodate P, then scenario A is also conceivable, unless A is mathematically impossible.
The move to a revised patchwork principle is called the “unsatisfiable pair diagnosis” of the original Grim Reaper story. There are two conditions that would have to be met in order to verify the Grim Reaper story (namely, that the Grim Reapers’ actions depend in a certain way on their predecessors’, and that the past is infinitely long), and we have seen that it is mathematically impossible for both conditions to be met.
I don’t believe that it is necessary to add this exception to the original principle. If A turns out to be mathematically impossible, that shows us that either S or P are not conceivable (or that it is not conceivable that copies of P be fit into all of the slots of P). The mathematical impossibility of A is a feature of my original argument, not a bug. We haven’t been given any good reason to weaken this Patchwork Principle in this way. The Revised Version arbitrarily picks out mathematical impossibility as the exception. What would happen if we excepted all inconceivable scenarios? In that case, the Patchwork Principle would become a mere tautology: all scenarios of a certain kind are conceivable, unless they’re not. Therefore, we have good reason to reject such a sweeping exception. But the defenders of the Unsatisfiable Pair Diagnosis have given us no reason to treat mathematical inconceivability as an exceptional case.
Nonetheless, I will not press this response further here. Instead, I will put forward a new version of the Grim Reaper story: the Chancy Grim Reaper story.
The Chancy Grim Reaper story. We change the program of each Grim Reaper. For each number n, when the Grim Reaper n receives the warrant on his assigned date of n B. C., a fair coin is flipped. If the coin comes up Heads on n BC, Grim Reaper n erases anything written on the warrant and signs it with his own numeral. If the coin comes up Tails, then Grim Reaper writes his number on the warrant only if the warrant is blank when he receives it. Otherwise, he passes it on unchanged to his successor, Reaper n – 1.
Here are two versions of the Chancy Grim Reaper story: the Lucky story and the Unlucky story. In the Lucky story, the coin comes up Heads an infinite number of times. This means that, for every Reaper n, the coin has come up Heads infinitely often in the past, and there will be a latest year m on which the coin had previously come up Heads. There is no logical or mathematical impossibility involved in the Lucky story. There will be a number on the warrant on January 2, 1 BC. The number will correspond with the last time that the coin landed Heads.
In the Unlucky story, the coin turns up Tails every time it is tossed. The Unlucky story is inconceivable for the same reason that the original Grim Reaper story was inconceivable However, the only difference between the Lucky and the Unlucky story concerns how the members of a set of completely independent events turned out.
The Independence Principle. If B is a conceivable scenario, X is a class of mutually independent chance events occurring in B, C is a scenario that is exactly like B except with reference to (i) events that are possible outcomes of members of X, and (ii) conceivable events that are causally downstream from these outcomes, then C is also a conceivable scenario.
In this case, it is certainly conceivable that the coin should turn up Heads infinitely many times. Since we’re supposing that the coin is a fair one, then any combination of Heads/Tails outcomes should be conceivable. We therefore should be able to conceive of a scenario in which the coin comes up Tails every time. This is an extremely unlikely outcome, but it is clearly conceivable. The conceivability of this outcome is a consequence of the concept of mutually independent trials. If we have a class of chancy events that are by hypothesis mutually independent, then this means that the probability of each event is not affected by the occurrence or non-occurrence of other events in the class. So, even if every other coin toss besides toss n turns up Tails, there is still a 50% chance that toss n will come up Tails. So, it is conceivable that all the coin tosses come up Tails. For similar reasons, the probability of a chance event cannot be affected by events that are causally posterior to it. So, the scenario in which every coin toss comes up Tails should be conceivable, no matter what the subsequent results of any of the tosses might be (so long as these results don’t affect the mutual independence of the coin tosses).
Here is an argument from the Independence Principle to the inconceivability of infinite regresses:
- Given the Independence Principle, if the Lucky story is conceivable, so is the Unlucky story.
- The Unlucky story is not conceivable (because it involves a mathematical impossibility)
- Therefore, the Lucky story is not conceivable. (From 1, 2)
- Given the Revised Patchwork Principle, if infinite regresses were conceivable, then the Lucky story would be also be conceivable (since it doesn’t involve any mathematical impossibility).
Therefore, infinite regresses are not conceivable. (From 3, 4)
So, even if we replace the Patchwork Principle with the Revised Principle (as the defenders of the unsatisfiable pair diagnosis advocate), we can still show that infinite regresses are inconceivable, so long as we have the Independence principle in hand. This means that we can know a priori that infinite regresses are impossible. Since we can also know that everything broadly causable has a cause, we can know that everything in the universe is caused by something strictly uncausable.
I have a second refutation of the conceivability of infinite regresses, one based on the idea of an infinite fair lottery. An infinite fair lottery is a lottery that will certainly produce a unique natural number (from 1 to infinity), providing each number with equal probability (hence, a fair lottery).
As Alexander Pruss has argued (Pruss 2018, 64-92), if an infinite regress is conceivable, then so is an infinite fair lottery. Imagine again an infinite series of coin tosses, stretching infinitely far into the past and numbered (in reverse) 1, 2, 3, and so on. Using our Revised Patchwork principle again, we can deduce the conceivability of such a series from the conceivability of an infinite causal regress. Using Independence, we can deduce that every combination of results from such a series is conceivable. In particular, we have the conceivability of a result in which the tosses all produce Tails except for a single Heads. This result can be treated as an infinite fair lottery, since each of the numbers had an equal chance of being selected as the unique correspondent to the single Heads result.
Here, then, is my final argument against the conceivability of an infinite regress:
- If infinite regresses are conceivable, then infinite fair lotteries are conceivable. (By Revised Patchwork Principle and Independence)
- If one such lottery is conceivable, so are two independent such lotteries.
- If two such lotteries are conceivable, then it is conceivable that fundamental principles of probability be violated.
- It is inconceivable that any fundamental principles of probability be violated.
Therefore, infinite regresses are inconceivable.
I take it that the fundamental principles of probability are, like the fundamental principles of logic and probability, known a priori to be necessary truths, hence premise 4.
Here is my case for the crucial premise 3. Let’s label the numbers selected by the two infinite fair lotteries L1 and L2. Suppose we are given the result of lottery L1, namely m. What is the probability that L2 is greater than L1, given that L1 = m? It is clear that this conditional probability is 1, or infinitely close to 1. There are only finitely many numbers less than or equal to m, while there are infinitely many numbers greater than m. Since L2 is an infinite fair lottery, each number greater than n has a chance of being selected equal to that of any number less than. I appeal here to a principle of Enumeration:
Enumeration. If the members of class C are mutually exclusive events of equal probability, and E is a subset of C, then the conditional probability Prpb(E/C) is equal to (or infinitely close to) the ratio #(E)/#(C).
If #E is finite and #C is infinite, then Prob(E/C) is infinitely close to 0, and P(~E/C) is infinitely close to 1.
In fact, by parity of reasoning, the conditional probability Prob((L2 > L1)/(L1 = x) will be infinitely close to 1, no matter what value L1 takes.
Take an arbitrary intelligent agent who knows with certainty that he will discover that L1 = n, for some particular number n. He also knows (by Enumeration) that he will conclude, after learning that value) that the probability of (L2 > L1) is infinitely close to 1. He can therefore conclude that the present rational probability of (L2 > L1) must already be infinitely close to 1. He also knows, however, by a principle of Isomorphism that the probability of (L2 > L1) cannot be significantly greater than ½, since the two lotteries are isomorphic in causal structure, and so the two inequalities must have approximately the same probability. Again, this means that an infinite fair lottery requires the rational agent to violate a fundamental principle of probability.
Dynamic Coherency. If a rational agent knows that he will continue to be rational and already knows that he will, after updating with some unknown piece of evidence), assign a probability p to some proposition p, then he must already assign probability p to p.
Isomorphism. If two situations are known by an agent causally isomorphic, and the agent is ignorant of the result of either situation, then the agent’s rational probabilities with respect to the two situations must also be isomorphic.
Violations of Dynamic Coherence produces a situation in which the agent is subject to a dynamic Dutch book: a set of bets, distributed over time, that are guaranteed to result in a net loss. For example, suppose the agent thinks initially that L1 has a 50% of being the larger of the two numbers, but the same agent will certainly judge (after learning the actual value of L1) that that probability is very close to 0. In that case, the agent will accept a fair bet on L1 initially but, after learning the actual value, will be willing to sell the bet back for a loss.
The Analytic Thomist 
I don’t see how causal finitism fixes infinite lotteries. The issue is mathematical. I don’t see how saying the lottery isn’t metaphysically possible fixes the problem. I think Daniel Rubio’s (2021) criticism of the argument has never been properly addressed.
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Thanks for the post, Rob! Apologies if I already posted this comment; WordPress is acting really weird for me! Here are my thoughts on your post:
(1) “The move to a revised patchwork principle is called the “unsatisfiable pair diagnosis of the original Grim Reaper story.”
This mischaracterizes the UPD. As we state in our paper, the UPD is simply the claim that the solution to Benardete paradoxes comes by way of recognizing that they instantiate a jointly logically unsatisfiable pair of conditions, and no substantive metaphysical hypothesis need be adopted to solve them (save the law of non-contradiction). From p. 7 of the paper: “The UPD is what Pruss (2018, p. 2) calls a ‘conservative’ analysis, as it does not postulate a revision in logic or any specific metaphysical thesis. It says that the paradoxical situations are simply (narrowly) logically impossible, end of story-no metaphysical thesis need be adopted.”
The move to a revised patchwork principle simply one dialectical move, among almost a half dozen published moves (and almost a half dozen more unpublished moves), that the UPDist can make in response to the patchwork argument. This, too, is noted in the p paper. From p. 14: “How might the UPD proponent respond? They could appeal to extant responses to the patchwork objection based on (i) branching actualism (Schmid and Malpass 2023), (ii) the endless future parody (Schmid Forthcoming), (iii) the problem that patchwork principles fail to secure signal transmission in the quilted world (Schmid 2024), (iv) the companions in quilt argument with finite Benardete-like paradoxes (Dana and Schmid Forthcoming), or (v) the problem that the realized powers/dispositions of Reapers are actually extrinsic (Dana and Schmid Forthcoming). Here we will suggest a different (but perhaps complementary) response to the patchwork objection.”
So the post mischaracterizes the UPD from the outset.
This is also crucial to something you later write: “So, even if we replace the Patchwork Principle with the Revised Principle (as the defenders of the unsatisfiable pair diagnosis advocate)”
But defenders of the UPD need not advocate the shift to the revised patchwork principle. For again, defenders of the UPD could instead make any of the almost dozen other responses to the patchwork argument. The UPDist can think Malpass and I went hopelessly wrong in proposing the revised patchwork principle, and the UPDist can even accept a full-blooded patchwork principle. For they have lots of other, independent responses to the patchwork argument up their sleeve — for instance, a paradoxical patched-together world requires successful signal or information transmission across regions, but this extrinsic fact is simply not guaranteed by the patchwork inference. So we cannot infer that a paradox actually arises in the patched-together world. That’s one response (among many others) they could give instead of our revised patchwork response (or, as we called it in the paper, the proviso problem’). So the UPDist is not wedded to the revised patchwork response.
(2) “I don’t believe that it is necessary to add this exception to the original principle. If A turns out to be mathematically impossible, that shows us that either S or P are not conceivable (or that it is not conceivable that copies of P be fit into all of the slots of P).”
Whether this is what is shown is the very question at issue, however. We argued against precisely this in the paper. Note, also, that we don’t focus on mathematical impossibility, but rather narrow logical impossibility. From p. 16: “provided that those regions being in that arrangement isn’t logically inconsistent (i.e., provided that no formal contradiction can be derived from those regions being in that arrangement)”.
(3) “The mathematical impossibility of A is a feature of my original argument, not a bug. We haven’t been given any good reason to weaken this Patchwork Principle in this way.”
But we gave two arguments in sections 8 and 9 to weaken the patchwork principle in precisely this way, and you haven’t addressed those here. It doesn’t seem appropriate, then, to lampoon our weakening of the principle for lacking good support.
(4) “The Revised Version arbitrarily picks out mathematical impossibility as the exception.”
Again, we didn’t pick out mathematical impossibility but rather narrow logical impossibility And we also gave independent, non-arbitrary motivation for the inclusion of our proviso in the paper — motivation which has not here been addressed. Moreover, we noted in the paper that the UPDist doesn’t need to independently motivate the proviso, since the UPDist can simply run the proviso problem as an undercutting defeater. This is elaborated in sections 8 and 9 of the paper, sol won’t elaborate it here.
(5) “What would happen if we excepted all inconceivable scenarios? In that case, the Patchwork Principle would become a mere tautology: all scenarios of a certain kind are conceivable, unless they’re not. Therefore, we have good reason to reject such a sweeping exception.”
There’s an odd shift here from talk of metaphysical possibility — which is what our paper is concerned with, and what the patchwork principle itself is concerned with — and conceivability. But setting that aside, we argue in section 9 of the paper that our proviso does not make the principle trivial or tautologous, and allows us to substantively extend our modal knowledge.
(6) Regarding the first new argument against the conceivability of infinite causal regresses, I would make at least two points in reply:
(a) The argument succumbs to many of the other objections to the patchwork argument apart from the proviso problem developed in sections 8 and 9 of our paper. For instance, the use of the revised patchwork principle still only allows us to infer that the intrinsic features of individual possibilities are preserved in the patched-together world. But to get a paradox in the Unlucky Story, we need to assume that some information carrier successfully propagates between regions, and we are not licensed to infer that this fact extrinsic to the Reapers and regions will be present in the patched-together world. In short, the Unlucky Story could play out entirely consistently, as the relevant information-bearing signal (e.g., the paper) can fail to successfully pass between the Reaper-containing regions. And there are other such objections applicable to this new argument as well — see the links above for more.
(b) I don’t see why we should accept (4) in the first argument against the conceivability of infinite causal regresses, which states, “Given the Revised Patchwork Principle, if infinite regresses were conceivable, then the Lucky story would be also be conceivable (since it doesn’t involve any mathematical impossibility).” In fact, it seems to me that this premise should be rejected. It seems to me that Lucky Story is straightforwardly logically inconsistent by virtue of logically entailing the possibility of an inconsistency. [Here I’m of course prescinding from the other objections, like the one mentioned in (a), which would remove the inconsistency but defeat the argument in any case. The Lucky Story, by stipulation, entails an infinite linearly ordered sequence of Reapers each of which has an objectively indeterministic fair coin independent of any other coins, such that if that Reaper’s coin lands on one of the coin’s objectively possible indeterministic outcomes, the Reaper φs iff no previous Reaper φs. But this story plainly logically entails, by virtue of the objective indeterminism and independence stipulated to hold of the coins, the possibility of an infinite linear set each member of which φs iff no previous member φs. In other words, the story logically entails the possibility of an inconsistency, and anything which logically entails the possibility of an inconsistency is itself inconsistent. So the Lucky Story, it seems to me, is inconsistent, and hence we cannot use the revised patchwork principle to infer its possibility (given the logical-inconsistency-debarring proviso). (Note also that if we don’t stipulate that the coins in the Unlucky Story are independent of one another, so as to render the scenario scenario logically consistent in itself, then we can no longer use the patchwork principle to infer the possibility, or conceivability, of a contradictory Unlucky scenario in the patched-together world, since it is then entirely consistent to suppose that the coins in the patched-together world are dependent on one another.)
(7) Regarding the second new argument against the conceivability of infinite causal regresses, I would make at least three points in reply:
(a) It seems to me that this argument can be paralleled to create an equally convincing argument for the impossibility of an endless future. Compare:
If an infinite progress is conceivable, then so is an infinite fair lottery. Imagine an infinite series of coin tosses, stretching infinitely far into the future and numbered 1, 2, 3, and so on. Using our Revised Patchwork principle, we can deduce the conceivability of such a series from the conceivability of an infinite causal progress. Using Independence, we can deduce that every combination of results from such a series is conceivable. In particular, we have the conceivability of a result in which the tosses all produce Tails except for a single Heads. This result can be treated as an infinite fair lottery, since each of the numbers has equal chance of being selected as the unique correspondent to the single Heads result.
Here, then, is an argument against the conceivability of an infinite progress:
1. If infinite progresses are conceivable, then infinite fair lotteries are conceivable. (By Revised Patchwork Principle and Independence)
2. If one such lottery is conceivable, so are two independent such lotteries.
3. If two such lotteries are conceivable, then it is conceivable that fundamental principles of probability be violated.
4. It is inconceivable that any fundamental principles of probability be violated.
Therefore, infinite progresses (and so endless futures) are inconceivable.
More generally, what a rational agent’s credences should be concerning which number is Heads in an infinite sequence of coins each of which is objectively indeterministically fair and independent is not affected by whether that sequence is stretched out across an infinite past, an infinite space, an infinite future, or even coins in isolated universes, etc. This would also mean (absurdly, in my view) that this argument appears to entail the impossibility of infinite space and an infinite multiverse. Quite a shocking conclusion to be committed to from the armchair when these are live empirical hypotheses!
(b) I also take issue with premise (2). Here is an inconsistent triad:
(i) an infinite (regressive) sequence of indeterministic and independent coins, only one of which comes up Heads, is conceivable;
(ii) if the sequence in (i) is conceivable, then two such sequences are conceivably conjoined;
(iii) two such sequences are not conceivably conjoined.
You seem to be urging accepting (ii) and (iii) and consequently rejecting (i). But I don’t see why I shouldn’t instead just accept (iii) and (i) and reject (ii). After all, (i) strikes me as very intuitively plausible — I think I can easily imagine an infinite sequence of coin flips, and I can flesh out the situation in detailed ways without finding any contradiction or evident absurdity. And if I go along with your case for (iii), then in the absence of even stronger support or intuition for (ii) than for (i), I should just reject (ii). And at least so far, we haven’t been given anything by way of support for (ii). Moreover, (ii) loses its intuitive force for me (which wasn’t much to begin with) when I see that its antecedent strikes me as very plausible and when I go along with your case against its consequent. So it seems to me that your case is underdeveloped here. We need to be given some compelling reason for solving this inconsistency by adopting (ii)&(iii) as opposed to adopting (i)&(iii).
(c) Finally, I don’t yet see why we shouldn’t just take this style of argument to be a reductio of the conjunction of (Isomorphism&Dynamic Coherence& Enumeration) as applied unrestrictedly (i.e., even in infinite contexts). This will also be an appealing resolution for those, like me, who find it independently very plausible that infinite regresses are conceivable.
(d) One final point: I also think Rubio’s (2021) objection to Pruss on infinite fair lotteries is compelling. (I discuss it with Rubio starting at 11:14 of this video, if you’re interested!)
All in all, this was a super interesting post, and I think it nicely carries the dialectic forward into fascinating new territory. Thanks for it!
Joe
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Joe,
Not to butt in, especially since you are more discussing the patchwork principle than the mysterious force objection, but after reading through your paper, I have a question.
On pp. 11-13 you set up a parody to Pruss’s argument by specifying the rules for 7 demons in Bridget’s bridge-crossing scenario but then argue that a scenario where they all flip heads and follow the rules cannot obtain. You then go on to say:
“We think there are two available moves here, and either move equips the UPD proponent with a successful rejoinder to Pruss’ mysterious force objection…First, one could grant that the scenario wherein the demons flip the coins and follow the rules is possible, but deny that it follows that all the coins in that scenario could (jointly) land heads…Second, one could deny that the scenario wherein the demons flip the seven coins and follow those rules is possible.”
Wouldn’t the proper option here be to adopt a modification of the second move and say that a scenario where the demons flip the seven coins, they all land heads, and they follow the rules is impossible? It seems to me that the actual problem is introduced in the rules themselves, particularly where the demons have to “ensure” or “prevent” certain movements by Bridget. We are asking the demons to ensure or prevent a mathematical possibility in that particular scenario, which would certainly be absurd. Specifically, one of the demons ensuring a certain movement would contradict another demon preventing that same movement.
It’s not clear to me that Pruss’s account suffers from the same problem such that the Reapers are, by rule, mandated to do contrary things. As such, it seems like this would break the symmetry between the scenario you provide and the one Pruss provides.
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Thanks for the question!
You say: “Wouldn’t the proper option here be to adopt a modification of the second move and say that a scenario where the demons flip the seven coins, they all land heads, and they follow the rules is impossible?”
The claim that those three conditions are jointly impossible is, of course, demonstrable, and we demonstrate precisely this in paper. So you’re correct that everyone should grant that a situation in which those three conditions are jointly satisfied is impossible. This doesn’t mean we have failed to provide a proper option in giving our two moves you quoted, since the disjunction of our two moves is logically equivalent to the impossibility of those three conditions being jointly satisfied.
Let p be <all the demons flip the seven coins>, q be <all those coins land heads>, and r be <all those demons follow the rules>.
The claim that p, q, and r are jointly impossible is the claim that ~◊(p&q&r). The two moves we suggested were: (i) grant that p and r are jointly possible, but deny that q is possibly conjoined with p and r; (ii) deny that p and r are jointly possible. So the disjunction of the two moves we suggested is the following disjunction: (◊(p&r)&~◊(p&q&r))∨(~◊(p&r)). This disjunction is provably logically equivalent to ~◊(p&q&r). So we are correct that these are the two options one has in response. For ~◊(p&q&r) is provable, and the disjunction of those two options is logically equivalent to that, and hence the disjunction of our two options is provable. And since the two options are mutually incompatible, one must take one or the other — one is forced to choose between them.
You continue: “It seems to me that the actual problem is introduced in the rules themselves, particularly where the demons have to “ensure” or “prevent” certain movements by Bridget.”
There is nothing impossible about the rules. It’s entirely possible that, e.g., a demon prevent or force a person from crossing a Konigsberg bridge. Each rule is clearly individually possible to follow, and the rules are even collectively possible to follow provided that, in any possible situation in which they’re followed, the demons’ coins don’t all land heads. For then the demons all ensure or prevent Bridget to cross the Konigsberg bridges in ways that are entirely mathematically consistent and entirely metaphysically possible. (Any combination of 6 or fewer bridges can be consistently crossed without doubling back on oneself. This is mathematically provable.)
You continue: “We are asking the demons to ensure or prevent a mathematical possibility in that particular scenario, which would certainly be absurd.”
I can prevent someone from eating a sandwich, even if their eating a sandwich is a mathematical possibility. There is no ‘absurdity’ in preventing someone from doing something mathematically possible. Perhaps you’re instead saying it’s absurd for the demons to ensure that Bridget do something mathematically impossible. Correct. But as explained, the demons are not ensuring that Bridget do something mathematically impossible just by virtue of following their rules, since they can follow their rules perfectly consistently so long as at least one of the coins fails to land heads. Any such situation is perfectly logically and mathematically consistent, since any such situation only involves Bridget crossing 6 or fewer bridges without doubling back on herself, and that’s provably mathematically possible.
You continue: “Specifically, one of the demons ensuring a certain movement would contradict another demon preventing that same movement.”
No, it wouldn’t. The rules for each demon specify different actions they’re ensuring or preventing Bridget from performing at different locations, at different bridges, at different times. There is clearly no contradiction whatsoever in, e.g., demon #1 preventing Bridget from crossing bridge #1 without doubling back on herself, while demon #4 ensuring that Bridget cross bridge #4 without doubling back on herself. (We are obviously to imagine that they are to ensure or prevent these things within some common, temporally extended timeframe.) That’s like a demon ensuring that I brush my teeth in the bathroom at some point during the day, and another demon preventing me from washing the dishes in the kitchen at any point during the day. These are clearly jointly possible. Ditto for the bridges. So there is no such symmetry breaker you purport to identify.
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Joe,
So the necessity of those two options is clear enough based upon your explanation; thank you for elaborating. That being said, I think we begin to run into issues when talking about whether or not the rules are obeyed.
For the sake of argument (even though you seem to reject this interpretation) if we say that the demons ensure Bridget cross their bridge if they flip heads, and we mean “ensure” in the strong sense of “They will make it obtain that Bridget cross their particular bridge, irrespective of where she is currently,” then this is where they would come into conflict with other demons, since if she must (of mathematical necessity) return over a bridge she has already crossed, then the first demon who ensures she cross his bridge will come into conflict with the one who will prevent her from crossing his own bridge, which she has already crossed.
Now, I realize you don’t intend this meaning of “ensure,” but this sets up my second point. If we take “ensure” in a more conditional sense, where it effectively amounts to “They will make it obtain that Bridget cross their particular bridge, if she is in fact present at their bridge,” then a different issue arises. In this case, the rule can be written as a material conditional (If Bridget is at demon n’s bridge, then it will ensure she crosses). But the truth-function for such a conditional statement, as we know, makes it such that if the antecedent is false, then the conditional as a whole is true. And if it’s true, then the rule is not actually violated. So if Bridget cannot in, fact, make it to a particular demon’s bridge, then by that very fact the relevant demon fulfills the rule.
Thus, depending upon how “ensure” is interpreted, either the rules themselves produce the contradiction (on the strong sense), or the rules are not actually violated (on the weak sense). Either way, the second move you proffer on p. 12 seems to be in trouble.
Granted, perhaps one might just be tempted to say “Okay, so we simply opt for the first move, where we deny the possibility that all the coins could jointly land heads.” For what it’s worth, I think this first move makes no sense at all. These coin flips, of themselves, do not result in any mathematical impossibility. It is only the coin flips when taken as part of this larger scenario that any contradiction results. It stands to reason that, if the demons simply flipped the coins without any rules attached, then any of the possible permutations might result. But if they immediately decide to implement the rules, then we hit a contradiction? Maybe it’s the rules that are the problem. And not to put too fine a point on it, but if the case you are making is an attempt to respond to the mysterious force objection, all this first move does is seem to reintroduce said mysterious force.
TL;DR:
– The issue with the second move outlined in the paper is that either the rules themselves introduce the contradiction (an issue which hasn’t been shown to afflict Pruss’s account), or the rules themselves, contra the hypothesis, are in fact fulfilled.
– The issue with the first move is that neither the demons flipping nor the landing of the coins on heads is impossible unless conjoined to the above rules. Much more sensible to say the rules introduce the impossibility than that the coins mysteriously are causally prevented from all landing heads. But if we opt for this, then the natural position to take really is the second move after all.
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Schmid’s co-author Troy Dana has also given a response to this post: https://open.substack.com/pub/fric/p/robbing-koons?r=4aqjw9&utm_medium=ios
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