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The gist of the puzzle is that every day, the Devil offers to flip a coin to see if you escape; one loss and you're guaranteed to be stuck forever, but each day the probability of a winning toss increases (see Logic: A Very Short Introduction, 99-101). The conclusion is "supposed" to be that you have reason to delay the coin-toss forever, meaning that you never escape anyway.

Now, where the reasoning goes wrong, it seems to me, is on the level of what we're comparing in order to decide on the perpetual delay. It seems to be that we start out with, "For any pair of days {today, tomorrow}, it is rational for me to wait until the day with the higher probability of winning assigned to it." Since {tomorrow} is always "better," we end up deferring to an eternal tomorrow. No dice.

So wouldn't it be more reasonable to think, "Out of n days, it would be most rational to wait until the specific day with the highest probability of winning," except then you'd note that if we're talking about infinitely many days, there is no one n that satisfies those parameters absolutely; so we'd realize that the question of waiting is not well-posed, ultimately?

I also feel like the A-time/B-time distinction might be relevant, but I'm having a hard time putting my feeling into words.

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    This may be solved with some form of temporal discounting. en.wikipedia.org/wiki/Time_preference#Temporal_discounting Essentially we view what happens in the distant future as less important than what happens more immediately. So when the chance of leaving hell rises above a threshold, then we judge the short-term reward worth not waiting for a higher chance.
    – causative
    May 14, 2022 at 2:10
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    If you're a finite living being, this obviously is an applied problem based on the escape probability increasing details, your tolerable period to suffer, attempt value threshold, and most importantly to weigh whether "being dead" is better than "in hell". If you're an infinite being then following Cantor's absolute infinity transcendental logic intuition, the finite timing most likely doesn't matter at all, thus makes no sense as a puzzle any more. A situation like this sounds very depressing in any scenario, you need something more than logic to deal with it, such as test the devil... May 14, 2022 at 4:39
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    The puzzle doesn't inherently require time; it could be set in terms of any countable infinite resource, apart from the intention to keep the subject restrained forever. If we consider the probability of failure to escape being P(N)=1/(2^N) where N is some number corresponding to time, money, or whatever else, it will always be the case that P(N+1) > P(N). Time-specific constraints therefore aren't relevant. The spoilsport answer to the question is that one would eventually say that a 99.9% chance of escape today is better than waiting for a 99.99% chance in 10 days' time.
    – Frog
    May 14, 2022 at 9:52
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    For an informal reasoner it's not clear this is really a problem, they might just make some intuitive decision like "I'll wait until my chance of remaining trapped in hell is less than 1 in 10^100 and then take the risk". It could be a paradox for a certain type of ideal reasoner, but in that case we may need more detail on how they reason--are we assuming that remaining in hell forever has infinite negative utility while escaping has only finite positive utility (you don't go to heaven, you just escape and live a finite life) and remaining any finite no. of days has finite neg. utility?
    – Hypnosifl
    May 14, 2022 at 20:44
  • @DoubleKnot If you're an infinite being then following Cantor's absolute infinity transcendental logic intuition, the finite timing most likely doesn't matter at all Are you defining "infinite being" to mean one that has days of experience corresponding to transfinite ordinals as well as finite ones? What if the being is "infinite" only in the sense that they have a day of experience corresponding to every finite ordinal, but they don't experience any transfinite-ordinal-numbered days like a ωth day, a ω+1th day, etc.?
    – Hypnosifl
    May 14, 2022 at 20:58

2 Answers 2

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The chapter is just Pascal's Wager and this, under the heading Decision Theory: Great Expectations, yet concludes with "So it looks as though the only rational thing to do is to be irrational!". Priest is a dialetheist (and a pluralist), he thinks there are "true" contradictions. And that logic is real. If logic is real, it says something about the world.

I'm not sure of his motivations for including it as he did. I think it's probably to prepare one for later seeing what motivates dialetheism and paraconsistent logic within pluralism. Priest believes in choosing the logic depending on the situation. I think he is trying to show a case where paraconsistent logic can apply by setting up a seeming contradiction. The only rational thing to do is to act irrationally.

I personally don't find this a great exercise. He denies the Principle of Explosion, so a true contradiction does not mean anything follows. But he doesn't tell us what follows besides not everything in the case of a contradiction. He doesn't tell us which specific logic should apply. Sure a paraconsistent or diaethalist one, but which specific one? There are many. Possibly infinitely many.

He is a pluralist and says the appropriate logic depends on the situation. But I don't think we can a priori catalog what empirical situations we will encounter. So how does he limit his pluralism to not be explosive? His meted pluralism seems unjustifiably a priori restrictive. There can be no a priori reasoning to doubt certain logics since the logic depends on the empirical setting.

I find his/Gracely's puzzle close to an even more irrational one:

It's possible infinitely-negative futures exist as maybe it only takes near-omnipotence to consign one to Heaven or Hell. There may be infinite near-omnipotent beings. Say each will ask a different question and if you get it wrong you go to Hell for eternity. You can't possibly do anything a head of time to prepare for this, as there will always be infinitely more questions you didn't have time to prepare for. It is so difficult to rationally strategize about or pick the correct logic for, it actually makes no difference whether you develop any strategy. Even getting as many questions as possible right is only a temporary reprieve while answering compared to infinite negativity in Hell. One could act in any way possible and it would be just as rational (explosion). I don't see the dividing line between this and Priest's/Gracely’s.

Since we have to deny this more irrational example since we are denying the PoE (in Priest's case), why exactly are we denying it? Where do we draw the line of which logics to accept in our pluralism?

Priest tells us we may act irrationally, but not in any manner. I like his refutation of classical logic being the only correct logic. But I don't see how to limit logical pluralism to the precise degree where we expect to have the right logic for every situation, yet not every logic.

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  • For the time being, my assessment of monism vs. pluralism re: logic is that talk of "the one true logic" is something like a category mistake. Rather, there are myriads of questions whose topic is logic (logic is not topic-neutral so much as omnitopical) and we can give answers to these questions for various reasons. Whether the answers are clear, compelling, strict, trivial, empty, iterable, etc. can be an open affair (just imagine trying to "prove" that second-order quantifiers "work better" than many-sorted ones, or something else along those lines). May 16, 2022 at 3:52
  • @KristianBerry I too think applying different logics is a very free exercise, but if logic is real these applications can’t be completely arbitrary or free I don’t think. Any application won’t do because then logic isn’t saying something about the world.
    – J Kusin
    May 16, 2022 at 4:41
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I have never run into Priest’s puzzle before. This is interesting.

I suppose the calculations would go like this: At some point the probability of success rises high enough that, although the odds in favor of escape are not certainty, they are still very good— say 95% or better. At that point a rational game player could call a halt to the repeated delays and announce that they would accept the result of the next toss.

One problem with the puzzle is that there is no result to compare to the bad result— eternal damnation. A game player who has a lover waiting might accept the result at a different set of odds; a game player who is only going to be late for work might not care what the result might be; and someone who would escape the puzzle only to re-enter an unhappy life might delay the final toss indefinitely.

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