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If the prisoner considers “I won’t be hung at all” from the start, his argument falls apart on its first day—on Friday he could still be surprised, since he doesn’t know if he’ll be hung at all. Thus getting hung would still be a surprise.
Of all those paradoxical thought experiments, this is still my favorite, or really high up at least. Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted.
> Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted. That rhyme definitely wasn't something I would've forecasted...
What, did you expect me to rhyme with the line that's last, Ted?
If the prisoner is alive on Thursday at noon he will either be surprised to find he is to be hanged on Thursday, or surprised to find it will be Friday. Before that he had 50% expectation for either, and there is bound to be a negative surprise with regard to one day and a positive surprise for the other.
Of all those paradoxical thought experiments, this is still my favorite, or really high up at least. Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted.
> Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted. That rhyme definitely wasn't something I would've forecasted...
But also I think Martin Gardner concluded the same thing.
Reminds me of the definition of a magic trick from "The Prestige":
“Every great magic trick consists of three parts or acts. The first part is called "The Pledge". The magician shows you something ordinary: a deck of cards, a bird or a man. He shows you this object. Perhaps he asks you to inspect it to see if it is indeed real, unaltered, normal. But of course... it probably isn't. The second act is called "The Turn". The magician takes the ordinary something and makes it do something extraordinary. Now you're looking for the secret... but you won't find it, because of course you're not really looking. You don't really want to know. You want to be fooled. But you wouldn't clap yet. Because making something disappear isn't enough; you have to bring it back. That's why every magic trick has a third act, the hardest part, the part we call "The Prestige".”
Kind of abstract but it's like...
- The Pledge: Hello convict, here is your sentence.
- The Turn: Actually just kidding, look at this giant loophole in my sentencing. Wow! Expectations subverted!
- The Prestige: Haha, no, I'm just messing with you. There's no loophole but you lacked the context to discern that in The Turn phase.
edit: This is pretty interesting. As the wiki says, how one defines a "surprise" is the whole bugger of the thing. If you use the magic trick metaphor, "surprise" is reframed as "context."
The reason it's so murky trying to pick apart what actually happened is hard is because that's the point of the system. It's designed to be un-figure-out-able. I would love to know how you'd formalize a system whose output is "lack of context"
edit2: Also if you zoom out a level, giving someone a link to the Unexpected Hanging paradox page is a repetition of the same Pledge->Turn->Prestige pattern
- The Pledge: Here's a wikipedia page. You've looked at information on wikipedia pages. You share my confidence this info on this page will adhere to logical rules.
- The Turn: Oh but buh-bam, now you are busy trying to unscramble a paradox.
- The Prestige: You can't figure out paradoxes, that's their point. I will, however, confess this is a terrible magic trick.
There is a simple resolution to this "paradox": it's only a paradox if you consider the judge's initial statement to be True. If the judge's initial statement is False (i.e. he's lying) then the whole situation is logically consistent. The truth table for modus ponens helps clarify, both F->T and F->F evaluate as T https://en.wikipedia.org/wiki/Modus_ponens#Justification_via... The judge is saying he can predict how…
Can this concept be generalized to the point where NOTHING is a surprise? For example if a surprise party is planned for someone at some point in $TIME_PERIOD, it can't occur during the very last hour/minute/second of $TIME_PERIOD because it wouldn't be a surprise. The surprise party can't occur during $TIME_PERIOD - 1 because this would be known as well. Repeat this ad infinum just like the weekdays in the article.
> if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday.
This belief is FALSE. The moment of surprise for a Friday hanging takes place at 12pm on Thursday. Of course, this also happens to be the moment of surprise for a Thursday hanging. In other words, at 11:59am on Thursday he will not know if his hanging takes place on Thursday or Friday. At 12pm he will be surprised to know for certain which one it is: it will be Thursday if he hears a knock and it will be Friday if there is no knock.
The prisoner holds this belief because he makes the FALSE assumption that his moment of surprise MUST occur at the sound of the knock ON THE DAY of the hanging. The prisoner correctly understands that there cannot be a moment of surprise on Friday but, due to this assumption, incorrectly reasons that a Friday hanging cannot be a surprise. It will be a surprise, but he will experience that moment on Thursday, not Friday.
The prisoner pictures himself at 12:01pm on Thursday wondering how he could possibly be surprised for a Friday hanging. He doesn't realize he has ALREADY been surprised by the news of his Friday hanging. That moment occurred one minute earlier (12:00pm)!
The remainder of the "paradox" text is rendered nonsensical because the reasoning is based on the apparent impossibility of being surprised by a Friday hanging.
TLDR
The prisoner makes the following faulty reasoning:
1) The moment of surprise must occur on the day of the hanging (FALSE)
2) There cannot be a moment of surprise on Friday (TRUE)
-> Therefore, a Friday hanging cannot elicit surprise
-> Therefore, Friday cannot be chosen as the hanging day
-> Therefore, Thursday cannot be chosen as the hanging day, etc
#1 is assumed to be TRUE by the prisoner. Unlike the other days, the moment of surprise for a Friday hanging occurs the day before when he does NOT hear a knock at 12pm. Therefore, #1 is false and leads to the false deductions