Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Unexpected hanging paradox

en.wikipedia.org

Post reply on HN
Print view View original

41–50 of 87 posts

Re: Unexpected hanging paradox

#41

Post by aekotra »

incompatible wrote: post #33
aekotra wrote: post #30

The prisoners reasoning is wrong. > 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 pla…

"He will not know the day of the hanging until the executioner knocks on his cell door at noon that day." The wording of the problem is everything.

I am very aware of that line. I've reworded most of the OP, to explain more clearly!

Re: Unexpected hanging paradox

#42

Post by p1esk »

jzwinck wrote: post #23

There is a surprise on Thursday (if not before). 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.

he will either be surprised to find he is to be hanged on Thursday, or surprised to find it will be Friday

The judge says the prisoner would be surprised exactly when the knock on the door happens (not before). As a consequence, the prisoner is right to believe he won't be hanged at all (following his reasoning), but as a consequence of that, he is surprised when the knock happens. In turn, as a consequence of that, the judge is right. Therefore, both the prisoner, and the judge are correct. Thus the paradox.

Re: Unexpected hanging paradox

#43

Post by p1esk »

aekotra wrote: post #40
p1esk wrote: post #38

Earlier quoted context omitted.

He will not know the day of the hanging until the executioner knocks on his cell door at noon Did you miss that line? Embarrassing indeed...

I don't follow. Can you please explain? I am very aware of that line.

The judge says the prisoner would be surprised exactly when the knock on the door happens (not before). As a consequence, the prisoner is right to believe he won't be hanged at all (following his reasoning), but as a consequence of that, he is surprised when the knock happens. In turn, as a consequence of that, the judge turns out to be right. Therefore, both the prisoner, and the judge are correct. Thus the paradox.

Re: Unexpected hanging paradox

#44

Post by colanderman »

throwaway2048 wrote: post #19

I've never understood why the resolution to this paradox isn't because of his confidence he expected not to be hung, thus any hanging would be unexpected/a suprise. Dosent seem terribly paradoxical to me.

Yes. The judge's statement is only valid because of the prisoner's short-sightedness (i.e., it is a prediction, not a logical statement). If the prisoner knows the punch line of the joke, the judge's pronouncement becomes undecidable and therefore meaningless.

It's no different than if the judge said, "you will be hung, and this statement is false", and the prisoner was surprised that they got hung.

Heck, even if the prisoner doesn't know the punch line, the judge's statement is invalid if they incorrectly judge the depth of the prisoner's thought. The prisoner could stop their logical reasoning after 2 days instead of 5, and be sure that they're being hanged Wednesday, making the judge's prediction false.

In summary: the judge made (intentionally or not) a prediction; the prisoner interpreted the prediction as a logical statement; the prisoner failed to see that the statement is undecidable, and drew a logical conclusion from it; the conclusion the prisoner drew happened to make the prediction true.

Re: Unexpected hanging paradox

#45

Post by p1esk »

pieguy wrote: post #10

Here's my take. The prisoner considers the statement "I will be hanged, and it will be a surprise", and after "proving" it false, takes its negation "Either I won't be hanged, or it won't be a surprise". But this is not correct. The prisoner actually disproves the statement "I will be hanged, and it will be a surprise no matter which day it happens". When you negate this statement, you obtain "Either I won't be hange…

The paradox is because the prisoner does indeed disprove the statement "I will be hanged, and it will be a surprise no matter which day it happens", but the statement turns out to be true (he is both hanged, and surprised no matter what day he's hanged).

Re: Unexpected hanging paradox

#46

Post by Gormisdomai »

mattdeboard wrote: post #27
tristanj wrote: post #4

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…

This was convincing at first. But this logic problem is predicated on the notion judges do not conceal their intentions (iow that his initial statement is True). There is just no point in even considering the question if the prisoner isn't actually going to be executed.

The Judge could be lying even if the prisoner is actually executed - in particular he could be lying about the "and it will be a surprise" bit.

Re: Unexpected hanging paradox

#47

Post by panic »

One way to think about this paradox is as a game with two players—a prisoner and an executioner. The game progresses over five turns, representing the days of the week.

On each turn, the prisoner secretly chooses whether or not they expect to be killed on that day. Then the executioner chooses whether or not to kill the prisoner. The game ends on the turn the executioner decides to kill the prisoner. If the prisoner wasn't expecting to die, the executioner wins; otherwise, the prisoner wins.

The important detail here (which is obscured by the logical perspective) is when the prisoner is allowed to expect to die. If they're allowed to expect their death on every single day, the prisoner can just do that and win automatically. If they can only expect to die once, there's a situation the usual paradoxical argument doesn't consider: that the prisoner has survived until Thursday but has already used their chance to expect to die. In this case, they know they're going to die on Friday, but there's nothing they can do about it.

Re: Unexpected hanging paradox

#48

Post by zaroth »

> He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

I have placed a ball under one of five cups. You can check only sequentially if the ball is under each cup.

Do you know which cup the ball is under?

If you start turning over the cups one by one, will you be surprised when you find it?

You won’t be surprised after you’ve turned over all but one cup, if you still haven’t found it, since you know the ball is under at least one of them.

You would be surprised if the ball was found under any of the other cups though.

And so there are only 4 days on which you would be surprised. The executioner knocks on Thursday and provides more data than on the prior 3 days.

The judge makes a promise he cannot keep, that no matter what day he schedules the execution that it will be a surprise.

But how can you schedule something for the end of a finite series and still have it be a surprise?

You can’t. So then does it follow if the usable tail of my sequence is marked out of range, what then becomes of the 2nd to last item in the sequence?

This assumes a sequential cascade leads to the state where all possible days have been marked out-of-range.

The simple reality is that the judge can only promise you a surprise Monday thru Thursday. By Thursday at noon the surprise will be spoiled. And there’s a chance that you will sit for a day knowing exactly when you will die.

Re: Unexpected hanging paradox

#49

Post by tristanj »

mattdeboard wrote: post #27
tristanj wrote: post #4

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…

This was convincing at first. But this logic problem is predicated on the notion judges do not conceal their intentions (iow that his initial statement is True). There is just no point in even considering the question if the prisoner isn't actually going to be executed.

I should clarify. The judge is not necessarily "lying" (i.e. making a false statement). He's making a statement which has no bearing on the future. The judge's statement could end up being True or False. Using that interpretation resolves the "paradox".

The question encourages you to start from a flawed premise to begin with. Why must the judge's first statement be True? A judge cannot guarantee how someone will feel the future. Further, why must the prisoner be surprised at the date of his execution? If the prisoner is not surprised at his date of execution, that is also a logically consistent situation.

The question tries to shoehorn you into an irreconcilable input state and output state. Yes, it is a paradox if you assume the question's implied start and end result. However if you remove the limitations implied by the question, in the bigger picture there is no paradox.

Re: Unexpected hanging paradox

#50

Post by p1esk »

zaroth wrote: post #48

> He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. I have placed a ball under one of five cups. You can check only sequentially if the ball is under each cup. Do you know which cup the ball is under? If you start turning over the cups one by one, will you be surprised when you find it? You won’t be surprised after you’ve turned over all but one cup, if you still…

The judge makes a promise that he cannot keep, but precisely because of that, he keeps it. Thus the paradox.
Post reply on HN