The Unexpected Hanging Paradox

Factual Questions
1

From Wikipedia’s article on The Unexpected Hanging Paradox:

This puzzle truly has me paradoxically confounded. I understand perfectly both sides of the puzzle, and simply can not resolve them, despite the alleged “explanations” contained within the Wikipedia article. I find it hard to believe that this paradox can not be resolved, in spite of Wikipedia’s claim that “Despite significant academic interest, no consensus on its correct resolution has yet been established”.

Help? :confused:

What is the resolution to this paradox?

2

I don’t know the answer, but I wanted to say that there are entire classes of logic puzzles that presume that two people are interacting with perfect logic, and therefore you can get some very convoluted meta-logic involving the two calculating that the other has calculated that the first has calculated… -> ∞. I suspect (but have absolutely no way to know) that such puzzles allow for the possibility of Russell’s Paradox-type condundrums

3

I’m afraid that I don’t see this as a paradox. I see it (given that he was surprised) as faulty reasoning by the prisoner.

4

Please explain his faulty reasoning.

5

It is not “resolved” only because some people refuse to accept the arguments of others. I know a guy who wrote a well-cited article on it and I disagree with him almost 100%. Which of us is right? I’ll give my point of view:

It is not a paradox at all. It’s just a statement of fact. Given two people, with the same information and full logical abilities, one person cannot make a rational choice that the other cannot deduce.

People are people, however. They make irrational choices all the time as well as having knowledge that others don’t have. E.g., maybe the hangman starts vacation on Wednesday and the judge decides to do the drop on Tuesday.

Once the prisoner realizes that the judge can make an irrational choice and/or has knowledge the prisoner does not have, the quasi-inductive argument goes out the window.

(It gets really interesting when the number of options is reduced. With a two-day time frame, the judge is going to make an extremely irrational decision. With a one-day time frame, it’s so irrational of a choice that it’s off the scale. Which means the prisoner cannot reliably know what the frak is going on with the judge and whether the sentence is going to be carried out at all.)

6

His logic is correct IF you assume that it is shared between the jailors and himself, AND IF you assume that that his definition of “surprise” is also shared, AND IF you assume that there would be no execution if he did know when he would be executed.

7

The judge knows that the prisoner will conclude from the information provided him that he can’t be hanged at all, therefor the judge is true to his word when he surprises the prisoner by hanging him on Wednesday.
maybe :smiley:

8

The flaw is in assuming that the prisoner will never learn the date of his execution. Obviously, this is false, since eventually he’ll be falling through that trapdoor, and he’ll certainly know then. But if we remove the silly assumption that he’ll never know, then what we’re really left with is just that he doesn’t know right now, but that he will eventually.

9

Well, that is pretty much what “not resolved” means.

Anyway, I think the prisoner’s argument is unsound. He starts his chain of argument with “If I have made it to Friday, I cannot be hung on Friday,” but if he is hung on any earlier day the antecedent of that conditional will be false and the chain of reasoning will not get off the ground. He is not entitled to draw the conclusion that he will not be hung on Friday until he has actually got to Friday and the antecedent has thus been proved true. (Also, if he gets to Friday and concludes he can’t be hung then, because it won’t be a surprise, then, if they do hang him that day it will be a surprise after all!)

10

I guess I don’t understand how this is even a paradox. Because the prisoner has convinced himself that he will not be hanged, any day on which his hanging occurs will be a surprise. The explanations are overthinking the problem to an absurd degree.

11

Isn’t the flaw that “surprised” doesn’t have a rigorous meaning in logic?

12

The paradox is that the knock is a surprise - not that the prisoner, contrary to his conclusion, is actually getting hanged.

13

I thinkt he only day which fulfills the original contract is Friday. Friday is logically impossible, therefore a total surprise if the hanging occurs on that day.

14

I don’t think that is a flaw. You could easily restate the paradox so that instead of being hanged, the prisoner is to be executed in some way that is so quick that he’s dead before he knows it.

It’s not just that it gets interesting when you reduce it to one day-- that is the very crux of the problem. Having more than one day is just a distraction. The basic paradox is this: “You will be hanged at noon today. You will not expect it.”

15

Yes, that was my understanding. The meaning of “surprise” or “unexpected” is used by the prisoner to logically conclude that he cannot be hanged. Then there is another level of “surprise” if he is indeed hanged on one of the days. So what is an unexpected surprise? Something expected? If so, then the judge’s edict was not doable. BTW, this paradox is easier to resolve if you just reduce it to one day instead of any day of the week. The judge could’ve said: “You will be hanged on noon on Friday. It will be a surprise to you.” The prisoner can reason that this edict cannot be carried out since it’s clearly contradictory. And then he gets executed on Friday, and thus surprised!

16

So what if we make it rigorous? Let’s restate the problem this way: The judge tells the prisoner that he will be hanged on one of the weekdays next week. The prisoner is given one piece of paper and a pencil. There is something like a mail slot in the cell door such that the prisoner can slip the piece of paper through the slot, the paper will drop into a bin, and then is not retrievable by the prisoner.

The rules are this: The hangman will come to the prisoner’s door at exactly noon on the day of the hanging. If the mail bin contains the piece of paper that was given to the prisoner (the one-and-only original piece, let’s say somehow unmistakably identifiable, and completely intact, i.e. not torn into smaller pieces) and on that paper is written the current day of the week (written by the prisoner himself, of course), then the hanging will not take place and the prisoner will be freed.

In other words, if the prisoner can correctly predict the day of the hanging, he goes free, otherwise he gets hung.

So, for example, if the hangman shows up at noon on Wednesday and the bin has the piece of paper with “Wednesday” written on it, it is clear that the prisoner predicted that the hanging would be attempted on Wednesday, he was correct in that prediction, and therefore no “surprise” will take place.

On the other hand, if the hangman arrives at noon on Wednesday and finds the paper with “Tuesday” written on it, the prisoner has predicted incorrectly and therefore we can count Wednesday as a “surprise” and he gets hung. And, of course, anytime the hangman arrives and there is no paper at all, we declare that a “surprise” and the hanging goes ahead.

So, now, does that change anything?

The prisoner can still reason that if Thursday noon passes with no visit from the hangman, he can write “Friday” on the paper, put it in the slot, and be guaranteed to be set free at noon on Friday. Therefore, he can reason that it would be stupid of the judge to wait until Friday to send the hangman, and so Friday can be ruled out as a possible day of the hanging. And now with Friday ruled out, he can use the same logic that if he is alive on Wednesday after noon, the only possible day is Thursday, he can write “Thursday” on the paper, put it in the slot, and when the hangman arrives on Thursday, he’ll be set free. And so on, ruling out all the days.

So then when the hangman actually shows up on Wednesday and the bin is empty, the prisoner gets hanged.

I think njtt hit it on the head: The prisoner’s chain of logic begins with “If I’ve made it to Friday…”, which is only a valid assumption on Friday. He ruled out Friday as a possible day of hanging only by virtue of making it to Friday. Ruling out Thursday based on having ruled out Friday which was based on having made it to Friday, just isn’t a valid chain of logic.

17

You can pretty easily tell who read the Wiki article in this thread.

18

I think the assumption in “it will be a suprise” is that it will be a surprise all the way up until the moment they knock on his cell door. In your scenario, once noon on Thursday has passed, he will know for 24 hours the hour of his death.

I don’t have a clue how to resolve this paradox.

19

I guess I’m not seeing the paradox. If the prisoner concludes the judge cannot show up and he does, wouldn’t he be surprised?

20

Isn’t it the case that if this is a paradox it is also a paradox that Peyton Manning is able to surprise the Tigers defensive line with a quick snap? After all, they know that if he hasn’t yet snapped the ball he must do so as the play clock hits zero. Since snapping exactly at zero would not be a surprise then the new deadline for losing surprise is just before that…and repeat onward until he doesn’t snap it all.

Therefore, if Peyton Manning’s goal is to surprise them he will incur a delay of penalty by never snapping the ball.