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Archive 1Archive 2

Should have hanged him at the weekend.

Should have hanged him at the weekend.

I'm not sure I agree with you on this one, LC:

The next week, he is hanged on Wednesday which, due to his reasoning, is a complete surprise. His reasoning had "proved" that he couldn't be hanged at all. Therefore, because of that reasoning, the hanging on Wednesday was very surprising. Everything the judge said turned out to be true. If it was all true, then where is the flaw in the prisoner's reasoning?

The hanging was "surpising" (i.e., unexpected) simply because it was, as the judge said, on a day he hadn't been told about. His earlier "deuction" that it wouldn't happen at all didn't make any difference. Had the prisoner made no such deduction, his hanging would have been just as unexpected. Our prisoner may have experienced some extra surprise because of his reasoning, but that is orthogonal to the paradox itself. The paradox is simply that the prisoner reasons thus: he is told the hanging event will have property X ("unexpectedness"); he reasons from that premise to the conclusion that it cannot happen at all; then, it happens, and it indeed has property X, just as advertised. --LDC

I do not disagree with you. As you say, the reasoning actually added a little extra surprise. That extra surprise is what I was referring to by the "complete" and "very" adjectives. That extra surprise was due to his reasoning. Saying "despite" doesn't sound quite right, since it suggests that his proof of no hanging should have reduced his surprise. I agree this whole concept is unrelated to the paradox. I'll just revert back to the original, and delete the clause in the middle of the sentence. --LC


I dunno

...This one could work out, if you consider the prisoner's reasoning. He reasons that he cannot be hanged on Friday, as he will have survived all but that one day. Now, working back to Tuesday, he must have realised to himself that they could only hang him on Monday. Think about it. He would think that they would hang him then because that is day 1. By his reasoning, days 2-5 are already accounted for. When he isn't hanged on Monday, his line of thought is wasted, and he is at a loss as to what day he will be hanged. He knows it can't be Friday, but apart from that, he does not know when. But there is also a flaw there. Considering the situation now, suppose that it is Tuesday. He must be now thinking that they did not hang him on the day before, so this is a new day 1. Now, they do not hang him on Tuesday, so he moves on to Wednesday knowing that it is either that day or Thursday. He has not yet survived Wednesday as yet, so he knows that he can either be hanged on that day or Thursday. He knows that if he survives Wednesday, then he will die on Thursday. So he reasons that he will definitely die on Wednesday. But he knows that the judge said that he will be surprised, so he must wonder whether he will really die on Wednesday. He must therefore think that he cannot die on Wednesday, but instead on Thursday. This leads to another contradiction, as he will know his day of execution. He knows that there is only two days left, and he reasons that his odds of dying on Wednesday must be 100%, as he will know what happens if he survives. But he also knows that the judge does not want him to know when he dies. Thus, he is left confused by the judges words, and does not truly know when he will die. That is how the executioner finds him. Upon the knock, he must know that the judge was very wise in his choice of words, as he did not really know what to expect...


I almost confused myself there, but it makes sense, I think.

Deletion of "simplest form" Passage

I deleted the passage purporting to give the "simplest form" of the paradox because it presupposes that the prisoner's reasoning involves self-referring premises. This is not only something that is disputed by scholars of the paradox, it is actually a minority view and I don't believe any published scholar of the paradox has endorsed it for a couple of decades at least. Both the Kirkham and the Wright & Sudbury papers in the bibliography give detailed versions that contain no self-reference. Even the informal versions in this article (as of 7/4/05) don't have any self-referring premises. --Nate Ladd July 4, 2005 20:58 (UTC)

Further thoughts and a Different View

The original statement (The execution will be a surprise to you: you won't know the day of the hanging until the executioner knocks on your cell door at noon that day) no longer exists when the prisoner reduces the possible days in which he can be hanged from five (M-F) to four (M-Th), and therefore cannot be tested for validity or lack thereof. It no longer exists because a requirement of the original 'paradox' is that the available days for execution are M-F.

It IS possible for the prisoner to CORRECTLY deduce that if he is not called to execution by noon on Thursday, then the execution will not be a suprise to him and he can therefore not be hanged. HOWEVER, he cannot go further into eliminating the other days because that changes the number of days the execution is possible, which revokes it from the original paradox. This makes new(?) paradox that the prisoner can be hanged on a day where hanging can't take place, and I believe is a separate paradox.

All I'm (probably not) doing is proving that the original paradox cannot exist, but that an equal one is made through re-wording.

Deletion of Murphy Reference

The following entry in the Annotated References section was deleted.

R. P. Murphy, "The Games Economists Play", Ludwig von Mises Institute Daily Articles

The reason is that it is not a peer-reviewed publication. Also it only repeats the paradox. The closest it comes to offering a solution is to quote another publication written by Martin Gardner which isn't itself much of a solution anyway. Add a reference to the Gardner publication if you want. --Nate Ladd 19:06, August 16, 2005 (UTC)

OK. Can this reference then be put into a 'related reading' or 'external links' section or something?
I could live with that, but Murphy doesn't say anything new about the paradox, so I wonder why you want a link to his article at all. I must say I have suspicions that this is only a way to generate traffic to the web site in question -- a kind of advertising. Add an 'external links' if you want, but don't be surprised if somebody else deletes it. --Nate Ladd 21:40, August 16, 2005 (UTC)


Surprise is guaranteed

Suppose, there is some line of reasoning that allows the prisoner to deduce the day of the execution.

Then the judge can repeat the same reasoning and schedule the execution on a different day.

An even better version: suppose, some line of reasoning that allows the prisoner to deduce that he will NOT be hanged on some particular day.
Then the judge can repeat the same reasoning and hang him on that day.

References

I know of a reference to this paradox in Sideways Stories from Wayside School... Should a References section be added to inclued it?

Hmmm

Is it not the case that for the prisoner to make the arguement that he will not be hanged on a given day, that he must first conclude that he will be hanged on that day?

Is it not impossible to believe both?

Doesn't that put an end to it?

A Sensible Interpretation

First, I will consider the 'surprise' to only occur if there are multiple possible days of execution at the time the analysis occurs. I am discounting notions of being surprised because the judge lied, or because his logical reasoning was false. I am taking the logic at face value.

A. The prisoner will be hanged at noon one day next week, Monday through Friday.

B. The execution will be a surprise to the prisoner: the prisoner won't know the day of the hanging until the executioner knocks on his cell door at noon that day.

'Common sense' provides the true meaning of the judge's statement B - the prisoner won't be informed of the day prior to the executioner knocking on the door. However, B literally stipulates that the prisoner won't know the day up until the moment of knocking, which includes the moment just prior to knocking.

To determine if the prisoner knows at this moment, the prisoner will make an analysis of the situational constraints. In light of this fact, the prisoner reaches the absurd conclusion that conditions A and B cannot both be met.

The contrast between the judge's intent and the prisoner's reasoning is due to the additional constraints that the stipulation of 'not knowing prior to knocking' implies. The judge's statement is only non-contradictory if not taken literally, or if the prisoner's ability to logically deduce the day is assumed to be outside of the prisoner's 'knowledge'.

Additionally, the seemingly apparent logical paradox is due to the self-referencing of one of the stipulations that includes the concept of 'knowledge' which includes the constraining results of deductive reasoning of the situational stipulations.

Turing Halting Problem

The structure of the paradox is the same as for Turing's Halting problem, the demonstration that there is no algorithm for deciding whether a provided algorithm halts. "Surprise" simply means "can't compute", the prisoner is the hypothetical algorithm being tested, and the judge provides an input to it which by definition can give an answer only if the prisoner can not. (To make the parallel clearer: Imagine you have a prisoner who claims to be able to determine whether a prisoner can foretell the date of their own hanging given the prisoner's identity and the sentence. They are given themself, and that the sentence is that they will die on Friday if and only if they cannot foretell it.) As with Turing's Halting problem, any given algorithm has a question that can break it, but that question can always be answered correctly by some other algorithm. The judge's sentence is perfectly consistent, and the prisoner might even be informally aware that it is, but any prediction the prisoner makes on that basis will be formally inconsistent. The result of the sentence is well defined but is not computable by the prisoner.


Confusing Article

The part about the Logical School is somewhat confusing... For example, the logic behind the following part isn't really understood:

"...the argument is blocked. This suggests that a better formulation would in fact be:..."

How does it suggest that? Why change the formulation simply to force the paradox into working (not that it helps much)? If the prisoner's argument is indeed blocked (as it seems) - then there is no paradox at all, and it would have a rather elegant solution as well (in a 2-day paradox, assuming the judge referred to formulation A and didn't lie, the prisoner won't be surprised to be hanged on the first day, and this still won't contradict the Judge, because each "surprise" (or lack of it) is defined differently. Also, the prisoner won't be able to reproduce any more stages of the argument (more days), and therefore really WILL be surprised in a more than a 2-day paradox)

Secondly, what does Fitch actually say? that the second formulation is a good one or that it isn't? The first sentence says some claim that it isn't, then it says Fitch showed it is actually ok, and then it says that he showed it isn't. Well - is it or isn't it? If it isn't, then we return to the first formulation or to the "objections" part, and the paradox is obviously solved.

As for the "objections" part, it seems to solve the objections being raised, so in the end there aren't really any objections left.

How is it a "significant problem in philosophy", when a complete solution appears to be in the article?

the link given is a major time-waster. It restates the assumptions that we should all recognize as obvious when reading the paradox itself across FOUR pages

then states it's argument across the last half a page, which is unhelpful too. In short, it says the man to be hanged is equally certain that he must not be hanged on any day, therefore he may be hanged on ANY say. This is the essence of the paradox, it does not undo anything!

You Cannot Say You Are Not Surprised After The Process of Giving Surprise Starts and You Observe It

The problem here is because of the elongation of time in the process of giving surprise to the prisoner. Consider Bob has 5 balls in his bag (red, green, blue, yellow, purple) and he says to Alice that he's going to give her the bag with one of these balls (like you'll be hanged) but which ball that would be is a surprise for her (like the day of the week). At this moment, it is indeed going to be a surprise for Alice which ball she gets. Now Bob starts removing ball from the bag one by one... and Alice observes it, so after 4 balls are removed (and say purple one is left there), she cannot say, "you see, I know now that it is purple ball there, so it is no surprise for me!" She already had a surprise by the time 4 balls are taken out. Similarly the prisoner already had a surprise if he has not been hanged for first four days. How can you have the same surprise again? Which day it would be is a surprise at the time sentence was announced but cannot be once the process of giving surprise is initiated and each day is eliminated one by one.

So if a prisoner says that his hanging on Friday is not a surprise for him, he is flawed in his statement, as he came to know about it just after Thursday noon, of course, surprisingly, as he could have been hanged on Thursday as well. If he says that before the start of the week (that he knows he is going to be hanged on Friday), then it is agreeable that it's not going to be a surprise for him.

So even if the prisoner is hanged on Friday, it is a surprise for him. That the surprise was revealed for him just after Thursday noon is another thing. It's just that there is some time gap before the surprise was open for him and he is hanged. But it is nevertheless, a surprise. If prisoner's argument is bought then there can be no surprises in the world, as just after any surprise is revealed (and before some act related to the surprise is completed), it can be said that now that we know it, it's not a surprise. Mani Sareen (talk) 07:29, 16 December 2012 (UTC)

No, "surprised" means that the prisoner won't know he's being hanged that day until the executioner comes to knock on his door. If it doesn't happen until Thursday, then the prisonner will know for sure that it's on Friday.

It doesn't matter that the prisoner decides he will escape

The judge makes 2 claims:

1) The prisoner will be hung on a weekday in the coming week

2) The prisoner will be surprised when summoned to the hanging

The prisoner's reasoning (in deciding that no day will be a surprise) is sound and it leads to the conclusion that claims 1 and 2 cannot both be true. But, why does that make the prisoner assume that claim 1 is false? Common sense would tell us that 2 is false, but 1 is true. If he accepts 1 as true (and accepts that he will be hung), and 2 as false, the prisoner no longer has reason to expect to be hung every day, because that theory was based on the presumption that 2 was true. He is therefore just as surprised to hear the knock at his door at noon on Wednesday as if the judge never said claim 2. The contradiction is removed, and the judge was right all along.

173.23.82.190 (talk) 11:24, 4 January 2014 (UTC)

Please see WP:NOTFORUM and WP:OR. If you want to suggest changes to the article, please state what should be changed, and which reliable sources support the change. Paradoctor (talk) 12:50, 4 January 2014 (UTC)

The View from 30,000 feet

The "paradox" here is because two contradictory things are asserted. Both can't be true. Here is a "resolution", light on "rigor" but heavy on "intuition".

By "complete information" I mean anything that the prisoner knows will happen with probability 1.

Thing 1: The prisoner has "complete information" about being hanged. When the prisoner is told "you will be hanged sometime next week", he has "complete information".

Thing 2: The prisoner does not have "complete information" about being hanged. When the prisoner is told "you will not know whether you will be hanged today", he does not have "complete information".

I don't have "complete information" if I will win the lotto today, even though it is possible I will. There is a small probability (say 1 in a billion trillion) I will suddenly have an urge to buy a lotto ticket that will win, but I do not know anything with probability 1. Similarly telling the prisoner "you may or may not hang today" implied lack of "complete information".

The "solution" for the paradox is that you can't both have and not have "complete information".

If you define the "events" rigorously using probability (measure theory etc.), define rigorously what it means to be "surprised" etc. etc. etc., any paradox goes away.

JS (talk) 19:04, 10 January 2014 (UTC)

WP:TPG: "Article talk pages should not be used by editors as platforms for their personal views on a subject." If you want to suggest changes to the article, please state what should be changed, and which reliable sources support the change. Paradoctor (talk) 19:07, 10 January 2014 (UTC)

this article should be deleted for lack of notability of the subject matter

This so-called paradox is not well known enough to warrant having its own article. Tweedledee2011 (talk) 01:02, 11 January 2014 (UTC)

WP:AfD, though you might want to check the weather report first, it might WP:Snow. Or read the article. Paradoctor (talk) 02:34, 11 January 2014 (UTC)

Lame Duck

I can't believe people are arguing this at all.

This "logic problem" is about as idiotic as the "arrow moves half the distance" problem. Anyone with half a brain realizes that there is no paradox, they just might not be able to articulate exactly why.

This problem breaks down the minute the timeframe is longer than 2 days. It's always a surprise except on the last day. This is exactly a case of the gambler's fallacy. Nothing more. Odds don't change due to expectations. If the Judge draws a random lot to choose the day (which is implied) then any expectations the prisoner makes are pure fantasy. He will be surprised by the day, in fact, he makes himself more surprised by the very concept of believing he can outwit the judge.

Think about it.

Lajekahr 15:15, 12 May 2007 (UTC)

Of course paradoxes do not exist, not in mathematics at least, but the point is to find situations which would be very hard to believe to not be paradoxical. The Unexpected Hanging Paradox is an excellent example in my opinion. To solve a "paradox" means to find an error in it. The point is not to decide whether the prisoner can outwit the judge, but to find where the prisoner made a mistake. By the way, the case of 2 days is no different from the case of 5 days, if the prisoner is to be hanged on the first. People are arguing, i think, because there are many possible solutions, but the explanation Lajekahr has given is not a complete solution. Do you think the prisoner made a mistake when he concluded that he would not be hanged on Thursday, but with Friday his conclusion was valid? Think about it. --Cokaban (talk) 15:30, 8 December 2007 (UTC)

I don't understand why this is considered a paradox at all. If he's alive on Thursday night, he would have to be hanged Friday, which means he expects it, which means it can't be the day. If he's alive Wednesday night, then the day can't be Thursday, because if it wasn't Thursday, it would have to be Friday, which he would expect, etc. However, on Monday morning, how does he know if it's going to be Monday, or Tuesday? Can't be Thursday because at 12:01 PM on Wednesday he would KNOW it's Thursday because it can't be Friday. If he's alive on Monday night, it could be Tuesday or Wednesday, which means they could kill him Monday and he would be surprised. — Preceding unsigned comment added by 207.199.253.60 (talk) 19:43, 18 February 2014 (UTC)

The Selfish Gene

Richard Dawkins brings up this very subject, although not by this name, and with regard to a different scenario, in The Selfish Gene. He talks about it in the context of the prisoner's dilemma, though in an iterated fashion, but where the number of rounds is unknown to either adversary, because of course if either of them knew that the game was going to end in a predictable number of rounds, the only rational thing to do, assuming the other person is also rational, is to defect.[1] The shadow of the future must be long in order for both adversaries to cooperate. Perhaps this would make an interesting addition to the page?24.6.187.181 (talk) 18:08, 7 April 2015 (UTC)

This is not a paradox at all

This paradox should be deleted as it arises because of the false assertion that on Thursday one minute past noon you are still alive. Only then you can say that on Friday you can’t be hung and build on the argument backwards till Monday… — Preceding unsigned comment added by Entropy1963 (talk • contribs) 07:54, 21 July 2015 (UTC)

Schools?

I'm a philosophy graduate student, and I wasn't aware that there are competing logic and epistemological "schools." Rather, there are different interpretations of this problem that suggest different resolutions.50.191.21.222 (talk) 02:14, 3 February 2016 (UTC)

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Really? I'm Not Sure

It seems to me that, if he thinks he cannot be hanged, using the reasoning that he employs, then using a basic principle of logic, the judge should be able to hang him any day. He does not think he can be hanged, so any hanging at all will be a surprise to him.

165.155.196.99 (talk) 12:47, 17 November 2016 (UTC)

Yes, that's the "apparently simple" solution. The problem is, if such is the logical conclusion, which step in the prisoner reasoning is wrong? Diego (talk) 12:53, 17 November 2016 (UTC)
Yes, that's the paradox. The prisoner's reasoning leads to a false conclusion. Clean Copytalk 15:12, 17 November 2016 (UTC)

I think the paradox arises from taking granted what the judge says to the prisoner, namely that he must be surprised by the timing of his execution. To cut the gordian knot non philosophically one could say to him, -So you werent surprised. Doesnt matter to me, hang you anyway. Or one could say that language is paradoxical in its nature, and that causes the confusion of the prisoner. (Like the judge said he WILL ne hanged, and also said he WILL be surprised. What if the judge never meant this to mean that he WILL NOT be hanged unless surprised.)

Anyway, each morning the prisoner must assume that he will be hanged the same day's noon, otherwise he would be surprised to be hanged that day and therefore can be executed in a legit way according to the judge's promise. If the prisoner does expect every day that he is going to be hanged that very day, then he still a legit target to hanging after the very first day passed, necessarily being a surprised man from there on, since he must have been surprised when he was not hanged the first day. Now if we allow the prisoner to take a gambit and just guess the time of his execution, only expecting to be hanged the day he picked on, that would mean that he can not be executed only that day. Whethet he guessed right and was hanged unsurprised, or not we actually may not know. The last day can only be eliminated as unsuitable because he would not be surprised by then, if the promise of surprise is taken for granted. In reality the judge's words mean: -I kind of try to surprise you, but if i fail at that, so bad, you still die when i want so. My word, so I take it back if I wish so. Hence the true uncertainity of the prisoner. It would be only a paradox if we assumed tbhat the judge must not be mistaken and also must not lie. On the other hand, the situation can lead to a paradox because of the self referencing nature of the prisoner's thinking. He is trying to guess whether he will be surprised, which depends on an objective (outside) factor (the actual day the judge wants to hang him) andf from a self-referring factor (whether he opts himself being expecting the execution or not). I think it is not defined in the original problem, whether the prisoner has one guess about the time of his execution (whic in case he guessed right would grant him a parole - either permanently, or only for that particular day - this is also not defined), or whether he is allowed to guess again every day. The latter would allow him to say every day when the hangman comes - Hey Joe, I just thought you would come today, - To which the hangman would have to reply -Oh, did you, okay, then I come back some other day. 176.63.176.112 (talk) 19:09, 1 December 2016 (UTC).

"which step in the prisoner reasoning is wrong?"

The prisoner's error is as follows:

  • he identifies that the judge's statement is paradoxical if he is to be hanged at all
  • he correctly recalls the basic principle that paradoxical situations cannot physically arise
  • he incorrectly assumes that the process of making a paradoxical statement is a paradoxical situation (but lots of people would make this error, so it's not his fault really)
  • this leads him to rule out that universe containing a paradoxical statement.

The error is fairly obvious when you realise that, of course people can make paradoxical statements, one need only read through Category:Paradoxes to see any number of them. The existence of these Wikipedia articles has not triggered the destruction of the universe. :-)

Correct thinking would have been to realise that the "surprise" portion of the judge's statement gave him no information at all. Perhaps he should have paid more attention to the first part of the statement instead. :-)

--2.101.114.143 (talk) 18:57, 21 April 2018 (UTC)

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