The judge knew that the prisoner would reason it out and therefore any day would be a surprise.

So, what was wrong with the prisoner's reasoning?

What's wrong with his reasoning? What philcore said.

So, if he hadn't been hanged by Thursday night, would he have been wrong to conclude that he couldn't be hanged on Friday?

Alright. If his reasoning is wrong, what would your reasoning be?

It seems to me that the word surprised is not well defined. How can the judge know for certain that the prisoner will be surprised? What does that even mean?

The key here is the nebulous concept of surprise. Since the judge (presumably) cannot read the prisoner's mind, whether or not the judge is able to follow through on his promise to surprise depends on his ability to make accurate assumptions about the prisoner's state of mind. The instant the prisoner concludes something about what would/would not surprise him, the opposite becomes true. If the prisoner was able to understand this dynamic in advance, he would be able to prevent the judge from succeeding in surprising him, simply by acknowledging that the execution could occur on any day of the week.

To be clear, the "problem" with the prisoner's reasoning is that he fails to realize that what will surprise him is determined by his reasoning, and that whether the judge's choice will surprise him--assuming the judge is indeed trying to surprise him--is determined by how well the judge can predict his reasoning. This is true for any belief he may develop through reasoning, EXCEPT for the belief that would logically follow from his ex ante deduction of this dynamic--the belief that the execution could occur on any day of the week.

@semck83: yes, he would be wrong to draw that conclusion, for two reasons: first, to conclude that the judge "cannot" hang him on a day that would not surprise him--even assuming that satisfaction of the surprise caveat is a necessary condition for the execution to proceed, which is not made explicit--assumes that the judge can read his mind, and is thus aware at all times of what would or would not surprise the prisoner. Second, even if the judge was capable of this type of telepathy, for the prisoner to draw the conclusion re: Friday would instantaneously render a Friday execution surprising.

The key is that the prisoner can't necessarily trust the judge, and even if that's stipulated, he can't trust that his interpretation of the judge's words is correct.

The sad irony here is that, assuming for the sake of argument the prisoner must be legitimately surprised for the execution to proceed, any attempt he makes to save himself through reasoning risks signing his death warrant, while eschewing any attempt at reasoning his way out of the sentence saves him. I'm renaming this the Quicksand Paradox, or, "you surprised yourself, but U.O.E.N.O. it."

You can only rule out Friday by making use of the information obtained living out Monday to Thursday. On Wednesday morning you do not have all the information required to rule out Friday (or Wednesday and Thursday) leaving lots of opportunity for fun surprises.

Of course, if prison food is anything like the food in the Holiday Inn, he wouldn't have survived the first night.

I really hate the Holiday Inn. Alas, work really loves sending me there...

Given the prisoner's reasoning it turned out that any day would have been a surprise to him. Seems like the judge was able to anticipate the prisoner's thought processes. It would have been better for the prisoner if he had managed to convince himself that he would definitely been hanged on one of those days, at least then he might had a chance of saying to judge "I knew you would hang me today".

I am not sure what you mean by resolve the paradox? Do you mean explain why the is a paradox exists, or demonstrate that it really isn't a paradox. Or do you mean the prisoner should have guessed that Wednesday would be the day and offer a reason for this?

I don't agree with the prisoners reasoning. Surely any day would be a surprise seen as he doesn't know which day it will occur.

Spyman, I mean that the prisoner engages in a thought process that seems correctly to prove that he will not be hanged, but he is hanged without contradiction; so locate the problem (either with his proof, with the situation, etc.)

The reasoning is based on the (false) assumption that the hanging hadn't take place in any day before Friday, starting a correct induction process excluding any other day.

dang antonio just beat me to it. I read the thing on my phone and ... ah well..

Well played antonio :-)

I am waiting for a confirmation from the thread starter but, thanks anyway, I don't find any other possible solution with the information given :)

I disagree, antonio: that reasoning is considering whether Friday is possible. For the hanging to take place on Friday, then it would definitely have to have failed to take place the first four days. So that is a legitimate hypothesis when analyzing the possibility of Friday.

Think of it like a proof by contradiction. You assume it takes place on Friday to get a contradiction, and once you've made that assumption, then it's definitely true that there will be a point Thursday evening when you're still alive and know that it didn't happen on a prior day.

antonion is correct

Yes, but the prisoner hasn't any right to start any reasoning assuming that he hadn't been hanged until Friday. As Landowner said, there is something unclear in the "surprise" word meaning, and adding the fuzzy nature of judge's statement -a bit misleading- makes a prisoner conduct a straight logic reasoning while the whole situation cannot be translated into a pure logic scheme.
Have you ever read "Gödel, Escher, Bach: An Eternal Golden Braid", by D. Hofstadter? :)

More important are the jurisprudence issues with this - why is the judge giving such a sadistic punishment? What was the prisoners offence? Is the "surprise" element retributional judgement?

These are the things we need to know.

@SYnapse, nothing has to do with jurisprudence: the prisoner (as we all) are inducted to logic reasoning due to our human nature on the basis of an unclear judge sentence. The prisoner assumes that "surprise" is the opposite of "certain" or "predicted", hence he leads his induction by elimination one day after another, and the key is on prisoner's interpretation of the sentence.

This is a fairly simple paradox that people above have explained well.

Moving on: what if at just before noon each day the prisoner stood by his door and said "I know you're about to knock". Would this frustrate the hangman?

Maniac - after a few days of this it would genuinely be a surprise for the knock to come, as he's been expecting it every day.

The surprise is not the day of the hanging, but that instead of the hanging killing him he is in fact expired via blood loss after a Yorkshire Terrier bites off his testicles whilst he's dangling from the rope.

The prisoner is both hanged and surprised, thus dies satisfied...

...more or less...

I suppose that it's still a surprise as he's now still surprised he was hung at all...?

He wouldn't be surprised by the hanging, as the judge gave him plenty of warning. Even if he was disappointed the hanging happened on a certain day, it would hardly be a surprise.

The Yorkshire Terrier, on the other hand, is very surprising. All the more so if it was trained by the dog handling department of the Spanish Inquisition. Astonishingly so if said dog went on to use this training to win America's Got Talent.

Fear not men for we have the element of surprise.. Surprise!!

@Octavious: Looks like a Monty Python sketch

Octavious, get them to put you in an Embassy Suites. It's definitely a step up.

The surprise is that he was actually executed on schedule. Death penalty executions always get delayed for one reason or another. Last minute appeal for a pardon, electrical system down, concerns about what happens if you lose the Karstarks, etc.

By the way, I was that prisoner.

^And were you surprised?

I think the prisoner's main problem is that by saying it might not happen he has now made 2 options on Friday: getting hanged then or never getting hanged. This makes it uncertain and so he can't rule it out, meaning he also can't rule any of the days out.

Anyone feeling like they are watching the scene between Wesley and Vincini the Sicilian in Princess Bride right now?

http://community.eflclassroom.com/video/princessbride

@antonio,

"Yes, but the prisoner hasn't any right to start any reasoning assuming that he hadn't been hanged until Friday. "

This is completely wrong. One can make whatever supposition one wants when attempting to get a proof by contradiction.

To those pursuing the "surprise is ambiguous" line:

I think you are on to something important, but would invite you to try reformulating the problem with the ambiguity gone and see if and why the paradox fails to obtain.

It is simply a flaw in the condemned man's logic. He took the judge's decision to ignore the "logical" out of the equation. The judge may have decided to execute him on a Friday knowing that, if he got to Friday, he would figure he wasn't being executed off the simple logic of "I clearly am expecting it today so it can't happen today" which would then come as a surprise. There is no paradox.

If the prisoner was actually convinced by his own inductive process that he wouldn't be hung, then yes, it probably came as a shock when he found out he would.

"One can make whatever supposition one wants when attempting to get a proof by contradiction."

Well, the prisoner is not trying to make supposition about the judge's contradiction...he goes straight to a logic conclusion through a inductive process, starting from the assumption that what the judge said was true, and ending with a conclusion that he was lying (no hanging could take place). There is the ambiguity: if we follow the prisoner going back on time, what he assumed to be true (the judge's statement) becomes false (there won't be a hanging on day7 if he is alive on day6 after noon, an so backwards until day0, the sentence's day). Time plays a role because the same scheme could be applied with a indefinite length of time (between the statement and the hanging), but it's not valid when this time lap is 24hours, and the judge cannot sentence the same statement involving the "surprise" factor.

(PS. I am not native English, and trying to explain logic is quite difficult. I do know I am on a good path, but I'd rather step back and lurk as I did in the past and wait for post #100 :-D)

@antonio,

The structure of the first step of his argument is proof-by-contradiction (reductio ad absurdum). It goes like this:

1) Assume (by way of contradiction) that I am hanged on Friday.
2) This yields a contradiction. Therefore I cannot be hanged on Friday.
3) NOW use induction; at each step of the induction, use proof by contradiction.

This is a completely acceptable form of argument. It is used all the time in mathematics.

"(PS. I am not native English, and trying to explain logic is quite difficult. I do know I am on a good path, but I'd rather step back and lurk as I did in the past and wait for post #100 :-D) "

lol, you're doing fine I think.

@Draug,

"The judge may have decided to execute him on a Friday knowing that, if he got to Friday, he would figure he wasn't being executed off the simple logic of "I clearly am expecting it today so it can't happen today" which would then come as a surprise. There is no paradox. "

There are too many "may have" type ambiguities in this for it to be a compelling analysis. However, I take it you are saying that the prisoner should have realized that if his reasoning was correct, then it would undercut his reasoning. This is what a paradox is. I am asking people to locate precisely the source of the paradox.

I think there have been some very good responses above about the ambiguity of surprise, etc. I'm much less impressed by attempts to write off the problem as uninteresting, though. It has spawned quite active discussion in the philosophy, logic, and math communities over the years. I am skeptical of those webdippers who think they have plumbed its depth in one or two short sentences.

@semck - I disagree. A true paradox is when the impossible happens. Not when a person uses, as has already been pointed out, reductio ad absurdum to deceive himself. The paradox would have been if he was expecting it each day at noon and was therefore never surprised.

Draug,

Indeed, perhaps apparent paradox would be a better term, since this paradox can be resolved by careful analysis. (I would still say that analysis has not been carried all the way through by anybody on this thread, but those who question the use of the word surprise or look to the analysis of information shared between judge and prisoner at different points are, at least, on the path to a complete, successful solution.)

We also have to assess this from two positions. The judge may simply have been.lying. That fact was pointed out as being.part of the logic flaw (the condemned assumes the judge is telling the truth about the surprise part but then assumes the judge was lying about the hanging taking.place) and we also need to consider not the ambiguity of surprise but the fact that it is an emotional response to an unexpected event and not a logical result of a process.

Well, Draug, the judge could have been lying, but one can't assume that. If he ends up being hanged unsurprised, *then* we can conclude that the judge is a liar. Since he isn't, we have to figure out what is wrong with his argument that assumed the judge is *not* a liar. (In particular: he assumed the judge was not a liar; and the judge's statement did not turn out to be a lie. So the problem should be, and is, resolvable without any assumption that the judge may have been lying).

The fact that surprise is an emotional response is precisely part of the ambiguity that is referred to. It is, on the one hand, an emotional response; and on the other hand, the learning that something is false which we believed to be true.

But his argument assumed the judge was a liar as well. He assumed he wouldn't be hung even though the judge stated clealry he would. Ergo, he assumed the judge or deduced the judge was a liar.

Yes, I suppose that's true, Draug. By his argument, the judge would have to be a liar no matter what happened.