Talk:Unexpected hanging paradox/Archive 1

Common Sense Intervenes

 * 'The prisoner reasons that he cannot be hanged on Friday because that wouldn't be a surprise.'

Nobody in their right mind would arrive at that conclusion - nobody who has lived in the real world that is. The prisoner might believe that it is highly unlikely that he would be hanged on Friday, but he wouldn't dare eliminate it as a possibility - no matter how emphatically the judge stated it. When it comes down to it, nobody has that amount of faith in such statements. The world goes on regardless of the misrepresentations of it.

The prisoner might harbour the small hope that he will survive until Friday, at which point he could point out that his hanging isn't a surprise and perhaps persuade the executioner to hang him next week instead. Unless he is pardoned in the meantime, he is still a dead man sooner or later.

For that reason, you cannot reduce the situation to a single day of the week because, every time, the prisoner might persuade the hangman to postpone to the next week on the grounds that he wasn't being surprised. So that breaks the possible equivalence to Gödel's theorem.

In summary, if you are going to make statements about reality, you can't ignore it. If you insist on ignoring it, you have to admit that there is no reason to suppose that anything you conclude from such statements will have any bearing on reality.

The true statement is


 * 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, unless he knocks on Friday; then it won't be a surprise.

Regardless of whether the hangman has any intention of hanging the prisoner on Friday, that statement is still true; the judge's isn't, as has been demonstrated by the arrival at a contradiction.

If we know what the true statement is, why insist on continuing with the untrue one. To do so is to indulge the error rather than tackle it. --Vibritannia 13:27, 17 July 2005 (UTC)

Relation to Gödel's Incompleteness Theorem
The Paradox may be reformulated as a : or more generally as: or even more generally as: Here we see a resemblance with Gödels theorem, here "I" is "the truth" and "you" is "a formal proof system". Indeed, I can say to anyone "You cannot prove this sentence", and that would be a true sentence (for me / in my world), but the other person would not be able to prove it (in his world).
 * 1) I know that day-of-hanging = Friday
 * 2) You cannot prove that day-of-hanging = Friday.
 * 1) I know that P
 * 2) You cannot prove P.
 * 1) You cannot prove 1.


 * Premises:


 * You cannot prove 1.
 * Gödel's theorem is in essence equivalent to premise 1.
 * Gödel's theorem is not a pointless one.


 * Reasoning:


 * Premise 1 says you cannot prove premise 1.
 * But nobody ever proves the premises. Premises are assumed to be true. If, from the premises, we can deduce a contradiction to any of the premises, we know that one, or more, or all of the premises are no good.
 * If 2 is true, Gödel's theorem amounts to no more than 'You cannot prove the premises'. In other words, Gödel's theorem tells us something about logical deduction that civilization has known ever since it discovered it.
 * If that is true, then Gödel's theorem is obviously a pointless one − since it is an obfuscated restatement of something that was already known.
 * And that is a contradiction of one of our premises.


 * Conclusion:


 * Since premise 1 is an assumption on which logical deduction is based, we can't then use deduction to reject it as false (because if we do that, the deduction that we used to reject it is also going to go out the window). So that leaves premises 2 and 3.


 * So premise 2, or premise 3, or both are false.


 * (I think Gödel's theorem probably is a pointless one. But it must have been a good trick since so many people fell for it.)


 * No, you are confusing Gödel (Incompleteness) Theorem with Gödel's Sentence. Gödel's Sentence states, informally speaking:
 * 1. You (i.e., a formal system F) cannot prove 1. (this very sentence).
 * This sentence corresponds to the Judge's Death Sentence (pun intended), from the criminal's (i.e. F's) point of view.
 * Gödel's Incompleteness Theorem states, informally:
 * 2. If a formal system F satisfies certain criteria of expressiveness, then the formal system F contains proposititions (such as Gödel's Sentence above) that are true, but unprovable by F.
 * These are two totally different things.


 * ...formal system F contains propositions (such as 'You can't prove the premises' or 'You cannot prove propositions that contain nothing that needs proving') that are true, but unprovable by F.


 * '1. You (i.e., a formal system F) cannot prove 1. (this very sentence).'


 * I couldn't disprove 1 by proving it, but I could disprove it by realizing that to prove it, it must contain something to prove. I could therefore make it a premise that 1 contains something to prove and show that this leads to a contradiction. That would show that it cannot both be true that 1 is true and 1 contains something to prove. So either 1 isn't true, or 1 doesn't contain anything to prove. If 1 doesn't contain anything to prove, then the content of 1 is absurd and 1 still isn't true. Either way, 1 isn't true, contrary to initial appearances, therefore 2 is false.


 * I don't actually have to find this proof. Just opening the possibility that one might exist is enough to remove the unquestioned truth of 1. The only way to restore it is to adopt 1 as an axiom. If that is done, 1 becomes


 * Axiom 1: You (i.e., a formal system F) cannot prove axiom 1. (this very sentence).


 * which is a simple consequence of the more general truth that 'You cannot prove any of the axioms'. --Vibritannia 08:51, 24 July 2005 (UTC)

None of this is relevant because the prisoner does not say anything about what the judge can prove. --Nate Ladd 01:25, July 14, 2005 (UTC)
 * The Judge says the prisoner cannot know on what day he will be executed, and he only way the prisoner can know on what day he will be executed, is for him to do a logical deduction (i.e. proof) from the statements that have been given to him by the Judge; so the Judge is in effect saying that the prisoner cannot prove that he will be executed on Friday.


 * (I read your comment again, and I think you have the words "prisoner" and "Judge" reversed.)


 * I was going to say that it is you who has the words reversed. It is the prisoner who engages in reasoning. So even if it were true that the judge is saying something about provability, that doesn't make the prisoner's reasoning in any way analogous to Gödel's reasoning. The prisoner does not use self-referring premises. (See the Wright & Sudbury and the Kirkham articles listed in the References section.) Finally, note that the paradox doesn't require that the judge say that the hanging will be a surprise in the sense that the prisoner will not "know" that the hanging will not take place. The judge can use "surprise" in the sense that the prisoner will not "believe" that the hanging is not going to take place. Again, see the Kirkham article for examples and details.


 * Yes, it is the prisoner who is engaging in the reasoning -- using the Judges statements as premises (The prisoner reasoning begins with "Assume that the hanging will be on Friday, and that I cannot deduce this..."). That they are self-referring (like the Gödel Sentence) and based on provability (not belief) I have already explained somewhat, but if you need a better explanation, or a reference, this article provides it: http://arxiv.org/abs/math/9903160 .  The paper "F. Fitch, "A Goedelized Formulation of the Prediction Paradox," Am.Phil.Q. 1:161-4, 1964" also looks promising, but I have not looked into it.


 * That's not true. The prisoner does NOT use the judge's statements as premises. The judge's two statements both have "you" in them and the "you" refers to the prisoner. The prisoner has premises that are ABOUT the judge's statements, but the statements themselves, the two "you" sentences are not the prisoner's premises. By the way, the Fitch article is on a different paradox. --Nate Ladd 04:42, July 24, 2005 (UTC)

Deletion of Egg Business
I deleted the following from the end of the "Simpler Version" section:
 * One weakness of this paradox (and the above, multi-day paradox) is the possibility, explicitly entertained by the prisoner, that the hanging will not take place. If this is allowed, then the paradox vanishes: the prisoner does not know whether or not he will be hanged on the Friday, and then must be surprised when he is in fact hanged.  It is possible to close this loophole by considering instead the "unexpected egg paradox" in which a person hides an egg in one of (say) five boxes, instructs you to open them in sequence, and announces that the egg will be found in an unexpected box.  The advantage of this formulation is that the egg must be in one box, but the paradox retains its sharpness.

The reasons for the deletion are: 1. I don't understand what is meant by saying that a paradox is "weak". 2. The prisoner does not entertain the possibility that the hanging will not take place. He concludes that it will not. 3. I'll be damned if I can see why this makes the paradox "vanish". We still have a case where there is a seemingly cogent argument to a conclusion that is latter falsified. So we still have a prima facie counterexample to logic itself. That is the essence of the paradox. 4. The egg version isn't different in any significant way. The prisoner is still going to conclude that the egg wasn't really hidden in any box, so he will still be surprised when he finds it in one. --Nate Ladd 00:52, 15 October 2005 (UTC)

Hi Nate. Thanks for these comments. (I tinkered with your numbering by putting hash signs in).

1. The paradox is weak because the prisoner thinks: I'll be hanged; but if I can convince myself that the conditions are inconsistent, maybe I won't be hanged. Then he gets hanged, and he didn't expect it (or at least failed to deduce that it would happen when it did). No paradox there: the prisoner is just not clever enough to see that the sentence is consistent.
 * The prisoner does not think what you say he thinks. If he did and these thoughts were relevant to the paradox, they would be a part of the paradox story. They aren't. You are just substituting a different story for the story of the paradox. We all know there are lots of stories that do not create a paradox for logic. What matters are the stories that do. The Unexpected Hanging is one of them. Your version, in which various thoughts are attributed to the prisoner is not that story, so the fact that your story doesn't have a paradox is irrelevant. By the way, the "prisoner" could well be a computer that doesn't have any "thoughts" in the usual sense. It could be only an unconscious machine that generates conclusions from premises by mechanically applying a set of inferential rules on sentences. The point of the paradox is that we have seemingly impeccable reasoning that leads to a conclusion that is falsified. The trick is to find out what is wrong with that reasoning: what premise is false or which inferential step is invalid. You are almost certainly missing the point of the paradox if you think the psychology of the prisoner has anything to do with it. --Nate Ladd 01:02, 17 October 2005 (UTC)

2. The prisoner (apparently) concludes that he will not be hung. Therefore it is at least a possibility in the prisoner's mind that he won't be hung. This is a good use of the word "entertains", surely?
 * No. This is very misleading. He doesn't merely conclude that it is possible he won't be hung. He concludes that he definitely won't be. He concludes that it is not possible that he will be hung. It is a misuse of "entertains" to say that this amounts to "entertaining the possibility" that he won't be hung. One who entertains the possibility of X is also, ipso facto, entertaining the possibility of not-X. He is someone who is unsure whether or not X is the case. But the prisoner is not unsure. He concludes he won't be hung, period. --Nate Ladd 01:02, 17 October 2005 (UTC)

3. The paradox vanishes (or at least becomes bogged down in second guessing) because the prisoner explicitly entertains the notion of not being hung.
 * I've just reread the paradox. There is no second guessing. The prisoner does not go back and forth on whether he will be hung. He concludes that he won't be and then he stops. There is no second guessing. (Once, again, you can make up a different story, if you want. A story in which the prisoner second guesses and is indecisive. But the fact that your new story is not a paradox, does not in any way indicate that the original story is not a paradox.) --Nate Ladd 01:02, 17 October 2005 (UTC)

4. The egg version is different (and, I maintain, stronger) because the prisoner knows that the egg is definitely in one of the boxes. There is no possibility of it not being somewhere. In the case of the hanging, he convinces himself that the hanging can't happen, which leaves him open to being surprised whenever it happens.
 * You need to spell out your egg version in detail, because it seems to me the prisoner does not know that the egg is definitely in one of the boxes. How could he? He didn't see it being put in. (He was told that it was "hidden" in one of them, remember? ) Moreover, he concludes that it is not in any box, just as he concludes that the hanging will not be on any day. These are essentially the same paradox. --Nate Ladd 01:02, 17 October 2005 (UTC)

best wishes Robinh 20:23, 15 October 2005 (UTC)

another version?
Consider the following example: You have a binary switch, and someone says: "The switch is always in other position than you expect." But then, since you know it is in other position, you expect it, and therefore, this sentence is both incorrect and correct. Is it another version of this paradox? Samohyl Jan 18:07, 8 December 2005 (UTC)

does anyone know the orgins of this paradox
I've seen another version before but anyone know who came up with it?


 * I first learned this as the Unexpected Examination paradox. But it was different than what this article says.  It has nothing to do with the prisoner's reasoning.  I completely agree with the writer at the top of this page - the article doesn't get at the heart of the paradox.


 * Let's say the prisoner is told he'll be executed sometime next year. I doubt anyone would reason that it can't happen because it wouldn't be surprising.  The prisoner would know ahead of time that if it happened, say, April 12, it would be a total surprise.  The prisoner's reasoning is irrelevant.  Even if the prisoner believes he will be hanged, it's still a surprise.


 * So the question is, what's the last possible date for the hanging that it wouldn't be expected? And the answer is that there can't be such a date, otherwise you would have expectation.  Maybe March 4, maybe July 22.  But then imagine thousands of hangings.  The judge might get close to December 31.  How close will the judge get?  Only the judge knows for sure.  And only over an infinite number of hangings (a rather bloody judge) could we determine, theoretically, the latest possible date, and then that date would, when it finally arose, be expected.


 * Doing it in a week produces the same results. Even if the prisoner does expect to be hung, it will be a surprise on a Monday.  Same on Tuesday.  Wednesday?  Depends how the judge thinks.  Thursday?  we're pretty sure we already what the judge thinks about Thursday, but maybe not.  Only over many executions do we get a sense of what day it should be "expected." Roger 07:42, 30 December 2005 (UTC)


 * The prisoner could save himself by screaming every morning "I will be hanged today !!!"  :)

Clarification (since there seems to be some confusion)
Three things I want to deal with.

Firstly, I want to give a simpler version of the paradox, to demonstrate more easily to people exactly what the paradox is. Imagine that instead of a week, there are two days in which the prisoner could be executed, Monday or Tuesday. Also, the judge has said that the execution will be unexpected (the definition of unexpected will be that he doesn't know he's going to be executed until the executioner knocks on his door). So, if the prisoner is sitting in his cell and the executioner does not come on Monday, the prisoner will immediately know that he is to be executed Tuesday. Therefore he cannot be executed Tuesday, as it couldn't be unexpected. Therefore he must be executed Monday. Which means he also couldn't be executed then, as that wouldn't be unexpected either. It's easy enough for the mind to deal with that, it's when you have a whole week and you're trying to extend backwards to Wednesday that it can get a bit fuzzy, and people can start coming up with points that don't really make sense.

Secondly, it's important to see what the paradox really is. The ideas that the paradox is merely the result of the judge making a statement that he can't back up with action or- as I've often seen argued- that the prisoner not expecting his execution when it actually does come somehow resolves the paradox are completely wrong. The paradox is, as follows:

-The prisoner has logically shown that no day can be unexpected. -When it does actually come to the day, it is unexpected.

What we clearly see as reality contradicts the prisoner's perfectly logical reasoning. THAT is the paradox. Not, as some people seem to think, that the prisoner somehow manages to outwit the judge by saying that he expects his execution when the judge said he couldn't.

Thirdly, and importantly, things in this paradox need to be better defined. Whenever people are presented with this kind of paradox or riddle, they immediately seem to like to deal not with the problem, but with minor loopholes in definitions. This phrase from the article, for example: Had the prisoner somehow guessed he would be hanged Wednesday, the judge would indeed have been wrong.. Now, that's not actually dealing with the issue, it's just using the fact that nobody's said that "unexpected" means that the prisoner is unable to logically and certainly know that he's going to be executed on that day until he hears the executioner knocking on his door.

Note By Ricky Mooston. I think that it would be better to describe this in terms of its tie in with the principle of math induction!!! I'm not so hot on the Goedel references and a few of the really high logic explanations. The essense of this paradox, on one level, is that the principle of mathematical induction that seems so obvious (I believe its axiom, forgot), is not indeed. The "paradox" is in fact that the induction criteria are satisfied and yet, we all know that the prisoner's logic is flawed. It is somewhat interesting to discuss why for a philosophy of logic point of view, but more than anything, that is the first essense of this paradox and why it is of interest to many people taking introductory math courses.

Now, in my opinion a large amount of the article written seems to go against the second and third points. Since there seems to be some controversy (especially about the second), I wouldn't want to go re-writing the article immediately, I'd just like at least some feedback first. Justdig 14:38, 13 April 2006 (UTC)


 * Actually, I lied. I have made one change, in removing a paragraph. If anybody thinks I'm wrong (keeping in mind the second point above) feel free to restore it:
 * "And another, and the simplest of all solutions is this one: The prisoner knows that he can't be hanged and surprised, because of the reasoning aforementioned. Therefore, he knows that he can't be hanged. Then the fact that he was hanged came as a surprise, thus covering both conditions imposed by the judge."
 * Justdig 14:42, 13 April 2006 (UTC)

=
=================================================== I neither agree.

The argument about Gödel is right. Even the whole set of mathematical theories is a theory and so incomplete.

And when we look critically very incomplete as well.

Mathematcs is build up from the numbers, and clever function and axioms. But the number we can get my the next principle. We also can build them more precise with set theory. Or even like Bourbaki did. So we are buiding the whole mathematical tool box.

In matrhematics we have niot one but two tyes of numbers. The cardinal numbers and the ordinal numbers, and everything is build on it. But...

Look... Cardinal numers are the things, you can have three ones. Man-made maybe they exist but in nature without man they don't exist.

Ordinal numbers are sequence numbers. For instance the third tree. When we look from a distance we see more trees. When we look from the back we see the ordinals in another order and as a relationship between observer and objects. So numbers don't exist in reality.

Fuctien are clever ideas. So the whole mathematical tool box is in our head.

Also numbers don't grow old, like biological cells do. They don't change. Otherwise we would be able to prove things. Like in CS we say a := a+1, you can't say that in mathematics. In fact our mathematical tool box is time-less.

When we look at time notions in mathematical theories. They are distance-like ones, totally not what physicists think.

Do you have a logic with time. Yesterday true, now wrong. In the reality that is possible. Do you want more arguments? I can show you urls about this. I'm spreading it already everwhere.

Very interesting is then why are our products sometimes so precise. Therefore we have to ask physicists.

Hidden assumption that prisoner knows the method the executioner decides
The article refers to several possible weaknesses in the situation that perhaps could be a way of finding a flaw in the logic of the paradox a couple of published ways of resolving the paradox. But there seems little evidence presented that these weaknesses are the actual flaw. I think I can see the actual flaw a possible alternative (or perhaps there are also other alternatives) and that it is that there is a hidden assumption that the prisoner knows the method that will be used to decide the day. Implicit in the judge's sentence is the assumption that there has to be a method of deciding the day (but this doesn't cause the flaw). If I am right about the flaw, then removing that assumption should make the paradox go away. So let's try that:

If the judge passes a sentence saying:


 * 1) You will be hanged at noon one day next week, Monday through Friday.
 * 2) The execution will be a surprise to you because the executioner will choose a method of deciding the day so that in no circumstance will the prisoner know the day of the hanging until the executioner knocks on your cell door at noon that day.
 * 3) In choosing a method of deciding, the executioner is to assume that the prisoner is able to guess all details about the method.

I believe the paradox has disappeared here and the prisoners logic works correctly. The executioner cannot find a method so at least one of the statements has to be broken.

Let's try some examples: If the method was to choose a random number up to 3, then the prisoner might be suprised on day 1 or 2. However if the random number was 3, the prisoner would not be suprised; so this method is no good and cannot be used by the executioner. Whatever method is chosen, there has to be a last possible day and this causes that method to be rejected as impossible. Therefore there is no inconsistency between being suprised and there being no method because the lack of the method prevents the suprise.

The paradox has disappeared with this 'know the method' assumption made explicit and clearly if the prisoner does not know the method he can be suprised. Therefore the flaw in the paradox is the hidden assumption that the prisoner knows the method. So what do you think, have I found the flaw an alternative and is it useful because it explains the situation where the prisoners reasoning would work? If so, should this be added to the article? crandles 16:34, 19 November 2005 (UTC)


 * You should not add it to the article, because it is your original research. If a published philosopher made the same point, then you can cite his/her work and summarize it in the article. --Nate Ladd 05:25, 20 November 2005 (UTC)


 * OK. I am also discussing it here. A blog discussion is obviously not a quality reference like a published philosopher. However sometimes things that are not published in peer reviewed journals are relevant enough and of good enough quality to be included in wikipedia. I am not presumptious enough to claim that this is one of those occassions. I am just suggesting that if a majority of people seeing this think it is useful, then it could be added. Replaced with question below. crandles 14:53, 20 November 2005 (UTC)


 * It's still original research, and thus not appropriate. And besides, it's quite wrong.  The problem with the prisoner's reasoning is that he takes the judge's statements as necessarily true, and argues from that assumption.  But the judge's statements aren't necessarily true -- for instance, it's possible that a prisoner who is convinced of the first statement but not the second could be hung on Friday, and be not at all surprised about that.  Since that's possible, there's no basis for claiming that he would not be surprised to be hung on Thursday, Wednesday, etc. -- those conclusions by the prisoner were all based on the claim that he can't be hung on Friday -- but he can.  And as for Friday, it depends on the prisoner's state of mind -- a prisoner who has convinced himself with the fallacious reasoning will be surprised, while a prisoner who is convinced of the judge's first statement but not necessarily his second will not be surprised.  As the judge's statements are not necessarily true, no conclusion can be reasoned from them. -- 68.6.73.60 01:03, 14 March 2006 (UTC)


 * I certainly got carried away with my thoughts. I have toned it down somewhat by striking out some text (some replacements added after stike outs). The question I want to replace some of it with is - 'Is this article about the paradox or is it only about published philosopher's views of the paradox?' FWIW my view remains that the simpler form of the paradox is a different paradox because my alternative resolution cannot be applied to it. Maybe this makes the simpler form a purer case of a Godel incompleteness theory paradox. I remain facinated by the logic of ruling out successive last days, and think it is helpful to show how that sort of logic can work. Maybe that is very much a minority view and everyone else is facinated by liar's paradox and Godels incompleteness theory paradoxes but TBH I would be suprised if that was the case. (Also the article seems rather short at present could it be expanded by having extra sections perhaps for why people find the paradox enticing and published philosophers' view of the paradox). crandles 15:13, 16 April 2006 (UTC)

A new suggestion?
Ok, let us examine the prisoner's line of reasoning more closely.

1) Assuming that I am still alive by Thursday night, I know that I will be hung on Friday, therefore it won't be a surprise.

2) Assuming that I am still alive by Wednesday night, I could be hung on Thursday or Friday.

NOW, the prisoner can only deduce that he won't be hung on Friday IF he's still alive on Thursday night. He needs to assume something that hasn't yet occured. Is this where the paradox breaks down? Andymc 09:39, 5 June 2006 (UTC)


 * "Assuming that I am still alive by Thursday night, I know that I will be hung on Friday, therefore it won't be a surprise" && "the execution will be a surprise" ==> "I won't be alive on Thursday night"


 * ...same for Wednesday night, Tuesday, Monday... ==> ??? still a paradox, only different...

Lovely work
It would be nice to see some of the above reasoning and discussion referenced in the article itself, but it seems only published philosophers are allowed to edit this article. 195.173.23.111 10:33, 16 June 2006 (UTC)

The problem with the paradox
I do not think this recent addition to the article provides any new material. It merely states that if the argument results in a contradiction it is meaningless.

I won't remove this section until I have some support from other contributors, mostly because I don't want my own contributions removed without discussion.

--Zfishwiki 01:29, 9 July 2006 (UTC)

Modal logic
If we let $$\Box$$P stand for 'the prisoner will believe P' and add the following laws :


 * Necessitation Rule: If p is a theorem of T, then so is $$\Box p$$.
 * Distribution Axiom: If $$ \Box (p \rightarrow q)$$ then $$ (\Box p \rightarrow \Box q) $$ (this is also known as axiom K)

we can reason symbolically about the simpler version of the paradox.

We are here supposing a theorem to be something the prisoner would believe at some time before friday, if he took the judges statements as axioms.

If we let $$ F = $$ 'The prisoner will be hanged on friday' so that $$F \land\neg\Box F$$ corresponds to what the judge said

We can then deduce as follows :

$$ F $$ axiom.

$$ \Box F $$ using necessitation rule.

$$ \neg \Box F $$ axiom.

$$ \Box F \land \neg \Box F $$

we now have a contradiction and can prove anything.

However in an alternative analysis we interpret expect to mean consistently expect, therefore the judges statement is better represented by

$$ F \land ( \neg \Box false \rightarrow \neg \Box F ) $$

Does this avoid the contradiction?

We can reason as follows -:

$$ F $$ axiom.

$$ \Box F $$ using necessitation rule.

$$ ( \neg \Box false \rightarrow \Box F ) $$ because $$ \Box F $$ is true.

$$ \neg \Box false \rightarrow \neg \Box F $$ axiom.

$$ \neg \Box false \rightarrow (\Box F \land \neg \Box F) $$ from last two theorems.

$$ \neg \Box false \rightarrow false $$.

$$ \neg \neg \Box false $$ contraction of previous.

$$ \Box false $$

We have shown the reasoner to believe that he is inconsistent and therefore the second half of the axiom is verified without contradiction.

Everything the judge said takes place.

--Zfishwiki 12:15, 9 July 2006 (UTC)

reductio ad absurdum
In the classic treatment of the paradox the prisoner seems to use reductio ad absurdum proofs in a very selective fashion. He treats the judges statements as having a greater status over suppositions such as he will not be hanged before friday

On Monday morning he has the choice of whether to dispense with both of the judges sentences by reductio ad absurdum, or accept the lesser evil of not expecting a surprise.

He prefers the former, as doing so does not require him to disbelieve in something without evidence of its absurdity. Is this really a rational argument. It seems the prisoner prefers some kinds of deduction over others.

Why does he not just simply prove a contradiction, when he is given the chance, then he can simultaneously expect to be hanged, expect not to be hanged, and expect pink elephants to fall from the sky.

Of course there is none of this trouble if you read expect as consistently expect :)

--Zfishwiki 17:53, 9 July 2006 (UTC)

Comparison with Godel's theorem
A simpler and slightly stronger version of the paradox is that where the judge simply says "You will not expect this statement to come true".

Like the original paradox, the prisoner cannot prove that it will come true. But whereas in the original, it is only possible that the judge will be proved right, in this simpler version as soon as the prisoner admits that he cannot consistently believe in the statement, at some higher level he has shown it to be true.

The paradox then has the same structure as a godelian sentence, that is one that asserts it's own non provability. To put it in a more rigorous fashion, Kurt Godel showed that in any formal theory 'F' of sufficient complexity there exists a statement 'S' in 'F' which asserts that 'S is not provable within F'.

see article on Godel's theorem

-Zfishwiki 23:16, 7 July 2006 (UTC)

[ My current view is that the way I phrased the judges sentence is not as interesting as if he had just said "You will not believe this sentence" in fact I don't believe the phrasing I used is actually paradoxical, the statement could come true without the prisoner expecting it to.

I think what makes the original paradox interesting is the fact that one can use backward induction of a discrete nature. As rephrasing the judges sentence as I have done makes it less like the original version, I am not so sure of the similarities of the two problems now. ]

--Zfishwiki 17:15, 11 July 2006 (UTC)

Answer to Hmmm
Well in the classic treatment, the prisoner considers the judges statements as a hypothesis to be tested. If a hypothesis results in a contradiction, the prisoner rejects the hypothesis. I can't see any problem with that, it is standard practice in mathematics.

However, the fact remains that it is still possible for the prisoner to be unexpectedly hanged, therefore there must be something wrong with the prisoner's reasoning.

Whilst it is difficult to imagine any real person believing in two contradictory results, If we assume the reasoner to have mathematically ideal properties, and to have total belief in the judges statements. I can't see how you can avoid a contradiction.

If you consider a real person however, he can know nothing for sure, and his beliefs about whether or not he would be hanged, would be down to his own personal prejudices. But where's the fun in considering that sort of argument?

--Zfishwiki 23:21, 23 July 2006 (UTC)

Consider changing to Unexpected Exam?
It seems to me that the unexpected hanging is a pretty lame version of this paradox. The idea that a prisoner will be hanged unexpectedly is far-fetched and its implausibility lends itself to misunderstanding of the paradox and, as mentioned above, the typical "find loopholes" technique (i.e. the Common Sense Intervenes section above). Meanwhile, the Unexpected Exam has a far more reasonable premise (Teacher tells the students, "I'll have a surprise test next week, but you won't know what day it's on until the class starts, so you can't just cram on the night before.") which helps avoid these issues, and keeps the focus on the logic rather than leaving people scratching their heads at ridiculous execution plans. I propose that the article be rewritten to utilise the unexpected exam version, any thoughts? Maelin 13:51, 24 July 2006 (UTC)

Wayside School
Should some mention be made of occurences of the Paradox in media? In the book Sideways Stories from Wayside School, by Louis Sachar, the Paradox makes an appearance in the form of a surprise exam, with the students arguing against its being a surprise. If I recall correctly, however, the story ends simply with the teacher giving up. -MBlume 00:47, 25 July 2006 (UTC)

Two observations
1) The "discussion" is longer than the article itself. 2) The article doesn't make any sense.

Come on -- less chit-chat, more editing. Sheesh! 163.192.21.44 20:07, 25 July 2006 (UTC)


 * If I knew more of logic, and actually understood the solution (if one exists?), I'd do it myself. In the meantime, I tried asking whether we should change the main version of the paradox to be the unexpected exam version, since that is a far more sensible and plausible story, but nobody seems to have responded to my comment. Maybe I'll just do it and see what people do. Maelin 15:42, 26 July 2006 (UTC)

The Prisoner fails to consider all the ways of being surprised.
Does this make sense?

If the Prisoner is certain the hanging will occur the moment that it does occur then he is not surprised, otherwise he is.

He can thus be surprised three ways:
 * 1) he has insufficient knowlege to predict the timing of the knock.
 * 2) the knock comes when he has concluded it will not.
 * 3) the knock comes at a moment about which he cannot come to a conclusion (the algorithm fails to stop).

The Prisoner's logic applies to case 1 only. The wednesday hanging is a case 2 surprise. If the Prisoner had made it to friday morning and thought of case 2 then the reasoning would go:


 * 1) This is the last available slot so they must hang me today.
 * 2) But I know this so it's not a surprise - so they can't.
 * 3) But I know this this so it would be a case 2 surprise - so they can.
 * 4) Back to 1.

And the knock comes when he's still going round in circles for a case 3 surprise.


 * Could this not be added to the article? I was still very confused after read the article, and it wasn't until I read this that the whole thing made sense to me. --jwandersTalk 10:27, 31 July 2006 (UTC)

Here's one possible solution
The paradox, as I understand it, is that the prisoner's logic appears sound, yet he is proven wrong. Thus, he must be making some sort of logical fallacy. As I read it, the prisoner is making one key assumption which is not based on any given; he assumes that he will be aware of the passage of time. His entire premise is based on knowing when each day's noon will come (or even the passage of days). Thus, "the prisoner is in solitary confinement" (or even just "the prisoner has no window or clock") is a conceivable situation where both the judge's statements could be true.

(Of course, I also don't understand why the executioner is knocking since, presumably, the prisoner is not able to open the door.)

Note that this only works for this specific version of the paradox; in the "pop quiz" version, it seems fair to assume that students would be aware of what day it was upon entering a classroom. I feel like there's something wrong with the whole thing, along the lines that he proved that each day was equally unlikely; thus, any one day being the real day would still be a surprise ... but I don't know how to express it properly. ThatGuamGuy 20:29, 3 November 2006 (UTC)sean

Actually, I have a better answer than that tongue-in-cheek one... it came to me in the shower. The flaw in his logic is that he inaccurately predicts future events. "If I am not executed on Monday, then I will know I'm going to be executed on Tuesday, because I've disproven Wednesday, Thursday, and Friday, so it will not be a surprise." But when he is not executed on Monday, he does not believe that he is going to be executed on Tuesday. (If he had been, he would've been surprised.) And, on Tuesday night, he did not believe that he was going to be executed on Wednesday and, thus, when he was, it was a surprise.

His problem is that he sets out to disprove B, and by proving that all days are equally unlikely for B, he concludes that (A+B) would be impossible. He then makes a further conclusion, that A is false. The rest of his actions are based on the belief that A is false, which is not really proven. ("If not B then not A" is, I suppose, implied by what the judge says ... except that B can only be disproven by assuming A is true.)

But B is "you will be surprised", and surprise is not an emotion which a person can plan to have, he can only attempt not to have it. He predicted that he would base his emotional response on logic, but, when actually in the situation, he did not base his emotional response on logic (or, at least, not the same logic he had predicted he would use, but rather the conclusion he had made based on those predictions). The reason he did this is that, since B is contingent on A, he neccessarily assumed that A was true in order to disprove B. For each day, he proves B ("surprise") is not true by assuming that A ("execution") is true. In disproving B, he concludes that he will not be executed (believing A to be false). Thus, when he is in the situation, his predicted behavior does not happen; he believes neither A nor B to be true. If and only if he had continued to believe A could B have been impossible.

It's funny, it's not a paradox if you don't see it (if you accept the two statements without thought, you will be surprised on Wednesday), and if you do see it, and go step by step logically, you wind up back where you started and, thus, again will be surprised by whichever specific day is chosen. It's like a self-defeating paradox. ThatGuamGuy 00:40, 4 November 2006 (UTC)sean

'paradox'?
I might be wrong, so if I am be sure to tell me...

"A judge tells a condemned prisoner that he will be hanged at noon on one day in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day."

You have to realize that this so-called 'paradox' depends entirely on the prisoner to convince himself that he won't be hanged, like in the example in this article.

However, it's ridiculous to think a prisoner would come to that conclusion. Much more likely is that the prisoner will simply come to the conclusion that he doesn't know when he'll be hanged, but is certain that he will be hanged one of those days.

Of course, in this case he's still surprised about the day he gets hanged, but even that can be taken care of. The prisoner might come to the conclusion that the hanging will happen on Monday, since that's the first day of the week, and therefor the most surprising. Monday comes and goes, and he's surprised to find he doesn't get hanged. So he assumes Tuesday and still doesn't get executed, so he assumes it must be Wednesday. When the executioner knocks on his door that afternoon, he isn't surprised at either being executed or the day of the week.

You can use the other sentence "The prisoner will be hanged next week and its date will not be deducible in advance using this statement as an axiom" for my first example.

Either way, using the initial sentence the end result is that the judge was simply wrong when he said it'd be a surprise, and using the second sentence the judge is correct, but it doesn't in any way constitute a paradox. —The preceding unsigned comment was added by 80.126.65.34 (talk) 18:13, 12 March 2007 (UTC).

Example
This paradox confused me for a long time moreso than any other paradox, but I have come to a solution that seems satisfactory to me. Of course it will only provide one angle on this issue, which can be seen from many. I'm not a logician, so I won't get into jargon, just an easy-to-understand example that might help provide some clarity:

The judge hangs a lot of people in a career -- say 100. He decides to tell them all it will be a surprise, and that in almost every case will hang between Monday and Thursday. Only once - the 87th hanging of his gruesome career - will he consider doing it on a Friday. It's possible he may even retire before #87. When #87 comes, he will roll 500 dice the week before, and only if they all come up sixes will the hanging occur on Friday.

Even if we assume the prisoner knows how the judge thinks, in 99 cases the hanging will be unexpected since it could occur on Friday (for all the prisoner knows), but won't. In case #87, it's still likely not to occur on Friday. The judge has succeeded in being true to his word by making Fridays extremely unlikely.

However, this doesn't let the judge off the hook. There is still one remote chance of an expected hanging. It seems, then, that there has to be a remote possibility the judge won't be right in order to make the assertion. He will probably always be right, but the slight chance of cheating still has to be there. Nonetheless, the hanging is almost sure to be always unexpected.

A slightly different angle could let the judge off the hook altogether. Let's say he decides to always hang Monday, Tuesday, or Wednesday, and only Thursday on the 87th hanging if he gets 500 sixes. Now there is no way any prisoner can expect their hanging, because they don't know he's making Thursday the last possible day instead of Friday. The prisoner's lack of knowledge of the judge's reasoning makes the hangings all guarenteed to be unexpected. If the prisoners knew the judge's method, then there would be no absolute guarentee.

Depending on your reasoning, you could still find fault with the judge. Making Thursday the absolute final day instead of Friday in some ways cheats the paradox - it doesn't purely address the problem of having a final day in the first place. Of course you could say the judge hasn't made his mind up about future hangings, so even he cannot absolutely predict them - "will I ever go to the Friday and ruin the whole thing? ...I'll decide later." There is still a nagging feeling, though, that the unexpected occurrences rely in some way on avoidance of something expected.

My main point is that, although they rely on the possibility of cheating, you don't necessarily actually have to cheat. Roger 22:02, 20 June 2007 (UTC)

Simple Refutation
This is perhaps no more than a simplification of arguments already made, but the paradox sounds more like a 'proof' that the judge/executioner etc. goes beyond his/her authority in claiming that the prisoner will be 'surprised' by the timing of his/her execution. At most, the judge/executioner can claim that the prisoner will not be told of the timing - this is a statement that can be made with some authority. The claim that the prisoner will be surprised cannot be made as surprise is a private emotional response that only the prisoner can claim to feel or have felt.

The use of the word surprise in this context therefore conflates a private emotional response with the very different condition of being without complete information. Being without complete information, of course, doesn't rule out fruitful deduction using the information that one is in possession of (of which the proof is an example). Perhaps the best summary of this position is offered in 'The epistemological school' section, that reduces the length of the week to one day, thus implying the Judge's comment on sentencing is, absurdly, "You will be hanged tomorrow, but you do not know that".

Simon Andrew - 20th Oct 2007 —Preceding unsigned comment added by 86.152.111.10 (talk) 15:46, 20 October 2007 (UTC)

What about this?
The prisoner is originally faced with five options for his execution: Monday - Friday. If these are his only choices, he is right to suspect that he cannot be surprised. But if there is the possibility that he will not be executed at all (which he begins to believe once his deductions are complete) then his execution on a Friday would come as a surprise, and the whole chain of logic collapses.

Put another way, the judge has said two things: 1) You are certain to be executed next week and 2) the day of your execution is certain to be a surprise.

Both of these two statements cannot simultaneously be true. But once the prisoner admits that either of these statements could be false, he is unable to make deductions based on them. MogTM —Preceding unsigned comment added by MogTM (talk • contribs) 02:52, 7 December 2007 (UTC)

Russian roulette
If you look at the problem as Russian Roulette the solution is realy simple.

You have a revolver with 6 chambers, one of them has a bullet, the other 5 is empty.

What the judge is saying: "I will trigger this revolver against you once every day without rolling the cylinder between each day. If you survive all the 5 empty chambers i will not fire the 6th and final shot containing the bullet."

/CrazyIvan 193.11.216.194 (talk) 14:14, 5 April 2008 (UTC)
 * Sorry but how does this help? The judge did not say anything equivalent to the last sentence.--Noe (talk) 15:16, 5 April 2008 (UTC)

Invalid form of an argument
It is an epistemological question, dealing with what the prisoner believes he knows and trying to translate those beliefs into what will actually happen. The only test for the validity of the decree is a real world test. Was he executed? Was it a surprise?

As to what’s going on in the prisoners beliefs he has made a simple error. Within the list of assumptions of any argument he may make to disprove that he will be executed on a specific date he must assert “I will be executed (on this date)” if other assumptions are also true (i.e. it is Friday and I have not been executed).

Let x = I will be executed.

No valid argument can hold an assumption x is true and a conclusion ~x is also true. This is a logically invalid form. This must be the case with the prisoner's beliefs. It is the very fact that he claims to know x to be true that he claims x is not true (or more accurately that ~ x is true) -- this is the "paradox". This is like saying I know it is raining therefore it is not raining. You can't say that the statement "I will be executed today" is true and use it to support the validity (or truth) of the statement "it is not the case that I will be executed today."

If the prisoner knows he will be executed on Friday, then he can not also know that he will not be executed on Friday. He can not hold both to be true at the same time. This isn't some trick or jargon, if you believe x to be true then you believe ~x to be false. Both can not be true.

It is a classic example of an invalid logical form and shows clearly how the rules of logic are stronger than our intuition.
 * I don't think it's necessary to prove the argument invalid, we know it to be invalid because it results in a paradox. Your logic is very good, using the same symbols we can arrive at this:  The judge says to the prisoner "If you know x then ~x" and the prisoner reduces this to "if x then ~x".  He takes "I know x" and "x is true" to be synonymous, when they are not.  that is his mistake.  The premise relies on what he knows, so as he thinks the logic the values that are inputted into the premise are changing.  The actual act of being logical causes the logic to fail!Dwarburton (talk) 14:58, 10 April 2008 (UTC)

My solution. Has this been covered?
The prisoner has to have some way of communicating the fact that he knows he is to be hanged on a particular day. If he is right, then they will let him go.

Let's say he has a bell which he can ring on the day he thinks he is to be hanged.

By inductive reasoning he would ring the bell on Monday, then again on Tuesday etc. etc.

Obviously it would be to his advantage to ring the bell every day -- he has to be right eventually.

But for this paradox to mean anything we have to impose a new rule-- he can only ring the bell once. It would be unfair to let him ring it every day.

Thus he rings the bell on Monday. However it happens that he's not killed until the very last day. On the last day he knows for sure he will be killed, but there's nothing he can do about it. The bell has already been rung once, and so he is hanged.

Christianjb

This doesn't really address the problem. The paradox is that the prisoner will, by logical process, expect to be executed on every day. He's not expecting it because it's "to his advantage", but because he logically should expect it (according to his reasoning). There's no reason why he should be restricted to only expecting it on one day. Justdig (talk) 01:52, 17 April 2008 (UTC)

Why Revert of "Contradictory Premises" solution
I reverted the complete rewrite for several reasons among them:

1. Many of the changes are pointless rewordings.

2. Some of them actually mess up the usage/grammar/style. For example, people don't "tell" statements, they make statements.

3. The claim that the judge's statements are contradictory does not work. If they were contradictory, the conjunction of them would be a necessary falsehood. But they both come true, so their conjunction cannot be false.


 * The judge's statements are not vindicated in the end (contrary to what this article says in its Objections section). Since a contradiction was present in the judge's statements, any conclusion can be reached.  Hence, it does not follow that the prisoner was necessarily surprised in the end.  The prisoner can logically deduce a hanging to occur on every day.  And so, there would be no surprise.

4. The solution to the paradox remains highly disputed by the professional logicians who write about it. The wikipedia article, accordingly, should discuss different proposed solutions, but it should not pronounce any one in particular as "the solution". --Nate Ladd July 1, 2005 05:14 (UTC)

One can rephrase the paradox to elucidate it
Premises:
 * 1) A can equal 1, 2, 4, 8, or 16.
 * 2) A cannot be the greatest number possible.

What can we conclude?


 * According to premise 1, 16 is the greatest number possible, so A cannot be 16.
 * If A cannot be 16, then 16 is no longer the greatest number possible.
 * 8 must now be the greatest number possible, so A cannot be 8 either.
 * If A cannot be 8, then etc. etc. until we arrive at A cannot be 1 (having eliminated 4 and 2 along the way).

Conclusion:
 * A cannot equal 1, 2, 4, 8, or 16 - in contradiction to premise 1.

What went wrong?

The problem was the word possible that sneaked into premise 2. If the premises had been simply then there wouldn't have been a problem. A couldn't have been 16, but it could still have been 1, 2, 4, or 8.
 * 1) A can equal 1, 2, 4, 8, or 16;
 * 2) A cannot be the greatest number,

The word possible made a big difference to the formal logic but not to our ear.

Likewise in the original version of the paradox, a seemingly innocuous omission by the judge created a problem. The judge's statements should have been
 * You will be hanged at noon one day next week, Monday through Friday;
 * The choice of day will be a surprise to you unless noon on Thursday passes and you still haven't been hanged (pedantic though this might have sounded)

because this was the truth of the situation he was describing. It was the judge's laxity of expression that caused the apparent paradox. But he could be forgiven because it would have been patently obvious to all that, in the unlikely event that the hanging fell on the Friday, the prisoner wouldn't in fact be surprised.

The important point is, the judge's statements are contradictory when examined formally, even if in our everyday sense they don't appear so.

Summing up:
 * The prisoner's conclusion that he would not be hanged does not follow from the premises (the judges statements). The premises are contradictory. There is no paradox.


 * Objection, Your Honor! If the premises were contradictory, then the prisoner's conclusion (as well as any other conclusion for that matter) would follow from them. The solution is simple --- in fact there are 3 of them, depending on what we do with the word "surprise":
 * either we agree that the meaning of "surprise" cannot be made precise, and then the claim of the judge is meaningless as well, neither true nor false, regardless of the reality;
 * or (which i believe vaguely coincides with the meaning of "surprise" for most people) we agree, that anything that happens in the life of the prisoner is eventually a surprise for him, and then there is no surprise that the judge was right;
 * or otherwise (the least satisfying and subjective interpretation of "surprise"), the judge was just lucky, it could have happened that the prisoner would truly believe, for one reason or another, that he would be hanged on Wednesday.
 * Ah, forgot to explain what's wrong with the reasoning of the prisoner. Apparently he did not make his mind which meaning of "surprise" to use. In the first two cases he would not have come to the conclusion that he would not be hanged. In the third he would not have come to any conclusion. My last remark: "paradox" would not have disappear even if the judge had said: "you will be hanged tomorrow at noon, by surprise".
 * --Cokaban 23:19, 15 November 2007 (UTC)--


 * Vibritannia,


 * 1. Let me make this as clear as I can: EVERYTHING THE JUDGE SAYS COMES TRUE. That means the things he says CANNOT be contradictory, because from a set of mutually contradictory statements, at least one has to be false. Any reasoning that makes you think the Judge's remarks are contradictory MUST be mistaken.


 * 2. Your "greatest number" analogy only parallel's the prisoner's reasoning, but the prisoner's reasoning is not the paradox, it is only a component part of the paradox. The paradox continues after the prisoner's argument is over: The continuation is that the prisoner is surprised anyway, so his conclusion is wrong. This shows that there is something wrong with his reasoning. There is nothing in your analogy that mirrors this continuation.


 * 3. Your reasoning about the Judge's sentence is actually just a variation of the prisoner's. HE is arguing that the sentence cannot be carried out because its terms are contradictory. BUT HE IS WRONG. The sentence is carried out in complete accordance with the Judge's restrictions: it is next week and it is a surprise.


 * 4. The prisoner's premises are ABOUT the Judge's remarks. The remarks themselves are not the premises.


 * 5. Your "greatest number" analogy is a DIFFERENT argument about a different subject, it is not a formalization of the prisoner's argument. (To see such a formalization consult C. Wright and A. Sudbury, "the Paradox of the Unexpected Examination," Australasian Journal of Philosophy, 1977, vol. 55, pages 41-58.)


 * 6. When this paradox first appeared in D. J. O'Connor's "Pragmatic Paradoxes", Mind, Vol. 57, pp. 358-9, the story ended after the judge's pronouncment. It was actually O'Connor himself who reasoned that the sentence could not be carried out. (Vibritannia, your reasoning is basically the same as O'Connor's.) It was then pointed out by M. Scriven (in "Paradoxical Announcements" Mind vol. 60, pp. 403-7) that in fact the sentence could be carried out and be a surprise. In the succeeding literature, the reasoning that O'Connor used is attributed to the prisoner. Scriven's point becomes the continuation of the story, the sentence is carried out and is a surprise. The issue is to figure out what is wrong with the prisoner's (that is, O'Connor's and your's, Vibritannia) reasoning. Thus, your thinking about the issue, Vibritannia, is 54 years out-of-date.


 * 7. As noted, the issue is to figure out what is wrong with the prisoner's reasoning. Finding things wrong in the judge's prouncement does not solve this paradox, because from a LOGICAL standpoint, there is nothing wrong with the prouncement: it comes true completely. The judge's statements, per se, don't cause a paradox. It is the prisoner's reasoning about them that is paradoxical. That was Scriven's point: Although O'Connor was wrong to say that there was something paradoxical with the judge's sentence, O'Connor's reasoning itself creates a genuine paradox, because it seems to be a sound argument to a conclusion that is falsfied.


 * --Nate Ladd July 4, 2005 19:44 (UTC)


 * The judge says
 * In a moment, I'm going to toss a coin;
 * The coin might land on its edge: If, and only if, the coin does not land on its edge, will an elephant not pop into existence out of nowhere.


 * The prisoner reasons
 * The judge is going to toss a coin in a moment.
 * If it lands on its edge, an elephant is going to pop into existence out of nowhere.
 * That's impossible. Therefore the judge cannot be going to toss in a moment (or the coin cannot possibly land on its edge { = or I can't be hanged on Friday} ).
 * My conclusion contradicts what the judge said. The premises of my reasoning are flawed, and so I can conclude nothing from them other than that the premises are no good.


 * A moment passes. The judge tosses his coin. It comes up heads. 'There!' the judge says. 'Everything I said came true.'


 * The prisoner thinks for a moment and then requests that a new judge preside over the court.'


 * ;-) Vibritannia


 * "Any reasoning that makes you think the Judge's remarks are contradictory MUST be mistaken." - Thats not true, Nate. In fact, from a logical standpoint, it is very much possible that the Prisoner is surprised AND the Judge's remarks are contradictory (or vise-versa). Why? Because the "Surprise" of the prisoner is not the one the Judge refers to in his remarks. In fact, thats exactly what happens.


 * Lets narrow the paradox into 2 days and look at the two following formulation, taken from the article (replace "week" with "2 days"):


 * 1. The prisoner will be hanged next week and its date will not be deducible from the assumption that the hanging will occur sometime during the week (A)


 * 2.The prisoner will be hanged next week and its date will not be deducible in advance using this statement as an axiom (B)


 * If the Judge refers to the first formulation, then the prisoner will reason:
 * -I can't be hanged on the second day, because tomorrow (day 1) evening I will know, from the assumption that the hanging will occur sometime during those 2 days, that I will be hanged on the second day, thus contradicting A (The Judge's formulation).
 * -He then figures: This means that the only day when I can be hanged and still be surprised is tomorrow. Therefore I will not be surprised.


 * This seems like a contradiction, but actually it isn't! Why? because the deduction which lead to the last conclusion is NOT based on A, but actually on B (which the Judge didn't mean)


 * Assuming the Judge didn't lie, he will be hanged on the first day, will be surprised according to formulation A, and won't be according to B. There is not paradox.


 * On this occasion, the Judge is correct and the prisoner is surprised, and there is no logical problem.


 * If we assume the Judge meant B, then the case is opposite:
 * The prisoner reasons he won't be hanged on day 2, and then reasons he won't be hanged on day 1 either, thus PROVING that the Judge's formulation is contradictory. The only possible situation which would still allow a hanging is if the Judge lied.


 * He than hanged on day 1 or 2 and surprised. Again - no contradiction, and the paradox is solved.


 * PavelR


 * I'm confused about how this solves the paradox. When we consider this paradox, we assume both A and B, both in testing the prisoner's logic, and in seeing whether or not he can be surprised. Sure, we can imagine a situation where the judge makes his proclamation, the executioner comes one day and the prisoner explains why, logically, he's not unexpected, then the executioner says "Oh, that's not really what the judge meant"... but that isn't the situation that's being examined in the paradox. We assume that everybody is reasoning by both A and B, and come to the conclusion both that the prisoner's logic is sound and that the judge's decleration that he can be executed unexpectedly is true.


 * By the time you come to the actual paradox, you just say that the prisoner proves that the judge's forumulation is contradictory, which is just restating one half of the paradox (the prisoner logically shows he can't be executed unexpectedly), then you say that he is surprised, which is just restating the other half of the paradox (the prisoner actually is executed unexpectedly). I don't see how that solves anything. Justdig (talk) 02:31, 17 April 2008 (UTC)

Logic
Premises:
 * 1) The judge's statements can form the premises of the prisoner's reasoning;
 * 2) The prisoner can reason about the real situation of his hanging from the premises;
 * 3) The events set in motion by the judge's statement do not necessarily reflect precisely what the judge stated, only what was understood to have been stated by the judge.

Reasoning:
 * If 1 is not true, the prisoner cannot reason anything about what the judge has stated. The prisoner accepts 1.
 * The only reason the prisoner wishes to accept 1 is because of his desire that 2 is true. He is not interested in abstract questions; he wishes to know about his hanging.
 * He knows that 3 is regrettably true. Only with very formal languages and a situation that is prescribed by that language (such as a programming language and its execution) can the truth of 3 be precluded.
 * He knows that if he finds evidence for 3, he cannot assert 2. Premise 2 will become uncertain.
 * Taking the judge's statements as his premises, he reasons to a conclusion that contradicts the premises, and therefore the judge's statements - (apparently either he can't be hanged, or he won't be surprised).
 * His arrival at the contradiction of his premises is evidence that 3 is unfortunately true - stupid judge.

Conclusion:

The prisoner cannot assert that he can reason about the real situation of his hanging based upon the judge's statements to him. That given, there is no basis for claiming a paradox.

--Vibritannia 11:25, 10 July 2005 (UTC)


 * This seems to be only a band-aid solution. It only requires the statement of that paradox is changed. Let's say that the judge is perfectly logical, or if we want a story-telling method, say it's some kind of computer run robo-judge. In fact, I think it's generally implied that all players in a paradox/thought experiemnt/logic riddle, etc. are perfectly logical, unless stated otherwise.


 * Either way, once you state that the judge is perfectly logical, the prisoner is perfectly logical, and that the prisoner and the judge are both aware that the other is perfectly logical, you're left with the original problem.

-Justdig (talk) 02:41, 17 April 2008 (UTC)

Hang him on Friday
Once the prisoner has concluded that he won't be executed Friday, a Friday execution will be a surprise. — Randall Bart (talk) 05:12, 4 August 2007 (UTC)

- I agree, I came here to write this possibility. I think that the surprise is that he would be hanged at all. Having the original story say that he was hanged on Friday would have depicted that. The judge could have thought that 1-the prisoner is not smart enough to deduce that Friday is an impossibility therefore it would be a surprise or 2-the prisoner would figure that Friday is impossible and end up using that same logic to remove all other days therefore rendering the hanging a surprise on any day. —Preceding unsigned comment added by 24.122.26.156 (talk) 13:23, 27 April 2008 (UTC)

Epimenides versus Unexpected Hanging
Does anybody think there are parallels with the 'Epimenides' paradox and this one.

Suppose Epimenides the cretan says "All cretans are liars"; are we justified in assuming that at least one other cretan exists who is not a liar, because if not there is a contradiction.

Like the 'Unexpected' paradox, there seems to be self reference causing a contradiction. In practice nobody who heard Epimenides would automatically assume there was another cretan, because of what he said. Also nobody who heard the judges statements would automatically come to the conclusion that it is either right or wrong.

Most people would assume it is meaningless because of it's circular nature. That is my belief anyway!

Zfishwiki (talk) 17:46, 7 May 2008 (UTC)

Why does it matter if the prisoners logic was flawed?
So long as the prisoner was suprised the judge's statment holds true. The prisoner managed to convince himself that his execution wouldn't happen, and when it did he was suprised. I also think that debating the definition of a suprise is rediculus, the judge said that the execution will be a surprise to the prisoner (it will be unexpected on the day it occurs). If he expects not to be hung at all but is anyway, then the hanging was unexpected (it was a suprise).

The only way the prisoner will escape is if he is not hung on Thursday, then he can be certian that his hanging will be on Friday (the only day left). He has correctly concluded this even if his reasoning is flawed from this point on. If he is not hung on friday he will be suprised, but will have no other oppertunity to be hung. —Preceding unsigned comment added by 207.112.127.48 (talk) 23:04, 6 November 2008 (UTC)

Added Expert template
I added an Expert template because this article simply isn't clear or understandable enough. It doesn't explain the solution to the paradox, and nobody in this talk page seems to really have a clear idea what's going on with it either. Meanwhile, I am going to see about asking a Logic lecturer at my uni about it. -Maelin 11:31, 7 August 2006 (UTC)


 * The reason the article isn't clear is because the resolutions aren't correct. I've written a paper giving a new resolution that I've submitted to Mathematics Magazine. Once my paper is accepted and published, I'll add the resolution to the article. --David Marcus (talk) 17:31, 14 December 2008 (UTC)

Argument from False Premise
> This is only a paradox is you accept these two incompatible sentences:

S1: The prisoner will be hanged at noon during the following week on a day such that at 11:59am of that day he will not be able to logically conclude that the hanging will take place the next minute at noon.

S2: Using the familiar logic, the prisoner will not be hanged on Friday, and therefore follows not Thursday, not Wednesday, etc.

> The problem is the S1 is not realistic (as S2 proves). We need to distinguish between saying:

a) You will be hanged on a day that today you will not be able to logically deduce to be the day of your hanging.

b) You will be hanged on a day that on that day (the minute before noon) you will not be able to logically deduce to be the day of your hanging.

> It is much more apparent when we reduced the possible days to two (although S2 is true for an even infinite number of days):

I) You will either be hanged on Monday or Tuesday, yet you will not know the minute before the hanging that you are about to be hanged.

II) Obviously it cannot be Tuesday because the minute before noon on Tuesday, Tuesday is the only possible day. So the prisoner must be hanged on Monday, which is the only other possible day. Yet it cannot therefore be Monday because he will know.

> The conclusion the prisoner should make is not that he will not be hanged, but rather that the judge provided him with false information. What the judge must have meant is something similar to b) above. Not because the judge did not say something like a), but rather because something like a) is simply not possibly [always] true.

--Kingabraham3 (talk) 20:06, 22 November 2008 (UTC)


 * How does that explain that after the prisoner is hanged, it appears that what the judge said would happen actually did happen? --David Marcus (talk) 06:47, 15 December 2008 (UTC)

One Solution
I have made sense of one form of the paradox: ''A teacher says that she will give a test tomorrow, and it will be a surprise when she does. A student rationalizes that no test can meet both of those conditions, therefore there will not be a test. The next day, the teacher gives the test, which surprises the student, who had convinced himself that it would not happen.''

I think in this case, the student is right to doubt the teacher's original claim, because it is impossible to guarantee that a student will always be surprised by a test, especially when the test can only occur on one day. If the teacher had only said she will attempt to surprise the student, rather than guaranteed it, then the student could have assigned it a high probability, and there wouldn't be a paradox.

The teacher's guarantee is an irrational one, given that a student might expect the test and not be surprised. The teacher effectively tells the student that there will be a test (which sets his expectations positively), and also that he won't expect it (which sets his expectations about his expectations negatively). I believe this statement is a contradiction. Now, the student is wrong to conclude from the teacher's irrational statement that there won't be a test tomorrow. Instead, I think he should just be confused, and possibly apply common sense and study for a test.

Is this good enough for the main article? Liron00 05:42, 2 January 2007 (UTC)


 * You are on the right track. Just because the prisoner/student was surprised, doesn't mean the judge/teacher can guarantee this. --David Marcus (talk) 07:13, 15 December 2008 (UTC)

Contradiction in its simplest form
Simplify: Judge says

1) You will be hanged now

2) It will be a surprise

Rest is verbiage. Obviously, under sensible definitions of "surprise", these two statements are contradictory, since second statement is saying that the prisoner does not know the truth of first statement, even after he is told. Judge might as well have said

1) X knows (that) Y (is true)

2) X does not know (that) Y (is true)

The contradiction is thus now transparent. -- SC —Preceding unsigned comment added by 68.146.129.4 (talk) 11:25, 3 January 2009 (UTC)

Solution to the Unexpected Hanging Paradox
First note that the prisoner assumes that he will be hanged (i.e. it is a premise of his argument). He assumes also that he is going to be surprised, and also that the last day he can be hanged is Friday. From these premises, he is able to conclude that he cannot be hanged on Friday (since he would know that he must be hanged that day and therefore could not be surprised in such an instance). In order not, at this point, to contradict the premises of his argument, he still assumes that a day does exist upon which he can be hanged and surprised. In the end however, the conclusion of his argument is that there is no such day, i.e. a contradiction of the premises from which he started his argument. In such a case, the rules of logic dictate that the only valid conclusion is that either the argument is incorrect, the premises contain a contradiction, or both. The prisoner cannot automatically, from the deduction of a contradiction, conclude that either he will not be hanged or he will not be surprised. That his argument is free from mistakes is something he did not demonstrate.

Common sense tells us that the prisoner can indeed be both hanged and surprised (and on a day before Friday). If such is the case, the premises of the prisoner's argument would seem to be beyond question. That leaves us with his argument.

The prisoner argues that he cannot be surprised on Thursday since he knows he cannot be hanged on Friday (if he is to be surprised). But if Thursday is reached, he must either be hanged that day or the next. If he is certain that it must be Thursday, he cannot expect to be surprised if he is indeed hanged that day. But if he is not, in the end, going to be surprised, what reason then does he have to suppose that Friday will not be the day of his hanging? He has none. Therefore, if he is certain that he will be hanged on Thursday, he cannot be certain that he will not be hanged on Friday – an absurdity. Therefore, because of the contradiction it implies, it is not logically possible for the prisoner to be certain that he will be hanged on Thursday. And if he cannot be certain, he can still be surprised.

And that reveals the flaw in the prisoner's argument.

--Vibritannia (talk) 10:17, 26 September 2008 (UTC)


 * I totally subscribe to this analysis. Can you give a reference to a reliable source (by definition, you are unreliable, and so am I!), so that it can be included without being "original research"? (Frankly, I haven't read the whole of our article recently; I'm not sure if your solution is essentially coverede allready.)--Noe (talk) 15:32, 26 September 2008 (UTC)


 * I don't think this analysis is correct. First the prisoner rules out Friday. Then, having ruled out Friday, he rules out Thursday. What is the flaw? You seem to be arguing that what the judge said is contradictory, hence the prisoner can't conclude anything. That may be, but if so, then you have to explain why it appears that what the judge said actually happened. --David Marcus (talk) 07:21, 15 December 2008 (UTC)


 * The flaw is he cannot rule out Thursday. --Vibritannia (talk) 13:10, 10 January 2009 (UTC)

next week/next year
instead of making the execution sometime next week one should make it sometime next year. then when the prisoner reasons backwark day by day it obviously becomes absurd. it becomes clear that some kind of reasoning is missing that would reverse the direction of the prisoners reasoning. unfortunately it isnt clear to me exacly what this missing reasoning is. one can also point out that the sheriff has only to choose a day at random to be able to fulfill the judges order therefore we know that the judges order isnt self contradictory. just-emery (talk) 00:44, 8 May 2009 (UTC)
 * moreover since it is self evident that the prisoner will not escape hanging (the sheriff has only to choose a day at random) the real question is 'what is the last day the prisoner can be hanged on. and it would be better if the prisoner asks himself 'what is the last day that I can be hanged on' since it is the prisoners knowledge of that fact that would change it, which is the basis for the whole paradox.
 * just-emery (talk) 01:09, 12 May 2009 (UTC)

Resolution: the judge can lie
Since this is supposed to be a potentially real-world situation, the prisoner should have an alternate conclusion: "The judge lied." His argument would then go like this:

If on Thursday night, the prisoner is still alive, his proper conclusion is not "I will die Friday." It wouldn't be a surprise if he died Friday, so the judge lied about the surprise, so perhaps the judge also lied about the execution entirely. So, if the prisoner is still alive Thursday night, his conclusion should be "the judge lied about something."

If on Wednesday night the prisoner is still alive, his proper conclusion is not "I will die Thursday." It's "I will die Thursday, OR I will survive Thursday because the judge lied." So on Wednesday night he can't be certain of his Thursday death.

Similarly, on Tuesday night, the proper conclusion would be, "I will die Wednesday, I will die Thursday, or the judge lied." On Monday night, the proper conclusion would be, "I will die Tuesday, Wednesday, or Thursday, or the judge lied." So on Sunday night (or any earlier time after hearing the Judge) the prisoner can only conclude "I will die Monday, Tuesday, Wednesday, or Thursday, or the judge lied." Because of the possibility of magisterial dishonesty, the element of surprise is preserved.

Halberdo (talk) 11:20, 17 July 2009 (UTC)

Another view
Hello, I wanted to share my view on this paradox. I think there are three states of mind the prisoner can have for each of the seven day. These are, "certain to be killed", "certain to be safe" and "unclear". Only in the first state he could not be surprised. He starts that by thursday he will be certain to be killed on friday which is true. He then uses this to deduce that each of the days he will be certain to be killed on the next day, which might be correct. But then he concludes that he cannot be surprised and thus not be killed at all, which sends him to the state of mind where he is certain to be safe and that finally opens him to be surprised. —Preceding unsigned comment added by 95.116.255.197 (talk) 08:59, 1 August 2009 (UTC)

Explanation
The "solution to the paradox," as requested by Maelin, is implied by the section of the article titled "The Problem with the Paradox." Spelled out, it is as follows.

The prisoner went wrong by accepting the judge's statements as true. The judge's statements turned out to be true, but this is not the same thing as the judge's statements being necessarily true.

Consider the judge's two statements: 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.

The prisoner accepted both A and B as being necessarily true—-that is, the prisoner “knows” both A and B to be true, “knows” that they cannot be otherwise. Knowing A and B, then, the prisoner deduced that the hanging could not be on Friday. The prisoner then deduced, from A and B, that the hanging could not be on any other day of the week.

The two premises that the prisoner was working with (that is, statements A and B) result in a reductio ad absurdum, which should lead the prisoner (and us) to conclude that at least one of the premises is untrue. If a set of premises leads to a conclusion that contradicts one or more of the premises, then one or more of the premises must be untrue. The prisoner’s conclusion that the hanging could not be any day of the week did indeed logically follow from premises A and B. If premises A and B were true, then the conclusion would be true (the prisoner’s reasoning is valid). However, the prisoner’s conclusion contradicted premise A. The prisoner’s conclusion was also disproved by the actual course of events. Either the logical contradiction or the practical disproof would be enough to conclude that at least one of the prisoner’s premises was untrue.

Confusion has typically arisen from the fact that the judge’s statements turned out to be, in fact, true. It is erroneously claimed that this means that the prisoner was not wrong in believing A and B to be true and basing subsequent logical deductions on the truth of A and B. However, there are two meanings of the word “true” which are at play in this scenario. Failure to distinguish between these two senses of the word “true” leads to the paradoxical conclusion that the prisoner’s reasoning was sound (i.e. valid with true premises) and yet his conclusion was somehow erroneous.

A and B are “true” in the sense that they actually occurred. They were not “true” before the hanging occurred in the sense that they could be “known.” The distinction is the same as the following: If I call “heads” before flipping a coin, and the coin indeed lands heads-up, my prediction was true after the fact, but it cannot be said that I had knowledge of the truth of my statement before the event.

The prisoner was basing his deductions in the knowledge that A and B must turn out to be true. Logical deductions cannot be made on the grounds of future events whose occurrences were not logically necessary; logical deductions are made from premises that are true in all cases, that must be true.

Some contributors to this talk page have emphatically rejected claims that the judge's statements were contradictory, arguing that since both of the judge's statements turned out to be true, there is no contradiction in the prisoner assuming the truth of these premises. But for the realization of assertion A to not contradict the realization of assertion B is an entirely different matter from whether or not the prisoner's "knowledge" of assertion A contradicts ("preempts" might be a better word) the prisoner's "knowledge" of assertion B. There has been much focus on the fact that the judge's statement came true; this, however, is completely irrelevant as to where the prisoner went wrong in his reasoning.

The judge’s statements turned out to be true. As such, there is obviously no contradiction between the truth of statement A and the truth of statement B. But there is a logical contradiction if the judge’s statements were known to be true by the prisoner before the week in question began.

If the prisoner can be said to “know” that the hanging will be between Monday and Friday, the prisoner must know why the hanging can only fall in that range of days. It is meaningless to say that the prisoner has knowledge of a future event if the prisoner has not been provided the basis for this knowledge. For example, if the prisoner witnessed the making of a computer program designed to select a day between Monday and Friday, the prisoner could reasonably be said to “know” that the hanging would occur in that range of days, assuming the program functions properly and the executioner administers the hanging on the day selected by the program (true "knowledge" of the fact that the hanging will occur between Monday and Friday would of course require that the prisoner have a way to be certain of these two assumptions, but a full explanation of the mechanism that provides the prisoner's "knowledge" is not necessary to illustrate the present point). If the prisoner can be said to "know" that the hanging will fall between Monday and Friday, then the prisoner must know why the hanging can only fall between Monday and Friday. The prisoner must know that there exists a mechanism restricting the options for the hanging day to Monday through Friday. Given this knowledge that the day of the hanging has a last possible day (Friday), the prisoner cannot rule out the possibility that the hanging will be administered on this last possible day, in which case the prisoner will have sufficient information the night before to logically deduce that the hanging will be the next day. So, if the prisoner knows that the hanging will occur between Monday and Friday, he cannot know it will be a surprise. It very well may turn out to be a surprise, but the prisoner cannot be certain that he will be surprised.

On the other hand, if the prisoner somehow "knows" that the hanging will be a surprise—-knows that in no case will he have enough information to logically deduce which day the hanging is to occur-—he cannot know the end-date by which the hanging must occur, if such a date exists. Knowledge of such an end-date creates the possibility that logical deduction will reveal the date of the hanging. An end-date by which the hanging must occur can exist if the prisoner knows that the hanging will be a surprise, but the prisoner cannot know the end-date.

So, it is not a contradiction if we were to say that the judge knows the truth of both statements A and B, even though the prisoner cannot have such knowledge. If the judge somehow knows that the hanging will occur, for example, on Tuesday, then he knows that the prisoner will have no way of deducing the date of the hanging and the judge also knows that the hanging will occur between Monday and Friday. It is possible for the judge to know A and B because B merely asserts that the prisoner will not know the date of the hanging until that day. If statement B asserted that the date of the hanging will not be known to anyone until the day of the hanging comes, then no one could know the truth of both A and B before the hanging actually takes place.

I hope this adequately explains what the article means by saying that "[t]he problem in the paradox is then a conflict in given/assumed information." For the prisoner, the given information that the judge's statement A is true directly contradicts the given information that the judge's statement B is true. Both statements turned out to be true, but nevertheless there is a contradiction between knowing the truth of statement A and knowing the truth of statement B. This contradiction can be seen when it is considered what it specifically means to know the truth of either statement A or statement B.  When the truth of both statements is assumed, valid reasoning leads to an erroneous conclusion simply because it cannot be that the prisoner knows the truth of both statements.

65.196.212.172 06:55, 29 August 2006 (UTC)Ben


 * Let me make sure I understand you correctly. Are you saying


 * If I know A and B are true,
 * then C must be true


 * is a valid logical premise => conclusion. But


 * If I assume A and B are true,
 * then I can assume C must be true


 * is an illogical premise => conclusion?


 * 67.49.243.3 (talk) 16:42, 21 May 2010 (UTC)Vincent

Cosmophases, and recursion.
Cosmophases

At the time of writing the article contains the sentence:

A further objection raised by some commentators is that the property of being a surprise may not be additive over cosmophases.

As yet I have been able to find only one definition of the word "cosmophases":

His [Lewis Carroll's] definition of "cosmophases" is "the state of the Universe at some particular moment" and his use of this concept is described in his essay, dated March 15, 1897 on DeMorgan's Problem (page 480 of Bartely's book).

If there is another definition of cosmophases it would be useful to link it to the article. If there is not then I suggest the sentence be changed to something more comprehensible.

Recursion

I believe flaw with the prisoners logic is that at any point at the prisoner draws a conclusion his conclusion is incorrect. The drawing of a conclusion defines his state of expectation, his state of expectation defines the conclusion he draws as opposite to his expectation, which is obviously recursive. As best I can determine it is the same problem as posed by "This statement is false.", while you reason about the statement the reasoning can be logical, but at any time you stop reasoning to draw a conclusion, the conclusion you draw is always illogical.

Mathonwy76 (talk) 02:33, 24 May 2010 (UTC)

Do not discuss the paradox or try to solve it here -- that's Original Research
The purpose of talk pages is to discuss how to improve the article within the Wikipedia framework of cited reliable sources, it is not to discuss the subject of the article, or to solve riddles posed by the article -- as a Wikipedia editor, your opinion about this paradox is irrelevant (and most likely wrong). -- 98.108.209.12 (talk) 04:15, 24 May 2010 (UTC)

Concerning section "The logical school"
The author wrote:
 * The prisoner will be hanged next week and the date (of the hanging) will not be deducible in advance from the assumption that the hanging will occur during the week (A).

Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week.

But I guess that the last is wrong. The prisoner CANNOT deduce that the hanging will not occur on the last day of the week. That's because when the last day only just started, it would become too late "to deduce IN ADVANCE".

I propose the next correction:
 * The prisoner will be hanged at a noon next week and the date (of the hanging) will not be deducible in advance from the assumption that the hanging will occur at a noon during the week (A).

Then in the last but one afternoon the prisoner would deduce the date of the hanging. And so he could deduce that the hanging would not occur on the last day of the week.

Eugepros (talk) 11:11, 30 July 2010 (UTC)

Additivity of surprise
This paragraph had a "clarify" note on it from 2009, contained weasel words, a gibberish sentence (containing a Lewis Carol nonsense word no less!), no citations and pretty much made even less sense than the rest of the page. I'm really surprised it stayed in as long as it did.Jmackaerospace (talk) 03:17, 27 August 2011 (UTC)

defining "surprise", and solution
Surprise is an event being true when expected not to be true.


 * "X is a surprise" = X ∧ ¬X_expected

This requires separation of the actual event variable from (uncertain/incorrect) expected/calculated outcome - X is the actual event outcome, X_expected is a parameter/variable of other, arbitrary origins.

The truth table for the "surprise function" would be:
 * 0=(X=false, X_expected=false) = nothing happened.
 * 0=(X=false, X_expected=true) = no event, despite expectations, "disappointment".
 * 0=(X=true, X_expected=true) = spoiled surprise, "I knew beforehand."
 * 1=(X=true, X_expected=false) = unexpected surprise!

Let's define the variables:
 * X: Prisoner will be killed on that day.
 * X_expected: Prisoner expects he will be killed on that day.

The paradox takes origin in a self-referring sentence, the prisoner trying to game his own expectations to disable the system: X_expected = "X is a surprise"


 * X_expected = X ∧ ¬X_expected

This is what the prisoner calculates, coming up with impossibility of being executed on various days. That sentence is true only for both X_real=false and X_expected=false.

This result lulls the prisoner into a false sense of security: first he "disproves" the possibility to be killed, whether he expects or not, then finds the condition will be only satisfied for X=false. This leads to final settling the result at X_expected=false (because none of the results leads to death of the prisoner).

Now the actual event follows the rule of: X = "X is a surprise", and has a different rule:


 * X = X ∧ ¬X_expected

Since the prisoner missed the difference, and settled for X_expected=false, the new condition is satisfied for any X, including true. 195.150.60.18 (talk) 01:24, 16 October 2011 (UTC)

Nasser Khan's Solution
The solution posted was uncited, had a bizarre listing of a copyright in the article, and was misplaced as part of the description of the paradox. I actually like the solution and think it's of value, but it shouldn't be posted until and unless it can be sourced. Reyemile (talk) 22:19, 15 January 2012 (UTC)

Comment by IP 163.1.72.74
This article should cite Kripke's paper 'Two Paradoxes of Knowledge' just published and exactly on this topic — Preceding unsigned comment added by 163.1.72.14 (talk) 15:01, 26 January 2012 (UTC)

this article is just an article title mistake,no paradox
the article formerly called vacuous truth in fact deal with ESPECIALLY false statements: http://en.wikipedia.org/wiki/Talk:Vacuous_truth#correct_resolution

I have left a resolution of this paradox there. Please incorporate it into this article. There is no paradox once you correctly see the series of arguments from ESPECIALLY false statements (which are considered 'true' due to the mistake in naming the article 'vacuous truth', which should be called something to reflect the fact that it concerns ESPECIALLY false statements.

This is not semantics. If I put an elf in one of six boxes, and order you to open them in sequence, guaranteeing that you will be surprised, you cannot conclude that you cannot be surprised. Your arguments about which box it 'must' be in are based on ESPECIALLY false statements that, due to the unfortunate article title, you have taken to be true.

Please update this page with teh correct resolution. If need be, you may publish it first in some glossy paper so you have a refecerence. I'm sorry I didn't reply sooner and prevent a great deal of spilled ink, but, better late than never. your friend, --188.157.169.36 (talk) 22:55, 20 March 2012 (UTC)

What the problem is really about
The mistake is that when the prisoner is reflecting on his sentence, he is making the hidden assumption that if the premises are true, then he will not arrive at a contradiction, which he eventually does. However, as he concludes his argument, this slightly changes the definition of "surprised" due to the fact that he is no longer assuming, for the sake of argument, that the judge's statement is true. Since he was not expecting this, he does not recognize that the reasoning behind what happens on Saturday etc. is no longer necessarily valid. If, from that point, he would still insist on the judge's statement being true, the proof remains valid, and he would be adopting an irrational and contradictory set of beliefs, and it would not be meaningful to classify him as either surprised or not surprised. Both statements would be true, which in turn makes the judge's assertion of him being surprised true and false at the same time, and therefore contradictory, in accordance with the prisoner's proof. Should the prisoner discard the initial premise, the proof doesn't work anymore, since he assumed that he would believe this on Monday through Saturday. --84.211.4.37 (talk) 00:00, 7 October 2012 (UTC)