The question isn’t why the prisoner would reason that way.
The point is, the prisoner does reason that way.
His mistake.
Try this: If the prisoner’s logic is not flawed and the judges logic is not flawed, but they lead to different outcomes there is a paradox. However, the prisoner’s logic is flawed, hence, no paradox.
Based on the prisoner’s “logic” the judge cannot surprise him by showing up on Monday (or any other day except Friday). So when the judge does show up on Monday the prisoner is “surprised”.
I think it’s a bad idea to get too hung up on the word “surprised”. Let’s instead say that the judge ruled “I am sentencing you to be hung next week on one of the five days between Monday and Friday inclusive. One of the conditions of this hanging is that you will not be able to prove by the use of formal logic which particular day it is you will be hanged on prior to the time of the hanging. If this condition cannot be met then the hanging will not occur.”
So to restate: If you can prove that you will be hanged on X day, then there will be no hanging?
I can sure rule out that I would be surprised on Friday if it was me.
On Friday, I would say to myself “looks like I am getting hanged today and I will not be surprised, the judge was incorrect when he said I would be surprised”.
Except, once you start countenancing the possibility that the judge was incorrect, then maybe the judge was actually incorrect when he said you would be hanged at all. So you don’t have certain knowledge that you will be getting hanged today. So when you do, it won’t be merely a foregone conclusion for you already.
I like this wording better but it will not change the fact that the conclusion the prisoner come to is flawed. Since the judge can surprise the prisoner based on the prisoner’s logic a better way to state this might be to just say: What is the flaw in the prisoner’s conclusions?
Again, we can simplify things and see the core of the problem by reducing to a 1-day week:
Little Nemo’s setup then becomes the judge/God saying “Here’s the deal. I’ll hang you tomorrow if and only if you cannot prove (today) that I will hang you tomorrow. (To mitigate concerns about my own reliability, you can take anything I say as already proven)”.
What does this do? Well, letting p be the claim that the prisoner is hung tomorrow, and A to mean that the prisoner can prove A today, the asserted setup is that p is equivalent to NOT(p).
[The setup is very similar to the old contradictory paradox where one posits a sentence p equivalent to NOT p. The only way this setup differs from that is by replacing a p with a p, and thus, in avoidance of contradiction, we see that p and p must differ in truth-value; the one is true if and only if the other isn’t.]
Thus, by design, the prisoner cannot reach the correct proof-status regarding p; if the prisoner can prove that he will be hung, then he won’t be hung (meaning the prisoner is able to prove something false), and if the prisoner cannot prove that he will be hung, then he will be hung (meaning there is something true which the prisoner is not able to prove).
Now, if, in fact, the prisoner really can’t prove anything false, then we know that the second of these cases has to hold; the prisoner will be unable to prove that he will be hung, and thus he will be hung.
But wait! Why can’t the prisoner carry out that very reasoning we just used, to himself deduce that he will be hung?
Well, he can, if he can prove the assumption we had to use; namely, the assumption that the prisoner cannot prove anything false. In that case, the prisoner can carry out the above argument and prove that he will be hung. Which means we land in the first case: the prisoner proves that he will be hung, and accordingly is not hung. Ergo, if the prisoner is able to prove that he cannot prove anything wrong, then he is wrong about it.
This is precisely the argument of Goedel’s Second Incompleteness Theorem (and more generally Loeb’s Theorem): in any system of proof (where provability is closed under logical consequence), if one can construct propositions whose truth provably depends in any way one likes upon their own provability (as in the judge’s stipulation that the hanging will occur just in case it is not possible to prove that the hanging will occur), then one can only prove “Nothing false is provable” if that proposition is itself false. [Thus, a sound proof system cannot prove its own soundness]
Weird, perhaps, but existent. (And, really, most of it isn’t that weird at all, once you think about it)
As I wrote above, I believe the flaw is assuming that the logical proof that worked on Friday can also be applied to all five days.
On Friday, four of the possible days have been eliminated as possibilities. Therefore Friday is the single possible day left for the hanging and it’s easy to prove that it’s the day of the hanging.
But the puzzle then claims you can eliminate all five days “by similar reasoning”. How? On Monday, there are five possible days. How do you claim to logically prove which day the hanging will be on out of the five possibilities? The logic only works on Friday because there’s only one possibility left.
No, this is not the same. This version again reduces the issue to a single posssiblity.
The real equivalent is saying that there’s five boxes and there’s a coin in one of them. You’re told that you can bet ten thousand dollars on whether or not a coin is in a box before it’s opened. The boxes will be opened one at a time and you can place the bet at any time.
You think to yourself “Ten thousand dollars is a lot of money. I don’t want to risk losing it. But I’ll wait and watch while he opens the boxes. If he opens four of them and I don’t see the coin, I’ll then jump in and place my bet that the coin is inside the fifth box before he opens it.”
Now I think everyone would agree that if you get to that fifth box without the coin being found in the first four, the bet becomes a sure thing. So your logic is valid - if you get to the situation where only one possibility remains, then you know the outcome with certainty.
The fallacy is thinking that because you can absolutely predict the outcome when there is only one possibility you can then go back and predict the outcome earlier “by similar reasoning” when there are multiple possibilities.
I can’t believe that this is still being debated. The paradox has already been explained and resolved, with references no less. Go back and read the entire thread. Ianzin has explained it very well, and I have tried to provide additional clarification. Enough already. Some people will never get this paradox the same way they will never get the Monty Hall Problem.
And again, all the essential features remain present in that case. The business about multiple days is a red herring. If the student can rule out a Friday quiz on Sunday night when there are still 5 days left, then the student’s same reasoning would rule out a Friday quiz on Thursday night when there’s only 1 day left [valid reasoning doesn’t sour and turn invalid]. And in both cases, the student’s reasoning would be erroneous, and the quiz could still be given on Friday. The student’s ability to reason erroneously in the posited manner, and our ability to appreciate this error, is not dependent on there being more than 1 day left; it is, in fact, only obscured by this.
Well, the wikipedia article states that academics have been debating the correct solution to this paradox for many years. Please forgive us mere mortals who don’t have the brain capacity to solve this in two pages of contradictory explanations.
That would only be analogous if the game show host guaranteed you that the coin was placed such that you wouldn’t know the coin was under the box before the box was lifted. That’s a guarantee that it’s not under the last box, so with that in mind, wouldn’t you want to put your money on the fourth box after the previous three boxes were lifted? But then that would make the fourth box predictable. Reading this thread and the Wikipedia page, I’m convinced there is no convincing answer to this problem.
If cites impress you, then get a copy of the Cambridge Dictionary of Philosophy.
“The prediction paradox takes a variety of forms. Suppose a teacher tells her students on Friday that the following week she will give a single quiz. But it will be a surprise: the students will not know the evening before that the quiz will take place the following day … Wednesday, Tuesday and Monday can be ruled out by similar reasoning. Convinced by this seemingly correct reasoning, the students do not study for the quiz. On Wednesday morning they are taken by surprise when the teacher distributes it. It has been pointed out that the students’ reasoning has this peculiar feature: in order to rule out any of the days, they must assume that the quiz will be given and that it will be a surprise. But their alleged conclusion is that it cannot be given or else will not be a surprise, undermining that very assumption.”
If you assume the event will happen, you can avoid being surprised by it.
If you assume it won’t happen, then you will be surprised by it.
The latter assumption makes the former impossible.
Correct, I chose the alternate situation, which really is no better, same result. But that is why I eliminated Friday, because on that day the judge can’t be correct. Now, I understand that the next thing you will probably say is that the chain of days really is the same as 1. But, it’s not identical, there are multiple opportunities for a selection, and some opportunities to satisfy both of the judges conditions. In real life, a choice of monday or tuesday would be a surprise, right? Seems like wednesday would also, and thurs, fri not so much.
Can you show me with predicate logic how the the judge’s statement is contradictory? I see lots of arguments, but nothing that satisfactorily condenses a 5 day choice to being equivalent to a 1 day choice.
Here is something to consider:
Judge says, “You will be hanged, and you will be surprised with which day I choose”, but he is allowed to tell the prisoner in advance.
On Monday, the prisoner would certainly be surprised by just about any choice, on Tuesday he would be surprised by any choice of Tues through Fri, etc.
This situation is slightly different and clearly there is no contradiction. What is it about these two slightly different situations that produce different results? How can we symbolically describe it?
As soon as you conclude that, he can surprise you by doing it on Friday.
That depends on which part of the judges statement you assume is going to be false, the hanging or the surprise.
Indeed, this has been the subject of long debate and Gardner’s proposed resolution hasn’t been widely accepted AFAICT. I first encountered this seven years ago in another SDMB discussion. See also here and here. Reviewing the first thread, though, I find that most of the links I found then have since rolled off the Internet.
Anyhoo, if we’re going to have cites, let’s do something that covers the range of views. Padding about this evening, I came up with the following, many of which I recognize from my first go round:
Ned Hall (prof. of phil. at MIT), How to Set a Surprise Exam (pdf). Probably the best article I’ve seen, though long. Among other things, explains very well why the one-day and five-day (or 100-day) versions of the paradox are different.
Edward Rozycki (Ed.D.), The Pseudo-Paradox of the Surprise Test. Similar to Hall’s argument, but simpler, less rigorous and easier to understand.
Stanford Encyclopedia of Philosophy, Epistemic Paradoxes. Usual survey of the literature, but more authoritative than Wiki.
Timothy Chow, The Surprise Examination or Unexpected Hanging Paradox (pdf download site), Amer. Math. Monthly 105 (1998) (rev. 2005). Widely cited article on the paradox, from a mostly-mathmatical perspective.
Paul Francheschi, A Dichotomic Analysis of the Surprise Examination Paradox (pdf download link) (note: for no good reason I can see, the document is coded to default to full screen mode, decline) (revised version of article originally published in 2002). Attempted resolution of Hall’s approach and Quine’s (which is similar to Gardner’s).
Halpern & Moses, Taken by Surprise, The Paradox of the Surprise Test Examined (pdf, scan). Article of unknown provenance or date, but posted to Cornell’s website, presumably by a philosophy professor. Nice discussion of how interpretation of the paradox depends on shades of meaning.
Ken Levy, The Solution to the Surprise Exam Paradox (pdf download link) (low resolution scan). Tackles the problem by arguing that even an exam on Friday is a surprise.Doubtless there’s more out there, but that’s what I came up with in one evening’s worth of work.
And, as per the OP, “he retires to his cell confident that the hanging will not occur at all.”