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I once wrote an answer to Unexpected hanging paradox that went something like this

"In the context of the unexpected hanging paradox, being hung by surprise is defined to mean he couldn't deduce the day before he was hanged that he was going to be hung on that day. It is not defined to mean he could deduce the day before he was hanged that he wasn't going to be hanged that day. If you assume the judge is telling the truth, you can deduce that if the prisoner can deduce on tuesday that he won't be hanged on wednesday, then he can't deduce on tuesday that he will. He thinks he can't be hanged on wednesday because it wouldn't be a surprize hanging because he can deduce that he will be hanged on wednesday. But he can also deduce that he won't be hanged on wednesday. In fact, since the judge was contradictory, he can deduce on tuesday that he will be hanged on wednesday. It does not follow from the fact that he could deduce on tuesday that he wasn't going to be hanged on wednesday that the hanging was a surprize. The hanging is defined not to be a surprise hanging because he could deduce that he was going to be hanged on wednesday. The fact that he could also deduce that he wasn't going to be hanged on wednesday doesn't change that. This makes sense." then it got deleted then I wrote another one about a year later a little differently then it also got deleted.

My question is is that one possible solution to the unexpected hanging paradox or could that be thought of as a solution? I know this is an XY question. It could mean I already have the answer so I'm making a mistake writing the answer in the body of the question rather than asking the original question and writing it in an answer. But apparently, there's controversy over whether that could be thought of as a solution to the unexpected hanging paradox. See https://meta.stackexchange.com/questions/7931/faq-for-stack-exchange-sites. Maybe some people want an answer that goes into the answer in depth explaining what's going on just like some people want an answer to https://math.stackexchange.com/questions/1839913/axiom-of-choice-where-does-my-argument-for-proving-the-axiom-of-choice-fail-he like AsafKaragila's does and it's not enough to just explain how the formal system of Zermelo-Fraenkel set theory doesn't prove the axiom of choice.

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    $\begingroup$ I think you copy-pasted the wrong link? ("FAQ for Stack Exchange sites" doesn't seem relevant here.) $\endgroup$ Sep 18 at 5:52
  • $\begingroup$ I think he's posted that by way of explanation of the term "XY question" a little earlier in the paragraph. $\endgroup$
    – Gareth McCaughan
    Sep 18 at 12:13
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I think there are two problems with your answer (neither of which is the reason given for deleting it, before, but that's understandable given problem #1).

The first problem is that your argument is really hard to follow. The second problem (if I have in fact successfully figured out what you mean) is that it doesn't resolve the paradox.

Problem 1

It's hard to follow for two reasons. The first, and more important, is that it doesn't state clearly how it supposedly resolves the paradox.

The reason why the paradox feels paradoxical is that it seems that the prisoner is able to prove that if the judge is telling the truth then there's no day on which they could be hanged unexpectedly -- and yet, then they get hanged and they didn't expect it, so it turns out the judge was telling the truth.

In order to resolve the paradox, you need to show that something is wrong with the reasoning described in the paragraph above. Maybe your answer does that, or at least tries to, but you never say explicitly what step in the apparently-paradoxical reasoning it finds a problem with.

The second problem is that it's laid out as a single paragraph and lacks "signposts" indicating how its steps fit together.

I don't know whether I have made my argument clear, above, so let me make it more concrete by offering what I think is a version of your argument that is easier to follow. I am not 100% sure I'm not distorting the argument because, as mentioned above, it's hard to be sure exactly what the argument is.

The point at which the "paradox" supposedly arises is where the prisoner, having proved that if the judge is telling the truth then a hanging on any day would be expected, then gets dragged from his cell on (say) Wednesday to a hanging he didn't in fact expect -- so that the judge was truthful after all.

But I say that his actual hanging wasn't unexpected. After all, the prisoner had already proved that if not already hanged by Tuesday, he must be hanged on Wednesday. And then, having not been hanged by Tuesday, he is hanged on Wednesday. That's the very opposite of "unexpected". He had an actual proof that it must happen!

(It's true that he also had a proof that he wouldn't be hanged on Wednesday, but so what? I say that if you have a proof that X will happen, then when X happens it is not "unexpected", no matter what other things you also have proofs of.)

So there isn't an unexpected hanging at the end after all, the prisoner was correct to conclude that there wouldn't be one, and there is no paradox, just a judge saying something that (as the prisoner predicted) turned out to be false.

As I say, I'm not 100% sure that the above is a presentation of the same argument as you are making; but I think it probably is, and at any rate I think it's much easier to follow, because it says exactly what it finds wrong with the "paradoxical" reasoning, and makes its argument explicitly.

Problem 2

Having done my best to clarify your argument, I don't find it convincing and don't think it does in fact resolve the paradox.

Your argument depends on a particular notion of what it means for something to be "unexpected" or a "surprise". I don't think that notion matches how these words are actually used in practice, nor is there anything specific to the unexpected-hanging scenario that calls for a nonstandard understanding of the term.

(To be concrete: you say "the hanging on Wednesday is defined not to be a surprise hanging"; no, it isn't. Not by how the word "surprise" is usually used; not by anything in the statement of the paradox.)

Yes, sure, the prisoner had a proof that if the judge was telling the truth (and if the prisoner knew that the judge was telling the truth) and he hadn't been hanged by Tuesday then he must be hanged by Wednesday. So far, so good. But the judge doesn't say "when we knock on your door at noon you will not have a proof that we were going to", he says "when we knock on your door at noon you will be surprised", and those two things are not the same, and simply declaring (as you do) that they are the same doesn't make them so. The story tells us that the prisoner is in fact surprised when the knock-on-the-door actually comes, and nothing in the story makes that difficult for us to believe.

In this case, indeed, the prisoner has a proof of that proposition. But he also has proofs (with the same hypotheses) of a bunch of other things, which add up to a contradiction. As a matter of psychology, when you prove a bunch of mutually contradictory things, it is perfectly possible to then be surprised when one of them turns out to be true. As a matter of mathematics, when you make some assumptions and use them to prove a bunch of mutually contradictory things, the correct conclusion is not that all those things are true, it's that something was wrong with your assumptions.

(In this case, it turns out that the prisoner should drop the assumption that he knows that what the judge said is true, or maybe the assumption that if he can prove that something will happen then he will be unsurprised when it happens; the first of those assumptions turns out to be untrue because he ends up not confident that what the judge said is true, and the second turns out to be untrue because he "proves" several things that turn out to be mutually contradictory.)

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  • $\begingroup$ Actually, the judge wasn't telling the truth. By definition, we say that it wasn't a surprise hanging because the prisoner could also deduce that he was going to be hanged on wednesday. $\endgroup$
    – Timothy
    Sep 18 at 18:05
  • $\begingroup$ I don't know why but apparently, this doesn't seem to quite be the type of question where I'm expecting an answer that completely answers it or could think of one as completely answering it and find it normal right now for this question. It's almost like the answer is something that goes on the chase as described in the video youtube.com/watch?v=68ZcE5GQP9c and it's normal. Some questions are able to get answers that give a complete answer that solves the problem like AsafKaragila's answer to $\endgroup$
    – Timothy
    Sep 18 at 18:33
  • $\begingroup$ math.stackexchange.com/questions/1839913/…. I upvoted it though. $\endgroup$
    – Timothy
    Sep 18 at 18:33
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    $\begingroup$ You say "by definition" but by what definition? It seems to be one you just made up. I'm afraid I have no idea what the relevance of AsafKaragila's answer to that question is meant to be. $\endgroup$
    – Gareth McCaughan
    Sep 18 at 19:24
  • $\begingroup$ (And I appreciate that part of your argument is that the judge wasn't telling the truth; that's why my attempt at paraphrasing your answer to make it more comprehensible includes the words "just a judge saying something that ... turned out to be false".) $\endgroup$
    – Gareth McCaughan
    Sep 18 at 19:25

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