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Link to statement of the paradox

My intuitive solution to the paradox would simply be that although the prisoner can "deduce" that he won't be hanged on Friday, he can only make that statement if it gets past Thursday. In other words, the paradox relies on the statement (statement A), "If I am not hanged by Thursday, then I can conclude that I will be hanged on Friday".

We can decompose this statement to two statements 1) and 2):

1) I am not hanged on Thursday, and 2) I am hanged on Friday

Then statement A is of the form 1 $\rightarrow$ 2. From elementary logic, this statement is true if 1 & 2 are both true, 1 is false + 2 is true, and 1 is false + 2 is false. However as of the time the prisoner makes the statement, 1 hasn't happened yet, so the prisoner doesn't know if 1 is true or false. Therefore, 2 can also be true or false, and so the prisoner can't make conclusions at all. The prisoner can deduce that he will be hanged on Friday only if the hangman doesn't knock on his door by Thursday noon, but right until Thursday 11:59:59am, he can't make conclusions.

The problem with my solution is, of course, if it were correct I'd have expected someone else to have thought of it and resolved the paradox (since the solution is so elementary). But that hasn't happened, and the logicians working on the paradox focus on the word 'surprise' instead. What's the error in my solution?

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    $\begingroup$ It's a surprisingly deep question. Many words have been expended on it. Here's a nice article about it that addresses your question and many others: jstor.org/stable/2024364 $\endgroup$ – dbx Dec 19 '17 at 1:20
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    $\begingroup$ As you suspected, it's not that easy. And the prisoner can make deductions just fine, well before Thursday noon in fact. For one, the prisoner can dedue that if he will be hanged by Friday, then Thursday noon will come and pass, and so he will know he will be hanged, contrary to the assumption that he doesnt. So, before it is Thursday noon he can already deduce they cant wait until past Thursday noon to hang him. ... or so at least the reasoning goes. $\endgroup$ – Bram28 Dec 19 '17 at 2:12
  • $\begingroup$ One interesting thing is that you can actually carry out the experiment if you change the problem to something more benign, like giving your friend 7 boxes numbered 1-7, only one of which contains a certain item, and tell them to open exactly one every day, in order. Before giving the boxes, flip a random coin to secretly place the item in either box 3 or 4 (but do not tell them that procedure). Tell them only "on the day you find the item, you will be surprised" and they actually will! $\endgroup$ – Michael Dec 19 '17 at 4:20
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I think the problem is at the very end: The prisoner starts with the assumption that he knows the judge tells the truth. From that assumption, he goes through the logical steps. But when at the end he reaches at the conclusion that, under this assumption, he cannot be hanged, the correct conclusion is that the assumption cannot be right, as the result explicitly contradicts the assumption (after all, the statement contains "you will be hanged"; if the conclusion of the prisoner that he will not be hanged were true, it would directly imply the judge's statement to be false).

Therefore after reaching that point, the prisoner must logically conclude that the assumption "I know that the judge's claim is true" must be wrong. But in that moment, the base case of his induction breaks down, as he no longer can deduce that he will be hanged on Friday if he hasn't been hanged on Thursday. He can't know in advance whether he will be hanged on Friday, since — as he deduced — he cannot know whether the judge's claim is true until the last possible time of execution has passed.

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Certainly it's true that he can't conclude from that single step of reasoning that he will be hanged on Friday. But, importantly, the judge's edict requires that the prisoner be surprised in the moment - it doesn't matter whether he knew days in advance when he would be hanged, only if he knew that morning. And it is certainly true that, at 12:01pm on Thursday, if he is still alive then he will know that he will be hanged on Friday. Which means that if the hanging is scheduled for Friday, then as of 12:01pm on Thursday it will not be a surprise, contradicting the judge. Which means that the hanging cannot be scheduled for Friday.

Now, this fact - that the hanging can't be scheduled for Friday - is a fact about now, not about Thursday afternoon. It's not saying "on Thursday we won't be able to decide to have the hanging on Friday", it's saying "the judge did not schedule the hanging for Friday". All of this deduction is deduction that the prisoner can do without actually waiting until Thursday afternoon, because it's all a "what if" situation. So now, Monday morning, the prisoner knows that the hanging was scheduled for some day other than Friday - so he asks himself, could it have been Thursday? Now, since he knows it can't be Friday, he now knows that if he survives to Wednesday afternoon, he will know that the hanging is scheduled for Thursday, making it not a surprise - so the hanging can't be Thursday.

Again, all of these deductions are "what-ifs" - I don't need to experience the condition of the what-if to know that it's true. For example, I carry a spare tire, because I know that if one of my tires goes flat then my spare tire will be useful. I don't have to know that my tires are going to go flat in order to reason about this situation; most simply, if someone tells me that my spare tire won't be useful (and I believe them), I immediately know that my tires aren't going to go flat.

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