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In the Unexpected Hanging Paradox, the prisoner tries to narrow down their date of execution using seemingly sound logical reasoning. They instead arrive at a contradiction. When the paradox is conveyed to many people, some people claim that the prisoner was justified in ruling out Friday as their date of execution, but no other day. No one I've spoken to has suggested a formal logic that incorporates such a restriction. I suspect that Girard's Linear Logic is such a logic.

The fact that the date of execution is supposed to be a surprise can get "spent" once the cut rule is applied. This should be sufficient to rule out Friday, but no other day - hence resolving the paradox.

Has linear logic been applied in such a way? Can substructural logics be applied to such things?

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  • $\begingroup$ Like @Steven Landsburg, I would put "paradox" in quotes. $\endgroup$ Nov 29, 2022 at 7:47
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    $\begingroup$ Perhaps a version with just one possibility is unconvincing, but how would this resolution work with two possibilities? $\endgroup$ Nov 29, 2022 at 11:18

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It's been some years since I updated my comprehensive bibliography on the surprise examination or unexpected hanging paradox, but I'm pretty sure there's no published paper that tries to enlist linear logic to try to resolve it. People have looked to dynamic epistemic logic, modal contrastive logic, Kripke's semantics, and a few other exotic logical ideas, but not linear logic to my knowledge.

The idea that we can resolve the paradox by declaring that Friday is correctly eliminated but that the rest of the argument is wrong is a decidedly minority view. For example, Lyon's 1959 article on the prediction paradox makes an argument about Friday being special, but most people find that Lyon evades the paradox rather than resolves it. In any case, even if you want to make this kind of argument about Friday, it seems strange to me to appeal to a substructural logic. What are you trying to argue, that the unexpected hanging paradox demonstrates that we shouldn't use classical logic for simple, everyday reasoning? That seems to be a radical move, especially when many less radical approaches to resolving the paradox are available.

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Here is why this won't work.

Consider the following "paradox":

I tell my class that "tomorrow, we will have a surprise exam. I promise that you will not know about this exam until you arrive in class tomorrow." The students go home and reason that I can't give the exam tomorrow, because if I do, then they already know about it, contrary to my promise. They are therefore very surprised when I hand out the exam tomorrow.

This is the Unexpected Hanging/Surprise Exam paradox in ultrasimplified form, stripped of the backward induction. But the backward induction was always just misdirection to begin with, put in there to deflect attention from the essence of the "paradox". (I am putting the word "paradox" in quotes to avoid getting into any discussion of whether this is a real paradox or a linguistic trick or something else. The argument I am about to make applies in any of those cases.)

Now: Pretty much any argument that allows us to "rule out Friday" in the original story --- e.g. the argument you are trying to construct --- will also allow us to "rule out tomorrow" in the simplified story. But when we rule out tomorrow in the simplified story, the "paradox" remains.

In other words, your proposed fix, however you fill in the details, can only work when it is surrounded by irrelevant fluff, and then fails completely to resolve the paradox when it is applied to a stripped-down story that captures everything relevant. That can't be much of a fix.

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    $\begingroup$ Your "simplified" version is oversimplified. In your version, the teacher says something clearly wrong. In the original version, the judge in fact tells the truth. The fact that the judge's words are true in the original version is what makes it interesting. I also don't appreciate you saying things like "irrelevant fluff" when it's just your opinion, which you haven't been able to justify. I feel like Timothy Chow's answer is closer to a summary of the different perspectives on the paradox, and how my proposal may relate to them. $\endgroup$
    – wlad
    Nov 29, 2022 at 11:02
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    $\begingroup$ @Tanner-reinstateLGBTpeople "then the judge's statement S is false" - I dispute that conclusion, and think the judge's words are true. I'm not talking about the condemned's emotions, as that would not be appropriate for MO. $\endgroup$
    – wlad
    Nov 29, 2022 at 11:22
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    $\begingroup$ @wlad: the judge (or professor) tells the truth in the new version in exactly the same sense that he tells the truth in the original version. Students do get an exam and they are surprised. $\endgroup$ Nov 29, 2022 at 12:47
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    $\begingroup$ "Surprised" means that they could not deduce the date of the exam. In your example, they can certainly deduce the date of the exam, should an exam happen. So it's simply non-sensical. In the original paradox, the prisoner was indeed unable to deduce the date of execution, so the judge told exactly the truth, and your teacher did not. I don't understand how MO gave this answer +9. It's disappointing. $\endgroup$
    – wlad
    Nov 29, 2022 at 16:49
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    $\begingroup$ @wlad "I dispute that conclusion, and think the judge's words are true." – If the judge's statement S is true, then the execution can't occur on Friday (because if it did, then on Friday morning, the prisoner could validly conclude that it would occur on Friday); and so it also can't occur on Thursday (because if it did, then on Thursday morning, the prisoner could validly conclude that it would occur on Thursday); and by analogous reasoning, it can't occur on Wednesday, Tuesday or Monday. The inescapable conclusion is that the judge's statement S is false. $\endgroup$ Nov 29, 2022 at 18:02

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