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This is a weird logic problem that was posed to me today. This question isn't about the actual logic problem (or the trivial paradox) but in its implications, so read to the end!

A man has been sentenced to death. The executioner comes to his cell on May 31st and tells him that he will die before June 30th. He also tells him that he will not be able to know when he is killed, and that if he is not killed before 9a.m. he will not be killed on that day.

On what day(s) can he be killed? The answer can be shown through induction. If the man isn't killed by June 29th, then he knows he will be killed on June 30th, hence He cannot be killed on the 30th because that would violate one of the logical axioms (he cannot know when he will be killed). The inductive step: The 29th now becomes the "last" day of the month, and using similiar logic, he cannot be killed on the 29th. Therefore the man CANNOT be killed without him knowing when it happens.

Logically we have a contradiction in the following statement. $A \land B \Rightarrow \sim A$

Therefore, if we assume we live in a logical world, such a system or situation should not realizable because if we have both $A$ and $\sim A$ true, then we can really prove anything to be true (this is a logic theorem).

The question then becomes, how can you reconcile the fact that we can easily do this in the physical world? Find a friend, and tell him that at some point in the next thirty days you will send him an email that says "you're dead", and that he will not be able to know when it will happen. It works out just fine...

So how can this be? How can we reconcile the physical realizability of the problem, with its obvious logical paradox.

And I don't know the answer, so don't expect one :P

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closed as not constructive by ShreevatsaR, Asaf Karagila, Jyotirmoy Bhattacharya, Rahul, Aryabhata Oct 15 '10 at 14:07

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Wikipedia (en.wikipedia.org/wiki/Unexpected_hanging_paradox) says that there is no academic consensus on the resolution to this paradox. –  Qiaochu Yuan Oct 15 '10 at 8:56
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Yes, there is a reason. math.SE is for asking questions with answers, not to have debates about anything. In any case, it is far from obvious that there is a logical paradox here, since you haven't formalized anything. –  Qiaochu Yuan Oct 15 '10 at 9:19
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I saw a nice talk that claims to resolve this paradox with some heavy duty dynamic epistemic logic. Sledgehammer-to-crack-a-nut-style –  Seamus Oct 15 '10 at 10:48
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The Stanford Encyclopedia of Philosophy also discusses this plato.stanford.edu/entries/epistemic-paradoxes/#SurTesPar But should be closed here: off-topic and cannot have a definitive answer. –  Jyotirmoy Bhattacharya Oct 15 '10 at 12:35
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I'm probably not fully appreciating the logical subtleties here, but is it cheating to simply conclude that the executioner is wrong or lying? This is exactly the way we prove things all the time using proof by contradiction. All this shows is that " A AND B " must be false. –  REDace0 Oct 15 '10 at 13:26

1 Answer 1

While I agree there is no academic consensus on the matter, to me it seems there is a pretty simple resolution. The man goes through all the logical steps to work out that he cannot possibly be hanged, because he would know it was going to happen, so he settles down to wait for the end of June. Then on the day he is actually killed his last thought will be "I didn't know that was going to happen".

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