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Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been 5 years since Martin Gardner passed away on May 22nd).

The essence of the paradox is whether it is possible to predict that an event will come unexpectedly, despite an inductive proof that that isn't possible.

What I would like to know is, what the mathematical interpretation of "expected/unexpected" is in the context of the paradoxon; specifically whether it is related to probability measures.

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You could formulate it in terms of modal logic. Say

  • $K_t$ means "the prisoner knows in the morning of day number $t\in\{1,\dots,5\}$ that",

  • $h_t$ means "the hanging takes places at noon of day $t$".

So the prison guard's promise is that $$ (\forall t)(h_t\rightarrow \neg K_t h_t), $$ which is taken to imply that $$ K_5 \neg h_5$$ and therefore also that $K_1 \neg h_5$. Moreover $$ (\forall t)((K_t \neg h_t) \rightarrow K_{t-1}\neg h_{t-1}).$$ Now you can play around with this, but in any case, no real contradiction from obviously true principles will be found.

One reason may be that it is not known that the prison guard's promise is actually honest, i.e., we don't know whether the prison guard will actually try to make sure that the hanging is a surprise.

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  • $\begingroup$ Sorry for being picky, but in your answer you use the word "knowing"; am I correct that it means that it is impossible to determine the day of hanging logically, mathematically or algorithmically and, that mere guessing or the outcome of a random experiment (like flipping a coin) are not covered by knowing? Another open question is, how often the prisoner may state that he expects the hanging at noon time of the present day? $\endgroup$
    – Manfred Weis
    Commented Jun 8, 2015 at 3:05
  • $\begingroup$ What I also don't understand, is the common conclusion that the prisoner will be still alive if the hanging doesn't take place on Friday; if he had been hung on a previous day, he also would not be hung on Friday and thus could not witness a contradiction in the judge's prediction. $\endgroup$
    – Manfred Weis
    Commented Jun 8, 2015 at 3:11
  • $\begingroup$ I think I will have to edit my question and state all my open questions with the hanging paradoxon, and there are some of them. $\endgroup$
    – Manfred Weis
    Commented Jun 8, 2015 at 3:33

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