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Judge's statement S: The prisoner will be hanged next week and its date will not be deducible the night before using this statement as an axiom.

Using an equivalent form of the paradox which reduces the length of the week to just two days, Fitch proved that the statement is self-contradictory.

I have three specific questions.

  1. I don't understand the Gödelian analysis done by Fitch. Is it possible to reach to the same conclusion using only propositional logic?

  2. What about the one day version, in which the statement reduces to T: X is true, and its truth is not deducible from T.

If we assume T is true, then we deduce from T that X is true, which means T is false. Since the truth of T leads to its falseness, we conclude T is self-contradictory. Is this argument sound?

  1. Chow states "the judge's announcement appears to be vindicated after the fact". Suppose the prisoner is hanged on Tuesday. Then compare S with the following statement P: The prisoner was hanged on Tuesday, and he couldn't deduce the date from S.

P is an objective truth. My question is, what is the difference between "the prisoner couldn't deduce" and "it is not deducible", if we assume the prisoner is a logician, let's say Fitch himself?

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    $\begingroup$ You have posted about the Hanging paradox several times ... What is your specific question now? $\endgroup$
    – Bram28
    Commented Apr 9, 2018 at 19:48
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    $\begingroup$ 1) I think not: we cannot "manage" self-reference in propositional logic only. The Gödelian way to manage self-ref is trough the "arithmetization of syntax" i.e. thorigh the encoding of expressions and sequences with numbers. To do this, we need FOL. $\endgroup$ Commented Apr 10, 2018 at 13:15
  • $\begingroup$ @Bram28, I've been trying a lot to derive the Fitch's conclusion using propositional logic, and it's been very confusing. I have three specific questions right now. An answer to each one of them is appreciated. $\endgroup$
    – asmani
    Commented Apr 11, 2018 at 10:32

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  1. week is 2 days $\implies$ hanging day 1 or day 2.

    • if hanging is day 2, then on day 1, the prisoner will be able to predict based on p, a contradiction, so hanging must be on day 1
    • if hanging must be day 1, then the prisoner can predict based on the statement, a contradiction, so it can't be day 1 either.
  2. When you reduce it down to 1 day, you get something similar to the liars paradox. You want your system of logic to prevent you from making such self referential statements

  3. the prisoner couldn't deduce in that case because execution happened to be before the last day. P is not true in general though (i.e. "not deducible"), because if the prisoner isn't executed before the last day, he can deduce when his execution is from statement P. Thus if the execution is on the last day, he knows that it is a contradiction. Now, the week where the prisoner can be executed has effectively been shortened. Continue this logic process until any day that the prisoner is killed is a contradiction with P.

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