Beliefs that Might Save Your Life

The first episode of Numb3rs: Season Six reminded me of the hangman’s paradox. Here is a one-day version of the hangman’s paradox:

Suppose you are in a prison and the guard says to you, “You will be hanged tomorrow at noon and it will be a surprise.” You presume that you can’t be surprised since they already told you, so there is a contradiction in what they’ve said. Therefore, you conclude that they can’t hang you and you relax. Next day at noon the guard comes for you, to take you to be hanged, and you are utterly surprised. Oops.

What I do not like about this paradox is that it assumes that you do not know about the paradox. I, on the other hand, imagine that you, my reader, are logical and intelligent. So the moment the guard tells you that you will be hanged tomorrow at noon and it will be a surprise, you realize that the situation depends on what you decide to believe in now. If you decide that you won’t be hanged tomorrow, then you will have a relatively relaxing day today and you will be caught by surprise tomorrow and die. If you decide that you will die tomorrow, then you will have a nerve-wracking day today, but the guard may release you, to save his honor, since you won’t be surprised.

The original hangman’s paradox in which the guard tells you that you will be hanged on a weekday the following week and that you will be caught by surprise, also assumes that you are not aware of the paradox. If you are aware of the paradox, you know that usually guards in this paradox come for you on Wednesday, so you can prepare yourself. Actually, to guarantee your survival, if not your feeling of moral superiority, you can daily persuade yourself to belief that you will be hanged at noon the next day. This way, you will never be caught by surprise. If you are a person who can control your own beliefs, you may be able to save your life.



  1. Akhil Mathew:

    “If you are a person who can control your own beliefs, you may be able to save your life.”

    There is an excellent if depressing exploration of related ideas in the short story “On Self-Delusion and Bounded Rationality.”

  2. colorblind:

    The first thing I thought of when I read this was the movie Memento, where the protagonist/anti-hero tattoos the (inaccurate) memories he wants to keep on his body. If he tattoos “The guards are going to kill you tomorrow!” on himself, he should be primed to neither take the tattoo for granted nor be surprised by it.

  3. TruePath:

    It has nothing to do with being aware of the paradox (also known as the surprise quiz paradox). It’s just an issue of properly preciscifying the claim.

    There are (at least) three different ways one can interpret the statement.

    1) As a matter of actual fact you (a fallible person of flesh and blood) will be surprised.

    In this case there is no paradox because nothing prevents you from being surprised regardless of whatever deductions can be made. Hell, they could just stick you with a drug that made directly surprised regardless of your expectations.

    2) Here is a formal system F. There is a predicate H in F and constants for each day you might be hung (just M if one day, M, T, W, TH, F etc..) and we will only hang you on day X if F does not prove H(X).

    In this case one needs to be very careful about the way the problem is setup. If F is sufficiently rich to capture the reasoning at issue (so includes the claim EXISTS(X) H(X) and the relevant deductions that you can’t be hung) then F is inconsistent.

    So the paradox rests on our confusion between the claim that F proves H(M) (or whatever day you are hung) and the claim that F is consistent and proves H(M). If they really won’t hang you as long as F proves H(X) for X the day you are hung then yes, indeed, you can’t be hung. If instead the issue is that they won’t hang you if F is consistent and proves H(X) then since F isn’t consistent there is no problem.

    In short there is no paradox once you try and suitably precisify what is being claimed.

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