The Unexpected Hanging Paradox

Epistemology logic

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Nick “The Slasher” McGurk was found guilty of no fewer than nine counts of first-degree murder. A jury took just ten minutes to reach a unanimous guilty verdict. The case was open and shut. The judge donned his black cap and passed sentence: “Nicholas James McGurk, the court sentences you to death by hanging. Your sentence shall be carried out on a weekday next week in a turn of events that meets the following description: You are to be taken from your cell at midday to be hanged, but you will not know the day of your execution until the executioner knocks on your door on that that fateful day. That moment shall come as a thief in the night, in a most unpleasant surprise.”

Staring death in the face, Nick was in despair. He was led back to his cell where he would await that dreadful knock at the door. It was Friday afternoon. In the wee hours of that night, the Slasher couldn’t sleep. He had the following conversation with himself:

Any day! I could die any day next week, and I won’t expect it until death comes knocking.

Or will I?

Let’s see if I can figure this out. Might it be Friday? No, it couldn’t be Friday. By Thursday afternoon, midday Thursday will have passed, so I will know for certain that it won’t be any day from Monday to Thursday, so that would only leave Friday. If I knew that it was going to be Friday, then it wouldn’t be a surprise. But the judge said that I wouldn’t expect it until that knock on the door and that it would be a surprise. So it’s not Friday. I can cross Friday off the list of possibilities.

What about Thursday then? It could be Thursday, right? But wait a minute, if I know that it’s definitely not Friday, then Thursday is the last possible day I could die. But that means that if noon on Monday, Tuesday and Wednesday have come and gone, then Thursday’s the only day left. That means that on Wednesday afternoon, it won’t be a surprise. I would know that I was going to die at lunchtime on Thursday. So it’s not Thursday. I can cross that one off the list too.

But this means that Wednesday is the last possible day that it could be, so on Tuesday afternoon that wouldn’t be a surprise either. Wednesday’s off the list. But that means that Tuesday is the last day I could die, so on Monday afternoon it wouldn’t be a surprise. So it can’t be Tuesday. And that leaves just Monday. So Monday’s the day! But wait… if I just figured out that it’s Monday, then it’s not a surprise. I guess I’m not going to be hanged at all. What a relief!

And with that, Nick “The Slasher” McGurk smiled, as though a weight had been lifted from his shoulders. He stretched out on his bed and slept like a baby.

Is Nick right? Is he really safe? Does the judge’s fearful announcement amount to the claim that the Slasher will not be hanged at all?

Let’s hear your thoughts!

Glenn Peoples

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{ 23 comments… add one }
  • Joey January 18, 2011, 8:44 pm

    He will be hanged.

    The judge said that he would not know the day and that it would be a terrible surprise. By the end of it, Nick “the Slasher” McGurk believes he will not be executed at all. Therefore, whenever the executioner comes, McGurk will have not known the day, and it will be a terrible surprise for him.

  • Glenn January 18, 2011, 9:19 pm

    So Joey, are you saying that the following two propositions are both true?

    1) Based on the judge’s advice, Nick is justified in believing that he will not be executed on any day next week.

    2) Based on the judge’s advice, Nick is justified in believing that he will be executed on one day next week.

    ?

  • Jason January 18, 2011, 10:15 pm

    I would say proposition 2 is true, after all the judge said he’d die one day that week.
    Proposition 1 is more tricky because really he’s just trying to rationalize away his situation.

  • Jeremy January 18, 2011, 10:18 pm

    Backward reasoning doesnt work here. Starting his reasoning process with Friday is just dumb, he may already have been executed by suprise on Wednesday.
    Working forwards he can never be sure about or eliminate the next day until it doesnt happen.
    Lastly both Sat and Sun are days of the week too.

  • Glenn January 18, 2011, 10:29 pm

    Jeremy, if he starts thinking about Monday then he will say sure, maybe monday, sure, maybe Tuesday, etc, until he gets to Friday. Then he will be forced into the reasoning I describe in this blog post. Surely! Or not?

    Oh, and by “Weekday,” the judge meant Monday to Friday. But the same thing happens if you include all seven days anyway.

  • bethyada January 18, 2011, 10:37 pm

    I am aware of this paradox, though a less macabre one. He will be executed, and it will be a surprise, but I am not certain of the solution.

    It seems that he may exclude Friday legitimately, but as he reasons the days backward he fails to include subsequent days again.

  • Colin January 19, 2011, 6:11 am

    This is a dress-up of the basic paradox of a man saying “I am lying”

    Nick has simply proven to himself that the Judge was lying to him, thus making it impossible for him to know anything on the basis of the judge’s statement, thus making the judge’s statement true etc etc

    In this case the paradox dissolves once events unfold.

    If I arrange a committee of people, each of whom is sequentially assigned a day and an hour (on that day) with the duty of convincing themselves that Jesus will return on that hour of that day, can we stop Jesus from returning? (and why would we)

  • Woland's cat January 19, 2011, 10:53 am

    Perhaps that’s what caused Jesus’ prophecy (Matt 24) to fail when he said that “Truly I tell you, this generation will certainly not pass away until all these things have happened” (v34). “But about that day or hour no one knows, not even the angels in heaven, nor the Son, but only the Father.” (v36). All that generation needed to do was estimate the maximum life expectancy of the youngest member of that generation and work their way backwards just like McGurk. Because “no one” could know then it wasn’t going to be on the last day of that generation, and so on and so on back to day one.

    (BTW, it seems the Son was not privy to information that the Father had. A dysfunctional Trinity??)

  • Glenn January 19, 2011, 12:38 pm

    Woland’s cat – you assume that unless the Father and the Son knew the same things, there was a malfunction. Since there’s no reason to think this, I guess I’ll just leave the comment alone.

    As for the comment about Jesus “failed” prophecy, I’m afraid you’re a little behind in your reading! I probably can’t help with that, but you might want to check out a talk I gave at our church a while ago which might help.

  • Tim January 19, 2011, 1:41 pm

    woohoo! Woland’s cat! just read that book!

    Back on topic…

    McGurk has no power over when he will be executed, he can only decide whether he will expect it.

    By ruling out expecting to be executed on a friday, he ensures that a friday execution will be unexpected – which is what the judge said it would be. Every day McGurk rules out becomes a day on which he would be surprised by his execution.

    Whether he will be executed or not depends on whether the judge went home and thought to himself:

    “hehe that’ll put that evil slasher on his toes… now.. what day shall we do it? It can’t be friday, because if we haven’t done it by friday he’ll know it’s coming then… damn he’ll know that too – he dropped out of philosophy last year… well if it definitely can’t be friday then it probably can’t be thursday either because… oh drat..”

    or whether he went home and took out his diary and wrote;

    “wednesday”

  • Woland's cat January 19, 2011, 2:11 pm

    Tim, I’m pleased you get the reference!

    Glenn, I’ve just listened to the podcast: very informative. Thank you.

    So, if you are Preterist, does that mean that the example of McGurk above is analogous to the situation you believe the early Christians found themselves in?

  • Joey January 19, 2011, 3:12 pm

    If you all, like me, are full of curiousity and are dying to know, you can just wikipedia “the unexpected hanging paradox” like I did 😉

  • Glenn January 19, 2011, 4:31 pm

    WC:

    “does that mean that the example of McGurk above is analogous to the situation you believe the early Christians found themselves in?”

    No. In the case of Jesus, he simply said it would happen within a given timeframe, but that nobody currently knows exactly when. He did say, however, that they would be able to see it about to happen (e.g. “when you see Jerusalem surrounded by armies” etc).

  • Richard P January 19, 2011, 6:59 pm

    The Bible would be more interesting (and relevant) if it had riddles like this one instead of talking about ‘miracles’ and ‘ethics.’

  • bossmanham January 19, 2011, 8:12 pm

    It seems to me that he obviously will be hanged. The judge is just telling him he isn’t going to reveal the exact time. It’s true, if the man is still in his cell Friday morning, he knows that’s the day, but it doesn’t seem like the rest of the reasoning is sound based on that.

  • Eugene Curry January 21, 2011, 1:28 pm

    I’ve been thinking it over.

    Glenn, what prevents us from challenging the paradox itself?

    The riddle seems to presuppose that the judge can have infallible knowledge of McGurk’s own thoughts, that he knows that McGurk cannot know the time of his hanging. But what possible foundation do we have for assuming this about the judge? In real, everyday life (situation A) it would be an absurd assumption (trans-temporal telepathy?) and, if challenged, the judge would likely retreat from his grander claim that McGurk “will not know the day” of his death to the more modest claim that McGurk “will not be informed as to the day” of his death and that he therefore “probably will not know.” And in that case there’s no paradox at all: McGurk could be executed Monday through Friday–and if he’s still around Thursday afternoon he could reasonably deduce he must be executed on Friday. Even though he knows what day he’ll die, no one told him and his knowing was unlikely (20% chance if there was an equal chance of death each day of the week)

    But if we insist on keeping the assumption of the judge’s infallibility–the judge is actually God, or something–then the paradox allows of two possible resolutions. Either (situation B) both of the judge’s statements (1. McGurk will die this week; 2. McGurk won’t know which day) are logically compatible and, thus, given the judge’s infallibility McGurk will die this week or (situation C) both of the statements are logically incompatible and, thus, such a situation as the riddle describes could never be instantiated in any possible world–that is, it’s a nonsense story so no actual McGurk would ever hear both of the judges statements.

    Therefore in all imaginable senarios a real McGurk can have no rational confidence that he will be spared: either (A) he dies, (B) he dies, or (C) he never existed in the first place.

    What do you think?

  • Glenn January 21, 2011, 1:31 pm

    “But what possible foundation do we have for assuming this about the judge?”

    The fact that I told you it’s true, and that is that. 🙂

  • Eugene Curry January 21, 2011, 1:56 pm

    Okay, so if it’s true that only situations B and C are live options. Which means that either way the answer to your initial question is “no, Nick is not right, he is not safe.”

  • Eugene Curry January 21, 2011, 1:56 pm

    *it’s true then

  • CPE Gaebler January 21, 2011, 5:00 pm

    Hmmm, it seems there are two answers to this problem. The smart-alec solution is that since McGurk has convinced himself that he will not be hanged, any day out of next week will suffice, as it will come as a surprise.

    However, I note that this answer has been posed before, and you responded in post 2 with a question of whether two contradictory statements are thus true. The answer is “no, no two contradictory statements are true,” and to elucidate I must first present the other answer, which assumes that the judges statement that McGurk “will not know” the day implies that he also “can not know” (in contrast to the first answer, where if the execution is on a Friday, McGurk COULD know Thursday evening but WOULD NOT, in this situation, know because he has convinced himself otherwise).

    In this case, no, McGurk is not justified in thinking he will not be hanged, because he has already proven that the judge’s statements are inherently contradictory. From only the premises stated by the judge, he has reasoned in a logically sound fashion that he will not be hanged – and yet, one of the judge’s statements was that McGurk WOULD be hanged. At least one of the judge’s statements must therefore be a false statement. However, McGurk makes an invalid assumption in concluding that he will not be hanged, which is obvious if we write his reasoning in syllogism form:

    1: “I will not be hanged according to the judge’s timing rules.”
    2: (omitted)
    C: “Therefore, I will not be hanged at all.”

    The omitted premise in his syllogism is the important step: he is implying the statement, “If I am hanged at all, it will be according to the judge’s timing rules.” However, he does not have justification for asserting this premise, since the only evidence he has for it is the judge’s own assertion, in a collection of statements he already knows to be inherently contradictory!

    Thus, I conclude that McGurk does not have sound justification for either being certain that he will be hanged or that he will not be hanged; he simply does not have sufficient knowledge to be certain of either case. The judge’s statement that he will be hanged at all may be false, or it could be true and the stated timing conditions may be false.

    Sleep well, McGurk. You might not have that luxury for long.

  • The Atheist Missionary January 22, 2011, 2:37 am

    I expect McGurk will conclude: “Don’t ask me how he’ll manage. That’s his job. Not mine.”

  • Garren April 9, 2011, 7:07 am

    This can be simplified somewhat by skipping forward to Friday morning and asking whether McGurk can still be hanged today. Monday through Thursday are not essential to the paradox; they just serve to make the result sound more counter-intuitive.

  • Marius January 3, 2013, 11:31 am

    “Is Nick right? Is he really safe? Does the judge’s fearful announcement amount to the claim that the Slasher will not be hanged at all?”

    depictions of this paradox in other places doesn’t always end by asking the question you ask, so i’m going to simplify my response and focus on your question about nick – otherwise i’d be here all day, formulating and reformulating my response (which i’ve already re-formulated for the last hour, because it is driving me crazy)

    nick failed to realise that the decree for his sentence must still be carried out, and therefore an attempt to reason the decree away fails. we can blame the judge for a confusing and contradictory statement, but your question pertains to nick, and nick’s naiveté prevails: for his very first conclusion about friday fails when he concludes not to expect his execution then – which is exactly what the judge said: that he will not expect it. the same failed reasoning applies to every day as he reasons backwards to monday from friday.

    we can ask numerous questions about the formulation of the problem, its logical errors, (both from the judge and nick the convict) but in this case those questions will be irrelevant since your question pertains to nick specifically.

    and the answer is simply no. he’s not right, he’s not really safe, and the judges announcement doesn’t amount to his conclusions.

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