Unexpected Hanging the Greatest Paradox

A man stands before a judge awaiting punishment. The judge deems his crimes to be heinous enough to deserve a unique sentence. The man will be put to death before the end of the week, and what's more, the day of his execution will be a surprise.

The man returns to his cell. With nothing to do, he contemplates his death. If his execution would happen before the end of the week, then the latest it could be would be Friday. But, if Friday morning comes and he is still alive, then he would know the day of his execution, and it wouldn't be a surprise. This means that he couldn't be killed on Friday, and the actual execution would take place between Monday and Thursday.

But wait! If he can't be killed on Friday, then when Thursday morning arrives, he would know that he would be executed on Thursday. That means that he cannot be killed on Thursday either. The man repeats this logic and rules out Wednesday, Tuesday, and Monday. He concludes that he could not be executed at all and relaxes.

When he was executed on Tuesday, it came as a total surprise.

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That is the Paradox of the Unexpected Hanging, the greatest paradox of all time. In my opinion, of course.

One of the reasons I like it so much is because you can tell it as a joke. The last bit about the execution being a surprise can be delivered like a punchline, and it usually gets a laugh.

The other reason I like it is because more than any other paradox, I have no idea how to resolve it.

Every step of the fictional criminal’s logic seems sound to me. If he did wake up on Friday, then he should be able to deduce the date of his death. It makes so much intuitive sense to me, that I fall back to thinking “Well, ok, maybe the entire line of reasoning doesn't work, but at the very least his first claim should hold”. The induction step seems correct as well though! If the judge had instead sentenced him to be executed before Thursday night, then the first claim would allow him to rule out Thursday in exactly the same way he ruled out Friday in our original problem. But isn't that the same situation as he’s in after the first claim!? Isn't saying “Monday through Thursday” exactly the same as saying “Monday through Friday, but not Friday”!? Then I use inductive reasoning to conclude the man can't be executed. But he can! So I start from the beginning and think “Well ok, the entire line of reasoning doesn't work, but he’s right about Friday” and my brain continues in circles.

That is, of course, why it's a paradox, but this one hits me a lot more viscerally than something more simple like the Liar’s Paradox.

There are lots of common methods to resolve paradoxes. One is that the original statement is contradictory. Imagine if the judge had said “Your execution will be tomorrow, and the day of your execution will be a total surprise.” That wouldn't feel like a paradox. It would just feel like the judge lied. Maybe that's what’s happening with the Unexpected Hanging Paradox, maybe you just cannot correctly claim that the date of the man’s death will come as a surprise. That seems wrong though. If the judge had only said “You will be put to death, and the date of your execution will be a surprise” then there would be no paradox and no contradiction. Does limiting the execution to this week really conflict with saying it will be a surprise? Even in the original situation, as the punchline goes, “when the man was executed on Thursday, it was a total surprise.” It seems like the judge was correct, and the man’s death will be a surprise no matter when it is. (Or at least, it will be as long as it's not in Friday. But wait, if it can't happen on Friday, then-aaarrrggh)

A solution I've read online is that the man cannot be sure he won't be killed on Friday until Friday morning. He is correct that if he knows he wasn't executed Monday through Thursday, then he can conclude that he would be executed on Friday which means it wouldn't be a surprise. But, he doesn't know he was not executed on Thursday until he wakes up on Friday. He can't make his induction argument, because he doesn't yet know that he wasn’t executed Monday through Thursday. This is reminiscent of Aristotle’s Sea Battle thought experiment. The truth value of future statements is an interesting topic, but it's one I'm not nearly equipped to address. Regardless, isn't the judge wrong if the execution is on Friday? If you sentenced a hundred prisoners this way, every one of them who were killed on Friday knew the date of their execution, right? Even if they didn't know it on Monday, they all would have learned it by Friday. If the original sentence was “you will be executed before Friday, but, right now, you don't know which day” there would be no paradox.

Maybe I'm just not smart enough to understand this argument, but it really seems like the man cannot be killed on Friday without first being able to deduce the date of his death, it doesn't matter when he is able to learn this. (And if he can't be killed on Friday, then surely when Thursday arrives he could-aaaaghhhh)

Another possible solution is to just accept the implications of the paradox. This is what happened to basically all of Zeno’s paradoxes. The paradox of the arrow, the paradox of the clap, Achilles vs the tortoise. All of these paradoxes show that motion (or change, really) implies that an infinite number of actions can be taken in a finite amount of time. The arrow must travel half the distance to its target an infinite amount of times before striking it. You can't do an infinite number of things, and yet you can clap your hands, therefore: paradox. This stumped philosophers for over a thousand years. Some of them thought that motion must be an illusion, and others proposed clever ways in which you didn't _really_ need to do infinite actions in order to achieve motion. Then Isaac Newton discovered Calculus, which shows that you can do infinite actions in a finite amount of time. There was no paradox at all. Zeno showed that in order for motion to be possible, an object would need to perform an infinite number of actions which is fine, because we now know that's a thing you can do.

Advances in science, philosophy, and mathematics can show that things we once thought were impossible actually aren't a problem at all. I think this is the most likely solution to the Unexpected Hanging Paradox. The thing about future discoveries though, is that they haven't happened yet. And me trying to comprehend a solution is as disorienting as an ancient thinker trying to understand infinite limits without Calculus.

That’s what makes paradoxes so fun. There's something immensely satisfying about the unknown. Think of a movie monster. It's much more interesting when it's off screen and mysterious. As soon as you see it in the light, it loses its appeal.