I recently came across the unexpected hanging paradox, and it’s been a surprisingly tricky one to work through. The paradox goes as follows (from Wikipedia):
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning, he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.
The paradoxical nature of the situation arises from a few areas (including “surprise” being a psychologically, rather than mathematically, defined concept), but its core resides in the bootstrapping approach the prisoner takes to eliminate the days one by one. Each of the eliminations appears to be justified, but when we actually put ourselves in the position of the prisoner on Sunday night, we realize we have no idea whether the hanging will occur the next day. To better understand the dynamics, we can consider the same situation with even fewer potential days for the hanging.
With one potential day (let’s say Friday), the situation simplifies into a form more akin to standard logical paradoxes. The judge states 1) that the prisoner will be hanged on Friday and 2) that it will be a surprise. As Friday is specified as the day, it won’t be a surprise, which means the judge’s two statements directly contradict each other. This does not mean the prisoner is necessarily off free (the judge may feel more strongly about his first statement!) but it does mean that one of the judge’s two statements is necessarily incorrect (assuming no gymnastics on the definition of “surprise”).
With two potential days (Thursday and Friday), some additional complexity creeps back in. The judge now states 1) that the prisoner will be hanged on Thursday or Friday and 2) that it will be a surprise. The prisoner can reason, as before, that the requirement for surprise means the hanging cannot occur on Friday, and can then lean on this observation to similarly exclude Thursday. Essentially, the prisoner is recognizing the same thing he saw with one potential day – either the judge’s first or second statement must be not be guaranteed. In the original framing of the paradox, he focused on the first statement (it’s the life or death one!), but what if we focus on the second instead?
With the second statement not guaranteed, we’re left only with the judge’s first statement – that the prisoner will be hanged on Thursday or Friday. Interestingly, without the second statement, we have no way of excluding either day, and so we’re back to the day of the hanging being a surprise! This is the key difference vs. one potential day (as with one day, even without the second statement there’s no way for the hanging to be a surprise), and it is the crack in the prisoner’s argument. He cannot rely on the judge’s statement that the hanging will be a surprise to prove that it will (or will not) be a surprise, much as Gödel could not rely on Principia Mathematica to prove certain truths about its own provability. This means that, although the hanging will need to occur on Thursday to be a surprise, the prisoner has no way of proving it will occur on Thursday.
This circular behavior is common to self-referential statements; we see the same type of thing happening with the two statements 1) this statement is true and 2) the previous statement is false (or more simply, “this statement is false”). We can hold either one in our mind individually, but when considering both they dance around each other and offer no place to establish a concrete footing. Taken together, these two statements are neither true nor false, but instead undefined – and due to a similar lack of definition, the prisoner is unable to say anything for certain about his day of hanging or the truthfulness of the judge.
One interesting aspect of the paradox which isn’t fully explained by the above line of thinking is the fact that the prisoner’s logic feels true for low numbers of days but empty for higher numbers. For myself, the cutoff of reasonability is three days. If a judge told me I’d be hanged on Wednesday, Thursday, or Friday, and that it would be a surprise, there’s a part of me that would know that Wednesday would be chosen, as on Thursday there’s no room left for surprises (if the judge’s statements are to hold true). I can picture myself looking out at the three possible days on Tuesday night and having 100% certainty that Wednesday needs to be chosen (though “needs to be” is slightly different from “will”). At four days, I’d feel significantly less certain (for some reason, my Wednesday certainty is not strong enough to fully extend to Tuesday), and at five or more days I’d have lost any illusion of certainty. It seems that we intuitively have some sense of the tenuousness of the prisoner’s logic, though the psychological cutoff is less rigid.