The Unexpected Hanging Paradox.
There’s no escaping death.
A judge sentences a prisoner to be hanged on a weekday during the coming week. He tells the prisoner that the execution will come as a surprise to him and that he will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
On returning to his cell the prisoner thinks about what the judge has just told him. After a while, he excitedly starts jumping up and down claiming that he’ll escape from his death sentence. He first concludes that the surprise hanging cannot take place on Friday, as if he already hasn’t been hanged by Thursday evening, then it can only happen on a Friday, which wouldn’t be a surprise.
He then concludes that the hanging cannot take place on Thursday either. As Friday has already been eliminated and if he isn’t hanged by Wednesday evening, the hanging must occur on a Thursday, which wouldn’t be a surprise either.
Using this logic, he concludes that a surprise hanging can also not take place on a Wednesday, Tuesday or Monday and so believes that he’ll escape his punishment.
To the prisoner’s surprise, the executioner knocks on his cell door at noon on Wednesday. Everything that the judge said comes true.
Despite the fact that nearly a hundred papers have been published on the Unexpected hanging paradox, there is still no consensus on what the correct solution is. There are broadly two approaches to tackle the problem: The logical school and the epistemological school.
The logical school attempts to formalize the argument so that the faulty logic then becomes apparent. You can formalize the judge’s announcement as statement A — The prisoner will be hanged next week and its date will not be deducible in advance from the assumption that the hanging will occur sometime during the week.
This formalization allows the prisoner to deduce that the execution will not occur on a Friday. However, it doesn’t allow the prisoner to them eliminate Thursday, as in order to do this, the prisoner has to show that his ability to deduce from statement A that the hanging will not occur on a Friday implies that a Friday hanging would not be surprising.
However, since statement A restricts “surprising” to mean “not deducible from the assumption that the hanging will occur sometime during the week”, instead of “not deducible from statement A”, the argument falls down.
To continue the prisoner’s argument and rule out the rest of the weekdays, we have to formalize the judge’s announcement as: “The prisoner will be hanged next week and it’s date will not be deducible in advance using this announcement as an axiom.”
This statement, however, is self-referential and exposes the flaw in the paradox.
The epistemological school takes a different approach and instead focuses on questions around what it is to know something. For example, the difference between the judge’s assertion that something is true and the prisoner’s knowing that it is true.
Some have argued that there are differences between imagining the future and experiencing it, or that a distinction can be made between the thought processes of a prisoner and those of an observer.
Others have argued that unexpected hanging paradox is actually a complicated version of Moore’s paradox, which can be expressed by reducing the number of possible days to just one. The judge would then tell the prisoner, “You’ll be hanged tomorrow but you do not know that”.
The paradox continues to divide mathematicians and philosophers and there is still no conclusive answer.
But what do you think, does the root of the paradox lie in the judge’s statement or is the prisoner’s reasoning to blame?
