# What are some popular ways of trying to resolve the "Hangman's Paradox"?

In a harsh totalitarian country an innocent person is arrested on Sunday evening and summarily condemned to execution, which they are told will take place on one of the following five mornings. To make matters worse, they are told that they will not know the day before which morning it will be. After several hours torment the prisoner fails into a peaceful sleep as they realize that such a threat cannot be carried out. They reasoned thus: The execution cannot take place on Friday morning; for if they are still alive on Thursday night then the execution must take place on Friday. But they were told that they would not know the day before which day it would be. So it cannot be Friday, and so Friday can be counted out as a possibility. But by the same reasoning it cannot be Thursday either. For if they are still alive on Wednesday night then the execution must take place on Thursday. But they were told that they would not know the day before which day it would be. So it cannot be Thursday, and so Thursday can be counted out as a possibility. But by the same reasoning it cannot be Wednesday either... The same reasoning covers Wednesday, Tuesday and Monday and so the prisoner can have a sound nights sleep.

The prisoner is, however, greatly surprised to find themselves facing the executioner on Wednesday (or indeed any other) morning.

### What went wrong with the prisoner’s reasoning?

I tried but I am not able to figure out how his reasoning is flawed.

• You should try to spell out what you've thought about so far, not only because it will make the problem you are having much clearer for those trying to help, but also because doing so often helps you solve the problem yourself. I can't count how many times I've had a question I started posting to one of the other SE sites and upon trying to explain the issue in detail, I stumbled upon the answer myself. :) As for your question, it seems there's a gap in the logic where he claims that the reasoning which discounts Friday as a possibility must also apply to Thursday... Apr 9, 2013 at 5:04
• I edited the title in an attempt to attract more informed answerers. Feel free to roll back my edit if you feel the new title is inappropriate. Apr 9, 2013 at 6:05

The prisoner's reasoning is forced to be paradoxical. The chain of days is just a distraction.

The judge tells the prisoner: you will die today unless you are sure of the date of your death.

The prisoner thinks: "Wait, that's ridiculous, he just told me I'd die today, and I know the date. So of course I am sure of the date. That means I can't be killed, because I'm sure! Yay! I won't die today! Wait, but if I know I can't die today and they kill me anyway...uh-oh...."

Pulling the paradox out via induction just makes it harder to notice the paradox (of the standard self-referential variety).

What you are describing seems to be a popular formulation of the Hangman's Paradox.

Here is a link to a paper which compares it to Newcomb's Paradox and gives a game theoretic treatment of the Hangman's paradox.

The comparison to Newcomb's Paradox suggests to me that this might be something that is susceptible to a Bayesian decision theoretic treatment. In the case of Newcomb's Paradox, it is often taken to divide Objective and Subjective Bayesian Epistemologists according to whether you endorse the one-box or two-box strategy, with objective bayesians advocating a one-box approach.

It apparently also goes by the "surprise inspection paradox" and the "unexpected examination paradox". This paper discusses one proposed resolution of the paradox (I gather the one discussed in: Margalit, A. & Bar-Hillel, M. (1983). "Expecting the Unexpected") and argues that it fails (unfortunately this one is behind a pay wall, so I don't know if you can access it; if you have access to a university's library database it is likely that Springer is something you can access through them).

If the wiki article is to be trusted here, it appears as though the attempts at solution divide into "logical" (which seem to approach the paradox as a form of a liar-style self-reference paradox) and "epistemological" (which seems to approach the paradox from a game theoretic or decision theoretic perspective).

I'll leave it to you or others to delve into these articles and appreciate the attempts at solution. If questions remain in a few days when I have more time, I'll try to do some research and summarizing myself. But as it stands, I'm not well enough acquainted with the paradox to discuss the attempts at solution meaningfully without further and more careful research.