Clarifications on the unexpected hanging paradox

Question

I read through the wiki on the unexpected hanging paradox .

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 night, 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.

I found the most of the arguments rely on the following facts.

1. The prisoner is a perfect logician.
2. The judge is a perfect logician.
3. The judge is truthful.
4. Statements 1, 2 and 3 are common knowledge.

Is there any reason these are not included as part of the problem statement?

Answer 1

As you have posed the question, the four conditions clearly don't work. The prisoner can't conclude that he won't be hanged because the judge said he would and the judge is truthful. As a perfect logician he would recognize that.

It would be quite tricky to come up with a problem statement that works well. The paradox rests, I believe, on a rather neat concealed Gödelization. If I can prove that an event would not be surprising, then it becomes surprising.

The circularity becomes clearer if we remove the part about the intervening days. Imagine if, at the conclusion of the trial, the judge pulled out a gun, pointed it at the prisoner and said: "I will shoot you in 10 seconds and it will be a surprise." The prisoner reasons that he is perfectly safe because there is nothing less surprising than being shot by a man who is holding a gun and has just said he will shoot you. Having reasoned this way, he will, of course, be surprised when he is shot.

Gödel's theorem (loosely rendered) states that any logical system powerful enough to express normal integer arithmetic is powerful enough to form the statement: "This statement is true and unprovable." Hence all such systems are either incomplete (there is at least one true statement which cannot be proved) or inconsistent (you can prove both a statement and its negation). For the latter, imagine that you could prove the Gödel statement. By proving it, you have made it false. If you can prove one false statement, you can prove anything.

Returning to the judge and the prisoner, if we replace the word "surprising" with the word "unprovable" then the parallel is clear. What does this mean for the set of four axioms? That is a bit murky. What do we really mean by "perfect logician"? Are all of the actions and emotions of a "perfect logician" defined by logical reasoning? Does this mean that "surprising" and "unprovable" are exactly the same for a perfect logician? If we accept this, then I think it implies that the judge must also have perfect knowledge of the prisoner's world-view or he could not say with certainty whether the prisoner would be surprised or not. In this case the axioms could be consistent in themselves but not consistent with the prisoner's reasoning. A perfect logician would accept that the judge had made a true yet unprovable statement about his future. Since the statement is unprovable, the prisoner would accept it as true but be unable to use it to form any expectations about his execution. Hence, he would still be surprised on his execution day but for a different reason.

Answer 2

The question is:

Is there any reason these are not included as part of the problem statement?

The problem with the suggested statements is that statement 1 and the portions of statement 4 related to the well-known status of statements 2 and 3 cannot both be true at the same time, in the posited scenario.

The problem is that the prisoner reaches a conclusion that an execution will not happen, when the judge has clearly stated that it will happen during the coming week.

The only way for the judge's statement regarding the execution to be false would be if the judge is either mistaken about future events (i.e.: not a perfect logician) or else lying (i.e.: not truthful).

Neither of these is the case as per statements 2 and 3, respectfully. Additionally, statement 4 tells us that the prisoner knows that neither of these are the case.

So taking statement 1 (prisoner is a perfect logician) and statement 4 together (The prisoner knows that the judge is a perfect logician and truthful), the prisoner cannot accept a conclusion which would require that the judge be either wrong or lying about the execution date.

Therefore, either statement 1 or statement 4 must be false. My money is on statement 1 being the false one, personally. That makes sense both logically and thematically, as the whole story is really a fairly simple morality play about a criminal who thinks they're more clever than they actually are, and gets their comeuppance.

But you can also make the story's logic work if you prefer to state that the prisoner didn't know the judge was infallible and truthful, and assumed that they were wrong and/or lying. It's not much of a bedtime story at that point, though.

Interestingly, neither statement 2 nor statement 3 is actually important to the key problem in the scenario; what's important is that we're told the prisoner believes that the judge is truthful and a perfect logician (statement 4), and that the prisoner is a perfect logician (statement 1), and yet somehow still comes to a conclusion which would require that the judge either has made a logical mistake or is lying. That's a pretty huge logical mistake for our "perfect logician" prisoner to make.

Answer 3

Is there any reason these are not included as part of the problem statement?

Simple! #1 is false, #2 is not necessary, and #3 is sufficiently implied.

In my answer to the other unexpected hanging paradox question, I pointed out that the prisoner is not a perfect logician. In short, if he was a perfect logician he would have realized that by believing he could not be hung on a certain day, he would be surprised by being hung on that day. So the story itself actually demonstrates that the prisoner is not a perfect logician.

As for #2, the judge doesn't need to be a perfect logician. He's not doing much logical reasoning. All he needs is to either know that the prisoner can't actually use what he knows to deduce any useful information, or know the prisoner well enough to know that the prisoner will convince himself that he won't be hung.

The judge's truthfulness is implied. If there was any doubt about it, the prisoner would be shown to consider that the judge may have lied. Also, it ends by saying "everything the judge said came true."

Points #2 and #3 are also not necessary because they are focused on the judge. The story itself focuses on the prisoner, and so instead of outright stating the points they are implied through what we see of the prisoner's reasoning. This is simply a case of good storytelling.