Monday, March 30, 2009

Resolution of the "Unexpected Hanging Paradox"

The Unexpected Hanging Paradox:

Here is a short version of a paradox that has intrigued me since I was in college: A condemned prisoner is told by the warden on a Sunday evening that he will be executed at noon on one of the upcoming weekdays, i.e., M-Tu-W-Th or F, but he will not know in advance which day it is until the time comes. The day of his hanging will be a “surprise”. The prisoner reasons that if Thursday noon passes and he has not yet been hung, then he can rule out Friday since that is the only day left and then he would know what day he was to die, in violation of the conditions specified by the warden. But having crossed Friday off the list, he quickly realizes that he can cross Thursday off too, since once Wednesday noon is past Thursday is the only day left (since he has already eliminated Friday). Going on this way, he realizes that he can eliminate all 5 weekdays, one by one, and he is happy to realize that he cannot therefore be hung (let’s say that he strongly believes the warden to be a truthful man). But here is the (sad) paradox…the executioner comes to his cell on (say) that Wednesday, and informs him that he is now to be hung. His last thoughts before dying are that the warden indeed spoke the truth: that he, the prisoner, was indeed going to be hung, and he was certainly surprised by it too. Where did his reasoning go wrong? It seemed so logical that he could not be hung if the warden was telling the truth……and yet, in the end, the warden did tell the truth. Seemingly, a paradox!

The Resolution:

Here is my understanding of the resolution of the paradox (and I think I follow, essentially, Martin Gardner’s explanation from his book with the paradox as the title). The prisoners’ reasoning goes wrong on the very first step. He cannot rule out Friday even if Thursday noon has come and gone. In somewhat formal logic terms, I see it as follows. Let the wardens statement to the prisoner be written as C =A + B, where A = “you will be hung”, and B = “you will not know in advance of the day of the deed”. First, suppose A = true. That would mean that B = false, and hence C = A + B = false. Next, suppose instead that B = true. Then this would mean that either A or “not A” can be true; if A is true, then C = A + B is true, and if instead "not A" is true, then C = A + B is false. This makes sense because even if he doesn’t know the day of the hanging, then he could still either be hung or not hung. Thus, considering all of the above possibilities, we see that C can be true or false. It could just as well turn out to be true as it could turn out to be false. So, if it turns out after the fact that C is true, this does not really entail a contradiction. What the prisoner should have realized is that the warden’s statement could turn out to be true or false, and since he could not be sure ahead of time what the truth value was, he could not logically deduce the outcome from it.

5 comments:

pjgriff said...

I agree with the general intent of your "resolution" but I disagree with the bottom line. I believe that the warden’s statement “C” is neither true nor false because it is a poorly formed self-contradictory statement – hence not subject to a logical analysis.

From my point of view, the intent/postulate of the puzzle is that the warden’s statement is “true”. This is a postulate to the puzzle and is not subject to question. Given the “truth” of the warden’s statement, your logic clearly shows that “C” can not be true in a logical sense consistent with the rules of inductive logic. So it is a poorly formed puzzle.

So, I disagree with your statement that “Thus, considering all of the above possibilities, we see that C can be true or false.”. “C” is a poorly formed statement and can be neither. I would restate your conclusion as “What the prisoner should have realized is that the warden’s statement was not ‘well-formed’ and could not be categorized as true or false, and since the warden’s statement had no logical (truth/false) value, he could not logically deduce the outcome from it.”

A related concept is addressed in a book by Heidegger called “What is a Thing”. As I read this book, the essence was that the answer to a question is essentially in the statement of the question itself. The difficulty for the listener is when the question is poorly formed and self-contradictory – often due to our confusing and contradictory use of words. This can also be expressed as “false problems are only poorly posed questions of interpretation”. I think that that is the issue here – and the puzzle “resolution” is to appreciate the poorly formed statement of the puzzle premise – the warden’s statement.

pjgriff said...

I agree with the general intent of your "resolution" but I disagree with the bottom line. I believe that the warden’s statement “C” is neither true nor false because it is a poorly formed self-contradictory statement – hence not subject to a logical analysis.

From my point of view, the intent/postulate of the puzzle is that the warden’s statement is “true”. This is a postulate to the puzzle and is not subject to question. Given the “truth” of the warden’s statement, your logic clearly shows that “C” can not be true in a logical sense, consistent with the rules of inductive logic. So it is a poorly formed puzzle.

So, I disagree with your statement that “Thus, considering all of the above possibilities, we see that C can be true or false.”. “C” is a poorly formed statement and can be neither true nor false. I would restate your conclusion as “What the prisoner should have realized is that the warden’s statement was not ‘well-formed’ and could not be categorized as true or false, and since the warden’s statement had no logical (truth/false) value, he could not logically deduce the outcome from it.”

A related concept is addressed in a book by Heidegger called “What is a Thing”. As I read this book, the essence was that the answer to a question is essentially in the statement/interpretation of the question itself. The difficulty for the listener is when the question is poorly formed and self-contradictory – often due to our confusing and contradictory use of words. This can also be expressed as “false problems are only poorly posed questions of interpretation”. I think that that is the issue here – and the puzzle “resolution” is to appreciate the poorly formed statement of the puzzle premise – the warden’s statement.

Tom said...

Thank you, Pat, for intriguing comments on my attempted “resolution” of the unexpected hanging puzzle. I especially found your brief discussion of Heidegger interesting, and I want to look into his views on this more when I get a chance. So I much appreciate your making me aware of that.

However, I do not fully agree with you that the warden’s statement cannot be true or false. And it was never stated in the puzzle that his statement had to be true. In fact, I do agree with you that it cannot consistently be taken to be true for certain. But—and here is the kicker that I think you may have missed (so many of the philosophical analyses of this puzzle have missed it, in fact)--- the wardens statement turns, after the fact, out to be true. He was hung, and it was unexpected! Thus “A and B”, in my earlier terminology.

My point was that the way the warden worded his statement it could turn out, before the week passes, to be true or false, and the prisoner should have realized that. Hence he should have realized that it did not provide any predictive basis for assuming that he would not be executed.

The inductive aspect of this puzzle is, I claim, a “red herring”. The same fallacy would be involved on the part of the prisoner if the week only contained one day. The reason is that if the prisoner assumes B is true (that the fact of his hanging is not predictable in advance), then he could be hung or not hung, thus making the combined statement A and B either true or false.

BTW, it is amusing to “Google” this puzzle. A search on “The Unexpected Hanging” yields about 260,000 hits. So maybe we should both do some more research on what the others are saying---there may well be a lot of additional insights into what is going on here---and revisit and discuss this more later.

efp said...

Hi Tom,

This looks to me like a version of the liar's paradox, which is just the problem of self-reference. Like you pointed out, imagining a one-day week, the problem is reduced to: you will be hung tomorrow only if you don't expect to be hung. For instance, A='you will be hung', B='you expect to be hung', then the problem is:

B iff A
A iff ~B

The joke of the "paradox" comes from the fact that the prisoner stops the circular reasoning. Once he concludes he cannot be hung, he should then conclude that he could be because he won't be expecting it, etc.

32oH2O said...

I have always found this to be a good puzzle. The warden's statement is not poorly formed or contradictory, but does have some uncertainty imbedded within it. What does "knowing" mean in this case. 100% certain, 20% certain, 80% certain? What are the win conditions for each player?

I presume (but there is an assumption here) that the warden can schedule or not for any day. The prisoner can predict or not for any day. Game ends positively for prisoner on Y,Y (scheduled hanging predicted and called off). Game ends positively for warden on Y,N (head cheese anyone?) Game goes to next day on N,N unless it is Friday in which case prisoner wins. However, to do the MECE analysis, you have to factor in what the game result is on a N,Y (I presume that the prisoner is hanged for guessing wrong.)

I think the puzzle at its deepest deals with making decisions in an uncertain environment (here is the real red herring--puzzlers in a uncertain environment). It devolves into a mixed strategy game, and when the prisoner deludes himself into believing it to be a set strategy game, he loses--always. Given that it is a mixed strategy game, the warden's statement is 100% true that the prisoner cannot "know" when the day will come. The warden might not even know-noose him up and roll a die perhaps.