The Paradox of the Unexpected Hanging, Part II

Zach and Anonymous commented on my last post on the paradox of the unexpected hanging. While their thinking is interesting, I doubt that they are on the right track toward a solution. Let me present an alternative version of the paradox (also drawn from Martin Gardner's article) and challenge them to see if their attempt at a solution applies to this form.

So a man is presented with ten boxes, each one numbered and all empty except for one, which contains an egg. He is told to open each box in sequence. He is also told that he will not know before opening the box that contains the egg that it contains the egg. So once again, the same reasoning applies. The egg cannot be in box 10 because if the man opens box 9 without having discovered the egg, he will know that it is in box 10. Elimination of the other boxes proceeds as before.

Maybe this less sinister and more basic formulation of the paradox will help in their search for a solution.

The Paradox of the Unexpected Hanging

You have been sentenced to death by hanging. The judge who condemns you to this fate informs you that your execution will satisfy two conditions. First it will take place some time in the early morning on one of the days of the following work week. Second you will not know which day you will die until the executioner appears in your jail cell to lead you to the gallows. You will be surprised. There does not appear to be any reason to think that your hanging cannot take place just as the judge has ordered. But a seemingly plausible argument leads to the conclusion that the judge's conditions are unrealizable.

In my last post I celebrated the birthday of Martin Gardner. Continuing in this theme, I now recount a paradox discussed by Gardner in one of his columns (and reprinted in his book The Unexpected Hanging).

Your execution must take place no later than Friday. Suppose you make it to Thursday afternoon. You now know that you will be executed on Friday, violating the second of the judge's pronouncements. So you cannot be executed on Friday. Now suppose you make it to Wednesday afternoon. Since Friday has been eliminated, you know on Wednesday that you will be executed on Thursday, again violating the surprise condition. So Thursday is out. The same reasoning will rule out Wednesday as well. Continue this line of reasoning until you have eliminated Monday. Therefore on no day of the week can you be surprised by the hangman.

As with all logical paradoxes, seemingly impeccable reasoning leads to a conclusion that is clearly at odds with reality. For having been convinced by your reasoning that you will not be executed, you are understandably surprised when the hangman arrives on Wednesday morning to carry out the orders of the judge.

So I leave it to you to resolve this paradox with the words of Bertrand Russell, who urged those who think about logic "to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science."