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."
Wednesday, October 28, 2009
The Paradox of the Unexpected Hanging
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2 comments:
Interesting quandary... while I'm sure that Gardner and others have provided excellent responses, I'll do my best to approach it with only the knowledge I already have, and see how well I do.
What's interesting to me is the interaction between the two premises. If "it will take place some time in the early morning on one of the days of the following work week", a necessary premise, is true, then it appears that "it will take place some time in the early morning on one of the days of the following work week" is false. However, here's the issue. Each day is contingent upon the previous day-- so, one might say,
If FRIDAY then THURSDAY. If THURSDAY then WEDNESDAY. If WEDNESDAY then TUESDAY. If TUESDAY then MONDAY. And so on and so forth. If I were executed on Thursday, I would not be alive on Friday, and thus "Friday" would be "false" for me in the sense that I cannot make decisions on Friday.
On this model, however, existence on Thursday presupposes existence on Friday-- that is, I know that I will be alive on Thursday because I know I cannot be hung on Friday. However, if that is the case, Thursday's existence is contingent upon Friday's existence. If Thursday's existence is contingent upon Friday's, and Friday's existence is contingent upon Thursday's, Thursday exists if and only if Friday exists-- for any day, it exists for you if and only if the following day exists for you.
This seems to me, at least, to state in clearer terms where the problem lies. If FRIDAY then THURSDAY seems valid, but if THURSDAY then FRIDAY seems invalid. I'll have to think some more about how exactly to deal with this, though.
To agree with Zach. The paradox does lie within the presupposition of my knowledge of Friday then Thursday then Wednesday and so forth. But to pick up where he left off... the existence of future day(s), for the purpose of this model, do not need to exist with the deadman, only the judge, who has the knowledge of the time of execution.
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