Here’s an interesting paradox you might not have heard before. It is called the paradox of the alleged impossibility of scheduling a surprise hanging, or fire drill, or appearance by Obama, or any such event that will be anticipated or not by people involved.
This paradox is found in William Poundstone’s book Labyrinths of Reason (Anchor books, 1988)
There are different ways of stating the paradox. Here’s one of them:
A man, call him Brad, has been convicted of murder and sentenced to hanging. The hanging must take place on the final week of the year, sometime during the final five-day work week. The judge who imposed the penalty also dictates that the convicted man will not know beforehand which day of the week he will be hanged, in short, he requires that the specific day of execution will come as a very bad surprise.
SENTENCE: Brad’s hanging will happen one day of the last week (M,T,W,Th,F) of the final week of the year. But we’re not saying which day.
Brad’s lawyer, Chris, happens also to be a logician. When he hears the judge pronounce sentence, he smiles. Brad is taken aback by this. Later, when he and Chris confer, Chris explains. “Relax, Brad,” he says, “the judge just contradicted himself. The hanging cannot take place.” “Why not?” asks Brad.
Chris explains: “The judge requires that the hanging day be a surprise, one which we cannot anticipate. But consider the possibilities. Let’s start by taking Friday as a possible hanging day. Well if we get through the first four days of the week (Mon, Tues, Wed., Thurs.) without a hanging, then we would know that the hanging would be on Friday, which denies the condition for the hanging. This implies that Friday cannot be the day; so eliminate Friday.
“Now consider Thursday as a possible day for the hanging. Well, since Friday has already been eliminated, and we get through the first three days (Mon, Tues, Wed) without a hanging, then by deductive inference we would know that Thursday was the day, since Friday has been eliminated. But we cannot anticipate the day; so Thursday cannot be the day. So eliminate Thursday, as well.
“Now let’s take Wednesday as a likely day for the hanging. Well, since Thursday and Friday have already been eliminated. Now suppose that we get through the first two days of the week (Mon, Tues) without a hanging, then by deductive inference we would know that Wednesday would be the day. But we cannot anticipate. So eliminate Wednesday, as well.
“Now consider Tuesday as a possibility. Well, Wednesday, Thursday and Friday have already been eliminated. Suppose we get through the Monday without a hanging; then by deductive inference we would know that Tuesday has to be the day. But this would anticipate Tuesday. So eliminate Tuesday as well.
So with Tuesday, Wednesday, Thursday, and Friday all eliminated as possible hanging days, only Monday remains. But then the possible hanging on Monday would not be a surprise, and thus cannot happen on Monday. So eliminate Monday.
Ergo, the surprise hanging cannot take place that week. Ergo, it won’t happen!
“So, chortles the triumphant Chris to the worried Brad, all five days of the week are logically eliminated! None of them can be the day of your hanging. You will not hang!”
But the final week of the year arrived and on Wednesday at 6 P.M. Brad was led to the gallows and hanged, contrary to the assurance given by defense lawyer-logician Chris that it could not occur.
What happened? Wasn’t Chris’s logic impeccable? How is it possible that the hanging took place and caught Chris and Brad by surprise, much Brad’s great disadvantage?
Can anyone give me a clear analysis of the paradox? Why the contradiction between the conclusion of a sound, deductive argument (hanging cannot happen) and the fact (hanging happened on Wednesday)?