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)?

The difference between a sound logical conclusion and the facts is that in logic, there's no difference between sound logical conclusions and the facts.

But given the story, isn't the deductive argument given by the lawyer, Chris, a sound argument? The argument is valid and —- within the context of the story, the premises are true: namely: the hanging had to be a surprise happening on one of five days, M,T,W.Th.Fr. And as the argument shows, given this background and the premises used — all which follow deductively from the initial premises; and the conclusion follows that the hanging — as one that could not be anticipated — could not happen. This "could not happen" applies to the facts of the case (within the context of the story).

Within the context of the story, the sound, valid argument 'proved' that — as a matter of fact — the hanging could not happen.

There is a flaw in the Lawyer's thinking. But it is not that he advanced an unsound argument. The argument is sound, but the lawyer assumed too much. It does show the difference between logic, even sound logic, and what can happen in the real world of facts and events. Maybe you or someone can spell out the fatal flaw in the lawyer's thinking?

The lawyer's argument is defective, in that he assumes Friday to be the day of hanging. That, by itself removes the randomness of the day of hanging.

This is not the defect of the lawyer's argument. He does not assume Friday to be the day of the hanging. In the process of developing his argument, he takes Friday as a possible day for the hanging, and then asks: "Could Friday be the day of the hanging and not be one we could anticipate? His answer is "No." Then the argument proceeds to consider another possible day for the hanging.

The paradox follows only if we assume a logical game being played by logical players: who make the correct deductions and also correctly infer the logical deductions made by the other player. The lawyer deduces what the warden must be deducing; and the warder deduces what the lawyer must be deducing. The logical conclusion is that the hanging, as a surprise hanging, cannot happen.

Consider an comparative logical game: Suppose that you (as puzzle master) have five boxes numbered '1' thru '5' and one marble to hide inside one of the boxes. The challenge to the player is to deduce which box has the marble without looking inside the box. The puzzle master must hide the marble in one of the boxes. The player must search each box in sequential order, starting with box #1.

Can the puzzle master hide the ball in one of those boxes so that the participant (call him Bill) cannot logically deduce which box contains the ball, prior to looking under the box? Let’s see.

Start with box 5: Suppose the marble is in box 5. If Bill looked through boxes 1 – 4, and did not find the ball, he would know that the ball must be in box 5. So box-5 is eliminated as a good box. Can it be hid in box 4? Well suppose it is. Now Bill can deduce that as well. If he looks through boxes 1-3 and does not find the ball, he knows that the ball must be in box-4, because the other remaining box, box-5, has already been eliminated. So Box-4 cannot hold the hidden ball. Eliminate Box-4 as the box for hiding the ball.

Continue the same process for boxes 3, 2, and 1. The conclusion is that the puzzle master cannot hide the ball in any of those boxes. For the puzzle master deduces that Bill is able to deduce (using deductive logic), for each of the five boxes, that the ball cannot be hidden in that box.

Therefore, the puzzle master concludes: the ball cannot be hidden from Bill.

But this shows nothing more than this. We can set up a logical game in which the puzzle master knowing — what Bill can deduce — cannot hide the ball from Bill. But this is just logical game, nothing more.