One day in class the teacher announced that there would be an exam the following week. It would be held at 9 am on either Monday, Tuesday or Wednesday. "However", he said, "you will not know which day the exam will be held until you come into class on that day." In other words, the exam would be unexpected. One of the students (Belinda) put up her hand and said "You've contradicted yourself!" Now why would she have said that?
Charlotte: It was Easter, and Monday and Tuesday were holidays.
No. It wasn't Easter. She had argued to herself as follows: "If I get to school on Tuesday and the exam still hasn't been held, then I'll know it must be on Wednesday, because that's the only day left. So a Wednesday exam wouldn't be unexpected. So the exam can't be on Wednesday."
Owen: I see. That leaves only Monday and Tuesday. By the same argument, Tuesday can be eliminated, leaving only Monday. Then Monday can be eliminated too.
Exactly. So Belinda thought the teacher had contradicted himself in saying that an exam would be held on one of the three days, and that it would be unexpected. Anyway, the teacher told Belinda that he was not interested in listening to that sort of nonsense. And sure enough, when the class arrived at school on Tuesday, they found the exam papers on their desks. The exam was, quite unexpectedly, held on that day.
This shows how important it is to make sure the meaning of statements is clear. Now, can you put what the teacher said into other words?
Charlotte: He said the exam would be next Monday, Tuesday or Wednesday. That's plain enough.
I agree. Let's call that Statement 1.
Charlotte: Then he said the class wouldn't know which day the exam would be until it happened.
Yes. That's a bit complicated when you think about it, isn't it. It doesn't really tell the students what they will know on any particular day. Let's split the statement up. What will the class know before school on Monday?
Owen: It could be any day.
Exactly. So Statement 2 could be "From now until school starts on Monday, you won't be able to work out whether the exam will be on Monday, Tuesday or Wednesday." I'll give you Statement 3. It is: "If the exam isn't held on Monday, then before school on Tuesday you won't be able to work out whether the exam will be on Tuesday or Wednesday." What is the next statement?
Charlotte: I don't know. This is too hard.
Statement 4 is similar: "If the exam isn't held on Monday or Tuesday, then before school on Wednesday you won't be able to work out whether the exam will be on Wednesday."
Owen: But that completes the contradiction!
Exactly. But the point is that the contradiction only occurs if the exam is actually held on Wednesday. Belinda was wrong in claiming that the teacher had contradicted himself right at the beginning. As you know, as it turned out the exam was held on Tuesday, so no contradiction occurred.
In fact, Statement 4 can be any statement which generates a contradiction. For example: "If the exam isn't held on Monday or Tuesday, then night is day."
Part of the problem is that the paradox is always stated in a way which gives the person predicting the unexpected event (in this case the teacher) great authority and credibility. Another common formulation involves a judge telling a prisoner he will be executed one day the following week. This great credibility of the predictor makes it harder to imagine that he would contradict himself, even in the conditional way described in the resolution.
It used to be thought that the problem was caused by statements which referred to themselves, as in the following formulation:
Statement 2 "You cannot tell from Statements 1 or 2 which day the exam will be, until you come to school on that day."
But it was then realised that the self-referring bit could be eliminated by adding extra statements.
The paradox has been discussed extensively by philosophers. See for example Smullyan's Forever Undecided at chapter 2, and also the following articles in the philosophical journal Mind:
Nerlich GC. Unexpected examinations and unprovable statements. Mind 1961; 70:503-513.
Janaway C. Knowing about surprises: A supposed antinomy revisited. Mind 1989; 98:391-409.