Notes from the underground
barang dot sg
Updated 2 December 2023
The surprise exam

1. A cheeky paradox
2. What’s the argument?
3. A better reading
4. Origin of the paradox
5. The birthday party
6. The one-day case
7. The two-day case
8. Faulty logic?
9. The heap
10. Sorensen’s objection
11. References

Other Sites

Paradox Lost (1971), by Ian Stewart. (archived copy)
7. The two-day case

In the case with only two school days, the teacher announces:
There’ll be an exam either tomorrow or the day after but you won’t know which day until it arrives.
We saw previously that the one-day announcement is a blindspot for the class, but what of the two-day announcement displayed above? Can the class accept these words to be true?

At first sight, it seems that if they do, then they must expect the exam to occur on the first day. For, clearly, if it were to occur on the second, then, having accepted that there would be an exam, they’d know about this by the end of the first day, in violation of the teacher’s word.

But now that they expect the exam on the first day, this violates her word as well!

So it seems that the class cannot sensibly accept the two-day announcement. As soon as they do, it will strike them as being false, whereupon they must abandon it at once. In other words, the two-day announcement defeats itself and seems to be a blindspot for the class no less than the one-day one.

This looks like the start of an unwelcome slippery slope for similar reasoning would presumably show that the three-day announcement is a blindspot for the class as well, and so forth, eventually vindicating our student. His target is the five-day announcement, but the essential features of his reasoning are already found in the two-day case above.

Just to be clear, the reasoning displayed above is a paraphrase of what would otherwise be the student’s mocking style. Thus, faced with the two-day announcement, he would have mocked it like this instead:
Ah, but what you say cannot be true! Obviously, the exam can’t be on the second day, for if it were, we’d know about it by the end of the first. Therefore, it must be on the first day, but that can’t be, because we know it now!
Taken at face value, this is to accuse the teacher’s words of being false, but, for reasons already explained, this isn’t his true intent. His (artful) point rather is that her words are a blindspot for the class. The paraphrase above is supposed to make this plain and I will henceforth ignore the mocking style of argument.

As we shall see, a residual mocking tone may still be found in the paraphrased reasoning but we can clean this up at the right time.

The question now is whether the student is right to claim that the two-day announcement is a blindspot for the class no less than the one-day one. Unsurprisingly, some people think that he is wrong and that the reasoning above contains a clear error. Let’s see what it’s supposed to be.