I’d like to see what you think about one (and a half) versions of the surprise examination paradox. I’m not actually sure this case is paradoxical; it’s a variant of one of the variants Williamson considers in Knowledge and Its Limits.
It goes like this:
Mr. Chips is teaching a class that meets Monday through Friday for the whole semester (which is, let’s say, ten weeks long). On the first day of class, he announces, “There will be a test one day during the semester; I’ve scheduled it already, but I won’t tell you when it is yet.”
On the second day of class, he announces, “About that test: It’s not on the last day; and if it is on a certain day, you can’t come to know now that it’s on the next day–unless yesterday you already knew that it wasn’t on the next day. And when I say ‘next day’, I’m counting weekends.”
Mr. Chips is extraordinarily reliable, such that it would be reasonable to take any ordinary assertion of his as conferring knowledge.
The students reason as follows:
Yesterday, we knew only that the exam would be on some weekday.
Today, we know that the exam won’t be on the last Friday, because Mr. Chips told us so.
The exam can’t be on the last Thursday, either. Because if it were on the last Thursday, we wouldn’t have come to know that the exam wasn’t on the next day. But we have come to know that the exam isn’t on the next day (the last Friday); and we didn’t know that yesterday.
Nor can the exam be on the last Wednesday. Because we have come to know (in the previous paragraph) that the exam isn’t on the last Thursday, we didn’t know that yesterday, and according to what Mr. Chips said we couldn’t have come to know that if the exam was on the last Wednesday.
Similary, the exam can’t be on the last Monday or Tuesday.
But the exam might be on the next-to-last Friday. Because we already knew yesterday that the exam wouldn’t be on the next day, since class doesn’t meet Saturday. So what Mr. Chips said doesn’t rule out that the exam isn’t on the last Friday, or any earlier day.
So we know that the exam doesn’t take place in the last week, but it might take place any other time.
My questions: Suppose the exam is in fact scheduled for the next-to-last Wednesday. Do the students come to know that the exam isn’t any day during the last week?
Suppose the exam is in fact scheduled for the last Wednesday. Do the students have the right to be annoyed at Mr. Chips? (They’d have the right to be annoyed at him if he told them a falsehood; but that might not be the only possible grounds for annoyance. I’m deliberately leaving this vague.)