Nothing new about the two-envelope paradox today, but I'd like to bring up another one: The Surprise Examination Paradox (aka the Unexpected Egg, Senior Sneak Week, the Suprise Hanging, the Class A Inspection--does anyone have a centralized list of all the variants?)
The basic set-up is as follows: I have a class that meets Monday through Friday. One Friday I tell the students, "There will be a pop quiz next week. The day it is given, five minutes before class, you will not know that the quiz will be that day."
The students say: The quiz can't be Friday. If they haven't had the quiz by Thursday, they'll know the quiz has to be Friday--and then it won't be a surprise, contradicting what I said.
What if I haven't given the quiz before Thursday? Well, Thursday morning, they reason as follows: "The quiz can't be Friday, as above. So it has to be today (that's the only day left)." But that means that, Thursday morning, they know the quiz it'll be Thursday--and that can't happen.
But on Wednesday, they can go through the same reasoning... the quiz can't be on Thursday or Friday, so it has to be today, so we know it'll be today, so it can't be today. And so on back through the week, no matter how many days it is.
Yet on Wednesday, when they get the quiz, it's a surprise. So I was right after all.
I'm going to argue that this is still paradoxical, even if the class only meets once a week.
Suppose I say:
[S] Class, you'll have a test next week. And you won't know that you have the test until you get it.
Seems obviously self-contradictory, right? I just told them that they would have the test.
Except... just because I told them they have the test, does that mean they know it? That assumes that they can gain knowledge by employing this rule:
[M] Accept what Matt tells them.
If Rule [M] yields knowledge, then the students know the truth of [S]:
(1) the students know that they will have a test next week, and
(2) the students know that they don't know that they have a test next week.
By the factivity of knowledge, (2) yields
(3) the students don't know that they will have a test next week
which is a flat contradiction of (1). So we have reduced to absurdity the premise that rule [M] yields knowledge.
(If you're familiar with Timothy Williamson's analysis of the surprise examination, you're looking around for an illicit use of the KK principle--that if you know p, you're in a position to know that you know p. I don't think there is one, but let me know if you find it.)
So, rule [M] does not yield knowledge. So the students have no way of coming to know that they have an examination next week. Which means, that when they get the test next week, they don't know about it until they get it. So my original statement [S] turned out to be true after all.
Williamson uses this paradox to make same deep points about margin-of-error requirements for knowledge. I've used it to make some cheap points about the epistemology of testimony (although not as cheap as in this post). It seems to me that it might be possible to turn my argument into something deeper about the nature of knowledge--in multi-day inspection paradoxes, the students can start the week knowing that what I said was true, and then lose that knowledge before the end of the week, even though it remains true all along.
But I'm not sure that works. As a great man once said, more on this later, maybe.Posted by Matt Weiner at January 28, 2004 11:58 AM