In the (or a) knowledge version of the surprise quiz paradox, we ask whether the student can know there will be a surprise during a certain finite period. Suppose the student knows at the outset (Monday). Suppose he then knows at the end of Thursday that there’s been no quiz. Can we then rule out Friday as a quiz day by deducing that the student knows there’ll be a quiz Friday?

The critical question is what the student knows on Thursday. Does the student know everything he knew on Monday? If just some, what specifically does he know?

The argument is alluring because we have just supposed the student knows various things on Monday. There’s probably some convention about hypotheticals that if you suppose an aspect of the setup it remains supposed. So the argument assumes that at the end of Thursday it remains supposed that the student knows there’ll be a surprise quiz this week.

Supposing the student knows on Thursday there’ll be a surprise quiz this week, and none has happened, it does follow that the student knows there’ll be a quiz on Friday. But it also follows that the student doesn’t know there’ll be a quiz Friday. Since the student knows the quiz will be a surprise, and knowledge that P entails P, it follows that the quiz will be a surprise. So a contradiction follows: the student knows there’ll be a quiz and not-(the student knows there’ll be a quiz).

That’s no skin off the student’s back. He’s trying to show knowledge of a surprise quiz is impossible. Deriving a contradiction from a supposition is a great way to way to show the supposition is false.

But the real lesson is that we’ve granted the student too much. It is a convention of made up situations that if they are made up some way, they remain made up that way. But there’s no requirement that suppositions remain constant. We just can’t add the supposition that the student knows there has been no quiz by Thursday to the supposition that he knows there’ll be a surprise quiz. The student can’t know all of that. The student can’t know [(there’ll be a quiz this week) & there hasn’t been a quiz by Thursday & I don’t know in advance the day the quiz will happen on].

Just b/c it’s possible to know there’ll be a surprise quiz on Monday, it doesn’t follow that it’s also possible to know there’s a surprise quiz etc. on Thursday.

Kripke says you tweak the scenario so that the student knows axiomatically that there’ll be a quiz. In that way he knows some of what he knew Monday, but not all. He knows the important part from the perspective of inferring that he knows there’ll be a quiz on Friday. He knows there’ll be a quiz this week. So then he knows there’ll be a quiz Friday.

Why should we care about this setup? What seemed intriguing to many of us was the possibility of proving lack of knowledge with a setup that more or less tracks life.