The student's logic seems good. But we know from experience that something's just not right. Do we accept what we know from intuition to be true or do we believe in the fairy tale the student conjured for us ?
"At some point during this class, there will be a surprise quiz. You will not know when the quiz is to be administered until I give it." Note that he never says that a student wouldn't know what day the quiz would happen on...this will be important later.
Let's play devil's advocate for a moment:
- We know that when the student reaches the final day of the class, she knows that she will be quizzed that day, so the quiz won't surprise her.
- We also know that when the student reaches the final term1 of the class, she knows that she will be quizzed that term, so the quiz won't surprise her.
That doesn't make much sense. Even though she knew what term the quiz would occur in, she considered the actual day (a smaller amount of time than a term) to be a surprise before she used her "logic."
In the same respect, if the student reaches the last day, she still will not know what time that day the quiz is to be administered. She might be lucky and have a few minutes to study hard -OR- she might have to take the quiz right away. But, she thinks those details are trivial; the exact time is not important enough for her to be surprised by it.
It all comes down to when, in a certain situation, surprise "disappears". If you wouldn't be surprised during the last day, simply because you know what day the quiz will happen on, you might still be surprised during a different day, if the quiz happened then instead; it just depends on your "surprise horizon." That horizon is a subjective, incredibly malleable, psychological concept that can change depending on the situation. But, it cannot be extended infinitely with logical trickery, as the student did.
If your surprise horizon is knowing what one day a quiz will happen on, it cannot be extended to two days, then three days, then four days, and so on until all days of the term are covered. You decide what surprises you, not some (il)logical formula.
You're no longer trapped in the web of deceit spun by the student. Congratulations.
But wait, there's more
Then again, you could look at it this way: surprise happens when you find out the answer. So, in the scenario above, you would be surprised that the quiz didn't happen on the day before last (you might have been in suspense, expecting it at any moment, and been surprised when it didn't happen). So, the surprise about the timing of the test is still there.
A good, related quote: Paradoxes don't exist in the real world, because things are rarely cut-and-dry into boolean logic. –robbway in the Reconciling Logical Paradoxes node.
1semester or year or whatever the term length is