1. A cheeky paradox
Epistemic paradoxes have been discussed for twenty five hundred years. Given their hardiness, some of these riddles will be discussed for the next twenty five hundred years.
The surprise exam is a cheeky and confusing paradox which dates to the 1940s. Easy to grasp but maddening to resolve, it is one of the most tantalizing paradoxes around.
A teacher tells her class that she will hold an exam some time next week, between Monday and Friday, on a day the class will be unable to anticipate beforehand. In other words, the class will know the exact day of the exam (e.g., Wednesday)
only when that day arrives.
Upon hearing this, a clever student thinks for a moment, then argues that
no such surprise exam can be held, i.e., that the teacher cannot keep her word!
He starts by pointing out that the teacher obviously can’t hold the exam on
Friday, the last day of the week. For if she did, the class would know by Thursday’s end that it was going to occur on Friday, since no exam would so far have occurred and only one day would be left for the exam to be held. But she said that they would
not know the day of the exam beforehand. So if she wants to keep her word, she can’t hold the exam on Friday.
But now it follows that she can’t hold the exam on
Thursday either. For if she did, the class would know by Wednesday’s end that it was going to occur on Thursday, since no exam would so far have occurred and only two days would be left, with one of them (Friday) already known to be ruled out! Again, they’d know the exam day beforehand. So if she wants to keep her word, she can’t hold the exam on Thursday either.
By the same reasoning, she can’t hold the exam on Wednesday, Tuesday or even Monday, without breaking her word. Apparently, the surprise exam can’t be held at all!
Something is obviously wrong with the student’s reasoning, but what? A precise explanation is not easy to achieve.