How to set a surprise exam

Abstract

The professor announces a surprise exam for the upcoming week; her clever student purports to demonstrate by reductio that she cannot possibly give such an exam. Diagnosing his puzzling argument reveals a deeper puzzle: Is the student justified in believing the announcement? It would seem so, particularly if the upcoming 'week' is long enough. On the other hand, a plausible principle states that if, at the outset, the student is justified in believing some proposition, then he is also justified in believing that he will continue to be justified in believing that proposition. It follows from this 'confidence' principle that the student is not justified in believing the announcement, regardless of the number of days in the week. I argue that the key to resolving this dilemma is to distinguish the confidence principle from a slightly weaker principle governing the student's justified degrees of belief. Representing these degrees of belief as probabilities, and taking 'justified belief' to mean 'justified degree of belief above a certain threshold', I show that we can uphold the weaker, probabilistic analog to the confidence principle, and maintain that, provided the 'week' is long enough, the student can justifiably believe the announcement. The resulting probabilistic analysis of the story leads to a new diagnosis of the logical flaw in the student's reasoning, and suggests, finally, that even those early stages of it which are logically impeccable exhibit another kind of flaw: circularity.

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