The “Hangman” puzzle, in its “Unexpected Examination” version, involves a Teacher announcing her students that the exam’s date (known only to be sometimes next week) will be a surprise: even in the evening before the exam, the students will still not be sure that the exam is tomorrow. Intuitively, one can prove (by backward induction, starting with Friday) that, if this announcement is true, then the exam cannot take place in any day of the week. So, using this argument, the students come to “know” that the announcement is false: the exam cannot be a surprise. Given this, they dismiss the announcement, and then, whenever the exam will come (say, on Tuesday) it will indeed be a complete surprise!
Jelle Gerbrandy proposed a nice solution to the puzzle, using the logic of public announcements. His answer is based on interpreting the expression “it will be a surprise” as: “before the Teacher’s announcement, it was the case that (if the Teacher didn’t make the announcement, then) the exam’s date would have been a surprise”. Gerbrandy interprets the announcement itself as an “update” (conveying “hard” information) with the above sentence. In this reading, the students’ correct conclusion should be only that (if the Teacher tells the truth, then) the exam won’t take place on Friday; but none of the previous days can be excluded.
I propose a different solution, which in my view better captures the intended meaning of Teacher’s announcement, namely: “The exam’s date will be a surprise, even after I’m telling you this!” Assuming that students trust their Teacher, I show that the belief revision induced by this self-referential announcement cannot be an “update”, but only an “upgrade” with “soft” information; i.e. the teacher is a trusted, but not infallible, source of information, and thus the students believe (though they do NOT know) that her announcements are true UNLESS they are already known to be false. I prove that in fact there exists only one possible doxastic transformation that models this announcement. Its effect is to make the content of the announcement known to be false, although the announcement itself (as a belief-revising action) is still “efficacious” (and thus it cannot be dismissed).
I show this first by using a combination of Temporal Logic and Conditional Doxastic Logic. But then I go back to (the belief-revision-friendly version of) Dynamic Epistemic Logic, and show that the effect of the announcement is equivalent to the students’ revising their beliefs with the sentence “surprise”, then revising again with the same sentence, then again … ad infinitum. This “iterated upgrade” perspective relates the Surprise Exam puzzle to the theory of fixed points of doxastic upgrades. Finally, I also give a Probabilistic analysis of the puzzle, using (infinite iterations of) Jeffrey’s Rule.
All the three proposed approaches come essentially to the same conclusion: if the students start by trusting the Teacher, then after the announcement they should come to regard as more plausible (or more probable) that the exam will take place in any particular day than that it will take place in the next day. As a result, every evening the students will believe the exam is tomorrow. So, whenever the exam comes, it won’t be a surprise! Thus, the Teacher’s announcement comes to be known to be false, but this doesn’t entitle the students to dismiss it. On the contrary, the only way for them to “prove” the teacher wrong is to be prepared for the exam at any given moment! I find it very pleasant that this solution agrees with (what should be every) Teacher’s true intentions: what more can she expect to achieve with this announcement?.