A judge tells a prisoner he will be hanged at noon on one weekday next week and that the day will be a surprise, meaning on the preceding evening he will not know it is coming. The prisoner argues by backward induction that no such day exists. Where does the argument break, and what happens?

A judge tells a prisoner he will be hanged at noon on one weekday next week and that the day will be a surprise, meaning on the preceding evening he will not know it is coming. The prisoner argues by backward induction that no such day exists. Where does the argument break, and what happens?

Approach: Write the judge's statement as a conjunction of a fact and a claim about what the prisoner can derive, then check whether each step of the elimination keeps that statement available as a premise.

The backward induction is invalid because it uses a premise its own conclusion destroys, and a hanging on Monday through Thursday is a genuine surprise. The prisoner rules out Friday, since surviving Thursday evening would reveal it. He then rules out Thursday only by assuming that on Thursday evening he still believes the judge, while the very argument he is running concludes the judge cannot be right. The statement is self referential: it asserts a fact together with the prisoner's inability to derive that fact, and no consistent reasoner can both accept it and prove the fact from it. Once the prisoner concludes no hanging is possible, a Wednesday hanging is unforeseen and the judge's claim is met. The elimination proves nothing about the calendar, only that the sentence cannot be used as a premise by the person it describes.

Follow-up: If the judge instead promises the hanging is a surprise only in the weak sense that the prisoner assigns it probability below 1/2, which days remain available?

Key concepts: backward induction, self reference, consistency.