Unexpected Hanging Paradox - The Logical School

The Logical School

Formulation of the judge's announcement into formal logic is made difficult by the vague meaning of the word "surprise". An attempt at formulation might be:

• The prisoner will be hanged next week and the date (of the hanging) will not be deducible in advance from the assumption that the hanging will occur during the week (A).

Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week. However, in order to reproduce the next stage of the argument, which eliminates the penultimate day of the week, the prisoner must argue that his ability to deduce, from statement (A), that the hanging will not occur on the last day, implies that a last-day hanging would not be surprising. But since the meaning of "surprising" has been restricted to not deducible from the assumption that the hanging will occur during the week instead of not deducible from statement (A), the argument is blocked.

This suggests that a better formulation would in fact be:

• The prisoner will be hanged next week and its date will not be deducible in advance using this statement as an axiom (B).

Some authors have claimed that the self-referential nature of this statement is the source of the paradox. Fitch has shown that this statement can still be expressed in formal logic. Using an equivalent form of the paradox which reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all circumstances, it is in this case because the statement is self-contradictory.