The unexpected hanging
A judge says a prisoner will be hanged next week on a day that will come as a surprise. The prisoner proves it cannot happen. Then it happens, and he is surprised.
- Reading time
- 4 min
- Difficulty
- Intermediate
- Branch
- Knowledge
- First posed
- Circulated in the 1940s; first published by D. J. O'Connor, Mind, 1948
The puzzle
On a Saturday, a judge tells a prisoner that he will be hanged at noon on one day of the coming week, Monday to Friday, and that the day will be a surprise: he will not know, the night before, that he is to be hanged the next day.
Back in his cell, the prisoner reasons. 'It cannot be Friday. If I am still alive on Thursday night, I will know it must be Friday, so it would not be a surprise. So Friday is ruled out. But then, if I am alive on Wednesday night, I will know it must be Thursday, the last remaining day, so Thursday is not a surprise either.' Working backwards in the same way, he eliminates Wednesday, Tuesday and Monday. 'So I cannot be hanged at all,' he concludes, relieved.
On Wednesday at noon, the executioner knocks on his door. The prisoner is completely surprised. The judge's statement turned out to be true.
The same puzzle is often told about a teacher who announces a surprise exam next week. Students prove it cannot happen; then it does. Where does the prisoner's reasoning go wrong?
Why it matters
The paradox seems to show that a perfectly sensible announcement can be both impossible to fulfil, by careful reasoning, and easy to fulfil in practice. It has generated a large literature on knowledge, belief and self-reference.
It also bears on backward induction, a reasoning method central to game theory. The same structure appears in finitely repeated games, where backward induction recommends cooperating on no round at all, though people often cooperate and do better.
Key terms
- Backward inductionReasoning from the last step of a sequence backwards to the first, ruling out options one at a time. It drives the unexpected hanging paradox and many results in game theory.
- Self-referenceA sentence, set or system referring to itself. Many paradoxes, such as the liar and Russell's, arise from it.
- Epistemic logicThe formal logic of knowledge and belief, which studies principles such as 'if you know that p, then p is true'.
- KnowledgeIn the sense philosophers usually study, knowing that something is the case. At minimum it requires true belief; exactly what else it needs is disputed.
Proposed solutions
The prisoner cannot know the announcement is true
The prisoner's reasoning assumes he knows the judge's statement is true. But that knowledge is exactly what the statement undermines.
The announcement is self-referential
The judge's statement refers to the prisoner's own knowledge of it, which gives it a self-referential structure similar to the liar.
Surprise comes in degrees
The prisoner's reasoning works for Friday, but each earlier step is weaker. By Wednesday, he cannot rationally be confident, so he can be surprised.
Each solution, tested
The prisoner cannot know the announcement is true
- How it resolves the paradox
- To rule out Friday, the prisoner must know, on Thursday night, that he will be hanged that week and that it will be a surprise. But if he still knew that on Thursday night, the hanging could not be a surprise, so the judge's statement would be false. So he cannot be sure on Thursday that the statement is true. Once he cannot rely on it, the backward chain never starts. W. V. O. Quine made an argument of this kind in 1953.
- Why it doesn’t fully work
- If the prisoner cannot know the judge's statement is true, then the judge seems to have announced something that cannot be known even though it turns out true. That is odd, but perhaps not contradictory.
The announcement is self-referential
- How it resolves the paradox
- 'You will not be able to deduce the day from this announcement' is a claim about what can be derived from itself. Like 'this sentence is false', such statements can be impossible to evaluate consistently. Some logicians formalise the paradox and show that the announcement is consistent only if the 'surprise' clause does not refer to deduction from the announcement itself.
- Why it doesn’t fully work
- The ordinary announcement does not feel self-referential. Teachers make surprise-exam announcements all the time, and they are clearly true when the exam comes.
Surprise comes in degrees
- How it resolves the paradox
- Friday really would not be a surprise. But eliminating Thursday requires being certain on Wednesday night that the judge will keep his word about Friday, which is less certain. Each step back reduces confidence further. By Monday, the reasoning has no force. Timothy Williamson argues that limits on knowing that we know explain why the induction fails.
- Why it doesn’t fully work
- This explains why the reasoning is unconvincing in practice, but does not say exactly where a logically valid-looking argument becomes invalid.
Who argued what
- The prisoner cannot know the announcement is true W. V. O. Quine
- The announcement is self-referential Robert Shaw, Frederic Fitch
- Surprise comes in degrees Timothy Williamson, Roy Sorensen
Variations
One day only
The judge says: 'You will be hanged tomorrow, and it will be a surprise.' Now it is clear the announcement cannot be true. With one day, the paradox collapses into a simple contradiction. Adding more days somehow creates room for the surprise. Why?
The repeated game
Two players play a prisoner's dilemma exactly 100 times. On the last round, both should defect, since there is no future to protect. So on round 99, both should defect too, and so on back to round 1. Backward induction says never cooperate. Yet in experiments, people often cooperate for many rounds and earn more. The same reasoning that dooms the prisoner's plan seems to make rational players worse off.
Go deeper
The paradox's mysterious origins
The puzzle was circulating by word of mouth in the early 1940s; one story connects it to a Swedish radio announcement of a surprise civil-defence exercise. It was published in 1948 by D. J. O'Connor and became famous through a 1963 column by Martin Gardner in Scientific American. Despite a vast literature, there is still no consensus solution.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: Epistemic Paradoxes
- WikipediaUnexpected hanging paradox
- BookThe Unexpected Hanging and Other Mathematical Diversions, Martin Gardner
- BookBlindspots, Roy Sorensen
Open questions
- Can something be true and impossible to know at the same time?
- When is it rational to reason backward from the end?
- Why does adding more days make a surprise possible?