A judge hands down the strangest sentence in the history of logic. The prisoner, he rules, will be hanged at noon on one weekday of the following week — and the hanging will come as a surprise: the prisoner will not know the day until the executioner knocks. In his cell, the condemned man reasons his way to freedom. The hanging cannot be on Friday, the last possible day, because by Thursday evening, with every other day spent, he would know it was coming — no surprise there. So Friday is out. But that makes Thursday the last possible day, and it falls to the same argument. So does Wednesday, then Tuesday, then Monday. The sentence, he concludes with relief, cannot be carried out at all. On Wednesday at noon the executioner knocks. The prisoner is entirely surprised. The judge's word has been kept to the letter.
That is the unexpected hanging paradox, and its origins lie not in a courtroom but in wartime Sweden. In the mid-1940s Swedish radio announced that a civil-defence exercise would be held the following week, on a day no one would be able to predict. Lennart Ekbom, a mathematics teacher in Stockholm, spotted the logical tangle and put it to his students; the puzzle spread by word of mouth until D. J. O'Connor gave it its first academic outing in the journal Mind in 1948. By 1951 the philosopher Michael Scriven was declaring that "a new and powerful paradox has come to light", and in 1953 the great logician W. V. Quine weighed in. Martin Gardner then carried it to millions in Scientific American, titling a 1969 collection of his columns The Unexpected Hanging. Dozens of papers later, philosophers still do not agree on what exactly goes wrong.
Where the argument leaks
Every link in the prisoner's chain looks solid, yet the conclusion is refuted by a knock on a Wednesday. Quine's diagnosis was that the prisoner smuggles in an assumption he has no right to: that he knows the sentence will be carried out as announced. Later writers point instead at the strange, self-swallowing character of the announcement itself, which is not a plain statement about the world but a statement about what the hearer will be able to deduce — a cousin of the absurdity philosophers call Moore's paradox, the oddity of saying "it is raining, but I don't believe it is". Backward induction, that impeccable tool, is being asked to chew on its own output. There is still no consensus resolution, which is precisely why logicians treasure the puzzle.
Two siblings from the same family
The hanging has company wherever crisp rules of decision collide. Take the two-envelopes problem, which descends from a necktie wager in Maurice Kraitchik's 1943 book Mathematical Recreations. You are handed two envelopes, one containing exactly twice as much money as the other, and you pick one. Before opening it, you are offered a swap. The other envelope, you reason, holds either half or double your amount with equal likelihood — an expected value of one and a quarter times whatever you hold. So you should swap. But the identical argument then tells you to swap back, and so on forever. Somewhere a plausible-looking step is illegitimate, and the debate over which one has filled journals for decades.
Then there is Newcomb's paradox, devised in 1960 by William Newcomb, a physicist at the Lawrence Livermore laboratory, and delivered to philosophy by Robert Nozick in 1969. A near-infallible predictor has placed £1,000,000 in an opaque box if it predicted you will take only that box, and nothing if it predicted you will take both it and a transparent box holding a visible £1,000. Do you take one box or two? One principle of rational choice says two — the money is already in the boxes, so taking both can never leave you poorer. Another says one — the people who take a single box are the ones who walk away rich. Nozick reported the truly maddening feature: almost everyone finds the answer obvious, and they divide into two nearly equal camps. A large 2009 survey of professional philosophers found two-boxers ahead of one-boxers by roughly three to two, with many undecided.
What unites the three puzzles is that none is a mere trick. Each takes principles we cannot do without — eliminate impossible cases, maximise expected value, never leave free money on the table — and arranges them into a machine that contradicts itself. The prisoner hangs, the envelopes stay sealed, the boxes divide the room. Paradoxes are where logic files its bug reports, and philosophy is still working through the queue.
Quiz nuggets
- The unexpected hanging paradox traces back to a 1940s Swedish radio announcement of a civil-defence exercise, noticed by the mathematics teacher Lennart Ekbom.
- D. J. O'Connor gave the paradox its first academic treatment in the journal Mind in 1948, and W. V. Quine analysed it in 1953.
- Martin Gardner's 1969 collection The Unexpected Hanging popularised the puzzle for a mass audience.
- The two-envelopes problem descends from the necktie paradox in Maurice Kraitchik's 1943 book Mathematical Recreations.
- Newcomb's paradox was devised in 1960 by Livermore physicist William Newcomb and published by the philosopher Robert Nozick in 1969.