Pedant’s Paradox: Gallows Humour

Pedant’s Paradox

About the series: Examining logic problems and paradoxes and dismantling them, because I am just that picky. Feel free to debate my answers. (Yes I am aware most of these have mathematical answers, but they’re dressed  in real-world examples so they can be looked at with real world logic).

The Problem: A man is told that he will be hung one day next week, but that he will not know until the day. He works out that he can’t be hung on Friday as he would know after noon on Thursday that only Friday 12:00pm remained. Likewise on Wednesday, he can rule out Thursday because he can rule out Friday, and so on through the week, so he can rule out being hung.

At this point the logic paradox stops, but the version I originally heard included a caveat missing here:

On Monday at 12:00 pm he is very surprised. Why?

The Answer: If you stop with the proof by induction, then he is correct: he cannot be hung by surprise. If you take the logic one step further however, the prisoner has just stated that he cannot be hung without knowing the day before and therefore cannot be hung. So, since he thinks he can’t be hung, whenever he is hung it will be a surprise.

The problem is that it assumes the prisoner can rule out each day in succession because he has ruled it out before – so on Thursday afternoon, he would know he was to be hung on Friday, on Wednesday afternoon it would have to be Thursday (because otherwise he would know on Thursday that it would be Friday) and so on.

He can be hung on:

He thinks: It is actually:
Monday t+1 t+4
Tuesday t+1 t+3
Wednesday t+1 t+2
Thursday t+1 t+1
Friday t t

On any day but Friday and Thursday there are multiple options so he cannot know for sure that he will be hung on the following day.

There is also the alternative, where he can be hung, but insists he cannot be hung by surprise. Again, he cannot know for sure which day he will be hung on until Thursday (because until Thursday pm he is not in possession of the certain knowledge that he has not been hung previously).

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