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Logical Lo: Hanging Around

Peter Cave has a new book just out, of philosophical puzzles old and new. Here, Peter tells the tale of Logical Lo and her reasoning.

Imagine is a wonderful Silly Offer: You’ll win a million pounds for doing something utterly silly. Perhaps you decide to go silly-walking in public, pink bananas on your head, clapping your ears, while singing hymns backwards. You claim the prize. But the provider insists that your behaviour was not utterly silly. It was designed to secure the million that you reasonably enough wanted; that wasn’t silly.

You now decide to do something highly rational. You lock your door on leaving your home. You take the umbrella; it looks like rain. You ask for the prize. After all, it was exceedingly silly to act highly rationally in order to win a prize, won only by silly doing.

The prize provider still refuses to hand over the million. Your action, she says, was not silly, given it should now secure the prize – hence, it won’t secure the prize. You retort that it was thus highly irrational, as it fails (it now transpires) to win the prize; hence it should win the prize. And so on… And so rationality can lead us into endless loopings. Consider the following plight.

Logical Lo knew it was a mistake to holiday in Wild West parts, where cowboys and sheriffs are laws unto themselves – and to visitors. Here sits Lo, sentenced to death by hanging – a strange twist in its noose. The sheriff had taken a dislike to her when she had rebuffed his advances and danced a jig on his ten-gallon hat.

’Tis now Sunday. The sheriff tells Lo that she will be hanged, at noon, one day this coming week, except Saturday. Saturday is siesta day; no one hangs around. The hanging will be a surprise: on the hanging’s morning, Lo will have no good reason to believe that the hanging will take place that day. Perhaps she will undergo the noonday noose on Tuesday: if so, on Tuesday morning she will have no good reason to expect the hanging that day.

Things look bleak; but then Lo starts to reason. After all, she is Logical Lo.

“If I get through to Friday morning, unhung and hence unhanged, then the hanging would have to occur that day at noon. No other days are left. So, for the hanging to surprise, it can’t occur on Friday. Mind you, there remain the other days. Hold on – with Friday ruled out, if I get through unhanged to Thursday morning, the hanging would have to be on that day’s noon. That’s the only day left, given Friday is no possibility; hence, I can rule out Thursday…”

And so Lo reasons, ruling out Friday, Thursday, Wednesday and so on down, Tuesday and Monday being similarly dismissed. She concludes that the sheriff’s sentence cannot be implemented. “My neck is safe,” smirks Lo, ravished by reason.

Then, one day that week, perhapsWednesday, or even Friday, the hangman intrudes upon Lo’s confident complacency and hangs her. She is both hanged and surprised or, more accurately, surprised and hanged.

Surprise hangings, surprise exams, can be given, yet if the possible dates are fixed and surprises guaranteed, has not Lo prove that they cannot occur? Or is there something silly here, as in the Silly Offer?