It was quite a few years ago – last century in fact – that through Martin Gardner’s ‘Mathematical Recreations’ column in Scientific American that I first learnt of Raymond Smullyan. It was in a review of either The Lady or the Tiger or What is the Name of this Book, two of Smullyan’s books of logic puzzles.
One of Smullyan’s creations is the island of Knights and Knaves, where the knights always tell the truth and knaves always lie. OK, so that’s just an adaptation of the well-known island with two tribes – a tribe of compulsive truth-tellers and another of compulsive liars – and the western (English-speaking) anthropologist who’s confronted with the task of deciding whether a native is from one tribe or another. Smullyan’s contribution to that genre of logic puzzle was to expand it by adding new categories and orthogonal complications: ‘Normals’ who tell the truth about 50% of the time, as it suits either their own interests or those of the puzzlist, and the sane and the insane.
Sane knaves know the facts and lie about them, insane knaves compulsively lie about their delusional view of the facts; sane knights compulsively tell the truth about the facts that they know while insane knights compulsively tell the truth about their delusions.
It should be pretty obvious why I consider Smullyan a logician for our time. But enough of that – let’s move on to the tribute poser. A little logic puzzle of my own making – one that’s by no means guaranteed to be hole-free but nonetheless might provide a little opportunity for some recreational thinking.
One day, when you’re feeling just a little tired and worn out by the mental effort of maintaining your sanity in a polity that values pleasing lies over unpleasant truths you read, in the travel supplement of your weekend paper of the fabulous archipelago where you might find the isles of Knights, and Knaves the Sane and the Insane. You decide (comfortable and relaxed, or merely complacent, as you are about your integrity and grip on reality) that the best place in the world to live would be the Isle of the Sane Knights where nobody is delusional and nobody is a liar.
So you sell up all your goods and chattels, your house if you have the good fortune to own one and invest the proceeds in an ocean going yacht in which you can (once you have mastered the rudiments of sailing, which ain’t that hard, and the rudiments of GPS navigation which are even easier) sail to this best of all places in the world.
After many days at sea you arrive within sight and hail of the coast of what you think might be the land you are looking for – the Isle of Sane Knights. As you approach you see three figures silhouetted in the moonlight on the pale sands of a beach.
“Ahoy,” you cry, “I am a weary traveller, come from the lands of bullshit and delusion in search of a land of honest, right thinking people, where I can find respite from the confused and the crazy.”
“Welcome pilgrim,” responds the first of the beachcombers, “Come forward! Here you have found what you seek – a land where all have a clear sighted view of facts and are honest in their dealings!”
“Bullshit!” cries the second, “There’s no-one here who isn’t a habitual liar!”
“Don’t listen to them!” cries the third, “They’re both mad.”
Now here’s the poser (in two and a bit parts):
- Which, of all the possible islands in the fabled archipelago of Knights and Knaves, the Sane and the Insane have you arrived at?
- Would you be willing to live there? Why?
‘Oh bugger’ moment: This problem doesn’t work out in its present form unless you assume that the two populations on the island are mutually exclusive with respect to sanity and knavery.