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Smullyan and the President’s sanity

Smullyan found himself in a very strange country indeed!

All the inhabitants of this country are completely truthful – they always tell you honestly what they believe, but the trouble is that about half of the population are totally mad, and all their beliefs are wrong!

The other half are totally sane and accurate in their judgments, all their beliefs are correct.

Smullyan felt honoured to be invited by the Presidential couple.

The President was the only one who said anything, and what he said was:

“My wife once said that I believe that she believes I am mad.”

What can be deduced about the President’s sanity, and that of his wife?

Reference: This is a minor adaptation of Problem 4.5 in Raymond Smullyan‘s book Logical Labyrinths.

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