On this island, could anyone honestly say 'I believe that I am insane'?
Raymond Smullyan's island has two kinds of people: the sane, who believe only true things, and the insane, who believe only false things, and everyone always says exactly what they believe. Could a sane person say 'I believe that I am insane'? Could an insane person say it?
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Neither one ever could. A sane person believes only true things, so believing 'I am insane' would have to be true, meaning they aren't sane after all — a contradiction. An insane person believes only false things, so believing 'I am insane' would have to be false, meaning they're actually sane — also a contradiction. It's one of Raymond Smullyan's cleanest self-referential puzzles, built from his 'sane vs insane' twist on knights and knaves.
— Raymond Smullyan, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles — 1978
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