Smullyan's Egocentric Bird
In a forest of talking birds, calling one bird's name to another makes it call back the name of some bird in reply. A 'mockingbird' always responds to any bird's name by calling back that very bird calling itself. Suppose the whole forest has this property: call any bird's name to any bird, and you'll always eventually land on some bird calling back its own name. Must there be a bird that, called by its own name, answers with exactly itself?
Reveal the answer
Yes, always. The forest's 'always land on a self-answer' property applies to the mockingbird too, so some bird x makes the mockingbird call back x. But the mockingbird's rule is that calling x to it returns 'x calling itself' — so 'x calling itself' equals x, meaning x is fond of itself. Raymond Smullyan built this exact argument into 'To Mock a Mockingbird' (1985) as a playful walk-through of the fixed-point theorem in combinatory logic, the same self-reference trick behind Gödel's incompleteness theorems.
— Raymond Smullyan, To Mock a Mockingbird and Other Logic Puzzles — 1985
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