Someone once said, "any egocentric Kestrel must be extremely lonely." Why is this true?
We want to show that a Kestrel that's egocentric must be, by definition, the only bird in the forest. My proof differs slightly from Smullyan's.
From problem 11, we know that if a Kestrel is egocentric, it is hopelessly egocentric:
K(x) === K; // For any bird `x`If we call both birds with any bird y, the left side of the equation will fixate on x:
K(x)(y) === x;But the right side (K(y)) will evaluate to K, because K is hopelessly egocentric. Therefore, for every bird x, x and K are the same bird. Thus, K is the only bird in the forest.
Smullyan's two proofs start from the same place: the notion that a Kestrel that's egocentric is hopelessly egocentric.
In the first proof, we take any two birds x and y, and note that K's response to both is the same (K).
K(x) => K;
K(y) => K;
K(x) === K(y);From problem 16, we know that if K(x) === K(y), then x === y.
The second proof is similar to mine, but doesn't need a second bird y. First, we consider K(x) for every bird x. It's fixated on x (because it's a Kestrel), and it evaluates to K (because it's hopelessly egocentric). Therefore, K is fixated on x (and, still, on K). From problem 17, we know that a bird cannot be fixated on more than one bird—thus, every value of x is the same bird as K, and K is the only bird in the forest.