Barber Paradox/Resolution 1

Paradox
There exists a community, one of whose members is a barber.

This barber operated under an unusual rule: his task was to shave every man in the community who did not shave himself, and only those men.

Who shaves the barber?

If he does not shave himself, then he must shave himself.

But if he shaves himself, he must not shave himself.

Resolution
Let $b$ be defined so that $b \notin \mathbb U$.

That is, suppose $b$ is not one of the men of the community.

This could be the case by, for example:
 * $(1): \quad$ The barber is a woman
 * $(2): \quad$ The barber is a boy too young to shave.

Then as $b \notin \mathbb U$, it is not necessarily the case that:
 * $\paren {\neg \map S b} \implies \map B b$

Thus $b$ is allowed not to be shaved, by himself or anyone else.