Barber Paradox/Resolution 1

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.