The barber's paradox

Analysis. If the barber shaves himself ($q$), then by the rule ($p$) he must belong to the set of men who do not shave themselves, i.e. $\bar q$: $(p\land q)\Rightarrow\bar q$. If he does not shave himself ($\bar q$), then by the rule he must shave himself, i.e. $q$: $(p\land\bar q)\Rightarrow q$.

Contradiction. These two implications imply that $p$ is false: no such barber exists.

Conclusion. The premise $p$ is self-contradictory; the barber does not exist.