Those of you who have had an encounter with logic (logicians,mathematicians,philosophers,just interested) probably know Russels paradox. If not, or if anyway, then the barber version goes as thus
There does not exist a barber who shaves exactly those who do not shave them selves.
To get a grip: say he exists, and then say he does not shave himself. But now he doesn't shave exactly those who does not shave them selves. Now suppose he does shave himself. But now he shaves someone who does shave himself. We can proof this using natural deduction:all ([x] S b x => ~ S x x) premise; [p1] all ([x] ~ S x x => S b x) premise; [p2] S b b \/ ~ S b b by lem ; [lem1] ( S b b assumption; [h1] S b b => ~ S b b by all_e b p1 ; [res2] ~ S b b by imp_e h1 res2 ; [res3] bot by neg_e h1 res3 ) ; [dis1] ( ~ S b b assumption; [h1] ~ S b b => S b b by all_e b p2 ; [res2] S b b by imp_e h1 res2 ; [res3] bot by neg_e res3 h1 ) ; [dis2] bot by dis_e lem1 dis1 dis2 .
Say that $b$ is our barber. We are quantifying over people, that is $x$. The predicate $S(x,y)$ is interpreted as "$x$ shaves $y$". So if $b$ encumber us with his existence, we can conclude absurdities. What a paradox.