Conceptual Analysis Problem

Here is a puzzle that requires some "conceptual analysis". You really need to think about the meaning of the words involved and the way this guy Eugene is described!! Good luck!!

RICK: Let me tell you about my hometown, Tattoopolis, and my favorite tattoo artist there, Eugene. As of today, everyone in my hometown has exactly one tattoo (though they could have more in the future). That of course makes it an interesting town. But Eugene is even more interesting! Eugene tattoos all and only those people in the town who do not give tattooes to themselves! That's right: he tattoos all people in the town who don't give tattooes to themselves, AND he only tattoos people in the town who don't give tattooes to themselves.

SLICK: Wow, that's quite a story there, Rick. Too bad it's false. I know for sure that there's no such person as Eugene.

Slick is right. There is no such person as Eugene. Slick knows this even though he has never been to Tattoopolis nor has he ever talked to anyone from Tattoopolis (other than Rick). As a matter of fact, Slick doesn't need to know anything else about Tattoopolis or who lives there to know that there is no such person as Eugene.
So how does he know that Eugene does not exist? (Hint: It has to do with the concept that supposedly applies to Eugene).