When an ontological paradox comes up in a time-travel story, that's usually my cue to stop paying attention. Not because they make my head hurt — though they do — but because they're usually just goddamn lazy writing.
Guy's walking down the street, hears an obviously crazy homeless guy singing a song. Gets it stuck in his head, goes on to write it down. Song becomes a big hit, guy gets rich. Then, decades later, the guy falls back through time for some reason, loses his mind, ends up singing to himself in an alley as his younger self walks by.
Crap like that makes me nuts. If you're doing it just to be clever, stop it, because it's on the same level as asking whether Jesus could beat Superman in a fight. And if you're basing your whole story on it, give up writing forever and go be a park ranger or something, 'cause you're not cut out for it.
Grumble grumble curse and swear.