Paradoxes are a normal, unavoidable part of time travel fiction. Just go back and watch Back to the Future and try counting them all. Trust me. It'll drive you nuts.
The "grandfather paradox" is perhaps the oldest of time travel conundrums: A man travels back in time and kills his grandfather, creating a paradox. If the man's grandfather is killed, then that would prevent the man's birth...which would prevent the grandfather from being killed...which would mean the man would be born and travel back in time to kill his grandfather...which would prevent the man's birth...
You get the idea. The result is what time travel theorists call a "closed timelike curve."
University of Queensland physicist Tim Ralph, drawing on a model proposed by theorist David Deutsch, believes such CTC paradoxes go away when considered in terms of quantum mechanics. (Well, duh.)
The quantum solution goes a like this:
Instead of a human being traversing a CTC to kill her ancestor, imagine that a fundamental particle goes back in time to flip a switch on the particle-generating machine that created it. If the particle flips the switch, the machine emits a particle—the particle—back into the CTC; if the switch isn't flipped, the machine emits nothing. In this scenario there is no a priori deterministic certainty to the particle's emission, only a distribution of probabilities. Deutsch's insight was to postulate self-consistency in the quantum realm, to insist that any particle entering one end of a CTC must emerge at the other end with identical properties. Therefore......blah, blah, blah, one-half probability, blah, blah, blah, causative loop, blah, blah, blah, doppelganger, blah, blah, blah, voila! Paradox solved. Or something like that.
As much as I like to nitpick time travel fiction, there is a point at which it becomes over-analyzed. Trying to tie up every paradox in a neat, little bow would make shows like Doctor Who virtually unwatchable.
So, I think I'll stick with a little bit of mystery, thank you.