Paradoxes are arguments that produce logical inconsistency. They're a good way of demolishing bad thinking, but some paradoxes remain problematic even when the smartest heads try to resolve them. After 2,500 years philosophers are still trying to figure out how the sentence 'this statement is a lie' can be neither true nor false. Mathematicians puzzle over a number of internally consistent logical puzzles that yet appear to be self-contradictory.
For fans of literature and film, the most entertaining and familiar brain-twister is the grandfather paradox. Imagine I travel back in time and kill my paternal grandfather as a child. Thus my father would never have been born, thus I would never have been born. So I'm not able to travel back in time in the first place to commit the murder. My grandfather lives, fathering my father, who then fathers me, enabling me to travel back in time to kill my grandfather...
Contemporary physics has answers to this paradox, among them the wonderfully-named Novikov self-consistency principle, the Huggins displacement theory, and Brane cosmology. Time-travel films generally attempt to respect these scientific responses and aim at internal consistency, or neglect considerations of consistency altogether.
Looper tries to do both things at once. Altering the past changes the future if it's convenient for the plot. What appears to be a clever, idiosyncratic and original ending, in which the film's hero works out a way of breaking the endless repetition of two destructive life-cycles, actually turns out to undermine the entire story by preventing the possibility of it ever having taken place.
To be fair, Bruce Willis warns us half-way through: 'I don't want to talk about time travel because if we start talking about
it then we're going to be here all day talking about it, making
diagrams with straws.' Fans of the brief-library-visit-as-harbinger-of-trouble scene will also have noticed Willis' impossibly-convenient break-in to an unconvingly-futuristic public library. Unfamiliar with thirty-year-old IT, he still manages to look up private medical records, cross-reference the individuals within with some kind of residency database, and print off giant maps of their present location before escaping undetected into the Kansas night. In the highly-surveilled future it's impossible to do anything without the powers-that-be noticing, unless a convulted way needs to be found of ensuring that the characters are brought together for the final gunfight. Call it the paradox paradox genre - a film that uses paradoxes to simultaneously drive and demolish the plot.