The paradox is this: Suppose a man traveled back in time and killed his biological grandfather before the latter met the traveler’s grandmother. As a result, one of the traveller’s parents and by extension, the traveler himself would never have been conceived. This would imply that he could not have traveled back in time after all, which in turn implies the grandfather would still be alive, and the traveler would have been conceived, allowing him to travel back in time and kill his grandfather. Thus each possibility seems to imply its own negation, a type of logical paradox. The grandfather paradox has been used to argue that backwards time travel must be impossible. However, other resolutions have also been advanced.
An equivalent paradox is known in philosophy as autoinfanticide — that is, going back in time and killing oneself as a baby — though when the word was first coined in a paper by Paul Horwich it was in the malformed version autofanticide.