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The Grandfather Paradox is a standard plot complication in time travel fiction. In the classic definition, as described by science fiction writer Rene Barjavel in 1943, a man travels back in time and kills his biological grandfather before the latter met the traveler's grandmother. As a result, one of the traveler'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 means 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. While this is the most famous example, the term is used to describe any situation where a time traveler does something in the past to prevent him traveling to the past in the first place.
The classic scenario is modified slightly in Part I when Marty McFly goes back in time and prevents his parents from marrying. However, Marty's paradox did involve interaction with his grandfather. In the original timeline, George McFly met Lorraine Baines when Lorraine's father hit George with the car when George fell out of the tree. Lorraine nursed him back to health and eventually married him, perhaps due to a Florence Nightingale effect. When Marty sees his future dad about to be hit by a car, he pushes him out of the way and inadvertently takes his dad's place in the story, as Lorraine's father hits Marty with the car and Lorraine becomes infatuated with Marty instead. Marty didn't kill his grandfather, but his grandfather could have killed him. This paradox is what drives the plot of the movie as Marty works to maneuver his parents into falling in love to undo the paradox and save his own existence.
Since Marty is eventually able to reverse the effects of his initial interactions with his parents, we never see what the full effect of the paradox would be in the setting, though it is clear that Marty was in imminent danger of being "erased from existence." It is also clear that changes to the timeline do not take effect immediately but propagate into the future gradually, as evidenced by the slow erasure of Marty's siblings in his photograph of the three of them. This propagation of changes to the timeline is described as a ripple effect.
The Grandfather Paradox comes into play again in Back to the Future Part II when Old Biff Tannen travels back to 1955 to give his younger self a copy of Grays Sports Almanac and instructions to look out for a wild-eyed scientist and a kid who will come asking about the book in the future. In 1985A, Doc Brown has been committed to an insane asylum and presumably never gets the chance to build the time machine in the first place. This paradox is not addressed in the films, though there is some discussion from The Bobs about the possibility of Old Biff causing his own demise in his own past by his actions and being erased from existence. This possibility did not make it into the final cut of the movie.