There are actually two very different temporal anomalies that are referenced by the name Grandfather Paradox, one in which a time traveler kills his own grandfather and so undoes his own existence, and one in which he becomes his own grandfather by impregnating his grandmother to become his own progenitor.  The latter of those was mentioned in our last article as being properly a subset of the Predestination Paradox.

It may seem at first glance that these are the same concept, but they are in significant ways diametrically opposite.  Becoming your own grandfather involves a causal loop, in which a necessary cause of a present reality is dependent on that present reality as its own necessary cause.  That is, if the time traveler does not travel to the past he will never become his father's father, and so he will never be born; but as long as he is born, he can become his own grandfather and so be the cause of his own existence.  Killing your own grandfather is a disruption in which a necessary cause of a present reality is undone, but that that present reality is a necessary cause of its undoing.  This is precisely opposite to the other.  In this case, if the time traveler successfully kills his own grandfather, he undoes the necessary cause of his own existence and ceases to exist; but if he ceases to exist, he cannot travel to the past to kill his grandfather, and so his act of undoing his own existence is undone and he then exists to undo his own existence.

Although the classic examples for which these paradoxes are named specifically involve an individual tampering with his own ancestry, both paradoxes can exist in similar forms but other contexts, including in relation to inanimate objects.  The existence of the T-800 in the original Terminator movie is an example of the machine effectively becoming its own grandfather, as Cyberdyne designs Skynet based on copying technology from the parts crushed in its assembly line.  In Deja Vu, Agent Doug Carlin prevents the murder of the girl whose murder investigation gives him the information he needs to identify the murderer and prevent her murder, effectively "killing his grandfather" by undoing the source of his knowledge.

These paradoxes do not occur at all under the parallel or divergent dimension theories, because in these cases the grandfather you kill or grandmother you impregnate is not actually your own, but is the counterpart to that person in another universe.  In that other universe, it does not matter whether you are ever born, or whether someone else is born instead of you, or no one is born at all when you would have been, because the existence of the person who has taken this action ("you") is completely independent of any event in the universe in which the action occurs.

These paradoxes are usually presented in connection with fixed time stories, in which the latter (in which you become your own grandfather) is treated somewhat inconsistently as a legitimate expression of the predestination paradox, while the former is generally said to be impossible by theorists but still explored in fiction.  Further consideration of these paradoxes and their resolutions under fixed time and under replacement theory, will be the subjects examined in our next two articles.