Grandfather Paradox

Grandfather Paradox was the nominal leader of Faction Paradox.

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The speaker's chair in the Eleven-Day Empire was left vacant to be taken by the Grandfather upon his appearance to the Faction Paradox. (EDA: Interference - Book One) The Faction Paradox held the Grandfather Paradox up as one of the ultimate forms of temporal paradox.

According to Faction Paradox legend, the Grandfather Paradox would have one arm, he removed one of his arms to remove the Time Lord branding/tattoo of a prisoner, and that he would have committed the ultimate paradox.

Resulting from a corrupted timeline, from Faction Paradox corrupted biodata, a future version of the the Doctor modelled himself as the Grandfather Paradox. (EDA: The Ancestor Cell). Although, as The Gallifrey Chronicles suggests, Grandfather Paradox appears as a twisted, morally bankrupt and broken version of whoever is looking at him, and given that the scene is from the Doctor's point-of-view, he (and we as the reader) see him as the Doctor, or "the ghost of Christmas cancelled" as the Doctor snidely puts it. However, if the Doctor was indeed the true Grandfather Paradox, then this version has indeed committed the ultimate paradox; he has cut off his arm not to remove the Time Lord branding/tattoo, but because he knew it was the arm which the Doctor would use to destroy Gallifrey. Before Gallifrey was destroyed he could not possibly have known this and after its destruction he ceased to have existed, so the very absence of his arm is the ultimate paradox that the legends speak of.

According to one account, Grandfather Paradox was a renegade Time Lord who willfully defied the Great Houses, lost his rights as the Head of a House and remade the House into a (or rather the) Faction. Later he wreathed that Faction with symbolism of magic, paradox and "base physicality" (hence the blood masks and the titles of his fellows in the Faction being that of "family members") just to torment the Great Houses and to play against their disgusts and fears. Everything he did was to offend the staid and hide-bound Time Lords. He later wrote himself out of history as a final insult to the Time Lords leaving Faction Paradox as his legacy and a constant thorn in the side of the Great Houses of Gallifrey. (FP: The Book of the War)

During the Second War in Heaven Cousin Justine possessed the shadow of the Grandfather, which made her accountable for all the crimes of the Grandafther to the Time Lords. (FP: The Eleven Day Empire)

Behind the scenes
The name Grandfather Paradox punningly refers to the familiar time travel concept from theoretical science and science fiction.

Isaac Asimov explained it in an episode of the PBS science series Cosmos as such: "If I go back in time and kill my grandfather as a young man, he will never marry, never have my parent, which means I will never be born. However, if I am never born, I will not be able to go back in time to kill him, so he will not die, so he will marry and have my parent, so I WILL be born. So I can go back and kill him, but then I will never be born, and so on."