Grandfather Paradox

Grandfather Paradox was more or less a theory who could nevertheless be considered as an individual. 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.

Another version of Grandfather Paradox turns up in the Book of the War. Here he is 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.

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.