Grandfather Paradox

Grandfather Paradox was the mythical founder and nominal leader of Faction Paradox. Even if a historical Grandfather "existed," he voluntarily stopped ever having existed, and was never under any obligation to be true to the legends about him. (PROSE: The Book of the War)

Biography
The "voodoo priest of House Lungbarrow" founded House Paradox 400 years before the War. Although any member of an Oldblood House was technically allowed to found a new House, the general consensus was that no more Great Houses were needed. When the Grandfather in quick succession founded House Paradox, began talking of alter-time structures, recruited members of other Houses and established the Eleven-Day Empire, the sluggish Homeworld was simply too shocked to react. They only got around to dealing with the Grandfather after the Imperator crisis forced them to start taking care of their errant members. Knowing that executing the Grandfather would only serve the purposes of his death imagery-obsessed House, the Great Houses simply locked him up on their prison planet. (PROSE: House Paradox)

241 years later, when the Carnival Queen threatened rationality throughout the universe, Lady President Romana released three hundred prisoners, including the Grandfather, during an epileptic fit. (PROSE: Christmas on a Rational Planet, The Book of the War) He promptly cut off his arm, removing the Houses' prison tattoo in the symbolic Act of Severance. (PROSE: Act of Severance, Interference - Book One) He redefined House Paradox as "Faction" Paradox, no longer protected by the Protocols of the Great Houses and recruiting from the lesser species instead of just Homeworlders. He then "retired" from history, erasing himself from it altogether. (PROSE: House Paradox)

The Speaker's Seat in the Eleven-Day Empire's Parliament was always left empty to symbolise the Grandfather-who-never-was. (PROSE: House Paradox)

Attack on Gallifrey
A one-armed person claiming to be the Grandfather led an invasion of Gallifrey at the start of the War. The Doctor initially believed that the man was actually a version of himself from the future, made into a surrogate Grandfather by the Faction (PROSE: The Ancestor Cell) but later came to believe that "Grandfather Paradox" appeared to everyone as their most twisted, cynical, even evil future self — the "Ghost of Christmas Cancelled." (PROSE: The Gallifrey Chronicles)

This Grandfather appeared to the Doctor as a bald man with a stump in place of his left arm, wearing black armour and robes, like the Doctor "if he'd spent twenty years in the navy before becoming a psycho." (PROSE: The Gallifrey Chronicles)

Other legends
The Doctor had memories of being a child with parents. In his memories, his mother read him a legend from a storybook. And in the legend, a young man (who also had parents) travelled back in time to before his parents were born (they had parents too) and murdered his grandfather. The man then became the Grandfather Paradox figure, with many more legends associated. He lost an arm — nobody could agree which, or how — and he was alive and dead, murderer and victim, all in one. (PROSE: The Gallifrey Chronicles)

Cousin Mila used a ritual to summon the spirits on Mercy, the most corporeal of which appeared as a man missing one of his arms — which arm was impossible to make out. (PROSE: Holding Pattern)

Cat
Gramps was a cat living in a retirement home. He had a dignified manner and appearance, with fur in patterns of dark brown and black that seemed to move and blur impossibly in the shadows, streaks of silver-grey around his face, and no front right leg. He led a group of other cats in a fight against a younger cat — apparently Gramps' past self — in which the younger cat lost a leg. (PROSE: Gramps)

Shadow
Supposedly, the knife which the Grandfather used to cut off his arm was held in the Stacks in the Eleven-Day Empire. The traces of blood it contained were thought to be the only physical remnant of the Grandfather. (PROSE: Act of Severance) More importantly, the knife held the shadow of the Grandfather. Godfather Morlock had it attached to the shadowless Cousin Justine, which gave her access to an infinite arsenal of every weapon imaginable, all contained in her shadow. Shortly afterwards, the Eleven-Day Empire was destroyed and Justine became the Faction's last scion, very literally bearing the Grandfather's spirit or quintessence. (AUDIO: The Eleven Day Empire, The Shadow Play) Of course, she turned out to not be quite as unique as she thought — Godfather Morlock had previously tried the exact same scheme with Cousin Shuncucker, who also bore the Grandfather's shadow and thought herself "last scion" after the destruction of the Faction's earlier homeworld. Morlock had been forced to abandon the project when it became obvious that Shuncucker was completely insane. Although it could also have been the shadow (directly or indirectly) which caused the insanity. (AUDIO: Movers, A Labyrinth of Histories)

Behind the scenes

 * The name Grandfather Paradox punningly refers to the familiar time travel concept from theoretical science and science fiction.
 * Lawrence Miles intended Grandfather Paradox to never actually appear, similar to the Doctor's true name or Judge Dredd's face.