Bootstrap paradox

A bootstrap paradox was a sequence of events in which an event was among the causes of another event, which in turn is among the causes of the first-mentioned event. A common example was an individual who grew with a piece of music, then travelled back in time and introduced the song into the past, with the resultant question being: Who composed the song? (TV: Before the Flood)

Examples
The Bad Wolf entity's existence was the result of a bootstrap paradox. Rose Tyler only looked into the heart of the TARDIS, creating the Bad Wolf, after she was motivated by seeing Bad Wolf graffiti, which she created after becoming the Bad Wolf herself. (TV: Rose, Aliens of London, The Parting of the Ways)

Joel Finch inadvertently inspired the pose of God in Michelangelo's Creation of Adam for the Sistine Chapel, after having been sent back to 1511 by a Weeping Angel. Joel studied the works of Michelangelo in the 21st century, and suggested this new pose based on his own recollections of the painting. Joel may have also invented the sandwich. (AUDIO: Fallen Angels)

It was the ArcHivist Hegelia's theory that the Isomorph Cybermen, the last remnant of the Neomorph Cyber-subspecies of 26th century Telos who were stranded in the 1980s after failing to change history by preventing the destruction of Mondas, made contact with the early CyberFaction on the dark side of the Moon. With the support and influence of the Isomorphs' cyber-technology, the CyberFaction would leave the Moon, splitting into seperate subspecies that would ultimately come together and create the first of the Neomorph Cybermen on Telos. (AUDIO: The Ultimate Cybermen)

A bootstrap paradox was created while the Twelfth Doctor and Clara Oswald were at the Drum: the Doctor gained the idea of using a hologram of a ghost version of himself, through Clara who had seen the ghost Doctor and told the Doctor about it. (TV: Before the Flood)

According to one account, the Fifth Doctor's recognition of the phrase wibbly wobbly, timey wimey, (TV: Time Crash) seemingly coined by his tenth incarnation, (TV: Blink) was as a result of hearing River Song say it during an adventure she had with that incarnation. (AUDIO: A Requiem for the Doctor) River herself learned the phrase from the Eleventh Doctor, (TV: The Wedding of River Song, The Angels Take Manhattan) and assured the Fifth Doctor that it would grow on him in time. (AUDIO: A Requiem for the Doctor)

River Song's own name also seemed to be, at least partially, the result of a paradox. Although her name was derived from a Gamma Forests translation of "Melody Pond", (TV: A Good Man Goes to War) River would not go on to assume that name until learning of her future self from the Eleventh Doctor and her parents. (TV: Let's Kill Hitler)

In New York City in 1985, Romana II approached Mrs Oppenheimer-Mason and made up a story about how the two of them had already met two years prior at the reopening of the Lincoln Center, which Oppenheimer-Mason remembered. Soon after, Romana and the Fourth Doctor travelled back to 1983 in order to meet Oppenheimer-Mason at the reopening, creating the backstory Romana had just made up. (COMIC: Time Lady of Means)