A predestination paradox, also called a causality loop, is a paradox of time travel. The idea is that when one travels back in time, one may influence events while in the past, such as causing, or undoing, the original reasons for the journey.
Suppose a man travels back in time and impregnates his great-great grandmother. The grandmother would thus give birth to one of the man's great grandparents, who would then give birth to his grandmother or father, who would then be able to give birth to one of the man's parents, and finally to the man himself. As a result, the man's very existence would be pre-determined by his time traveling adventure, and therein lies the paradox.
The Predestination paradox is very closely related to the Grandfather paradox.
One of the possible examples of this paradox is the death of Edith Keeler: if Doctor McCoy does not travel back in time accidentally and change the timeline by saving her, Captain Kirk does not follow him, does not fall in love with Keeler and does not leave her in the way of that certain truck.
A predestination paradox may exist in Star Trek IV: The Voyage Home when McCoy and Scotty give Dr. Nichols the formula for transparent aluminum. McCoy comments that if they do so, they may alter the future but Scotty asks how do they know he didn't invent it? McCoy seems satisfied with that.
Another paradox in Star Trek IV: The Voyage Home is Kirk's glasses. He hands them to an antiques dealer, and when Spock protests that they were a present from McCoy, Kirk says "They can be again". This means that when Kirk received them, they could have been a lot older than they appeared - although the damage caused by Khan's attacks had been repaired.
In order to stop him making advances on her, Beverly Crusher told Berlinghoff Rasmussen that she could be his 4th great grandmother. (TNG: "A Matter of Time")
On Earth, when evidence of an alien presence in the nineteenth century is discovered, along with Data's 500-year-old severed head, this causes the Enterprise-D to intervene, eventually leading to the original incident where Data's head was severed. (TNG: "Time's Arrow", "Time's Arrow, Part II")
The Temporal Investigations agents Dulmur and Lucsly hated predestination paradoxes. (DS9: "Trials and Tribble-ations")
A predestination paradox exists when Henry Starling reverse-engineers the time ship that crashed on Earth in the 1960s. When USS Voyager travels back in time, Captain Janeway discovers that the computer revolution of the late 20th century should never have actually happened if the time ship had not crashed. (VOY: "Future's End")
References
- TOS: "Tomorrow is Yesterday"
- TOS: "The City on the Edge of Forever"
- TOS: "Assignment: Earth"
- TOS: "All Our Yesterdays"
- TAS: "Yesteryear"
- TNG: "Time's Arrow", "Time's Arrow, Part II"
- DS9: "Trials and Tribble-ations"
- VOY: "Relativity"