Predestination paradox
A predestination paradox, or time loop, was a phenomenon in which an event of time travel could become part of events which had already occurred, and could even lead to the initial event of time travel in the first place.

Examples
In 2268, after having traveled back in time from 2373, Doctor Julian Bashir speculated that the events in which he was participating constituted a predestination paradox when he met  crewmember Lieutenant . Realizing that his great-grandmother's name was , and that nobody ever met his great-grandfather, Bashir began to suspect that he was destined to fall in love with Watley and become his own ancestor. Bashir's suspicions were later proven incorrect, however, in that he did not pursue a relationship with her, a decision which had no effect on the timeline. Agents Dulmur and Lucsly of the Department of Temporal Investigations, who were sent to investigate the incident, professed to hate predestination paradoxes and were relieved to find out that the incident was not one of them. ()

In 1968, the crew of the Enterprise, who had traveled back in time from 2268 to perform historical research, encountered a Human named Gary Seven, who was attempting to sabotage a nuclear warhead platform. Fearing that Seven was an alien from the future attempting to manipulate the timeline, the Enterprise crew attempted to stop him, but Seven was eventually able to convince them that he was indigenous to the 20th century, and that he was in the employ of a benevolent race of aliens attempting to protect Earth. Later, upon checking their historical records, Commander Spock discovered that the crew of the Enterprise was, in fact, always a part of this series of events, and that, by interfering, they were simply fulfilling their role in history. ()

In 1986, needing money after travelling back in time from the year 2286, James Kirk sold a pair of glasses which had been given to him in 2285 by Leonard McCoy as a birthday gift. When Spock expressed his surprise at Kirk's actions, Kirk asserted his belief that they would wind up in his possession again. ()

On several occasions Captain Jean-Luc Picard attempted to prevent a predestination paradox from forming by ordering the  to stay on its current course, fearing that changing course to avoid a time loop may be the very thing that got the ship stuck in the loop in the first place. ()

In another predestination paradox, two time-traveling Vorgon thieves from the 27th century arrived in 2366 to obtain the Tox Uthat from Captain Picard, who was known to have discovered and destroyed the device on his holiday on Risa. Their attempt to obtain the device was, in the end, the reason why Picard destroyed the Uthat, so that it would not fall into the Vorgons' hands, thus completing the paradox. ()

A predestination paradox occurred on Earth in the year 1893, when Enterprise-D crewmember Lieutenant Commander Data, who had traveled back in time from 2368, became trapped in a temporal vortex and had his body separated from his head, which remained in a cavern under the Presidio in San Francisco until 2368. In that year, Data's head was discovered, leading the crew of the Enterprise to mount an investigation. This investigation led to the very time travel event which transported Data back in time in the first place, thereby completing the time loop. ()

In 2371, the devastation of a planet in the Delta Quadrant due to a polaric ion explosion occurred due to a predestination paradox. Kathryn Janeway and Tom Paris were transported back through time by a subspace fracture to the day the explosion occurred. There, a subspace fracture opened and intersected with a polaric conduit and resulted in an explosion. The shock wave from the explosion dissipated into the past, which shattered subspace and created the subspace fractures in the first place. The following day, the  arrived at the planet, where the fractures were floating across its surface like icebergs. After Janeway and Paris disappeared through one of the fractures, the crew of Voyager constructed a device that could open the subspace fractures as a means of rescuing their crew. Ultimately, the crew opened a subspace fracture at the location of the explosion, which lead to the explosion that created the fractures and lead to the deaths of Janeway and the planet's natives. This time loop was broken when Kes experienced a vision of what happened. This allowed Janeway to close the fracture before it intersected with the conduit, resulting in the creation of an alternate timeline where the explosion never occurred and Voyager never visited the planet. ()Captain Braxton, a Starfleet officer from the 29th century, described this type of paradox as "A leads to B leads to C leads to A," when attempting to explain the series of events which led to the destruction of Earth's solar system in his time to the crew of Voyager. Finding debris from Voyager in the aftermath of the temporal explosion that destroyed the solar system, Braxton came to believe that Voyager was somehow responsible for the incident. In an attempt to change history and prevent the explosion, Braxton traveled back in time to 2373 to destroy Voyager. This action caused both ships to be transported back to 20th century Earth, where Braxton's timeship was stolen by an unscrupulous industrialist named Henry Starling. It was at this point that Braxton realized that it would be Starling who would cause the explosion, by taking the timeship to the 29th century without properly recalibrating its temporal matrix. This time loop was eventually broken when the crew of Voyager'' was able to destroy the timeship moments before it traveled to the future. ()

See also
Moebius
Pogo paradox
Temporal causality loop

External links
 

fr:Paradoxe de la prédestination
ja:プレデスティネーション・パラドックス
Time travel
Philosophy