In philosophy, a conviction was a firmly held belief or opinion. When applied to law, a conviction was a verdict handed down by court which deemed a defendant guilty of a crime.
Philosophy[]
In 2368, after Berlinghoff Rasmussen believed that Captain Jean-Luc Picard was faced with a dilemma that questioned his convictions, Picard responded that he'd never been afraid to reevaluate his convictions, but now he had twenty million reasons to do so, as Penthara IV was slowly turning to ice which would result in thousands of deaths if not prevented, but Lieutenant Commander Geordi La Forge's solution to stop it could kill all life on the planet if it were not successful. (TNG: "A Matter Of Time")
Law[]
In Akritirian law, criminal convictions cannot be reversed, "no matter what new evidence is brought to light". While by Federation standards that was considered "an outrageous policy", they firmly believed that "it has proved to be a most effective deterrent."
In 2372, Harry Kim and Tom Paris were placed in an Akritirian maximum security detention facility for the terrorist bombing of the Laktivia recreational facility and the resulting deaths of forty seven off duty patrollers; a fact was later revealed to Captain Kathryn Janeway by Akritirian Ambassador Liria. Janeway assured the ambassador that they had nothing to do with the crime, and after Liria presented the evidence: chemical traces of the trilithium-based explosive were found on their hands and clothing. Janeway argued that that was likely the case of anyone who was found near the explosion, contending that that was "hardly enough evidence for a conviction." (VOY: "The Chute")