An argument was a conversation in which individuals gave their opinions in support of or against something. The goal of an argument was to convince an individual of something. Arguments were largely supported by evidence. The relationship between argument and evidence might be counterargued.
Among many known species, Tellarite especially enjoyed a good argument, and they even considered debate a sport on Tellar. (ENT: "Bounty") Tellarites often began an interaction with a series of complaints; this was how they started arguments with someone they had recently met. If they had nothing to complain about, they would simply insult the person. Because of their ability to argue, and hence their skills in debate, Tellarites made excellent politicians. (ENT: "Babel One")
In 2265, the USS Enterprise was probing the galaxy's edge when they discovered the disaster recorder of the SS Valiant. After recovering what could be recovered from the memory banks of the recorder, the only sure fact was that the Valiant was destroyed. Captain James T. Kirk saw this as the "best argument" for continuing the probe, because he reasoned that other vessels might come this way and they would need to know what dangers they might encounter. (TOS: "Where No Man Has Gone Before")