c. 1300, "to make reasoned statements to prove or refute a proposition," from Old French arguer "maintain an opinion or view; harry, reproach, accuse, blame" (12c.), ultimately from Latin arguere "make clear, make known, prove, declare, demonstrate" (from  a suffixed form of PIE root *arg- "to shine; white"). The transmission to French might be via arguere in a Medieval Latin sense of "to argue," or from Latin argutare "to prattle, prate," frequentative of arguere.

De Vaan says arguere is probably "a denominative verb 'to make bright, enlighten' to an adj. *argu- 'bright' as continued in argutus and outside Italic." He cites a closely similar formation in Hittite arkuuae- "to make a plea." The meaning "to oppose, dispute, contend in argument" is from late 14c. Related: Argued; arguing.

mid-15c., "presentation of formal arguments," from Old French argumentacion (14c.), from Latin argumentationem (nominative argumentatio) "the bringing forth of a proof," noun of action from past-participle stem of argumentari "adduce proof, draw a conclusion," from argumentum (see argument). The meaning "debate, wrangling, argument back and forth" is from 1530s.

mid-15c., "pertaining to arguments," from Old French argumentatif "able to argue or reason well," or directly from Medieval Latin argumentat-, past-participle stem of argumentari "adduce proof, draw a conclusion," from argumentum (see argument) + -ive. The meaning "fond of arguing" is recorded from 1660s. Related: Argumentatively; argumentativeness.

also counterargument, "argument set forth to oppose or refute another argument," 1812, from counter- + argument. Counter-arguing is attested from 1660s.

also re-argument, "renewed argument," as of a case in court, by 1811; see re- "back, again" + argument.