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. 

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). Meaning "debate, wrangling, argument back and forth" is from 1530s.

1540s, refutacion, "act of disproving; overthrowing of an argument" (by countervailing argument or proof), from French réfutation (16c.) and directly from Latin refutationem (nominative refutatio) "disproof of a claim or argument," noun of action from past-participle stem of refutare ""drive back; rebut, disprove" (see refute).

c. 1600, "dispute (something), oppose by argument" (a sense now obsolete); 1610s, "to make the subject of verbal contention, debate, discuss; contend against (someone) in argument," probably a back-formation from controversy. Related: Controverted; controverting; controvertible.

"the opposite point" (in an argument), 1590s, from counter- + point (n.1). As a verb from 1940s.