c. 1300, "engage in argumentation or discussion," from Old French desputer (12c.) "dispute, fight over, contend for, discuss" and directly from Latin disputare "weigh, examine, discuss, argue, explain," from dis- "separately, apart" (see dis-) + putare "to count, consider," originally "to prune, make clean, clear up" (from PIE root *pau- (2) "to cut, strike, stamp").
The Latin word was used in Vulgate in sense of "to argue, contend with words." In English, transitive sense of "argue against, attempt to disprove, deny" is from 1510s. Related: Disputable; disputed; disputing.
c. 1300, "argumentative contention," from dispute (v.). Rare before c. 1600 (disputacioun in that sense is from late 14c.). Meaning "contention, strife, quarrel" is from 1610s.