Origin and history of irrefutable

1510s, "refuse, reject" someone or something, a sense now obsolete, from French réfuter (16c.) and directly from Latin refutare "to drive back; rebut, disprove; to repress, repel, resist, oppose," from re- "back" (see re-) + *futare "to beat" (from PIE root *bhau- "to strike").

The meaning "prove (someone) wrong, prove (someone) to be in error, disprove and overthrow by argument or countervailing proof" is from 1540s; of statements, opinions, etc., by 1590s. Many have frowned on the subtle shift in meaning towards "to deny," which occurred as the word came to be used in connection with allegation. Related: Refuted; refuting.

For people who still use the word in its older sense it is rather shocking to hear on the B.B.C., which has a reputation for political impartiality, a news-report that Politician A has refuted the arguments of Politician B. [Charles L. Barber, "Linguistic Change in Present-day English," 1964]

word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.