Etymology
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approve (v.)
c. 1300, apreven, approven, "to demonstrate, prove," from Old French aprover (Modern French approuver) "approve, agree to," from Latin approbare "to assent to as good, regard as good," from ad "to" (see ad-) + probare "to try, test something (to find if it is good)," from probus "honest, genuine" (see prove).

The meaning was extended late 14c. to "regard or assent to (something) as good or superior; commend; sanction, endorse, confirm formally," especially in reference to the actions of authorities, parliaments, etc. Related: Approved; approving.
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certify (v.)

mid-14c., "to declare the truth of," also "to vouch for or confirm" (an official record, etc.), from Old French certefiier "make certain, witness the truth of" (12c.), from Late Latin certificare "to certify, to make certain," from Latin certus "fixed, sure" (see certain) + root of facere "to make, do" (from PIE root *dhe- "to set, put"). Also used in Middle English in broader senses of "inform, give notice to; instruct, to direct; to designate." Related: Certified; certifying.

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grant (v.)
in early use also graunt, early 13c., "to allow, permit (something); consent to (a prayer, request, etc.)," from Old French graanter, variant of creanter "assure, promise, guarantee, swear; confirm, authorize, approve (of)," from Latin credentem (nominative credens), present participle of credere "to believe, to trust" (see credo). From c. 1300 as "transfer possession of in any formal way." Meaning "admit to be true, acknowledge" in English is from c. 1300; hence to take (something) for granted "regard as not requiring proof" (1610s). The irregular change of -c- to -g- in Old French is perhaps from influence of garantir. Related: Granted; granting.
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amen (interj.)
Old English, from Late Latin amen, from Ecclesiastical Greek amen, from Hebrew amen "truth," used adverbially as an expression of agreement (as in Deuteronomy xxvii.26, I Kings i.36), from Semitic root a-m-n "to be trustworthy, confirm, support."

Compare similar use of Modern English certainly, absolutely. Used in Old English only at the end of Gospels, otherwise translated as Soðlic! or Swa hit ys, or Sy! As an expression of concurrence after prayers, it is recorded from early 13c.
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affirmation (n.)

early 15c., "assertion that something is true," from Old French afermacion "confirmation" (14c.), from Latin affirmationem (nominative affirmatio) "an affirmation, solid assurance," noun of action from past-participle stem of affirmare "to make steady; strengthen; confirm," from ad "to" (see ad-) + firmare "strengthen, make firm," from firmus "strong" (see firm (adj.)). In law, as the word for the conscientious objector alternative to oath-taking (Quakers, Moravians, etc.), it is attested from 1690s; if false, it incurs the same penalty as perjury.

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authorize (v.)

late 14c., auctorisen, autorisen, "give formal approval or sanction to," also "confirm as authentic or true; regard (a book) as correct or trustworthy," from Old French autoriser, auctoriser "authorize, give authority to" (12c.) and directly from Medieval Latin auctorizare, from auctor (see author (n.)). Meaning "give authority or legal power to" is from mid-15c. Modern spelling from late 16c. Related: Authorized; authorizing. Authorized Version as a popular name for the 1611 ("King James") English Bible is by 1811.

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trim (v.)
mid-15c., probably from Old English trymian, trymman "strengthen, fortify, confirm; comfort; incite; set in order, arrange, prepare, make ready; become strong," from trum "strong, stable," from Proto-Germanic *trum-, from PIE *dru-mo-, suffixed form of root *deru- "be firm, solid, steadfast." Examples in Middle English are wanting.

Original sense is preserved in nautical phrase in fighting trim (see trim (n.)); where the verb meant "distribute the load of a ship so she floats on an even keel" (1570s). Meaning "make neat by cutting" is first recorded 1520s; that of "decorate, adorn" is from 1540s. Sense of "reduce" is attested from 1966.
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affirmative (adj.)
"answering 'yes,' " mid-15c., from use in logic; from Old French affirmatif, earlier afirmatif (13c.), from Latin affirmativus, from affirmat-, past participle stem of affirmare "to make steady; strengthen; confirm," from ad "to" (see ad-) + firmare "strengthen, make firm," from firmus "strong" (see firm (adj.)).

As a noun from early 15c., "that which affirms or asserts." American English affirmative action "positive or corrective effort by employers to prevent discrimination in hiring or promotion" is attested from 1935 with regard to labor unions (reinstatement of fired members, etc.). The specific racial sense is attested from 1961; by late 1970s the sense had shifted toward pro-active methods such as hiring quotas. Related: Affirmatively.
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plausible (adj.)

1540s, "acceptable, agreeable; deserving applause or approval" (senses now obsolete), from Latin plausibilis "praiseworthy, pleasing, acceptable," from plaus-, past-participle stem of plaudere "to applaud" (see plaudit). Meaning "having the appearance of truth, apparently right, seemingly worthy of acceptance or approval" is recorded from 1560s; especially "having a specious or superficial appearance of trustworthiness." Related: Plausibly.

The expression plausible deniability emerged during the Watergate scandal (1973) but is said to be from CIA jargon in the 1950s (Allen Dulles sometimes is credited with the first public use); the thing itself is older: "the situation that allows senior officials or powers to deny responsibility for discreditable actions by others in their hierarchy because no one can confirm the deniers knew of the actions."

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*re- 

*rē-, Proto-Indo-European root meaning "to reason, count;" a variant of PIE root *ar-, also arə-, "to fit together." 

It forms all or part of: Alfred; arraign; arithmetic; Conrad; dread; Eldred; Ethelred; hatred; hundred; kindred; logarithm; Ralph; rate (n.) "estimated value or worth;" rathskeller; ratify; ratio; ration; read; reason; rede; rhyme; riddle (n.1) "word-game;" rite; ritual.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit radh- "to succeed, accomplish;" Greek arithmos "number, amount;" Latin reri "to consider, confirm, ratify," ritus "rite, religious custom;" Old Church Slavonic raditi "to take thought, attend to;" Old Irish im-radim "to deliberate, consider;" Old English rædan "to advise, counsel, persuade; read;" Old English, Old High German rim "number;" Old Irish rim "number," dorimu "I count."

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