Etymology
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Words related to ab-

*apo- 
also *ap-, Proto-Indo-European root meaning "off, away."

It forms all or part of: ab-; abaft; ablaut; aft; after; apanthropy; aperitif; aperture; apo-; apocalypse; apocryphal; Apollyon; apology; apoplexy; apostle; apostrophe; apothecary; apotheosis; awk; awkward; ebb; eftsoons; of; off; offal; overt.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit apa "away from," Avestan apa "away from," Greek apo "from, away from; after; in descent from," Latin ab "away from, from," Gothic af, Old English of "away from."
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of (prep.)

Old English of, unstressed form of æf (prep., adv.) "away, away from," from Proto-Germanic *af (source also of Old Norse af, Old Frisian af, of "of," Dutch af "off, down," German ab "off, from, down"), from PIE root *apo- "off, away."

The primary sense in Old English still was "away," but it shifted in Middle English with use of the word to translate Latin de, ex, and especially Old French de, which had come to be the substitute for the genitive case. "Of shares with another word of the same length, as, the evil glory of being accessory to more crimes against grammar than any other." [Fowler]

Also by 1837 of in print could be a non-standard or dialectal representation of have as pronounced in unstressed positions (could of, must of, etc.)

off (prep., adv.)

by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.

Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.

apo- 
before vowels ap-, word-forming element meaning "of, from, away from; separate, apart from, free from," from Greek apo "from, away from; after; in descent from," in compounds, "asunder, off; finishing, completing; back again," of time, "after," of origin, "sprung from, descended from; because of," from PIE root *apo- "off, away" (source also of Sanskrit apa "away from," Avestan apa "away from," Latin ab "away from, from," Gothic af, Old English of "away from," Modern English of, off).
a- (2)
word-forming element meaning "away," from Latin a "off, of, away from," the usual form of Latin ab before consonants (see ab-). As in avert, avocation. It is also the a in a priori and the à in Thomas à Kempis, Thomas à Becket.
a- (1)

prefix or inseparable particle, a relic of various Germanic and Latin elements.

In words derived from Old English, it commonly represents Old English an "on, in, into" (see on (prep.)), as in alive, above, asleep, aback, abroad, afoot, ashore, ahead, abed, aside, obsolete arank "in rank and file," etc., forming adjectives and adverbs from nouns, with the notion "in, at; engaged in." In this use it is identical to a (2).

It also can represent Middle English of (prep.) "off, from," as in anew, afresh, akin, abreast. Or it can be a reduced form of the Old English past participle prefix ge-, as in aware.

Or it can be the Old English intensive a-, originally ar- (cognate with German er- and probably implying originally "motion away from"), as in abide, arise, awake, ashamed, marking a verb as momentary, a single event. Such words sometimes were refashioned in early modern English as though the prefix were Latin (accursed, allay, affright are examples).

In words from Romanic languages, often it represents reduced forms of Latin ad "to, toward; for" (see ad-), or ab "from, away, off" (see ab-); both of which by about 7c. had been reduced to a in the ancestor of Old French. In a few cases it represents Latin ex.

[I]t naturally happened that all these a- prefixes were at length confusedly lumped together in idea, and the resultant a- looked upon as vaguely intensive, rhetorical, euphonic, or even archaic, and wholly otiose. [OED]
a posteriori 
17c., in reference to reasoning from a consequent to its antecedent, from an effect to its cause; Latin, literally "from what comes after;" from a "off, away from," usual form of ab before consonants (see ab-) + posteriori, neuter ablative of posterius, comparative of posterus "after, subsequent," from post "after" (see post-). Opposed to a priori. In modern use (from c. 1830, based on Kant) roughly equivalent to "from experience."
a priori 
1710, "from cause to effect," a Latin term in logic from c. 1300, in reference to reasoning from antecedent to consequent, based on causes and first principles, literally "from what comes first," from priori, ablative of prior "first" (see prior (adj.)). Opposed to a posteriori. Since c. 1840, based on Kant, used more loosely for "cognitions which, though they may come to us in experience, have their origin in the nature of the mind, and are independent of experience" [Century Dictionary]. Related: Apriorist; apriorism; aprioristic. The a is the usual form of Latin ab "off, of, away from" before consonants (see ab-).
ab initio 
c. 1600, Latin, literally "from the beginning," from ab "from" (see ab-) + ablative of initium "entrance, beginning," which is from or related to the verb inire "to go into, enter upon, begin" (see initial).
ab ovo 
"from the beginning," Latin, literally "from the egg," from ab "from, away from" (see ab-) + ablative of ovum "egg" (see ovum). The expression is said to refer to the Roman custom of beginning the meal with eggs, as also in the expression ab ovo usque ad mala, "from the egg to the apples" (Horace), hence "from the beginning to the end" (compare early 20c. soup to nuts).