Etymology
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just (adv.)

c. 1400, "precisely, exactly;" late 15c., "fittingly, snugly;" c. 1500, "immediately;" from just (adj.) and paralleling the adverbial use of French juste (also compare Dutch juist, German just, from the adjectives).

The original sense of "exactly" in space, time, kind, or degree; "precisely, without interval, deviation, or variation" is preserved in just so"exactly that, in that very way" (1751), just as I thought, etc. But the sense decayed, as it often does in general words for exactness (compare anon, soon), from "exactly, precisely, punctually" to "within a little; with very little but a sufficient difference; nearly; almost exactly;" then by 1660s to "merely, barely, by or within a narrow margin (as in just missed). Hence just now as "a short time ago" (1680s). Also "very lately, within a brief period of time" (18c.). It is also used intensively, "quite" (by 1855).

Just-so story is attested 1902 in Kipling, from just so "exactly that, in that very way."

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just (adj.)

late 14c., "morally upright, righteous  in the eyes of God" ("Now chiefly as a Biblical archaism" - OED); also "equitable, fair, impartial in one's dealings;" also "fitting, proper, conforming to standards or rules;" also "justifiable, reasonable;" from Old French juste "just, righteous; sincere" (12c.) and directly from Latin iustus "upright, righteous, equitable; in accordance with law, lawful; true, proper; perfect, complete" (source also of Spanish and Portuguese justo, Italian giusto), from ius "a right," especially "legal right, law" (see jurist; from Latin ius also come English jury (n.), injury, etc.).

From c. 1400 as "right-minded, good in intention;" from early 15c. as "legal, lawful, right in law." Also "exact, precise; marked or characterized by precision; having correct dimensions" (late 14c.); of narrations, calculations, etc., "accurate, correct" (early 15c.). The sense in music, "harmonically pure, correct, and exact" is by 1850.

The more mundane Latin law-word lex covered specific laws as opposed to the body of laws. The noun meaning "righteous person or persons; Christ" is from late 14c. (The neuter adjective in Latin was used as a noun, iustum, "what is right or just").

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justness (n.)
"quality or fact of being equitable or by right," early 15c., from just (adj.) + -ness.
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unjust (adj.)
late 14c., of persons, "sinful; perpetrating injustice," from un- (1) "not" + just (adj.). Of actions, from c. 1400. Related: Unjustly.
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Justin 
masc. proper name, from Latin Iustinus, literally "just," from iustus (see just (adj.)) + common name-forming element -inus (see -ine (1)). The Justinian Code was a compilation made by Justinian, emperor of the East, in 529.
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de jure 

Latin, literally "of law," thus "legitimate, lawful, by right of law, according to law." Jure is ablative of ius "law" (see de +  just (adj.)).

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justly (adv.)
early 14c., "in an adjacent position, closely" (obsolete except in dialect), from just (adj.) + -ly (2). Meanings "truthfully, honestly" and "equitably, with justice, fairly" are from late 14c. Sense of "justifiably, with good reason, accurately" is from c. 1400; that of "legally, legitimately, rightfully" is early 15c.
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injustice (n.)
late 14c., from Old French injustice "unfairness, injustice" (14c.), from Latin iniustitia "unfairness, injustice," from iniustus "unjust, wrongful, unreasonable, improper, oppressive," from in- "not" (see in- (1)) + iustus "just" (see just (adj.)). Injust (adj.) is attested from late 15c., from French, but unjust is the usual English word.
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justify (v.)
c. 1300, "to administer justice;" late 14c., "to show (something) to be just or right," from Old French justifiier "submit to court proceedings" (12c.), from Late Latin iustificare "act justly toward; make just," from Latin iustificus "dealing justly, righteous," from iustus "just" (see just (adj.)) + combining form of facere "to make, to do" (from PIE root *dhe- "to set, put").

Meaning "declare to be innocent or blameless" is from 1520s. Of circumstances, "to afford justification," from 1630s. Meaning "to make exact" (now largely restricted to typesetting) is from 1550s. Related: Justified; justifier; justifying.
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judicial (adj.)
late 14c., "of or pertaining to a judge; pertaining to the administration of justice," from Latin iudicalis "of or belonging to a court of justice," from iudicium "judgment, decision of a court of justice," also the court itself, from iudex "a judge," a compound of ius "right, law" (see just (adj.)) + root of dicere "to say" (from PIE root *deik- "to show," also "pronounce solemnly"). Related: Judicially.
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