From Vine's New Testament Dictionary:
Noun: elegmos (G1650), "a reproof" (akin to B), is found in the best texts in 2Ti_3:16 (some mss. have elenchos, which denotes "a proof, proving, test," as in Heb_11:1, "proving," RV marg., "test"). Cf. elenxis, "rebuke," 2Pe_2:16 (lit., "had rebuke").
Verb. elencho (G1651), "to convict, rebuke, reprove," is translated "to reprove" in Luk_3:19; Joh_3:20, RV marg., "convicted"; the real meaning here is "exposed" (KJV marg., "discovered"); Eph_5:11, Eph_5:13, where "to expose" is again the significance; in Joh_16:8, KJV, "will reprove" (RV, "will convict"); in 1Co_14:24, RV, "reproved" (KJV, "convinced"); in the following the RV has "to reprove," for KJV, "to rebuke," 1Ti_5:20; Tit_2:15; Heb_12:5; Rev_3:19; for synonymous words see CONVICT and REBUKE.
It means to verbally correct someone who is doing something wrong thereby bringing the wrongdoing to light, rebuking the person for wrong doing, and encouraging them to turn from their wrongdoing and do what is right.
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