Commentaries:
Jamieson, Fausset, and Brown
not by commandment—"not by way of commandment."
but by occasion of the forwardness of others, and etc.—rather, "But by (mention of) the forwardness of others (as an inducement to you), and to prove (literally, 'proving') the sincerity of your love." The Greek is "by means of," not "on account of the forwardness," etc. BENGEL, ELLICOTT, and others translate, "By means of the forwardness of others, proving the sincerity of your love ALSO." The former is the simpler construction in the Greek.
Other Jamieson, Fausset, and Brown entries containing 2 Corinthians 8:8:
2 Corinthians 8:13
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