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consequence (n.)

late 14c., "logical inference, conclusion," from Old French consequence "result" (13c., Modern French conséquence), from Latin consequentia, abstract noun from present-participle stem of consequi "to follow after," from assimilated form of com "with, together" (see con-) + sequi "to follow" (from PIE root *sekw- (1) "to follow").

Meaning "that which follows from or grows out of any act or course" is from c. 1400. Sense of "importance, significance" (1590s) is from notion of being "full of consequences."