Past tense of mince; cut into very small pieces, or walked with short, delicate steps.
From Old French 'mincier', possibly from Vulgar Latin 'minutiare' (to make small). The culinary and walking senses developed from the same idea of making small, precise movements.
The phrase 'not to mince words' comes from the delicate, careful chopping motion of mincing - when you don't mince words, you're speaking bluntly rather than cutting your thoughts into polite, small pieces. Minced meat was historically a way to make tough cuts more palatable and digestible.
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