The ability to be sensitive and thoughtful when dealing with other people's feelings, especially in difficult situations.
From Latin 'tactus' (sense of touch, from 'tangere' to touch). The metaphorical meaning evolved from the idea of 'touching' someone's feelings carefully and appropriately.
Tact is valued so highly in cultures with high power distance and formality, but in very egalitarian cultures it's sometimes seen as unnecessary—showing that what counts as 'tactful' is actually cultural, not universal.
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