Present participle of 'to axe'; removing or eliminating something, or using an axe to chop.
From Old English 'æx' (axe), combining with present participle '-ing' suffix. The figurative sense (to eliminate) arose in 20th-century English from cutting things away.
When someone says 'they're axing that project,' they're using a perfectly logical metaphor—just like chopping down a tree removes it permanently, cutting a project kills it completely.
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