Unwilling to talk or share information; keeping secrets or refusing to gossip.
From 'close' (tight, shut) plus 'lipped' (relating to lips). It literally suggests lips that are closed tightly, preventing words from escaping.
While 'closelipped' means refusing to share secrets, the phrase 'loose lips sink ships' was a WWII propaganda slogan showing that information control was literally a matter of national security!
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