I've sometimes used JN Crossley's What is Mathematical Logic? to good effect for students who needed a lightning introduction. Also, it's short (~90ish pages) and cheap (Dover edition, usually under ten bucks). (Disclaimer: I haven't been in that sort of classroom for decades, though.)
Just reading the first couple chapters is enough, since the rest is determined to get you to Godel's theorem which they usually don't need.
It's not for non-mathematicians especially, but for those who need to be able to read and manipulate first-order classical logic.