What prerequisites / requirements are needed to have "Mathematical Sophistication"?

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Certain courses use this term as a prerequisite, but the term is vague. Should I assume the term generally means having an understanding of logic and proofs?

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Usually this means that you will have developed your skills in not only proof-writing, but also proof-reading$^\dagger$. Moreover, depending on the level of the course, this means that you will be able to fill in omitted bits of reasoning on your own.

$\dagger$ In particular, you should be able to actually extract the reasoning behind the construction of a well-written proof, rather than simply the steps.