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You look at some more advanced logic books. Many of them have an introductory chapter summarizing set theory, which turns out to be a prerequisite. You need logic to understand set theory, but it seems you also need set theory to understand logic! These books jump right into proving rather advanced theorems about logic, without offering the faintest clue about where the logic came from [that allows them to prove these theorems].

Norman D. Megill, David A. Wheeler. Metamath: A Computer Language for Mathematical Proofs. Lulu Press, North Carolina, 2019. Available on the Web as http://us.metamath.org/downloads/metamath.pdf.

Is the clause "where the logic came from" the antecedent of the bracketed that-clause above? I'm not even sure if this is grammatically possible. I think the antecedent is "the logic."

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    Not quite: the relative clause "that allows them to prove these theorems" is postposed. The antecedent is "logic".
    – BillJ
    Oct 3, 2022 at 7:30

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'That' refers to the logic.

If it had been written this way it might have been easier for you to recognise:

...without offering the faintest clue about where the logic that allows them to prove these theorems came from.

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