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In a well known book I am reading about mathematical logic, in the chapter of propositional calculus, he uses induction and contrapositive proof to prove things about propositional calculus. But I am wondering, why is this allowed?

The reason I find this unclear is that contrapositive proof is also formulated inside propositional calculus as $(A\rightarrow B) \leftrightarrow(\neg B\rightarrow \neg A )$. But when he proves things about logic he is not inside the language?

The same way he uses induction. But we also have that induction comes after first-order logic when you create ZFC and create the natural numbers?

So why are we allowed to use these techniques that are inside logic on the language itself? Is not a point of creating logic to formalize these proof techniques?, and hence why does it make sense to use them when they haven yet "been created"?

user394334
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  • This is very closely related; if this answers your question, please click on the "close" link and close your question as a duplicate. – MJD Nov 04 '21 at 19:37
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    Another related discussion. We formalize logic and proof as mathematical objects so that we can reason about them, not as a prerequisite for using them. There's no "building logic from scratch" in the sense you're imagining; we necessarily always start with some assumed agreement on what constitutes valid reasoning. – Karl Nov 04 '21 at 20:28

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