Can you use two kinds of logic for a given proof?

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Can you use two kinds of logic for a given proof? Can you use classical and paraconsistent logic at the same time in order to prove a theorem or you can only use one? Why is it allowed or not allowed?

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The theorems you prove using logic or math are things that are true for some kind of domain or, if you want, for some (class of) world.

Thus, for example, using the axioms of Euclidian geometry, you prove theorems that hold for Euclidian worlds. Theorems proven from non-Euclidian axioms would hold for non-Euclidian worlds.

Likewise, different logics pertain to worlds that, logically, behave differently. Any theorems you prove are therefore also implicitly constrained to those kinds of worlds, and the meaning of the theorem should always be understood as such. So no, as soon as you use two different kinds of logic, with different kinds of assumptions built in their inference rules or semantics, then we are really no longer talking about the same result.