Typically, classical logic, or extensions of classical logic, are used in all scientific and mathematical contexts to justify conclusions. By doing this, aren't we implicitly assuming that all other systems of logic that provide contrary conclusions, are "wrong" in some way?
It seems much deeper than a matter of interpretation, because, for example, in classical logic, if the premises contain a contradiction, then any argument containing those premises is valid, and thus, it's possible to "prove" anything true with those premises. Whereas, in paraconsistent logical systems, contradictions don't necessarily imply everything.
So, if a contradiction (i.e. something being both true and false in the same respect at the same time) were discovered in the real world, would that imply every possible statement about the real world is true, or would it mean that there are still false statements, and we should have been using a paraconsistent logical system instead?
More generally, for any valid argument you could make using classical logic, I could imagine an alternative logical system in which that argument is invalid. So, then, how could you say, that an argument is really "valid" or "true" in any meaningful or universal sense? What makes us so confident that classical logic is the "correct" logical system to use?
On top of this, if one were to argue that one logical system was "better" than another, you would have to be using some system of logic to make such an argument, which implicitly assumes that the system of logic you were using was the "correct" one from the onset. How can I even talk about this subject without presupposing a system of logic?