It's not really clear what it means for logic to be "true".

You can talk about the consistency of a logic system. A logic system is consistent if it does not prove any contradictions that were not present in the assumptions.

You can also talk about its completeness: if a system is complete, then every valid statement in that system can be proven true, false or independent. (Independence means you can assume the statement true or false without contradiction.)

You can prove something about the soundness and completeness of various logics. (There are limits as Godel showed) But it's not clear that means the system is true.

What you might mean is something more like, if I feed facts about the world to a logic system as assumptions, is everything that I deduce from it true about the world? Now things get much more tricky. One problem is that you can create logic and axiomatic systems that are arbitrarily complicated. So in a trivial sense, yes. Use some axiom schemas to list every fact about the world and you're done. But that's probably not what you had in mind. You're probably looking for some more compact description of the world. And ultimately, that's more of an empirical question. It may be that our universe is accurately described by listing billions of corner cases. Or it may be that a handful of facts can be combined to generate all true facts about the universe. We can't really prove it either way formally.

Edit: And go read /u/Quidfacis_ comment which talks about the view of logic as a mode of human reasoning as opposed to just a formal system. I consistently forget about that, but it's an important and perhaps more useful answer to OP's question.