Nope, there is no such universal logic. Godel helped show the opposite actually (incompleteness). I think Scott Aaronson's post is very fascinating explanation of this stuff. https://www.scottaaronson.com/papers/bb.pdf

I don't think incompleteness disproves my idea, at least not trivially. Let's take the case of ZF vs ZFC. I would say that ZF can simulate ZFC. To create a simulation we want a function f from statements in ZFC to statements in some subset of ZF so that valid inferences in one correspond to valid inferences in the other.

This is quite simple. f(p) = C implies p does the job quite elegantly.

Interestingly it's harder to do the opposite, to simulate ZF in ZFC, because there is no way to express "forget that you know about C". Such a construction cannot be possible in general because if a contradictory axiom is added, then everything is true, and a theory where everything is true is useless and can't simulate anything. However for C in particular I believe it should be possible to make such a construction but I can't immediately think of how I would do it.