The first step in sorting out your question is to admit that a body of logic is made up of axioms and that there are multiple interesting potentially "correct" bodies of logic (I am unfamiliar with the jargon logicians use for "body of logic", formal system perhaps).
Next is to understand a theorem called the Principle of Explosion, which asserts that a contradiction in a body of logic implies that any statement is true. You can easily find proofs online for this statement. Such a body of logic, one where any statement is true, might be called the trivial one. It would be "valid" in the sense that it's a body of logic but invalid relative to the body of logic we like most; this last phrase being a loaded one.
So in summary, a paradox cannot exist in a given body of logic unless it is the trivial one. Since humans tend not to believe that every statement is true, we believe that there are no paradoxes in our reality.
Edit: I want to add the disclaimer that the Principle of Explosion might depend on some axiom that doesn't exist in some body of logic and that perhaps there is an interesting logic in which it doesn't hold. I hope someone might comment on this post to let me know.
Re-edit: Looks like there is a concept called paraconsistent logic which rejects the principle of explosion (either by removing the law of excluded middle it seems or some other way). So if you subscribe to a paraconsistent logic then you can have a universe in which paradoxes exist.