Prove it doesn’t...

Which fallacy describes counter-arguing with the negative of the proposition being posed? As an example, when arguing that there is no evidence or “proof” that a given conspiracy exists, the counter argument always seems to arise “prove it doesn’t exist”, as if proving the negative will invalidate the lack of evidence for the positive. Which fallacy is this?

Person A: Makes a claim.

Person B: Prove it's true.

Person A: No, you prove it's false!

This is Shifting of the Burden of Proof. The onus probandi is on Person A (the proponent), not Person B (the skeptic) to provide evidence or a logical argument. Once this is done, the burden then shifts to the skeptic to refute the evidence or argument.

Where no evidence is given, there is no reason to accept a claim that is not self-evident or at least conceivable, so there's no basis for reversing the burden of proof on Person A's part.

"Person A: x conspiracy theory bla bla

Person B: “There’s no proof that x is true.”

Person A: “well, prove x is not true.”

That would be an Argument from Ignorance Fallacy:

Logical Forms:

X is true because you cannot prove that X is false.

X is false because you cannot prove that X is true.