I think not the whole of logic, but this particular distinction -- what changes in the world if one is true versus the other? Is there even a difference that is manifest?

It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?

It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.