Logic can prove mathematical truths, and with evidence, physical truths.So let me get this straight; logic is not sufficient to prove the existence of a thing, but but yet logic is sufficient to rule out the existence of a thing?
Logic can also rule out some claims.
Nope.The possessor of that logic is doomed to suffer because survival favours the positive, not those that rule things out; because it is possibility that things are and will be, that supports survival.
You're referring to a primitive survival strategy. If you assign agency to an unfamiliar sensation or other phenomenon, you will survive longer than if you don't. Hearing a leaf crunch causes one animal to run, but not the other, and the runner lives whenever that sound was caused by a potential predator.
But we can do better than that, can't we?