Just curious, can you think of a non-contrived, moderately difficult, non-mathematical problem that has or even in principle could be settled through abscence of evidence? Especially a positive assertion. Closest I could come up with is say confirming a diagnosis by elimination, through abscence of evidence of other candidate diagnoses. But even then there's a non-zero possibility that it's a hitherto unknown disease, or that there was some false negative somewhere.