Well, no it isn’t philosophically impossible… read on:
It is commonly thought that one cannot prove a negative, but of course I can. If I say “there are no weasels in my right pocket”, all I need to do is enumerate the objects in my right pocket and find a dearth of weasels among them to prove that negative claim. So why do people think one can’t prove a negative?
Negative claims are of the form
Or, in English, “NOT Thereis an x such that x Fs”… oh, OK, it asserts that no x is F.
Now to prove this claim, you need something that logicians call “the Universe of Discourse”, or “the Domain”. That is, the totality of the world or worlds that the claim ranges across. In the pocket example, that is my right pocket. If the domain or universe is small enough, and all the objects in it accessible in a reasonable time, we certainly do think that we can make proof claims. Consider the extinction of the Yellow River dolphin. Pretty well all areas in which that animal can exist are under constant observation by a very large population that has the means to report its existence. So we can safely say that it no longer exists.
So why is it a common claim? This has to do with the development of the medieval logics, and ambiguity (errors in logic are nearly always due to ambiguity in some way or another). The medievals had what they called “the Square of Opposition”. It went like this:
Propositions of the form E are negative claims. But if the universe is not defined, as it wasn’t (the medievals thought that logical possibility ranged across all the universe and possible universes unrestrictedly), one cannot find out if something is false until one encounters an existing contradiction to the claim (an x that is F). Because they had an unrestricted domain, they could never prove that negative if none were ever encountered.
I suspect, though, that it is easier to prove that Obama is not a Muslim or a terrorist than to prove that no gods are green, for example. The domain is smaller, and more manageable.