It's often been said that you cannot
prove a negative, and among those negatives would
be an attempt to
prove that something does not exist. So, is such a thing even possible?
In general, there are in fact two conditions where you can successfully prove the
non-existence of something.
First, if you can constrain the possibilities of where something must exist to a point
where actual tests can be done to verify the existence/non-existence of that item, then you
have a possibility. For example, if someone were to claim that an elephant exists on my desk,
then I have a means to verify or disprove that claim. After all, my desk has not only a
finite surface, but a quite small one. It is rather straightforward for me to view
my desk to check for the presence of elephants. I could even cover my desk with a sheet to
be certain that there are no invisible pygmy elephants on there, unable to be seen.
You can also extend this to items that would create effects that could be observed as a means
checking for existence/non-existence. We can verify that there is not a red giant occupying
the same orbit as Jupiter, based simply on the gravitational effects that would be
observed by such a star's existence.
Second is the case where there is a logical proof that clearly demonstrates that something
does not or cannot exist. For example, there is no need to scour the universe to see if
we can find an example of a square circle to show that one cannot exist - the definitions
logically preclude such a thing from existing. Showing that a theoretical entity would
have contradictory is a clear means of proving non-existence. Such a method has been
attempted by some philosophically-minded skeptics/atheists as a means of proving that
certain descriptions of gods cannot exist, though the value in such is questionable.