I know the myth of "you can't prove a negative" circulates throughout the nontheist community, and it is good to dispel myths whenever we can. As it happens, there really isn't such a thing as a "purely" negative statement, because every negative entails a positive, and vice versa. Thus, "there are no crows in this box" entails "this box contains something other than crows" (in the sense that even "no things" is something, e.g. a vacuum). "Something" is here a set restricted only by excluding crows, such that for every set S there is a set Not-S, and vice versa, so every negative entails a positive and vice versa. And to test the negative proposition one merely has to look in the box: since crows being in the box (p) entails that we would see crows when we look in the box (q), if we find q false, we know that p is false. Thus, we have proved a negative. Of course, we could be mistaken about what we saw, or about what a crow is, or things could have changed after we looked, but within the limits of our knowing anything at all, and given a full understanding of what a proposition means and thus entails, we can easily prove a negative in such a case. This is not "proof" in the same sense as a mathematical proof, which establishes that something is inherent in the meaning of something else (and that therefore the conclusion is necessarily true), but it is proof in the scientific sense and in the sense used in law courts and in everyday life. So the example holds because when p entails q, it means that q is included in the very meaning of p. Whenever you assert p, you are also asserting q (and perhaps also r and s and t). In other words, q is nothing more than an element of p. Thus, all else being as we expect, "there are big green Martians in my bathtub" means if you look in your bathtub you will see big green Martians, so not seeing them means the negative of "there are big green Martians in my bathtub."
Negative statements often make claims that are hard to prove because they make predictions about things we are in practice unable to observe in a finite time. For instance, "there are no big green Martians" means "there are no big green Martians in this or any universe," and unlike your bathtub, it is not possible to look in every corner of every universe, thus we cannot completely test this proposition--we can just look around within the limits of our ability and our desire to expend time and resources on looking, and prove that, where we have looked so far, and within the limits of our knowing anything at all, there are no big green Martians. In such a case we have proved a negative, just not the negative of the sweeping proposition in question.
