Of Spiders and Elephants

As I said, there's a lot more to how hard it is to prove something than just whether it's positive or negative, universal or existential. Sticking with just mathematical statements and a strong notion of proof, consider the proof that earned Kurt Goedel his fame. Goedel proved "there is at least one true statement of arithmetic that is unprovable." Notice this is an existential statement, but it turned out to be very difficult to prove. On the other hand, a universal statement such as "all triangles have interior angles that sum to 180 degrees" is easy enough that we routinely have high school students do the proof.

Let's move to some examples that are easy to understand without knowing a lot of math. Consider the following two statements:
A) There are no elephants in this room.
B) There are no spiders in this room.
Clearly, A is a lot easier to prove than B is. A quick glance around any average size room is enough to prove that there are no elephants in it. However it is much more difficult to prove that there are no spiders in the room. The difference in this case has nothing at all to do with grammatical differences between A and B, it has to do with physical differences between elephants and spiders. Elephants are just the sorts of things that are hard to conceal, whereas spiders are easy to conceal. In many many cases, the grammar of a statement just isn't enough to decide how difficult a proof of the statement might be, you have to take account of the content.

So where does that leave us with respect to proving whether or not there is a god? Well, it means we have to start digging into the concept of god before we know how hard it will be to prove whether or not such a being exists.