Universal and Existential

Last time, I talked about positive and negative claims. I argued that neither was inherently harder to prove than the other. But of course I don't mean to claim that all statements are equally hard or easy to prove.

Let's start by thinking about the "strength" of a statement. All I mean by "strength" is how much is being claimed. For example, consider the difference between "There is at least one egg roll on my plate" and "there are at least two eggrolls on my plate." The second statement is stronger than the first. It is stronger because it tells us more about the world. There are more ways to make the first sentence true, than there are to make the second statement true, so merely knowing the first statement is true tells us less about the world than knowing the second statement is true. Of course an exact statement like "There are exactly three eggrolls on my plate" is even stronger - it tells us more about the world.

In general, stronger statements are harder to prove than weaker ones. For example, if you can't see my plate, all I need to do to prove that there is at least one egg roll on it is hold up one egg roll. But to prove that there are at least two egg rolls on it, I have to hold up two egg rolls. It's harder (ok, not much harder) to hold up two egg rolls than one.

Logicians typically deal with existential and universal statements. Existential statements start out something like "there is at least one thing that ....". They make a claim that can be satisfied very easily. Universal claims are about everything that exists, or perhaps about everything of a certain type. So you can surmise that universal statements are generally stronger than existential statements.

Consider, for example, the difference between "All ravens are black" and "at least one raven is black." To prove the existential statement, I just need to produce one black raven. But to prove the universal statement, I have to do more. Not only do I have to somehow check the color of each and every raven, I also have to find some way to show that I haven't missed any. It's not sufficient to simply note that every raven I've seen is black. After all, perhaps almost all ravens are black except for an isolated colony that manages to cling to life in the snowy wilds of Greenland.

Ok, so now we turn to the question of theism. It's easy to see that the theistic claim "there is a god" is existential. All we need to do is show that there is at least one god. It's a bit more difficult to see that the atheistic claim "there is no god" is universal. To do so, we restate it as "Each thing is not god." It turns out that the denial of an existential statement is a universal. Similarly, the denial of a universal statement is existential. To see this general structure, let's go back to ravens.

If I claim that all ravens are black, you can prove me wrong by producing one raven that isn't black. You show I'm wrong by proving an existential statement. On the other hand, if you claim that there is at least one black raven, I can only prove you wrong by checking all of the ravens and showing that none of them are black. To show you're wrong, I have to prove a universal statement.

So on the face of it, it seems that the atheist is at a bit of a disadvantage. There is something inherently more difficult in proving of each and every thing that it is not god, than there is in proving that there is at least one thing that is not god. But the atheist's claim is hard to prove because it's universal, not because it's negative.

So, why does all this matter, aren't we in roughly the same position as we were when we thought that it was impossible to prove a negative? Not quite. I'll go into more detail in the next post, but it turns out that a lot more goes into how hard something is to prove than merely whether it's universal or existential. It turns out that some universal statements are easy to prove, and some existential statements are extremely difficult to prove.