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.

On Proving Negatives

So where does an atheist start to talk about god? We could start with the classical arguments, but those have already gotten a lot of press. I do have some things to say about them, but I think I'd rather start elsewhere.

How about this? One of the comments I often hear when the subject of atheism comes up in a conversation is that atheism is an untenable position because, as everyone knows, "you can't prove a negative". Atheism's central claim is, of course a negative - "There is no god." So the argument is that atheism cannot be proven. This puts it at a disadvantage to both theism and agnosticism. Simple agnosticism makes no claim about god's existence. As theism's central claim is positive - god exists - it is at least possible that it might be proved. Thus the theist can offer positive arguments for his position while the atheist is reduced to merely finding fault with the theist's arguments rather than offering positive proof.

Or so the argument goes. The most glaring problem with this argument is that it rests on a false premise. It turns out that it is possible to prove negatives. Not only that, it's not even especially difficult. Let me start out by noting an oddity in the premise - if it were true, it would be unprovable! That's right the premise, "You can't prove a negative" is itself a negative statement and hence unprovable by it's own lights. We have to be a bit careful here. The fact that the premise is unproveable if true, or false if provable, does not show that the premise is false. Goedel made a lot of his fame in mathematics on the basis of just such a sentence - one that was unprovable if true. But it is a weird to use a negative premise as the central claim in criticizing atheism for having a negative as its central claim. The the extent that the argument works against atheism, it undercuts itself.

Luckily there is a more direct way to show the argument doesn't work. We simply note that proofs of negatives are readily available. For this we can turn to that most proof-laden of disciplines, mathematics. Here's an old one: "there is no greatest prime number." This is Euclid's theorem which can also be stated as "there are infinitely many primes." The proof is pretty straightforward. Or how about "there is no triangle whose internal angles sum to more than 180 degrees." It turns out that we routinely rquire high school students and undergraduates to prove negative mathematical statements. Ok, what about statements outside of math. Here's an easy one "There are no elephants in this room." It seems pretty easy to prove that, just have a quick look around. Elephants in rooms are the sort of thing that are easily discoverable with a quick visual inspection. If you look earnestly for an elephant in your room and don't find one, that proves there isn't one.

The bigger issue here is that statements don't divide neatly into negative and positive one except on purely grammatical grounds. Any claim can be made by using either a positive or a negative statement. Sometimes one is more natural, but there is always someway to state the claim positively and some way to state it negatively. We saw this with Euclid's theorem. The formulation I initially gave was negative, but the second one is positive. It's not hard to show that the two statements are equivalent. Try it. I'll try to show you the answer in a later post. So how would we state the atheist's central claim positively? Well, one way to do it is to divide things into to groups, the divine and the mundane. All gods are in the divine group, everything else is in the mundane group. Now atheism amounts to the claim that everything that exists is mundane. Theism can be expressed as the negation of atheism, i.e. as the claim that not everything is mundane. There are other ways too, but they often depend on the specific concept of god being employed, and we haven't gotten that far yet.

There, I think that's enough for now. In my next posts I want to talk about universal and existential claims, and also about the notion of proof.