Friday, April 01, 2005

You Can't Prove a Negative

Actually, the above statement is false. Let me explain.

Oftentimes, in the course of debate, we find ourselves in the awkward position of claiming that a certain assertion is false. Are aliens visiting Earth? Is there a Santa Claus? Or, the ever popular, were there weapons of mass destruction present in Iraq immediately preceding the American-led invasion? I don't know about you, but I want to answer "no" to each of these questions. The problem arises when your adversary responds by saying, "Oh yeah? Prove it!"

I generally try to avoid this predicament by refusing to provide an outright denial. Instead, I try to shift the burden of proof to the individual making the claim. "Is there a Santa Claus? Well, I can't say for certain, but I don't see any affirmative evidence demonstrating his existence. Come back when you have some and we'll talk."

But, some people just won't let up. For them, this expression of healthy skepticism comes across as weasel-speak. Or, it is used to claim that the positive and negative position are equivalent; neither proved nor disproved and thus equal. This can be extremely frustrating, especially in situations where there have been extensive investigations that have failed to prove the assertion. For example, for years, children across the globe have attempted to catch a glimpse of Santa Claus, often going to extraordinary measures to achieve discovery. To my knowledge, no child has yet succeeded. So, when the adversary claims that neither the "Santa Claus exists" nor the "Santa Claus does not exist" proposition has been proved and, therefore, each position has equal merit, I tend to get a little annoyed.

It is at this point that I tend to lose it and scream (or type in all-caps), "You idiot! You can't prove a negative! Therefore, the burden is yours!" Then, I tend to feel a little better.

However, as I thought about this some more, I realize that this isn't exactly true.

There are lots of negatives that can be proved. I am not a woman. Babe Ruth never played for the Atlanta Falcons. There is no beached orca decaying in my living room. You are not reading this article in today's edition of the New York Times. I could go on. These are all negatives and they are all easily disproved. Therefore, I've been a little disingenuous on this point from time to time.

That said, I'm not forced to endure the triumphalism of fools. I just need to clarify what I actually mean.

The problem isn't specifically with the negativity of the assertion I am making. It actually has to do with the size of the domain in which such a proof would have to occur. If the domain is large enough, many positive assertions are also impossible to demonstrate definitively. Jim Lippard offers us a clear explanation of the problem.
This is really the idea behind the claim that "you can't prove a negative"--that we don't have the resources or ability to exhaustively enumerate all examples over the entire universe. But notice that this is an issue whether the proposition is positive or negative, and that all positive propositions have equivalent negative propositions, and vice versa. Also notice that, if the scope of the domain is sufficiently small, proof can be quite easy.

Let's take our domain to be swans, and look at the property of being purple. Which of these statements is supposed to be impossible to prove? (1) All swans are purple. (2) Not all swans are purple. (3) There is a purple swan. (4) There is no purple swan. Now, statements (1) and (2) are easy to disprove and prove, respectively--each requires only a single non-purple swan to demonstrate. The former is a universal positive statement, the latter is a negated universal. Statement (3), a positive existential statement, requires only a single purple swan to prove, but takes a lot of enumeration to disprove. Statement (4), a negative existential, is a statement of the type that is supposed to be impossible to prove, and it clearly requires the most work to demonstrate, but only a single purple swan to disprove. (This statement is equivalent to the universal negative, "All swans are not-purple.")

All that lies behind the more precise statement, "you can't prove a universal negative (or negative existential) statement" is that the most straight-forward, direct manner of proof--exhaustive enumeration--is not always available due to practical limitations.
Therefore, to all of you out there that I have admonished with the "you can't prove a negative" reprimand, I apologize. I was wrong, and I take credit for this failing. I should've said, “It is impossible to enumerate through all possibilities within the domain in a reasonable amount of time -- and thus, the burden of proof is yours!" It doesn't have quite the same zing, but at least it's accurate.

And, when you get down to it, it doesn't change a thing. So, for everyone out there who believes in aliens visiting Earth, Santa Claus, or Baathist WMD programs -- the burden is still yours. Step up if you can. Otherwise, don't let the door hit your ass on the way out.
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