Tuesday, June 20, 2006

Proving a Negative

Ed Brayton at Dispatches from the Culture Wars:

My guess is that there is no evidence because it is nearly impossible to prove a negative. It's a bit like someone claiming that there is an invisible leprauchan that makes it rain; we can point to all sorts of "naturalistic" theories and the data that support them on how rain is produced, but the question will still remain, "Well, what evidence do you have that the invisible leprachaun doesn't guide the whole process?" And the answer is "none", because there is no hypothetical evidence to prove such a negative - there just isn't any reason to believe it to be the case, and Occam's razor would certainly be germane here.

This has started me thinking about the matter. It's often said that it is nearly impossible to prove a negative, but I think it's a case of a cliché running on without adequate basis. Even if we ratchet up the level of proof required to rigorous demonstration, there is a straightforward way to prove a negative: show that what's being negated and something known to be true imply a contradiction. In reality, we usually don't require the standard of proof to be anywhere near so strict, since we usually allow for defeasible proofs. If you want to prove that there is no ordinary cat on the desk in front of you, look and see whether there is a cat on the desk in front of you. It's barely possible that there's an invisible cat on the desk in front of you, either because of something to do with the cat (like the one in H. G. Wells's Invisible Man) or because of something to do with your eyes. If you want to prove that there is no invisible cat in front of you, feel around and check it out. If someone suggests that there is an invisible, intangible cat on the desk in front of you, you should be able to prove that an invisible, intangible cat implies a contradiction, unless the word 'cat' is being used in an odd way. And so forth.

It is curious that we tend to assume this sort of asymmetry between affirmations and negations.It has been pointed out before that affirmations and negations are convertible -- every affirmation can be stated in an negative way and every negation can be stated in an affirmative way. If you can prove an affirmative claim, you can prove infinitely many negative claims. This, of course, is a purely formal issue; one might think that it's just an artefact of the formal system, i.e., that the formal system fails to model real affirmations and negations on this point. There's some plausibility to that, but even setting aside the formal issue there are problems with the claim that you can't prove a negative. In particular, if you treated affirmations in the way negations are treated by the cliché, it seems you couldn't prove an affirmation, either. If you aren't accepting the testimony of your senses as proof that there is no cat on the desk, why would you accept the testimony of your senses as proof that there is a cat on the desk? If you can't prove that rain isn't caused by an unobservable cause, what is the basis for thinking you can prove that rain is caused by an observable one while using the same standard of proof?

I think one reason for the long life of the cliché is that it gets confused with considerations of irrelevance. Most of the cases that people propose as instances showing the difficulty of proving a negative are actually just cases showing the difficulty of proving something irrelevant to the topic at hand. Brayton's invisible leprechaun is a good example. Unless the existence of the invisible leprechaun is suggested by relevant evidence (either pertaining to the causal processes of rain, or external to but associated with them), there is no way to link it to the phenomenon as relevant one way or another. And if you can't link it to the phenomenon as relevant, you can't (short of showing 'invisible leprechaun' self-contradictory) say what would prove or disprove its involvement in that domain at all. If you can't lay down any conditions of proof for a claim, under any standard of proof short of rigorous demonstration, you can't prove or disprove the claim except by rigorous demonstration. So the problem with proving that invisible leprechauns who guide the rain don't exist is not that the claim is negative; it's that the claim has no straightforward relevance to the actual phenomena.

This sort of issue arises in a lot of skeptical scenarios in philosophy. Someone suggests that I am a brain in a vat, and that what I think is the external world really isn't. The challenge, then, is to prove that I'm not a brain in a vat. I look around me; all my senses show me a world (and a me) that seems to exclude that supposition. If we're accepting this level of proof as the standard, I've proven that I'm not a BIV. If, as is more often the case, the skeptic wishes to say, by an additional supposition, that my senses are deceived and that I am a BIV in a way that cannot be registered by any means of sensory proof or disproof, the relevance of the BIV supposition to what I experience is no longer well-defined. Now it's difficult to prove that I'm not a BIV, because there isn't any clear way in which such a supposition is relevant to anything I experience. This is what I sometimes call the Berkeleyan solution to external world skepticism. By becoming irrelevant to anything, the claim has become harder to refute; it has also become clear that it's not in need of any refutation. To handle a given domain of thought adequately, you don't need to be bothered about things irrelevant to it.

It's also likely that the cliché gains some of its plausibility due to the problem of exhaustive division. How do you know that your inductive process covered all of the possibilities? You can't, unless you can show that it divided the field of possibilities completely. While this is possible, in practical cases it's often prohibitively difficult, because you have to show that it is a contradiction for there to be a possibility you did not cover. This is a high standard of proof we can't usually meet. Thus, it's very difficult to prove that there is nothing you've left out -- some hidden factor that you haven't recognized yet. However, even here we can still often show (and sometimes very easily) that a given candidate cannot be this hidden factor; so we can still prove negatives, although there are negatives that are prohibitively difficult to prove at this level of proof. (This is not exclusive to negatives, it should be pointed out; there are positive statements that are prohibitively difficult to prove at this level of proof.) If the relevance problem for the invisible leprechaun could be fixed (there is excellent reason to think it can't), and if we can't show that our induction is an exhaustive search over the possibilities, and if we can't show that our evidence rules it out, the only reasonable thing to do would be to admit the possibility that the invisible leprechaun is a hidden factor, if there are any hidden factors. The problem here, though, is much like the ones we've already noted. The problem is not that negatives are difficult to prove but that for this particular negative we've ruled out our means of proof (in the above cases arbitrarily, in this case in a principled way).

So the real maxim should be that it is nearly impossible to prove a negative when either you or the facts have made it nearly impossible. As a maxim it's not as catchy, though.

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