Tuesday, April 06, 2010

Proving a Negative

I've noted before that much of what passes for principles of critical thinking is actually folklore -- some of it having some real foundation, much of it not. One of my bêtes noires in this regard has been the claim, "You can't prove a negative," which I've repeatedly found in recent times, so I thought I'd consolidate some of my comments on the subject into a single post.

It's difficult to say why this particular claim has such staying power; it has been around for at least two hundred years now, despite the fact that you can quite literally find philosophers and logicians in every generation who point out that it is false. It's also difficult to determine exactly where it came from; one possibility is that it first arose in the context of the British court system. Court systems, of course, typically have conventions defining various kinds of proof requirements for the purposes of protecting the innocent and bringing the guilty to justice, and relative to particular proof requirements it is indeed true that you might not be able to prove a negative. But this is not merely true of negatives; it's necessarily true of affirmatives, as well. And this, in fact, is quite general: you can only get the claim by creating an artificial asymmetry between negative and affirmative claims.

It's easy enough to think through. 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. Suppose someone says that rain clouds are guided by invisible leprechauns, and this is clearly something they believe rather than just made up for some reason. Unless the existence of the invisible leprechaun is suggested by specific 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 we have no clear idea of how the two are supposed to be related.

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. Depending on what kind of possibilities you are considering, however, this can sometimes be prohibitively difficult as a practical matter, 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 also not exclusive to negatives, however; there are affirmative statements that are prohibitively difficult to prove at this level of proof, for exactly the same reason. (The problem of division is closely related to Wilkins's suggestion that it's a problem with lack of caution with regard to universes of discourse.)

We need some good serious study of critical thinking folklore; it's an area of folklore that is very common, but it's overlooked because when people think of folklore they think of savages with feathers and not of themselves. What are the origins of principles of folk-logic like these? What keeps them in currency? There is so much about this area that we just don't fully understand. And when you don't understand what underlies these principles, it's hard to say how to make them extinct when they need to be made so.

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