Evidence vs The Dayalektik

Intuitionistic proof

Hands up if you're aware that logicians talk about all sorts of different logics. Emphasis on the plural. If you are currently afflicted with an elevated hand, you probably have heard about intuitionistic logic, and can safely skip down a ways. And put your hand down. Unless it's enjoying the view, of course.

For all the rest of you, I'll begin by apologizing (mildly) for the name. I didn't come up with the name, so I can't apologize much for it, but given how nearly opposite its original meaning (and usage in this term) is from any of the current meanings that you may already be puzzling through... well, someone ought to apologize, and I'm not afraid to throw that first stone.

This isn't feminine intuition, it's not mathematical intuition, it doesn't have to do with intuitive people... heck, I'm not even going to explain what kind of intuition it is, because it's not really all that necessary to understanding the concept. I'll just recommend this: whenever I say "intuitionistic", pretend I instead said "evidential" or "monotonic".

If 'monotonic' makes you think of medicine taken by mononucleosis sufferers, just go with 'evidential' instead.

So, how does intuitionistic logic differ from run-of-the-mill everyday logic? In one sense, it all boils down to rejecting the law of the excluded middle. That is to say, it rejects Sherlock Holmes' maxim that "when you have excluded the impossible, whatever remains, however improbable, must be the truth." On the face of it, this might seem a disastrous thing to do (or, at least, I thought it was) but consider this not-quite-accurate-but-illuminating example:

A man is told there may be a pot of gold buried under his farmland. He goes out and digs a hole. If he finds a pot in the hole, then he has established that, yes, there is a pot of gold buried under his farmland. If, on the other hand, the hole has nothing but dirt, he hasn't proven that there's no gold under his land! He could try again, in a different spot. If he finds gold, he's proven that the statement was true, but if he fails, he has not proved it false, but again he could dig another hole. In this manner, any 'true' hole can prove the statement true, but no 'false' hole proves the statement false.

Perhaps farmlands are small enough, in your mind, to be able to dig up the whole place looking for gold, in which case I simply ask that you change 'farmland' to 'nation' and 'gold' to 'WMDs', which should be enough to convince you that there are relevant issues here.

Now, proving something doesn't exist is notoriously more difficult than proving something does exist, but even when it comes to proving that something does exist, intuitionistic logic holds more stringent standards. Here's another example which has recently entertained me:

Someone wants to know if there are any two numbers, x and y, such that both of them are irrational numbers (numbers like Pi, which can't be exactly expressed as the ratio of two whole numbers) but that x^y (x to the power y) is a rational number. Here's a non-intuitionistic proof that this is true: The square root of 2 is an irrational number, so perhaps x=sqrt(2), y=sqrt(2) satisfies the property that x^y is rational, and maybe it doesn't. If it does, well then, the claim is true. If it doesn't, that means that sqrt(2)^sqrt(2) is an irrational number... which isn't so useful, except that this in turn means that perhaps x=sqrt(2), y=sqrt(2)^sqrt(2) satisfies the property. Indeed, the value of x^y turns out to be 2, which is a rational number!

So, either x=sqrt(2), y=sqrt(2) satisfies the property, or (if that's not the case) x=sqrt(2), y=sqrt(2)^sqrt(2) satisfies the property, so the property is true. Any way you cut it, there are two numbers that satisfy the property.

There's a proof that there are such numbers, but... uh... what ARE those numbers? Is it the first pair? Is it the second pair? We've just proved that some pair of numbers exists without nailing down what the numbers actually are, and while that's allowed in everyday logic, it's not kosher by intuitionistic rules. If we want an intuitionistic proof that something exists, it's necessary to actually go out and find an actual specimen, not just show that it's impossible for one to fail to exist.

It's provable (intuitionistically and otherwise) that 'normal' (i.e. classical) logic and intuitionistic logic are equally powerful. What you can do with one, you can do with the other, so it wouldn't make sense to say that one is better than the other in terms of being able to prove more, or failing to prove less, or the suchlike. There are, however, ways in which using a different (and equally capable) system of logic can be handy.

For an example, I turn to C.S. Lewis. Mere Christianity is a fun read, and some parts of it appear tremendously persuasive. Alas, it's an excellent example of what some folks I know call "The Dayalektik" (any resemblance to 'dialectic' that your ears may detect should be ignored, as The Dayalektik is a method for obfuscating truth and making people agree with things by pushing them through a series of subtle false dichotomies, rather the opposite of the meaning of Socratic dialectic, and I'm not sure how it relates to Marxist dialectic.) Lewis sets up a series of arguments where each major step essentially ran "Either A or B or C must be true, A and B are clearly false, so C is true." Then, of course, going on with C assumed to be true, another step was taken, and so forth.

While the method of "List all the cases, eliminate the impossible ones, and say the remaining one is true" is logically impeccable, the method of "List some of the cases, eliminate all but one, and say the remaining one is true" isn't. If you haven't listed all the cases, it might be that you haven't even listed the 'true' case, so you can end up 'proving' that something false is true by elimination. It's a powerful method for clouding minds, evading the truth, and coming to unsound conclusions, and the worst part is that it's very, very, very easy to use this method completely by accident and with the best of intentions. Yes, people who want to cloud the truth have good reason to use this method, but without discipline and training, people who want to reveal the truth can blunder into it even while trying to avoid it.

I raise this example because it's a place where intuitionistic reasoning nicely fails to present a big danger. To show, intuitionistically, that C is true, it's necessary to give evidence that it is true, not simply accumulate evidence that everything else isn't true. False dichotomies aren't a big problem, because knocking out a strawman collection of opposition isn't intuitionistically enough to support a pet view... there actually has to be something substantial supporting it before you can claim victory.

Now, to those who raised their hand way back at the beginning, I am aware that there's a certain amount of proof-by-eliminating-alternatives that's still intuitionistically acceptable, but I still assert that it requires enough work to prevent the false dichotomy from being a pitfall that's excessively easy for the untrained to stumble into. And if your hand is still up at this point, you might consider giving it a rest.

So, the take-home of today is that if you're worried you're dealing with a false dichotomy, consider taking a hiatus from the law of the excluded middle and demand positive proof of an assertion, not just negative evidence against its negation.