Tuesday, September 20, 2005

Paradox as a counter-example to logic, part 2

The second problem with using a paradox to disprove the laws of logic is that without the laws of logic, there wouldn’t even be such a thing as a paradox. So contrary to disproving logic, paradoxes actually prove logic. Take the barber paradox for example. The Barber paradox is where there’s this [ship] with a barber, and the barber shaves all and only those who do not shave themselves. But if the barber shaves all and only those who do not shave themselves, then who shaves the barber? If you think about that for a while, you’ll see the paradox. I’ll try to illustrate it, though. Let’s draw a box and say that the box represents everybody in town. We’ll draw a line down the middle of the box so that we divide it into two parts. There are only two kinds of people [on the ship]. There are those who shave themselves, and those who do not shave themselves.

Box A: Those who shave themselves.
Box B: Those who do not shave themselves.

The question now becomes: Which box does the barber go in? The barber shaves everybody in Box B, but he doesn’t shave anybody in Box A. If we put him in Box A, that means he shaves himself. But that can’t be because the Barber only shaves people in Box B, not Box A. But if we put him in Box B, then he does not shave himself. But that can’t be either, because he shaves everybody in Box B. See how we’ve got a paradox now? We’ve got nowhere to put the barber. Does he shave himself or not? Now notice that the only reason we’ve got a paradox on our hands is because we’re assuming the law of excluded-middle. Either people shave themselves, or they don’t. That’s why we’ve only got two boxes. Without the law of excluded-middle, we could have a third box labeled, “Those who both shave themselves and don’t shave themselves,” and then we’d have no paradox. So we can see that the existence of the paradox, far from disproving logic, actually shows us that logic is true. All the barber paradox shows us is that such a scenario could not be instantiated in the real world.

The same is the case with the liar paradox. [i.e. This statement is false.] The reason it’s a paradox is because logic is true. If logic were not true, the statement might make sense. As it is, the statement is meaningless and absurd because it’s neither true nor false. The reason we puzzle over it and can’t make sense out of it is because of logic.

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