## Tuesday, August 09, 2005

### Logical theory versus illogical logicians

There's this guy named Russell who put forward what he thought was a logical paradox:

See http://www.iep.utm.edu/p/par-russ.htm

or

http://www.cut-the-knot.org/selfreference/russell.shtml

One variant of it is called the "Barber's Paradox" and I use it for laypeople as it might be easier to understand than starting with the full abstraction:

Consider a small village where all men are shaved, the barber is a man, and the barber shaves all and only those men who do not shave themselves.

But who shaves the barber?

The answer is that it's impossible for the barber to shave himself and not shave himself, which is what those conditions would require.

That is, the supposed paradox is with assertions that would require that the barber shave himself and not shave himself, which is a direct contradiction.

More abstractly, consider a set made up only of sets that exclude themselves that includes all such sets, but stop, before you run around with that idea, as it reduces to: consider a set that includes and excludes itself.

So why is this a big deal in logic?

Because, I guess, people a while back didn't deal much with exceptions.

But we're modern, well-educated people so let's toss in exceptions:

Consider a small village where all men are shaved, the barber is a man, and the barber shaves all and only those men who do not shave themselves, except himself, as the barber shaves himself

Easy.

So let's handle the abstraction: consider a set that includes all and only sets that exclude themselves, except itself.

And just like that I've handled a big enough issue in logical theory that you can find quite a bit on it in books and on the web, and look at them to see that they do not provide the correct solution.

What? Yup. Look at them and they do not provide the correct solution.

Why don't they?

Best guess is that we have the benefit of something modern: computers.