Tuesday, August 11, 2009

Limits of Logic

In yesterday's Demented Diary entry I posed a riddle and promised the answer today, but the DiaryLand entry page is down, so the answer will have to wait until tomorrow.

Here's the riddle.

Once upon a time a certain king offered to open his castle to any visitor who could provide the doorkeeper with a logically meaningful statement of what he wanted to do in the castle. If the visitor's statement was true, he was tobe granted the privilege of completing his business. If the visitor had lied, he was to be executed.

A certain visitor told the doorkeeper, "I'm coming to the castle to be executed for lying about why I'm here." The doorkeeper let him in.

The executioner was puzzled. If the visitor is not executed, he would have lied so he would need to be executed. But if he is executed, he would have been telling the truth so he can't be executed. So the executioner consulted the royal wizard.

Here was the wizard's advice. Execute the doorkeeper for admitting the visitor who had not provided a logically meaningful statement of his purpose.

Now, as pointed out by a couple of loyal readers of my Demented Diary, a better solution would have been to get the king to take everybody out for a round of margaritas.

I think everyone knows by now of Kurt Godel's Incompleteness Theorem proved in 1931, even though nobody understands it and everybody tries to draw implications that simply aren't there. It states that there's no such thing as a premise set that is both complete (all statements within the bounds of the premises can be proved either true or false) and consistent (no statement within the bounds of the premises can be proved both true and false).

An obvious corollary is that within the bounds of all human knowledge there are truths that can't be proved. Theologians are welcome to make any sort of hay they please out of this. Theology is nothing but hay that's already been used by livestock anyhow. If you can't know something, the best thing to do is quit worrying and go join your buddies for a drink.