Monday, June 4, 2007

Chalmer's Argument of Perfections

Consider whether statements (1) and (2) are true:

(1) If X is true, then I believe X.
If one does not accept this, then one accepts the opposite: one would be willing to state "X is true and I do not believe X." But that is clearly a contradiction for any particular X. Therefore, we must accept that (1) is true for all X.

(2) If I believe X, then X is true.
The negation of this would be saying "I believe X and X is false." Again, this is impossible to state consistently for any X. So statement (2) is also true for all X.

But (1) means I am omniscient, and (2) means I am omnipotent. Cool, I never knew that!


mike said...

There are plenty of true things that I don't believe. There are party games where someone gives a bunch of weird statements and you have to pick which one is true, i.e. which one you find most believable.

"If I know X is true then I believe X."

D said...

The phrase "If X is true" in philosophical language means "Assume X is true" or "Take X as an axiom." So one automatically knows X is true in the context of the argument, that's given. And if you know X is true, then you believe X, by your own admission.

mike said...

Yes, classical logic assumes an omniscient, omnipotent "I". Small wonder that there are those who feel it's not as useful as some other flavors of logic!

If your definition of truth includes the possibility that something may be "true" without you knowing it, then you're working in intuitionistic logic. There, they define truth to be "known truth."

Then the only true things are those that you have a proof for and the law of the excluded middle doesn't hold: if something is not provable, that doesn't imply that it's false.

In this context, omniscient means "I know all proven things," which is hardly surprising.

Statement 2 translates as "If I believe X, then I have a proof for X," which implies omnipotence, as before. Its negation is "It is not the case that (I do not believe X or I have a proof for X)." For example, I could believe X when X is unproven.

D said...

That's a good answer to the paradox! It's always nice when they lead to a deeper understanding.