Tuesday, February 12, 2008

A refutation of the knowability principle?

The knowability principle: For all p, if p is true, then it is possible to know p.

But it seems that the following statement is a counterexample.

Ψ: It is impossible to know Ψ.


Claim.
Ψ is true (and, thus, it is impossible to know Ψ).

Proof. Assume (to reach a contradiction) that Ψ is not true. Then it is possible to know Ψ. So, there is a world w such that Ψ is known at w. Since Ψ is known at w, Ψ is true at w. But then it is impossible to know Ψ at w. Hence, (assuming T or stronger) Ψ is not known at w. Contradiction.


[Does this work? It is just a "modalized" version of the Knower. Has this already been done?]


3 comments:

Brian Rabern said...

Well...I guess I just proved that \psi is true, so should know it! Is this a proof or a paradox? Now I'm confused...again.

Aaron Cotnoir said...

Check out Peter Milne's "Omniscient Beings Are Dialetheists" in Analysis. There is a similar (though not the same) paradox.

Leon said...

Check out "an unsolved puzzle about knowledge" by Tymoczko, Phil Quarterly 1984. I think it's very similar. He claims its a paradox.