Re: Fitch's paradox and OWA
Date: 31 Dec 2009 09:47:07 -0800
Message-ID: <hhio2r01a8v_at_drn.newsguy.com>
Okay, I've thought about it a little more, and I have come to the conclusion that Fitch's paradox is invalid. Or perhaps the statement of the knowability principle is wrong.
Here's the proof of the contradiction:
- (Knowability principle) For all p: p -> <> K(p)
where <>Phi means "Phi is possibly true" and K(Phi) means "Phi is known".
2. (Non-omniscience principle) For some p: p & ~K(p)
3. Letting p0 be the true but unknown proposition, we have p0 & ~K(p0)
4. From 1&3, we have <>K(p0 & ~K(p0))
At this point, let me switch to possible world semantics: <> Phi means "Phi is true in some world". So let's switch to the world in which K(p0 & ~K(p0)) is true. In that world we have
5. K(p0 & ~K(p0))
From this it follows:
6. K(p0) & K(~K(p0))
But only true things are knowable, so from K(~K(p0)) it follows that ~K(p0). So we have
7. K(p0) & ~K(p0)
which is a contradiction.
The mistake becomes clearer if we explicitly introduce possible worlds. Let's use w ||- Phi to mean "Phi is true in world w" and K_w(Phi) to mean "Phi is known in world w". Let's introduce w0 to mean "our world". Then the proof becomes the following:
- (Knowability principle) for all p: (w0 ||- p) -> exists w, K_w(p)
In other words, if p is true in our world, then there exists another world in which p is knowable.
2. (Non-omniscience principle) for some p: w0 ||- p & ~K_w0(p)
3. Introducing the constant p0 for this unknown proposition, we have: w0 ||- p0 & ~K_w0(p0)
4. From 1&3, we have exists w, K_w(p0 & ~K_w0(p0))
5. Letting w' be a name for some world making the existential true, we have: K_w'(p0 & ~K_w0(p0))
From this it follows:
6. K_w'(p0) & K_w'(~K_w0(p0))
Since only true things are knowable, we have:
7. K_w'(p0) & ~K_w0(p0)
That's no contradiction at all! The proposition p0 is known in one world, w', but not in another world, w0. It only becomes a contradiction when you erase the world suffixes.
-- Daryl McCullough Ithaca, NYReceived on Thu Dec 31 2009 - 18:47:07 CET