# Re: Fitch's paradox and OWA

Date: Fri, 1 Jan 2010 05:07:41 -0800 (PST)

Message-ID: <1110713c-d1cb-4d72-ade0-26ff82301b58_at_m26g2000yqb.googlegroups.com>

On 31 dec 2009, 18:47, stevendaryl3..._at_yahoo.com (Daryl McCullough)
wrote:

*> 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:
**>
**> 1. (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:
**>
**> 1. (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.
*

If you assume the model theory that I presented earlier the inferences can be verified to be in fact all correct (with apologies for copying your words):

Let W be the set of all possible worlds, and w0 the element of W that is our world in W. Since W will be fixed I will omit it in (W, w) ||- Phi and simply write w ||- Phi.

- (Knowability principle) for all p : (w0 ||- p) -> exists w in W, w ||- K(p)
- (Non-omniscience principle) For some p: w0 ||- p & ~K(p)
- Introducing the constant p0 for this unknown proposition, we have: w0 ||- p0 & ~K(p0)
- From 1&3, we have: Exists w in W, w ||- K(p0 & ~K(p0))
- Letting w' be a name for some world making the existential true, we have: w' ||- K(p0 & ~K(p0))

From this by principle (A) it follows:

6. w' ||- K(p0) and w' ||- K(~K(p0))

Since only true things are knowable by principle (B), we have:

7. w' ||- K(p0) and w' ||- ~K(p0)

From the semantics of K in the model theory it then follows that:

8. K(p0) in w' and K(p0) not in w'

Which is indeed a contradiction.

- Jan Hidders