Re: Fitch's paradox and OWA
Date: Fri, 1 Jan 2010 05:07:41 -0800 (PST)
On 31 dec 2009, 18:47, stevendaryl3..._at_yahoo.com (Daryl McCullough)
> 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.
True, but you have now fundamentally changed the semantics of the K operator in the sense that the model theory now looks very different. You have essentially turned K from a unary operator K(p) to a binary operator K(w,p).
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