Re: Fitch's paradox and OWA

From: Daryl McCullough <>
Date: 31 Dec 2009 09:47:07 -0800
Message-ID: <>

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.

Daryl McCullough
Ithaca, NY 
Received on Thu Dec 31 2009 - 18:47:07 CET

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