# Re: Fitch's paradox and OWA

From: Jan Hidders <hidders_at_gmail.com>
Date: Thu, 31 Dec 2009 02:15:38 -0800 (PST)

On 31 dec, 01:07, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
> Jan Hidders says...
>
> >If we reformulate the meaning of (C) in the model theory we get:
>
> >(mC)   If (W,w) |- f then (W,w) |- []f.
>
> >Given the semantics of []f this is equivalent with:
>
> >(mC')   If (W,w) |- f then (W,w') |- f for all w' in W.
>
> I don't think that that is correct. Rule (C) says that
> if p is a *theorem* (that is, p is provable) then it is
> necessarily true (and so is true in all worlds).

My apologies. Everywhere where I wrote (W,w) |- f I actually meant (W,w) ||- f.

(C) if |- f then |- []f

```(1) p & ~Kp   (assumption, for arbitrary variable p)
(2) <>Kp   (from (1) using KP)
(3) []~Kp   (from (1) using (C))
(4) ~<>Kp   (from (3) using (D)
(5) ~(p & ~Kp)   (from (1) and contradicting (2) and (4))
(6) Forall p (~(p & ~Kp))   (forall introduction)
(7) Forall p (p -> Kp)   (propositional reasoning)

```

The error in the reasoning is caused by the omission of |- before each formula. If you add that, it is clear that at step (5) it is concluded erroneously that |- ~(p & ~Kp) but it should have said that "it is not true that |- (p & ~Kp)", which is of course not the same thing.

• Jan Hidders
Received on Thu Dec 31 2009 - 11:15:38 CET

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