# Re: Fitch's paradox and OWA

Date: Thu, 31 Dec 2009 02:15:38 -0800 (PST)

Message-ID: <d02a389d-777e-42e5-992d-7fe4989d77b3_at_22g2000yqr.googlegroups.com>

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.

Their reasoning can be simplified to this:

(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