Re: Fitch's paradox and OWA

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.

So what I wanted to say with the above is the following. You are of course right that what (C) really says is:

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

And, assuming that for all f it holds that |- f iff ||- f, this is in fact confirmed by the model theory. However, in the inference process of the paradox as described on the Stanford page the rule is used as if it says f |- []f or |- f->[]f, and that would have the much stronger model-theoretic meaning that I described.

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