Re: Fitch's paradox and OWA

Jan Hidders says...

>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.

I don't see a rule saying f |- []f. Where did you see that?

I don't think that's a sensible modal logic rule. That is essentially saying that there is no difference between f and []f. (Usually, the accessibility relation on worlds is set up so that []f -> f. So if we add f -> []f, then f and []f are logically equivalent.)

>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))

No, we don't have |- ~Kp. We only have (W,w) ||- ~Kp. So we can't conclude |- []~Kp.

--
Daryl McCullough
Ithaca, NY