Re: Fitch's paradox and OWA

On 3 jan, 18:32, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
>  Jan Hidders says...
>
>
>
>
>
>
>
> >On 3 jan, 14:09, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
> >> Jan Hidders says...
>
> >> >The purpose of (3) was only to illustrate the translation of formulas
> >> >in the original logic to your logic. You are right that by itself it
> >> >does not show the paradox. But if this translation exists then all
> >> >formulas used in the proof of the paradox will have their equivalents
> >> >in your logic.
>
> >> Yes, but my point is that in the more expressive logic, the
> >> knowability principle can be expressed as
>
> >> forall p:P, exists w:W, k(w,p)
>
> >Er, I think you forgot the part where it requires that p is true.
>
> Right. Thanks.
>
> >But if you fix that, then this is indeed equivalent with the one used in
> >the Stanford page. This one will still lead to the conclusion that all
> >truths are known.
>
> No, it doesn't. I already went through this. In the Stanford logic,
> if p is some proposition such that p & ~K(p), then the application
> of the knowability principle gives (in some world w')
>
> K(p & ~K(p))
>
> which is a contradiction. My rule does *not* lead to that
> conclusion. Instead, we have, for some world w,
>
> p & ~k(w,p)
>
> If we apply my version of the knowability principle, we get,
> for some world w'
>
> k(w',(p & ~k(w,p)))
>
> which is *not* a contradiction. Proposition p is known in world w',
> but not in world w.

Hmm. That only shows that in that particular way you don't get a contradiction. But my claim is that you do get a contradiction for the simple reason that your logic contains the old logic.

> >> The original knowability principle, when translated into this
> >> new logic, would look something like this:
>
> >> forall f:W -> P, forall w:W, f(w) -> exists w':W, f(w') & k(w',f(w'))
>
> >> The "propositions" of modal logic are actually functions on worlds.
>
> >Really? This is actually a stronger principle that implies the
> >previous one since as a particular case I can take for f the function
> >that maps each world to the same predicate p in P.
>
> Right. It's *too* strong, which is why it leads to a contradiction
> (together with the principle of non-omniscience).

But it doesn't correspond in any way to the semantics of the knowability principle in the old logic. The model theory there says something very different. So in what sense is this the semantics of the old knowability principle?

> >Also, I don't understand what you mean by "propositions are actually
> >functions on world" except that the same proposition can have a
> >different semantics in different worlds,
>
> That's exactly what I mean. For each modal proposition p
> (which varies from world to world) we can associate a function
> f_p from worlds to nonmodal propositions as follows:
>
> f_p(w) == the nonmodal proposition "p is true in world w"
>
> >and that was already taken into account in the old semantics.
>
> Yes. It's the old *syntax* that was inadequate to express a
> reasonable knowability principle.

But until now you have only shown that in the new syntax you can express an equivalent one (you can verify that by looking at the model theories) and one that's even stronger.

• Jan Hidders