Re: Fitch's paradox and OWA

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. 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.

> 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.

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, and that was already taken into account in the old semantics.

• Jan Hidders