Re: Fitch's paradox and OWA
Date: 2 Jan 2010 13:52:44 -0800
Message-ID: <hhof7c010cg_at_drn.newsguy.com>
Jan Hidders says...
>But I'm afraid I don't think that will work. The reason is that in
Yes, but with the correct axiomatization of knowability
predicate, the corresponding proposition will not be true.
>You can do this by using a predicate CW(w) that expresses
I don't see how it is a paradox. Your proposition (3) will
>your logic you can still express the same things that could be
>expressed in the old logic. Take for example the following proposition
>in the old model theory:
>
>(1) K(p & ~K(p))
>
>You can still express this in your logic.
>that w is (equivalent to) the current world. You can express
>this as follows:
>
>(2) CW(w) =def= For all p, ( t(w,p) <-> p )
>
>With that you can write (1) in your logic as:
>
>(3) Forall w : W, ( CW(w) -> k(w, (p & ~k(w, p))) )
>
>This can be done for all for all formulas in the old logic and so it
>seems to me that you will still have the same paradox but written down
>differently.
The only reason in the original proof of Fitch's paradox to believe (1) (the claim K(p & ~K(p))) is because it follows from the knowability principle and the principle of non-omniscience. In the logic that I sketched, I don't believe it follows from those.
- Knowability principle: forall p:P, p -> exists w:W (k(w,p))
- Non-omniscience principle: forall w:W, exists p:P, p & ~k(w,p)
Your statement (3) above does not follow from my 1. and 2. At least, I don't see how.
-- Daryl McCullough Ithaca, NYReceived on Sat Jan 02 2010 - 22:52:44 CET