Re: Fitch's paradox and OWA

From: Daryl McCullough <stevendaryl3016_at_yahoo.com>
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
>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.

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
>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.

I don't see how it is a paradox. Your proposition (3) will (with the appropriate axiomatization of the knowability predicate) be provably false.

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.

  1. Knowability principle: forall p:P, p -> exists w:W (k(w,p))
  2. 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, NY
Received on Sat Jan 02 2010 - 15:52:44 CST

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