Re: Fitch's paradox and OWA
From: Daniel Pitts <newsgroup.spamfilter_at_virtualinfinity.net>
Date: Fri, 01 Jan 2010 14:11:31 -0800
Message-ID: <oqu%m.1905$Sk4.734_at_newsfe10.iad>
>
> I haven't worked through the semantic details (at least not recently),
> but the proof clearly "works" and the intuition behind the proof seems
> plausible enough.
>
> Suppose that p is true, but I don't know it. Then p & ~Kp is true.
> But surely, I could not know p & ~Kp. That is, I couldn't know "p is
> true, but I don't know that p is true."
>
> After all, if I know that conjunction, then I know that p is true, so
> how could I know that I don't know that p is true?
>
> The argument seems perfectly clear to me, both formally and
> informally.
It actually is possible for "p & ~Kp" may be true, in which case "~K(p & ~Kp)" would also be true. There are a whole islands of truths that can't be known to be true, but are indeed true. This is basically Gödel's theorem.
Date: Fri, 01 Jan 2010 14:11:31 -0800
Message-ID: <oqu%m.1905$Sk4.734_at_newsfe10.iad>
Jesse F. Hughes wrote:
> stevendaryl3016_at_yahoo.com (Daryl McCullough) writes:
>
>> That's not a change of the *semantics*. That's a change of the >> *syntax*. My claim is that in the possible worlds semantics, >> every predicate (and operator) that can vary from world to world >> implicitly is a function of the world. That complexity can usually >> be avoided because implicitly we assume that everything is talking >> the same world. But when you nest <> and K, it is no longer possible >> to make that assumption. Not without restrictions on what can be >> said. My point is that the knowability principle doesn't make >> any sense without explicit mention of possible worlds. >> >> It might make sense if we restrict the principle to propositions >> p that don't involve the knowability operator. But if we restrict >> it that way, we can't carry out Fitch's proof.
>
> I haven't worked through the semantic details (at least not recently),
> but the proof clearly "works" and the intuition behind the proof seems
> plausible enough.
>
> Suppose that p is true, but I don't know it. Then p & ~Kp is true.
> But surely, I could not know p & ~Kp. That is, I couldn't know "p is
> true, but I don't know that p is true."
>
> After all, if I know that conjunction, then I know that p is true, so
> how could I know that I don't know that p is true?
>
> The argument seems perfectly clear to me, both formally and
> informally.
It actually is possible for "p & ~Kp" may be true, in which case "~K(p & ~Kp)" would also be true. There are a whole islands of truths that can't be known to be true, but are indeed true. This is basically Gödel's theorem.
Fitch's proof (at least by your description) is using the proof as its own premise. p & ~Kp can be true without knowing it, therefore you still don't know p is true.
-- Daniel Pitts' Tech Blog: <http://virtualinfinity.net/wordpress/>Received on Fri Jan 01 2010 - 23:11:31 CET