Re: Fitch's paradox and OWA

From: Mr. Scott <do_not_reply_at_noone.com>
Date: Thu, 17 Dec 2009 19:40:05 -0500
Message-ID: <bZOdneC2hM_7UrfWnZ2dnUVZ_uqdnZ2d_at_giganews.com>


"Reinier Post" <rp_at_raampje.lan> wrote in message news:4b2ab63e$0$28714$703f8584_at_news.kpn.nl...
> Nilone wrote:
>
>>Does Fitch's paradox prove an inherent contradiction in the open-world
>>assumption?
>
> I don't understand the paradox.
>
> http://plato.stanford.edu/entries/fitch-paradox/
>
> explains:
>
> suppose that
>
> (KP) all truths are knowable, i.e. can be known by somebody at some time
>
> and
>
> (NonO) not all truths are known now
>
> then
>
> (1) there is an unknown truth p
>
> and then
>
> (2) p is true and unknown is itself a truth
>
> and hence, by KP,
>
> (3) (p is true and unknown) can be known by somebody at some time
>
> "However, it can be shown independently that it is impossible
> to know this conjunction. Line 3 is false."
>

Your addition of 'now' to (NonO) is the cause of your confusion. Go back and re-read what you cited.

If K is the epistemic operator meaning 'it is known by someone at some time that,' then not K would have to deny that, meaning 'it is not known by anyone at any time that,' so with that in mind....

(KP) forall p(p implies possibly Kp): all truths are knowable by somebody at some time.

(NonO) exists p(p and not Kp): there is a truth that is not known by anybody at any time.

These are contradictory. If all truths are knowable by somebody at some time then there can't be a truth that is not known by anybody at any time.

> I'm eager to see that demonstration. Clearly, if p is unknown (1),
> then so is the truth than p is true and unknown (2), but KP doesn't
> contradict that - all it says is that p may be known, perhaps at some
> other time, and indeed, at that same time, the statement that p was true
> but unknown will also be known to be true.
>
> They seem to mess up the scoping of K and P, simplifying their language
> until paradoxes become inevitable; the paradox results from that,
> as far as I can see. In any case, the paradox depends
> on the exact formalization on K and P, which isn't given.
> Once again, I'm left with the feeling that these Stanford guys
> could use some field experience in database design.
>
> --
> Reinier
Received on Thu Dec 17 2009 - 18:40:05 CST

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