Re: Fitch's paradox and OWA
Date: Sat, 26 Dec 2009 04:24:51 -0800 (PST)
On 25 dec, 10:45, Jan Hidders <hidd..._at_gmail.com> wrote:
> On 24 dec, 11:58, "Mr. Scott" <do_not_re..._at_noone.com> wrote:
> > "Reinier Post" <r..._at_raampje.lan> wrote in message
> > > Mr. Scott wrote:
> > >>"Reinier Post" <r..._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:
> > > [...]
> > >>Your addition of 'now' to (NonO) is the cause of your confusion. †Go back
> > >>and re-read what you cited.
> > > I just did.
> > >>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....
> > > Yes indeed, that's how they define K ... I should have seen that
> > > the first time around. †Thank you.
> > >>(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.
> > > Now, the contradiction is direct: KP says that all
> > > truths are knowable while NonO says that some truth isn't.
> > > There is no paradox, just a contradiction.
> > > But that's not how NonO is introduced:
> > > | And suppose that collectively we are non-omniscient, that there is an
> > > unknown truth:
> > > |
> > > | † †(NonO) ‚^fp(p ‚^ß ¬¨Kp).
> > > See: here they say 'unknown', not 'unknowable'. †Hence my confusion.
> > > Once it's merely 'unknown' I believe the scoping issue arises.
> > Now I'm getting confused. †I agree that there is something that doesn't sit
> > right: it doesn't make sense for something being knowable to imply that it
> > is known, or for something being unknown to imply that it is unknowable.
> > But supposing that (NonO) is true throws a wrench into the works. †If one
> > supposes that there is a truth that is not known by anybody at any time,
> > then it is necessary that there is a truth that is not known by anybody at
> > any time, and that if it is necessary that there is a truth that is not
> > known by anybody at any time, then it is not possible for that truth to be
> > known by somebody at some time. †On the page you cited, these are instances
> > of lines (C) and (D), respectively. †When something is actually true, it is
> > necessary that it is possible, but when something is supposed to be true, it
> > is necessary that it is not just possible but true.
> I agree with Reinier that there is a scoping issue here that causes
> the strange conclusion. Since the logic includes the notion of
> "possible" we need to take into account that the model theory changes
> and therefore also what it means for a proposition represented by a
> formula to be true. In the setting of the paradox the truth of a
> formula is necessarily defined wrt. to a model that describes all
> possible situations, or "the set of all possible worlds" as it is
> usually called. For example, <>f means that in at least one possible
> world f holds, and f means that it holds in all possible worlds. As
> usual a proposition is assumed to be true iff it holds in all possible
> worlds. If it holds in only some possible worlds it is assumed to be
> only possibly true.
> So what happens to the paradox if we take that into account? Let's
> look at what KP says:
> KP: For all propositions p it holds that if p is true in all possible
> worlds then there is at least one possible world in which it is known.
> (NOTE1: That is actually not what the formula on the Stanford page
> says. It has the meaning "In all possible worlds it holds that if p is
> true in that world then there a possible world in which p is known."
> which is a slightly stronger statement.)
> And NonO:
> NonO: There is a proposition p such that it holds in all possible
> worlds but is not known in any of them.
> (NOTE2: Also here the formula in the Stanford page says something
> subtly different: "In all possible worlds there is a formula p that is
> true in that world but not known in that world." This is actually a
> weaker statement and does not contradict KP. The formula should have
> said: Exists p (p -> ~Kp). )
> It will be clear that KP and NonO as I've stated them immediately
> contradict each other. The reasoning of the paradox now proceeds as
> follows: if you accept KP then you cannot accept NonO and therefore
> must accept it's negation. Fair enough. Let's write this negation
> NotNonO: There is no proposition p such that it holds in all possible
> worlds but is not known in any of them.
> Or, equivalently:
> NotNonO2: For all propositions p it holds that if it holds in all
> possible worlds then it is known in at least one possible world.
> But wait! Did we not see that before? Of course we did, this is
> exactly KP. So basically what we have concluded until now is that if
> you accept KP then you also accept KP. At least it has the benefit of
> being very probably correct. :-)
> Anyway, on with the reasoning of the paradox. How does it proceed from
> NotNonO? If you look on the Stanford page you see that this is
> formulated in formula (10). But what this formula says is:
> StanfordNotNonO: In all possible worlds there is no proposition p that
> holds in that world and is not known in that world.
> But this is of course very different NotNonO, which is what it should
> have said.
In a possible-worlds setting (which you have to be in if you talk about "possible" and "necessary") the inference rule "reductio ad absurdum" is not always valid, i.e., you cannot conclude from the observation that "the assumption of P leads to a contradiction" that ~P.
The reason for this is very simple. The truth of P means in a possibleworlds setting that P is true in all possible worlds, i.e., it is a necessary truth. So if the assumption of P leads to a contradiction then we can conclude that P is not a necessary truth, but *not* that P is necessarily false. In other words, if P (or equivalently, P) leads to a contradiction then we can conclude that ~P (or equivalently, <>~P) but not that ~P).
Without the "reductio ad absurdum" inference rule we cannot derive proposition (10) on the Stanford page and the chain of inference of the the paradox is broken.
Fitch's paradox is no more. :-)
- Jan Hidders