Re: Fitch's paradox and OWA
Date: 3 Jan 2010 11:55:20 -0800
Message-ID: <hhqsn801q0_at_drn.newsguy.com>
Jan Hidders says...
>
>On 3 jan, 18:32, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
>> In the Stanford logic,
>> if p is some proposition such that p & ~K(p), then the application
>> of the knowability principle gives (in some world w')
>>
>> K(p & ~K(p))
>>
>> which is a contradiction. My rule does *not* lead to that
>> conclusion. Instead, we have, for some world w,
>>
>> p & ~k(w,p)
>>
>> If we apply my version of the knowability principle, we get,
>> for some world w'
>>
>> k(w',(p & ~k(w,p)))
>>
>> which is *not* a contradiction. Proposition p is known in world w',
>> but not in world w.
>
>Hmm. That only shows that in that particular way you don't get a
>contradiction.
Well, the point is that the contradiction derived in Fitch's paradox does not go through. It's certainly possible that some other paradox can be derived, but I don't see any evidence of that.
>But my claim is that you do get a contradiction for the
>simple reason that your logic contains the old logic.
It doesn't contain the same *axioms*. In particular, I'm rejecting the "knowability principle" in favor of a variant principle that is (as far as I can see) consistent.
>> >> The original knowability principle, when translated into this
>> >> new logic, would look something like this:
>>
>> >> forall f:W -> P, forall w:W, f(w) -> exists w':W, f(w') & k(w',f(w'))
>>
>> >> The "propositions" of modal logic are actually functions on worlds.
>>
>> >Really? This is actually a stronger principle that implies the
>> >previous one since as a particular case I can take for f the function
>> >that maps each world to the same predicate p in P.
>>
>> Right. It's *too* strong, which is why it leads to a contradiction
>> (together with the principle of non-omniscience).
>
>But it doesn't correspond in any way to the semantics of the
>knowability principle in the old logic.
It certainly does. It's just a translation of the principle into a more expressive logic.
>The model theory there says
>something very different. So in what sense is this the semantics of
>the old knowability principle?
It's the same semantics!
Let's try to make this more explicit.
You have a set W of possible worlds, a set MP of
modal propositions, and for each world w, a set S_w of
the elements of MP true in world w. The set S_w is constrained
by the following rules:
Now, to capture this semantics in type theory, we use
the following translations:
>> >Also, I don't understand what you mean by "propositions are actually
No, the new "knowability principle" is *not* equivalent.
Look, once again, I'm formalizing the knowability principle
as:
forall p:P, p -> exists w:W, k(w,p)
I'm formalizing the non-omniscience principle as:
forall w:W, exists p:P, ~k(w,p)
These axioms do *not* lead to a contradiction.
functions from W into P.
>> >functions on world" except that the same proposition can have a
>> >different semantics in different worlds,
>>
>> That's exactly what I mean. For each modal proposition p
>> (which varies from world to world) we can associate a function
>> f_p from worlds to nonmodal propositions as follows:
>>
>> f_p(w) == the nonmodal proposition "p is true in world w"
>>
>> >and that was already taken into account in the old semantics.
>>
>> Yes. It's the old *syntax* that was inadequate to express a
>> reasonable knowability principle.
>
>But until now you have only shown that in the new syntax you can
>express an equivalent one (you can verify that by looking at the model
>theories) and one that's even stronger.
-- Daryl McCullough Ithaca, NYReceived on Sun Jan 03 2010 - 20:55:20 CET