# 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:

- If Kp is in S_w, then p is in S_w (you can only know true statements)
- And(p,q) is in S_w iff p is in S_w and q is in S_w
- Or(p,q) is in S_w iff p is in S_w or q is in S_w.
- Not(p) is in S_w iff p is not in S_w
- Implies(p,q) is in S_w iff p is not in S_w or q is in S_w
- <>p is in S_w iff for some w', p is in w'
- []p is in S_w iff for all w', p is in w'

Now, to capture this semantics in type theory, we use the following translations:

- Introduce a type, W, of all possible worlds.
- Introduce a type, A, of all atoms (atomic modal propositions).
- Introduce the predicate t(w,a) saying which atoms are true in which possible worlds.
- Introduce a predicate k(w,p) saying which propositions are known in which worlds.
- Define MP, the type of all modal propositions, to be the type of functions from W into P.
- For each atom a, we associate a corresponding element of MP: p_a == that function f such that f(w) = t(w,a).
- Define the operator K as follows: Kf == that function g such that g(w) = k(w,p)
- Define the operator And as follows: And(f,g) == that function h such that h(w) = f(w) & g(w)
- Similarly for Or, Implies, Not
- Define the operator <> as follows: <>f == that function g such that g(w) = exists w':W, f(w')
- Define the operator [] as follows: []f == that function g such that g(w) = forall w':W, f(w')

>> >Also, I don't understand what you mean by "propositions are actually

*>> >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.
*

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.

-- Daryl McCullough Ithaca, NYReceived on Sun Jan 03 2010 - 20:55:20 CET