Re: Fitch's paradox and OWA

On 3 jan, 20:55, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
> Jan Hidders says...
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> >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)))
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> >> 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.

Fair enough. But I think I do.

> >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.

Well, I'm not so sure. Your new variant look very similar to how the principle is formulated in my model theory. And there I got the contradiction.

> 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:
>
> 1. If Kp is in S_w, then p is in S_w (you can only know true
> statements)
> 2. And(p,q) is in S_w iff p is in S_w and q is in S_w
> 3. Or(p,q) is in S_w iff p is in S_w or q is in S_w.
> 4. Not(p) is in S_w iff p is not in S_w
> 5. Implies(p,q) is in S_w iff p is not in S_w or q is in S_w
> 6. <>p is in S_w iff for some w', p is in w'
> 7. []p is in S_w iff for all w', p is in w'

That already looks close enough to a model theory to me. A model could be a pair (W, S) with W the set of possible worlds and S : W -> 2^F where F is the set of formulas and satisfies the rules 1-7. I strongly conjecture that those models would be isomorphic to the models in my formulation of the model theory and lead to the same formulas being true.

Your mapping to type theory is a bit hard for me to get my head around, so I'll assume for the moment that the above is your model theory.

> Now, to capture this semantics in type theory, we use
> the following translations:
>
> 1. Introduce a type, W, of all possible worlds.
> 2. Introduce a type, A, of all atoms (atomic modal propositions).
> 3. Introduce the predicate t(w,a) saying which atoms are true in
> which possible worlds.
> 4. Introduce a predicate k(w,p) saying which propositions
> are known in which worlds.
> 5. Define MP, the type of all modal propositions, to be the type of
> functions from W into P.

You didn't define / postulate P yet. But a deeper problem is that I don't see why you let modal propositions be different propositions in different worlds. Why is it not enough that their truth value can be different in different worlds? It also makes it hard for me to see whether this formulation is equivalent withe the above one that it is supposed to capture.

> 6. For each atom a, we associate a corresponding element of MP:
> p_a == that function f such that f(w) = t(w,a).
> 7. Define the operator K as follows:
> Kf == that function g such that g(w) = k(w,p)

Kf should be Kp?

> 8. Define the operator And as follows:
> And(f,g) == that function h such that h(w) = f(w) & g(w)
> 9. Similarly for Or, Implies, Not
> 10. Define the operator <> as follows:
> <>f == that function g such that g(w) = exists w':W, f(w')
> 11. 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)

In my model theory the semantics of the formula that represented it can be formulated as: (with M being the set/class of valid models)

Forall (W,w_1) in M, forall w_2 in W, forall f in F, (W,w_2)||-f -> exists w_3 in W, (W,w_3)||-Kf

If you fix W we can simplify this to:

(JH-KP) forall w_2 in W, forall f in F, w_2||-f -> exists w_3 in W, w_3||-Kf

Doesn't that look similar to you?

> I'm formalizing the non-omniscience principle as:
>
> forall w:W, exists p:P, ~k(w,p)

I think you forgot that p has to be true in at least one possible world.

And the semantics of the NonO formula in my model theory was:

(JH-NonO) forall w in W, exists f in F, w||-f and not w||-Kf

Again, quite similar, no? In my model theory JH-KP and JH-nonO lead to a contradiction. As far as I can tell yours is very similar to mine.

• Jan Hidders