Re: Fitch's paradox and OWA

On 26 dec, 22:39, Marshall <marshall.spi..._at_gmail.com> wrote:
> On Dec 26, 4:36 am, Jan Hidders <hidd..._at_gmail.com> wrote:
>
>
>
> > Fitch's paradox is no more. :-)
>
> From plato.stanford.edu:
>
> "As for the knowability proof itself, there continues to be no
> consensus on whether and where it goes wrong."
>
> So ... you gonna publish that? Give those modal logic guys
> a kick in the pants? :-)
>
> Marshall

Only after checking carefully whether I'm right. :-) Unfortunately it turns out that I was wrong. But in the process I did find what I think is really the problem with the paradox.

You can "check" the reasoning by writing down a model theory for the logic and see if all the inferences are actually valid there. In this case the model theory is a bit more complicated than usual because we are reasoning about possible worlds.

Basically a model will now look like a pair (W,w) where W is the set of all possible worlds and w is the world we are in right now which must of course be an element of W. In normal propositional logic a single world is described by the set of true atomic propositions. In the setting of the paradox it also contains propositions of the form Kf where f is some formula, which say that f is known in that world. I will call those atomic propositions and Kf propositions collectively basic propositions.

So a world could for example be {a, b, Ka} where a and b are true, but only a is actually known. A complete model could for example be ({w1, w2, w3}, w1) where w1 = {a,b,Ka}, w2 = {a,Ka} and w3 = {b,Kb}.

For normal logic operators the model theory is straightforward: - (W,w) |- f1 & f2 iff (W,w) |- f1 and (W,w) |- f2 - (W,w) |- ~f1 iff not (W,w) |- f1
etc.

For the basic proposition and the K-facts: - (W,w) |- a iff a in w
- (W,w) |- Kf iff Kf in w

For the modal operators:
- (W,w) |- []f iff (W,w') |- f for all w' in W - (W,w) |- <>f iff (W,w') |- f for some w' in W

It can be verified that in this model theory all the inference steps used in the paradox are always valid. For the assumptions (A), (B), (C) and (D) on the Stanford page <http://plato.stanford.edu/entries/ fitch-paradox/> it holds that (D) is always valid in the above model theory. So we can question the validity of (A), (B) and (C). As it turns out (I will not explain that here) you don't need (A) and (B) to get the result of the paradox, so I'm going to focus on (C).

If we reformulate the meaning of (C) in the model theory we get:

(mC) If (W,w) |- f then (W,w) |- []f.

Given the semantics of []f this is equivalent with:

(mC') If (W,w) |- f then (W,w') |- f for all w' in W.

Note that in particular this will hold for f's that are basic propositions or negations of basic propositions. Since the basic propositions hold for (W,w') iff they are elements of w', and their negation only holds if they are not elements of w', it follows that w and w' must always contain exactly the same elements, and therefore in fact be the same world. In other words, (C) says effectively that there is always only one possible world. Knowing this, it is not surprising we get such unintuitive results, because it means that everything that is possible, i.e., true in one of the possible worlds, is actually necessary, i.e., true in all possible worlds. This also explains how, from the assumption that every truth is possibly known, we can come to the conclusion that every truth is necessarily known.

Conclusion: axiom (C) is not a modest modal assumption at all, and in fact quite absurd.

• Jan Hidders