And another thing ...

Here is, perhaps, a better way of wording the argument:

Remember that we are introducing this possibility - that someone may believe that it is not possible to talk - into our present converstaion.  To do this, we need to be able articulate this attribution coherently.

If I say "Mary knows how to talk but doesn't believe that it works", I am either saying something false, or showing that I don't understand what 'talk' means.

The only properly complete answer to "What does 'talk' mean'?" is recursive, and the root is ostensive:  it is what we are doing now, in having this conversation, in writing and reading this post.  And we are doing this successfully (on the whole), or we are not doing it at all.

Since I would be saying of Mary (in the example) that she does not understand this - that she does not accept that we are talking now - I am also saying that she does not mean by 'talk' what we mean by 'talk'; and so that my statement about her does not mean what it initially appears to mean.

It is, in fact, another Moorean 'paradox'.

Some other things we can't not believe in ...

Of course there are others that are peculiar, and almost certainly related.

Could we say, for instance, that someone understood basic arithmetic if they also believed that it didn't work?  This is a tricky one.

They would need to give us an explanation with made arithmetic sense.

If they said, for instance, "I can see how adding up works for the examples you have shown me, but I don't think you can extrapolate from these cases to all possible cases"; we would know they were making a mistake of some kind.

On the other hand, Gödel's proofs are, exactly, proofs that arithmetic does not work in the way that many Arithmeticians previously thought.  Certain basic expectations cannot be extended to the set of all possible numbers.

No-one, however, would draw the conclusion from Gödel's argument that there must be something fundamentally wrong with arithmetical computation - or, for instance, with the fundamental theorem (unique factoring into primes).

We're just changing out a bit of structure here, not rebuilding the boat.  Even if the change makes some aspects of the navigation appear more mysterious than they did before...

Why we can't believe it is not possible to talk ...

What does 'talk' mean in 'We can talk to one another?'

It means our participation in the functioning language game we are presently engaged in (with all its relevant contiguities), within which we are interlocutors.  And it only means this - there isn't some further way of defining what we mean by 'talk'.  No other account can be complete or unambiguous (for Kripkean reasons).

This means that a belief that we cannot talk is internally incoherent - it is a belief that the functioning language game in which we are engaged is not a functioning language game.

It is not just that we cannot reliably attribute this believe, we cannot coherently attribute it.