That part doesn't follow from anything prior to it, and if you're trying to argue that language isn't private, you can't just jump to the conclusion you're shooting for in the middle of the argument. — Terrapin Station

You're correct, this is still part of the set up.

I don't think this part works, either. For one, we don't say that someone is speaking a language or not based on whether we understand what they're saying. We say that someone is speaking (or writing) a language if it seems to function as a language to them--it's used to communicate, to record information, etc. As it is, there are a handful of ancient languages that we haven't been able to crack yet, and maybe we'll never crack them. We don't say that they're not languages because of this. — Terrapin Station

Please substitute "impossible to learn" for unintelligible. That's what I meant.

That part I'm not sure I understand. Could you explain it in other words? — Terrapin Station

The point of my set up was to exemplify the relative nature of language.
To one person it's intelligible, while it maybe completely impossible to learn for the other,
since they do not share any experiences (except for that one person living in both worlds, so that's a weak point of the argument).
Two worlds A,B and people a,x,b. a lives n A, b lives in B, x lives in both.
In A and B different languages are spoken, also A and B do not share anything but x.
The meta view is exactly this formalization of the two worlds. In this meta view we understand that from the point of view of a, x does speak weird gibberish, but we also know that this gibberish is indeed an actual language. One could say the language spoken in B is a private language in A used only by x.
However as we can choose only the viewpoint of a,b or x in real life, there are only these possibilities:
If we choose a: x is talking gibberish, it's not a language.
If we choose x: I know all languages, they all are public, as I talk to a and b.
If we choose b: same as a.
So there is no private language here, even with the more permissive use of the term above.

To repeat my alleged relation to Wittgensteins argument:
Wittgenstein talks about someone relating a sign to an experience privately,
here I'm replacing the single sign with a complete language spoken to people, that
are either imagined or simply not considered their own people, at least from the point of view of a or b.
Thus x's experience of talking to people "from the other world" is a private one from the point of view of a or b. I do this because I do not consider it sensible to divide people into real und unreal and then consider talk to unreal people private, as from any point of view the unreal people simply do not exist.
The distinction exists only in a "meta-view" which is not clearly connected to the actual problem, but instead a game of words, played by rules which maybe do not bear any connection to the actual life of a,b or x.

My understanding of the "Private Language Argument" is that it would be impossible for a lone individual to create a language if they did not already speak a public language, and that if they did create a language they would need to translate their new words for things, actions, qualities and so on into their native language equivalents in order to be able to specify to themselves what they mean by them. But this would mean that their language is not genuinely private, because it is understandable only in terms of an existing public language. — Janus

This would work for me, if I would not believe it to be possible for someone to imagine a whole world. But maybe exactly that is my mistake. In this made up world, there would be actions and things that one could talk about, you see? But again, maybe one can not really make up things.

@Luke

Solipsism or not, if we suppose that you are talking-in-a-dream to other people in a language that is understood by your dream participants, then it is not a private language.

I'm rather surprised by this statement. So talking to people I'm imagining in a language that can not be taught to anyone real would not be considered a private language by you?
This would make things a lot more sensible for me, however very uninteresting too.
Because now every language or sign is obviously public now, as I can simply imagine (that's an ok substitute for dreaming right?) somebody to talk to in this language.
I find it hard to believe that people have come to argue about anything, assuming they agree on this definition of a private language.