PI doesn't really have a private language argument, but rather many, and these have been interpreted in different ways because PI is written is a fairly obscure and discursive style.

We'll start with this section from PI:

The words of this language are to refer to what only the speaker can know — to his immediate private sensations. So another person cannot understand the language

For clarification:

This is not intended to cover (easily imaginable) cases of recording one’s experiences in a personal code, for such a code, however obscure in fact, could in principle be deciphered. What Wittgenstein had in mind is a language conceived as necessarily comprehensible only to its single originator because the things which define its vocabulary are necessarily inaccessible to others.

The possibility of such a language is dismissed. This clashes with theories of consciousness that propose "mentalese" a sort of "native language of thought."

Often called Mentalese, the mental language resembles spoken language in several key respects: it contains words that can combine into sentences; the words and sentences are meaningful; and each sentence’s meaning depends in a systematic way upon the meanings of its component words and the way those words are combined. For example, there is a Mentalese word whale that denotes whales, and there is a Mentalese word mammal that denotes mammals. These words can combine into a Mentalese sentence whales are mammals, which means that whales are mammals. To believe that whales are mammals is to bear an appropriate psychological relation to this sentence. During a prototypical deductive inference, I might transform the Mentalese sentence whales are mammals and the Mentalese sentence Moby Dick is a whale into the Mentalese sentence Moby Dick is a mammal. As I execute the inference, I enter into a succession of mental states that instantiate those sentences.

I am not a fan of mentalese theories, except in that they seem to get something right regarding the syntactical structure and combinatorial nature of thought.

It's easy to see how the private language argument against mentalese can be taken by its advocates as question begging. For the crux of the issue here seems to come down to how "language" is defined. If "language" is necessarily a vehicle for communication between two minds, then, by definition, the private language is excluded. Or, if it is defined by intersubjective rules, then mentalese also seems on the chopping block.

However, advocates of mentalese have two responses here. First, they can reject the definition of language, and define it in some other way, or they can say that mentalese isn't such a private language, because it can be translated into natural languages and shared, and that this is enough of a linkage to make it non-private.

Anyhow, if occured to me that this is really a question of essence. What properly defines a language? It reminds me over the controversy over neurodarwinism. Neurodarwinism likens the process by which synapses and neurons are pruned to natural selection, and indeed the processes do share many things in common.

Many biologists objected to neurodarwinism. Natural selection, they claim, has nothing to do with intentionality. It occurs at the level of thoughtless genes. Hebbian learning and neuronal pruning is ineluctably bound up in concious decisions and intentionality. What a toddler chooses to do, and what a parent chooses to do with them, plays an enormous role in which neurons survive and thrive. Therefore, natural selection can't be at work in neuronal pruning of synapse formation.

Leaving aside if the objectors' position is a good way of thinking about natural selection in the first place, this seems also be a question of essence. The question is about what defines "natural selection," and if the lack of intentionality is essential, a property in Porphyry's sense, or accidental. Similar debates have sprung up regarding if languages or corporations undergo "natural selection."

This has led to the term "selection-like." To return to the original question, we could ask if advocates of mentalese can simply sidestep the issue by declaring mentalese "language-like?"

I thought these two cases were interesting ones because they get into substance/accidents, and the identification of "general principles" in terms of genera.

Elucidations of the private language argument have focused on definition. This is an interesting area, because it would seem that mentalese wouldn't have any problems with definitions because the substance of mentalese has intrinsic meaning, being thought itself. But the question would be, does this make it unlike language, because it doesn't refer in the same way? Or is mentalese more the paradigmatic language (what advocates claim) precisely because there is no distance between its "words" and their meanings? That is, is natural language just a work around for sharing mentalese, and it only needs rules because it lacks the inherit content of mentalese terms?