First, I am working on a paper that deals with Wittgenstein's arguments against the possibility of a private language. He begins by defining a private language as a one that can only be known by a person that speaks it. The purpose of a private language is to communicate to one's self inner thoughts, feelings, and sensations. The key is that a private language is only accessible to the person that speaks it. To this Wittgenstein seeks to argue against. The different ways that Wittgenstein argues is in my judgment by three arguments. I would term these arguments as follows: the reductio argument, the presuppositional argument, and the ordinary language argument.
The reductio argument shows that from the assumption of a private language it entails the absurd conclusion of solipsism.
The presuppositional argument refers back to Wittgenstein's assumption that language is essentially social. Wittgenstein assumes that all language is a "language game." And that in order to follow a rule (e.g. grammar, language) one must first play the game. Language games must be prior to rule following in order to know what to follow, and how to interpret the rules. With this in mind Wittgenstein dismantles the possibility of a private language by showing that the preconditions for a private language is a public language. So ones private language is dependent upon the public language. And since this is the case, a private language i.e., one's inner thoughts and sensations can be communicated to others with public language. Therefore, there is no such thing as a private language.
Lastly, Wittgenstein's ordinary language argument. It simply states that ordinary language is sufficient to communicate one's inner thoughts, thus refuting the possibility of a private language.