In his "Thinking it Through" textbook, Appiah writes
It is a big step from saying that some of our mental states are things that other people can know about, to saying, with the behaviorists, that all of them must be in this way public. Yet one of the most influential philosophical arguments of recent years has just this conclusion. The argument was made by the Austrian-born philosopher Ludwig Wittgenstein, whose work we will discuss again in the chapter on language. Wittgenstein began by supposing that anyone who believed in the essentially private thoughts of Descartes’ philosophy of mind would find it quite acceptable to suppose that someone could name a private experience—one, that is, that nobody else could know about. And indeed, as we shall see in Chapter 3, Thomas Hobbes, who was an English philosopher who reacted against some of Descartes’ ideas, thought that we used words as names of our private thoughts in order to remember them. He called them “marks” of our thoughts. To use marks in this way, someone would have to have a rule that they should use the name just on the occasions where that private experience occurred. Wittgenstein argued that obeying such a rule required more than that there should be both circumstances when it was and circumstances when it wasn’t appropriate to use the name. He thought that it also required that it should be possible to check whether you were using the name in accordance with the rule. And he offered a very ingenious argument that was supposed to show that such checking was impossible. If Wittgenstein was right, there could be no such “private languages.” And his argument is called, for that reason, the private-language argument.
I am trying to follow the line of argumentation. Is this meant to be a reductio argument? Is Appiah saying "suppose we accept that such private thoughts exist. Then this implies (according to Hobbes) that we have private 'marks'. Wittgenstein then says that these 'marks' can only exist if we have a way of checking(i.e. the existence of 'marks' implies the ability to check), and yet such checking is impossible -- therefore private mental events are impossible"?