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In Philosophical Investigations 244-254, before talking about private signs, Wittgenstein is talking about sensations.
He seems to divide this section into addressing in what way words refer to sensations, and in what way sensations themselves can be said to be private.
What is a brief outline of the points Wittgenstein is making about these two questions?
How does this section relate to his private language argument?
These paragraphs are, in a nutshell, Wittgenstein's account of public language, which precedes the core of his private language argument. Its purpose is preparatory, to dispel the traditional idea that names are associated with some "content" or "meaning", which represents or "describes" the named thing (Frege-Russell view). Wittgenstein's view is in a slogan that "meaning is use", what is learned is linguistic behavior, not mental "descriptions":
But how is the connexion between the name and the thing named set up? This question is the same as: how does a human being learn the meaning of the names of sensations?—of the word "pain" for example. Here is one possibility: words are connected with the primitive, the natural, expressions of the sensation and used in their place. A child has hurt himself and he cries; and then adults talk to him and teach him exclamations and, later, sentences. They teach the child new pain-behaviour. "So you are saying that the word 'pain' really means crying?" — On the contrary: the verbal expression of pain replaces crying and does not describe it.
He then goes on to explain how some common sentences, that encourage thinking of sensations as "stuff" when viewed in context support no such substantivation. This is similar in spirit to Russell's paraphrases of expressions like "the current king of France", which show that such sentences can be understood without presupposing existence of (potentially incoherent) entities, despite the apparent suggestion in their grammatic form. Except Wittgenstein "paraphrases" not apparent names into descriptions, but descriptions into verbal behavior. "Sensations are private" is compared to "one plays patience [card game] by oneself", i.e. it is a comment on how a game is played. "Another person can't have my pains" is personalized into a person striking herself on the chest and saying "surely another person can't have THIS pain!" To be discharged with "one does not define a criterion of identity [of a thing] by emphatic stressing of the word "this"".
What Wittgenstein argues is quite counterintuitive: that the traditional view of words as pointers to "mental stuff" in the head or mind is philosophically naive, and linked to taking surface grammar at face value. Instead, he suggests, the words are just verbalized parts of our actions and responses, verbal signals, "the verbal expression of pain replaces crying and does not describe it". Like bearerless names dissolve into descriptions, meanings dissolve into inferential roles, see more under Do Wittgenstein and Quine give the same criticisms of semantics? At the end of §254 he compares the traditional view to mathematical Platonism:"...what a mathematician is inclined to say about the objectivity and reality of mathematical facts, is not a philosophy of mathematics, but something for philosophical treatment."
With the nature of public language clarified Wittgenstein can move directly to the core of his private language argument. It turns out that the idea of private language is only plausible on the traditional "descriptive" view of language. With that view out of the way it is easier to explain why the conditions that enable verbalized public behavior are not met in the private setting, making private language problematic. See Did Wittgenstein consider the possibility of a private language with public content?
