1. The meaning of a word is the object the word stands for (PU 1).
2. Sensations are private.
Hence a language that refers to private objects could only be understood by the owner of the sensations (PU 243). Nobody else could understand it, because he has no access to the inner world of another person. (Besides: This is an similar paradoxical result to that of rule-following (PU 201) and the whole treatment of it resembles that of the preceding paragraphs concerning meaning and understanding, rule-following etc. If you have understood that part, you should also understand this; it is only an application to the “inner”!).
To resolve the puzzle LW proceeds by asking how words relate to sensations, i.e. how we learn the meaning of sensation-words (PU 244), respectively by asking after the criteria for ascribing sensation to someone. (Of whom would we say he understands the word pain? Of whom would we say he is in pain ?). LW starts by giving an example of how we could learn the meaning of the word pain. One possibility is described in PU 244 (replace pain behavior by a word), others in PU 288 (by gestures, by pricking with a needle). This explanations could be understood or misunderstood as any other (ostensive) definition, but what is the case is shown how the word is subsequently used (PU 288, PU 28). This is in short form LW’s solution. If you want to know what the word for sensations means (= What are sensations?) it is an misunderstanding to start by looking at one’s headache (PU 314), because the meaning of the word is (in many cases) shown by it’s use (PU 48, PU 264, PU 278).
The answer of someone entangled by the private-language-argument to the same question would probably consist in an objection to the above method, because (A) sensations are private and that (B) the connection is made by focusing the attention on the sensation, giving to oneself something like an ostensive private definition (PU 258, PU 263). This implies that the language-game begins with the d e s c r i p t i o n of the sensation as it is perceived in an inner space (PU 290).
(1) By the conventional use of the words “knowing”, “doubting” and “the same” in relation to that of sensation, we can very well say that we know or doubt that someone else is in pain (PU 246-247), resp. say that he has the same pain as we do (PU 253-254). This is the way we use this words!
(2) The idea of the private sensations arises from the confusion of an sentence about experiences (Erfahrungssatz) with an grammatical convention (PU 247). We tend to believe, that we are saying something about the nature of sensations, while we only bump against grammatical conventions.
(1) To name a sensation (or to make notes of their occurrence) there must already be prepared quite a lot in our (public) language (i.e. the grammar of sensations (257), function of a word (260), technique of use (262)).
Hence the meaning of the word would not be private (PU 257, 258, 259, 260).
(2) Defining (giving rules for the use of) an sensation-word by focusing the attention (pointing inwardly) on a sensation is idle, because impression of rules are no rules, or consulting an table in the mind is not really consulting an table (PU 259, 265, 266, 267). Hence there is no criterion for correctness of the definition, hence no possibility for any definition. An “internal process” needs external criteria (PU 580).
(3) The language game with sensation-words does not begin with the d e s c r i p t i o n of the sensation, but with the expression of them (PU 290, PU291). The child first expresses pain and only then can we teach it the word.
(1) Of whom would we say he u n d e r s t a n d s the word pain? We would say it of someone how uses the word in the same way as we do (PU 271), i.e. what happens inside of him is irrelevant. The beetle in the box (= private sensation) is of no relevance for the correct use of the word pain. (This is crux of the beetle-in-the-box parable!). The thing in the box is not part of the language game, respectively we can speak of pains without actually having them (PU 270, 293). To know what the word pain means I must not every time I use it pinch myself! Only of someone who knows the technique of application of the sensation-word, we would say he understands it (PU 288).
Yet some language-games would lose their point if we for example would all get color blind (PU 142, but also PUII xii). “The image of pain certainly enters into the language-game in a sense; only not as a picture” (PU 300) and “An image is not a picture, but a picture can correspond to it” (PI 301).
(2) Of whom would we say he h a s pain? Only of someone how behaves like a living man, we would say he is in pain. That is the way we use the word. Applying the word to objects is only a metaphor, a secondary use, i.e. without the primary use this would be senseless. (PU 281-284, 286, 288).
(3) I (first person) don’t need criteria to identify my pains, I simply express them, therefore I can’t doubt to have them (246, 290).
The origin of the confusion derives from the typical way philosophers tend to approach the problem, (e.g. by introspection, the headache-approach) (PU 273-277) or by falling victim to the pictures (inner space) offered by our language (PU 295). LW does not want to deny that there are sensations, but only (and this is the important point in the whole discussion!) that the picture of an inner process (grammatical fiction) could give us the right idea for the use of an word (PU 305-308). LW is not interested in ontology but in grammar, because grammar reveals the nature of our sensations (PU 371, 373). If you want to know what pain is, you should not introspect yourself, but examine the meaning of the word and that is it’s use (see beginning Blue Book and PU).