Significance
The private language argument is of central importance to debates about the nature of language. One compelling theory about language is that language maps words to ideas, concepts or representations in each person's mind. On this account, the concepts in my head are distinct from the concepts in your head. But I can match my concepts to a word in our common language, and then speak the word. You then match the word to a concept in your mind. So our concepts in effect form a private language which we translate into our common language and so share. This account is found for example in An Essay Concerning Human Understanding, and more recently in Jerry Fodor's language of thought theory.
Wittgenstein shows, in his later work, that this account of private language is inconsistent. If the idea of a private language is inconsistent, then a logical conclusion would be that all language serves a social function. This would have profound implications for other areas of philosophical and psychological study. For example, if one cannot have a private language, it might not make any sense to talk of private experiences or of private mental states.
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