Private language argument
The idea of a private language was made famous in philosophy by Ludwig Wittgenstein, who in section 243 of his book Philosophical Investigations explains it thus: ‘The words of this language are to refer to what can be known only to the speaker; to his immediate, private, sensations. So another cannot understand the language.’ [My translation.] This is not intended to cover (easily imaginable) cases of recording one's experiences in a personal code, for such a code, however obscure in fact, could in principle be deciphered. What Wittgenstein had in mind is a language conceived as necessarily comprehensible only to its single originator because the things which define its vocabulary are necessarily inaccessible to others.
Immediately after introducing the idea, Wittgenstein goes on to argue that there cannot be such a language. The importance of drawing philosophers' attention to a largely unheard-of notion and then arguing that it is unrealizable lies in the fact that an unformulated reliance on the possibility of a private language is arguably essential to mainstream epistemology, philosophy of mind and metaphysics from Descartes to versions of the representational theory of mind which became prominent in late twentieth century cognitive science.