So, here's a weird idea. Suppose that you lived in a small village. And everyone in this village had an odd habit -- they all carried around a little wooden box. Whenever someone said the word "beetle" to someone else, the person who heard it would determine what it meant by looking at their own box.
Now, if everyone had exactly the same thing in their box, then communication would be smooth and easy. In fact, indistinguishable from how communication works for us in reality. But, suppose further than the objects in the box differ from person to person. So, in the box I carry, there might be an actual beetle, but in the box you carry, there might be a small piece of cake, and in the box someone else carries, there might be nothing at all.
Obviously, we'd figure out something was up pretty soon. If I say "Beetles have six legs", you'd think I was crazy; if you said "Beetles have light pink frosting", I'd think you were insane; in short, unless all our words work like this, we can triangulate on the unusual word -- the one where we individually assign the same sounds (or letters) to different meanings.
So, suppose further that this is actually the case. Whenever we hear any word, we look up the meaning in something that is private and individual. Probably not a little box -- given the number of words in English, little boxes would get a mite cumbersome -- so, let's say it's something mental. A mental dictionary, containing meanings for every set of sounds that you or I hear. And everyone has their own, such that whenever someone speaks to you, you quickly (mentally) look up each word in your mental dictionary, and determine the meaning of their utterance. And in constructing an utterance of your own, you would reverse the procedure: look up the meanings in your mental dictionary, elicit the related terms, and then speak the words.
Sound possible? According to Ludwig Wittgenstein, it's not. Language, understood not just as a set of symbols but a set of symbols associated with meanings, must be public. A language that relied upon private meanings, like the mental dictionary -- what's called a "private language" -- is not a language at all.
The beetle-in-the-box example is deceiving because it overlooks an important feature of language, namely the ability to be wrong or right -- to use the language correctly or incorrectly. If language is wholly private, then the only standard possible is one's own determination of whether the language is used correctly or incorrectly. But that's not actually how we do determine whether language is being used rightly or wrongly.
Consider another example, a fairly simple one involving numbers. (This is being used analogously, BTW.) We all know that, given the sequence 1, 2, 3, the next number is 4. But how could you correct someone who incorrectly continued the sequence with something else -- say, 5?
You could point out that there is a rule underlying the sequence, namely the rule "+1". Every item in the sequence is generated by applying "+1" to the previous item.
That would work for some, but suppose that the hypothetical mistaken person insists that the rule "+1" isn't the rule being applied in the sequence. Or, worse, suppose that the hypothetical mistaken person insists that when you apply "+1" to 3 you get 5, not 4. How do you justify or explain the rule?
Wittgenstein holds that you can't do much. The best you can do is note that this isn't the behaviour that is expected. Given the sequence 1, 2, 3, we expect 4. And anyone who puts down something else is wrong because they are in violation of the expectations of the surrounding social group.
Language, he said, works the same way. There probably are some rules regarding meanings, particularly rules about compositionality -- that is, how larger chunks of language, like sentences, derive their meanings from the meanings of their parts, namely words. But what do we do with someone who just isn't following the rules? Well, we indicate that this is not the expected behaviour, and insist on conformity to expectations.
The thing to notice here is that this is a picture of language as a set of publicly-observable -- and -correctable -- behaviours. If we contrast it with the picture of private language, where language is primarily a set of internal states, we find there is no sense in which an individual's use of words can ever be wrong. Inconsistent over time, maybe; but why wrong? Corrected how?
It should be noted that the point here isn't just about language that is spoken/heard or written/read. It's also a point about thought. After all, when we think, we think often in a sort of language. A purely private thought would therefore have to be fairly primitive -- an image or a sound, without any linguistic content.
If there is no such thing as a private language, then it follows that whatever portion of our thoughts is based upon language is also public.