The fact that I have no problem imagining a private language probably implies that I don't understand the notion of private language. My understanding is private language is a language understandable by only a single individual.
Is it a claim about the existing natural languages or any conceivable language?
Perhaps it has to do with what qualifies as a language. Otherwise I see no difficulty in making up an artificial language contra Wittgenstein's claim.
At the simplest level the PL is "incoherent" in the manner of an oxymoron. A little like saying it is possible to "communicate without communicating."
More to the point, it might be harder than you think to create your truly "private" language. In the first place, it would have be absolutely unique and, in theory, indecipherable. So you can't just "translate" the grammar and words of some language you already know by some term-grinding equation. Easily reverse engineered.
You might "randomize" or "scramble" an existing language by some operation. But if even you can find no "meanings" in the result, is it really your "language"? Finally, how would this language distinguish itself from the continuity of your "private" thoughts? It must somehow become discontinuous with your own thoughts while also modeling them... and then reappearing as "recognizable" to them. Hard to do entirely on your own.
And should you ever succeed, how would you convince others of your unique epistemological accomplishment?
I think Wittgenstein's point was, as usual, to draw attention to something rather obvious that gets lost only in philosophy, where so many projects begin in some sort of isolated inner dialogue. This is the epistemological version of what Marx called "Robinsonism," the propensity of modern thinkers to construct theories around some atomic Robison Crusoe who just appears there, spontaneously generated, with all his ideas and tools.
Let us try to understand how a private language could come into existence.
Suppose that someone, while writing in his diary, for example, draws an S whenever he feels in a certain way (whenever he has a private sensation). In this case, the only one who can judge the exactitude with which S has been employed should be only the one keeping the diary, the "inventor" of S (suppose S cannot be defined in words). We can say about this that he is following a rule. The question is now whether the writing of S in conjunction with the private sensation can pass as a sort of private language.
Wittgenstein's answer is negative because whenever the inventor of S writes down S it means that he experiences the "same" sensation, but in what sense is S the same? Is there a criterion for distinguishing between S being the same and S seeming to be the same? If he makes an S and by that he means the sensation he has at this very moment, how can he know that it is the same sensation to which S is assigned? Isn't it similar to saying "This is a red apple!" and by that defining "redness"?
But if this means that he refers to a certain kind of sensations, then he has to have a rule which entails knowledge of a former conjunction of S with a kind of sensation. There is, however, nothing that can warrant the fact that the person follows a rule "privately". The most we can say about this is that the person "is licensed to follow the rule as it strikes him".
Following a rule is something that happens in a community, where the others can keep tabs on how you follow the rule. Hence the impossibility of having a truly private language.
They are incoherent pragmatically and essentially.
Pragmatically, because we see all natural languages are learnt publicly.
Essentially, because language is essentially for communication; and an essentially private language cannot be used to communicate between two distinct persons; but also for the single self when he speaks to himself reflexively; so it's oxymoronic ie incoherent.
It's interesting to note though people do talk about private languages - for example the private symbolism and language in Blakes prophetic poems, but this usually means that new senses have been overlaid in a much more complex manner than a pragmatic or instrumental use of language would entail.
