Typically, when I think that P, I can express this thought by uttering "P". We expect the thought and its expression to have the same truth value. For example, if I believe that grass is green, and I say to you "Grass is green," then presumably my belief and the corresponding utterance are both true. But let's try something a bit tricky. Suppose I start thinking about the very thought I am having right now. I might think to myself, this is a thought token. Can I express this thought?
Suppose I follow the usual formula, and proclaim aloud: "This is a thought token." That statement would be false. It is a speech token, not a thought token! But the aforementioned thought was certainly true. My thought was a thought token, but my utterance was not. So we must either say that the utterance failed to express the thought, or else it is a case where a thought and its expression can differ in truth values. Perhaps this is a merely terminological difference, but I think the former sounds more sensible.
It is plausible, after all, that the indexical nature of the statements leads them to differ in semantic content. (Compare: if you and I were both to utter "I am Richard Chappell", we would be saying different things, with different truth values.) So let us label the original thought 'T'. Perhaps the way to express T is not merely to verbalize it (by saying "This is a thought token") as before, but instead to substitute out the indexical and say: "T is a thought token".
That seems inadequate. We have lost the indexical information, and are thus saying something that is importantly different in some sense. Let T2 refer to my thought that this token is token T2. Now, there is something far more trivial about the statement "Token T2 is token T2" -- it lacks the cognitive significance of the former thought. They differ in primary intension, with only the latter statement being 1-necessary (or a priori).
Let us call a token "tricky" if it refers to its own tokenage in the sort of tricky way exemplified in the above examples. I think we should then conclude that tricky thoughts are strictly inexpressible. The best we can do is express a closely related thought that has been stripped of the self-referential indexical information.
I don't know whether there's any great point to noticing this, but I found it kind of fun. Well, I guess it is a significant result to note that the "typical" formula described in the first sentence of this post does not work universally.
A final (mostly unrelated) question about expressing mental states: suppose I assert "I believe that P". As I understand it, this does not express my belief that P, but instead expresses my second-order belief that I believe that P. (You can see this by following the typical formula.) But then how do we directly express desires? I can't say "I desire that P", because that merely expresses my belief that I desire that P. It seems that expressions of desire must be non-cognitive, rather than statements (i.e. what you say must not be truth-evaluable). Perhaps it should instead be an imperative, e.g. "Make P true!". Or even an action, such as doing what you believe will bring about P. Anyway, I'm sure someone who's given the issue more thought must've written something about this at some stage. If you know more, leave a comment!