A number of folks seem to think that there is some innate "pragmatic contradiction" in assertions of the form: "p and I don't believe that p". Certainly, whenever I've heard these Moorean sentences mentioned, the mentioner assumed this. Yet, there are counterexamples to the "pragmatic contradiction" thesis. And this fact seems to be pretty well-known to people in the relevant field. I mentioned that I had some counterexamples to an ethicist and he found it surprising and interesting. But I then mentioned it to some epistemologists, and they were quite unimpressed. So, here, we have a case where inter-area communication in philosophy has failed: the people in the relevant area know that a thesis is false, while folks in other areas act as if the thesis were uncontroversially true.
For what it's worth, here are some of my counterexamples to the thesis. These counterexamples provide cases where one quite sincerely and unproblematically utters an instance of "p and I don't believe that p". Nobody I've met finds all the examples compelling. In all the examples below, "p" is a sentence and the quotation marks are meant to be right-angle quotes so one can substitute within them.
1. An expert tells me "p" and adds that ordinary people like me don't believe that p. But "p" is a sentence so replete with technical vocabulary that not only do I not know what all the words mean, I cannot even parse its grammar. I sincerely tell someone else: "p and I don't believe that p". In this case, I believe that "p" is true, but I don't believe that p. There are two responses I hear to cases like this. Some people say that the distinction between believing that p and believing that "p" is true is specious, and hence the sentence embodies a pragmatic contradiction. These people have a very low bar for what counts as belief and assertion. They will have to accept the next counterexample. Others say that the sentence is not an assertion if I don't understand it. I worry that this sets the bar for assertions too high. These folks may reject the next example for the same reason, but some of the others might still work for them.
2. An expert tells me: "p and you don't believe that p. Work out the consequences for yourself." I'm not very good at logic, so I have to do this step by step. I thus say: "p and I don't believe that p. By conjunction elimination, p. Hey, that's cool! I didn't know that p, and now I do."
3. Suppose I believe that one has no beliefs when one is in the afterlife, because the afterlife is an undifferentiated beliefless mist of joy. I write you a letter to be opened after my death. In the letter I say: "p. And I don't believe that p. I don't believe it, because right now I am an undifferentiated beliefless mist of joy. Therefore: p and I don't believe that p."
4. I write a paper. I think everything in the paper is true. I present the paper at a conference. When I present it, I am really tired. I am reading the sentences outloud, and sincerely, but I cannot parse all of them, nor do I believe their content. Some of the sentences are intermediate steps in the argument, and I've completely forgotten them (or I never believed them in the first place, though I believed them to be true; sometimes, I write down things in the course of a proof by copying and pasting an existing sentence and transforming it by rules of inference—that's just a matter of syntactic manipulation—without bothering to figure out what the new sentence means). But I still believe that whatever I am saying is true. One of the sentences is "p". So I read the sentence: "p". I then add, surprised at myself: "You know: p and I don't believe that p. I don't believe it because it's too complex to parse, and I remember that this is one of those steps that I've forgotten completely."
5. I program a robot to bring you a drink whenever I say to you: "I don't believe that the robot will bring you the drink, and the robot will bring you the drink" (this example works better with the Moore sentence re-ordered in this way). Now, you keep on interrupting me, never letting me say a whole sentence. I say the sentence sincerely; I don't believe that I will finish the sentence, and hence I don't believe the robot will bring you the drink, but the sentence is so constructed that if I do manage to say it, it'll be true, and that's all that sincerity requires.
6. I'm deaf and have been learning out how to vocalize. I do not believe I can do so yet. I know you're standing somewhere where you can't see my lips and I can't see your reactions. I say to you: "I can speak and I don't believe I can speak." I can say this sincerely, because although it's true that I don't believe I can speak, I also know that if I do succeed in saying it, it is true.
7. I have a mental inertia on which once I form the intention to do an action such as saying a sentence, often I am unable to stop even if I change my mind prior to beginning the action. (Unlike in #6, this is actually the case for me.) So, while the sentence is proceeding from my mouth, it need no longer be true that I have any intention of saying it—though I had to have had that intention. Suppose that I know that as soon as I fully form the intention to speak, Fred will (e.g., by neural manipulation) bring it about that I do not believe that p. So, right now I believe that p, but I know that at the time of utterance I won't. I say: "p but I do not believe that p." This example rests on taking the present tense in a sentence to refer to the time of utterance, not the time of deliberating whether to speak. I think this is correct: think of sentences like "It's 12 o'clock", which you utter while watching the clock—you time yourself to begin to speak so that the clock strikes 12 while you're speaking (or maybe at the very end; Richard Gale has an argument that "now" refers to the time at the end of a sentence, by reference to sports announcers who say things like "He's got the ball, no Jones has now taken it from him, but, wait, no, now he's got it back!")