A. J. Ayer was a non-cognitivist who thought that moral judgments are expressions of an agent's feelings of approval or disapproval. According to him, what one does when one makes a moral judgment is express one's moral sentiments--one doesn't report or describe some fact. Ayer's version of non-cognitivism is known as emotivism.
Consider a moral judgment of the form: "X is wrong" Emotivists like Ayer think that this should be understood to mean something like: "Boo X!" or "X, meh!" Just as I'm not asserting something that could be true or false when I say "Go Badgers!," I am not asserting something that could be true or false when I say "X is wrong." This is because what I'm really doing by saying this is expressing my feelings about X. Such an expression--it's a sort of exclamation or rooting--isn't the sort of claim that could be true or false, or so Ayer's argument goes.
One objection to emotivism involves moral disagreement. If I say "X is wrong" and you say "X is morally permissible," it appears that we have contradicted each other and that there's some fact about which we are in dispute (i.e., whether it is the case that X is wrong).
Here's (part of) Ayer's reply to this objection:
"When someone disagrees with us about the moral value of a certain type of action, we do admittedly resort to argument in order to win him over to our way of thinking. But we do not attempt to show by our arguments that he has the 'wrong' ethical feeling towards a situation whose nature he has correctly apprehended. What we attempt to show is that he is mistaken about the facts of the case..."
(Language, Truth, and Logic, 1946, 111)
So he thought that moral disagreement isn't a case of having contradictory beliefs (as the cognitivist would see things). Rather, it's a matter of having feelings that clash or do not coincide. I feel that X is wrong and you feel that X is morally permissible. These feelings are not contradictory. What would it even mean to say that feelings are contradictory? Given this, Ayer thought that there really isn't any such thing as moral disagreement. There's only disagreement about certain facts (e.g., whether Mary stabbed Paula on purpose or whether a fetus is a person) and this is what explains any intuitions we have about there being a conflict between the claim that X is wrong and the claim that X is morally permissible.
I don't find Ayer's reply to the "moral disagreement objection" very satisfying. I'm not going to offer much of an argument here, since I'm primarily interested in what my students think about this issue. One general worry I have about emotivism has to do with a divergence in how things appear and how things really are (assuming that emotivism is true). That is, for example, the emotivist denies that a moral claim like, "X is wrong" functions as a subject-predicate sentence. When one says something of the form, "a is b," she usually means to ascribe a predicate/property (b) to a subject/thing (a). Emotivists deny that this is true of moral claims. They think that "X is wrong" is not a subject-predicate sentence. The "surface grammar" of such a claim is misleading.
A similar point can be made regarding moral disagreement. It appears that you and I actually disagree when I say that X is wrong and you say that X is morally permissible. That is, it appears from this exchange that we disagree about a fact--whether X is wrong--and that what we're doing is more substantive than simply expressing our feelings (which happen to differ) regarding X. It seems different than when I say something like, "I love dark ales" and you say, "I prefer lighter lagers." There's a sense in which we're disagreeing, but it's not the same sense as when we have a moral dispute. It seems to me that moral disputes are often genuine disagreements, much like when I say that "Wellington is the capital of New Zealand" and you say "No, it's Auckland." As indicated above, however, Ayer denies that things are as they appear.
So, the emotivist denies that appearances should be taken to indicate what's really going on. Given this, I think the emotivist has some work to do to justify why we ought to accept her theory over one that doesn't have such counterintuitive commitments. To his credit, Ayer does offer arguments for why one should accept emotivism. But only a few of these speak to this issue of why we should view things quite differently than they appear--and, as you can tell, I find these to be rather weak. Comments about this and other issues related to emotivism are most welcome.