Cognitivism is compatible with the claim that moral statements have truth values that vary with the speaker. (despite lack of explicit indexicals, yadda yadda). The contrary claim is that they don't. I don't see why the one claim should be more readily comprehensible that its opposite.
The contrary claim is often called realism, although that muddies the water, since in addition to the epistemological claim it can be used to state the claim that moral terms have real referents.
"Cognitivism encompasses all forms of moral realism, but cognitivism can also agree with ethical irrealism or anti-realism. Aside from the subjectivist branch of cognitivism, some cognitive irrealist theories accept that ethical sentences can be objectively true or false, even if there exist no natural, physical or in any way real (or "worldly") entities or objects to make them true or false.
There are a number of ways of construing how a proposition can be objectively true without corresponding to the world:
By the coherence rather than the correspondence theory of truth
In a figurative sense: it can be true that I have a cold, but that doesn't mean that the word "cold" corresponds to a distinct entity.
In the way that mathematical statements are true for mathematical anti-realists. This would typically be the idea that a proposition can be true if it is a entailment of some intuitively appealing axiom — in other words, apriori anayltical reasoning.
Crispin Wright, John Skorupski and some others defend normative cognitivist irrealism. Wright asserts the extreme implausibility of both J. L. Mackie's error-theory and non-cognitivism (including S. Blackburn's quasi-realism) in view of both everyday and sophisticated moral speech and argument. The same point is often expressed as the Frege-Geach Objection. Skorupski distinguishes between receptive awareness, which is not possible in normative matters, and non-receptive awareness (including dialogical knowledge), which is possible in normative matters.
Hilary Putnam's book Ethics without ontology (Harvard, 2004) argues for a similar view, that ethical (and for that matter mathematical) sentences can be true and objective without there being any objects to make them so.
Cognitivism points to the semantic difference between imperative sentences and declarative sentences in normative subjects. Or to the different meanings and purposes of some superficially declarative sentences. For instance, if a teacher allows one of her students to go out by saying "You may go out", this sentence is neither true or false. It gives a permission. But, in most situations, if one of the students asks one of his classmates whether she thinks that he may go out and she answers "Of course you may go out", this sentence is either true or false. It does not give a permission, it states that there is a permission.
Another argument for ethical cognitivism stands on the close resemblance between ethics and other normative matters, such as games. As much as morality, games consist of norms (or rules), but it would be hard to accept that it be not true that the chessplayer who checkmates the other one wins the game. If statements about game rules can be true or false, why not ethical statements? One answer is that we may want ethical statements to be categorically true, while we only need statements about right action to be contingent on the acceptance of the rules of a particular game - that is, the choice to play the game according to a given set of rules." -- WP
Nothing in this is at all illuminating as to what on earth realism is supposed to be.
Continuing my quest to untangle people's confusions about Eliezer's metaethics... I've started to wonder if maybe some people have the intuition that the orthogonality thesis is at odds with moral realism.
I personally have a very hard time seeing why anyone would think that, perhaps in part because of my experience in philosophy of religion. Theistic apologists would love to be able to say, "moral realism, therefore a sufficiently intelligent being would also be good." It would help patch some obvious holes in their arguments and help them respond to things like Stephen Law's Evil God Challenge. But they mostly don't even try to argue that, for whatever reason.
You did see philosophers claiming things like that back in the bad old days before Kant, which raises the question of what's changed. I suspect the reason is fairly mundane, though: before Kant (roughly), it was not only dangerous to be an atheist, it was dangerous to question that the existence of God could be proven through reason (because it would get you suspected of being an atheist). It was even dangerous to advocated philosophical views that might possibly undermine the standard arguments for the existence of God. That guaranteed that philosophers could used whatever half-baked premises they wanted in constructing arguments for the existence of God, and have little fear of being contradicted.
Besides, even if you think an all-knowing would also necessarily be perfectly good, it still seems perfectly possible to have an otherwise all-knowing being with a horrible blind spot regarding morality.
On the other hand, in the comments of a post on the orthogonality thesis, Stuart Armstrong mentions that:
This is not super-enlightening, partly because Stuart is talking about people whose views he admits he doesn't understand... but on the other hand, maybe Stuart agrees that there is some kind of conflict there, since he seems to imply that he himself rejects moral realism.
I realize I'm struggling a bit to guess what people could be thinking here, but I suspect some people are thinking it, so... anyone?