The contemporary naturalist's best bet for an account of conscious states seems to be representationalism: reducing conscious states to representational states. For independent reasons, I am very friendly to this reduction. Let us assume representationalism.
Representational states represent reality (including the epistemic agent) as being a certain way. But now consider different kinds of aesthetic awareness, say aesthetic awe or aesthetic repugnance or beauty-appreciation. If representationalism is true, each of these states represents reality as being a certain way. But aesthetic statements like "This is sublime", "This is hideous" and "This is beautiful" are connected to kinds of aesthetic awareness. For instance it seems that aesthetic awe gives rise to statements (or exclamations) of "This is sublime", aesthetic repugnance gives rise to "This is hideous" and beauty-appreciation gives rise to "This is beautiful." But once it is granted that a state of aesthetic awareness have representational content, surely the aesthetic statement that it is naturally connected with expresses that representational content. When I am visually aware of a red cube, I say "That's a red cube" and what I say expresses at least a part of the representational content of my awareness.
Thus, once we accept representationalism in the philosophy of mind, we should accept cognitivism in aesthetics. The representational content of aesthetic awe is surely something like the sublime, the representational content of aesthetic repugnance is surely something like the hideous and the representational content of beauty-appreciation is the beautiful.
Granted, the above argument is compatible with the sublime, the hideous and the beautiful being indexical or mind-dependent. That's a matter for further investigation. But the argument does make it difficult to be a non-cognitivist if one is a representationalism.
And the same argument applies in the moral sphere. If representationalism is true, moral admiration and moral repugnance have representational content, and sure if they have representational content, that content is something's being morally admirable or repugnant, respectively.
Objection: Although the aesthetic or ethical feelings have representational content, that content is an inner state of the individual, and not the sort of thing that could be the content of aesthetic or ethical claims. Imagine the content of aesthetic awe is the fluttering of the heart (no doubt it's something more subtle). Then representationalism is satisfied. But plainly the fluttering of the heart is not the object of statements of (say) sublimity. We have two constraints on what could be the object of a statement of sublimity: (1) it has to be appropriately connected to the right sorts of aesthetic consciousness, and (2) it has to fit with enough other things we say. Fluttering of the heart fits with (1)--it is on this toy theory the representational content of aesthetic awe--but not with (2).
Reply: It's certainly true that the fluttering of the heart is not at all a plausible content for aesthetic statements. But likewise it is not a plausible content for aesthetic awareness. Suppose I am having a state of aesthetic awe at a performance of King Lear. The representational content of that state is not the fluttering of the heart. For then there would be no difference in representational content between aesthetic awe at one performance of King Lear and aesthetic awe at another performance of the same play. But there is, since it is a part of the awe, qua conscious state, that it is awe at this performance, and so when the performance is different, the representational state is different. In other words, the fluttering of the heart does not capture the directedness of the awe.
I think there are two routes for the non-cognitivist now. The first is to say that the object is something like the performance's causing fluttering of one's heart. But once we do this, it is hard to resist saying that the sublime just is whatever causes one's heart to flutter. I don't say that the latter is a plausible theory--but is just as plausible as saying that the representational content of aesthetic awe at the performance is the performance's causing of the fluttering of one's heart. (Interestingly, on this view, we seem to have perception of causal features of the world, pace Hume.)
The other route is to entrench and say that the conscious component of aesthetic awe just is an experience of the fluttering of the heart (say), but that we mistakenly describe this by saying it is awe at the performance. Rather, it is awe caused by the performance. I think this is not true to the phenomenology, however.