Given that higher-order theories of consciousness are committed to the claim that there are unconscious sensory states (like pains, and seeings of red, etc) and that such unconscious states are not like anything for the creature that has them, they need a way to identify the sensory qualitative properties independently of our access to those properties (i.e. independent of their being conscious). This is where homomorphism theory comes in.
Rosenthal begins by noting that we characterize our sensory qualities in terms of their resemblances and differences within families of properties. These families of properties are in turn specified by reference to the perceptible properties of things in the world. For example we can characterize red as more similar to pink than to brown and so on and these resembelances and differences are homomorphic to the family of perceptible properties (presuambly wavelength reflective properties) that give rise to the mental qualities. What we get from doing this systematically is a ‘quality space’ which is homomorphic to the quality space of the perceptible properties. Our being aware of the qualitative properties of sensory states explains how it is that we have mental access to the perceptible properties. An unconscious pain state, then, will be one that resembles and differs other pain states in ways that are homomorphic to a family of perceptible properties, and via which we gain mental access to those properties. Though there may be other ways to independently specify the qualitative properties all higher-order theories need some way to do it and homomorphism theory looks promising. It is, at the very least, an illustration that it can be done. How can we extend this to cover the requirement that there is something that it is like for a creature to have a conscious thought?
I have elsewhere argued (The Qualitative Character of Conscious Thoughts) that the propositional attitudes can be modeled as taking some specific mental attitude towards some represented proposition and that the mental attitude just is some particular way of feeling about the represented proposition. So, for instance having a belief consists in feeling convinced, that is, it is the subjective feeling of certainty that one has with respect to the truth of the represented propositon. This model of the propositional attitudes actually fits very nicely with homomorphism theory. In the sensory case we become aware of the sensory qualities, which are the properties that mental states have in virtue of which they resemble and differ each other, and which resemblances and differences are homomorphic to the resemblances and differences that hold between the family of perceptible worldly properties. Our being conscious of these properties explains how it is that we have mental access to colors. So too in thought we become conscious of the cognitive qualities and this gives us access to our thoughts. To have a conscious belief is to be conscious of oneself as having a certain cognitive quality with respect to some content. And, these cognitive qualities (that is the mental attitudes themselves) will stand in various patterns of resemblances and differences from each other in just the same way that the sensory qualities do.
What are we to say about the actual homomorphism to perceptible properties? Is there any set of properties that the mental attitudes are homomorphic to? That is, is there a set of properties that have similarities and differences which resemble and differ in a way that preserves the similarities and differences between the mental attitudes? This is important since we need a way to specify the attitudes apart from their qualitative component. Yes; we can hypothesize that the homomorphic properties are the illocutionary forces of utterances. So the differences between beliefs that p and desires that p are homomorphic to the differences between the illocutionary force of the utterance of some linguistic item in the process of expressing the belief or desire.
This even may even turn out to be an explanation of why it is that having language allows us to have more fine-grained thoughts, if we could defend the claim that being conscious of our thoughts in respect of their qualitative attitude towards some represented content gives us mental access to the properties of the language that we would use to express that thought. If this were the case then the cognitive qualities would be exactly like the sensory qualities and our theory of one could be used to explain the other. Obviously more work needs to be done to flesh this out completely, but this line of thought seems to be a promising way of extending homomorphism theory to cover propositional attitudes and so this account of the propositional attitudes should be very attractive to anyone who accepts a higher-order theory of consciousness.