By way of background, I take consciousness to be awareness of present, typically neurophysiologically encoded, intelligibility. I see qualia as of minor interest, being merely the contingent forms of awareness.
I am not a dualist. I hold that human beings are fully natural unities, but that we can, via abstraction, separate various notes of intelligibility found in unified substances. Such separation is mental, not based on ontological separation. As a result, we can maintain a two-subsystem theory of mind without resort to ontological dualism.
Here are the reasons I see intentional reality as irreducible to material reality.
1. Neurophysiological data processing cannot be the explanatory invariant of our awareness of contents. If A => B, then every case of A entails a case of B. So, if there is any case of neurophysiological data processing which does not result in awareness of the processed data (consciousness) then neurophysiological data processing alone cannot explain awareness. Clearly, we are not aware of all the data we process.
2. All knowledge is a subject-object relation. There is always a knowing subject and a known object. At the beginning of natural science, we abstract the object from the subject — we choose to attend to physical objects to the exclusion of the mental acts by which the subject knows those objects. In natural science care what Ptolemy, Brahe, Galileo, and Hubble saw, not the act by which the intelligibility of what they saw became actually known. Thus, natural science is, by design, bereft of data and concepts relating to the knowing subject and her acts of awareness. Lacking these data and concepts, it has no way of connecting what it does know of the physical world, including neurophysiology, to the act of awareness. Thus it is logically impossible for natural science, as limited by its Fundamental Abstraction, to explain the act of awareness. Forgetting this is a prime example of Whitehead’s Fallacy of Misplaced Concreteness (thinking what exists only in abstraction is the concrete reality in its fullness).
3. The material and intentional aspects of reality are logically orthogonal. That is to say, that, though they co-occur and interact, they do not share essential, defining notes. Matter is essentially extended and changeable. It is what it is because of intrinsic characteristics. As extended, matter has parts outside of parts, and so is measurable. As changeable, the same matter can take on different forms. As defined by intrinsic characteristics, we need not look beyond a sample to understand its nature.
Intentions do not have these characteristics. They are unextended, having no parts outside of parts. Instead they are indivisible unities. Further, there is no objective means of measuring them. They are not changeable. If you change your intent, you no longer have the same intention, but a different intention. As Franz Brentano noted, an essential characteristic of intentionality is its aboutness, which is to to say that they involve some target that they are about. We do not just know, will or hope, we know, will and hope something. Thus, to fully understand/specify an intention we have to go beyond its intrinsic nature, and say what it is about. (To specify a desire, we have to say what is desired.) This is clearly different from what is needed to specify a sample of matter.
4. Intentional realities are information based. What we know, will, desire, etc. is specified by actual, not potential, information. By definition, information is the reduction of (logical) possibility. If a message is transmitted, but not yet fully received, then it is not physical possibility that is reduced in the course of its reception, but logical possibility. As each bit is received, the logical possibility that it could be other than it is, is reduced.
The explanatory invariant of information is not physical. The same information can be encoded in a panoply of physical forms that have only increased in number with the advance of technology. Thus, information is not physically invariant. So, we have to look beyond physicality to understand information, and so the intentional realities that are essentially dependent on information.