1. Many philosophers of mind who eschew substance dualism opt for a property dualism. Allowing only one category of substances, material substances, they allow at least two categories of properties, mental and physical. An example of a mental property is sensing red, or to put it adverbially, the property of sensing redly, or in a Chisholmian variant, being-appeared-to-redly. Any sensory quale would serve as an example of a mental property. Their irreducibility to physical properties is the reason for thinking of them as irreducibly mental properties. This post, taking for granted this irreducibility, focuses on the question whether it is coherent to suppose that a mental property could be had by a physical substance. Before proceeding, I will note that it is not just qualia, but also the phenomena of intentionality that supply us with putative mental properties. Recalling as I am right now a particular dark and rainy night in Charlottesville, Virginia, I am in an intentional state. So one can reasonably speak of my now instantiating an intentional mental property.
In sum, there are (instantiated) mental properties and there are (instantiated) physical properties, and the former are irreducible to the latter.
2. Now could a physical thing such as a (functioning) brain, or a part thereof, be the possessor of a mental property? Finding this incoherent, I suggest that if there are instantiated mental properties, then there are irreducibly mental subjects. Or perhaps you prefer the contrapositive: If there are no irreducibly mental subjects, then there are no irreducibly mental properties. But it all depends on what exactly we mean by mental and physical properties.
3. What is a physical property? An example is the property of weighing 10 kg. Although there are plenty of things that weigh 10 kg, the property of weighing 10 kg does not itself weigh 10 kg. Physical properties are not themselves physical. So in what sense are physical properties 'physical'? It seems we must say that physical properties are physical in virtue of being properties of physical items. And what would the latter be? Well, tables and chairs, and their parts, and their parts, all the way down to celluose molecules, and their atomic parts, and so on, together with the fields and forces pertaining to them, with chemistry and physics being the ultimate authorities as to what exactly counts as physical.
So I'm not saying that a physical property is a property of a physical thing where a physical thing is a thing having physical properties. That would be circular. I am saying that a physical property is a property of a physical item where physical items are (i) obvious meso- and macro-particulars such as tables and turnips and planets, and (ii) the much less obvious micro-particulars that natural science tells us all these things are ultimately made of. Taking a stab at a definition:
D1. P is a physical property =df P is such that, if it is instantiated, then it is instantiated by a physical item.
Admirably latitudinarian, this definition allows a property to be physical even if no actual item possesses it. This is is as it should be.
4. Now if a physical property is a property of physical items, then a mental property is a property of mental items. After all, no mental property is itself a mind. No mental property feels anything, or thinks about any thing or wants anything. Just as no physical property is a body, no mental property is a mind. So, in parallel with (D1), we have
D2. P is a mental property =df P is such that, if it is instantiated, then it is instantiated by a mental item.
(D2) implies that if there are any instantiated mental properties, there there are irreducibly mental items, i.e., minds or mental subjects. Now there are instantiated mental properties. Therefore, there are irreducibly mental subjects. For all I have shown, these subjects might be momentary entities, hence not substances in the full sense of the term, where this implies being a continuant. The main point, however, is that what instantiates mental properties must be irreducibly mental and so cannot be physical. Therefore, brains could not have mental properties.
This flies in the face of much current opinion. So let's think about it some more. If you countenance irreducibly mental properties being instantiated by brains, do you also countenance irreducibly physical properties being instantiated by nonphysical items such as minds or abstracta? Do you consider it an open question whether some numbers have mass, density, velocity? How fast, and in what direction, is that mathematical function moving? If physical properties cannot be instantiated by nonphysical items, but mental properties can be instantited by nonmental items, then we are owed an explanation of this asymmetry. It is difficult to see what that explanation could be.
5. My argument, then, is this:
a) If there are any instantiated mental properties, then there are irreducibly mental subjects.
b) There are some instantiated mental properties.
c) There are irreducibly mental subjects.
(a) rests on (D2).
The attempt to combine property dualism with substance monism is a failure. If all substances are physical, then all properties of these substances are physical. If, on the other hand, there are both mental and physical properties, then there must be both mental and physical subjects, if not substances. A physical item can no more instantiate a mental property than a mental item can instantiate a physical property.