The setup is that physical tropes (that are causally relevant) fit into a class of similar tropes to form a homogeneous property. A mental property is a higher-level (not 'higher-order') property that contains classes of these same physical tropes, and other physical tropes that instantiate the same trope-functional mental trope, e.g. pain is c-fibers and/or h-fibers and/or o-fibers... (pg550-1) Since the physical trope that is causally relevant falls into both a physical property and a mental property, the problem looks solvable. However, authors throw up the following objections: if a physical trope is causally relevant, in virtue of what? Prima facie, it seems relevant because it is a physical property, not because it is also a mental one. (pg552) Secondly, just because you call a higher-level property 'mental' doesn't make it mental-- there are lots of higher-level properties that are also physical. (pg553) Finally, authors claim that logically there is no connection between a causally relevant physical trope that is a physical property and a mental property, even if that causally relevant physical trope also inhabits that mental class. After all, there are several other heterogeneous classes which that physical trope will also inhabit that should not be considered causally relevant. (pg553-4)
A way out for the trope theorist is to claim not that properties are classes of tropes but instead just the tropes within the classes. (pg554) Authors attribute this view to Heil & Robb. Here is where authors level their biggest objection: the trope theorist misses the point of causal relevance: it isn't that a P property and an M property are identical therefore both relevant; causal relevance is the problem of: what is effected in virtue of what property? (pg555) This problem authors claim the trope theorist fails to address. Instead, authors offer up the Property Exemplification Account (PEA). PEA says that events like having a pain right now not only has the property of 'being a pain event' but also is an exemplifying of a property like 'has-pain'. (pg556). Here's how it works: objects are the subjects of events. In objects, property exemplifyings occur. When a property is an exemplifying in an event, it is actually exemplifying in the subject. A property exemplifying in a subject at a time is constitutive of an event, (pg556-7) though does not 'constitute' the event the same way e.g. a chair parts constitute a chair. (pg559)
Authors then posit that not only do events have constitutive properties (the properties of the objects), but events also have 'characterizing properties' as well. Characterizing properties have exemplifyings in events, and constitutive properties have exemplifyings in objects, the subjects of those events. (pg560) This sets up two sets of properties that an event can have. Kim argues that mind-body identity in events must be between constitutive properties of events, but authors consider instead that the identity should be between the properties of the events, not of their objects (subjects).
From here authors elaborate what a property is according to the PEA, and claim that two distinct properties can have exemplifyings in the same object of an event. In this case, you can have a mental property and a physical property exemplify in the same object of an event. They claim the mental-physical co-instantiation is a supervenience relation that is similar to the metaphysical relation between 'being colored' and 'being red' (pg561). So mental properties and physical ones are both exemplified in the same subject in the same event. Authors then argue that the 'universalist understanding' of properties forces the causal efficacy of mental events, since when a mental property is exemplified in an object of a physical event, that event is constitutively a mental event as well.(pg562) The result is that all properties that are exemplified in the subject of an event become efficacious. The immediate objection arises: too many properties are efficacious! Authors argue that this isn't a problem-- the only problem is if too many properties are causally relevant. (pg563)
To save causal relevance from this objection, authors introduce another thesis that works off the 'is colored'/'is red' relation by talking about different levels of mental and physical properties. Mental properties are 'higher-level' than their lower-level physical ones, but related in that when the lower-level one is exemplified, the higher-level one automatically is. Authors call this the Property-Dependence thesis (pg564). Crucial to this is understanding that a mental property of 'thinking of Vienna' is a higher-level property of 'neuro-state x'. The causal relevance of the lower-level physical property then can become the causal relevance of the higher-level mental one of the same object in the same event. Yet not every property becomes causally relevant (though every one could be considered causally efficacious) since not every property of the object is a higher-level property of the lower-level, causally relevant physical property. Lastly, mental properties aren't considered constitutive to the event (I guess they are part of the 'characterizing properties' of the event). This is a supervenience relation that authors analyze (pg565).
The next step is to show that mental properties can be causally relevant qua mental properties, not because they supervene on causally relevant physical ones. Authors claim that the framing of this problem by Kim is hostile to this possibility, so if they can show that the causal relevance of the mental no more problematic than any other causal relevance claim, they have done enough (pg567). At this point they draw the distinction between property instances and properties themselves. Causal efficacy is about property instances; causal relevance is about the properties themselves (that are instanced in objects of events). This distinction serves to show that there can be many physical properties being instanced in an event that will not be causally relevant to some of the effects. Authors claim this is a by-product of having a metaphysics that allows for multiple properties to be exemplified in the same event. In other words, there are some properties being relevant to some effects, other properties relevant to other effects, and so on. So it isn't just mental properties but also other physical ones that may fail to be causally relevant (for a particular effect property exemplifying). Given that this is context-dependent and empirical, authors insist it would be 'churlish' to reject the mental. (pg568)
The last objection is one that claims that Davidson's Principle of the Nomological Character of Causality (PNCC), combined with the position that the mental is anomalous (the mental doesn't figure in causal laws), makes mental property relevance impossible. (pg571) Authors first argue that PNCC isn't the only enduring causal theory, and that proposals from the likes of Lewis have suggested co-variance as a theory of causation. (pg572) These new propoals remove the 'covering-law' as necessary for causation, therefore still leaving open the possibility of mental property relevance along with physical property relevance. The lesson here is that 'overdetermination has to do with causal instances-- efficacy, not relevance' (pg574) Authors argue that two 'co-variation relationships' (causally relvant properties) can exist 'harmoniously'.