Describing mental properties as supervening on physical ones is an attractive proposal. It eliminates some possibilities about the mental (such as a complete Cartesian independence with respect to the physical) and, almost as importantly, encapsulates what might be messy metaphysical details about what exactly properties are, ect. Unfortunately supervenience, although a step in the right direction, has its failings. Most significantly it admits the possibility that the mental properties are some kind of epiphenomenal stuff that is simply correlated with the right kinds of physical properties. And if we can admit such possibilities then clearly describing the mental properties as supervenient can’t resolve questions about mental causation or reduction.
To fix these problems we must revise our definition of what supervenience is, making it narrower so that some possibilities are eliminated, and thus we can deduce from it a definite position on issues such as mental causation. Such a redefinition, however, is basically replacing our existing concept of supervenience with some new idea. In my opinion this “new” idea should be dependence. Dependence is by no means a new way of describing connections between objects/properties. For example, Husserl described essences as dependant on the objects they were abstracted from, which I take to mean that without the objects the “essence” wouldn’t exist, since it is not something extra above and beyond the objects. We can also understand dependence, in a less ontologically loaded sense, in terms of descriptions. For example, the description “red book” depends on the description “red”, since “red book” can only be applied when the object meets the requirements to be “red” (and “book”, as I define it below we would have to say that “red book” depends on “red” and “book”, not just one of them alone). Thus dependence, like supervenience, can be seen as sweeping at least some of the metaphysical framework under the rug, so to speak, since there are a number of ways to interpret what dependence means metaphysically.
No matter how we interpret it there are certain conditions a relationship between properties must meet if it is to be described as one of dependence. When defining it here I will call properties that are in the same role of the mental ones (i.e. dependant on some simpler or “lower” level of properties) the abstract properties, and the properties they are dependant upon the concrete properties. Please note that this is simply a convenient way of labeling them, and not a borrowing of the well developed philosophical notions of abstract and concrete. After all, the concrete properties may very well depend on some other properties, and thus be abstract in a different context. The first requirement that a dependence relationship must meet is the supervenience requirement, which means that determining if the abstract property holds or doesn’t for a particular object can be determined solely by examining other properties, although not necessarily the same properties in all cases. This collection of other properties is the concrete properties. Secondly, for any given instance of the abstract property there will be subset of the concrete properties that are “realizing” it. This act of realization can also be called instantiation, meaning that that instance of the abstract property is identical with that collection of realizing concrete properties. I must point out that the realizing concrete properties in any given instance are a subset of all the concrete properties (in many cases) because there may be more than one way, in fact a great number of ways, that an abstract property can be realized. For example the property of being a wheel is an abstract property that is can be realized by a vast collection of physical properties. Finally there is the non-circularity requirement, which is that no concrete property that an abstract property depends on may itself depend on that abstract property.
It is the second requirement, that the abstract property be instantiated by the concrete properties in specific cases, which allows dependence to say more about mental properties than supervenience does. For starters, because the abstract property is being instantiated by some concrete properties we can say that it is a description or way of looking at those properties, which in turn means that if that collection of concrete properties has causal powers so does the associated instance of the abstract property. For the same reasons we can say that a specific instance of the abstract property is reducible to the concrete properties that are realizing it, even though the abstract property as a whole may not be so reducible, since it can be realized in many different ways.
Hopefully it is obvious how dependence can be applied to describe the relationship between physical properties and mental properties (I’ll give you a hint, mental properties are the abstract ones). However, even though it is correct to say that the mental properties depend on the physical properties I think it is more informative to say that the mental properties depend on certain functional/computational properties, which then in turn depend on the physical. I claim this is more informative because there are fewer ways in which functional/computational properties can instantiate mental properties, and it is thus easier to reduce and explain mental properties in terms of them than it would be to construct such explanations in terms of the physical properties alone. Ultimately, since dependence is transitive, adding a layer of properties between the mental and the physical doesn’t change the relationship between the mental and physical properties, although it would still require an independent argument to show that such a layer was a good description of reality (see here).