Defining what, exactly, is a property is a harder task than you might think. Of course on the macroscopic level the task is relatively easy, a property is just a kind of similarly between objects. And an instance of such a property is simply whatever features in a particular object are responsible for that similarity. For example, things are macroscopically red because they reflect similar wavelengths of light, and the instantiation of redness in each of them is whatever is making them reflect those wavelengths (usually some feature of their surfaces).
But ultimately these macroscopic similarities and macroscopic properties are reducible to more and more fundamental properties, which we appeal to in describing the similarity between two objects. Ultimately we reach the fundamental physical properties, and since these fundamental physical properties are not reducible further they are not so easily explained by an appeal to similarity, as we can’t give a reason as to why two fundamental constituents are similar except by appeal to the very property we were trying to explain in terms of that similarity. What makes the property of having a certain spin value the same property in two different fundamental particles?
One way to resolve this problem is to define a property as the set of objects which we say have that property. Thus the property of having a certain spin value would actually be the set containing all the fundamental particles with that property. And we could thus explain why two particles have the same property, or at least systematize it, by appealing to the fact that they are members of the same property set. Of course why the particles belong to the sets that they do may still seem like an open question, but at a certain point our explanations must simply bottom out. Ideally we would like to appeal to natural law, but unfortunately natural law doesn’t seem to deal with sets.
The other option is to identify a property with its causal effects. Under this definition of properties two particles have the same spin because they have the same causal effects, or, more precisely, the same causal dispositions. Under this definition of properties natural law does, in some sense, underlie the fundamental physical properties, because all it means to have such a property is to have certain causal effects, which means being part of a certain kind of regularity, which is what we call natural law.
Neither definition may seem inherently better than the other at this point, so let’s consider how we can tell if two objects have the same property given these definitions. If we can’t determine whether two objects have the same property then clearly we are dealing with a poor understanding of what a property is, because if properties can’t be distinguished from each other then it would seem that they aren’t doing any explanatory work, and hence aren’t a good description of the world. And indeed defining a property as a set of objects fails by this criterion, because there is no way for us to determine if two objects belong to the same set, since we don’t have access to these theoretical sets. In theory there could be a nearly infinite number of properties that all had the same causal effects, and we wouldn’t be able to tell them apart. Thus identifying properties with causal effects may seem like a better choice. But how can we tell when two things have the same causal effects? Certainly we can’t appeal to a similarity in results, since that relies on us being able to tell what properties those results have, and here we are trying to establish that we can tell properties apart from each other, so such a definition would be circular. But there is a way out of this problem, by appealing to an observer of those effects. An observer, in this context, is simply a system with a memory, the ability to compare its internal states, and that ability to be put in various states by external events. We can define what it means to have the same causal effects then by saying that an observer sensitive to those effects would think that they effects of both were the same (the observer judges that the state it is put into is the same in both cases). Of course in real life any observer may make mistakes, but here in our definition we can appeal to a hypothetical ideal observer. And thus we have a way to tell causal effects apart, and thus a way to tell properties apart given that they are identified with causal effects.
Thus I favor identifying properties, at least the fundamental ones, with causal effects. Such a definition, as shown above, allows the distinctions between properties be meaningful instead of arbitrary. Furthermore, it makes the properties themselves a byproduct of the regularities of nature instead of a cause of them, meaning that properties would thus be a description of the world, as they should be. In contrast, identifying properties with sets would seem to make them a kind of framework that the world is built on, and, moreover, would make the sets themselves physical objects, since they would part of the causal explanation of why particles behave as they do. Not only does this seem implausible, but physics just doesn’t seem to be heading in that direction. And thus the identification of properties with causal effects seems like a better treatment of them on several levels.