Recall that the metaphysical realist thinks that universals exist: when two objects agree in attribute, that is because there is some one thing they have in common. For example, if two objects are both red, then that is because they both possess the same property (universal) - that of "redness".
By contrast, the austere nominalist denies that universals exist - only particular objects exist. If two objects agree in attribute, then that is just a basic unanalysable fact. There is no property of "redness" floating out there in the world. When people refer to "redness", they're really just refering to the set of all the red objects in the world.
Now, there is a nice compromise between these two extremes, called Trope theory. The Trope theorist basically takes universals, and de-universalises them. That is, he agrees with the realist that we can analyse objects in terms of the properties they possess, BUT he denies that it is possible for two objects to share the same property. Properties are understood not as being multiply-exemplifiable 'universals', but rather, as being individual, singular 'tropes'.
If two objects are both red, then that just means that they both possess red tropes. It is important to note that each trope is distinct - even if they are exactly resembling (e.g. the exact same colour), they nevertheless are numerically distinct tropes. In contrast to realism, then, the two objects do not share the same constituent property (trope). Instead, they are made up of two distinct tropes, two different constituents, though the two happen to be exactly resembling.
Trope theory was portrayed rather unsympathetically in class, since both our lecturer and tutor find it counterintuitive ("crazy" might have been the word they used). But I disagree - for although I'm more inclined towards nominalism regarding properties (they don't really exist, do they?), I at least think Trope theory is better than metaphysical realism. And it's not really as counter-intuitive as they suggested.
Consider two jerseys churned out by a factory, which look exactly alike. Are they made of the same material? Well, no, not exactly. They're made of the same type of material (cotton, wool, or whatever), but if you pull out a thread from each jersey, you will nevertheless agree that they are two different pieces (or 'tokens') of material. So trope theory is just like that, but with all the other properties too. Are the jerseys made of the same colour? The same shape? The same size? No, in each case, the shape/size/colour is exactly resembling between the two jerseys, but they nevertheless are distinct tropes.
Problems with sets:
Now, according to the trope theorist, "redness" does not refer to some ethereal universal (unlike the realist), nor the set of all red objects (unlike the austere nominalist), but instead, it refers to the set of all red tropes. But, as I've discussed before, set-based accounts raise certain problems:
One obvious problem with set-based accounts is that properties which are exemplified by all the same objects would refer to the same set, and so, according to this account, would mean the same thing. For example, austere nominalism considers "being a featherless biped" and "being human" to be the same property, which is obviously an unacceptable result. Possible worlds nominalism improves this, since obviously the sets will differ in other possible worlds (where, say, chickens have no feathers). But there will still be some properties which are co-exemplified (i.e. exemplified by the same set of objects) in all possible worlds - Loux gives the example of being triangular and being trilateral (all objects with three angles also have three sides, and vice versa).Tropes overcome this to some degree (it doesn't matter if properties are possessed by all the same objects, for we are comparing sets of tropes, not sets of objects). But what about non-existent tropes? For example, in the real world, no tropes of either "elven" or "dwarven" actually exist. Now, supposedly "being a dwarf" refers to the set of all "dwarven" tropes, which is the empty set. Yet "being an elf" refers to the set of all "elven" tropes, which is also the empty set. Hence, the trope theorist concludes, being an elf and being a dwarf are actually the same thing. This, of course, is an unacceptable result.
There are two ways to solve this problem:
1) Expand the set of tropes to cover those in all possible worlds - an analogous solution to that of the austere nominalist in the quoted paragraph above.
2) Deny that non-exemplified properties refer to anything meaningful. I don't much like this answer, but it's acceptable enough, since it's precisely what the Aristotelian realist believes about universals anyway (in contrast to the Platonist).
Another set-based objection is that mathematical sets have their members necessarily. So if there was one more red trope in the world, that would be a different set - "redness" would refer to something different from what it does now! And that seems odd. This is a powerful objection, but I think it can be overcome by distinguishing strict (philosophical) identity from loose (common-sense) uses of the word. I'll discuss that more in a future post.