What follows is another short essay I wrote for my Epistemology & Metaphysics exam several months ago. This one provides an overview of the 'universals debate' in metaphysics.
Metaphysical realists posit the existence of universals - multiply-exemplifiable properties that are shared by various particular objects. Their existence is posited primarily to explain three phenomena: attribute agreement, predication, and abstract reference.
When two objects agree in attribute (e.g. both are red), the realist suggests that this is because they both exemplify the same universal (e.g. that of redness). As for predication: to say that an object "is red", is just to say that it has the universal in question (redness). Abstract reference is the realist's strongest support. It seems that we are able to meaningfully refer using abstract nouns such as "redness" and "wisdom". The realist's explanation is that we are simply referring to the appropriate universal.
There are problems with realism, however. These abstract objects seem very odd; it is difficult to see precisely how - or where - they are supposed to exist. Presumably they exist outside of space and time (at least, so says the Platonist, who allows for the existence of unexemplified universals, contrary to the Aristotelian), but then how do we interact with them? This seems to raise serious epistemic difficulties.
Furthermore, it could be argued that the very notion of unrestricted properties (universals) is inherently inconsistent. For consider the apparently meaningful property of Being Non-Self-Exemplifying (BNSE) - the property which is held by all and only those properties that do not exemplify themselves. We can ask: is this property itself self-exemplifying?
Suppose it is. It thus has the property of BNSE, so it is not (by the definition of that property). Conversely, suppose it is not self-exemplifying. It thus has the property of being non-self-exemplifying - which is of course precisely the property required for us to say that it is self-exemplifying! So either way, there is a contradiction.
The realist must therefore deny that there is any such property as BNSE. Perhaps exemplification is merely a 'tie', not a relation. But such responses seem implausibly ad hoc.
For these reasons, and Ockham's Razor, nominalists suggest that we had best not posit such entities (i.e. universals) unless absolutely necessary. They tend to recommend we take predication and attribute agreement as basic, unanalysible facts about particulars. The difficulty is in explaining abstract reference. To deal with this, several different variants of nominalism have arisen.
According to austere nominalism, talk about universals is really just disguised talk about particular, concrete objects. For example, to say "John prefers red to blue" is translated to "John prefers red objects to blue objects". A problem is that such translations are not always accurate - John could like the colour red whilst hating the objects which happen to be that colour. So the austere nominalist must appeal to ceteris paribus clauses, and other such complications, all of which are deemed to be unanalysible.
It is also unfortunate that we cannot achieve any genuine abstract reference with this theory. For suppose we took the set of F-objects as being the referent of F-ness. Then we would get the unacceptable result that the properties of being human and being a featherless biped were one and the same, as they apply to identical sets of particulars.
This last problem can be overcome if we extend the sets to cover all possible worlds. Then, because there is a possible world where chickens have no feathers, those properties would be appropriately defined as distinct. Possible-worlds nominalism is thus quite a powerful option. But there is still the problem of sets which are necessarily co-referential, e.g. the set of 3-angled objects and that of 3-sided objects. We would not want to say that triangularity and trilaterality are one and the same property.
An intriguing solution to this problem would be to extend our ontology to include impossible worlds. That impossible particular which has three sides but four angles would differentiate the two properties for us.
Alternatively, one might say these properties are built out of distinct component parts, for example "being sided" + "being 3 of ___", as opposed to "being angled" + "being 3 of ___".
For an entirely different nominalist approach, one might appeal to tropes: unrepeatable attributes. This approach fits nicely with our perceptions - we can see the redness of a particular object, existing in space and time right there with it. We might also see an exactly resembling redness on another object, but it is nevertheless numerically distinct - just like two jerseys off a factory line might be made of exactly resembling material, yet of course the particular threads composing each are numerically distinct.
A possible objection to tropes is again a problem with sets. Trope theory takes abstract reference to point to sets of tropes (not objects). Though this avoids the previously discussed set problems, there is still the matter of unexemplified tropes, all of which point to the same set: the empty set. The simplest response is just to join the Aristotelian in denying that unexemplified properties exist. Alternatively, one could plot a parallel course to that discussed above, where we extend our sets to cover all possible worlds (or even impossible ones, if we wish to distinguish between impossible tropes).
Another popular form of nominalism is metalinguistic, translating talk about universals into talk about language. Two immediate problems with this are achieving universality across languages (i.e. making properties multi-lingual), and dealing with only tokens of words, not types (which would be a form of universal).
Both objections are met by Sellars' dot notation. Just as quotes around a word refer to that word in English, so dot-quotes refer to that word and all its functional equivalents, in any language. As for the type/token problem, Sellars suggests his dot-quoted words function as common nouns - singular distributive terms that refer to multiple tokens simulaneously.
So just like "the lion is tawny" refers to many particular lions (according to his analysis of singular distributive terms), so "the *courageous* is a virtue predicate" refers to many particular tokens of "courageous" (and its functional equivalents in other languages).
It is worth noting that Sellars' "functional rules" or "linguistic roles" sound suspiciously like universals - though Sellars insists that they need not be understood as such. Another possible objection is that it just seems counterintuitive that our talk about abstract concepts is merely talk about words.
This raises the interesting possibility of a concept-based nominalism (which I understand was more popular with medieval philosophers than modern ones). A realist's universals are mind- and language-independent. Concepts are in many ways similar, except that they are entirely mind-dependent: they exist only in people's minds, not out there in the external world. This strikes me as an intriguing and intuitive possibility, though there are probably difficulties inherent in the notion of a concept. Concepts may, for example, share a similar type/token problem to that the meta-linguistic nominalist faces with words.
Of all the alternatives on offer here, trope theory would seem to be the least problematic. Realists are stuck with a bloated and implausible "two-worlds" ontology, and have difficulty explaining away the paradox of self-exemplification. Austere nominalists have the opposite problems: though ontologically simple, they must appeal to much theoretical complexity to justify their various translations. Metalinguistic nominalism has difficulty escaping universal notions, and the same may be true of concept-based nominalism. Possible-worlds nominalism has a lot of potential, but the expansion to impossible worlds (which may be required on technical grounds) might strike many philosophers as overly implausible. I think it is an interesting (im)possibility, though!