Nominalism and realism

In the previous lecture we saw that unrestricted realism leads to difficulties, including contradictions. Yet another argument can be presented in support of this claim. As we know, the realist interpretation of subject-predicate sentences relies on the notion of exemplification. Thus, the meaning of the statement “Socrates is courageous” is explicated as “Socrates exemplifies courage”. But now we can observe that the last statement can receive a treatment similar to that we applied to the sentence “Socrates is the teacher of Plato”. Namely, we should interpret it as stating that the pair (Socrates, courage) exemplifies the two-argument relation of exemplification E. But this manoeuvre can be repeated: now the last statement is explained as follows: the triple (Socrates, courage, exemplification) exemplifies the three-argument relation of exemplification E’. Clearly, this procedure can be repeated indefinitely, and hence it leads to an infinite regress (this regress is related to the so-called third-man argument considered by Plato and subscribed to by Aristotle, as well as to Bradley’s argument against the existence of irreducible relations). It is open to a debate whether this regress is vicious, but it can be maintained that it actually is, for according to the semantic rules accepted by the realist, the initial statement “Socrates is courageous” does not have a definite meaning unless it is explicated in terms of the meaningful sentence “Socrates exemplifies courage”. However, this last sentence is meaningful only if it can be explicated in terms of yet another meaningful sentence and so on. Consequently, sentences of natural language will never receive their proper meaning.

Another restriction that can be placed on realism stems from the relation between the universals and the particulars that exemplify them. A radical version of realism, called Platonism, insists that universals exist independently of whether they are exemplified. A more moderate view, originally proposed by Aristotle, rejects universals that are not exemplified. Thanks to this restriction, Aristotelians can maintain that universals exist in things (in space-time), and that the way we can discover them is through abstraction from individual objects. On the other hand, Platonists must assume that universals exist beyond space-time, because unexemplified universals do not have specific locations. Aristotelians argue that postulating things that have no power in the spatiotemporal world is useless. Platonists reply to this that unexemplified universals, such as the property of being a unicorn, are necessary to account for the meaning of certain statements (for instance the false sentence “This animal is a unicorn”). They also point out that the fact that some universals are not exemplified is a contingent matter (there could be unicorns), and the existence of universals should not be contingent.

Nominalists reject the existence of universals for many reasons. They point out that postulating universals violates the principle of ontological parsimony (Ockham’s razor). Nominalists emphasise that universals are troublesome entities. One particular problem is their location. If we agree with the Aristotelians that universals are located where the objects that exemplify them are located, then we have to accept some unintuitive consequences, such as the multilocation of properties. For instance, one property can be said to be located at a certain distance from itself. The problem of locating relations is even more difficult (they cannot be said to be wholly located where separate relata are located, or even where the mereological sums of the relata are located). If, on the other hand, we follow the Platonists in their view that universals have no location at all, the question arises how we can have any knowledge about them (without being able to causally interact with universals). Another criticism of universals is that they don’t admit clear-cut criteria of numerical identity. This follows from the fact that universals are not extensional. Two properties can be instantiated by exactly the same particulars, and yet be numerically distinct. An example can be: the property of being the greatest planet in the solar system, and the property of being the fifth planet from the Sun. It is quite clear that these are distinct properties, and yet they are exemplified by exactly the same object: the planet Jupiter.

Several solutions to the problem of the criterion of identity/distinctness for universals can be proposed. One proposal involves possible worlds: two properties are considered identical if they have the same extension in all possible worlds. But this solution still has an unintuitive consequence: two properties that are necessarily empty (such as being a square circle and being a triangular circle) will be treated as one. Another possibility is to stipulate that properties defined with the help of distinct fundamental universals are numerically distinct. According to this criterion the two above-mentioned properties are distinct because the property of being a square and the property of being a triangle are distinct (they are exemplified by distinct individuals). But the following case remains problematic: the property defined as being divisible by 4 and being divisible by 3, and the property of being divisible by 2 and being divisible by 6, are both the same property of being divisible by 12.

The most radical version of nominalism, austere nominalism, insists that there are only particular, individual objects: individual people, tables, trees. The phenomena of attribution agreement, and of objective similarities between individuals, do not require any explanation. The subject-predicate sentences are treated as primitive, not explainable with the help of any further statements. The main problem of the austere nominalist is how to account for abstract expressions in natural language. The only available strategy is to paraphrase the sentences involving abstract reference in the form of statements that are about individual objects only. A simple example of that sort of paraphrase is as follows: instead of saying that triangularity is a shape, we can express the same idea in the sentence “All triangular objects are shaped objects”. But austere nominalism does not offer any systematic way of making nominalist paraphrases. Each sentence has to be approached individually, on a case-by-case basis, and there is no guarantee that a satisfactory solution will be found. Examples of troublesome cases are: “Courage is a moral virtue”, “This tulip and that rose have the same colour”, “Each object possesses a property that we will never know”. The first one cannot be simply interpreted as the statement that all courageous people are morally virtuous, for this last sentence is obviously false. It may be true that all courageous people are morally virtuous ceteris paribus, but this just means that if two people possess the same moral virtues except that one is courageous and the other is not, then the one that is courageous is more virtuous than the other one. It is unclear how a nominalist can express this thought without relying on properties.   
Reading:
 
M.J. Loux, The problem of universals I, pp. 30-43; The problem of universals II, pp. 46-62 (Metaphysics, A Contemporary Introduction).