Versions of nominalism

Metalinguistic nominalism proposes a more uniform method of paraphrasing statements containing abstract terms. The general idea is to replace terms referring to putative universals (properties, relations, kinds) by terms describing linguistic expressions. Thus the statement “This ball is red” can be explicated as “This ball satisfies predicate ‘red’”. “Triangularity is a shape” becomes “’Triangular’ is a shape predicate”, and the troublesome sentence “Courage is a moral virtue” gets translated into “’Courageous’ is a virtue predicate” (note that the word “virtue” is clearly ambiguous: in the first sentence it serves as a noun, and hence carries an unwanted commitment to properties, whereas in the second sentence it becomes an adjective, modifying the noun “predicate”). Similarly we can treat the sentence “This tulip and that rose have the same colour”, rephrasing it as “This tulip and that rose satisfy the same colour predicate”. But it is unclear what the ontological status of linguistic expressions is, and whether a nominalist can accept them in his ontology. First we have to make a distinction between types and tokens. A token of an expression is an individual inscription or utterance. Hence each word has more than one token which belong to one and the same type. In the above examples of metalinguistic paraphrases the subject terms are singular, not general, hence it looks like they refer to types, not tokens. But types resemble universals in all relevant aspects: they are entities that are common to all individual tokens of a given expression, hence they can be interpreted as the common property of all inscriptions (utterances).

Another problem with metalinguistic nominalism is that it trades the objective, independent notion of property for a language-dependent notion of predicate. But what with properties that are not expressed in any language? There are examples of properties that we discovered and named only recently, such as spin or charm. It is quite natural to expect that there are more properties of that sort which have yet to be discovered. Consequently, a metalinguistic nominalist can’t offer a satisfactory translation for the sentence “Every object has a property that we will never know”.

The initial motivation for metaphysical realism was provided by the existence of objective similarities between particulars. Resemblance nominalism tries to develop and apply the notion of similarity without any recourse to universals. It may be said for instance that to be red is to be sufficiently similar to a paradigmatic red object. But this simple interpretation won’t do. Clearly there may be non-red objects that are similar to a selected red thing (with respect to every property other than colour). A more sophisticated attempt to explain away the attribution of properties may be as follows. The resemblance nominalist may try to define resemblance classes which will roughly correspond to the realist’s properties. For instance, a resemblance class can be defined as a maximal class such that any two objects in this class are more similar to one another than they are to any object outside of the class (more formally this condition can be spelled out as follows: for all x, y, and z, if x and y belong to class K but z does not belong to K, then x is more similar to y than to z). The condition of maximality is needed, because we don’t want to qualify the class of two red objects as a resemblance class. However, three fundamental objections can be made against such a solution.

(1) As we have already indicated, it can be argued that you can find a non-red object which is more similar to a particular red object than this object is to another red thing. Think for example of a green sphere, a red sphere of exactly the same dimensions, and a red cube twice as big as the sphere. It can be argued that the spheres resemble each other more that the red one resembles the red cube.

(2) Let’s consider two properties P and Q such that all objects that have P have Q but not vice versa. In such a case P will not define a resemblance class, for the condition of maximality fails.

(3) The universal class (the class of all particulars) satisfies the condition of being a resemblance class. But it is debatable whether there is a (non-trivial) property that is common to all particulars.

It should be clear that the problems (1) and (2) are a direct consequence of the fact that the nominalist cannot distinguish between various aspects of the similarity relation (for instance we would like to say that the class of red objects is defined by the relation of similarity with respect to colour). One solution that promises to evade this difficulty is known as trope theory. It postulates a new kind of objects – tropes – that may be acceptable to nominalists. Tropes are individual properties: the redness of that rose, the shape of that tree. Two numerically distinct individuals can never share any tropes. However, their tropes can be similar. The idea is that resemblance classes can be defined on tropes, and not on particulars, so that the resemblance class corresponding to redness will contain all tropes of redness. It is easy to notice that problems (1) and (2) disappear in this approach. No non-red trope can be more similar to a particular trope of red than a different trope of red, because tropes don’t have any ‘aspects’: they are themselves aspects. If two tropes are similar, they are always similar in precisely one respect. Problem (2) disappears, because the class of tropes that correspond to one property is always disjoint from the class of tropes corresponding to a numerically distinct property, even if all objects that posses one property possess the other one as well.

An interesting question arises whether the Principle of the Identity of Indiscernibles can be reinterpreted in trope theory. A simple replacement of properties with tropes results in a trivialisation of the PII. In virtue of the definition of tropes, if two individuals share at least one trope, they are numerically identical. A more promising strategy is to reformulate the PII in the form of the requirement that if each trope of object x is similar to a trope of object y, and vice versa, then x is numerically identical with y. Another point worth mentioning is that trope theory cannot accommodate unexemplified universals, hence it is more appropriate for a reinterpretation of the Aristotelian version of realism rather than the Platonist one.

Further readings:

M.J. Loux, The Problem of Universals II, pp. 62-79 (Metaphysics. A Contemporary Introduction)

E.J. Lowe, Realism Versus Nominalism, pp. 355-365 (A Survey of Metaphysics)