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).

Universals

The concept of universals arises very naturally from the observation that things are similar to one another and that thanks to these similarities they can be grouped in various categories. Classification of objects is the foundation of all knowledge, and without it we wouldn’t be able to discover and describe any regularities. Most probably even the language that we speak would not be possible without things being similar in different respects. A natural metaphysical explanation of the fact that two things of a given kind (trees, chairs, electrons) have “something in common” is that there exists a certain object that stands in a particular relation to those things. For instance, two red flowers can be said to resemble one another because there is a third object, redness, that is somehow present in both flowers.

Thus the notion of a general object – a universal – is born. The fundamental relation that is supposed to hold between ordinary things (which we will now refer to as “particulars”) and universals can be called “instantiation” or “exemplification”. The colour red is instantiated, or exemplified, by various red objects. Hence we can broadly characterize universals as entities that are exemplified (more formally: u is a universal if and only if there is an x such that x exemplifies u), while particulars can exemplify but are never exemplified themselves. We can distinguish two categories of universals: monadic and polyadic. Monadic universals are instantiated by individual objects, while polyadic universals are exemplified by pairs, or more generally n-tuples of objects. Monadic universals are primarily properties, although some insist to identify a separate category of monadic universals, namely kinds. Polyadic universals, on the other hand, comprise relations of many arguments (such as being the father of, or being a child of two parents). We can also distinguish universals of the first order, which are instantiated by particulars only, and universals of higher orders, instantiated by other universals.

Postulating the existence of universals helps us analyzing some semantic features of basic subject-predicate statements of the natural language. If we consider the statement “Socrates is courageous”, it can be said to be true when the object described as “Socrates” exemplifies the property represented by the predicate “is courageous” (i.e. courage). However, the predicate “is courageous” cannot be said to refer to (to be the name of) the property of being courageous. Two different semantic functions have to be distinguished: that of denoting (referring, naming) and that of connoting. The predicate “is courageous” denotes all courageous people, but connotes courage. On the other hand, the name “Socrates” denotes an individual, but connotes nothing (there is no property with respect to which we call an individual “Socrates”). Thus, the semantic analysis of statements of the form “a is P” is as follows: this statement is true if the object denoted by “a” exemplifies the universal connoted by “P”.

The assumption of the existence of universals can also account for the phenomenon of abstract reference of natural language. Abstract reference is visible in sentences of the following sort: “Redness is a colour”, “Courage is a moral virtue”, “This rose and this tulip have the same colour”, “Each object has a property that no one will ever know”, “The relation of being an ancestor is the smallest relation satisfying the following conditions: if x is a parent of y, x is an ancestor of y, and if x is an ancestor of y and y is an ancestor of z, then x is an ancestor of z”. The terms “redness”, “courage”, “colour”, “moral virtue”, “property”, “the relation of being an ancestor”, can be naturally interpreted as denoting (not connoting!) universals. (Exercise: read all the past lectures and find sentences containing abstract reference there.)

The ontological position accepting the existence of universals is called metaphysical realism (realism in short), and its negation is referred to as nominalism. The unrestricted version of realism maintains that for every meaningful predicate of natural language there is a universal connoted by it. But it is highly unlikely that this version of realism could be true. Let us consider the following example: the predicate “has inertial mass m or electric charge q”. While it is natural to expect that the separate predicates “has mass m” and “has charge q” indeed connote some properties, the disjunctive predicate is unlikely to pick a new property. After all, it is unintuitive to assume that an object that has mass m and no electric charge, and another object that has a different mass m’ but electric charge q”, have something in common by virtue of this very fact. Moreover, disjunctive properties typically do not figure in causal explanations or laws of nature.

There is a more formal argument that can be made against the unrestrained version of realism. Let us consider the predicate “is not exemplified by itself”. If this predicate connotes a property, then a contradiction ensues, for it will be true that this property exemplifies itself if and only if it does not exemplify itself. This argument is a version of Russell’s famous paradox of the class of all classes that are not their own elements.

This lecture is based on Chapter 1 "The Problem of Universals I" of Michael Loux's book Metaphysics: A Contemporary Introduction. You are advised to read the entire chapter.

Identity and Indiscernibility

What logical relations connect qualitative identity (indiscernibility) and numerical identity? First of all, there is Leibniz's law, or the principle of the indiscernibility of identicals, which states that numerical identity implies indiscernibility. More formally, this principle can be written as follows: For all x, y, if x = y, then for all P, if x has P, then y has P. (If you are worried that this formalization can only ensure that all x's properties will be y's properties, note that you can swap x with y and get the conclusion that all y's properties are x's properties as well). Alternatively, we may put this idea like that: if x has a property that y doesn't have, x must be distinct from y. Leibniz's law seems to be rather uncontroversial; however some possible counterexamples to it are conceivable. Let us consider the property of number 8 that it is necessarily greater than 6 (as a matter of mathematical necessity, 8 has to be greater than 6). But 8 is numerically identical to the number of planets in the solar system (excluding Pluto). But is it really necessary that the number of planets is greater than 6? (We will analyse this problem in more detail later in the course.) This example teaches us that perhaps not every proposition true of a given object picks out its property. The expression 'it is necessary that' is an example of an intensional context, which means that the truth of a statement built with its help depends not only on the reference of the terms included in the statement, but also on their meaning (the meaning of 'number 8' is obviously different from the meaning of 'the number of planets in the solar system').

A more controversial principle connecting numerical identity with qualitative identity is the principle of the identity of indiscernibles (PII for short). It states that all indiscernible objects are identical, or more formally that for all x and y, if x and y have the same properties, x is numerically identical with y. Alternatively, PII can be presented as the principle of the discernibility of distinct, i.e. if x is not numerically identical with y, x and y are discernible (there is at least one property that they don't have in common). It is important to notice that the meaning and status of PII depends on how broad the category of properties which can discern individual objects is. For instance, suppose that we agree that for any object a being identical with a is a property of a. In such a case PII becomes trivially satisfied, for if a and b agree with respect to all their properties, b will have to possess the property of being identical with a which settles the issue of their numerical identity. In order to avoid this trivialisation, we should restrict the range of acceptable properties to so-called qualitative properties (properties that do not involve numerical identity, or any form of 'labelling'). On the other hand, it is acceptable to discern objects with the help of relational properties (for instance, being at a given distance from a third object c).

At first sight it looks like PII should be satisfied at least by all material, spatiotemporal objects, for numerically distinct things are discernible by their different space-time locations. This conclusion can be accepted if we assume that material objects are impenetrable, i.e. that they cannot occupy the same space at the same time. In spite of this argument, it turns out that plausible counterexamples to PII can be presented. One of them is the famous Max Black example involving two spheres. Suppose that the universe consists only of two objects which happen to be two perfect spheres made of pure iron of exactly the same dimensions. In such a case the spheres seem to be indiscernible even by their location, because there is no third object in this entirely symmetrical universe which would stand in different spatiotemporal relations to the spheres. Black's counterexample does not show that PII is actually false (obviously our universe is not even similar to this impoverished world), but at least proves that PII can be only contingently true (that the falsity of PII is conceivable).

But a stronger claim can be made that PII is actually false in our world. According to modern physics, subatomic particles do not possess definite trajectories, thus their discernibility in terms of individual locations is not guaranteed. Moreover, quantum physics introduces the so-called symmetrization postulate which puts severe constraints on possible states of many-particles systems of the same type. The postulate requires that the state of many particles be either a symmetric, or an antisymmetric function* (the first case is when the particles in questions are bosons, e.g. photons; the second when they are fermions). From the postulate it follows that all measurable properties of particles of the same type assume the same values (either deterministic or probabilistic). This fact is interpreted by many as a clear sign that PII is violated in the case of elementary particles of the same type.

*) A symmetric function is a function that gets transformed onto itself when we swap its arguments. An antisymmetric function changes its sign under this operation.

Appendix

Recently an argument in favour of the discernibility of quantum particles has been proposed which involves the notion of weak discernibility. Two objects x and y are said to be weakly discernible if there is an irreflexive relation R (a relation that does not hold between an object and itself) which connects these objects. It can be argued that fermions are connected by an irreflexive relation "having opposite spin directions". But it is doubtful whether weak discernibility presents a case of genuine discernibility. It can be easily verified that the weak discernibility of two objects x and y can only guarantee that they can be discerned in the following way: the relation R connects x with y, but does not connect x with x, hence x discerns relationally x from y. But this is the same sort of 'discernibility' as in the case of the property of being identical with x, and therefore should not be accepted.

Identity

Two notions of identity have to be distinguished: qualitative identity and numerical identity. Qualitative identity can be defined in a relatively simple way: two objects are qualitatively identical if and only if they have all properties in common. Formally, we can write this as follows: x is qualitatively identical with y iff for all properties P, x has P if and only if y has P. It remains to be decided what is the range of the property variable P, but let's put this aside for the moment. However, the notion of numerical identity presents a challenge. It is not trivial to come up with a direct definition of this concept. One suggestion may be to characterize numerical identity as the relation that holds between each object and itself, and only itself. But this definition only appears to be correct. In fact it is not, since the condition that every object is identical only to itself is in fact empty, as it can be presented equivalently as follows: if y is distinct from x, then x is not identical with y. But this last expression is tautologous, hence does not restrict the notion of identity in any way. Consequently, we are left with the characteristic that identity holds between every object and itself, and this is definitely too broad.


Another possible definitional characteristic of numerical identity is similarly bound to fail. Suppose that we stipulate that for x to be numerically identical with y there has to be exactly one object that is both x and y. But the expression "exactly one object" contains hidden reference to the notion of identity. The standard way of interpreting the expression "There is exactly one x such that Px" is "There is an x such that Px and for all y, if Py, then y is identical to x". The proposed definition of identity turns out to be circular.


It is commonly accepted that the relation of numerical identity should satisfy the following conditions:


1. Reflexivity. For all x, x is identical to x
2. Symmetricity. For all x, y, if x is identical to y, y is identical to x
3. Transitivity. For all x, y, z, if x is identical to y, and y is identical to z, then x is identical to z


But obviously not all relations satisfying 1-3 are identities. The conditions 1-3 define what is called equivalence relations. However, equivalence relations are closely connected with identity. It turns out that each equivalence relation can be turned into numerical identity if we change appropriately the domain of the relation. To give an example, let us consider the relation of being parallel defined on the set of all straight lines on a plane. This relation is clearly not identity, but if we introduce the abstract notion of a direction (meant either as the property common to all mutually parallel lines, or as their set), then it can be claimed that this relation is reducible to identity when defined on the set of all directions (two "parallel" directions are actually one and the same direction).


In order to obtain the relation of identity, we have to add to 1-3 the following condition:


4. Identity is the smallest relation satisfying conditions 1-3


where the condition 4 is understood in such a way that the relation of identity must be included in any relation satisfying 1-3.