As should already be evident from the previous sections, there are intimate con- nections between the issues under discussion here—the existence and nature of universals and of mathematical objects—and the ways that we use language. Among other things, it is natural to think that universals—if such exist—are the semantic values of, or somehow correspond to, our predicates (or, at least some of our predicates)—hence REDNESS , if such a universal exists, is of interest to us because of the usefulness of the predicate expression “is red.” Along similar lines, one of the reasons we might be tempted to believe in abstract objects is the fact that mathematical language isfilled with singular terms that seem to refer to abstract objects, if they refer at all. This later intuition, in fact, can be strengthened to provide an argument, srcinally due to Go tlob Frege (1884), for the claim that the subject mater of mathematics is abstract.
The Singular Term Argument for Abstracta
(1) If a simple subject–predicate statement of the form “t is P” is true, then t must refer.4
(2) Arithmetic contains an in fi
nite sequence of distinct such true state- ments (e.g., “0 is a number,” “1 is a number,” “2 is a number,” etc.).
∴(3) The terms of arithmetic refer (and refer to infinitely many distinct
objects).
(4) Only abstract objects could be the referents of these terms.
∴(5) Abstract objects exist.5
The Singular Term Argument for Abstracta, in both its srcinal Fregean form and in more sophisticated contemporary variants (see, e.g., Wright, 1983), underlies most contemporary accounts of mathematics as being about abstract objects. Furthermore, we can modify the argument so that the new version con- cludes that universals exist and are abstract. The modified argument proceeds as follows.
The Predicate Argument for Universals
(1) If a simple subject–predicate statement of the form “t is P” is true, then P must have a semantic value.6
Universals and Abstract Objects
(2) Everyday (and scientific) discourse contains a multitude of true state- ments of this form (e.g., “The ball is red,” “The salt shaker is red,” “The stoplight is red,” etc.).
∴(3) These predicates must have semantic values.
(4) Only (abstract) universals could be the semantic values of these terms.
∴(5) Universals exist (and are abstract).
Interestingly, the Predicate Argument for Universals is typically taken to be much less compelling than the superficially quite similar Singular Term
Argument for Abstracta. As we shall see in the next section, W. V. O. Quine— one of the most vocal defenders of one form of the Singular Term Argument for Abstracta—is also one of the most vocal critics of the Predicate Argument for Universals. Before moving on to that discussion, however, it is worth noting one additional issue regarding universals that is highlighted by atending to the Predicate Argument for Universals: the abundance or sparseness of these entities.
An account of universals is abundant if there is a universal corresponding to every arbitrary collection of objects, and is sparse if there are stricter criteria for universal existence. For example, a sparse account of universals might claim that only those universals that “carve reality at the joints” exist, or only those universals that correspond to a simple predicate exist, etc.7 Even if one accepts
the validity of the Predicate Argument for Universals, it is worth noting that, depending on how one works out the details, the argument need not entail the existence of a universal for each arbitrary collection of particulars—or even one for each complex predicate—in the same manner that the Singular Term Argument for Abstracta implies the existence of an object foreach and every gen- uine singular term.8 The issue, of course, hinges on what predicates can occur
in statements of the form “t is P” such that the statement in question is simple in the sense relevant to the Predicate Argument for Universals. To see the issue, consider the following two statements:
Mr. Decatur is tall.
Mr. Decatur is tall and handsome.
If we analyze these statements as having the following logical forms:
T(d)
T(d)∧ H(d)
then the Predicate Argument for Universals entails the existence of the abstract universal TALLNESS (and would, of course, entail the existence of a second uni- versal, HANDSOMENESS), but does not entail the existence of a “conjunctive”
The Continuum Companion to Metaphysics
universal TALL-AND-HANDSOMENESS. If, on the other hand, we analyze the logical form of the second statement as:
λx[T(x) ∧ H(x)](d)
where λx[T(x) ∧ H(x)] denotes a single complex predicate that holds of exactly those objects that are both tall and handsome, then the Predicate Argument for Universals would apply to the predicate:
λx[T(x) ∧ H(x)]
and we would obtain, in addition to REDNESS and TALLNESS , a universal TALL-AND-REDNESS. In short, whether the Predicate Argument for Universals entails the existence of a complex univeral TALL-AND-REDNESS depends on whether we analyze the sentence as one with two distinct predicates T and H, in which case the expression is not simple in the relevant sense, or as one involv- ing a single predicate expressing tall-and-redness, in which case the sentence is simple.
While typical extant arguments over the sparseness vs abundance issue tend to hinge on nonlinguistic issues (e.g., Lewis, 1986), the above emphasizes that the connections between universals and predicates are crucial to our under- standing not only of what universals are, but of the number and nature of the universals that do exist (if any).