The problem of abstract reference is the problem of how to best understand the reference of singular terms. This is a worry that, like the linguistic problem of predication, originates in language. One feature of predicate expressions is they can, with little trouble, be turned into singular terms— usually with the addition of some suffix such as “-ness” or “-ity.” The problem of abstract reference revolves around the question of how best to understand the use of certain terms that prima facie name or refer to properties— terms such as “redness”, “triangularity”, or “honesty.”
Call these terms, for lack of a better name, “abstract singular terms.” Both the metaphysical realist and trope nominalist claim these terms are genuine singular terms. Hence, they are said to derive their meanings from a relation of naming. They stand, within the sentences in which they appear, for the objects they designate. And each term is singular in that it is held to have but one referent. Such is the role singular terms are said to play within semantic theory. Each genuine singular term is object denoting.
The existence of terms like “redness”, “triangularity” and “honesty” has in recent years been the focus of the debate over universals. Metaphysical realists and trope nominalists have argued that nominalism is unable to provide a satisfactory account of abstract singular terms. The classic articulation of this argument is found in Arthur Pap’s “Nominalism, Empiricism, and Universals.” Elaborations on this argument can be found in Frank Jackson’s “Statements about Universals”, Loux’s Substance and Attribute, and Armstrong’s Nominalism and Realism.
In discussing the features of this line of argument, we will stay closest to the version of the argument Loux articulates in Substance and Attribute. We do this not because his discussion differs in important ways from that of the others but, rather, because he makes explicit the important role the notion of a singular term plays within the argument. Even so, we will not be offering a reconstruction of his argument so much as a way to conceptualize the structure of that argument.
In its details, the argument Loux presents is quite simple. It can be summarized as follows: 1) Either terms such as “redness” and “triangularity” are genuine singular terms or they are not; 2) Assume, to start, that they are genuine singular terms; 3)Then, from the definition of such a term, each such term must designate one and only one object; 4) Hence, it is the requirement for any adequate theory of abstract reference that, within the theory, one be able to fix the reference of all such terms; 5) Nominalism purports to be such a theory. 6) But within nominalism one cannot fix the reference of such terms; 7) Hence, either nominalism must be false or such terms must not be genuine singular terms; 8) Such terms are genuinely singular; 9) Hence, nominalism is false.
Obviously much of the weight of this argument falls on premises six and eight. If “redness” and “triangularity” are genuine singular terms and we can secure the truth of premise six, then it would seem that we are in a position to deal a serious blow to the nominalist program. Before we can evaluate this argument, however, we will first have to say more on the notion of a singular term. What we require is criteria of singularity articulating necessary and sufficient conditions for the delineation of the class of genuine singular terms.
We begin this paper with a consideration of the problem. In articulating the problem, we defend a criterion of singular term-hood. According to this criterion, terms such as “redness” or “triangularity” qualify as genuine singular terms. Hence, since such terms appear within true sentences and singular terms appearing within true sentences are held to be object denoting, there is a need to say to what it is such terms refer. It is here that nominalism stumbles. Nominalism, we argue, is unable to answer the problem of abstract reference — it lacks the semantic resources necessary for fixing the reference of abstract singular terms.
- Singular Termhood
Initially, it seems plausible to think that the sort of criteria we demand may be formulated from grammatical considerations alone. Criteria of singularity must differentiate singular terms from general terms— differentiate, that is, terms such as the name “Jeff Crowe” which purport to name objects singularly from common names like “the Crowes.” In Word and Object, Quine offers the following definition: “if a term admits the definite and indefinite article and the plural ending, then normally under our perfected adult usage it is a general term.” Conversely, if a term, e.g. ‘mama’, admits only the singular grammatical form and admits of no article, then it is singular. Yet there are problems with the criteria Quine advocates. It fails to correctly classify some terms.
Consider, for example, the term from which “mama” is derived, i.e., “mother.” “Mother” admits of both indefinite and definite articles as well as a plural ending. Consider, for instance, the following sentences: “A mother is a valuable thing to have growing up”; “mothers are honoured on mother’s day”; “The mother of my friend is still young.” By Quine’s criteria the term “mother” must be classified as a general term and, hence, should never be thought to function as a singular referring expression. But surely there are instances in which it does function as a singular referring expression. I was in a store recently when a child who had been separated from his mother screamed “Mother, where are you?” The store was crowded. The child was near a number of adult females some of whom were undoubtedly mothers. Yet if they and not his mother had come to him, he would not have been comforted.
In such a context the child’s utterance of “mother” functions as a singular term. It is used by the child to denote a particular person, namely his mother and no other. Though this example may sound contrived, it points to a critical failing in the criteria Quine has offered. Quine accepts a distinction between singular terms and general terms. He defines a general term as any term that “admits the definite and indefinite article and the plural ending.” According to this criteria “mother” must be thought a general term and never a singular term. But as our example shows there are occasions in which the term “mother” functions as a singular referring expression.
The problem with Quine’s criteria is this: It tails to be sensitive to the context of a term’s utterance. The very notion of asking of a term whether it admits of indefinite and definite articles or plural endings suggests that the term in question maybe plucked from the context in which it appears and be considered separately. But surely this is a mistake. The term “mother” though it admits of both indefinite and definite articles and a plural ending is itself, considered by itself, neither a singular term nor a general term, though relative to some context it is capable of being either.
It is general as it occurs in the sentence “A mother is a wonderful thing to have” and it is a singular term when used by the child to call out to his mother as when he yells “Mother where are you?” We should reject the attempt to draw a distinction between general and singular terms based on very general grammatical distinctions. What matters most to whether a term is general or singular is the inferential role it plays within the contexts in which it used. Moreover, we recognize that orthographically identical terms may differ on the matter of being singular or general depending upon the occasions of their use.
Better than the criteria offered by Quine are those offered by Michael Dummett in Frege: Philosophy of Language and improved upon by Bob Hale in Abstract Objects. Both Dummett and Hale reject purely grammatical criteria of singular termhood in favour of criteria that take into account the inferential roles such terms play within the sentences and contexts in which they appear. The basic idea behind both authors’ discussions of singular terms is this: Certain inferences are valid only when certain terms occupy certain positions in the premises. And though no single inferential pattern is sufficient for a delineation of the class of singular terms, Hale maintains that together several such patterns do jointly yield a set of necessary and sufficient conditions for delimiting the class of such terms.
Hale’s criteria are as follows:
T functions as a singular term in a use A(t) if:
1) In that use of A(t), the inference from A(t) to ‘there is something such that A(it)’ is valid.
2) In that use of A(t) and some use of some sentence B(t), the inference from A(t), B(t) to ‘there is something such that A(it) and B(it)’ is valid.
3) In that use of A(t) and some use of some sentence B(t), the inference from ‘It is true of t that A (it) or B (it)’ to ‘A(t) or B(t)’ is valid and;
4) the conclusions reached in (1) and (2) cannot be such that a point may be reached where a grammatically well formed request for further specification can be rejected as illegitimate.
II. Singular Terms and Nominalism
After presenting of argument, we are now in a position to revisit the concerns of this paper. Recall we concerned the possibility of constructing an argument against nominalism based on the occurrence of so called “abstract singular terms” — terms such as “redness” or “triangularity.” In representing the structure of such arguments, we constructed the following nine step argument that we take to be representative of this general line of argument:
1) Either terms such as “redness” and “triangularity” are genuine singular term o r they are not.
2) Assume, to start, that they are genuine singular terms.
3) Then, from the definition of a singular term, each such term must designate or denote one and only one object.
4) Hence, it is a requirement for any adequate theory of abstract reference that, within the theory, one be able to fix the reference of all such terms.
5) Nominalism purports to be such a theory.
6) But within nominalism one cannot fix the reference of such terms.
7) Hence, either nominalism must be false or such terms must not be genuine singular terms.
8) Such terms are genuinely singular.
9) Hence, Nominalism is false.
Such an argument is clearly valid. It is not, however, problem free. Two premises demand our attention. The first is premise six: the claim that nominalism is incapable of offering an adequate semantic treatment of terms such as “redness” or “triangularity” when these terms are assumed to be genuine singular terms. The second is premise eight: the claim that such terms are, in fact, genuine singular terms. It was in preparation for a discussion of these two premises that we engaged in an exposition and critical examination of Hale’s criteria of singular termhood. We cannot evaluate the nominalist’s responses to this argument outside of an informed opinion concerning the notion of a singular term. Hence, it is only now that we are in a position to give this argument the sort o f consideration it deserves.
The importance of premises six and eight lie in the fact that these are the premises the nominalist will attack in offering a response to this argument. In order to save his theory, the nominalist must deny one or both of these premises. That is, he must show us either that, pace premise six, he does have the semantic resources to secure the reference of abstract singular terms within his ontology or he must convince us, pace premise eight, that these terms are not the singular terms they appear to be. In what follows, we will consider whether either of these two strategies may be carried out successfully. In considering the nominalist’s possibilities for response we will implicitly assume that my nominalist has available to him all of the resources of any of the nominalist theories we have touched on thus far (i.e. he has available to him the resources not only of the ostrich nominalist but also those of the class nominalist, the predicate nominalist… ).
Consider first the nominalist’s attempt to show premise six false. The adoption of this strategy can be regarded as an implicit acceptance of the claim that premise eight is true and that terms such as “redness” and “triangularity” are genuine singular terms. As such, the nominalist accepts that there must be an entity to which each of these terms refers — that is what it is to be a genuine singular term. Genuine singular terms occurring within true sentences are object denoting and there is little doubt that “redness” and “triangularity” appear in true sentences. But to accept that “redness” and “triangularity” are singular terms and, thus, object denoting is to say nothing about the objects that they denote. The metaphysical realist of course insists that they denote abstract properties of colour and shape or universals. The nominalist, however, need not accept the metaphysical realist’s answer. He or she might respond that the metaphysical realist is fooled or seduced by language into positing entities for which there is no need. We can secure the reference of these terms, the nominalist maintains, simply within the world of physical objects or, at worst, the world of physical objects with the addition of classes.
Such is the strategy Quine explores in “Identity, Ostension, and Hypostasis.” In this work, Quine outlines how, consistent with the acceptance of “redness” as a singular term, the nominalist might, nonetheless, avoid a commitment to universals. He or she can do so, Quine writes, by allowing “redness’ or “red” to designate the spatio-temporally scattered individual that is the scattered total of all red things. Red becomes on this view “the largest red thing in the universe.” It is, if you will, the mereological sum of all rednesses. Such a strategy, however attractive it may be, cannot be consistently maintained. Quine himself acknowledges as much when he turns his attention away from colour terms to shape terms.
Consider, for instance, the denotatum of “triangularity”. If we follow the strategy Quine suggests for “red” or “redness” we will say that the denotatum of “triangularity” simply is the spatio-temporally scattered sum of all triangular things. But this is problematic. There is no guarantee when we do this that we will not generate the identical sum for “squareness” as well. Suppose, for instance, that all isosceles right triangles are created by the vertical halving of a square, then it is possible that the sum о f all such triangles will simply be the sum of all squares. But this leaves us in the intolerable position of concluding that squareness is identical to triangularity. As a general strategy for dealing with the problem of abstract reference the attempt to fix the reference of terms like “redness” and “triangularity” to the mereological sums of spatio-temporal individuals fails. Such an answer cannot be consistently maintained.
Mereological sums of spatio-temporal objects, however, are not the only objects the nominalist might suggest as being the denotata оf abstract singular terms. The class nominalist might well maintain that the denotata of such terms are classes and that sentences containing such terms are but veiled ways of referring to classes. Consider, for instance, the sentence “Red resembles orange more than blue.” “Red, “orange”, and “blue” each seem to (unction in this sentence as singular terms. What a nominalist who accepts this strategy is likely to claim about this sentence is that once it is properly understood one will see that it really concerns itself with classes of individuals and the relations that obtain between those classes. So, on reinterpretation, the following might be suggested as a perspicuous translation of the above sentence: “The class of red objects resembles the class of orange objects more than it resembles the class of blue objects.”
Again, though, there are problems, it is not clear that such a translation is truth preserving. To see this, suppose that the only red, orange, and blue objects that exist are a red and blue writing pen and an orange dance-floor. Then the class of red objects would contain but one red pen, the class of orange objects would contain but one orange dance-floor, and the class of blue objects a blue pen. But then, what we have claimed to be a perspicuous translation fails to be truth preserving, for, even though red resembles orange more than it does blue, the class of red objects does not resemble the class of orange objects more than it does the class of blue objects. Two pens resemble one another to a greater degree, whatever their colour, than does a pen and a dance-floor.
Worse still is the trouble coextensional properties pose for such an account of abstract reference. Classes come with well-defined identity conditions. Two classes are identical if and only if they have the same members. But now imagine that we live in a world where all and only triangular objects are red, then if we identify the class of red objects as the referent of the term “redness” and the class of triangular objects as the referent of the term “triangularity”, then we must say that these two terms are, at least in transparent contexts, synonyms and that a certain shape is identical to a certain colour.
Now for the above two problems, there is a solution. It is got at by going modal. We may solve these difficulties provided we allow classes to contain as their members not only all actual objects answering to a certain description but also all possible objects answering to that description. Once such modalisation is accepted, the above two problems evaporate. First, consider again the sentence “red resembles orange more than it resembles blue.” Once we allow classes to contain possible entities as well as actual ones we can then offer the following interpretation of this sentence: What it really says is, “it is necessarily the case that the class of red objects resembles the class of orange objects more than it resembles the class of blue objects.” And this sentence will be true in all and only those case where the original sentence is true. To see this we need only recognize that colour is a contingent property.
Hence, for any member a of the class of red objects there will be a counterpart to a in both the class of blue objects and the class of orange objects that differs from a only in respect to its colour. The same will be true, respectively, of any member of the class of blue objects and the class of orange objects — each will have a counterpart in the other classes that differs only in respect to its colour. Hence, if there is a difference in the resemblance of any two of these classes to the third, then it must be the colour of the objects within the class that accounts for that difference. Hence, red resembles orange more than it resembles blue if and only if the class of red things (taken modally across all possible worlds) resembles the class of orange objects (taken modally) more than it resembles the class of blue objects (taken modally).
Similarly, the modalisation of our classes seems to allow us to solve the problem of co-extensional properties. We said one problem with the above account was the possibility that all and only red objects were triangular. But if this is so and all red objects are triangular, this is merely a contingent and not a necessary relationship that obtains between triangularity and redness. It is a coincidental (and somewhat strange) fact about our world. It is not a fact about all worlds. Hence, once we allow classes to be constructed from possible individuals as well as actual ones these two classes pull apart in a way that frees us from having to identify them one with another. Once classes are construed modally the problem of co-extensional properties largely dissolves and the attractiveness of identifying classes as the referents o f abstract singular terms increases.
Still there lingers a residue of our original problem. There will still exist a subset of properties for which it will be true that the classes of individuals having those properties are identical. Intuitively, there are properties that, though different, are necessarily related. Triangularity and trilateralness are two such properties. It is necessarily the case that anything that is trilateral is also triangular and vice versa. Hence, if we adopt the strategy of regarding abstract singular terms as having as their referents classes of individuals, then we will be forced to say that, at least in transparent contexts, “triangularity” and “trilateralness” are synonyms which they are not.
Moreover, we are forced into this position even if we construct our classes from possible individuals as well as actual ones. We must conclude that this strategy, too, fails. Moreover, having now examined the two most plausible ways in which the nominalist might agree that so called “abstract singular terms” are genuine singular terms, we are forced to conclude that the nominalist, if he or she is to salvage his or her theory without emendation, must show that such is going to salvage his or her theory, then he or she must show us that it is a mistake to accept premise eight of the realist’s argument.
It is here, in the effort to show that so called “abstract singular terms” are not genuine singular terms at all, that our examination of Hale’s criteria of singular termhood becomes important. Terms such as “redness” or “triangularity” meet the first four of our criteria for being singular terms. “Redness is a colour” licenses an inference to “there is something such that it is a colour”. Equally, from “redness is a colour” and “redness resembles orangeness” we may validly infer “there is something such that it is a colour and it resembles orangeness.” So too, does “redness” meet our third criterion. From “it is true of redness that it is a colour or it is not a colour” the inference to “either redness is a colour or redness is not a colour” is valid.
And finally, the occurrence of the term “redness” is such that in the above contexts the term operates at the first level of generality. No point is ever reached where a grammatically well-constructed request for further specification may be rejected as being illegitimate in the sense we outlined earlier. Consequently, the focus of the nominalist’s rebuttal must be our fifth criterion. What the nominalist must argue is that all such terms are eliminable via perspicuous paraphrase in a fashion that makes clear that the ostensible reference to an object is merely a stylistic element of the sentence and not a necessary feature of the proposition the sentence expresses.
It is this very thing, however, that the metaphysical realist and the trope nominalist has claimed the nominalist can not show. Consider, again, the sentence “red is a colour.” According to the suggestion we are now considering, “red”, as it occurs in this sentence, is non-denoting. It is not a genuine singular term. Instead, its occurrence in the sentence is a stylistic shorthand for talking about red objects. Such sentences, the nominalist maintains, are but abbreviations of sentences that talk of objects having certain properties. Thus, “red is a colour,” under this suggestion, is regarded as equivalent to the sentence “for all x, if x is red, then x is coloured.” And as we have seen, sentences of this form commit us to nothing more than the existence of red objects — the predicates of the sentence are treated syncategorematically or as non-referring terms.
While such an answer may seem plausible, or perhaps even warranted, in the case of a sentence such a “red is a colour” there exists a subset of seemingly descriptive statements in which the apparent reference to objects cannot be eliminated via paraphrase. Instead of “red is a colour” consider the sentence “red resembles orange more than blue.” According to the nominalist’s suggestion, this sentence is to be interpreted as an abbreviated way of talking about red, orange and blue objects. Hence, the sentence will be claimed equivalent to the following sentence: “for any x, y, and z, if x is red and y is orange and z is blue, then x resembles y more than it resembles blue.”
But this is clearly inadequate. The problem, of course, is one with which we are already familiar. It is that red objects, blue objects, and orange objects may resemble one another in respects other than colour. Hence, while red may resemble orange more than blue, a red object may well resemble a blue object more than it resembles an orange o ne. Think again of our two pens, one red and one blue, and the orange dance-floor. As a paraphrase of our original sentence, the above will not suffice.
Nor are mere plausible strategies for resolving this difficulty. One manner in which one might try is this: One could attempt to build into one’s paraphrase the respect of resemblance that is said to hold between the various objects. According to this attempt, our resulting paraphrase would be as follows: “For any x, y, and z, if x is red, y is orange and z is blue, then x colour resembles y more than it colour resembles z.” But when we unpack this paraphrase what we have to say is that the notion of colour resemblance smuggles into our paraphrase a new singular term that itself will have to be excised from our analysis. Colour resemblance just is resemblance in respect to colour. And “colour” as it appears in the phrase “resemblance in respect to colour” is a singular term.
Hence, we have not excised the so called “abstract singular terms” from our analysis; we have merely replaced the existing ones with a new one that itself will have to be excised if our strategy is to succeed. A more detailed explanation is perhaps called for here. What we are saying is this: If colour resemblance was a two place predicate, then all would be fine with our proposed paraphrase for we could treat the predicate syncategorematically and thus excise from the original sentence all unwanted reference to suspicious objects. But, unlike resemblance simpliciter, colour resemblance is not plausibly construed as being a two place-predicate. It is best construed as a three-place predicate. Resemblance in some respect is resemblance in respect to something. That something must in the case of colour resemblance be colour.
Hence, our paraphrase, though it suppresses this information, involves an implicit reference to colour that, when made explicit, involves us in the use of yet another suspicious singular term, i.e., the term “colour”. Hence, we conclude that this attempt at linguistic reconstruction fails. More generally, we can say the nominalist’s strategy for eliminating such singular terms is itself highly dubious. There seems no systematic strategy for the elimination of abstract singular terms from all of the contexts within which they appear. Nominalism, considered by itself, seems unable to answer the problem of abstract reference. We must concede, we think, that nominalism, by itself, cannot answer the problem of universals.
In this paper, we have considered the phenomenon of abstract reference and how it relates to the notion of a singular term. So called “abstract singular terms”, we have argued, pose considerable difficulty for any nominalist attempt to solve the problem of universals. The nominalist has two strategies for treating such terms. He may either attempt to secure their reference within his own minimal ontology or he may try to deny of them that they are genuine singular terms and thus are object denoting. Neither strategy, we have argued, may be successfully carried out. Some of these terms must be regarded as genuine singular terms that have as their denotata objects the nominalist refuses to countenance. Nominalism by itself is thus found to be inadequate.
Such a conclusion is often regarded as a victory for the metaphysical realist. Pap, Loux, and Armstrong all regard the above conclusion as evidence for the existence of universals. Universals it is claimed are well suited to filling the role of being the denotata of abstract singular terms. With this we agree — at least, to an extent. Universals indeed seem well suited to this role. We disagree, however, that nominalism’s inadequacies constitute a de facto argument in favour of metaphysical realism. Even if we set aside the fact that trope nominalists seem able to deal with these problems, we still disagree.
Suppose that it is the case that nominalism by itself fails. Suppose as well that universals are well suited to the theoretical role of being the denotata of abstract singular terms. This alone does not constitute an argument for universals being uniquely suited to this role. It may still be the case that nominalism may be ontologically supplemented in some way that both preserves its claim to being the more parsimonious theory while, nonetheless, allowing it to answer the problem of abstract reference and, hence, the problem of universals.
Armstrong, D.M. Universals and Scientific Realism, 2 vols, Cambridge: Cambridge Univeristy Press, 1978
Armstrong, D.M. Universals, Boulder Westview Press, 1989
Dummett, Michael. Frege: Philosophy of Language, Cambirdge: Harvard University Press, 1981
Hale, Bob. Abstract Objects, Oxford: Basil Blackwell, 1987.
Jackson, Frank. “Statements about Universals”, Mind 76 (1977): 427-45
Loux, Michael. Substance and Attribute, Boston: D. Reidel Publishing Co., 1978
Quine, W.V. Word and Object, Cambridge: MIT Press, 1960
Quine, W.V. “Identity, ostension, and hypostasis”, in From a Logical Point of View, second edition (revised). Cambridge: Harvard University Press, 1980
Pap, Arthur. “Nominalism, Empricism, and Universals,” Philosophical Quarterly 9 (1959): 334-355.
Vendler. Zeno. Linguistics in Philosophy, Ithaca: Cornell University Press, 1967
 Michael J. Loux. Substance and Attribute. Boston. D.Reidel Publishing Co.. 1978, pp.61-65.
 Zeno Vendler. Linguistics in Philosophy. Ithaca. Cornell University Press, 1967. pp.33-69: Hob Hale. Abstract Objects. Oxford, Basil Blackwcll. 1987. p. 15.
 Arthur Pap. “Nominalism, empiricism, and universals. The Philosophical Quarterly Vol.9, No.37 (1959). pp.330-350
 C. Jackson. “Statements about Universals.” Mind 86 (1977) , pp.427-429; M.J. Loux. Substance and Attribute. Boston. D.Reidel Publishing Co., 1978, pp.54-87; David M. Armstrong. Nominalism and Realism: Universals and Scientific Realism. Cambridge, Cambridge University Press, 1978, pp.58-59.
 Quine. Word and Object. Cambridge. The MIT Press. 1960, p.90.
 Ibid., p. 90.
 Ibid., p. 90.
 Michael Dummett. Frege: Philosophy of language. Cambridge. Harvard University Press. 1981. p 54-80; Bob Hale. Abstract Objects. New York. Basil Blackwell, 1987 pp. 15-44.
 Bob Hale, Abstract Objects. New York. Basil Blackwell Press. 1987, pp. 15-21.
 W.Y.O. Quine. “Identity. Ostension. and Hypostasis.” in From a logical Point of View. Cambridge, Harvard University Press, 1980 (revised), pp.65-79.
 Ibid., p.72.
 there are many who would regard this response as itself being a violation of nominalism. Arthur Pap, for instance, in “Nominalism, Empiricism, and Universals” (The Philosophical Quarterly, 1959) writes that nominalists notoriously frown on “subjunctive and modalities as much as on names of attributes”(p.334). But we are inclined to take a less skeptical view of these matters. The focus of nominalism, as we interpret it, is to provide a more parsimonious answer to the problem of universals than that which the metaphysical realist provides. If this can be done consistent with the acceptance of modalities, then we see no reason to rule out such a manoeuvre on the part of the nominalist. And since universals should themselves be considered modal entities, we see no reason for thinking a nominalism that accepts some form of modality must necessarily forfeit its claim to having still provided a more parsimonious answer than that which the metaphysical realist provides.
 Arthur Pap, “Nominalism”, p.334