Types and Tokens: On Abstract Objects

Introduction

Peirce (1931–58, vol. 4, p. 423) illustrated the type–token distinction by means of the definite article: there is only one word type ‘the’, but there are likely to be about twenty tokens of it on this page.¹ Not all tokens are inscriptions; some are sounds, whispered or shouted, and some are smoke signals. And some, as David Kaplan (1990) pointed out, are empty space (e.g., in a piece of cardboard after letters have been cut out of it). The type ‘the’ is neither written ink nor spoken sound. In fact, it is no physical object at all; it is an abstract object.

Or consider the grizzly (or brown) bear, Ursus arctos horribilis. At one time its U.S. range was most of the area west of the Missouri River, and it numbered 10,000 in California alone. Today its U.S. range is Montana, Wyoming, and Idaho, and it numbers fewer than 1,000. Of course no particular bear numbers 1,000, and no particular bear ever had a range comprising most of the area west of the Missouri. It is a type of bear, a species of bear, that has both properties. For one more example, consider Mozart’s Coronation Concerto (K.537), the penultimate of his twenty-seven piano concerti. There are scores of the work, performances of it, recordings of performances of it, compact discs that contain a recording of a performance of it, and playings of the compact discs; but none of these is identical to the concerto (type) itself.

This book is, first, an attempt to make the case that types exist, and second, to explore and answer some of the questions that naturally arise if they do. It is therefore an essay in ontology, urging that there are abstract objects. Traditionally, an object is said to be abstract if it lacks a spatial and a temporal location.² So numbers, sets and propositions are said to be abstract objects. What makes types of special interest is that, unlike abstract objects of other sorts—sets and numbers, for example—types have tokens; they are in some sense repeatable.³ Are they universals, then? Many authors just assume that they are (see, for example, the quotes from Goodman in chapter 5), although some do not (see the quote from Richard Wollheim below). There is no need to take a stand on whether types are universals here, since nothing hinges on it in what follows. But I will anyway. Having instances is to my mind the hallmark of a universal, and since types are the sort of thing that have instances, they are universals. That is, the tokening relationship is a sort of instantiation relationship. Defending this claim, however, would result in a different, much longer essay, and since nothing hinges on it in what follows, I will bracket the question here. Readers who do not agree that types are universals should just view this essay as an essay on abstract objects, for that is what it is.

Whether or not types are universals, there certainly are important differences between types and such classic examples of universals as the property of being white or the relation of being between. Wollheim (1968) helpfully mentions three differences. First, he says, the relationship between a type and its tokens is more intimate than that between (a classic example of) a property and its instances. By this he means that not merely is the type present in all its tokens like the [property] in all its instances, but for much of the time we think and talk of the type as though it were itself a kind of token, though a peculiarly important or pre-eminent one (p. 76). The way I would put this last point is that types are objects. (The point will be defended in chapter 2.) Second, Wollheim notes that although types and the classic examples of properties often satisfy the same predicates, there are many more predicates shared between a type and its tokens than between a classic example of a property and its instances (p. 77). Third, he argues that predicates true of tokens in virtue of being tokens of the type are therefore true of the type (Old Glory is rectangular), but this is never the case with classic properties (being white is not white) (p. 77). The relation between types and their tokens will be taken up in more detail in chapter 6. For now it suffices to note the apparent differences between types and other abstract objects, on the one hand, and between types and other universals, on the other. Again, this is not to say that types are not universals; I think they are. Nor is it to insist that there is no concept of property under which types could turn out to be properties in the final analysis. But I am not here identifying types as properties, in view of the important differences mentioned above and below between types and the classic examples of properties such as being white.

One of the chief differences between types and such classic properties as being white has to do with the sorts of arguments advanced in favor of their existence. In the debate over classic properties and relations, it is traditional to concentrate on predicates. However, ever since Quine 1961a was published in 1948, there seems to have emerged a philosophical consensus that Quine neutralized the arguments for universals and abstract objects based on the meaningfulness of predicates—on the grounds that one need not infer from Some dogs are white that whiteness or being white exists. That is, many philosophers today share Quine’s view that predicates do not need a reference—that "The letter ‘A’ was Phoenician" can be true even if there is no property of being Phoenician. (David Armstrong [1978a, p. 16] calls Quine’s form of nominalism ostrich nominalism, I suppose because it puts its head in the sand on the question of what makes the sentence true.) Whatever the merits of the Quinean view, I shall not argue against it in this essay. There is a better argument (for abstract objects, anyway). Few think that "The letter ‘A’ was Phoenician" can be true if there is no unique letter ‘A’, or that there are exactly twenty six letters of the English alphabet can be true if there are no letters, or more than twenty six of them. Unlike such properties as whiteness and being Phoenician, types are quintessentially objects.

Types are quintessentially objects in the Fregean and Quinean senses. For Gottlob Frege, (roughly) an object is anything that can be referred to with a singular term (e.g., ‘A’, ‘the letter A ’, ‘the first letter of the English alphabet’). For W.V. Quine, (roughly) an object is anything that can be the value of a bound variable of quantification (e.g., There are exactly twenty six letters of the English alphabet). We use singular terms to refer to types and we quantify over them, not only in our everyday language but also in our best scientific theories of reality. Just how very often we do so will be made clear in chapter 1. The linguistic data presented there suggest that, far from being the exception, apparent reference to and quantification over types is the norm or nearly so in our speaking and writing habits. Types have to be reckoned with, and cannot simply be swept under the rug. In chapter 2, therefore, it is urged that if we take either Quine’s or Frege’s criterion of ontological commitment for objects seriously, we must countenance types.

Another hallmark of objecthood that types have associated with them are criteria of identity—rules that give criteria for when x and y are numerically identical and when they are different. It is commonly held that physical objects of the same sort are different if and only if they do not occupy the same spatiotemporal location. But abstract objects have criteria of identity too. Sets are different if and only if they have different members. Symphonies are different if and only if they are composed of quite different notes. Chemical elements are different if and only if they have different atomic numbers. Sometimes the criteria of identity are not so easily formulated and are a matter of theoretical debate, as with species and words. But the problem with such cases, as we will see in chapter 6, is not a lack of criteria of identity, but the existence of several (not unlike the standard problem afflicting the concept of a person).

Nominalists, of course, want to sweep abstract objects under the rug, or, rather, analyze them away. The epistemological motivation for this is explored in chapter 2. Paul Benacerraf (1983) has posed a challenge to any realist philosopher of abstract objects to explain how spatiotemporal creatures like ourselves, caught up in the causal nexus, could have knowledge of them. In recent years, the motivation for nominalism has largely come down to a felt need for some sort of causal requirement on knowledge. Although the nominalist seems at first blush to have an advantage here, I argue that recent philosophical efforts have shown that no reasonable causal requirement on knowledge is likely to rule out knowledge involving abstract objects. And in chapter 5 I show that whatever epistemological advantage the nominalist might be thought to have in virtue of a less abstract ontology is offset by the epistemological disadvantages that accompany nominalism.

For the nominalist, type talk is just a harmless façon de parler for talk about tokens. Essential to the nominalist program, then, is analyzing away all apparent references to, or quantification over, types. The usual nominalist attitude seems to be that there are a few bothersome sentences that need paraphrasing—for example, the color red resembles the color orange more than the color green, or ‘Paris’ consists of five letters—but that they are easily paraphrased in terms of sentences referring to and quantifying over tokens (i.e., spatiotemporal particulars). In chapter 3 I consider the popular suggestion that an adequate paraphrase for ‘The T is P’ is ‘Every (token) t is P’. I focus on the case of words to see if there is anything all tokens of a word have in common and show that there isn’t. Chapter 4 further explores the prospects of providing adequate paraphrasing for ‘T is P’, rejecting such promising possibilities as ‘Every normal t is P’, ‘Most ts are P’, ‘Average ts are P’, and ‘Either every (token) t is P, every normal t is P, most ts are P, or average ts are P.’ Borrowing from research in linguistics, I suggest that the best paraphrase would be ‘ts are P’ where this is a generic, or characterizing sentence, but show that even this does not work. I argue that in view of the fact shown in chapter 1 that we are up to our necks in apparent references to, and quantifications over, types, only a systematic reduction could assure us of adequate paraphrasing. The thrust of chapters 3 and 4 are that prospects for such a systematic reduction are slim. Chapter 5 explores the consequences of taking Quine and Goodman’s form of nominalism as applied to linguistics seriously to show how counterintuitive and epistemologically problematic it is.

Assuming, then, that realism about types can be shown to be more attractive than nominalism as a philosophy, and that therefore we ought to take types seriously, certain questions naturally arise. What are types? What makes a token a token of one type rather than another? (This is not to assume that a token can’t be a token of more than one type; grizzly bears are still bears.) How do we know it is a token of that type? Do some types fail to have tokens? What, if anything, do all and only tokens of a particular type have in common other than being tokens of that type? This last question is answered in chapter 3 (Nothing beyond being a token of the type); sketches of answers to the other questions are in chapter 6.

The final chapter concerns a puzzle that arises if we do take types seriously. Consider the line ‘Macavity, Macavity, there’s no one like Macavity’. The word ‘Macavity’ occurs three times in the line. The line itself occurs three times in T. S. Eliot’s (1952, p. 163) poem Macavity: The Mystery Cat, so the line, we may assume, is a type. It consists of seven words. Seven word types, or seven word tokens? Not seven word tokens, since tokens are concrete and the line is abstract. So it must consist of seven word types. But this too is impossible because there are only five word types of which it might consist! I offer a solution to this problem for words and expressions. Then I extend the puzzle to other so-called structural types (e.g., flags, molecules, and sonatas)—types that structurally involve other types (stars and stripes, atoms, and notes, respectively)—and I offer a solution to it also. David Lewis (1986a) surveys several plausible accounts of what a structural universal is, and argues that none of them work. I argue that his objections do not work against the account of structural universals that depends upon my account of occurrences.

The style of doing metaphysics in this book is not revisionist—I am not proposing that we embark on a bold new metaphysics. Although I am a great fan of Quine (and Goodman), there is a noticeable tension in Quine. He preaches ontological relativity, but also speaks of acquiescing in our mother tongue and not rocking the boat. Of course it is fun to rock the boat ontologically, as do the delightful and bizarre theories, for example, which hold that everything is a number, or that only the spacetime field and its wrinkles exist, or that there is just one very busy subatomic particle zipping back and forth in time, comprising all the matter there is. But although desert landscapes are lovely, so are other landscapes. Even the overpopulated urban one in which I find myself has a good deal of charm. The task I set myself is to figure out what inhabits the local landscape—that is, what we are committed to when we acquiesce in our mother tongue. When we do so we are committed to types.⁴

1 The Data

The distinction between types and tokens has widespread application. The present chapter will show just how widespread it is. Reference to types occurs not only in philosophy, logic, zoology and linguistics, but in most other disciplines. First we will look at the important role the type–token distinction plays in philosophy; then I will present the data that show that talk of types is thoroughly ensconced in ordinary and scientific language and theory. I will not argue in this chapter that, as a result of the truth of this type-talk, types exist; that is the thesis of chapter 2.

Nor will I argue here or anywhere else that there are no type-free statements logically equivalent to any of these type statements; I assume there often are. Chapters 3 and 4 will examine whether all of them can be so paraphrased. The purpose of chapter 1 is simply to present the multitudinous data that would require this nominalistic paraphrasing.

Type–Token Use in Philosophy

In philosophy of language, linguistics, and logic the type–token distinction is clearly important because of the central role played in all three by expressions, which come in types and tokens. Especially noteworthy is the debate concerning the relation between the meaning of a sentence type and the meaning of a sentence token (a relation that figures prominently in Grice 1969). So, for example, the sentence type ‘John loves opera’ means that John loves opera, but a speaker might say it sarcastically meaning by her token that John loathes opera.

In philosophy of mind, the distinction yields two versions of the identity theory of mind (each of which is separately criticized in Kripke 1972, for example). The type version of the identity theory (defended by Smart [1959] and Place [1956], among others) identifies types of mental events/ states/processes with types of physical events/states/processes. So, for example, it says that just as lightning turned out to be electrical discharge, so pain might turn out to be C-fiber stimulation, and consciousness might turn out to be brain waves of 40 cycles per second. On this type view, thinking and feeling are certain types of neurological processes, so absent those processes, there is no thinking or feeling. The token identity theory (defended by Kim [1966] and Davidson [1980], among others) maintains that every token mental event is some token physical event or other, but it denies that a type matchup must be expected. So, for example, even if pain in humans turns out to be realized by C-fiber stimulation, there may be other life-forms that lack C-fibers but have pains too. And if consciousness in humans turns out to be brain waves that occur 40 times per second, perhaps androids have consciousness even if they lack such brain waves.

In aesthetics, it is generally necessary to distinguish works of art themselves (types) from their physical incarnations (tokens). (See, e.g., Wollheim 1968; Wolterstorff 1980; Davies 2001.) This is not the case with respect to oil paintings like da Vinci’s Mona Lisa where there is and perhaps can be only one token, but it seems to be the case for many other works of art. There can be more than one token of a sculpture made from a mold, more than one elegant building made from a blueprint, more than one copy of a film, and more than one performance of a musical work. Beethoven wrote nine symphonies, but although he conducted the first performance of Symphony No. 9, he never heard the Ninth, whereas the rest of us have all heard it; we have all heard tokens of it.

In ethics, actions are said to be right or wrong—but is it action types or only action tokens? There is a dispute about this. Most ethicists from Mill (1979) to Ross (1988) hold that the hallmark of ethical conduct is universalizability, so that a particular action is right/wrong only if it is right/wrong for anyone else in similar circumstances—in other words, only if it is the right/wrong type of action. If certain types of actions are right and others wrong, then there may be general indefeasible ethical principles (however complicated they may be to state, and whether they can be stated at all). But some ethicists hold that there are no general ethical principles that hold come what may—that there is always some circumstance in which such principles would prescribe the wrong action—and such ethicists go on to add that only particular (token) actions are right or wrong, not types of actions. See, for example, Murdoch 1970 and Dancy 2004.

The Data

The type–token distinction is also widely applicable outside of philosophy. The main point of the current chapter—apt to be denied, ignored or understated by most philosophers (even myself, before I did the research)—is that:

Talk of types is thoroughly ensconced in ordinary and scientific language and theory.

That is, we seem to refer to and quantify over types (and other abstract objects) with an astounding level of frequency in ordinary language, science, and art—whenever generality is sought. In such contexts, apparent references to types may well be the rule, rather than the exception. Even when the reference is ambiguous as to a type or a token—for example, in the title of the Pratt (1998) article From the Andes to Epcot, the Adventures of an 8,000-Year-Old Bean—it usually turns out it is the type being referred to. Witness how in the bean case the ambiguity was not cleared up until the fourth sentence of this paragraph:

What may be the world’s oldest bean has driven Daniel Debouck, a Belgian plant geneticist, in and out of Andean mountains and valleys from Bolivia to Ecuador for 20 years. Recently, the bean has embarked on an international circuit, from Peru’s Sacred Valley of the Incas to Epcot Center at Disney World, with stops in the Midwest and Japan. This is the story of the nuna, a 8,000-year-old bean. For most of its history, the nuna has been a bit player on the agricultural stage.

Since the ubiquity of type talk is somewhat contrary to what one’s philosophical training would lead one to expect, this entire chapter has been devoted to presenting it.

The ubiquity of type talk will be shown by exhibiting many examples from various sources and disciplines. To assuage the reader’s concern that perhaps these examples are not representative but constitute merely carefully culled anecdotal evidence, we shall use as our starting point an arbitrary copy of a science periodical. Science magazine might seem a good choice, because it is a general science periodical that covers numerous areas of science. A quick read of the July 7, 1995, issue (randomly selected) reveals that at least two-thirds of the articles clearly have apparent references to types. Unfortunately, it would take the rest of this book to wade through all twenty-eight articles.¹ And, with titles like p34cdc2 and Apoptosis, or In Vivo Transfer of GPI-Linked Complement Restriction Factors from Erythrocytes to the Endothelium, there is a risk of distraction by too much biomedical jargon. Hence my second choice: one issue of the New York Times’s Science Times. It also covers general science—often relying on the same sources as periodicals such as Science—but it usually contains fewer than a dozen articles. Examining it will be a more manageable job. The selection was random since I had not looked at it before I selected it. What topics are touched on will dictate what topics we pursue in more detail after we’ve scrutinized the newspaper’s presentation. I hope the reader will be impressed by the sheer volume of the examples, their ubiquitousness, and how ordinary, familiar and harmless they seem. Those for whom this point is obvious would do well to read the section on phonology in this chapter (for it will be useful later) and then proceed to chapter 2, where it is urged that we ought to conclude from the data that types exist. (If the reader is straining at the bit to offer type-free paraphrases of all the data, she should proceed at once to chapters 3 and 4, where it is shown that there is reason to think this cannot be done.)

The January 2, 1996, copy of the New York Times’s Science Times contains nine articles on a variety of topics: one on historical linguistics, two on environmental biology, two on human genetics (four altogether in biology), one on physics, two on computers, and one on chess. Even a casual reading shows that eight of the articles contain many apparent references to, or quantifications over, types (and the ninth is not without them). The references are apparent in that they appear in the surface structure. That is, in the surface structures of the sentences, there are many quantifiers that, if they quantify over anything, must be construed as quantifying over types rather than tokens (particulars with unique spatiotemporal locations)—unless we are to impute nonsense or falsity to the sentence.