The problem of universals

THE PROBLEM OF UNIVERSALS Edited and with an Introduction by CHARLES LANDESMAN Basic Books, Inc., Publishers NEW YORK : LONDON

@ 1971 by Basic Books, Inc. Library of Congress Catalog Card Number 72-158443 SBN 465-06361-6 Manufactured in the United States of America DESIGNED BY THE INKWELL STUDIO

CONTENTS

Introduction: The Problem of Universals PART ONE: The Theory of Universals 1. On the Relations of Universals and Particulars 2. Universals and Resemblances 3. On Concept and Object 4. Frege's Hidden Nominalism 5. Universals 6. Universals and Metaphysical Realism 7. Universals and Family Resemblances 8. Particular and General PART TWO: The Theory of Abstract Particulars 9. The Nature of Universals and Propositions 10. Are the Characteristics of Particular Things Universal or Particular?

11. The Relation of Resemblance 12. Qualities PART THREE Abstract Entities, Meaning, and Language 13. On What There Is 14. Empiricism, Semantics, and Ontology 15. The Languages of Realism and Nominalism 16. Grammar and Existence: A Preface to Ontology 17. A World of Individuals Bibliographical Notes Index -vi-

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Introduction THE PROBLEM OF UNIVERSALS IF one were to list the sorts of things that philosophers have characteristically disagreed about, one would find disagreements not only over the solutions to antecedently formulated problems but also over the very terms in which the problems are stated. Often, the very existence of a problem which has moved one generation of philosophers is doubted by the next. The problem of universals is no exception to this philosophical ambivalence. It is and has been a paradigm case of a metaphysical problem; yet in our time there has been a great deal of skepticism over the very possibility of metaphysics. Reasons that many philosophers have found persuasive have been offered to show that metaphysical speculation is either trival or fruitless or meaningless. In offering a collection of essays on the problem of universals, one has an obligation to justify the claim that there is such a problem. In what follows, the nature of the problem is explained and a defense of metaphysics is offered against some of the arguments of the skeptics.

I It is a truism that there is such a thing as recurrence or repetition in nature. The same colors, shapes, and sounds occur over and over again. We are continuously presented not only with novel things but with qualities and features of things that we have observed time and time again. And even the novel things very often belong to types—that is, to species and genera— which are quite familiar. As an example of natural recurrence, consider two leaves which are exactly alike in all their qualities; they have the same green color, the exact same shape, size, and feel. We have no difficulty in distinguishing between their qualities; a color, after all, is not a shape, a shape not a size; in other objects the same color coexists with different shapes, the same shapes with different sizes. In addition, we can easily distinguish between any one of these qualities and the thing itself—the leaf—which possesses it; we can distinguish between the leaf and its shape or its color; and we can imagine changes in quality—the color turning from green to brown in autumn or the shape being modified by a storm—though the object remains the same. The distinction between a particular thing and its qualities is well-entrenched in

our everyday thinking and speaking about the world. Things persist through space and time; qualities repeat themselves in different things at various times and places. The phenomenon of natural recurrence seems easily explained and accounted for. It consists of the fact that different objects have qualities in common: our two leaves, for example, have their color, shape, and size in common. It is the existence of common qualities which allows us to say that recurrence is literally repetition; the same quality literally appears over and over again. Common qualities have been given the technical name universals because of the fact that they are repeatable and can be possessed by diverse things. The spatiotemporal things which have the qualities are often called particulars or individuals or concrete entities; these are not repeatable at least in just the same way that universals are; no particular can be in totally different places at the same time, whereas a universal such as a shade of color is subject to no such restriction. The explanation of natural recurrence in terms of common qualities or universals, we shall call the theory of universals; it is also often called realism, because it asserts that universals are real. The opposite view, which tries to get by without universals, we shall call particularism. 1 It may be thought that the explanation of natural recurrence in terms of common qualities consists of a series of truisms which no sane man would want to deny. Nevertheless, at the very beginning of the western philosophical tradition, Plato found that, upon reflection, the truisms led to certain puzzles. In his dialogue Parmenides, Parmenides receives an affirmative

answer from Socrates to this question: But I should like to know whether you mean that there are certain ideas of which all other things partake, and from which they derive their names; that similars for example, become similar, because they partake of similarity; and great things become great, because they partake of greatness; and that just and beautiful things become just and beautiful, because they partake of justice and beauty. In this and other passages the Platonic ideas seem to be common qualities; a thing which has a certain quality is said to partake of or participate in the corresponding idea. Parmenides then proceeds to find difficulties in Socrates's theory. One difficulty he comes up with is especially relevant to the notion of a common quality. Then each individual partakes either of the whole of the idea or else of a part of the idea? Can there be any other mode of participation? There cannot be, Socrates said. Then do you think that the whole idea is one, and yet, being one, is in each one of the many? Why not, Parmenides? said Socrates. ____________________ 1The terminology of metaphysics is not standardized: different philosophers give different titles to the views they espouse or oppose. Other terms for realism are Platonism and the identity theory. The term nominalism is often used instead of particularism, but we shall use it later to designate one form that particularism can take. -4-

Because one and the same thing will exist as a whole at the same time in many separate individuals, and will therefore be in a state of separation from itself. Nay, but the idea may be like the day which is one and the same in many places at once, and yet continuous with itself; in this way each idea may be one and the same in all at the same time. I like your way, Socrates, of making one in many places at once. You mean to say, that if I were to spread out a sail and cover a number of men, there would be one whole including many—is not that your meaning? I think so. And would you say that the whole sail includes each man, or a part of it only, and different parts different men? The latter. Then, Socrates, the ideas themselves will be divisible, and things which participate in them will have a part of them only and not the whole idea existing in each of them? That seems to follow. Then would you like to say, Socrates, that the one idea is really divisible and yet remains one? Certainly not, he said. Suppose you divide absolute greatness, and that of the many great things, each one is great in virtue of a portion of greatness less than absolute

greatness— is that conceivable? No.... Then in what way, Socrates, will all things participate in the ideas, if they are unable to participate in them either as parts or as wholes? 2 Parmenides's argument has the form of a dilemma. The two leaves are both green. There are two alternatives for understanding this fact in terms of common qualities. Either the color green is possessed by each leaf as a whole or each possesses only a part of the color. But each alternative is absurd. The first is absurd because it implies that the same entity, the color green, can be in different and separated places at the same time, that it will be "in a state of separation from itself." The second is likewise absurd because it is unintelligible how something can be green by possessing not the quality green but only a part of the quality. In fact, it is not even clear what a part of a simple quality like green can be. The fact that realism is still a living option proves that philosophers have not taken Parmenides's argument to be conclusive. Realists usually accept the first alternative. Some deny that the allegedly absurd proposition that the same color can be in two or more places at once follows from it by asserting the universals do not have spatiotemporal location except insofar as the things that participate in them do. Others accept the implication but deny ____________________ 2Plato Parmenides, 131, Jowett translation. -5-

its absurdity. The supposition that it is absurd for a quality to be in two places at once comes from confusing qualities with particulars. For particulars, spatial separation is a criterion of nonidentity, but since qualities are different from particulars there is no reason to think that they are bound by the same identity criteria. Notice that this last response invokes a certain kind of argument which appears over and over in metaphysics. It is a category explanation. The question was raised, "How can the color green be in two different places at the same time?" The answer the realist gave was that green, being a quality, is not subject to spatial separation as a criterion of nonidentity. The question is answered first by specifying the category or ontological type to which green belongs and then pointing out that certain things are or fail to be true of it just because it belongs to that type. In general a category explanation of the question "How can some entity x have the feature F?" has the form: "x belongs to category C and its belonging to C entails that it has F." Someone might argue that a category explanation is not a final answer to the question since one can always ask how it is that something has F just because it belongs to C. But this would be a mistake; to raise this further question is like asking, "How is it that something is colored just because it is green?" or "How is it that something is rectangular just because it is square?" There are no answers to these questions other than pointing out that, given what these are, these relationships hold.

II Given this sketch of what realism amounts to, we can describe the problem of universals, or at least one form of it, in the history of philosophy as the question of whether realism provides an adequate account of natural recurrence. As it stands, however, the theory of universals hardly seems worthy of being classified as a theory. Despite the apparent difficulties that Parmenides found in it, it seems to consist of obvious truisms which are expressed in a somewhat technical vocabulary. Who can deny that things possess qualities in common, that many things can be red or green or square at the same time? The claim that this triviality is a metaphysical assertion seems pretentious. In order to explain why realism is not an uninteresting truism, it is necessary to distinguish between three things: the phenomenon to be explained, natural recurrence; our description of this phenomenon; and an explanation of it, the theory of universals. It is not easy to keep

these things separate, for the reason that almost every set of terms we use to point out the phenomenon of natural recurrence suggests one of the explanatory theories. The description and the explanation may both be formulated in the same vocabulary. For example, in indicating what we meant by natural recurrence at the very beginning of this essay, we used the vocabulary of common qualities. Thus our -6-

description is linguistically indistinguishable from the theory of universals. But there are other terms we could have used to point to the phenomenon. We could have said not that things have qualities in common, but that things resemble one another to a great degree. Or, instead of speaking of qualities or of resemblances, we could just as well have mentioned the fact that we often apply the same word to different things: we describe different things with the word "green" or with the word "square." That there is such a thing as natural recurrence is a truism; no one wishes to deny it. It is that which various metaphysical theories try to explain. The theory of universals, on the other hand, is not itself to be identified with the assertion of the existence of natural recurrence, but rather is one account of it. The situation is not unlike that in the sciences, in which the very phenomenon to be explained tends to be identified in terms of a current explanatory theory. To take an example from psychology, there are several theories attempting to explain the phenomenon of mental illness. But the very term "mental illness" already suggests a certain theoretical and practical point of view. A psychologist who rejects the point of view would naturally refuse to use the term "mental illness" to identify what he wants to explain; he will have to use another term such as "behavior problem" to indicate what he is after. If he should say that there is no such thing as mental illness, it sounds as if he is uttering a paradox, although in fact he is not rejecting the phenomenon but the theory. Similarly, a philosopher who refuses to accept the theory of universals is not necessarily denying that there is such a thing as natural recurrence, although he may wish to describe it in terms which suggest a different theory. These distinctions will not satisfy the skeptic who insists that realism is an uninteresting truism. He will say that even as an explanation of natural recurrence one cannot sensibly deny that different things are square because each possesses the shape square. That there is such a thing as the color red in virtue of which all red things are red can be ascertained just by opening one's eyes and looking. In order to demonstrate that the skeptic is again incorrect in his evaluation of the issue, it is necessary to introduce another distinction: that between commonsense and philosophical existence-claims. In everyday discourse such sentences as "The red color of that book is darker than the red color of this table" and "Both these leaves possess the same shape" may be used to make true statements which imply or presuppose the existence of certain abstract entities—colors and shapes. But these commonsense existence-claims do not yet constitute a philosophical theory, nor are they of any significance to metaphysics other than as data upon which to reflect. A philosopher may claim that statements which mention certain types of entity, such as qualities, and ostensibly presuppose their existence, are equivalent to statements which do not explicitly mention them; perhaps statements ostensibly about qualities are equivalent to statements asserting resemblances among particulars. If he can make out such a claim, then he is entitled to assert that, in a sense, there are no such things as qualities; there are only particulars. -7-

In general, if a philosopher can show that things of type A are reducible to things of type B, that A's are nothing but B's, or that statements which mention things of type A are translatable into statements which mention only things of type B, he is then entitled to say that, in a sense, things of type A do not exist but things of type B do. In a philosophical system, the types of entities out of which all other types of entities are "composed" are the system's ultimate categories. A philosophical existence-statement asserts the existence of members of the ultimate categories. The statement that qualities —e.g., colors and shapes —exist, if it is intended as a commonsense existence - statement, does not contradict the statement that qualities do not exist, if that is intended as the claim that qualities are not an

ultimate category. From the standpoint of the philosophical claim, our commonsense reference to qualities is a mere manner of speaking with which one can dispense for philosophical purposes. Therefore, the statements that the skeptic finds truistic and uninteresting are not the ones that the philosopher asserts; those that he asserts, although they often are compatible with commonsense existence-claims, are not identical with them. The skeptic is not yet finished; he may now make the notion of metaphysical explanation the target of his doubts. There are explanations of various sorts in everyday life and science. There are causal explanations when we explain a phenomenon by mentioning one of its causes, as when we explain why water turns to ice by citing the drop in temperature. There are motive explanations, as when we explain why a person did something by mentioning the aims and goals that he wanted to achieve by doing it. That there are these and other types of explanations is noncontroversial. But that there is something more we can do with a phenomenon, such as providing a metaphysical account of it, is problematic. Once we have stated something's causes or reasons, what else is there to know about it which a further explanation can accomplish? One source of the resistance to the very notion of metaphysical explanation is that such explanations are often couched in terms which make them appear as if they were rivals to the causal explanations in everyday life and science. In the passage from Plato quoted earlier there occurs the sentence, "Just and beautiful things become just and beautiful, because they partake of justice and beauty." This sounds as if Plato were offering a causal explanation of how things become just or beautiful. Yet clearly if we wanted instructions as to how to make a person just or how to make a painting beautiful, what Plato wrote would be useless. The metaphysical explanation sounds as if it belongs to primitive science which now can be replaced by sophisticated modern science. Thus Pears has written: "Because universals exist" is the answer to at least two general questions: "Why are things what they are?" and "Why are we able to name things as we do?" Though Plato and Aristotle sometimes distinguished these two questions, it was characteristic of Greek thought to confuse them. Yet they can be clearly distinguished, the first requiring a dynamic answer from scientists, and the second -8-

a static answer from logicians. Now philosophy has often staked premature claims in the territory of science by giving quick comprehensive answers to questions which really required laborious detailed answers. And clearly this is what happened to the first of the two questions. When detailed causal answers were provided to it, the comprehensive answer "Because universals exist" was no longer acceptable or necessary. 3 For skeptics like Pears, science has replaced metaphysics and has shown it to be superfluous. Now this skepticism would be justified only if its understanding of the function of metaphysics is correct; only if metaphysics had the same explanatory function as science could one claim that science has replaced metaphysics and shown it to be unnecessary. But if metaphysics can be shown to be doing something different from what science accomplishes, if its explanations can be shown not to be rivals of those provided by science, then this criticism misses the target. What then is a metaphysical explanation? The theory of universals provides us with a way of answering this question. It attempts to account for a general feature of our experience —natural recurrence—by stating the entities that that feature consists of—universals occurring in diverse particulars—where the entities are classified in terms of ultimate categories. In general, a metaphysical account is a redescription of some general and pervasive feature of our experience and of the world in terms of ultimate categories. These redescriptions are not rival causal explanations; they are not alternatives to the accounts provided by science; they give not antecedent causes but the ultimate categories from which our world (and any possible world) is constituted. In this sense, metaphysics tells about the nature of reality.

This account of metaphysics makes use of the notion of an ultimate category. Yet a commonsense philosopher who is skeptical of reductionism, who doubts philosophical claims of the form "A's are nothing but B's," may very well question whether this notion is meaningful or useful. He may say something like this: "Of course there are qualities; we speak about them all the time. Of course there are physical particulars; we constantly refer to them. Of course there are numbers, sets, propositions, events, relations; all these things are subjects of discourse. But there is no reason to think that any one of these sorts of thing can be 'reduced' to any other sort of thing. The assumption that there are any ultimate categories is an attempt to impose a simplified conception of discourse upon the multifarious uses and concepts of ordinary discourse." Now this argument of the antireductionist, rather than establishing that there are no ultimate categories, really shows, if it is correct, that there are many more such categories than those for which traditional systems allowed. If qualities, physical things, numbers, sets, propositions, events, and relations are types of entity independent of one another, then they are all ultimate categories. The antireductionist is a metaphysician despite himself. 3

D. F. Pears, "Universals," reprinted in A. G. N. Flew, ed., Logic and Language, Second Series (Oxford: Basil Blackwell, 1955), p. 52. -9-

III There is one more skeptical argument that is important enough to consider. Even if there is such a thing as a metaphysical account of things which is not simply a truistic commonsense existence-claim, nevertheless such accounts are uninteresting because it is pointless to deny them. Metaphysical theories are well known for the length of time they persist throughout the history of philosophy; they are also well known for their recalcitrance to clear-cut verification or refutation. The theory of universals and its alleged rivals have endured for most of the course of western philosophy. Each view still has its stanch advocates. This suggests that in fact the different theories are not really rivals after all. Perhaps they are different, though compatible, descriptions of the world; perhaps the actual ground for choosing one theory over another is not rational argumentation but something nonrational, such as aesthetic preference or interest of one sort or another. Just as the Eskimos have more terms in their language than we in our language for classifying a variety of types of snow because they have a greater interest in so doing, so different metaphysical systems are different, though compatible, ways of classifying the very same phenomena. 4 If this should be correct, then metaphysical systems would become uninteresting, in the sense that there would be no point in rejecting them or arguing about them. We do not reject the Eskimo classificatory scheme for snow; we simply fail to use it because it is not useful to us. Conceivably, for some purposes it might be useful to adopt a metaphysical scheme. But that one turns out to be useful and others not, does not mean that one is correct or true and the others incorrect or false. 5 This form of skepticism incorporates two distinct theses. First, it asserts that different metaphysical theories are not rivals because they are logically compatible with one another; second, it claims that such theories are not the sorts of things which can be verified or refuted. In order to rebut the first claim, it is sufficient to provide formulations of the various theories concerning qualities in such a way as to demonstrate that they are logically incompatible with one another. It is possible to provide a philosophically neutral identification of the sorts of entities the various theories of universals are about by pointing to examples of them. For instance, take any physical object and point to its color or its shape; these are the kinds of things meant by "quality." Realism asserts that qualities are universals. This claim can be interpreted as an identity criterion for qualities: the assertion that statements of the form "x is identical with y" or "x is numerically the ____________________ 4This view of metaphysics is suggested in certain writings of the linguist Benjamin Whorf and the Neo-Kantian philosopher Ernst Cassirer. See Whorf's Language, Thought and

Reality (New York: John Wiley and Sons, 1959) and Cassirer's The Philosophy of Symbolic Forms, Vol. I (New Haven: Yale University Press, 1953). point of view has received a sharp formulation in Carnap's "Empiricism, Semantics, and Ontology," reprinted in this volume.

5This

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same as y" can be true when "x" and "y" are names of qualities of different objects. This formulation of the theory of universals suggests a way of specifying an alternative account which we shall call the theory of abstract particulars. It says that when x and y are qualities of different objects, then it follows that x is not identical with y—that x is not numerically the same as y. In the words of G. F. Stout, qualities are abstract particulars; they are as particular as the things they qualify. 6 The theory of universals and the theory of abstract particulars are rivals in this precise sense: although both accept qualities as an ultimate category, the former asserts and the latter denies a certain identity criterion for qualities. They are saying different things about the same thing. We have made sure they are talking about the same thing by specifying what they are talking about in a philosophically neutral manner: a manner which does not commit one either to the existence of qualities as an ultimate category or to an identity criterion for qualities. They are, thus, logically incompatible with one another. This analysis yields another way in which metaphysical theories can be rivals. Instead of merely disagreeing about the identity conditions of some ultimate category, a philosopher may deny that this is an ultimate category in the first place; he may assert that there are no such things. And this is exactly what several forms of particularism do. Nominalism is based on the fact that there is such a thing as linguistic recurrence; there are general words in the vocabulary of any language which are applied to different particulars. The words "green," "square," and "leaf" are general in the sense that each applies to a variety of distinct individuals: green things, square things, and leaves. Nominalism explains natural recurrence as simply being linguistic recurrence ; qualities are reduced to general words, in the sense that the fact that two different things have the same quality is explained as consisting in the fact that the same word applies to both of them. 7 For the nominalist, general words are not names of qualities; rather, they are applied to or predicated on the concrete things that are commonsensically said to have the qualities. Consequently, for the nominalist the use of a general term does not presuppose the existence of qualities. Another version of particularism that rejects the category of quality is the resemblance theory. 8 In this view there is an objective basis—the resemblances between particulars—for applying general words to diverse particulars: we apply "green" to various things because they resemble one another in being green; we apply "square" to those things that resemble one another in being square. The resemblance theory, by providing an explanation of how linguistic recurrence is possible in terms of resemblances among particulars, could be looked upon as a corollary to nominalism. But it could also be formulated as a rival to nominalism in the following way. Suppose a philosopher thinks that the use of a general term in a sentence such as "This leaf is green" in some way ____________________ 6See the articles by Stout reprinted in this volume. 7See Quine's essay "On What There Is," reprinted in this volume. 8See H. H. Price's formulation in the selection reprinted in this volume. -11-

commits one to the existence of the quality green. But, as a particularist, he wants to avoid a philosophical commitment to qualities. He might then claim that the sentence can be translated into another one which does not presuppose qualities. For example, select some green object as an exemplar of green. Call the exemplar A. Then "This leaf is green" becomes something like "This leaf resembles A." 9 Qualities then are reduced to resembling particulars. The nominalist, having a different view of general terms, finds no need for such a reduction. In any case we now have four theories—realism and three forms of

particularism—which can be so formulated as to entail or presuppose logically incompatible theses. This dispenses with one of the claims of the skeptic. His second claim, that theories about universals are not the sorts of things that can be verified or refuted, is based on the fact that the problem has lasted a long time and disagreement still persists. Now there is a dialectical argument against this version of skepticism. The skeptic is presenting one theory about philosophy, but there are other theories as well which have endured through lengthy stretches of the history of philosophy. So, given the skeptic's own argument, his view is also not the sort of thing that can be verified or refuted. Thus we have no reason to accept it. Moreover, the fact that a problem has persisted a long time may be due to a variety of factors. It is characteristic of most philosophical problems, including the problem of universals, that they have a systematic nature: this means that what one says about one problem affects what one says about a variety of problems. One cannot solve philosophical problems piecemeal; one problem is not independent of others. This distinguishes philosophical problems from many of those in the sciences. Moreover, even in the sciences the fundamental problems, such as the ultimate constitution of matter and the nature of the human mind, have persisted as long as the problem of universals. While there tends to be more agreement at any given time within the scientific community than within the philosophical community, yet if one compares the sciences over a period of time one finds changes and disagreements as fundamental as in philosophy.

IV We have identified the problem of universals with the question of whether realism provides a satisfactory account of natural recurrence. A deeper insight into the nature of the problem depends on understanding the sorts of difficulties that particularists have found with realism. We shall now consider several of the more fundamental issues, examine the criticisms, and see how the realist would reply to them. ____________________ 9See Price's formulation for the details of the theory and for the various qualifications that have to be introduced. -12-

(a) Linguistic recurrence consists in the fact that there are general terms in language. In trying to understand the way language works and the way words apply to the world, realism tends to argue that general words are names of common qualities—of universals. 10 Consider the sentence: (1) This leaf is green. when it is used to make a true statement. The subject term "this leaf" is used to name (or stand for or refer to or denote) a given particular; similarly, the general term in the predicate position "green" is used to name a given universal. We apply the word "green" to the leaf by naming a quality of the leaf. General words apply to things by naming their qualities. Critics of the theory of universals argue that this semantical theory misunderstands how predicates actually function. The function of predicates is being confused with the function of subjects. This can be shown through the following argument. If general terms are names of qualities, then it follows that: (2) "Green" is a name of the quality green. Note also that: (3) "The quality green" is a name of the quality green. Since "green" and "the quality green" name the same thing, it follows that: (4) Green is identical with the quality green.

There is a rule of inference—the law of substitutivity—which says that words which name the same thing can be substituted for one another in sentences in which they occur, without turning a true statement into a false one. 11 One should be able, then, to substitute "the quality green" for "green" in (1). But if we do we come up with: (5) This leaf is the quality green. which is manifestly false. What went wrong? Neither (1) nor (3) nor the law of substitutivity are in doubt here. (4) follows from (2) and (3). The point of difficulty must be (2). To reject this is to say at least that in (1) the word "green" is not functioning as a name. Thus realism appears to rely upon an incorrect semantics. This is a persuasive argument. But it does not directly establish particularism. Realism is compatible with other views about the functioning of predicates. Someone who knows the meaning of (1) knows which quality the statement asserts the leaf to have. In general, someone who knows the meaning of a general word knows which quality something is asserted to have in a statement formulated with the general word as its predicate. For the realist ____________________ 10See Bergmann's "Frege's Hidden Nominalism," reprinted in this volume. 11There have been doubts whether this law holds unreservedly. But the context in which it is used in this discussion is not among those for which doubts have arisen. -13-

this implies that in some sense qualities enter into the meanings of general words. But it does not follow that the way they have meaning is by being names of qualities. The realist could say, for example, that in (1) both the singular and general terms function to introduce 12 certain entities into discourse : "this leaf" introduces a particular and "green" a universal. But there are different species of introduction, so that the way a singular term introduces an entity is not the same as the way a general term does so. "This leaf" introduces a particular by helping to single out which particular the sentence is about: it refers to or names that particular. Referring or naming is the function of singling out the subject of the sentence. On the other hand, "green" introduces a quality not by naming it but by ascribing it to the particular singled out by the subject term. Here ascribing is a different function from naming. This approach, which is here merely sketched, avoids the particularist's argument in this way: the law of substitutivity is interpreted to apply only to words which name or refer and not to words which ascribe. Since "green" in (1) is interpreted as ascribing a quality and not as naming it, then one cannot use the law to go from (1) to (5). (b) We have discussed the functions of the subject and predicate in (1), but not that of the copula "is." One of the functions of the copula is obviously to indicate tense. Another function, due to its position, is to signify that the sentence (1) is used to make an assertion or statement; if its position should be changed, as in "Is this leaf green?" instead of an assertion we have a question. In one version of realism, since the subject and predicate introduce distinct entities, the copula must also introduce that entity which connects them together. For this function, let us use Bergmann's term, exemplification. Sentence (1) asserts that the leaf exemplifies the quality green. Exemplification is the relation that holds between a quality and the things which have the quality. 13 Because exemplification connects entities of one ultimate category—universals—to those of another—particulars—it may be classified as a metaphysical tie. One motive for introducing the notion of exemplification is to answer the question, "How are universals related to particulars?" The question is answered by specifying the entity that does the relating. According to this way of thinking, whenever there is one entity related to a second entity, one can always ask for the entity which relates them. Thus we are entitled to raise the question of how exemplification is itself related to the entities it connects. This new question presents the realist with a dilemma. Either there is an entity which relates exemplification to the things it connects or there isn't. If there isn't, then exemplification fails to do the job it was introduced to do: to connect universals to particulars. But if there

is, then we can ask of this entity what relates it to the things it connects. And thus an infinite regress is gen____________________ 12The notion of introduction is Strawson's. See his Individuals, an Essay in Descriptive Metaphysics (London: Methuen & Co., 1959), p. 146. 13In systems employing the symbolism of mathematical logic, exemplification is represented not by the copula, since there isn't any, but by the juxtaposition of the predicate and subject terms. -14-

erated. To the particularist this dilemma indicates a fundamental difficulty with realism, which he solves by rejecting universals and thus avoiding the need for a metaphysical tie such as exemplification. The realist may try to respond by pointing out that not every infinite regress refutes the theory which generates it. There is no contradiction in the mathematical theory which says that there is no last number. Similarly, there is no contradiction in the supposition that, for every relation which connects entities to one another, there is another relation which connects the former to its terms. Although this reply suffices to rebut the charge of inconsistency, it is not satisfactory. For if the regress does not violate the laws of logic, it does seem to violate the principle of economy or simplicity sometimes known as Occam's razor. All things being equal, a theory without such a regress is more plausible than one that generates a regress. The regress is a complexity which is very difficult to understand. Particularism has the advantage, then, of not implying this bit of unintelligibility. But there are other ways out for realism. In the first place, a realist could argue that the infinite regress is merely verbal. It is a feature of English grammar that allows us to reiterate the verb "exemplifies" over and over again, as in "This leaf exemplifies the property of exemplifying green." Endless reiteration is a feature of other phrases in English, such as "it is true that." But just as "It is true that it is true that this is green" says no more than "It is true that this is green," so "This leaf exemplifies the property of exemplifying green" says no more than "This leaf exemplifies green." The regress is not one of an endless series of entities but of an endless series of sentences. As a corollary to this reply, the realist might claim in addition that the question which appeared to generate the regress is itself illegitimate. The question was, "What entity relates exemplification to the entities that it connects?" But, says the realist, exemplification is the sort of entity whose nature it is to connect other entities; it needs no further thing to achieve the connection. Here is another example of a category explanation; it is because exemplification is just the sort of thing it is that it has the ability to make the relevant connections. In the second place, there is another version of realism, one suggested by some of the writings of Frege. 14 In this view, qualities are unsaturated or incomplete ; it is their nature to be joined to particulars; a special tie is not required to make the connection. To the question, "How are qualities connected to particulars?" this view answers that by their very nature qualities are always qualities of something. If exemplification were indeed required as a tie, then it would always be an open question whether a given quality belonged to some particular. But this is absurd, since there cannot be colors or shapes without things which have that color or are of that shape. The dispute between these alternatives parallels the ancient dispute between Plato and Aristotle whether universals can exist independently of particulars. (c) In our earlier discussion of Parmenides's argument we noted that one ____________________ 14See Frege's essay "On Concept and Object," reprinted in this volume. -15-

way out of the dilemma was to deny that universals have spatiotemporal location in any direct sense. The color green, for example, may be said to be in a given place at a given time only in the sense that a green thing is in that place at that time. The spatiotemporal location of qualities is derived from that of the physical particulars that exemplify them, whereas the location of the physical particulars is direct. The tendency to deny location to universals reinforced another tendency one finds in Plato's philosophy: to claim that our knowledge of universals is based not on sense perception but on reason. Another road to this conclusion is that the theory of universals was often invoked 15 to explain how a priori knowledge is possible. Certain truths are known independently of experience; an example is, "Whatever is green is colored." These are said to assert connections among universals; our ability to know these truths is based on the fact that we can become acquainted with these universals by means of a nonsensory form of intuition. A distinguished logician has written: The extreme demand for a simple prohibition of abstract entities under all circumstances perhaps arises from a desire to maintain the connection between theory and observation. But the preference of (say) seeing over understanding as a method of observation seems to me capricious. For just as an opaque body may be seen, so a concept may be understood or grasped. 16 Realists have thus tended to be rationalists as well, claiming that there is a source of factual knowledge other than sense-perception. Particularists have tended to be empiricists, restricting our factual knowledge to what can be derived from sense. The bad reputation that rationalism has acquired as a result of doubts about the existence of a priori forms of cognition has tended to extend itself to realism. Thus the theory of universals has often been rejected on the grounds that it conflicts with an empiricist foundation of knowledge. The historical connection between realism and rationalism does not prove that they are logically interwoven. Quine has suggested that the fact that something is observable does not settle the question of to which metaphysical category it belongs. 17 That one can see qualities such as shapes, leaves open the question of their classification. Whether a quality is a universal or particular, whether it is reducible to some other type of entity depends upon systematic considerations and philosophical argument. Since a theory about qualities presupposes the ability to identify qualities through observation as a way of specifying what the theory is about, no adequate theory can end up by denying the possibility of observing them. (d) Earlier we considered two leaves which are exactly alike in every respect. According to realism, they possess the same qualities. But then, how ____________________ 15By Plato and by Bertrand Russell. See Russell's Problems of Philosophy (New York: Oxford University Press, 1912), Ch. 9 and 10. 16Alonzo Church, "The Need for Abstract Entities in Semantic Analysis," reprinted in J. A. Fodor and J. J. Katz, The Structure of Language (Englewood Cliffs: Prentice-Hall, 1964), p. 442. 17Word and Object (Cambridge: The M.I.T. Press, 1960), p. 236. -16-

is it that they are two, not one? This question, sometimes called the problem of individuation, is suggested by a principle apparently presupposed by realism that (P 1 ) it is logically possible for two different things to have the same qualities. This principle, together with what seems to be an obvious truism—that (P 2 ) things which are the same in every way are the same— makes us wonder how we have two leaves after all. Whereas the problem of universals is to explain how different things can have common qualities, the problem of individuation is to explain how things with all their qualities in common can be different. Since particularism does not accept (P 1 ), it does not generate the problem of individuation in just this form. Realists have adopted different solutions to the problem. One group of solutions accepts (P 1 ) and claims that when two objects have all their qualities in common, then there is something else about them which explains their difference. The theory of bare particulars 18 asserts that in addition to its qualities, a thing contains an individual which exemplifies them.

Different things, even if they should have the same qualities, contain different individuals. Many realists find it difficult to accept the idea of bare particulars because, although they can discern qualities in the objects of experience, they claim to be unable to find the individual which has them. For this group, a thing is just a bundle of qualities. If they accept (P 1 ) then what individuates can be neither a quality nor a bare particular but is rather a relation. Two things with the same qualities will stand in different spatiotemporal relations to other things; e.g., one will be to the left and the other to the right of some object in space. 19 Some realists have decided to reject (P 1 ) and argue that different particulars always have some difference in quality. One way of achieving this is to claim that the places and times at which a particular is located are among its qualities. 20 Another way is to hold that for every relation that holds between particulars, there is a relational quality in each of the particulars. These, then, are some of the issues that are implicit in the problem of universals. Since their resolution brings in questions about language, meaning, knowledge, space, and time, it is clear that the problem cannot be solved in any simple or neat way. But this explains why the problem is so interesting —why it has held the attention of philosophers throughout the entire history of philosophy. ____________________ 18Bergmann has advocated this view. See also Edwin B. Allaire, "Bare Particulars," Philosophical Studies (1963). 19This view is defended by Jack Meiland in "Do Relations Individuate?" Philosophical Studies (1966). Allaire's reply is contained in "Relations and the Problem of Individuation," Philosophical Studies (1968). 20Nelson Goodman, The Structure of Appearance (Indianapolis: Bobbs-Merrill, 1966), pp. 194-200. -17-

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PART ONE The Theory of Universals

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:1: ON THE RELATIONS OF UNIVERSALS AND PARTICULARS 1 BERTRAND RUSSELL THE purpose of the following paper is to consider whether there is a fundamental division of the objects with which metaphysics is concerned into two classes, universals and particulars, or whether there is any method of overcoming this dualism. My own opinion is that the dualism is ultimate; on the other hand, many men with whom, in the main, I am in close agreement, hold that it is not ultimate. I do not feel the grounds in favour of its ultimate nature to be very conclusive, and in what follows I should lay stress rather on the distinctions and considerations introduced during the argument than on the conclusion at which the argument arrives.

It is impossible to begin our discussion with sharp definitions of universals and particulars, though we may hope to reach such definitions in the end. At the beginning, we can only roughly indicate the kind of facts that we wish to analyse and the kind of distinctions that we wish to examine. There are several cognate distinctions which produce confusion by intermingling, and which it is important to disentangle before advancing into the heart of our problem. The first distinction that concerns us is the distinction between percepts and concepts, i.e., between objects of acts of perception and objects of acts of conception. If there is a distinction between particulars and universals, percepts will be among particulars, while concepts will be among universals. Opponents of universals, such as Berkeley and Hume, will maintain that concepts are derivable from percepts, as faint copies, or in some other way. Opponents of particulars will maintain that the apparent particularity of percepts is illusory, and that, though the act of perception may differ from the act of con ____________________ Reprinted from the Proceedings of The Aristotelian Society, XII (1911-1912), by courtesy of the Editor of The Aristotelian Society. Copyright © 1912, The Aristotelian Society. 1The thesis of the present paper is closely similar to that of Mr. Moore's paper "Identity," read before this Society in 1900-1901. My chief reason for thinking that the question demands a fresh discussion is that the statement of the grounds for the thesis appears to require some examination of the nature of sensible space as opposed to physical space.

ception, yet its objects differ only by their greater complexity, and are really composed of constituents which are, or might be, concepts. But the distinction of percepts and concepts is too psychological for an ultimate metaphysical distinction. Percepts and concepts are respectively the relata of two different relations, perception and conception, and there is nothing in their definitions to show whether, or how, they differ. Moreover, the distinction of percepts and concepts, in itself, is incapable of being extended to entities which are not objects of cognitive acts. Hence we require some other distinction expressing the intrinsic difference which we seem to feel between percepts and concepts. A cognate distinction, which effects part at least of what we want, is the distinction between things which exist in time and things which do not. In order to avoid any question as to whether time is relative or absolute, we may say that an entity x "exists in time" provided x is not itself a moment or part of time, and some such proposition as "x before y or simultaneous with y or after y" is true of x. (It is not to be assumed that before, simultaneous, and after are mutually exclusive: if x has duration, they will not be so.) Prima facie, a percept exists in time, in the above sense, while a concept does not. The object of perception is simultaneous with the act of perception, while the object of conception seems indifferent to the time of conceiving and to all time. Thus, prima facie, we have here the non-psychological distinction of which we were in search. But the same controversies will break out as in the case of percepts and concepts. The man who reduces concepts to percepts will say that nothing is really out of time, and that the appearance of this in the case of concepts is illusory. The man who reduces percepts to concepts may either, like most idealists, deny that anything is in time, or, like some realists, maintain that concepts can and do exist in time. In addition to the above distinction as regards time, there is a distinction as regards space which, as we shall find, is very important in connexion with our present question. Put as vaguely as possible, this is a distinction which divides entities into three classes: (a) those which are not in any place, (b) those which are in one place at one time, but never in more than one, (c) those which are in many places at once. To make this threefold division precise, we should have to discuss what we mean by a place, what we mean by "in," and how the different kinds of space—visual, tactile, physical—produce different forms of this threefold division. For the present I will merely illustrate what I mean by examples. Relations, obviously, do not exist anywhere in space. Our bodies, we think, exist in one place at a time, but not in more than one. General qualities, such as whiteness, on the contrary, may be said to be in many places at once: we may say, in a sense, that whiteness is

in every place where there is a white thing. This division of entities will be discussed later; for the present I merely wish to indicate that it requires examination. In addition to the above psychological and metaphysical distinctions, there are two logical distinctions which are relevant in the present enquiry. In the -22-

first place, there is the distinction between relations and entities which are not relations. It has been customary for philosophers to ignore or reject relations, and speak as if all entities were either subjects or predicates. But this custom is on the decline, and I shall assume without further argument that there are such entities as relations. Philosophy has, so far as I know, no common name for all entities which are not relations. Among such entities are included not only all the things that would naturally be called particulars, but also all the universals that philosophers are in the habit of considering when they discuss the relation of particulars to universals, for universals are generally conceived as common properties of particulars, in fact, as predicates. For our purpose it is hardly worth while to invent a technical term ad hoc; I shall therefore speak of entities which are not relations simply as nonrelations.

The second logical distinction which we require is one which may or may not be identical in extension with that between relations and non-relations, but is certainly not identical in intension. It may be expressed as the distinction between verbs and substantives, or, more correctly, between the objects denoted by verbs and the objects denoted by substantives. 2 (Since this more correct expression is long and cumbrous, I shall generally use the shorter phrase to mean the same thing. Thus, when I speak of verbs, I mean the objects denoted by verbs, and similarly for substantives.) The nature of this distinction emerges from the analysis of complexes. In most complexes, if not in all, a certain number of different entities are combined into a single entity by means of a relation. "A's hatred for B," for example, is a complex in which hatred combines A and B into one whole; "C's belief that A hates B" is a complex in which belief combines A and B and C and hatred into one whole, and so on. A relation is distinguished as dual, triple, quadruple, etc., or dyadic, triadic, tetradic, etc., according to the number of terms which it unites in the simplest complexes in which it occurs. Thus in the above examples, hatred is a dual relation and belief is a quadruple relation. The capacity for combining terms into a single complex is the defining characteristic of what I call verbs. The question now arises: Are there complexes which consist of a single term and a verb? "A exists" might serve as an example of what is possibly such a complex. It is the possibility that there may be complexes of this kind which makes it impossible to decide off-hand that verbs are the same as relations. There may be verbs which are philosophically as well as grammatically intransitive. Such verbs, if they exist, may be called predicates, and the propositions in which they are attributed may be called subjectpredicate propositions. If there are no such verbs as those whose possibility we have been considering, i.e., if all verbs are relations, it will follow that subject-predicate propositions, if there are any, will express a relation of subject to predicate. ____________________ 2This is the distinction which I formerly spoke of as the distinction between things and concepts, but these terms no longer seem to me appropriate, Cf. Principles of Mathematics, § 48. -23-

Such propositions will then be definable as those that involve a certain relation called predication. Even if there are subject-predicate propositions in which the predicate is the verb, there will still be equivalent propositions in which the predicate is related to the subject; thus

"A exists," for example, will be equivalent to "A has existence." Hence the question whether predicates are verbs or not becomes unimportant. The more important question is whether there is a specific relation of predication, or whether what are grammatically subjectpredicate propositions are really of many different kinds, no one of which has the characteristics one naturally associates with subject- predicate propositions. This question is one to which we shall return at a later stage. The above logical distinctions are relevant to our enquiry because it is natural to regard particulars as entities which can only be subjects or terms of relations, and cannot be predicates or relations. A particular is naturally conceived as a this or something intrinsically analogous to a this; and such an entity seems incapable of being a predicate or a relation. A universal, on this view, will be anything that is a predicate or a relation. But if there is no specific relation of predication, so that there is no class of entities which can properly be called predicates, then the above method of distinguishing particulars and universals fails. The question whether philosophy must recognize two ultimately distinct kinds of entities, particulars and universals, turns, as we shall see more fully later on, on the question whether non-relations are of two kinds, subjects and predicates, or rather terms which can only be subjects and terms which may be either subjects or predicates. And this question turns on whether there is an ultimate simple asymmetrical relation which may be called predication, or whether all apparent subject-predicate propositions are to be analysed into propositions of other forms, which do not require a radical difference of nature between the apparent subject and the apparent predicate. The decision of the question whether there is a simple relation of predication ought perhaps to be possible by inspection, but for my part I am unable to come to any decision in this way. I think, however, that it can be decided in favour of predication by the analysis of things and by considerations as to spatio-temporal diversity. This analysis and these considerations will also show the way in which our purely logical question is bound up with the other questions as to particulars and universals which I raised at the beginning of this paper. The common-sense notion of things and their qualities is, I suppose, the source of the conception of subject and predicate, and the reason why language is so largely based on this conception. But the thing, like other common-sense notions, is a piece of half-hearted metaphysics, which neither gives crude data nor gives a tenable hypothesis as to a reality behind the data. A thing, of the everyday sort, is constituted by a bundle of sensible qualities belonging to various senses, but supposed all to coexist in one continuous por -24-

tion of space. But the common space which should contain both visual and tactile qualities is not the space of either visual or tactile perception: it is a constructed "real" space, belief in which has, I suppose, been generated by association. And in crude fact, the visual and tactile qualities of which I am sensible are not in a common space, but each in its own space. Hence if the thing is to be impartial as between sight and touch, it must cease to have the actual qualities of which we are sensible, and become their common cause or origin or whatever vaguer word can be found. Thus the road is opened to the metaphysical theories of science and to the metaphysical theories of philosophy: the thing may be a number of electric charges in rapid motion, or an idea in the mind of God, but it is certainly not what the senses perceive. The argument against things is trite, and I need not labour it. I introduce it here only in order to illustrate a consequence which is sometimes overlooked. Realists who reject particulars are apt to regard a thing as reducible to a number of qualities coexisting in one place. But, apart from other objections to this view, it is doubtful whether the different qualities in question ever do coexist in one place. If the qualities are sensible, the place must be in a sensible space; but this makes it necessary that the qualities should belong to only one sense, and it is not clear that genuinely different qualities belonging to one sense ever coexist in a single place in a perceptual space. If, on the other hand, we consider what may be called "real" space, i.e. the inferred space containing the "real" objects which we suppose to be the causes of our perceptions, then we no longer know what is the nature of the qualities, if any, which exist in this "real" space, and it is natural to replace the bundle of qualities by a collection of pieces of matter having whatever characteristics the science of the moment

may prescribe. Thus in any case the bundle of coexisting qualities in the same place is not an admissible substitute for the thing. For our purposes, the "real" object by which science or philosophy replaces the thing is not important. We have rather to consider the relations of sensible objects in a single sensible space, say that of sight. The theory of sensible qualities which dispenses with particulars will say, if the same shade of colour is found in two different places, that what exists is the shade of colour itself, and that what exists in the one place is identical with what exists in the other. The theory which admits particulars will say, on the contrary, that two numerically different instances of the shade of colour exist in the two places: in this view, the shade of colour itself is a universal and a predicate of both the instances, but the universal does not exist in space and time. Of the above two views, the first, which does not introduce particulars, dispenses altogether with predication as a fundamental relation: according to this view, when we say "this thing is white," the fundamental fact is that whiteness exists here. According to the other view, which admits particulars, what exists here is something of which whiteness is a predicate—not, -25-

as for common sense, the thing with many other qualities, but an instance of whiteness, a particular of which whiteness is the only predicate except shape and brightness and whatever else is necessarily connected with whiteness. Of the above two theories, one admits only what would naturally be called universals, while the other admits both universals and particulars. Before examining them, it may be as well to examine and dismiss the theory which admits only particulars, and dispenses altogether with universals. This is the theory advocated by Berkeley and Hume in their polemic against "abstract ideas." Without tying ourselves down to their statements, let us see what can be made of this theory. The general name "white," in this view, is defined for a given person at a given moment by a particular patch of white which he sees or imagines; another patch is called white if it has exact likeness in colour to the standard patch. In order to avoid making the colour a universal, we have to suppose that "exact likeness" is a simple relation, not analysable into community of predicates; moreover, it is not the general relation of likeness that we require, but a more special relation, that of colour-likeness, since two patches might be exactly alike in shape or size but different in colour. Thus, in order to make the theory of Berkeley and Hume workable, we must assume an ultimate relation of colour-likeness, which holds between two patches which would commonly be said to have the same colour. Now, prima facie, this relation of colour-likeness will itself be a universal or an "abstract idea," and thus we shall still have failed to avoid universals. But we may apply the same analysis to colour-likeness. We may take a standard particular case of colour-likeness, and say that anything else is to be called a colour-likeness if it is exactly like our standard case. It is obvious, however, that such a process leads to an endless regress: we explain the likeness of two terms as consisting in the likeness which their likeness bears to the likeness of two other terms, and such a regress is plainly vicious. Likeness at least, therefore, must be admitted as a universal, and, having admitted one universal, we have no longer any reason to reject others. Thus the whole complicated theory, which had no motive except to avoid universals, falls to the ground. Whether or not there are particulars, there must be relations which are universals in the sense that (a) they are concepts, not percepts; (b) they do not exist in time; (c) they are verbs, not substantives. It is true that the above argument does not prove that there are universal qualities as opposed to universal relations. On the contrary, it shows that universal qualities can, so far as logic can show, be replaced by exact likenesses of various kinds between particulars. This view has, so far as I know, nothing to recommend it beyond its logical possibility. But from the point of view of the problem whether there are particulars, it has no bearing on the argument. It is a view which is only possible if there are particulars, and it demands only an easy re-statement of subject-predicate propositions: instead of saying that an entity has such and such a predicate, we shall have to say that there are entities to which it has such and such a specific likeness. I shall therefore in future ignore this view, which in any case assumes our main

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thesis, namely, the existence of particulars. To the grounds in favour of this thesis we must now return. When we endeavoured to state the two theories as to sensible qualities, we had occasion to consider two white patches. On the view which denies particulars, whiteness itself exists in both patches: a numerically single entity, whiteness, exists in all places that are white. Nevertheless, we speak of two white patches, and it is obvious that, in some sense, the patches are two, not one. It is this spatial plurality which makes the difficulty of the theory that denies particulars. Without attempting, as yet, to introduce all the necessary explanations and distinctions, we may state the argument for particulars roughly as follows. It is logically possible for two exactly similar patches of white, of the same size and shape, to exist simultaneously in different places. Now, whatever may be the exact meaning of "existing in different places," it is self-evident that, in such a case, there are two different patches of white. Their diversity might, if we adopted the theory of absolute position, be regarded as belonging, not to the white itself which exists in the two places, but to the complexes "whiteness in this place" and "whiteness in that place." This would derive their diversity from the diversity of this place and that place; and since places cannot be supposed to differ as to qualities, this would require that the places should be particulars. But if we reject absolute position, it will become impossible to distinguish the two patches as two, unless each, instead of being the universal whiteness, is an instance of whiteness. It might be thought that the two might be distinguished by means of other qualities in the same place as the one but not in the same place as the other. This, however, presupposes that the two patches are already distinguished as numerically diverse, since otherwise what is in the same place as the one must be in the same place as the other. Thus the fact that it is logically possible for precisely similar things to coexist in two different places, but that things in different places at the same time cannot be numerically identical, forces us to admit that it is particulars, i.e., instances of universals, that exist in places, and not universals themselves. The above is the outline of our argument. But various points in it have to be examined and expanded before it can be considered conclusive. In the first place, it is not necessary to assert that there ever are two exactly similar existents. It is only necessary to perceive that our judgment that this and that are two different existents is not necessarily based on any difference of qualities, but may be based on difference of spatial position alone; and that difference of qualities, whether or not it always in fact accompanies numerical difference, is not logically necessary in order to insure numerical difference where there is difference of spatial position. Again, it is not easy to state exactly what sort of spatial distribution in perceived space warrants us in asserting plurality. Before we can use space as an argument for particulars, we must be clear on this point. We are accustomed to concede that a thing cannot be in two places at once, but this -27-

common-sense maxim, unless very carefully stated, will lead us into inextricable difficulties. Our first business, therefore, is to find out how to state this maxim in an unobjectionable form. In rational dynamics, where we are concerned with matter and "real" space, the maxim that nothing can be in two places at once is taken rigidly, and any matter occupying more than a point of space is regarded as at least theoretically divisible. Only what occupies a bare point is simple and single. This view is straightforward, and raises no difficulties as applied to "real" space. But as applied to perceived space, such a view is quite inadmissible. The immediate object of (say) visual perception is always of finite extent. If we suppose it to be, like the matter

corresponding to it in "real" space, composed of a collection of entities, one for each point which is not empty, we shall have to suppose two things, both of which seem incredible, namely: (1) that every immediate object of visual (or tactile) perception is infinitely complex; (2) that every such object is always composed of parts which are by their very nature imperceptible. It seems quite impossible that the immediate object of perception should have these properties. Hence we must suppose that an indivisible object of visual perception may occupy a finite extent of visual space. In short, we must, in dividing any complex object of visual perception, reach, after a finite number of steps, a minimum sensible, which contains no plurality although it is of finite extent. Visual space may, in a sense, be infinitely divisible, for, by attention alone, or by the microscope, the immediate object of perception can be changed in a way which introduces complexity where formerly there was simplicity; and to this process no clear limit can be set. But this is a process which substitutes a new immediate object in place of the old one, and the new object, though more subdivided than the old one, will still consist of only a finite number of parts. We must therefore admit that the space of perception is not infinitely divided, and does not consist of points, but is composed of a finite though constantly varying number of surfaces or volumes, continually breaking up or joining together according to the fluctuations of attention. If there is a "real" geometrical space corresponding to the space of perception, an infinite number of points in the geometrical space will have to correspond to a single simple entity in the perceived space. It follows from this that, if we are to apply to the immediate objects of perception the maxim that a thing cannot be in two places at once, a "place" must not be taken to be a point, but must be taken to be the extent occupied by a single object of perception. A white sheet of paper, for example, may be seen as a single undivided object, or as an object consisting of two parts, an upper and a lower or a right hand and a left hand part, or again as an object consisting of four parts, and so on. If we on this account consider that, even when the sheet appeared as an undivided object, its upper and lower halves were in different places, then we shall have to say that the undivided object was in both these places at once. But it is better to say that, when the -28-

sheet appeared as an undivided object, this object occupied only one "place," though the place corresponded to what were afterwards two places. Thus a "place" may be defined as the space occupied by one undivided object of perception. With this definition, the maxim that a thing cannot be in two places at once might seem to reduce to a tautology. But this maxim, though it may need rewording, will still have a substantial significance, to be derived from the consideration of spatial relations. It is obvious that perceived spatial relations cannot hold between points, but must hold between the parts of a single complex object of perception. When the sheet of paper is perceived as consisting of two halves, an upper and a lower, these two halves are combined into a complex whole by means of a spatial relation which holds directly between the two halves, not between supposed smaller subdivisions which in fact do not exist in the immediate object of perception. Perceived spatial relations, therefore, must have a certain roughness, not the neat smooth properties of geometrical relations between points. What, for example, shall we say of distance? The distance between two simultaneously perceived objects will have to be defined by the perceived objects between them; in the case of two objects which touch, like the two halves of the sheet of paper, there is no distance between them. What remains definite is a certain order; by means of right and left, up and down, and so on, the parts of a complex object of perception acquire a spatial order, which is definite, though not subject to quite the same laws as geometrical order. The maxim that a thing cannot be in two places at once will then become the maxim that every spatial relation implies diversity of its terms, i.e., that nothing is to the right of itself, or above itself, and so on. In that case, given two white patches, one of which is to the right of the other, it will follow that there is not a single thing, whiteness, which is to the right of itself, but that there are two different things, instances of whiteness, of which one is to the right of the other. In this way our maxim will support the conclusion that there must be particulars as well as universals. But the above outline of an argument needs some amplification before it can be considered conclusive. Let us therefore examine, one by one, the steps of the argument.

Let us suppose, for the sake of definiteness, that within one field of vision we perceive two separated patches of white on a ground of black. It may then be taken as quite certain that the two patches are two and not one. The question is: Can we maintain that there are two if what exists in each is the universal whiteness? If absolute space is admitted, we can of course say that it is the difference of place that makes the patches two; there is whiteness in this place, and whiteness in that place. From the point of view of our main problem, which is as to the existence of particulars, such a view would prove our thesis, since this place and that place would be or imply particulars consituting absolute space. But from the point of view of our immediate problem, which is concerned with plurality in perceived space, we might reject the above view on -29-

the ground that, whatever may be the case with "real" space, perceived space is certainly not absolute, i.e., absolute positions are not among objects of perception. Thus the whiteness here and the whiteness there cannot be distinguished as complexes of which this place and that place are respectively constituents. Of course the whiteness may be of different shapes, say one round and one square, and then they could be distinguished by their shapes. It will be observed that, with the view adopted above as to the nature of perceived space, it is perfectly possible for a simple object of perception to have a shape: the shape will be a quality like another. Since a simple object of perception may be of finite extent, there is no reason to suppose that a shape must imply spatial divisibility in the object of perception. Hence our two patches may be respectively round and square, and yet not be spatially divisible. It is obvious, however, that this method of distinguishing the two patches is altogether inadequate. The two patches are just as easily distinguished if both are square or both are round. So long as we can see both at once, no degree of likeness between them causes the slightest difficulty in perceiving that there are two of them. Thus difference of shape, whether it exists or not, is not what makes the patches two entities instead of one. It may be said that the two patches are distinguished by the difference in their relations to other things. For example, it may happen that a patch of red is to the right of one and to the left of the other. But this does not imply that the patches are two unless we know that one thing cannot be both to the right and to the left of another. This, it might be said, is obviously false. Suppose a surface of black with a small white space in the middle. Then the whole of the black may form only one simple object of perception, and would seem to be both to the right and to the left of the white space which it entirely surrounds. I think it would be more true to say, in this case, that the black is neither to the right nor to the left of the white. But right and left are complicated relations involving the body of the percipient. Let us take some other simpler relation, say that of surrounding, which the black surface has to the white patch in our example. Suppose we have another white patch, of exactly the same size and shape, entirely surrounded by red. Then, it may be said, the two patches of white are distinguished by difference of relation, since one is surrounded by black and the other by red. But if this ground of distinction is to be valid, we must know that it is impossible for one entity to be both wholly and immediately surrounded by black and wholly and immediately surrounded by red. I do not mean to deny that we do know this. But two things deserve notice—first, that it is not an analytic proposition; second, that it presupposes the numerical diversity of our two patches of white. We are so accustomed to regarding such relations as "inside" and "outside" as incompatible that it is easy to suppose a logical incompatibility, although in fact the incompatibility is a characteristic of space, not a result of logic. I do not know what are the unanalysable spatial relations of objects of perception, whether visual or tactile, but whatever they are they must have -30-

the kind of characteristics which are required in order to generate an order. They, or some of them, must be asymmetrical, i.e., such that they are incompatible with their converses: for example, supposing "inside" to be one of them, a thing which is inside another must not also be outside it. They, or some of them, must also be transitive, i.e., such that, for example, if x is inside y and y is inside z, then x is inside z—supposing, for the sake of illustration, "inside" to be among fundamental spatial relations. Probably some further properties will be required, but these at least are essential, in view of the fact that there is such a thing as spatial order. It follows that some at least of the fundamental spatial relations must be such as no entity can have to itself. It is indeed self-evident that spatial relations fulfil these conditions. But these conditions are not demonstrable by purely logical considerations: they are synthetic properties of perceived spatial relations. It is in virtue of these self-evident properties that the numerical diversity of the two patches of white is self-evident. They have the relation of being outside each other, and this requires that they should be two, not one. They may or may not have intrinsic differences—of shape, or size, or brightness, or any other quality—but whether they have or not they are two, and it is obviously logically possible that they should have no intrinsic differences whatever. It follows from this that the terms of spatial relations cannot be universals, but must be particulars capable of being exactly alike and yet numerically diverse. It is very desirable, in such discussions as that on which we are at present engaged, to be able to talk of "places" and of things or qualities "occupying" places, without implying absolute position. It must be understood that, on the view which adopts relative position, a "place" is not a precise notion. But its usefulness arises as follows: Suppose a set of objects, such as the walls and furniture of a room, to retain their spatial relations unchanged for a certain length of time, while a succession of other objects, say people who successively sit in a certain chair, have successively a given set of spatial relations to the relatively fixed objects. Then the people have, one after the other, a given set of properties, consisting in spatial relations to the walls and furniture. Whatever has this given set of properties at a given moment is said to "occupy" a certain place, the "place" itself being merely a fixed set of spatial relations to certain objects whose spatial relations to each other do not change appreciably during the time considered. Thus when we say that one thing can only be in one place at one time, we mean that it can only have one set of spatial relations to a given set of objects at one time, It might be argued that, since we have admitted that a simple object of perception may be of finite extent, we have admitted that it may be in many places at once, and therefore may be outside itself. This, however, would be a misunderstanding. In perceived space, the finite extent occupied by a simple object of perception is not divided into many places. It is a single place occupied by a single thing. There are two different ways in which this place may "correspond" to many places. First, if there is such a thing as -31-

"real" space with geometrical properties, the one place in perceived space will correspond to an infinite number of points in "real" space, and the single entity which is the object of perception will correspond to many physical entities in "real" space. Secondly, there is a more or less partial correspondence between perceived space at one time and perceived space at another. Suppose that we attend closely to our white patch, and meanwhile no other noticeable changes occur in the field of vision. Our white patch may, and often does, change as the result of attention—we may perceive differences of shade of other differentiations, or, without differences of quality, we may merely observe parts in it which make it complex and introduce diversity and spatial relations within it. We consider, naturally, that we are still looking at the same thing as before, and that what we see now was there all along. Thus we conclude that our apparently simple white patch was not really simple. But, in fact, the object of perception is not the same as it was before; what may be the same is the physical object supposed to correspond to the object of perception. This physical object is, of course, complex. And the perception which results from attention will be in one sense more correct than that which perceived a simple object, because, if attention reveals previously unnoticed differences, it may be assumed that there are corresponding differences in the "real" object which corresponds to the object of perception. Hence the perception resulting from attention gives more information about the "real" object than the

other perception did: but the object of perception itself is no more and no less real in the one case than in the other—that is to say, in both cases it is an object which exists when perceived, but which there is no reason to believe existent except when it is perceived. In perceived space, the spatial unit is not a point, but a simple object of perception or an ultimate constituent in a complex object of perception. This is the reason why, although two patches of white which are visibly separated from each other must be two, a continuous area of white may not be two. A continuous area, if not too large, may be a single object of perception not consisting of parts, which is impossible for two visibly separated areas. The spatial unit is variable, constantly changing its size, and subject to every fluctuation of attention, but it must occupy a continuous portion of perceived space, since otherwise it would be perceived as plural. The argument as to numerical diversity which we have derived from perceived space may be reinforced by a similar argument as regards the contents of different minds. If two people are both believing that two and two are four, it is at least theoretically possible that the meanings they attach to the words two and and and are and four are the same, and that therefore, so far as the objects of their beliefs are concerned, there is nothing to distinguish the one from the other. Nevertheless, it seems plain that there are two entities, one the belief of the one man and the other the belief of the other. A particular belief is a complex of which something which we may call a subject is a constituent; in our case, it is the diversity of the subjects that produces the diversity of the beliefs. But these subjects cannot be mere bundles of general -32-

qualities. Suppose one of our men is characterized by benevolence, stupidity, and love of puns. It would not be correct to say: "Benevolence, stupidity, and love of puns believe that two and two are four." Nor would this become correct by the addition of a larger number of general qualities. Moreover, however many qualities we add, it remains possible that the other subject may also have them; hence qualities cannot be what constitutes the diversity of the subjects. The only respect in which two different subjects must differ is in their relations to particulars: for example, each will have to the other relations which he does not have to himself. But it is not logically impossible that everything concerning one of the subjects otherwise only concerning universals might be true of the other subject. Hence, even when differences in regard to such propositions occur, it is not these differences that constitute the diversity of the two subjects. The subjects, therefore, must be regarded as particulars, and as radically different from any collection of those general qualities which may be predicated of them. It will be observed that, according to the general principles which must govern any correspondence of real things with objects of perception, any principle which introduces diversity among objects of perception must introduce a corresponding diversity among real things. I am not now concerned to argue as to what grounds exist for assuming a correspondence, but, if there is such a correspondence, it must be supposed that diversity in the effects—i.e., the perceived objects—implies diversity in the causes—i.e., the real objects. Hence if I perceive two objects in the field of vision, we must suppose that at least two real objects are concerned in causing my perception. The essential characteristic of particulars, as they appear in perceived space, is that they cannot be in two places at once. But this is an unsatisfactory way of stating the matter, owing to the doubt as to what a "place" is. The more correct statement is that certain perceptible spatial relations imply diversity of their terms; for example, if x is above y, x and y must be different entities. So long, however, as it is understood that this is what is meant, no harm is done by the statement that a thing cannot be in two places at once. We may now return to the question of particulars and universals with a better hope of being able to state precisely the nature of the opposition between them. It will be remembered that we began with three different oppositions : (1) that of percept and concept, (2) that of entities existing in time and entities not existing in time, (3) that of substantives and verbs. But in the course of our discussion a different opposition developed itself, namely, (4) that between entities which can be in one place, but not in more than one, at a given time, and

entities which either cannot be anywhere or can be in several places at one time. What makes a particular patch of white particular, whereas whiteness is universal, is the fact that the particular patch cannot be in two places simultaneously, whereas the whiteness, if it exists at all, exists wherever there are white things. This opposition, as stated, might be -33-

held not to apply to thoughts. We might reply that a man's thoughts are in his head; but without going into this question, we may observe that there certainly is some relation between a man's thoughts and his head (or some part of it) which there is not between his thoughts and other things in space. We may extend our definition of particulars so as to cover this relation. We may say that a man's thought "belongs to" the place where his head is. We may then define a particular in our fourth sense as an entity which cannot be in or belong to more than one place at one time, and a universal as an entity which either cannot be in or belong to any place, or can be in or belong to many places at once. This opposition has certain affinities with the three earlier oppositions, which must be examined. (1) Owing to the admission of particulars in our fourth sense, we can make an absolute division between percepts and concepts. The universal whiteness is a concept, whereas a particular white patch is a percept. If we had not admitted particulars in our fourth sense, percepts would have been identical with certain concepts. (2) For the same reason, we are able to say that such general qualities as whiteness never exist in time, whereas the things that do exist in time are all particulars in our fourth sense. The converse, that all particulars in our fourth sense exist in time, holds in virtue of their definition. Hence the second and fourth senses of the opposition of particulars and universals are co- extensive. (3) The third opposition, that of substantives and verbs, presents more difficulties, owing to the doubt whether predicates are verbs or not. In order to evade this doubt, we may substitute another opposition, which will be co - extensive with substantives and verbs if predicates are verbs, but not otherwise. This other opposition puts predicates and relations on one side, and everything else on the other. What is not a predicate or relation is, according to one traditional definition, a substance. It is true that, when substance was in vogue, it was supposed that a substance must be indestructible, and this quality will not belong to our substances. For example, what a man sees when he sees a flash of lightning is a substance in our sense. But the importance of indestructibility was metaphysical, not logical. As far as logical properties are concerned, our substances will be fairly analogous to traditional substances. Thus we have the opposition of substances on the one hand and predicates and relations on the other hand. The theory which rejects particulars allows entities commonly classed as predicates—e.g. white—to exist; thus the distinction between substances and predicates is obliterated by this theory. Our theory, on the contrary, preserves the distinction. In the world we know, substances are identical with particulars in our fourth sense, and predicates and relations with universals. It will be seen that, according to the theory which assumes particulars, there is a specific relation of subject to predicate, unless we adopt the view— considered above in connexion with Berkeley and Hume—that common sensible qualities are really derivative from specific kinds of likeness. Assum -34-

ing this view to be false, ordinary sensible qualities will be predicates of the particulars which are instances of them. The sensible qualities themselves do not exist in time in the same sense in which the instances do. Predication is a relation involving a fundamental logical difference between its two terms. Predicates may themselves have predicates, but the predicates of predicates will be radically different from the predicates of substances. The predicate, on this view, is never part of the subject, and thus no true subject-predicate proposition is analytic. Propositions of the form "All A is B" are not really subject-predicate propositions, but express relations of predicates; such propositions may be analytic, but the

traditional confusion of them with true subject-predicate propositions has been a disgrace to formal logic. The theory which rejects particulars, and assumes that, e.g., whiteness itself exists wherever (as common sense would say) there are white things, dispenses altogether with predication as a fundamental relation. "This is white," which, on the other view, expresses a relation between a particular and whiteness, will, when particulars are rejected, really state that whiteness is one of the qualities in this place, or has certain spatial relations to certain other qualities. Thus the question whether predication is an ultimate simple relation may be taken as distinguishing the two theories; it is ultimate if there are particulars, but not otherwise. And if predication is an ultimate relation, the best definition of particulars is that they are entities which can only be subjects of predicates or terms of relations, i.e., that they are (in the logical sense) substances. This definition is preferable to one introducing space or time, because space and time are accidental characteristics of the world with which we happen to be acquainted, and therefore are destitute of the necessary universality belonging to purely logical categories. We have thus a division of all entities into two classes: (1) particulars, which enter into complexes only as the subjects of predicates or the terms of relations, and, if they belong to the world of which we have experience, exist in time, and cannot occupy more than one place at one time in the space to which they belong; (2) universals, which can occur as predicates or relations in complexes, do not exist in time, and have no relation to one place which they may not simultaneously have to another. The ground for regarding such a division as unavoidable is the self-evident fact that certain spatial relations imply diversity of their terms, together with the self-evident fact that it is logically possible for entities having such spatial relations to be wholly indistinguishable as to predicates. -35-

:2: UNIVERSALS AND RESEMBLANCES H. H. PRICE WHEN we consider the world around us, we cannot help noticing that there is a great deal of recurrence or repetition in it. The same colour recurs over and over again in ever so many things. Shapes repeat themselves likewise. Over and over again we notice oblong-shaped things, hollow things, bulgy things. Hoots, thuds, bangs, rustlings occur again and again. There is another and very important sort of recurrence which we also notice. The same pattern or mode of arrangement is found over and over again in many sets of things, in many different pairs of things, or triads, or quartets, as the case may be. When A is above B, and C is above D, and E is above F, the above-and-below pattern or mode of arrangement recurs in three pairs of things, and in ever so many other pairs of things as well. Likewise we repeatedly notice one thing inside another, one preceding another, one thing between two others. These recurrent features sometimes recur singly or separately. The same colour recurs in this tomato, that sunset sky, and this blushing face; there are few other features, if any, which repeat themselves in all three. But it is a noteworthy fact about the world that there are conjoint recurrences as well as separate ones. A whole group of features recurs again and again as a whole in many objects. Examine twenty dandelions, and you will find that they have many features in common; likewise fifty cats have very many features in common, or two hundred lumps of lead. In such cases as these there is conjoint recurrence of many different features. Again and again they recur together in a clump or block. This is how it comes about that many of the objects in the world group themselves together into Natural Kinds. A Natural Kind is a group of objects which have many (perhaps indefinitely many) features in common. From observing that an object has some of these features, we can infer with a good deal of probability that it has the rest.

These constant recurrences or repetitions, whether separate or conjoint ones, are what make the world a dull or stale or boring place. The same old ____________________ This selection consists of Chapter 1 of H. H. Price's Thinking and Experience (London: Hutchinson's University Library, 1953), reprinted by permission of the publisher.

features keep turning up again and again. The best they can do is to present themselves occasionally in new combinations, as in the black swan or the duck-billed platypus. There is a certain monotony about the world. The extreme case of it is found where the same old feature repeats itself in all parts of a single object, as when something is red all over, or sticky all through, or a noise is uniformly shrill throughout its entire duration. Nevertheless, this perpetual repetition, this dullness or staleness, is also immensely important, because it is what makes conceptual cognition possible. In a world of incessant novelty, where there was no recurrence at all and no tedious repetitions, no concepts could ever be acquired; and thinking, even of the crudest and most primitive kind, could never begin. For example, in such a world nothing would ever be recognizable. Or again, in so far as there is novelty in the world, non-recurrence, absence of repetition, so far the world cannot be thought about, but only experienced. Hitherto I have been trying to use entirely untechnical language, so that we may not commit ourselves unawares to any philosophical theory. But it is at any rate not unnatural, it is not a very wild piece of theorizing, to introduce the words "quality" and "relation" for referring to those facts about the world to which I have been trying to draw the reader's attention. A quality, we say, is a recurrent feature of the world which presents itself in individual objects or events taken singly. Redness or bulginess or squeakiness are examples. A relation, on the other hand, is a recurrent feature of the world which presents itself in complexes of objects or events, such as this beside that, this preceding that, or B between A and C. It is also convenient sometimes to speak of relational properties. If A precedes B, we may say that A has the relational property of preceding B, and B has the converse relational property of succeeding A. One further remark may be made on the distinction between qualities and relations. I said just now that a quality presents itself in individual objects or events taken singly, and a relation in complexes of objects or events. But it must not be forgotten that an individual object or event usually (perhaps always) has an internal complexity. In its history there is a plurality of temporal phases, and often it has a plurality of spatial parts as well. And there are relations between these parts, or these phases, which it has. Such relations within an individual object or event are sometimes said to constitute the "structure" of the object or event. For scientific purposes, and even for purposes of ordinary common sense prediction, what we most need to know about any object or process is its structure. And from this point of view the chief importance of qualities, such as colour or hardness or stickiness, is that they often enable us to infer the presence of a structure more minute than our unaided senses would reveal. It has often been maintained that sensible qualities are "subjective." But subjective or not, they have a most important function. They give us a clue to what the minute structure of objects and events is. If a gas smells like rotten eggs, we can infer that it is sulphuretted hydrogen. -37-

The terms "quality" and "relation" enable us to give a simple analysis of change. The notion of change has puzzled some philosophers greatly, ever since Heracleitus, or some disciple of his, remarked long ago that πàντa ̕ρ∊î, "all things flow." Indeed, it has sometimes led them to suppose that this world is a world of perpetual novelty after all, and not the tedious or boring or repetitious world which it has to be, if conceptual cognition is to be possible. They

have, therefore, concluded—rightly, from their premises—that all conceptual cognition is radically erroneous or illusory, a kind of systematic distortion of Reality; so that whatever we think, however intelligent or however stupid we may be, we are in error. On this view, only non-conceptual cognition—immediate experience or direct intuition—can be free from error. These conclusions are so queer that we suspect something is wrong with the premises. We can now see what it is. The notion of Change, as Plato pointed out, has itself to be analysed in terms of the notions of Quality and Relation. In qualitative change, as when an apple changes from being green to being red, an object has quality q 1 at one time and a different quality q 2 at a later time. In relational change, an object A has a relation R 1 to another object B at one time, and a different relation R 2 to B at a later time. At 12 o'clock, for example, it is six inches away from B, at 12.5 it is a mile away from B; at one time the relation it has to B is the relation "hotter than," at another the relation "as hot as," at another the relation "cooler than." It is not necessary for our present purposes to enquire whether there are other recurrent features of the world which are neither qualities nor relations nor analysable in terms of these. Some philosophers have thought, for instance, that causality (in its various determinate forms, hitting, bending, pushing, pulling, attracting, repelling, etc.) was such an ultimate and irreducible feature of the world, recurring or repeating itself in many situations. Others have undertaken to give a purely relational analysis of causality (the "Regularity" theory). For our present purpose, however—which is merely to explain why philosophers have thought it worth while to talk about recurrent features of the world at all—it is not necessary to decide how many irreducibly different types of recurrents there might be. It will be enough to consider just qualities and relations. We may now sum up the results of this ontological discussion so far by introducing another technical term, again not so very technical, the term "characteristic." Characteristics, we say, are of at least two different types, qualities and relations. What has been said so far then comes to this: there are recurrent characteristics in the world, which repeat themselves over and over again in many different contexts. Is it not just an obvious fact about the world, something we cannot help noticing whether we like it or not, that there are recurrent characteristics? Now these recurrent characteristics have been called by some philosophers universals. And the line of thought we have been pursuing leads very naturally to the traditional Aristotelian doctrine of universalia in rebus, "universals in things." (To provide for universals of relation, "things" must be understood to cover complexes as well as individuals. -38-

The res which the universal "beside" is in is not this, nor that, but this-and - that.) I do not propose to discuss the Platonic doctrine of universalia ante rem, "universals anterior to (or independent of) things." This is not because I think it uninteresting or unimportant, but merely because it is more remote from common sense and our ordinary everyday habits of thought than the Aristotelian theory of universalia in rebus. It is a sufficiently difficult task in these days to convince people that there is any sense in talking of universals at all, even in the mild and moderate Aristotelian way. The doctrine of universalia in rebus may, of course, be mistaken, or gravely misleading. There certainly are objections to it, as we shall find presently. But I cannot see that it is in the least absurd or silly, as the most approved thinkers nowadays seem to suppose. Nor can I see that it arises entirely from erroneous views about language, as the same thinkers seem to suppose; for example, from the superstition that all words are names, from which it would follow that general or abstract words must be names of general or abstract entities. On the contrary, this philosophy seems to me to be the result, and the very natural result, of certain ontological reflections. It seems to me to arise from reflections about the world; from consideration of what things are, and not— or certainly not merely—from consideration of the way we talk about them. On the contrary, it could be argued that we talk in the way we do, using general terms and abstract terms, because of what we find the world to be; because we find or notice recurrences in it.

Let us now consider how the doctrine of universalia in rebus might mislead us, although it arises in this natural and plausible way from the ontological considerations which we have been discussing. One danger of it obviously is that universals may be regarded as a sort of things or entities, over and above the objects or situations in which they recur. We may indeed emphasize the word "in." We may insist that universals are in things, and not apart from them as the doctrine of universalia ante rem maintains. But is the danger of supposing that they are themselves things or quasi-things entirely removed? Does it not arise over again as soon as we reflect upon the implications of the word "in" itself? If it is our profession to be misled—as, of course, it is the profession of philosophers—we shall be liable to suppose that redness is in the tomato somewhat as juice is in it, or as a weevil is in it. And if so, what can be meant by saying that redness is recurrent? How can it be in thousands of other tomatoes as well, or hundreds of post boxes, or dozens of blushing faces? It does not make sense to say that a weevil is in many places at once. Again, when the tomato begins to decay and turns brown, where has the redness gone to, which used to be in it? (The weevil has gone somewhere else; you will find him in the potato basket.) Likewise, where has the brownness come from? If we prefer to say that the tomato has redness, rather than "redness is in -39-

it," we shall again mislead these literal-minded persons, and in the same sort of way. Does the tomato have redness as Jones has a watch? If so, how can millions of other things have it too? I confess that I do not think much of these difficulties. The meaning of "in" and "have" in this context can be quite easily exhibited by examples, just as their literal meaning can, when we say that there is a weevil in the tomato, or I have a watch. Surely we all know quite well what is being referred to when two things are said to have the same colour? And is it really so very difficult to recognize what is meant by saying that the same colour is in them both? It is true no doubt that the words "in" and "have" are here being used in a metaphorical sense, though not, I think, extravagantly metaphorical. But we must use metaphorical words, or else invent new and technical terms (which are themselves usually metaphorical words taken from a dead language, Greek or Latin). Our ordinary language exists for practical ends, and it has to be "stretched" in one way or other if we are to use it for purposes of philosophical analysis. And if our metaphors can be cashed quite easily by examples, as these can, no harm whatever is done. It could, however, be argued that the terminology of "characteristics," which was current in the last philosophical epoch, some twenty years ago, is better than the more ancient terminology of "universals." A characteristic is pretty obviously a characteristic of something or other, and cannot easily be supposed to be an independent entity, like the weevil. Nor can we be easily misled into supposing that when something "has" a characteristic, i.e. is characterized by it, this is at all analogous to the having of a watch. In the technical symbolism of Formal Logic, the most appropriate expression for referring to one of these recurrent features of the world is not a single letter, such as ϕ or R, which might possibly be mistaken for the name of an entity, but a propositional function, such as ϕx, or xRy, or R(x,y,z). Here the x, y and z are variables, so that the propositional function is an overtly incomplete expression. To complete it, one must replace the variable by a constant, denoting some object which satisfies the function; or if there are several variables, as in xRy or R(x,y,z), each of these must be replaced by a constant. The terminology of characteristics is an approximate equivalent in words to the non-verbal symbolism of propositional functions, and has much the same advantages; whereas if we use the more traditional terminology of universals, there is some danger (though not, I think, unavoidable) that we may be led to speak of them as though they were in themselves complete and independent entities. Henceforth, the Aristotelian theory of universalia in rebus will be called "the Philosophy of Universals" for short. If our argument so far has been correct, the Philosophy of Universals is drawing our attention to certain important facts about the world. Yet it is at the same time

proposing an analysis of those facts. We cannot dispute the facts, nor can we dispute their funda -40-

mental importance. We cannot deny that something which may be called "the recurrence of characteristics" is genuinely there. We must also admit that if it were not there, conceptual cognition could not exist. If the world were not like this, if there were no recurrence in it, it could be neither thought about nor spoken about. We could never have acquired any concepts; and even if we had them innately (without needing to acquire them) they could never have been applied to anything. But though we cannot dispute the facts, nor their importance, we may, nevertheless, have doubts about the analysis of them which the Philosophy of Universals proposes. At any rate, another and quite different analysis of them appears to be possible. It is the analysis offered by what one may call the Philosophy of Ultimate Resemblances. (Henceforth I shall call this "the Philosophy of Resemblances" for short.) This is the analysis which most contemporary philosophers accept, so far as they consider the ontological side of the Problem of Universals at all. It is also accepted by Conceptualists, like Locke. The Philosophy of Resemblances is more complicated than the Philosophy of Universals, and more difficult to formulate. It involves one in long and cumbrous circumlocutions. Yet it claims, not unplausibly, that it keeps closer to the facts which have to be analysed. The unkind way of putting this, the one its critics prefer, is to say that it is "more naturalistic." Let us now consider the Philosophy of Resemblances in more detail. When we say that a characteristic, e.g. whiteness, recurs, that it presents itself over and over again, that it characterizes ever so many numerically different objects, what we say is admittedly in some sense true. But would it not be clearer, and closer to the facts, if we said that all these objects resemble each other in a certain way? Is not this the rock-bottom fact to which the Philosophy of Universals is drawing our attention, when it uses this rather inflated language of "recurrent characteristics"? The Philosophy of Universals of course agrees that all the objects characterized by whiteness do resemble one another. But according to it, resemblance is always derivative, and is just a consequence of the fact that the very same characteristic—whiteness, in this case—characterizes all these objects. To use more traditional language, it says that when A resembles B, this is because they are both instances of the same universal. Now this is all very well where the resemblance is exact, but what are we to say when it is not? Let us consider the following series of examples: a patch of freshly fallen snow; a bit of chalk; a piece of paper which has been used for wrapping the meat in; the handkerchief with which I have been dusting a rather dirty mantelpiece; a full evening dress bow tie which has been left lying about for several years on the floor. All these, we say, are white objects. But are they exactly alike in their colour, if white may be counted as a colour for the purpose of this discussion? Clearly they are not. They are, of course, more or less alike. In fact there is a very considerable degree of colour-likeness between them. But certainly they are not exactly alike in -41-

colour. And yet if the very same characteristic, whiteness, is present in them all (as the Philosophy of Universals, apparently, says it is) ought it not to follow that they are exactly alike in colour? To make quite clear what the point at issue is, we shall have to distinguish, rather pedantically perhaps, between exact resemblance in this or that respect and total or complete resemblance. To put it in another way, resemblance has two dimensions of variation. It may vary in intensity; it may also vary in extent. For example, a piece of writing paper and an envelope, before one has written on either of them, may be exactly alike in colour, and perhaps also in texture. These likenesses between them have the maximum degree of intensity. But the two objects are not completely or totally alike. For one thing, they are unlike in shape. Moreover, the envelope is stuck together with gum and has gum on its flap,

while the piece of writing paper has no gum on it. It might perhaps be thought that two envelopes from the same batch are completely alike; and certainly they come nearer to it than the envelope and the piece of notepaper. All the same, there is unlikeness in respect of place. At any given time, envelope A is in one place and envelope B is in a different place. On the Relational Theory of Space, this is equivalent to saying that at any given time A and B are related in unlike ways to something else, e.g. the North Pole, or Greenwich Observatory. According to Leibniz's Principle of the Identity of Indiscernibles, complete or total likeness is an ideal limit which can never quite be reached, though some pairs of objects (the two envelopes, for example) come closer to it than others. For if per impossible two objects were completely alike, place and date included, there would no longer be two objects, but only one. Whether Leibniz's Principle is correct, has been much disputed. But we need not concern ourselves with this dispute. It is sufficient to point out that if there were two objects which resembled each other completely, in date and place as well as in all other ways, and this complete resemblance continued throughout the whole of the histories of both, there could not possibly be any evidence for believing there were two of them. So in this discussion we need not concern ourselves any more with complete or total resemblance, though it is of course an important fact about resemblances that they vary in extent, as well as in degree of intensity. What does concern us is intensity of resemblance. The maximum intensity of it is what I called "exact resemblance in this or that respect." Now some people appear to think that even this is an ideal limit. They seem to think that no two objects are ever exactly alike even in one way (e.g. colour, or shape) though, of course, many objects are closely alike in one way or in several. I do not see what evidence we could have for believing such a sweeping negative generalization. It is true that sometimes, when we thought at first that there was an exact likeness in one or more respects between two objects, we may find on more careful examination that there was not. We may have thought that two twins were exactly alike in the conformation of -42-

their faces. We look more closely, and find that John's nose is a millimetre longer than William's. But still, there are many cases where there is no discoverable inexactness in a resemblance. We often find that two pennies are indistinguishable in shape, or two postage stamps indistinguishable in colour. And we should not confine ourselves to cases where two or more objects are being compared. There is such a thing as monotony or uniformity within a single object. For example, a certain patch of sky is blue, and the same shade of blue, all over. It is monotonously ultramarine. In other words, all its discernible parts are exactly like each other in colour; at any rate, we can discover no unlikeness of colour between them. Again, there is often no discoverable unlikeness of pitch between two successive phases of the same sound. Will it be said that such monotony is only apparent, not real? But what ground could we have for thinking that no entity is ever really "monotonous" in this sense, not even in the smallest part of its extent, or throughout the smallest phase of its duration? Thus there is no good ground for maintaining that resemblance of maximum intensity never occurs at all, still less for maintaining that it never could occur. Nevertheless, it is not so very common for two objects to be exactly alike even in one way, though monotony within a single object or event is more frequent. What we most usually find in two or more objects which are said to be "alike" is close resemblance in one respect or in several. We can now return to the controversy between the Philosophy of Resemblances and the Philosophy of Universals. It is argued that if the Philosophy of Universals were right, exact resemblance in one or several respects (resemblance of maximum intensity) ought to be much more common than it is; indeed, that inexact resemblance in a given respect, say colour or shape, ought not to exist at all. Of course, there could still be incomplete or partial resemblance, resemblance between two objects in one respect or in several, and lack of resemblance in others. But whenever two objects do resemble each other in a certain respect, it would appear that the resemblance ought to be exact (of maximum intensity), if the Philosophy of Universals were right; either it should be exact, or else it should not exist at all. The Philosophy of Universals tells us that resemblance is derivative, not ultimate; that when two objects resemble each other in a given respect, it is because the very same

universal is present in them both. This seems to leave no room for inexact resemblance. Now if we consider the various white objects I mentioned before—the whole series of them, from the freshly fallen snow to the unwashed bow-tie— how can anyone maintain that the very same characteristic, whiteness, recurs in all of them? Clearly it does not. If it did, they must be exactly alike in their colour; and quite certainly they are not. If we are to use the language of universals or characteristics, shall we not have to say that each of the objects in this series, from the snow to the unwashed tie, is characterized by a different characteristic, or is an instance of a different universal? In this case, -43-

then, the resemblance seems to be ultimate and underivative, not dependent on the presence of a single universal in all these objects, although they certainly do resemble each other. Let us consider another example. Two pennies may be exactly alike in their shape. If so, one may plausibly say that the very same characteristic, roundness, is present in both of them, and that their resemblance is dependent on this. But what about a penny and a sixpence? They certainly are alike in shape; but not exactly, because the sixpence has a milled edge and the penny a smooth one. So here again, it would seem, there is no single characteristic present in them both, upon which the resemblance could be dependent. This resemblance again seems to be ultimate and underivative. Thus the Philosophy of Universals, when it makes all resemblance derivative, appears to forget that resemblances have degrees of intensity. Resemblance is treated as if it were degreeless, either present in its maximum degree or else not present at all. In practice, the Philosopher of Universals concentrates his attention on close resemblances, and averts his attention from the awkward circumstance that few of them are exact; and resemblances of a lower degree than this (small or moderate ones, not intense enough to be called "close") are just neglected altogether. But is it not a glaringly obvious fact that resemblances do differ in degree or intensity? That being so, shall we not be inclined to reverse this alleged dependence- relation between "being alike" and "being characterized by"? Surely we shall be inclined to say that it is resemblance which is more fundamental than characterization, rather than the other way round. We shall, of course, be willing to go on using terms like "characteristic" and "characterized by"; they are part of ordinary language, and everyone has a sufficient understanding of them. But we shall define "characteristic" in terms of resemblance, and not conversely. Where a number of objects do happen to resemble each other exactly in one respect or three or fifteen, there, and in consequence, we shall be quite willing to say that they have one, or three, or fifteen "characteristics in common." But in other cases, where the resemblance is less than exact, we shall not be willing to say this. We shall just say that they resemble each other in such and such a degree, and stop there. In a given set of objects there is whatever degree or resemblance there is. Let us be content to take the facts as we find them. Turning for a moment to the epistemological side of the matter, surely it is obvious that the applicability of concepts does not require an exact resemblance in the objects which a concept applies to? Of course there does have to be a considerable degree of resemblance between all the objects which "satisfy" a given concept. As we say, there has to be a sufficient likeness between them, e.g. between all the objects to which the concept White applies. What degree of likeness is sufficient, and where the borderline comes between something which falls just within the concept's sphere of application and something else which just falls outside it, is often difficult to decide. For instance, one may wonder whether the very dirty bow-tie is white at all. In -44-

deed, it is difficult to see how such a question can be definitely answered, at least in the case of whiteness and many other familiar concepts. The right way to tackle it, perhaps, is to refuse to answer it as it stands. Perhaps we should rather say that a concept may be

"satisfied" in many different degrees; or, in more commonsensical language, that there are good instances and bad instances, better and worse ones, and some so bad that it is arbitrary whether one counts them as instances or not. Thus the piece of chalk is a better instance of whiteness than the rather dirty handkerchief is. The patch of freshly fallen snow is a better instance still, perhaps a perfect one. We may give it the mark a (+). Then αβ is about the right mark for the piece of chalk, and we will give the unwashed bow-tie γ=, to denote that it is just on the borderline between "pass" and "failure." It is not easy to see how the doctrine of universalia in rebus can make any room for this important and familiar notion of degrees of instantiation. But there is plenty of room for it in Conceptualism, which is the epistemological counterpart of the ontological Philosophy of Resemblances. We must add, in fairness, that there is also plenty of room for it in the Platonic doctrine of universalia ante rem. Indeed Plato, or perhaps Socrates, was the first philosopher who noticed that there are degrees of instantiation. This is one of the points, and a good point, which Conceptualism and Platonic Realism have in common. 1 In the last few pages, I have been discussing the difficulties which the Resemblance Philosophers find in the Philosophy of Universals. But the Philosophy of Resemblances has its difficulties too. The most important ones are concerned with resemblance itself. I shall discuss two of them, and the solutions proposed for them. The first arises from the phrase "resemblance in respect of...." It is obvious that we must distinguish between different resemblances. Objects resemble each other in different respects, as well as in different degrees. Red objects resemble each other in one respect, round objects in another respect. The members of a natural kind, for instance cats or oak trees, resemble each other in many respects at once. Thus it would be much too vague if we said that red objects, for example, are just a set of objects which resemble one another, or sufficiently resemble each other. That would not distinguish them from blue objects, or round objects, or any other class of objects one cares to name. We must specify what resemblance it is. Red objects are those which resemble each other "in a certain respect." But in what respect ? And now it looks as if we should have to introduce universals again. Our first answer would probably be that they resemble each other in respect of colour; and this looks very like saying that they are all instances of the universal Colouredness. That is bad enough; but we shall be driven to go ____________________ 1In Christian Platonism, where Plato's transcendent "forms" become concepts in the mind of God, the differences between Platonic Realism and Conceptualism are still further diminished, though they do not disappear altogether. -45-

farther, because we have not yet said enough to distinguish red objects from blue ones or green ones. Can we stop short of saying that red objects are just those objects which resemble each other in respect of redness? And here we seem to be admitting the very point which the Philosophy of Universals is so anxious to maintain; namely that the resemblance between these objects is after all derivative, dependent upon the presence of a single universal, Redness, in them all. To generalize the argument: whenever we say that A, B and C resemble each other in a certain respect, we shall be asked "In what respect?" And how can we answer, except by saying "in respect of being instances of the universal ϕ" or "in respect of being characterized by the characteristic ϕ"? We may try to get round the difficulty by saying that they resemble each other in a certain way (avoiding the word "respect"), or that there is a certain sort of resemblance between them. But when we are asked to specify in what way they resemble each other, or what sort of resemblance there is between them, surely we shall still have to answer by mentioning such and such a universal or characteristic? "The way in which red objects resemble each other is that all of them are instances of the universal Redness, or all of them are characterized by the characteristic Redness." This is one of the classical objections to the Philosophy of Resemblances. The argument purports to show that resemblance is not after all ultimate or underivative, but is dependent on the presence of a universal or characteristic which is common to the things which

resemble each other. There is something about this objection which arouses our suspicions. It comes perilously near to the tautology "red things are the things which are red." The Resemblance philosophers were not undertaking to deny this tautology. They do not deny that x is red entails x is red. They are only concerned to offer an analysis of x is red itself. Let us now consider the answer they might make to this celebrated objection. Roughly, it consists in substituting "resemblance towards ..." for "resemblance in respect of ..." Resemblance towards what? Towards certain standard objects, or exemplars as I shall call them—certain standard red objects, or standard round objects, or whatever it may be. It is agreed by both parties that there is a class of red objects. The question is, what sort of a structure does a class have? That is where the two philosophies differ. According to the Philosophy of Universals, a class is so to speak a promiscuous or equalitarian assemblage. All its members have, as it were, the same status in it. All of them are instances of the same universal, and no more can be said. But in the Philosophy of Resemblances a class has a more complex structure than this; not equalitarian, but aristocratic. Every class has, as it were, a nucleus, an inner ring of key-members, consisting of a small group of standard objects or exemplars. The exemplars for the class of red things might be a certain tomato, a certain brick and a certain British post- box. Let us call them A, B and C for short. Then a red object is any object which resembles A, B and C as closely as they resemble one another. The resemblance between the exemplars need not itself be a very close one, -46-

though it is of course pretty close in the example just given. What is required is only that every other member of the class should resemble the class - exemplars as closely as they resemble one another. Thus the exemplars for a class might be a summer sky, a lemon, a post-box, and a lawn. These do resemble one another, though not very closely. Accordingly there is a class consisting of everything which resembles these four entities as closely as they resemble each other. It is the class of coloured things, whereas the previous one was the class of red things. It may be thought that there is still a difficulty about the resemblance between the exemplar objects themselves. In what respect do the tomato, the brick and the post-box resemble each other? Surely this question still arises, even though it does not arise about the other members of the class? And how can one answer it, except by saying that these three objects resemble each other in respect of being red, or of being characterized by redness? But this assumes that we know beforehand what "being red" is, or what "being characterized by redness" amounts to. And this begs the question against the Resemblance Philosophy. The Resemblance Philosophers maintain that our knowledge of what it is for something to be red just consists in a capacity to compare any particular object X with certain standard objects, and thereby to discover whether X does or does not resemble these standard objects as closely as they resemble each other. It does not make sense to speak of comparing the standard objects with themselves, or to ask whether they resemble one another as closely as they do resemble one another. Yet that is just what we should be trying to do, if we tried to say "in what respect" they are alike. To say that they are red, or are characterized by redness, would not be an informative statement, but a tautology. This objection does however draw our attention to an important point. According to the Philosophy of Resemblances, there cannot be a class unless there are exemplar objects to hold the class together. Nevertheless, the same class may have alternative sets of exemplars. The class of red things, we said, consists of everything which resembles the post-box, the tomato and the brick as closely as they resemble each other. It could equally be said to consist of everything which resembles a certain bit of sealing wax, a certain blushing face and a certain sunset sky as closely as they resemble each other. In that case, it does make sense to ask whether the post-box, the tomato and the brick are red, or are characterized by redness. And the answer "Yes, they are" is now no longer tautologous. We are no longer trying, absurdly, to compare them with themselves. We are comparing them with three other things, and discovering that they do all have a sufficient degree of resemblance to these other things. But because there are (within limits) alternative sets of standard objects for the same class, we are led to suppose, erroneously, that a class can exist without any standard

objects at all. This or that set of standard objects can be deposed from its privileged position without destroying the unity of the class; and we then suppose, by a process of illegitimate generalization, that the class would still remain what it is if privilege -47-

were abolished altogether. There must be a set of standard objects for each class, though within limits it does not matter which set of objects have this status. Thus in the Philosophy of Resemblances, as well as the Philosophy of Universals, there does after all have to be something which holds a class together, if one may put it so. Where the two philosophies differ is, in their view of what that something is. In the Philosophy of Universals, what holds a class together is a universal, something of a different ontological type from the members. In the Philosophy of Resemblances there is no question of different ontological types. There are just particular objects, and there is nothing non-particular which is "in" them, in the way that a universal is supposed to be "in" the particulars which are its instances. What holds the class together is a set of nuclear or standard members. Anything which has a sufficient degree of resemblance to these is thereby a member of the class; and "resembling them sufficiently" means "resembling them as closely as they resemble each other." Again, to turn for a moment to epistemological considerations, it is their relationship to the standard objects or exemplars which enables all these objects to satisfy the same concept, e.g. the concept Red, and likewise enables the same word or other symbol, e.g. the word "red," to apply to them all. But this is to anticipate. The Philosophy of Resemblances is an ontological doctrine, though it may be used as the starting point for certain epistemological theories (Conceptualism, Imagism and Nominalism), just as the Philosophy of Universals may be used as the starting-point of a Realist epistemology. If the Philosophy of Resemblances is true at all, it might still have been true even if there had been no thinkers and no speakers. As it happens, there are thinkers and speakers too. But there may be many classes in the world, which do exist (because the requisite resemblances do happen to be there) although no mind happens to have formed the corresponding class - concepts, and no speaker has acquired the habit of using the corresponding class-symbols. Thus there is nothing subjectivist or anthropocentric about the Philosophy of Resemblances. It denies that there are universals in rebus, but it asserts that there are resemblances inter res. Certain objects really are as like the objects A, B and C as these are to one another, whether anyone notices the fact or not. Known or not, spoken of or not, the relationship is there; just as in the Philosophy of Universals objects are instances of universals whether they are known to be so or not. In this respect, both these philosophies are equally "realistic." We must now turn to the second of the classical objections against the Philosophy of Resemblances, an objection so familiar that one might almost call it notorious. It is concerned with resemblance itself. Surely resemblance is itself a universal, present in many pairs or groups of resemblant objects? It is of course, a universal of relation. The instances of it are not individual -48-

objects taken singly, but complexes, and each of these complexes is composed of two objects or more. In their attempt to "get rid of universals," the Philosophers of Resemblance seem to concentrate their attention on universals of quality (e.g. redness, colour, shape) and say little or nothing about universals of relation. Hence they have failed to notice that resemblance itself is one of them. But if we are obliged to admit that resemblance at any rate is a genuine universal, a relation which does literally recur in many different situations or complexes, what ground have we for denying that there are other universalia in rebus as well? It may seem audacious to question this formidable argument, which has convinced many illustrious men. But is it as strong as it looks? The Resemblance philosophers might very well reply that it begs the question at issue, that it just assumes what it purports to prove. For after all, what reason is given for the crucial assertion that resemblance is a universal? Apparently none. It is not enough just to say "surely resemblance at any rate is a universal."

Could any reason be given? We might perhaps try to find one by starting from the linguistic side of the matter. The word "resemblance," we might say, is an abstract word, like the words "redness" and "proximity"; therefore it must stand for a universal or characteristic (a relational one, of course). But if this is the argument, it seems to beg the question. For if one does start from a linguistic point of view, the very question at issue is whether abstract words and general words do stand for universals. And if the argument is to be cogent, it ought to be an argument about the noun "resemblance" in particular, or about the verb "to resemble" in particular. We ought to be shown that it is somehow peculiarly obvious that this word at any rate (or this pair of words) stands for a universal, even though it may be less obvious that other general words do. Perhaps it will be said, the peculiar obviousness consists in this, that even the people who try to get rid of universals have to use this general word at least, or equivalent general words such as "similar," "like." True enough, one cannot speak in a language consisting entirely of proper names and demonstratives. One cannot say anything at all without using some general words. As an observation about the nature of language, this is perfectly indisputable. But the question is, what are its implications? Does it follow that because we must use general words, there are, therefore, general somethings in rerum natura which they stand for? That is just the point at issue. One cannot just assume that the answer is "Yes." Of course the Philosophy of Resemblances admits that we do use general words, and cannot avoid using them if we are to speak at all. It does not at all deny the fact. But it does deny the conclusion which the Philosophy of Universals draws from it— namely that because we use general words, there must be general somethings (universals) which they mean. Has anything been done to show that this denial is mistaken? Nothing. The Philosophy of Universals has just repeated over again the principle which has to be proved, the principle that every -49-

general word stands for a universal; adding—what is obvious—that if this principle is true, the word "resemblance" is an illustration of it. Of course. But is the principle true? If the Philosopher of Resemblances is asked to explain how the general word "resemblance" is used, or what kind of meaning it has, he will presumably point out that there are resemblances of different orders. Two cats, A and B, resemble each other, and two sounds, C and D, also resemble each other. These are first-order resemblances. But it is also true that the two-cat situation resembles the two-sound situation, and resembles many other situations too. This is a second-order resemblance. The A-B situation and the C-D situation really are alike, though the constituents of the one are unlike the constituents of the other. In virtue of this second-order likeness (a likeness between likeness-situations) we may apply the same general word to both of them; and the word we happen to use for this purpose is the word "resemblance," in a second-order sense. There is nothing wrong or unintelligible in the notion of second-order resemblance. Or if it be said that there is, we can reply with the tu quoque argument that universality must itself be a universal. When it is said that "cathood is a universal" the word "universal" is itself a general word, just as "cat" is when we say "Pussy is a cat." So according to the Philosophy of Universals, there must be a universal called "universality." And if it is a universal, universality must accordingly be an instance of itself. But this is a contradiction. For according to this Philosophy, anything which is an instance of a universal is ipso facto a particular, and not a universal. To get out of this difficulty, the Philosophy of Universals must introduce the notion of "different orders" too. The word "universal," it has to say, stands for a second-order universal, whereas "green" or "cat" or "in" stand for first-order ones. This is equivalent to saying that the expression "a universal," or the propositional function " ϕ is a universal," can occur only in a metalanguage. This suggests another way in which the Philosophy of Resemblances might reply to the objection that "resemblance is itself a universal." The objection assumes that resemblance is just one relation among others: a relation of the same type as "on top of," or "near to," or "side by side with." But according to the Philosophy of Resemblances, resemblance is not just one relation among others. Indeed, according to this philosophy, it would be misleading to call it "a relation" at all. It is too fundamental to be called so. For what we ordinarily call "relations" (as well as what we call "qualities") are themselves founded upon or analysable into resemblances. For example, the relation "being inside of" is founded upon the resemblance between the Jonahwhale complex, the room-house complex, the match-

matchbox complex, etc. Moreover, the Philosophy of Universals itself does not really hold that resemblance is just one relation among others, and in pretending that it does, it is abandoning one of its own fundamental principles; indeed it is abandoning the very one which this argument ("resemblance is itself a universal") is ultimately intended to establish, the principle, namely, that all resemblance -50-

is derivative. In the Philosophy of Universals itself, resemblance has a status quite different from relations like "side by side with" or "on top of." Resembling is connected with being an instance of ... in a way that ordinary relations are not. When A resembles B and C, this is supposed to be a direct consequence of the fact that A, B and C are all instances of the same universal ; and not only when A, B and C are individual objects (in which case the universal is a universal of quality) but also when they are complexes, so that the universal they are instances of is a relational one, such as "being inside of." If resemblance, in the Philosophy of Universals, is to be called a relation at all, it is a relation of a very special sort, quite different from anything to which the word "relation" is ordinarily applied. We should have to say that it is a "formal" or "metaphysical" relation (as opposed to a "natural" or empirical one) just as the relation of instantiation is, if that can be called a relation at all. So much for the reply the Philosophy of Resemblances might make to this celebrated argument that "resemblance is itself a universal." First, it might be objected that the argument begs the question, by just assuming (what it ought to prove) that because "resemblance" is admittedly a general word, it must stand for a universal. Secondly, the argument overlooks the fact that there are resemblances of different orders. Thirdly, it treats resemblance as one relation among others, parallel in principle to "side by side with" or "on top of," whereas the Philosophy of Resemblances maintains that it is too fundamental to be called a relation at all, in the ordinary sense of the word "relation." Fourthly, the Philosophy of Universals itself admits, in its own way, that resemblance does not have the same status as other relations, in spite of maintaining in this argument that it has. Thus the Philosophy of Resemblances has an answer to these two classical objections, the one about "resemblance in respect of" and the one we have just discussed "that resemblance is itself a universal." But the Philosophy of Universals also has an answer to the objection about inexact resemblances, and to the complaint that it ignores the different degrees of intensity which resemblances may have. We must consider this answer if we are to do justice to both parties. The first step is to distinguish between determinable and determinate characteristics. Universals or characteristics, it is said, have different degrees of determinateness. The adjectives "determinable" and "determinate" are too fundamental to be defined. But their meaning can be illustrated. Thus the characteristic of being coloured is a determinable, and the characteristic of being red is a determinate of it. Being red is again a sub-determinable, and has under it the determinates being scarlet, being brick-red, being cherry-red, etc. Likewise, being a mammal is a determinable characteristic, a highly complex one this time. There are many different ways of being a mammal. Being a dog, being a whale, being a man are some of the determinates of this determinable. -51-

Now whenever two objects resemble each other with less than the maximum intensity (i.e. whenever they have what was called an "inexact" resemblance) we can always say that the same determinable characteristic characterizes them both, though not the same determinate one. Two objects may each have a different shade of red. A is scarlet, and B is brick-red. They resemble each other fairly closely, but by no means exactly. That is because redness itself is a determinable characteristic, a sub-determinable falling under the higher determinable colouredness. The two objects do have this determinable characteristic in common, though each of them has a different determinate form of it. So we can still maintain that this resemblance, though inexact, is a derivative, dependent on the presence of the same determinable universal in both objects.

Let us apply these considerations to the two examples given earlier: (1) the various white objects; (2) the penny and the sixpence. It may now be maintained that all my different white objects—from the freshly fallen snow at one end of the series to the unwashed bow-tie at the other—do have a determinable characteristic in common; though "whitish," rather than "white," would be the appropriate word for it. "White" might be taken to mean pure white. And pure white is only one determinate of the determinable whitish. We certainly should not say that all the objects in this series are pure white. At the most, only the freshly fallen snow is pure white, but not the piece of chalk, or the rather messy bit of paper, or the rather dirty handkerchief, or the very dirty unwashed bow-tie. But we should admit that all of them are "whitish." Let us now consider my other example, the penny and the sixpence, which resemble each other in shape, but inexactly. The penny with its smooth edge and the sixpence with its milled (slightly serrated) edge have different determinate shapes. How is it, then, that they do still resemble each other in shape, though inexactly, and both would be called "round coins" in ordinary speech? Because the same determinable shape—we might more appropriately call it "roundish"—characterizes both of them; and it characterizes many other things as well, e.g. slightly buckled bicycle wheels, cogwheels with not too large teeth, which resemble each other a good deal less closely than the penny and the sixpence do. By this expedient the Philosophy of Universals is able to maintain its thesis that all resemblances, inexact ones too, are derivative, and not ultimate, as the Philosophy of Resemblances would have them. Inexact resemblance, we are invited to say, depends upon or is derived from the presence of the same determinable characteristic in a number of objects; exact resemblance (resemblance of maximum intensity) depends upon their being characterized by the same determinate characteristic. Perhaps this will also enable us to dispense with the notion of "degrees of instantiation" which was mentioned earlier. It was not easy to see what could be meant in the Philosophy of universalia in rebus by saying that one object is a better instance of so-and-so than another, though this notion fits well -52-

enough into the Platonic theory of universalia ante rem, and into Conceptualism too. Perhaps it could now be suggested that the determinates of some determinables, e.g. "whitish," "roundish," are serially ordered. Thus the various determinates of whitishness which characterize the patch of snow, the piece of chalk, the paper, etc., may be arranged in a series beginning with "pure white." After this comes "nearly pure white" (the colour the piece of chalk has), then "farther from pure white" and then "farther still from pure white," until we come to a characteristic which is as far from pure whiteness as it can be without ceasing to be a determinate of whitishness at all. The system of marking (a+, a, a—, β+, etc.) which we suggested for indicating the "goodness" or "badness" of instances can still be used: only it is now applied not to the objects themselves, but to the determinate characteristics by which they are respectively characterized. Thus this objection to the Philosophy of Universals, that it can make no room for inexact resemblances (resemblances of less than the maximum intensity), turns out after all to be indecisive, although it looked so convincing at first sight. The facts to which this argument draws our attention are of course perfectly genuine, and important too. It is, for example, an important fact about language that most of our general words apply to sets of objects which inexactly resemble one another; and it is an important fact about thinking that the various objects which "satisfy" a given concept, e.g. the concept of Crow, do not have to be exactly alike. Nevertheless, this argument does not at all refute the Philosophy of Universals, as it is often supposed to do. All it does is to point out what was lacking in our first rough-and ready formulation of that philosophy. Certainly the Philosophy of Universals would be quite unworkable without the distinction between determinable and determinate universals. The doctrine that universals or characteristics have different degrees of determinacy is an indispensable part of it. But the distinction between determinables and determinates is perfectly consistent with the contention that there are recurrent characteristics in the world, and with the accompanying doctrine that resemblances are derivative, not ultimate. Indeed,

it could be argued, the fact that recurrent characteristics do differ in their degree of determinateness is just as obvious as the fact of recurrence itself. Finally, it is worth repeating that the phrases "inexact resemblance" "not exactly alike" are sometimes used in another way, to mean incomplete or partial resemblance. If A and B are closely alike in a large number of respects, but unlike or not closely alike in one or two, we sometimes say that they are very like each other but not exactly like each other. For example, within the same species of bird we often find that there are slight differences of size or colouring between two individual specimens, although they also resemble each other closely in very many ways. It is obvious that if the phrase "inexact resemblance" is used in this sense, the Philosophy of Universals has no difficulty at all about inexact resemblances. We merely have to say that many universals are common to the two birds, or recur in both of them; and consequently the -53-

two individuals resemble each other in a great many respects. We then add that bird A is also an instance of a certain universal ϕ, while bird B is not an instance of this, but of a certain other universal ψ; and consequently there is a respect in which they are not alike. (It may be found, of course, and in this example it almost certainly will be, that though ϕ and ψ are different determinate universals, they are determinates of the same determinable universal, say "mottled.") It must not be forgotten that every individual object is an instance of several universals at once, and often of very many at once. When we compare it with another object, we may easily find that some universals are common to both of them, and other universals are not. It would be a strange misunderstanding of the Philosophy of Universals to suppose that in this philosophy every particular is held to be an instance of only one universal. When we say that something is a cat, we are saying that it is an instance of many universals conjointly, and not just of one. Our discussion has been long and complicated. What conclusion shall we draw from it? It would seem that there is nothing to choose between these two philosophies, the Philosophy of Universals or characteristics (universalia in rebus) 2 on the one hand, and the Philosophy of Ultimate Resemblances on the other. At any rate, it would seem that there is nothing to choose between them so long as they are considered as purely ontological doctrines, which is the way we have been considering them in this chapter. Both seem to cover the facts, though only when both are stated with sufficient care. Moreover, they both cover the same facts. This strongly suggests that they are two different (systematically different) terminologies, two systematically different ways of saying the same thing. It does not follow that both alike are just pieces of solemn and elaborate trifling. On the contrary, the thing which they both say is of the first importance, and we do need a way of saying it. The efforts which each party has made to provide us with a systematic terminology for saying it have not been a waste of time. For if there were no recurrent characteristics, or no resemblances between different objects— whichever way you choose to put it—there could be no conceptual cognition, and no use of general symbols either. Now if there is only a (systematic) difference of terminology between these two philosophies, it is well to be familiar with both. Each of them may have its misleading features; and when we are in danger of being misled by the one, we may save ourselves by changing over to the other. The danger of the terminology of Universals has been pointed out already. If we can only do our philosophizing in this terminology, we may be led to regard universals as things or entities. We reduce this danger by using the word "characteristic" instead; or by using phrases like "being red" "being a cat" "being side by side with ..." instead of noun-substantives like "redness," ____________________ 2It may be worth while to remind the reader that the phrase "the Philosophy of Universals," as it has been used in this chapter, is not intended to cover the Platonic doctrine of universalia ante rem. -54-

"cathood," "side-by-sideness," which do look like names for entities; or by using the propositional function notation ϕx, xRy, R(x,y,z), etc., where "x," "y" and "z" are variables. But perhaps we do not avoid the danger altogether, especially when we make very general statements, as we have to in philosophy; for example "characteristics are divided into two sorts, qualities and relations" or even "the characteristic of being red entails the characteristic of being coloured." Such statements may mislead us into supposing that "there are" characteristics in the sense in which "there are" dogs, or planets. We can avoid these dangers by changing over to the terminology of Resemblances, and by recalling that everything which can be said in the language of Universals or Characteristics can also be said (though usually less elegantly) in the language of Resemblances. Perhaps there is another danger as well. The Philosophy of Universals may tend to make us think that the world is a more neat and tidy place than it is. If one may say so, there is sometimes a certain air of infallibility or omni - competence about its exponents, as if the basic structure of the universe were perfectly clear to them, and only a few rather unimportant details remained to be settled. The Philosophy of Resemblances delivers us from this danger, by reminding us that most of the resemblances we think and talk of are by no means exact ones. It restores to human thought and language that fuzziness or haziness, that absence of hard and fast boundaries, which do belong to them, and even in a way to the world itself. On the other hand, the terminology of Resemblances has its defects too. It is clumsy, complicated, and difficult to handle. Moreover, it tends to make us too much preoccupied with the inexactitude of resemblances; and so we may come to forget the vastly important fact that after all they are resemblances, and some of them pretty close ones too. There is such a thing as too much attention to "marginal cases." Attention to them is a philosophical virtue, but exclusive preoccupation with them is a philosophical vice. If that is our temptation, we may escape it by changing over to the terminology of Universals. In this terminology, we remind ourselves, there are determinable characteristics and not only determinate ones; so that even where objects resemble each other inexactly, there is still recurrence. -55-

:3: ON CONCEPT AND OBJECT GOTTLOB FREGE IN a series of articles on intuition and its psychical elaboration, Benno Kerry has several times referred to my Grundlagen der Arithmetik and other works of mine, sometimes agreeing and sometimes disagreeing with me. I cannot but be pleased at this, and I think the best way I can show my appreciation is to take up the discussion of the points he contests. This seems to me all the more necessary, because his opposition is at least partly based on a misunderstanding, which might be shared by others, of what I say about the concept; and because, even apart from this special occasion, the matter is important and difficult enough for a more thorough treatment than seemed to me suitable in my Grundlagen. The word "concept" is used in various ways; its sense is sometimes psychological, sometimes logical, and sometimes perhaps a confused mixture of both. Since this licence exists, it is natural to restrict it by requiring that when once a usage is adopted it shall be maintained. What I decided was to keep strictly to a purely logical use; the question whether this or that use is more appropriate is one that I should like to leave on one side, as of minor importance. Agreement about the mode of expression will easily be reached when once it is recognized that there is something that deserves a special term. It seems to me that Kerry's misunderstanding results from his unintentionally confusing his own usage of the word "concept" with mine. This readily gives rise to contradictions, for

which my usage is not to blame. Kerry contests what he calls my definition of "concept." I would remark, in the first place, that my explanation is not meant as a proper definition. One cannot require that everything shall be defined, any more than one can require that a chemist shall decompose every substance. What is simple cannot be decomposed, and what is logically simple cannot have a proper definition. Now something logically simple is no more given us at the outset than most of the chemical elements are; it is reached only by means of scientific work. If something has been discovered that is simple, or at least must count as ____________________ First published in the Vierteljahrsschrift fiir wissenschaftliche Philosophie (1892); this translation is reprinted from Translations from the Philosophical Writings of Gottlob Frege, edited by Peter Geach and Max Black (Oxford: Basil Blackwell, 1952) by permission of the publisher.

simple for the time being, we shall have to coin a term for it, since language will not originally contain an expression that exactly answers. On the introduction of a name for something logically simple, a definition is not possible; there is nothing for it but to lead the reader or hearer, by means of hints, to understand the words as is intended. Kerry wants to make out that the distinction between concept and object is not absolute. "In a previous passage," he says, "I have myself expressed the opinion that the relation between the content of the concept and the concept-object is, in a certain respect, a peculiar and irreducible one; but this was in no way bound up with the view that the properties of being a concept and of being an object are mutually exclusive. The latter view no more follows from the former than it would follow, if, e.g., the relation of father and son were one that could not be further reduced, that a man could not be at once a father and a son (though of course not, e.g., father of the man whose son he was)." Let us fasten on this simile! If there were, or had been, beings that were fathers but could not be sons, such beings would obviously be quite different in kind from all men who are sons. Now it is something like this that happens here. The concept (as I understand the word) is predicative. 1 On the other hand, a name of an object, a proper name, is quite incapable of being used as a grammatical predicate. This admittedly needs elucidation, otherwise it might appear false. Surely one can just as well assert of a thing that it is Alexander the Great, or is the number four, or is the planet Venus, as that it is green or is a mammal? If anybody thinks this, he is not distinguishing the usages of the word "is." In the last two examples it serves as a copula, as a mere verbal sign of predication. (In this sense [the German word ist] can sometimes be replaced by the mere personal suffix: cf. dies Blatt ist grün and dies Blatt grünt.) We are here saying that something falls under a concept, and the grammatical predicate stands for this concept. In the first three examples, on the other hand, "is" is used like the "equals" sign in arithmetic, to express an equation. 2 In the sentence "The morning star is Venus," we have two proper names, "morning star" and "Venus," for the same object. In the sentence "the morning star is a planet" we have a proper name, "the morning star," and a concept-word, "planet." So far as language goes, no more has happened than that "Venus" has been replaced by "a planet"; but really the relation has become wholly different. An equation is reversible; an object's falling under a concept is an irreversible relation. In the sentence "the morning star is Venus," "is" is obviously not the mere copula; its con____________________ 1It is, in fact, the reference of a grammatical predicate. 2I use the word "equal" and the symbol " = " in the sense "the same as," "no other than" "identical with." Cf. E. Schroeder Vorlesungen ueber die Algebra der Logik (Leipzig 1890), Vol. 1, No. 1. Schroeder must however be criticized for not distinguishing two fundamentally different relations; the relation of an object to a concept it falls under, and the subordination of one concept to another. His remarks on the Vollwurzel are likewise open to objection. Schroeder's symbol

does not simply take the place of the copula. -57-

tent is an essential part of the predicate, so that the word "Venus" does not constitute the whole of the predicate. 3 One might say instead: "the morning star is no other than Venus"; what was previously implicit in the single word "is" is here set forth in four separate words, and in "is no other than" the word "is" now really is the mere copula. What is predicated here is thus not Venus but no other than Venus. These words stand for a concept; admittedly only one object falls under this, but such a concept must still always be distinguished from the object. 4 We have here a word "Venus" that can never be a proper predicate, although it can form part of a predicate. The reference 5 of this word is thus something that can never occur as a concept, but only as an object. Kerry, too, would probably not wish to dispute that there is something of this kind. But this would mean admitting a distinction, which it is very important to recognize, between what can occur only as an object, and everything else. And this distinction would not be effaced even if it were true, as Kerry thinks it is, that there are concepts that can also be objects.

There are, indeed, cases that seem to support his view. I myself have indicated (in Grundlagen, § 53, ad fin.) that a concept may fall under a higher concept—which, however, must not be confused with one concept's being subordinate to another. Kerry does not appeal to this; instead, he gives the following example: "the concept 'horse' is a concept easily attained," and thinks that the concept "horse" is an object, in fact one of the objects that fall under the concept "concept easily attained." Quite so; the three words "the concept 'horse' " do designate an object, but on that very account they do not designate a concept, as I am using the word. This is in full accord with the criterion I gave—that the singular definite article always indicates an object, whereas the indefinite article accompanies a concept-word. 6 Kerry holds that no logical rules can be based on linguistic distinctions; but my own way of doing this is something that nobody can avoid who lays down such rules at all; for we cannot come to an understanding with one another apart from language, and so in the end we must always rely on other people's understanding words, inflexions, and sentenceconstruction in essentially the same way as ourselves. As I said before, I was not trying to give a definition, but only hints; and to this end I appealed to the general feeling for the German language. It is here very much to my advantage that there is such good accord between the linguistic distinction and the real one. As regards the indefinite article there are probably no exceptions to our rule at all for us to remark, apart from obsolete formulas like Ein edler Rath ["Councillor"]. The matter is not so simple for the definite article, especially in the plural; but then my criterion does not relate to this case. In the singular, so far as I can see, the matter is doubtful only when a singular takes the place of a ____________________ 3Cf. my Grundlagen, § 66, footnote. 4Cf. my Grundlagen, § 51. 5Cf. my paper, "On Sense and Reference" (Ueber Sinn und Bedeutung), shortly to appear in the Zeitschrift fiir Phil. und phil. Kritik. 6Grundlagen, § 51; § 66, footnote; § 68, footnote on p. 80. -58-

plural, as in the sentence, "the Turk besieged Vienna," "the horse is a four - legged animal." These cases are so easily recognizable as special ones that the value of our rule is hardly impaired by their occurrence. It is clear that in the first sentence "the Turk" is the proper name of a people. The second sentence is probably best regarded as expressing a universal judgment, say "all horses are four-legged animals" or "all properly constituted horses are four-legged animals"; these will be discussed later. 7 Kerry calls my criterion unsuitable; for surely, he says, in the sentence "the concept that I am now talking about is an individual concept" the name composed of the first eight words stands for a concept; but he is not taking the word "concept" in my sense, and it is not in what I have laid down that the contradiction lies. But nobody can require that my mode of expression shall agree with Kerry's. It must indeed be recognized that here we are confronted by an awkwardness of language, which I admit cannot be avoided, if we say that the concept horse is not a concept, 8 whereas, e.g., the city of Berlin is a city, and the volcano Vesuvius is a volcano. Language is here in a predicament that justifies the departure from custom. The peculiarity of our case is indicated by Kerry himself, by means of the quotation-marks around "horse"; I use italics to the same end. There was no reason to mark out the words "Berlin" and "Vesuvius" in a similar way. In logical discussions one quite often needs to assert something about a concept, and to express this in the form usual for such assertions—viz. to make what is asserted of the concept into the content of the grammatical predicate. Consequently, one would expect that the reference of the grammatical subject would be the concept; but the concept as such cannot play this part, in view of its predicative nature; it must first be converted into an object, 9 or, speaking more precisely, represented by an object. We designate this object by prefixing the words "the concept"; e.g.: ____________________

7Nowadays

people seem inclined to exaggerate the scope of the statement that different linguistic expressions are never completely equivalent, that a word can never be exactly translated into another language. One might perhaps go even further, and say that the same word is never taken in quite the same way even by men who share a language. I will not enquire as to the measure of truth in these statements; I would only emphasize that nevertheless different expressions quite often have something in common, which I call the sense, or, in the special case of sentences, the thought. In other words, we must not fail to recognize that the same sense, the same thought, may be variously expressed; thus the difference does not here concern the sense, but only the apprehension, shading, or colouring of the thought, and is irrelevant for logic. It is possible for one sentence to give no more and no less information than another; and, for all the multiplicity of languages, mankind has a common stock of thoughts. If all transformation of the expression were forbidden on the plea that this would alter the content as well, logic would simply be crippled; for the task of logic can hardly be performed without trying to recognize the thought in its manifold guises. Moreover, all definitions would then have to be rejected as false. 8A similar thing happens when we say as regards the sentence "this rose is red": The grammatical predicate "is red" belongs to the subject "this rose." Here the words "The grammatical predicate 'is red' " are not a grammatical predicate but a subject. By the very act of explicitly calling it a predicate, we deprive it of this property. 9Cf. my Grundlagen, p. x. -59-

"The concept man is not empty." Here the first three words are to be regarded as a proper name, 10 which can no more be used predicatively than "Berlin" or "Vesuvius." When we say "Jesus falls under the concept man," then, setting aside the copula, the predicate is: "someone falling under the concept man" and this means the same as: "a man." But the phrase "the concept man" is only part of this predicate. Somebody might urge, as against the predicative nature of the concept, that nevertheless we speak of a subject-concept. But even in such cases, e.g. in the sentence "all mammals have red blood" we cannot fail to recognize the predicative nature 11 of the concept; for we could say instead: "whatever is a mammal has red blood" or: "if anything is a mammal, then it has red blood." When I wrote my Grundlagen der Arithmetik, I had not yet made the distinction between sense and reference; 12 and so, under the expression "a possible content of judgment," I was combining what I now designate by the distinctive words "thought" and "truth-value." Consequently, I no longer entirely approve of the explanation I then gave (Grundlagen, p. 77), as regards its wording; my view is, however, still essentially the same. We may say in brief, taking "subject" and "predicate" in the linguistic sense: A concept is the reference of a

predicate; an object is something that can never be the whole reference of a predicate, but can be the reference of a subject. It must here be remarked that the words "all," "any," "no," "some," are prefixed to concept-words. In universal and particular affirmative and negative sentences, we are expressing relations between concepts; we use these words to indicate ____________________ 10I call anything a proper name if it is a sign for an object. 11What I call here the predicative nature of the concept is just a special case of the need of supplementation, the "unsaturatedness," that I gave as the essential feature of a function in my work Funktion und Begriff (Jena, 1891). It was there scarcely possible to avoid the expression "the function f(x)," although there too the difficulty arose that what this expression stands for is not a function. 12Cf. my essay "On Sense and Reference" in the Zeitschrift fiir Phil. und phil. Kritik. -60-

the special kind of relation. They are thus, logically speaking, not to be more closely associated with the concept-words that follow them, but are to be related to the sentence as a whole. It is easy to see this in the case of negation. If in the sentence "all mammals are land-dwellers" the phrase "all mammals" expressed the logical subject of the predicate are land-dwellers, then in order to negate the whole sentence we should have to negate the predicate: "are not land-dwellers." Instead, we must put the "not" in front of "all"; from which it follows that "all" logically belongs with the predicate. On the other hand, we do negate the sentence "The concept mammal is subordinate to the concept land-dweller" by negating the predicate: "is not subordinate to the concept land-dweller." If we keep it in mind that in my way of speaking expressions like "the concept F" designate not concepts but objects, most of Kerry's objections already collapse. If he thinks (cf. p. 281) that I have identified concept and extension of concept, he is mistaken; I merely expressed my view that in the expression "the number that applies to the concept F is the extension of the concept like-numbered to the concept F" the words "extension of the concept" could be replaced by "concept." Notice carefully that here the word "concept" is combined with the definite article. Besides, this was only an incidental remark; I did not base anything upon it. Thus Kerry does not succeed in filling the gap between concept and object. Someone might attempt, however, to make use of my own statements in this sense. I have said that to assign a number involves an assertion about a concept; 13 I speak of properties asserted of a concept, and I allow that a concept may fall under a higher one. 14 I have called existence a property of a concept. How I mean this to be taken is best made clear by an example. In the sentence "there is at least one square root of 4," we have an assertion, not about (say) the definite number 2, nor about —2, but about a concept, square root of 4; viz. that it is not empty. But if I express the same thought thus: "The concept square root of 4 is realized," then the first six words form the proper name of an object, and it is about this object that something is asserted. But notice carefully that what is asserted here is not the same thing as was asserted about the concept. This will be surprising only to somebody who fails to see that a thought can be split up in many ways, so that now one thing, now another, appears as subject or predicate. The thought itself does not yet determine what is to be regarded as the subject. If we say "the subject of this judgment," we do not designate anything definite unless at the same time we indicate a definite kind of analysis; as a rule, we do this in connexion with a definite wording. But we must never forget that different sentences may express the same thought. For example, the thought we are considering could also be taken as an assertion about the number 4: ____________________ 13Grundlagen, § 46. 14Grundlagen, § 53. -61-

"The number 4 has the property that there is something of which it is the square." Language has means of presenting now one, now another, part of the thought as the subject; one of the most familiar is the distinction of active and passive forms. It is thus not impossible that one way of analysing a given thought should make it appear as a singular judgment; another, as a particular judgment; and a third, as a universal judgment. It need not then surprise us that the same sentence may be conceived as an assertion about a concept and also as an assertion about an object; only we must observe that what is asserted is different. In the sentence "there is at least one square root of 4" it is impossible to replace the words "square root of 4" by "the concept square root of 4"; i.e. the assertion that suits the concept does not suit the object. Although our sentence does not present the concept as a subject, it asserts something about it; it can be regarded as expressing the fact that a concept falls under a higher one. 15 But this does not in any way efface the distinction between

object and concept. We see to begin with that in the sentence "there is at least one square root of 4" the predicative nature of the concept is not belied; we could say "there is something that has the property of giving the result 4 when multiplied by itself." Hence what is here asserted about a concept can never be asserted about an object; for a proper name can never be a predicative expression, though it can be part of one. I do not want to say it is false to assert about an object what is asserted here about a concept; I want to say it is impossible, senseless, to do so. The sentence "there is Julius Caesar" is neither true nor false but senseless; the sentence "there is a man whose name is Julius Caesar" has a sense, but here again we have a concept, as the indefinite article shows. We get the same thing in the sentence "there is only one Vienna." We must not let ourselves be deceived because language often uses the same word now as a proper name, now as a concept-word; in our example, the numeral indicates that we have the latter; "Vienna" is here a concept-word, like "metropolis." Using it in this sense, we may say: "Trieste is no Vienna." If, on the other hand, we substitute "Julius Caesar" for the proper name formed by the first six words of the sentence "the concept square root of 4 is realized," we get a sentence that has a sense but is false; for the assertion that something is realized (as the word is being taken here) can be truly made only about a quite special kind of objects, viz. such as can be designated by proper names of the form "the concept F." Thus the words "the concept square root of 4" have an essentially different behaviour, as regards possible substitutions, from the words "square root of 4" in our original sentence; i.e. the reference of the two phrases is essentially different. 16 What has been shown here in one example holds good generally; the behaviour of the concept is essentially predicative, even where something is ____________________ 15In my Grundlagen I called such a concept a second-order concept; in my work Funktion und Begriff I called it a second-level concept, as I shall do here. 16Cf. my essay "On Sense and Reference" (cited above). -62-

being asserted about it; consequently it can be replaced there only by another concept, never by an object. Thus the assertion that is made about a concept does not suit an object. Second-level concepts, which concepts fall under, are essentially different from first-level concepts, which objects fall under. The relation of an object to a first-level concept that it falls under is different from the (admittedly similar) relation of a first-level to a second-level concept. (To do justice at once to the distinction and to the similarity, we might perhaps say: An object falls under a first-level concept; a concept falls within a second-level concept.) The distinction of concept and object thus still holds, with all its sharpness. With this there hangs together what I have said (Grundlagen, § 53) about my usage of the words "property" and "mark"; Kerry's discussion gives me occasion to revert once more to this. The words serve to signify relations, in sentences like " Φ is a property of r" and "Φ is a mark of Ω." In my way of speaking, a thing can be at once a property and a mark, but not of the same thing. I call the concepts under which an object falls its properties; thus "to be Φ is a property of Γ" is just another way of saying: "r falls under the concept of a Φ." If the object r has the properties Φ, X, and Ψ, I may combine them into Ω; so that it is the same thing if I say that Γ has the property Ω, or, that Γ has the properties Φ, X, and Ψ. I then call Φ, X, and Ψ marks of the concept Ω, and, at the same time, properties of Γ. It is clear that the relations of Φ to r and to Ω are quite different, and that consequently different terms are required. r falls under the concept Φ; but Ω, which is itself a concept, cannot fall under the first-level concept Φ; only to a second-level concept could it stand in a similar relation. Ω is, on the other hand, subordinate to Φ. Let us consider an example. Instead of saying:

"2 is a positive number" and "2 is a whole number" and "2 is less than 10" we may also say "2 is a positive whole number less than 10." Here to be a positive number, to be a whole number, to be less than 10, appear as properties of the object 2, and also as marks of the concept positive whole number less than 10. -63-

This is neither positive, nor a whole number, nor less than 10. It is indeed subordinate to the concept whole number, but does not fall under it. Let us now compare with this what Kerry says in his second article (p. 224). "By the number 4 we understand the result of additively combining 3 and 1. The concept object here occurring is the numerical individual 4; a quite definite number in the natural number-series. This object obviously bears just the marks that are named in its concept, and no others besides— provided we refrain, as we surely must, from counting as propria of the object its infinitely numerous relations to all other individual numbers; 'the' number 4 is likewise the result of additively combining 3 and 1." We see at once that my distinction between property and mark is here quite slurred over. Kerry distinguishes here between the number 4 and "the" number 4. I must confess that this distinction is incomprehensible to me. The number 4 is to be a concept; "the" number 4 is to be a concept-object, and none other than the numerical individual 4. It needs no proof that what we have here is not my distinction between concept and object. It almost looks as though what was floating (though very obscurely) before Kerry's mind were my distinction between the sense and the reference of the words "the number 4." But it is only of the reference of the words that we can say: this is the result of additively combining 3 and 1. Again, how are we to take the word "is" in the sentences "the number 4 is the result of additively combining 3 and 1" and " 'the' number 4 is the result of additively combining 3 and I"? Is it a mere copula, or does it help to express a logical equation? In the first case, "the" would have to be left out before "result," and the sentences would go like this: "The number 4 is a result of additively combining 3 and 1"; " 'The' number 4 is a result of additively combining 3 and 1." In that case, the objects that Kerry designates by "the number 4" and " 'the' number 4" would both fall under the concept result of additively combining 3 and 1. And then the only question would be what difference there was between these objects. (I am here using the words "object" and "concept" in my accustomed way.) I should express as follows what Kerry is apparently trying to say: "The number 4 has those properties, and those alone, which are marks of the concept: result of additively combining 3 and 1."

I should then express as follows the sense of the first of our two sentences: "To be a number 4 is the same as being a result of additive combination of 3 and 1." -64-

In that case, what I conjectured just now to have been Kerry's intention could also be put thus: "The number 4 has those properties, and those alone, which are marks of the concept a number 4." (We need not here decide whether this is true.) The inverted commas around the definite article in the words " 'the' number 4" could in that case be omitted. But in these attempted interpretations we have assumed that in at least one of the two sentences the definite articles in front of "result" and "number 4" were inserted only by an oversight. If we take the words as they stand, we can only regard them as having the sense of a logical equation, like: "The number 4 is none other than the result of additively combining 3 and 1." The definite article in front of "result" is here logically justified only if it is known (i) that there is such a result; (ii) that there is not more than one. In that case, the phrase designates an object, and is to be regarded as a proper name. If both of our sentences were to be regarded as logical equations, then, since their right sides are identical, it would follow from them that the number 4 is "the" number 4, or, if you prefer, that the number 4 is no other than "the" number 4; and so Kerry's distinction would have been proved untenable. However, it is not my present task to point out contradictions in his exposition ; his way of taking the words "object" and "concept" is not properly my concern here. I am only trying to set my own usage of these words in a clearer light, and incidentally show that in any case it differs from his, whether that is consistent or not. I do not at all dispute Kerry's right to use the words "concept" and "object" in his own way, if only he would respect my equal right, and admit that with my use of terms I have got hold of a distinction of the highest importance. I admit that there is a quite peculiar obstacle in the way of an understanding with my reader. By a kind of necessity of language, my expressions, taken literally, sometimes miss my thought; I mention an object, when what I intend is a concept. I fully realize that in such cases I was relying upon a reader who would be ready to meet me halfway—who does not begrudge a pinch of salt. Somebody may think that this is an artificially created difficulty; that there is no need at all to take account of such an unmanageable thing as what I call a concept; that one might, like Kerry, regard an object's falling under a concept as a relation, in which the same thing could occur now as object, now as concept. The words "object" and "concept" would then serve only to indicate the different positions in the relation. This may be done; but anybody who thinks the difficulty is avoided this way is very much mistaken; it is only shifted. For not all the parts of a thought can be complete; at least one must be "unsaturated," or predicative; otherwise they would not hold to -65-

gether. For example, the sense of the phrase "the number 2" does not hold together with that of the expression "the concept prime number" without a link. We apply such a link in the sentence "the number 2 falls under the concept prime number"; it is contained in the words "falls under," which need to be completed in two ways—by a subject and an accusative; and only because their sense is thus "unsaturated" are they capable of serving as a link. Only when they have been supplemented in this twofold respect do we get a complete sense, a thought. I say that such words or phrases stand for a relation. We now get the same

difficulty for the relation that we were trying to avoid for the concept. For the words "the relation of an object to the concept it falls under" designate not a relation but an object; and the three proper names "the number 2," "the concept prime number," "the relation of an object to a concept it falls under," hold aloof from one another just as much as the first two do by themselves; however we put them together, we get no sentence. It is thus easy for us to see that the difficulty arising from the "unsaturatedness" of one part of the thought can indeed be shifted, but not avoided. "Complete" and "unsaturated" are of course only figures of speech; but all that I wish or am able to do here is to give hints. It may make it easier to come to an understanding if the reader compares my work Funktion und Begriff. For over the question what it is that is called a function in Analysis, we come up against the same obstacle; and on thorough investigation it will be found that the obstacle is essential, and founded on the nature of our language; that we cannot avoid a certain inappropriateness of linguistic expression; and that there is nothing for it but to realize this and always take it into account. -66-

:4: FREGE'S HIDDEN NOMINALISM * GUSTAV BERGMANN SOME philosophical pieces are like symphonies, others like quartets. This one is merely an ontological theme with variations. After I have introduced the theme, it will be seen that Exemplification versus Mapping is a very good name for it. The phrase does not signify, though, except to one already familiar with the theme. That is why I did not choose it as a title. All but the last of the variations are comments on Frege's ontology, though as such they are highly selective. 1 This is one reason for the title I chose. The other is that I hope to draw expository advantage from its shock value.

I In ontological discourse two clusters of very ordinary words are used philosophically. First of all, therefore, I shall state how I propose to handle these words. One cluster contains "thing," "object," "entity," "existent." When I do not wish to indicate anybody's ontological commitment, I use entity. When I wish to speak of what philosophers, speaking philosophically, assert to "exist," I use existent. Frege uses both "object" and "function" philosophically; and he holds, either explicitly, or very nearly so, that every entity (not, existent!) is either an object or a function. 2 The other cluster contains "naming," "denoting," "designating," "referring." Nor are the philosophical uses of the two clusters independent. Some philosophers, for instance, maintain that an existent is what is or could be named (denoted, ____________________ Reprinted with the permission of the author and the editor from The Philosophical Review, LXVII (1958). *I have profited from discussions with Mr. Reinhardt Grossmann. 1An excellent detailed exposition may be found in R. Wells, "Frege's Ontology," Rev. Metaphys., IV (1957), 537-573. References to this study are by page number, preceded by the letter W. 2In rendering Frege's terms I follow the well-known Translations from the Philosophical Writings of Gottlob Frege by M. Black and P. Geach (Oxford, 1952). References to this volume are by page number, preceded by the letter F.

designated) by a word or expression. When I wish to speak without indicating ontological commitment, I avoid all these verbs and borrow instead Frege's standing for. If they followed this use, the philosophers just mentioned could say that a name is a word or expression which stands for an existent.

Consider "This is green." On one occasion, "this" may stand for an apple; "green," for its color. Some hold that, on another occasion, the two words could stand for what they call a sensum and one of its qualities, respectively. Others disagree. The difference makes no difference for my purpose. So I shall entirely ignore it in calling the sort of thing "this" and "green" stand for, on all occasions, an individual and a character, respectively. Also, I shall in this essay use the two words without ontological commitment even though, in the case of "individual," that is admittedly unusual. For Frege, individuals are one kind of object; characters, one kind of function. But there are in his world still other kinds of objects and of functions. Some ontologists, including Russell (and myself) though not Frege, make much of a distinction between simple and complex characters. For what I am about it is irrelevant, so I shall disregard it. There is also the distinction between types of characters and functions. Both Frege and Russell pay attention to it. But again, with one glancing exception, it is irrelevant for what I am about, so I shall virtually ignore it. This shows how selective I shall be in my remarks about Frege. Even in ontological discourse the terms realism and nominalism are used in two ways; once strictly, once broadly. The ontologist's first business is to list all kinds of existents (not, all existents). If he discerns many kinds, perhaps too many, one calls him a realist. If he lists but few, perhaps too few, one calls him a nominalist. This is the broad use. In the strict sense, an ontologist is a realist if he counts characters, or at least some characters (for example, simple ones), as a kind of existent. A nominalist in the strict sense holds, conversely, that no characters are existents. In this essay, unless there be a qualification to the contrary, both terms are always used strictly. Two things about Frege are beyond reasonable doubt. Had he used "existent" as I do (as far as I know he did not use it at all), he would have agreed that everything he calls an object is an existent. This is the first thing. Of objects there are in his world many kinds. Nor are the distinctions among them "ordinary," like that between cats and dogs. They are even more sweeping than that between physical and phenomenal objects, which surely is sweeping enough and, according to some, anything but ordinary. His distinctions are so sweeping indeed that, if the word is to have any meaning at all, one cannot but call them ontological. This is the second thing beyond reasonable doubt. Specifically, Frege distinguishes (at least) the following (ontological) kinds (of objects): individuals, numbers, truth values, value ranges (classes of objects), senses, propositions (thoughts), concept correlates. 3 Many philosophers think that with the sole exception of individuals all these kinds are odd. Or, to say the same thing differently, they refuse to consider ____________________ 3As far as I know, the term is Wells's. -68-

the odd kinds of entities as existents. Still differently, these philosophers (including myself) reject Frege's exaggerated realism (broad sense). It is not my purpose to rehearse their arguments, or to improve on them, or to invent new ones; even though I shall permit myself, en passant, to call attention to two oddities about truth values. As far as Frege is concerned, my purpose is rather to give some reasons for my belief that, for all his exuberant realism in the broad sense, he was in the strict sense at least implicity a nominalist. My primary concern, though, is neither biographical nor textual. What I shall really argue, therefore, is that the structure of Frege's ontology though not, as will transpire, of Church's emendation of it, 4 is nominalistic. Everything said so far is merely preliminary. I shall proceed as follows. In the next section the theme is introduced and used to exhibit what I take to be the root of Frege's nominalism. In the third section I shall show that in one case this very nominalism forces upon Frege that multiplication of entities (or rather of existents) which is so characteristic of his ontology. In the fourth section I shall show that what is, broadly speaking, the most serious as well as the most obvious intrinsic flaw of the system is but another consequence of its author's hidden nominalism. I say intrinsic because this flaw is of course not that

exuberant realism (broad sense) which, as I said, I shall not question except once and incidentally. In the last section I shall vary my theme, very briefly and very sketchily, by sounding it as a background for the siren songs of a more recent nominalism.

II Where one arrives depends in part on where he starts. A philosopher's starting point depends in part on his basic paradigm. Frege's is very different from the realist's. My theme builds on this difference or, rather, contrast. I present first the realist, because it helps to bring out the contrast more forcefully. Some realists, including myself, propose an explication of the philosophical use of "exist" upon which all individuals and (some of) their characters but no higher characters are existents. 5 The realist I present is of this sort. My case does not depend on the limitation. But again, it helps to bring out the theme more clearly by freeing it from some bywork that otherwise would have to be introduced. The realist starts from individuals and their characters. That is, he starts from entities Frege calls objects (and from their characters), though not from those objects which with Frege's critics I called odd. "Peter is blond" may thus serve as the realist's paradigm. What claims does it suggest to him? What do and what don't these claims imply? What reasons can he give for them? I shall take up the three questions in this order. ____________________ 4A. Church, Introduction to Mathematical Logic (Princeton, 1957). 5See "Elementarism," J. Phil, and Phen. Res., XVIII (1957), 107-114. -69-

First. "Peter" and "blond" both stand for existents. Generally, both individuals and characters are existents. This is one major claim. Many realists, including myself, also hold that every existent is either an individual or a character; but for the purpose at hand that does not matter. What "Peter is blond" stands for comes about if two existents, one of each kind, enter into a certain "relation," or, as I would rather say, nexus. This nexus the realist calls exemplification. Obviously it is a very fundamental feature of his world. This is another major claim. To see that it is not independent of the first, notice that to say of something that it does or may enter into a nexus is to presuppose that it exists. Notice also that when I spoke of a sentence "standing for" something, I quite deliberately paid with clumsiness for the agreed- upon neutrality of the phrase. Second. (a) An individual may or may not exemplify a character. To say that a character exemplifies an individual is nonsense. The very nexus between the two kinds is thus asymmetrical. This alone shows that individuals and characters are not alike in all respects which concern the ontologist. Nor do the realist's claims imply that they are. Obviously not, I should say. For, if they were, what point would there be in distinguishing between them? (b) In the paradigm, the copula or, as I prefer to say, the predicative "is" stands for exemplification. In a much larger number of cases, to say the least, the verbal image of this nexus between things is predication, that is, the grammatical relation between subject and predicate. That follows from the realist's claims. Third. An articulate realist has of course many reasons for the position he takes. Fortunately they do not all matter for my purpose. What matters is that every time he discovers some respect in which individuals and characters are alike, he has discovered a reason. I shall state two such reasons, the two which I think carry most weight. (a) Just as we are never presented with an individual that is not qualitied, that is, does not exemplify a character, so we are never presented with a character that is not exemplified by some individual with which we are also presented. Notice that I speak of entities being presented to us, that is, as one says, epistemologically. Had I not promised to keep out the distinction between simple and complex characters and had I limited myself to the former, I could have spoken ontologically: Just as there is no individual that is not qualitied, so there is no character that is not exemplified. This is one fundamental likeness. (b) Consider the three entities that "Peter," "blond," and "Peter is blond" stand for. The differences among them are ontologically significant. With that Frege agrees. For, if he did not, he would not, as an ontologist, either distinguish concepts from objects or set aside truth values as a kind of

object. The realist says the same thing differently. Neither an individual nor a character is the kind of entity (notice the noncommittal word!) a sentence stands for. This is a second likeness. (b) The reader is assumed to be familiar with Frege's distinction between saturated and unsaturated expressions. "Blond" or, as he would really have to say, "is blond," is unsaturated. "Peter" is saturated. (So is, very importantly, "Peter is blond"; but that does not yet enter into my argument.) Primarily at -70-

least, the dichotomy is between expressions, not between entities. If one wants to apply it to entities, one must specify what in this case the two terms are to mean. For instance, one may propose "Individuals and characters are equally unsaturated" as an alternative way of stating the second fundamental likeness. An unsophisticated realist may object to this on the ground that, since individuals and characters are both existents, they really are both equally saturated. We know that he merely proposes another meaning for one (!) of the two terms as applied to entities. But his proposal has the merit of showing not only that this application serves no purpose but also the dangers that beset it. 6 The realist's gambit has a further consequence, which gives rise to an objection, which in turn leads to a clarification. It will pay later if in introducing the matter I change the paradigm to "Peter is a boy." The objector points out that one who takes the predicative "is" to stand for exemplification ought to admit that the paradigm and the sentence "Peter exemplifies the character of being a boy" stand for one and the same entity. After a fashion, the realist does admit that. He even admits it for the sentence "The individual (object) Peter exemplifies the character (concept) of being a boy," and, if he used the two words in parentheses, he would also admit it for the further variations they make possible. Now the objector taunts the realist with a Bradleyan regress. It seems, he says, that in order to grasp what exemplification is, he must first grasp what it means for the (relational) character of being exemplified to be exemplified. The realist answers as follows. Exemplification is a very peculiar character. (That is why I called it a nexus and only once, in quotation marks, a relation.) Its peculiarity is the same as that of such "characters" as being an individual or a character. To be an individual, for instance, is not to be of an "ordinary" kind, as is being a cat or a dog, but to be the sort of entity expressions of a certain grammatical kind stand for. The expressions for these peculiar kinds are therefore all expendable and the several alternatives for the paradigm need not be considered unless one speaks in a language containing them about another language and what its expressions stand for. I need not on this occasion endorse this answer. Nor would I on any occasion endorse it in this crude form. (That is why I said the realist admits what I made him admit "after a fashion.") I merely mention this answer because, being familiar and thus permitting me to communicate quickly about what does not matter for what I am about, it helps me to prepare the ground for what does. So much for the realist. Frege starts from numbers and their functions. "x2" may thus serve as his paradigm. That numbers are objects and therefore existents he takes for granted. Questionable as that is, I need not question it. My concern is with functions, mathematical and otherwise. The current mathematical name for the crucial idea is mapping. The square function, for instance, maps each number onto another, namely its square. Generally, given two classes of entities which may but need not coincide or overlap, a function is a mapping rule, mapping each member of one of the two classes upon one (and, in the ____________________ 6See "Propositional Functions," Analysis, XVII (1956), 43-48. -71-

paradigmatic case, only one) member of the other. Let me now for a moment speak as a poet might, using some very loaded words very freely. A rule is a thing totally different from the things to which it applies. A mapping rule, in particular, is a thing much more shadowy, much less real, less palpable, less substantial than the things mapped and mapped upon. This more and these less, rather than the questionable status of numbers, is the heart of the

matter. So I shall try to state it, not as a poet but as a philosopher might. Numbers and their functions differ from each other in the two fundamental respects in which, as we saw, individuals and characters are alike. (a) Just as there is no unexemplified character, so there is no unqualified individual. But there are of course numbers whether or not they be either arguments or values of functions. (b) The two notions of an individual and of a character, containing or presupposing each other to exactly the same extent, are equally "saturated" or "unsaturated." The notion of a number neither contains nor presupposes that of a function. The latter, however, contains and presupposes that of the two ranges (of numbers). (c) The realist's basic paradigm involves two entities, an individual and a (nonrelational) character. So does a function of one variable, which is Frege's basic paradigm. Or, rather, this is so after an argument has been chosen and the corresponding value computed by means of the function. The need for this additional step, which Frege never tires of emphasizing, increases the "ontological distance" between objects and functions. In Frege's paradigm, moreover, though not in the realist's, the two existents involved, the argument and "its" value, are existents of the same kind. This further increases the impression of disparity between them and the entity which is the function. Still another circumstance subtly undermines the ontological status of functions. What mathematicians say about the latter often has a subjective ring. As they speak, it is they who do the mapping or, as it is often put, establish the correlation. "Rule" itself, in most of its uses, has the same ring or tinge. This is an occasion for comment. First. Dangerous as that subjective ring or tinge may be philosophically, it is of course quite harmless as long as the mathematicians attend to their own business. What makes it even more harmless is that in fact mathematicians are not all busy establishing or making functions. The vast bulk of their work consists in demonstrating what further properties a function has, assuming that it has some others. Second. The realist pulls the fangs of this potentially dangerous talk by construing functional expressions as, in the Principia sense, indefinite descriptions, 7 either mathematical or otherwise (the square of x, the father of x). Descriptions in turn contain expressions standing for characters. And there is nothing subjective about characters (including, of course, relations), least of all in the case of numbers. As Russell put it, in a justly celebrated passage about the order relation, "we ____________________ 7"The father of x" is in this sense an indefinite description; "a son of Peter" is not, irrespective of how many sons, if any, Peter has. A. J. Ayer's "Individuals," reprinted in his Philosophical Essays (New York, 1954), makes one wonder whether he appreciates the distinction. What he calls indefinite descriptions seem to be predicative expressions which happen to be true for several subjects. -72-

can no more 'arrange' the natural numbers than we can the starry heavens." 8 Notice, though, that even if starting from numbers in one sense, one who accepts this clarification starts in another sense from the realist's paradigm. Third. I spoke quite deliberately of the subjective tinge or blur from which some of the mathematicians' phrases suffer. My purpose was to give an objector his opportunity. Frege, this objector reminds us, insists over and over again that not only his odd objects but also functions, though they are not objects, are yet objective. I know that this is one of his guiding ideas. And I admire the steadfastness with which he wielded it as a weapon against the psychologism rampant in the Germany of his day. One may appreciate all this, as of course I do, and yet consistently hold, as I also do, that while within his system at least Frege succeeded in securing full ontological status for his odd objects, he did not so succeed, even within the system, in the case of functions. That is what I mean by his hidden nominalism. Notice that I call it hidden or implicit. Remember, too, that in this section I merely undertook to state it and to trace it to what I take to be its root, namely, the contrast between exemplification and mapping. The evidence will be found in the next two sections. But I am not quite done with the business at hand. Nominalism is a thesis about characters. Nothing will be lost if we limit ourselves to nonrelational ones. Frege calls them concepts. What, then, does he have to say about concepts? The realist, we just saw, construes functions in terms of characters (concepts). Frege, proceeding in the opposite direction, as it were, construes concepts as a kind of

function. In this way, the nominalism I have shown to be implicit in any analysis that starts from mapping is spread to concepts (characters). This is the point. Or, if you please, this is the last bar of my theme. Frege's execution of the idea is familiar. If "Peter is blond" is to be construed in analogy to "the square of 3," then we must look for an object that goes with Peter as the value 9 goes with the argument 3. This object is the truth value of the sentence. This is one motive for Frege's "creation" of the two odd objects T and F. It fits nicely with another. He wants every saturated expression to stand for an object; sentences are saturated expressions; so the truth value of a sentence can serve as the object it stands for. Though Frege had still other motives for the "reification" of T and F, structurally these two are undoubtedly the most important. I should like to suggest that the first, that is, the need to make good the precarious analogy between concepts and functions, goes even deeper than the second. I turn from things to words. The alleged analogy between characters and functions is not reflected in our language. If it were, we would have to say "blond of Peter," just as we say "the square of 3," and not, as we do, "Peter is blond." (The reader who can anticipate the sequel will be struck by the irony of our propensity, if we consider such verbal violence at all, to say "the blond[-ness] of Peter.") The culprit in the case is the predicative "is." ____________________ 8B. Russell, Introduction to Mathematical Philosophy, p. 30 of the 1919 edition. The whole page repays reading in this context. -73-

From where the realist stands, we remember, it reflects very nicely the nexus of exemplification and is a, if not perhaps the, fundamental use of "is." 9 For Frege it is but a clumsily disguised "of." In "Peter is the father of John," on the other hand, as in "3 2 = 9" and, alas, in "Peter is blond = T," "is" reflects very nicely, from where Frege stands, what it is used to speak about. To express identity thus becomes the or at least a fundamental use of "is." In the last section I shall give some reasons why I believe that all nominalists are forced to consider it, more or less covertly, the fundamental use of "is." This is the place to call attention to two oddities about truth values. First, let "P" stand for a sentence and consider the series is true, so are all members of the series. If the arithmetical analogue were to hold, the

If P

members of the series would also all be true. Yet they are not even well formed. This is odd. Second, senses and propositions (thoughts) are in the system two kinds of objects, kindred in that a proposition is, as it were, the sense of a sentence. This is familiar. 10 Now there is a passage (F 64) in which I take Frege to assert, as I believe consistently he must, that "P" and "P = T" have (express) the same sense. If the arithmetical analogue held, so would therefore "3 2" and "32 = 9." Yet the former denotes a number, the second, T. Thus they do not even denote the same kind of object. This, too, is odd. If one has already accepted the system he will probably not boggle at such oddities. But if he hasn't, they may make him even more averse to the reification of T and F and to what, as we saw, is at the bottom of it all, Frege's analysis of exemplification in terms of mapping.

III A nodding acquaintance with Frege's work, or even with what is currently being said about it, leaves two impressions. The first is of the multiplication of entities due to his distinction between reference and sense, that is, between the objects denoted and the objects expressed by such saturated expressions as, say, "the Morning Star" and "Peter is blond." The second impression is that the main intellectual motive for the multiplication is the hope that, by means of it, it will be possible to conquer the difficulties that arise in inten____________________ 10See Sections III and IV.

9It

becomes the fundamental use if the Leibniz-Russell explication of identity is considered adequate. -74-

sional contexts, that is, in contexts mentioning either modalities or propositional attitudes, such as believing, knowing, and so on. Considering the system as a whole, that certainly is a motive. But if the two impressions are as widely spread as I believe them to be, then it is probably worth pointing out that Frege was first forced into that business of multiplying entities by the very logic of his nominalism in a case much simpler and more fundamental, which has nothing whatsoever to do with either modalities or propositional attitudes. This is the case of classes ( extensions, value ranges). Corresponding to each concept, which according to him is not an object and (if I am right) at least implicitly not an existent, Frege "creates" another entity, which according to him is an object, namely, the class of all objects which, as he says, fall under the concept. 11 In this section I shall give two reasons why his nominalism, made explicit, forces him to do just that. Such reasons, if sound, are of course the kind of structural evidence one must accumulate in order to establish that the system is, at least implicitly, nominalistic. Consider the two functional expressions "x 2" and "x2 — x + x" and the equation "x 2 = x2 — x + x." After a fashion, we all know what the equation stands for. Practicing mathematicians, without giving the matter much thought, say either that the two functions have the same extension, that is, the class of ordered pairs [(1,1), (2,4) ... ], or, alternatively, that the two functions are equal (the same, identical). Strictly speaking, from where Frege stands, the second alternative makes no sense. A function is a mapping rule. Following one of the two rules mentioned, one obtains the value by squaring the argument. Following the other, one first squares the argument, then subtracts it from the square, then adds it to the intermediate result. Considered as rules, the two functions are thus not the same. Generally, two rules are never the same, in a sense both strict and intelligible, unless they are, as one says, two tokens of the same type. All one can mean by calling them so, is, therefore, that they yield the same result (have the same extension). I conclude that, if he wants to be consistent, Frege cannot, as in fact he does not, specify conditions of identity for concepts and functions. Or, rather, he would have to hold that an assertion of identity between two (concepts or) functions is true if and only if the two expressions mentioned are different tokens of the same type. Of this more later. For the moment we notice that had he stopped at this point, Frege could not have preserved what as a mathematician he surely wanted to preserve, namely, the equation "x 2 = x2 — x + x" and the truism for which it stands. His way out is to interpret it as an identity not between the two functions but, rather, between their extensions. (Technically, he introduces a special notation for the latter and rewrites the equation as a statement of identity between them.) The possibility of this interpretation he thinks is indemonstrable. So he appeals to a fundamental law of logic. The words and the phrase italicized are his (F 26). This is the first reason I undertook to adduce. It shows how the nominalism implicit in any analysis that starts ____________________ 11The limitation to first-level concepts which this formulation entails does not affect my purpose. -75-

from mapping creates the need for the new entities. The second reason will show that they must be objects. Frege does not specify the fundamental law of logic to which he appeals. What then is it? Does he use the phrase merely to dignify what is done in this case? Or is there a general principle involved? Even though his words at this place seem to suggest the first alternative, the second is I believe the right one. Consider the following three propositions. 1. To be a name and to be an expression standing for an existent is one and the same thing. 2. For a statement of identity, "α = β" to be true, "α" and "β" must denote existents. 3. For "α =β" to be a statement (to be well-formed, or to be meaningful), "α" and "β" must denote (or

purport to denote) existents. Obviously, 3 is stronger than 2. Many ontologists endorse either 2 or 3, depending on the stand they take on issues which do not concern us. I would not even endorse 2. 12 That, however, is beside the point. The point is that either 1 and 3 or, perhaps, 1 and 2 follow deductively from the way many ontologists tended and still tend to use "existent," "name," and "identity." Add now, as a fourth proposition, that every saturated expression is a name; in the first three propositions replance "existent" by "object," and you obtain four principles, or four aspects of one principle, which underlie, either explicitly or very nearly so, all of Frege's analysis. This, I believe, is quite uncontroversial. Assume now that this principle is the fundamental law of logic to which Frege appealed. It follows that extensions (value ranges, classes) must be objects. This is the second reason I promised to adduce. Moreover, I have made it plausible, to say the least, that Frege uses "object" as many ontologists use "existent." Concepts and functions in general, we remember, though objective, are not objects. It follows that, at least implicitly, the system is nominalistic. This concludes the main argument of the section. I proceed to four comments. First. Let a 1 , a 2 , ... be all the objects that are blond; b 1 , b 2 , ... all those that are not. The class one ordinarily associates with the character blond is that of the a, [a 1 , a 2 , ... ]. It can be argued that the class Frege associates with the concept blond is that of all ordered pairs (a, T) and (b, F). If so, in his world what sort of object is an ordered pair of objects? Probing deeply in some such directions, one is eventually led to the Russell paradox and the question as to what, if anything, Frege can do about it. I realize all that. Clearly, these subtleties do not matter for what I am about. That is why I proceeded as I did. But it may hurt my thesis if I appear ignorant where I am merely selective. That is why I mention the matter. Second. Frege (F 50) feels the need to distinguish "the relation of an object to a first-level concept that it falls under ... from the (admittedly similar) relation of a first-level concept to a second-level concept." So he proposes that in the former case we speak of falling under, in the latter of falling within. His purpose, he tells us, is to preserve the distinction of con____________________ 12To this point I hope to return on another occasion. See "Sameness, Meaning, and Identity," in the Proceedings of the Twelfth International Congress of Philosophy (Venice, 1958). Reprinted in Meaning and Existence. -76-

cept and object "with all its sharpness" or, as I would put it, to increase the ontological distance between them. Taken by itself, the passage may be read as merely a plea for pervasive type distinctions. In context it provides subsidiary evidence for my thesis. Third. An objector might argue as follows: "True, Frege does not specify conditions of identity for functions. But, then, neither does he specify such conditions or criteria for senses. (I follow you in using 'sense' for propositions (thoughts) as well as for what is expressed by nonsentential saturated expressions such as 'the Morning Star'.) Yet senses are objects. This greatly weakens your argument." Wells (W 544) shrewdly anticipates part of the answer. Frege, he reminds us, had an additional reason, which does not, or at least not directly, apply to functions, for not specifying criteria for the identity of senses. To grasp this reason, remember that one of Frege's intellectual motives was to solve the problems of intentional contexts. In this enterprise the reification of senses is merely a first step. The decisive second step is to specify criteria of identity among senses. This second step Frege never took, simply because (or so I believe) he could not think of any that were acceptable to him and did the job. In any case he examined and rejected two. By one of them, two expressions have (express) the same sense if and only if they are analytically equivalent. This criterion he recognized as too broad. By the other, two expressions have the same sense if and only if they are different tokens of the same type. If we accepted this criterion, logic, he thinks (F 46), "would simply be crippled," if only because "all definitions would then have to be rejected as false." 13 However that may be, Frege had an additional reason for not specifying a criterion of identity for senses. This is part of the answer I would give to the objector. For the rest, I would remind him of what was pointed out at the beginning of this section, namely, that its main argument is completely independent of what does or does not hold for senses.

Fourth. Frege, who started from mapping, was forced to reify classes. Does the realist who starts from exemplification find himself compelled? If the answer were no, it would greatly add to the poignancy of my theme. The answer, I believe, is no. However, this is not the place to go into the reasons for that belief. So I shall merely hint at what is rather familiar. Our realist may so explicate the philosophical use of "exist" that only what undefined terms stand for exists in this very peculiar sense. (This is an issue I promised to keep out of the main argument.) When Russell supported his contention that classes do not exist by defining the expressions standing for them in terms of predicative expressions, he was at least implicitly guided by this explication of "exist." 14 A realist who accepts Russell's analysis of the class notion together with the ____________________ 13I do not subscribe to the dogma. This, though, is another story. See "Intentionality," in Semantica (Archivio di Filosofia, 1955), 177-216; "Elementarism," I.c.; and "Concepts," Phil. Studies, VII (1957), 19-27 (jointly with H. Hochberg). Reprinted in Meaning and Existence. 14See "Particularity and the New Nominalism," Methodos, VI (1954), 131-148. Reprinted in Meaning and Existence. -77-

explication of "exist" it implies or at least suggests need not therefore, like the Fregean nominalist, reify classes. The contrast adds depth to that between mapping and exemplification.

IV Consider the phrase "the concept blond" and the sentence "The concept blond is a concept." As Wells points out (W 550), one would think that the latter is not only true but truistic. Frege, as he must, disagrees. Like every saturated expression beginning with "the," the phrase is a name. A name denotes an object (existent). These two principles, we saw, Frege never questions. But a concept is not an object (existent). Hence, the supposed truism is false. Perhaps it is even nonsensical. That leaves two alternatives. One is to declare the sentence expendable. The other is to create a new kind of odd objects. Frege chose the second. The new objects are the concept correlates. A concept correlate "represents" its concept. Or it is obtained by "converting" the concept into an object. These are Frege's words. Nor must the correlate of a concept be confused with its extension. Starting with one entity, the concept, which as I argue does not exist, Frege thus ends up with two more which indubitably do exist, namely, the concept's correlate and its extension. The need for the creation of this further kind of odd object is, to my mind, the most obvious intrinsic flaw of the system as it stands. By calling it intrinsic I indicate, as before, that I do not on this occasion wish to challenge the odd kinds merely because to some of us they seem odd. Nor do I call it a flaw merely because this particular reification is patently ad hoc. For so is that of the two truth values. My point is that, once the new kind, the concept correlate, has been introduced, one cannot escape answering the question in what relation, or connection—use any word you wish—it stands to the other two, the concept itself and its extension. Yet there is no answer. This is the flaw I have in mind. It is as obvious as it is serious. One cue to its being a flaw is the opacity of the two metaphors, representation and conversion. This is not to say that I blame Frege just for speaking metaphorically. Many philosophers sometimes do, and sometimes it helps. I merely wish to say that these two metaphors do not help me in the least. Nothing comes through. The way I presented the matter leaves no doubt that the need for concept correlates is a consequence of what I claim is the implicit nominalism of the system. In this respect there is no difference between concept correlates and classes (extensions). But there is in another. Notice that I spoke of the system "as it stands." In the case of concept correlates, though not of classes, a very slight emendation eliminates the need for their reification as a further kind of objects. I shall next use this circumstance to argue that the nominalism which, -78-

if it were explicitly present in the system, would necessitate the reification, actually is present in it, at least implicitly. Take (α) "Peter," (β) "Peter is blond," (γ) "blond." They exemplify Frege's three basic grammatical categories. Let the Greek letters stand for these categories. Every α and β expression has a reference and a sense. It denotes the former, expresses the latter. The reference of a β expression is a truth value, its sense a proposition (thought). Senses are objects. Hence they can be named. In the case of the paradigm, their names are "the sense Peter" 15 and "the proposition Peter is blond" 16 respectively. Being a name, each of these two expressions has in turn a sense. We have entered upon an infinite regress. Within the system, though, that is no difficulty. Nor does it disturb the lucidity of the pattern. The difficulty is, rather, that the pattern does not apply to the unsaturated expressions γ. There (I limit myself to concepts) it is disturbed by that dangling third entity, the concept correlate. Assume now that concepts are existents. If so, they can be named and the pattern can be extended as follows. A γ expression, say "blond," denotes its extension and expresses its sense, which is the concept itself, which in turn is denoted by "the concept blond." With this emendation the same pattern applies to all three categories. Moreover, the need for concept correlates has disappeared. The emendation is in substance the one Church proposed. With all the respect due to him, it

does not seen very far-fetched, at least by hindsight. Nor is it overdoing the respect we owe to Frege's ingenuity to believe that it was not, even by foresight, beyond his grasp. Why, then, one must ask, did he not take this almost obvious step? The answer I propose will not come as a surprise. He balked at the one assumption which as I have shown the step implies, namely, that concepts, though not objects, are yet full-fledged existents. This concludes the main argument of the section. I proceed to two comments. First. Wells reports (W 546) Church to have argued as early as 1939, in a paper not generally accessible, that Frege's concepts are full-fledged existents. In other words, he disagreed sharply with my thesis. I am neither surprised nor disturbed. As emended by Church, the system is indeed no longer nominalistic in structure. That merely proves that Church is not a nominalist. It proves nothing about the system as it stands. I also grant and even insist that the emendation is nearly obvious. That explains why Church did not want to charge his master with what he must have considered a flaw. But it does not explain why Frege himself put up with an obvious and serious flaw rather than take Church's nearly obvious step. My thesis does explain that. Second. What can the realist do about Frege's problem? To answer, I first state the problem in the way it impressed itself upon Frege. Consider "Fido is a dog." If "dog" and "the concept dog" stood for the same entity, they ____________________ 15Or, as one says rather, the concept Peter. It is obvious why I avoid that locution. 16Or, in intentional contexts, that Peter is blond. -79-

would have to be interchangeable salva veritate. This is another principle Frege never questioned. Yet, "Fido is the concept dog" is nonsense. (Frege, everyone knows, was tremendously and of course quite rightly impressed with the contrast between the definite and the indefinite article. That is why I changed the paradigm.) The concept correlates are an obvious way out of the difficulty. The realist does not need them. Admitting, for the sake of the argument, that "dog" and "the concept dog" stand for the same entity, he need not therefore abandon the principle. He merely adds the proviso, which I introduced in Section II, that whenever such a substitution is made, "exemplifies" must, in the nature of things, be substituted for "is." In this way he obtains, quite smoothly, "Fido exemplifies the concept dog." The reason the realist finds this answer is that, unlike Frege who starts from mapping, he can do justice to that fundamental feature of our world from which he starts. Thus we are once more led back to the contrast between mapping and exemplification.

V Consider "This is red." Frege and the realist agree that the demonstrative denotes an existent. By the realist's account, so does the adjective. Frege, we saw, disagrees, at least implicitly. The root of the matter, we also saw, is that he starts from mapping. The inspiration of this alternative to exemplification is mathematical. In this his nominalism is unique. All other varieties I know of—I am tempted to call them the ordinary varieties —operate with the doctrine of common names. Or, as I would rather say, there are really only two kinds of nominalism, Frege's and the doctrine of common names. The difference between the two is by no means negligible. Both kinds, though, in addition to being nominalisms, which to my mind is a weakness, share still another weakness. Neither does justice to the predicative "is," which stands for exemplification. Both, therefore, more or less covertly take identity to be the fundamental meaning of "is." In Section II, I showed this for Frege. In this section I shall show it for the doctrine of common names, first generally, then by analyzing an essay of Quine's, 17 Assume that when I said "This is red" I pointed at an individual, say, a red apple. The demonstrative and the adjective are both names of the apple. The only difference is that while the former is (serves as) a "proper name," the latter is a "common name." A proper name is a label arbitrarily attached to one and only one individual. A common name applies indifferently to each of several individuals, namely, all those sharing a character. This is the doctrine of common names. It runs into an objection and a difficulty. ____________________ 17"Identity, Ostension, and Hypostasis," reprinted in From a Logical Point of View (Cambridge, Mass., 1953). References to this volume are by page number, preceded by the letter Q. -80-

A common name applies to an individual if and only if that individual has a certain character. Hence it is not an arbitrary label, in the sense in which a proper name is one, unless it be, as the realist insists and the nominalist denies, the name or label of the character itself. What, then, the realist asks the defender of the doctrine, is there "objectively" in or about each of the several individuals by virtue of which the common name is properly applied to each of them? Some nominalists answer (Q 68) that upon hearing and seeing a common name applied, we learn to apply it ourselves "by induction." The realist retorts that for such learning to occur, there must be a clue common to all individuals to which the learner hears and sees the common name applied. That leaves the issue where it was before. The doctrine of common names has no answer to this objection. Frege answered it, after a fashion, by insisting that functions, though not objects (existents), are yet "objective." That is why his nominalism is so superior to the other kind and, being superior, can remain "hidden." That much for the objection. The difficulty relates to what I am about. Assume that an individual has two names, say "Napoleon" and "Bonaparte." Consider "Napoleon is Bonaparte." In this sentence "is" stands for identity. Generally, if an individual has two names, what way is there of combining them into a sentence except by the "is" of identity? This suggests that proponents of the doctrine may be tempted to

assimilate the predicative "is" to that of identity. Notice that I just spoke of names, without distinction between common and proper ones. I did this because I do not really understand what it means for a word to be a common name. Or, to say the same thing differently, a name, in the only use of the term I understand, is a word attached as a label to one and only one entity. Or, still differently, in the manner of speaking I wish to discourage, every name, whatever it may name, is a proper name. Notice, second, that I spoke of suggestion. In other words, not every proponent of the doctrine asserts that in "This is red" the copula stands for identity. To claim anything of the sort would be unreasonable indeed. For is it not the very purpose of the doctrine of common names to prevent this collapse of the two uses of "is?" Only its proponents still somehow think of "common names" as "names." Therein, I claim, lies a temptation, or a suggestion, or perhaps even a compulsion in the direction of that collapse. This claim is not at all unreasonable. To substantiate it, I shall present two series of comments, the first about classical (Aristotelian) logic, the second about the essay by Quine. 1. For Aristotle, there is an important difference between "This is green" and "Socrates is a man." For my purpose the difference does not matter. Even so, consider the second sentence, if only because everyone knows that classical logic cannot cope with it except by the device of subsuming it, rather artificially, under the A-sentence. To do that is to construe the predicative "is" as the "is" (or "are") of the A-, E-, I-, and O-sentences. For Aristotle, this third use of "is" is the fundamental one. It is, if I may so put it, the only one for which he can account. (Frege and the realist both construe it as a com -81-

bination of two predicative uses with a quantifier.) Be that as it may, formally or logically the device does the trick. But it does not even touch the heart of the difficulty, which is ontological rather than logical. Within the hylomorphic scheme, the problem of individuation is insoluble. 18 One may of course abandon individuals. That is the way Scotus took. But it can be argued that in taking it he also abandoned hylomorphism. The original terminists did not wish to go that far. Yet they faced up to the Parmenidean illusion, which is one of the roots, if not perhaps the root, of the classical difficulty, that every occurrence of the copula indicates an identity. Moreover, they insisted, like Frege, that any two names of an individual must be substitutable for each other salva veritate in all contexts; and they noticed that, say, "this" and "red" are not so substitutable. Thus they were led to the distinction which is the core of their doctrine of signification. Connotatively, they held, the adjective signifies indifferently each of the several red things; denotatively it, or, rather, its abstractum "redness," signifies the character itself. Deny now that characters are existents. Then there is nothing for the adjective (or its abstractum) to signify denotatively and you arrive at the doctrine of common names. This is the step Ockham took. 2. An apple is, in a familiar sense, spatio-temporally extended. Its color and shape, as the realist conceives them, are not. For Quine, to be an existent and to be spatio-temporally extended, or, for brevity's sake I shall say, to be extended or an extension, are one and the same thing. This is the guiding idea of his ontology. It has an important corollary. The sum of any number of extensions is itself extended. Roughly, sum here means set-theoretical sum. Precisely, it is the function axiomatized in the so-called meromorphic calculus. This subtlety we can safely ignore. Notice, though, that the notion of a function, in the Fregean sense of the term, is needed to state the intuitive core of this ontology. Assume now once more that, pointing at an apple, I say "This is red." Or for the matter, assume that, pointing at a certain volume of water, I say "This is the Iowa River." Quine holds (Q 69) that there is in principle no difference between the two "ostensions." Just as the Iowa River is the sum of certain watery extensions, so the color may be thought of as the sum of all red ones. Which extensions we are meant to sum we learn in either case "by induction" from watching what is being pointed at ( Q 68). Pointing, however, is ambiguous (Q 67). Taking advantage of the ambiguity, Quine says that one who speaks and points as by assumption I do "identifies" for the purpose of the discourse what he points at with the sum in question (Q 71). Notice how subtle it all is. Quine does not tell us that the copula in "This is red" ____________________

18This

is by now at least a respectable opinion. See, for all this, "Some Remarks on the Ontology of Ockham," Philos. Rev., LIII (1954), 560-571; "Russell's Examination of Leibniz Examined," Philosophy of Science, XXIII (1956), 175-203; "Some Remarks on the Philosophy of Malebranche," Rev. Metaphys., X (1956), 207-226. Reprinted in Meaning and Existence. Leibniz, it is true, manages to accommodate both kinds of existent. But he pays the price of having to maintain that in a sense every predication is analytic (Predicatum inest subjecto). -82-

stands for identity. Of course he doesn't. As a logician, he knows better than that. Yet he says obliquely, by means of the opaque metaphor of "identification," that what I really point at is the sum. If this were so and if the color could be conceived as the sum in question, then "This is red" would indeed state an identity. Quine is convinced that nobody in his right mind would "hypostatize" characters as the realist does. So he must explain why such hypostasis ever seemed plausible. The explanation takes the form of an anthropological fable. Its hero is misled by a faulty analogy (Q 73). As it happens, some adjectives may be thought of as standing for a sum of extensions. So he is led to hope that this is so for all adjectives. Quine constructs a simple universe in which the sum of all triangles coincides with the sum of all squares ( Q 72). Negatively, this frustrates that hope. Positively, we are told (Q 75) that "in ostensively explaining 'square' ... we say each time 'This is square' without imputing identity of indicated object from one occasion to the next." Again the metaphor is opaque. I do not really know what it means to impute identity. I do know, though, that we are left with two alternatives. Either "square" stands for a character as the realist conceives it; or, even though there is no such character, we learn "inductively" how to use the common name. Quine rejects the first alternative, chooses the second (Q 75). Thus he lays himself open to the classical objection which I rehearsed earlier. Quine's nominalism is clearly a doctrine of common names. That makes it very different from Frege's. Yet there are also two points of contact. The first is that, sums being functions, the Fregean notion of function is an ingredient of the intuitive core of the doctrine. The other point of contact seems at first rather verbal. Quine is fond of the formula that while sentences are either true or false, a predicate is either true or false of something. For Frege, we remember, the predicative "is" is merely a clumsily disguised "of." Ofness, if I may coin a word, thus plays a crucial role in both systems. One may wonder whether this similarity is merely a chance product of the idiom. -83-

:5: UNIVERSALS F. P. RAMSEY THE purpose of this paper is to consider whether there is a fundamental division of objects into two classes, particulars and universals. This question was discussed by Mr. Russell in a paper printed in the Aristotelian Society's Proceedings for 1911. 1 His conclusion that the distinction was ultimate was based upon two familiar arguments, directed against the two obvious methods of abolishing the distinction by holding either that universals are collections of particulars, or that particulars are collections of their qualities. These arguments, perfectly sound as far as they go, do not however seem to me to settle the whole question. The first, which appears again in The Problems of Philosophy, shows as against the nominalists that such a proposition as "This sense-datum is white" must have as one constituent something, such as whiteness or similarity, which is not of the same logical type as the sense- datum itself. The second argument, also briefly expounded in McTaggart's The Nature of Existence, proves that a man cannot be identified with the sum of his qualities. But although a man cannot be one of his own qualities, that is no reason why he should not be a quality of something else. In fact material objects are described by Dr. Whitehead as "true Aristotelian adjectives"; so that we cannot regard these two arguments as rendering the

distinction between particular and universal secure against all criticism. What then, I propose to ask, is the difference between a particular and a universal? What can we say about one which will not also be true of the other? If we follow Mr. Russell we shall have to investigate three kinds of distinction, psychological, physical and logical. First we have the difference between a percept and a concept, the objects of two different kinds of mental acts; but this is unlikely to be a distinction of any fundamental importance, since a difference in two mental acts may not correspond to any difference whatever in their objects. Next we have various distinctions between objects based on their relations to space and time; for instance, some objects can only be in one place at a time, others, like the colour red, can be in many. Here again, in spite of the importance of the subject, I do not think we can have reached the essence of the matter. For when, for instance, Dr. Whitehead ____________________ Reprinted from The Foundations of Mathematics (London: Routledge & Kegan Paul Ltd. and New York: Humanities Press, Inc., 1950) by permission of the publishers. Original source: Mind, n.s. XXXIV (1925), Gilbert Ryle, editor. 1[See the paper by Russell reprinted in this volume.—ED.].

says that a table is an adjective, and Mr. Johnson that it is a substantive, they are not arguing about how many places the table can be in at once, but about its logical nature. And so it is with logical distinctions that our inquiry must mainly deal. According to Mr. Russell the class of universals is the sum of the class of predicates and the class of relations; but this doctrine has been denied by Dr. Stout. 2 But Dr. Stout has been already sufficiently answered. 3 So I shall only discuss the more usual opinion to which Mr. Russell adheres. According to him terms are divided into individuals or particulars, qualities and relations, qualities and relations being grouped together as universals; and sometimes qualities are even included among relations as one-termed relations in distinction from two-, three-, or many-termed relations. Mr. Johnson also divides terms into substantives and adjectives, including relations as transitive adjectives, and he regards the distinction between substantive and adjective as explaining that between particular and universal. But between these authorities, who agree so far, there is still an important difference. Mr. Johnson holds that although the nature of a substantive is such that it can only function in a proposition as subject and never as predicate, yet an adjective can function either as predicate or as a subject of which a secondary adjective can be predicated. For example, in "Unpunctuality is a fault" the subject is itself an adjective—the quality of unpunctuality. There is thus a want of symmetry between substantives and adjectives, for while a predicate must be an adjective, a subject may be either a substantive or an adjective, and we must define a substantive as a term which can only be a subject, never a predicate. Mr. Russell, on the other hand, in his lectures on Logical Atomism, 4 has denied this. He says that about an adjective there is something incomplete, some suggestion of the form of a proposition; so that the adjective-symbol can never stand alone or be the subject of a proposition, but must be completed into a proposition in which it is the predicate. Thus, he says, the appropriate symbol for redness is not the word "red" but the function "x is red," and red can only come into a proposition through the values of this function. So Mr. Russell would say "Unpunctuality is a fault" really means something like "For all x, if x is unpunctual, x is reprehensible"; and the adjective unpunctuality is not the subject of the proposition but only comes into it as the predicate of those of its parts which are of the form "x is unpunctual." This doctrine is the basis of new work in the Second Edition of Principia Mathematica. Neither of these theories seems entirely satisfactory, although neither could ____________________ 2"The Nature of Universals and Propositions," Proc. British Academy, 1921-22 (reprinted in Studies in Philosophy and Psychology, 1930). [Reprinted in this volume—ED.] 3See the symposium between G. E. Moore, G. F. Stout & G. Dawes Hicks in Aristotelian Society Supplementary Volume III, 1923. [The contributions of Moore and Stout are reprinted

in this volume—ED.] and 1919.

4The Monist, 1918

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be disproved. Mr. Russell's view does, indeed, involve difficulties in connection with our cognitive relations to universals, for which reason it was rejected in the First Edition of Principia; but these difficulties seem to me, as now to Mr. Russell, by no means insurmountable. But I could not discuss them here without embarking upon innumerable questions irrelevant to the main points which I wish to make. Neither theory, then, can be disproved, but to both objections can be raised which may seem to have some force. For instance, Mr. Russell urges that a relation between two terms cannot be a third term which comes between them, for then it would not be a relation at all, and the only genuinely relational element would consist of the connections between this new term and the two original terms. This is the kind of consideration from which Mr. Bradley deduced his infinite regress, of which Mr. Russell apparently now approves. Mr. Johnson might reply that for him the connectional or structural element is not the relation but the characterizing and coupling ties; but these ties remain most mysterious objects. It might also be objected that Mr. Johnson does not make particulars and universals different enough, or take into account the peculiar incompleteness of adjectives which appears in the possibility of prefixing to them the auxiliary "being"; "being red," "being a man" do not seem real things like a chair and a carpet. Against Mr. Russell it might be asked how there can be such objects as his universals, which contain the form of a proposition and so are incomplete. In a sense, it might be urged, all objects are incomplete; they cannot occur in facts except in conjunction with other objects, and they contain the forms of propositions of which they are constituents. In what way do universals do this more than anything else? Evidently, however, none of these arguments are really decisive, and the position is extremely unsatisfactory to any one with real curiosity about such a fundamental question. In such cases it is a heuristic maxim that the truth lies not in one of the two disputed views but in some third possibility which has not yet been thought of, which we can only discover by rejecting something assumed as obvious by both the disputants. Both the disputed theories make an important assumption which, to my mind, has only to be questioned to be doubted. They assume a fundamental antithesis between subject and predicate, that if a proposition consists of two terms copulated, these two terms must be functioning in different ways, one as subject, the other as predicate. Thus in "Socrates is wise," Socrates is the subject, wisdom the predicate. But suppose we turn the proposition round and say "Wisdom is a characteristic of Socrates," then wisdom, formerly the predicate, is now the subject. Now it seems to me as clear as anything can be in philosophy that the two sentences "Socrates is wise," "Wisdom is a characteristic of Socrates" assert the same fact and express the same proposition. They are not, of course, the same sentence, but they have the same meaning, just as two sentences in two different languages can have the same meaning. Which sentence we use is a matter either of literary style, or of the point of view from which we approach the fact. If the centre of our interest -86-

is Socrates we say "Socrates is wise," if we are discussing wisdom we may say "Wisdom is a characteristic of Socrates"; but whichever we say we mean the same thing. Now of one of these sentences "Socrates" is the subject, of the other "wisdom"; and so which of the two is subject, which predicate, depends upon what particular sentence we use to express our proposition, and has nothing to do with the logical nature of Socrates or wisdom, but is a matter entirely for grammarians. In the same way, with a sufficiently elastic language any proposition can be so expressed that any of its terms is the subject. Hence there is no essential distinction between the subject of a proposition and its predicate, and no fundamental classification of objects can be based upon such a distinction. I do not claim that the above argument is immediately conclusive; what I claim is that it throws doubt upon the whole basis of the distinction between particular and universal as deduced from that between subject and predicate, and that the question requires a new

examination. It is a point which has often been made by Mr. Russell that philosophers are very liable to be misled by the subject-predicate construction of our language. They have supposed that all propositions must be of the subject-predicate form, and so have been led to deny the existence of relations. I shall argue that nearly all philosophers, including Mr. Russell himself, have been misled by language in a far more far-reaching way than that; that the whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality what is merely a characteristic of language. Let us, therefore, examine closely this distinction of subject and predicate, and for simplicity let us follow Mr. Johnson and include relations among predicates and their terms among subjects. The first question we have to ask is this: what propositions are they that have a subject or subjects and a predicate? Is this the case with all propositions or only with some? Before, however, we go on to answer this question, let us remind ourselves that the task on which we are engaged is not merely one of English grammar; we are not school children analysing sentences into subject, extension of the subject, complement and so on, but are interested not so much in sentences themselves, as in what they mean, from which we hope to discover the logical nature of reality. Hence we must look for senses of subject and predicate which are not purely grammatical, but have a genuine logical significance. Let us begin with such a proposition as "Either Socrates is wise or Plato is foolish." To this, it will probably be agreed, the conception of subject and predicate is inapplicable; it may be applicable to the two parts "Socrates is wise," "Plato is foolish," but the whole "Either Socrates is wise or Plato is foolish" is an alternative proposition and not one with a subject or predicate. But to this someone may make the following objection: In such a proposition we can take any term we please, say Socrates, to be the subject. The predicate will then be "being wise unless Plato is foolish" or the propositional function " ̂x is wise or Plato is foolish." The phrase "being wise unless Plato is foolish" will then stand for a complex universal which is asserted to characterize -87-

Socrates. Such a view, though very frequently held, seems to me nevertheless certainly mistaken. In order to make things clearer let us take a simpler case, a proposition of the form "aRb"; then this theory will hold that there are three closely related propositions; one asserts that the relation R holds between the terms a and b, the second asserts the possession by a of the complex property of "having R to b," while the third asserts that b has the complex property that a has R to it. These must be three different propositions because they have different sets of constituents, and yet they are not three propositions, but one proposition, for they all say the same thing, namely that a has R to b. So the theory of complex universals is responsible for an incomprehensible trinity, as senseless as that of theology. This argument can be strengthened by considering the process of definition, which is as follows. For certain purposes "aRb" may be an unnecessarily long symbol, so that it is convenient to shorten it into " ϕb." This is done by definition, ϕx = aRx, signifying that any symbol of the form ϕx is to be interpreted as meaning what is meant by the corresponding symbol aRx, for which it is an abbreviation. In more complicated cases such an abbreviation is often extremely useful, but it could always be dispensed with if time and paper permitted. The believer in complex universals is now confronted with a dilemma: is "ϕ," thus defined, a name for the complex property of x which consists in a having R to x? If so, then ϕx will be the assertion that x has this property; it will be a subject-predicate proposition whose subject is x and predicate ϕ; and this is not identical with the relational proposition aRx. But as ϕx is by hypothesis defined to be short for aRx this is absurd. For if a definition is not to be interpreted as signifying that the definiendum and the definiens have the same meaning, the process of definition becomes unintelligible and we lose all justification for interchanging definiens and definiendum at will, on which depends its whole utility. Suppose on the other hand " ϕ," as defined above, is not a name for the complex property; then how can the complex property ever become an object of our contemplation, and how can we ever speak of it, seeing that " ϕ," its only possible name, is not a name for it at all but short for something else? And then what reason can there be to postulate the existence of this thing?

In spite of this reductio ad absurdum of the theory, it may still be worth while to inquire into its origin, and into why it is held by so many people, including formerly myself, without its occurring to them to doubt it. The chief reason for this is I think to be found in linguistic convenience; it gives us one object which is "the meaning" of " ϕ." We often want to talk of "the meaning of 'ϕ'," and it is simpler to suppose that this is a unique object than to recognize that it is a much more complicated matter, and that "ϕ" has a relation of meaning not to one complex object but to the several simple objects which are named in its definition. There is, however, another reason why this view is so popular, and that is the imaginary difficulty which would otherwise be felt in the use of a variable propositional function. How, it might be asked, are we to interpret such a statement as "a has all the properties of b," except on the supposition that there are properties? The answer -88-

is that it is to be interpreted as being the logical product of all propositions which can be constructed in the following way: take a proposition in which a occurs, say ϕa, change a into b and obtain ϕb, and then form the proposition ϕb . ⊃. ϕa. It is not really quite so simple as that, but a more accurate account of it would involve a lot of tiresome detail, and so be out of place here; and we can take it as a sufficient approximation that "a has all the properties of b" is the joint assertion of all propositions of the form ϕb . ⊃ . ϕa, where there is no necessity for ϕ to be the name of a universal, as it is merely the rest of a proposition in which a occurs. Hence the difficulty is entirely imaginary. It may be observed that the same applies to any other case of apparent variables some of whose values are incomplete symbols, and this may explain the tendency to assert that some of Mr. Russell's incomplete symbols are not really incomplete but the names of properties or predicates. I conclude, therefore, that complex universals are to be rejected; and that such a proposition as "Either Socrates is wise or Plato foolish" has neither subject nor predicate. Similar arguments apply to any compound proposition, that is any proposition containing such words as "and," "or," "not," "all," "some"; and hence if we are to find a logical distinction between subject and predicate anywhere it will be in atomic propositions, as Mr. Russell calls them, which could be expressed by sentences containing none of the above words, but only names and perhaps a copula. The distinction between subject and predicate will then arise from the several names in an atomic proposition functioning in different ways; and if this is not to be a purely grammatical distinction it must correspond to a difference in the functioning of the several objects in an atomic fact, so that what we have primarily to examine is the construction of the atomic fact out of its constituents. About this three views might be suggested; first there is that of Mr. Johnson according to whom the constituents are connected together by what he calls the characterizing tie. The nature of this entity is rather obscure, but I think we can take it as something which is not a constituent of the fact but represented in language by the copula "is," and we can describe this theory as holding that the connection is made by a real copula. Next there is the theory of Mr. Russell that the connection is made by one of the constituents; that in every atomic fact there must be one constituent which is in its own nature incomplete or connective and, as it were, holds the other constituents together. This constituent will be a universal, and the other particulars. Lastly there is Mr. Wittgenstein's theory that neither is there a copula, nor one specially connected constituent, but that, as he expresses it, the objects hang one in another like the links of a chain. From our point of view it is the second of these theories that demands most attention; for the first and third do not really explain any difference in the mode of functioning of subject

and predicate, but leave this a mere dogma. Only on Mr. Russell's theory will there be an intelligible difference between particular and universal, grounded on the necessity for there to be in each fact a copulating term or universal, corresponding to the need for every -89-

sentence to have a verb. So it is Mr. Russell's theory that we must first consider. The great difficulty with this theory lies in understanding how one sort of object can be specially incomplete. There is a sense in which any object is incomplete; namely that it can only occur in a fact by connection with an object or objects of suitable type; just as any name is incomplete, because to form a proposition we have to join to it certain other names of suitable type. As Wittgenstein says: "The thing is independent, in so far as it can occur in all possible circumstances, but this form of independence is a form of connection with the atomic fact, a form of dependence. (It is impossible for words to occur in two different ways, alone and in the proposition.)" 5 And Johnson: "Ultimately a universal means an adjective that may characterize a particular, and a particular means a substantive that may be characterized by a universal." 6 Thus we may admit that "wise" involves the form of a proposition, but so does "Socrates," and it is hard to see any ground for distinguishing between them. This is the substance of Mr. Johnson's criticism that Mr. Russell will not let the adjective stand alone, and in treating "s is p" as a function of two variables takes the arguments to be not s and p, but s and "̂x is p." In reply to this criticism Mr. Russell would, I imagine, use two lines of argument, whose validity we must examine. The first would dwell on the great convenience in mathematical logic of his functional symbolism, of which he might say there was no explanation except that this symbolism corresponded to reality more closely than any other. His second line of argument would be that everyone can feel a difference between particulars and universals ; that the prevalence of nominalism showed that the reality of universals was always suspected, and that this was probably because they did in fact differ from particulars by being less independent, less self-contained. Also that this was the only account of the difference between particulars and universals which made them really different kinds of objects, as they evidently were, and not merely differently related to us or to our language. For instance, Mr. Johnson describes the particular as presented to thought for its character to be determined in thought, and others might say a particular was what was meant by the grammatical subject of a sentence; and on these views what was particular, what universal would depend on unessential characteristics of our psychology or our language. Let us take these lines of argument in reverse order, beginning with the felt difference between particular and universal, and postponing the peculiar symbolic convenience of propositional functions. Anyone, it may be said, sees a difference between Socrates and wisdom. Socrates is a real independent entity, wisdom a quality and so essentially a quality of something else. The first thing to remark about this argument is that it is not really about objects at all. "Socrates is wise" is not an atomic proposition, and the symbols ____________________ 5Tractatus Logico-Philosophicus, 2.0122. 6Logic Part I, p. 11. -90-

"Socrates" and "wise" are not the names of objects but incomplete symbols. And according to Wittgenstein, with whom I agree, this will be the case with any other instance that may be suggested, since we are not acquainted with any genuine objects or atomic propositions, but merely infer them as presupposed by other propositions. Hence the distinction we feel is one between two sorts of incomplete symbols, or logical constructions, and we cannot infer without further investigation that there is any corresponding distinction between two sorts of names or objects. We can, I think, easily obtain a clearer idea of the difference between these two sorts of

incomplete symbols (Wittgenstein calls them "expressions") typified by "Socrates" and "wise." Let us consider when and why an expression occurs, as it were, as an isolated unit. For instance "aRb" does not naturally divide into "a" and "Rb," and we want to know why anyone should so divide it and isolate the expression "Rb." The answer is that if it were a matter of this proposition alone, there would be no point in dividing it in this way, but that the importance of expressions arises, as Wittgenstein points out, just in connection with generalization. It is not "aRb" but "(x) . xRb" which makes Rb prominent. In writing (x) . xRb we use the expression Rb to collect together the set of propositions xRb which we want to assert to be true; and it is here that the expression Rb is really essential because it is this which is common to this set of propositions. If now we realize that this is the essential use of expressions, we can see at once what is the difference between Socrates and wise. By means of the expression "Socrates" we collect together all the propositions in which it occurs, that is, all the propositions which we should ordinarily say were about Socrates, such as "Socrates is wise," "Socrates is just," "Socrates is neither wise nor just." These propositions are collected together as the values of " ϕ Socrates," where ϕ is a variable. Now consider the expression "wise"; this we use to collect together the propositions "Socrates is wise," "Plato is wise," and so on, which are values of "x is wise." But this is not the only collection we can use "wise" to form; just as we used "Socrates" to collect all the propositions in which it occurred, we can use "wise" to collect all those in which it occurs, including not only ones like "Socrates is wise" but also ones like "Neither Socrates nor Plato is wise," which are not values of "x is wise" but only of the different function " ϕ wise," where ϕ is variable. Thus whereas Socrates gives only one collection of propositions, wise gives two: one analogous to that given by Socrates, namely the collection of all propositions in which wise occurs; and the other a narrower collection of propositions of the form "x is wise." This is obviously the explanation of the difference we feel between Socrates and wise which Mr. Russell expresses by saying that with wise you have to bring in the form of a proposition. Since all expressions must be completed to form a proposition, it was previously hard to understand how wise could be more incomplete than Socrates. Now we can see that the reason for this is that whereas with "Socrates" we only have the idea of completing it in any manner into a proposition, with "wise" we have not only this but also an -91-

idea of completing it in a special way, giving us not merely any proposition in which wise occurs but also one in which it occurs in a particular way, which we may call its occurrence as predicate, as in "Socrates is wise." What is this difference due to, and is it a real difference at all? That is to say, can we not do with "Socrates" what we do with "wise," and use it to collect a set of propositions narrower than the whole set in which it occurs? Is this impossible, or is it merely that we never in fact do it? These are the questions we must now try to answer. The way to do it would seem to be the following. Suppose we can distinguish among the properties of Socrates a certain subset which we can call qualities, the idea being roughly that only a simple property is a quality. Then we could form in connection with "Socrates" two sets of propositions just as we can in connection with "wise." There would be the wide set of propositions in which "Socrates" occurs at all, which we say assert properties of Socrates, but also there would be the narrower set which assert qualities of Socrates. Thus supposing justice and wisdom to be qualities, "Socrates is wise," "Socrates is just" would belong to the narrower set and be values of a function "Socrates is q." But "Socrates is neither wise nor just" would not assert a quality of Socrates but only a compound characteristic or property, and would only be a value of the function "ϕ Socrates," not of "Socrates is q." But although such a distinction between qualities and properties may be logically possible, we do not seem ever to carry it out systematically. Some light may be thrown on this fact by a paragraph in Mr. Johnson's Logic in which he argues that, whereas "we may properly construct a compound adjective out of simple adjectives, yet the nature of any term functioning as a substantive is such that it is impossible to construct a genuine compound

substantive." 7 Thus from the two propositions "Socrates is wise," "Socrates is just" we can form the proposition "Neither is Socrates wise nor is Socrates just," or, for short, "Socrates is neither wise nor just"; which still, according to Mr. Johnson, predicates an adjective of Socrates, is a value of "ϕ Socrates" and would justify "(∃ϕ). ϕ Socrates," or "Socrates has some property." If, on the other hand, we take the two propositions "Socrates is wise," "Plato is wise" and form from them "Neither Socrates is wise nor Plato is wise"; this is not a value of "x is wise" and would not justify "(∃x) . x is wise," or "Someone is wise." So

inasmuch as "Socrates is neither wise nor just" justifies "Socrates has some adjective" we can say that "neither wise nor just" is a compound adjective; but since "Neither Socrates nor Plato is wise" does not justify "something is wise," "neither Socrates nor Plato" cannot be a compound substantive any more than nobody is a compound man. If, however, we could form a range of qualities as opposed to properties, "Socrates is neither wise nor just" would not justify "Socrates has some quality" and "neither wise nor just" would not be a quality. Against this Mr. Johnson says that there is no universally valid criterion by which we can distinguish qualities from other properties; and this is certainly a very ____________________ 7Part II, p. 61. -92-

plausible contention when we are talking, as we are now, of qualities and properties of logical constructions such as Socrates. For the distinction is only really clear in connection with genuine objects; then we can say that ϕ represents a quality when ϕa is a two-termed atomic proposition, and this would distinguish qualities from other propositional functions or properties. But when the subject a is a logical construction and ϕa a compound proposition of which we do not know the analysis, it is hard to know what would be meant by asking if ϕ were simple, and calling it, if simple, a quality. It would clearly have to be a matter not of absolute but of relative simplicity. Yet it is easy to see that, in theory, an analogous distinction can certainly be made for incomplete symbols also. Take any incomplete symbol "a"; this will be defined not in isolation but in conjunction with any symbol of a certain sort x. Thus we might define ax to mean aRx. Then this incomplete symbol "a" will give us two ranges of propositions: the range ax obtained by completing it in the way indicated in its definition; and the general range of propositions in which a occurs at all, that is to say, all truth-functions of the propositions of the preceding range and constant propositions not containing a. Thus in the two famous cases of descriptions and classes, as treated in Principia Mathematica, the narrower range will be that in which the description or class has primary occurrence, the wider range that in which it has any sort of occurrence primary or secondary, where the terms "primary" and "secondary" occurrence have the meanings explained in Principia. In brief with regard to any incomplete symbol we can distinguish its primary and secondary occurrences, and this is fundamentally the same distinction which we found to be characteristic in the adjective. So that any incomplete symbol is really an adjective, and those which appear substantives only do so in virtue of our failing whether through inability or neglect to distinguish their primary and secondary occurrences. As a practical instance let us take the case of material objects; these we are accustomed to regard as substantives, that is to say we use them to define ranges of propositions in one way only, and make no distinction between their primary and secondary occurrences. At least no one made such a distinction until Dr. Whitehead declared that material objects are adjectives of the events in which they are situated, so that the primary occurrence of a material object A is in a

proposition "A is situated in E." From such propositions as this we can construct all other propositions in which A occurs. Thus "A is red" will be "For all E, A is situated in E implies redness is situated in E," in which A has secondary occurrence. So the distinction between primary and secondary occurrence is not merely demonstrated as logically necessary, but for this case effected practically. The conclusion is that, as regards incomplete symbols, the fundamental distinction is not between substantive and adjective but between primary and secondary occurrence; and that a substantive is simply a logical construction between whose primary and secondary occurrences we fail to distinguish. So that to be a substantive is not an objective but a subjective property in the -93-

sense that it depends not indeed on any one mind but on the common elements in all men's minds and purposes. This is my first conclusion, which is I think of some importance in the philosophy of nature and of mind; but it is not the conclusion which I most want to stress, and it does not answer the question with which I began my paper. For it is a conclusion about the method and possibility of dividing certain logical constructions into substantives and adjectives, it being in connection with these logical constructions that the idea of substantive and adjective traditionally originated. But the real question at issue is the possibility of dividing not logical constructions but genuine objects into particulars and universals, and to answer this we must go back and pick up the thread of the argument, where we abandoned it for this lengthy digression about logical constructions. We saw above that the distinction between particular and universal was derived from that between subject and predicate which we found only to occur in atomic propositions. We then examined the three theories of atomic propositions or rather of atomic facts, Mr. Johnson's theory of a tie, Mr. Russell's that the copulation is performed by universals, of which there must be one and only one in each atomic fact, and Mr. Wittgenstein's that the objects hang in one another like the links of a chain. We observed that of these theories only Mr. Russell's really assigned a different function to subject and predicate and so gave meaning to the distinction between them, and we proceeded to discuss this theory. We found that to Mr. Johnson's criticisms Mr. Russell had two possible answers; one being to argue that his theory alone took account of the difference we feel there to be between Socrates and wisdom, the other that his notation was far more convenient than any other and must therefore correspond more closely to the facts. We then took the first of these arguments, and examined the difference between Socrates and wisdom. This we found to consist in the fact that whereas Socrates determined only one range of propositions in which it occurred, wise determined two such ranges, the complete range "f wise," and the narrower range "x is wise." We then examined the reason for this difference between the two incomplete symbols Socrates and wise, and decided that it was of a subjective character and depended on human interests and needs. What we have now to consider is whether the difference between Socrates and wise has any such bearing on the composition of atomic facts as Mr. Russell alleges it to have. This we can usefully combine with the consideration of Mr. Russell's other possible argument from the superior convenience of his symbolism. The essence of this symbolism, as Mr. Johnson has observed, consists in not letting the adjective stand alone, but making it a propositional function by attaching it to a variable x. A possible advantage of this procedure at once suggests itself in terms of our previous treatment of the difference between substantive and adjective; namely, that attaching the variable x helps us to make the distinction we require to make in the case of the adjective, but not in the case of the substantive, between the values of ϕx and -94-

those of f (ϕ̂Z) where f is variable. Only so, it might be said, can we distinguish (x) . ϕx from

( f) . f(ϕ̂Z) But very little consideration is required to see that this advantage is very slight and of no fundamental importance. We could easily make the distinction in other ways; for instance by determining that if the variable came after the ϕ it should mean what we now express by ϕx, but if before the ϕ what we express by f (ϕ̂Z); or simply by deciding to use the letters "x," "y," "z," in one case, "f," "g," "h," in the other. But, although this supposed advantage in the functional symbolism is imaginary, there is a reason which renders it absolutely indispensable. Take such property as "either having R to a, or having S to b"; it would be absolutely impossible to represent this by a simple symbol "ϕ." For how then could we define ϕ? We could not put ϕ = Ra . v . Sb because we should not know whether the blanks were to be filled with the same or different arguments, and so whether ϕ was to be a property or relation. Instead we must put ϕx. = . xRa . v . xSb; which explains not what is meant by ϕ by itself but that followed by any symbol x it is short for xRa . v . xSb. And this is the reason which makes inevitable the introduction of propositional functions. It simply means that in such a case " ϕ" is not a name but an incomplete symbol and cannot be defined in isolation or allowed to stand by itself. But this conclusion about xRa . v . xSb will not apply to all propositional functions. If ϕa is a two-termed atomic proposition, "ϕ" is a name of the term other than a, and can perfectly well stand by itself; so, it will be asked, why do we write " ϕx" instead of "ϕ" in this case also? The reason for this lies in a fundamental characteristic of mathematical logic, its extensionality, by which I mean its primary interest in classes and relations in extension. Now if in any proposition whatever we change any individual name into a variable, the resulting propositional function defines a class; and the class may be the same for two functions of quite different forms, in one of which " ϕ" is an incomplete symbol, in the other a name. So mathematical logic, being only interested in functions as a means to classes, sees no need to distinguish these two sorts of functions, because the difference between them, though all-important to philosophy, will not correspond to any difference between the classes they define. So because some ϕ's are incomplete and cannot stand alone, and all ϕ's are to be treated alike in order to avoid useless complication, the only solution is to allow none to stand alone. Such is the justification of Mr. Russell's practice; but it is also the refutation of his theory, which fails to appreciate the distinction between those functions which are names and those which are incomplete symbols, a distinction which, as remarked above, though immaterial for mathematics is essential for philosophy. I do not mean that Mr. Russell would now deny this distinction; on the contrary it is clear from the Second Edition of Principia that he would accept it; but I think that his present theory of universals is the relic of his previous failure to appreciate it. It will be remembered that we found two possible arguments for his theory of universals. One was from the efficiency of the functional notation; this -95-

clearly lapses because, as we have seen, the functional notation merely overlooks an essential distinction which happens not to interest the mathematician, and the fact that some functions cannot stand alone is no argument that all cannot. The other argument was from the difference we feel between Socrates and wise, which corresponds to a difference in his logical system between individuals and functions. Just as Socrates determines one range of propositions, but wise two, so a determines the one range ϕa, but ϕ̂Z the two ranges ϕx and f (ϕ̂Z). But what is this difference between individuals and functions due to? Again simply to the fact that certain things do not interest the mathematician. Anyone who was interested not only in classes of things, but also in their qualities, would want to distinguish from among the others those functions which were names; and if we called the objects of which they are names qualities, and denoted a variable quality by q, we should have not only the range ϕa but also the narrower range qa, and the difference analogous to that between "Socrates" and "wisdom" would have disappeared. We should have complete symmetry

between qualities and individuals; each could have names which could stand alone, each would determine two ranges of propositions, for a would determine the ranges qa and ϕa, where q and ϕ are variables, and q would determine the ranges qx and fq, where x and f are variables. So were it not for the mathematician's biassed interest he would invent a symbolism which was completely symmetrical as regards individuals and qualities; and it becomes clear that there is no sense in the words individual and quality; all we are talking about is two different types of objects, such that two objects, one of each type, could be sole constituents of an atomic fact. The two types being in every way symmetrically related, nothing can be meant by calling one type the type of individuals and the other that of qualities, and these two words are devoid of connotation. To this, however, various objections might be made which must be briefly dealt with. First it might be said that the two terms of such an atomic fact must be connected by the characterizing tie and/or the relation of characterization, which are asymmetrical, and distinguish their relata into individuals and qualities. Against this I would say that the relation of characterization is simply a verbal fiction. "q characterizes a" means no more and no less than "a is q," it is merely a lengthened verbal form; and since the relation of characterization is admittedly not a constituent of "a is q" it cannot be anything at all. As regards the tie, I cannot understand what sort of a thing it could be, and prefer Wittgenstein's view that in the atomic fact the objects are connected together without the help of any mediator. This does not mean that the fact is simply the collection of its constituents but that is consists in their union without any mediating tie. There is one more objection suggested by Mr. Russell's treatment in the new edition of Principia. He there says that all atomic propositions are of the form R 1 (x), R 2 (x, y), R 3 (x, y, z), etc., and so can define individuals as terms which can occur in propositions with any number of terms; whereas of course an n-termed relation could only occur in a proposition with n + 1 terms. But this assumes his theory as to the con -96-

stitution of atomic facts, that each must contain a term of a special kind, called a universal; a theory we found to be utterly groundless. The truth is that we know and can know nothing whatever about the forms of atomic propositions; we do not know whether some or all objects can occur in more than one form of atomic proposition; and there is obviously no way of deciding any such question. We cannot even tell that there are not atomic facts consisting of two terms of the same type. It might be thought that this would involve us in a vicious circle contradiction, but a little reflection will show that it does not, for the contradictions due to letting a function be its own argument only arise when we take for argument a function containing a negation which is therefore an incomplete symbol not the name of an object. In conclusion let us describe from this new point of view the procedure of the mathematical logician. He takes any type of objects whatever as the subject of his reasoning, and calls them individuals, meaning by that simply that he has chosen this type to reason about, though he might equally well have chosen any other type and called them individuals. The results of replacing names of these individuals in propositions by variables he then calls functions, irrespective of whether the constant part of the function is a name or an incomplete symbol, because this does not make any difference to the class which the function defines. The failure to make this distinction has led to these functional symbols, some of which are names and some incomplete, being treated all alike as names of incomplete objects or properties, and is responsible for that great muddle the theory of universals. Of all philosophers Wittgenstein alone has seen through this muddle and declared that about the forms of atomic propositions we can know nothing whatever. -97-

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UNIVERSALS AND METAPHYSICAL REALISM ALAN DONAGAN The point of philosophy is to start with something so simple as not to seem worth stating, and end with something so paradoxical that no one will believe it. —BERTRAND RUSSELL, The Monist (1918) THE late Friedrich Waismann once remarked that, while you may confute and kill a scientific theory, a philosophy dies only of old age. The realist theory of universals, which G. E. Moore and Bertrand Russell revived in the brilliant fifteen years which preceded the first World War, 1 seems to have aged more rapidly than its authors, and to have died, or fallen into oblivion, during the 'forties. In the United States, the very different conception of realism propounded by Professor Quine and Goodman, 2 and nicknamed by Quine "Plato's Beard," has displaced it, leaving Professor Bergmann almost alone to defend it. 3 In Britain, a polished essay by Mr. Pears seems to have been received as its epitaph. 4 In this paper I propose to re-examine Moore's and Russell's principal argument for the reality of universals, in order to determine whether any spark of life remains in it. Is it truly dead, or only neglected? ____________________ Reprinted from The Monist, 47, no. 2 (1963), with the permission of The Open Court publishing Co., La Salle, Illinois, and of the author. 1G. E. Moore, "The Nature of Judgment," Mind, 8 (1899), esp. pp. 178-83; "Identity," Proc. Aris. Soc., 1 (1900-1), esp. pp. 105-15; Some Main Problems of Philosophy (London, 1953), hereafter cited as Main Problems, pp. 301-5, 312-77 (composed in 1910-1); and Bertrand Russell, "On the Relations of Universals to Particulars" (composed 1911) in Logic and Knowledge, ed. Robert C. Marsh (London, 1956); The Problems of Philosophy (London, 1912), chs. 8-10. 2W. V. Quine, From a Logical Point of View (Cambridge, Mass., 1951), chs. 1, 6; Nelson Goodman, The Structure of Appearance (Cambridge, Mass., 1951), ch. 2. 3Gustav Bergmann, Meaning and Existence (Madison, Wis., 1959), esp. chs. 4, 13. My debts in this essay to Professor Bergmann, particularly in what I say about the attempts of Professors Quine and Goodman to shave Plato's Beard, are heavy and obvious, though no doubt he would reject most of my conclusions. 4D. F. Pears, "Universals," in Logic and Language, 2nd series, ed. Antony Flew (Oxford, 1953).

I Russell's The Problems of Philosophy is a convenient and familiar point of departure. Lucidly and simply, it states the position which Moore and Russell held, and their reason for holding it. In its eighth chapter, Russell wrote this: Suppose, for instance, that I am in my room. I exist, and my room exists; but does "in" exist? Yet obviously the word "in" has a meaning; it denotes a relation which holds between me and my room. This relation is something, although we cannot say that it exists in the same sense in which I and my room exist. The relation "in" is something which we can think about and understand, for, if we could not understand it, we could not understand the sentence "I am in my room." 5 The conclusion that we are to investigate is that the relation denoted by "in" is, or is real. Russell's distinction between being and existence, according to which the relation denoted by "in" has being (is or is real) but does not exist, is notoriously difficult, and we shall defer investigating it. Yet, even apart from that distinction, Russell's argument and conclusion are puzzling.His reasoning seems to have been as follows: i. Some propositions of the form "x is in y," where "x" and "y" deputize for names or descriptions of things which in a familiar sense exist, can be thought about and

understood. They could not be thought about or understood unless the word "in" were thought about and understood; i.e. "in" is not redundant. iii. Some propositions in the form "x is in y" are true. (I take this to be presupposed in Russell's opening injunction: "Suppose, for instance, that....") iv. The non-redundant elements of true propositions denote things that are real or have being, if not things that exist. v. Therefore, "in" denotes something which is or is real, if not something which exists; and since if "in" denotes anything at all it is a relation, it follows that at least one relation is or is real. ii.

If relations are real, then universals are real: for "a universal will be anything which may be shared by many particulars"; 6 and at least two pairs of particulars, namely, Russell and his room, and Moore and his room, may share the relation denoted by "in." Neither Russell nor Moore believed that all universals were relational. In The Problems of Philosophy Russell had much to say of justice and whiteness, which he considered to be non-relational qualities; and in Some Main Problems of Philosophy Moore strove to demonstrate that in some sense ____________________ 5Bertrand Russell, The Problems of Philosophy (London, reset edn. 1946), hereafter cited as Problems, p. 90. 6Russell, Problems, p. 93; cf. Moore, Main Problems, p. 304. -99-

whiteness is a universal which is neither a relation (like in) nor a relational property (like in Russell's room). Yet both Moore and Russell considered the being of relations and relational properties to be far more evident than that of non-relational (qualitative) universals; and both ascribed the nominalist tendencies in the work of Berkeley and Hume to their error that, unlike qualities, relations are evidently the work of the mind. 7 Russell plainly agreed with Moore that "it is ... comparatively easy to distinguish universals of both these two sorts [relations and relational properties]; and if it were quite clear that they were the only sorts, the whole question about universals would be ... comparatively simple." 8 Simple or not, it is the question we are to investigate. In doing so, I shall