Originally published in Mind October 1925.

UNIVERSALS*

By F. P. RAMSEY
(1925)
 
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. 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 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 'two 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 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 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.1 But Dr. Stout has been already sufficiently answered.2 So I shall only discuss the more usual opinion to which 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 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

1 "The Nature of Universals and Propositions," Proc. British Academy, 1921-22 (reprinted in Studies in Philosophy and Psychology, 1930).

2 See the symposium between G. E. Moore, G. F. Stout & G. Dawes Hicks in Aristotelian Society Supplementary Volume III 1923.

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,1 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 somethng 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 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

1 The Monist, 1918 and 1919.

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 in 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 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 auxillary 'being'; 'being red', '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 a subject, the other as predicate. Thus in 'Socrates is wise', Socrates is the subject, wisdom the predicate. But suppose we turn the proposition around 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 no, 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 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 constructions of our language. They have supposed that all propositions must be of a 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? Is this the case 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 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. 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 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 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 contents. 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 consituent 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 others particulars. Lastly there is Mr. Wittgenstein's theory that neither is there 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 more 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 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 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; 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)."1 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."2 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 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 evidentally 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

1 Tractatus Logico-Philosophicus, 2.0122.

2 Logic Part 1, p. 11.

views what was particular, what universal would depend on unessential characteristics of 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 'Socrates' and 'wise' are not the names of 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 itwere, 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 the different function 'φ is 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 different 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 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 the 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 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".1 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 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

1 Part II. p. 61.

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 of 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 occurrence we fail to distinguish. So that to be a substantive is not an objective but a subjective property in the 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 Jr. 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 those of f(φz^) where f is variable. Only so, it might be said, can we distinguish (x) . φx from (f) . f(φz^). 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(φz^); 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 indispensible. Take such a 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 φx . = . 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 propostion, '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 functon 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 for 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 to 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 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 range φa, but φz^ the two ranges φx and f(φz^). But what is this difference between individuals and functions due to? Again simply to the fact that certain things do not interest the mathematicians. Anyone who was interested not only in classses of things, but also in their qualities, would want to distinguish from among the others those function 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 symmetrical, 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 it 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 forms R1(x), R2(x,y),R3(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 constitution 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 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.

NOTE ADDED 1926

When I wrote my article I was sure that it was impossible to discover atomic propositions by actual analysis. Of this I am now very doubtful, and I cannot therefore be sure that they may not be discovered to be all of one or other of a series of forms which can be expressed by R1(x), R2(x,y),R3(x,y,z) etc., in which case we could, as Mr. Russell has suggested, define individuals as terms which can occur in propositions of any of these forms, universals as terms which can only occur in one form. This I admit may be found to be the case, but as no one can as yet be certain what sort of atomic propositions there are, it cannot be positively asserted; and there is no strong presumption in its favour, for I think that the argument of my article establishes that nothing of the sort can be known a priori.

And this is a matter of some importance, for philosophers such as Mr. Russell have thought that, although they did not know into what ultimate terms propositions are analysable, these terms must nevertheless be divisible into universals and particulars, categories which are used in philosophical investigations as if it were certain a priori that they would be applicable. This certainly seems to be derived primarily from the supposition that there must be a difference between ulitmate objects analogous to one felt to subsist between such terms as Socrates and wise; and to see if this can reasonably be maintained, we must discover what difference there is between Socrates and wise analogous to the distinction made in Mr. Russell's system between particulars and universals.

If we consider the development of Mr. Russell's system of logic, as expounded in the Introduction to the Second Edition of Principia Mathematica, we can see what difference there is in his treatment of particulars and universals. We find that universals always occur as propositional functions, which serve to determine ranges of propositions, especially the range of the function φx, and the range of functions of the function f(φx^) (where f is variable). Individuals also serve to determine ranges of propositions, but in this case there is only one principal range, the range of functions of the individual φa (φ variable). We could make a narrower range, as Mr. Russell points out, by using a variable quality, but we have no need to do so. Now this is the only difference between the way individuals and universals function in his system, and as we find that there is precisely similar difference between Socrates and wise, it is probable that we have here the essence of the matter. Wise, like a φx in Mr. Russell's system, determines the narrower range of propositions 'x is wise' and the wider one 'f wise', where the last range includes all propositions whatever in which wise occurs. Socrates, on the other hand, is only used to determine the wider range of propositions in which it occurs in any manner; we have no precise way of singling out any narrower range. We cannot do it by limiting it to propositions in which Socrates occurs as subject, because in any proposition in which he occurs he can be regarded as subject: we can always regard the proposition in which he occurs he can be regarded as the subject: we can always regard the proposition as saying 'It is true of Socrates that --'. the point is that with Socrates the narrower range is missing...

Nevertheless this difference between Socrates and wise is illusory, because it can be shown to be theoretically possible to make a similar narrower range for Socrates, though we have never needed to do this. Nevertheless, once this fact is observed, the difference between Socrates and wise lapses, and we begin, like Dr. Whitehead, to call Socrates an adjective. If you think all or nearly all propositions about material objects are truth-functions of propositions about their location in events, then, on my view, you will regard material objects as adjectives of events. For that is the real meaning of the distinction between adjective and substantive. I do not say that the distinction has arisen from explicit reflection about the difference in regard to ranges of propositions, but that this difference obscurely felt is the source of the distinction. My view is strikingly confirmed by the case of Dr. Whitehead, who, having made material objects analogous to wise in the way in question, then declared that they were adjectives.