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The Philosophy of Logical Atomism (1918-19)

 Lecture 6: Descriptions and Incomplete Symbols*

By Bertrand Russell

I am proposing to deal this time with the subject of descriptions, and what I call “incomplete symbols,” and the existence of described individuals. You will remember that last time I dealt with the existence of kinds of things, what you mean by saying “There are men” or “There are Greeks” or phrases of that sort, where you have an existence which may be plural. I am going to deal today with an existence which is asserted to be singular, such as “The man with the iron mask existed” or some phrase of that sort, where you have some object described by the phrase “The so-and-so” in the singular, and I want to discuss the analysis of propositions in which phrases of that kind occur.

There are, of course, a great many propositions very familiar in metaphysics which are of that sort: “I exist” or “God exists” or “Homer existed,” and other such statements are always occurring in metaphysical discussions, and are, I think, treated in ordinary metaphysics in a way which embodies a simple logical mistake that we shall be concerned with today, the same sort of mistake that I spoke of last week in connection with the existence of kinds of things. One way of examining a proposition of that sort is to ask yourself what would happen if it were false. If you take such a proposition as “Romulus existed,” probably most of us think that Romulus did not exist. It is obviously a perfectly significant statement, whether true or false, to say that Romulus existed. If Romulus himself entered into our statement, it would be plain that the statement that he did not exist would be nonsense, because you cannot have a constituent of a proposition which is nothing at all. Every constituent has got to be there as one of the things in the world, and therefore if Romulus himself entered into the propositions that he existed or that he did not exist, both these propositions could not only not be true, but could not be even significant, unless he existed. That is obviously not the case, and the first conclusion one draws is that, although it looks as if Romulus were a constituent of that proposition, that is really a mistake. Romulus does not occur in the proposition “Romulus did not exist.”

Suppose you try to make out what you do mean by that proposition. You can take, say, all the things that Livy has to say about Romulus, all the properties he ascribes to him, including the only one probably that most of us remember, namely, the fact that he was called “Romulus.” You can put all this together, and make a propositional function saying “x has such-and-such properties,” the properties being those you find enumerated in Livy. There you have a propositional function, and when you say that Romulus did not exist you are simply saying that that propositional function is never true, that it is impossible in the sense I was explaining last time, i.e., that there is no value of x that makes it true. That reduces the non-existence of Romulus to the sort of non-existence I spoke of last time, where we had the non-existence of unicorns. But it is not a complete account of this kind of existence or non-existence, because there is one other way in which a described individual can fail to exist, and that is where the description applies to more than one person. You cannot, e.g., speak of “The inhabitant of London,” not because there are none, but because there are so many.

You see, therefore, that this proposition “Romulus existed” or “Romulus did not exist” does introduce a propositional function, because the name “Romulus” is not really a name but a sort of truncated description. It stands for a person who did such-and-such things, who killed Remus, and founded Rome, and so on. It is short for that description; if you like, it is short for “the person who was called ‘Romulus.’ ” If it were really a name, the question of existence could not arise, because a name has got to name something or it is not a name, and if there is no such person as Romulus there cannot be a name for that person who is not there, so that this single word “Romulus” is really a sort of truncated or telescoped description, and if you think of it as a name you will get into logical errors. When you realize that it is a description, you realize therefore that any proposition about Romulus really introduces the propositional function embodying the description, as (Say) “x was called ‘Romulus.’ ” That introduces you at once to a propositional function, and when you say “Romulus did not exist,” you mean that this propositional function is not true for one value of x.

There are two sorts of descriptions, what one may call “ambiguous descriptions,” when we speak of “a so-and-so,” and what one may call “definite descriptions,” when we speak of “the so-and-so” (in the singular). Instances are:

Ambiguous: A man, a dog, a pig, a Cabinet Minister
Definite:      The man with the iron mask
                      The last person who came into this room
                      The only Englishman who ever occupied the Papal See
                      The number of the inhabitants of London
                      The sum of 43 and 34

(It is not necessary for a description that it should describe an individual: it may describe a predicate or a relation or anything else.)

It is phrases of that sort, definite descriptions, that I want to talk about today. I do not want to talk about ambiguous descriptions, as what there was to say about them was said last time.

I want you to realize that the question whether a phrase is a definite description turns only upon its form, not upon the question whether there is a definite individual so described. For instance, I should call “The inhabitant of London” a definite description, although it does not in fact describe any definite individual.

The first thing to realize about a definite description is that it is not a name. We will take “The author of Waverley.” That is a definite description, and it is easy to see that it is not a name. A name is a simple symbol (i.e., a symbol which does not have any parts that are symbols), a simple symbol used to designate a certain particular or by extension an object which is not a particular but is treated for the moment as if it were, or is falsely believed to be a particular, such as a person. This sort of phrase, “The author of Waverley,” is not a name because it is a complex symbol. It contains parts which are symbols. It contains four words, and the meanings of those four words are already fixed and they have fixed the meaning of “The author of Waverley” in the only sense in which that phrase does have any meaning. In that sense, its meaning is already determinate, i.e., there is nothing arbitrary or conventional about the meaning of that whole phrase, when the meanings of “the,” “author,” “of,” and “Waverley” have already been fixed. In that respect, it differs from “Scott,” because when you have fixed the meaning of all the other words in the language, you have done nothing toward fixing the meaning of the name “Scott.” That is to say, if you understand the English language, you would understand the meaning of the phrase “The author of Waverley” if you had never heard it before, whereas you would not understand the meaning of “Scott” if you had never heard the word before because to know the meaning of a name is to know who it is applied to.

You sometimes find people speaking as if descriptive phrases were names, and you will find it suggested, e.g., that such a proposition as “Scott is the author of Waverley” really asserts that “Scott” and “the author of Waverley” are two names for the same person. That is an entire delusion; first of all, because “the author of Waverley” is not a name, and, secondly, because, as you can perfectly well see, if that were what is meant, the proposition would be one like “Scott is Sir Walter,” and would not depend upon any fact except that the person in question was so called, because a name is what a man is called. As a matter of fact, Scott was the author of Waverley at a time when no one called him so, when no one knew whether he was or not, and the fact that he was the author was a physical fact, the fact that he sat down and wrote it with his own hand, which does not have anything to do with what he was called. It is in no way arbitrary. You cannot settle by any choice of nomenclature whether he is or is not to be the author of Waverley, because in actual fact he chose to write it and you cannot help yourself.

That illustrates how “the author of Waverley” is quite a different thing from a name. You can prove this point very clearly by formal arguments. In “Scott is the author of Waverley” the “is,” of course, expresses identity, i.e., the entity whose name is Scott is identical with the author of Waverley. But, when I say “Scott is mortal” this “is” is the “is” of predication, which is quite different from the “is” of identity. It is a mistake to interpret “Scott is mortal” as meaning “Scott is identical with one among mortals,” because (among other reasons) you will not be able to say what “mortals” are except by means of the propositional function “x is mortal,” which brings back the “is” of predication. You cannot reduce the “is” of predication to the other “is.” But the “is” in “Scott is the author of Waverley” is the “is” of identity and not of predication.1

If you were to try to substitute for “the author of Waverley” in that proposition any name whatever, say “c,” so that the proposition becomes “Scott is c,” then if “c” is a name for anybody who is not Scott, that proposition would become false, while if, on the other hand, “c” is a name for Scott, then the proposition will become simply a tautology. It is at once obvious that if “c” were “Scott” itself, “Scott is Scott” is just a tautology. But if you take any other name which is just a name for Scott, then if the name is being used as a name and not as a description, the proposition will still be a tautology. For the name itself is merely a means of pointing to the thing, and does not occur in what you are asserting, so that if one thing has two names, you make exactly the same assertion whichever of the two names you use, provided they are really names and not truncated descriptions.

So there are only two alternatives. If “c” is a name, the proposition “Scott is c” is either false or tautologous. But the proposition “Scott is the author of Waverley” is neither, and therefore is not the same as any proposition of the form “Scott is c,” where “c” is a name. That is another way of illustrating the fact that a description is quite a different thing from a name.

I should like to make clear what I was saying just now, that if you substitute another name in place of “Scott” which is also a name of the same individual, say, “Scott is Sir Walter,” then “Scott” and “Sir Walter” are being used as names and not as descriptions, your proposition is strictly a tautology. If one asserts “Scott is Sir Walter,” the way one would mean it would be that one was using the names as descriptions. One would mean that the person called “Scott” is the person called “Sir Walter,” and “the person called ‘Scott’ ” is a description, and so is “the person called ‘Sir Walter.’ ” So that would not be a tautology. It would mean that the person called “Scott” is identical with the person called “Sir Walter.” But if you are using both as names, the matter is quite different. You must observe that the name does not occur in that which you assert when you use the name. The name is merely that which is a means of expressing what it is you are trying to assert, and when I say “Scott wrote Waverley” the name “Scott” does not occur in the thing I am asserting. The thing I am asserting is about the person, not about the name. So if I say “Scott is Sir Walter,” using these two names as names, neither “Scott” nor “Sir Walter” occurs in what I am asserting, but only the person who has these names, and thus what I am asserting is a pure tautology.

It is rather important to realize this about the two different uses of names or of any other symbols: the one when you are talking about the symbol and the other when you are using it as a symbol, as a means of talking about something else. Normally, if you talk about your dinner, you are not talking about the word “dinner” but about what you are going to eat, and that is a different thing altogether. The ordinary use of words is as a means of getting through to things, and when you are using words in that way the statement “Scott is Sir Walter” is a pure tautology, exactly on the same level as “Scott is Scott.”

That brings me back to the point that when you take “Scott is the author of Waverley” and you substitute for “the author of Waverley” a name in the place of a description, you get necessarily either a tautology or a falsehood – a tautology if you substitute “Scott” or some other name for the same person, and a falsehood if you substitute anything else. But the proposition itself is neither a tautology nor a falsehood, and that shows you that the proposition “Scott is the author of Waverley” is a different proposition from any that can be obtained if you substitute a name in the place of “the author of Waverley.” That conclusion is equally true of any other proposition in which the phrase “the author of Waverley” occurs. If you take any proposition in which that phrase occurs and substitute for that phrase a proper name, whether that name be “Scott” or any other, you will get a different proposition. Generally speaking, if the name that you substitute is “Scott,” your proposition, if it was true before will remain true, and if it was false before will remain false. But it is a different proposition. It is not always true that it will remain true or false, as may be seen by the example: “George IV wished to know if Scott was the author of Waverley.” It is not true that George IV wished to know if Scott was Scott. So it is even the case that the truth or the falsehood of a proposition is sometimes changed when you substitute a name of an object for a description of the same object. But in any case it is always a different proposition when you substitute a name for a description.

Identity is a rather puzzling thing at first sight. When you say “Scott is the author of Waverley,” you are half-tempted to think there are two people, one of whom is Scott and the other the author of Waverley, and they happen to be the same. That is obviously absurd, but that is the sort of way one is always tempted to deal with identity.

When I say “Scott is the author of Waverley” and that “is” expresses identity, the reason that identity can be asserted there truly and without tautology is just the fact that the one is a name and the other a description. Or they might both be descriptions. If I say “The author of Waverley is the author of Marmion,” that, of course, asserts identity between two descriptions.

Now the next point that I want to make clear is that when a description (when I say “description” I mean, for the future, a definite description) occurs in a proposition, there is no constituent of that proposition corresponding to that description as a whole. In the true analysis of the proposition, the description is broken up and disappears. That is to say, when I say “Scott is the author of Waverley” it is a wrong analysis of that to suppose that you have there three constituents, “Scott,” “is,” and “the author of Waverley.” That, of course, is the sort of way you might think of analyzing. You might admit that “the author of Waverley” was complex and could be further cut up, but you might think the proposition could be split into those three bits to begin with. That is an entire mistake. “The author of Waverley” is not a constituent of the proposition at all. There is no constituent really there corresponding to the descriptive phrase. I will try to prove that to you now.

The first and most obvious reason is that you can have significant propositions denying the existence of “the so-and-so.” “The unicorn does not exist.” “The greatest finite number does not exist.” Propositions of that sort are perfectly significant, are perfectly sober, true, decent propositions, and that could not possibly be the case if the unicorn were a constituent of the proposition, because plainly it could not be a constituent as long as there were not any unicorns. Because the constituents of propositions, of course, are the same as the constituents of the corresponding facts, and since it is a fact that the unicorn does not exist, it is perfectly clear that the unicorn is not a constituent of that fact, because if there were any fact of which the unicorn was a constituent, there would be a unicorn, and it would not be true that it did not exist. That applies in this case of descriptions particularly. Now since it is possible for “the so-and-so” not to exist and yet for propositions in which “the so-and-so” occurs to be significant and even true, we must try to see what is meant by saying that the so-and-so does exist.

The occurrence of tense in verbs is an exceedingly annoying vulgarity due to our preoccupation with practical affairs. It would be much more agreeable if they had no tense, as I believe is the case in Chinese, but I do not know Chinese. You ought to be able to say “Socrates exists in the past,” “Socrates exists in the present,” or “Socrates exists in the future,” or simply “Socrates exists,” without any implication of tense, but language does not allow that, unfortunately. Nevertheless, I am going to use language in this tenseless way: when I say “The so-and-so exists,” I am not going to mean that it exists in the present or in the past or in the future, but simply that it exists, without implying anything involving tense.

“The author of Waverley exists”: there are two things required for that. First of all, what is “the author of Waverley”? It is the person who wrote Waverley, i.e., we are coming now to this, that you have a propositional function involved, viz., “x writes Waverley,” and the author of Waverley is the person who writes Waverley, and in order that the person who writes Waverley may exist, it is necessary that this propositional function should have two properties:

1. It must be true for at least one x.
2. It must be true for at most one x.

If nobody had ever written Waverley the author could not exist, and if two people had written it, the author could not exist. So you want these two properties, the one that it is true for at least one x, and the other that it is true for at most one x, both of which are required for existence.

The property of being true for at least one x is the one we dealt with last time: what I expressed by saying that the propositional function is possible. Then we come on to the second condition, that it is true for at most one x, and that you can express in this way: “If x and y wrote Waverley, then x is identical with y, whatever x and y may be.” That says that at most one person wrote it. It does not say that anybody wrote Waverley at all, because if nobody had written it, that statement would still be true. It only says that at most one person wrote it.

The first of these conditions for existence fails in the case of the unicorn, and the second in the case of the inhabitant of London.

We can put these two conditions together and get a portmanteau expression including the meaning of both. You can reduce them both down to this, that: “(‘x wrote Waverley’ is equivalent to ‘x is c’ whatever x may be) is possible in respect of c.” That is as simple, I think, as you can make the statement.

You see that means to say that there is some entity c, we may not know what it is, which is such that when x is c, it is true that x wrote Waverley, and when x is not c, it is not true that x wrote Waverley, which amounts to saying that c is the only person who wrote Waverley; and I say there is a value of c which makes that true. So that this whole expression, which is a propositional function about c, is possible in respect of c (in the sense explained last time).

That is what I mean when I say that the author of Waverley exists. When I say “the author of Waverley exists,” I mean that there is an entity c such that “x wrote Waverley” is true when x is c, and is false when x is not c. “The author of Waverley” as a constituent has quite disappeared there, so that when I say “The author of Waverley exists” I am not saying anything about the author of Waverley. You have instead this elaborate to-do with propositional functions, and “the author of Waverley” has disappeared. That is why it is possible to say significantly “The author of Waverley did not exist.” It would not be possible if “the author of Waverley” were a constituent of propositions in whose verbal expression this descriptive phrase occurs.

The fact that you can discuss the proposition “God exists” is a proof that “God,” as used in that proposition, is a description and not a name. If “God” were a name, no question as to existence could arise.

I have now defined what I mean by saying that a thing described exists. I have still to explain what I mean by saying that a thing described has a certain property. Supposing you want to say “The author of Waverley was human.” That will be represented thus: “(‘x wrote Waverley’ is equivalent to ‘x is c’ whatever x may be, and c is human) is possible with respect to c.”

You will observe that what we gave before as the meaning of “The author of Waverley exists” is part of this proposition. It is part of any proposition in which “the author of Waverley” has what I call a “primary occurrence.” When I speak of a “primary occurrence” I mean that you are not having a proposition about the author of Waverley occurring as a part of some larger proposition, such as “I believe that the author of Waverley was human” or “I believe that the author of Waverley exists.” When it is a primary occurrence, i.e., when the proposition concerning it is not just part of a larger proposition, the phrase which we defined as the meaning of “The author of Waverley exists” will be part of that proposition. If I say the author of Waverley was human, or a poet, or a Scotsman, or whatever I say about the author of Waverley in the way of a primary occurrence, always this statement of his existence is part of the proposition. In that sense all these propositions that I make about the author of Waverley imply that the author of Waverley exists, so that any statement in which a description has a primary occurrence implies that the object described exists. If I say “The present King of France is bald,” that implies that the present King of France exists. If I say, “The present King of France has a fine head of hair,” that also implies that the present King of France exists. Therefore unless you understand how a proposition containing a description is to be denied, you will come to the conclusion that it is not true either that the present King of France is bald or that he is not bald, because if you were to enumerate all the things that are bald you would not find him there, and if you were to enumerate all the things that are not bald, you would not find him there either. The only suggestion I have found for dealing with that on conventional lines is to suppose that he wears a wig. You can only avoid the hypothesis that he wears a wig by observing that the denial of the proposition “The present King of France is bald” will not be “The present King of France is not bald,” if you mean by that “There is such a person as the King of France and that person is not bald.” The reason of this is that when you state that the present King of France is bald you say “There is a c such that c is now King of France and c is bald” and the denial is not “There is a c such that c is now King of France and c is not bald.” It is more complicated. It is: “Either there is not a c such that c is now King of France, or, if there is such a c, then c is not bald.” Therefore you see that, if you want to deny the proposition “The present King of France is bald,” you can do it by denying that he exists, instead of by denying that he is bald. In order to deny this statement that the present King of France is bald, which is a statement consisting of two parts, you can proceed by denying either part. You can deny the one part, which would lead you to suppose that the present King of France exists but is not bald, or the other part, which will lead you to the denial that the present King of France exists; and either of those two denials will lead you to the falsehood of the proposition “The present King of France is bald.” When you say “Scott is human” there is no possibility of a double denial. The only way you can deny “Scott is human” is by saying “Scott is not human.” But where a descriptive phrase occurs, you do have the double possibility of denial.

It is of the utmost importance to realize that “the so-and-so” does not occur in the analysis of propositions in whose verbal expression it occurs, that when I say “The author of Waverley is human,” “the author of Waverley” is not the subject of that proposition, in the sort of way that Scott would be if I said “Scott is human,” using “Scott” as a name. I cannot emphasize sufficiently how important this point is, and how much error you get in metaphysics if you do not realize that when I say “The author of Waverley is human” that is not a proposition of the same form as “Scott is human.” It does not contain a constituent “the author of Waverley.” The importance of that is very great for many reasons, and one of them is this question of existence. As I pointed out to you last time, there is a vast amount of philosophy that rests upon the notion that existence is, so to speak, a property that you can attribute to things, and that the things that exist have the property of existence and the things that do not exist do not. That is rubbish, whether you take kinds of things, or individual things described. When I say, e.g., “Homer existed,” I am meaning by “Homer” some description, say “the author of the Homeric poems,” and I am asserting that those poems were written by one man, which is a very doubtful proposition; but if you could get hold of the actual person who did actually write those poems (supposing there was such a person), to say of him that he existed would be uttering nonsense, not a falsehood but nonsense, because it is only of persons described that it can be significantly said that they exist. Last time I pointed out the fallacy in saying “Men exist, Socrates is a man, therefore Socrates exists.” When I say “Homer exists, this is Homer, therefore this exists,” that is a fallacy of the same sort. It is an entire mistake to argue: “This is the author of the Homeric poems and the author of the Homeric poems exists, therefore this exists.” It is only where a propositional function comes in that existence may be significantly asserted. You can assert “The so-and-so exists,” meaning that there is just one c which has those properties, but when you get hold of a c that has them, you cannot say of this c that it exists, because that is nonsense: it is not false, but it has no meaning at all.

So the individuals that there are in the world do not exist, or rather it is nonsense to say that they exist and nonsense to say that they do not exist. It is not a thing you can say when you have named them, but only when you have described them. When you say “Homer exists,” you mean “Homer” is a description which applies to something. A description when it is fully stated is always of the form “the so-and-so.”

The sort of things that are like these descriptions in that they occur in words in a proposition, but are not in actual fact constituents of the proposition rightly analyzed, things of that sort I call “incomplete symbols.” There are a great many sorts of incomplete symbols in logic, and they are sources of a great deal of confusion and false philosophy, because people get misled by grammar. You think that the proposition “Scott is mortal” and the proposition “The author of Waverley is mortal” are of the same form. You think that they are both simple propositions attributing a predicate to a subject. That is an entire delusion: one of them is (or rather might be) and one of them is not. These things, like “the author of Waverley” which I call incomplete symbols, are things that have absolutely no meaning whatsoever in isolation but merely acquire a meaning in a context. “Scott” taken as a name has a meaning all by itself. It stands for a certain person, and there it is. But “the author of Waverley” is not a name, and does not all by itself mean anything at all, because when it is rightly used in propositions, those propositions do not contain any constituent corresponding to it.

There are a great many other sorts of incomplete symbols besides descriptions. These are classes, which I shall speak of next time, and relations taken in extension, and so on. Such aggregations of symbols are really the same thing as what I call “logical fictions,” and they embrace practically all the familiar objects of daily life: tables, chairs, Piccadilly, Socrates, and so on. Most of them are either classes, or series, or series of classes. In any case they are all incomplete symbols, i.e., they are aggregations that only have a meaning in use and do not have any meaning in themselves.

It is important, if you want to understand the analysis of the world, or the analysis of facts, or if you want to have any idea what there really is in the world, to realize how much of what there is in phraseology is of the nature of incomplete symbols. You can see that very easily in the case of “the author of Waverley” because “the author of Waverley” does not stand simply for Scott, nor for anything else. If it stood for Scott, “Scott is the author of Waverley” would be the same proposition as “Scott is Scott,” which it is not, since George IV wished to know the truth of the one and did not wish to know the truth of the other. If “the author of Waverley” stood for anything other than Scott, “Scott is the author of Waverley” would be false, which it is not. Hence you have to conclude that “the author of Waverley” does not, in isolation, really stand for anything at all; and that is the characteristic of incomplete symbols.

[to be concluded]

Bertrand Russell
London, England

*  Bertrand Russell, The Philosophy of Logical Atomism, Lecture 6, “Descriptions and Incomplete Symbols,” The Monist 29 (Apr 1919), 206-22

1  The confusion of these two meanings of “is” is essential to the Hegelian conception of identity-in-difference