THE THEORY OF TYPES AND SYMBOLISM: CLASSES

Before I begin today the main subject of my lecture, I should like to make a few remarks in explanation and amplification of what I have said about existence in my previous two lectures. This is chiefly on account of a letter I have received from a member of the class, raising many points which, I think, were present in other minds too.

The first point I wish to clear up is this: I did not mean to say that when one says a thing exists, one means the same as when one says it is possible. What I meant was, that the fundamental logical idea, the primitive idea, out of which both those are derived is the same. That is not quite the same thing as to say that the statement that a thing exists is the same as the statement that it is possible, which I do not hold. I used the word “possible” in perhaps a somewhat strange sense, because I wanted some word for a fundamental logical idea for which no word exists in ordinary language, and therefore if one is to try to express in ordinary language the idea in question, one has to take some word and make it convey the sense that I was giving to the word

“possible”, which is by no means the only sense that it has but is a sense that was convenient for my purpose. We say of a propositional function that it is possible, where there are cases in which it is true. That is not exactly the same thing as what one ordinarily means, for instance, when one says that it is possible it may rain tomorrow. But what I contend is, that the ordinary uses of the word “possible” are derived from this notion by a process.

E.g. normally when you say of a proposition that it is possible, you mean something like this: first of all it is implied that you do not know whether it is true or false; and I think it is implied, secondly, that it is one of a class of propositions, some of which are known to be true. When I say, e.g., “It is possible that it may rain tomorrow”—“It will rain tomorrow” is one of the class of propositions “It rains at time t”, where t is different times. We mean partly that we do not know whether it will rain or whether it will not, but also that we do know that that is the sort of proposition that is quite apt to be true, that it is a value of a propositional function of which we know some value to be true.

Many of the ordinary uses of “possible” come under that head, I think you will find. That is to say, that if you say of a proposition that it is possible, what you have is this: “There is in this prop-osition some constituent, which, if you turn it into a variable, will give you a propositional function that is sometimes true.”

You ought not therefore to say of a proposition simply that it is possible, but rather that it is possible in respect of such-and-such a constituent. That would be a more full expression.

When I say, for instance, that “Lions exist”, I do not mean the same as if I said that lions were possible; because when you say

“Lions exist”, that means that the propositional function “x is a lion” is a possible one in the sense that there are lions, while when you say “Lions are possible” that is a different sort of statement altogether, not meaning that a casual individual ani-mal may be a lion, but rather that a sort of aniani-mal may be the sort that we call “lions”. If you say “Unicorns are possible”, e.g., you

would mean that you do not know any reason why there should not be unicorns, which is quite a different proposition from

“Unicorns exist”. As to what you would mean by saying that unicorns are possible, it would always come down to the same thing as “It is possible it may rain tomorrow”. You would mean, the proposition “There are unicorns” is one of a certain set of propositions some of which are known to be true, and that the description of the unicorn does not contain in it anything that shows there could not be such beasts.

When I say a propositional function is possible, meaning there are cases in which it is true, I am consciously using the word “possible” in an unusual sense, because I want a single word for my fundamental idea, and cannot find any word in ordinary language that expresses what I mean.

Secondly, it is suggested that when one says a thing exists, it means that it is in time, or in time and space, at any rate in time.

That is a very common suggestion, but I do not think that really there is much to be said for that use of words; in the first place, because if that were all you meant, there would be no need for a separate word. In the second place, because after all in the sense, whatever that sense may be, in which the things are said to exist that one ordinarily regards as existing, one may very well wish to discuss the question whether there are things that exist with-out being in time. Orthodox metaphysics holds that whatever is really real is not in time, that to be in time is to be more or less unreal, and that what really exists is not in time at all. And orthodox theology holds that God is not in time. I see no reason why you should frame your definition of existence in such a way as to preclude that notion of existence. I am inclined to think that there are things that are not in time, and I should be sorry to use the word existence in that sense when you have already the phrase “being in time” which quite sufficiently expresses what you mean.

Another objection to that definition is that it does not in the

least fit the sort of use of “existence” which was underlying my discussion, which is the common one in mathematics. When you take existence-theorems, for instance, as when you say “An even prime exists”, you do not mean that the number two is in time but that you can find a number of which you can say “This is even and prime”. One does ordinarily in mathematics speak of propositions of that sort as existence-theorems, i.e. you establish that there is an object of such-and-such a sort, that object being, of course, in mathematics a logical object, not a particular, not a thing like a lion or a unicorn, but an object like a function or a number, something which plainly does not have the property of being in time at all, and it is that sort of sense of existence-theorems that is relevant in discussing the meaning of existence as I was doing in the last two lectures. I do, of course, hold that that sense of existence can be carried on to cover the more ordinary uses of existence, and does in fact give the key to what is underlying those ordinary uses, as when one says that “Homer existed” or “Romulus did not exist”, or whatever we may say of that kind.

I come now to a third suggestion about existence, which is also a not uncommon one, that of a given particular “this” you can say “This exists” in the sense that it is not a phantom or an image or a universal. Now I think that use of existence involves confusions which it is exceedingly important to get out of one’s mind, really rather dangerous mistakes. In the first place, we must separate phantoms and images from universals; they are on a different level. Phantoms and images do undoubtedly exist in that sense, whatever it is, in which ordinary objects exist. I mean, if you shut your eyes and imagine some visual scene, the images that are before your mind while you are imagining are undoubt-edly there. They are images, something is happening, and what is happening is that the images are before your mind, and these images are just as much part of the world as tables and chairs and anything else. They are perfectly decent objects, and you

only call them unreal (if you call them so), or treat them as non-existent, because they do not have the usual sort of relations to other objects. If you shut your eyes and imagine a visual scene and you stretch out your hand to touch what is imaged, you won’t get a tactile sensation, or even necessarily a tactile image.

You will not get the usual correlation of sight and touch. If you imagine a heavy oak table, you can remove it without any muscular effort, which is not the case with oak tables that you actually see. The general correlations of your images are quite different from the correlations of what one chooses to call “real”

objects. But that is not to say images are unreal. It is only to say they are not part of physics. Of course, I know that this belief in the physical world has established a sort of reign of terror. You have got to treat with disrespect whatever does not fit into the physical world. But that is really very unfair to the things that do not fit in. They are just as much there as the things that do. The physical world is a sort of governing aristocracy, which has somehow managed to cause everything else to be treated with disrespect. That sort of attitude is unworthy of a philosopher. We should treat with exactly equal respect the things that do not fit in with the physical world, and images are among them.

“Phantoms”, I suppose, are intended to differ from “images”

by being of the nature of hallucinations, things that are not merely imagined but that go with belief. They again are perfectly real; the only odd thing about them is their correlations.

Macbeth sees a dagger. If he tried to touch it, he would not get any tactile sensation, but that does not imply that he was not seeing a dagger, it only implies that he was not touching it. It does not in any way imply that the visual sensation was not there. It only means to say that the sort of correlation between sight and touch that we are used to is the normal rule but not a universal one. In order to pretend that it is universal, we say that a thing is unreal when it does not fit in. You say, “Any man who is a man will do such-and-such a thing.” You then find a man who will

not, and you say, he is not a man. That is just the same sort of thing as with these daggers that you cannot touch.

I have explained elsewhere the sense in which phantoms are unreal.⁸ When you see a “real” man, the immediate object that you see is one of a whole system of particulars, all of which belong together and make up collectively the various “appear-ances” of the man to himself and others. On the other hand, when you see a phantom of a man, that is an isolated particular, not fitting into a system as does a particular which one calls an appearance of the “real” man. The phantom is in itself just as much part of the world as the normal sense-datum, but it lacks the usual correlation and therefore gives rise to false inferences and becomes deceptive.

As to universals, when I say of a particular that it exists, I certainly do not mean the same thing as if I were to say that it is not a universal. The statement concerning any particular that it is not a universal is quite strictly nonsense—not false, but strictly and exactly nonsense. You never can place a particular in the sort of place where a universal ought to be, and vice versa. If I say “a is not b”, or if I say “a is b”, that implies that a and b are of the same logical type. When I say of a universal that it exists, I should be meaning it in a different sense from that in which one says that particulars exist. E.g. you might say, “Colours exist in the spectrum between blue and yellow.” That would be a perfectly respectable statement, the colours being taken as universals. You mean simply that the propositional function “x is a colour between blue and yellow” is one which is capable of truth. But the x which occurs there is not a particular, it is a universal.

So that you arrive at the fact that the ultimate important notion involved in existence is the notion that I developed in the lecture before last, the notion of a propositional function being

8See Our Knowledge of the External World, Chap. III. Also Section XII of “Sense-Data and Physics” in Mysticism and Logic.

sometimes true, or being, in other words, possible. The distinc-tion between what some people would call real existence, and existence in people’s imagination or in my subjective activity, that distinction, as we have just seen, is entirely one of correl-ation. I mean that anything which appears to you, you will be mistakenly inclined to say has some more glorified form of existence if it is associated with those other things I was talking of in the way that the appearance of Socrates to you would be associated with his appearance to other people. You would say he was only in your imagination if there were not those other correlated appearances that you would naturally expect. But that does not mean that the appearance to you is not exactly as much a part of the world as if there were other correlated appearances.

It will be exactly as much a part of the real world, only it will fail to have the correlations that you expect. That applies to the question of sensation and imagination. Things imagined do not have the same sort of correlations as things sensated. If you care to see more about this question, I wrote a discussion in The Monist for January 1915, and if any of you are interested, you will find the discussion there.

I come now to the proper subject of my lecture, but shall have to deal with it rather hastily. It was to explain the theory of types and the definition of classes. Now first of all, as I suppose most of you are aware, if you proceed carelessly with formal logic, you can very easily get into contradictions. Many of them have been known for a long time, some even since the time of the Greeks, but it is only fairly recently that it has been discovered that they bear upon mathematics, and that the ordinary mathematician is liable to fall into them when he approaches the realms of logic, unless he is very cautious. Unfortunately the mathematical ones are more difficult to expound, and the ones easy to expound strike one as mere puzzles or tricks.

You can start with the question whether or not there is a greatest cardinal number. Every class of things that you can

choose to mention has some cardinal number. That follows very easily from the definition of cardinal numbers as classes of similar classes, and you would be inclined to suppose that the class of all the things there are in the world would have about as many members as a class could be reasonably expected to have.

The plain man would suppose you could not get a larger class than the class of all the things there are in the world. On the other hand, it is very easy to prove that if you take selections of some of the members of a class, making those selections in every conceivable way that you can, the number of different selections that you can make is greater than the original numbers of terms.

That is easy to see with small numbers. Suppose you have a class with just three members, a, b, c. The first selection that you can make is the selection of no terms. The next of a alone, b alone, c alone. Then bc, ca, ab, abc, which makes in all 8 (i.e. 2³) selections.

Generally speaking, if you have n terms, you can make 2ⁿ selec-tions. It is very easy to prove that 2ⁿ is always greater than n, whether n happens to be finite or not. So you find that the total number of things in the world is not so great as the number of classes that can be made up out of those things. I am asking you to take all these propositions for granted, because there is not time to go into the proofs, but they are all in Cantor’s work.

Therefore you will find that the total number of things in the world is by no means the greatest number. On the contrary, there is a hierarchy of numbers greater than that. That, on the face of it, seems to land you in a contradiction. You have, in fact, a perfectly precise arithmetical proof that there are fewer things in heaven or earth than are dreamt of in our philosophy. That shows how philosophy advances.

You are met with the necessity, therefore, of distinguishing between classes and particulars. You are met with the necessity of saying that a class consisting of two particulars is not itself in turn a fresh particular, and that has to be expanded in all sorts of ways; i.e. you will have to say that in the sense in which there

are particulars, in that sense it is not true to say there are classes.

The sense in which there are classes is a different one from the sense in which there are particulars, because if the senses of the two were exactly the same, a world in which there are three particulars and therefore eight classes, would be a world in which there are at least eleven things. As the Chinese phil-osopher pointed out long ago, a dun cow and a bay horse make three things: separately they are each one, and taken together they are another, and therefore three.

I pass now to the contradiction about classes that are not members of themselves. You would say generally that you would not expect a class to be a member of itself. For instance, if you take the class of all the teaspoons in the world, that is not in itself a teaspoon. Or if you take all the human beings in the world, the

I pass now to the contradiction about classes that are not members of themselves. You would say generally that you would not expect a class to be a member of itself. For instance, if you take the class of all the teaspoons in the world, that is not in itself a teaspoon. Or if you take all the human beings in the world, the