GENERAL PROPOSITIONS AND EXISTENCE

I am going to speak today about general propositions and existence. The two subjects really belong together; they are the same topic, although it might not have seemed so at the first glance. The propositions and facts that I have been talking about hitherto have all been such as involved only perfectly definite particulars, or relations, or qualities, or things of that sort, never involved the sort of indefinite things one alludes to by such

words as “all”, “some”, “a”, “any”, and it is propositions and facts of that sort that I am coming on to today.

Really all the propositions of that sort that I mean to talk of today collect themselves into two groups—the first that are about

“all”, and the second that are about “some”. These two sorts belong together; they are each other’s negations. If you say, for instance, “All men are mortal”, that is the negative of “Some men are not mortal”. In regard to general propositions, the distinction of affirmative and negative is arbitrary. Whether you are going to regard the propositions about “all” as the affirma-tive ones and the propositions about “some” as the negaaffirma-tive ones, or vice versa, is purely a matter of taste. For example, if I say “I met no one as I came along”, that, on the face of it, you would think is a negative proposition. Of course, that is really a proposition about “all”, i.e. “All men are among those whom I did not meet”. If, on the other hand, I say “I met a man as I came along”, that would strike you as affirmative, whereas it is the negative of “All men are among those I did not meet as I came along”. If you consider such propositions as “All men are mortal” and “Some men are not mortal”, you might say it was more natural to take the general propositions as the affirmative and the existence-propositions as the negative, but, simply because it is quite arbitrary which one is to choose, it is better to forget these words and to speak only of general propositions and propositions asserting existence. All general propositions deny the existence of something or other. If you say “All men are mortal”, that denies the existence of an immortal man, and so on.

I want to say emphatically that general propositions are to be interpreted as not involving existence. When I say, for instance,

“All Greeks are men”, I do not want you to suppose that that implies that there are Greeks. It is to be considered emphatic-ally as not implying that. That would have to be added as a separate proposition. If you want to interpret it in that sense,

you will have to add the further statement “and there are Greeks”. That is for purposes of practical convenience. If you include the fact that there are Greeks, you are rolling two propositions into one, and it causes unnecessary confusion in your logic, because the sorts of propositions that you want are those that do assert the existence of something and general propositions which do not assert existence. If it happened that there were no Greeks, both the proposition that “All Greeks are men” and the proposition that “No Greeks are men” would be true. The proposition “No Greeks are men” is, of course, the proposition “All Greeks are not-men”. Both propositions will be true simultaneously if it happens that there are no Greeks.

All statements about all the members of a class that has no members are true, because the contradictory of any general statement does assert existence and is therefore false in this case. This notion, of course, of general propositions not involv-ing existence is one which is not in the traditional doctrine of the syllogism. In the traditional doctrine of the syllogism, it was assumed that when you have such a statement as “All Greeks are men”, that implies that there are Greeks, and this produced fallacies. For instance, “All chimeras are animals, and all chimeras breathe flame, therefore some animals breathe flame.” This is a syllogism in Darapti, but that mood of the syllogism is fallacious, as this instance shows. That was a point, by the way, which had a certain historical interest, because it impeded Leibniz in his attempts to construct a mathematical logic. He was always engaged in trying to construct such a mathematical logic as we have now, or rather such a one as Boole constructed, and he was always failing because of his respect for Aristotle. Whenever he invented a really good system, as he did several times, it always brought out that such moods as Darapti are fallacious. If you say “All A is B and all A is C, there-fore some B is C”—if you say this you incur a fallacy, but he could not bring himself to believe that it was fallacious, so

he began again. That shows you that you should not have too much respect for distinguished men.⁶

Now when you come to ask what really is asserted in a general proposition, such as “All Greeks are men” for instance, you find that what is asserted is the truth of all values of what I call a propositional function. A propositional function is simply any expression containing an undetermined constituent, or several undetermined constituents, and becoming a proposition as soon as the undetermined constituents are determined. If I say “x is a man” or “n is a number”, that is a propositional function; so is any formula of algebra, say (x+y)(x−y) = x²−y². A propositional function is nothing, but, like most of the things one wants to talk about in logic, it does not lose its importance through that fact. The only thing really that you can do with a propositional function is to assert either that it is always true, or that it is sometimes true, or that it is never true. If you take:

“If x is a man, x is mortal”,

that is always true (just as much when x is not a man as when x is a man); if you take:

“x is a man”,

that is sometimes true; if you take:

“x is a unicorn”, that is never true.

One may call a propositional function necessary, when it is always true;

possible, when it is sometimes true;

6Cf. Couturat, La logique de Leibniz.

impossible, when it is never true.

Much false philosophy has arisen out of confusing propositional functions and propositions. There is a great deal in ordinary traditional philosophy which consists simply in attributing to propositions the predicates which only apply to propositional functions, and, still worse, sometimes in attributing to indi-viduals predicates which merely apply to propositional functions.

This case of necessary, possible, impossible, is a case in point. In all traditional philosophy there comes a heading of “modality”, which discusses necessary, possible, and impossible as properties of propositions, whereas in fact they are properties of propositional functions. Propositions are only true or false.

If you take “x is x”, that is a propositional function which is true whatever “x” may be, i.e. a necessary propositional function. If you take “x is a man”, that is a possible one. If you take “x is a unicorn”, that is an impossible one.

Propositions can only be true or false, but propositional functions have these three possibilities. It is important, I think, to realize that the whole doctrine of modality only applies to propositional functions, not to propositions.

Propositional functions are involved in ordinary language in a great many cases where one does not usually realize them. In such a statement as “I met a man”, you can understand my statement perfectly well without knowing whom I met, and the actual person is not a constituent of the proposition. You are really asserting there that a certain propositional function is sometimes true, namely the propositional function “I met x and x is human”. There is at least one value of x for which that is true, and that therefore is a possible propositional function. Whenever you get such words as “a”, “some”, “all”, “every”, it is always a mark of the presence of a propositional function, so that these things are not, so to speak, remote or recondite: they are obvious and familiar.

A propositional function comes in again in such a statement as

“Socrates is mortal”, because “to be mortal” means “to die at some time or other”. You mean there is a time at which Socrates dies, and that again involves a propositional function, namely that “t is a time, and Socrates dies at t” is possible. If you say

“Socrates is immortal”, that also will involve a propositional function. That means that “If t is any time whatever, Socrates is alive at time t”, if we take immortality as involving existence throughout the whole of the past as well as throughout the whole of the future. But if we take immortality as only involving existence throughout the whole of the future, the interpretation of “Socrates is immortal” becomes more complete, viz., “There is a time t, such that if t′ is any time later than t, Socrates is alive at t′.” Thus when you come to write out properly what one means by a great many ordinary statements, it turns out a little complicated. “Socrates is mortal” and “Socrates is immortal” are not each other’s contradictories, because they both imply that Socrates exists in time, otherwise he would not be either mortal or immortal. One says, “There is a time at which he dies”, and the other says, “Whatever time you take, he is alive at that time”, whereas the contradictory of “Socrates is mortal” would be true if there is not a time at which he lives.

An undetermined constituent in a propositional function is called a variable.

Existence. When you take any propositional function and assert of it that it is possible, that it is sometimes true, that gives you the fundamental meaning of “existence”. You may express it by saying that there is at least one value of x for which that prop-ositional function is true. Take “x is a man”, there is at least one value of x for which this is true. That is what one means by saying that “There are men”, or that “Men exist”. Existence is essentially a property of a propositional function. It means that that propositional function is true in at least one instance. If you say “There are unicorns”, that will mean that “There is an x, such

that x is a unicorn”. That is written in phrasing which is unduly approximated to ordinary language, but the proper way to put it would be “(x is a unicorn) is possible”. We have got to have some idea that we do not define, and one takes the idea of

“always true”, or of “sometimes true”, as one’s undefined idea in this matter, and then you can define the other one as the negative of that. In some ways it is better to take them both as undefined, for reasons which I shall not go into at present. It will be out of this notion of sometimes, which is the same as the notion of possible, that we get the notion of existence. To say that unicorns exist is simply to say that “(x is a unicorn) is possible”.

It is perfectly clear that when you say “Unicorns exist”, you are not saying anything that would apply to any unicorns there might happen to be, because as a matter of fact there are not any, and therefore if what you say had any application to the actual individuals, it could not possibly be significant unless it were true. You can consider the proposition “Unicorns exist” and can see that it is false. It is not nonsense. Of course, if the proposition went through the general conception of the unicorn to the individual, it could not be even significant unless there were unicorns. Therefore when you say “Unicorns exist”, you are not saying anything about any individual things, and the same applies when you say “Men exist”. If you say that “Men exist, and Socrates is a man, therefore Socrates exists”, that is exactly the same sort of fallacy as it would be if you said “Men are numerous, Socrates is a man, therefore Socrates is numerous”, because existence is a predicate of a propositional function, or derivatively of a class. When you say of a propositional function that it is numerous, you will mean that there are several values of x that will satisfy it, that there are more than one; or, if you like to take “numerous” in a larger sense, more than ten, more than twenty, or whatever number you think fitting. If x, y, and z all satisfy a propositional function, you may say that that prop-osition is numerous, but x, y, and z severally are not numerous.

Exactly the same applies to existence, that is to say that the actual things that there are in the world do not exist, or, at least, that is putting it too strongly, because that is utter nonsense. To say that they do not exist is strictly nonsense, but to say that they do exist is also strictly nonsense.

It is of propositional functions that you can assert or deny existence. You must not run away with the idea that this entails consequences that it does not entail. If I say “The things that there are in the world exist”, that is a perfectly correct statement, because I am there saying something about a certain class of things; I say it in the same sense in which I say “Men exist”. But I must go on to “This is a thing in the world, and therefore this exists”. It is there the fallacy comes in, and it is simply, as you see, a fallacy of transferring to the individual that satisfies a propositional function a predicate which only applies to a prop-ositional function. You can see this in various ways. For instance, you sometimes know the truth of an existence-proposition without knowing any instance of it. You know that there are people in Timbuctoo, but I doubt if any of you could give me an instance of one. Therefore you clearly can know existence-propositions without knowing any individual that makes them true. Existence-propositions do not say anything about the actual individual but only about the class or function.

It is exceedingly difficult to make this point clear as long as one adheres to ordinary language, because ordinary language is rooted in a certain feeling about logic, a certain feeling that our primeval ancestors had, and as long as you keep to ordinary language you find it very difficult to get away from the bias which is imposed upon you by language. When I say, e.g., “There is an x such that x is a man”, that is not the sort of phrase one would like to use. “There is an x” is meaningless. What is “an x”

anyhow? There is not such a thing. The only way you can really state it correctly is by inventing a new language ad hoc, and making the statement apply straight off to “x is a man”, as when

one says “(x is a man) is possible”, or invent a special symbol for the statement that “x is a man” is sometimes true.

I have dwelt on this point because it really is of very funda-mental importance. I shall come back to existence in my next lecture: existence as it applies to descriptions, which is a slightly more complicated case than I am discussing here. I think an almost unbelievable amount of false philosophy has arisen through not realizing what “existence” means.

As I was saying a moment ago, a propositional function in itself is nothing: it is merely a schema. Therefore in the inven-tory of the world, which is what I am trying to get at, one comes to the question: What is there really in the world that corresponds with these things? Of course, it is clear that we have general propositions, in the same sense in which we have atomic propositions. For the moment I will include existence-propositions with general existence-propositions. We have such proposi-tions as “All men are mortal” and “Some men are Greeks”. But you have not only such propositions; you have also such facts, and that, of course, is where you get back to the inventory of the world: that, in addition to particular facts, which I have been talking about in previous lectures, there are also general facts and existence-facts, that is to say, there are not merely propositions of that sort but also facts of that sort. That is rather an important point to realize. You cannot ever arrive at a general fact by infer-ence from particular facts, however numerous. The old plan of complete induction, which used to occur in books, which was always supposed to be quite safe and easy as opposed to ordinary induction, that plan of complete induction, unless it is accom-panied by at least one general proposition, will not yield you the result that you want. Suppose, for example, that you wish to prove in that way that “All men are mortal”, you are supposed to proceed by complete induction, and say “A is a man that is mortal”, “B is a man that is mortal”, C is a man that is mortal”, and so on until you finish. You will not be able, in that way, to

arrive at the proposition “All men are mortal” unless you know when you have finished. That is to say that, in order to arrive by this road at the general proposition “All men are mortal”, you must already have the general proposition “All men are among those I have enumerated”. You never can arrive at a general proposition by inference from particular propositions alone. You will always have to have at least one general proposition in your premisses. That illustrates, I think, various points. One, which is epistemological, is that if there is, as there seems to be, knowledge of general propositions, then there must be primitive knowledge of general propositions (I mean by that, knowledge of general propositions which is not obtained by inference), because if you can never infer a general proposition except from premisses of which one at least is general, it is clear that you can never have knowledge of such propositions by inference unless there is knowledge of some general proposi-tions which is not by inference. I think that the sort of way such knowledge—or rather the belief that we have such know-ledge—comes into ordinary life is probably very odd. I mean to say that we do habitually assume general propositions which are exceedingly doubtful; as, for instance, one might, if one were counting up the people in this room, assume that one could see all of them, which is a general proposition, and very doubtful

arrive at the proposition “All men are mortal” unless you know when you have finished. That is to say that, in order to arrive by this road at the general proposition “All men are mortal”, you must already have the general proposition “All men are among those I have enumerated”. You never can arrive at a general proposition by inference from particular propositions alone. You will always have to have at least one general proposition in your premisses. That illustrates, I think, various points. One, which is epistemological, is that if there is, as there seems to be, knowledge of general propositions, then there must be primitive knowledge of general propositions (I mean by that, knowledge of general propositions which is not obtained by inference), because if you can never infer a general proposition except from premisses of which one at least is general, it is clear that you can never have knowledge of such propositions by inference unless there is knowledge of some general proposi-tions which is not by inference. I think that the sort of way such knowledge—or rather the belief that we have such know-ledge—comes into ordinary life is probably very odd. I mean to say that we do habitually assume general propositions which are exceedingly doubtful; as, for instance, one might, if one were counting up the people in this room, assume that one could see all of them, which is a general proposition, and very doubtful