** The Philosophy of Logical Atomism (1918)**

**5. 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) = x*^{2}−*y*^{2}.
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;*

6*Cf. 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