philosophical importance except in its beginnings. After the beginnings,
it belongs rather to mathematics than to philosophy. Of its beginnings,
which are the only part of it that can properly be called
_philosophical_ logic, I shall speak shortly. But even the later
developments, though not directly philosophical, will be found of great
indirect use in philosophising. They enable us to deal easily with more
abstract conceptions than merely verbal reasoning can enumerate; they
suggest fruitful hypotheses which otherwise could hardly be thought of;
and they enable us to see quickly what is the smallest store of
materials with which a given logical or scientific edifice can be
constructed. Not only Frege's theory of number, which we shall deal with
in Lecture VII., but the whole theory of physical concepts which will be
outlined in our next two lectures, is inspired by mathematical logic,
and could never have been imagined without it.
In both these cases, and in many others, we shall appeal to a certain
principle called "the principle of abstraction." This principle, which
might equally well be called "the principle which dispenses with
abstraction," and is one which clears away incredible accumulations of
metaphysical lumber, was directly suggested by mathematical logic, and
could hardly have been proved or practically used without its help. The
principle will be explained in our fourth lecture, but its use may be
briefly indicated in advance. When a group of objects have that kind of
similarity which we are inclined to attribute to possession of a common
quality, the principle in question shows that membership of the group
will serve all the purposes of the supposed common quality, and that
therefore, unless some common quality is actually known, the group or
class of similar objects may be used to replace the common quality,
which need not be assumed to exist. In this and other ways, the indirect
uses of even the later parts of mathematical logic are very great; but
it is now time to turn our attention to its philosophical foundations.
In every proposition and in every inference there is, besides the
particular subject-matter concerned, a certain _form_, a way in which
the constituents of the proposition or inference are put together. If I
say, "Socrates is mortal," "Jones is angry," "The sun is hot," there is
something in common in these three cases, something indicated by the
word "is." What is in common is the _form_ of the proposition, not an
actual constituent. If I say a number of things about Socrates--that he
was an Athenian, that he married Xantippe, that he drank the
hemlock--there is a common constituent, namely Socrates, in all the
propositions I enunciate, but they have diverse forms. If, on the other
hand, I take any one of these propositions and replace its constituents,
one at a time, by other constituents, the form remains constant, but no
constituent remains. Take (say) the series of propositions, "Socrates
drank the hemlock," "Coleridge drank the hemlock," "Coleridge drank
opium," "Coleridge ate opium." The form remains unchanged throughout
this series, but all the constituents are altered. Thus form is not
another constituent, but is the way the constituents are put together.
It is forms, in this sense, that are the proper object of philosophical
logic.
It is obvious that the knowledge of logical forms is something quite
different from knowledge of existing things. The form of "Socrates drank
the hemlock" is not an existing thing like Socrates or the hemlock, nor
does it even have that close relation to existing things that drinking
has. It is something altogether more abstract and remote. We might
understand all the separate words of a sentence without understanding
the sentence: if a sentence is long and complicated, this is apt to
happen. In such a case we have knowledge of the constituents, but not of
the form. We may also have knowledge of the form without having
knowledge of the constituents. If I say, "Rorarius drank the hemlock,"
those among you who have never heard of Rorarius (supposing there are
any) will understand the form, without having knowledge of all the
constituents. In order to understand a sentence, it is necessary to have
knowledge both of the constituents and of the particular instance of the
form. It is in this way that a sentence conveys information, since it
tells us that certain known objects are related according to a certain
known form. Thus some kind of knowledge of logical forms, though with
most people it is not explicit, is involved in all understanding of
discourse. It is the business of philosophical logic to extract this
knowledge from its concrete integuments, and to render it explicit and
pure.
In all inference, form alone is essential: the particular subject-matter
is irrelevant except as securing the truth of the premisses. This is one
reason for the great importance of logical form. When I say, "Socrates
was a man, all men are mortal, therefore Socrates was mortal," the
connection of premisses and conclusion does not in any way depend upon
its being Socrates and man and mortality that I am mentioning. The
general form of the inference may be expressed in some such words as,
"If a thing has a certain property, and whatever has this property has a
certain other property, then the thing in question also has that other
property." Here no particular things or properties are mentioned: the
proposition is absolutely general. All inferences, when stated fully,
are instances of propositions having this kind of generality. If they
seem to depend upon the subject-matter otherwise than as regards the
truth of the premisses, that is because the premisses have not been all
explicitly stated. In logic, it is a waste of time to deal with
inferences concerning particular cases: we deal throughout with
completely general and purely formal implications, leaving it to other
sciences to discover when the hypotheses are verified and when they are
not.
But the forms of propositions giving rise to inferences are not the
simplest forms: they are always hypothetical, stating that if one
proposition is true, then so is another. Before considering inference,
therefore, logic must consider those simpler forms which inference
presupposes. Here the traditional logic failed completely: it believed
that there was only one form of simple proposition (_i.e._ of
proposition not stating a relation between two or more other
propositions), namely, the form which ascribes a predicate to a subject.
This is the appropriate form in assigning the qualities of a given
thing--we may say "this thing is round, and red, and so on." Grammar
favours this form, but philosophically it is so far from universal that
it is not even very common. If we say "this thing is bigger than that,"
we are not assigning a mere quality of "this," but a relation of "this"
and "that." We might express the same fact by saying "that thing is
smaller than this," where grammatically the subject is changed. Thus
propositions stating that two things have a certain relation have a
different form from subject-predicate propositions, and the failure to
perceive this difference or to allow for it has been the source of many
errors in traditional metaphysics.
The belief or unconscious conviction that all propositions are of the
subject-predicate form--in other words, that every fact consists in some
thing having some quality--has rendered most philosophers incapable of
giving any account of the world of science and daily life. If they had
been honestly anxious to give such an account, they would probably have
discovered their error very quickly; but most of them were less anxious
to understand the world of science and daily life, than to convict it of
unreality in the interests of a super-sensible "real" world. Belief in
the unreality of the world of sense arises with irresistible force in
certain moods--moods which, I imagine, have some simple physiological
basis, but are none the less powerfully persuasive. The conviction born
of these moods is the source of most mysticism and of most metaphysics.
When the emotional intensity of such a mood subsides, a man who is in
the habit of reasoning will search for logical reasons in favour of the
belief which he finds in himself. But since the belief already exists,
he will be very hospitable to any reason that suggests itself. The
paradoxes apparently proved by his logic are really the paradoxes of
mysticism, and are the goal which he feels his logic must reach if it is
to be in accordance with insight. It is in this way that logic has been
pursued by those of the great philosophers who were mystics--notably
Plato, Spinoza, and Hegel. But since they usually took for granted the
supposed insight of the mystic emotion, their logical doctrines were
presented with a certain dryness, and were believed by their disciples
to be quite independent of the sudden illumination from which they
sprang. Nevertheless their origin clung to them, and they remained--to
borrow a useful word from Mr Santayana--"malicious" in regard to the
world of science and common sense. It is only so that we can account for
the complacency with which philosophers have accepted the inconsistency
of their doctrines with all the common and scientific facts which seem
best established and most worthy of belief.
The logic of mysticism shows, as is natural, the defects which are
inherent in anything malicious. While the mystic mood is dominant, the
need of logic is not felt; as the mood fades, the impulse to logic
reasserts itself, but with a desire to retain the vanishing insight, or
at least to prove that it _was_ insight, and that what seems to
contradict it is illusion. The logic which thus arises is not quite
disinterested or candid, and is inspired by a certain hatred of the
daily world to which it is to be applied. Such an attitude naturally
does not tend to the best results. Everyone knows that to read an author
simply in order to refute him is not the way to understand him; and to
read the book of Nature with a conviction that it is all illusion is
just as unlikely to lead to understanding. If our logic is to find the
common world intelligible, it must not be hostile, but must be inspired
by a genuine acceptance such as is not usually to be found among
metaphysicians.
Traditional logic, since it holds that all propositions have the
subject-predicate form, is unable to admit the reality of relations: all
relations, it maintains, must be reduced to properties of the apparently
related terms. There are many ways of refuting this opinion; one of the
easiest is derived from the consideration of what are called
"asymmetrical" relations. In order to explain this, I will first explain
two independent ways of classifying relations.
Some relations, when they hold between A and B, also hold between B and
A. Such, for example, is the relation "brother or sister." If A is a
brother or sister of B, then B is a brother or sister of A. Such again
is any kind of similarity, say similarity of colour. Any kind of
dissimilarity is also of this kind: if the colour of A is unlike the
colour of B, then the colour of B is unlike the colour of A. Relations
of this sort are called _symmetrical_. Thus a relation is symmetrical
if, whenever it holds between A and B, it also holds between B and A.
All relations that are not symmetrical are called _non-symmetrical_.
Thus "brother" is non-symmetrical, because, if A is a brother of B, it
may happen that B is a _sister_ of A.
A relation is called _asymmetrical_ when, if it holds between A and B,
it _never_ holds between B and A. Thus husband, father, grandfather,
etc., are asymmetrical relations. So are _before_, _after_, _greater_,
_above_, _to the right of_, etc. All the relations that give rise to
series are of this kind.
Classification into symmetrical, asymmetrical, and merely
non-symmetrical relations is the first of the two classifications we had
to consider. The second is into transitive, intransitive, and merely
non-transitive relations, which are defined as follows.
A relation is said to be _transitive_, if, whenever it holds between A
and B and also between B and C, it holds between A and C. Thus _before_,
_after_, _greater_, _above_ are transitive. All relations giving rise to
series are transitive, but so are many others. The transitive relations
just mentioned were asymmetrical, but many transitive relations are
symmetrical--for instance, equality in any respect, exact identity of
colour, being equally numerous (as applied to collections), and so on.
A relation is said to be _non-transitive_ whenever it is not transitive.
Thus "brother" is non-transitive, because a brother of one's brother may
be oneself. All kinds of dissimilarity are non-transitive.
A relation is said to be _intransitive_ when, if A has the relation to
B, and B to C, A never has it to C. Thus "father" is intransitive. So is
such a relation as "one inch taller" or "one year later."
Let us now, in the light of this classification, return to the question
whether all relations can be reduced to predications.
In the case of symmetrical relations--_i.e._ relations which, if they
hold between A and B, also hold between B and A--some kind of
plausibility can be given to this doctrine. A symmetrical relation which
is transitive, such as equality, can be regarded as expressing
possession of some common property, while one which is not transitive,
such as inequality, can be regarded as expressing possession of
different properties. But when we come to asymmetrical relations, such
as before and after, greater and less, etc., the attempt to reduce them
to properties becomes obviously impossible. When, for example, two
things are merely known to be unequal, without our knowing which is
greater, we may say that the inequality results from their having
different magnitudes, because inequality is a symmetrical relation; but
to say that when one thing is _greater_ than another, and not merely
unequal to it, that means that they have different magnitudes, is
formally incapable of explaining the facts. For if the other thing had
been greater than the one, the magnitudes would also have been
different, though the fact to be explained would not have been the same.
Thus mere _difference_ of magnitude is not _all_ that is involved,
since, if it were, there would be no difference between one thing being
greater than another, and the other being greater than the one. We shall
have to say that the one magnitude is _greater_ than the other, and thus
we shall have failed to get rid of the relation "greater." In short,
both possession of the same property and possession of different
properties are _symmetrical_ relations, and therefore cannot account for
the existence of _asymmetrical_ relations.
Asymmetrical relations are involved in all series--in space and time,
greater and less, whole and part, and many others of the most important
characteristics of the actual world. All these aspects, therefore, the
logic which reduces everything to subjects and predicates is compelled
to condemn as error and mere appearance. To those whose logic is not
malicious, such a wholesale condemnation appears impossible. And in fact
there is no reason except prejudice, so far as I can discover, for
denying the reality of relations. When once their reality is admitted,
all _logical_ grounds for supposing the world of sense to be illusory
disappear. If this is to be supposed, it must be frankly and simply on
the ground of mystic insight unsupported by argument. It is impossible
to argue against what professes to be insight, so long as it does not
argue in its own favour. As logicians, therefore, we may admit the
possibility of the mystic's world, while yet, so long as we do not have
his insight, we must continue to study the everyday world with which we
are familiar. But when he contends that our world is impossible, then
our logic is ready to repel his attack. And the first step in creating
the logic which is to perform this service is the recognition of the
reality of relations.
Relations which have two terms are only one kind of relations. A
relation may have three terms, or four, or any number. Relations of two
terms, being the simplest, have received more attention than the others,
and have generally been alone considered by philosophers, both those who
accepted and those who denied the reality of relations. But other
relations have their importance, and are indispensable in the solution
of certain problems. Jealousy, for example, is a relation between three
people. Professor Royce mentions the relation "giving": when A gives B
to C, that is a relation of three terms.[15] When a man says to his
wife: "My dear, I wish you could induce Angelina to accept Edwin," his
wish constitutes a relation between four people, himself, his wife,
Angelina, and Edwin. Thus such relations are by no means recondite or
rare. But in order to explain exactly how they differ from relations of
two terms, we must embark upon a classification of the logical forms of
facts, which is the first business of logic, and the business in which
the traditional logic has been most deficient.
[15] _Encyclopaedia of the Philosophical Sciences_, vol. i. p. 97.
The existing world consists of many things with many qualities and
relations. A complete description of the existing world would require
not only a catalogue of the things, but also a mention of all their
qualities and relations. We should have to know not only this, that, and
the other thing, but also which was red, which yellow, which was earlier
than which, which was between which two others, and so on. When I speak
of a "fact," I do not mean one of the simple things in the world; I mean
that a certain thing has a certain quality, or that certain things have
a certain relation. Thus, for example, I should not call Napoleon a
fact, but I should call it a fact that he was ambitious, or that he
married Josephine. Now a fact, in this sense, is never simple, but
always has two or more constituents. When it simply assigns a quality to
a thing, it has only two constituents, the thing and the quality. When
it consists of a relation between two things, it has three constituents,
the things and the relation. When it consists of a relation between
three things, it has four constituents, and so on. The constituents of
facts, in the sense in which we are using the word "fact," are not other
facts, but are things and qualities or relations. When we say that there
are relations of more than two terms, we mean that there are single
facts consisting of a single relation and more than two things. I do not
mean that one relation of two terms may hold between A and B, and also
between A and C, as, for example, a man is the son of his father and
also the son of his mother. This constitutes two distinct facts: if we
choose to treat it as one fact, it is a fact which has facts for its
constituents. But the facts I am speaking of have no facts among their
constituents, but only things and relations. For example, when A is
jealous of B on account of C, there is only one fact, involving three
people; there are not two instances of jealousy, but only one. It is in
such cases that I speak of a relation of three terms, where the simplest
possible fact in which the relation occurs is one involving three things
in addition to the relation. And the same applies to relations of four
terms or five or any other number. All such relations must be admitted
in our inventory of the logical forms of facts: two facts involving the
same number of things have the same form, and two which involve
different numbers of things have different forms.
Given any fact, there is an assertion which expresses the fact. The fact
itself is objective, and independent of our thought or opinion about it;
but the assertion is something which involves thought, and may be either
true or false. An assertion may be positive or negative: we may assert
that Charles I. was executed, or that he did _not_ die in his bed. A
negative assertion may be said to be a _denial_. Given a form of words
which must be either true or false, such as "Charles I. died in his
bed," we may either assert or deny this form of words: in the one case
we have a positive assertion, in the other a negative one. A form of
words which must be either true or false I shall call a _proposition_.
Thus a proposition is the same as what may be significantly asserted or
denied. A proposition which expresses what we have called a fact, _i.e._
which, when asserted, asserts that a certain thing has a certain
quality, or that certain things have a certain relation, will be called
an atomic proposition, because, as we shall see immediately, there are
other propositions into which atomic propositions enter in a way
analogous to that in which atoms enter into molecules. Atomic
propositions, although, like facts, they may have any one of an infinite
number of forms, are only one kind of propositions. All other kinds are
more complicated. In order to preserve the parallelism in language as
regards facts and propositions, we shall give the name "atomic facts" to
the facts we have hitherto been considering. Thus atomic facts are what
determine whether atomic propositions are to be asserted or denied.
Whether an atomic proposition, such as "this is red," or "this is before
that," is to be asserted or denied can only be known empirically.
Perhaps one atomic fact may sometimes be capable of being inferred from
another, though this seems very doubtful; but in any case it cannot be
inferred from premisses no one of which is an atomic fact. It follows
that, if atomic facts are to be known at all, some at least must be
known without inference. The atomic facts which we come to know in this
way are the facts of sense-perception; at any rate, the facts of
sense-perception are those which we most obviously and certainly come to
know in this way. If we knew all atomic facts, and also knew that there
were none except those we knew, we should, theoretically, be able to
infer all truths of whatever form.[16] Thus logic would then supply us
with the whole of the apparatus required. But in the first acquisition
of knowledge concerning atomic facts, logic is useless. In pure logic,
no atomic fact is ever mentioned: we confine ourselves wholly to forms,
without asking ourselves what objects can fill the forms. Thus pure
logic is independent of atomic facts; but conversely, they are, in a
sense, independent of logic. Pure logic and atomic facts are the two
poles, the wholly _a priori_ and the wholly empirical. But between the
two lies a vast intermediate region, which we must now briefly explore.
[16] This perhaps requires modification in order to include such facts
as beliefs and wishes, since such facts apparently contain
propositions as components. Such facts, though not strictly atomic,
must be supposed included if the statement in the text is to be true.
"Molecular" propositions are such as contain conjunctions--_if_, _or_,
_and_, _unless_, etc.--and such words are the marks of a molecular
proposition. Consider such an assertion as, "If it rains, I shall bring
my umbrella." This assertion is just as capable of truth or falsehood as
the assertion of an atomic proposition, but it is obvious that either
the corresponding fact, or the nature of the correspondence with fact,
must be quite different from what it is in the case of an atomic
proposition. Whether it rains, and whether I bring my umbrella, are each
severally matters of atomic fact, ascertainable by observation. But the
connection of the two involved in saying that _if_ the one happens,
_then_ the other will happen, is something radically different from
either of the two separately. It does not require for its truth that it
should actually rain, or that I should actually bring my umbrella; even
if the weather is cloudless, it may still be true that I should have
brought my umbrella if the weather had been different. Thus we have here
a connection of two propositions, which does not depend upon whether
they are to be asserted or denied, but only upon the second being
inferable from the first. Such propositions, therefore, have a form
which is different from that of any atomic proposition.