 The
Problems of Philosophy
Bertrand Russell
CHAPTER X
ON OUR KNOWLEDGE OF UNIVERSALS
IN regard to one man's knowledge at a given time, universals,
like particulars, may be divided into those known by acquaintance,
those known only by description, and those not known either
by acquaintance or by description.
Let us consider first the knowledge of universals by acquaintance.
It is obvious, to begin with, that we are acquainted with
such universals as white, red, black, sweet, sour, loud,
hard, etc., i.e. with qualities which are exemplified in
sense-data. When we see a white patch, we are acquainted,
in the first instance, with the particular patch; but by
seeing many white patches, we easily learn to abstract the
whiteness which they all have in common, and in learning
to do this we are learning to be acquainted with whiteness.
A similar process will make us acquainted with any other
universal of the same sort. Universals of this sort may
be called 'sensible qualities'. They can be apprehended
with less effort of abstraction than any others, and they
seem less removed from particulars than other universals
are.
We come next to relations. The easiest relations to apprehend
are those which hold between the different parts of a single
complex sense-datum. For example, I can see at a glance
the whole of the page on which I am writing; thus the whole
page is included in one sense-datum. But I perceive that
some parts of the page are to the left of other parts, and
some parts are above other parts. The process of abstraction
in this case seems to proceed somewhat as follows: I see
successively a number of sense-data in which one part is
to the left of another; I perceive, as in the case of different
white patches, that all these sense-data have something
in common, and by abstraction I find that what they have
in common is a certain relation between their parts, namely
the relation which I call 'being to the left of'. In this
way I become acquainted with the universal relation.
In like manner I become aware of the relation of before
and after in time. Suppose I hear a chime of bells: when
the last bell of the chime sounds, I can retain the whole
chime before my mind, and I can perceive that the earlier
bells came before the later ones. Also in memory I perceive
that what I am remembering came before the present time.
From either of these sources I can abstract the universal
relation of before and after, just as I abstracted the universal
relation 'being to the left of'. Thus time-relations, like
space-relations, are among those with which we are acquainted.
Another relation with which we become acquainted in much
the same way is resemblance. If I see simultaneously two
shades of green, I can see that they resemble each other;
if I also see a shade of red at the same time, I can see
that the two greens have more resemblance to each other
than either has to the red. In this way I become acquainted
with the universal resemblance or similarity.
Between universals, as between particulars, there relations
of which we may be immediately aware. We have just seen
that we can perceive that the resemblance between two shades
of green is greater than the resemblance between a shade
of red and a shade of green. Here we are dealing with a
relation, namely 'greater than', between two relations.
Our knowledge of such relations, though it requires more
power of abstraction than is required for perceiving the
qualities of sense-data, appears to be equally immediate,
and (at least in some cases) equally indubitable. Thus there
is immediate knowledge concerning universals well as concerning
sense-data.
Returning now to the problem of a priori knowledge,
which we left unsolved when we began the consideration of
universals, we find ourselves in a position to deal with
it in a much more satisfactory manner than was possible
before. Let us revert to the proposition 'two and two are
four'. It is fairly obvious, in view of what has been said,
that this proposition states a relation between the universal
'two' and the universal 'four'. This suggests a proposition
which we shall now endeavour to establish: namely, All
a priori knowledge deals exclusively with the relations
of universals. This proposition is of great importance,
and goes a long way towards solving our previous difficulties
concerning a priori knowledge.
The only case in which it might seem, at first sight, as
if our proposition were untrue, is the case in which an
a priori proposition states that all of one class
of particulars belong to some other class, or (what comes
the same thing) that all particulars having some one property
also have some other. In this case it might seem as though
we were dealing with the particulars that have the property
rather than with the property. The proposition 'two and
two are four' is really a case in point, for this may be
stated in the form 'any two and any other two are four',
or 'any collection formed of two twos is a collection of
four'. If we can show that such statements as this really
deal only with universals, our proposition may be regarded
as proved.
One way of discovering what a proposition deals with is
to ask ourselves what words we must understand -- in other
words, what objects we must be acquainted with -- in order
to see what the proposition means. As soon as we see what
the proposition means, even if we do not yet know whether
it is true or false, it is evident that we must have acquaintance
with whatever is really dealt with by the proposition. By
applying this test, it appears that many propositions which
might seem to be concerned with particulars are really concerned
only with universals. In the special case of 'two and two
are four', even when we interpret it as meaning 'any collection
formed of two twos is a collection of four', it is plain
that we can understand the proposition, i.e. we
can see what it is that it asserts, as soon as we know what
is meant by 'collection' and 'two' and 'four' . It is quite
unnecessary to know all the couples in the world: if it
were necessary, obviously we could never understand the
proposition, since the couples are infinitely numerous and
therefore cannot all be known to us. Thus although our general
statement implies statements about particular couples,
as soon as we know that there are such particular couples,
yet it does not itself assert or imply that there are such
particular couples, and thus fails to make any statement
whatever about actual particular couple. The statement made
is about 'couple', the universal, and not about this or
that couple.
Thus the statement 'two and two are four' deals exclusively
with universals, and therefore may be known by anybody who
is acquainted with the universals concerned and can perceive
the relation between them which the statement asserts. It
must be taken as a fact, discovered by reflecting upon our
knowledge, that we have the power of sometimes perceiving
such relations between universals, and therefore of sometimes
knowing general a priori propositions such as those
of arithmetic and logic. The thing that seemed mysterious,
when we formerly considered such knowledge, was that it
seemed to anticipate and control experience. This, however,
we can now see to have been an error. No fact concerning
anything capable of being experienced can be known independently
of experience. We know a priori that two things
and two other things together make four things, but we do
not know a priori that if Brown and Jones
are two, and Robinson and Smith are two, then Brown and
Jones and Robinson and Smith are four. The reason is that
this proposition cannot be understood at all unless we know
that there are such people as Brown and Jones and Robinson
and Smith, and this we can only know by experience. Hence,
although our general proposition is a priori, all
its applications to actual particulars involve experience
and therefore contain an empirical element. In this way
what seemed mysterious in our a priori knowledge
is seen to have been based upon an error.
It will serve to make the point dearer if we contrast our
genuine a priori judgement with an empirical generalization,
such as 'all men are mortals'. Here as before, we can understand
what the proposition means as soon as we understand the
universals involved, namely man and mortal.
It is obviously unnecessary to have an individual acquaintance
with the whole human race in order to understand what our
proposition means. Thus the difference between an a
priori general proposition and an empirical generalization
does not come in the meaning of the proposition;
it comes in the nature of the evidence for it.
In the empirical case, the evidence consists in the particular
instances. We believe that all men are mortal because we
know that there are innumerable instances of men dying,
and no instances of their living beyond a certain age. We
do not believe it because we see a connexion between the
universal man and the universal mortal.
It is true that if physiology can prove, assuming the general
laws that govern living bodies, that no living organism
can last for ever, that gives a connexion between man
and mortality which would enable us to assert our
proposition without appealing to the special evidence of
men dying. But that only means that our generalization
has been subsumed under a wider generalization, for which
the evidence is still of the same kind, though more extensive.
The progress of science is constantly producing such subsumptions,
and therefore giving a constantly wider inductive basis
for scientific generalizations. But although this gives
a greater degree of certainty, it does not give
a different kind: the ultimate ground remains inductive,
i.e. derived from instances, and not an a priori
connexion of universals such as we have in logic and arithmetic.
Two opposite points are to be observed concerning a
priori general propositions. The first is that, if
many particular instances are known, our general proposition
may be arrived at in the first instance by induction, and
the connexion of universals may be only subsequently perceived.
For example, it is known that if we draw perpendiculars
to the sides of a triangle from the opposite angles, all
three perpendiculars meet in a point. It would be quite
possible to be first led to this proposition by actually
drawing perpendiculars in many cases, and finding that they
always met in a point; this experience might lead us to
look for the general proof and find it. Such cases are common
in the experience of every mathematician.
The other point is more interesting, and of more philosophical
importance. It is, that we may sometimes know a general
proposition in cases where we do not know a single instance
of it. Take such a case as the following: We know that any
two numbers can be multiplied together, and will give a
third called their product. We know that all pairs
of integers the product of which is less than 100 have been
actually multiplied together, and the value of the product
recorded in the multiplication table. But we also know that
the number of integers is infinite, and that only a finite
number of pairs of integers ever have been or ever will
be thought of by human beings. Hence it follows that there
are pairs of integers which never have been and never will
be thought of by human beings, and that all of them deal
with integers the product of which is over 100. Hence we
arrive at the proposition: 'All products of two integers,
which never have been and never will be thought of by any
human being, are over 100.' Here is a general proposition
of which the truth is undeniable, and yet, from the very
nature of the case, we can never give an instance; because
any two numbers we may think of are excluded by the terms
of the proposition.
This possibility, of knowledge of general propositions
of which no instance can be given, is often denied, because
it is not perceived that the knowledge of such propositions
only requires a knowledge of the relations of universals,
and does not require any knowledge of instances of the universals
in question. Yet the knowledge of such general propositions
is quite vital to a great deal of what is generally admitted
to be known. For example, we saw, in our early chapters,
that knowledge of physical objects, as opposed to sense-data,
is only obtained by an inference, and that they are not
things with which we are acquainted. Hence we can never
know any proposition of the form 'this is a physical object',
where 'this' is something immediately known. It follows
that all our knowledge concerning physical objects is such
that no actual instance can be given. We can give instances
of the associated sense-data, but we cannot give instances
of the actual physical objects. Hence our knowledge as to
physical objects depends throughout upon this possibility
of general knowledge where no instance can be given. And
the same applies to our knowledge of other people's minds,
or of any other class of things of which no instance is
known to us by acquaintance.
We may now take a survey of the sources of our knowledge,
as they have appeared in the course of our analysis. We
have first to distinguish knowledge of things and knowledge
of truths. In each there are two kinds, one immediate and
one derivative. Our immediate knowledge of things, which
we called acquaintance, consists of two sorts,
according as the things known are particulars or universals.
Among particulars, we have acquaintance with sense-data
and (probably) with ourselves. Among universals, there seems
to be no principle by which we can decide which can be known
by acquaintance, but it is clear that among those that can
be so known are sensible qualities, relations of space and
time, similarity, and certain abstract logical universals.
Our derivative knowledge of things, which we call knowledge
by description, always involves both acquaintance
with something and knowledge of truths. Our immediate knowledge
of truths may be called intuitive knowledge,
and the truths so known may be called self-evident
truths. Among such truths are included those which merely
state what is given in sense, and also certain abstract
logical and arithmetical principles, and (though with less
certainty) some ethical propositions. Our derivative
knowledge of truths consists of everything that we can deduce
from self-evident truths by the use of self-evident principles
of deduction.
If the above account is correct, all our knowledge of truths
depends upon our intuitive knowledge. It therefore becomes
important to consider the nature and scope of intuitive
knowledge, in much the same way as, at an earlier stage,
we considered the nature and scope of knowledge by acquaintance.
But knowledge of truths raises a further problem, which
does not arise in regard to knowledge of things, namely
the problem of error. Some of our beliefs turn
out to be erroneous, and therefore it becomes necessary
to consider how, if at all, we can distinguish knowledge
from error. This problem does not arise with regard to knowledge
by acquaintance, for, whatever may be the object of acquaintance,
even in dreams and hallucinations, there is no error involved
so long as we do not go beyond the immediate object: error
can only arise when we regard the immediate object, i.e.
the sense-datum, as the mark of some physical object. Thus
the problems connected with knowledge of truths are more
difficult than those connected with knowledge of things.
As the first of the problems connected with knowledge of
truths, let us examine the nature and scope of our intuitive
judgements.
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