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 are 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 as 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 to 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 any 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 clearer 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.
