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.