ON OUR KNOWLEDGE OF GENERAL PRINCIPLES
We saw in the preceding chapter that the principle of induction, while
necessary to the validity of all arguments based on experience, is
itself not capable of being proved by experience, and yet is
unhesitatingly believed by every one, at least in all its concrete
applications. In these characteristics the principle of induction
does not stand alone. There are a number of other principles which
cannot be proved or disproved by experience, but are used in arguments
which start from what is experienced.
Some of these principles have even greater evidence than the principle
of induction, and the knowledge of them has the same degree of
certainty as the knowledge of the existence of sense-data. They
constitute the means of drawing inferences from what is given in
sensation; and if what we infer is to be true, it is just as necessary
that our principles of inference should be true as it is that our data
should be true. The principles of inference are apt to be overlooked
because of their very obviousness--the assumption involved is assented
to without our realizing that it is an assumption. But it is very
important to realize the use of principles of inference, if a correct
theory of knowledge is to be obtained; for our knowledge of them
raises interesting and difficult questions.
In all our knowledge of general principles, what actually happens is
that first of all we realize some particular application of the
principle, and then we realize that the particularity is irrelevant,
and that there is a generality which may equally truly be affirmed.
This is of course familiar in such matters as teaching arithmetic:
'two and two are four' is first learnt in the case of some particular
pair of couples, and then in some other particular case, and so on,
until at last it becomes possible to see that it is true of any pair
of couples. The same thing happens with logical principles. Suppose
two men are discussing what day of the month it is. One of them says,
'At least you will admit that _if_ yesterday was the 15th to-day must
be the 16th.' 'Yes', says the other, 'I admit that.' 'And you know',
the first continues, 'that yesterday was the 15th, because you dined
with Jones, and your diary will tell you that was on the 15th.' 'Yes',
says the second; 'therefore to-day _is_ the 16th.'
Now such an argument is not hard to follow; and if it is granted that
its premisses are true in fact, no one will deny that the conclusion
must also be true. But it depends for its truth upon an instance of a
general logical principle. The logical principle is as follows:
'Suppose it known that _if_ this is true, then that is true. Suppose
it also known that this _is_ true, then it follows that that is true.'
When it is the case that if this is true, that is true, we shall say
that this 'implies' that, and that that 'follows from' this. Thus our
principle states that if this implies that, and this is true, then
that is true. In other words, 'anything implied by a true proposition
is true', or 'whatever follows from a true proposition is true'.
This principle is really involved--at least, concrete instances of it
are involved--in all demonstrations. Whenever one thing which we
believe is used to prove something else, which we consequently
believe, this principle is relevant. If any one asks: 'Why should I
accept the results of valid arguments based on true premisses?' we can
only answer by appealing to our principle. In fact, the truth of the
principle is impossible to doubt, and its obviousness is so great that
at first sight it seems almost trivial. Such principles, however, are
not trivial to the philosopher, for they show that we may have
indubitable knowledge which is in no way derived from objects of
sense.
The above principle is merely one of a certain number of self-evident
logical principles. Some at least of these principles must be granted
before any argument or proof becomes possible. When some of them have
been granted, others can be proved, though these others, so long as
they are simple, are just as obvious as the principles taken for
granted. For no very good reason, three of these principles have been
singled out by tradition under the name of 'Laws of Thought'.
They are as follows:
(1) _The law of identity_: 'Whatever is, is.'
(2) _The law of contradiction_: 'Nothing can both be and not be.'
(3) _The law of excluded middle_: 'Everything must either be or not
be.'
These three laws are samples of self-evident logical principles, but
are not really more fundamental or more self-evident than various
other similar principles: for instance, the one we considered just
now, which states that what follows from a true premiss is true. The
name 'laws of thought' is also misleading, for what is important is
not the fact that we think in accordance with these laws, but the fact
that things behave in accordance with them; in other words, the fact
that when we think in accordance with them we think _truly_. But this
is a large question, to which we must return at a later stage.
In addition to the logical principles which enable us to prove from a
given premiss that something is _certainly_ true, there are other
logical principles which enable us to prove, from a given premiss,
that there is a greater or less probability that something is true.
An example of such principles--perhaps the most important example is
the inductive principle, which we considered in the preceding chapter.
One of the great historic controversies in philosophy is the
controversy between the two schools called respectively 'empiricists'
and 'rationalists'. The empiricists--who are best represented by the
British philosophers, Locke, Berkeley, and Hume--maintained that all
our knowledge is derived from experience; the rationalists--who are
represented by the Continental philosophers of the seventeenth
century, especially Descartes and Leibniz--maintained that, in
addition to what we know by experience, there are certain 'innate
ideas' and 'innate principles', which we know independently of
experience. It has now become possible to decide with some confidence
as to the truth or falsehood of these opposing schools. It must be
admitted, for the reasons already stated, that logical principles are
known to us, and cannot be themselves proved by experience, since all
proof presupposes them. In this, therefore, which was the most
important point of the controversy, the rationalists were in the
right.
On the other hand, even that part of our knowledge which is
_logically_ independent of experience (in the sense that experience
cannot prove it) is yet elicited and caused by experience. It is on
occasion of particular experiences that we become aware of the general
laws which their connexions exemplify. It would certainly be absurd
to suppose that there are innate principles in the sense that babies
are born with a knowledge of everything which men know and which
cannot be deduced from what is experienced. For this reason, the word
'innate' would not now be employed to describe our knowledge of
logical principles. The phrase '_a priori_' is less objectionable,
and is more usual in modern writers. Thus, while admitting that all
knowledge is elicited and caused by experience, we shall nevertheless
hold that some knowledge is _a priori_, in the sense that the
experience which makes us think of it does not suffice to prove it,
but merely so directs our attention that we see its truth without
requiring any proof from experience.
There is another point of great importance, in which the empiricists
were in the right as against the rationalists. Nothing can be known
to _exist_ except by the help of experience. That is to say, if we
wish to prove that something of which we have no direct experience
exists, we must have among our premisses the existence of one or more
things of which we have direct experience. Our belief that the
Emperor of China exists, for example, rests upon testimony, and
testimony consists, in the last analysis, of sense-data seen or heard
in reading or being spoken to. Rationalists believed that, from
general consideration as to what must be, they could deduce the
existence of this or that in the actual world. In this belief they
seem to have been mistaken. All the knowledge that we can acquire _a
priori_ concerning existence seems to be hypothetical: it tells us
that if one thing exists, another must exist, or, more generally, that
if one proposition is true, another must be true. This is exemplified
by the principles we have already dealt with, such as '_if_ this is
true, and this implies that, then that is true', or '_if_ this and
that have been repeatedly found connected, they will probably be
connected in the next instance in which one of them is found'. Thus
the scope and power of _a priori_ principles is strictly limited. All
knowledge that something exists must be in part dependent on
experience. When anything is known immediately, its existence is
known by experience alone; when anything is proved to exist, without
being known immediately, both experience and _a priori_ principles
must be required in the proof. Knowledge is called _empirical_ when
it rests wholly or partly upon experience. Thus all knowledge which
asserts existence is empirical, and the only _a priori_ knowledge
concerning existence is hypothetical, giving connexions among things
that exist or may exist, but not giving actual existence.
_A priori_ knowledge is not all of the logical kind we have been
hitherto considering. Perhaps the most important example of
non-logical _a priori_ knowledge is knowledge as to ethical value. I
am not speaking of judgements as to what is useful or as to what is
virtuous, for such judgements do require empirical premisses; I am
speaking of judgements as to the intrinsic desirability of things. If
something is useful, it must be useful because it secures some end;
the end must, if we have gone far enough, be valuable on its own
account, and not merely because it is useful for some further end.
Thus all judgements as to what is useful depend upon judgements as to
what has value on its own account.
We judge, for example, that happiness is more desirable than misery,
knowledge than ignorance, goodwill than hatred, and so on. Such
judgements must, in part at least, be immediate and _a priori_. Like
our previous _a priori_ judgements, they may be elicited by
experience, and indeed they must be; for it seems not possible to
judge whether anything is intrinsically valuable unless we have
experienced something of the same kind. But it is fairly obvious that
they cannot be proved by experience; for the fact that a thing exists
or does not exist cannot prove either that it is good that it should
exist or that it is bad. The pursuit of this subject belongs to
ethics, where the impossibility of deducing what ought to be from what
is has to be established. In the present connexion, it is only
important to realize that knowledge as to what is intrinsically of
value is _a priori_ in the same sense in which logic is _a priori_,
namely in the sense that the truth of such knowledge can be neither
proved nor disproved by experience.
All pure mathematics is _a priori_, like logic. This was strenuously
denied by the empirical philosophers, who maintained that experience
was as much the source of our knowledge of arithmetic as of our
knowledge of geography. They maintained that by the repeated
experience of seeing two things and two other things, and finding that
altogether they made four things, we were led by induction to the
conclusion that two things and two other things would _always_ make
four things altogether. If, however, this were the source of our
knowledge that two and two are four, we should proceed differently, in
persuading ourselves of its truth, from the way in which we do
actually proceed. In fact, a certain number of instances are needed
to make us think of two abstractly, rather than of two coins or two
books or two people, or two of any other specified kind. But as soon
as we are able to divest our thoughts of irrelevant particularity, we
become able to see the general principle that two and two are four;
any one instance is seen to be _typical_, and the examination of other
instances becomes unnecessary.[1]
[1] Cf. A. N. Whitehead, _Introduction to Mathematics_ (Home
University Library).
The same thing is exemplified in geometry. If we want to prove some
property of _all_ triangles, we draw some one triangle and reason
about it; but we can avoid making use of any property which it does
not share with all other triangles, and thus, from our particular
case, we obtain a general result. We do not, in fact, feel our
certainty that two and two are four increased by fresh instances,
because, as soon as we have seen the truth of this proposition, our
certainty becomes so great as to be incapable of growing greater.
Moreover, we feel some quality of necessity about the proposition 'two
and two are four', which is absent from even the best attested
empirical generalizations. Such generalizations always remain mere
facts: we feel that there might be a world in which they were false,
though in the actual world they happen to be true. In any possible
world, on the contrary, we feel that two and two would be four: this
is not a mere fact, but a necessity to which everything actual and
possible must conform.
The case may be made clearer by considering a genuinely-empirical
generalization, such as 'All men are mortal.' It is plain that we
believe this proposition, in the first place, because there is no
known instance of men living beyond a certain age, and in the second
place because there seem to be physiological grounds for thinking that
an organism such as a man's body must sooner or later wear out.
Neglecting the second ground, and considering merely our experience of
men's mortality, it is plain that we should not be content with one
quite clearly understood instance of a man dying, whereas, in the case
of 'two and two are four', one instance does suffice, when carefully
considered, to persuade us that the same must happen in any other
instance. Also we can be forced to admit, on reflection, that there
may be some doubt, however slight, as to whether _all_ men are mortal.
This may be made plain by the attempt to imagine two different worlds,
in one of which there are men who are not mortal, while in the other
two and two make five. When Swift invites us to consider the race of
Struldbugs who never die, we are able to acquiesce in imagination.
But a world where two and two make five seems quite on a different
level. We feel that such a world, if there were one, would upset the
whole fabric of our knowledge and reduce us to utter doubt.
The fact is that, in simple mathematical judgements such as 'two and
two are four', and also in many judgements of logic, we can know the
general proposition without inferring it from instances, although some
instance is usually necessary to make clear to us what the general
proposition means. This is why there is real utility in the process
of _deduction_, which goes from the general to the general, or from
the general to the particular, as well as in the process of
_induction_, which goes from the particular to the particular, or from
the particular to the general. It is an old debate among philosophers
whether deduction ever gives _new_ knowledge. We can now see that in
certain cases, at least, it does do so. If we already know that two
and two always make four, and we know that Brown and Jones are two,
and so are Robinson and Smith, we can deduce that Brown and Jones and
Robinson and Smith are four. This is new knowledge, not contained in
our premisses, because the general proposition, 'two and two are
four', never told us there were such people as Brown and Jones and
Robinson and Smith, and the particular premisses do not tell us that
there were four of them, whereas the particular proposition deduced
does tell us both these things.
But the newness of the knowledge is much less certain if we take the
stock instance of deduction that is always given in books on logic,
namely, 'All men are mortal; Socrates is a man, therefore Socrates is
mortal.' In this case, what we really know beyond reasonable doubt is
that certain men, A, B, C, were mortal, since, in fact, they have
died. If Socrates is one of these men, it is foolish to go the
roundabout way through 'all men are mortal' to arrive at the
conclusion that _probably_ Socrates is mortal. If Socrates is not one
of the men on whom our induction is based, we shall still do better to
argue straight from our A, B, C, to Socrates, than to go round by the
general proposition, 'all men are mortal'. For the probability that
Socrates is mortal is greater, on our data, than the probability that
all men are mortal. (This is obvious, because if all men are mortal,
so is Socrates; but if Socrates is mortal, it does not follow that all
men are mortal.) Hence we shall reach the conclusion that Socrates is
mortal with a greater approach to certainty if we make our argument
purely inductive than if we go by way of 'all men are mortal' and then
use deduction.
This illustrates the difference between general propositions known _a
priori_ such as 'two and two are four', and empirical generalizations
such as 'all men are mortal'. In regard to the former, deduction is
the right mode of argument, whereas in regard to the latter, induction
is always theoretically preferable, and warrants a greater confidence
in the truth of our conclusion, because all empirical generalizations
are more uncertain than the instances of them.
We have now seen that there are propositions known _a priori_, and
that among them are the propositions of logic and pure mathematics, as
well as the fundamental propositions of ethics. The question which
must next occupy us is this: How is it possible that there should be
such knowledge? And more particularly, how can there be knowledge of
general propositions in cases where we have not examined all the
instances, and indeed never can examine them all, because their number
is infinite? These questions, which were first brought prominently
forward by the German philosopher Kant (1724-1804), are very
difficult, and historically very important.
