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.