 The
Problems of Philosophy
Bertrand Russell
CHAPTER VII
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 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 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 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, 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 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
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 principles we have already dealt with, such as 'if
this is true, and this implies that, then that is true',
of 'ifthis 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 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 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 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.*
---------------
* 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, 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.
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