TRANSITION TO INDUCTION
Sec. 1. Having now discussed Terms, Propositions, Immediate and Mediate
Inferences, and investigated the conditions of formal truth or
consistency, we have next to consider the conditions of material truth:
whether (or how far) it is possible to arrive at propositions that
accurately represent the course of nature or of human life. Hitherto we
have dealt with no sort of proof that gives any such assurance. A valid
syllogism guarantees the truth of its conclusion, provided the premises
be true: but what of the premises? The relation between the premises of
a valid syllogism and its conclusion is the same as the relation between
the antecedent and consequent of a hypothetical proposition. If A is B,
C is D: grant that A is B, and it follows that C is D; and, similarly,
grant the premises of a syllogism, and the conclusion follows. Again,
grant that C is not D, and it follows that A is not B; and, similarly,
if the conclusion of a valid syllogism be false, it follows that one, or
other, or both of the premises must be false. But, once more, grant that
C is D, and it does not follow that A is B; so neither, if the
conclusion of a syllogism be true, does it follow that the premises are.
For example:--
Sociology is an exact science;
Mathematics is a branch of Sociology:
.'. Mathematics is an exact science.
Here the conclusion is true although the premises are absurd. Or
again:--
Mathematics is an exact science;
Sociology is a branch of Mathematics:
.'. Sociology is an exact science.
Here the major premise is true, but the minor is false, and the
conclusion is false. In both cases, however, whether the conclusion be
true or false, it equally follows from the premises, if there is any
cogency in Barbara. The explanation of this is, that Barbara has only
formal cogency; and that whether the conclusion of that, or any other
valid mood, shall be true according to fact and experience, depends upon
how the form is filled up. How to establish the premises, then, is a
most important problem; and it still remains to be solved.
Sec. 2. We may begin by recalling the distinction between the denotation
and connotation of a general term: the denotation comprising the things
or events which the term is a name for; the connotation comprising the
common qualities on account of which these things are called by the same
name. Obviously, there are very few general terms whose denotation is
exhaustively known; since the denotation of a general term comprises all
the things that have its connotation, or that ever have had, or that
ever will have it, whether they exist here, or in Australia, or in the
Moon, or in the utmost stars. No one has examined all men, all mammoths,
all crystals, all falling bodies, all cases of fever, all revolutions,
all stars--nor even all planets, since from time to time new ones are
discerned. We have names for animals that existed long before there were
men to observe them, and of which we know only a few bones, the remains
of multitudinous species; and for others that may continue to exist when
men have disappeared from the earth.
If, indeed, we definitely limit the time, or place, or quantity of
matter to be explored, we may sometimes learn, within the given limits,
all that there is to know: as all the bones of a particular animal, or
the list of English monarchs hitherto, or the names of all the members
of the House of Commons at the present time. Such cases, however, do not
invalidate the above logical truth that few general terms are
exhaustively known in their denotation; for the very fact of assigning
limits of time and place impairs the generality of a term. The bones of
a certain animal may be all examined, but not the bones of all animals,
nor even of one species. The English monarchs that have reigned hitherto
may be known, but there may be many still to reign.
The general terms, then, with which Logic is chiefly concerned, the
names of Causes and Kinds, such as gravitation, diseases, social events,
minerals, plants and animals, stand for some facts that are, or have
been, known, and for a great many other similar ones that have not been,
and never will be, known. The use of a general term depends not upon our
direct knowledge of everything comprised in its denotation, but upon our
readiness to apply it to anything that has its connotation, whether we
have seen the thing or not, and even though we never can perceive it; as
when a man talks freely of the ichthyosaurus, or of the central heat of
planets, or of atoms and ether.
Hence Universal Propositions, which consist of general terms, deceive
us, if we suppose that their predicates are directly known to be related
to all the facts denoted by their subjects. In exceptional cases, in
which the denotation of a subject is intentionally limited, such
exhaustive direct knowledge may be possible; as that "all the bones of a
certain animal consist of phosphate of lime," or that every member of
the present Parliament wears a silk hat. But what predication is
possible concerning the hats of all members of Parliament from the
beginning? Ordinarily, then, whilst the relation of predicate to subject
has been observed in some cases, in much the greater number of cases
our belief about it depends upon something besides observation, or may
be said (in a certain sense) to be taken on trust.
'All rabbits are herbivorous': why do we believe that? We may have seen
a few wild rabbits feeding: or have kept tame ones, and tried
experiments with their diet; or have read of their habits in a book of
Natural History; or have studied the anatomy and physiology of the
digestive system in many sorts of animals: but with whatever care we add
testimony and scientific method to our own observation, it still remains
true that the rabbits observed by ourselves and others are few in
comparison with those that live, have lived and will live. Similarly of
any other universal proposition; that it 'goes beyond the evidence' of
direct observation plainly follows from the fact that the general terms,
of which such propositions consist, are never exhaustively known in
their denotation. What right have we then to state Universal
Propositions? That is the problem of Inductive Logic.
Sec. 3. Universal Propositions, of course, cannot always be proved by
syllogisms; because to prove a universal proposition by a syllogism, its
premises must be universal propositions; and, then, these must be proved
by others. This process may sometimes go a little way, thus: _All men
are mortal_, because _All animals are_; and _All animals are mortal_,
because _All composite bodies are subject to dissolution._ Were there no
limit to such sorites, proof would always involve a _regressus ad
infinitum_, for which life is too short; but, in fact, prosyllogisms
soon fail us.
Clearly, the form of the Syllogism must itself be misleading if the
universal proposition is so: if we think that premises prove the
conclusion because they themselves have been established by detailed
observation, we are mistaken. The consideration of any example will show
this. Suppose any one to argue:
All ruminants are herbivorous;
Camels are ruminants:
.'. Camels are herbivorous.
Have we, then, examined all ruminants? If so, we must have examined all
camels, and cannot need a syllogism to prove their herbivorous nature:
instead of the major premise proving the conclusion, the proof of the
conclusion must then be part of the proof of the major premise. But if
we have not examined all ruminants, having omitted most giraffes, most
deer, most oxen, etc., how do we know that the unexamined (say, some
camels) are not exceptional? Camels are vicious enough to be
carnivorous; and indeed it is said that Bactrian camels will eat flesh
rather than starve, though of course their habit is herbivorous.
Or, again, it is sometimes urged that--
All empires decay:
.'. Britain will decay.
This is manifestly a prediction: at present Britain flourishes, and
shows no signs of decay. Yet a knowledge of its decay seems necessary,
to justify any one in asserting the given premise. If it is a question
whether Britain will decay, to attempt (while several empires still
flourish) to settle the matter by asserting that _all_ empires decay,
seems to be 'a begging of the question.' But although this latter case
is a manifest prediction, it does not really differ from the former one;
for the proof that camels are herbivorous has no limits in time. If
valid, it shows not only that they are, but also that they will be,
herbivorous.
Hence, to resort to a dilemma, it may be urged: If _all_ the facts of
the major premise of any syllogism have been examined, the syllogism is
needless; and if _some_ of them have not been examined, it is a _petitio
principii_. But either all have been examined, or some have not.
Therefore; the syllogism is either useless or fallacious.
Sec. 4. A way of escape from this dilemma is provided by distinguishing
between the formal and material aspects of the syllogism considered as a
means of proof. It begs the question formally, but not materially; that
is to say, if it be a question whether camels are herbivorous, and to
decide it we are told that '_all_ ruminants are,' laying stress upon the
'all,' as if all had been examined, though in fact camels have not been,
then the question as to camels is begged. The form of a universal
proposition is then offered as evidence, when in fact the evidence has
not been universally ascertained. But if in urging that 'all ruminants
are herbivorous' no more is meant than that so many other ruminants of
different species are known to be herbivorous, and that the ruminant
stomach is so well adapted to a coarse vegetable diet, that the same
habit may be expected in other ruminants, such as camels, the argument
then rests upon material evidence without unfairly implying the case in
question. Now the nature of the material evidence is plainly this, that
the resemblance of camels to deer, oxen, etc., in chewing the cud,
justifies us in believing that they have a further resemblance in
feeding on herbs; in other words, we assume that _resemblance is a
ground of inference_.
Another way of putting this difficulty which we have just been
discussing, with regard to syllogistic evidence, is to urge that by the
Laws of Syllogism a conclusion must never go beyond the premises, and
that therefore no progress in knowledge can ever be established, except
by direct observation. Now, taking the syllogism formally, this is true:
if the conclusion go beyond the premises, there must be either four
terms, or illicit process of the major or minor term. But, taking it
materially, the conclusion may cover facts which were not in view when
the major premise was laid down; facts of which we predicate something
not as the result of direct observation, but because they resemble in a
certain way those facts which had been shown to carry the predicate when
the major premise was formed.
'What sort of resemblance is a sufficient ground of inference?' is,
therefore, the important question alike in material Deduction and in
Induction; and in endeavouring to answer it we shall find that the
surest ground of inference is resemblance of causation. For example, it
is due to causation that ruminants are herbivorous. Their instincts make
them crop the herb, and their stomachs enable them easily to digest it;
and in these characters camels are like the other ruminants.
Sec. 5. In ch. ix, Sec. 3, the _Dictum de omni et nullo_ was stated: 'Whatever
may be predicated of a term distributed may be predicated of anything
that can be identified with that term.' Nothing was there said (as
nothing was needed) of the relations that might be implied in the
predication. But now that it comes to the ultimate validity of
predication, we must be clear as to what these relations are; and it
will also be convenient to speak no longer of terms, as in Formal Logic,
but of the things denoted. What relations, then, can be determined
between concrete facts or phenomena (physical or mental) with the
greatest certainty of general truth; and what axioms are there that
sanction mediate inferences concerning those relations?
In his _Logic_ (B. II. c. 2, Sec. 3) Mill gives as the axiom of syllogistic
reasoning, instead of the _Dictum_: "A thing which co-exists with
another thing, which other co-exists with a third thing, also co-exists
with that third thing." Thus the peculiar properties of Socrates
co-exist with the attributes of man, which co-exist with mortality:
therefore, Socrates is mortal. But, again, he says that the ground of
the syllogism is Induction; that man is mortal is an induction. And,
further, the ground of Induction is causation; the law of causation is
the ultimate major premise of every sound induction. Now causation is
the principle of the succession of phenomena: how, then, can the
syllogism rest on an axiom concerning co-existence? On reflection, too,
it must appear that 'Man is mortal' predicates causation: the human
constitution issues in death.
The explanation of this inconsistency may perhaps be found in the
history of Mill's work. Books I. and II. were written in 1831; but being
unable at that time to explain Induction, he did not write Book III.
until 1837-8. Then, no doubt, he revised the earlier Books, but not
enough to bring his theory of the syllogism into complete agreement with
the theory of Induction; so that the axiom of co-existence was allowed
to stand.
Mill also introduced the doctrine of Natural Kinds as a ground of
Induction supplementary, at least provisionally, to causation; and to
reasoning about Kinds, or Substance and Attribute, his axiom of
co-existence is really adapted. Kinds are groups of things that agree
amongst themselves and differ from all others in a multitude of
qualities: these qualities co-exist, or co-inhere, with a high degree of
constancy; so that where some are found others may be inferred. Their
co-inherence is not to be considered an ultimate fact; for, "since
everything which occurs is determined by laws of causation and
collocations of the original causes, it follows that the co-existences
observable amongst effects cannot themselves be the subject of any
similar set of laws distinct from laws of causation" (B. III. c. 5, Sec.
9). According to the theory of evolution (worked out since Mill wrote),
Kinds--that is, species of plants, animals and minerals--with their
qualities are all due to causation. Still, as we can rarely, or never,
trace the causes with any fullness or precision, a great deal of our
reasoning, as, e.g., about men and camels, does in fact trust to the
relative permanence of natural Kinds as defined by co-inhering
attributes.
To see this more clearly, we should consider that causation and natural
Kinds are not at present separable; propositions about causation in
concrete phenomena (as distinct from abstract 'forces') always involve
the assumption of Kinds. For example--'Water rusts iron,' or the oxygen
of water combines with iron immersed in it to form rust: this statement
of causation assumes that water, oxygen, iron, and iron-rust are known
Kinds. On the other hand, the constitution of every concrete thing, and
manifestly of every organised body, is always undergoing change, that
is, causation, upon which fact its properties depend.
How, then, can we frame principles of mediate reasoning, about such
things? So far as we consider them as Kinds, it is enough to say:
_Whatever can be identified as a specimen of a known substance or Kind
has the properties of that Kind_. So far as we consider them as in the
relation of causation, we may say: _Whatever relation of events can be
identified with the relation of cause and effect is constant_. And these
principles may be generalised thus: _Whatever is constantly related to a
phenomenon (cause or Kind), determined by certain characters, is related
in the same way to any phenomenon, that has the same characters_. Taking
this as axiom of the syllogism materially treated, we see that
herbivorousness, being constantly related to ruminants, is constantly
related to camels; mortality to man and, therefore, to Socrates; rusting
to the immersion of iron in water generally and, therefore, to this
piece of iron. _Nota notae, nota rei ipsius_ is another statement of the
same principle; still another is Mill's axiom, "Whatever has a mark has
what it is a mark of." A mark is anything (A) that is never found
without something else (B)--a phenomenon constantly related to another
phenomenon--so that wherever A is found, B may be expected: human nature
is a mark of mortality.
Sec. 6. The Syllogism has sometimes been discarded by those who have only
seen that, as formally stated, it is either useless or fallacious: but
those who also perceive its material grounds retain and defend it. In
fact, great advantages are gained by stating an argument as a formal
syllogism. For, in the first place, we can then examine separately the
three conditions on which the validity of the argument depends:
(1) Are the Premises so connected that, _if they are true_, the
Conclusion follows? This depends upon the formal principles of chap. x.
(2) Is the Minor Premise true? This question can only arise when the
minor premise is a real proposition; and then it may be very difficult
to answer. Water rusts iron; but is the metal we are now dealing with a
fair specimen of iron? Few people, comparatively, know how to determine
whether diamonds, or even gold or silver coins, are genuine. That
_Camels are ruminants_ is now a verbal proposition to a Zoologist, but
not to the rest of us; and to the Zoologist the ascertaining of the
relation in which camels stand to such ruminants as oxen and deer, was
not a matter of analysing words but of dissecting specimens. What a long
controversy as to whether the human race constitutes a Family of the
Primates! That 'the British Empire is an empire' affords no matter for
doubt or inquiry; but how difficult to judge whether the British Empire
resembles Assyria, Egypt, Rome, Spain in those characters and
circumstances that caused their downfall!
(3) Is the Major Premise true? Are all ruminants herbivorous? If there
be any exceptions to the rule, camels are likely enough to be among the
exceptions. And here the need of Inductive Logic is most conspicuous:
how can we prove our premises when they are universal propositions?
Universal propositions, however, are also involved in proving the minor
premise: to prove a thing to be iron, we must know the constant
reactions of iron.
A second advantage of the syllogism is, that it makes us fully aware of
what an inference implies. An inference must have some grounds, or else
it is a mere prejudice; but whatever the grounds, if sufficient in a
particular case, they must be sufficient for all similar cases, they
must admit of being generalised; and to generalise the grounds of the
inference, is nothing else than to state the major premise. If the
evidence is sufficient to justify the argument that camels are
herbivorous _because_ they are ruminants, it must also justify the major
premise, _All ruminants are herbivorous_; for else the inference cannot
really depend merely upon the fact of ruminating. To state our evidence
syllogistically, then, must be possible, if the evidence is mediate and
of a logical kind; and to state it in this formal way, as depending on
the truth of a general principle (the major premise), increases our
sense of responsibility for the inference that is thus seen to imply so
much; and if any negative instances lie within our knowledge, we are the
more likely to remember them. The use of syllogisms therefore tends to
strengthen our reasonings.
A third advantage is, that to formulate an accurate generalisation may
be useful to others: it is indeed part of the systematic procedure of
science. The memoranda of our major premises, or reasons for believing
anything, may be referred to by others, and either confirmed or refuted.
When such a memorandum is used for further inferences, these inferences
are said, in the language of Formal Logic, to be drawn _from_ it, as if
the conclusion were contained in our knowledge of the major premise;
but, considering the limited extent of the material evidence, it is
better to say that the inference is drawn _according to_ the memorandum
or major premise, since the grounds of the major premise and of the
conclusion are in fact the same (Mill: _Logic_, B. II. c. 3). Inductive
proofs may be stated in Syllogisms, and inductive inferences are drawn
_according to_ the Law of Causation.
Sec. 7. To assume that resemblance is a ground of inference, and that
substance and attribute, or cause and effect, are phenomena constantly
related, implies belief in the Uniformity of Nature. The Uniformity of
Nature cannot be defined, and is therefore liable to be misunderstood.
In many ways Nature seems not to be uniform: there is great variety in
the sizes, shapes, colours and all other properties of things: bodies
falling in the open air--pebbles, slates, feathers--descend in different
lines and at different rates; the wind and weather are proverbially
uncertain; the course of trade or of politics, is full of surprises. Yet
common maxims, even when absurd, testify to a popular belief that the
relations of things are constant: the doctrine of St. Swithin and the
rhyme beginning 'Evening red and morning grey,' show that the weather is
held to be not wholly unpredictable; as to human affairs, it is
said that 'a green Yule makes a fat churchyard,' that 'trade follows the
flag,' and that 'history repeats itself'; and Superstition knows that
witches cannot enter a stable-door if a horse-shoe is nailed over it,
and that the devil cannot cross a threshold inscribed with a perfect
pentagram. But the surest proof of a belief in the uniformity of nature
is given by the conduct of men and animals; by that adherence to habit,
custom and tradition, to which in quiet times they chiefly owe their
safety, but which would daily disappoint and destroy them, if it were
not generally true that things may be found where they have been left
and that in similar circumstances there are similar events.
Now this general belief, seldom distinctly conceived, for the most part
quite unconscious (as a principle), merely implied in what men do, is
also the foundation of all the Sciences; which are entirely occupied in
seeking the Laws (that is, the Uniformities) of Nature. As the
uniformity of nature cannot be defined, it cannot be proved; the most
convincing evidence in its favour is the steady progress made by Science
whilst trusting in it. Nevertheless, what is important is not the
comprehensive but indeterminate notion of Uniformity so much as a number
of First Principles, which may be distinguished in it as follows:
(1) The Principles of Contradiction and Excluded Middle (ch. vi. Sec. 3)
declare that in a given relation to a given phenomenon any two or more
other phenomena are incompatible (_B is not A and a_); whilst the given
phenomenon either stands related to another phenomenon or not (_B is
either A or a_). It is not only a matter of Logic but of fact that, if a
leaf is green, it is not under the same conditions red or blue, and that
if it is not green it is some other colour.
(2) Certain Axioms of Mediate Evidence: as, in Mathematics, 'that
magnitudes equal to the same magnitude are equal to one another'; and,
in Logic, the _Dictum_ or its material equivalent.
(3) That all Times and all Spaces are commensurable; although in certain
relations of space (as [pi]) the unit of measurement must be infinitely
small.--If Time really trotted with one man and galloped with another,
as it seems to; if space really swelled in places, as De Quincey dreamed
that it did; life could not be regulated, experience could not be
compared and science would be impossible. The Mathematical Axioms would
then never be applicable to space or time, or to the objects or
processes that fill them.
(4) The Persistence of Matter and Energy: the physical principle that,
in all changes of the universe, the quantities of Matter and Energy
(actual and potential, so-called) remain the same.--For example, as to
matter, although dew is found on the grass at morning without any
apparent cause, and although a candle seems to burn away to a scrap of
blackened wick, yet every one knows that the dew has been condensed from
vapour in the air, and that the candle has only turned into gas and
smoke. As to energy, although a stone thrown up to the housetop and
resting there has lost actual energy, it has gained such a position that
the slightest touch may bring it to the earth again in the same time as
it took to travel upwards; so on the house-top it is said to have
potential energy. When a boiler works an engine, every time the piston
is thrust forward (mechanical energy), an equivalent in heat (molecular
energy) is lost. But for the elucidation of these principles, readers
must refer to treatises of Chemistry and Physics.
(5) Causation, a special form of the foregoing principles of the
persistence of matter and energy, we shall discuss in the next chapter.
It is not to be conceived of as anything occult or noumenal, but merely
as a special mode of the uniformity of Nature or experience.
(6) Certain Uniformities of Co-existence; but for want of a general
principle of Co-existence, corresponding to Causation (the principle of
Succession), we can only classify these uniformities as follows:
(a) The Geometrical; as that, in a four-sided figure, if the opposite
angles are equal, the opposite sides are equal and parallel.--Countless
similar uniformities of co-existence are disclosed by Geometry. The
co-existent facts do not cause one another, nor are they jointly caused
by something else; they are mutually involved: such is the nature of
space.
(b) Universal co-inherences among the properties of concrete
things.--The chief example is the co-inherence of gravity with inertia
in all material bodies. There is, I believe, no other entirely
satisfactory case; but some good approximations to such uniformity are
known to physical science.
(c) Co-existence due to Causation; such as the positions of objects in
space at any time.--The houses of a town are where they are, because
they were put there; and they remain in their place as long as no other
causes arise strong enough to remove or destroy them. Similarly, the
relative positions of rocks in geological strata, and of trees in a
forest, are due to causes.
(d) The co-inherence of properties in Natural Kinds; which we call the
constitution, defining characters, or specific nature of such
things.--Oxygen, platinum, sulphur and the other elements; water, common
salt, alcohol and other compounds; the various species of plants and
animals: all these are known to us as different groups of co-inherent
properties. It may be conjectured that these groupings of properties are
also due to causation, and sometimes the causes can be traced: but very
often the causes are still unknown; and, until resolved into their
causes, they must be taken as necessary data in the investigation of
nature. Laws of the co-inherence of the properties of Kinds do not, like
laws of causation, admit of methodical proof upon their own principles,
but only by constancy in experience and statistical probability (c. xix,
Sec. 4).
(e) There are also a few cases in which properties co-exist in an
unaccountable way, without being co-extensive with any one species,
genus, or order: as most metals are whitish, and scarlet flowers are
wanting in fragrance. (On this Sec. 7, see Venn's _Empirical Logic_, c. 4.)
Sec. 8. Inasmuch as Axioms of Uniformity are ultimate truths, they cannot
be deduced; and inasmuch as they are universal, no proof by experience
can ever be adequate. The grounds of our belief in them seem to be
these:
(1) Every inference takes for granted an order of Nature corresponding
with it; and every attempt to explain the origin of anything assumes
that it is the transformation of something else: so that uniformity of
order and conservation of matter and energy are necessary
presuppositions of reasoning.
(2) On the rise of philosophic reflection, these tacit presuppositions
are first taken as dogmas, and later as postulates of scientific
generalisation, and of the architectonic unification of science. Here
they are indispensable.
(3) The presuppositions or postulates are, in some measure, verifiable
in practical life and in scientific demonstration, and the better
verifiable as our methods become more exact.
(4) There is a cause of this belief that cannot be said to contain any
evidence for it, namely, the desire to find in Nature a foundation for
confidence in our own power to foresee and to control events.