THE CANONS OF DIRECT INDUCTION
Sec. 1. Let me begin by borrowing an example from Bain (_Logic_: B. III. c.
6). The North-East wind is generally detested in this country: as long
as it blows few people feel at their best. Occasional well-known causes
of a wind being injurious are violence, excessive heat or cold,
excessive dryness or moisture, electrical condition, the being laden
with dust or exhalations. Let the hypothesis be that the last is the
cause of the North-East wind's unwholesome quality; since we know it is
a ground current setting from the pole toward the equator and bent
westward by the rotation of the earth; so that, reaching us over
thousands of miles of land, it may well be fraught with dust, effluvia,
and microbes. Now, examining many cases of North-East wind, we find that
this is the only circumstance in which all the instances agree: for it
is sometimes cold, sometimes hot; generally dry, but sometimes wet;
sometimes light, sometimes violent, and of all electrical conditions.
Each of the other circumstances, then, can be omitted without the N.E.
wind ceasing to be noxious; but one circumstance is never absent,
namely, that it is a ground current. That circumstance, therefore, is
probably the cause of its injuriousness. This case illustrates:--
(I) THE CANON OF AGREEMENT.
_If two or more instances of a phenomenon under investigation have only
one other circumstance (antecedent or consequent) in common, that
circumstance is probably the cause (or an indispensable condition) or
the effect of the phenomenon, or is connected with it by causation._
This rule of proof (so far as it is used to establish direct causation)
depends, first, upon observation of an invariable connection between the
given phenomenon and one other circumstance; and, secondly, upon I. (a)
and II. (b) among the propositions obtained from the unconditionality of
causation at the close of the last chapter.
To prove that A is causally related to _p_, suppose two instances of the
occurrence of A, an antecedent, and _p_, a consequent, with concomitant
facts or events--and let us represent them thus:
Antecedents: A B C A D E
Consequents: _p q r_ _p s t_;
and suppose further that, in this case, the immediate succession of
events can be observed. Then A is probably the cause, or an
indispensable condition, of _p_. For, as far as our instances go, A is
the invariable antecedent of _p_; and _p_ is the invariable consequent
of A. But the two instances of A or _p_ agree in no other circumstance.
Therefore A is (or completes) the unconditional antecedent of _p_. For B
and C are not indispensable conditions of _p_, being absent in the
second instance (Rule II. (b)); nor are D and E, being absent in the
first instance. Moreover, _q_ and _r_ are not effects of A, being absent
in the second instance (Rule II. (d)); nor are _s_ and _t_, being absent
in the first instance.
It should be observed that the cogency of the proof depends entirely
upon its tending to show the unconditionality of the sequence A-_p_, or
the indispensability of A as a condition of _p_. That _p_ follows A,
even immediately, is nothing by itself: if a man sits down to study and,
on the instant, a hand-organ begins under his window, he must not infer
malice in the musician: thousands of things follow one another every
moment without traceable connection; and this we call 'accidental.' Even
invariable sequence is not enough to prove direct causation; for, in
our experience does not night invariable follow day? The proof requires
that the instances be such as to show not merely what events _are_ in
invariable sequence, but also what _are not_. From among the occasional
antecedents of _p_ (or consequents of A) we have to eliminate the
accidental ones. And this is done by finding or making 'negative
instances' in respect of each of them. Thus the instance
A D E
_p s t_
is a negative instance of B and C considered as supposable causes of _p_
(and of _q_ and _r_ as supposable effects of A); for it shows that they
are absent when _p_ (or A) is present.
To insist upon the cogency of 'negative instances' was Bacon's great
contribution to Inductive Logic. If we neglect them, and merely collect
examples of the sequence A-_p_, this is 'simple enumeration'; and
although simple enumeration, when the instances of agreement are
numerous enough, may give rise to a strong belief in the connection of
phenomena, yet it can never be a methodical or logical proof of
causation, since it does not indicate the unconditionalness of the
sequence. For simple enumeration of the sequence A-_p_ leaves open the
possibility that, besides A, there is always some other antecedent of
_p_, say X; and then X may be the cause of _p_. To disprove it, we must
find, or make, a negative instance of X--where _p_ occurs, but X is
absent.
So far as we recognise the possibility of a plurality of causes, this
method of Agreement cannot be quite satisfactory. For then, in such
instances as the above, although D is absent in the first, and B in the
second, it does not follow that they are not the causes of _p_; for they
may be alternative causes: B may have produced _p_ in the first
instance, and D in the second; A being in both cases an accidental
circumstance in relation to _p_. To remedy this shortcoming by the
method of Agreement itself, the only course is to find more instances of
_p_. We may never find a negative instance of A; and, if not, the
probability that A is the cause of _p_ increases with the number of
instances. But if there be no antecedent that we cannot sometimes
exclude, yet the collection of instances will probably give at last all
the causes of _p_; and by finding the proportion of instances in which
A, B, or X precedes _p_, we may estimate the probability of any one of
them being the cause of _p_ in any given case of its occurrence.
But this is not enough. Since there cannot really be vicarious causes,
we must define the effect (_p_) more strictly, and examine the cases to
find whether there may not be varieties of _p_, with each of which one
of the apparent causes is correlated: A with _p_^{1} B with _p_^{11}, X
with _p_^{111}. Or, again, it may be that none of the recognised
antecedents is effective: as we here depend solely on observation, the
true conditions may be so recondite and disguised by other phenomena as
to have escaped our scrutiny. This may happen even when we suppose that
the chief condition has been isolated: the drinking of foul water was
long believed to cause dysentery, because it was a frequent antecedent;
whilst observation had overlooked the bacillus, which was the
indispensable condition.
Again, though we have assumed that, in the instances supposed above,
immediate sequence is observable, yet in many cases it may not be so, if
we rely only on the canon of Agreement; if instances cannot be obtained
by experiment, and we have to depend on observation. The phenomena may
then be so mixed together that A and _p_ seem to be merely concomitant;
so that, though connection of some sort may be rendered highly probable,
we may not be able to say which is cause and which is effect. We must
then try (as Bain says) to trace the expenditure of energy: if _p_ gains
when A loses, the course of events if from A to _p_.
Moreover, where succession cannot be traced, the method of Agreement may
point to a connection between two or more facts (perhaps as co-effects
of a remote cause) where direct causation seems to be out of the
question: e.g., that Negroes, though of different tribes, different
localities, customs, etc., are prognathous, woolly-haired and
dolichocephalic.
The Method of Agreement, then, cannot by itself prove causation. Its
chief use (as Mill says) is to suggest hypotheses as to the cause; which
must then be used (if possible) experimentally to try if it produces the
given effect. A bacillus, for example, being always found with a certain
disease, is probably the chief condition of it: give it to a guinea-pig,
and observe whether the disease appears in that animal.
Men often use arguments which, if they knew it, might be shown to
conform more or less to this canon; for they collect many instances to
show that two events are connected; but usually neglect to bring out the
negative side of the proof; so that their arguments only amount to
simple enumeration. Thus Ascham in his _Toxophilus_, insisting on the
national importance of archery, argues that victory has always depended
on superiority in shooting; and, to prove it, he shows how the Parthians
checked the Romans, Sesostris conquered a great part of the known world,
Tiberius overcame Arminius, the Turks established their empire, and the
English defeated the French (with many like examples)--all by superior
archery. But having cited these cases to his purpose, he is content;
whereas he might have greatly strengthened his proof by showing how one
or the other instance excludes other possible causes of success. Thus:
the cause was not discipline, for the Romans were better disciplined
than the Parthians; nor yet the boasted superiority of a northern
habitat, for Sesostris issued from the south; nor better manhood, for
here the Germans probably had the advantage of the Romans; nor superior
civilisation, for the Turks were less civilised than most of those they
conquered; nor numbers, nor even a good cause, for the French were more
numerous than the English, and were shamefully attacked by Henry V. on
their own soil. Many an argument from simple enumeration may thus be
turned into an induction of greater plausibility according to the Canon
of Agreement.
Still, in the above case, the effect (victory) is so vaguely conceived,
that a plurality of causes must be allowed for: although, e.g.,
discipline did not enable the Romans to conquer the Parthians, it may
have been their chief advantage over the Germans; and it was certainly
important to the English under Henry V. in their war with the French.
Here is another argument, somewhat similar to the above, put forward by
H. Spencer with a full consciousness of its logical character. States
that make war their chief object, he says, assume a certain type of
organisation, involving the growth of the warrior class and the
treatment of labourers as existing solely to sustain the warriors; the
complete subordination of individuals to the will of the despotic
soldier-king, their property, liberty and life being at the service of
the State; the regimentation of society not only for military but also
for civil purposes; the suppression of all private associations, etc.
This is the case in Dahomey and in Russia, and it was so at Sparta, in
Egypt, and in the empire of the Yncas. But the similarity of
organisation in these States cannot have been due to race, for they are
all of different races; nor to size, for some are small, some large; nor
to climate or other circumstances of habitat, for here again they differ
widely: the one thing they have in common is the military purpose; and
this, therefore, must be the cause of their similar organisation.
(_Political Institutions._)
By this method, then, to prove that one thing is causally connected with
another, say A with _p_, we show, first, that in all instances of _p_, A
is present; and, secondly, that any other supposable cause of _p_ may be
absent without disturbing _p_. We next come to a method the use of which
greatly strengthens the foregoing, by showing that where _p_ is absent
A is also absent, and (if possible) that A is the only supposable cause
that is always absent along with _p_.
Sec. 2. THE CANON OF THE JOINT METHOD OF AGREEMENT IN PRESENCE AND IN
ABSENCE.
_If_ (1) _two or more instances in which a phenomenon occurs have only
one other circumstance (antecedent or consequent) in common, while_ (2)
_two or more instances in which it does not occur (though in important
points they resemble the former set of instances) have nothing else in
common save the absence of that circumstance--the circumstance in which
alone the two sets of instances differ throughout (being present in the
first set and absent in the second) is probably the effect, or the
cause, or an indispensable condition of the phenomenon._
The first clause of this Canon is the same as that of the method of
Agreement, and its significance depends upon the same propositions
concerning causation. The second clause, relating to instances in which
the phenomenon is absent, depends for its probative force upon Prop. II.
(a), and I. (b): its function is to exclude certain circumstances (whose
nature or manner of occurrence gives them some claim to consideration)
from the list of possible causes (or effects) of the phenomenon
investigated. It might have been better to state this second clause
separately as the Canon of the Method of Exclusions.
To prove that A is causally related to _p_, let the two sets of
instances be represented as follows:
Instances of Presence. Instances of Absence.
A B C C H F
_p q r_ _r x v_
A D E B D K
_p s t_ _q y s_
A F G E G M
_p u v_ _t f u_
Then A is probably the cause or a condition of _p_, or _p_ is dependent
upon A: first, by the Canon of Agreement in Presence, as represented by
the first set of instances; and, secondly, by Agreement in Absence in
the second set of instances. For there we see that C, H, F, B, D, K, E,
G, M occur without the phenomenon _p_, and therefore (by Prop. II. (a))
are not its cause, or not the whole cause, unless they have been
counteracted (which is a point for further investigation). We also see
that _r, v, q, s, t, u_ occur without A, and therefore are not the
effects of A. And, further, if the negative instances represent all
possible cases, we see that (according to Prop. I. (b)) A is the cause
of _p_, because it cannot be omitted without the cessation of _p_. The
inference that A and _p_ are cause and effect, suggested by their being
present throughout the first set of instances, is therefore strengthened
by their being both absent throughout the second set.
So far as this Double Method, like the Single Method of Agreement,
relies on observation, sequence may not be perceptible in the instances
observed, and then, direct causation cannot be proved by it, but only
the probability of causal connection; and, again, the real cause, though
present, may be so obscure as to evade observation. It has, however, one
peculiar advantage, namely, that if the second list of instances (in
which the phenomenon and its supposed antecedent are both absent) can be
made exhaustive, it precludes any hypothesis of a plurality of causes;
since all possible antecedents will have been included in this list
without producing the phenomenon. Thus, in the above symbolic example,
taking the first set of instances, the supposition is left open that B,
C, D, E, F, G may, at one time or another, have been a condition of _p_;
but, in the second list, these antecedents all occur, here or there,
without producing _p_, and therefore (unless counteracted somehow)
cannot be a condition of _p_. A, then, stands out as the one thing that
is present whenever _p_ is present, and absent whenever _p_ is absent.
Stated in this abstract way, the Double Method may seem very elaborate
and difficult; yet, in fact, its use may be very simple. Tyndall, to
prove that dispersed light in the air is due to motes, showed by a
number of cases (1) that any gas containing motes is luminous; (2) that
air in which the motes had been destroyed by heat, and any gas so
prepared as to exclude motes, are not luminous. All the instances are of
gases, and the result is: motes--luminosity; no motes--no luminosity.
Darwin, to show that cross-fertilisation is favourable to flowers,
placed a net about 100 flower-heads, and left 100 others of the same
varieties exposed to the bees: the former bore no seed, the latter
nearly 3,000. We must assume that, in Darwin's judgment, the net did not
screen the flowers from light and heat sufficiently to affect the
result.
There are instructive applications of this Double Method in Wallace's
_Darwinism_. In chap. viii., on _Colour in Animals_, he observes, that
the usefulness of their coloration to animals is shown by the fact that,
"as a rule, colour and marking are constant in each species of wild
animal, while, in almost every domesticated animal, there arises great
variability. We see this in our horses and cattle, our dogs and cats,
our pigeons and poultry. Now the essential difference between the
conditions of life of domesticated and wild animals is, that the former
are protected by man, while the latter have to protect themselves." Wild
animals protect themselves by acquiring qualities adapted to their mode
of life; and coloration is a very important one, its chief, though not
its only use, being concealment. Hence a useful coloration having been
established in any species, individuals that occasionally may vary from
it, will generally, perish; whilst, among domestic animals, variation of
colour or marking is subject to no check except the taste of owners. We
have, then, two lists of instances; first, innumerable species of wild
animals in which the coloration is constant and which depend upon their
own qualities for existence; secondly, several species of domestic
animals in which the coloration is _not_ constant, and which do _not_
depend upon their own qualities for existence. In the former list two
circumstances are present together (under all sorts of conditions); in
the latter they are absent together. The argument may be further
strengthened by adding a third list, parallel to the first, comprising
domestic animals in which coloration is approximately constant, but
where (as we know) it is made a condition of existence by owners, who
only breed from those specimens that come up to a certain standard of
coloration.
Wallace goes on to discuss the colouring of arctic animals. In the
arctic regions, he says, some animals are wholly white all the year
round, such as the polar bear, the American polar hare, the snowy owl
and the Greenland falcon: these live amidst almost perpetual snow.
Others, that live where the snow melts in summer, only turn white in
winter, such as the arctic hare, the arctic fox, the ermine and the
ptarmigan. In all these cases the white colouring is useful, concealing
the herbivores from their enemies, and also the carnivores in
approaching their prey; this usefulness, therefore, is a condition of
the white colouring. Two other explanations have, however, been
suggested: first, that the prevalent white of the arctic regions
directly colours the animals, either by some photographic or chemical
action on the skin, or by a reflex action through vision (as in the
chameleon); secondly, that a white skin checks radiation and keeps the
animals warm. But there are some exceptions to the rule of white
colouring in arctic animals which refute these hypotheses, and confirm
the author's. The sable remains brown throughout the winter; but it
frequents trees, with whose bark its colour assimilates. The musk-sheep
is brown and conspicuous; but it is gregarious, and its safety depends
upon its ability to recognise its kind and keep with the herd. The raven
is always black; but it fears no enemy and feeds on carrion, and
therefore does not need concealment for either defence or attack. The
colour of the sable, then, though not white, serves for concealment; the
colour of the musk-sheep serves a purpose more important than
concealment; the raven needs no concealment. There are thus two sets of
instances:--in one set the animals are white (a) all the year, (b) in
winter; and white conceals them (a) all the year, (b) in winter; in the
other set, the animals are _not_ white, and to them either whiteness
would _not_ give concealment, or concealment would _not_ be
advantageous. And this second list refutes the rival hypotheses: for the
sable, the musk-sheep and the raven are as much exposed to the glare of
the snow, and to the cold, as the other animals are.
Sec. 3. THE CANON OF DIFFERENCE.
_If an instance in which a phenomenon occurs, and an instance in which
it does not occur, have every other circumstance in common save one,
that one (whether consequent or antecedent) occurring only in the
former; the circumstance in which alone the two instances differ is the
effect, or the cause, or an indispensable condition of the phenomenon._
This follows from Props. I (a) and (b), in chapter xv. Sec. 7. To prove
that A is a condition of _p_, let two instances, such as the Canon
requires, be represented thus:
A B C B C
_p q r_ _q r_
Then A is the cause or a condition of _p_. For, in the first instance, A
being introduced (without further change), _p_ arises (Prop. I. (a));
and, in the second instance, A having been removed (without other
change), _p_ disappears (Prop. I. (b)). Similarly we may prove, by the
same instances, that _p_ is the effect of A.
The order of the phenomena and the immediacy of their connection is a
matter for observation, aided by whatever instruments and methods of
inspection and measurement may be available.
As to the invariability of the connection, it may of course be tested by
collecting more instances or making more experiments; but it has been
maintained, that a single perfect experiment according to this method is
sufficient to prove causation, and therefore implies invariability
(since causation is uniform), though no other instances should ever be
obtainable; because it establishes once for all the unconditionality of
the connection
A B C
_p q r_.
Now, formally this is true; but in any actual investigation how shall we
decide what is a satisfactory or perfect experiment? Such an experiment
requires that in the negative instance
B C
_q r_,
BC shall be the least assemblage of conditions necessary to co-operate
with A in producing _p_; and that it is so cannot be ascertained without
either general prior knowledge of the nature of the case or special
experiments for the purpose. So that invariability will not really be
inferred from a single experiment; besides that every prudent inquirer
repeats his experiments, if only to guard against his own liability to
error.
The supposed plurality of causes does not affect the method of
Difference. In the above symbolic case, A is clearly _one_ cause (or
condition) of _p_, whatever other causes may be possible; whereas with
the Single Method of Agreement, it remained doubtful (admitting a
plurality of causes) whether A, in spite of being always present with
_p_, was ever a cause or condition of it.
This method of Difference without our being distinctly aware of it, is
oftener than any other the basis of ordinary judgments. That the sun
gives light and heat, that food nourishes and fire burns, that a stone
breaks a window or kills a bird, that the turning of a tap permits or
checks the flow of water or of gas, and thousands of other propositions
are known to be true by rough but often emphatic applications of this
method in common experience.
The method of Difference may be applied either (1) by observation, on
finding two instances (distinct assemblages of conditions) differing
only in one phenomenon together with its antecedent or consequent; or
(2) by experiment, and then, either (a) by preparing two instances that
may be compared side by side, or (b) by taking certain conditions, and
then introducing (or subtracting) some agent, supposed to be the cause,
to see what happens: in the latter case the "two instances" are the same
assemblage of conditions considered before and, again, after, the
introduction of the agent. As an example of (a) there is an experiment
to show that radium gives off heat: take two glass tubes, in one put
some chloride of radium, in both thermometers, and close them with
cotton-wool. Soon the thermometer in the tube along with radium reads
54 deg. F. higher than the other one. The tube without the radium, whose
temperature remains unaltered, is called the "control" experiment. Most
experiments are of the type (b); and since the Canon, which describes
two co-existing instances, does not readily apply to this type, an
alternative version may be offered: _Any agent whose introduction into
known circumstances (without further change) is immediately followed by
a definite phenomenon is a condition of the occurrence of that
phenomenon._
The words _into known circumstances_ are necessary to emphasise what is
required by this Method, namely, that the two instances differ in only
one thing; for this cannot be ascertained unless all the other
conditions are known; and this further implies that they have been
prepared. It is, therefore, not true (as Sigwart asserts) that this
method determines only one condition of a phenomenon, and that it is
then necessary to inquire into the other conditions. If they were not
known they must be investigated; but then the experiment would not have
been made upon this method. Practically, experiments have to be made in
all degrees of imperfection, and the less perfect they are, that is, the
less the circumstances are known beforehand, the more remains to be
done. A common imperfection is delay, or the occurrence of a latent
period between the introduction of an agent and the manifestation of its
effects; it cannot then be the unconditional cause; though it may be an
indispensable remote condition of whatever change occurs. If, feeling
out of sorts, you take a drug and some time afterwards feel better, it
is not clear on this ground alone that the drug was the cause of
recovery, for other curative processes may have been active
meanwhile--food, or sleep, or exercise.
Any book of Physics or of Chemistry will furnish scores of examples of
the method of Difference: such as Galileo's experiment to show that air
has weight, by first weighing a vessel filled with ordinary air, and
then filling it with condensed air and weighing it again; when the
increased weight can only be due to the greater quantity of air
contained. The melting-point of solids is determined by heating them
until they do melt (as silver at 1000 deg. C., gold at 1250 deg., platinum at
2000 deg.); for the only difference between bodies at the time of melting
and just before is the addition of so much heat. Similarly with the
boiling point of liquids. That the transmission of sound depends upon
the continuity of an elastic ponderable medium, is proved by letting a
clock strike in a vacuum (under a glass from which the air has been
withdrawn by an air pump), and standing upon a non-elastic pedestal:
when the clock be seen to strike, but makes only such a faint sound as
may be due to the imperfections of the vacuum and the pedestal.
The experiments by which the chemical analysis or synthesis of various
forms of matter is demonstrated are simple or compound applications of
this method of Difference, together with the quantitative mark of
causation (that cause and effect are equal); since the bodies resulting
from an analysis are equal in weight to the body analysed, and the body
resulting from a synthesis is equal in weight to the bodies synthesised.
That an electric current resolves water into oxygen and hydrogen may be
proved by inserting the poles of a galvanic battery in a vessel of
water; when this one change is followed by another, the rise of bubbles
from each pole and the very gradual decrease of the water. If the
bubbles are caught in receivers placed over them, it can be shown that
the joint weight of the two bodies of gas thus formed is equal to the
weight of the water that has disappeared; and that the gases are
respectively oxygen and hydrogen may then be shown by proving that they
have the properties of those gases according to further experiments by
the method of Difference; as (e.g.) that one of them is oxygen because
it supports combustion, etc.
When water was first decomposed by the electric current, there appeared
not only oxygen and hydrogen, but also an acid and an alkali. These
products were afterwards traced to impurities of the water and of the
operator's hands. Mill observes that in any experiment the effect, or
part of it, may be due, not to the supposed agent, but to the means
employed in introducing it. We should know not only the other conditions
of an experiment, but that the agent or change introduced is nothing
else than what it is supposed to be.
In the more complex sciences the method of Difference is less easily
applicable, because of the greater difficulty of being sure that only
one circumstance at a time has altered; still, it is frequently used.
Thus, if by dividing a certain nerve certain muscles are paralysed, it
is shown that normally that nerve controls those muscles. That the sense
of smell in flies and cockroaches is connected with the antennae has
been shown by cutting them off: whereupon the insects can no longer find
carrion. In his work on _Earthworms_, Darwin shows that, though
sensitive to mechanical tremors, they are deaf (or, at least, not
sensitive to sonorous vibrations transmitted through the air), by the
following experiment. He placed a pot containing a worm that had come to
the surface, as usual at night, upon a table, whilst close by a piano
was violently played; but the worm took no notice of the noise. He then
placed the pot upon the piano, whilst it was being played, when the
worm, probably feeling mechanical vibrations, hastily slid back into its
burrow.
When, instead of altering one circumstance in an instance (which we have
done our best not otherwise to disturb) and then watching what follows,
we try to find two ready-made instances of a phenomenon, which only
differ in one other circumstance, it is, of course, still more difficult
to be sure that there is only one other circumstance in which they
differ. It may be worth while, however, to look for such instances.
Thus, that the temperature of ocean currents influences the climate of
the shores they wash, seems to be shown by the fact that the average
temperature of Newfoundland is lower than that of the Norwegian coast
some 15 deg. farther north. Both regions have great continents at their
back; and as the mountains of Norway are higher and capped with
perennial snow, we might expect a colder climate there: but the shore of
Norway is visited by the Gulf Stream, whilst the shore of Newfoundland
is traversed by a cold current from Greenland. Again, when in 1841 the
railway from Rouen to Paris was being built, gangs of English and gangs
of French workmen were employed upon it, and the English got through
about one-third more work per man than the French. It was suspected that
this difference was due to one other difference, namely, that the
English fed better, preferring beef to thin soup. Now, logically, it
might have been objected that the evidence was unsatisfactory, seeing
that the men differed in other things besides diet--in 'race' (say),
which explains so much and so easily. But the Frenchmen, having been
induced to try the same diet as the English, were, in a few days, able
to do as much work: so that the "two instances" were better than they
looked. It often happens that evidence, though logically questionable,
is good when used by experts, whose familiarity with the subject makes
it good.
Sec. 4. THE CANON OF CONCOMITANT VARIATIONS.
_Whatever phenomenon varies in any manner whenever another phenomenon
(consequent or antecedent) varies in some particular manner [no other
change having concurred] is either the cause or effect of that
phenomenon [or is connected with it through some fact of causation]._
This is not an entirely fresh method, but may be regarded as a special
case either of Agreement or of Difference, to prove the cause or effect,
not of a phenomenon as a whole, but of some increment of it (positive or
negative). There are certain forces, such as gravitation, heat,
friction, that can never be eliminated altogether, and therefore can
only be studied in their degrees. To such phenomena the method of
Difference cannot be applied, because there are no negative instances.
But we may obtain negative instances of a given quantity of such a
phenomenon (say, heat), and may apply the method of Difference to that
quantity. Thus, if the heat of a body increases 10 degrees, from 60 to
70, the former temperature of 60 was a negative instance in respect of
those 10 degrees; and if only one other circumstance (say, friction)
has altered at the same time, that circumstance (if an antecedent) is
the cause. Accordingly, if in the above Canon we insert, after
'particular manner,' "[no other change having concurred,]" it is a
statement of the method of Difference as applicable to the increment of
a phenomenon, instead of to the phenomenon as a whole; and we may then
omit the last clause--"[or is connected, etc.]." For these words are
inserted to provide for the case of co-effects of a common cause (such
as the flash and report of a gun); but if no other change (such as the
discharge of a gun) has concurred with the variations of two phenomena,
there cannot have been a common cause, and they are therefore cause and
effect.
If, on the other hand, we omit the clause "[no other change having
concurred,]" the Canon is a statement of the method of Agreement as
applicable to the increment of a phenomenon instead of to the phenomenon
as a whole; and it is then subject to the imperfections of that method:
that is to say, it leaves open the possibilities, that an inquirer may
overlook a plurality of causes; or may mistake a connection of two
phenomena, which (like the flash and report of a gun) are co-effects of
a common cause, for a direct relation of cause and effect.
It may occur to the reader that we ought also to distinguish Qualitative
and Quantitative Variations as two orders of phenomena to which the
present method is applicable. But, in fact, Qualitative Variations may
be adequately dealt with by the foregoing methods of Agreement, Double
Agreement, and Difference; because a change of quality or property
entirely gets rid of the former phase of that quality, or substitutes
one for another; as when the ptarmigan changes from brown to white in
winter, or as when a stag grows and sheds its antlers with the course of
the seasons. The peculiar use of the method of Variations, however, is
to formulate the conditions of proof in respect of those causes or
effects which cannot be entirely got rid of, but can be obtained only in
greater or less amount; and such phenomena are or course, quantitative.
Even when there are two parallel series of phenomena the one
quantitative and the other qualitative--like the rate of air-vibration
and the pitch of sound, or the rate of ether-vibration and the
colour-series of the spectrum--the method of Variations is not
applicable. For (1) two such series cannot be said to vary together,
since the qualitative variations are heterogeneous: 512: 576 is a
definite ratio; but the corresponding notes, C, D, in the treble clef,
present only a difference. Hence (2) the correspondence of each note
with each number is a distinct fact. Each octave even is a distinct
fact; there is a difference between C 64 and C 128 that could never have
been anticipated without the appropriate experience. There is,
therefore, no such law of these parallel series as there is for
temperature and change of volume (say) in mercury. Similar remarks apply
to the physical and sensitive light-series.
We may illustrate the two cases of the method thus (putting a dash
against any letter, A' or _p_', to signify an increase or decrease of
the phenomenon the letter stands for): Agreement in Variations (other
changes being admissible)--
A B C A' D E A'' F G
_p q r_ _p' s t_ _p'' u v_
Here the accompanying phenomena (_B C q r, D E s t, F G u v_) change
from time to time, and the one thing in which the instances agree
throughout is that any increase of A (A' or A'') is followed or
accompanied by an increase of _p (p' or p'')_: whence it is argued that
A is the cause of _p_, according to Prop. III. (a) (ch. xv. Sec. 7). Still,
it is supposable that, in the second instance, D or E may be the cause
of the increment of _p_; and that, in the third instance, F or G may be
its cause: though the probability of such vicarious causation decreases
rapidly with the increase of instances in which A and _p_ vary together.
And, since an actual investigation of this type must rely on
observation, it is further possible that some undiscovered cause, X, is
the real determinant of both A and _p_ and of their concomitant
variations.
Professor Ferri, in his _Criminal Sociology_, observes: "I have shown
that in France there is a manifest correspondence of increase and
decrease between the number of homicides, assaults and malicious
wounding, and the more or less abundant vintage, especially in the years
of extraordinary variations, whether of failure of the vintage (1853-5,
1859, 1867, 1873, 1878-80), attended by a remarkable diminution of crime
(assaults and wounding), or of abundant vintages (1850, 1856-8, 1862-3,
1865, 1868, 1874-5), attended by an increase of crime" (p. 117, Eng.
trans.). And earlier he had remarked that such crimes also "in their
oscillations from month to month display a characteristic increase
during the vintage periods, from June to December, notwithstanding the
constant diminution of other offences" (p. 77). This is necessarily an
appeal to the canon of Concomitant Variations, because France is never
without her annual vintage, nor yet without her annual statistics of
crime. Still, it is an argument whose cogency is only that of Agreement,
showing that probably the abuse of the vintage is a cause of crimes of
violence, but leaving open the supposition, that some other circumstance
or circumstances, arising or varying from year to year, may determine
the increase or decrease of crime; or that there is some unconsidered
agent which affects both the vintage and crimes of violence. French
sunshine, it might be urged, whilst it matures the generous grape, also
excites a morbid fermentation in the human mind.
Difference in Variations may be symbolically represented thus (no other
change having concurred):
A B A' B A'' B
_p q_, _p' q_, _p'' q_.
Here the accompanying phenomena are always the same B/q; and the only
point in which the successive instances differ is in the increments of A
(A', A'') followed by corresponding increments of _p_ (_p', p''_): hence
the increment of A is the cause of the increment of _p_.
For examples of the application of this method, the reader should refer
to some work of exact science. He will find in Deschanel's _Natural
Philosophy_, c. 32, an account of some experiments by which the
connection between heat and mechanical work has been established. It is
there shown that "whenever work is performed by the agency of heat" [as
in driving an engine], "an amount of heat disappears equivalent to the
work performed; and whenever mechanical work is spent in generating
heat" [as in rubbing two sticks together], "the heat generated is
equivalent to the work thus spent." And an experiment of Joule's is
described, which consisted in fixing a rod with paddles in a vessel of
water, and making it revolve and agitate the water by means of a string
wound round the rod, passed over a pulley and attached to a weight that
was allowed to fall. The descent of the weight was measured by a
graduated rule, and the rise of the water's temperature by a
thermometer. "It was found that the heat communicated to the water by
the agitation amounted to one pound-degree Fahrenheit for every 772
foot-pounds of work" expended by the falling weight. As no other
material change seems to take place during such an experiment, it shows
that the progressive expenditure of mechanical energy is the cause of
the progressive heating of the water.
The thermometer itself illustrates this method. It has been found that
the application of heat to mercury expands it according to a law; and
hence the volume of the mercury, measured by a graduated index, is used
to indicate the temperature of the air, water, animal body, etc., in
which the thermometer is immersed, or with which it is brought into
contact. In such cases, if no other change has taken place, the heat of
the air, water, or body is the cause of the rise of the mercury in its
tube. If some other substance (say spirit) be substituted for mercury in
constructing a thermometer, it serves the same purpose, provided the
index be graduated according to the law of the expansion of that
substance by heat, as experimentally determined.
Instances of phenomena that do not vary together indicate the exclusion
of a supposed cause (by Prop. III (c)). The stature of the human race
has been supposed to depend on temperature; but there is no
correspondence. The "not varying together," however, must not be
confused with "varying inversely," which when regular indicates a true
concomitance. It is often a matter of convenience whether we regard
concomitant phenomena as varying directly or inversely. It is usual to
say--'the greater the friction the less the speed'; but it is really
more intelligible to say--'the greater the friction the more rapidly
molar is converted into molecular motion.'
The Graphic Method exhibits Concomitant Variations to the eye, and is
extensively used in physical and statistical inquiries. Along a
horizontal line (the abscissa) is measured one of the conditions (or
agents) with which the inquiry is concerned, called the Variable; and
along perpendiculars (ordinates) is measured some phenomenon to be
compared with it, called the Variant.
Thus, the expansion of a liquid by heat may be represented by measuring
degrees of temperature along the horizontal, and the expansion of a
column of the liquids in units of length along the perpendicular.
[Illustration: FIG. 9.]
In the next diagram (Fig. 10), reduced from one given by Mr. C.H. Denyer
in an article on the Price of Tea (_Economic Journal_, No. 9), the
condition measured horizontally is Time; and, vertically, three variants
are measured simultaneously, so that their relations to one another from
time to time may be seen at a glance. From this it is evident that, as
the duty on tea falls, the price of tea falls, whilst the consumption of
tea rises; and, in spite of some irregularity of correspondence in the
courses of the three phenomena, their general causal connection can
hardly be mistaken. However, the causal connection may also be inferred
by general reasoning; the statistical Induction can be confirmed by a
Deduction; thus illustrating the combined method of proof to be
discussed in the next chapter. Without such confirmation the proof by
Concomitant Variations would not be complete; because, from the
complexity of the circumstances, social statistics can only yield
evidence according to the method of Agreement in Variations. For,
besides the agents that are measured, there may always be some other
important influence at work. During the last fifty years, for example,
crime has decreased whilst education has increased: true, but at the
same time wages have risen and many other things have happened.
[Illustration: FIG. 10.]
It will be noticed that in the diagram the three lines, especially those
of Price and Consumption (which may be considered _natural_ resultants,
in contrast with the arbitrary fixation of a Tax), do not depart widely
from regular curves; and accordingly, assuming the causes at work to
vary continuously during the intervals between points of measurement,
curves may be substituted. In fact, a curve often represents the course
of a phenomenon more truthfully than can be done by a line that zigzags
along the exact measurements; because it is less influenced by temporary
and extraordinary causes that may obscure the operation of those that
are being investigated. On the other hand, the abrupt deviations of a
punctilious zigzag may have their own logical value, as will appear in
the next section.
In working with the Method of Variations one must allow for the
occurrence in a series of 'critical points,' at which sudden and
sometimes heterogeneous changes may take place. Every substance exists
at different temperatures in three states, gaseous, liquid, solid; and
when the change takes place, from one state to another, the series of
variations is broken. Water, e.g., follows the general law that
cooling is accompanied by decrease of volume between 212 deg. and 39 deg. F.:
but above 212 deg., undergoes a sudden expansion in becoming a gas; and
below 39 deg. begins to expand, until at 32 deg. the expansion is considerable
on its becoming solid. This illustrates a common experience that
concomitant variations are most regular in the 'median range,' and are
apt to become irregular at the extremities of the series, where new
conditions begin to operate.
The Canon of Variations, again, deals not with sudden irruptions of a
cause, force or agent, but with some increase or decrease of an agent
already present, and a corresponding increase or decrease of some other
phenomenon--say an increase of tax and a rise of price. But there are
cases in which the energy of a cause is not immediately discharged and
dissipated. Whilst a tax of 6_d._ per lb. on tea raises the price per
lb. by about 6_d._, however long it lasts, the continuous application of
friction to a body may gradually raise its temperature to the point of
combustion; because heat is received faster than it is radiated, and
therefore accumulates. Such cases are treated by Mill under the title of
'progressive effects' (_Logic_: B. III., c. 15): he gives as an example
of it the acceleration of falling bodies. The storage of effects is a
fact of the utmost importance in all departments of nature, and is
especially interesting in Biology and Sociology, where it is met with as
heredity, experience, tradition. Evolution of species of plants and
animals would (so far as we know) be impossible, if the changes (however
caused) that adapt some individuals better than others to the conditions
of life were not inherited by, and accumulated in, their posterity. The
eyes in the peacock's tail are supposed to have reached their present
perfection gradually, through various stages that may be illustrated by
the ocelli in the wings of the Argus pheasant and other genera of
_Phasianidae_. Similarly the progress of societies would be impossible
without tradition, whereby the improvements made in any generation may
be passed on to the next, and the experience of mankind may be gradually
accumulated in various forms of culture. The earliest remains of culture
are flint implements and weapons; in which we can trace the effect of
tradition in the lives of our remote forefathers, as they slowly through
thousands of years learnt to improve the chipping of flints, until the
first rudely shaped lumps gave place to works of unmistakable design,
and these to the beautiful weapons contemporary with the Bronze Age.
The Method of Gradations, the arranging of any phenomena to be studied
in series, according to the degree in which some character is exhibited,
is, perhaps, the most definite device in the Art of Discovery. (Bain:
_Induction_, c. 6, and App. II.) If the causes are unknown it is likely
to suggest hypotheses: and if the causes are partly known, variation in
the character of the series is likely to indicate a corresponding
variation of the conditions.
Sec. 5. THE CANON OF RESIDUES.
_Subduct from any phenomenon such part as previous inductions have shown
to be the effect of certain antecedents, and the residue of the
phenomenon is the effect of the remaining antecedents_.
The phenomenon is here assumed to be an effect: a similar Canon may be
framed for residuary causes.
This also is not a fresh method, but a special case of the method of
Difference. For if we suppose the phenomenon to be _p q r_, and the
antecedent to be A B C, and that we already know B and C to have (either
severally or together) the consequents _q r_, in which their efficacy is
exhausted; we may regard
B C
_q r_
as an instance of the absence of _p_ obtained deductively from the whole
phenomenon
A B C
_p q r_
by our knowledge of the laws of B and C; so that
A B C
_p q r_
is an instance of the presence of _p_, differing otherwise from
B C
_q r_
in nothing except that A is also present. By the Canon of Difference,
therefore A is the cause of _p_. Or, again, when phenomena thus treated
are strictly quantitative, the method may be based on Prop. III. (b),
ch. xv. Sec. 7.
Of course, if A can be obtained apart from B C and directly experimented
with so as to produce _p_, so much the better; and this may often be
done; but the special value of the method of Residues appears, when some
complex phenomenon has been for the most part accounted for by known
causes, whilst there remains some excess, or shortcoming, or deviation
from the result which those causes alone would lead us to expect, and
this residuary fact has to be explained in relation to the whole. Here
the negative instance is constituted by deduction, showing what would
happen but for the interference of some unknown cause which is to be
investigated; and this prominence of the deductive process has led some
writers to class the method as deductive. But we have seen that all the
Canons involve deduction; and, considering how much in every experiment
is assumed as already known (what circumstances are 'material,' and when
conditions may be called 'the same'), the wonder is that no one has
insisted upon regarding every method as concerned with residues. In
fact, as scientific explanation progresses, the phenomena that may be
considered as residuary become more numerous and the importance of this
method increases.
Examples: The recorded dates of ancient eclipses having been found to
differ from those assigned by calculation, it appears that the average
length of a day has in the meanwhile increased. This is a residuary
phenomenon not accounted for by the causes formerly recognised as
determining the rotation of the earth on its axis; and it may be
explained by the consideration that the friction of the tides reduces
the rate of the earth's rotation, and thereby lengthens the day.
Astronomy abounds in examples of the method of Residues, of which the
discovery of Neptune is the most famous.
Capillarity seems to be a striking exception to the principle that water
(or any liquid) 'finds its level,' that being the condition of
equilibrium; yet capillarity proves to be only a refined case of
equilibrium when account is taken of the forces of adhesion exerted by
different kinds of bodies in contact.
"Many of the new elements of Chemistry," says Herschel, "have been
detected in the investigation of residual phenomena." Thus, Lord
Rayleigh and Sir W. Ramsay found that nitrogen from the atmosphere was
slightly heavier than nitrogen got from chemical sources; and, seeking
the cause of this difference, discovered argon.
The Economist shows that when a country imports goods the chief means of
paying for them is to export other goods. If this were all, imports and
exports would be of equal value: yet the United Kingdom imports about
L400,000,000 annually, and exports about L300,000,000. Here, then, is a
residuary phenomenon of L100,000,000 to be accounted for. But foreign
countries owe us about L50,000,000 for the use of shipping, and
L70,000,000 as interest on the capital we have lent them, and
L15,000,000 in commissions upon business transacted for them. These sums
added together amount to L135,000,000; and that is L35,000,000 too much.
Thus another residuary phenomenon emerges; for whilst foreigners seem to
owe us L435,000,000 they only send us L400,000,000 of imports. These
L35,000,000 are accounted for by the annual investment of our capital
abroad, in return for which no immediate payment is due; and, these
being omitted, exports and imports balance. Since this was written the
figures of our foreign trade have greatly risen; but the character of
the explanation remains the same.
When, in pursuing the method of Variations, the phenomena compared do
not always correspond in their fluctuations, the irregular movements of
that phenomenon which we regard as the effect may often be explained by
treating them as residuary phenomena, and then seeking for exceptional
causes, whose temporary interference has obscured the influence of the
general cause. Thus, returning to the diagram of the Price of Tea in Sec.
4, it is clear that generally the price falls as the duty falls; but in
Mr. Denyer's more minutely wrought diagram, from which this is reduced,
it may be seen that in 1840 the price of tea rose from 3_s._ 9_d._ to
4_s._ 9_d._ without any increase of duty. This, however, is readily
explained by the Chinese War of that year, which checked the supply.
Again, from 1869 to 1889 the duty was constant, whilst the price of tea
fell as much as 8_d._ per lb.; but this residuary phenomenon is
explained by the prodigiously increased production of tea during that
period in India and Ceylon.
The above examples of the method of Residues are all quantitative; but
the method is often employed where exact estimates are unobtainable.
Thus Darwin, having found certain modifications of animals in form,
coloration and habits, that were not clearly derivable from their
struggle for existence in relation to other species or to external
conditions, suggested that they were due to Sexual Selection.
The 'vestiges' and 'survivals' so common in Biology and Sociology are
residuary phenomena. It is a general inference from the doctrine of
Natural Selection that every organ of a plant, animal, or society is in
some way useful to it. There occur, however, organs that have at present
no assignable utility, are at least wasteful, and sometimes even
injurious. And the explanation is that formerly they were useful; but
that, their uses having lapsed, they are now retained by the force of
heredity or tradition. Either they are not injurious enough to be
eliminated by natural selection; or they are correlated with other
organs, whose utility outweighs their disutility.