Of The Deductive Method.
§ 1. The mode of investigation which, from the proved inapplicability of
direct methods of observation and experiment, remains to us as the main
source of the knowledge we possess or can acquire respecting the
conditions and laws of recurrence, of the more complex phenomena, is
called, in its most general expression, the Deductive Method; and consists
of three operations: the first, one of direct induction; the second, of
ratiocination; the third, of verification.
I call the first step in the process an inductive operation, because there
must be a direct induction as the basis of the whole; though in many
particular investigations the place of the induction may be supplied by a
prior deduction; but the premises of this prior deduction must have been
derived from induction.
The problem of the Deductive Method is, to find the law of an effect, from
the laws of the different tendencies of which it is the joint result. The
first requisite, therefore, is to know the laws of those tendencies; the
law of each of the concurrent causes: and this supposes a previous process
of observation or experiment upon each cause separately; or else a
previous deduction, which also must depend for its ultimate premises on
observation or experiment. Thus, if the subject be social or historical
phenomena, the premises of the Deductive Method must be the laws of the
causes which determine that class of phenomena; and those causes are human
actions, together with the general outward circumstances under the
influence of which mankind are placed, and which constitute man’s position
on the earth. The Deductive Method, applied to social phenomena, must
begin, therefore, by investigating, or must suppose to have been already
investigated, the laws of human action, and those properties of outward
things by which the actions of human beings in society are determined.
Some of these general truths will naturally be obtained by observation and
experiment, others by deduction: the more complex laws of human action,
for example, may be deduced from the simpler ones; but the simple or
elementary laws will always, and necessarily, have been obtained by a
directly inductive process.
To ascertain, then, the laws of each separate cause which takes a share in
producing the effect, is the first desideratum of the Deductive Method. To
know what the causes are which must be subjected to this process of study,
may or may not be difficult. In the case last mentioned, this first
condition is of easy fulfillment. That social phenomena depend on the acts
and mental impressions of human beings, never could have been a matter of
any doubt, however imperfectly it may have been known either by what laws
those impressions and actions are governed, or to what social consequences
their laws naturally lead. Neither, again, after physical science had
attained a certain development, could there be any real doubt where to
look for the laws on which the phenomena of life depend, since they must
be the mechanical and chemical laws of the solid and fluid substances
composing the organized body and the medium in which it subsists, together
with the peculiar vital laws of the different tissues constituting the
organic structure. In other cases, really far more simple than these, it
was much less obvious in what quarter the causes were to be looked for: as
in the case of the celestial phenomena. Until, by combining the laws of
certain causes, it was found that those laws explained all the facts which
experience had proved concerning the heavenly motions, and led to
predictions which it always verified, mankind never knew that those _were_
the causes. But whether we are able to put the question before, or not
until after, we have become capable of answering it, in either case it
must be answered; the laws of the different causes must be ascertained,
before we can proceed to deduce from them the conditions of the effect.
The mode of ascertaining those laws neither is, nor can be any other than
the fourfold method of experimental inquiry, already discussed. A few
remarks on the application of that method to cases of the Composition of
Causes are all that is requisite.
It is obvious that we can not expect to find the law of a tendency by an
induction from cases in which the tendency is counteracted. The laws of
motion could never have been brought to light from the observation of
bodies kept at rest by the equilibrium of opposing forces. Even where the
tendency is not, in the ordinary sense of the word, counteracted, but only
modified, by having its effects compounded with the effects arising from
some other tendency or tendencies, we are still in an unfavorable position
for tracing, by means of such cases, the law of the tendency itself. It
would have been scarcely possible to discover the law that every body in
motion tends to continue moving in a straight line, by an induction from
instances in which the motion is deflected into a curve, by being
compounded with the effect of an accelerating force. Notwithstanding the
resources afforded in this description of cases by the Method of
Concomitant Variations, the principles of a judicious experimentation
prescribe that the law of each of the tendencies should be studied, if
possible, in cases in which that tendency operates alone, or in
combination with no agencies but those of which the effect can, from
previous knowledge, be calculated and allowed for.
Accordingly, in the cases, unfortunately very numerous and important, in
which the causes do not suffer themselves to be separated and observed
apart, there is much difficulty in laying down with due certainty the
inductive foundation necessary to support the deductive method. This
difficulty is most of all conspicuous in the case of physiological
phenomena; it being seldom possible to separate the different agencies
which collectively compose an organized body, without destroying the very
phenomena which it is our object to investigate:
——following life, in creatures we dissect,
We lose it, in the moment we detect.
And for this reason I am inclined to the opinion that physiology (greatly
and rapidly progressive as it now is) is embarrassed by greater natural
difficulties, and is probably susceptible of a less degree of ultimate
perfection, than even the social science; inasmuch as it is possible to
study the laws and operations of one human mind apart from other minds,
much less imperfectly than we can study the laws of one organ or tissue of
the human body apart from the other organs or tissues.
It has been judiciously remarked that pathological facts, or, to speak in
common language, diseases in their different forms and degrees afford in
the case of physiological investigation the most valuable equivalent to
experimentation properly so called; inasmuch as they often exhibit to us a
definite disturbance in some one organ or organic function, the remaining
organs and functions being, in the first instance at least, unaffected. It
is true that from the perpetual actions and reactions which are going on
among all parts of the organic economy, there can be no prolonged
disturbance in any one function without ultimately involving many of the
others; and when once it has done so, the experiment for the most part
loses its scientific value. All depends on observing the early stages of
the derangement; which, unfortunately, are of necessity the least marked.
If, however, the organs and functions not disturbed in the first instance
become affected in a fixed order of succession, some light is thereby
thrown upon the action which one organ exercises over another: and we
occasionally obtain a series of effects which we can refer with some
confidence to the original local derangement; but for this it is necessary
that we should know that the original derangement _was_ local. If it was
what is termed constitutional; that is, if we do not know in what part of
the animal economy it took its rise, or the precise nature of the
disturbance which took place in that part, we are unable to determine
which of the various derangements was cause and which effect; which of
them were produced by one another, and which by the direct, though perhaps
tardy, action of the original cause.
Besides natural pathological facts, we can produce pathological facts
artificially: we can try experiments, even in the popular sense of the
term, by subjecting the living being to some external agent, such as the
mercury of our former example, or the section of a nerve to ascertain the
functions of different parts of the nervous system. As this
experimentation is not intended to obtain a direct solution of any
practical question, but to discover general laws, from which afterward the
conditions of any particular effect may be obtained by deduction, the best
cases to select are those of which the circumstances can be best
ascertained: and such are generally not those in which there is any
practical object in view. The experiments are best tried, not in a state
of disease, which is essentially a changeable state, but in the condition
of health, comparatively a fixed state. In the one, unusual agencies are
at work, the results of which we have no means of predicting: in the
other, the course of the accustomed physiological phenomena would, it may
generally be presumed, remain undisturbed, were it not for the disturbing
cause which we introduce.
Such, with the occasional aid of the Method of Concomitant Variations (the
latter not less encumbered than the more elementary methods by the
peculiar difficulties of the subject), are our inductive resources for
ascertaining the laws of the causes considered separately, when we have it
not in our power to make trial of them in a state of actual separation.
The insufficiency of these resources is so glaring, that no one can be
surprised at the backward state of the science of physiology; in which
indeed our knowledge of causes is so imperfect, that we can neither
explain, nor could without specific experience have predicted, many of the
facts which are certified to us by the most ordinary observation.
Fortunately, we are much better informed as to the empirical laws of the
phenomena, that is, the uniformities respecting which we can not yet
decide whether they are cases of causation, or mere results of it. Not
only has the order in which the facts of organization and life
successively manifest themselves, from the first germ of existence to
death, been found to be uniform, and very accurately ascertainable; but,
by a great application of the Method of Concomitant Variations to the
entire facts of comparative anatomy and physiology, the characteristic
organic structure corresponding to each class of functions has been
determined with considerable precision. Whether these organic conditions
are the whole of the conditions, and in many cases whether they are
conditions at all, or mere collateral effects of some common cause, we are
quite ignorant; nor are we ever likely to know, unless we could construct
an organized body and try whether it would live.
Under such disadvantages do we, in cases of this description, attempt the
initial, or inductive step, in the application of the Deductive Method to
complex phenomena. But such, fortunately, is not the common case. In
general, the laws of the causes on which the effect depends may be
obtained by an induction from comparatively simple instances, or, at the
worst, by deduction from the laws of simpler causes, so obtained. By
simple instances are meant, of course, those in which the action of each
cause was not intermixed or interfered with, or not to any great extent,
by other causes whose laws were unknown. And only when the induction which
furnished the premises to the Deductive method rested on such instances
has the application of such a method to the ascertainment of the laws of a
complex effect, been attended with brilliant results.
§ 2. When the laws of the causes have been ascertained, and the first
stage of the great logical operation now under discussion satisfactorily
accomplished, the second part follows; that of determining from the laws
of the causes what effect any given combination of those causes will
produce. This is a process of calculation, in the wider sense of the term;
and very often involves processes of calculation in the narrowest sense.
It is a ratiocination; and when our knowledge of the causes is so perfect
as to extend to the exact numerical laws which they observe in producing
their effects, the ratiocination may reckon among its premises the
theorems of the science of number, in the whole immense extent of that
science. Not only are the most advanced truths of mathematics often
required to enable us to compute an effect, the numerical law of which we
already know; but, even by the aid of those most advanced truths, we can
go but a little way. In so simple a case as the common problem of three
bodies gravitating toward one another, with a force directly as their mass
and inversely as the square of the distance, all the resources of the
calculus have not hitherto sufficed to obtain any general solution, but an
approximate one. In a case a little more complex, but still one of the
simplest which arise in practice, that of the motion of a projectile, the
causes which affect the velocity and range (for example) of a cannon-ball
may be all known and estimated: the force of the gunpowder, the angle of
elevation, the density of the air, the strength and direction of the wind;
but it is one of the most difficult of mathematical problems to combine
all these, so as to determine the effect resulting from their collective
action.
Besides the theorems of number, those of geometry also come in as
premises, where the effects take place in space, and involve motion and
extension, as in mechanics, optics, acoustics, astronomy. But when the
complication increases, and the effects are under the influence of so many
and such shifting causes as to give no room either for fixed numbers, or
for straight lines and regular curves (as in the case of physiological, to
say nothing of mental and social phenomena), the laws of number and
extension are applicable, if at all, only on that large scale on which
precision of details becomes unimportant. Although these laws play a
conspicuous part in the most striking examples of the investigation of
nature by the Deductive Method, as for example in the Newtonian theory of
the celestial motions, they are by no means an indispensable part of every
such process. All that is essential in it is reasoning from a general law
to a particular case, that is, determining by means of the particular
circumstances of that case, what result is required in that instance to
fulfill the law. Thus in the Torricellian experiment, if the fact that air
has weight had been previously known, it would have been easy, without any
numerical data, to deduce from the general law of equilibrium, that the
mercury would stand in the tube at such a height that the column of
mercury would exactly balance a column of the atmosphere of equal
diameter; because, otherwise, equilibrium would not exist.
By such ratiocinations from the separate laws of the causes, we may, to a
certain extent, succeed in answering either of the following questions:
Given a certain combination of causes, what effect will follow? and, What
combination of causes, if it existed, would produce a given effect? In the
one case, we determine the effect to be expected in any complex
circumstances of which the different elements are known: in the other case
we learn, according to what law—under what antecedent conditions—a given
complex effect will occur.
§ 3. But (it may here be asked) are not the same arguments by which the
methods of direct observation and experiment were set aside as illusory
when applied to the laws of complex phenomena, applicable with equal force
against the Method of Deduction? When in every single instance a
multitude, often an unknown multitude, of agencies, are clashing and
combining, what security have we that in our computation _a priori_ we
have taken all these into our reckoning? How many must we not generally be
ignorant of? Among those which we know, how probable that some have been
overlooked; and, even were all included, how vain the pretense of summing
up the effects of many causes, unless we know accurately the numerical law
of each—a condition in most cases not to be fulfilled; and even when it is
fulfilled, to make the calculation transcends, in any but very simple
cases, the utmost power of mathematical science with all its most modern
improvements.
These objections have real weight, and would be altogether unanswerable,
if there were no test by which, when we employ the Deductive Method, we
might judge whether an error of any of the above descriptions had been
committed or not. Such a test, however, there is: and its application
forms, under the name of Verification, the third essential component part
of the Deductive Method; without which all the results it can give have
little other value than that of conjecture. To warrant reliance on the
general conclusions arrived at by deduction, these conclusions must be
found, on careful comparison, to accord with the results of direct
observation wherever it can be had. If, when we have experience to compare
with them, this experience confirms them, we may safely trust to them in
other cases of which our specific experience is yet to come. But if our
deductions have led to the conclusion that from a particular combination
of causes a given effect would result, then in all known cases where that
combination can be shown to have existed, and where the effect has not
followed, we must be able to show (or at least to make a probable surmise)
what frustrated it: if we can not, the theory is imperfect, and not yet to
be relied upon. Nor is the verification complete, unless some of the cases
in which the theory is borne out by the observed result are of at least
equal complexity with any other cases in which its application could be
called for.
If direct observation and collation of instances have furnished us with
any empirical laws of the effect (whether true in all observed cases, or
only true for the most part), the most effectual verification of which the
theory could be susceptible, would be, that it led deductively to those
empirical laws; that the uniformities, whether complete or incomplete,
which were observed to exist among the phenomena, were accounted for by
the laws of the causes—were such as could not but exist if those be really
the causes by which the phenomena are produced. Thus it was very
reasonably deemed an essential requisite of any true theory of the causes
of the celestial motions, that it should lead by deduction to Kepler’s
laws; which, accordingly, the Newtonian theory did.
In order, therefore, to facilitate the verification of theories obtained
by deduction, it is important that as many as possible of the empirical
laws of the phenomena should be ascertained, by a comparison of instances,
conformably to the Method of Agreement: as well as (it must be added) that
the phenomena themselves should be described, in the most comprehensive as
well as accurate manner possible; by collecting from the observation of
parts, the simplest possible correct expressions for the corresponding
wholes: as when the series of the observed places of a planet was first
expressed by a circle, then by a system of epicycles, and subsequently by
an ellipse.
It is worth remarking, that complex instances which would have been of no
use for the discovery of the simple laws into which we ultimately analyze
their phenomena, nevertheless, when they have served to verify the
analysis, become additional evidence of the laws themselves. Although we
could not have got at the law from complex cases, still when the law, got
at otherwise, is found to be in accordance with the result of a complex
case, that case becomes a new experiment on the law, and helps to confirm
what it did not assist to discover. It is a new trial of the principle in
a different set of circumstances; and occasionally serves to eliminate
some circumstance not previously excluded, and the exclusion of which
might require an experiment impossible to be executed. This was strikingly
conspicuous in the example formerly quoted, in which the difference
between the observed and the calculated velocity of sound was ascertained
to result from the heat extricated by the condensation which takes place
in each sonorous vibration. This was a trial, in new circumstances, of the
law of the development of heat by compression; and it added materially to
the proof of the universality of that law. Accordingly, any law of nature
is deemed to have gained in point of certainty, by being found to explain
some complex case which had not previously been thought of in connection
with it; and this indeed is a consideration to which it is the habit of
scientific inquirers to attach rather too much value than too little.
To the Deductive Method, thus characterized in its three constituent
parts, Induction, Ratiocination, and Verification, the human mind is
indebted for its most conspicuous triumphs in the investigation of nature.
To it we owe all the theories by which vast and complicated phenomena are
embraced under a few simple laws, which, considered as the laws of those
great phenomena, could never have been detected by their direct study. We
may form some conception of what the method has done for us from the case
of the celestial motions: one of the simplest among the greater instances
of the Composition of Causes, since (except in a few cases not of primary
importance) each of the heavenly bodies may be considered, without
material inaccuracy, to be never at one time influenced by the attraction
of more than two bodies, the sun and one other planet or satellite;
making, with the reaction of the body itself, and the force generated by
the body’s own motion and acting in the direction of the tangent, only
four different agents on the concurrence of which the motions of that body
depend; a much smaller number, no doubt, than that by which any other of
the great phenomena of nature is determined or modified. Yet how could we
ever have ascertained the combination of forces on which the motions of
the earth and planets are dependent, by merely comparing the orbits or
velocities of different planets, or the different velocities or positions
of the same planet? Notwithstanding the regularity which manifests itself
in those motions, in a degree so rare among the effects of concurrence of
causes; and although the periodical recurrence of exactly the same effect,
affords positive proof that all the combinations of causes which occur at
all, recur periodically; we should not have known what the causes were, if
the existence of agencies precisely similar on our own earth had not,
fortunately, brought the causes themselves within the reach of
experimentation under simple circumstances. As we shall have occasion to
analyze, further on, this great example of the Method of Deduction, we
shall not occupy any time with it here, but shall proceed to that
secondary application of the Deductive Method, the result of which is not
to prove laws of phenomena, but to explain them.