Of Empirical Laws.
§ 1. Scientific inquirers give the name of Empirical Laws to those
uniformities which observation or experiment has shown to exist, but on
which they hesitate to rely in cases varying much from those which have
been actually observed, for want of seeing any reason _why_ such a law
should exist. It is implied, therefore, in the notion of an empirical law,
that it is not an ultimate law; that if true at all, its truth is capable
of being, and requires to be, accounted for. It is a derivative law, the
derivation of which is not yet known. To state the explanation, the _why_,
of the empirical law, would be to state the laws from which it is
derived—the ultimate causes on which it is contingent. And if we knew
these, we should also know what are its limits; under what conditions it
would cease to be fulfilled.
The periodical return of eclipses, as originally ascertained by the
persevering observation of the early Eastern astronomers, was an empirical
law, until the general laws of the celestial motions had accounted for it.
The following are empirical laws still waiting to be resolved into the
simpler laws from which they are derived: the local laws of the flux and
reflux of the tides in different places; the succession of certain kinds
of weather to certain appearances of sky; the apparent exceptions to the
almost universal truth that bodies expand by increase of temperature; the
law that breeds, both animal and vegetable, are improved by crossing; that
gases have a strong tendency to permeate animal membranes; that substances
containing a very high proportion of nitrogen (such as hydrocyanic acid
and morphia) are powerful poisons; that when different metals are fused
together the alloy is harder than the various elements; that the number of
atoms of acid required to neutralize one atom of any base is equal to the
number of atoms of oxygen in the base; that the solubility of substances
in one another depends,(171) at least in some degree, on the similarity of
their elements.
An empirical law, then, is an observed uniformity, presumed to be
resolvable into simpler laws, but not yet resolved into them. The
ascertainment of the empirical laws of phenomena often precedes by a long
interval the explanation of those laws by the Deductive Method; and the
verification of a deduction usually consists in the comparison of its
results with empirical laws previously ascertained.
§ 2. From a limited number of ultimate laws of causation, there are
necessarily generated a vast number of derivative uniformities, both of
succession and co-existence. Some are laws of succession or of
co-existence between different effects of the same cause; of these we had
examples in the last chapter. Some are laws of succession between effects
and their remote causes, resolvable into the laws which connect each with
the intermediate link. Thirdly, when causes act together and compound
their effects, the laws of those causes generate the fundamental law of
the effect, namely, that it depends on the co-existence of those causes.
And, finally, the order of succession or of co-existence which obtains
among effects necessarily depends on their causes. If they are effects of
the same cause, it depends on the laws of that cause; if on different
causes, it depends on the laws of those causes severally, and on the
circumstances which determine their co-existence. If we inquire further
when and how the causes will co-exist, that, again, depends on _their_
causes; and we may thus trace back the phenomena higher and higher, until
the different series of effects meet in a point, and the whole is shown to
have depended ultimately on some common cause; or until, instead of
converging to one point, they terminate in different points, and the order
of the effects is proved to have arisen from the collocation of some of
the primeval causes, or natural agents. For example, the order of
succession and of co-existence among the heavenly motions, which is
expressed by Kepler’s laws, is derived from the co-existence of two
primeval causes, the sun, and the original impulse or projectile force
belonging to each planet.(172) Kepler’s laws are resolved into the laws of
these causes and the fact of their co-existence.
Derivative laws, therefore, do not depend solely on the ultimate laws into
which they are resolvable; they mostly depend on those ultimate laws, and
an ultimate fact; namely, the mode of co-existence of some of the
component elements of the universe. The ultimate laws of causation might
be the same as at present, and yet the derivative laws completely
different, if the causes co-existed in different proportions, or with any
difference in those of their relations by which the effects are
influenced. If, for example, the sun’s attraction, and the original
projectile force, had existed in some other ratio to one another than they
did (and we know of no reason why this should not have been the case), the
derivative laws of the heavenly motions might have been quite different
from what they are. The proportions which exist happen to be such as to
produce regular elliptical motions; any other proportions would have
produced different ellipses, or circular, or parabolic, or hyperbolic
motions, but still regular ones; because the effects of each of the agents
accumulate according to a uniform law; and two regular series of
quantities, when their corresponding terms are added, must produce a
regular series of some sort, whatever the quantities themselves are.
§ 3. Now this last-mentioned element in the resolution of a derivative
law, the element which is not a law of causation, but a collocation of
causes, can not itself be reduced to any law. There is, as formerly
remarked,(173) no uniformity, no _norma_, principle, or rule, perceivable
in the distribution of the primeval natural agents through the universe.
The different substances composing the earth, the powers that pervade the
universe, stand in no constant relation to one another. One substance is
more abundant than others, one power acts through a larger extent of space
than others, without any pervading analogy that we can discover. We not
only do not know of any reason why the sun’s attraction and the force in
the direction of the tangent co-exist in the exact proportion they do, but
we can trace no coincidence between it and the proportions in which any
other elementary powers in the universe are intermingled. The utmost
disorder is apparent in the combination of the causes, which is consistent
with the most regular order in their effects; for when each agent carries
on its own operations according to a uniform law, even the most capricious
combination of agencies will generate a regularity of some sort; as we see
in the kaleidoscope, where any casual arrangement of colored bits of glass
produces by the laws of reflection a beautiful regularity in the effect.
§ 4. In the above considerations lies the justification of the limited
degree of reliance which scientific inquirers are accustomed to place in
empirical laws.
A derivative law which results wholly from the operation of some one
cause, will be as universally true as the laws of the cause itself; that
is, it will always be true except where some one of those effects of the
cause, on which the derivative law depends, is defeated by a counteracting
cause. But when the derivative law results not from different effects of
one cause, but from effects of several causes, we can not be certain that
it will be true under any variation in the mode of co-existence of those
causes, or of the primitive natural agents on which the causes ultimately
depend. The proposition that coal-beds rest on certain descriptions of
strata exclusively, though true on the earth, so far as our observation
has reached, can not be extended to the moon or the other planets,
supposing coal to exist there; because we can not be assured that the
original constitution of any other planet was such as to produce the
different depositions in the same order as in our globe. The derivative
law in this case depends not solely on laws, but on a collocation; and
collocations can not be reduced to any law.
Now it is the very nature of a derivative law which has not yet been
resolved into its elements, in other words, an empirical law, that we do
not know whether it results from the different effects of one cause, or
from effects of different causes. We can not tell whether it depends
wholly on laws, or partly on laws and partly on a collocation. If it
depends on a collocation, it will be true in all the cases in which that
particular collocation exists. But, since we are entirely ignorant, in
case of its depending on a collocation, what the collocation is, we are
not safe in extending the law beyond the limits of time and place in which
we have actual experience of its truth. Since within those limits the law
has always been found true, we have evidence that the collocations,
whatever they are, on which it depends, do really exist within those
limits. But, knowing of no rule or principle to which the collocations
themselves conform, we can not conclude that because a collocation is
proved to exist within certain limits of place or time, it will exist
beyond those limits. Empirical laws, therefore, can only be received as
true within the limits of time and place in which they have been found
true by observation; and not merely the limits of time and place, but of
time, place, and circumstance; for, since it is the very meaning of an
empirical law that we do not know the ultimate laws of causation on which
it is dependent, we can not foresee, without actual trial, in what manner
or to what extent the introduction of any new circumstance may affect it.
§ 5. But how are we to know that a uniformity ascertained by experience is
only an empirical law? Since, by the supposition, we have not been able to
resolve it into any other laws, how do we know that it is not an ultimate
law of causation?
I answer that no generalization amounts to more than an empirical law when
the only proof on which it rests is that of the Method of Agreement. For
it has been seen that by that method alone we never can arrive at causes.
The utmost that the Method of Agreement can do is, to ascertain the whole
of the circumstances common to all cases in which a phenomenon is
produced; and this aggregate includes not only the cause of the
phenomenon, but all phenomena with which it is connected by any derivative
uniformity, whether as being collateral effects of the same cause, or
effects of any other cause which, in all the instances we have been able
to observe, co-existed with it. The method affords no means of determining
which of these uniformities are laws of causation, and which are merely
derivative laws, resulting from those laws of causation and from the
collocation of the causes. None of them, therefore, can be received in any
other character than that of derivative laws, the derivation of which has
not been traced; in other words, empirical laws: in which light all
results obtained by the Method of Agreement (and therefore almost all
truths obtained by simple observation without experiment) must be
considered, until either confirmed by the Method of Difference, or
explained deductively; in other words, accounted for _a priori_.
These empirical laws may be of greater or less authority, according as
there is reason to presume that they are resolvable into laws only, or
into laws and collocations together. The sequences which we observe in the
production and subsequent life of an animal or a vegetable, resting on the
Method of Agreement only, are mere empirical laws; but though the
antecedents in those sequences may not be the causes of the consequents,
both the one and the other are doubtless, in the main, successive stages
of a progressive effect originating in a common cause, and therefore
independent of collocations. The uniformities, on the other hand, in the
order of superposition of strata on the earth, are empirical laws of a
much weaker kind, since they not only are not laws of causation, but there
is no reason to believe that they depend on any common cause; all
appearances are in favor of their depending on the particular collocation
of natural agents which at some time or other existed on our globe, and
from which no inference can be drawn as to the collocation which exists or
has existed in any other portion of the universe.
§ 6. Our definition of an empirical law, including not only those
uniformities which are not known to be laws of causation, but also those
which are, provided there be reason to presume that they are not ultimate
laws; this is the proper place to consider by what signs we may judge that
even if an observed uniformity be a law of causation, it is not an
ultimate, but a derivative law.
The first sign is, if between the antecedent _a_ and the consequent _b_
there be evidence of some intermediate link; some phenomenon of which we
can surmise the existence, though from the imperfection of our senses or
of our instruments we are unable to ascertain its precise nature and laws.
If there be such a phenomenon (which may be denoted by the letter _x_), it
follows that even if _a_ be the cause of _b_, it is but the remote cause,
and that the law, _a_ causes _b_, is resolvable into at least two laws,
_a_ causes _x_, and _x_ causes _b_. This is a very frequent case, since
the operations of nature mostly take place on so minute a scale, that many
of the successive steps are either imperceptible, or very indistinctly
perceived.
Take, for example, the laws of the chemical composition of substances; as
that hydrogen and oxygen being combined, water is produced. All we see of
the process is, that the two gases being mixed in certain proportions, and
heat or electricity being applied, an explosion takes place, the gases
disappear, and water remains. There is no doubt about the law, or about
its being a law of causation. But between the antecedent (the gases in a
state of mechanical mixture, heated or electrified), and the consequent
(the production of water), there must be an intermediate process which we
do not see. For if we take any portion whatever of the water, and subject
it to analysis, we find that it always contains hydrogen and oxygen; nay,
the very same proportions of them, namely, two-thirds, in volume, of
hydrogen, and one-third oxygen. This is true of a single drop; it is true
of the minutest portion which our instruments are capable of appreciating.
Since, then, the smallest perceptible portion of the water contains both
those substances, portions of hydrogen and oxygen smaller than the
smallest perceptible must have come together in every such minute portion
of space; must have come closer together than when the gases were in a
state of mechanical mixture, since (to mention no other reasons) the water
occupies far less space than the gases. Now, as we can not see this
contact or close approach of the minute particles, we can not observe with
what circumstances it is attended, or according to what laws it produces
its effects. The production of water, that is, of the sensible phenomena
which characterize the compound, may be a very remote effect of those
laws. There may be innumerable intervening links; and we are sure that
there must be some. Having full proof that corpuscular action of some kind
takes place previous to any of the great transformations in the sensible
properties of substances, we can have no doubt that the laws of chemical
action, as at present known, are not ultimate, but derivative laws;
however ignorant we may be, and even though we should forever remain
ignorant, of the nature of the laws of corpuscular action from which they
are derived.
In like manner, all the processes of vegetative life, whether in the
vegetable properly so called or in the animal body, are corpuscular
processes. Nutrition is the addition of particles to one another,
sometimes merely replacing other particles separated and excreted,
sometimes occasioning an increase of bulk or weight so gradual that only
after a long continuance does it become perceptible. Various organs, by
means of peculiar vessels, secrete from the blood fluids, the component
particles of which must have been in the blood, but which differ from it
most widely both in mechanical properties and in chemical composition.
Here, then, are abundance of unknown links to be filled up; and there can
be no doubt that the laws of the phenomena of vegetative or organic life
are derivative laws, dependent on properties of the corpuscles, and of
those elementary tissues which are comparatively simple combinations of
corpuscles.
The first sign, then, from which a law of causation, though hitherto
unresolved, may be inferred to be a derivative law, is any indication of
the existence of an intermediate link or links between the antecedent and
the consequent. The second is, when the antecedent is an extremely complex
phenomenon, and its effects, therefore, probably in part at least,
compounded of the effects of its different elements; since we know that
the case in which the effect of the whole is not made up of the effects of
its parts is exceptional, the Composition of Causes being by far the more
ordinary case.
We will illustrate this by two examples, in one of which the antecedent is
the sum of many homogeneous, in the other of heterogeneous, parts. The
weight of a body is made up of the weights of its minute particles; a
truth which astronomers express in its most general terms when they say
that bodies, at equal distances, gravitate to one another in proportion to
their quantity of matter. All true propositions, therefore, which can be
made concerning gravity, are derivative laws; the ultimate law into which
they are all resolvable being, that every particle of matter attracts
every other. As our second example, we may take any of the sequences
observed in meteorology; for instance, a diminution of the pressure of the
atmosphere (indicated by a fall of the barometer) is followed by rain. The
antecedent is here a complex phenomenon, made up of heterogeneous
elements; the column of the atmosphere over any particular place
consisting of two parts, a column of air, and a column of aqueous vapor
mixed with it; and the change in the two together manifested by a fall of
the barometer, and followed by rain, must be either a change in one of
these, or in the other, or in both. We might, then, even in the absence of
any other evidence, form a reasonable presumption, from the invariable
presence of both these elements in the antecedent, that the sequence is
probably not an ultimate law, but a result of the laws of the two
different agents; a presumption only to be destroyed when we had made
ourselves so well acquainted with the laws of both, as to be able to
affirm that those laws could not by themselves produce the observed
result.
There are but few known cases of succession from very complex antecedents
which have not either been actually accounted for from simpler laws, or
inferred with great probability (from the ascertained existence of
intermediate links of causation not yet understood) to be capable of being
so accounted for. It is, therefore, highly probable that all sequences
from complex antecedents are thus resolvable, and that ultimate laws are
in all cases comparatively simple. If there were not the other reasons
already mentioned for believing that the laws of organized nature are
resolvable into simpler laws, it would be almost a sufficient reason that
the antecedents in most of the sequences are so very complex.
§ 7. In the preceding discussion we have recognized two kinds of empirical
laws: those known to be laws of causation, but presumed to be resolvable
into simpler laws; and those not known to be laws of causation at all.
Both these kinds of laws agree in the demand which they make for being
explained by deduction, and agree in being the appropriate means of
verifying such deduction, since they represent the experience with which
the result of the deduction must be compared. They agree, further, in
this, that until explained, and connected with the ultimate laws from
which they result, they have not attained the highest degree of certainty
of which laws are susceptible. It has been shown on a former occasion that
laws of causation which are derivative, and compounded of simpler laws,
are not only, as the nature of the case implies, less general, but even
less certain, than the simpler laws from which they result; not in the
same degree to be relied on as universally true. The inferiority of
evidence, however, which attaches to this class of laws, is trifling,
compared with that which is inherent in uniformities not known to be laws
of causation at all. So long as these are unresolved, we can not tell on
how many collocations, as well as laws, their truth may be dependent; we
can never, therefore, extend them with any confidence to cases in which we
have not assured ourselves, by trial, that the necessary collocation of
causes, whatever it may be, exists. It is to this class of laws alone that
the property, which philosophers usually consider as characteristic of
empirical laws, belongs in all its strictness—the property of being unfit
to be relied on beyond the limits of time, place, and circumstance in
which the observations have been made. These are empirical laws in a more
emphatic sense; and when I employ that term (except where the context
manifestly indicates the reverse) I shall generally mean to designate
those uniformities only, whether of succession or of co-existence, which
are not known to be laws of causation.