Of The Extension Of Derivative Laws To Adjacent Cases.
§ 1. We have had frequent occasion to notice the inferior generality of
derivative laws, compared with the ultimate laws from which they are
derived. This inferiority, which affects not only the extent of the
propositions themselves, but their degree of certainty within that
extent, is most conspicuous in the uniformities of co-existence and
sequence obtaining between effects which depend ultimately on different
primeval causes. Such uniformities will only obtain where there exists
the same collocation of those primeval causes. If the collocation
varies, though the laws themselves remain the same, a totally different
set of derivative uniformities may, and generally will, be the result.
Even where the derivative uniformity is between different effects of the
same cause, it will by no means obtain as universally as the law of the
cause itself. If _a_ and _b_ accompany or succeed one another as effects
of the cause A, it by no means follows that A is the only cause which
can produce them, or that if there be another cause, as B, capable of
producing _a_, it must produce _b_ likewise. The conjunction, therefore,
of _a_ and _b_ perhaps does not hold universally, but only in the
instances in which _a_ arises from A. When it is produced by _a_ cause
other than A, _a_ and _b_ may be dissevered. Day (for example) is always
in our experience followed by night; but day is not the cause of night;
both are successive effects of _a_ common cause, the periodical passage
of the spectator into and out of the earth’s shadow, consequent on the
earth’s rotation, and on the illuminating property of the sun. If,
therefore, day is ever produced by a different cause or set of causes
from this, day will not, or at least may not, be followed by night. On
the sun’s own surface, for instance, this may be the case.
Finally, even when the derivative uniformity is itself a law of
causation (resulting from the combination of several causes), it is not
altogether independent of collocations. If a cause supervenes, capable
of wholly or partially counteracting the effect of any one of the
conjoined causes, the effect will no longer conform to the derivative
law. While, therefore, each ultimate law is only liable to frustration
from one set of counteracting causes, the derivative law is liable to it
from several. Now, the possibility of the occurrence of counteracting
causes which do not arise from any of the conditions involved in the law
itself depends on the original collocations.
It is true that, as we formerly remarked, laws of causation, whether
ultimate or derivative, are, in most cases, fulfilled even when
counteracted; the cause produces its effect, though that effect is
destroyed by something else. That the effect may be frustrated, is,
therefore, no objection to the universality of laws of causation. But it
is fatal to the universality of the sequences or co-existences of
effects, which compose the greater part of the derivative laws flowing
from laws of causation. When, from the law of a certain combination of
causes, there results a certain order in the effects; as from the
combination of a single sun with the rotation of an opaque body round
its axis, there results, on the whole surface of that opaque body, an
alternation of day and night; then, if we suppose one of the combined
causes counteracted, the rotation stopped, the sun extinguished, or a
second sun superadded, the truth of that particular law of causation is
in no way affected; it is still true that one sun shining on an opaque
revolving body will alternately produce day and night; but since the sun
no longer does shine on such a body, the derivative uniformity, the
succession of day and night on the given planet, is no longer true.
Those derivative uniformities, therefore, which are not laws of
causation, are (except in the rare case of their depending on one cause
alone, not on a combination of causes) always more or less contingent on
collocations; and are hence subject to the characteristic infirmity of
empirical laws—that of being admissible only where the collocations are
known by experience to be such as are requisite for the truth of the
law; that is, only within the conditions of time and place confirmed by
actual observation.
§ 2. This principle, when stated in general terms, seems clear and
indisputable; yet many of the ordinary judgments of mankind, the
propriety of which is not questioned, have at least the semblance of
being inconsistent with it. On what grounds, it may be asked, do we
expect that the sun will rise to-morrow? To-morrow is beyond the limits
of time comprehended in our observations. They have extended over some
thousands of years past, but they do not include the future. Yet we
infer with confidence that the sun will rise to-morrow; and nobody
doubts that we are entitled to do so. Let us consider what is the
warrant for this confidence.
In the example in question, we know the causes on which the derivative
uniformity depends. They are: the sun giving out light, the earth in a
state of rotation and intercepting light. The induction which shows
these to be the real causes, and not merely prior effects of a common
cause, being complete, the only circumstances which could defeat the
derivative law are such as would destroy or counteract one or other of
the combined causes. While the causes exist and are not counteracted,
the effect will continue. If they exist and are not counteracted
to-morrow, the sun will rise to-morrow.
Since the causes, namely, the sun and the earth, the one in the state of
giving out light, the other in a state of rotation, will exist until
something destroys them, all depends on the probabilities of their
destruction, or of their counteraction. We know by observation (omitting
the inferential proofs of an existence for thousands of ages anterior)
that these phenomena have continued for (say) five thousand years.
Within that time there has existed no cause sufficient to diminish them
appreciably, nor which has counteracted their effect in any appreciable
degree. The chance, therefore, that the sun may not rise to-morrow
amounts to the chance that some cause, which has not manifested itself
in the smallest degree during five thousand years, will exist to-morrow
in such intensity as to destroy the sun or the earth, the sun’s light or
the earth’s rotation, or to produce an immense disturbance in the effect
resulting from those causes.
Now, if such a cause will exist to-morrow, or at any future time, some
cause, proximate or remote, of that cause must exist now, and must have
existed during the whole of the five thousand years. If, therefore, the
sun do not rise to-morrow, it will be because some cause has existed,
the effects of which, though during five thousand years they have not
amounted to a perceptible quantity, will in one day become overwhelming.
Since this cause has not been recognized during such an interval of time
by observers stationed on our earth, it must, if it be a single agent,
be either one whose effects develop themselves gradually and very
slowly, or one which existed in regions beyond our observation, and is
now on the point of arriving in our part of the universe. Now all causes
which we have experience of act according to laws incompatible with the
supposition that their effects, after accumulating so slowly as to be
imperceptible for five thousand years, should start into immensity in a
single day. No mathematical law of proportion between an effect and the
quantity or relations of its cause could produce such contradictory
results. The sudden development of an effect of which there was no
previous trace always arises from the coming together of several
distinct causes, not previously conjoined; but if such sudden
conjunction is destined to take place, the causes, or _their_ causes,
must have existed during the entire five thousand years; and their not
having once come together during that period shows how rare that
particular combination is. We have, therefore, the warrant of a rigid
induction for considering it probable, in a degree undistinguishable
from certainty, that the known conditions requisite for the sun’s rising
will exist to-morrow.
§ 3. But this extension of derivative laws, not causative, beyond the
limits of observation can only be to _adjacent_ cases. If, instead of
to-morrow, we had said this day twenty thousand years, the inductions
would have been any thing but conclusive. That a cause which, in
opposition to very powerful causes, produced no perceptible effect
during five thousand years, should produce a very considerable one by
the end of twenty thousand, has nothing in it which is not in conformity
with our experience of causes. We know many agents, the effect of which
in a short period does not amount to a perceptible quantity, but by
accumulating for a much longer period becomes considerable. Besides,
looking at the immense multitude of the heavenly bodies, their vast
distances, and the rapidity of the motion of such of them as are known
to move, it is a supposition not at all contradictory to experience that
some body may be in motion toward us, or we toward it, within the limits
of whose influence we have not come during five thousand years, but
which in twenty thousand more may be producing effects upon us of the
most extraordinary kind. Or the fact which is capable of preventing
sunrise may be, not the cumulative effect of one cause, but some new
combination of causes; and the chances favorable to that combination,
though they have not produced it once in five thousand years, may
produce it once in twenty thousand. So that the inductions which
authorize us to expect future events, grow weaker and weaker the further
we look into the future, and at length become inappreciable.
We have considered the probabilities of the sun’s rising to-morrow, as
derived from the real laws; that is, from the laws of the causes on
which that uniformity is dependent. Let us now consider how the matter
would have stood if the uniformity had been known only as an empirical
law; if we had not been aware that the sun’s light and the earth’s
rotation (or the sun’s motion) were the causes on which the periodical
occurrence of daylight depends. We could have extended this empirical
law to cases adjacent in time, though not to so great a distance of time
as we can now. Having evidence that the effects had remained unaltered
and been punctually conjoined for five thousand years, we could infer
that the unknown causes on which the conjunction is dependent had
existed undiminished and uncounteracted during the same period. The same
conclusions, therefore, would follow as in the preceding case, except
that we should only know that during five thousand years nothing had
occurred to defeat perceptibly this particular effect; while, when we
know the causes, we have the additional assurance that during that
interval no such change has been noticeable in the causes themselves as
by any degree of multiplication or length of continuance could defeat
the effect.
To this must be added, that when we know the causes, we may be able to
judge whether there exists any known cause capable of counteracting
them, while as long as they are unknown, we can not be sure but that if
we did know them, we could predict their destruction from causes
actually in existence. A bed-ridden savage, who had never seen the
cataract of Niagara, but who lived within hearing of it, might imagine
that the sound he heard would endure forever; but if he knew it to be
the effect of a rush of waters over a barrier of rock which is
progressively wearing away, he would know that within a number of ages
which may be calculated it will be heard no more. In proportion,
therefore, to our ignorance of the causes on which the empirical law
depends, we can be less assured that it will continue to hold good; and
the further we look into futurity, the less improbable is it that some
one of the causes, whose co-existence gives rise to the derivative
uniformity, may be destroyed or counteracted. With every prolongation of
time the chances multiply of such an event; that is to say, its
non-occurrence hitherto becomes a less guarantee of its not occurring
within the given time. If, then, it is only to cases which in point of
time are adjacent (or nearly adjacent) to those which we have actually
observed, that _any_ derivative law, not of causation, can be extended
with an assurance equivalent to certainty, much more is this true of a
merely empirical law. Happily, for the purposes of life it is to such
cases alone that we can almost ever have occasion to extend them.
In respect of place, it might seem that a merely empirical law could not
be extended even to adjacent cases; that we could have no assurance of
its being true in any place where it has not been specially observed.
The past duration of a cause is a guarantee for its future existence,
unless something occurs to destroy it; but the existence of a cause in
one or any number of places is no guarantee for its existence in any
other place, since there is no uniformity in the collocations of
primeval causes. When, therefore, an empirical law is extended beyond
the local limits within which it has been found true by observation, the
cases to which it is thus extended must be such as are presumably within
the influence of the same individual agents. If we discover a new planet
within the known bounds of the solar system (or even beyond those
bounds, but indicating its connection with the system by revolving round
the sun), we may conclude, with great probability, that it revolves on
its axis. For all the known planets do so; and this uniformity points to
some common cause, antecedent to the first records of astronomical
observation; and though the nature of this cause can only be matter of
conjecture, yet if it be, as is not unlikely, and as Laplace’s theory
supposes, not merely the same kind of cause, but the same individual
cause (such as an impulse given to all the bodies at once), that cause,
acting at the extreme points of the space occupied by the sun and
planets, is likely, unless defeated by some counteracting cause, to have
acted at every intermediate point, and probably somewhat beyond; and
therefore acted, in all probability, upon the supposed newly-discovered
planet.
When, therefore, effects which are always found conjoined can be traced
with any probability to an identical (and not merely a similar) origin,
we may with the same probability extend the empirical law of their
conjunction to all places within the extreme local boundaries within
which the fact has been observed, subject to the possibility of
counteracting causes in some portion of the field. Still more
confidently may we do so when the law is not merely empirical; when the
phenomena which we find conjoined are effects of ascertained causes,
from the laws of which the conjunction of their effects is deducible. In
that case, we may both extend the derivative uniformity over a larger
space, and with less abatement for the chance of counteracting causes.
The first, because instead of the local boundaries of our observation of
the fact itself, we may include the extreme boundaries of the
ascertained influence of its causes. Thus the succession of day and
night, we know, holds true of all the bodies of the solar system except
the sun itself; but we know this only because we are acquainted with the
causes. If we were not, we could not extend the proposition beyond the
orbits of the earth and moon, at both extremities of which we have the
evidence of observation for its truth. With respect to the probability
of counteracting causes, it has been seen that this calls for a greater
abatement of confidence, in proportion to our ignorance of the causes on
which the phenomena depend. On both accounts, therefore, a derivative
law which we know how to resolve, is susceptible of a greater extension
to cases adjacent in place, than a merely empirical law.