Of Chance And Its Elimination.
§ 1. Considering, then, as empirical laws only those observed uniformities
respecting which the question whether they are laws of causation must
remain undecided until they can be explained deductively, or until some
means are found of applying the Method of Difference to the case, it has
been shown in the preceding chapter that until a uniformity can, in one or
the other of these modes, be taken out of the class of empirical laws, and
brought either into that of laws of causation or of the demonstrated
results of laws of causation, it can not with any assurance be pronounced
true beyond the local and other limits within which it has been found so
by actual observation. It remains to consider how we are to assure
ourselves of its truth even within those limits; after what quantity of
experience a generalization which rests solely on the Method of Agreement
can be considered sufficiently established, even as an empirical law. In a
former chapter, when treating of the Methods of Direct Induction, we
expressly reserved this question,(174) and the time is now come for
endeavoring to solve it.
We found that the Method of Agreement has the defect of not proving
causation, and can, therefore, only be employed for the ascertainment of
empirical laws. But we also found that besides this deficiency, it labors
under a characteristic imperfection, tending to render uncertain even such
conclusions as it is in itself adapted to prove. This imperfection arises
from Plurality of Causes. Although two or more cases in which the
phenomenon _a_ has been met with may have no common antecedent except A,
this does not prove that there is any connection between _a_ and A, since
_a_ may have many causes, and may have been produced, in these different
instances, not by any thing which the instances had in common, but by some
of those elements in them which were different. We nevertheless observed,
that in proportion to the multiplication of instances pointing to A as the
antecedent, the characteristic uncertainty of the method diminishes, and
the existence of a law of connection between A and _a_ more nearly
approaches to certainty. It is now to be determined after what amount of
experience this certainty may be deemed to be practically attained, and
the connection between A and _a_ may be received as an empirical law.
This question may be otherwise stated in more familiar terms: After how
many and what sort of instances may it be concluded that an observed
coincidence between two phenomena is not the effect of chance?
It is of the utmost importance for understanding the logic of induction,
that we should form a distinct conception of what is meant by chance, and
how the phenomena which common language ascribes to that abstraction are
really produced.
§ 2. Chance is usually spoken of in direct antithesis to law; whatever, it
is supposed, can not be ascribed to any law is attributed to chance. It
is, however, certain that whatever happens is the result of some law; is
an effect of causes, and could have been predicted from a knowledge of the
existence of those causes, and from their laws. If I turn up a particular
card, that is a consequence of its place in the pack. Its place in the
pack was a consequence of the manner in which the cards were shuffled, or
of the order in which they were played in the last game; which, again,
were effects of prior causes. At every stage, if we had possessed an
accurate knowledge of the causes in existence, it would have been
abstractedly possible to foretell the effect.
An event occurring by chance may be better described as a coincidence from
which we have no ground to infer a uniformity—the occurrence of a
phenomenon in certain circumstances, without our having reason on that
account to infer that it will happen again in those circumstances. This,
however, when looked closely into, implies that the enumeration of the
circumstances is not complete. Whatever the fact be, since it has occurred
once, we may be sure that if _all_ the same circumstances were repeated it
would occur again; and not only if all, but there is some particular
portion of those circumstances on which the phenomenon is invariably
consequent. With most of them, however, it is not connected in any
permanent manner; its conjunction with those is said to be the effect of
chance, to be merely casual. Facts casually conjoined are separately the
effects of causes, and therefore of laws; but of different causes, and
causes not connected by any law.
It is incorrect, then, to say that any phenomenon is produced by chance;
but we may say that two or more phenomena are conjoined by chance, that
they co-exist or succeed one another only by chance; meaning that they are
in no way related through causation; that they are neither cause and
effect, nor effects of the same cause, nor effects of causes between which
there subsists any law of co-existence, nor even effects of the same
collocation of primeval causes.
If the same casual coincidence never occurred a second time, we should
have an easy test for distinguishing such from the coincidences which are
the results of a law. As long as the phenomena had been found together
only once, so long, unless we knew some more general laws from which the
coincidence might have resulted, we could not distinguish it from a casual
one; but if it occurred twice, we should know that the phenomena so
conjoined must be in some way connected through their causes.
There is, however, no such test. A coincidence may occur again and again,
and yet be only casual. Nay, it would be inconsistent with what we know of
the order of nature to doubt that every casual coincidence will sooner or
later be repeated, as long as the phenomena between which it occurred do
not cease to exist, or to be reproduced. The recurrence, therefore, of the
same coincidence more than once, or even its frequent recurrence, does not
prove that it is an instance of any law; does not prove that it is not
casual, or, in common language, the effect of chance.
And yet, when a coincidence can not be deduced from known laws, nor proved
by experiment to be itself a case of causation, the frequency of its
occurrence is the only evidence from which we can infer that it is the
result of a law. Not, however, its absolute frequency. The question is not
whether the coincidence occurs often or seldom, in the ordinary sense of
those terms; but whether it occurs more often than chance will account
for; more often than might rationally be expected if the coincidence were
casual. We have to decide, therefore, what degree of frequency in a
coincidence chance will account for; and to this there can be no general
answer. We can only state the principle by which the answer must be
determined; the answer itself will be different in every different case.
Suppose that one of the phenomena, A, exists always, and the other
phenomenon, B, only occasionally; it follows that every instance of B will
be an instance of its coincidence with A, and yet the coincidence will be
merely casual, not the result of any connection between them. The fixed
stars have been constantly in existence since the beginning of human
experience, and all phenomena that have come under human observation have,
in every single instance, co-existed with them; yet this coincidence,
though equally invariable with that which exists between any of those
phenomena and its own cause, does not prove that the stars are its cause,
nor that they are in anywise connected with it. As strong a case of
coincidence, therefore, as can possibly exist, and a much stronger one in
point of mere frequency than most of those which prove laws, does not here
prove a law; why? because, since the stars exist always, they _must_
co-exist with every other phenomenon, whether connected with them by
causation or not. The uniformity, great though it be, is no greater than
would occur on the supposition that no such connection exists.
On the other hand, suppose that we were inquiring whether there be any
connection between rain and any particular wind. Rain, we know,
occasionally occurs with every wind; therefore, the connection, if it
exists, can not be an actual law; but still rain may be connected with
some particular wind through causation; that is, though they can not be
always effects of the same cause (for if so they would regularly
co-exist), there may be some causes common to the two, so that in so far
as either is produced by those common causes, they will, from the laws of
the causes, be found to co-exist. How, then, shall we ascertain this? The
obvious answer is, by observing whether rain occurs with one wind more
frequently than with any other. That, however, is not enough; for perhaps
that one wind blows more frequently than any other; so that its blowing
more frequently in rainy weather is no more than would happen, although it
had no connection with the causes of rain, provided it were not connected
with causes adverse to rain. In England, westerly winds blow during about
twice as great a portion of the year as easterly. If, therefore, it rains
only twice as often with a westerly as with an easterly wind, we have no
reason to infer that any law of nature is concerned in the coincidence. If
it rains more than twice as often, we may be sure that some law is
concerned; either there is some cause in nature which, in this climate,
tends to produce both rain and a westerly wind, or a westerly wind has
itself some tendency to produce rain. But if it rains less than twice as
often, we may draw a directly opposite inference: the one, instead of
being a cause, or connected with causes of the other, must be connected
with causes adverse to it, or with the absence of some cause which
produces it; and though it may still rain much oftener with a westerly
wind than with an easterly, so far would this be from proving any
connection between the phenomena, that the connection proved would be
between rain and an easterly wind, to which, in mere frequency of
coincidence, it is less allied.
Here, then, are two examples: in one, the greatest possible frequency of
coincidence, with no instance whatever to the contrary, does not prove
that there is any law; in the other, a much less frequency of
coincidence, even when non-coincidence is still more frequent, does
prove that there is a law. In both cases the principle is the same. In
both we consider the positive frequency of the phenomena themselves, and
how great frequency of coincidence that must of itself bring about,
without supposing any connection between them, provided there be no
repugnance; provided neither be connected with any cause tending to
frustrate the other. If we find a greater frequency of coincidence than
this, we conclude that there is some connection; if a less frequency,
that there is some repugnance. In the former case, we conclude that one
of the phenomena can under some circumstances cause the other, or that
there exists something capable of causing them both; in the latter, that
one of them, or some cause which produces one of them, is capable of
counteracting the production of the other. We have thus to deduct from
the observed frequency of coincidence as much as may be the effect of
chance, that is, of the mere frequency of the phenomena themselves; and
if any thing remains, what does remain is the residual fact which proves
the existence of a law.
The frequency of the phenomena can only be ascertained within definite
limits of space and time; depending as it does on the quantity and
distribution of the primeval natural agents, of which we can know
nothing beyond the boundaries of human observation, since no law, no
regularity, can be traced in it, enabling us to infer the unknown from
the known. But for the present purpose this is no disadvantage, the
question being confined within the same limits as the data. The
coincidences occurred in certain places and times, and within those we
can estimate the frequency with which such coincidences would be
produced by chance. If, then, we find from observation that A exists in
one case out of every two, and B in one case out of every three; then,
if there be neither connection nor repugnance between them, or between
any of their causes, the instances in which A and B will both exist,
that is to say will co-exist, will be one case in every six. For A
exists in three cases out of six; and B, existing in one case out of
every three without regard to the presence or absence of A, will exist
in one case out of those three. There will therefore be, of the whole
number of cases, two in which A exists without B; one case of B without
A; two in which neither B nor A exists, and one case out of six in which
they both exist. If, then, in point of fact, they are found to co-exist
oftener than in one case out of six; and, consequently, A does not exist
without B so often as twice in three times, nor B without A so often as
once in every twice, there is some cause in existence which tends to
produce a conjunction between A and B.
Generalizing the result, we may say that if A occurs in a larger
proportion of the cases where B is than of the cases where B is not,
then will B also occur in a larger proportion of the cases where A is
than of the cases where A is not; and there is some connection, through
causation, between A and B. If we could ascend to the causes of the two
phenomena, we should find, at some stage, either proximate or remote,
some cause or causes common to both; and if we could ascertain what
these are, we could frame a generalization which would be true without
restriction of place or time; but until we can do so, the fact of a
connection between the two phenomena remains an empirical law.
§ 3. Having considered in what manner it may be determined whether any
given conjunction of phenomena is casual, or the result of some law, to
complete the theory of chance it is necessary that we should now
consider those effects which are partly the result of chance and partly
of law, or, in other words, in which the effects of casual conjunctions
of causes are habitually blended in one result with the effects of a
constant cause.
This is a case of Composition of Causes; and the peculiarity of it is,
that instead of two or more causes intermixing their effects in a
regular manner with those of one another, we have now one constant
cause, producing an effect which is successively modified by a series of
variable causes. Thus, as summer advances, the approach of the sun to a
vertical position tends to produce a constant increase of temperature;
but with this effect of a constant cause, there are blended the effects
of many variable causes, winds, clouds, evaporation, electric agencies
and the like, so that the temperature of any given day depends in part
on these fleeting causes, and only in part on the constant cause. If the
effect of the constant cause is always accompanied and disguised by
effects of variable causes, it is impossible to ascertain the law of the
constant cause in the ordinary manner by separating it from all other
causes and observing it apart. Hence arises the necessity of an
additional rule of experimental inquiry.
When the action of a cause A is liable to be interfered with, not
steadily by the same cause or causes, but by different causes at
different times, and when these are so frequent, or so indeterminate,
that we can not possibly exclude all of them from any experiment, though
we may vary them; our resource is, to endeavor to ascertain what is the
effect of all the variable causes taken together. In order to do this,
we make as many trials as possible, preserving A invariable. The results
of these different trials will naturally be different, since the
indeterminate modifying causes are different in each; if, then, we do
not find these results to be progressive, but, on the contrary, to
oscillate about a certain point, one experiment giving a result a little
greater, another a little less, one a result tending a little more in
one direction, another a little more in the contrary direction; while
the average or middle point does not vary, but different sets of
experiments (taken in as great a variety of circumstances as possible)
yield the same mean, provided only they be sufficiently numerous; then
that mean, or average result, is the part, in each experiment, which is
due to the cause A, and is the effect which would have been obtained if
A could have acted alone; the variable remainder is the effect of
chance, that is, of causes the co-existence of which with the cause A
was merely casual. The test of the sufficiency of the induction in this
case is, when any increase of the number of trials from which the
average is struck does not materially alter the average.
This kind of elimination, in which we do not eliminate any one
assignable cause, but the multitude of floating unassignable ones, may
be termed the Elimination of Chance. We afford an example of it when we
repeat an experiment, in order, by taking the mean of different results,
to get rid of the effects of the unavoidable errors of each individual
experiment. When there is no permanent cause, such as would produce a
tendency to error peculiarly in one direction, we are warranted by
experience in assuming that the errors on one side will, in a certain
number of experiments, about balance the errors on the contrary side. We
therefore repeat the experiment, until any change which is produced in
the average of the whole by further repetition, falls within limits of
error consistent with the degree of accuracy required by the purpose we
have in view.(175)
§ 4. In the supposition hitherto made, the effect of the constant cause
A has been assumed to form so great and conspicuous a part of the
general result, that its existence never could be a matter of
uncertainty, and the object of the eliminating process was only to
ascertain _how much_ is attributable to that cause; what is its exact
law. Cases, however, occur in which the effect of a constant cause is so
small, compared with that of some of the changeable causes with which it
is liable to be casually conjoined, that of itself it escapes notice,
and the very existence of any effect arising from a constant cause is
first learned by the process which in general serves only for
ascertaining the quantity of that effect. This case of induction may be
characterized as follows: A given effect is known to be chiefly, and not
known not to be wholly, determined by changeable causes. If it be wholly
so produced, then if the aggregate be taken of a sufficient number of
instances, the effects of these different causes will cancel one
another. If, therefore, we do not find this to be the case, but, on the
contrary, after such a number of trials has been made that no further
increase alters the average result, we find that average to be, not
zero, but some other quantity, about which, though small in comparison
with the total effect, the effect nevertheless oscillates, and which is
the middle point in its oscillation; we may conclude this to be the
effect of some constant cause; which cause, by some of the methods
already treated of, we may hope to detect. This may be called _the
discovery of a residual phenomenon by eliminating the effects of
chance_.
It is in this manner, for example, that loaded dice may be discovered.
Of course no dice are so clumsily loaded that they must always throw
certain numbers; otherwise the fraud would be instantly detected. The
loading, a constant cause, mingles with the changeable causes which
determine what cast will be thrown in each individual instance. If the
dice were not loaded, and the throw were left to depend entirely on the
changeable causes, these in a sufficient number of instances would
balance one another, and there would be no preponderant number of throws
of any one kind. If, therefore, after such a number of trials that no
further increase of their number has any material effect upon the
average, we find a preponderance in favor of a particular throw; we may
conclude with assurance that there is some constant cause acting in
favor of that throw, or, in other words, that the dice are not fair; and
the exact amount of the unfairness. In a similar manner, what is called
the diurnal variation of the barometer, which is very small compared
with the variations arising from the irregular changes in the state of
the atmosphere, was discovered by comparing the average height of the
barometer at different hours of the day. When this comparison was made,
it was found that there was a small difference, which on the average was
constant, however the absolute quantities might vary, and which
difference, therefore, must be the effect of a constant cause. This
cause was afterward ascertained, deductively, to be the rarefaction of
the air, occasioned by the increase of temperature as the day advances.
§ 5. After these general remarks on the nature of chance, we are
prepared to consider in what manner assurance may be obtained that a
conjunction between two phenomena, which has been observed a certain
number of times, is not casual, but a result of causation, and to be
received, therefore, as one of the uniformities of nature, though (until
accounted for _a priori_) only as an empirical law.
We will suppose the strongest case, namely, that the phenomenon B has
never been observed except in conjunction with A. Even then, the
probability that they are connected is not measured by the total number
of instances in which they have been found together, but by the excess
of that number above the number due to the absolutely frequency of A.
If, for example, A exists always, and therefore co-exists with every
thing, no number of instances of its co-existence with B would prove a
connection; as in our example of the fixed stars. If A be a fact of such
common occurrence that it may be presumed to be present in half of all
the cases that occur, and therefore in half the cases in which B occurs,
it is only the proportional excess above half that is to be reckoned as
evidence toward proving a connection between A and B.
In addition to the question, What is the number of coincidences which,
on an average of a great multitude of trials, may be expected to arise
from chance alone? there is also another question, namely, Of what
extent of deviation from that average is the occurrence credible, from
chance alone, in some number of instances smaller than that required for
striking a fair average? It is not only to be considered what is the
general result of the chances in the long run, but also what are the
extreme limits of variation from the general result, which may
occasionally be expected as the result of some smaller number of
instances.
The consideration of the latter question, and any consideration of the
former beyond that already given to it, belong to what mathematicians
term the doctrine of chances, or, in a phrase of greater pretension, the
Theory of Probabilities.