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 an 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 cannot 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,[15] and the time
is now come for endeavouring 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
labours 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 connexion
between _a_ and A, since _a_ may have many causes, and may have been
produced, in these different instances, not by anything 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 connexion 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
connexion 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) cannot 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 an 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 coexist 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 coexistence, 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 cannot 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 connexion 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, coexisted 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_ coexist 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
connexion exists.
On the other hand, suppose that we were inquiring whether there be any
connexion between rain and any particular wind. Rain, we know,
occasionally occurs with every wind; therefore the connexion, if it
exists, cannot be an actual law; but still, rain may be connected with
some particular wind through causation; that is, though they cannot be
always effects of the same cause (for if so they would regularly
coexist), 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 coexist. 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 connexion 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 connexion between the phenomena, that the
connexion 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 connexion 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 connexion; 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 anything 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 connexion 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 coexist, 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 coexist 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 connexion, 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
connexion 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 cannot possibly exclude all of them from any experiment, though
we may vary them; our resource is, to endeavour 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 coexistence 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.[16]
§ 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 learnt, 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 favour of a particular throw; we may
conclude with assurance that there is some constant cause acting in
favour 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 afterwards 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 _à 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 absolute frequency of A. If,
for example, A exists always, and therefore coexists with everything, no
number of instances of its coexistence with B would prove a connexion;
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 towards proving a connexion 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.