OF APPROXIMATE GENERALIZATIONS, AND PROBABLE EVIDENCE.
§ 1. In our inquiries into the nature of the inductive process, we must
not confine our notice to such generalizations from experience as
profess to be universally true. There is a class of inductive truths
avowedly not universal; in which it is not pretended that the predicate
is always true of the subject; but the value of which, as
generalizations, is nevertheless extremely great. An important portion
of the field of inductive knowledge does not consist of universal
truths, but of approximations to such truths; and when a conclusion is
said to rest on probable evidence, the premises it is drawn from are
usually generalizations of this sort.
As every certain inference respecting a particular case, implies that
there is ground for a general proposition, of the form, Every A is B; so
does every probable inference suppose that there is ground for a
proposition of the form, Most A are B: and the degree of probability of
the inference in an average case, will depend on the proportion between
the number of instances existing in nature which accord with the
generalization, and the number of those which conflict with it.
§ 2. Propositions in the form, Most A are B, are of a very different
degree of importance in science, and in the practice of life. To the
scientific inquirer they are valuable chiefly as materials for, and
steps towards, universal truths. The discovery of these is the proper
end of science: its work is not done if it stops at the proposition that
a majority of A are B, without circumscribing that majority by some
common character, fitted to distinguish them from the minority.
Independently of the inferior precision of such imperfect
generalizations, and the inferior assurance with which they can be
applied to individual cases, it is plain that, compared with exact
generalizations, they are almost useless as means of discovering
ulterior truths by way of deduction. We may, it is true, by combining
the proposition Most A are B, with an universal proposition, Every B is
C, arrive at the conclusion that Most A are C. But when a second
proposition of the approximate kind is introduced,--or even when there
is but one, if that one be the major premise,--nothing can in general be
positively concluded. When the major is Most B are D, then, even if the
minor be Every A is B, we cannot infer that most A are D, or with any
certainty that even some A are D. Though the majority of the class B
have the attribute signified by D, the whole of the sub-class A may
belong to the minority.[34]
Though so little use can be made, in science, of approximate
generalizations, except as a stage on the road to something better, for
practical guidance they are often all we have to rely on. Even when
science has really determined the universal laws of any phenomenon, not
only are those laws generally too much encumbered with conditions to be
adapted for every-day use, but the cases which present themselves in
life are too complicated, and our decisions require to be taken too
rapidly, to admit of waiting till the existence of a phenomenon can be
proved by what have been scientifically ascertained to be universal
marks of it. To be indecisive and reluctant to act, because we have not
evidence of a perfectly conclusive character to act on, is a defect
sometimes incident to scientific minds, but which, wherever it exists,
renders them unfit for practical emergencies. If we would succeed in
action, we must judge by indications which, though they do not
generally mislead us, sometimes do; and must make up, as far as
possible, for the incomplete conclusiveness of any one indication, by
obtaining others to corroborate it. The principles of induction
applicable to approximate generalization are therefore a not less
important subject of inquiry, than the rules for the investigation of
universal truths; and might reasonably be expected to detain us almost
as long, were it not that these principles are mere corollaries from
those which have been already treated of.
§ 3. There are two sorts of cases in which we are forced to guide
ourselves by generalizations of the imperfect form, Most A are B. The
first is, when we have no others; when we have not been able to carry
our investigation of the laws of the phenomena any farther; as in the
following propositions: Most dark-eyed persons have dark hair; Most
springs contain mineral substances; Most stratified formations contain
fossils. The importance of this class of generalizations is not very
great; for, though it frequently happens that we see no reason why that
which is true of most individuals of a class is not true of the
remainder, nor are able to bring the former under any general
description which can distinguish them from the latter, yet if we are
willing to be satisfied with propositions of a less degree of
generality, and to break down the class A into subclasses, we may
generally obtain a collection of propositions exactly true. We do not
know why most wood is lighter than water, nor can we point out any
general property which discriminates wood that is lighter than water
from that which is heavier. But we know exactly what species are the one
and what the other. And if we meet with a specimen not conformable to
any known species (the only case in which our previous knowledge affords
no other guidance than the approximate generalization), we can generally
make a specific experiment, which is a surer resource.
It often happens, however, that the proposition, Most A are B, is not
the ultimatum of our scientific progress, though the knowledge we
possess beyond it cannot conveniently be brought to bear upon the
particular instance. In such a case we know well enough what
circumstances distinguish the portion of A which has the attribute B
from the portion which has it not, but have no means, or have not time,
to examine whether those characteristic circumstances exist or not in
the individual case. This is the situation we are generally in when the
inquiry is of the kind called moral, that is, of the kind which has in
view to predict human actions. To enable us to affirm anything
universally concerning the actions of classes of human beings, the
classification must be grounded on the circumstances of their mental
culture and habits, which in an individual case are seldom exactly
known; and classes grounded on these distinctions would never precisely
accord with those into which mankind are divided for social purposes.
All propositions which can be framed respecting the actions of human
beings as ordinarily classified, or as classified according to any kind
of outward indications, are merely approximate. We can only say, Most
persons of a particular age, profession, country, or rank in society,
have such and such qualities; or, Most persons when placed in certain
circumstances act in such and such a way. Not that we do not often know
well enough on what causes the qualities depend, or what sort of persons
they are who act in that particular way; but we have seldom the means of
knowing whether any individual person has been under the influence of
those causes, or is a person of that particular sort. We could replace
the approximate generalizations by propositions universally true; but
these would hardly ever be capable of being applied to practice. We
should be sure of our majors, but we should not be able to get minors to
fit: we are forced, therefore, to draw our conclusions from coarser and
more fallible indications.
§ 4. Proceeding now to consider, what is to be regarded as sufficient
evidence of an approximate generalization; we can have no difficulty in
at once recognising that when admissible at all, it is admissible only
as an empirical law. Propositions of the form, Every A is B, are not
necessarily laws of causation, or ultimate uniformities of coexistence;
propositions like Most A are B _cannot_ be so. Propositions hitherto
found true in every observed instance, may yet be no necessary
consequence of laws of causation, or of ultimate uniformities, and
unless they are so, may, for aught we know, be false beyond the limits
of actual observation: still more evidently must this be the case with
propositions which are only true in a mere majority of the observed
instances.
There is some difference, however, in the degree of certainty of the
proposition, Most A are B, according as that approximate generalization
composes the whole of our knowledge of the subject, or not. Suppose,
first, that the former is the case. We know only that most A are B, not
why they are so, nor in what respect those which are, differ from those
which are not. How then did we learn that most A are B? Precisely in the
manner in which we should have learnt, had such happened to be the fact,
that all A are B. We collected a number of instances sufficient to
eliminate chance, and having done so, compared the number of instances
in the affirmative with the number in the negative. The result, like
other unresolved derivative laws, can be relied on solely within the
limits not only of place and time, but also of circumstance, under which
its truth has been actually observed; for as we are supposed to be
ignorant of the causes which make the proposition true, we cannot tell
in what manner any new circumstance might perhaps affect it. The
proposition, Most judges are inaccessible to bribes, would be found true
of Englishmen, Frenchmen, Germans, North Americans, and so forth; but if
on this evidence alone we extended the assertion to Orientals, we should
step beyond the limits, not only of place but of circumstance, within
which the fact had been observed, and should let in possibilities of the
absence of the determining causes, or the presence of counteracting
ones, which might be fatal to the approximate generalization.
In the case where the approximate proposition is not the ultimatum of
our scientific knowledge, but only the most available form of it for
practical guidance; where we know, not only that most A have the
attribute B, but also the causes of B, or some properties by which the
portion of A which has that attribute is distinguished from the portion
which has it not; we are rather more favourably situated than in the
preceding case. For we have now a double mode of ascertaining whether it
be true that most A are B; the direct mode, as before, and an indirect
one, that of examining whether the proposition admits of being deduced
from the known cause, or from any known criterion, of B. Let the
question, for example, be whether most Scotchmen can read? We may not
have observed, or received the testimony of others respecting, a
sufficient number and variety of Scotchmen to ascertain this fact; but
when we consider that the cause of being able to read is the having been
taught it, another mode of determining the question presents itself,
namely, by inquiring whether most Scotchmen have been sent to schools
where reading is effectually taught. Of these two modes, sometimes one
and sometimes the other is the more available. In some cases, the
frequency of the effect is the more accessible to that extensive and
varied observation which is indispensable to the establishment of an
empirical law; at other times, the frequency of the causes, or of some
collateral indications. It commonly happens that neither is susceptible
of so satisfactory an induction as could be desired, and that the
grounds on which the conclusion is received are compounded of both. Thus
a person may believe that most Scotchmen can read, because, so far as
his information extends, most Scotchmen have been sent to school, and
most Scotch schools teach reading effectually; and also because most of
the Scotchmen whom he has known or heard of, could read; though neither
of these two sets of observations may by itself fulfil the necessary
conditions of extent and variety.
Although the approximate generalization may in most cases be
indispensable for our guidance, even when we know the cause, or some
certain mark, of the attribute predicated; it needs hardly be observed
that we may always replace the uncertain indication by a certain one, in
any case in which we can actually recognise the existence of the cause
or mark. For example, an assertion is made by a witness, and the
question is, whether to believe it. If we do not look to any of the
individual circumstances of the case, we have nothing to direct us but
the approximate generalization, that truth is more common than
falsehood, or, in other words, that most persons, on most occasions,
speak truth. But if we consider in what circumstances the cases where
truth is spoken differ from those in which it is not, we find, for
instance, the following: the witness's being an honest person or not;
his being an accurate observer or not; his having an interest to serve
in the matter or not. Now, not only may we be able to obtain other
approximate generalizations respecting the degree of frequency of these
various possibilities, but we may know which of them is positively
realized in the individual case. That the witness has or has not an
interest to serve, we perhaps know directly; and the other two points
indirectly, by means of marks; as, for example, from his conduct on some
former occasion; or from his reputation, which, though a very uncertain
mark, affords an approximate generalization (as, for instance, Most
persons who are believed to be honest by those with whom they have had
frequent dealings, are really so) which approaches nearer to an
universal truth than the approximate general proposition with which we
set out, viz. Most persons on most occasions speak truth.
As it seems unnecessary to dwell further on the question of the evidence
of approximate generalizations, we shall proceed to a not less important
topic, that of the cautions to be observed in arguing from these
incompletely universal propositions to particular cases.
§ 5. So far as regards the direct application of an approximate
generalization to an individual instance, this question presents no
difficulty. If the proposition, Most A are B, has been established, by a
sufficient induction, as an empirical law, we may conclude that any
particular A is B with a probability proportioned to the preponderance
of the number of affirmative instances over the number of exceptions. If
it has been found practicable to attain numerical precision in the data,
a corresponding degree of precision may be given to the evaluation of
the chances of error in the conclusion. If it can be established as an
empirical law that nine out of every ten A are B, there will be one
chance in ten of error in assuming that any A, not individually known to
us, is a B: but this of course holds only within the limits of time,
place, and circumstance, embraced in the observations, and therefore
cannot be counted on for any sub-class or variety of A (or for A in any
set of external circumstances) which were not included in the average.
It must be added, that we can guide ourselves by the proposition, Nine
out of every ten A are B, only in cases of which we know nothing except
that they fall within the class A. For if we know, of any particular
instance _i_, not only that it falls under A, but to what species or
variety of A it belongs, we shall generally err in applying to _i_ the
average struck for the whole genus, from which the average corresponding
to that species alone would, in all probability, materially differ. And
so if _i_, instead of being a particular sort of instance, is an
instance known to be under the influence of a particular set of
circumstances. The presumption drawn from the numerical proportions in
the whole genus would probably, in such a case, only mislead. A general
average should only be applied to cases which are neither known, nor can
be presumed, to be other than average cases. Such averages, therefore,
are commonly of little use for the practical guidance of any affairs but
those which concern large numbers. Tables of the chances of life are
useful to insurance offices, but they go a very little way towards
informing any one of the chances of his own life, or any other life in
which he is interested, since almost every life is either better or
worse than the average. Such averages can only be considered as
supplying the first term in a series of approximations; the subsequent
terms proceeding on an appreciation of the circumstances belonging to
the particular case.
§ 6. From the application of a single approximate generalization to
individual cases, we proceed to the application of two or more of them
together to the same case.
When a judgment applied to an individual instance is grounded on two
approximate generalizations taken in conjunction, the propositions may
co-operate towards the result in two different ways. In the one, each
proposition is separately applicable to the case in hand, and our object
in combining them is to give to the conclusion in that particular case
the double probability arising from the two propositions separately.
This may be called joining two probabilities by way of Addition; and the
result is a probability greater than either. The other mode is, when
only one of the propositions is directly applicable to the case, the
second being only applicable to it by virtue of the application of the
first. This is joining two probabilities by way of Ratiocination or
Deduction; the result of which is a less probability than either. The
type of the first argument is, Most A are B; most C are B; this thing is
both an A and a C; therefore it is probably a B. The type of the second
is, Most A are B; most C are A; this is a C; therefore it is probably an
A, therefore it is probably a B. The first is exemplified when we prove
a fact by the testimony of two unconnected witnesses; the second, when
we adduce only the testimony of one witness that he has heard the thing
asserted by another. Or again, in the first mode it may be argued that
the accused committed the crime, because he concealed himself, and
because his clothes were stained with blood; in the second, that he
committed it because he washed or destroyed his clothes, which is
supposed to render it probable that they were stained with blood.
Instead of only two links, as in these instances, we may suppose chains
of any length. A chain of the former kind was termed by Bentham[35] a
self-corroborative chain of evidence; the second, a self-infirmative
chain.
When approximate generalizations are joined by way of addition, we may
deduce from the theory of probabilities laid down in a former chapter,
in what manner each of them adds to the probability of a conclusion
which has the warrant of them all.
In the early editions of this treatise, the joint probability arising
from the sum of two independent probabilities was estimated in the
following manner. If, on an average, two of every three As are Bs, and
three of every four Cs are Bs, the probability that something which is
both an A and a C is a B, will be more than two in three, or than three
in four. Of every twelve things which are As, all except four are Bs by
the supposition; and if the whole twelve, and consequently those four,
have the characters of C likewise, three of these will be Bs on that
ground. Therefore, out of twelve which are both As and Cs, eleven are
Bs. To state the argument in another way; a thing which is both an A and
a C, but which is not a B, is found in only one of three sections of the
class A, and in only one of four sections of the class C; but this
fourth of C being spread over the whole of A indiscriminately, only
one-third part of it (or one-twelfth of the whole number) belongs to the
third section of A; therefore a thing which is not a B occurs only once,
among twelve things which are both As and Cs. The argument would in the
language of the doctrine of chances, be thus expressed: the chance that
an A is not a B is 1/3, the chance that a C is not a B is 1/4; hence if
the thing be both an A and a C, the chance is 1/3 of 1/4 = 1/12.
It has, however, been pointed out to me by a mathematical friend, that
in this statement the evaluation of the chances is erroneous. The
correct mode of setting out the possibilities is as follows. If the
thing (let us call it T) which is both an A and a C, is a B, something
is true which is only true twice in every thrice, and something else
which is only true thrice in every four times. The first fact being true
eight times in twelve, and the second being true six times in every
eight, and consequently six times in those eight; both facts will be
true only six times in twelve. On the other hand if T, although it is
both an A and a C, is not a B, something is true which is only true once
in every thrice, and something else which is only true once in every
four times. The former being true four times out of twelve, and the
latter once in every four, and therefore once in those four; both are
only true in one case out of twelve. So that T is a B six times in
twelve, and T is not a B, only once: making the comparative
probabilities, not eleven to one, as I had previously made them, but six
to one.
It may be asked, what happens in the remaining cases? since in this
calculation seven out of twelve cases seem to have exhausted the
possibilities. If T is a B in only six cases of every twelve, and a
not-B in only one, what is it in the other five? The only supposition
remaining for those cases is that it is neither a B nor not a B, which
is impossible. But this impossibility merely proves that the state of
things supposed in the hypothesis does not exist in those cases. They
are cases that do not furnish anything which is both an A and a C.
To make this intelligible, we will substitute for our symbols a concrete
case. Let there be two witnesses, M and N, whose probabilities of
veracity correspond with the ratios of the preceding example: M speaks
truth twice in every thrice, N thrice in every four times. The question
is, what is the probability that a statement, in which they both concur,
will be true. The cases may be classed as follows. Both the witnesses
will speak truly six in every twelve times; both falsely once in twelve
times. Therefore, if they both agree in an assertion, it will be true
six times, for once that it will be false. What happens in the remaining
cases is here evident; there will be five cases in every twelve in which
the witnesses will not agree. M will speak truth and N falsehood in two
cases of every twelve; N will speak truth and M falsehood in three
cases, making in all five. In these cases, however, the witnesses will
not agree in their testimony. But disagreement between them is excluded
by the supposition. There are, therefore, only seven cases which are
within the conditions of the hypothesis; of which seven, veracity exists
in six, and falsehood in one. Resuming our former symbols, in five cases
out of twelve T is not both an A and a C, but an A only, or a C only.
The cases in which it is both are only seven, in six of which it is a B,
in one not a B, making the chance six to one, or 6/7 and 1/7
respectively.
In this correct, as in the former incorrect computation, it is of course
presupposed that the probabilities arising from A and C are independent
of each other. There must not be any such connexion between A and C,
that when a thing belongs to the one class it will therefore belong to
the other, or even have a greater chance of doing so. Otherwise the
not-Bs which are Cs may be, most or even all of them, identical with the
not-Bs which are As; in which last case the probability arising from A
and C together will be no greater than that arising from A alone.
When approximate generalizations are joined together in the other mode,
that of deduction, the degree of probability of the inference, instead
of increasing, diminishes at each step. From two such premises as Most A
are B, Most B are C, we cannot with certainty conclude that even a
single A is C; for the whole of the portion of A which in any way falls
under B, may perhaps be comprised in the exceptional part of it. Still,
the two propositions in question afford an appreciable probability that
any given A is C, provided the average on which the second proposition
is grounded, was taken fairly with reference to the first; provided the
proposition, Most B are C, was arrived at in a manner leaving no
suspicion that the probability arising from it is otherwise than fairly
distributed over the section of B which belongs to A. For though the
instances which are A _may_ be all in the minority, they may, also, be
all in the majority; and the one possibility is to be set against the
other. On the whole, the probability arising from the two propositions
taken together, will be correctly measured by the probability arising
from the one, abated in the ratio of that arising from the other. If
nine out of ten Swedes have light hair, and eight out of nine
inhabitants of Stockholm are Swedes, the probability arising from these
two propositions, that any given inhabitant of Stockholm is
light-haired, will amount to eight in ten; though it is rigorously
possible that the whole Swedish population of Stockholm might belong to
that tenth section of the people of Sweden who are an exception to the
rest.
If the premises are known to be true not of a bare majority, but of
nearly the whole, of their respective subjects, we may go on joining one
such proposition to another for several steps, before we reach a
conclusion not presumably true even of a majority. The error of the
conclusion will amount to the aggregate of the errors of all the
premises. Let the proposition, Most A are B, be true of nine in ten;
Most B are C, of eight in nine: then not only will one A in ten not be
C, because not B, but even of the nine-tenths which are B, only
eight-ninths will be C: that is, the cases of A which are C will be only
8/9 of 9/10, or four-fifths. Let us now add Most C are D, and suppose
this to be true of seven cases out of eight; the proportion of A which
is D will be only 7/8 of 8/9 of 9/10, or 7/10. Thus the probability
progressively dwindles. The experience, however, on which our
approximate generalizations are grounded, has so rarely been subjected
to, or admits of, accurate numerical estimation, that we cannot in
general apply any measurement to the diminution of probability which
takes place at each illation; but must be content with remembering that
it does diminish at every step, and that unless the premises approach
very nearly indeed to being universally true, the conclusion after a
very few steps is worth nothing. A hearsay of a hearsay, or an argument
from presumptive evidence depending not on immediate marks but on marks
of marks, is worthless at a very few removes from the first stage.
§ 7. There are, however, two cases in which reasonings depending on
approximate generalizations may be carried to any length we please with
as much assurance, and are as strictly scientific, as if they were
composed of universal laws of nature. But these cases are exceptions of
the sort which are currently said to prove the rule. The approximate
generalizations are as suitable, in the cases in question, for purposes
of ratiocination, as if they were complete generalizations, because they
are capable of being transformed into complete generalizations exactly
equivalent.
First: If the approximate generalization is of the class in which our
reason for stopping at the approximation is not the impossibility, but
only the inconvenience, of going further; if we are cognizant of the
character which distinguishes the cases that accord with the
generalization from those which are exceptions to it; we may then
substitute for the approximate proposition, an universal proposition
with a proviso. The proposition, Most persons who have uncontrolled
power employ it ill, is a generalization of this class, and may be
transformed into the following:--All persons who have uncontrolled power
employ it ill, provided they are not persons of unusual strength of
judgment and rectitude of purpose. The proposition, carrying the
hypothesis or proviso with it, may then be dealt with no longer as an
approximate, but as an universal proposition; and to whatever number of
steps the reasoning may reach, the hypothesis, being carried forward to
the conclusion, will exactly indicate how far that conclusion is from
being applicable universally. If in the course of the argument other
approximate generalizations are introduced, each of them being in like
manner expressed as an universal proposition with a condition annexed,
the sum of all the conditions will appear at the end as the sum of all
the errors which affect the conclusion. Thus, to the proposition last
cited, let us add the following:--All absolute monarchs have
uncontrolled power, unless their position is such that they need the
active support of their subjects (as was the case with Queen Elizabeth,
Frederick of Prussia, and others). Combining these two propositions, we
can deduce from them an universal conclusion, which will be subject to
both the hypotheses in the premises; All absolute monarchs employ their
power ill, unless their position makes them need the active support of
their subjects, or unless they are persons of unusual strength of
judgment and rectitude of purpose. It is of no consequence how rapidly
the errors in our premises accumulate, if we are able in this manner to
record each error, and keep an account of the aggregate as it swells up.
Secondly: there is a case in which approximate propositions, even
without our taking note of the conditions under which they are not true
of individual cases, are yet, for the purposes of science, universal
ones; namely, in the inquiries which relate to the properties not of
individuals, but of multitudes. The principal of these is the science of
politics, or of human society. This science is principally concerned
with the actions not of solitary individuals, but of masses; with the
fortunes not of single persons, but of communities.
For the statesman, therefore, it is generally enough to know that
_most_ persons act or are acted upon in a particular way; since his
speculations and his practical arrangements refer almost exclusively to
cases in which the whole community, or some large portion of it, is
acted upon at once, and in which, therefore, what is done or felt by
_most_ persons determines the result produced by or upon the body at
large. He can get on well enough with approximate generalizations on
human nature, since what is true approximately of all individuals is
true absolutely of all masses. And even when the operations of
individual men have a part to play in his deductions, as when he is
reasoning of kings, or other single rulers, still, as he is providing
for indefinite duration, involving an indefinite succession of such
individuals, he must in general both reason and act as if what is true
of most persons were true of all.
The two kinds of considerations above adduced are a sufficient
refutation of the popular error, that speculations on society and
government, as resting on merely probable evidence, must be inferior in
certainty and scientific accuracy to the conclusions of what are called
the exact sciences, and less to be relied on in practice. There are
reasons enough why the moral sciences must remain inferior to at least
the more perfect of the physical: why the laws of their more complicated
phenomena cannot be so completely deciphered, nor the phenomena
predicted with the same degree of assurance. But though we cannot attain
to so many truths, there is no reason that those we can attain should
deserve less reliance, or have less of a scientific character. Of this
topic, however, I shall treat more systematically in the concluding
Book, to which place any further consideration of it must be deferred.