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
toward 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 a 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 can not 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.(194)
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 everyday 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 further; 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 sub-classes, 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 attainments, though the knowledge we possess
beyond it can not conveniently be brought to bear upon the particular
instance. We may know well enough what circumstances distinguish the
portion of A which has the attribute B from the portion which has it not,
but may have no means, or may not have 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 any thing 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 recognizing 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 co-existence; propositions
like Most A are B, _can not_ 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 learned, 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 can not tell in what manner any
new circumstance might perhaps affect it. The proposition, Most judges are
inaccessible to bribes, would probably 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 favorably 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
fulfill 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 recognize 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 a 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 can not 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 instances _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 toward 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
cooperate toward 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(195) 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.
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 ⅓, the chance that a
C is not a B is ¼; hence if the thing be both an A and a C, the chance
is ⅓ of ¼ = ¹⁄₁₂.(196)
In this computation it is of course supposed that the probabilities
arising from A and C are independent of each other. There must not be any
such connection 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 can not 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 ⅞ 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 can not 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, a 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 a 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 a 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 a 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 can not
be so completely deciphered, nor the phenomena predicted with the same
degree of assurance. But though we can not 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.