_INDUCTION AND ITS CONNECTION WITH PROBABILITY._
1. We were occupied, during the last chapter, with the
examination of a rule, the object of which was to enable
us to make inferences about instances as yet unexamined.
It was professedly, therefore, a rule of an inductive character.
But, in the form in which it is commonly expressed,
it was found to fail utterly. It is reasonable therefore to
enquire at this point whether Probability is entirely a formal
or deductive science, or whether, on the other hand, we are
able, by means of it, to make valid inferences about instances
as yet unexamined. This question has been already in part
answered by implication in the course of the last two chapters.
It is proposed in the present chapter to devote a fuller
investigation to this subject, and to describe, as minutely as
limits will allow, the nature of the connection between Probability
and Induction. We shall find it advisable for clearness
of conception to commence our enquiry at a somewhat
early stage. We will travel over the ground, however, as
rapidly as possible, until we approach the boundary of what
can properly be termed Probability.
2. Let us then conceive some one setting to work to
investigate nature, under its broadest aspect, with the view
of systematizing the facts of experience that are known, and
thence (in case he should find that this is possible) discovering
others which are at present unknown. He observes a
multitude of phenomena, physical and mental, contemporary
and successive. He enquires what connections are there
between them? what rules can be found, so that some of
these things being observed I can infer others from them?
We suppose him, let it be observed, deliberately resolving to
investigate the things themselves, and not to be turned
aside by any prior enquiry as to there being laws under
which the mind is compelled to judge of the things. This
may arise either from a disbelief in the existence of any
independent and necessary mental laws, and a consequent
conviction that the mind is perfectly competent to observe
and believe anything that experience offers, and should
believe nothing else, or simply from a preference for investigations
of the latter kind. In other words, we suppose him
to reject Formal Logic, and to apply himself to a study of
objective existences.
It must not for a moment be supposed that we are here
doing more than conceiving a fictitious case for the purpose
of more vividly setting before the reader the nature of the
inductive process, the assumptions it has to make, and the
character of the materials to which it is applied. It is not
psychologically possible that any one should come to the study
of nature with all his mental faculties in full perfection, but
void of all materials of knowledge, and free from any bias as
to the uniformities which might be found to prevail around
him. In practice, of course, the form and the matter--the
laws of belief or association, and the objects to which they
are applied--act and react upon one another, and neither
can exist in any but a low degree without presupposing the
existence of the other. But the supposition is perfectly legitimate
for the purpose of calling attention to the requirements
of such a system of Logic, and is indeed nothing more
than what has to be done at almost every step in psychological
enquiry.[1]
3. His task at first might be conceived to be a slow
and tedious one. It would consist of a gradual accumulation
of individual instances, as marked out from one another
by various points of distinction, and connected with one
another by points of resemblance. These would have to be
respectively distinguished and associated in the mind, and
the consequent results would then be summed up in general
propositions, from which inferences could afterwards be
drawn. These inferences could, of course, contain no new
facts, they would only be repetitions of what he or others
had previously observed. All that we should have so far
done would have been to make our classifications of things
and then to appeal to them again. We should therefore be
keeping well within the province of ordinary logic, the processes
of which (whatever their ultimate explanation) may
of course always be expressed, in accordance with Aristotle's
Dictum, as ways of determining whether or not we can show
that one given class is included wholly or partly within
another, or excluded from it, as the case may be.
4. But a very short course of observation would suggest
the possibility of a wide extension of his information.
Experience itself would soon detect that events were connected
together in a regular way; he would ascertain that
there are 'laws of nature.' Coming with no _à priori_ necessity
of believing in them, he would soon find that as a matter
of fact they do exist, though he could not feel any certainty
as to the extent of their prevalence. The discovery of this
arrangement in nature would at once alter the plan of his proceedings,
and set the tone to the whole range of his methods
of investigation. His main work now would be to find out
by what means he could best discover these laws of nature.
An illustration may assist. Suppose I were engaged in
breaking up a vast piece of rock, say slate, into small pieces.
I should begin by wearily working through it inch by inch.
But I should soon find the process completely changed owing
to the existence of _cleavage_. By this arrangement of things
a very few blows would do the work--not, as I might possibly
have at first supposed, to the extent of a few inches--but
right through the whole mass. In other words, by the
process itself of cutting, as shown in experience, and by
nothing else, a constitution would be detected in the things
that would make that process vastly more easy and extensive.
Such a discovery would of course change our tactics.
Our principal object would thenceforth be to ascertain the
extent and direction of this cleavage.
Something resembling this is found in Induction. The
discovery of laws of nature enables the mind to dart with its
inferences from a few facts completely through a whole class
of objects, and thus to acquire results the successive individual
attainment of which would have involved long and
wearisome investigation, and would indeed in multitudes
of instances have been out of the question. We have no
demonstrative proof that this state of things is universal;
but having found it prevail extensively, we go on with the
resolution at least to try for it everywhere else, and we are
not disappointed. From propositions obtained in this way,
or rather from the original facts on which these propositions
rest, we can make _new_ inferences, not indeed with absolute
certainty, but with a degree of conviction that is of the
utmost practical use. We have gained the great step of
being able to make trustworthy generalizations. We conclude,
for instance, not merely that John and Henry die,
but that all men die.
5. The above brief investigation contains, it is hoped,
a tolerably correct outline of the nature of the Inductive
inference, as it presents itself in Material or Scientific Logic.
It involves the distinction drawn by Mill, and with which
the reader of his _System of Logic_ will be familiar, between
an inference drawn _according_ to a formula and one drawn
_from_ a formula. We do in reality make our inference from
the data afforded by experience directly to the conclusion;
it is a mere arrangement of convenience to do so by passing
through the generalization. But it is one of such extreme
convenience, and one so necessarily forced upon us when we
are appealing to our own past experience or to that of others
for the grounds of our conclusion, that practically we find it
the best plan to divide the process of inference into two
parts. The first part is concerned with establishing the
generalization; the second (which contains the rules of ordinary
logic) determines what conclusions can be drawn from
this generalization.
6. We may now see our way to ascertaining the province
of Probability and its relation to kindred sciences.
Inductive Logic gives rules for discovering such generalizations
as those spoken of above, and for testing their correctness.
If they are expressed in universal propositions it is
the part of ordinary logic to determine what inferences can
be made from and by them; if, on the other hand, they are
expressed in proportional propositions, that is, propositions
of the kind described in our first chapter, they are handed
over to Probability. We find, for example, that three infants
out of ten die in their first four years. It belongs to Induction
to say whether we are justified in generalizing our observation
into the assertion, All infants die in that proportion.
When such a proposition is obtained, whatever may be the
value to be assigned to it, we recognize in it a series of a
familiar kind, and it is at once claimed by Probability.
In this latter case the division into two parts, the inductive
and the ratiocinative, seems decidedly more than one of
convenience; it is indeed imperatively necessary for clearness
of thought and cogency of treatment. It is true that in
almost every example that can be selected we shall find
both of the above elements existing together and combining
to determine the degree of our conviction, but when we come
to examine them closely it appears to me that the grounds
of their cogency, the kind of conviction they produce, and
consequently the rules which they give rise to, are so entirely
distinct that they cannot possibly be harmonized into
a single consistent system.
The opinion therefore according to which certain Inductive
formulæ are regarded as composing a portion of
Probability, and which finds utterance in the Rule of Succession
criticised in our last chapter, cannot, I think, be
maintained. It would be more correct to say, as stated
above, that Induction is quite distinct from Probability, yet
co-operates in almost all its inferences. By Induction we
determine, for example, whether, and how far, we can safely
generalize the proposition that four men in ten live to be
fifty-six; supposing such a proposition to be safely generalized,
we hand it over to Probability to say what sort of inferences
can be deduced from it.
7. So much then for the opinion which tends to regard
pure Induction as a subdivision of Probability. By the
majority of philosophical and logical writers a widely different
view has of course been entertained. They are mostly disposed
to distinguish these sciences very sharply from, not to
say to contrast them with, one another; the one being
accepted as philosophical or logical, and the other rejected
as mathematical. This may without offence be termed the
popular prejudice against Probability.
A somewhat different view, however, must be noticed
here, which, by a sort of reaction against the latter, seems
even to go beyond the former; and which occasionally finds
expression in the statement that all inductive reasoning of
every kind is merely a matter of Probability. Two examples
of this may be given.
Beginning with the older authority, there is an often
quoted saying by Butler at the commencement of his _Analogy_,
that 'probability is the very guide of life'; a saying
which seems frequently to be understood to signify that the
rules or principles of Probability are thus all-prevalent when
we are drawing conclusions in practical life. Judging by
the drift of the context, indeed, this seems a fair interpretation
of his meaning, in so far of course as there could
be said to be any such thing as a science of Probability in
those days. Prof. Jevons, in his _Principles of Science_
(p. 197), has expressed a somewhat similar view, of course
in a way more consistent with the principles of modern
science, physical and mathematical. He says, "I am convinced
that it is impossible to expound the methods of induction
in a sound manner, without resting them on the
theory of Probability. Perfect knowledge alone can give
certainty, and in nature perfect knowledge would be infinite
knowledge, which is clearly beyond our capacities. We have,
therefore, to content ourselves with partial knowledge,--knowledge
mingled with ignorance, producing doubt."[2]
8. There are two senses in which this disposition to
merge the two sciences into one may be understood. Using
the word Probability in its vague popular signification,
nothing more may be intended than to call attention to the
fact, that in every case alike our conclusions are nothing
more than 'probable,' that is, that they are not, and cannot
be, absolutely certain. This must be fully admitted, for of
course no one acquainted with the complexity of physical
and other evidence would seriously maintain that absolute
ideal certainty can be attained in any branch of applied
logic. Hypothetical certainty, in abstract science, may be
possible, but not absolute certainty in the domain of the
concrete. This has been already noticed in a former chapter,
where, however, it was pointed out that whatever justification
may exist, on the subjective view of logic, for regarding
this common prevalence of absence of certainty as warranting
us in fusing the sciences into one, no such justification is
admitted when we take the objective view.
9. What may be meant, however, is that the _grounds_
of this absence of certainty are always of the same general
character. This argument, if admitted, would have real
force, and must therefore be briefly noticed. We have seen
abundantly that when we say of a conclusion within the
strict province of Probability, that it is not certain, all that
we mean is that in some proportion of cases only will such
conclusion be right, in the other cases it will be wrong.
Now when we say, in reference to any inductive conclusion,
that we feel uncertain about its absolute cogency, are we
conscious of the same interpretation? It seems to me that
we are not. It is indeed quite possible that on ultimate
analysis it might be proved that experience of failure in
the past employment of our methods of investigation was
the main cause of our present want of perfect confidence in
them. But this, as we have repeatedly insisted, does not
belong to the province of logical, but to that of Psychological
enquiry. It is surely not the case that we are, as a rule,
consciously guided by such occasional or repeated instances
of past failure. In so far as they are at all influential, they
seem to do their work by infusing a vague want of confidence
which cannot be referred to any statistical grounds for
its justification, at least not in a quantitative way. Part of
our want of confidence is derived sympathetically from those
who have investigated the matter more nearly at first hand.
Here again, analysis might detect that a given proportion of
past failures lay at the root of the distrust, but it does not
show at the surface. Moreover, one reason why we cannot
feel perfectly certain about our inductions is, that the
_memory_ has to be appealed to for some of our data; and will
any one assert that the only reason why we do not place
absolute reliance on our memory of events long past is that
we have been deceived in that way before?
In any other sense, therefore, than as a needful protest
against attaching too great demonstrative force to the conclusions
of Inductive Logic, it seems decidedly misleading to
speak of its reasonings as resting upon Probability.
10. We may now see clearly the reasons for the
limits within which causation[3] is necessarily required, but
beyond which it is not needed. To be able to generalize
a formula so as to extend it from the observed to the unobserved,
it is clearly essential that there should be a certain
permanence in the order of nature; this permanence is one
form of what is implied in the term causation. If the
circumstances under which men live and die remaining the
same, we did not feel warranted in inferring that four men
out of ten would continue to live to fifty, because in the case
of those whom we had observed this proportion had hitherto
done so, it is clear that we should be admitting that the
same antecedents need not be followed by the same consequents.
This uniformity being what the Law of Causation
asserts, the truth of the law is clearly necessary to enable us
to obtain our generalizations: in other words, it is necessary
for the Inductive part of the process. But it seems to be
equally clear that causation is not necessary for that part of
the process which belongs to Probability. Provided only
that the truth of our generalizations is secured to us, in the
way just mentioned, what does it matter to us whether or
not the individual members are subject to causation? For
it is not in reality about these individuals that we make
inferences. As this last point has been already fully treated
in Chapter VI., any further allusion to it need not be made here.
11. The above description, or rather indication, of the
process of obtaining these generalizations must suffice for the
present. Let us now turn and consider the means by which
we are practically to make use of them when they are obtained.
The point which we had reached in the course of the
investigations entered into in the sixth and seventh chapters
was this:--Given a series of a certain kind, we could draw
inferences about the members which composed it; inferences,
that is, of a peculiar kind, the value and meaning of which
were fully discussed in their proper place.
We must now shift our point of view a little; instead of
starting, as in the former chapters, with a determinate series
supposed to be given to us, let us assume that the individual
only is given, and that the work is imposed upon us of finding
out the appropriate series. How are we to set about the
task? In the former case our data were of this kind:--Eight
out of ten men, aged fifty, will live eleven years more,
and we ascertained in what sense, and with what certainty,
we could infer that, say, John Smith, aged fifty, would live
to sixty-one.
12. Let us then suppose, instead, that John Smith
presents himself, how should we in this case set about obtaining
a series for him? In other words, how should we
collect the appropriate statistics? It should be borne in
mind that when we are attempting to make real inferences
about things as yet unknown, it is in this form that the
problem will practically present itself.
At first sight the answer to this question may seem to be
obtained by a very simple process, viz. by counting how
many men of the age of John Smith, respectively do and do
not live for eleven years. In reality however the process is
far from being so simple as it appears. For it must be remembered
that each individual thing has not one distinct
and appropriate class or group, to which, and to which alone,
it properly belongs. We may indeed be practically in the
habit of considering it under such a single aspect, and it may
therefore seem to us more familiar when it occupies a place
in one series rather than in another; but such a practice is
merely customary on our part, not obligatory. It is obvious
that every individual thing or event has an indefinite
number of properties or attributes observable in it, and
might therefore be considered as belonging to an indefinite
number of different classes of things. By belonging to any
one class it of course becomes at the same time a member of
all the higher classes, the genera, of which that class was a
species. But, moreover, by virtue of each accidental attribute
which it possesses, it becomes a member of a class
intersecting, so to say, some of the other classes. John Smith
is a consumptive man say, and a native of a northern climate.
Being a man he is of course included in the class of vertebrates,
also in that of animals, as well as in any higher
such classes that there may be. The property of being consumptive
refers him to another class, narrower than any of
the above; whilst that of being born in a northern climate
refers him to a new and distinct class, not conterminous with
any of the rest, for there are things born in the north which
are not men.
13. When therefore John Smith presents himself to
our notice without, so to say, any particular label attached to
him informing us under which of his various aspects he is to
be viewed, the process of thus referring him to a class becomes
to a great extent arbitrary. If he had been indicated
to us by a general name, that, of course, would have been
some clue; for the name having a determinate connotation
would specify at any rate a fixed group of attributes within
which our selection was to be confined. But names and
attributes being connected together, we are here supposed
to be just as much in ignorance what name he is to be
called by, as what group out of all his innumerable attributes
is to be taken account of; for to tell us one of these things
would be precisely the same in effect as to tell us the other.
In saying that it is thus arbitrary under which class he is
placed, we mean, of course, that there are no logical grounds
of decision; the selection must be determined by some extraneous
considerations. Mere inspection of the individual
would simply show us that he could equally be referred
to an indefinite number of classes, but would in itself give
no inducement to prefer, for our special purpose, one of these
classes to another.
This variety of classes to which the individual may be
referred owing to his possession of a multiplicity of attributes,
has an important bearing on the process of inference
which was indicated in the earlier sections of this chapter,
and which we must now examine in more special reference
to our particular subject.
14. It will serve to bring out more clearly the nature
of some of those peculiarities of the step which we are now
about to take in the case of Probability, if we first examine
the form which the corresponding step assumes in the case
of ordinary Logic. Suppose then that we wished to ascertain
whether a certain John Smith, a man of thirty, who is
amongst other things a resident in India, and distinctly
affected with cancer, will continue to survive there for
twenty years longer. The terms in which the man is thus
introduced to us refer him to different classes in the way
already indicated. Corresponding to these classes there will
be a number of propositions which have been obtained by
previous observations and inductions, and which we may
therefore assume to be available and ready at hand when
we want to make use of them. Let us conceive them to
be such as these following:--Some men live to fifty; some
Indian residents live to fifty; no man suffering thus from
cancer lives for five years. From the first and second of these
premises nothing whatever can be inferred, for they are both[4]
particular propositions, and therefore lead to no conclusion
in this case. The third answers our enquiry decisively.
To the logical reader it will hardly be necessary to point
out that the process here under consideration is that of
finding middle terms which shall serve to connect the
subject and predicate of our conclusion. This subject and
predicate in the case in question, are the individual before
us and his death within the stated period. Regarded by
themselves there is nothing in common between them, and
therefore no link by which they may be connected or disconnected
with each other. The various classes above
referred to are a set of such middle terms, and the propositions
belonging to them are a corresponding set of major
premises. By the help of any one of them we are enabled,
under suitable circumstances, to connect together the subject
and predicate of the conclusion, that is, to infer whether the
man will or will not live twenty years.
15. Now in the performance of such a logical process
there are two considerations to which the reader's attention
must for a moment be directed. They are simple enough in
this case, but will need careful explanation in the corresponding
case in Probability. In the first place, it is clear that
whenever we can make any inference at all, we can do so
with absolute certainty. Logic, within its own domain,
knows nothing of hesitation or doubt. If the middle term
is appropriate it serves to connect the extremes in such a
way as to preclude all uncertainty about the conclusion;
if it is not, there is so far an end of the matter: no conclusion
can be drawn, and we are therefore left where we were.
Assuming our premises to be correct, we either know our
conclusion for certain, or we know nothing whatever about
it. In the second place, it should be noticed that none of
the possible alternatives in the shape of such major premises
as those given above can ever contradict any of the others,
or be at all inconsistent with them. Regarded as isolated
propositions, there is of course nothing to secure such harmony;
they have very different predicates, and may seem
quite out of each other's reach for either support or opposition.
But by means of the other premise they are in each
case brought into relation with one another, and the general
interests of truth and consistency prevent them therefore
from contradicting one another. As isolated propositions
it might have been the case that all men live to fifty, and
that no Indian residents do so, but having recognised that
some men are residents in India, we see at once that these
premises are inconsistent, and therefore that one or other
of them must be rejected. In all applied logic this necessity
of avoiding self-contradiction is so obvious and imperious
that no one would think it necessary to lay down the formal
postulate that all such possible major premises are to be
mutually consistent. To suppose that this postulate is not
complied with, would be in effect to make two or more contradictory
assumptions about matters of fact.
16. But now observe the difference when we attempt
to take the corresponding step in Probability. For ordinary
propositions, universal or particular, substitute statistical
propositions of what we have been in the habit of calling
the 'proportional' kind. In other words, instead of asking
whether the man will live for twenty years, let us ask whether
he will live for one year? We shall be unable to find any
universal propositions which will cover the case, but we may
without difficulty obtain an abundance of appropriate proportional
ones. They will be of the following description:--Of
men aged 30, 98 in 100 live another year; of residents in
India a smaller proportion survive, let us for example say
90 in 100; of men suffering from cancer a smaller proportion
still, let us say 20 in 100.
Now in both of the respects to which attention has just
been drawn, propositions of this kind offer a marked contrast
with those last considered. In the first place, they do
not, like ordinary propositions, either assert unequivocally
yes or no, or else refuse to open their lips; but they give
instead a sort of qualified or hesitating answer concerning
the individuals included in them. This is of course nothing
more than the familiar characteristic of what may be called
'probability propositions.' But it leads up to, and indeed
renders possible, the second and more important point;
viz. that these various answers, though they cannot directly
and formally contradict each other (this their nature as proportional
propositions, will not as a rule permit), may yet, in
a way which will now have to be pointed out, be found to be
more or less in conflict with each other.
Hence it follows that in the attempt to draw a conclusion
from premises of the kind in question, we may be placed
in a position of some perplexity; but it is a perplexity
which may present itself in two forms, a mild and an aggravated
form. We will notice them in turn.
17. The mild form occurs when the different classes
to which the individual case may be appropriately referred
are successively included one within another; for here our
sets of statistics, though leading to different results, will
not often be found to be very seriously at variance with
one another. All that comes of it is that as we ascend in the
scale by appealing to higher and higher genera, the statistics
grow continually less appropriate to the particular
case in point, and such information therefore as they afford
becomes gradually less explicit and accurate.
The question that we originally wanted to determine,
be it remembered, is whether John Smith will die within
one year. But all knowledge of this fact being unattainable,
owing to the absence of suitable inductions, we felt
justified (with the explanation, and under the restrictions
mentioned in Chap VI.), in substituting, as the only available
equivalent for such individual knowledge, the answer to the
following statistical enquiry, What proportion of men in his
circumstances die?
18. But then at once there begins to arise some doubt
and ambiguity as to what exactly is to be understood by his
circumstances. We may know very well what these circumstances
are in themselves, and yet be in perplexity as to
how many of them we ought to take into account when
endeavouring to estimate his fate. We might conceivably,
for a beginning, choose to confine our attention to those
properties only which he has in common with all animals.
If so, and statistics on the subject were attainable, they
would presumably be of some such character as this, Ninety-nine
animals out of a hundred die within a year. Unusual as
such a reference would be, we should, logically speaking, be
doing nothing more than taking a wider class than the one
we were accustomed to. Similarly we might, if we pleased,
take our stand at the class of vertebrates, or at that of
mammalia, if zoologists were able to give us the requisite
information. Of course we reject these wide classes and
prefer a narrower one. If asked why we reject them, the
natural answer is that they are so general, and resemble the
particular case before us in so few points, that we should be
exceedingly likely to go astray in trusting to them. Though
accuracy cannot be insured, we may at least avoid any needless
exaggeration of the relative number and magnitude of
our errors.
19. The above answer is quite valid; but whilst cautioning
us against appealing to too wide a class, it seems to
suggest that we cannot go wrong in the opposite direction,
that is in taking too narrow a class. And yet we do avoid
any such extremes. John Smith is not only an Englishman;
he may also be a native of such a part of England, be living
in such a Presidency, and so on. An indefinite number of
such additional characteristics might be brought out into
notice, many of which at any rate have some bearing upon
the question of vitality. Why do we reject any consideration
of these narrower classes? We do reject them, but it is for
what may be termed a practical rather than a theoretical
reason. As was explained in the first chapters, it is essential
that our series should contain a considerable number of terms
if they are to be of any service to us. Now many of the
attributes of any individual are so rare that to take them
into account would be at variance with the fundamental
assumption of our science, viz. that we are properly concerned
only with the averages of large numbers. The more special
and minute our statistics the better, provided only that we
can get enough of them, and so make up the requisite large
number of instances. This is, however, impossible in many
cases. We are therefore obliged to neglect one attribute
after another, and so to enlarge the contents of our class; at
the avowed risk of somewhat increased variety and unsuitability
in the members of it, for at each step of this kind we
diverge more and more from the sort of instances that we
really want. We continue to do so, until we no longer gain
more in quantity than we lose in quality. We finally take
our stand at the point where we first obtain statistics drawn
from a sufficiently large range of observation to secure the
requisite degree of stability and uniformity.
20. In such an example as the one just mentioned,
where one of the successive classes--man--is a well-defined
natural kind or species, there is such a complete break in
each direction at this point, that every one is prompted to
take his stand here. On the one hand, no enquirer would
ever think of introducing any reference to the higher classes
with fewer attributes, such as animal or organized being:
and on the other hand, the inferior classes, created by our
taking notice of his employment or place of residence, &c.,
do not as a rule differ sufficiently in their characteristics
from the class _man_ to make it worth our while to attend
to them.
Now and then indeed these characteristics do rise into
importance, and whenever this is the case we concentrate
our attention upon the class to which they correspond, that
is, the class which is marked off by their presence. Thus,
for instance, the quality of consumptiveness separates any
one off so widely from the majority of his fellow-men in all
questions pertaining to mortality, that statistics about the
lives of consumptive men differ materially from those which
refer to men in general. And we see the result; if a consumptive
man can effect an insurance at all, he must do it
for a much higher premium, calculated upon his special
circumstances. In other words, the attribute is sufficiently
important to mark off a fresh class or series. So with insurance
against accident. It is not indeed attempted to
make a special rate of insurance for the members of each
separate trade, but the differences of risk to which they are
liable oblige us to take such facts to some degree into
account. Hence, trades are roughly divided into two or
three classes, such as the ordinary, the hazardous, and the
extra-hazardous, each having to pay its own rate of premium.
21. Where one or other of the classes thus corresponds
to natural kinds, or involves distinctions of co-ordinate importance
with those of natural kinds, the process is not
difficult; there is almost always some one of these classes
which is so universally recognised to be the appropriate one,
that most persons are quite unaware of there being any
necessity for a process of selection. Except in the cases
where a man has a sickly constitution, or follows a dangerous
employment, we seldom have occasion to collect statistics for
him from any class but that of men in general of his age in
the country.
When, however, these successive classes are not ready
marked out for us by nature, and thence arranged in easily
distinguishable groups, the process is more obviously arbitrary.
Suppose we were considering the chance of a man's
house being burnt down, with what collection of attributes
should we rest content in this instance? Should we include
all kinds of buildings, or only dwelling-houses, or confine
ourselves to those where there is much wood, or those which
have stoves? All these attributes, and a multitude of others
may be present, and, if so, they are all circumstances which
help to modify our judgment. We must be guided here by
the statistics which we happen to be able to obtain in
sufficient numbers. Here again, rough distinctions of this
kind are practically drawn in Insurance Offices, by dividing
risks into ordinary, hazardous, and extra-hazardous. We
examine our case, refer it to one or other of these classes,
and then form our judgment upon its prospects by the statistics
appropriate to its class.
22. So much for what may be called the mild form in
which the ambiguity occurs; but there is an aggravated form
in which it may show itself, and which at first sight seems
to place us in far greater perplexity.
Suppose that the different classes mentioned above are
not included successively one within the other. We may
then be quite at a loss which of the statistical tables to
employ. Let us assume, for example, that nine out of ten
Englishmen are injured by residence in Madeira, but that
nine out of ten consumptive persons are benefited by such a
residence. These statistics, though fanciful, are conceivable
and perfectly compatible. John Smith is a consumptive
Englishman; are we to recommend a visit to Madeira in his
case or not? In other words, what inferences are we to
draw about the probability of his death? Both of the statistical
tables apply to his case, but they would lead us to
directly contradictory conclusions. This does not mean, of
course, contradictory precisely in the logical sense of that
word, for one of these propositions does not assert that an
event must happen and the other deny that it must; but
contradictory in the sense that one would cause us in some
considerable degree to believe what the other would cause us
in some considerable degree to disbelieve. This refers, of
course, to the individual events; the statistics are by supposition
in no degree contradictory. Without further data,
therefore, we can come to no decision.
23. Practically, of course, if we were forced to a decision
with only these data before us, we should make our
choice by the consideration that the state of a man's lungs
has probably more to do with his health than the place of
his birth has; that is, we should conclude that the duration
of life of consumptive Englishmen corresponds much more
closely with that of consumptive persons in general than
with that of their healthy countrymen. But this is, of
course, to import empirical considerations into the question.
The data, as they are given to us, and if we confine ourselves
to them, leave us in absolute uncertainty upon the
point. It may be that the consumptive Englishmen almost
all die when transported into the other climate; it may be
that they almost all recover. If they die, this is in obvious
accordance with the first set of statistics; it will be found in
accordance with the second set through the fact of the
foreign consumptives profiting by the change of climate in
more than what might be termed their due proportion.
A similar explanation will apply to the other alternative,
viz. to the supposition that the consumptive Englishmen
mostly recover. The problem is, therefore, left absolutely
indeterminate, for we cannot here appeal to any general rule
so simple and so obviously applicable as that which, in a
former case, recommended us always to prefer the more
special statistics, when sufficiently extensive, to those which
are wider and more general. We have no means here of
knowing whether one set is more special than the other.
And in this no difficulty can be found, so long as we
confine ourselves to a just view of the subject. Let me
again recall to the reader's mind what our present position
is; we have substituted for knowledge of the individual
(finding that unattainable) a knowledge of what occurs in
the average of similar cases. This step had to be taken the
moment the problem was handed over to Probability. But
the conception of similarity in the cases introduces us to a
perplexity; we manage indeed to evade it in many instances,
but here it is inevitably forced upon our notice.
There are here two aspects of this similarity, and they
introduce us to two distinct averages. Two assertions are
made as to what happens in the long run, and both of these
assertions, by supposition, are verified. Of their truth there
need be no doubt, for both were supposed to be obtained
from experience.
24. It may perhaps be supposed that such an example
as this is a _reductio ad absurdum_ of the principle upon which
Life and other Insurances are founded. But a moment's
consideration will show that this is quite a mistake, and
that the principle of insurance is just as applicable to
examples of this kind as to any other. An office need find
no difficulty in the case supposed. They _might_ (for a reason
to be mentioned presently, they probably _would_ not) insure
the individual without inconsistency at a rate determined
by either average. They might say to him, "You are an
Englishman. Out of the multitude of English who come to
us nine in ten die if they go to Madeira. We will insure
you at a rate assigned by these statistics, knowing that in
the long run all will come right so far as we are concerned.
You are also consumptive, it is true, and we do not know
what proportion of the English are consumptive, nor what
proportion of English consumptives die in Madeira. But
this does not really matter for our purpose. The formula,
nine in ten die, is in reality calculated by taking into account
these unknown proportions; for, though we do not know
them in themselves, statistics tell us all that we care to
know about their results. In other words, whatever unknown
elements may exist, must, in regard to all the effects
which they can produce, have been already taken into
account, so that our ignorance about them cannot in the
least degree invalidate such conclusions as we are able to
draw. And this is sufficient for our purpose." But precisely
the same language might be held to him if he presented
himself as a consumptive man; that is to say, the office
could safely carry on its proceedings upon either alternative.
This would, of course, be a very imperfect state for the
matter to be left in. The only rational plan would be to
isolate the case of consumptive Englishmen, so as to make
a separate calculation for their circumstances. This calculation
would then at once supersede all other tables so
far as they were concerned; for though, _in the end_, it could
not arrogate to itself any superiority over the others, it
would in the mean time be marked by fewer and slighter
aberrations from the truth.
25. The real reason why the Insurance office could
not long work on the above terms is of a very different
kind from that which some readers might contemplate, and
belongs to a class of considerations which have been much
neglected in the attempts to construct sciences of the different
branches of human conduct. It is nothing else than
that annoying contingency to which prophets since the time
of Jonah have been subject, of uttering _suicidal_ prophecies;
of publishing conclusions which are perfectly certain when
every condition and cause but one have been taken into
account, that one being the effect of the prophecy itself
upon those to whom it refers.
In our example above, the office (in so far as the particular
cases in Madeira are concerned) would get on very well
until the consumptive Englishmen in question found out
what much better terms they could make by announcing
themselves as consumptives, and paying the premium appropriate
to that class, instead of announcing themselves as
Englishmen. But if they did this they would of course be
disturbing the statistics. The tables were based upon the
assumption that a certain fixed proportion (it does not
matter what proportion) of the English lives would continue
to be consumptive lives, which, under the supposed circumstances,
would probably soon cease to be true. When it is
said that nine Englishmen out of ten die in Madeira, it is
meant that of those who come to the office, as the phrase is,
at random, or in their fair proportions, nine-tenths die. The
consumptives are supposed to go there just like red-haired
men, or poets, or any other special class. Or they might go
in any proportions greater or less than those of other classes,
so long as they adhered to the same proportion throughout.
The tables are then calculated on the continuance of this
state of things; the practical contradiction is in supposing
such a state of things to continue after the people had once
had a look at the tables. If we merely make the assumption
that the publication of these tables made no such alteration
in the conduct of those to whom it referred, no hitch of
this kind need occur.
26. The assumptions here made, as has been said, are
not in any way contradictory, but they need some explanation.
It will readily be seen that, taken together, they are
inconsistent with the supposition that each of these classes is
homogeneous, that is, that the statistical proportions which
hold of the whole of either of them will also hold of any
portion of them which we may take. There are certain
individuals (viz. the consumptive Englishmen) who belong
to each class, and of course the two different sets of statistics
cannot both be true of them taken by themselves. They
might coincide in their characteristics with either class, but
not with both; probably in most practical cases they will
coincide with neither, but be of a somewhat intermediate
character. Now when it is said of any such heterogeneous
body that, say, nine-tenths die, what is meant (or rather
implied) is that the class might be broken up into smaller
subdivisions of a more homogeneous character, in some of
which, of course, more than nine-tenths die, whilst in others
less, the differences depending upon their character, constitution,
profession, &c.; the number of such divisions and the
amount of their divergence from one another being perhaps
very considerable.
Now when we speak of either class as a whole and say
that nine-tenths die, the most natural and soundest meaning
is that that would be the proportion if all without
exception went abroad, or (what comes to the same thing) if
each of these various subdivisions was represented in fair
proportion to its numbers. Or it might only be meant that
they go in some other proportion, depending upon their
tastes, pursuits, and so on. But whatever meaning be adopted
one condition is necessary, viz. that the proportion of each
class that went at the time the statistics were drawn up
must be adhered to throughout. When the class is homogeneous
this is not needed, but when it is heterogeneous the
statistics would be interfered with unless this condition were
secured.
We are here supposed to have two sets of statistics, one
for the English and one for the consumptives, so that the
consumptive English are in a sense counted twice over. If
their mortality is of an intermediate amount, therefore, they
serve to keep down the mortality of one class and to keep
up that of the other. If the statistics are supposed to be
exhaustive, by referring to the whole of each class, it follows
that actually the same individuals must be counted each
time; but if representatives only of each class are taken, the
same individuals need not be inserted in each set of tables.
27. When therefore they come to insure (our remarks
are still confined to our supposed Madeira case), we have
some English consumptives counted as English, and paying
the high rate; and others counted as consumptives and paying
the low rate. Logically indeed we may suppose them all
entered in each class, and paying therefore each rate. What
we have said above is that any individual may be conceived
to present himself for either of these classes. Conceive that
some one else pays his premium for him, so that it is a
matter of indifference to him personally at which rate he
insures, and there is nothing to prevent some of the class (or
for that matter all) going to one class, and others (or all
again) going to the other class.
So long therefore as we make the logically possible
though practically absurd supposition that some men will
continue to pay a higher rate than they need, there is nothing
to prevent the English consumptives (some or all) from
insuring in each category and paying its appropriate premium.
As soon as they gave any thought to the matter, of
course they would, in the case supposed, all prefer to insure
as consumptives. But their doing this would disturb each set
of statistics. The English mortality in Madeira would instantly
become heavier, so far as the Insurance company was
concerned, by the loss of all their best lives; whilst the consumptive
statistics (unless _all_ the English consumptives had
already been taken for insurance) would be in the same way
deteriorated.[5] A slight readjustment therefore of each scale
of insurance would then be needed; this is the disturbance
mentioned just above. It must be clearly understood, however,
that it is not our original statistics which have proved
to be inconsistent, but simply that there were practical
obstacles to carrying out a system of insurance upon them.
28. Examples subject to the difficulty now under
consideration will doubtless seem perplexing to the student
unacquainted with the subject. They are difficult to reconcile
with any other view of the science than that insisted on
throughout this Essay, viz. that we are only concerned with
averages. It will perhaps be urged that there are two
different values of the man's life in these cases, and that
they cannot both be true. Why not? The 'value' of his
life is simply the number of years to which men in his
circumstances do, on the average, attain; we have the man
set before us under two different circumstances; what wonder,
therefore, that these should offer different averages? In such
an objection it is forgotten that we have had to substitute
for the unattainable result about the individual, the really
attainable result about a set of men as much like him as
possible. The difficulty and apparent contradiction only
arise when people will try to find some justification for their
belief in the individual case. What can we possibly conclude,
it may be asked, about this particular man John
Smith's prospects when we are thus offered two different
values for his life? Nothing whatever, it must be replied;
nor could we in reality draw a conclusion, be it remembered,
in the former case, when we were practically confined to one
set of statistics. There also we had what we called the
'value' of his life, and since we only knew of one such value,
we came to regard it as in some sense appropriate to him as
an individual. Here, on the other hand, we have two values,
belonging to different series, and as these values are really
different it may be complained that they are discordant, but
such a complaint can only be made when we do what we
have no right to do, viz. assign a value to the individual
which shall admit of individual justification.
29. Is it then perfectly arbitrary what series or class
of instances we select by which to judge? By no means; it
has been stated repeatedly that in choosing a series, we must
seek for one the members of which shall resemble our individual
in as many of his attributes as possible, subject only
to the restriction that it must be a sufficiently extensive
series. What is meant is, that in the above case, where we
have two series, we cannot fairly call them contradictory; the
only valid charge is one of incompleteness or insufficiency for
their purpose, a charge which applies in exactly the same
sense, be it remembered, to all statistics which comprise
genera unnecessarily wider than the species with which we
are concerned. The only difference between the two different
classes of cases is, that in the one instance we are on a
path which we know will lead at the last, through many
errors, towards the truth (in the sense in which truth can be
attained here), and we took it for want of a better. In the
other instance we have two such paths, perfectly different
paths, either of which however will lead us towards the truth
as before. Contradiction can only seem to arise when it is
attempted to justify each separate step on our paths, as well
as their ultimate tendency.
Still it cannot be denied that these objections are a
serious drawback to the completeness and validity of any
anticipations which are merely founded upon statistical frequency,
at any rate in an early stage of experience, when
but few statistics have been collected. Such knowledge as
Probability can give is not in any individual case of a high
order, being subject to the characteristic infirmity of repeated
error; but even when measured by its own standard
it commences at a very low stage of proficiency. The
errors are then relatively very numerous and large compared
with what they may ultimately be reduced to.
30. Here as elsewhere there is a continuous process
of specialization going on. The needs of a gradually widening
experience are perpetually calling upon us to subdivide
classes which are found to be too heterogeneous. Sometimes
the only complaint that has to be made is that the class to
which we are obliged to refer is found to be somewhat too
broad to suit our purpose, and that it might be subdivided
with convenience. This is the case, as has been shown above,
when an Insurance office finds that its increasing business
makes it possible and desirable to separate off the men who
follow some particular trades from the rest of their fellow-countrymen.
Similarly in every other department in which
statistics are made use of. This increased demand for specificness
leads, in fact, as naturally in this direction, as does
the progress of civilization to the subdivision of trades in
any town or country. So in reference to the other kind of
perplexity mentioned above. Nothing is more common in
those sciences or practical arts, in which deduction is but
little available, and where in consequence our knowledge is
for the most part of the empirical kind, than to meet with
suggestions which point more or less directly in contrary
directions. Whenever some new substance is discovered or
brought into more general use, those who have to deal with
it must be familiar with such a state of things. The medical
man who has to employ a new drug may often find himself
confronted by the two distinct recommendations, that on
the one hand it should be employed for certain diseases, and
that on the other hand it should not be tried on certain constitutions.
A man with such a constitution, but suffering
from such a disease, presents himself; which recommendation
is the doctor to follow? He feels at once obliged to
set to work to collect narrower and more special statistics,
in order to escape from such an ambiguity.
31. In this and a multitude of analogous cases
afforded by the more practical arts it is not of course necessary
that numerical data should be quoted and appealed
to; it is sufficient that the judgment is more or less consciously
determined by them. All that is necessary to make
the examples appropriate is that we should admit that in
their case statistical data are our ultimate appeal in the
present state of knowledge. Of course if the empirical
laws can be resolved into their component causes we may
appeal to direct deduction, and in this case the employment
of statistics, and consequently the use of the theory of
Probability, may be superseded.
In this direction therefore, as time proceeds, the advance
of statistical refinement by the incessant subdivision of classes
to meet the developing wants of man is plain enough. But
if we glance backwards to a more primitive stage, we shall
soon see in what a very imperfect state the operation commences.
At this early stage, however, Probability and Induction
are so closely connected together as to be very apt to
be merged into one, or at any rate to have their functions
confounded.
32. Since the generalization of our statistics is found to
belong to Induction, this process of generalization may be
regarded as prior to, or at least independent of, Probability.
We have, moreover, already discussed (in Chapter VI.) the
step corresponding to what are termed immediate inferences,
and (in Chapter VII.) that corresponding to syllogistic inferences.
Our present position therefore is that in which we
may consider ourselves in possession of any number of generalizations,
but wish to employ them so as to make inferences
about a given individual; just as in one department of
common logic we are engaged in finding middle terms to
establish the desired conclusion. In this latter case the
process is found to be extremely simple, no accumulation of
different middle terms being able to lead to any real ambiguity
or contradiction. In Probability, however, the case is
different. Here, if we attempt to draw inferences about the
individual case before us, as often is attempted--in the Rule
of Succession for example--we shall encounter the full force
of this ambiguity and contradiction. Treat the question,
however, fairly, and all difficulty disappears. Our inference
really is not about the individuals as individuals, but about
series or successions of them. We wished to know whether
John Smith will die within the year; this, however, cannot
be known. But John Smith, by the possession of many
attributes, belongs to many different series. The multiplicity
of middle terms, therefore, is what ought to be
expected. We _can_ know whether a succession of men, residents
in India, consumptives, &c. die within a year. We
may make our selection, therefore, amongst these, and in the
long run the belief and consequent conduct of ourselves and
other persons (as described in Chapter VI.) will become
capable of justification. With regard to choosing one of
these series rather than another, we have two opposing
principles of guidance. On the one hand, the more special
the series the better; for, though not more right in the end,
we shall thus be more nearly right all along. But, on the
other hand, if we try to make the series too special, we shall
generally meet the practical objection arising from insufficient
statistics.
1. Some of my readers may be familiar with a very striking digression
in Buffon's Natural History (_Natural Hist. of Man_, §VIII.), in
which he supposes the first man in full possession of his faculties,
but with all his experience to gain, and speculates on the gradual
acquisition of his knowledge. Whatever may be thought of his
particular conclusions the passage is very interesting and
suggestive to any student of Psychology.
2. See also Dugald Stewart (Ed. by Hamilton; VII. pp. 115-119).
3. Required that is for purposes of logical inference within the
limits of Probability; it is not intended to imply any doubts as to
its actual universal prevalence, or its all-importance for
scientific purposes. The subject is more fully discussed in a future
chapter.
4. As particular propositions they are both of course identical in
form. The fact that the 'some' in the former corresponds to a
larger proportion than in the latter, is a distinction alien to pure
Logic.
5. The reason is obvious. The healthiest English lives in Madeira
(viz. the consumptive ones) have now ceased to be reckoned as
English; whereas the worst consumptive lives there (viz. the
English) are now increased in relative numbers.