_THE RULE OF SUCCESSION._[1]
1. A word of apology may be offered here for the introduction of a new
name. The only other alternative would have been to entitle the rule
one of _Induction_. But such a title I cannot admit, for reasons
which will be almost immediately explained.
1. In the last chapter we discussed at some length the
nature of the kinds of inference in Probability which correspond
to those termed, in Logic, immediate and mediate inferences.
We ascertained what was the meaning of saying, for
example, that the chance of any given man A. B. dying
in a year is 1/3, when concluded from the general proposition
that one man out of three in his circumstances dies. We
also discussed the nature and evidence of rules of a more
completely inferential character. But to stop at this point
would be to take a very imperfect view of the subject. If
Probability is a science of real inference about things, it
must surely lead up to something more than such merely
formal conclusions; we must be able, if not by means of it, at
any rate by some means, to step beyond the limits of what
has been actually observed, and to draw conclusions about
what is as yet unobserved. This leads at once to the question,
What is the connection of Probability with Induction?
This is a question into which it will be necessary to enter
now with some minuteness.
That there is a close connection between Probability and
Induction, must have been observed by almost every one
who has treated of either subject; I have not however seen
any account of this connection that seemed to me to be
satisfactory. An explicit description of it should rather be
sought in treatises upon the narrower subject, Probability;
but it is precisely here that the most confusion is to be
found. The province of Probability being somewhat narrow,
incursions have been constantly made from it into the adjacent
territory of Induction. In this way, amongst the
arithmetical rules discussed in the last chapter, others have
been frequently introduced which ought not in strictness to
be classed with them, as they rest on an entirely different
basis.
2. The origin of such confusion is easy of explanation;
it arises, doubtless, from the habit of laying undue
stress upon the _subjective_ side of Probability, upon that
which treats of the quantity of our belief upon different
subjects and the variations of which that quantity is susceptible.
It has been already urged that this variation of
belief is at most but a constant accompaniment of what is
really essential to Probability, and is moreover common to
other subjects as well. By defining the science therefore
from this side these other subjects would claim admittance
into it; some of these, as Induction, have been accepted, but
others have been somewhat arbitrarily rejected. Our belief
in a wider proposition gained by Induction is, prior to verification,
not so strong as that of the narrower generalization
from which it is inferred. This being observed, a so-called
rule of probability has been given by which it is supposed
that this diminution of assent could in many instances be
calculated.
But _time_ also works changes in our conviction; our belief
in the happening of almost every event, if we recur to it long
afterwards, when the evidence has faded from the mind, is
less strong than it was at the time. Why are not rules of
oblivion inserted in treatises upon Probability? If a man is
told how firmly he ought to expect the tide to rise again,
because it has already risen ten times, might he not also ask
for a rule which should tell him how firm should be his belief
of an event which rests upon a ten years' recollection?[1] The
infractions of a rule of this latter kind could scarcely be more
numerous and extensive, as we shall see presently, than those
of the former confessedly are. The fact is that the agencies,
by which the strength of our conviction is modified, are so
indefinitely numerous that they cannot all be assembled into
one science; for purposes of definition therefore the quantity
of belief had better be omitted from consideration, or at any
rate regarded as a mere appendage, and the science, defined
from the other or statistical side of the subject, in which,
as has been shown, a tolerably clear boundary-line can be
traced.
3. Induction, however, from its importance does merit
a separate discussion; a single example will show its bearing
upon this part of our subject. We are considering the prospect
of a given man, A. B. living another year, and we find
that nine out of ten men of his age do survive. In forming
an opinion about his surviving, however, we shall find that
there are in reality two very distinct causes which aid in
determining the strength of our conviction; distinct, but in
practice so intimately connected that we are very apt to
overlook one, and attribute the effect entirely to the other.
(I.) There is that which strictly belongs to Probability;
that which (as was explained in Chap VI.) measures our
belief of the individual case as deduced from the general
proposition. Granted that nine men out of ten of the kind
to which A. B. belongs do live another year, it obviously
does not follow at all necessarily that _he_ will. We describe
this state of things by saying, that our belief of his surviving
is diminished from certainty in the ratio of 10 to 9, or, in
other words, is measured by the fraction 9/10.
(II.) But are we certain that nine men out of ten like
him _will_ live another year? we know that they have so survived
in time past, but will they continue to do so? Since
A. B. is still alive it is plain that this proposition is to a
certain extent assumed, or rather obtained by Induction.
We cannot however be as certain of the inductive inference
as we are of the data from which it was inferred. Here,
therefore, is a second cause which tends to diminish our
belief; in practice these two causes always accompany each
other, but in thought they can be separated.
The two distinct causes described above are very liable
to be confused together, and the class of cases from which
examples are necessarily for the most part drawn increases
this liability. The step from the statement 'all men have
died in a certain proportion' to the inference 'they will continue
to die in that proportion' is so slight a step that it is
unnoticed, and the diminution of conviction that should
accompany it is unsuspected. In what are called _à priori_
examples the step is still slighter. We feel so certain about
the permanence of the laws of mechanics, that few people
would think of regarding it as an inference when they
believe that a die will in the long run turn up all its faces
equally often, because other dice have done so in time
past.
4. It has been already pointed out (in Chapter VI.)
that, so far as concerns that definition of Probability which
regards it as the science which discusses the degree and
modifications of our belief, the question at issue seems to be
simply this:--Are the causes alluded to above in (II.) capable
of being reduced to one simple coherent scheme, so that any
universal rules for the modification of assent can be obtained
from them? If they are, strong grounds will have been
shown for classing them with (I.), in other words, for considering
them as rules of probability. Even then they
would be rules practically of a very different kind, contingent
instead of necessary (if one may use these terms without
committing oneself to any philosophical system), but this
objection might perhaps be overruled by the greater simplicity
secured by classing them together. This view is, with
various modifications, generally adopted by writers on Probability,
or at least, as I understand the matter, implied by
their methods of definition and treatment. Or, on the other
hand, must these causes be regarded as a vast system, one
might almost say a chaos, of perfectly distinct agencies;
which may indeed be classified and arranged to some extent,
but from which we can never hope to obtain any rules of
perfect generality which shall not be subject to constant
exception? If so, but one course is left; to exclude them
all alike from Probability. In other words, we must assume
the general proposition, viz. that which has been described
throughout as our starting-point, to be given to us; it may
be obtained by any of the numerous rules furnished by
Induction, or it may be inferred deductively, or given by our
own observation; its value may be diminished by its depending
upon the testimony of witnesses, or its being recalled by
our own memory. Its real value may be influenced by
these causes or any combinations of them; but all these are
preliminary questions with which we have nothing directly
to do. We assume our statistical proposition to be true,
neglecting the diminution of its value by the process of
attainment; we take it up first at this point and then apply
our rules to it. We receive it in fact, if one may use the
expression, _ready-made_, and ask no questions about the process
or completeness of its manufacture.
5. It is not to be supposed, of course, that any writers
have seriously attempted to reduce to one system of calculation
all the causes mentioned above, and to embrace in one
formula the diminution of certainty to which the inclusion of
them subjects us. But on the other hand, they have been
unwilling to restrain themselves from all appeal to them.
From an early period in the study of the science attempts
have been made to proceed, by the Calculus of Probability,
from the observed cases to adjacent and similar cases. In
practice, as has been already said, it is not possible to avoid
some extension of this kind. But it should be observed,
that in these instances the divergence from the strict ground
of experience is not in reality recognized, at least not as a
part of our logical procedure. We have, it is true, wandered
somewhat beyond it, and so obtained a wider proposition
than our data strictly necessitated, and therefore one of less
certainty. Still we assume the conclusion given by induction
to be equally certain with the data, or rather omit all
notice of the divergence from consideration. It is assumed
that the unexamined instances will resemble the examined,
an assumption for which abundant warrant may exist; the
theory of the calculation rests upon the supposition that
there will be no difference between them, and the practical
error is insignificant simply because this difference is small.
6. But the rule we are now about to discuss, and
which may be called the Rule of Succession, is of a very
different kind. It not only recognizes the fact that we are
leaving the ground of past experience, but takes the consequences
of this divergence as the express subject of its calculation.
It professes to give a general rule for the measure of
expectation that we should have of the reappearance of a
phenomenon that has been already observed any number of
times. This rule is generally stated somewhat as follows:
"To find the chance of the recurrence of an event already
observed, divide the number of times the event has been
observed, increased by one, by the same number increased
by two."
7. It will be instructive to point out the origin of
this rule; if only to remind the reader of the necessity of
keeping mathematical formulæ to their proper province,
and to show what astonishing conclusions are apt to
be accepted on the supposed warrant of mathematics.
Revert then to the example of Inverse Probability on p. 182.
We saw that under certain assumptions, it would
follow that when a single white ball had been drawn
from a bag known to contain 10 balls which were white
or black, the chance could be determined that there was
only one white ball in it. Having done this we readily
calculate 'directly' the chance that this white ball will
be drawn next time. Similarly we can reckon the chances
of there being two, three, &c. up to ten white balls in it,
and determine on each of these suppositions the chance
of a white ball being drawn next time. Adding these
together we have the answer to the question:--a white
ball has been drawn once from a bag known to contain
ten balls, white or black; what is the chance of a second
time drawing a white ball?
So far only arithmetic is required. For the next step we
need higher mathematics, and by its aid we solve this
problem:--A white ball has been drawn m times from a
bag which contains any number, we know not what, of
balls each of which is white or black, find the chance of
the next drawing also yielding a white ball. The answer is
(m + 1)/(m + 2).
Thus far mathematics. Then comes in the physical
assumption that the universe may be likened to such a bag
as the above, in the sense that the above rule may be
applied to solve this question:--an event has been observed
to happen m times in a certain way, find the chance that
it will happen in that way next time. Laplace, for instance,
has pointed out that at the date of the writing of his _Essai
Philosophique_, the odds in favour of the sun's rising again
(on the old assumption as to the age of the world) were
1,826,214 to 1. De Morgan says that a man who standing
on the bank of a river has seen ten ships pass by with flags
should judge it to be 11 to 1 that the next ship will also
carry a flag.
8. It is hard to take such a rule as this seriously, for
there does not seem to be even that moderate confirmation
of it which we shall find to hold good in the case of the
application of abstract formulæ to the estimation of the
evidence of witnesses. If however its validity is to be discussed
there appear to be two very distinct lines of enquiry
along which we may be led.
(1) In the first place we may take it for what it professes
to be, and for what it is commonly understood to
be, viz. a rule which assigns the measure of expectation
we ought to entertain of the recurrence of the event under
the circumstances in question. Of course, on the view
adopted in this work, we insist on enquiring whether it is
really true that on the average events do thus repeat their
performance in accordance with this law. Thus tested, no
one surely would attempt to defend such a formula. So
far from past occurrence being a ground for belief in future
recurrence, there are (as will be more fully pointed out
in the Chapter on Fallacies) plenty of cases in which the
direct contrary holds good. Then again a rule of this kind
is subject to the very serious perplexity to be explained
in our next chapter, arising out of the necessary arbitrariness
of such inverse reference. That is, when an event has
happened but a few times, we have no certain guide; and
when it has happened but once,[2] we have no guide whatever,
as to the class of cases to which it is to be referred. In
the example above, about the flags, why did we stop short at
this notion simply, instead of specifying the size, shape, &c.
of the flags?
De Morgan, it must be remembered, only accepts this rule
in a qualified sense. He regards it as furnishing a
_minimum_ value for the amount of our expectation. He
terms it "the rule of probability of a _pure induction_," and
says of it, "The probabilities shown by the above rules
are merely _minima_ which may be augmented by other
sources of knowledge." That is, he recognizes only those
instances in which our belief in the Uniformity of Nature
and in the existence of special laws of causation comes in
to supplement that which arises from the mere frequency
of past occurrence. This however does not meet those cases
in which past occurrence is a positive ground of disbelief in
future recurrence.
9. (2) There is however another and very different
view which might be taken of such a rule. It is one, an
obscure recognition of which has very likely had much to
do with the acceptance which the rule has received.
What we might suppose ourselves to be thus expressing
is,--not the measure of rational expectation which might be
held by minds sufficiently advanced to be able to classify
and to draw conscious inferences, but,--the law according to
which the primitive elements of belief were started and
developed. Of course such an interpretation as this would
be equivalent to quitting the province of Logic altogether
and crossing over into that of Psychology; but it would be a
perfectly valid line of enquiry. We should be attempting
nothing more than a development of the researches of
Fechner and his followers in psychophysical measurement.
Only then we ought, like them, not to start with any analogy
of a ballot box and its contents, but to base our enquiry
on careful determination of the actual mental phenomena
experienced. We know how the law has been determined
in accordance with which the intensity of the feeling of
_light_ varies with that of its objective source. We see how it
is possible to measure the growth of _memory_ according to
the number of repetitions of a sentence or a succession
of mere syllables. In this latter case, for instance, we just
try experiments, and determine how much better a man can
remember any utterances after eight hearings than after
seven.[3]
Now this case furnishes a very close parallel to our
supposed attempt to measure the increase of intensity of
belief after repeated recurrence. That is, if it were possible
to experiment in this order of mental phenomena, we ought
simply to repeat a phenomenon a certain number of times
and then ascertain by actual introspection or by some simple
test, how fast the belief was increasing. Thus viewed the
problem seems to me a hopeless one. The difficulties are
serious enough, when we are trying to measure our simple
sensations, of laying aside the effects of past training, and of
attempting, as it were, to leave the mind open and passive
to mere reception of stimuli. But if we were to attempt
in this way to measure our belief these difficulties would
become quite insuperable. We can no more divest ourselves
of past training here than we can of intelligence or
thought. I do not see how any one could possibly avoid
classing the observed recurrences with others which he had
experienced, and of being thus guided by special analogies
and inductions instead of trusting solely to De Morgan's
'pure induction'. The same considerations tend to rebut
another form of defence for the rule in question. It is
urged, for instance, that we may at least resort to it in
those cases in which we are in entire ignorance as to the
number and nature of the antecedents. This is a position to
which I can hardly conceive it possible that we should ever
be reduced. However remote or exceptional may be the
phenomenon selected we may yet bring it into relation with
some accepted generalizations and thus draw our conclusions
from these rather than from purely _à priori_ considerations.
10. Since then past acquisitions cannot be laid aside
or allowed for, the only remaining resource would be to
experiment upon the infant mind. One would not like
to pronounce that any line of enquiry is impossible; but
the difficulties would certainly be enormous. And interesting
as the facts would be, supposing that we had succeeded
in securing them, they would not be of the slightest importance
in Logic. However the question were settled:--whether,
for instance, we proved that the sentiment or
emotion of belief grew up slowly and gradually from a sort
of zero point under the impress of repetition of experience;
or whether we proved that a single occurrence produced
complete belief in the repetition of the event, so that
experience gradually untaught us and weakened our convictions;--in
no case would the mature mind gain any aid
as to what it ought to believe.
I cannot but think that some such view as this must
occasionally underlie the acceptance which this rule has received.
For instance, Laplace, though unhesitatingly adopting
it as a real, that is, objective rule of inference, has gone
into so much physiological and psychological matter towards
the end of his discussion (_Essai philosophique_) as to suggest
that what he had in view was the natural history of belief
rather than its subsequent justification.
Again, the curious doctrine adopted by Jevons, that the
principles of Induction rest entirely upon the theory of
Probability,--a very different doctrine from that which is
conveyed by saying that all knowledge of facts is _probable_
only, i.e. not necessary,--seems unintelligible except on some
such interpretation. We shall have more to say on this
subject in our next chapter. It will be enough here to
remark that in our present reflective and rational stage
we find that every inference in Probability involves some
appeal to, or support from, Induction, but that it is impossible
to base either upon the other. However far back
we try to push our way, and however disposed we might be
to account for our ultimate beliefs by Association, it seems
to me that so long as we consider ourselves to be dealing
with rules of inference we must still distinguish between
Induction and Probability.
1. John Craig, in his often named work, _Theologiæ Christianæ
Principia Mathematica_ (Lond. 1699) attempted something in this
direction when he proposed to solve such problems as:--Quando
evanescet probabilitas cujusvis Historiæ, cujus subjectum est
transiens, vivâ tantum voce transmissæ, determinare.
2. When m = 1 the fraction becomes 2/3; i.e. the odds are 2 to 1 in
favour of recurrence. And there are writers who accept this
result. For instance, Jevons (_Principles of Science_ p. 258) says
"Thus on the first occasion on which a person sees a shark, and
notices that it is accompanied by a little pilot fish, the odds are
2 to 1 that the next shark will be so accompanied." To say nothing
of the fact that recognizing and naming the fish implies that they
have often been seen before, how many of the observed
characteristics of that single 'event' are to be considered
essential? Must the pilot precede; and at the same distance? Must
we consider the latitude, the ocean, the season, the species of
shark, as matter also of repetition on the next occasion? and so
on. I cannot see how the Inductive problem can be even intelligibly
stated, for quantitative purposes, on the first occurrence of any
event.
3. See in _Mind_ (x. 454) Mr Jacob's account of the researches of Herr
Ebbinghaus as described in his work _Ueber das Gedächtniss_.