138 A THEORY OF PROBABLE
INFERENCE.
of M's are probably P's in
about the same proportion,
and though this may happen not to be so, yet
at any
rate, on continuing the drawing
sufficiently, our pre
diction of the ratio will be vindicated at
last. On the
other hand, in induction we say that the
proportion ρ of
the sample being P's, probably there is
about the same
proportion in the whole lot; or at least, if
this happens
not to be so, then on continuing the
drawings the in
ference will be, not vindicated as
in the other case, but
modified so as to become true. The
deduction, then,
is probable in this sense, that though its
conclusion may
in a particular case be falsified, yet
similar conclusions
(with the same ratio ρ) would generally
prove approxi-
mately true; while the induction is probable
in this
sense, that though it may happen to give a
false con
clusion, yet in most cases in which the same
precept of
inference was followed, a different and
approximately
true inference (with the right value of ρ)
would be
drawn.
IV.
Before going any further with
the study of Form V.,
I wish to join to it another extremely
analogous form.
We often speak of one thing being very much
like
another, and thus apply a vague quantity to
resemblance.
Even if qualities are not subject to exact
numeration,
we may conceive them to be approximately
measurable.
We may then measure resemblance by a scale
of num-
bers from zero up to unity. To say that S
has a
1-likeness to a P will mean that it has
every character
of a P, and consequently is a P. To
say that it has a
0-likeness will imply total dissimilarity.
We shall then
be able to reason as follows :
A THEORY OF PROBABLE
INFERENCE. 139
It would be difficult, perhaps impossible,
to adduce an
example of such kind of inference, for the
reason that
simple marks are not known to us. We may,
however,
illustrate the complex probable deduction in
depth (the
general form of which it is not worth while
to set down)
as follows: I forget whether, in the
ritualistic churches,
a bell is tinkled at the elevation of the
Host or not.
Knowing, however, that the services resemble
somewhat
decidedly those of the Roman Mass, I think
that it is not
unlikely that the bell is used in the
ritualistic, as in the
Roman, churches.
We shall also have the following :
For example, we know that the French and
Italians
are a good deal alike in their ideas,
characters, tempera-
ments, genius, customs, institutions, etc.,
while they also
differ very markedly in all these respects.
Suppose, then,
that I know a boy who is going to make a
short trip
through France and Italy; I can safely
predict that
among the really numerous though relatively
few res-
140 A THEORY OF PROBABLE
INFERENCE.
pects in which he will be able
to compare the two people,
about the same degree of resemblance will be
found.
Both these modes of inference are clearly
deductive.
When r = 1, they reduce to Barbara.
1
Corresponding to induction, we have the
following
mode of inference:
Thus, we know, that the ancient
Mound-builders of
North America present, in all those respects
in which we
have been able to make the comparison, a
limited degree
of resemblance with the Pueblo Indians. The
inference
is, then, that in all respects there is
about the same de-
gree of resemblance between these races.
If I am permitted the extended sense which I
have
given to the word "induction," this argument
is simply
an induction respecting qualities instead of
respecting
A THEORY OF PROBABLE
INFERENCE. 141
things. In point of fact P
, P", P", etc. constitute a
random sample of the characters of M,
and the ratio r
of them being found to belong to S,
the same ratio of all
the characters of M are concluded to
belong to S. This
kind of argument, however, as it actually
occurs, differs
very much from induction, owing to the
impossibility
of simply counting qualities as individual
things are
counted. Characters have to be weighed
rather than
counted. Thus, antimony is bluish-gray :
that is a char
acter. Bismuth is a sort of rose-gray; it is
decidedly
different from antimony in color, and yet
not so very
different as gold, silver, copper, and tin
are.
I call this induction of characters hypothetic
inference,
or, briefly, hypothesis. This is
perhaps not a very happy
designation, yet it is difficult to find a
better. The term
"hypothesis" has many well established and
distinct
meanings. Among these is that of a
proposition believed
in because its consequences agree with
experience. This
is the sense in which Newton used the word
when he
said, Hypotheses non fingo. He meant
that he was merely
giving a general formula for the motions of
the heavenly
bodies, but was not undertaking to mount to
the causes
of the acceleration they exhibit. The
inferences of
Kepler, on the other hand, were hypotheses
in this sense;
for he traced out the miscellaneous
consequences of the
supposition that Mars moved in an ellipse,
with the sun
at the focus, and showed that both the
longitudes and the
latitudes resulting from this theory were
such as agreed
with observation. These two components of
the motion
were observed; the third, that of approach
to or regression
from the earth, was supposed. Now, if in
Form V. (bis)
we put r = 1, the inference is the
drawing of a hypothesis
in this sense. I take the liberty of
extending the use of
the word by permitting r to have any value
from zero to
142 A THEORY OF PROBABLE
INFERENCE.
unity. The term is certainly not all that
could be de-
sired; for the word hypothesis, as
ordinarily used, carries
with it a suggestion of uncertainty, and of
something to
be superseded, which does not belong at all
to my use of
it. But we must use existing language as
best we may,
balancing the reasons for and against any
mode of ex
pression, for none is perfect; at least the
term is not
so utterly misleading as "analogy" would be,
and with
proper explanation it will, I hope, be
understood.
V.
The following examples will
illustrate the distinction
between statistical deduction, induction,
and hypothesis.
If I wished to order a font of type
expressly for the
printing of this book, knowing, as I do,
that in all Eng-
lish writing the letter e occurs
oftener than any other
letter, I should want more e's in my
font than other
letters. For what is true of all other
English writing is
no doubt true of these papers. This is a
statistical de-
duction. But then the words used in logical
writings are
rather peculiar, and a good deal of use is
made of single
letters. I might, then, count the number of
occurrences
of the different letters upon a dozen or so
pages of the
manuscript, and thence conclude the relative
amounts of
the different kinds of type required in the
font. That
would be inductive inference. If now I were
to order
the font, and if, after some days, I were to
receive a box
containing a large number of little paper
parcels of very
different sizes, I should naturally infer
that this was the
font of types I had ordered; and this would
be hypothetic
inference. Again, if a dispatch in cipher is
captured, and
it is found to be written with twenty-six
characters, one
of which occurs much more frequently than
any of the
A THEORY OF PROBABLE
INFERENCE. 143
others, we are at once led to suppose that
each charac-
ter represents a letter, and that the one
occurring so fre
quently stands fer e. This is also
hypothetic inference.
We are thus led to divide all probable
reasoning into
deductive and ampliative, and further to
divide ampliative
reasoning into induction and hypothesis. In
deductive
reasoning, though the predicted ratio may be
wrong in a
limited number of drawings, yet it will be
approximately
verified in a larger number. In ampliative
reasoning the
ratio may be wrong, because the inference is
based on but
a limited number of instances; but on
enlarging the
sample the ratio will be changed till it
becomes approxi
mately correct. In induction, the instances
drawn at
random are numerable things; in hypothesis
they are
characters, which are not capable of strict
enumeration,
but have to be otherwise estimated.
This classification of probable inference is
connected
with a preference for the copula of
inclusion over those
used by Miss Ladd and by Mr. Mitchell. 1
De Morgan
established eight forms of simple
propositions; and from
a purely formal point of view no one of
these has a right
to be considered as more fundamental than
any other.
But formal logic must not be too purely
formal; it must
represent a fact of psychology, or else it
is in danger of
degenerating into a mathematical recreation.
The cate
gorical proposition, "every man is mortal,"
is but a modifi
cation of the hypothetical proposition, "if
humanity, then
mortality;" and since the very first
conception from which
logic springs is that one proposition
follows from another,
I hold that "if A, then B" should be taken
as the typical
form of judgment. Time flows; and, in time,
from one
state of belief (represented by the premises
of an argu-
1 I do not here speak of
Mr. Jevons, because my objection to the
copula of identity is of a somewhat
different kind.
144 A THEORY OF PROBABLE
INFERENCE.
ment) another (represented by
its conclusion) is de-
veloped. Logic arises from this
circumstance, without
which we could not learn anything nor
correct any
opinion. To say that an inference is correct
is to say
that if the premises are true the conclusion
is also true;
or that every possible state of things in
which the prem
ises should be true would be included among
the possible
states of things in which the conclusion
would be true.
We are thus led to the copula of inclusion.
But the
main characteristic of the relation of
inclusion is that it
is transitive, that is, that what is
included in some
thing included in anything is itself
included in that
thing; or, that if A is B
and B is C, then A is C.
We
thus get Barbara as the primitive
type of inference.
Now in Barbara we have a Rule, a Case
under the Rule,
and the inference of the Result of
that rule in that case.
For example :
Rule. All men are
mortal;
Case. Enoch was a man.
Result. Enoch was mortal.
The cognition of a rule is not
necessarily conscious,
but is of the nature of a habit, acquired or
congenital.
The cognition of a case is of the general
nature of a
sensation; that is to say, it is something
which comes
up into present consciousness. The cognition
of a result
is of the nature of a decision to act in a
particular way
on a given occasion. 1 In point
of fact, a syllogism, in
Barbara virtually takes place when we
irritate the foot
of a decapitated frog. The connection
between the af-
ferent and efferent nerve, whatever it may
be, constitutes
a nervous habit, a rule of action, which is
the physio-
1 See my paper on "How
to make our ideas clear."Popular Science
Monthly, January, 1878.
A THEORY OF PROBABLE
INFERENCE. 145
logical analogue of the major
premise. The disturbance
of the ganglionic equilibrium, owing to the
irritation, is
the physiological form of that which,
psychologically con-
sidered, is a sensation; and, logically
considered, is the
occurrence of a case. The explosion through
the efferent
nerve is the physiological form of that
which psychologi-
cally is a volition, and logically the
inference of a result.
When we pass from the lowest to the highest
forms of
inervation, the physiological equivalents
escape our ob
servation; but, psychologically, we still
have, first, habit,
- which in its highest form is
understanding, and which
corresponds to the major premise of Barbara;
we have,
second, feeling, or present consciousness,
corresponding
to the minor premise of Barbara; and we
have, third,
volition, corresponding to the conclusion of
the same
mode of syllogism. Although these analogies,
like all
very broad generalizations, may seem very
fanciful at
first sight, yet the more the reader
reflects upon them
the more profoundly true I am confident they
will appear.
They give a significance to the ancient
system of formal
logic which no other can at all share.
Deduction proceeds from Rule and Case to
Result; it
is the formula of Volition. Induction
proceeds from Case
and Result to Rule; it is the formula of the
formation of
a habit or general conception, - a process
which, psycho-
logically as well as logically, depends on
the repetition of
instances or sensations. Hypothesis proceeds
from Rule
and Result to Case; it is the formula of the
acquirement
of secondary sensation, - a process by which
a confused
concatenation of predicates is brought into
order under
a synthetizing predicate.
We usually conceive Nature to be perpetually
making
deductions in Barbara. This is our natural
and anthro-
pomorphic metaphysics. We conceive that
there are
146 A THEORY OF PROBABLE
INFERENCE.
Laws of Nature, which are her
Rules or major premises.
We conceive that Cases arise under these
laws; these
cases consist in the predication, or
occurrence, of causes,
which are the middle terms of the
syllogisms. And,
finally, we conceive that the occurrence of
these causes,
by virtue of the laws of Nature, result in
effects which
are the conclusions of the syllogisms.
Conceiving of
nature in this way, we naturally conceive of
science as
having three tasks, - (1) the discovery of
Laws, which
is accomplished by induction; (2) the
discovery of Causes,
which is accomplished by hypothetic
inference; and (3)
the prediction of Effects, which is
accomplished by de
duction. It appears to me to be highly
useful to select
a system of logic which shall preserve all
these natural
conceptions.
It may be added that, generally speaking,
the conclu-
sions of Hypothetic Inference cannot be
arrived at in
ductively, because their truth is not
susceptible of direct
observation in single cases. Nor can the
conclusions of
Inductions, on account of their generality,
be reached by
hypothetic inference. For instance, any
historical fact,
as that Napoleon Bonaparte once lived, is a
hypothesis;
we believe the fact, because its effects - I
mean current
tradition, the histories, the monuments,
etc. - are ob-
served. But no mere generalization of
observed facts
could ever teach us that Napoleon lived. So
we induc-
tively infer that every particle of matter
gravitates toward
every other. Hypothesis might lead to this
result for
any given pair of particles, but it never
could show that
the law was universal.
VI.
We now come to the consideration of the
Rules which
have to be followed in order to make valid
and strong
A THEORY OF PROBABLE
INFERENCE. 147
Inductions and Hypotheses.
These rules can all be re-
duced to a single one; namely, that the
statistical deduc-
tion of which the Induction or Hypothesis is
the inversion,
must be valid and strong.
We have seen that Inductions and Hypotheses
are in-
ferences from the conclusion and one premise
of a sta-
tistical syllogism to the other premise. In
the case of
hypothesis, this syllogism is called the explanation.
Thus
in one of the examples used above, we
suppose the cryp-
tograph to be an English cipher, because, as
we say, this
explains the observed phenomena that
there are about
two dozen characters, that one occurs more
frequently
than the rest, especially at the ends of
words, etc. The
explanation is,
This explanation is present to the mind of
the reasoner,
too; so much so, that we commonly say that
the hypo
thesis is adopted for the sake of
the explanation. Of
induction we do not, in ordinary language,
say that it
explains phenomena; still, the statistical
deduction, of
which it is the inversion, plays, in a
general way, the
same part as the explanation in hypothesis.
From a
barrel of apples, that I am thinking of
buying, I draw
out three or four as a sample. If I find the
sample some
what decayed, I ask myself, in ordinary
language, not
"Why is this?" but "How is this?" And I
answer
that it probably comes from nearly all the
apples in the
barrel being in bad condition. The
distinction between
the "Why" of hypothesis and the "How" of
induction
is not very great; both ask for a
statistical syllogism, of
which the observed fact shall be the
conclusion, the
148 A THEORY OF PROBABLE
INFERENCE.
known conditions of the
observation one premise, and
the inductive or hypothetic inference the
other. This
statistical syllogism may be conveniently
termed the ex
planatory syllogism.
In order that an induction or hypothesis
should have
any validity at all, it is requisite that
the explanatory
syllogism should be a valid statistical
deduction. Its
conclusion must not merely follow from the
premises,
but follow from them upon the principle of
probability.
The inversion of ordinary syllogism
does not give rise
to an induction or hypothesis. The
statistical syllogism
of Form IV. is invertlble, because it
proceeds upon the
principle of an approximate equality
between the ratio
of P's in the whole class and the
ratio in a well-drawn
sample, and because equality is a
convertible relation.
But ordinary syllogism is based upon the
property of the
relation of containing and contained, and
that is not a
convertible relation. There is, however, a
way in which
ordinary syllogism may be inverted; namely,
the con
clusion and either of the premises may be
interchanged
by negativing each of them. This is the way
in which
the indirect, or apagogical, 1
figures of syllogism are de-
rived from the first, and in which the modus
tollens is
derived from the modus ponens. The following
schemes
show this :
A THEORY OF PROBABLE
INFERENCE. 149
Now suppose we ask ourselves what would be
the re-
sult of thus apagogically inverting a
statistical deduction.
Let us take, for example, Form IV :
The ratio r, as we have already
noticed, is not neces-
sarily perfectly definite; it may be only
known to have
a certain maximum or minimum; in fact, it
may have
any kind of indeterminacy. Of all possible
values be
tween and 1, it admits of some and excludes
others.
The logical negative of the ratio r
is, therefore, itself a
ratio, which we may name ρ; it admits of
every value
which r excludes, and excludes every
value of which r
admits. Transposing, then, the major premise
and con-
clusion of our statistical deduction, and at
the same time
denying both, we obtain the following
inverted form:-
150 A THEORY OF PROBABLE
INFERENCE.
But this coincides with the formula of
Induction.
Again, let us apagogically invert the
statistical deduction
of Form IV. (bis). This form is,-
Transposing the minor premise and
conclusion, at the
same time denying both, we get the inverted
form,
This coincides with the formula of
Hypothesis. Thus
we see that Induction and Hypothesis are
nothing but
the apagogical inversions of statistical
deductions. Ac
cordingly, when r is taken as 1, so that ρ
is "less than 1,"
or when r is taken as 0, so that ρ
is "more than 0,"the
induction degenerates into a syllogism of
the third figure
and the hypothesis into a syllogism of the
second figure.
A THEORY OF PEOBABLE INFERENCE. 151
In these special cases, there
is no very essential difference
between the mode of reasoning in the direct
and in the
apagogical form. But, in general, while the
probability
of the two forms is precisely the same, in
this sense,
that for any fixed proportion of P's
among the M's
(or of marks of S's among the marks
of the M's) the
probability of any given error in the
concluded value is
precisely the same in the indirect as it is
in the direct
form, yet there is this striking difference,
that a multi-
plication of instances will in the one case
confirm, and
in the other modify, the concluded value of
the ratio.
We are thus led to another form for our rule
of validity
of ampliative inference; namely, instead of
saying that
the explanatory syllogism must be a
good probable de
duction, we may say that the syllogism of
which the
induction or hypothesis is the apagogical
modification
(in the traditional language of logic, the
reduction) must
be valid.
Probable inferences, though valid, may still
differ in
their strength. A probable deduction has a
greater or
less probable error in the concluded ratio.
When r is a
definite number the probable error is also
definite; but
as a general rule we can only assign maximum
and mini-
mum values of the probable error. The
probable error
is, in fact,
where n is the number of independent
instances. The
same formula gives the probable error of an
induction or
hypothesis; only that in these cases, r
being wholly inde-
terminate, the minimum value is zero, and
the maximum
is obtained by putting r = 1/2.
152 A THEORY OF PROBABLE
INFERENCE.
VII.
Although the rule given above
really contains all the
conditions to which Inductions and
Hypotheses need to
conform, yet inasmuch as there are many
delicate ques-
tions in regard to the application of it,
and particularly
since it is of that nature that a violation
of it, if not
too gross, may not absolutely destroy the
virtue of the
reasoning, a somewhat detailed study of its
requirements
in regard to each of the premises of the
argument is still
needed.
The first premise of a scientific inference
is that certain
things (in the case of induction) or certain
characters
(in the case of hypothesis) constitute a
fairly chosen
sample of the class of things or the
run of characters
from which they have been drawn.
The rule requires that the sample should be
drawn at
random and independently from the whole lot
sampled.
That is to say, the sample must be taken
according to a
precept or method which, being applied over
and over
again indefinitely, would in the long run
result in the
drawing of any one set of instances as often
as any other
set of the same number.
The needfulness of this rule is obvious; the
difficulty
is to know how we are to carry it out. The
usual method
is mentally to run over the lot of objects
or characters to
be sampled, abstracting our attention from
their peculi
arities, and arresting ourselves at this one
or that one
from motives wholly unconnected with those
peculiarities.
But this abstention from a further
determination of our
choice often demands an effort of the will
that is beyond
our strength; and in that case a mechanical
contrivance
may be called to our aid. We may, for
example, number
all the objects of the lot, and then draw
numbers by
A THEORY OF PROBABLE
INFERENCE. 153
means of a roulette, or other
such instrument. We may
even go so far as to say that this method is
the type of
all random drawing; for when we abstract our
attention
from the peculiarities of objects, the
psychologists tell us
that what we do is to substitute for the
images of sense
certain mental signs, and when we proceed to
a random
and arbitrary choice among these abstract
objects we are
governed by fortuitous determinations of the
nervous sys-
tem, which in this case serves the purpose
of a roulette.
The drawing of objects at random is an act
in which
honesty is called for; and it is often hard
enough to be
sure that we have dealt honestly with
ourselves in the
matter, and still more hard to be satisfied
of the honesty
of another. Accordingly, one method of
sampling has
come to be preferred in argumentation;
namely, to take
of the class to be sampled all the objects
of which we
have a sufficient knowledge. Sampling is,
however, a
real art, well deserving an extended study
by itself: to
enlarge upon it here would lead us aside
from our main
purpose.
Let us rather ask what will be the effect
upon inductive
inference of an imperfection in the strictly
random char
acter of the sampling. Suppose that, instead
of using
such a precept of selection that any one M
would in the
long run be chosen as often as any other, we
used a
precept which would give a preference to a
certain half
of the M's, so that they would be
drawn twice as often
as the rest. If we were to draw a numerous
sample by
such a precept, and if we were to find that
the proportion
ρ of the sample consisted of M's,
the inference that we
should be regularly entitled to make would
be, that among
all the M's, counting the preferred
half for two each, the
proportion p would be P's. But this
regular inductive
inference being granted, from it we could
deduce by
154 A THEORY OF PROBABLE
INFERENCE.
arithmetic the further
conclusion that, counting the M's
for one each, the proportion of P's
among them must
(ρ being over 2/3) lie between 3/4ρ + 1/4
and 3/4ρ - 1/2. Hence, if more than two
thirds of the instances drawn by the use of
the false precept were found to be P's,
we should be
entitled to conclude that more than half of
all the M's
were P's. Thus, without allowing
ourselves to be led
away into a mathematical discussion, we can
easily see
that, in general, an imperfection of that
kind in the
random character of the sampling will only
weaken the
inductive conclusion, and render the
concluded ratio less
determinate, but will not necessarily
destroy the force
of the argument completely. In particular,
when p ap
proximates towards 1 or 0, the effect of the
imperfect
sampling will be but slight.
Nor must we lose sight of the constant
tendency of the
inductive process to correct itself. This is
of its essence.
This is the marvel of it. The probability of
its conclusion
only consists in the fact that if the true
value of the ratio
sought has not been reached, an extension of
the induc
tive process will lead to a closer
approximation. Thus,
even though doubts may be entertained
whether one se-
lection of instances is a random one, yet a
different se-
lection, made by a different method, will be
likely to vary
from the normal in a different way, and if
the ratios
derived from such different selections are
nearly equal,
they may be presumed to be near the truth.
This con-
sideration makes it extremely advantageous
in all ampli-
ative reasoning to fortify one method of
investigation by
another. 1 Still we must not
allow ourselves to trust so
A THEORY OF PROBABLE
INFERENCE. 155
much to this virtue of
induction as to relax our efforts
towards making our drawings of instances as
random
and independent as we can. For if we infer a
ratio from
a number of different inductions, the
magnitude of its
probable error will depend very much more on
the worst
than on the best inductions used.
We have, thus far, supposed that although
the selection
of instances is not exactly regular, yet the
precept fol
lowed is such that every unit of the lot
would eventually
get drawn. But very often it is
impracticable so to draw
our instances, for the reason that a part of
the lot to be
sampled is absolutely inaccessible to our
powers of obser
vation. If we want to know whether it will
be profit
able to open a mine, we sample the ore; but
in advance
of our mining operations, we can obtain only
what ore
lies near the surface. Then, simple
induction becomes
worthless, and another method must be
resorted to. Sup
pose we wish to make an induction regarding
a series
of events extending from the distant past to
the distant
future; only those events of the series
which occur within
the period of time over which available
history extends
can be taken as instances. Within this
period we may
find that the events of the class in
question present some
uniform character; yet how do we know but
this uni
formity was suddenly established a little
while before the
history commenced, or will suddenly break up
a little
while after it terminates ? Now, whether the
uniformity
156 A THEORY OF PROBABLE
INFERENCE.
observed consists (1) in a
mere resemblance between all
the phenomena, or (2) in their consisting of
a disorderly
mixture of two kinds in a certain constant
proportion, or
(3) in the character of the events being a
mathematical
function of the time of occurrence, - in any
of these cases
we can make use of an apagoge from the
following proba
ble deduction :
Inverting this deduction, we have the
following ampli-
ative inference :
The probability of the conclusion consists
in this, that
we here follow a precept of inference,
which, if it is very
often applied, will more than half the time
lead us right.
Analogous reasoning would obviously apply to
any por
tion of an unidimensional continuum, which
might be
similar to periods of time. This is a sort
of logic which
is often applied by physicists in what is
called extrapola-
tion of an empirical law. As compared
with a typical
induction, it is obviously an excessively
weak kind of in
ference. Although indispensable in almost
every branch
of science, it can lead to no solid
conclusions in regard to
what is remote from the field of direct
perception, unless
it be bolstered up in certain ways to which
we shall have
occasion to refer further on.
A THEORY OF PROBABLE
INFERENCE. 157
Let us now consider another
class of difficulties in
regard to the rule that the samples must be
drawn at
random and independently. In the first
place, what if
the lot to be sampled be infinite in number
? In what
sense could a random sample be taken from a
lot like
that ? A random sample is one taken
according to a
method that would, in the long run, draw any
one object
as often as any other. In what sense can
such drawing
be made from an infinite class ? The answer
is not far
to seek. Conceive a cardboard disk revolving
in its own
plane about its centre, and pretty
accurately balanced,
so that when put into rotation it shall be
about 1 as likely
to come to rest in any one position as in
any other; and
let a fixed pointer indicate a position on
the disk: the
number of points on the circumference is
infinite, and on
rotating the disk repeatedly the pointer
enables us to
make a selection from this infinite number.
Tbis means
merely that although the points are
innumerable, yet
there is a certain order among them that
enables us to
run them through and pick from them as from
a very
numerous collection. In such a case, and in
no other,
can an infinite lot be sampled. But it would
be equally
true to say that a finite lot can be sampled
only on
condition that it can be regarded as
equivalent to an
infinite lot. For the random sampling of a
finite class
supposes the possibility of drawing out an
object, throw-
ing it back, and continuing this process
indefinitely; so
that what is really sampled is not the
finite collection of
things, but the unlimited number of possible
drawings.
But though there is thus no insuperable
difficulty in
sampling an infinite lot, yet it must be
remembered that
the conclusion of inductive reasoning only
consists in the
158 A THEORY OF PROBABLE
INFERENCE.
approximate evaluation of a ratio,
so that it never can
authorize us to conclude that in an infinite
lot sampled
there exists no single exception to a rule.
Although all
the planets are found to gravitate toward
one another,
this affords not the slightest direct reason
for denying
that among the innumerable orbs of heaven
there may
be some \vhich exert no such force. Although
at no
point of space where we have yet been have
we found
any possibility of motion in a fourth
dimension, yet this
does not tend to show (by simple induction,
at least)
that space has absolutely but three
dimensions. Although
all the bodies we have had the opportunity
of examining
appear to obey the law of inertia, this does
not prove
that atoms and atomicules are subject to the
same law.
Such conclusions must be reached, if at all,
in some
other way than by simple induction. This
latter may
show that it is unlikely that, in my
lifetime or yours,
things so extraordinary should be found, but
do not war
rant extending the prediction into the
indefinite future.
And experience shows it is not safe to
predict that such
and such a fact will never be met
with.
If the different instances of the lot
sampled are to
be drawn independently, as the rule
requires, then the
fact that an instance has been drawn once
must not
prevent its being drawn again. It is true
that if the
objects remaining unchosen are very much
more numer
ous than those selected, it makes
practically no difference
whether they have a chance of being drawn
again or not,
since that chance is in any case very small.
Proba-
bility is wholly an affair of approximate,
not at all of
exact, measurement; so that when the class
sampled is
very large, there is no need of considering
whether ob-
jects can be drawn more than once or not.
But in what
is known as "reasoning from analogy," the
class sam-
A THEORY OF PROBABLE
INFERENCE. 159
pled is small, and no instance
is taken twice. For ex
ample : we know that of the major planets
the Earth,
Mars, Jupiter, and Saturn revolve on their
axes, and
we conclude that the remaining four,
Mercury, Venus,
Uranus, and Neptune, probably do the like.
This is
essentially different from an inference from
what has
been found in drawings made hitherto, to
what will be
found in indefinitely numerous drawings to
be made
hereafter. Our premises here are that the
Earth, Mars,
Jupiter, and Saturn are a random sample of a
natural
class of major planets, - a class which,
though (so far
as we know) it is very small, yet may
be very extensive,
comprising whatever there may be that
revolves in a
circular orbit around a great sun, is nearly
spherical,
shines with reflected light, is very large,
etc. Now the
examples of major planets that we can
examine all ro-
tate on their axes; whence we suppose that
Mercury,
Venus, Uranus, and Neptune, since they
possess, so far
as we know, all the properties common to the
natural
class to which the Earth, Mars, Jupiter, and
Saturn be
long, possess this property likewise. The
points to be
observed are, first, that any small class of
things may be
regarded as a mere sample of an actual or
possible large
class having the same properties and subject
to the same
conditions; second, that while we do not
know what all
these properties and conditions are, we do
know some of
them, which some may be considered as a
random sam
ple of all; third, that a random selection
without re
placement from a small class may be regarded
as a true
random selection from that infinite class of
which the
finite class is a random selection. The
formula of the
analogical inference presents, therefore,
three premises,
thus: -
160 A THEORY OF PROBABLE
INFERENCE.
We have evidently here an induction and an
hypothe-
sis followed by a deduction; thus,
An argument from analogy may be strengthened
by
the addition of instance after instance to
the premises,
until it loses its ampliative character by
the exhaustion
of the class and becomes a mere deduction of
that kind
called complete induction, in which,
however, some shadow
A THEORY OF PROBABLE INFERENCE. 161
of the inductive character
remains, as this name im-
plies.
VIII.
Take any human being, at
random, - say Queen Eliz-
abeth. Now a little more than half of all
the human
beings who have ever existed have been
males; but it
does not follow that it is a little more
likely than not
that Queen Elizabeth was a male, since we
know she was
a woman. Nor, if we had selected Julius
Caesar, would
it be only a little more likely than not
that he was a
male. It is true that if we were to go on
drawing at
random an indefinite number of instances of
human be
ings, a slight excess over one-half would be
males. But
that which constitutes the probability of an
inference is
the proportion of true conclusions among all
those which
could be derived from the same precept.
Now a precept
of inference, being a rule which the mind is
to follow,
changes its character and becomes different
when the
case presented to the mind is essentially
different. When,
knowing that the proportion r of all
M's are P's, I draw
an instance, S, of an M,
without any other knowledge of
whether it is a P or not, and infer
with probability, r,
that it is P, the case presented to
my mind is very
different from what it is if I have such
other knowledge.
In short, I cannot make a valid probable
inference with
out taking into account whatever knowledge I
have (or,
at least, whatever occurs to my mind) that
bears upon
the question.
The same principle may be applied to the
statistical
deduction of Form IV. If the major premise,
that the
proportion r of the M's are
P's, be laid down first,
before the instances of Ms are
drawn, we really draw our
inference concerning those instances (that
the proper-
162 A THEORY OF PROBABLE
INFERENCE.
tion r of them will be
P's) in advance of the drawing,
and therefore before we know whether they
are P s or
not. But if we draw the instances of the M B
first, and
after the examination of them decide what we
will select
for the predicate of our major premise, the
inference
will generally be completely fallacious. In
short, we
have the rule that the major term P must be
decided
upon in advance of the examination of the
sample; and
in like manner in Form IV. (bis) the
minor term S must
be decided upon in advance of the drawing.
The same rule follows us into the logic of
induction
and hypothesis. If in sampling any class,
say the M's,
we first decide what the character P
is for which we
propose to sample that class, and also how
many instan-
ces we propose to draw, our inference is
really made
before these latter are drawn, that the
proportion of P's
in the whole class is probably about the
same as among
the instances that are to be drawn, and the
only thing
we have to do is to draw them and observe
the ratio.
But suppose we were to draw our inferences
without
the predesignation of the character P;
then we might in
every case find some recondite character in
which those
instances would all agree. That, by the
exercise of
sufficient ingenuity, we should be sure to
be able to do
this, even if not a single other object of
the class M
possessed that character, is a matter of
demonstration.
For in geometry a curve may be drawn through
any
given series of points, without passing
through any one
of another given series of points, and this
irrespective of
the number of dimensions. Now, all the
qualities of
objects may be conceived to result from
variations of a
number of continuous variables; hence any
lot of ob-
jects possesses some character in common,
not possessed
by any other. It is true that if the
universe of quality
A THEORY OF PROBABLE
INFERENCE. 163
is limited, this is not
altogether true; but it remains
true that unless we have some special
premise from
which to infer the contrary, it always may
be possible
to assign some common character of the
instances S', S",
S'", etc., drawn at random from among
the M s, which
does not belong to the M's
generally. So that if the
character P were not predesignate, the
deduction of
which our induction is the apagogical
inversion would
not be valid; that is to say, we could not
reason that if
the M's did not generally possess
the character P, it
would not be likely that the S's
should all possess this
character.
I take from a biographical dictionary the
first five
names of poets, with their ages at death.
They are,
These five ages have the following
characters in com-
mon :
1. The difference of the two digits
composing the
number, divided by three, leaves a remainder
of one.
2. The first digit raised to the power
indicated by the
second, and then divided by three, leaves a
remainder of
one.
3. The sum of the prime factors of each age,
including
one as a prime factor, is divisible by three.
Yet there is not the smallest reason to
believe that the
next poet s age would possess these
characters.
Here we have a conditio sine qua non
of valid induc-
tion which has been singularly overlooked by
those who
have treated of the logic of the subject,
and is very fre-
164 A THEORY OF PROBABLE
INFERENCE.
quently violated by those who
draw inductions. So ac
complished a reasoner as Dr. Lyon Playfair,
for instance,
has written a paper of which the following
is an abstract.
He first takes the specific gravities of the
three allotropic
forms of carbon, as follows :
He now seeks to find a
uniformity connecting these three
instances; and he discovers that the atomic
weight of
carbon, being 12,
This, he thinks, renders it probable that
the specific
gravities of the allotropic forms of other
elements would,
if we knew them, be found to equal the
different roots of
their atomic weight. But so far, the
character in which
the instances agree not having been
predesignated, the
induction can serve only to suggest a
question, and ought
not to create any belief. To test the
proposed law, he
selects the instance of silicon, which like
carbon exists
in a diamond and in a graphitoidal
condition. He finds
for the specific gravities
A THEORY OF PROBABLE
INFERENCE. 165
Now, the atomic weight of
silicon, that of carbon being
12, can only be taken as 28. But 2.47 does
not approx
imate to any root of 28. It is, however,
nearly the
cube root of 14, 
, while 2.33 is
nearly
the fourth root of 28 
. Dr. Playfair
claims
that silicon is an instance satisfying his
formula. But
in fact this instance requires the formula
to be modified;
and the modification not being predesignate,
the instance
cannot count. Boron also exists in a diamond
and a
graphitoidal form; and accordingly Dr.
Playfair takes
this as his next example. Its atomic weight
is 10.9, and
its specific gravity is 2.68; which is the
square root of
f X 10.9. There seems to be here a further
modification
of the formula not predesignated, and
therefore this in
stance can hardly be reckoned as
confirmatory. The
next instances which would occur to the mind
of any
chemist would be phosphorus and sulphur,
which exist
in familiarly known allotropic forms. Dr.
Playfair ad
mits that the specific gravities of
phosphorus have no
relations to its atomic weight at all
analogous to those
of carbon. The different forms of sulphur
have nearly
the same specific gravity, being
approximately the fifth
root of the atomic weight 32. Selenium also
has two
.allotropic forms, whose specific gravities
are 4.8 and 4.3;
one of these follows the law, while the
other does not.
For tellurium the law fails altogether; but
for bromine
and iodine it holds. Thus the number of
specific gravi
ties for which the law was predesignate are
8; namely,
2 for phosphorus, 1 for sulphur, 2 for
selenium, 1 for
tellurium, 1 for bromine, and 1 for iodine.
The law
holds for 4 of these, and the proper
inference is that
about half the specific gravities of
metalloids are roots
of some simple ratio of their atomic
weights.
Having thus determined this ratio, we
proceed to
166 A THEORY OF PROBABLE
INFERENCE.
inquire whether an agreement
half the time with the
formula constitutes any special connection
between the
specific gravity and the atomic weight of a
metalloid.
As a test of this, let us arrange the
elements in the order
of their atomic weights, and compare the
specific gravity
of the first with the atomic weight of the
last, that of
the second with the atomic weight of the
last but one,
and so on. The atomic weights are -
There are three specific gravities given for
carbon, and
two each for silicon, phosphorus, and
selenium. The
question, therefore, is, whether of the
fourteen specific
gravities as many as seven are in Playfair s
relation
with the atomic weights, not of the same
element, but
of the one paired with it. Now, taking the
original
formula of Playfair we find
or five such relations without counting that
of sulphur
to itself. Next, with the modification
introduced by Play-
fair, we have
A THEORY OF PROBABLE
INFERENCE. 167
It thus appears that there is
no more frequent agree
ment with Playfair s proposed law than what
is due to
chance. 1
Another example of this fallacy was "Bode's
law" of
the relative distances of the planets, which
was shattered
by the first discovery of a true planet
after its enuncia
tion. In fact, this false kind of induction
is extremely
common in science and in medicine. 2
In the case of
hypothesis, the correct rule has often been
laid down;
namely, that a hypothesis can only be
received upon the
ground of its having been verified
by successful prediction.
The term predesignation used in this
paper appears to be
more exact, inasmuch as it is not at all
requisite that the
ratio ρ should be given in advance of the
examination of
the samples. Still, since ρ is equal to 1 in
all ordinary
hypotheses, there can be no doubt that the
rule of pre-
diction, so far as it goes, coincides with
that here laid
down.
We have now to consider an important
modification of
the rule. Suppose that, before sampling a
class of objects,
we have predesignated not a single character
but n char-
acters, for which we propose to examine the
samples.
This is equivalent to making n different
inductions from
the same instances. The probable error in
this case is
that error whose probability for a simple
induction is only
(1/2)n , and the theory of
probabilities shows that it in-
168 A THEORY OF PROBABLE
INFERENCE.
creases but slowly with n;
in fact, for n = 1000 it is only
about five times as great as for n =
1, so that with only
25 times as many instances the inference
would be as
secure for the former value of n as
with the latter; with
100 times as many instances an induction in
which n =
10,000,000,000 would be equally secure. Now
the whole
universe of characters will never contain
such a number
as the last; and the same may be said of the
universe of
objects in the case of hypothesis. So that,
without any
voluntary predesignation, the limitation of
our imagina
tion and experience amounts to a
predesignation far
within those limits; and we thus see that if
the number
of instances be very great indeed, the
failure to predes-
ignate is not an important fault. Of
characters at all
striking, or of objects at all familiar, the
number will
seldom reach 1,000; and of very striking
characters or
very familiar objects the number is still
less. So that if
a large number of samples of a class are
found to have
some very striking character in common, or
if a large
number of characters of one object are found
to be pos-
sessed by a very familiar object, we need
not hesitate to
infer, in the first case, that the same
characters belong
to the whole class, or, in the second case,
that the two
objects are practically identical;
remembering only that
the inference is less to be relied upon than
it would be
had a deliberate predesignation been made.
This is no
doubt the precise significance of the rule
sometimes laid
down, that a hypothesis ought to be simple,
simple
here being taken in the sense of familiar.
This modification of the rule shows that,
even in the
absence of voluntary predesignation, some
slight weight
is to be attached to an induction or
hypothesis. And
perhaps when the number of instances is not
very small,
it is enough to make it worth while to
subject the in-
A THEORY OF PROBABLE
INFERENCE. 169
ference to a regular test. But
our natural tendency will
be to attach too much importance to such
suggestions,
and we shall avoid waste of time in passing
them by
without notice until some stronger
plausibility presents
itself.
IX.
In almost every case in which
we make an induction
or a hypothesis, we have some knowledge
which renders
our conclusion antecedently likely or
unlikely. The ef-
fect of such knowledge is very obvious, and
needs no
remark. But what also very often happens is
that we
have some knowledge, which, though not of
itself bearing
upon the conclusion of the scientific
argument, yet serves
to render our inference more or less
probable, or even
to alter the terms of it. Suppose, for
example, that we
antecedently know that all the M's
strongly resemble
one another in regard to characters of a
certain order.
Then, if we find that a moderate number of M's
taken
at random have a certain character, P,
of that order, we
shall attach a greater weight to the
induction than we
should do if we had not that antecedent
knowledge.
Thus, if we find that a certain sample of
gold has a
certain chemical character, - since we have
very strong
reason for thinking that all gold is alike
in its chemical
characters, - we shall have no hesitation in
extending
the proposition from the one sample to gold
in general.
Or if we know that among a certain people,
say the
Icelanders, - an extreme uniformity prevails
in regard
to all their ideas, then, if we find that
two or three in-
dividuals taken at random from among them
have all
any particular superstition, we shall be the
more ready
to infer that it belongs to the whole people
from what
we know of their uniformity. The influence
of this sort
170 A THEORY OF PROBABLE
INFERENCE.
of uniformity upon inductive
conclusions was strongly in
sisted upon by Philodemus, and some very
exact concep
tions in regard to it may be gathered from
the writings
of Mr. Galton. Again, suppose we know of a
certain
character, P, that in whatever
classes of a certain des-
cription it is found at all, to those it
usually belongs as
a universal character; then any induction
which goes
toward showing that all the M's are
P will be greatly
strengthened. Thus it is enough to find that
two or
three individuals taken at random from a
genus of ani-
mals have three toes on each foot, to prove
that the same
is true of the whole genus; for we know that
this is a
generic character. On the other hand, we
shall be slow
to infer that all the animals of a genus
have the same
color, because color varies in almost every
genus. This
kind of uniformity seemed to J. S. Mill to
have so con
trolling an influence upon inductions, that
he has taken
it as the centre of his whole theory of the
subject.
Analogous considerations modify our
hypothetic infer-
ences. The sight of two or three words will
be sufficient
to convince me that a certain manuscript was
written by
myself, because I know a certain look is
peculiar to it.
So an analytical chemist, who wishes to know
whether a
solution contains gold, will be completely
satisfied if it
gives a precipitate of the purple of cassius
with chloride
of tin; because this proves that either gold
or some hith
erto unknown substance is present. These are
examples
of characteristic tests. Again, we may know
of a certain
person, that whatever opinions he holds he
carries out
with uncompromising rigor to their utmost
logical con
sequences; then, -if we find his views bear
some of the
marks of any ultra school of thought, we
shall readily
conclude that he fully adheres to that
school.
There are thus four different kinds of
uniformity and
A THEORY OF PROBABLE
INFERENCE. 171
non-uniformity which may
influence our ampliative in-
ferences: -
1. The members of a class may present a
greater or
less general resemblance as regards a
certain line of char
acters.
2. A character may have a greater or less
tendency
to be present or absent throughout the whole
of whatever
classes of certain kinds.
3. A certain set of characters may be more
or less
intimately connected, so as to be probably
either present
or absent together in certain kinds of
objects.
4. An object may have more or less tendency
to
possess the whole of certain sets of
characters when it
possesses any of them.
A consideration of this sort may be so
strong as to
amount to demonstration of the conclusion.
In this case,
the inference is mere deduction, - that is,
the application
of a general rule already established. In
other cases, the
consideration of uniformities will not
wholly destroy the
inductive or hypothetic character of the
inference, but
will only strengthen or weaken it by the
addition of a
new argument of a deductive kind.
X.
We have thus seen how, in a general way, the
processes
of inductive and hypothetic inference are
able to afford
answers to our questions, though these may
relate to
matters beyond our immediate ken. In short,
a theory
of the logic of verification has been
sketched out. This
theory will have to meet the objections of
two opposing
schools of logic.
The first of these explains induction by
what is called
the doctrine of Inverse Probabilities, of
which the follow-
172 A THEORY OF PROBABLE
INFERENCE.
ing is an example : Suppose an
ancient denizen of the
Mediterranean coast, who had never heard of
the tides,
had wandered to the shore of the Atlantic
Ocean, and
there, on a certain number m of successive
days had
witnessed the rise of the sea. Then, says
Quetelet, he
would have been entitled to conclude that
there was a
probability equal to ((m+1)/(m+2)) that the
sea would rise on the next following day. 1
Putting m = 0, it is seen that
this view assumes that the probability of a
totally un-
known event is 1/2; or that of all theories
proposed for
examination one half are true. In point of
fact, we
know that although theories are not proposed
unless
they present some decided plausibility,
nothing like one
half turn out to be true. But to apply
correctly the
doctrine of inverse probabilities, it is
necessary to know
the antecedent probability of the event
whose proba-
bility is in question. Now, in pure
hypothesis or induc-
tion, we know nothing of the conclusion
antecedently
to the inference in hand. Mere ignorance,
however,
cannot advance us toward any knowledge;
therefore it
is impossible that the theory of inverse
probabilities
should rightly give a value for the
probability of a pure
inductive or hypothetic conclusion. For it
cannot do
this without assigning an antecedent
probability to this
conclusion; so that if this antecedent
probability rep-
resents mere ignorance (which never aids
us), it cannot
do it at all.
The principle which is usually assumed by
those who
seek to reduce inductive reasoning to a
problem in in
verse probabilities is, that if nothing
whatever is known
about the frequency of occurrence of an
event, then any
one frequency is as probable as any other.
But Boole
1 See Laplace, "Théorie
Analitique des Probabilités,"livre ii. chap.
vi.
A THEORY OF PROBABLE
INFERENCE. 173
has shown that there is no
reason whatever to prefer this
assumption, to saying that any one
"constitution of the
universe"is as probable as any other.
Suppose, for
instance, there were four possible occasions
upon which
an event might occur. Then there would be 16
"con-
stitutions of the universe," or possible
distributions of
occurrences and non-occurrences. They are
shown in
the following table, where Y stands
for an occurrence
and N for a non-occurrence.
It will be seen that different frequencies
result some
from more and some from fewer different
"constitutions
of the universe," so that it is a very
different thing to
assume that all frequencies are equally
probable from
what it is to assume that all constitutions
of the universe
are equally probable.
Boole says that one assumption is as good as
the other.
But I will go further, and say that the
assumption that
all constitutions of the universe are
equally probable is
far better than the assumption that all
frequencies are
equally probable. For the latter
proposition, though it
may be applied to any one unknown event,
cannot be
applied to all unknown events without
inconsistency.
Thus, suppose all frequencies of the event
whose occur-
rence is represented by Y in the
above table are equally
probable. Then consider the event which
consists in a
Y following a Y or an N
following an N. The possible
174 A THEORY OF PROBABLE
INFERENCE.
ways in which this event may
occur or not are shown in
the following table :
It will be found that assuming the different
frequencies
of the first event to be equally probable,
those of this new
event are not so, - the probability of three
occurrences
being half as large again as that of two, or
one. On the
other hand, if all constitutions of the
universe are equally
probable in the one case, they are so in the
other; and
this latter assumption, in regard to
perfectly unknown
events, never gives rise to any
inconsistency.
Suppose, then, that we adopt the assumption
that any
one constitution of the universe is as
probable as any
other; how will the inductive inference then
appear, con-
sidered as a problem in probabilities? The
answer is
extremely easy; 1 namely, the
occurrences or non-occur-
rences of an event in the past in no way
affect the proba-
bility of its occurrence in the future.
Boole frequently finds a problem in
probabilities to be
indeterminate. There are those to whom the
idea of an
unknown probability seems an absurdity.
Probability,
they say, measures the state of our
knowledge, and ig
norance is denoted by the probability 1/2.
But I appre-
hend that the expression "the probability of
an event "
is an incomplete one. A probability is a
fraction whose
1 See Boole, "Laws of
Thought."
A THEORY OF PROBABLE
INFERENCE. 175
numerator is the frequency of
a specific kind of event,
while its denominator is the frequency of a
genus embrac
ing that species. Now the expression in
question names
the numerator of the fraction, but omits to
name the de
nominator. There is a sense in which it is
true that the
probability of a perfectly unknown event is
one half;
namely, the assertion of its occurrence is
the answer to
a possible question answerable by "yes" or
"no," and
of all such questions just half the possible
answers are
true. But if attention be paid to the
denominators of
the fractions, it will be found that this
value of 1/2 is one
of which no possible use can be made in the
calculation
of probabilities.
The theory here proposed does not assign any
proba-
bility to the inductive or hypothetic
conclusion, in the
sense of undertaking to say how frequently that
conclu-
sion would be found true. It does not
propose to look
through all the possible universes, and say
in what pro
portion of them a certain uniformity occurs;
such a
proceeding, were it possible, would be quite
idle. The
theory here presented only says how
frequently, in this
universe, the special form of induction or
hypothesis
would lead us right. The probability given
by this theory
is in every way different in meaning,
numerical value,
and form from that of those who would apply
to am-
pliative inference the doctrine of inverse
chances.
Other logicians hold that if inductive and
hypothetic
premises lead to true oftener than to false
conclusions,
it is only because the universe happens to
have a certain
constitution. Mill and his followers
maintain that there
is a general tendency toward uniformity in
the universe,
as well as special uniformities such as
those which we
have considered. The Abbé Gratry believes
that the
tendency toward the truth in induction is
due to a mirac-
176 A THEORY OF PROBABLE
INFERENCE.
ulous intervention of Almighty
God, whereby we are led
to make such inductions as happen to be
true, and are
prevented from making those which are false.
Others
have supposed that there is a special
adaptation of the
mind to the universe, so that we are more
apt to make
true theories than we otherwise should be.
Now, to say
that a theory such as these is necessary
to explaining the
validity of induction and hypothesis is to
say that these
modes of inference are not in themselves
valid, but that
their conclusions are rendered probable by
being probable
deductive inferences from a suppressed (and
originally
unknown) premise. But I maintain that it has
been
shown that the modes of inference in
question are neces-
sarily valid, whatever the constitution of
the universe, so
long as it admits of the premises being
true. Yet I am
willing to concede, in order to concede as
much as possi
ble, that when a man draws instances at
random, all that
he knows is that he tries to follow
a certain precept; so
that the sampling process might be rendered
generally
fallacious by the existence of a mysterious
and malign
connection between the mind and the
universe, such that
the possession by an object of an unperceived
character
might influence the will toward choosing it
or rejecting
it. Such a circumstance would, however, be
as fatal to
deductive as to ampliative inference.
Suppose, for exam
ple, that I were to enter a great hall where
people were
playing rouge et noir at many
tables; and suppose that
I knew that the red and black were turned up
with equal
frequency. Then, if I were to make a large
number of
mental bets with myself, at this table and
at that I might,
by statistical deduction, expect to win
about half of them,
- precisely as I might expect, from the
results of these
samples, to infer by induction the probable
ratio of fre-
quency of the turnings of red and black in
the long run,
A THEORY OF PROBABLE
INFERENCE. 177
if I did not know it. But
could some devil look at each
card before it was turned, and then
influence me mentally
to bet upon it or to refrain therefrom, the
observed ratio
in the cases upon which I had bet might be
quite different
from the observed ratio in those cases upon
which I had
not bet. I grant, then, that even upon my
theory some
fact has to be supposed to make induction
and hypothe
sis valid processes; namely, it is supposed
that the su-
pernal powers withhold their hands and let
me alone,
and that no mysterious uniformity or
adaptation inter
feres with the action of chance. But then
this negative
fact supposed by my theory plays a totally
different part
from the facts supposed to be requisite by
the logicians
of whom I have been speaking. So far as
facts like those
they suppose can have any bearing, they
serve as major
premises from which the fact inferred by
induction or
hypothesis might be deduced; while the
negative fact
supposed by me is merely the denial of any
major premise
from which the falsity of the inductive or
hypothetic con
clusion could in general be deduced. Nor is
it necessary
to deny altogether the existence of
mysterious influences
adverse to the validity of the inductive and
hypothetic
processes. So long as their influence were
not too over-
whelming, the wonderful self-correcting
nature of the
ampliative inference would enable us, even
if they did
exist, to detect and make allowance for
them.
Although the universe need have no peculiar
consti-
tution to render ampliative inference valid,
yet it is worth
while to inquire whether or not it has such
a constitu-
tion; for if it has, that circumstance must
have its effect
upon all our inferences. It cannot any
longer be denied
that the human intellect is peculiarly
adapted to the
comprehension of the laws and facts of
nature, or at
least of some of them; and the effect of
this adaptation
178 A THEORY OF PROBABLE
INFERENCE.
upon our reasoning will be
briefly considered in the next
section. Of any miraculous interference by
the higher
powers, we know absolutely nothing; and it
seems in
the present state of science altogether
improbable. The
effect of a knowledge of special
uniformities upon ampli-
ative inferences has already been touched
upon. That
there is a general tendency toward
uniformity in nature
is not merely an unfounded, it is an
absolutely absurd,
idea in any other sense than that man is
adapted to his
surroundings. For the universe of marks is
only limited
by the limitation of human interests and
powers of ob
servation. Except for that limitation, every
lot of objects
in the universe would have (as I have
elsewhere shown)
some character in common and peculiar to it.
Conse-
quently, there is but one possible
arrangement of charac-
ters among objects as they exist, and there
is no room
for a greater or less degree of uniformity
in nature. If
nature seems highly uniform to us, it is
only because our
powers are adapted to our desires.
XI.
The questions discussed in
this essay relate to but a
small part of the Logic of Scientific
Investigation. Let
us just glance at a few of the others.
Suppose a being, from some remote part of
the uni
verse, where the conditions of existence are
inconceivably
different from ours, to be presented with a
United States
Census Report, - which is for us a mine of
valuable in-
ductions, so vast as almost to give that
epithet a new signi-
fication. He begins, perhaps, by comparing
the ratio of
indebtedness to deaths by consumption in
counties whose
names begin with the different letters of
the alphabet.
It is safe to say that he would find the
ratio everywhere
A THEORY OF PROBABLE
INFERENCE. 179
the same, and thus his inquiry
would lead to nothing.
For an induction is wholly unimportant
unless the pro-
portions of P's among the M's
and among the non-M's
differ; and a hypothetic inference is
unimportant unless
it be found that S has either a
greater or a less propor-
tion of the characters of M than it
has of other charac
ters. The stranger to this planet might go
on for some
time asking inductive questions that the
Census would
faithfully answer, without learning anything
except that
certain conditions were independent of
others. At length,
it might occur to him to compare the January
rain-fall
with the illiteracy. What he would find is
given in the
following table 1 :
180 A THEORY OF PROBABLE
INFERENCE.
He would infer that in places
that are drier in January
there is, not always but generally, less
illiteracy than
in wetter places. A detailed comparison
between Mr.
Schott's map of the winter rain-fall with
the map of
illiteracy in the general census, would
confirm the result
that these two conditions have a partial
connection.
This is a very good example of an induction
in which
the proportion of P's among the M's
is different, but
not very different, from the proportion
among the non-
M's. It is unsatisfactory; it
provokes further inquiry;
we desire to replace the M by some different
class, so
that the two proportions may be more widely
separated.
Now we, knowing as much as we do of the
effects of
winter rain-fall upon agriculture, upon
wealth, etc., and
of the causes of illiteracy, should come to
such an inquiry
furnished with a large number of appropriate
conceptions;
so that we should be able to ask intelligent
questions not
unlikely to furnish the desired key to the
problem. But
the strange being we have imagined could
only make his
inquiries hap-hazard, and could hardly hope
ever to find
the induction of which he was in search.
Nature is a far vaster and less clearly
arranged reper-
tory of facts than a census report; and if
men had not
come to it with special aptitudes for
guessing right, it
may well be doubted whether in the ten or
twenty thou-
sand years that they may have existed their
greatest
mind would have attained the amount of
knowledge
which is actually possessed by the lowest
idiot. But,
in point of fact, not man merely, but all
animals derive
by inheritance (presumably by natural
selection) two
classes of ideas which adapt them to their
environment.
In the first place, they all have from.
birth some notions,
however crude and concrete, of force,
matter, space, and
time; and, in the next place, they have some
notion of
A THEORY OF PROBABLE
INFERENCE. 181
what sort of objects their
fellow-beings are, and of how
they will act on given occasions. Our innate
mechanical
ideas were so nearly correct that they
needed but slight
correction. The fundamental principles of
statics were
made out by Archimedes. Centuries later
Galileo began
to understand the laws of dynamics, which in
our times
have been at length, perhaps, completely
mastered. The
other physical sciences are the results of
inquiry based
on guesses suggested by the ideas of
mechanics. The
moral sciences, so far as they can be called
sciences,
are equally developed out of our instinctive
ideas about
human nature. Man has thus far not attained
to any
knowledge that is not in a wide sense either
mechanical
or anthropological in its nature, and it may
be reasonably
presumed that he never will.
Side by side, then, with the well
established propo
sition that all knowledge is based on
experience, and
that science is only advanced by the
experimental verifi
cations of theories, we have to place this
other equally
important truth, that all human knowledge,
up to the
highest flights of science, is but the
development of our
inborn animal instincts.
NOTE A.
BOOLE, De Morgan, and their
followers, frequently
speak of a "limited universe of discourse
"in logic. An
unlimited universe would comprise the whole
realm of the
logically possible. In such a universe,
every universal
proposition, not tautologous, is false;
every particular
proposition, not absurd, is true. Our
discourse seldom
relates to this universe : we are either
thinking of the
physically possible, or of the historically
existent, or of
the world of some romance, or of some other
limited
universe.
But besides its universe of objects, our
discourse also
refers to a universe of characters. Thus, we
might
naturally say that virtue and an orange have
nothing
in common. It is true that the English word
for each
is spelt with six letters, but this is not
one of the marks
of the universe of our discourse.
A universe of things is unlimited in which
every com
bination of characters, short of the whole
universe of
characters, occurs in some object. In like
manner, the
universe of characters is unlimited in case
every aggre
gate of things short of the whole universe
of things
possesses in common one of the characters of
the uni
verse of characters. The conception of
ordinar}^ syllo
gistic is so unclear that it would hardly be
accurate to
say that it supposes an unlimited universe
of characters;
ON A LIMITED UNIVERSE OF
MARKS. 183
but it comes nearer to that
than to any other consistent
view. The non-possession of any character is
regarded
as implying the possession of another
character the nega
tive of the first.
In our ordinary discourse, on the other
hand, not only
are both universes limited, but, further
than that, we
have nothing to do with individual objects
nor simple
marks; so that we have simply the two
distinct universes
of things and marks related to one another,
in general, in
a perfectly indeterminate manner. The
consequence is, 4
that a proposition concerning the relations
of two groups
of marks is not necessarily equivalent to
any proposition
concerning classes of things; so that the
distinction
between propositions in extension and
propositions in
comprehension is a real one, separating two
kinds of
facts, whereas in the view of ordinary
syllogistic the
distinction only relates to two modes of
considering any
fact. To say that every object of the class
S is included
among the class of P's, of course
must imply that every
common character of the P's is a
common character of
the S's. But the converse
implication is by no means
necessary, except with an unlimited universe
of marks.
The reasonings in depth of which I have
spoken, suppose,
of course, the absence of any general
regularity about the
relations of marks and things.
I may mention here another respect in which
this view
differs from that of ordinary logic,
although it is a point
which has, so far as I am aware, no bearing
upon the
theory of probable inference. It is that
under this view
there are propositions of which the subject
is a class of
things, while the predicate is a group of
marks. Of such
propositions there are twelve species,
distinct from one
another in the sense that any fact capable
of being ex
pressed by a proposition of one of these
species cannot
184 ON A LIMITED UNIVERSE OF
MARKS.
be expressed by any
proposition of another species. The
following are examples of six of the twelve
species :
The remaining six species of propositions
are like the
above, except that they speak of objects wanting
charac-
ters instead of possessing
characters.
But the varieties of proposition do not end
here; for
we may have, for example, such a form as
this : "Some
object of the class S possesses
every character not want
ing to any object of the class P." In
short, the relative
term "possessing as a character," or its
negative, may
enter into the proposition any number of
times. We
may term this number the order of the
proposition.
An important characteristic of this kind of
logic is the
part that immediate inference plays in it.
Thus, the
proposition numbered 3, above, follows from
No. 2, and
No. 5 from No. 4. It will be observed that
in both cases
a universal proposition (or one that states
the non-
existence of something) follows from a
particular propo-
sition (or one that states the existence of
something).
All the immediate inferences are essentially
of that
nature. A particular proposition is never
immediately
inferable from a universal one. (It is true
that from
ON A LIMITED UNIVERSE OF
MARKS. 185
"no A exists" we can
infer that "something not A
exists;" but this is not properly an
immediate infer-
ence, it really supposes the additional
premise that
"something exists.") There are also
immediate in-
ferences raising and reducing the order
of propositions.
Thus, the proposition of the second order
given in the
last paragraph follows from "some S is a P."
On the
other hand, the inference holds,
The necessary and sufficient condition of
the existence
of a syllogistic conclusion from two
premises is simple
enough. There is a conclusion if, and only
if, there is
a middle term distributed in one premise and
undistribu
ted in the other. But the conclusion is of
the kind called
spurious 1 by De Morgan if, and
only if, the middle term
is affected by a "some" in both premises.
For exam-
ple, let the two premises be,
The middle term μ is distributed in the
second premise,
but not in the first; so that a conclusion
can be drawn.
But, though both propositions are universal,
μ is under
a "some" in both; hence only a spurious
conclusion
can be drawn, and in point of fact we can
infer both of
the following :
186 ON A LIMITED UNIVERSE OF
MARKS.
Every object of the class S
wants a character other than
some character common to the class P;
Every object of the class P
possesses a character other
than some character wanting to every object
of the class S.
The order of the conclusion is always the
sum of the
orders of the premises; but to draw up a
rule to deter
mine precisely what the conclusion is, would
be difficult.
It would at the same time be useless,
because the prob
lem is extremely simple when considered in
the light of
the logic of relatives.
NOTE B.
A DUAL relative term, such as
"lover," "benefactor,"
"servant," is a common name signifying a
pair of ob-
jects. Of the two members of the pair, a
determinate
one is generally the first, and the other
the second; so
that if the order is reversed, the pair is
not considered as
remaining the same.
Let A, B, C, D, etc., be all the individual
objects in
the universe; then all the individual pairs
may be arrayed
in a block, thus :
A general relative may be conceived as a
logical aggre-
gate of a number of such individual
relatives. Let l de-
note "lover;" then we may write
where (l)ij is a numerical
coefficient, whose value is 1 in
case I is a lover of J, and
0 in the opposite case, and
where the sums are to be taken for all
individuals in the
universe.
. It is defined by
the 
