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 *M*s 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.