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.