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Sep 1, 2011           

C.S. Peirce: A theory of probable inference


Having mentioned William James in the previous Nederlog, I had reason to search the internet to see what there is available by his friend Charles Sanders Peirce, a much greater philosopher than James was (as James very probably would have admitted), and found this:

It consists of seven essays, the last one by Peirce, the others not, and also two appendices to Peirce's essay, called resp. A Theory of Probable Inference, namely Note A and Note B, with latter an outline of Peirce's theory of relations (or relatives, as he said).

The link is to the pdf-copy of the original by Internet Archive.org and the associated txt-version, both of which have been used to produce the following html-version of Peirce's essay, and of Note A and Note B.

Peirce's essay is important in being the first version of statistical hypotheses testing, later rediscovered by Sir Ronald Fisher, and Note B is important in being the first worked out mathematical logic of relations, later rediscovered by Frege.

The following is mostly html, but I have at various places inserted copies of parts of the pdf-file, namely where it would be quite difficult to render formulas as html.

I reproduce it here because (1) I wanted some of Peirce's texts on my site for a long time, holding him to be one of the greatest philosophers and logicians I know of, and because (2) I was much impressed by Note B, which is an outline of Peirce's logic of relations, that I first read between 1970 and 1980. (Readers wholly naive to logic and Peirce, will vey probably find it mostly incomprehensible, but it is really great stuff.)

And besides this, the main text,  A Theory of Probable Inference is an excellent explanation of the main ideas involved in the statistical testing of hypotheses.

There are some more references at the end, and also a warning about the present edition:




THE following is an example of the simplest kind of
probable inference :

About two per cent of persons wounded in the liver recover;
This man has been wounded in the liver;
Therefore, there are two chances out of a hundred that he will recover.

Compare this with the simplest of syllogisms, say the
following :

Every man dies;
Enoch was a man;
Hence, Enoch must have died.

The latter argument consists in the application of a
general rule to a particular case. The former applies to
a particular case a rule not absolutely universal, but sub-
ject to a known proportion of exceptions. Both may
alike be termed deductions, because they bring informa-
tion about the uniform or usual course of things to bear
upon the solution of special questions; and the probable
argument may approximate indefinitely to demonstration
as the ratio named in the first premise approaches to
unity or to zero.


Let us set forth the general formula of the two kinds
of inference in the manner of formal logic.

It is to be observed that the ratio ρ need not be exactly
specified. We may reason from the premise that not
more than two per cent of persons wounded in the liver
recover, or from "not less than a certain proportion of
the M's are P's,"or from "no very large nor very
small proportion, etc." In short, ρ is subject to every
kind of indeterminacy; it simply excludes some ratios
and admits the possibility of the rest.

The analogy between syllogism and what is here called
probable deduction is certainly genuine and important;
yet how wide the differences between the two modes of
inference are, will appear from the following considera-

1. The logic of probability is related to ordinary syllo-
gistic as the quantitative to the qualitative branch of the
same science. Necessary syllogism recognizes only the
inclusion or non-inclusion of one class under another;
but probable inference takes account of the proportion


of one class which is contained under a second. It is
like the distinction between projective geometry, which
asks whether points coincide or not, and metric geome-
try, which determines their distances.

2. For the existence of ordinary syllogism, all that is
requisite is that we should be able to say, in some sense,
that one term is contained in another, or that one object
stands to a second in one of those relations : "better
than," "equivalent to," etc., which are termed transitive
because if A is in any such relation to B, and B is in
the same relation to C, then A is in that relation to C.
The universe might be all so fluid and variable that
nothing should preserve its individual identity, and that
no measurement should be conceivable; and still one
portion might remain inclosed within a second, itself
inclosed within a third, so that a syllogism would be
possible. But probable inference could not be made in
such a universe, because no signification would attach to
the words "quantitative ratio." For that there must be
counting; and consequently units must exist, preserving
their identity and variously grouped together.

3. A cardinal distinction between the two kinds of
inference is, that in demonstrative reasoning the con-
clusion follows from the existence of the objective facts
laid down in the premises; while in probable reasoning
these facts in themselves do not even render the con-
clusion probable, but account has to be taken of various
subjective circumstances, of the manner in which the
premises have been obtained, of there being no counter-
vailing considerations, etc.; in short, good faith and hon-
esty are essential to good logic in probable reasoning.

When the partial rule that the proposition ρ of the
M's are P's is applied to show with probability ρ that
S is a P, it is requisite, not merely that S should be an


M, but also that it should be an instance drawn at ran-
from among the M's. Thus, there being four aces
in a picquet pack of thirty-two cards, the chance is one
eighth that a given card not looked at is an ace; but
this is only on the supposition that the card has been
drawn at random from the whole pack. If, for instance,
it had been drawn from the cards discarded by the
players at piquet or euchre, the probability would be
quite different. The instance must be drawn at ran-
dom. Here is a maxim of conduct. The volition of
the reasoner (using what machinery it may) has to
choose S so that it shall be an M; but he ought to
restrain himself from all further preference, and not
allow his will to act in any way that might tend to
settle what particular M is taken, but should leave that
to the operation of chance. Willing and wishing, like
other operations of the mind, are general and imperfectly
determinate. I wish for a horse, for some particular
kind of horse perhaps, but not usually for any individual
one. I will to act in a way of which I have a general
conception; but so long as my action conforms to that
general description, how it is further determined I do
not care. Now in choosing the instance S, the general
intention (including the whole plan of action) should
be to select an M, but beyond that there should be no
preference; and the act of choice should be such that if
it were repeated many enough times with the same in
tention, the result would be that among the totality of
selections the different sorts of M's would occur with
the same relative frequencies as in experiences in which
volition does not intermeddle at all. In cases in which
it is found difficult thus to restrain the will by a direct
effort, the apparatus of games of chance, - a lottery-
wheel, a roulette, cards, or dice, - may be called to our


aid. Usually, however, in making a simple probable
deduction, we take that instance in which we happen at
the time to be interested. In such a case, it is our
interest that fulfils the function of an apparatus for
random selection; and no better need be desired, so
long as we have reason to deem the premise "the pro-
portion ρ of the M's are P's" to be equally true in
regard to that part of the M s which are alone likely
ever to excite our interest.

Nor is it a matter of indifference in what manner the
other premise has been obtained. A card being drawn
at random from a picquet pack, the chance is one-eighth
that it is an ace, if we have no other knowledge of it.
But after we have looked at the card, we can no longer
reason in that way. That the conclusion must be drawn
in advance of any other knowledge on the subject is
a rule that, however elementary, will be found in the
sequel to have great importance.

4. The conclusions of the two modes of inference like
wise differ. One is necessary; the other only probable.
Locke, in the "Essay concerning Human Understanding,"
hints at the correct analysis of the nature of probability.
After remarking that the mathematician positively knows
that the sum of the three angles of a triangle is equal to
two right angles because he apprehends the geometrical
proof, he then continues : "But another man who never
took the pains to observe the demonstration, hearing a
mathematician, a man of credit, affirm the three angles
of a triangle to be equal to two right ones, assents to it;
that is, receives it for true. In which case, the founda-
tion of his assent is the probability of the thing, the proof
being such as, for the most part, carries truth with it;
the man on whose testimony he receives it not being wont
to affirm anything contrary to or besides his knowledge,


especially in matters of this kind." Those who know
Locke are accustomed to look for more meaning in his
words than appears at first glance. There is an allusion
in this passage to the fact that a probable argument is
always regarded as belonging to a genus of arguments.
This is, in fact, true of any kind of argument. For the
belief expressed by the conclusion is determined or caused
by the belief expressed by the premises. There is, there-
fore, some general rule according to which the one suc-
ceeds the other. But, further, the reasoner is conscious
of there being such a rule, for otherwise he would not
know he was reasoning, and could exercise no attention
or control; and to such an involuntary operation the
name reasoning is very properly not applied. In all
cases, then, we are conscious that our inference belongs
to a general class of logical forms, although we are not
necessarily able to describe the general class. The dif-
ference between necessary and probable reasoning is that
in the one case we conceive that such facts as are ex-
pressed by the premises are never, in the whole range of
possibility, true, without another fact, related to them as
our conclusion is to our premises, being true likewise;
while in the other case we merely conceive that, in rea
soning as we do, we are following a general maxim that
will usually lead us to the truth.

So long as there are exceptions to the rule that all
men wounded in the liver die, it does not necessarily
follow that because a given man is wounded in the liver
he cannot recover. Still, we know that if we were to
reason in that way, we should be following a mode of
inference which would only lead us wrong, in the long
run, once in fifty times; and this is what we mean when
we say that the probability is one out of fifty that the
man will recover. To say, then, that a proposition has



the probability p means that to infer it to be true would
be to follow an argument such as would carry truth with
it in the ratio of frequency p.

It is plainly useful that we should have a stronger
feeling of confidence about a sort of inference which will
oftener lead us to the truth than about an inference that
will less often prove right, and such a sensation we do
have. The celebrated law of Fechner is, that as the
force acting upon an organ of sense increases in geo-
metrical progression, the intensity of the sensation in
creases in arithmetical progression. In this case the
odds (that is, the ratio of the chances in favor of a
conclusion to the chances against it) take the place of
the exciting cause, while the sensation itself is the feel
ing of confidence. When two arguments tend to the
same conclusion, our confidence in the latter is equal to
the sum of what the two arguments separately would
produce; the odds are the product of the odds in favor
of the two arguments separately. When the value of the
odds reduces to unity, our confidence is null; when the
odds are less than unity, we have more or less confidence
in the negative of the conclusion.


The principle of probable deduction still applies when
S, instead of being a single M, is a set of M's, n
in number. The reasoning then takes the following
form :



In saying that S, S', S", etc. form a set drawn at ran-
dom, we here mean that not only are the different in
dividuals drawn at random, but also that they are so
drawn that the qualities which may belong to one have
no influence upon the selection of any other. In other
words, the individual drawings are independent, and the
set as a whole is taken at random from among all possi
ble sets of n M's. In strictness, this supposes that the
same individual may be drawn several times in the same
set, although if the number of M's is large compared
with n, it makes no appreciable difference whether this
is the case or not.

The following formula expresses the proportion, among
all sets of n M's, of those which consist of m P's and
n-m not-P's. The letter r denotes the proportion of
P's among the M's, and the sign of admiration is used
to express the continued product of all integer numbers
from 1 to the number after which it is placed. Thus,
4 ! = 1 . 2 . 3 . 4 = 24, etc. The formula is

As an example, let us assume the proportion r = 2/3
and the number of M's in a set n = 15. Then the
values of the probability q for different numbers, m, of
P's, are fractions having for their common denominator
14,348,907, and for their numerators as follows:-


A very little mathematics would suffice to show that,
r and n being fixed, q always reaches its maximum value
with that value of m that is next less than (n + 1)r, * and
that q is very small unless m has nearly this value.

Upon these facts is based another form of inference to
which I give the name of statistical deduction. Its gen
eral formula is as follows :

As an example, take this :

A little more than half of all human births are males;
Hence, probably a little over half of all the births in New
York during any one year are males.

We have now no longer to deal with a mere probable
inference, but with a probable approximate inference.

In case (n+1)r is a whole number, q has equal values for m =
(n+1)r and for m = (n+1)r - 1.



This conception is a somewhat complicated one, meaning
that the probability is greater according as the limits of
approximation are wider, conformably to the mathemati
cal expression for the values of q.

This conclusion has no meaning at all unless there be
more than one instance; and it has hardly any meaning
unless the instances are somewhat numerous. When
this is the case, there is a more convenient way of ob
taining (not exactly, but quite near enough for all practi
cal purposes) either a single value of q or the sum of
successive values from m = m1 to m = m2 inclusive. The
rule is first to calculate two quantities which may con
veniently be called t1 and t2 according to these form-
ulae :

where m2 > m1. Either or both the quantities  t1 and t2
may be negative. Next with each of these quantities
enter the table below, and take out 1/2 Θ1 and 1/2 Θ2 and
give each the same sign as the t from which it is derived



In rough calculations we may take Θt equal to t for t
less than 0.7, and as equal to unity for any value above
t = 1.4. 

The principle of statistical deduction is that these two
proportions, namely, that of the P's among the M's,
and that of the P's among the S's, are probably and
approximately equal. If, then, this principle justifies our
inferring the value of the second proportion from the
known value of the first, it equally justifies our inferring
the value of the first from that of the second, if the first



is unknown but the second has been observed. We
thus obtain the following form of inference :

The following are examples. From a bag of coffee a
handful is taken out, and found to have nine tenths of
the beans perfect; whence it is inferred that about nine-
tenths of all the beans in the bag are probably perfect.
The United States Census of 1870 shows that of native
white children under one year old, there were 478,774
males to 463,320 females; while of colored children of
the same age there were 75,985 males to 76,637 females.
We infer that generally there is a larger proportion of
female births among negroes than among whites.

When the ratio ρ is unity or zero, the inference is an
ordinary induction; and I ask leave to extend the term
induction to all such inference, whatever be the value of
ρ. It is, in fact, inferring from a sample to the whole
lot sampled. These two forms of inference, statistical
deduction and induction, plainly depend upon the same
principle of equality of ratios, so that their validity is the
same. Yet the nature of the probability in the two cases
is very different. In the statistical deduction, we know
that among the whole body of M s the proportion of P s
is ρ; we say, then, that the S's being random drawings



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


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 :



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-



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



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



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.


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


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.



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
, January, 1878.


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


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

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.


We now come to the consideration of the Rules which
have to be followed in order to make valid and strong


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



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 :



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:-


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.


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.



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

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


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

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



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



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



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-
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.



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


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-



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: -


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



of the inductive character remains, as this name im-


Take any human being, at random, - say Queen Eliz-
abeth. Now a little more than half of all the human
beings who have ever existed have been males; but it
does not follow that it is a little more likely than not
that Queen Elizabeth was a male, since we know she was
a woman. Nor, if we had selected Julius Caesar, would
it be only a little more likely than not that he was a
male. It is true that if we were to go on drawing at
random an indefinite number of instances of human be
ings, a slight excess over one-half would be males. But
that which constitutes the probability of an inference is
the proportion of true conclusions among all those which
could be derived from the same precept. Now a precept
of inference, being a rule which the mind is to follow,
changes its character and becomes different when the
case presented to the mind is essentially different. When,
knowing that the proportion r of all M's are P's, I draw
an instance, S, of an M, without any other knowledge of
whether it is a P or not, and infer with probability, r,
that it is P, the case presented to my mind is very
different from what it is if I have such other knowledge.
In short, I cannot make a valid probable inference with
out taking into account whatever knowledge I have (or,
at least, whatever occurs to my mind) that bears upon
the question.

The same principle may be applied to the statistical
deduction of Form IV. If the major premise, that the
proportion r of the M's are P's, be laid down first,
before the instances of Ms are drawn, we really draw our
inference concerning those instances (that the proper-



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



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",
, 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

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

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-



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



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



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



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

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-



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-



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


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



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



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

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.


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-



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.



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


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."



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-
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-



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,



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



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.


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


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 :




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



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.





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

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;



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



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



"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 :


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.



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



Every relative term has a negative (like any other
term) which may be represented by drawing a straight
line over the sign for the relative itself. The negative
of a relative includes every pair that the latter excludes,
and vice versa. Every relative has also a converse, pro-
duced by reversing the order of the members of the pair.
Thus, the converse of u "lover" is "loved." The con-
verse may be represented by drawing a curved line over
the sign for the relative, thus: . It is defined by the

The following formulae are obvious, but important :

Relative terms can be aggregated and compounded like
others. Using + for the sign of logical aggregation, and
the comma for the sign of logical composition (Boole's
multiplication, here to be called non-relative or internal
multiplication), we have the definitions

The first of these equations, however, is to be understood
in a peculiar way : namely, the + in the second member
is not strictly addition, but an operation by which

Instead of (l)ij+ (b)ij , we might with more accuracy



The main formulas of aggregation and composition are

The subsidiary formulas need not be given, being the
same as in non-relative logic.

We now come to the combination of relatives. Of
these, we denote two by special symbols; namely, we


The former is called a particular combination, because
it implies the existence of something loved by its relate
and a benefactor of its correlate. The second combina
tion is said to be universal, because it implies the non-
existence of anything except what is either loved by its
relate or a benefactor of its correlate. The combination



Relative addition and multiplication are subject to the
associative law. That is,

Two formulae so constantly used that hardly anything
can be done without them are

The former asserts that whatever is lover of an object
that is benefactor of everything but a servant, stands to
everything but servants in the relation of lover of a
benefactor. The latter asserts that whatever stands to
any servant in the relation of lover of everything but its
benefactors, is a lover of everything but benefactors of
servants. The following formulas are obvious and triv

Unobvious and important, however, are these :

There are a number of curious development formulae.
Such are

The summations and multiplications denoted by ^ and IT
are to be taken non-relatively, and all relative terms are
to be successively substituted for p.


The negatives of the combinations follow these rules :

The converses of combinations are as follows :

Individual dual relatives are of two types,

Relatives containing no pair of an object with itself are
called alio-relatives as opposed to self-relatives. The
negatives of alio-relatives pair every object with itself.
Relatives containing no pair of an object with anything
but itself are called concurrents as opposed to opponents.
The negatives of concurrents pair every object with every

There is but one relative which pairs every object with
itself and with every other. It is the aggregate of all
pairs, and is denoted by ∞. It is translated into ordi-
nary language by "coexistent with." Its negative is 0.
There is but one relative which pairs every object with
itself and none with any other. It is

is denoted by 1, and in ordinary language is "identical
with - ." Its negative, denoted by n, is "other than,"
or "not."

No matter what relative term x may be, we have




and that

The effect of these peculiarities is that this algebra can
not be subjected to hard and fast rules like those of
the Boolian calculus; and all that can be done in this
place is to give a general idea of the way of working with
it. The student must at the outset disabuse himself of
the notion that the chief instruments of algebra are the
inverse operations. General algebra hardly knows any
inverse operations. When an inverse operation is iden
tical with a direct operation with an inverse quantity
(as subtraction is the addition of the negative, and as
division is multiplication by the reciprocal), it is useful;
otherwise it is almost always useless. In ordinary alge
bra, we speak of the "principal value" of the logarithm,
etc., which is a direct operation substituted for an in-
definitely ambiguous inverse operation. The elimination
and transposition in this algebra really does depend,
however, upon formulae quite analogous to the

of arithmetical algebra. These formulas are

For example, to eliminate s from the two propositions

we relatively multiply them in such an order as to bring
the two s s together, and then apply the second of the
above formulas, thus :















For more
on Peirce see

I have done the best I could to make the present edition correspond to the original, but it probably contains a few typos, and you should also realize that the whole thing in this edition is a html-textfile of 229 KB plus 88 jpg files. (The latter have a yellow background in the above and have been
included to avoid a lot of hassle trying to put mathematics into html.)

For the few enlightened minds who deeply care for probability and logic, this should be a treat if it so far was unknown, while it should also explain, by implication, why I have such a high opinion of Peirce.

P.S. Corrections, if any are necessary, have to be made later.
-- Sep 2, 2011: Corrected some typos and some scanner errors and also inserted some links at the beginning.

As to ME/CFS (that I prefer to call ME):

1.  Anthony Komaroff Ten discoveries about the biology of CFS (pdf)
3.  Hillary Johnson The Why
4.  Consensus of M.D.s Canadian Consensus Government Report on ME (pdf)
5.   Eleanor Stein Clinical Guidelines for Psychiatrists (pdf)
6.  William Clifford The Ethics of Belief
7.  Paul Lutus

Is Psychology a Science?

8.  Malcolm Hooper Magical Medicine (pdf)
 Maarten Maartensz
ME in Amsterdam - surviving in Amsterdam with ME (Dutch)
 Maarten Maartensz Myalgic Encephalomyelitis

Short descriptions of the above:                

1. Ten reasons why ME/CFS is a real disease by a professor of medicine of Harvard.
2. Long essay by a professor emeritus of medical chemistry about maltreatment of ME.
3. Explanation of what's happening around ME by an investigative journalist.
4. Report to Canadian Government on ME, by many medical experts.
5. Advice to psychiatrist by a psychiatrist who understands ME is an organic disease
6. English mathematical genius on one's responsibilities in the matter of one's beliefs:

7. A space- and computer-scientist takes a look at psychology.
8. Malcolm Hooper puts things together status 2010.
9. I tell my story of surviving (so far) in Amsterdam with ME.
10. The directory on my site about ME.

See also: ME -Documentation and ME - Resources
The last has many files, all on my site to keep them accessible.

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