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Title: The Logic of Chance, 3rd edition

Author: John Venn

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Most recently updated: February 15, 2019

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*** START OF THE PROJECT GUTENBERG EBOOK THE LOGIC OF
CHANCE, 3RD EDITION ***

THE

LOGIC OF CHANCE
AN ESSAY
ON THE FOUNDATIONS AND PROVINCE OF
THE THEORY OF PROBABILITY,

WITH ESPECIAL REFERENCE TO ITS LOGICAL BEARINGS
AND ITS APPLICATION TO

MORAL AND SOCIAL SCIENCE AND TO STATISTICS,

BY
JOHN VENN, Sc.D., F.R.S.,
FELLOW AND LECTURER IN THE MORAL SCIENCES, GONVILLE AND CAIUS COLLEGE,
CAMBRIDGE.

LATE EXAMINER IN LOGIC AND MORAL PHILOSOPHY IN THE
UNIVERSITY OF LONDON.

“So careful of the type she seems
So careless of the single life.”

THIRD EDITION, RE-WRITTEN AND ENLARGED.

London:
MACMILLAN AND CO.
AND NEW YORK
1888

[All Rights reserved.]

First Edition printed 1866.
Second Edition 1876.
Third Edition 1888.

PREFACE TO FIRST EDITION.
Any work on Probability by a Cambridge man will be so likely to have its
scope and its general treatment of the subject prejudged, that it may be well
to state at the outset that the following Essay is in no sense mathematical.
Not only, to quote a common but often delusive assurance, will ‘no
knowledge of mathematics beyond the simple rules of Arithmetic’ be
required to understand these pages, but it is not intended that any such
knowledge should be acquired by the process of reading them. Of the two
or three occasions on which algebraical formulæ occur they will not be
found to form any essential part of the text.

The science of Probability occupies at present a somewhat anomalous
position. It is impossible, I think, not to observe in it some of the marks and
consequent disadvantages of a sectional study. By a small body of ardent
students it has been cultivated with great assiduity, and the results they have
obtained will always be reckoned among the most extraordinary products of
mathematical genius. But by the general body of thinking men its principles
seem to be regarded with indifference or suspicion. Such persons may
admire the ingenuity displayed, and be struck with the profundity of many
of the calculations, but there seems to them, if I may so express it, an
unreality about the whole treatment of the subject. To many persons the
mention of Probability suggests little else than the notion of a set of rules,
very ingenious and profound rules no doubt, with which mathematicians
amuse themselves by setting and solving puzzles.

It must be admitted that some ground has been given for such an opinion.
The examples commonly selected by writers on the subject, though very
well adapted to illustrate its rules, are for the most part of a special and
peculiar character, such as those relating to dice and cards. When they have
searched for illustrations drawn from the practical business of life, they
have very generally, but unfortunately, hit upon just the sort of instances
which, as I shall endeavour to show hereafter, are among the very worst that

could be chosen for the purpose. It is scarcely possible for any unprejudiced
person to read what has been written about the credibility of witnesses by
eminent writers, without his experiencing an invincible distrust of the
principles which they adopt. To say that the rules of evidence sometimes
given by such writers are broken in practice, would scarcely be correct; for
the rules are of such a kind as generally to defy any attempt to appeal to
them in practice.

This supposed want of harmony between Probability and other branches of
Philosophy is perfectly erroneous. It arises from the belief that Probability
is a branch of mathematics trying to intrude itself on to ground which does
not altogether belong to it. I shall endeavour to show that this belief is
unfounded. To answer correctly the sort of questions to which the science
introduces us does generally demand some knowledge of mathematics,
often a great knowledge, but the discussion of the fundamental principles
on which the rules are based does not necessarily require any such
qualification. Questions might arise in other sciences, in Geology, for
example, which could only be answered by the aid of arithmetical
calculations. In such a case any one would admit that the arithmetic was
extraneous and accidental. However many questions of this kind there
might be here, those persons who do not care to work out special results for
themselves might still have an accurate knowledge of the principles of the
science, and even considerable acquaintance with the details of it. The same
holds true in Probability; its connection with mathematics, though certainly
far closer than that of most other sciences, is still of much the same kind. It
is principally when we wish to work out results for ourselves that
mathematical knowledge is required; without such knowledge the student
may still have a firm grasp of the principles and even see his way to many
of the derivative results.

The opinion that Probability, instead of being a branch of the general
science of evidence which happens to make much use of mathematics, is a
portion of mathematics, erroneous as it is, has yet been very
disadvantageous to the science in several ways. Students of Philosophy in
general have thence conceived a prejudice against Probability, which has
for the most part deterred them from examining it. As soon as a subject
comes to be considered ‘mathematical’ its claims seem generally, by the

mass of readers, to be either on the one hand scouted or at least courteously
rejected, or on the other to be blindly accepted with all their assumed
consequences. Of impartial and liberal criticism it obtains little or nothing.

The consequences of this state of things have been, I think, disastrous to the
students themselves of Probability. No science can safely be abandoned
entirely to its own devotees. Its details of course can only be studied by
those who make it their special occupation, but its general principles are
sure to be cramped if it is not exposed occasionally to the free criticism of
those whose main culture has been of a more general character. Probability
has been very much abandoned to mathematicians, who as mathematicians
have generally been unwilling to treat it thoroughly. They have worked out
its results, it is true, with wonderful acuteness, and the greatest ingenuity
has been shown in solving various problems that arose, and deducing
subordinate rules. And this was all that they could in fairness be expected to
do. Any subject which has been discussed by such men as Laplace and
Poisson, and on which they have exhausted all their powers of analysis,
could not fail to be profoundly treated, so far as it fell within their province.
But from this province the real principles of the science have generally been
excluded, or so meagrely discussed that they had better have been omitted
altogether. Treating the subject as mathematicians such writers have
naturally taken it up at the point where their mathematics would best come
into play, and that of course has not been at the foundations. In the works of
most writers upon the subject we should search in vain for anything like a
critical discussion of the fundamental principles upon which its rules rest,
the class of enquiries to which it is most properly applicable, or the relation
it bears to Logic and the general rules of inductive evidence.

This want of precision as to ultimate principles is perfectly compatible here,
as it is in the departments of Morals and Politics, with a general agreement
on processes and results. But it is, to say the least, unphilosophical, and
denotes a state of things in which positive error is always liable to arise
whenever the process of controversy forces us to appeal to the foundations
of the science.

With regard to the remarks in the last few paragraphs, prominent exceptions
must be made in the case of two recent works at least.[1] The first of these is

Professor de Morgan's Formal Logic. He has there given an investigation
into the foundations of Probability as conceived by him, and nothing can be
more complete and precise than his statement of principles, and his
deductions from them. If I could at all agree with these principles there
would have been no necessity for the following essay, as I could not hope to
add anything to their foundation, and should be far indeed from rivalling his
lucid statement of them. But in his scheme Probability is regarded very
much from the Conceptualist point of view; as stated in the preface, he
considers that Probability is concerned with formal inferences in which the
premises are entertained with a conviction short of absolute certainty. With
this view I cannot agree. As I have entered into criticism of some points of
his scheme in one of the following chapters, and shall have occasion
frequently to refer to his work, I need say no more about it here. The other
work to which I refer is the profound Laws of Thought of the late Professor
Boole, to which somewhat similar remarks may in part be applied. Owing
however to his peculiar treatment of the subject, I have scarcely anywhere
come into contact with any of his expressed opinions.

The view of the province of Probability adopted in this Essay differs so
radically from that of most other writers on the subject, and especially from
that of those just referred to, that I have thought it better, as regards details,
to avoid all criticism of the opinions of others, except where conflict was
unavoidable. With regard to that radical difference itself Bacon's remark
applies, behind which I must shelter myself from any change of
presumption.—“Quod ad universalem istam reprehensionem attinet,
certissimum vere est rem reputanti, eam et magis probabilem esse et magis
modestam, quam si facta fuisset ex parte.”

Almost the only writer who seems to me to have expressed a just view of
the nature and foundation of the rules of Probability is Mr Mill, in his
System of Logic.[2] His treatment of the subject is however very brief, and a
considerable portion of the space which he has devoted to it is occupied by
the discussion of one or two special examples. There are moreover some
errors, as it seems to me, in what he has written, which will be referred to in
some of the following chapters.

The reference to the work just mentioned will serve to convey a general
idea of the view of Probability adopted in this Essay. With what may be
called the Material view of Logic as opposed to the Formal or
Conceptualist,—with that which regards it as taking cognisance of laws of
things and not of the laws of our own minds in thinking about things,—I am
in entire accordance. Of the province of Logic, regarded from this point of
view, and under its widest aspect, Probability may, in my opinion, be
considered to be a portion. The principal objects of this Essay are to
ascertain how great a portion it comprises, where we are to draw the
boundary between it and the contiguous branches of the general science of
evidence, what are the ultimate foundations upon which its rules rest, what
the nature of the evidence they are capable of affording, and to what class
of subjects they may most fitly be applied. That the science of Probability,
on this view of it, contains something more important than the results of a
system of mathematical assumptions, is obvious. I am convinced moreover
that it can and ought to be rendered both interesting and intelligible to
ordinary readers who have any taste for philosophy. In other words, if the
large and growing body of readers who can find pleasure in the study of
books like Mill's Logic and Whewell's Inductive Sciences, turn with
aversion from a work on Probability, the cause in the latter case must lie
either in the view of the subject or in the manner and style of the book.

I take this opportunity of thanking several friends, amongst whom I must
especially mention Mr Todhunter, of St John's College, and Mr
H. Sidgwick, of Trinity College, for the trouble they have kindly taken in
looking over the proof-sheets, whilst this work was passing through the
Press. To the former in particular my thanks are due for thus adding to the
obligations which I, as an old pupil, already owed him, by taking an amount
of trouble, in making suggestions and corrections for the benefit of another,
which few would care to take for anything but a work of their own. His
extensive knowledge of the subject, and his extremely accurate judgment,
render the service he has thus afforded me of the greatest possible value.

Gonville and Caius College,
September, 1866.
1
I am here speaking, of course, of those only who have expressly treated of the foundations of the
science. Mr Todhunter's admirable work on the History of the Theory of Probability being, as the

name denotes, mainly historical, such enquiries have not directly fallen within his province.
2
This remark, and that at the commencement of the last paragraph, having been misunderstood, I
ought to say that the only sense in which originality is claimed for this Essay is in the thorough
working out of the Material view of Logic as applied to Probability. I have given a pretty full
discussion of the general principles of this view in the tenth chapter, and have there pointed out some
of the peculiarities to which it leads.

PREFACE TO SECOND EDITION.
The principal reason for designating this volume a second edition consists
in the fact that the greater portion of what may be termed the first edition is
incorporated into it. Besides various omissions (principally where the
former treatment has since seemed to me needlessly prolix), I have added
new matter, not much inferior in amount to the whole of the original work.
In addition, moreover, to these alterations in the matter, the general
arrangement of the subject as regards the successive chapters has been
completely changed; the former arrangement having been (as it now seems
to me) justly objected to as deficient and awkward in method.

After saying this, it ought to be explained whether any change of general
view or results will be found in the present treatment.

The general view of Probability adopted is quite unchanged, further reading
and reflection having only confirmed me in the conviction that this is the
soundest and most fruitful way of regarding the subject. It is the more
necessary to say this, as to a cursory reader it might seem otherwise; owing
to my having endeavoured to avoid the needlessly polemical tone which, as
is often the case with those who are making their first essay in writing upon
any subject, was doubtless too prominent in the former edition. I have not
thought it necessary, of course, except in one or two cases, to indicate
points of detail which it has seemed necessary to correct.

A number of new discussions have been introduced upon topics which were
but little or not at all treated before. The principal of these refer to the
nature and physical origin of Laws of Error (Ch. II.); the general view of
Logic, and consequently of Probability, termed the Material view, adopted
here (Ch. X.); a brief history and criticism of the various opinions held on
the subject of Modality (Ch. XII.); the logical principles underlying the
method of Least Squares (Ch. XIII.); and the practices of Insurance and
Gambling, so far as the principles involved in them are concerned
(Ch. XV.). The Chapter on the Credibility of Extraordinary Stories is also
mainly new; this was the portion of the former work which has since
seemed to me the least satisfactory, but owing to the extreme intricacy of
the subject I am far from feeling thoroughly satisfied with it even now.

I have again to thank several friends for the assistance they have so kindly
afforded. Amongst these I must prominently mention Mr C. J. Monro, late
fellow of Trinity. It is only the truth to say that I have derived more
assistance from his suggestions and criticisms than has been consciously
obtained from all other external sources together. Much of this criticism has
been given privately in letters, and notes on the proof-sheets; but one of the
most elaborate of his discussions of the subject was communicated to the
Cambridge Philosophical Society some years ago; as it was not published,
however, I am unfortunately unable to refer the reader to it. I ought to add
that he is not in any way committed to any of my opinions upon the subject,
from some of which in fact he more or less dissents. I am also much
indebted to Mr J. W. L. Glaisher, also of Trinity College, for many hints and
references to various publications upon the subject of Least Squares, and
for careful criticism (given in the midst of much other labour) of the chapter
in which that subject is treated.

I need not add that, like every one else who has had to discuss the subject of
Probability during the last ten years, I have made constant use of Mr
Todhunter's History.

I may take this opportunity of adding that a considerable portion of the
tenth chapter has recently appeared in the January number of Mind, and that
the substance of several chapters, especially in the more logical parts, has
formed part of my ordinary lectures in Cambridge; the foundation and

logical treatment of Probability being now expressly included in the
Schedule of Subjects for the Moral Sciences Tripos.

March, 1876.

PREFACE TO THIRD EDITION.
The present edition has been revised throughout, and in fact rewritten.
Three chapters are new, viz. the fifth (On the conception of Randomness)
and the eighteenth and nineteenth (On the nature, and on the employment,
of Averages). The eighth, tenth, eleventh, and fifteenth chapters have been
recast, and much new matter added, and numerous alterations made in the
remaining portions.[1] On the other hand three chapters of the last edition
have been nearly or entirely omitted.

These alterations do not imply any appreciable change of view on my part
as to the foundations and province of Probability. Some of them are of
course due to the necessary changes involved in the attempt to write up to
date upon a subject which has not been stationary during the last eleven
years. For instance the greatly increased interest now taken in what may be
called the Theory of Statistics has rendered it desirable to go much more
fully into the Nature and treatment of Laws of Error. The omissions are
mainly due to a wish to avoid increasing the bulk of this volume more than
is actually necessary, and to a feeling that the portions treating specially of
Inductive Logic (which occupied some space in the last edition) would be
more suitable to a regular work on that subject. I am at present engaged on
such a work.

The publications which I have had occasion to notice have mostly appeared
in various scientific journals. The principal authors of these have been Mr
F. Galton and Mr F. Y. Edgeworth: to the latter of whom I am also
personally much obliged for many discussions, oral and written, and for his

kindness in looking through the proof-sheets. His published articles are too
numerous for separate mention here, but I may say generally, in addition to
the obligations specially noticed, that I have been considerably indebted to
them in writing the last two chapters. Two authors of works of a somewhat
more substantial character, viz. Prof. Lexis and Von Kries, only came under
my notice unfortunately after this work was already in the printer's hands.
With the latter of these authors I find myself in closer agreement than with
most others, in respect of his general conception and treatment of
Probability.

December, 1887.
1
I have indicated the new chapters and sections by printing them in italics in the Table of Contents.

TABLE OF CONTENTS.[*]
*
Chapters and sections which are nearly or entirely new are printed in italics.

PART I.
PHYSICAL FOUNDATIONS OF THE SCIENCE OF
PROBABILITY. Chh. I–V.

CHAPTER I.
THE SERIES OF PROBABILITY.

§§ 1, 2. Distinction between the proportional propositions of Probability,
and the propositions of Logic.

3, 4. The former are best regarded as presenting a series of individuals,

5. Which may occur in any order of time,

6, 7. And which present themselves in groups.

8. Comparison of the above with the ordinary phraseology.

9, 10. These series ultimately fluctuate,

11. Especially in the case of moral and social phenomena,

12. Though in the case of games of chance the fluctuation is practically
inappreciable.

13, 14. In this latter case only can rigorous inferences be drawn.

15, 16. The Petersburg Problem.

CHAPTER II.
ARRANGEMENT AND FORMATION OF THE SERIES. LAWS OF
ERROR.

§§ 1, 2. Indication of the nature of a Law of Error or Divergence.

3. Is there necessarily but one such law,

4. Applicable to widely distinct classes of things?

5, 6. This cannot be proved directly by statistics;

7, 8. Which in certain cases show actual asymmetry.

9, 10. Nor deductively;

11. Nor by the Method of Least Squares.

12. Distinction between Laws of Error and the Method of Least Squares.

13. Supposed existence of types.

14–16. Homogeneous and heterogeneous classes.

17, 18. The type in the case of human stature, &c.

19, 20. The type in mental characteristics.

21, 22. Applications of the foregoing principles and results.

CHAPTER III.
ORIGIN OR PROCESS OF CAUSATION OF THE SERIES.

§ 1. The causes consist of (1) ‘objects,’

2, 3. Which may or may not be distinguishable into natural kinds,

4–6. And (2) ‘agencies.’

7. Requisites demanded in the above:

8, 9. Consequences of their absence.

10. Where are the required causes found?

11, 12. Not in the direct results of human will.

13–15. Examination of apparent exceptions.

16–18. Further analysis of some natural causes.

CHAPTER IV.
HOW TO DISCOVER AND PROVE THE SERIES.

§ 1. The data of Probability are established by experience;

2. Though in practice most problems are solved deductively.

3–7. Mechanical instance to show the inadequacy of any à priori proof.

8. The Principle of Sufficient Reason inapplicable.

9. Evidence of actual experience.

10, 11. Further examination of the causes.

12, 13. Distinction between the succession of physical events and the
Doctrine of Combinations.

14, 15. Remarks of Laplace on this subject.

16. Bernoulli's Theorem;

17, 18. Its inapplicability to social phenomena.

19. Summation of preceding results.

CHAPTER V.

THE CONCEPTION OF RANDOMNESS.

§ 1. General Indication.

2–5. The postulate of ultimate uniform distribution at one stage or another.

6. This area of distribution must be finite:

7, 8. Geometrical illustrations in support:

9. Can we conceive any exception here?

10, 11. Experimental determination of the random character when the
events are many:

12. Corresponding determination when they are few.

13, 14. Illustration from the constant π.

15, 16. Conception of a line drawn at random.

17. Graphical illustration.

PART II.
LOGICAL SUPERSTRUCTURE ON THE ABOVE PHYSICAL
FOUNDATIONS. Chh. VI–XIV.

CHAPTER VI.
MEASUREMENT OF BELIEF.

§§ 1, 2. Preliminary remarks.

3, 4. Are we accurately conscious of gradations of belief?

5. Probability only concerned with part of this enquiry.

6. Difficulty of measuring our belief;

7. Owing to intrusion of emotions,

8. And complexity of the evidence.

9. And when measured, is it always correct?

10, 11. Distinction between logical and psychological views.

12–16. Analogy of Formal Logic fails to show that we can thus detach and
measure our belief.

17. Apparent evidence of popular language to the contrary.

18. How is full belief justified in inductive enquiry?

19–23. Attempt to show how partial belief may be similarly justified.

24–28. Extension of this explanation to cases which cannot be repeated in
experience.

29. Can other emotions besides belief be thus measured?

30. Errors thus arising in connection with the Petersburg Problem.

31, 32. The emotion of surprise is a partial exception.

33, 34. Objective and subjective phraseology.

35. The definition of probability,

36. Introduces the notion of a ‘limit’,

37. And implies, vaguely, some degree of belief.

CHAPTER VII.
THE RULES OF INFERENCE IN PROBABILITY.

§ 1. Nature of these inferences.

2. Inferences by addition and subtraction.

3. Inferences by multiplication and division.

4–6. Rule for independent events.

7. Other rules sometimes introduced.

8. All the above rules may be interpreted subjectively, i.e. in terms of belief.

9–11. Rules of so-called Inverse Probability.

12, 13. Nature of the assumption involved in them:

14–16. Arbitrary character of this assumption.

17, 18. Physical illustrations.

CHAPTER VIII.
THE RULE OF SUCCESSION.

§ 1. Reasons for desiring some such rule:

2. Though it could scarcely belong to Probability.

3. Distinction between Probability and Induction.

4, 5. Impossibility of reducing the various rules of the latter under one head.

6. Statement of the Rule of Succession;

7. Proof offered for it.

8. Is it a strict rule of inference?

9. Or is it a psychological principle?

CHAPTER IX.
INDUCTION.

§§ 1–5. Statement of the Inductive problem, and origin of the Inductive
inference.

6. Relation of Probability to Induction.

7–9. The two are sometimes merged into one.

10. Extent to which causation is needed in Probability.

11–13. Difficulty of referring an individual to a class:

14. This difficulty but slight in Logic,

15, 16. But leads to perplexity in Probability:

17–21. Mild form of this perplexity;

22, 23. Serious form.

24–27. Illustration from Life Insurance.

28, 29. Meaning of ‘the value of a life’.

30, 31. Successive specialization of the classes to which objects are
referred.

32. Summary of results.

CHAPTER X.
CHANGE, CAUSATION AND DESIGN.

§ 1. Old Theological objection to Chance.

2–4. Scientific version of the same.

5. Statistics in reference to Free-will.

6–8. Inconclusiveness of the common arguments here.

9, 10. Chance as opposed to Physical Causation.

11. Chance as opposed to Design in the case of numerical constants.

12–14. Theoretic solution between Chance and Design.

15. Illustration from the dimensions of the Pyramid.

16, 17. Discussion of certain difficulties here.

18, 19. Illustration from Psychical Phenomena.

20. Arbuthnott's Problem of the proportion of the sexes.

21–23. Random or designed distribution of the stars.

(Note on the proportion of the sexes.)

CHAPTER XI.
MATERIAL AND FORMAL LOGIC.

§§ 1, 2. Broad distinction between these views;

2, 3. Difficulty of adhering consistently to the objective view;

4. Especially in the case of Hypotheses.

5. The doubtful stage of our facts is only occasional in Inductive Logic.

6–9. But normal and permanent in Probability.

10, 11. Consequent difficulty of avoiding Conceptualist phraseology.

CHAPTER XII.
CONSEQUENCES OF THE DISTINCTIONS OF THE PREVIOUS
CHAPTER.

§§ 1, 2. Probability has no relation to time.

3, 4. Butler and Mill on Probability before and after the event.

5. Other attempts at explaining the difficulty.

6–8. What is really meant by the distinction.

9. Origin of the common mistake.

10–12. Examples in illustration of this view,

13. Is Probability relative?

14. What is really meant by this expression.

15. Objections to terming Probability relative.

16, 17. In suitable examples the difficulty scarcely presents itself.

CHAPTER XIII.
ON MODALITY.

§ 1. Various senses of Modality;

2. Having mostly some relation to Probability.

3. Modality must be recognized.

4. Sometimes relegated to the predicate,

5, 6. Sometimes incorrectly rejected altogether.

7, 8. Common practical recognition of it.

9–11. Modal propositions in Logic and in Probability.

12. Aristotelian view of the Modals;

13, 14. Founded on extinct philosophical views;

15. But long and widely maintained.

16. Kant's general view.

17–19. The number of modal divisions admitted by various logicians.

20. Influence of the theory of Probability.

21, 22. Modal syllogisms.

23. Popular modal phraseology.

24–26. Probable and Dialectic syllogisms.

27, 28. Modal difficulties occur in Jurisprudence.

29, 30. Proposed standards of legal certainty.

31. Rejected formally in English Law, but possibly recognized practically.

32. How, if so, it might be determined.

CHAPTER XIV.
FALLACIES.

§§ 1–3. (I.) Errors in judging of events after they have happened.

4–7. Very various judgments may be thus involved.

8, 9. (II.) Confusion between random and picked selections.

10, 11. (III.) Undue limitation of the notion of Probability.

12–16. (IV.) Double or Quits: the Martingale.

17, 18. Physical illustration.

19, 20. (V.) Inadequate realization of large numbers.

21–24. Production of works of art by chance.

25. Illustration from doctrine of heredity.

26–30. (VI.) Confusion between Probability and Induction.

31–33. (VII.) Undue neglect of small chances.

34, 35. (VIII.) Judging by the event in Probability and in Induction.

PART III.
VARIOUS APPLICATIONS OF THE THEORY OF
PROBABILITY. Chh. XV–XIX.

CHAPTER XV.
INSURANCE AND GAMBLING.

§§ 1, 2. The certainties and uncertainties of life.

3–5. Insurance a means of diminishing the uncertainties.

6, 7. Gambling a means of increasing them.

8, 9. Various forms of gambling.

10, 11. Comparison between these practices.

12–14. Proofs of the disadvantage of gambling:—

(1) on arithmetical grounds:

15, 16. Illustration from family names.

17. (2) from the ‘moral expectation’.

18, 19. Inconclusiveness of these proofs.

20–22. Broader questions raised by these attempts.

CHAPTER XVI.
APPLICATION OF PROBABILITY TO TESTIMONY.

§§ 1, 2. Doubtful applicability of Probability to testimony.

3. Conditions of such applicability.

4. Reasons for the above conditions.

5, 6. Are these conditions fulfilled in the case of testimony?

7. The appeal here is not directly to statistics.

8, 9. Illustrations of the above.

10, 11. Is any application of Probability to testimony valid?

CHAPTER XVII.
CREDIBILITY OF EXTRAORDINARY STORIES.

§ 1. Improbability before and after the event.

2, 3. Does the rejection of this lead to the conclusion that the credibility of a
story is independent of its nature?

4. General and special credibility of a witness.

5–8. Distinction between alternative and open questions, and the usual rules
for application of testimony to each of these.

9. Discussion of an objection.

10, 11. Testimony of worthless witnesses.

12–14. Common practical ways of regarding such problems.

15. Extraordinary stories not necessarily less probable.

16–18. Meaning of the term extraordinary, and its distinction from
miraculous.

19, 20. Combination of testimony.

21, 22. Scientific meaning of a miracle.

23, 24. Two distinct prepossessions in regard to miracles, and the logical
consequences of these.

25. Difficulty of discussing by our rules cases in which arbitrary
interference can be postulated.

26, 27. Consequent inappropriateness of many arguments.

CHAPTER XVIII.
ON THE NATURE AND USE OF AN AVERAGE, AND ON THE
DIFFERENT KINDS OF AVERAGE.

§:nbsp;1. Preliminary rude notion of an average,

2. More precise quantitative notion, yielding

(1) the Arithmetical Average,

3. (2) the Geometrical.

4. In asymmetrical curves of error the arithmetic average must be
distinguished from,

5. (3) the Maximum Ordinate average,

6. (4) and the Median.

7. Diagram in illustration.

8–10. Average departure from the average, considered under the above
heads, and under that of

11. (5) The (average of) Mean Square of Error,

12–14. The objects of taking averages.

15. Mr Galton's practical method of determining the average.

16, 17. No distinction between the average and the mean.

18–20. Distinction between what is necessary and what is experimental
here.

21, 22. Theoretical defects in the determination of the ‘errors’.

23. Practical escape from these.

(Note about the units in the exponential equation and integral.)

CHAPTER XIX.
THE THEORY OF THE AVERAGE AS A MEANS OF
APPROXIMATION TO THE TRUTH.

§§ 1–4. General indication of the problem: i.e. an inverse one requiring the
previous consideration of a direct one.

[I. The direct problem:—given the central value and law of dispersion of the
single errors, to determine those of the averages. §§ 6–20.]

6. (i) The law of dispersion may be determinable à priori,

7. (ii) or experimentally, by statistics.

8, 9. Thence to determine the modulus of the error curve.

10–14. Numerical example to illustrate the nature and amount of the
contraction of the modulus of the average-error curve.

15. This curve is of the same general kind as that of the single errors;

16. Equally symmetrical,

17, 18. And more heaped up towards the centre.

19, 20. Algebraic generalization of the foregoing results.

[II. The inverse problem:—given but a few of the errors to determine their
centre and law, and thence to draw the above deductions. §§ 21–25.]

22, 23. The actual calculations are the same as before,

24. With the extra demand that we must determine how probable are the
results.

25. Summary.

[III. Consideration of the same questions as applied to certain peculiar
laws of error. §§ 26–37.]

26. (i) All errors equally probable.

27, 28. (ii) Certain peculiar laws of error.

29, 30. Further analysis of the reasons for taking averages.

31–35. Illustrative examples.

36, 37. Curves with double centre and absence of symmetry.

38, 39. Conclusion.

THE LOGIC OF CHANCE.

CHAPTER I.

ON CERTAIN KINDS OF GROUPS OR SERIES AS
THE FOUNDATION OF PROBABILITY.
§ 1. It is sometimes not easy to give a clear definition of a science at the
outset, so as to set its scope and province before the reader in a few words.
In the case of those sciences which are more immediately and directly
concerned with what are termed objects, rather than with what are termed
processes, this difficulty is not indeed so serious. If the reader is already
familiar with the objects, a simple reference to them will give him a
tolerably accurate idea of the direction and nature of his studies. Even if he
be not familiar with them, they will still be often to some extent connected
and associated in his mind by a name, and the mere utterance of the name
may thus convey a fair amount of preliminary information. This is more or
less the case with many of the natural sciences; we can often tell the reader
beforehand exactly what he is going to study. But when a science is
concerned, not so much with objects directly, as with processes and laws, or
when it takes for the subject of its enquiry some comparatively obscure
feature drawn from phenomena which have little or nothing else in
common, the difficulty of giving preliminary information becomes greater.
Recognized classes of objects have then to be disregarded and even broken
up, and an entirely novel arrangement of the objects to be made. In such
cases it is the study of the science that first gives the science its unity, for
till it is studied the objects with which it is concerned were probably never
thought of together. Here a definition cannot be given at the outset, and the
process of obtaining it may become by comparison somewhat laborious.

The science of Probability, at least on the view taken of it in the following
pages, is of this latter description. The reader who is at present
unacquainted with the science cannot be at once informed of its scope by a
reference to objects with which he is already familiar. He will have to be
taken in hand, as it were, and some little time and trouble will have to be
expended in directing his attention to our subject-matter before he can be
expected to know it. To do this will be our first task.

§ 2. In studying Nature, in any form, we are continually coming into
possession of information which we sum up in general propositions. Now in
very many cases these general propositions are neither more nor less certain
and accurate than the details which they embrace and of which they are
composed. We are assuming at present that the truth of these generalizations
is not disputed; as a matter of fact they may rest on weak evidence, or they
may be uncertain from their being widely extended by induction; what is
meant is, that when we resolve them into their component parts we have
precisely the same assurance of the truth of the details as we have of that of
the whole. When I know, for instance, that all cows ruminate, I feel just as
certain that any particular cow or cows ruminate as that the whole class
does. I may be right or wrong in my original statement, and I may have
obtained it by any conceivable mode in which truths can be obtained; but
whatever the value of the general proposition may be, that of the particulars
is neither greater nor less. The process of inferring the particular from the
general is not accompanied by the slightest diminution of certainty. If one
of these ‘immediate inferences’ is justified at all, it will be equally right in
every case.

But it is by no means necessary that this characteristic should exist in all
cases. There is a class of immediate inferences, almost unrecognized indeed
in logic, but constantly drawn in practice, of which the characteristic is, that
as they increase in particularity they diminish in certainty. Let me assume
that I am told that some cows ruminate; I cannot infer logically from this
that any particular cow does so, though I should feel some way removed
from absolute disbelief, or even indifference to assent, upon the subject; but
if I saw a herd of cows I should feel more sure that some of them were
ruminant than I did of the single cow, and my assurance would increase
with the numbers of the herd about which I had to form an opinion. Here
then we have a class of things as to the individuals of which we feel quite in
uncertainty, whilst as we embrace larger numbers in our assertions we
attach greater weight to our inferences. It is with such classes of things and
such inferences that the science of Probability is concerned.

§ 3. In the foregoing remarks, which are intended to be purely preliminary,
we have not been able altogether to avoid some reference to a subjective
element, viz. the degree of our certainty or belief about the things which we

are supposed to contemplate. The reader may be aware that by some writers
this element is regarded as the subject-matter of the science. Hence it will
have to be discussed in a future chapter. As however I do not agree with the
opinion of the writers just mentioned, at least as regards treating this
element as one of primary importance, no further allusion will be made to it
here, but we will pass on at once to a more minute investigation of that
distinctive characteristic of certain classes of things which was introduced
to notice in the last section.

In these classes of things, which are those with which Probability is
concerned, the fundamental conception which the reader has to fix in his
mind as clearly as possible, is, I take it, that of a series. But it is a series of a
peculiar kind, one of which no better compendious description can be given
than that which is contained in the statement that it combines individual
irregularity with aggregate regularity. This is a statement which will
probably need some explanation. Let us recur to an example of the kind
already alluded to, selecting one which shall be in accordance with
experience. Some children will not live to thirty. Now if this proposition is
to be regarded as a purely indefinite or, as it would be termed in logic,
‘particular’ proposition, no doubt the notion of a series does not obviously
present itself in connection with it. It contains a statement about a certain
unknown proportion of the whole, and that is all. But it is not with these
purely indefinite propositions that we shall be concerned. Let us suppose
the statement, on the contrary, to be of a numerical character, and to refer to
a given proportion of the whole, and we shall then find it difficult to
exclude the notion of a series. We shall find it, I think, impossible to do so
as soon as we set before us the aim of obtaining accurate, or even
moderately correct inferences. What, for instance, is the meaning of the
statement that two new-born children in three fail to attain the age of sixty-
three? It certainly does not declare that in any given batch of, say, thirty, we
shall find just twenty that fail: whatever might be the strict meaning of the
words, this is not the import of the statement. It rather contemplates our
examination of a large number, of a long succession of instances, and states
that in such a succession we shall find a numerical proportion, not indeed
fixed and accurate at first, but which tends in the long run to become so. In
every kind of example with which we shall be concerned we shall find this

reference to a large number or succession of objects, or, as we shall term it,
series of them.

A few additional examples may serve to make this plain.

Let us suppose that we toss up a penny a great many times; the results of
the successive throws may be conceived to form a series. The separate
throws of this series seem to occur in utter disorder; it is this disorder which
causes our uncertainty about them. Sometimes head comes, sometimes tail
comes; sometimes there is a repetition of the same face, sometimes not. So
long as we confine our observation to a few throws at a time, the series
seems to be simply chaotic. But when we consider the result of a long
succession we find a marked distinction; a kind of order begins gradually to
emerge, and at last assumes a distinct and striking aspect. We find in this
case that the heads and tails occur in about equal numbers, that similar
repetitions of different faces do so also, and so on. In a word,
notwithstanding the individual disorder, an aggregate order begins to
prevail. So again if we are examining the length of human life, the different
lives which fall under our notice compose a series presenting the same
features. The length of a single life is familiarly uncertain, but the average
duration of a batch of lives is becoming in an almost equal degree familiarly
certain. The larger the number we take out of any mixed crowd, the clearer
become the symptoms of order, the more nearly will the average length of
each selected class be the same. These few cases will serve as simple
examples of a property of things which can be traced almost everywhere, to
a greater or less extent, throughout the whole field of our experience. Fires,
shipwrecks, yields of harvest, births, marriages, suicides; it scarcely seems
to matter what feature we single out for observation.[1] The irregularity of
the single instances diminishes when we take a large number, and at last
seems for all practical purposes to disappear.

In speaking of the effect of the average in thus diminishing the irregularities
which present themselves in the details, the attention of the student must be
prominently directed to the point, that it is not the absolute but the relative
irregularities which thus tend to diminish without limit. This idea will be
familiar enough to the mathematician, but to others it may require some
reflection in order to grasp it clearly. The absolute divergences and

irregularities, so far from diminishing, show a disposition to increase, and
this (it may be) without limit, though their relative importance shows a
corresponding disposition to diminish without limit. Thus in the case of
tossing a penny, if we take a few throws, say ten, it is decidedly unlikely
that there should be a difference of six between the numbers of heads and
tails; that is, that there should be as many as eight heads and therefore as
few as two tails, or vice versâ. But take a thousand throws, and it becomes
in turn exceedingly likely that there should be as much as, or more than, a
difference of six between the respective numbers. On the other hand the
proportion of heads to tails in the case of the thousand throws will be very
much nearer to unity, in most cases, than when we only took ten. In other
words, the longer a game of chance continues the larger are the spells and
runs of luck in themselves, but the less their relative proportions to the
whole amounts involved.

§ 4. In speaking as above of events or things as to the details of which we
know little or nothing, it is not of course implied that our ignorance about
them is complete and universal, or, what comes to the same thing, that
irregularity may be observed in all their qualities. All that is meant is that
there are some qualities or marks in them, the existence of which we are not
able to predicate with certainty in the individuals. With regard to all their
other qualities there may be the utmost uniformity, and consequently the
most complete certainty. The irregularity in the length of human life is
notorious, but no one doubts the existence of such organs as a heart and
brains in any person whom he happens to meet. And even in the qualities in
which the irregularity is observed, there are often, indeed generally, positive
limits within which it will be found to be confined. No person, for instance,
can calculate what may be the length of any particular life, but we feel
perfectly certain that it will not stretch out to 150 years. The irregularity of
the individual instances is only shown in certain respects, as e.g. the length
of the life, and even in these respects it has its limits. The same remark will
apply to most of the other examples with which we shall be concerned. The
disorder in fact is not universal and unlimited, it only prevails in certain
directions and up to certain points.

§ 5. In speaking as above of a series, it will hardly be necessary to point out
that we do not imply that the objects themselves which compose the series

must occur successively in time; the series may be formed simply by their
coming in succession under our notice, which as a matter of fact they may
do in any order whatever. A register of mortality, for instance, may be made
up of deaths which took place simultaneously or successively; or, we might
if we pleased arrange the deaths in an order quite distinct from either of
these. This is entirely a matter of indifference; in all these cases the series,
for any purposes which we need take into account, may be regarded as
being of precisely the same description. The objects, be it remembered, are
given to us in nature; the order under which we view them is our own
private arrangement. This is mentioned here simply by way of caution, the
meaning of this assertion will become more plain in the sequel.

I am aware that the word ‘series’ in the application with which it is used
here is liable to some misconstruction, but I cannot find any better word, or
indeed any as suitable in all respects. As remarked above, the events need
not necessarily have occurred in a regular sequence of time, though they
often will have done so. In many cases (for instance, the throws of a penny
or a die) they really do occur in succession; in other cases (for instance, the
heights of men, or the duration of their lives), whatever may have been the
order of their actual occurrence, they are commonly brought under our
notice in succession by being arranged in statistical tables. In all cases alike
our processes of inference involve the necessity of examining one after
another of the members which compose the group, or at least of being
prepared to do this, if we are to be in a position to justify our inferences.
The force of these considerations will come out in the course of the
investigation in Chapter VI.

The late Leslie Ellis[2] has expressed what seems to me a substantially
similar view in terms of genus and species, instead of speaking of a series.
He says, “When individual cases are considered, we have no conviction that
the ratios of frequency of occurrence depend on the circumstances common
to all the trials. On the contrary, we recognize in the determining
circumstances of their occurrence an extraneous element, an element, that
is, extraneous to the idea of the genus and species. Contingency and
limitation come in (so to speak) together; and both alike disappear when we
consider the genus in its entirety, or (which is the same thing) in what may
be called an ideal and practically impossible realization of all which it

potentially contains. If this be granted, it seems to follow that the
fundamental principle of the Theory of Probabilities may be regarded as
included in the following statement,—The conception of a genus implies
that of numerical relations among the species subordinated to it.” As
remarked above, this appears a substantially similar doctrine to that
explained in this chapter, but I do not think that the terms genus and species
are by any means so well fitted to bring out the conception of a tendency or
limit as when we speak of a series, and I therefore much prefer the latter
expression.

§ 6. The reader will now have in his mind the conception of a series or
group of things or events, about the individuals of which we know but little,
at least in certain respects, whilst we find a continually increasing
uniformity as we take larger numbers under our notice. This is definite
enough to point out tolerably clearly the kind of things with which we have
to deal, but it is not sufficiently definite for purposes of accurate thought.
We must therefore attempt a somewhat closer analysis.

There are certain phrases so commonly adopted as to have become part of
the technical vocabulary of the subject, such as an ‘event’ and the ‘way in
which it can happen.’ Thus the act of throwing a penny would be called an
event, and the fact of its giving head or tail would be called the way in
which the event happened. If we were discussing tables of mortality, the
former term would denote the mere fact of death, the latter the age at which
it occurred, or the way in which it was brought about, or whatever else in it
might be the particular circumstance under discussion. This phraseology is
very convenient, and will often be made use of in this work, but without
explanation it may lead to confusion. For in many cases the way in which
the event happens is of such great relative importance, that according as it
happens in one way or another the event would have a different name; in
other words, it would not in the two cases be nominally the same event. The
phrase therefore will have to be considerably stretched before it will
conveniently cover all the cases to which we may have to apply it. If for
instance we were contemplating a series of human beings, male and female,
it would sound odd to call their humanity an event, and their sex the way in
which the event happened.

If we recur however to any of the classes of objects already referred to, we
may see our path towards obtaining a more accurate conception of what we
want. It will easily be seen that in every one of them there is a mixture of
similarity and dissimilarity; there is a series of events which have a certain
number of features or attributes in common,—without this they would not
be classed together. But there is also a distinction existing amongst them; a
certain number of other attributes are to be found in some and are not to be
found in others. In other words, the individuals which form the series are
compound, each being made up of a collection of things or attributes; some
of these things exist in all the members of the series, others are found in
some only. So far there is nothing peculiar to the science of Probability; that
in which the distinctive characteristic consists is this;—that the occasional
attributes, as distinguished from the permanent, are found on an extended
examination to tend to exist in a certain definite proportion of the whole
number of cases. We cannot tell in any given instance whether they will be
found or not, but as we go on examining more cases we find a growing
uniformity. We find that the proportion of instances in which they are found
to instances in which they are wanting, is gradually subject to less and less
comparative variation, and approaches continually towards some apparently
fixed value.

The above is the most comprehensive form of description; as a matter of
fact the groups will in many cases take a far simpler form; they may appear,
e.g. simply as a succession of things of the same kind, say human beings,
with or without an occasional attribute, say that of being left-handed. We
are using the word attribute, of course, in its widest sense, intending it to
include every distinctive feature that can be observed in a thing, from
essential qualities down to the merest accidents of time and place.

§ 7. On examining our series, therefore, we shall find that it may best be
conceived, not necessarily as a succession of events happening in different
ways, but as a succession of groups of things. These groups, on being
analysed, are found in every case to be resolvable into collections of
substances and attributes. That which gives its unity to the succession of
groups is the fact of some of these substances or attributes being common to
the whole succession; that which gives their distinction to the groups in the
succession is the fact of some of them containing only a portion of these

substances and attributes, the other portion or portions being occasionally
absent. So understood, our phraseology may be made to embrace every
class of things of which Probability can take account.

§ 8. It will be easily seen that the ordinary expression (viz. the ‘event,’ and
the ‘way in which it happens’) may be included in the above. When the
occasional attributes are unimportant the permanent ones are sufficient to
fix and appropriate the name, the presence or absence of the others being
simply denoted by some modification of the name or the addition of some
predicate. We may therefore in all such cases speak of the collection of
attributes as ‘the event,’—the same event essentially, that is—only saying
that it (so as to preserve its nominal identity) happens in different ways in
the different cases. When the occasional attributes however are important,
or compose the majority, this way of speaking becomes less appropriate;
language is somewhat strained by our implying that two extremely different
assemblages are in reality the same event, with a difference only in its mode
of happening. The phrase is however a very convenient one, and with this
caution against its being misunderstood, it will frequently be made use of
here.

§ 9. A series of the above-mentioned kind is, I apprehend, the ultimate basis
upon which all the rules of Probability must be based. It is essential to a
clear comprehension of the subject to have carried our analysis up to this
point, but any attempt at further analysis into the intimate nature of the
events composing the series, is not required. It is altogether unnecessary, for
instance, to form any opinion upon the questions discussed in metaphysics
as to the independent existence of substances. We have discovered, on
examination, a series composed of groups of substances and attributes, or of
attributes alone. At such a series we stop, and thence investigate our rules of
inference; into what these substances or attributes would themselves be
ultimately analysed, if taken in hand by the psychologist or metaphysician,
it is no business of ours to enquire here.

§ 10. The stage then which we have now reached is that of having
discovered a quantity of things (they prove on analysis to be groups of
things) which are capable of being classified together, and are best regarded
as constituting a series. The distinctive peculiarity of this series is our

finding in it an order, gradually emerging out of disorder, and showing in
time a marked and unmistakeable uniformity.

The impression which may possibly be derived from the description of such
a series, and which the reader will probably already entertain if he have
studied Probability before, is that the gradual evolution of this order is
indefinite, and its approach therefore to perfection unlimited. And many of
the examples commonly selected certainly tend to confirm such an
impression. But in reference to the theory of the subject it is, I am
convinced, an error, and one liable to lead to much confusion.

The lines which have been prefixed as a motto to this work, “So careful of
the type she seems, so careless of the single life,” are soon after corrected
by the assertion that the type itself, if we regard it for a long time, changes,
and then vanishes and is succeeded by others. So in Probability; that
uniformity which is found in the long run, and which presents so great a
contrast to the individual disorder, though durable is not everlasting. Keep
on watching it long enough, and it will be found almost invariably to
fluctuate, and in time may prove as utterly irreducible to rule, and therefore
as incapable of prediction, as the individual cases themselves. The full
bearing of this fact upon the theory of the subject, and upon certain
common modes of calculation connected with it, will appear more fully in
some of the following chapters; at present we will confine ourselves to very
briefly establishing and illustrating it.

Let us take, for example, the average duration of life. This, provided our
data are sufficiently extensive, is known to be tolerably regular and
uniform. This fact has been already indicated in the preceding sections, and
is a truth indeed of which the popular mind has a tolerably clear grasp at the
present day. But a very little consideration will show that there may be a
superior as well as an inferior limit to the extent within which this
uniformity can be observed; in other words whilst we may fall into error by
taking too few instances we may also fail in our aim, though in a very
different way and from quite different reasons, by taking too many. At the
present time the average duration of life in England may be, say, forty
years; but a century ago it was decidedly less; several centuries ago it was
presumably very much less; whilst if we possessed statistics referring to a

still earlier population of the country we should probably find that there has
been since that time a still more marked improvement. What may be the
future tendency no man can say for certain. It may be, and we hope that it
will be the case, that owing to sanitary and other improvements, the
duration of life will go on increasing steadily; it is at least conceivable,
though doubtless incredible, that it should do so without limit. On the other
hand, and with much more likelihood, this duration might gradually tend
towards some fixed length. Or, again, it is perfectly possible that future
generations might prefer a short and a merry life, and therefore reduce their
average longevity. The duration of life cannot but depend to some extent
upon the general tastes, habits and employments of the people, that is upon
the ideal which they consciously or unconsciously set before them, and he
would be a rash man who should undertake to predict what this ideal will be
some centuries hence. All that it is here necessary however to indicate is,
that this particular uniformity (as we have hitherto called it, in order to
mark its relative character) has varied, and, under the influence of future
eddies in opinion and practice, may vary still; and this to any extent, and
with any degree of irregularity. To borrow a term from Astronomy, we find
our uniformity subject to what might be called an irregular secular
variation.

§ 11. The above is a fair typical instance. If we had taken a less simple
feature than the length of life, or one less closely connected with what may
be called by comparison the great permanent uniformities of nature, we
should have found the peculiarity under notice exhibited in a far more
striking degree. The deaths from small-pox, for example, or the instances of
duelling or accusations of witchcraft, if examined during a few successive
decades, might have shown a very tolerable degree of uniformity. But these
uniformities have risen possibly from zero; after various and very great
fluctuations seem tending towards zero again, at least in this century; and
may, for anything we know, undergo still more rapid fluctuations in future.
Now these examples must be regarded as being only extreme ones, and not
such very extreme ones, of what is the almost universal rule in nature. I
shall endeavour to show that even the few apparent exceptions, such as the
proportions between male and female births, &c., may not be, and probably
in reality are not, strictly speaking, exceptions. A type, that is, which shall
be in the fullest sense of the words, persistent and invariable is scarcely to

be found in nature. The full import of this conclusion will be seen in future
chapters. Attention is only directed here to the important inference that,
although statistics are notoriously of no value unless they are in sufficient
numbers, yet it does not follow but that in certain cases we may have too
many of them. If they are made too extensive, they may again fall short, at
least for any particular time or place, of their greatest attainable accuracy.

§ 12. These natural uniformities then are found at length to be subject to
fluctuation. Now contrast with them any of the uniformities afforded by
games of chance; these latter seem to show no trace of secular fluctuation,
however long we may continue our examination of them. Criticisms will be
offered, in the course of the following chapters, upon some of the common
attempts to prove à priori that there must be this fixity in the uniformity in
question, but of its existence there can scarcely be much doubt. Pence give
heads and tails about equally often now, as they did when they were first
tossed, and as we believe they will continue to do, so long as the present
order of things continues. The fixity of these uniformities may not be as
absolute as is commonly supposed, but no amount of experience which we
need take into account is likely in any appreciable degree to interfere with
them. Hence the obvious contrast, that, whereas natural uniformities at
length fluctuate, those afforded by games of chance seem fixed for ever.

§ 13. Here then are series apparently of two different kinds. They are alike
in their initial irregularity, alike in their subsequent regularity; it is in what
we may term their ultimate form that they begin to diverge from each other.
The one tends without any irregular variation towards a fixed numerical
proportion in its uniformity; in the other the uniformity is found at last to
fluctuate, and to fluctuate, it may be, in a manner utterly irreducible to rule.

As this chapter is intended to be little more than explanatory and illustrative
of the foundations of the science, the remark may be made here (for which
subsequent justification will be offered) that it is in the case of series of the
former kind only that we are able to make anything which can be
interpreted into strict scientific inferences. We shall be able however in a
general way to see the kind and extent of error that would be committed if,
in any example, we were to substitute an imaginary series of the former
kind for any actual series of the latter kind which experience may present to

us. The two series are of course to be as alike as possible in all respects,
except that the variable uniformity has been replaced by a fixed one. The
difference then between them would not appear in the initial stage, for in
that stage the distinctive characteristics of the series of Probability are not
apparent; all is there irregularity, and it would be as impossible to show that
they were alike as that they were different; we can only say generally that
each shows the same kind of irregularity. Nor would it appear in the next
subsequent stage, for the real variability of the uniformity has not for some
time scope to make itself perceived. It would only be in what we have
called the ultimate stage, when we suppose the series to extend for a very
long time, that the difference would begin to make itself felt.[3] The
proportion of persons, for example, who die each year at the age of six
months is, when the numbers examined are on a small scale, utterly
irregular; it becomes however regular when the numbers examined are on a
larger scale; but if we continued our observation for a very great length of
time, or over a very great extent of country, we should find this regularity
itself changing in an irregular way. The substitution just mentioned is really
equivalent to saying, Let us assume that the regularity is fixed and
permanent. It is making a hypothesis which may not be altogether
consistent with fact, but which is forced upon us for the purpose of securing
precision of statement and definition.

§ 14. The full meaning and bearing of such a substitution will only become
apparent in some of the subsequent chapters, but it may be pointed out at
once that it is in this way only that we can with perfect strictness introduce
the notion of a ‘limit’ into our account of the matter, at any rate in reference
to many of the applications of the subject to purely statistical enquiries. We
say that a certain proportion begins to prevail among the events in the long
run; but then on looking closer at the facts we find that we have to express
ourselves hypothetically, and to say that if present circumstances remain as
they are, the long run will show its characteristics without disturbance.
When, as is often the case, we know nothing accurately of the
circumstances by which the succession of events is brought about, but have
strong reasons to suspect that these circumstances are likely to undergo
some change, there is really nothing else to be done. We can only introduce
the conception of a limit, towards which the numbers are tending, by
assuming that these circumstances do not change; in other words, by

substituting a series with a fixed uniformity for the actual one with the
varying uniformity.[4]

§ 15. If the reader will study the following example, one well known to
mathematicians under the name of the Petersburg[5] problem, he will find
that it serves to illustrate several of the considerations mentioned in this
chapter. It serves especially to bring out the facts that the series with which
we are concerned must be regarded as indefinitely extensive in point of
number or duration; and that when so regarded certain series, but certain
series only (the one in question being a case in point), take advantage of the
indefinite range to keep on producing individuals in it whose deviation from
the previous average has no finite limit whatever. When rightly viewed it is
a very simple problem, but it has given rise, at one time or another, to a
good deal of confusion and perplexity.

The problem may be stated thus:—a penny is tossed up; if it gives head I
receive one pound; if heads twice running two pounds; if heads three times
running four pounds, and so on; the amount to be received doubling every
time that a fresh head succeeds. That is, I am to go on as long as it
continues to give a succession of heads, to regard this succession as a ‘turn’
or set, and then take another turn, and so on; and for each such turn I am to
receive a payment; the occurrence of tail being understood to yield nothing,
in fact being omitted from our consideration. However many times head
may be given in succession, the number of pounds I may claim is found by
raising two to a power one less than that number of times. Here then is a
series formed by a succession of throws. We will assume,—what many
persons will consider to admit of demonstration, and what certainly
experience confirms within considerable limits,—that the rarity of these
‘runs’ of the same face is in direct proportion to the amount I receive for
them when they do occur. In other words, if we regard only the occasions
on which I receive payments, we shall find that every other time I get one
pound, once in four times I get two pounds, once in eight times four
pounds, and so on without any end. The question is then asked, what ought
I to pay for this privilege? At the risk of a slight anticipation of the results
of a subsequent chapter, we may assume that this is equivalent to asking,
what amount paid each time would on the average leave me neither winner
nor loser? In other words, what is the average amount I should receive on

the above terms? Theory pronounces that I ought to give an infinite sum:
that is, no finite sum, however great, would be an adequate equivalent. And
this is really quite intelligible. There is a series of indefinite length before
me, and the longer I continue to work it the richer are my returns, and this
without any limit whatever. It is true that the very rich hauls are extremely
rare, but still they do come, and when they come they make it up by their
greater richness. On every occasion on which people have devoted
themselves to the pursuit in question, they made acquaintance, of course,
with but a limited portion of this series; but the series on which we base our
calculation is unlimited; and the inferences usually drawn as to the sum
which ought in the long run to be paid for the privilege in question are in
perfect accordance with this supposition.

The common form of objection is given in the reply, that so far from paying
an infinite sum, no sensible man would give anything approaching to £50
for such a chance. Probably not, because no man would see enough of the
series to make it worth his while. What most persons form their practical
opinion upon, is such small portions of the series as they have actually seen
or can reasonably expect. Now in any such portion, say one which
embraces 100 turns, the longest succession of heads would not amount on
the average to more than seven or eight. This is observed, but it is forgotten
that the formula which produced these, would, if it had greater scope, keep
on producing better and better ones without any limit. Hence it arises that
some persons are perplexed, because the conduct they would adopt, in
reference to the curtailed portion of the series which they are practically
likely to meet with, does not find its justification in inferences which are
necessarily based upon the series in the completeness of its infinitude.

§ 16. This will be more clearly seen by considering the various possibilities,
and the scope required in order to exhaust them, when we confine ourselves
to a limited number of throws. Begin with three. This yields eight equally
likely possibilities. In four of these cases the thrower starts with tail and
therefore loses: in two he gains a single point (i.e. £1); in one he gains two
points, and in one he gains four points. Hence his total gain being eight
pounds achieved in four different contingencies, his average gain would be
two pounds.

Now suppose he be allowed to go as far as n throws, so that we have to
contemplate 2n possibilities. All of these have to be taken into account if we
wish to consider what happens on the average. It will readily be seen that,
when all the possible cases have been reckoned once, his total gain will be
(reckoned in pounds),

2n−2 + 2n−3·2 + 2n−4·22 + … + 2·2n−3 + 2n−2 + 2n−1,
viz.
(n + 1) 2n−2.

This being spread over 2n−1 different occasions of gain his average gain will
be 12(n + 1).

Now when we are referring to averages it must be remembered that the
minimum number of different occurrences necessary in order to justify the
average is that which enables each of them to present itself once. A man
proposes to stop short at a succession of ten heads. Well and good. We tell
him that his average gain will be £5. 10s. 0d.: but we also impress upon him
that in order to justify this statement he must commence to toss at least
1024 times, for in no less number can all the contingencies of gain and loss
be exhibited and balanced. If he proposes to reach an average gain of £20,
he will require to be prepared to go up to 39 throws, To justify this payment
he must commence to throw 239 times, i.e. about a million million times.
Not before he has accomplished this will he be in a position to prove to any
sceptic that this is the true average value of a ‘turn’ extending to
39 successive tosses.

Of course if he elects to toss to all eternity we must adopt the line of
explanation which alone is possible where questions of infinity in respect of
number and magnitude are involved. We cannot tell him to pay down ‘an
infinite sum,’ for this has no strict meaning. But we tell him that, however
much he may consent to pay each time runs of heads occur, he will attain at
last a stage in which he will have won back his total payments by his total
receipts. However large n may be, if he perseveres in trying 2n times he may

have a true average receipt of 12 (n + 1) pounds, and if he continues long
enough onwards he will have it.

The problem will recur for consideration in a future chapter.

1 The following statistics will give a fair idea of the wide range of
experience over which such regularity is found to exist: “As illustrations of
equal amounts of fluctuation from totally dissimilar causes, take the deaths
in the West district of London in seven years (fluctuation 13.66), and
offences against the person (fluctuation 13.61); or deaths from apoplexy
(fluctuation 5.54), and offences against property, without violence
(fluctuation 5.48); or students registered at the College of Surgeons
(fluctuation 1.85), and the number of pounds of manufactured tobacco
taken for home consumption (fluctuation 1.89); or out-door paupers
(fluctuation 3.45) and tonnage of British vessels entered in ballast
(fluctuation 3.43), &c.” [Extracted from a paper in the Journal of the
Statistical Society, by Mr Guy, March, 1858; the ‘fluctuation’ here given is
a measure of the amount of irregularity, that is of departure from the
average, estimated in a way which will be described hereafter.]
2
Transactions of the Cambridge Philosophical Society, Vol. IX. p. 605.
Reprinted in the collected edition of his writings, p. 50.
3 We might express it thus:—a few instances are not sufficient to display a

law at all; a considerable number will suffice to display it; but it takes a
very great number to establish that a change is taking place in the law.
4 The mathematician may illustrate the nature of this substitution by the

analogies of the ‘circle of curvature’ in geometry, and the ‘instantaneous
ellipse’ in astronomy. In the cases in which these conceptions are made use
of we have a phenomenon which is continuously varying and also changing
its rate of variation. We take it at some given moment, suppose its rate at
that moment to be fixed, and then complete its career on that supposition.

5 So called from its first mathematical treatment appearing in the
Commentarii of the Petersburg Academy; a variety of notices upon it will
be found in Mr Todhunter's History of the Theory of Probability.

CHAPTER II.

FURTHER DISCUSSION UPON THE NATURE OF
THE SERIES MENTIONED IN THE LAST
CHAPTER.
§ 1. In the course of the last chapter the nature of a particular kind of series,
that namely, which must be considered to constitute the basis of the science
of Probability, has received a sufficiently general explanation for the
preliminary purpose of introduction. One might indeed say more than this;
for the characteristics which were there pointed out are really sufficient in
themselves to give a fair general idea of the nature of Probability, and of the
sort of problems with which it deals. But in the concluding paragraphs an
indication was given that the series of this kind, as they actually occur in
nature or as the results of more or less artificial production, are seldom or
never found to occur in such a simple form as might possibly be expected
from what had previously been said; but that they are almost always seen to
be associated together in groups after a somewhat complicated fashion. A
fuller discussion of this topic must now be undertaken.

We will take for examination an instance of a kind with which the
investigations of Quetelet will have served to familiarize some readers.
Suppose that we measure the heights of a great many adult men in any town
or country. These heights will of course lie between certain extremes in
each direction, and if we continue to accumulate our measures it will be
found that they tend to lie continuously between these extremes; that is to
say, that under those circumstances no intermediate height will be found to
be permanently unrepresented in such a collection of measurements. Now
suppose these heights to be marshalled in the order of their magnitude.
What we always find is something of the following kind;—about the middle
point between the extremes, a large number of the results will be found
crowded together: a little on each side of this point there will still be an
excess, but not to so great an extent; and so on, in some diminishing scale
of proportion, until as we get towards the extreme results the numbers thin
off and become relatively exceedingly small.

The point to which attention is here directed is not the mere fact that the
numbers thus tend to diminish from the middle in each direction, but, as
will be more fully explained directly, the law according to which this
progressive diminution takes place. The word ‘law’ is here used in its
mathematical sense, to express the formula connecting together the two
elements in question, namely, the height itself, and the relative number that
are found of that height. We shall have to enquire whether one of these
elements is a function of the other, and, if so, what function.

§ 2. After what was said in the last chapter, it need hardly be insisted upon
that the interest and significance of such investigations as these are almost
entirely dependent upon the statistics being very extensive. In one or other
of Quetelet's works on Social Physics[1] will be found a selection of
measurements of almost every element which the physical frame of man
can furnish:—his height, his weight, the muscular power of various limbs,
the dimensions of almost every part and organ, and so on. Some of the most
extensive of these express the heights of 25,000 Federal soldiers from the
Army of the Potomac, and the circumferences of the chests of 5738 Scotch
militia men taken many years ago. Those who wish to consult a large
repertory of such statistics cannot be referred to any better sources than to
these and other works by the same author.[2]

Interesting and valuable, however, as are Quetelet's statistical investigations
(and much of the importance now deservedly attached to such enquiries is,
perhaps, owing more to his efforts than to those of any other person), I
cannot but feel convinced that there is much in what he has written upon the
subject which is erroneous and confusing as regards the foundations of the
science of Probability, and the philosophical questions which it involves.
These errors are not by any means confined to him, but for various reasons
they will be better discussed in the form of a criticism of his explicit or
implicit expression of them, than in any more independent way.

§ 3. In the first place then, he always, or almost always, assumes that there
can be but one and the same law of arrangement for the results of our
observations, measurements, and so on, in these statistical enquiries. That
is, he assumes that whenever we get a group of such magnitudes clustering
about a mean, and growing less frequent as they depart from that mean, we

shall find that this diminution of frequency takes place according to one
invariable law, whatever may be the nature of these magnitudes, and
whatever the process by which they may have been obtained.

That such a uniformity as this should prevail amongst many and various
classes of phenomena would probably seem surprising in any case. But the
full significance of such a fact as this (if indeed it were a fact) only becomes
apparent when attention is directed to the profound distinctions in the nature
and origin of the phenomena which are thus supposed to be harmonized by
being brought under one comprehensive principle. This will be better
appreciated if we take a brief glance at some of the principal classes into
which the things with which Probability is chiefly concerned may be
divided. These are of a three-fold kind.

§ 4. In the first place there are the various combinations, and runs of luck,
afforded by games of chance. Suppose a handful, consisting of ten coins,
were tossed up a great many times in succession, and the results were
tabulated. What we should obtain would be something of the following
kind. In a certain proportion of cases, and these the most numerous of all,
we should find that we got five heads and five tails; in a somewhat less
proportion of cases we should have, as equally frequent results, four heads
six tails, and four tails six heads; and so on in a continually diminishing
proportion until at length we came down, in a very small relative number of
cases, to nine heads one tail, and nine tails one head; whilst the least
frequent results possible would be those which gave all heads or all tails.[3]
Here the statistical elements under consideration are, as regards their origin
at any rate, optional or brought about by human choice. They would,
therefore, be commonly described as being mainly artificial, but their
results ultimately altogether a matter of chance.

Again, in the second place, we might take the accurate measurements—i.e.
the actual magnitudes themselves,—of a great many natural objects,
belonging to the same genus or class; such as the cases, already referred to,
of the heights, or other characteristics of the inhabitants of any district. Here
human volition or intervention of any kind seem to have little or nothing to
do with the matter. It is optional with us to collect the measures, but the
things measured are quite outside our control. They would therefore be

commonly described as being altogether the production of nature, and it
would not be supposed that in strictness chance had anything whatever to
do with the matter.

In the third place, the result at which we are aiming may be some fixed
magnitude, one and the same in each of our successive attempts, so that if
our measurements were rigidly accurate we should merely obtain the same
result repeated over and over again. But since all our methods of attaining
our aims are practically subject to innumerable imperfections, the results
actually obtained will depart more or less, in almost every case, from the
real and fixed value which we are trying to secure. They will be sometimes
more wide of the mark, sometimes less so, the worse attempts being of
course the less frequent. If a man aims at a target he will seldom or never
hit it precisely in the centre, but his good shots will be more[4] numerous
than his bad ones. Here again, then, we have a series of magnitudes (i.e. the
deflections of the shots from the point aimed at) clustering about a mean,
but produced in a very different way from those of the last two cases. In this
instance the elements would be commonly regarded as only partially the
results of human volition, and chance therefore as being only a co-agent in
the effects produced. With these must be classed what may be called
estimates, as distinguished from measurements. By the latter are generally
understood the results of a certain amount of mechanism or manipulation;
by the former we may understand those cases in which the magnitude in
question is determined by direct observation or introspection. The interest
and importance of this class, so far as scientific principles are concerned,
dates mainly from the investigations of Fechner. Its chief field is naturally
to be found amongst psychological data.

Other classes of things, besides those alluded to above, might readily be
given. These however are the classes about which the most extensive
statistics are obtainable, or to which the most practical importance and
interest are attached. The profound distinctions which separate their origin
and character are obvious. If they all really did display precisely the same
law of variation it would be a most remarkable fact, pointing doubtless to
some deep-seated identity underlying the various ways, apparently so
widely distinct, in which they had been brought about. The questions now
to be discussed are: Is it the case, with any considerable degree of rigour,

that only one law of distribution does really prevail? and, in so far as this is
so, how does it come to pass?

§ 5. In support of an affirmative answer to the former of these two
questions, several different kinds of proof are, or might be, offered.

(I.) For one plan we may make a direct appeal to experience, by collecting
sets of statistics and observing what is their law of distribution. As
remarked above, this has been done in a great variety of cases, and in some
instances to a very considerable extent, by Quetelet and others. His
researches have made it abundantly convincing that many classes of things
and processes, differing widely in their nature and origin, do nevertheless
appear to conform with a considerable degree of accuracy to one and the
same[5] law. At least this is made plain for the more central values, for those
that is which are situated most nearly about the mean. With regard to the
extreme values there is, on the other hand, some difficulty. For instance in
the arrangements of the heights of a number of men, these extremes are
rather a stumbling-block; indeed it has been proposed to reject them from
both ends of the scale on the plea that they are monstrosities, the fact being
that their relative numbers do not seem to be by any means those which
theory would assign.[6] Such a plan of rejection is however quite
unauthorized, for these dwarfs and giants are born into the world like their
more normally sized brethren, and have precisely as much right as any
others to be included in the formulæ we draw up.

Besides the instance of the heights of men, other classes of observations of
a somewhat similar character have been already referred to as collected and
arranged by Quetelet. From the nature of the case, however, there are not
many appropriate ones at hand; for when our object is, not to illustrate a law
which can be otherwise proved, but to obtain actual direct proof of it, the
collection of observations and measurements ought to be made upon such a
large scale as to deter any but the most persevering computers from
undergoing the requisite labour. Some of the remarks made in the course of
the note on the opposite page will serve to illustrate the difficulties which
would lie in the way of such a mode of proof.

We are speaking here, it must be understood, only of symmetrical curves: if
there is asymmetry, i.e. if the Law of Error is different on different sides of
the mean,—a comparatively very small number of observations would
suffice to detect the fact. But, granted symmetry and rapid decrease of
frequency on each side of the mean, we could generally select some one
species of the exponential curve which should pretty closely represent our
statistics in the neighbourhood of the mean. That is, where the statistics are
numerous we could secure agreement; and where we could not secure
agreement the statistics would be comparatively so scarce that we should
have to continue the observations for a very long time in order to prove the
disagreement.

§ 6. Allowing the various statistics such credit as they deserve, for their
extent, appropriateness, accuracy and so on, the general conclusion which
will on the whole be drawn by almost every one who takes the trouble to
consult them, is that they do, in large part, conform approximately to one
type or law, at any rate for all except the extreme values. So much as this
must be fully admitted. But that they do not, indeed we may say that they
cannot, always do so in the case of the extreme values, will become obvious
on a little consideration. In some of the classes of things to which the law is
supposed to apply, for example, the successions of heads and tails in the
throws of a penny, there is no limit to the magnitude of the fluctuations
which may and will occur. Postulate as long a succession of heads or of tails
as we please, and if we could only live and toss long enough for it we
should succeed in getting it at length. In other cases, including many of the
applications of Probability to natural phenomena, there can hardly fail to be
such limits. Deviations exceeding a certain range may not be merely
improbable, that is of very rare occurrence, but they may often from the
nature of the case be actually impossible. And even when they are not
actually impossible it may frequently appear on examination that they are
only rendered possible by the occasional introduction of agencies which are
not supposed to be available in the production of the more ordinary or
intermediate values. When, for instance, we are making observations with
any kind of instrument, the nature of its construction may put an absolute
limit upon the possible amount of error. And even if there be not an absolute
limit under all kinds of usage it may nevertheless be the case that there is
one under fair and proper usage; it being the case that only when the

instrument is designedly or carelessly tampered with will any new causes of
divergence be introduced which were not confined within the old limits.

Suppose, for instance, that a man is firing at a mark. His worst shots must
be supposed to be brought about by a combination of such causes as were
acting, or prepared to act, in every other case; the extreme instance of what
we may thus term ‘fair usage’ being when a number of distinct causes have
happened to conspire together so as to tend in the same direction, instead of,
as in the other cases, more or less neutralizing one another's work. But the
aggregate effect of such causes may well be supposed to be limited. The
man will not discharge his shot nearly at right angles to the true line of fire
unless some entirely new cause comes in, as by some unusual circumstance
having distracted his attention, or by his having had some spasmodic
seizure. But influences of this kind were not supposed to have been
available before; and even if they were we are taking a bold step in
assuming that these occasional great disturbances are subject to the same
kind of laws as are the aggregates of innumerable little ones.

We cannot indeed lay much stress upon an example of this last kind, as
compared with those in which we can see for certain that there is a fixed
limit to the range of error. It is therefore offered rather for illustration than
for proof. The enormous, in fact inconceivable magnitude of the numbers
expressive of the chance of very rare combinations, such as those in
question, has such a bewildering effect upon the mind that one may be
sometimes apt to confound the impossible with the higher degrees of the
merely mathematically improbable.

§ 7. At the time the first edition of this essay was composed writers on
Statistics were, I think, still for the most part under the influence of
Quetelet, and inclined to overvalue his authority on this particular subject:
of late however attention has been repeatedly drawn to the necessity of
taking account of other laws of arrangement than the binomial or
exponential.

Mr Galton, for instance,—to whom every branch of the theory of statistics
owes so much,—has insisted[7] that the “assumption which lies at the basis
of the well-known law of ‘Frequency of Error’… is incorrect in many

groups of vital and social phenomena…. For example, suppose we
endeavour to match a tint; Fechner's law, in its approximative and simplest
form of sensation = log stimulus, tells us that a series of tints, in which the
quantities of white scattered on a black ground are as 1, 2, 4, 8, 16, 32, &c.,
will appear to the eye to be separated by equal intervals of tint. Therefore,
in matching a grey that contains 8 portions of white, we are just as likely to
err by selecting one that has 16 portions as one that has 4 portions. In the
first case there would be an error in excess, of 8; in the second there would
be an error, in deficiency, of 4. Therefore, an error of the same magnitude in
excess or in deficiency is not equally probable.” The consequences of this
assumption are worked out in a remarkable paper by Dr D. McAlister, to
which allusion will have to be made again hereafter. All that concerns us
here to point out is that when the results of statistics of this character are
arranged graphically we do not get a curve which is symmetrical on both
sides of a central axis.

§ 8. More recently, Mr F. Y. Edgeworth (in a report of a Committee of the
British Association appointed to enquire into the variation of the monetary
standard) has urged the same considerations in respect of prices of
commodities. He gives a number of statistics “drawn from the prices of
twelve commodities during the two periods 1782–1820, 1820–1865. The
maximum and minimum entry for each series having been noted, it is found
that the number of entries above the ‘middle point,’ half-way between the
maximum and minimum,[8] is in every instance less than half the total
number of entries in the series. In the twenty-four trials there is not a single
exception to the rule, and in very few cases even an approach to an
exception. We may presume then that the curves are of the lop-sided
character indicated by the accompanying diagram.” The same facts are also
ascertained in respect to place variations as distinguished from time
variations. To these may be added some statistics of my own, referring to
the heights of the barometer taken at the same hour on more than 4000
successive days (v. Nature, Sept. 2, 1887). So far as these go they show a
marked asymmetry of arrangement.

In fact it appears to me that this want of symmetry ought to be looked for in
all cases in which the phenomena under measurement are of a ‘one-sided’
character; in the sense that they are measured on one side only of a certain

fixed point from which their possibility is supposed to start. For not only is
it impossible for them to fall below this point: long before they reach it the
influence of its proximity is felt in enhancing the difficulty and importance
of the same amount of absolute difference.

Look at a table of statures, for instance, with a mean value of 69 inches. A
diminution of three feet (were this possible) is much more influential,—
counts for much more, in every sense of the term,—than an addition of the
same amount; for the former does not double the mean, while the latter
more than halves it. Revert to an illustration. If a vast number of petty
influencing circumstances of the kind already described were to act upon a
swinging pendulum we should expect the deflections in each direction to
display symmetry; but if they were to act upon a spring we should not
expect such a result. Any phenomena of which the latter is the more
appropriate illustration can hardly be expected to range themselves with
symmetry about a mean.[9]

§ 9. (II.) The last remarks will suggest another kind of proof which might be
offered to establish the invariable nature of the law of error. It is of a direct
deductive kind, not appealing immediately to statistics, but involving an
enquiry into the actual or assumed nature of the causes by which the events
are brought about. Imagine that the event under consideration is brought to
pass, in the first place, by some fixed cause, or group of fixed causes. If this
comprised all the influencing circumstances the event would invariably
happen in precisely the same way: there would be no errors or deflections
whatever to be taken account of. But now suppose that there were also an
enormous number of very small causes which tended to produce
deflections; that these causes acted in entire independence of one another;
and that each of the lot told as often, in the long run, in one direction as in
the opposite. It is easy[10] to see, in a general way, what would follow from
these assumptions. In a very few cases nearly all the causes would tell in the
same direction; in other words, in a very few cases the deflection would be
extreme. In a greater number of cases, however, it would only be the most
part of them that would tell in one direction, whilst a few did what they
could to counteract the rest; the result being a comparatively larger number
of somewhat smaller deflections. So on, in increasing numbers, till we
approach the middle point. Here we shall have a very large number of very

small deflections: the cases in which the opposed influences just succeed in
balancing one another, so that no error whatever is produced, being, though
actually infrequent, relatively the most frequent of all.

Now if all deflections from a mean were brought about in the way just
indicated (an indication which must suffice for the present) we should
always have one and the same law of arrangement of frequency for these
deflections or errors, viz. the exponential[11] law mentioned in § 5.

§ 10. It may be readily admitted from what we know about the production
of events that something resembling these assumptions, and therefore
something resembling the consequences which follow from them, is really
secured in a very great number of cases. But although this may prevail
approximately, it is in the highest degree improbable that it could ever be
secured, even artificially, with anything approaching to rigid accuracy. For
one thing, the causes of deflection will seldom or never be really
independent of one another. Some of them will generally be of a kind such
that the supposition that several are swaying in one direction, may affect the
capacity of each to produce that full effect which it would have been
capable of if it had been left to do its work alone. In the common example,
for instance, of firing at a mark, so long as we consider the case of the
tolerably good shots the effect of the wind (one of the causes of error) will
be approximately the same whatever may be the precise direction of the
bullet. But when a shot is considerably wide of the mark the wind can no
longer be regarded as acting at right angles to the line of flight, and its
effect in consequence will not be precisely the same as before. In other
words, the causes here are not strictly independent, as they were assumed to
be; and consequently the results to be attributed to each are not absolutely
uninfluenced by those of the others. Doubtless the effect is trifling here, but
I apprehend that if we were carefully to scrutinize the modes in which the
several elements of the total cause conspire together, we should find that the
assumption of absolute independence was hazardous, not to say
unwarrantable, in a very great number of cases. These brief remarks upon
the process by which the deflections are brought about must suffice for the
present purpose, as the subject will receive a fuller investigation in the
course of the next chapter.

According, therefore, to the best consideration which can at the present
stage be afforded to this subject, we may draw a similar conclusion from
this deductive line of argument as from the direct appeal to statistics. The
same general result seems to be established; namely, that approximately,
with sufficient accuracy for all practical purposes, we may say that an
examination of the causes by which the deflections are generally brought
about shows that they are mostly of such a character as would result in
giving us the commonly accepted ‘Law of Error,’ as it is termed.[12] The
two lines of enquiry, therefore, within the limits assigned, afford each other
a decided mutual confirmation.

§ 11. (III.) There still remains a third, indirect and mathematical line of
proof, which might be offered to establish the conclusion that the Law of
Error is always one and the same. It may be maintained that the recognized
and universal employment of one and the same method, that known to
mathematicians and astronomers as the Method of Least Squares, in all
manner of different cases with very satisfactory results, is compatible only
with the supposition that the errors to which that method is applied must be
grouped according to one invariable law. If all ‘laws of error’ were not of
one and the same type, that is, if the relative frequency of large and small
divergences (such as we have been speaking of) were not arranged
according to one pattern, how could one method or rule equally suit them
all?

In order to preserve a continuity of treatment, some notice must be taken of
this enquiry here, though, as in the case of the last argument, any thorough
discussion of the subject is impossible at the present stage. For one thing, it
would involve too much employment of mathematics, or at any rate of
mathematical conceptions, to be suitable for the general plan of this treatise:
I have accordingly devoted a special chapter to the consideration of it.

The main reason, however, against discussing this argument here, is, that to
do so would involve the anticipation of a totally different side of the science
of Probability from that hitherto treated of. This must be especially insisted
upon, as the neglect of it involves much confusion and some error. During
these earlier chapters we have been entirely occupied with laying what may
be called the physical foundations of Probability. We have done nothing

else than establish, in one way or another, the existence of certain groups or
arrangements of things which are found to present themselves in nature; we
have endeavoured to explain how they come to pass, and we have
illustrated their principal characteristics. But these are merely the
foundations of Inference, we have not yet said a word upon the logical
processes which are to be erected upon these foundations. We have not
therefore entered yet upon the logic of chance.

§ 12. Now the way in which the Method of Least Squares is sometimes
spoken of tends to conceal the magnitude of this distinction. Writers have
regarded it as synonymous with the Law of Error, whereas the fact is that
the two are not only totally distinct things but that they have scarcely even
any necessary connection with each other. The Law of Error is the
statement of a physical fact; it simply assigns, with more or less of
accuracy, the relative frequency with which errors or deviations of any kind
are found in practice to present themselves. It belongs therefore to what
may be termed the physical foundations of the science. The Method of
Least Squares, on the other hand, is not a law at all in the scientific sense of
the term. It is simply a rule or direction informing us how we may best
proceed to treat any group of these errors which may be set before us, so as
to extract the true result at which they have been aiming. Clearly therefore
it belongs to the inferential or logical part of the subject.

It cannot indeed be denied that the methods we employ must have some
connection with the arrangement of the facts to which they are applied; but
the two things are none the less distinct in their nature, and in this case the
connection does not seem at all a necessary one, but at most one of
propriety and convenience. The Method of Least Squares is usually applied,
no doubt, to the most familiar and common form of the Law of Error,
namely the exponential form with which we have been recently occupied.
But other forms of laws of error may exist, and, if they did, the method in
question might equally well be applied to them. I am not asserting that it
would necessarily be the best method in every case, but it would be a
possible one; indeed we may go further and say, as will be shown in a
future chapter, that it would be a good method in almost every case. But its
particular merits or demerits do not interfere with its possible employment
in every case in which we may choose to resort to it. It will be seen

therefore, even from the few remarks that can be made upon the subject
here, that the fact that one and the same method is very commonly
employed with satisfactory results affords little or no proof that the errors to
which it is applied must be arranged according to one fixed law.

§ 13. So much then for the attempt to prove the prevalence, in all cases, of
this particular law of divergence. The next point in Quetelet's treatment of
the subject which deserves attention as erroneous or confusing, is the
doctrine maintained by him and others as to the existence of what he terms
a type in the groups of things in question. This is a not unnatural
consequence from some of the data and conclusions of the last few
paragraphs. Refer back to two of the three classes of things already
mentioned in § 4. If it really were the case that in arranging in order a series
of incorrect observations or attempts of our own, and a collection of natural
objects belonging to some one and the same species or class, we found that
the law of their divergence was in each case identical in the long run, we
should be naturally disposed to apply the same expression ‘Law of Error’ to
both instances alike, though in strictness it could only be appropriate to the
former. When we perform an operation ourselves with a clear consciousness
of what we are aiming at, we may quite correctly speak of every deviation
from this as being an error; but when Nature presents us with a group of
objects of any kind, it is using a rather bold metaphor to speak in this case
also of a law of error, as if she had been aiming at something all the time,
and had like the rest of us missed her mark more or less in almost every
instance.[13]

Suppose we make a long succession of attempts to measure accurately the
precise height of a man, we should from one cause or another seldom or
never succeed in doing so with absolute accuracy. But we have no right to
assume that these imperfect measurements of ours would be found so to
deviate according to one particular law of error as to present the precise
counterpart of a series of actual heights of different men, supposing that
these latter were assigned with absolute precision. What might be the actual
law of error in a series of direct measurements of any given magnitude
could hardly be asserted beforehand, and probably the attempt to determine
it by experience has not been made sufficiently often to enable us to
ascertain it; but upon general grounds it seems by no means certain that it

would follow the so-called exponential law. Be this however as it may, it is
rather a licence of language to talk as if nature had been at work in the same
way as one of us; aiming (ineffectually for the most part) at a given result,
that is at producing a man endowed with a certain stature, proportions, and
so on, who might therefore be regarded as the typical man.

§ 14. Stated as above, namely, that there is a fixed invariable human type to
which all individual specimens of humanity may be regarded as having
been meant to attain, but from which they have deviated in one direction or
another; according to a law of deviation capable of à priori determination,
the doctrine is little else than absurd. But if we look somewhat closer at the
facts of the case, and the probable explanation of these facts, we may see
our way to an important truth. The facts, on the authority of Quetelet's
statistics (the great interest and value of which must be frankly admitted),
are very briefly as follows: if we take any element of our physical frame
which admits of accurate measurement, say the height, and determine this
measure in a great number of different individuals belonging to any
tolerably homogeneous class of people, we shall find that these heights do
admit of an orderly arrangement about a mean, after the fashion which has
been already repeatedly mentioned. What is meant by a homogeneous
class? is a pertinent and significant enquiry, but applying this condition to
any simple cases its meaning is readily stated. It implies that the mean in
question will be different according to the nationality of the persons under
measurement. According to Quetelet,[14] in the case of Englishmen the
mean is about 5 ft. 9 in.; for Belgians about 5 ft. 7 in.; for the French about
5 ft. 4 in. It need hardly be added that these measures are those of adult
males.

§ 15. It may fairly be asked here what would have been the consequence,
had we, instead of keeping the English and the French apart, mixed the
results of our measurements of them all together? The question is an
important one, as it will oblige us to understand more clearly what we mean
by homogeneous classes. The answer that would usually be given to it,
though substantially correct, is somewhat too decisive and summary. It
would be said that we are here mixing distinctly heterogeneous elements,
and that in consequence the resultant law of error will be by no means of
the simple character previously exhibited. So far as such an answer is to be

admitted its grounds are easy to appreciate. In accordance with the usual
law of error the divergences from the mean grow continuously less
numerous as they increase in amount. Now, if we mix up the French and
English heights, what will follow? Beginning from the English mean of
5 feet 9 inches, the heights will at first follow almost entirely the law
determined by these English conditions, for at this point the English data
are very numerous, and the French by comparison very few. But, as we
begin to approach the French mean, the numbers will cease to show that
continual diminution which they should show, according to the English
scale of arrangement, for here the French data are in turn very numerous,
and the English by comparison few. The result of such a combination of
heterogeneous elements is illustrated by the figure annexed, of course in a
very exaggerated form.

§ 16. In the above case the nature of the heterogeneity, and the reasons why
the statistics should be so collected and arranged as to avoid it, seemed
tolerably obvious. It will be seen still more plainly if we take a parallel case
drawn from artificial proceedings. Suppose that after a man had fired a few
thousand shots at a certain spot, say a wafer fixed somewhere on a wall, the
position of the spot at which he aims were shifted, and he fired a few
thousand more shots at the wafer in its new position. Now let us collect and
arrange all the shots of both series in the order of their departure from either
of the centres, say the new one. Here we should really be mingling together
two discordant sets of elements, either of which, if kept apart from the
other, would have been of a simple and homogeneous character. We should
find, in consequence, that the resultant law of error betrayed its composite
or heterogeneous origin by a glaring departure from the customary form,
somewhat after the fashion indicated in the above diagram.

The instance of the English and French heights resembles the one just
given, but falls far short of it in the stringency with which the requisite
conditions are secured. The fact is we have not here got the most suitable
requirements, viz. a group consisting of a few fixed causes supplemented by
innumerable little disturbing influences. What we call a nation is really a
highly artificial body, the members of which are subject to a considerable
number of local or occasional disturbing causes. Amongst Frenchmen were
included, presumably, Bretons, Provençals, Alsatians, and so on, thus
commingling distinctions which, though less than those between French
and English, regarded as wholes, are very far from being insignificant. And
to these differences of race must be added other disturbances, also highly
important, dependent upon varying climate, food and occupation. It is plain,
therefore, that whatever objections exist against confusing together French
and English statistics, exist also, though of course in a less degree, against
confusing together those of the various provincial and other components
which make up the French people.

§ 17. Out of the great variety of important causes which influence the height
of men, it is probable that those which most nearly fulfil the main
conditions required by the ‘Law of Error’ are those about which we know
the least. Upon the effects of food and employment, observation has
something to say, but upon the purely physiological causes by which the
height of the parents influences the height of the offspring, we have
probably nothing which deserves to be called knowledge. Perhaps the best
supposition we can make is one which, in accordance with the saying that
‘like breeds like’, would assume that the purely physiological causes
represent the constant element; that is, given a homogeneous race of people
to begin with, who freely intermarry, and are subject to like circumstances
of climate, food, and occupation, the standard would remain on the whole
constant.[15] In such a case the man who possessed the mean height, mean
weight, mean strength, and so on, might then be called, in a sort of way, a
‘type’. The deviations from this type would then be produced by
innumerable small influences, partly physiological, partly physical and
social, acting for the most part independently of one another, and resulting
in a Law of Error of the usual description. Under such restrictions and
explanations as these, there seems to be no reasonable objection to speaking
of a French or English type or mean. But it must always be remembered
that under the present circumstances of every political nation, these
somewhat heterogeneous bodies might be subdivided into various smaller
groups, each of which would frequently exhibit the characteristics of such a
type in an even more marked degree.

§ 18. On this point the reports of the Anthropometrical Committee, already
referred to, are most instructive. They illustrate the extent to which this
subdivision could be carried out, and prove,—if any proof were necessary,
—that the discovery of Quetelet's homme moyen would lead us a long
chase. So far as their results go the mean ‘English’ stature (in inches) is
67.66. But this is composed of Scotch, Irish, English and Welsh
constituents, the separate means of these being, respectively; 68.71, 67.90,
67.36, and 66.66. But these again may be subdivided; for careful
observation shows that the mean English stature is distinctly greater in
certain districts (e.g. the North-Eastern counties) than in others. Then again
the mean of the professional classes is considerably greater than that of the
labourers; and that of the honest and intelligent is very much greater than

that of the criminal and lunatic constituents of the population. And, so far as
the observations are extensive enough for the purpose, it appears that every
characteristic in respect of the grouping about a mean which can be
detected in the more extensive of these classes can be detected also in the
narrower. Nor is there any reason to suppose that the same process of
subdivision could not be carried out as much farther as we chose to prolong
it.

§ 19. It need hardly be added to the above remarks that no one who gives
the slightest adhesion to the Doctrine of Evolution could regard the type, in
the above qualified sense of the term, as possessing any real permanence
and fixity. If the constant causes, whatever they may be, remain unchanged,
and if the variable ones continue in the long run to balance one another, the
results will continue to cluster about the same mean. But if the constant
ones undergo a gradual change, or if the variable ones, instead of balancing
each other suffer one or more of their number to begin to acquire a
preponderating influence, so as to put a sort of bias upon their aggregate
effect, the mean will at once begin, so to say, to shift its ground. And having
once begun to shift, it may continue to do so, to whatever extent we
recognize that Species are variable and Development is a fact. It is as if the
point on the target at which we aim, instead of being fixed, were slowly
changing its position as we continue to fire at it; changing almost certainly
to some extent and temporarily, and not improbably to a considerable extent
and permanently.

§ 20. Our examples throughout this chapter have been almost exclusively
drawn from physical characteristics, whether of man or of inanimate things;
but it need not be supposed that we are necessarily confined to such
instances. Mr Galton, for instance, has proposed to extend the same
principles of calculation to mental phenomena, with a view to their more
accurate determination. The objects to be gained by so doing belong rather
to the inferential part of our subject, and will be better indicated further on;
but they do not involve any distinct principle. Like other attempts to apply
the methods of science in the region of the mind, this proposal has met with
some opposition; with very slight reason, as it seems to me. That our mental
qualities, if they could be submitted to accurate measurement, would be
found to follow the usual Law of Error, may be assumed without much

hesitation. The known extent of the correlation of mental and bodily
characteristics gives high probability to the supposition that what is proved
to prevail, at any rate approximately, amongst most bodily elements which
have been submitted to measurement, will prevail also amongst the mental
elements.

To what extent such measurements could be carried out practically, is
another matter. It does not seem to me that it could be done with much
success; partly because our mental qualities are so closely connected with,
indeed so run into one another, that it is impossible to isolate them for
purposes of comparison.[16] This is to some extent indeed a difficulty in
bodily measurements, but it is far more so in those of the mind, where we
can hardly get beyond what can be called a good guess. The doctrine,
therefore, that mental qualities follow the now familiar law of arrangement
can scarcely be grounded upon anything more than a strong analogy. Still
this analogy is quite strong enough to justify us in accepting the doctrine
and all the conclusions which follow from it, in so far as our estimates and
measurements can be regarded as trustworthy. There seems therefore
nothing unreasonable in the attempt to establish a system of natural
classification of mankind by arranging them into a certain number of groups
above and below the average, each group being intended to correspond to
certain limits of excellency or deficiency.[17] All that is necessary for such a
purpose is that the rate of departure from the mean should be tolerably
constant under widely different circumstances: in this case throughout all
the races of man. Of course if the law of divergence is the same as that
which prevails in inanimate nature we have a still wider and more natural
system of classification at hand, and one which ought to be familiar, more
or less, to every one who has thus to estimate qualities.

§ 21. Perhaps one of the best illustrations of the legitimate application of
such principles is to be found in Mr Galton's work on Hereditary Genius.
Indeed the full force and purport of some of his reasonings there can hardly
be appreciated except by those who are familiar with the conceptions which
we have been discussing in this chapter. We can only afford space to notice
one or two points, but the student will find in the perusal, of at any rate the
more argumentive parts, of that volume[18] an interesting illustration of the
doctrines now under discussion. For one thing it may be safely asserted,

that no one unfamiliar with the Law of Error would ever in the least
appreciate the excessive rapidity with which the superior degrees of
excellence tend to become scarce. Every one, of course, can see at once, in
a numerical way at least, what is involved in being ‘one of a million’; but
they would not at all understand, how very little extra superiority is to be
looked for in the man who is ‘one of two million’. They would confound
the mere numerical distinction, which seems in some way to imply double
excellence, with the intrinsic superiority, which would mostly be
represented by a very small fractional advantage. To be ‘one of ten million’
sounds very grand, but if the qualities under consideration could be
estimated in themselves without the knowledge of the vastly wider area
from which the selection had been made, and in freedom therefore from any
consequent numerical bias, people would be surprised to find what a very
slight comparative superiority was, as a rule, thus obtained.

§ 22. The point just mentioned is an important one in arguments from
statistics. If, for instance, we find a small group of persons, connected
together by blood-relationship, and all possessing some mental
characteristic in marked superiority, much depends upon the comparative
rarity of such excellence when we are endeavouring to decide whether or
not the common possession of these qualities was accidental. Such a
decision can never be more than a rough one, but if it is to be made at all
this consideration must enter as a factor. Again, when we are comparing one
nation with another,[19] say the Athenian with any modern European people,
does the popular mind at all appreciate what sort of evidence of general
superiority is implied by the production, out of one nation, of such a group
as can be composed of Socrates, Plato, and a few of their contemporaries?
In this latter case we are also, it should be remarked, employing the ‘Law of
Error’ in a second way; for we are assuming that where the extremes are
great so will also the means be, in other words we are assuming that every
amount of departure from the mean occurs with a (roughly) calculable
degree of relative frequency. However generally this truth may be accepted
in a vague way, its evidence can only be appreciated by those who know the
reasons which can be given in its favour.

But the same principles will also supply a caution in the case of the last
example. They remind us that, for the mere purpose of comparison, the

average man of any group or class is a much better object for selection than
the eminent one. There may be greater difficulties in the way of detecting
him, but when we have done so we have got possession of a securer and
more stable basis of comparison. He is selected, by the nature of the case,
from the most numerous stratum of his society; the eminent man from a
thinly occupied stratum. In accordance therefore with the now familiar laws
of averages and of large numbers the fluctuations amongst the former will
generally be very few and small in comparison with those amongst the
latter.

1 Essai de Physique Sociale, 1869. Anthropométrie, 1870.

2
As regards later statistics on the same subject the reader can refer to the
Reports of the Anthropometrical Committee of the British Association
(1879, 1880, 1881, 1883;—especially this last). These reports seem to me to
represent a great advance on the results obtained by Quetelet, and fully to
justify the claim of the Secretary (Mr C. Roberts) that their statistics are
“unique in range and numbers”. They embrace not merely military recruits
—like most of the previous tables—but almost every class and age, and
both sexes. Moreover they refer not only to stature but to a number of other
physical characteristics.
3 As every mathematician knows, the relative numbers of each of these

possible throws are given by the successive terms of the expansion of
(1 + 1)10, viz. 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.
4
That is they will be more densely aggregated. If a space the size of the
bull's-eye be examined in each successive circle, the number of shot marks
which it contains will be successively less. The actual number of shots
which strike the bull's-eye will not be the greatest, since it covers so much
less surface than any of the other circles.
5 Commonly called the exponential law; its equation being of the form
2
y = Ae−hx . The curve corresponding to it cuts the axis of y at right angles

(expressing the fact that near the mean there are a large number of values
approximately equal); after a time it begins to slope away rapidly towards
the axis of x (expressing the fact that the results soon begin to grow less
common as we recede from the mean); and the axis of x is an asymptote in
both directions (expressing the fact that no magnitude, however remote
from the mean, is strictly impossible; that is, every deviation, however
excessive, will have to be encountered at length within the range of a
sufficiently long experience). The curve is obviously symmetrical,
expressing the fact that equal deviations from the mean, in excess and in
defect, tend to occur equally often in the long run.

A rough graphic representation of the curve is given above. For the benefit
of those unfamiliar with mathematics one or two brief remarks may be here
appended concerning some of its properties. (1) It must not be supposed
that all specimens of the curve are similar to one another. The dotted lines
are equally specimens of it. In fact, by varying the essentially arbitrary units
in which x and y are respectively estimated, we may make the portion
towards the vertex of the curve as obtuse or as acute as we please. This
consideration is of importance; for it reminds us that, by varying one of
these arbitrary units, we could get an ‘exponential curve’ which should
tolerably closely resemble any symmetrical curve of error, provided that
this latter recognized and was founded upon the assumption that extreme
divergences were excessively rare. Hence it would be difficult, by mere
observation, to prove that the law of error in any given case was not
exponential; unless the statistics were very extensive, or the actual results
departed considerably from the exponential form. (2) It is quite impossible
by any graphic representation to give an adequate idea of the excessive

rapidity with which the curve after a time approaches the axis of x. At the
point R, on our scale, the curve would approach within the fifteen-
thousandth part of an inch from the axis of x, a distance which only a very
good microscope could detect. Whereas in the hyperbola, e.g. the rate of
approach of the curve to its asymptote is continually decreasing, it is here
just the reverse; this rate is continually increasing. Hence the two, viz. the
curve and the axis of x, appear to the eye, after a very short time, to merge
into one another.
6 As by Quetelet: noted, amongst others, by Herschel, Essays, page 409.

7
Proc. R. Soc. Oct. 21, 1879.
8 We are here considering, remember, the case of a finite amount of
statistics; so that there are actual limits at each end.
9 It must be admitted that experience has not yet (I believe) shown this

asymmetry in respect of heights.
10
The above reasoning will probably be accepted as valid at this stage of
enquiry. But in strictness, assumptions are made here, which however
justifiable they may be in themselves, involve somewhat of an anticipation.
They demand, and in a future chapter will receive, closer scrutiny and
criticism.
11 A definite numerical example of this kind of concentration of frequency

about the mean was given in the note to § 4. It was of a binomial form,
consisting of the successive terms of the expansion of (1 + 1)m. Now it may
be shown (Quetelet, Letters, p. 263; Liagre, Calcul des Probabilités, § 34)
that the expansion of such a binomial, as m becomes indefinitely great,
approaches as its limit the exponential form; that is, if we take a number of
equidistant ordinates proportional respectively to 1, m, m(m1·2− 1) &c., and
connect their vertices, the figure we obtain approximately represents some
2
form of the curve y = Ae−hx , and tends to become identical with it, as m is
increased without limit. In other words, if we suppose the errors to be
produced by a limited number of finite, equal and independent causes, we

have an approximation to the exponential Law of Error, which merges into
identity as the causes are increased in number and diminished in magnitude
without limit. Jevons has given (Principles of Science, p. 381) a diagram
drawn to scale, to show how rapid this approximation is. One point must be
carefully remembered here, as it is frequently overlooked (by Quetelet, for
instance). The coefficients of a binomial of two equal terms—as (1 + 1)m, in
the preceding paragraph—are symmetrical in their arrangement from the
first, and very speedily become indistinguishable in (graphical) outline from
the final exponential form. But if, on the other hand, we were to consider
the successive terms of such a binomial as (1 + 4)m (which are proportional
to the relative chances of 0, 1, 2, 3, … failures in m ventures, of an event
which has one chance in its favour to four against it) we should have an
unsymmetrical succession. If however we suppose m to increase without
limit, as in the former supposition, the unsymmetry gradually disappears
and we tend towards precisely the same exponential form as if we had
begun with two equal terms. The only difference is that the position of the
vertex of the curve is no longer in the centre: in other words, the likeliest
term or event is not an equal number of successes and failures but successes
and failures in the ratio of 1 to 4.
12 ‘Law of Error’ is the usual technical term for what has been elsewhere

spoken of above as a Law of Divergence from a mean. It is in strictness
only appropriate in the case of one, namely the third, of the three classes of
phenomena mentioned in § 4, but by a convenient generalization it is
equally applied to the other two; so that we term the amount of the
divergence from the mean an ‘error’ in every case, however it may have
been brought about.
13 This however seems to be the purport, either by direct assertion or by

implication, of two elaborate works by Quetelet, viz. his Physique Sociale
and his Anthropométrie.
14
He scarcely, however, professes to give these as an accurate measure of
the mean height, nor does he always give precisely the same measure.
Practically, none but soldiers being measured in any great numbers, the
English stature did not afford accurate data on any large scale. The statistics
given a few pages further on are probably far more trustworthy.

15 This statement will receive some explanation and correction in the next

chapter.
16
I am not speaking here of the now familiar results of Psychophysics,
which are mainly occupied with the measurement of perceptions and other
simple states of consciousness.
17 Perhaps the best brief account of Mr Galton's method is to be found in a

paper in Mind (July, 1880) on the statistics of Mental Imagery. The subject
under comparison here—viz. the relative power, possessed by different
persons, of raising clear visual images of objects no longer present to us—is
one which it seems impossible to ‘measure’, in the ordinary sense of the
term. But by arranging all the answers in the order in which the faculty in
question seems to be possessed we can, with some approach to accuracy,
select the middlemost person in the row and use him as a basis of
comparison with the corresponding person in any other batch. And similarly
with those who occupy other relative positions than that of the middlemost.
18 I refer to the introductory and concluding chapters: the bulk of the book

is, from the nature of the case, mainly occupied with statistical and
biographical details.
19
See Galton's Hereditary Genius, pp. 336–350, “On the comparative
worth of different races.”

CHAPTER III.

ON THE CAUSAL PROCESS BY WHICH THE
GROUPS OR SERIES OF PROBABILITY ARE
BROUGHT ABOUT.
§ 1. In discussing the question whether all the various groups and series
with which Probability is concerned are of precisely one and the same type,
we made some examination of the process by which they are naturally
produced, but we must now enter a little more into the details of this
process. All events are the results of numerous and complicated
antecedents, far too numerous and complicated in fact for it to be possible
for us to determine or take them all into account. Now, though it is strictly
true that we can never determine them all, there is a broad distinction
between the case of Induction, in which we can make out enough of them,
and with sufficient accuracy, to satisfy a reasonable certainty, and
Probability, in which we cannot do so. To Induction we shall return in a
future chapter, and therefore no more need be said about it here.

We shall find it convenient to begin with a division which, though not
pretending to any philosophical accuracy, will serve as a preliminary guide.
It is the simple division into objects, and the agencies which affect them.
All the phenomena with which Probability is concerned (as indeed most of
those with which science of any kind is concerned) are the product of
certain objects natural and artificial, acting under the influence of certain
agencies natural and artificial. In the tossing of a penny, for instance, the
objects would be the penny or pence which were successively thrown; the
agencies would be the act of throwing, and everything which combined
directly or indirectly with this to make any particular face come uppermost.
This is a simple and intelligible division, and can easily be so extended in
meaning as to embrace every class of objects with which we are concerned.

Now if, in any two or more cases, we had the same object, or objects
indistinguishably alike, and if they were exposed to the influence of
agencies in all respects precisely alike, we should expect the results to be
precisely similar. By one of the applications of the familiar principle of the
uniformity of nature we should be confident that exact likeness in the

antecedents would be followed by exact likeness in the consequents. If the
same penny, or similar pence, were thrown in exactly the same way, we
should invariably find that the same face falls uppermost.

§ 2. What we actually find is, of course, very far removed from this. In the
case of the objects, when they are artificial constructions, e.g. dice, pence,
cards, it is true that they are purposely made as nearly as possible
indistinguishably alike. We either use the same thing over and over again or
different ones made according to precisely the same model. But in natural
objects nothing of the sort prevails. In fact when we come to examine them,
we find reproduced in them precisely the same characteristics as those
which present themselves in the final result which we were asked to
explain, so that unless we examine them a stage further back, as we shall
have to do to some extent at any rate, we seem to be merely postulating
again the very peculiarity of the phenomena which we were undertaking to
explain. They will be found, for instance, to consist of large classes of
objects, throughout all the individual members of which a general
resemblance extends. Suppose that we were considering the length of life.
The objects here are the human beings, or that selected class of them, whose
lives we are considering. The resemblance existing among them is to be
found in the strength and soundness of their principal vital organs, together
with all the circumstances which collectively make up what we call the
goodness of their constitutions. It is true that most of these circumstances
do not admit of any approach to actual measurement; but, as was pointed
out in the last chapter, very many of the circumstances which do admit of
such measurement have been measured, and found to display the
characteristics in question. Hence, from the known analogy and correlation
between our various organs, there can be no reasonable doubt that if we
could arrange human constitutions in general, or the various elements which
compose them in particular, in the order of their strength, we should find
just such an aggregate regularity and just such groupings about the mean, as
the final result (viz. in this case the length of their lives) presents to our
notice.

§ 3. It will be observed therefore that for this purpose the existence of
natural kinds or groups is necessary. In our games of chance of course the
same die may be thrown, or a card be drawn from the same pack, as often

as we please; but many of the events which occur to human beings either
cannot be repeated at all, or not often enough to secure in the case of the
single individual any sufficient statistical uniformity. Such regularity as we
trace in nature is owing, much more than is often suspected, to the
arrangement of things in natural kinds, each of them containing a large
number of individuals. Were each kind of animals or vegetables limited to a
single pair, or even to but a few pairs, there would not be much scope left
for the collection of statistical tables amongst them. Or to take a less violent
supposition, if the numbers in each natural class of objects were much
smaller than they are at present, or the differences between their varieties
and sub-species much more marked, the consequent difficulty of extracting
from them any sufficient length of statistical tables, though not fatal, might
be very serious. A large number of objects in the class, together with that
general similarity which entitles the objects to be fairly comprised in one
class, seem to be important conditions for the applicability of the theory of
Probability to any phenomenon. Something analogous to this excessive
paucity of objects in a class would be found in the attempt to apply special
Insurance offices to the case of those trades where the numbers are very
limited, and the employment so dangerous as to put them in a class by
themselves. If an insurance society were started for the workmen in
gunpowder mills alone, a premium would have to be charged to avoid
possible ruin, so high as to illustrate the extreme paucity of appropriate
statistics.

§ 4. So much (at present) for the objects. If we turn to what we have termed
the agencies, we find much the same thing again here. By the adjustment of
their relative intensity, and the respective frequency of their occurrence, the
total effects which they produce are found to be also tolerably uniform. It is
of course conceivable that this should have been otherwise. It might have
been found that the second group of conditions so exactly corrected the
former as to convert the merely general uniformity into an absolute one; or
it might have been found, on the other hand, that the second group should
aggravate or disturb the influence of the former to such an extent as to
destroy all the uniformity of its effects. Practically neither is the case. The
second condition simply varies the details, leaving the uniformity on the
whole of precisely the same general description as it was before. Or if the
objects were supposed to be absolutely alike, as in the case of successive

throws of a penny, it may serve to bring about a uniformity. Analysis will
show these agencies to be thus made up of an almost infinite number of
different components, but it will detect the same peculiarity that we have so
often had occasion to refer to, pervading almost all these components. The
proportions in which they are combined will be found to be nearly, though
not quite, the same; the intensity with which they act will be nearly though
not quite equal. And they will all unite and blend into a more and more
perfect regularity as we proceed to take the average of a larger number of
instances.

Take, for instance, the length of life. As we have seen, the constitutions of a
very large number of persons selected at random will be found to present
much the same feature; general uniformity accompanied by individual
irregularity. Now when these persons go out into the world, they are
exposed to a variety of agencies, the collective influence of which will
assign to each the length of life allotted to him. These agencies are of
course innumerable, and their mutual interaction complicated beyond all
power of analysis to extricate. Each effect becomes in its turn a cause, is
interwoven inextricably with an indefinite number of other causes, and
reacts upon the final result. Climate, food, clothing, are some of these
agencies, or rather comprise aggregate groups of them. The nature of a
man's work is also important. One man overworks himself, another follows
an unhealthy trade, a third exposes himself to infection, and so on.

The result of all this interaction between what we have thus called objects
and agencies is that the final outcome presents the same general
characteristics of uniformity as may be detected separately in the two
constituent elements. Or rather, as we shall proceed presently to show, it
does so in the great majority of cases.

§ 5. It may be objected that such an explanation as the above does not really
amount to anything deserving of the name, for that instead of explaining
how a particular state of things is caused it merely points out that the same
state exists elsewhere. There is a uniformity discovered in the objects at the
stage when they are commonly submitted to calculation; we then grope
about amongst the causes of them, and after all only discover a precisely
similar uniformity existing amongst these causes. This is to some extent

true, for though part of the objection can be removed, it must always remain
the case that the foundations of an objective science will rest in the last
resort upon the mere fact that things are found to be of such and such a
character.

§ 6. This division, into objects and the agencies which affect them, is
merely intended for a rough practical arrangement, sufficient to point out to
the reader the immediate nature of the causes which bring about our
familiar uniformities. If we go back a step further, it might fairly be
maintained that they may be reduced to one, namely, to the agencies. The
objects, as we have termed them, are not an original creation in the state in
which we now find them. No one supposes that whole groups or classes
were brought into existence simultaneously, with all their general
resemblances and particular differences fully developed. Even if it were the
case that the first parents of each natural kind had been specially created,
instead of being developed out of pre-existing forms, it would still be true
that amongst the numbers of each that now present themselves the
characteristic differences and resemblances are the result of what we have
termed agencies. Take, for instance, a single characteristic only, say the
height; what determines this as we find it in any given group of men?
Partly, no doubt, the nature of their own food, clothing, employment, and so
on, especially in the earliest years of their life; partly also, very likely,
similar conditions and circumstances on the part of their parents at one time
or another. No one, I presume, in the present state of knowledge, would
attempt to enumerate the remaining causes, or even to give any indication
of their exact nature; but at the same time few would entertain any doubt
that agencies of this general description have been the determining causes at
work.

If it be asked again, Into what may these agencies themselves be ultimately
analysed? the answer to this question, in so far as it involves any detailed
examination of them, would be foreign to the plan of this essay. In so far as
any general remarks, applicable to nearly all classes alike of such agencies,
are called for, we are led back to the point from which we started in the
previous chapter, when we were discussing whether there is necessarily one
fixed law according to which all our series are formed. We there saw that
every event might be regarded as being brought about by a comparatively

few important causes, of the kind which comprises all of which ordinary
observation takes any notice, and an indefinitely numerous group of small
causes, too numerous, minute, and uncertain in their action for us to be able
to estimate them or indeed to take them individually into account at all. The
important ones, it is true, may also in turn be themselves conceived to be
made up of aggregates of small components, but they are still best regarded
as being by comparison simple and distinct, for their component parts act
mostly in groups collectively, appearing and disappearing together, so that
they possess the essential characteristics of unity.

§ 7. Now, broadly speaking, it appears to me that the most suitable
conditions for Probability are these: that the important causes should be by
comparison fixed and permanent, and that the remaining ones should on the
average continue to act as often in one direction as in the other. This they
may do in two ways. In the first place we may be able to predicate nothing
more of them than the mere fact that they act[1] as often in one direction as
the other; what we should then obtain would be merely the simple statistical
uniformity that is described in the first chapter. But it may be the case, and
in practice generally is so more or less approximately, that these minor
causes act also in independence of one another. What we then get is a group
of uniformities such as was explained and illustrated in the second chapter.
Every possible combination of these causes then occurring with a regular
degree of frequency, we find one peculiar kind of uniformity exhibited, not
merely in the mere fact of excess and defect (of whatever may be the
variable quality in question), but also in every particular amount of excess
and defect. Hence, in this case, we get what some writers term a ‘mean’ or
‘type,’ instead of a simple average. For instance, suppose a man throwing a
quoit at a mark. Here our fixed causes are his strength, the weight of the
quoit, and the intention of aiming at a given point. These we must of course
suppose to remain unchanged, if we are to obtain any such uniformity as we
are seeking. The minor and variable causes are all those innumerable little
disturbing influences referred to in the last chapter. It might conceivably be
the case that we were only able to ascertain that these acted as often in one
direction as in the other; what we should then find was that the quoit tended
to fall short of the mark as often as beyond it. But owing to these little
causes being mostly independent of one another, and more or less equal in
their influence, we find also that every amount of excess and defect presents

the same general characteristics, and that in a large number of throws the
quantity of divergences from the mark, of any given amount, is a tolerably
determinate function, according to a regular law, of that amount of
divergence.[2]

§ 8. The necessity of the conditions just hinted at will best be seen by a
reference to cases in which any of them happen to be missing. Thus we
know that the length of life is on the whole tolerably regular, and so are the
numbers of those who die in successive years or centuries of most of the
commoner diseases. But it does not seem to be the case with all diseases.
What, for instance, of the Sweating Sickness, the Black Death, the Asiatic
Cholera? The two former either do not recur, or, if they do, recur in such a
mild form as not to deserve the same name. What in fact of any of the
diseases which are epidemic rather than endemic? All these have their
causes doubtless, and would be produced again by the recurrence of the
conditions which caused them before. But some of them apparently do not
recur at all. They seem to have depended upon such rare conditions that
their occurrence was almost unique. And of those which do recur the course
is frequently so eccentric and irregular, often so much dependent upon
human will or want of will, as to entirely deprive their results (that is, the
annual number of deaths which they cause) of the statistical uniformity of
which we are speaking.

The explanation probably is that one of the principal causes in such cases is
what we commonly call contagion. If so, we have at once a cause which so
far from being fixed is subject to the utmost variability. Stringent caution
may destroy it, carelessness may aggravate it to any extent. The will of
man, as finding its expression either on the part of government, of doctors,
or of the public, may make of it pretty nearly what is wished, though
against the possibility of its entrance into any community no precautions
can absolutely insure us.

§ 9. If it be replied that this want of statistical regularity only arises from the
fact of our having confined ourselves to too limited a time, and that we
should find irregularity disappear here, as elsewhere, if we kept our tables
open long enough, we shall find that the answer will suggest another case in
which the requisite conditions for Probability are wanting. Such a reply

would only be conclusive upon the supposition that the ways and thoughts
of men are in the long run invariable, or if variable, subject to periodic
changes only. On the assumption of a steady progress in society, either for
the better or the worse, the argument falls to the ground at once. From what
we know of the course of the world, these fearful pests of the past may be
considered as solitary events in our history, or at least events which will not
be repeated. No continued uniformity would therefore be found in the
deaths which they occasion, though the registrar's books were kept open for
a thousand years. The reason here is probably to be sought in the gradual
alteration of those indefinitely numerous conditions which we term
collectively progress or civilization. Every little circumstance of this kind
has some bearing upon the liability of any one to catch a disease. But when
a kind of slow and steady tide sets in, in consequence of which these
influences no longer remain at about the same average strength, warring on
about equal terms with hostile influences, but on the contrary show a steady
tendency to increase their power, the statistics will, with consequent
steadiness and permanence, take the impress of such a change.

§ 10. Briefly then, if we were asked where the distinctive characteristics of
Probability are most prominently to be found, and where they are most
prominently absent, we might say that (1) they prevail principally in the
properties of natural kinds, both in the ultimate and in the derivative or
accidental properties. In all the characteristics of natural species, in all they
do and in all which happens to them, so far as it depends upon their
properties, we seldom fail to detect this regularity. Thus in men; their
height, strength, weight, the age to which they live, the diseases of which
they die; all present a well-known uniformity. Life insurance tables offer the
most familiar instance of the importance of these applications of
Probability.

(2) The same peculiarity prevails again in the force and frequency of most
natural agencies. Wind and weather are seen to lose their proverbial
irregularity when examined on a large scale. Man's work therefore, when
operated on by such agencies as these, even though it had been made in
different cases absolutely alike to begin with, afterwards shows only a
general regularity. I may sow exactly the same amount of seed in my field
every year. The yield may one year be moderate, the next year be abundant

through favourable weather, and then again in turn be destroyed by hail. But
in the long run these irregularities will be equalized in the result of my
crops, because they are equalized in the power and frequency of the
productive agencies. The business of underwriters, and offices which insure
the crops against hail, would fall under this class; though, as already
remarked, there is no very profound distinction between them and the
former class.

The reader must be reminded again that this fixity is only temporary, that is,
that even here the series belong to the class of those which possess a
fluctuating type. Those indeed who believe in the fixity of natural species
will have the best chance of finding a series of the really permanent type
amongst them, though even they will admit that some change in the
characteristic is attainable in length of time. In the case of the principal
natural agencies, it is of course incontestable that the present average is
referable to the present geological period only. Our average temperature and
average rainfall have in former times been widely different from what they
now are, and doubtless will be so again.

Any fuller investigation of the process by which, on the Theory of
Evolution, out of a primeval simplicity and uniformity the present variety
was educed, hardly belongs to the scope of the present work: at most, a few
hints must suffice.

§ 11. The above, then, are instances of natural objects and natural agencies.
There seems reason to believe that it is in such things only, as distinguished
from things artificial, that the property in question is to be found. This is an
assertion that will need some discussion and explanation. Two instances, in
apparent opposition, will at once occur to the mind of some readers; one of
which, from its great intrinsic importance, and the other, from the frequency
of the problems which it furnishes, will demand a few minutes' separate
examination.

(1) The first of these is the already mentioned case of instrumental
observations. In the use of astronomical and other instruments the utmost
possible degree of accuracy is often desired, a degree which cannot be
reasonably hoped for in any one single observation. What we do therefore
in these cases is to make a large number of successive observations which

are naturally found to differ somewhat from each other in their results; by
means of these the true value (as explained in a future chapter, on the
Method of Least Squares) is to be determined as accurately as possible. The
subjects then of calculation here are a certain number of elements, slightly
incorrect elements, given by successive observations. Are not these
observations artificial, or the direct product of voluntary agency? Certainly
not: or rather, the answer depends on what we understand by voluntary.
What is really intended and aimed at by the observer, is of course, perfect
accuracy, that is, the true observation, or the voluntary steps and
preliminaries on which this observation depends. Whether voluntary or not,
this result only can be called intentional. But this result is not obtained.
What we actually get in its place is a series of deviations from it, containing
results more or less wide of the truth. Now by what are these deviations
caused? By just such agencies as we have been considering in some of the
earlier sections in this chapter. Heat and its irregular warping influence,
draughts of air producing their corresponding effects, dust and consequent
friction in one part or another, the slight distortion of the instrument by
strains or the slow uneven contraction which continues long after the metal
was cast; these and such as these are some of the causes which divert us
from the truth. Besides this group, there are others which certainly do
depend upon human agency, but which are not, strictly speaking, voluntary.
They are such as the irregular action of the muscles, inability to make our
various organs and members execute precisely the purposes we have in
mind, perhaps different rates in the rapidity of the nervous currents, or in
the response to stimuli, in the same or different observers. The effect
produced by some of these, and the allowance that has in consequence to be
made, are becoming familiar even to the outside world under the name of
the ‘personal equation’ in astronomical, psychophysical, and other
observations.

§ 12. (2) The other example, alluded to above, is the stock one of cards and
dice. Here, as in the last case, the result is remotely voluntary, in the sense
that deliberate volition presents itself at one stage. But subsequently to this
stage, the result is produced or affected by so many involuntary agencies
that it owes its characteristic properties to these. The turning up, for
example, of a particular face of a die is the result of voluntary agency, but it
is not an immediate result. That particular face was not chosen, though the

fact of its being chosen was the remote consequence of an act of choice.
There has been an intermediate chaos of conflicting agencies, which no one
can calculate before or distinguish afterwards. These agencies seem to show
a uniformity in the long run, and thence to produce a similar uniformity in
the result. The drawing of a card from a pack is indeed more directly
volitional, as in cutting for partners in a game of whist. But no one
continues to do this long without having the pack well shuffled in the
interval, whereby a host of involuntary influences are let in.

§ 13. The once startling but now familiar uniformities exhibited in the cases
of suicides and misdirected letters, do not belong to the same class. The
final resolution, or want of it, which leads to these results, is in each case
indeed an important ingredient in the individual's action or omission; but, in
so far as volition has anything to do with the results as a whole, it instantly
disturbs them. If the voice of the Legislature speaks out, or any great
preacher or moralist succeeds in deterring, or any impressive example in
influencing, our moral statistics are instantly tampered with. Some further
discussion will be devoted to this subject in a future chapter; it need only be
remarked here that (always excluding such common or general influence as
those just mentioned) the average volition, potent as it is in each separate
case, is on the whole swayed by non-voluntary conditions, such as those of
health, the casualties of employment, &c., in fact the various circumstances
which influence the length of a man's life.

§ 14. Such distinctions as those just insisted on may seem to some persons
to be needless, but serious errors have occasionally arisen from the neglect
of them. The immediate products of man's mind, so far indeed as we can
make an attempt to obtain them, do not seem to possess this essential
characteristic of Probability. Their characteristic seems rather to be, either
perfect mathematical accuracy or utter want of it, either law unfailing or
mere caprice. If, e.g., we find the trees in a forest growing in straight lines,
we unhesitatingly conclude that they were planted by man as they stand. It
is true on the other hand, that if we find them not regularly planted, we
cannot conclude that they were not planted by man; partly because the
planter may have worked without a plan, partly because the subsequent
irregularities brought on by nature may have obscured the plan. Practically
the mind has to work by the aid of imperfect instruments, and is subjected

to many hindrances through various and conflicting agencies, and by these
means the work loses its original properties. Suppose, for instance, that a
man, instead of producing numerical results by imperfect observations or by
the cast of dice, were to select them at first hand for himself by simply
thinking of them at once; what sort of series would he obtain? It would be
about as difficult to obtain in this way any such series as those appropriate
to Probability as it would be to keep his heart or pulse working regularly by
direct acts of volition, supposing that he had the requisite control over these
organs. But the mere suggestion is absurd. A man must have an object in
thinking, he must think according to a rule or formula; but unless he takes
some natural series as a copy, he will never be able to construct one
mentally which shall permanently imitate the originals. Or take another
product of human efforts, in which the intention can be executed with
tolerable success. When any one builds a house, there are many slight
disturbing influences at work, such as shrinking of bricks and mortar,
settling of foundations, &c. But the effect which these disturbances are able
to produce is so inappreciably small, that we may fairly consider that the
result obtained is the direct product of the mind, the accurate realization of
its intention. What is the consequence? Every house in the row, if designed
by one man and at one time, is of exactly the same height, width, &c. as its
neighbours; or if there are variations they are few, definite, and regular. The
result offers no resemblance whatever to the heights, weights, &c. of a
number of men selected at random. The builder probably had some regular
design in contemplation, and he has succeeded in executing it.

§ 15. It may be replied that if we extend our observations, say to the houses
of a large city, we shall then detect the property under discussion. The
different heights of a great number, when grouped together, might be found
to resemble those of a great number of human beings under similar
treatment. Something of this kind might not improbably be found to be the
case, though the resemblance would be far from being a close one. But to
raise this question is to get on to different ground, for we were speaking (as
remarked above) not of the work of different minds with their different
aims, but of that of one mind. In a multiplicity of designs, there may be that
variable uniformity, for which we may look in vain in a single design. The
heights which the different builders contemplated might be found to group
themselves into something of the same kind of uniformity as that which

prevails in most other things which they should undertake to do
independently. We might then trace the action of the same two conditions,
—a uniformity in the multitude of their different designs, a uniformity also
in the infinite variety of the influences which have modified those designs.
But this is a very different thing from saying that the work of one man will
show such a result as this. The difference is much like that between the
tread of a thousand men who are stepping without thinking of each other,
and their tread when they are drilled into a regiment. In the former case
there is the working, in one way or another, of a thousand minds; in the
latter, of one only.

The investigations of this and the former chapter constitute a sufficiently
close examination into the detailed causes by which the peculiar form of
statistical results with which we are concerned is actually produced, to
serve the purpose of a work which is occupied mainly with the methods of
the Science of Probability. The great importance, however, of certain
statistical or sociological enquiries will demand a recurrence in a future
chapter to one particular application of these statistics, viz. to those
concerned with some classes of human actions.

§ 16. The only important addition to, or modification of, the foregoing
remarks which I have found occasion to make is due to Mr Galton. He has
recently pointed out,—and was I believe the first to do so,—that in certain
cases some analysis of the causal processes can be effected, and is in fact
absolutely necessary in order to account for the facts observed. Take, for
instance, the heights of the population of any country. If the distribution or
dispersion of these about their mean value were left to the unimpeded
action of those myriad productive agencies alluded to above, we should
certainly obtain such an arrangement in the posterity of any one generation
as had already been exhibited in the parents. That is, we should find
repeated in the previous stage the same kind of order as we were trying to
account for in the following stage.

But then, as Mr Galton insists, if such agencies acted freely and
independently, though we should get the same kind of arrangement or
distribution, we should not get the same degree of it: there would, on the
contrary, be a tendency towards further dispersion. The ‘curve of facility’

(v. the diagram on p. 29) would belong to the same class, but would have a
different modulus. We shall see this at once if we take for comparison a
case in which similar agencies work their way without any counteraction
whatever. Suppose, for instance, that a large number of persons, whose
fortunes were equal to begin with, were to commence gambling or betting
continually for some small sum. If we examine their circumstances after
successive intervals of time, we should expect to find their fortunes
distributed according to the same general law,—i.e. the now familiar law in
question,—but we should also expect to find that the poorest ones were
slightly poorer, and the richest ones slightly richer, on each successive
occasion. We shall see more about this in a future chapter (on Gambling),
but it may be taken for granted here that there is nothing in the laws of
chance to resist this tendency towards intensifying the extremes.

Now it is found, on the contrary, in the case of vital phenomena,—for
instance in that of height, and presumably of most of the other qualities
which are in any way characteristic of natural kinds,—that there is, through
a number of successive generations, a remarkable degree of fixity. The tall
men are not taller, and the short men are not shorter, per cent. of the
population in successive generations: always supposing of course that some
general change of circumstances, such as climate, diet, &c. has not set in.
There must therefore here be some cause at work which tends, so to say, to
draw in the extremes and thus to check the otherwise continually increasing
dispersion.

§ 17. The facts were first tested by careful experiment. At the date of Mr
Galton's original paper on the subject,[3] there were no available statistics of
heights of human beings; so a physical element admitting of careful
experiment (viz. the size or weight of certain seeds) was accurately
estimated. From these data the actual amount of reversion from the
extremes, that is, of the slight pressure continually put upon the extreme
members with the result of crowding them back towards the mean, was
determined, and this was compared with what theory would require in order
to keep the characteristics of the species permanently fixed. Since then,
statistics have been obtained to a large extent which deal directly with the
heights of human beings.

The general conclusion at which we arrive is that there are several causes at
work which are neither slight nor independent. There is, for instance, the
observed fact that the extremes are as a rule not equally fertile with the
means, nor equally capable of resisting death and disease. Hence as regards
their mere numbers, there is a tendency for them somewhat to thin out.
Then again there is a distinct positive cause in respect of ‘reversion.’ Not
only are the offspring of the extremes less numerous, but these offspring
also tend to cluster about a mean which is, so to say, shifted a little towards
the true centre of the whole group; i.e. towards the mean offspring of the
mean parents.

§ 18. For a full discussion of these characteristics, and for a variety of most
ingenious illustrations of their mode of agency and of their comparative
efficacy, the reader may be referred to Mr Galton's original articles. For our
present purpose it will suffice to say that these characteristics tend towards
maintaining the fixity of species; and that though they do not affect what
may be called the general nature of the ‘probability curve’ or ‘law of
facility’, they do determine its precise value in the cases in question. If,
indeed, it be asked why there is no need for any such corrective influence in
the case of, say, firing at a mark: the answer is that there is no opening for it
except where a cumulative influence is introduced. The reason why the
fortunes of our betting party showed an ever increasing divergency, and
why some special correction was needed in order to avert such a tendency
in the case of vital phenomena, was that the new starting-point at every step
was slightly determined by the results of the previous step. The man who
has lost a shilling one time starts, next time, worse off by just a shilling;
and, but for the corrections we have been indicating, the man who was born
tall would, so to say, throw off his descendants from a vantage ground of
superior height. The true parallel in the case of the marksmen would be to
suppose that their new points of aim were always shifted a little in the
direction of the last divergence. The spreading out of the shot-marks would
then continue without limit, just as would the divergence of fortunes of the
supposed gamblers.

1 As stated above, this is really little more than a re-statement, a stage

further back, of the existence of the same kind of uniformity as that which
we are called upon to explain in the concrete details presented to us in
experience.
2
“It would seem in fact that in coarse and rude observations the errors
proceed from a very few principal causes, and in consequence our
hypothesis [as to the Exponential Law of Error] will probably represent the
facts only imperfectly, and the frequency of the errors will only
approximate roughly and vaguely to the law which follows from it. But
when astronomers, not content with the degree of accuracy they had
reached, prosecuted their researches into the remaining sources of error,
they found that not three or four, but a great number of minor sources of
error of nearly co-ordinate importance began to reveal themselves, having
been till then masked and overshadowed by the graver errors which had
been now approximately removed…. There were errors of graduation, and
many others in the contraction of instruments; other errors of their
adjustments; errors (technically so called) of observation; errors from the
changes of temperature, of weather, from slight irregular motions and
vibrations; in short, the thousand minute disturbing influences with which
modern astronomers are familiar.” (Extracted from a paper by Mr Crofton
in the Vol. of the Philosophical Transactions for 1870, p. 177.)
3 Typical Laws of Heredity; read before the Royal Institution, Feb. 9, 1877.

See also Journal of the Anthrop. Inst. Nov. 1885.

CHAPTER IV.

ON THE MODES OF ESTABLISHING AND
DETERMINING THE EXISTENCE AND
NUMERICAL PROPORTIONS OF THE
CHARACTERISTIC PROPERTIES OF OUR SERIES
OR GROUPS.
§ 1. At the point which we have now reached, we are supposed to be in
possession of series or groups of a certain kind, lying at the bottom, as one
may say, and forming the foundation on which the Science of Probability is
to be erected. We have described with sufficient particularity the
characteristics of such a series, and have indicated the process by which it
is, as a rule, actually brought about in nature. The next enquiries which have
to be successively made are, how in any particular case we are to establish
their existence and determine their special character and properties? and
secondly,[1] when we have obtained them, in what mode are they to be
employed for logical purposes?

The answer to the former enquiry does not seem difficult. Experience is our
sole guide. If we want to discover what is in reality a series of things, not a
series of our own conceptions, we must appeal to the things themselves to
obtain it, for we cannot find much help elsewhere. We cannot tell how many
persons will be born or die in a year, or how many houses will be burnt or
ships wrecked, without actually counting them. When we thus speak of
‘experience’ we mean to employ the term in its widest signification; we
mean experience supplemented by all the aids which inductive or deductive
logic can afford. When, for instance, we have found the series which
comprises the numbers of persons of any assigned class who die in
successive years, we have no hesitation in extending it some way into the
future as well as into the past. The justification of such a procedure must be
sought in the ordinary canons of Induction. As a special discussion will be
given upon the connection between Probability and Induction, no more
need be said upon this subject here; but nothing will be found there at
variance with the assertion just made, that the series we employ are
ultimately obtained by experience only.

§ 2. In many cases it is undoubtedly true that we do not resort to direct
experience at all. If I want to know what is my chance of holding ten
trumps in a game of whist, I do not enquire how often such a thing has
occurred before. If all the inhabitants of the globe were to divide themselves
up into whist parties they would have to keep on at it for a great many
years, if they wanted to settle the question satisfactorily in that way. What
we do of course is to calculate algebraically the proportion of possible
combinations in which ten trumps can occur, and take this as the answer to
our problem. So again, if I wanted to know the chance of throwing six with
a die whose faces were unequal, it would be a question if my best way
would not be to calculate geometrically the solid angle subtended at the
centre of gravity by the opposite face, and the ratio of this to the whole
surface of a sphere would represent sufficiently closely the chance required.

It is quite true that in such examples as the above, especially the former
one, nobody would ever think of appealing to statistics. This would be a
tedious process to adopt when, as here, the mechanical and other conditions
upon which the production of the events depend are comparatively few,
determinate, and admit of isolated consideration, whilst the enormous
number of combinations which can be constructed out of them causes an
enormous consequent multiplicity of ways in which the events can possibly
happen. Hence, in practice, à priori determination is often easy, whilst
à posteriori appeal to experience would be not merely tedious but utterly
impracticable. This, combined with the frequent simplicity and
attractiveness of such examples when deductively treated, has made them
very popular, and produced the impression in many quarters that they are
the proper typical instances to illustrate the theory of chance. Whereas, had
the science been concerned with those kinds of events only which in
practice are commonly made subjects of insurance, probably no other view
would ever have been taken than that it was based upon direct appeal to
experience.

§ 3. When, however, we look a little closer, we find that there is no occasion
for such a sharp distinction as that apparently implied between the two
classes of examples just indicated. In such cases as those of dice and cards,
even, in which we appear to reason directly from the determining
conditions, or possible variety of the events, rather than from actual

observation of their occurrence, we shall find that this procedure is only
valid by the help of a tacit assumption which can never be determined
otherwise than by direct experience. It is, no doubt, an exceedingly natural
and obvious assumption, and one which is continually deriving fresh weight
from every-day observation, but it is one which ought not to be admitted
without consideration. As this is a very important matter, not so much in
itself as in connection with the light which it throws upon the theory of the
subject, we will enter into a somewhat detailed examination of it.

Let us take a very simple example, that of tossing up a penny. Suppose that
I am contemplating a succession of two throws; I can see that the only
possible events are[2] HH, HT, TH, TT. So much is certain. We are
moreover tolerably well convinced from experience that these events occur,
in the long run, about equally often. This is of course admitted on all hands.
But on the view commonly maintained, it is contended that we might have
known the fact beforehand on grounds which are applicable to an indefinite
number of other and more complex cases. The form in which this view
would generally be advanced is, that we are enabled to state beforehand that
the four throws above mentioned are equally likely. If in return we ask what
is meant by the expression ‘equally likely’, it appears that there are two and
only two possible forms of reply. One of these seeks the explanation in the
state of mind of the observer, the other seeks it in some characteristic of the
things observed.

(1) It might, for instance, be said on the one hand, that what is meant is that
the four events contemplated are equally easy to imagine, or, more
accurately, that our expectation or belief in their occurrence is equal. We
could hardly be content with this reply, for the further enquiry would
immediately be urged, On what ground is this to be believed? What are the
characteristics of events of which our expectation is equal? If we consented
to give an answer to this further enquiry, we should be led to the second
form of reply, to be noticed directly; if we did not consent we should, it
seems, be admitting that Probability was only a portion of Psychology,
confined therefore to considering states of mind in themselves, rather than
in their reference to facts, viz. as being true or false. We should, that is, be
ceasing to make it a science of inference about things. This point will have
to be gone into more thoroughly in another chapter; but it is impossible to

direct attention too prominently to the fact that Logic (and therefore
Probability as a branch of Logic) is not concerned with what men do
believe, but with what they ought to believe, if they are to believe correctly.

(2) In the other form of reply the explanation of the phrase in question
would be sought, not in a state of mind, but in a quality of the things
contemplated. We might assign the following as the meaning, viz. that the
events really would occur with equal frequency in the long run. The ground
of this assertion would probably be found in past experience, and it would
doubtless be impossible so to frame the answer as to exclude the notion of
our belief altogether. But still there is a broad distinction between seeking
an equality in the amount of our belief, as before, and in the frequency of
occurrence of the events themselves, as here.

§ 4. When we have got as far as this it can readily be shown that an appeal
to experience cannot be long evaded. For can the assertion in question (viz.
that the throws of the penny will occur equally often) be safely made
à priori? Those who consider that it can seem hardly to have fully faced the
difficulties which meet them. For when we begin to enquire seriously
whether the penny will really do what is expected of it, we find that
restrictions have to be introduced. In the first place it must be an ideal coin,
with its sides equal and fair. This restriction is perfectly intelligible; the
study of solid geometry enables us to idealize a penny into a circular or
cylindrical lamina. But this condition by itself is not sufficient, others are
wanted as well. The penny was supposed to be tossed up, as we say ‘at
random.’ What is meant by this, and how is this process to be idealized? To
ask this is to introduce no idle subtlety; for it would scarcely be maintained
that the heads and tails would get their fair chances if, immediately before
the throwing, we were so to place the coin in our hands as to start it always
with the same side upwards. The difference that would result in
consequence, slight as its cause is, would tend in time to show itself in the
results. Or, if we persisted in starting with each of the two sides alternately
upwards, would the longer repetitions of the same side get their fair chance?

Perhaps it will be replied that if we think nothing whatever about these
matters all will come right of its own accord. It may, and doubtless will be
so, but this is falling back upon experience. It is here, then, that we find

ourselves resting on the experimental assumption above mentioned, and
which indeed cannot be avoided. For suppose, lastly, that the circumstances
of nature, or my bodily or mental constitution, were such that the same side
always is started upwards, or indeed that they are started in any arbitrary
order of our own? Well, it will be replied, it would not then be a fair trial. If
we press in this way for an answer to such enquiries, we shall find that
these tacit restrictions are really nothing else than a mode of securing an
experimental result. They are only another way of saying, Let a series of
actions be performed in such a way as to secure a sequence of a particular
kind, viz., of the kind described in the previous chapters.

§ 5. An intermediate way of evading the direct appeal to experience is
sometimes found by defining the probability of an event as being measured
by the ratio which the number of cases favourable to the event bears to the
total number of cases which are possible. This seems a somewhat loose and
ambiguous way of speaking. It is clearly not enough to count the number of
cases merely, they must also be valued, since it is not certain that each is
equally potent in producing the effect. This, of course, would never be
denied, but sufficient importance does not seem to be attached to the fact
that we have really no other way of valuing them except by estimating the
effects which they actually do, or would produce. Instead of thus appealing
to the proportion of cases favourable to the event, it is far better (at least as
regards the foundation of the science, for we are not at this moment
discussing the practical method of facilitating our calculations) to appeal at
once to the proportion of cases in which the event actually occurs.

§ 6. The remarks above made will apply, of course, to most of the other
common examples of chance; the throwing of dice, drawing of cards, of
balls from bags, &c. In the last case, for instance, one would naturally be
inclined to suppose that a ball which had just been put back would thereby
have a better chance of coming out again next time, since it will be more in
the way for that purpose. How is this to be prevented? If we designedly
thrust it to the middle or bottom of the others, we may overdo the
precaution; and are in any case introducing human design, that element so
essentially hostile to all that we understand by chance. If we were to trust to
a good shake setting matters right, we may easily be deceived; for shaking
the bag can hardly do more than diminish the disposition of those balls

which were already in each other's neighbourhood, to remain so. In the
consequent interaction of each upon all, the arrangement in which they start
cannot but leave its impress to some extent upon their final positions. In all
such cases, therefore, if we scrutinize our language, we shall find that any
supposed à priori mode of stating a problem is little else than a
compendious way of saying, Let means be taken for obtaining a given
result. Since it is upon this result that our inferences ultimately rest, it seems
simpler and more philosophical to appeal to it at once as the groundwork of
our science.

§ 7. Let us again take the instance of the tossing of a penny, and examine it
somewhat more minutely, to see what can be actually proved about the
results we shall obtain. We are willing to give the pence fair treatment by
assuming that they are perfect, that is, that in the long run they show no
preference for either head or tail; the question then remains, Will the
repetitions of the same face obtain the proportional shares to which they are
entitled by the usual interpretations of the theory? Putting then, as before,
for the sake of brevity, H for head, and HH for heads twice running, we are
brought to this issue;—Given that the chance of H is 12, does it follow
necessarily that the chance of HH (with two pence) is 14? To say nothing of
‘H ten times’ occurring once in 1024 times (with ten pence), need it occur at
all? The mathematicians, for the most part, seem to think that this
conclusion follows necessarily from first principles; to me it seems to rest
upon no more certain evidence than a reasonable extension by Induction.

Taking then the possible results which can be obtained from a pair of pence,
what do we find? Four different results may follow, namely, (1) HT, (2) HH,
(3) TH, (4) TT. If it can be proved that these four are equally probable, that
is, occur equally often, the commonly accepted conclusions will follow, for
a precisely similar argument would apply to all the larger numbers.

§ 8. The proof usually advanced makes use of what is called the Principle of
Sufficient Reason. It takes this form;—Here are four kinds of throws which
may happen; once admit that the separate elements of them, namely, H
and T, happen equally often, and it will follow that the above combinations

will also happen equally often, for no reason can be given in favour of one
of them that would not equally hold in favour of the others.

To a certain extent we must admit the validity of the principle for the
purpose. In the case of the throws given above, it would be valid to prove
the equal frequency of (1) and (3) and also of (2) and (4); for there is no
difference existing between these pairs except what is introduced by our
own notation.[3] TH is the same as HT, except in the order of the occurrence
of the symbols H and T, which we do not take into account. But either of
the pair (1) and (3) is different from either of the pair (2) and (4). Transpose
the notation, and there would still remain here a distinction which the mind
can recognize. A succession of the same thing twice running is
distinguished from the conjunction of two different things, by a distinction
which does not depend upon our arbitrary notation only, and would remain
entirely unaltered by a change in this notation. The principle therefore of
Sufficient Reason, if admitted, would only prove that doublets of the two
kinds, for example (2) and (4), occur equally often, but it would not prove
that they must each occur once in four times. It cannot be proved indeed in
this way that they need ever occur at all.

§ 9. The formula, then, not being demonstrable à priori, (as might have
been concluded,) can it be obtained by experience? To a certain extent it
can; the present experience of mankind in pence and dice seems to show
that the smaller successions of throws do really occur in about the
proportions assigned by the theory. But how nearly they do so no one can
say, for the amount of time and trouble to be expended before we could feel
that we have verified the fact, even for small numbers, is very great, whilst
for large numbers it would be simply intolerable. The experiment of
throwing often enough to obtain ‘heads ten times’ has been actually
performed by two or three persons, and the results are given by De Morgan,
and Jevons.[4] This, however, being only sufficient on the average to give
‘heads ten times’ a single chance, the evidence is very slight; it would take a
considerable number of such experiments to set the matter nearly at rest.

Any such rule, then, as that which we have just been discussing, which
professes to describe what will take place in a long succession of throws, is
only conclusively proved by experience within very narrow limits, that is,

for small repetitions of the same face; within limits less narrow, indeed, we
feel assured that the rule cannot be flagrantly in error, otherwise the
variation would be almost sure to be detected. From this we feel strongly
inclined to infer that the same law will hold throughout. In other words, we
are inclined to extend the rule by Induction and Analogy. Still there are so
many instances in nature of proposed laws which hold within narrow limits
but get egregiously astray when we attempt to push them to great lengths,
that we must give at best but a qualified assent to the truth of the formula.

§ 10. The object of the above reasoning is simply to show that we cannot be
certain that the rule is true. Let us now turn for a minute to consider the
causes by which the succession of heads and tails is produced, and we may
perhaps see reasons to make us still more doubtful.

It has been already pointed out that in calculating probabilities à priori, as it
is called, we are only able to do so by introducing restrictions and
suppositions which are in reality equivalent to assuming the expected
results. We use words which in strictness mean, Let a given process be
performed; but an analysis of our language, and an examination of various
tacit suppositions which make themselves felt the moment they are not
complied with, soon show that our real meaning is, Let a series of a given
kind be obtained; it is to this series only, and not to the conditions of its
production, that all our subsequent calculations properly apply. The
physical process being performed, we want to know whether anything
resembling the contemplated series really will be obtained.

Now if the penny were invariably set the same side uppermost, and thrown
with the same velocity of rotation and to the same height, &c.—in a word,
subjected to the same conditions,—it would always come down with the
same side uppermost. Practically, we know that nothing of this kind occurs,
for the individual variations in the results of the throws are endless. Still
there will be an average of these conditions, about which the throws will be
found, as it were, to cluster much more thickly than elsewhere. We should
be inclined therefore to infer that if the same side were always set
uppermost there would really be a departure from the sort of series which
we ordinarily expect. In a very large number of throws we should probably
begin to find, under such circumstances, that either head or tail was having

a preference shown to it. If so, would not similar effects be found to be
connected with the way in which we started each successive pair of throws?
According as we chose to make a practice of putting HH or TT uppermost,
might there not be a disturbance in the proportion of successions of two
heads or two tails? Following out this train of reasoning, it would seem to
point with some likelihood to the conclusion that in order to obtain a series
of the kind we expect, we should have to dispose the antecedents in a
similar series at the start. The changes and chances produced by the act of
throwing might introduce infinite individual variations, and yet there might
be found, in the very long run, to be a close similarity between these two
series.

§ 11. This is, to a certain extent, only shifting the difficulty, I admit; for the
claim formerly advanced about the possibility of proving the proportions of
the throws in the former series, will probably now be repeated in favour of
those in the latter. Still the question is very much narrowed, for we have
reduced it to a series of voluntary acts. A man may put whatever side he
pleases uppermost. He may act consciously, as I have said, or he may think
nothing whatever about the matter, that is, throw at random; if so, it will
probably be asserted by many that he will involuntarily produce a series of
the kind in question. It may be so, or it may not; it does not seem that there
are any easily accessible data by which to decide. All that I am concerned
with here is to show the likelihood that the commonly received result does
in reality depend upon the fulfilment of a certain condition at the outset, a
condition which it is certainly optional with any one to fulfil or not as he
pleases. The short successions doubtless will take care of themselves, owing
to the infinite complications produced by the casual variations in throwing;
but the long ones may suffer, unless their interest be consciously or
unconsciously regarded at the outset.

§ 12. The advice, ‘Only try long enough, and you will sooner or later get
any result that is possible,’ is plausible, but it rests only on Induction and
Analogy; mathematics do not prove it. As has been repeatedly stated, there
are two distinct views of the subject. Either we may, on the one hand, take a
series of symbols, call them heads and tails; H, T, &c.; and make the
assumption that each of these, and each pair of them, and so on, will occur
in the long run with a regulated degree of frequency. We may then calculate

their various combinations, and the consequences that may be drawn from
the data assumed. This is a purely algebraical process; it is infallible; and
there is no limit whatever to the extent to which it may be carried. This way
of looking at the matter may be, and undoubtedly should be, nothing more
than the counterpart of what I have called the substituted or idealized series
which generally has to be introduced as the basis of our calculation. The
danger to be guarded against is that of regarding it too purely as an
algebraical conception, and thence of sinking into the very natural errors
both of too readily evolving it out of our own consciousness, and too freely
pushing it to unwarranted lengths.

Or on the other hand, we may consider that we are treating of the behaviour
of things;—balls, dice, births, deaths, &c.; and drawing inferences about
them. But, then, what were in the former instance allowable assumptions,
become here propositions to be tested by experience. Now the whole theory
of Probability as a practical science, in fact as anything more than an
algebraical truth, depends of course upon there being a close
correspondence between these two views of the subject, in other words,
upon our substituted series being kept in accordance with the actual series.
Experience abundantly proves that, between considerable limits, in the
example in question, there does exist such a correspondence. But let no one
attempt to enforce our assent to every remote deduction that
mathematicians can draw from their formulæ. When this is attempted the
distinction just traced becomes prominent and important, and we have to
choose our side. Either we go over to the mathematics, and so lose all right
of discussion about the things; or else we take part with the things, and so
defy the mathematics. We do not question the formal accuracy of the latter
within their own province, but either we dismiss them as somewhat
irrelevant, as applying to data of whose correctness we cannot be certain, or
we take the liberty of remodelling them so as to bring them into accordance
with facts.

§ 13. A critic of any doctrine can hardly be considered to have done much
more than half his duty when he has explained and justified his grounds for
objecting to it. It still remains for him to indicate, if only in a few words,
what he considers its legitimate functions and position to be, for it can
seldom happen that he regards it as absolutely worthless or unmeaning. I

should say, then, that when Probability is thus divorced from direct
reference to objects, as it substantially is by not being founded upon
experience, it simply resolves itself into the common algebraical or
arithmetical doctrine of Permutations and Combinations.[5] The
considerations upon which these depend are purely formal and necessary,
and can be fully reasoned out without any appeal to experience. We there
start from pure considerations of number or magnitude, and we terminate
with them, having only arithmetical calculations to connect them together. I
wish, for instance, to find the chance of throwing heads three times running
with a penny. All I have to do is first to ascertain the possible number of
throws. Permutations tell me that with two things thus in question (viz. head
and tail) and three times to perform the process, there are eight possible
forms of the result. Of these eight one only being favourable, the chance in
question is pronounced to be one-eighth.

Now though it is quite true that the actual calculation of every chance
problem must be of the above character, viz. an algebraical or arithmetical
process, yet there is, it seems to me, a broad and important distinction
between a material science which employs mathematics, and a formal one
which consists of nothing but mathematics. When we cut ourselves off from
the necessity of any appeal to experience, we are retaining only the
intermediate or calculating part of the investigation; we may talk of dice, or
pence, or cards, but these are really only names we choose to give to our
symbols. The H's and T's with which we deal have no bearing on objective
occurrences, but are just like the x's and y's with which the rest of algebra
deals. Probability in fact, when so treated, seems to be absolutely nothing
else than a system of applied Permutations and Combinations.

It will now readily be seen how narrow is the range of cases to which any
purely deductive method of treatment can apply. It is almost entirely
confined to such employments as games of chance, and, as already pointed
out, can only be regarded as really trustworthy even there, by the help of
various tacit restrictions. This alone would be conclusive against the theory
of the subject being rested upon such a basis. The experimental method, on
the other hand, is, in the same theoretical sense, of universal application. It
would include the ordinary problems furnished by games of chance, as well
as those where the dice are loaded and the pence are not perfect, and also

the indefinitely numerous applications of statistics to the various kinds of
social phenomena.

§ 14. The particular view of the deductive character of Probability above
discussed, could scarcely have intruded itself into any other examples than
those of the nature of games of chance, in which the conditions of
occurrence are by comparison few and simple, and are amenable to accurate
numerical determination. But a doctrine, which is in reality little else than
the same theory in a slightly disguised form, is very prevalent, and has been
applied to truths of the most purely empirical character. This doctrine will
be best introduced by a quotation from Laplace. After speaking of the
irregularity and uncertainty of nature as it appears at first sight, he goes on
to remark that when we look closer we begin to detect “a striking regularity
which seems to suggest a design, and which some have considered a proof
of Providence. But, on reflection, it is soon perceived that this regularity is
nothing but the development of the respective probabilities of the simple
events, which ought to occur more frequently according as they are more
probable.”[6]

If this remark had been made about the succession of heads and tails in the
throwing up of a penny, it would have been intelligible. It would simply
mean this: that the constitution of the body was such that we could
anticipate with some confidence what the result would be when it was
treated in a certain way, and that experience would justify our anticipation
in the long run. But applied as it is in a more general form to the facts of
nature, it seems really to have but little meaning in it. Let us test it by an
instance. Amidst the irregularity of individual births, we find that the male
children are to the female, in the long run, in about the proportion of 106
to 100. Now if we were told that there is nothing in this but “the
development of their respective probabilities,” would there be anything in
such a statement but a somewhat pretentious re-statement of the fact already
asserted? The probability is nothing but that proportion, and is
unquestionably in this case derived from no other source but the statistics
themselves; in the above remark the attempt seems to be made to invert this
process, and to derive the sequence of events from the mere numerical
statement of the proportions in which they occur.

§ 15. It will very likely be replied that by the probability above mentioned is
meant, not the mere numerical proportion between, the births, but some fact
in our constitution upon which this proportion depends; that just as there
was a relation of equality between the two sides of the penny, which
produced the ultimate equality in the number of heads and tails, so there
may be something in our constitution or circumstances in the proportion of
106 to 100, which produces the observed statistical result. When this
something, whatever it might be, was discovered, the observed numbers
might be supposed capable of being determined beforehand. Even if this
were the case, however, it must not be forgotten that there could hardly fail
to be, in combination with such causes, other concurrent conditions in order
to produce the ultimate result; just as besides the shape of the penny, we had
also to take into account the nature of the ‘randomness’ with which it was
tossed. What these may be, no one at present can undertake to say, for the
best physiologists seem indisposed to hazard even a guess upon the subject.
[7] But without going into particulars, one may assert with some confidence

that these conditions cannot well be altogether independent of the health,
circumstances, manners and customs, &c. (to express oneself in the vaguest
way) of the parents; and if once these influencing elements are introduced,
even as very minute factors, the results cease to be dependent only on fixed
and permanent conditions. We are at once letting in other conditions, which,
if they also possess the characteristics that distinguish Probability (an
exceedingly questionable assumption), must have that fact specially proved
about them. That this should be the case indeed seems not merely
questionable, but almost certainly impossible; for these conditions partaking
of the nature of what we term generally, Progress and Civilization, cannot
be expected to show any permanent disposition to hover about an average.

§ 16. The reader who is familiar with Probability is of course acquainted
with the celebrated theorem of James Bernoulli. This theorem, of which the
examples just adduced are merely particular cases, is generally expressed
somewhat as follows:—in the long run all events will tend to occur with a
relative frequency proportional to their objective probabilities. With the
mathematical proof of this theorem we need not trouble ourselves, as it lies
outside the province of this work; but indeed if there is any value in the
foregoing criticism, the basis on which the mathematics rest is faulty, owing

to there being really nothing which we can with propriety call an objective
probability.

If one might judge by the interpretation and uses to which this theorem is
sometimes exposed, we should regard it as one of the last remaining relics
of Realism, which after being banished elsewhere still manages to linger in
the remote province of Probability. It would be an illustration of the
inveterate tendency to objectify our conceptions, even in cases where the
conceptions had no right to exist at all. A uniformity is observed;
sometimes, as in games of chance, it is found to be so connected with the
physical constitution of the bodies employed as to be capable of being
inferred beforehand; though even here the connection is by no means so
necessary as is commonly supposed, owing to the fact that in addition to
these bodies themselves we have also to take into account their relation to
the agencies which influence them. This constitution is then converted into
an ‘objective probability’, supposed to develop into the sequence which
exhibits the uniformity. Finally, this very questionable objective probability
is assumed to exist, with the same faculty of development, in all the cases in
which uniformity is observed, however little resemblance there may be
between these and games of chance.

§ 17. How utterly inappropriate any such conception is in most of the cases
in which we find statistical uniformity, will be obvious on a moment's
consideration. The observed phenomena are generally the product, in these
cases, of very numerous and complicated antecedents. The number of
crimes, for instance, annually committed in any society, is a function
amongst other things, of the strictness of the law, the morality of the people,
their social condition, and the vigilance of the police, each of these
elements being in itself almost infinitely complex. Now, as a result of all
these agencies, there is some degree of uniformity; but what has been called
above the change of type, which it sooner or later tends to display, is
unmistakeable. The average annual numbers do not show a steady gradual
approach towards what might be considered in some sense a limiting value,
but, on the contrary, fluctuate in a way which, however it may depend upon
causes, shows none of the permanent uniformity which is characteristic of
games of chance. This fact, combined with the obvious arbitrariness of
singling out, from amongst the many and various antecedents which

produced the observed regularity, a few only, which should constitute the
objective probability (if we took all, the events being absolutely
determined, there would be no occasion for an appeal to probability in the
case), would have been sufficient to prevent any one from assuming the
existence of any such thing, unless the mistaken analogy of other cases had
predisposed him to seek for it.

There is a familiar practical form of the same error, the tendency to which
may not improbably be derived from a similar theoretical source. It is that
of continuing to accumulate our statistical data to an excessive extent. If the
type were absolutely fixed we could not possibly have too many statistics;
the longer we chose to take the trouble of collecting them the more accurate
our results would be. But if the type is changing, in other words, if some of
the principal causes which aid in their production have, in regard to their
present degree of intensity, strict limits of time or space, we shall do harm
rather than good if we overstep these limits. The danger of stopping too
soon is easily seen, but in avoiding it we must not fall into the opposite
error of going on too long, and so getting either gradually or suddenly under
the influence of a changed set of circumstances.

§ 18. This chapter was intended to be devoted to a consideration, not of the
processes by which nature produces the series with which we are
concerned, but of the theoretic basis of the methods by which we can
determine the existence of such series. But it is not possible to keep the two
enquiries apart, for here, at any rate, the old maxim prevails that to know a
thing we must know its causes. Recur for a minute to the considerations of
the last chapter. We there saw that there was a large class of events, the
conditions of production of which could be said to consist of (1) a
comparatively few nearly unchangeable elements, and (2) a vast number of
independent and very changeable elements. At least if there were any other
elements besides these, we are assumed either to make special allowance
for them, or to omit them from our enquiry. Now in certain cases, such as
games of chance, the unchangeable elements may without practical error be
regarded as really unchangeable throughout any range of time and space.
Hence, as a result, the deductive method of treatment becomes in their case
at once the most simple, natural, and conclusive; but, as a further
consequence, the statistics of the events, if we choose to appeal to them,

may be collected ad libitum with better and better approximation to truth.
On the other hand, in all social applications of Probability, the
unchangeable causes can only be regarded as really unchangeable under
many qualifications. We know little or nothing of them directly; they are
often in reality numerous, indeterminate, and fluctuating; and it is only
under the guarantee of stringent restrictions of time and place, that we can
with any safety attribute to them sufficient fixity to justify our theory.
Hence, as a result, the deductive method, under whatever name it may go,
becomes totally inapplicable both in theory and practice; and, as a further
consequence, the appeal to statistics has to be made with the caution in
mind that we shall do mischief rather than good if we go on collecting too
many of them.

§ 19. The results of the last two chapters may be summed up as follows:—
We have extended the conception of a series obtained in the first chapter;
for we have found that these series are mostly presented to us in groups.
These groups are found upon examination to be formed upon approximately
the same type throughout a very wide and varied range of experience; the
causes of this agreement we discussed and explained in some detail. When,
however, we extend our examination by supposing the series to run to a
very great length, we find that they may be divided into two classes
separated by important distinctions. In one of these classes (that containing
the results of games of chance) the conditions of production, and
consequently the laws of statistical occurrence, may be practically regarded
as absolutely fixed; and the extent of the divergences from the mean seem
to know no finite limit. In the other class, on the contrary (containing the
bulk of ordinary statistical enquiries), the conditions of production vary
with more or less rapidity, and so in consequence do the results. Moreover it
is often impossible that variations from the mean should exceed a certain
amount. The former we may term ideal series. It is they alone which show
the requisite characteristics with any close approach to accuracy, and to
make the theory of the subject tenable, we have really to substitute one of
this kind for one of the less perfect ones of the other class, when these latter
are under treatment. The former class have, however, been too exclusively
considered by writers on the subject; and conceptions appropriate only to
them, and not always even to them, have been imported into the other class.

It is in this way that a general tendency to an excessive deductive or à priori
treatment of the science has been encouraged.

1 This latter enquiry belongs to what may be termed the more purely logical

part of this volume, and is entered on in the course of Chapter VI.
2
For the use of those not acquainted with the common notation employed
in this subject, it may be remarked that HH is simply an abbreviated way of
saying that the two successive throws of the penny give head; HT that the
first of them gives head, and the second tail; and so on with the remaining
symbols.
3 I am endeavouring to treat this rule of Sufficient Reason in a way that

shall be legitimate in the opinion of those who accept it, but there seem very
great doubts whether a contradiction is not involved when we attempt to
extract results from it. If the sides are absolutely alike, how can there he any
difference between the terms of the series? The succession seems then
reduced to a dull uniformity, a mere iteration of the same thing many times;
the series we contemplated has disappeared. If the sides are not absolutely
alike, what becomes of the applicability of the rule?
4 Formal Logic, p. 185. Principles of Science, p. 208.

5
The close connection between these subjects is well indicated in the title
of Mr Whitworth's treatise, Choice and Chance.
6 Essai Philosophique. Ed. 1825, p. 74.

7 An opinion prevailed rather at one time (quoted and supported by Quetelet

amongst others) that the relative ages of the parents had something to do
with the sex of the offspring. If this were so, it would quite bear out the
above remarks. As a matter of fact, it should be observed, that the
proportion of 106 to 100 does not seem by any means universal in all

countries or at all times. For various statistical tables on the subject see
Quetelet, Physique Sociale, Vol. I. 166, 173, 238.

CHAPTER V.

THE CONCEPTION RANDOMNESS AND ITS
SCIENTIFIC TREATMENT.
§ 1. There is a term of frequent occurrence in treatises on Probability, and
which we have already had repeated occasion to employ, viz. the
designation random applied to an event, as in the expression ‘a random
distribution’. The scientific conception involved in the correct use of this
term is, I apprehend, nothing more than that of aggregate order and
individual irregularity (or apparent irregularity), which has been already
described in the preceding chapters. A brief discussion of the requisites in
this scientific conception, and in particular of the nature and some of the
reasons for the departure from the popular conception, may serve to clear
up some of the principal remaining difficulties which attend this part of our
subject.

The original,[1] and still popular, signification of the term is of course
widely different from the scientific. What it looks to is the origin, not the
results, of the random performance, and it has reference rather to the single
action than to a group or series of actions. Thus, when a man draws a bow
‘at a venture’, or ‘at random’, we mean only to point out the aimless
character of the performance; we are contrasting it with the definite
intention to hit a certain mark. But it is none the less true, as already pointed
out, that we can only apply processes of inference to such performances as
these when we regard them as being capable of frequent, or rather of
indefinitely extended repetition.

Begin with an illustration. Perhaps the best typical example that we can
give of the scientific meaning of random distribution is afforded by the
arrangement of the drops of rain in a shower. No one can give a guess
whereabouts at any instant a drop will fall, but we know that if we put out a
sheet of paper it will gradually become uniformly spotted over; and that if
we were to mark out any two equal areas on the paper these would
gradually tend to be struck equally often.

§ 2. I. Any attempt to draw inferences from the assumption of random
arrangement must postulate the occurrence of this particular state of things
at some stage or other. But there is often considerable difficulty, leading
occasionally to some arbitrariness, in deciding the particular stage at which
it ought to be introduced.

(1) Thus, in many of the problems discussed by mathematicians, we look as
entirely to the results obtained, and think as little of the actual process by
which they are obtained, as when we are regarding the arrangement of the
drops of rain. A simple example of this kind would be the following. A
pawn, diameter of base one inch, is placed at random on a chess-board, the
diameter of the squares of which is one inch and a quarter: find the chance
that its base shall lie across one of the intersecting lines. Here we may
imagine the pawns to be so to say rained down vertically upon the board,
and the question is to find the ultimate proportion of those which meet a
boundary line to the total of those which fall. The problem therefore
becomes a merely geometrical one, viz. to determine the ratio of a certain
area on the board to the whole area. The determination of this ratio is all
that the mathematician ever takes into account.

Now take the following. A straight brittle rod is broken at random in two
places: find the chance that the pieces can make a triangle.[2] Since the only
condition for making a triangle with three straight lines is that each two
shall be greater than the third, the problem seems to involve the same
general conception as in the former case. We must conceive such rods
breaking at one pair of spots after another,—no one can tell precisely
where,—but showing the same ultimate tendency to distribute these spots
throughout the whole length uniformly. As in the last case, the
mathematician thinks of nothing but this final result, and pays no heed to
the process by which it may be brought about. Accordingly the problem is
again reduced to one of mensuration, though of a somewhat more
complicated character.

§ 3. (2) In another class of cases we have to contemplate an intermediate
process rather than a final result; but the same conception has to be
introduced here, though it is now applied to the former stage, and in
consequence will not in general apply to the latter.

For instance: a shot is fired at random from a gun whose maximum range
(i.e. at 45° elevation) is 3000 yards: what is the chance that the actual range
shall exceed 2000 yards? The ultimately uniform (or random) distribution
here is commonly assumed to apply to the various directions in which the
gun can be pointed; all possible directions above the horizontal being
equally represented in the long run. We have therefore to contemplate a
surface of uniform distribution, but it will be the surface, not of the ground,
but of a hemisphere whose centre is occupied by the man who fires. The
ultimate distribution of the bullets on the spots where they strike the ground
will not be uniform. The problem is in fact to discover the law of variation
of the density of distribution.

The above is, I presume, the treatment generally adopted in solving such a
problem. But there seems no absolute necessity for any such particular
choice. It is surely open to any one to maintain[3] that his conception of the
randomness of the firing is assigned by saying that it is likely that a man
should begin by facing towards any point of the compass indifferently, and
then proceed to raise his gun to any angle indifferently. The stage of
ultimately uniform distribution here has receded a step further back. It is not
assigned directly to the surface of an imaginary hemisphere, but to the lines
of altitude and azimuth drawn on that surface. Accordingly, the distribution
over the hemisphere itself will not now be uniform,—there will be a
comparative crowding up towards the pole,—and the ultimate distribution
over the ground will not be the same as before.

§ 4. Difficulties of this kind, arising out of the uncertainty as to what stage
should be selected for that of uniform distribution, will occasionally present
themselves. For instance: let a book be taken at random out of a bookcase;
what is the chance of hitting upon some assigned volume? I hardly know
how this question would commonly be treated. If we were to set our man
opposite the middle of the shelf and inquire what would generally happen in
practice, supposing him blindfolded, there cannot be much doubt that the
volumes would not be selected equally often. On the contrary, it is likely
that there would be a tendency to increased frequency about a centre
indicated by the height of his shoulder, and (unless he be left-handed) a
trifle to the right of the point exactly opposite his starting point.

If the question were one which it were really worth while to work out on
these lines we should be led a long way back. Just as we imagined our
rifleman's position (on the second supposition) to be determined by two
independent coordinates of assumed continuous and equal facility, so we
might conceive our making the attempt to analyse the man's movements
into a certain number of independent constituents. We might suppose all the
various directions from his starting point, along the ground, to be equally
likely; and that when he reaches the shelves the random motion of his hand
is to be regulated after the fashion of a shot discharged at random.

The above would be one way of setting about the statement of the problem.
But the reader will understand that all which I am here proposing to
maintain is that in these, as in every similar case, we always encounter,
under this conception of ‘randomness’, at some stage or other, this postulate
of ultimate uniformity of distribution over some assigned magnitude: either
time; or space, linear, superficial, or solid. But the selection of the stage at
which this is to be applied may give rise to considerable difficulty, and even
arbitrariness of choice.

§ 5. Some years ago there was a very interesting discussion upon this
subject carried on in the mathematical part of the Educational Times (see,
especially, Vol. VII.). As not unfrequently happens in mathematics there
was an almost entire accord amongst the various writers as to the
assumptions practically to be made in any particular case, and therefore as
to the conclusion to be drawn, combined with a very considerable amount
of difference as to the axioms and definitions to be employed. Thus Mr
M. W. Crofton, with the substantial agreement of Mr Woolhouse, laid it
down unhesitatingly that “at random” has “a very clear and definite
meaning; one which cannot be better conveyed than by Mr Wilson's
definition, ‘according to no law’; and in this sense alone I mean to use it.”
According to any scientific interpretation of ‘law’ I should have said that
where there was no law there could be no inference. But ultimate tendency
towards equality of distribution is as much taken for granted by Mr Crofton
as by any one else: in fact he makes this a deduction from his definition:
—“As this infinite system of parallels are drawn according to no law, they
are as thickly disposed along any part of the [common] perpendicular as
along any other” (VII. p. 85). Mr Crofton holds that any kind of unequal

distribution would imply law,—“If the points [on a plane] tended to become
denser in any part of the plane than in another, there must be some law
attracting them there” (ib. p. 84). The same view is enforced in his paper on
Local Probability (in the Phil. Trans., Vol. 158). Surely if they tend to
become equally dense this is just as much a case of regularity or law.

It may be remarked that wherever any serious practical consequences turn
upon duly securing the desired randomness, it is always so contrived that no
design or awkwardness or unconscious one-sidedness shall disturb the
result. The principal case in point here is of course afforded by games of
chance. What we want, when we toss a die, is to secure that all numbers
from 1 to 6 shall be equally often represented in the long run, but that no
person shall be able to predict the individual occurrence. We might, in our
statement of a problem, as easily postulate ‘a number thought of at random’
as ‘a shot fired at random’, but no one would risk his chances of gain and
loss on the supposition that this would be done with continued fairness.
Accordingly, we construct a die whose sides are accurately alike, and it is
found that we may do almost what we like with this, at any previous stage
to that of its issue from the dice box on to the table, without interfering with
the random nature of the result.

§ 6. II. Another characteristic in which the scientific conception seems to
me to depart from the popular or original signification is the following. The
area of distribution which we take into account must be a finite or limited
one. The necessity for this restriction may not be obvious at first sight, but
the consideration of one or two examples will serve to indicate the point at
which it makes itself felt. Suppose that one were asked to choose a number
at random, not from a finite range, but from the inexhaustible possibilities
of enumeration. In the popular sense of the term,—i.e. of uttering a number
without pausing to choose,—there is no difficulty. But a moment's
consideration will show that no arrangement even tending towards
ultimately uniform distribution can be secured in this way. No average
could be struck with ever increasing steadiness. So with spatial infinity. We
can rationally speak of choosing a point at random in a given straight line,
area, or volume. But if we suppose the line to have no end, or the selection
to be made in infinite space, the basis of ultimate tendency towards what
may be called the equally thick deposit of our random points fails us utterly.

Similarly in any other example in which one of the magnitudes is unlimited.
Suppose I fling a stick at random in a horizontal plane against a row of iron
railings and inquire for the chance of its passing through without touching
them. The problem bears some analogy to that of the chessmen, and so far
as the motion of translation of the stick is concerned (if we begin with this)
it presents no difficulty. But as regards the rotation it is otherwise. For any
assigned linear velocity there is a certain angular velocity below which the
stick may pass through without contact, but above which it cannot. And
inasmuch as the former range is limited and the latter is unlimited, we
encounter the same impossibility as before in endeavouring to conceive a
uniform distribution. Of course we might evade this particular difficulty by
beginning with an estimate of the angular velocity, when we should have to
repeat what has just been said, mutatis mutandis, in reference to the linear
velocity.

§ 7. I am of course aware that there are a variety of problems current which
seem to conflict with what has just been said, but they will all submit to
explanation. For instance; What is the chance that three straight lines, taken
or drawn at random, shall be of such lengths as will admit of their forming a
triangle? There are two ways in which we may regard the problem. We may,
for one thing, start with the assumption of three lines not greater than a
certain length n, and then determine towards what limit the chance tends as
n increases unceasingly. Or, we may maintain that the question is merely
one of relative proportion of the three lines. We may then start with any
magnitude we please to represent one of the lines (for simplicity, say, the
longest of them), and consider that all possible shapes of a triangle will be
represented by varying the lengths of the other two. In either case we get a
definite result without need to make an attempt to conceive any random
selection from the infinity of possible length.

So in what is called the “three-point problem”:—Three points in space are
selected at random; find the chance of their forming an acute-angled
triangle. What is done is to start with a closed volume,—say a sphere, from
its superior simplicity,—find the chance (on the assumption of uniform
distribution within this volume); and then conceive the continual
enlargement without limit of this sphere. So regarded the problem is
perfectly consistent and intelligible, though I fail to see why it should be

termed a random selection in space rather than in a sphere. Of course if we
started with a different volume, say a cube, we should get a different result;
and it is therefore contended (e.g. by Mr Crofton in the Educational Times,
as already referred to) that infinite space is more naturally and appropriately
regarded as tended towards by the enlargement of a sphere than by that of a
cube or any other figure.

Again: A group of integers is taken at random; show that the number thus
taken is more likely to be odd than even. What we do in answering this is to
start with any finite number n, and show that of all the possible
combinations which can be made within this range there are more odd than
even. Since this is true irrespective of the magnitude of n, we are apt to
speak as if we could conceive the selection being made at random from the
true infinity contemplated in numeration.

§ 8. Where these conditions cannot be secured then it seems to me that the
attempt to assign any finite value to the probability fails. For instance, in the
following problem, proposed by Mr J. M. Wilson, “Three straight lines are
drawn at random on an infinite plane, and a fourth line is drawn at random
to intersect them: find the probability of its passing through the triangle
formed by the other three” (Ed. Times, Reprint, Vol. V. p. 82), he offers the
following solution: “Of the four lines, two must and two must not pass
within the triangle formed by the remaining three. Since all are drawn at
random, the chance that the last drawn should pass through the triangle
formed by the other three is consequently 12.”

I quote this solution because it seems to me to illustrate the difficulty to
which I want to call attention. As the problem is worded, a triangle is
supposed to be assigned by three straight lines. However large it may be, its
size bears no finite ratio whatever to the indefinitely larger area outside it;
and, so far as I can put any intelligible construction on the supposition, the
chance of drawing a fourth random line which should happen to intersect
this finite area must be reckoned as zero. The problem Mr Wilson has
solved seems to me to be a quite different one, viz. “Given four intersecting
straight lines, find the chance that we should, at random, select one that
passes through the triangle formed by the other three.”

The same difficulty seems to me to turn up in most other attempts to apply
this conception of randomness to real infinity. The following seems an exact
analogue of the above problem:—A number is selected at random, find the
chance that another number selected at random shall be greater than the
former;—the answer surely must be that the chance is unity, viz. certainty,
because the range above any assigned number is infinitely greater than that
below it. Or, expressed in the only language in which I can understand the
term ‘infinity’, what I mean is this. If the first number be m and I am
restricted to selecting up to n (n > m) then the chance of exceeding m is
n − m : n; if I am restricted to 2n then it is 2n − m : 2n and so on. That is,
however large n and m may be the expression is always intelligible; but,
m being chosen first, n may be made as much larger than m as we please:
i.e. the chance may be made to approach as near to unity as we please.

I cannot but think that there is a similar fallacy in De Morgan's admirably
suggestive paper on Infinity (Camb. Phil. Trans. Vol. 11.) when he is
discussing the “three-point problem”:—i.e. given three points taken at
random find the chance that they shall form an acute-angled triangle. All
that he shows is, that if we start with one side as given and consider the
subsequent possible positions of the opposite vertex, there are infinitely as
many such positions which would form an acute-angled triangle as an
obtuse: but, as before, this is solving a different problem.

§ 9. The nearest approach I can make towards true indefinite randomness, or
random selection from true indefiniteness, is as follows. Suppose a circle
with a tangent line extended indefinitely in each direction. Now from the
centre draw radii at random; in other words, let the semicircumference
which lies towards the tangent be ultimately uniformly intersected by the
radii. Let these radii be then produced so as to intersect the tangent line, and
consider the distribution of these points of intersection. We shall obtain in
the result one characteristic of our random distribution; i.e. no portion of
this tangent, however small or however remote, but will find itself in the
position ultimately of any small portion of the pavement in our supposed
continual rainfall. That is, any such elementary patch will become more and
more closely dotted over with the points of intersection. But the other
essential characteristic, viz. that of ultimately uniform distribution, will be
missing. There will be a special form of distribution,—what in fact will

have to be discussed in a future chapter under the designation of a ‘law of
error’,—by virtue of which the concentration will tend to be greatest at a
certain point (that of contact with the circle), and will thin out from here in
each direction according to an easily calculated formula. The existence of
such a state of things as this is quite opposed to the conception of true
randomness.

§ 10. III. Apart from definitions and what comes of them, perhaps the most
important question connected with the conception of Randomness is this:
How in any given case are we to determine whether an observed
arrangement is to be considered a random one or not? This question will
have to be more fully discussed in a future chapter, but we are already in a
position to see our way through some of the difficulties involved in it.

(1) If the events or objects under consideration are supposed to be
continued indefinitely, or if we know enough about the mode in which they
are brought about to detect their ultimate tendency,—or even, short of this,
if they are numerous enough to be beyond practical counting,—there is no
great difficulty. We are simply confronted with a question of fact, to be
settled like other questions of fact. In the case of the rain-drops, watch two
equal squares of pavement or other surfaces, and note whether they come to
be more and more densely uniformly and evenly spotted over: if they do,
then the arrangement is what we call a random one. If I want to know
whether a tobacco-pipe really breaks at random, and would therefore serve
as an illustration of the problem proposed some pages back, I have only to
drop enough of them and see whether pieces of all possible lengths are
equally represented in the long run. Or, I may argue deductively, from what
I know about the strength of materials and the molecular constitution of
such bodies, as to whether fractures of small and large pieces are all equally
likely to occur.

§ 11. The reader's attention must be carefully directed to a source of
confusion here, arising out of a certain cross-division. What we are now
discussing is a question of fact, viz. the nature of a certain ultimate
arrangement; we are not discussing the particular way in which it is brought
about. In other words, the antithesis is between what is and what is not
random: it is not between what is random and what is designed. As we shall

see in a few moments it is quite possible that an arrangement which is the
result,—if ever anything were so,—of ‘design’, may nevertheless present
the unmistakeable stamp of randomness of arrangement.

Consider a case which has been a good deal discussed, and to which we
shall revert again: the arrangement of the stars. The question here is rather
complicated by the fact that we know nothing about the actual mutual
positions of the stars, all that we can take cognizance of being their
apparent or visible places as projected upon the surface of a supposed
sphere. Appealing to what alone we can thus observe, it is obvious that the
arrangement, as a whole, is not of the random sort. The Milky Way and the
other resolvable nebulæ, as they present themselves to us, are as obvious an
infraction of such an arrangement as would be the occurrence here and
there of patches of ground in a rainfall which received a vast number more
drops than the spaces surrounding them. If we leave these exceptional areas
out of the question and consider only the stars which are visible by the
naked eye or by slight telescopic power, it seems equally certain that the
arrangement is, for the most part, a fairly representative random one. By
this we mean nothing more than the fact that when we mark off any number
of equal areas on the visible sphere these are found to contain
approximately the same number of stars.

The actual arrangement of the stars in space may also be of the same
character: that is, the apparently denser aggregation may be apparent only,
arising from the fact that we are looking through regions which are not
more thickly occupied but are merely more extensive. The alternative
before us, in fact, is this. If the whole volume, so to say, of the starry
heavens is tolerably regular in shape, then the arrangement of the stars is
not of the random order; if that volume is very irregular in shape, it is
possible that the arrangement within it may be throughout of that order.

§ 12. (2) When the arrangement in question includes but a comparatively
small number of events or objects, it becomes much more difficult to
determine whether or not it is to be designated a random one. In fact we
have to shift our ground, and to decide not by what has been actually
observed but by what we have reason to conclude would be observed if we
could continue our observation much longer. This introduces what is called

‘Inverse Probability’, viz. the determination of the nature of a cause from
the nature of the observed effect; a question which will be fully discussed in
a future chapter. But some introductory remarks may be conveniently made
here.

Every problem of Probability, as the subject is here understood, introduces
the conception of an ultimate limit, and therefore presupposes an indefinite
possibility of repetition. When we have only a finite number of occurrences
before us, direct evidence of the character of their arrangement fails us, and
we have to fall back upon the nature of the agency which produces them.
And as the number becomes smaller the confidence with which we can
estimate the nature of the agency becomes gradually less.

Begin with an intermediate case. There is a small lawn, sprinkled over with
daisies: is this a random arrangement? We feel some confidence that it is so,
on mere inspection; meaning by this that (negatively) no trace of any
regular pattern can be discerned and (affirmatively) that if we take any
moderately small area, say a square yard, we shall find much about the
same number of the plants included in it. But we can help ourselves by an
appeal to the known agency of distribution here. We know that the daisy
spreads by seed, and considering the effect of the wind and the continued
sweeping and mowing of the lawn we can detect causes at work which are
analogous to those by which the dealing of cards and the tossing of dice are
regulated.

In the above case the appeal to the process of production was subsidiary,
but when we come to consider the nature of a very small succession or
group this appeal becomes much more important. Let us be told of a certain
succession of ‘heads’ and ‘tails’ to the number of ten. The range here is far
too small for decision, and unless we are told whether the agent who
obtained them was tossing or designing we are quite unable to say whether
or not the designation of ‘random’ ought to be applied to the result obtained.
The truth must never be forgotten that though ‘design’ is sure to break down
in the long run if it make the attempt to produce directly the semblance of
randomness,[4] yet for a short spell it can simulate it perfectly. Any short
succession, say of heads and tails, may have been equally well brought
about by tossing or by deliberate choice.

§ 13. The reader will observe that this question of randomness is being here
treated as simply one of ultimate statistical fact. I have fully admitted that
this is not the primitive conception, nor is it the popular interpretation, but
to adopt it seems the only course open to us if we are to draw inferences
such as those contemplated in Probability. When we look to the producing
agency of the ultimate arrangement we may find this very various. It may
prove itself to be (a few stages back) one of conscious deliberate purpose,
as in drawing a card or tossing a die: it may be the outcome of an extremely
complicated interaction of many natural causes, as in the arrangement of the
flowers scattered over a lawn or meadow: it may be of a kind of which we
know literally nothing whatever, as in the case of the actual arrangement of
the stars relatively to each other.

This was the state of things had in view when it was said a few pages back
that randomness and design would result in something of a cross-division.
Plenty of arrangements in which design had a hand, a stage or two back,
can be mentioned, which would be quite indistinguishable in their results
from those in which no design whatever could be traced. Perhaps the most
striking case in point here is to be found in the arrangement of the digits in
one of the natural arithmetical constants, such as π or e, or in a table of
logarithms. If we look to the process of production of these digits, no
extremer instance can be found of what we mean by the antithesis of
randomness: every figure has its necessarily pre-ordained position, and a
moment's flagging of intention would defeat the whole purpose of the
calculator. And yet, if we look to results only, no better instance can be
found than one of these rows of digits if it were intended to illustrate what
we practically understand by a chance arrangement of a number of objects.
Each digit occurs approximately equally often, and this tendency develops
as we advance further: the mutual juxtaposition of the digits also shows the
same tendency, that is, any digit (say 5) is just as often followed by 6 or 7 as
by any of the others. In fact, if we were to take the whole row of hitherto
calculated figures, cut off the first five as familiar to us all, and contemplate
the rest, no one would have the slightest reason to suppose that these had
not come out as the results of a die with ten equal faces.

§ 14. If it be asked why this is so, a rather puzzling question is raised.
Wherever physical causation is involved we are generally understood to

have satisfied the demand implied in this question if we assign antecedents
which will be followed regularly by the event before us; but in geometry
and arithmetic there is no opening for antecedents. What we then commonly
look for is a demonstration, i.e. the resolution of the observed fact into
axioms if possible, or at any rate into admitted truths of wider generality. I
do not know that a demonstration can be given as to the existence of this
characteristic of statistical randomness in such successions of digits as those
under consideration. But the following remarks may serve to shift the onus
of unlikelihood by suggesting that the preponderance of analogy is rather in
favour of the existence.

Take the well-known constant π for consideration. This stands for a quantity
which presents itself in a vast number of arithmetical and geometrical
relations; let us take for examination the best known of these, by regarding
it as standing for the ratio of the circumference to the diameter of a circle.
So regarded, it is nothing more than a simple case of the measurement of a
magnitude by an arbitrarily selected unit. Conceive then that we had before
us a rod or line and that we wished to measure it with absolute accuracy. We
must suppose—if we are to have a suitable analogue to the determination
of π to several hundred figures,—that by the application of continued higher
magnifying power we can detect ever finer subdivisions in the graduation.
We lay our rod against the scale and find it, say, fall between 31 and
32 inches; we then look at the next division of the scale, viz. that into tenths
of an inch. Can we see the slightest reason why the number of these tenths
should be other than independent of the number of whole inches? The
“piece over” which we are measuring may in fact be regarded as an entirely
new piece, which had fallen into our hands after that of 31 inches had been
measured and done with; and similarly with every successive piece over, as
we proceed to the ever finer and finer divisions.

Similar remarks may be made about most other incommensurable
quantities, such as irreducible roots. Conceive two straight lines at right
angles, and that we lay off a certain number of inches along each of these
from the point of intersection; say two and five inches, and join the
extremities of these so as to form the diagonal of a right-angled triangle. If
we proceed to measure this diagonal in terms of either of the other lines we
are to all intents and purposes extracting a square root. We should expect,

rather than otherwise, to find here, as in the case of π, that
incommensurability and resultant randomness of order in the digits was the
rule, and commensurability was the exception. Now and then, as when the
two sides were three and four, we should find the diagonal commensurable
with them; but these would be the occasional exceptions, or rather they
would be the comparatively finite exceptions amidst the indefinitely
numerous cases which furnished the rule.

§ 15. The best way perhaps of illustrating the truly random character of such
a row of figures is by appealing to graphical aid. It is not easy here, any
more than in ordinary statistics, to grasp the import of mere figures;
whereas the arrangement of groups of points or lines is much more readily
seized. The eye is very quick in detecting any symptoms of regularity in the
arrangement, or any tendency to denser aggregation in one direction than in
another. How then are we to dispose our figures so as to force them to
display their true character? I should suggest that we set about drawing a
line at random; and, since we cannot trust our own unaided efforts to do
this, that we rely upon the help of such a table of figures to do it for us, and
then examine with what sort of efficiency they can perform the task. The
problem of drawing straight lines at random, under various limitations of
direction or intersection, is familiar enough, but I do not know that any one
has suggested the drawing of a line whose shape as well as position shall be
of a purely random character. For simplicity we suppose the line to be
confined to a plane.

The definition of such a line does not seem to involve any particular
difficulty. Phrased in accordance with the ordinary language we should
describe it as the path (i.e. any path) traced out by a point which at every
moment is as likely to move in any one direction as in any other. That we
could not ourselves draw such a line, and that we could not get it traced by
any physical agency, is certain. The mere inertia of any moving body will
always give it a tendency, however slight, to go on in a straight line at each
moment, instead of being instantly responsive to instantaneously varying
dictates as to its direction of motion. Nor can we conceive or picture such a
line in its ultimate or ideal condition. But it is easy to give a graphical
approximation to it, and it is easy also to show how this approximation may
be carried on as far as we please towards the ideal in question.

We may proceed as follows. Take a sheet of the ordinary ruled paper
prepared for the graphical exposition of curves. Select as our starting point
the intersection of two of these lines, and consider the eight ‘points of the
compass’ indicated by these lines and the bisections of the contained right
angles.[5] For suggesting the random selection amongst these directions let
them be numbered from 0 to 7, and let us say that a line measured due
‘north’ shall be designated by the figure 0, ‘north-east’ by 1, and so on. The
selection amongst these numbers, and therefore directions, at every corner,
might be handed over to a die with eight faces; but for the purpose of the
illustration in view we select the digits 0 to 7 as they present themselves in
the calculated value of π. The sort of path along which we should travel by
a series of such steps thus taken at random may be readily conceived; it is
given at the end of this chapter.

For the purpose with which this illustration was proposed, viz. the graphical
display of the succession of digits in any one of the incommensurable
constants of arithmetic or geometry, the above may suffice. After actually
testing some of them in this way they seem to me, so far as the eye, or the
theoretical principles to be presently mentioned, are any guide, to answer
quite fairly to the description of randomness.

§ 16. As we are on the subject, however, it seems worth going farther by
enquiring how near we could get to the ideal of randomness of direction. To
carry this out completely two improvements must be made. For one thing,
instead of confining ourselves to eight directions we must admit an infinite
number. This would offer no great difficulty; for instead of employing a
small number of digits we should merely have to use some kind of circular
teetotum which would rest indifferently in any direction. But in the next
place instead of short finite steps we must suppose them indefinitely short.
It is here that the actual unattainability makes itself felt. We are familiar
enough with the device, employed by Newton, of passing from the
discontinuous polygon to the continuous curve. But we can resort to this
device because the ideal, viz. the curve, is as easily drawn (and, I should
say, as easily conceived or pictured) as any of the steps which lead us
towards it. But in the case before us it is otherwise. The line in question will
remain discontinuous, or rather angular, to the last: for its angles do not
tend even to lose their sharpness, though the fragments which compose

them increase in number and diminish in magnitude without any limit. And
such an ideal is not conceivable as an ideal. It is as if we had a rough body
under the microscope, and found that as we subjected it to higher and
higher powers there was no tendency for the angles to round themselves off.
Our ‘random line’ must remain as ‘spiky’ as ever, though the size of its
spikes of course diminishes without any limit.

The case therefore seems to be this. It is easy, in words, to indicate the
conception by speaking of a line which at every instant is as likely to take
one direction as another. It is easy moreover to draw such a line with any
degree of minuteness which we choose to demand. But it is not possible to
conceive or picture the line in its ultimate form.[6] There is in fact no ‘limit’
here, intelligible to the understanding or picturable by the imagination
(corresponding to the asymptote of a curve, or the continuous curve to the
incessantly developing polygon), towards which we find ourselves
continually approaching, and which therefore we are apt to conceive
ourselves as ultimately attaining. The usual assumption therefore which
underlies the Newtonian infinitesimal geometry and the Differential
Calculus, ceases to apply here.

§ 17. If we like to consider such a line in one of its approximate stages, as
above indicated, it seems to me that some of the usual theorems of
Probability, where large numbers are concerned, may safely be applied. If it
be asked, for instance, whether such a line will ultimately tend to stray
indefinitely far from its starting point, Bernoulli's ‘Law of Large Numbers’
may be appealed to, in virtue of which we should say that it was excessively
unlikely that its divergence should be relatively great. Recur to our
graphical illustration, and consider first the resultant deviation of the point
(after a great many steps) right or left of the vertical line through the
starting point. Of the eight admissible motions at each stage two will not
affect this relative position, whilst the other six are equally likely to move
us a step to the right or to the left. Our resultant ‘drift’ therefore to the right
or left will be analogous to the resultant difference between the number of
heads and tails after a great many tosses of a penny. Now the well-known
outcome of such a number of tosses is that ultimately the proportional
approximation to the à priori probability, i.e. to equality of heads and tails,

is more and more nearly carried out, but that the absolute deflection is more
and more widely displayed.

Applying this to the case in point, and remembering that the results apply
equally to the horizontal and vertical directions, we should say that after
any very great number of such ‘steps’ as those contemplated, the ratio of
our distance from the starting point to the whole distance travelled will
pretty certainly be small, whereas the actual distance from it would be large.
We should also say that the longer we continued to produce such a line the
more pronounced would these tendencies become. So far as concerns this
test, and that afforded by the general appearance of the lines drawn,—this
last, as above remarked, being tolerably trustworthy,—I feel no doubt as to
the generally ‘random’ character of the rows of figures displayed by the
incommensurable or irrational ratios in question.

As it may interest the reader to see an actual specimen of such a path I
append one representing the arrangement of the eight digits from 0 to 7 in
the value of π. The data are taken from Mr Shanks' astonishing performance
in the calculation of this constant to 707 places of figures (Proc. of R. S.,
XXI. p. 319). Of these, after omitting 8 and 9, there remain 568; the
diagram represents the course traced out by following the direction of these
as the clue to our path. Many of the steps have of course been taken in
opposite directions twice or oftener. The result seems to me to furnish a
very fair graphical indication of randomness. I have compared it with
corresponding paths furnished by rows of figures taken from logarithmic
tables, and in other ways, and find the results to be much the same.

1 According to Prof. Skeat (Etymological Dictionary) the earliest known

meaning is that of furious action, as in a charge of cavalry. The etymology,
he considers, is connected with the Teutonic word rand (brim), and implies
the furious and irregular action of a river full to the brim.
2
See the problem paper of Jan. 18, 1854, in the Cambridge Mathematical
Tripos.
3 As, according to Mr H. Godfray, the majority of the candidates did assume

when the problem was once proposed in an examination. See the
Educational Times (Reprint, Vol. VII. p. 99.)
4 Vide p. 68.

5 It would of course be more complete to take ten alternatives of direction,

and thus to omit none of the digits; but this is much more troublesome in
practice than to confine ourselves to eight.
6
Any more than we picture the shape of an equiangular spiral at the centre.

CHAPTER VI.[*]

THE SUBJECTIVE SIDE OF PROBABILITY.
MEASUREMENT OF BELIEF.
*
Originally written in somewhat of a spirit of protest against what seemed to me the prevalent
disposition to follow De Morgan in taking too subjective a view of the science. In reading it through
now I cannot find any single sentence to which I could take distinct objection, though I must admit
that if I were writing it entirely afresh I should endeavour to express myself with less emphasis, and I
have made alterations in that direction. The reader who wishes to see a view not substantially very
different from mine, but expressed with a somewhat opposite emphasis, can refer to Mr F. Y.
Edgeworth's article on “The Philosophy of Chance” (Mind, Vol. IX.)

§ 1. Having now obtained a clear conception of a certain kind of series, the
next enquiry is, What is to be done with this series? How is it to be
employed as a means of making inferences? The general step that we are
now about to take might be described as one from the objective to the
subjective, from the things themselves to the state of our minds in
contemplating them.

The reader should observe that a substitution has, in a great number of
cases, already been made as a first stage towards bringing the things into a
shape fit for calculation. This substitution, as described in former chapters,
is, in a measure, a process of idealization. The series we actually meet with
are apt to show a changeable type, and the individuals of them will
sometimes transgress their licensed irregularity. Hence they have to be
pruned a little into shape, as natural objects almost always have before they
are capable of being accurately reasoned about. The form in which the
series emerges is that of a series with a fixed type. This imaginary or ideal
series is the basis of our calculation.

§ 2. It must not be supposed that this is at all at variance with the assertion
previously made, that Probability is a science of inference about real things;
it is only by a substitution of the above kind that we are enabled to reason
about the things. In nature nearly all phenomena present themselves in a
form which departs from that rigorously accurate one which scientific
purposes mostly demand, so we have to introduce an imaginary series,
which shall be free from any such defects. The only condition to be fulfilled
is, that the substitution is to be as little arbitrary, that is, to vary from the

truth as slightly, as possible. This kind of substitution generally passes
without notice when natural objects of any kind are made subjects of exact
science. I direct distinct attention to it here simply from the apprehension
that want of familiarity with the subject-matter might lead some readers to
suppose that it involves, in this case, an exceptional deflection from
accuracy in the formal process of inference.

It may be remarked also that the adoption of this imaginary series offers no
countenance whatever to the doctrine criticised in the last chapter, in
accordance with which it was supposed that our series possessed a fixed
unchangeable type which was merely the “development of the
probabilities” of things, to use Laplace's expression. It differs from anything
contemplated on that hypothesis by the fact that it is to be recognized as a
necessary substitution of our own for the actual series, and to be kept in as
close conformity with facts as possible. It is a mere fiction or artifice
necessarily resorted to for the purpose of calculation, and for this purpose
only.

This caution is the more necessary, because in the example that I shall
select, and which belongs to the most favourite class of examples in this
subject, the substitution becomes accidentally unnecessary. The things, as
has been repeatedly pointed out, may sometimes need no trimming, because
in the form in which they actually present themselves they are almost
idealized. In most cases a good deal of alteration is necessary to bring the
series into shape, but in some—prominently in the case of games of chance
—we find the alterations, for all practical purposes, needless.

§ 3. We start then, from such a series as this, upon the enquiry, What kind of
inference can be made about it? It may assist the logical reader to inform
him that our first step will be analogous to one class of what are commonly
known as immediate inferences,—inferences, that is, of the type,—‘All men
are mortal, therefore any particular man or men are mortal.’ This case,
simple and obvious as it is in Logic, requires very careful consideration in
Probability.

It is obvious that we must be prepared to form an opinion upon the
propriety of taking the step involved in making such an inference. Hitherto
we have had as little to do as possible with the irregular individuals; we

have regarded them simply as fragments of a regular series. But we cannot
long continue to neglect all consideration of them. Even if these events in
the gross be tolerably certain, it is not only in the gross that we have to deal
with them; they constantly come before us a few at a time, or even as
individuals, and we have to form some opinion about them in this state. An
insurance office, for instance, deals with numbers large enough to obviate
most of the uncertainty, but each of their transactions has another party
interested in it—What has the man who insures to say to their proceedings?
for to him this question becomes an individual one. And even the office
itself receives its cases singly, and would therefore like to have as clear
views as possible about these single cases. Now, the remarks made in the
preceding chapters about the subjects which Probability discusses might
seem to preclude all enquiries of this kind, for was not ignorance of the
individual presupposed to such an extent that even (as will be seen
hereafter) causation might be denied, within considerable limits, without
affecting our conclusions? The answer to this enquiry will require us to turn
now to the consideration of a totally distinct side of the question, and one
which has not yet come before us. Our best introduction to it will be by the
discussion of a special example.

§ 4. Let a penny be tossed up a very great many times; we may then be
supposed to know for certain this fact (amongst many others) that in the
long run head and tail will occur about equally often. But suppose we
consider only a moderate number of throws, or fewer still, and so continue
limiting the number until we come down to three or two, or even one? We
have, as the extreme cases, certainty or something undistinguishably near it,
and utter uncertainty. Have we not, between these extremes, all gradations
of belief? There is a large body of writers, including some of the most
eminent authorities upon this subject, who state or imply that we are
distinctly conscious of such a variation of the amount of our belief, and that
this state of our minds can be measured and determined with almost the
same accuracy as the external events to which they refer. The principal
mathematical supporter of this view is De Morgan, who has insisted
strongly upon it in all his works on the subject. The clearest exposition of
his opinions will be found in his Formal Logic, in which work he has made
the view which we are now discussing the basis of his system. He holds that
we have a certain amount of belief of every proposition which may be set

before us, an amount which in its nature admits of determination, though
we may practically find it difficult in any particular case to determine it. He
considers, in fact, that Probability is a sort of sister science to Formal Logic,
[1] speaking of it in the following words: “I cannot understand why the

study of the effect, which partial belief of the premises produces with
respect to the conclusion, should be separated from that of the
consequences of supposing the former to be absolutely true.”[2] In other
words, there is a science—Formal Logic—which investigates the rules
according to which one proposition can be necessarily inferred from
another; in close correspondence with this there is a science which
investigates the rules according to which the amount of our belief of one
proposition varies with the amount of our belief of other propositions with
which it is connected.

The same view is also supported by another high authority, the late Prof.
Donkin, who says (Phil. Mag. May, 1851), “It will, I suppose, be generally
admitted, and has often been more or less explicitly stated, that the subject-
matter of calculation in the mathematical theory of Probabilities is quantity
of belief.”

§ 5. Before proceeding to criticise this opinion, one remark may be made
upon it which has been too frequently overlooked. It should be borne in
mind that, even were this view of the subject not actually incorrect, it might
be objected to as insufficient for the purpose of a definition, on the ground
that variation of belief is not confined to Probability. It is a property with
which that science is concerned, no doubt, but it is a property which meets
us in other directions as well. In every case in which we extend our
inferences by Induction or Analogy, or depend upon the witness of others,
or trust to our own memory of the past, or come to a conclusion through
conflicting arguments, or even make a long and complicated deduction by
mathematics or logic, we have a result of which we can scarcely feel as
certain as of the premises from which it was obtained. In all these cases
then we are conscious of varying quantities of belief, but are the laws
according to which the belief is produced and varied the same? If they
cannot be reduced to one harmonious scheme, if in fact they can at best be
brought to nothing but a number of different schemes, each with its own

body of laws and rules, then it is vain to endeavour to force them into one
science.

This opinion is strengthened by observing that most of the writers who
adopt the definition in question do practically dismiss from consideration
most of the above-mentioned examples of diminution of belief, and confine
their attention to classes of events which have the property discussed in
Chap I., viz. ‘ignorance of the few, knowledge of the many.’ It is quite true
that considerable violence has to be done to some of these examples, by
introducing exceedingly arbitrary suppositions into them, before they can
be forced to assume a suitable form. But still there is little doubt that, if we
carefully examine the language employed, we shall find that in almost
every case assumptions are made which virtually imply that our knowledge
of the individual is derived from propositions given in the typical form
described in Chap I. This will be more fully proved when we come to
consider some common misapplications of the science.

§ 6. Even then, if the above-mentioned view of the subject were correct, it
would yet, I consider, be insufficient for the purpose of a definition; but it is
at least very doubtful whether it is correct. Before we could properly assign
to the belief side of the question the prominence given to it by De Morgan
and others, certainly before the science could be defined from that side, it
would be necessary, it appears, to establish the two following positions,
against both of which strong objections can be brought.

(1) That our belief of every proposition is a thing which we can, strictly
speaking, be said to measure; that there must be a certain amount of it in
every case, which we can realize somehow in consciousness and refer to
some standard so as to pronounce upon its value.

(2) That the value thus apprehended is the correct one according to the
theory, viz. that it is the exact fraction of full conviction that it should be.
This statement will perhaps seem somewhat obscure at first; it will be
explained presently.

§ 7. (I.) Now, in the first place, as regards the difficulty of obtaining any
measure of the amount of our belief. One source of this difficulty is too
obvious to have escaped notice; this is the disturbing influence produced on

the quantity of belief by any strong emotion or passion. A deep interest in
the matter at stake, whether it excite hope or fear, plays great havoc with the
belief-meter, so that we must assume the mind to be quite unimpassioned in
weighing the evidence. This is noticed and acknowledged by Laplace and
others; but these writers seem to me to assume it to be the only source of
error, and also to be of comparative unimportance. Even if it were the only
source of error I cannot see that it would be unimportant. We experience
hope or fear in so very many instances, that to omit such influences from
consideration would be almost equivalent to saying that whilst we profess
to consider the whole quantity of our belief we will in reality consider only
a portion of it. Very strong feelings are, of course, exceptional, but we
should nevertheless find that the emotional element, in some form or other,
makes itself felt on almost every occasion. It is very seldom that we cannot
speak of our surprise or expectation in reference to any particular event.
Both of these expressions, but especially the former, seem to point to
something more than mere belief. It is true that the word ‘expectation’ is
generally defined in treatises on Probability as equivalent to belief; but it
seems doubtful whether any one who attends to the popular use of the terms
would admit that they were exactly synonymous. Be this however as it may,
the emotional element is present upon almost every occasion, and its
disturbing influence therefore is constantly at work.

§ 8. Another cause, which co-operates with the former, is to be found in the
extreme complexity and variety of the evidence on which our belief of any
proposition depends. Hence it results that our actual belief at any given
moment is one of the most fugitive and variable things possible, so that we
can scarcely ever get sufficiently clear hold of it to measure it. This is not
confined to the times when our minds are in a turmoil of excitement
through hope or fear. In our calmest moments we shall find it no easy thing
to give a precise answer to the question, How firmly do I hold this or that
belief? There may be one or two prominent arguments in its favour, and one
or two corresponding objections against it, but this is far from comprising
all the causes by which our state of belief is produced. Because such
reasons as these are all that can be practically introduced into oral or written
controversies, we must not conclude that it is by these only that our
conviction is influenced. On the contrary, our conviction generally rests
upon a sort of chaotic basis composed of an infinite number of inferences

and analogies of every description, and these moreover distorted by our
state of feeling at the time, dimmed by the degree of our recollection of
them afterwards, and probably received from time to time with varying
force according to the way in which they happen to combine in our
consciousness at the moment. To borrow a striking illustration from
Abraham Tucker, the substructure of our convictions is not so much to be
compared to the solid foundations of an ordinary building, as to the piles of
the houses of Rotterdam which rest somehow in a deep bed of soft mud.
They bear their weight securely enough, but it would not be easy to point
out accurately the dependence of the different parts upon one another.
Directly we begin to think of the amount of our belief, we have to think of
the arguments by which it is produced—in fact, these arguments will
intrude themselves without our choice. As each in turn flashes through the
mind, it modifies the strength of our conviction; we are like a person
listening to the confused hubbub of a crowd, where there is always
something arbitrary in the particular sound we choose to listen to. There
may be reasons enough to suffice abundantly for our ultimate choice, but on
examination we shall find that they are by no means apprehended with the
same force at different times. The belief produced by some strong argument
may be very decisive at the moment, but it will often begin to diminish
when the argument is not actually before the mind. It is like being dazzled
by a strong light; the impression still remains, but begins almost
immediately to fade away. I think that this is the case, however we try to
limit the sources of our conviction.

§ 9. (II.) But supposing that it were possible to strike a sort of average of
this fluctuating state, should we find this average to be of the amount
assigned by theory? In other words, is our natural belief in the happening of
two different events in direct proportion to the frequency with which those
events happen in the long run? There is a lottery with 100 tickets and ten
prizes; is a man's belief that he will get a prize fairly represented by one-
tenth of certainty? The mere reference to a lottery should be sufficient to
disprove this. Lotteries have flourished at all times, and have never failed to
be abundantly supported, in spite of the most perfect conviction, on the part
of many, if not of most, of those who put into them, that in the long run all
will lose. Deductions should undoubtedly be made for those who act from
superstitious motives, from belief in omens, dreams, and so on. But apart

from these, and supposing any one to come fortified by all that mathematics
can do for him, it is difficult to believe that his natural impressions about
single events would be always what they should be according to theory. Are
there many who can honestly declare that they would have no desire to buy
a single ticket? They would probably say to themselves that the sum they
paid away was nothing worth mentioning to lose, and that there was a
chance of gaining a great deal; in other words, they are not apportioning
their belief in the way that theory assigns.

What bears out this view is, that the same persons who would act in this
way in single instances would often not think of doing so in any but single
instances. In other words, the natural tendency here is to attribute too great
an amount of belief where it is or should be small; i.e. to depreciate the risk
in proportion to the contingent advantage. They would very likely, when
argued with, attach disparaging epithets to this state of feeling, by calling it
an unaccountable fascination, or something of that kind, but of its existence
there can be little doubt. We are speaking now of what is the natural
tendency of our minds, not of that into which they may at length be
disciplined by education and thought. If, however, educated persons have
succeeded for the most part in controlling this tendency in games of chance,
the spirit of reckless speculation has scarcely yet been banished from
commerce. On examination, this tendency will be found so prevalent in all
ages, ranks, and dispositions, that it would be inadmissible to neglect it in
order to bring our supposed instincts more closely into accordance with the
commonly received theories of Probability.

§ 10. There is another aspect of this question which has been often
overlooked, but which seems to deserve some attention. Granted that we
have an instinct of credence, why should it be assumed that this must be just
of that intensity which subsequent experience will justify? Our instincts are
implanted in us for good purposes, and are intended to act immediately and
unconsciously. They are, however, subject to control, and have to be
brought into accordance with what we believe to be true and right. In other
departments of psychology we do not assume that every spontaneous
prompting of nature is to be left just as we find it, or even that on the
average, omitting individual variations, it is set at that pitch that will be
found in the end to be the best when we come to think about it and assign

its rules. Take, for example, the case of resentment. Here we have an
instinctive tendency, and one that on the whole is good in its results. But
moralists are agreed that almost all our efforts at self-control are to be
directed towards subduing it and keeping it in its right direction. It is
assumed to be given as a sort of rough protection, and to be set, if one
might so express oneself, at too high a pitch to be deliberately and
consciously acted on in society. May not something of this kind be the case
also with our belief? I only make a passing reference to this point here, as
on the theory of Probability adopted in this work it does not appear to be at
all material to the science. But it seems a strong argument against the
expediency of commencing the study of the science from the subjective
side, or even of assigning any great degree of prominence to this side.

That men do not believe in exact accordance with this theory must have
struck almost every one, but this has probably been considered as mere
exception and irregularity; the assumption being made that on the average,
and in far the majority of cases, they do so believe. As stated above, it is
very doubtful whether the tendency which has just been discussed is not so
widely prevalent that it might with far more propriety be called the rule than
the exception. And it may be better that this should be so: many good
results may follow from that cheerful disposition which induces a man
sometimes to go on trying after some great good, the chance of which he
overvalues. He will keep on through trouble and disappointment, without
serious harm perhaps, when the cool and calculating bystander sees plainly
that his ‘measure of belief’ is much higher than it should be. So, too, the
tendency also so common, of underrating the chance of a great evil may
also work for good. By many men death might be looked upon as an almost
infinite evil, at least they would so regard it themselves; suppose they kept
this contingency constantly before them at its right value, how would it be
possible to get through the practical work of life? Men would be stopping
indoors because if they went out they might be murdered or bitten by a mad
dog. To say this is not to advocate a return to our instincts; indeed when we
have once reached the critical and conscious state, it is hardly possible to do
so; but it should be noticed that the advantage gained by correcting them is
at best but a balanced one.[3] What is most to our present purpose, it
suggests the inexpediency of attempting to found an exact theory on what

may afterwards prove to be a mere instinct, unauthorized in its full extent
by experience.

§ 11. It may be replied, that though people, as a matter of fact, do not
apportion belief in this exact way, yet they ought to do so. The purport of
this remark will be examined presently; it need only be said here that it
grants all that is now contended for. For it admits that the degree of our
belief is capable of modification, and may need it. But in accordance with
what is the belief to be modified? obviously in accordance with experience;
it cannot be trusted to by itself, but the fraction at which it is to be rated
must be determined by the comparative frequency of the events to which it
refers. Experience then furnishing the standard, it is surely most reasonable
to start from this experience, and to found the theory of our processes upon
it.

If we do not do this, it should be observed that we are detaching Probability
altogether from the study of things external to us, and making it nothing
else in effect than a portion of Psychology. If we refuse to be controlled by
experience, but confine our attention to the laws according to which belief
is naturally or instinctively compounded and distributed in our minds, we
have no right then to appeal to experience afterwards even for illustrations,
unless under the express understanding that we do not guarantee its
accuracy. Our belief in some single events, for example, might be correct,
and yet that in a compound of several (if derived merely from our
instinctive laws of belief) very possibly might not be correct, but might lead
us into practical mistakes if we determined to act upon it. Even if the two
were in accordance, this accordance would have to be proved, which would
lead us round, by what I cannot but think a circuitous process, to the point
which has been already chosen for commencing with.

§ 12. De Morgan seems to imply that the doctrine criticised above finds a
justification from the analogy of Formal Logic. If the laws of necessary
inference can be studied apart from all reference to external facts (except by
way of illustration), why not those of probable inference? There does not,
however, seem to be much force in any such analogy. Formal Logic, at any
rate under its modern or Kantian mode of treatment, is based upon the
assumption that there are laws of thought as distinguished from laws of

things, and that these laws of thought can be ascertained and studied
without taking into account their reference to any particular object. Now so
long as we are confined to necessary or irreversible laws, as is of course the
case in ordinary Formal Logic, this assumption leads to no special
difficulties. We mean by this, that no conflict arises between these
subjective and objective necessities. The two exist in perfect harmony side
by side, the one being the accurate counterpart of the other. So precise is the
correspondence between them, that few persons would notice, until study of
metaphysics had called their attention to such points, that there were these
two sides to the question. They would make their appeal to either with equal
confidence, saying indifferently, ‘the thing must be so,’ or, ‘we cannot
conceive its being otherwise.’ In fact it is only since the time of Kant that
this mental analysis has been to any extent appreciated and accepted. And
even now the dominant experience school of philosophy would not admit
that there are here two really distinct sides to the phenomenon; they
maintain either that the subjective necessity is nothing more than the
consequence by inveterate association of the objective uniformity, or else
that this so-called necessity (say in the Law of Contradiction) is after all
merely verbal, merely a different way of saying the same thing over again
in other words. Whatever the explanation adopted, the general result is that
fallacies, as real acts of thought, are impossible within the domain of pure
logic; error within that province is only possibly by a momentary lapse of
attention, that is of consciousness.

§ 13. But though this perfect harmony between subjective and objective
uniformities or laws may exist within the domain of pure logic, it is far
from existing within that of probability. The moment we make the quantity
of our belief an integral part of the subject to be studied, any such invariable
correspondence ceases to exist. In the former case, we could not
consciously think erroneously even though we might try to do so; in the
latter, we not only can believe erroneously but constantly do so. Far from
the quantity of our belief being so exactly adjusted in conformity with the
facts to which it refers that we cannot even in imagination go astray, we
find that it frequently exists in excess or defect of that which subsequent
judgment will approve. Our instincts of credence are unquestionably in
frequent hostility with experience; and what do we do then? We simply
modify the instincts into accordance with the things. We are constantly

performing this practice, and no cultivated mind would find it possible to do
anything else. No man would think of divorcing his belief from the things
on which it was exercised, or would suppose that the former had anything
else to do than to follow the lead of the latter. Hence it results that that
separation of the subjective necessity from the objective, and that
determination to treat the former as a science apart by itself, for which a
plausible defence could be made in the case of pure logic, is entirely
inadmissible in the case of probability. However we might contrive to
‘think’ aright without appeal to facts, we cannot believe aright without
incessantly checking our proceedings by such appeals. Whatever then may
be the claims of Formal Logic to rank as a separate science, it does not
appear that it can furnish any support to the theory of Probability at present
under examination.

§ 14. The point in question is sometimes urged as follows. Suppose a man
with two, and only two, alternatives before him, one of which he knows
must involve success and the other failure. He knows nothing more about
them than this, and he is forced to act. Would he not regard them with
absolutely similar and equal feelings of confidence, without the necessity of
referring them to any real or imaginary series? If so, is not this equivalent to
saying that his belief of either, since one of them must come to pass, is
equal to that of the other, and therefore that his belief of each is one-half of
full confidence? Similarly if there are more than two alternatives: let it be
supposed that there are any number of them, amongst which no distinctions
whatever can be discerned except in such particulars as we know for certain
will not affect the result; should we not feel equally confident in respect of
each of them? and so here again should we riot have a fractional estimate of
our absolute amount of belief? It is thus attempted to lay the basis of a pure
science of Probability, determining the distribution and combination of our
belief hypothetically; viz. if the contingencies are exactly alike, then our
belief is so apportioned, the question whether the contingencies are equal
being of course decided as the objective data of Logic or Mathematics are
decided.

To discuss this question fully would require a statement at some length of
the reasons in favour of the objective or material view of Logic, as opposed
to the Formal or Conceptualist. I shall have to speak on this subject in

another chapter, and will not therefore enter upon it here. But one
conclusive objection which is applicable more peculiarly to Probability may
be offered at once. To pursue the line of enquiry just indicated, is, as already
remarked, to desert the strictly logical ground, and to take up that
appropriate to psychology; the proper question, in all these cases, being not
what do men believe, but what ought they to believe? Admitting, as was
done above, that in the case of Formal Logic these two enquiries, or rather
those corresponding to them, practically run into one, owing to the fact that
men cannot consciously ‘think’ wrongly; it cannot be too strongly insisted
on that in Probability the two are perfectly separable and distinct. It is of no
use saying what men do or will believe, we want to know what they will be
right in believing; and this can never be settled without an appeal to the
phenomena themselves.

§ 15. But apart from the above considerations, this way of putting the case
does not seem to me at all conclusive. Take the following example. A man[4]
finds himself on the sands of the Wash or Morecambe Bay, in a dense mist,
when the spring-tide is coming in; and knows therefore that to be once
caught by the tide would be fatal. He hears a church-bell at a distance, but
has no means of knowing whether it is on the same side of the water with
himself or on the opposite side. He cannot tell therefore whether by
following its sound he will be led out into the mid-stream and be lost, or led
back to dry land and safety. Here there can be no repetition of the event, and
the cases are indistinguishably alike, to him, in the only circumstances
which can affect the issue: is not then his prospect of death, it will be said,
necessarily equal to one-half? A proper analysis of his state of mind would
be a psychological rather than a logical enquiry, and in any case, as above
remarked, the decision of this question does not touch our logical position.
But according to the best introspection I can give I should say that what
really passes through the mind in such a case is something of this kind: In
most doubtful positions and circumstances we are accustomed to decide our
conduct by a consideration of the relative advantages and disadvantages of
each side, that is by the observed or inferred frequency with which one or
the other alternative has succeeded. In proportion as these become more
nearly balanced, we are more frequently mistaken in the individual cases;
that is, it becomes more and more nearly what would be called ‘a mere toss
up’ whether we are right or wrong. The case in question seems merely the

limiting case, in which it has been contrived that there shall be no
appreciable difference between the alternatives, by which to decide in
favour of one or other, and we accordingly feel no confidence in the
particular result. Having to decide, however, we decide according to the
precedent of similar cases which have occurred before. To stand still and
wait for better information is certain death, and we therefore appeal to and
employ the only rule we know of; or rather we feel, or endeavour to feel, as
we have felt before when acting in the presence of alternatives as nearly
balanced as possible. But I can neither perceive in my own case, nor feel
convinced in that of others, that this appeal, in a case which cannot be
repeated,[5] to a rule acted on and justified in cases which can be and are
repeated, at all forces us to admit that our state of mind is the same in each
case.

§ 16. This example serves to bring out very clearly a point which has been
already mentioned, and which will have to be insisted upon again, viz. that
all which Probability discusses is the statistical frequency of events, or, if
we prefer so to put it, the quantity of belief with which any one of these
events should be individually regarded, but leaves all the subsequent
conduct dependent upon that frequency, or that belief, to the choice of the
agents. Suppose there are two travellers in the predicament in question:
shall they keep together, or separate in opposite directions? In either case
alike the chance of safety to each is the same, viz. one-half, but clearly their
circumstances must decide which course it is preferable to adopt. If they are
husband and wife, they will probably prefer to remain together; if they are
sole depositaries of an important state secret, they may decide to part. In
other words, we have to select here between the two alternatives of the
certainty of a single loss, and the even chance of a double loss; alternatives
which the common mathematical statement of their chances has a decided
tendency to make us regard as indistinguishable from one another. But
clearly the decision must be grounded on the desires, feelings, and
conscience of the agents. Probability cannot say a word upon this question.
As I have pointed out elsewhere, there has been much confusion on this
matter in applications of the science to betting, and in the discussion of the
Petersburg problem.

We have thus examined the doctrine in question with a minuteness which
may seem tedious, but in consequence of the eminence of its supporters it
would have been presumptuous to have rejected it without the strongest
grounds. The objections which have been urged might be summarised as
follows:—the amount of our belief of any given proposition, supposing it to
be in its nature capable of accurate determination (which does not seem to
be the case), depends upon a great variety of causes, of which statistical
frequency—the subject of Probability—is but one. That even if we confine
our attention to this one cause, the natural amount of our belief is not
necessarily what theory would assign, but has to be checked by appeal to
experience. The subjective side of Probability therefore, though very
interesting and well deserving of examination, seems a mere appendage of
the objective, and affords in itself no safe ground for a science of inference.

§ 17. The conception then of the science of Probability as a science of the
laws of belief seems to break down at every point. We must not however
rest content with such merely negative criticism. The degree of belief we
entertain of a proposition may be hard to get at accurately, and when
obtained may be often wrong, and may need therefore to be checked by an
appeal to the objects of belief. Still in popular estimation we do seem to be
able with more or less accuracy to form a graduated scale of intensity of
belief. What we have to examine now is whether this be possible, and, if so,
what is the explanation of the fact?

That it is generally believed that we can form such a scale scarcely admits
of doubt. There is a whole vocabulary of common expressions such as, ‘I
feel almost sure,’ ‘I do not feel quite certain,’ ‘I am less confident of this
than of that,’ and so on. When we make use of any one of these phrases we
seldom doubt that we have a distinct meaning to convey by means of it. Nor
do we feel much at a loss, under any given circumstances, as to which of
these expressions we should employ in preference to the others. If we were
asked to arrange in order, according to the intensity of the belief with which
we respectively hold them, things broadly marked off from one another, we
could do it from our consciousness of belief alone, without a fresh appeal to
the evidence upon which the belief depended. Passing over the looser
propositions which are used in common conversation, let us take but one
simple example from amongst those which furnish numerical data. Do I not

feel more certain that some one will die this week in the whole town, than
in the particular street in which I live? and if the town is known to contain a
population one hundred times greater than that in the street, would not
almost any one be prepared to assert on reflection that he felt a hundred
times more sure of the first proposition than of the second? Or to take a
non-numerical example, are we not often able to say unhesitatingly which
of two propositions we believe the most, and to some rough degree how
much more we believe one than the other, at a time when all the evidence
upon which each rests has faded from the mind, so that each has to be
judged, as we may say, solely on its own merits?

Here then a problem proposes itself. If popular opinion, as illustrated in
common language, be correct,—and very considerable weight must of
course be attributed to it,—there does exist something which we call partial
belief in reference to any proposition of the numerical kind described
above. Now what we want to do is to find some test or justification of this
belief, to obtain in fact some intelligible answer to the question, Is it
correct? We shall find incidentally that the answer to this question will
throw a good deal of light upon another question nearly as important and far
more intricate, viz. What is the meaning of this partial belief?

§ 18. We shall find it advisable to commence by ascertaining how such
enquiries as the above would be answered in the case of ordinary full belief.
Such a step would not offer the slightest difficulty. Suppose, to take a
simple example, that we have obtained the following proposition,—whether
by induction, or by the rules of ordinary deductive logic, does not matter for
our present purpose,—that a certain mixture of oxygen and hydrogen is
explosive. Here we have an inference, and consequent belief of a
proposition. Now suppose there were any enquiry as to whether our belief
were correct, what should we do? The simplest way of settling the matter
would be to find out by a distinct appeal to experience whether the
proposition was true. Since we are reasoning about things, the justification
of the belief, that is, the test of its correctness, would be most readily found
in the truth of the proposition. If by any process of inference I have come to
believe that a certain mixture will explode, I consider my belief to be
justified, that is to be correct, if under proper circumstances the explosion
always does occur; if it does not occur the belief was wrong.

Such an answer, no doubt, goes but a little way, or rather no way at all,
towards explaining what is the nature of belief in itself; but it is sufficient
for our present purpose, which is merely that of determining what is meant
by the correctness of our belief, and by the test of its correctness. In all
inferences about things, in which the amount of our belief is not taken into
account, such an explanation as the above is quite sufficient; it would be the
ordinary one in any question of science. It is moreover perfectly intelligible,
whether the conclusion is particular or universal. Whether we believe that
‘some men die’, or that ‘all men die’, our belief may with equal ease be
tested by the appropriate train of experience.

§ 19. But when we attempt to apply the same test to partial belief, we shall
find ourselves reduced to an awkward perplexity. A difficulty now emerges
which has been singularly overlooked by those who have treated of the
subject. As a simple example will serve our purpose, we will take the case
of a penny. I am about to toss one up, and I therefore half believe, to adopt
the current language, that it will give head. Now it seems to be overlooked
that if we appeal to the event, as we did in the case last examined, our belief
must inevitably be wrong, and therefore the test above mentioned will fail.
For the thing must either happen or not happen: i.e. in this case the penny
must either give head, or not give it; there is no third alternative. But
whichever way it occurs, our half-belief, so far as such a state of mind
admits of interpretation, must be wrong. If head does come, I am wrong in
not having expected it enough; for I only half believed in its occurrence. If
it does not happen, I am equally wrong in having expected it too much; for I
half believed in its occurrence, when in fact it did not occur at all.

The same difficulty will occur in every case in which we attempt to justify
our state of partial belief in a single contingent event. Let us take another
example, slightly differing from the last. A man is to receive £1 if a die
gives six, to pay 1s. if it gives any other number. It will generally be
admitted that he ought to give 2s. 6d. for the chance, and that if he does so
he will be paying a fair sum. This example only differs from the last in the
fact that instead of simple belief in a proposition, we have taken what
mathematicians call ‘the value of the expectation’. In other words, we have
brought into a greater prominence, not merely the belief, but the conduct
which is founded upon the belief. But precisely the same difficulty recurs

here. For appealing to the event,—the single event, that is,—we see that one
or other party must lose his money without compensation. In what sense
then can such an expectation be said to be a fair one?

§ 20. A possible answer to this, and so far as appears the only possible
answer, will be, that what we really mean by saying that we half believe in
the occurrence of head is to express our conviction that head will certainly
happen on the average every other time. And similarly, in the second
example, by calling the sum a fair one it is meant that in the long run
neither party will gain or lose. As we shall recur presently to the point
raised in this form of answer, the only notice that need be taken of it at this
point is to call attention to the fact that it entirely abandons the whole
question in dispute, for it admits that this partial belief does not in any strict
sense apply to the individual event, since it clearly cannot be justified there.
At such a result indeed we cannot be surprised; at least we cannot on the
theory adopted throughout this Essay. For bearing in mind that the
employment of Probability postulates ignorance of the single event, it is not
easy to see how we are to justify any other opinion or statement about the
single event than a confession of such ignorance.

§ 21. So far then we do not seem to have made the slightest approximation
to a solution of the particular question now under examination. The more
closely we have analysed special examples, the more unmistakeably are we
brought to the conclusion that in the individual instance no justification of
anything like quantitative belief is to be found; at least none is to be found
in the same sense in which we expect it in ordinary scientific conclusions,
whether Inductive or Deductive. And yet we have to face and account for
the fact that common impressions, as attested by a whole vocabulary of
common phrases, are in favour of the existence of this quantitative belief.
How are we to account for this? If we appeal to an example again, and
analyse it somewhat more closely, we may yet find our way to some
satisfactory explanation.

In our previous analysis (§ 18) we found it sufficient to stop at an early
stage, and to give as the justification of our belief the fact of the proposition
being true. Stopping however at that stage, we have found this explanation
fail altogether to give a justification of partial belief; fail, that is, when

applied to the individual instance. The two states of belief and disbelief
correspond admirably to the two results of the event happening and not
happening respectively, and unless for psychological purposes we saw no
reason to analyse further; but to partial belief there is nothing corresponding
in the result, for the event cannot partially happen in such cases as we are
concerned with. Suppose then we advance a step further in the analysis, and
ask again what is meant by the proposition being true? This introduces us,
of course, to a very long and intricate path; but in the short distance along it
which we shall advance, we shall not, it is to be hoped, find any very
serious difficulty. As before, we will illustrate the analysis by first applying
it to the case of ordinary full belief.

§ 22. Whatever opinion then may be held about the essential nature of
belief, it will probably be admitted that a readiness to act upon the
proposition believed is an inseparable accompaniment of that state of mind.
There can be no alteration in our belief (at any rate in the case of sane
persons) without a possible alteration in our conduct, nor anything in our
conduct which is not connected with something in our belief. We will first
take an example in connection with the penny, in which there is full belief;
we will analyse it a step further than we did before, and then attempt to
apply the same analysis to an example of a similar kind, but one in which
the belief is partial instead of full.

Suppose that I am about to throw a penny up, and contemplate the prospect
of its falling upon one of its sides and not upon its edge. We feel perfectly
confident that it will do so. Now whatever else may be implied in our
belief, we certainly mean this; that we are ready to stake our conduct upon
its falling thus. All our betting, and everything else that we do, is carried on
upon this supposition. Any risk whatever that might ensue upon its falling
otherwise will be incurred without fear. This, it must be observed, is equally
the case whether we are speaking of a single throw or of a long succession
of throws.

But now let us take the case of a penny falling, not upon one side or the
other, but upon a given side, head. To a certain extent this example
resembles the last. We are perfectly ready to stake our conduct upon what
comes to pass in the long run. When we are considering the result of a large

number of throws, we are ready to act upon the supposition that head comes
every other time. If e.g. we are betting upon it, we shall not object to
paying £1 every time that head comes, on condition of receiving £1 every
time that head does not come. This is nothing else than the translation, as
we may call it, into practice, of our belief that head and tail occur equally
often.

Now it will be obvious, on a moment's consideration, that our conduct is
capable of being slightly varied: of being varied, that is, in form, whilst it
remains identical in respect of its results. It is clear that to pay £1 every
time we lose, and to get £1 every time we gain, comes to precisely the same
thing, in the case under consideration, as to pay ten shillings every time
without exception, and to receive £1 every time that head occurs. It is so,
because heads occur, on the average, every other time. In the long run the
two results coincide; but there is a marked difference between the two
cases, considered individually. The difference is two-fold. In the first place
we depart from the notion of a payment every other time, and come to that
of one made every time. In the second place, what we pay every time is half
of what we get in the cases in which we do get anything. The difference
may seem slight; but mark the effect when our conduct is translated back
again into the subjective condition upon which it depends, viz. into our
belief. It is in consequence of such a translation, as it appears to me, that the
notion has been acquired that we have an accurately determinable amount
of belief as to every such proposition. To have losses and gains of equal
amount, and to incur them equally often, was the experience connected with
our belief that the two events, head and tail, would occur equally often. This
was quite intelligible, for it referred to the long run. To find that this could
be commuted for a payment made every time without exception, a payment,
observe, of half the amount of what we occasionally receive, has very
naturally been interpreted to mean that there must be a state of half-belief
which refers to each individual throw.

§ 23. One such example, of course, does not go far towards establishing a
theory. But the reader will bear in mind that almost all our conduct tends
towards the same result; that it is not in betting only, but in every course of
action in which we have to count the events, that such a numerical
apportionment of our conduct is possible. Hence, by the ordinary principles

of association, it would appear exceedingly likely that, not exactly a
numerical condition of mind, but rather numerical associations, become
inseparably connected with each particular event which we know to occur
in a certain proportion of times. Once in six times a die gives ace; a
knowledge of this fact, taken in combination with all the practical results to
which it leads, produces, one cannot doubt, an inseparable notion of one-
sixth connected with each single throw. But it surely cannot be called belief
to the amount of one-sixth; at least it admits neither of justification nor
explanation in these single cases, to which alone the fractional belief, if
such existed, ought to apply.

It is in consequence, I apprehend, of such association that we act in such an
unhesitating manner in reference to any single contingent event, even when
we have no expectation of its being repeated. A die is going to be thrown up
once, and once only. I bet 5 to 1 against ace, not, as is commonly asserted,
because I feel one-sixth part of certainty in the occurrence of ace; but
because I know that such conduct would be justified in the long run of such
cases, and I apply to the solitary individual the same rule that I should apply
to it if I knew it were one of a long series. This accounts for my conduct
being the same in the two cases; by association, moreover, we probably
experience very similar feelings in regard to them both.

§ 24. And here, on the view of the subject adopted in this Essay, we might
stop. We are bound to explain the ‘measure of our belief’ in the occurrence
of a single event when we judge solely from the statistical frequency with
which such events occur, for such a series of events was our starting-point;
but we are not bound to inquire whether in every case in which persons
have, or claim to have, a certain measure of belief there must be such a
series to which to refer it, and by which to justify it. Those who start from
the subjective side, and regard Probability as the science of quantitative
belief, are obliged to do this, but we are free from the obligation.

Still the question is one which is so naturally raised in connection with this
subject, that it cannot be altogether passed by. I think that to a considerable
extent such a justification as that mentioned above will be found applicable
in other cases. The fact is that we are very seldom called upon to decide and
act upon a single contingency which cannot be viewed as being one of a

series. Experience introduces us, it must be remembered, not merely to a
succession of events neatly arranged in a single series (as we have hitherto
assumed them to be for the purpose of illustration), but to an infinite
number belonging to a vast variety of different series. A man is obliged to
be acting, and therefore exercising his belief about one thing or another,
almost the whole of every day of his life. Any one person will have to
decide in his time about a multitude of events, each one of which may never
recur again within his own experience. But by the very fact of there being a
multitude, though they are all of different kinds, we shall still find that order
is maintained, and so a course of conduct can be justified. In a plantation of
trees we should find that there is order of a certain kind if we measure them
in any one direction, the trees being on an average about the same distance
from each other. But a somewhat similar order would be found if we were
to examine them in any other direction whatsoever. So in nature generally;
there is regularity in a succession of events of the same kind. But there may
also be regularity if we form a series by taking successively a number out of
totally distinct kinds.

It is in this circumstance that we find an extension of the practical
justification of the measure of our belief. A man, say, buys a life annuity,
insures his life on a railway journey, puts into a lottery, and so on. Now we
may make a series out of these acts of his, though each is in itself a single
event which he may never intend to repeat. His conduct, and therefore his
belief, measured by the result in each individual instance, will not be
justified, but the reverse, as shown in § 19. Could he indeed repeat each
kind of action often enough it would be justified; but from this, by the
conditions of life, he is debarred. Now it is perfectly conceivable that in the
new series, formed by his successive acts of different kinds, there should be
no regularity. As a matter of fact, however, it is found that there is
regularity. In this way the equalization of his gains and losses, for which he
cannot hope in annuities, insurances, and lotteries taken separately, may yet
be secured to him out of these events taken collectively. If in each case he
values his chance at its right proportion (and acts accordingly) he will in the
course of his life neither gain nor lose. And in the same way if, whenever he
has the alternative of different courses of conduct, he acts in accordance
with the estimate of his belief described above, i.e. chooses the event whose
chance is the best, he will in the end gain more in this way than by any

other course. By the existence, therefore, of these cross-series, as we may
term them, there is an immense addition to the number of actions which
may be fairly considered to belong to those courses of conduct which offer
many successive opportunities of equalizing gains and losses. All these
cases then may be regarded as admitting of justification in the way now
under discussion.

§ 25. In the above remarks it will be observed that we have been giving
what is to be regarded as a justification of his belief from the point of view
of the individual agent himself. If we suppose the existence of an enlarged
fellow-feeling, the applicability of such a justification becomes still more
extensive. We can assign a very intelligible sense to the assertion that it is
999 to 1 that I shall not get a prize in a lottery, even if this be stated in the
1
form that my belief in my so doing is represented by the fraction 1000 th of
certainty. Properly it means that in a very large number of throws I should
gain once in 1000 times. If we include other contingencies of the same kind,
as described in the last section, each individual may be supposed to reach to
something like this experience within the limits of his own life. He could
not do it in this particular line of conduct alone, but he could do it in this
line combined with others. Now introduce the possibility of each man
feeling that the gain of others offers some analogy to his own gains, which
we may conceive his doing except in the case of the gains of those against
whom he is directly competing, and the above justification becomes still
more extensively applicable.

The following would be a fair illustration to test this view. I know that I
must die on some day of the week, and there are but seven days. My belief,
therefore, that I shall die on a Sunday is one-seventh. Here the contingent
event is clearly one that does not admit of repetition; and yet would not the
belief of every man have the value assigned it by the formula? It would
appear that the same principle will be found to be at work here as in the
former examples. It is quite true that I have only the opportunity of dying
once myself, but I am a member of a class in which deaths occur with
frequency, and I form my opinion upon evidence drawn from that class. If,
for example, I had insured my life for £1000, I should feel a certain
propriety in demanding £7000 in case the office declared that it would only
pay in the event of my dying on a Sunday. I, indeed, for my own private

part, might not find the arrangement an equitable one; but mankind at large,
in case they acted on such a principle, might fairly commute their aggregate
gains in such a way, whilst to the Insurance Office it would not make any
difference at all.

§ 26. The results of the last few sections might be summarised as follows:—
the different amounts of belief which we entertain upon different events,
and which are recognized by various phrases in common use, have
undoubtedly some meaning. But the greater part of their meaning, and
certainly their only justification, are to be sought in the series of
corresponding events to which they belong; in regard to which it may be
shown that far more events are capable of being referred to a series than
might be supposed at first sight. The test and justification of belief are to be
found in conduct; in this test applied to the series as a whole, there is
nothing peculiar, it differs in no way from the similar test when we are
acting on our belief about any single event. But so applied, from the nature
of the case it is applied successively to each of the individuals of the series;
here our conduct generally admits of being separately considered in
reference to each particular event; and this has been understood to denote a
certain amount of belief which should be a fraction of certainty. Probably
on the principles of association, a peculiar condition of mind is produced in
reference to each single event. And these associations are not unnaturally
retained even when we contemplate any one of these single events isolated
from any series to which it belongs. When it is found alone we treat it, and
feel towards it, as we do when it is in company with the rest of the series.

§ 27. We may now see, more clearly than we could before, why it is that we
are free from any necessity of assuming the existence of causation, in the
sense of necessary invariable sequence, in the case of the events which
compose our series. Against such a view it might very plausibly be urged,
that we constantly talk of the probability of a single event; but how can this
be done, it may reasonably be said, if we once admit the possibility of that
event occurring fortuitously? Take an instance from human life; the average
duration of the lives of a batch of men aged thirty will be about thirty-four
years. We say therefore to any individual of them, Your expectation of life
is thirty-four years. But how can this be said if we admit that the train of
events composing his life is liable to be destitute of all regular sequence of

cause and effect? To this it may be replied that the denial of causation
enables us to say neither more nor less than its assertion, in reference to the
length of the individual life, for of this we are ignorant in each case alike.
By assigning, as above, an expectation in reference to the individual, we
mean nothing more than to make a statement about the average of his class.
Whether there be causation or not in these individual cases does not affect
our knowledge of the average, for this by supposition rests on independent
experience. The legitimate inferences are the same on either hypothesis, and
of equal value. The only difference is that on the hypothesis of non-
causation we have forced upon our attention the impropriety of talking of
the ‘proper’ expectation of the individual, owing to the fact that all
knowledge of its amount is formally impossible; on the other hypothesis the
impropriety is overlooked from the fact of such knowledge being only
practically unattainable. As a matter of fact the amount of our knowledge is
the same in each case; it is a knowledge of the average, and of that only.[6]

§ 28. We may conclude, then, that the limits within which we are thus able
to justify the amount of our belief are far more extensive than might appear
at first sight. Whether every case in which persons feel an amount of belief
short of perfect confidence could be forced into the province of Probability
is a wider question. Even, however, if the belief could be supposed capable
of justification on its principles, its rules could never in such cases be made
use of. Suppose, for example, that a father were in doubt whether to give a
certain medicine to his sick child. On the one hand, the doctor declared that
the child would die unless the medicine were given; on the other, through a
mistake, the father cannot feel quite sure that the medicine he has is the
right one. It is conceivable that some mathematicians, in their conviction
that everything has its definite numerical probability, would declare that the
man's belief had some ‘value’ (if they could only find out what it is), say
nine-tenths; by which they would mean that in nine cases out of ten in
which he entertained a belief of that particular value he proved to be right.
So with his belief and doubt on the other side of the question. Putting the
two together, there is but one course which, as a prudent man and a good
father, he can possibly follow. It may be so, but when (as here) the
identification of an event in a series depends on purely subjective
conditions, as in this case upon the degree of vividness of his conviction, of

which no one else can judge, no test is possible, and therefore no proof can
be found.

§ 29. So much then for the attempts, so frequently made, to found the
science on a subjective basis; they can lead, as it has here been endeavoured
to show, to no satisfactory result. Still our belief is so inseparably connected
with our action, that something of a defence can be made for the attempts
described above; but when it is attempted, as is often the case, to import
other sentiments besides pure belief, and to find a justification for them also
in the results of our science, the confusion becomes far worse. The
following extract from Archbishop Thomson's Laws of Thought (§ 122,
Ed. II.) will show what kind of applications of the science are contemplated
here: “In applying the doctrine of chances to that subject in connexion with
which it was invented—games of chance,—the principles of what has been
happily termed ‘moral arithmetic’ must not be forgotten. Not only would it
be difficult for a gamester to find an antagonist on terms, as to fortune and
needs, precisely equal, but also it is impossible that with such an equality
the advantage of a considerable gain should balance the harm of a serious
loss. ‘If two men,’ says Buffon, ‘were to determine to play for their whole
property, what would be the effect of this agreement? The one would only
double his fortune, and the other reduce his to naught. What proportion is
there between the loss and the gain? The same that there is between all and
nothing. The gain of the one is but a moderate sum,—the loss of the other is
numerically infinite, and morally so great that the labour of his whole life
may not perhaps suffice to restore his property.’ ”

As moral advice this is all very true and good. But if it be regarded as a
contribution to the science of the subject it is quite inappropriate, and seems
calculated to cause confusion. The doctrine of chances pronounces upon
certain kinds of events in respect of number and magnitude; it has
absolutely nothing to do with any particular person's feelings about these
relations. We might as well append a corollary to the rules of arithmetic, to
point out that although it is very true that twice two are four it does not
follow that four, horses will give twice as much pleasure to the owner as
two will. If two men play on equal terms their chances are equal; in other
words, if they were often to play in this manner each would lose as
frequently as he would gain. That is all that Probability can say; what under

the circumstances may be the determination and opinions of the men in
question, it is for them and them alone to decide. There are many persons
who cannot bear mediocrity of any kind, and to whom the prospect of
doubling their fortune would outweigh a greater chance of losing it
altogether. They alone are the judges.

If we will introduce such a balance of pleasure and pain the individual must
make the calculation for himself. The supposition is that total ruin is very
painful, partial loss painful in a less proportion than that assigned by the
ratio of the losses themselves; the inference is therefore drawn that on the
average more pain is caused by occasional great losses than by frequent
small ones, though the money value of the losses in the long run may be the
same in each case. But if we suppose a country where the desire of
spending largely is very strong, and where owing to abundant production
loss is easily replaced, the calculation might incline the other way. Under
such circumstances it is quite possible that more happiness might result
from playing for high than for low stakes. The fact is that all emotional
considerations of this kind are irrelevant; they are, at most, mere
applications of the theory, and such as each individual is alone competent to
make for himself. Some more remarks will be made upon this subject in the
chapter upon Insurance and Gambling.

§ 30. It is by the introduction of such considerations as these that the
Petersburg Problem has been so perplexed. Having already given some
description of this problem we will refer to it very briefly here. It presents
us with a sequence of sets of throws for each of which sets I am to receive
something, say a shilling, as the minimum receipt. My receipts increase in
proportion to the rarity of each particular kind of set, and each kind is
observed or inferred to grow more rare in a certain definite but unlimited
order. By the wording of the problem, properly interpreted, I am supposed
never to stop. Clearly therefore, however large a fee I pay for each of these
sets, I shall be sure to make it up in time. The mathematical expression of
this is, that I ought always to pay an infinite sum. To this the objection is
opposed, that no sensible man would think of advancing even a large finite
sum, say £50. Certainly he would not; but why? Because neither he nor
those who are to pay him would be likely to live long enough for him to
obtain throws good enough to remunerate him for one-tenth of his outlay; to

say nothing of his trouble and loss of time. We must not suppose that the
problem, as stated in the ideal form, will coincide with the practical form in
which it presents itself in life. A carpenter might as well object to Euclid's
second postulate, because his plane came to a stop in six feet on the plank
on which he was at work. Many persons have failed to perceive this, and
have assumed that, besides enabling us to draw numerical inferences about
the members of a series, the theory ought also to be called upon to justify
all the opinions which average respectable men might be inclined to form
about them, as well as the conduct they might choose to pursue in
consequence. It is obvious that to enter upon such considerations as these is
to diverge from our proper ground. We are concerned, in these cases, with
the actions of men only, as given in statistics; with the emotions they
experience in the performance of these actions we have no direct concern
whatever. The error is the same as if any one were to confound, in political
economy, value in use with value in exchange, and object to measuring the
value of a loaf by its cost of production, because bread is worth more to a
man when he is hungry than it is just after his dinner.

§ 31. One class of emotions indeed ought to be excepted, which, from the
apparent uniformity and consistency with which they show themselves in
different persons and at different times, do really present some better claim
to consideration. In connection with a science of inference they can never
indeed be regarded as more than an accident of what is essential to the
subject, but compared with other emotions they seem to be inseparable
accidents.

The reader will remember that attention was drawn in the earlier part of this
chapter to the compound nature of the state of mind which we term belief. It
is partly intellectual, partly also emotional; it professes to rest upon
experience, but in reality the experience acts through the distorting media of
hopes and fears and other disturbing agencies. So long as we confine our
attention to the state of mind of the person who believes, it appears to me
that these two parts of belief are quite inseparable. Indeed, to speak of them
as two parts may convey a wrong impression; for though they spring from
different sources, they so entirely merge in one result as to produce what
might be called an indistinguishable compound. Every kind of inference,
whether in probability or not, is liable to be disturbed in this way. A timid

man may honestly believe that he will be wounded in a coming battle, when
others, with the same experience but calmer judgments, see that the chance
is too small to deserve consideration. But such a man's belief, if we look
only to that, will not differ in its nature from sound belief. His conduct also
in consequence of his belief will by itself afford no ground of
discrimination; he will make his will as sincerely as a man who is
unmistakeably on his death-bed. The only resource is to check and correct
his belief by appealing to past and current experience.[7] This was advanced
as an objection to the theory on which probability is regarded as concerned
primarily with laws of belief. But on the view taken in this Essay in which
we are supposed to be concerned with laws of inference about things, error
and difficulty from this source vanish. Let us bear clearly in mind that we
are concerned with inferences about things, and whatever there may be in
belief which does not depend on experience will disappear from notice.

§ 32. These emotions then can claim no notice as an integral portion of any
science of inference, and should in strictness be rigidly excluded from it.
But if any of them are uniform and regular in their production and
magnitude, they may be fairly admitted as accidental and extraneous
accompaniments. This is really the case to some extent with our surprise.
This emotion does show a considerable degree of uniformity. The rarer any
event is the more am I, in common with most other men, surprised at it
when it does happen. This surprise may range through all degrees, from the
most languid form of interest up to the condition which we term ‘being
startled’. And since the surprise seems to be pretty much the same, under
similar circumstances, at different times, and in the case of different
persons, it is free from that extreme irregularity which is found in most of
the other mental conditions which accompany the contemplation of
unexpected events. Hence our surprise, though, as stated above, having no
proper claim to admission into the science of Probability, is such a constant
and regular accompaniment of that which Probability is concerned with,
that notice must often be taken of it. References will occasionally be found
to this aspect of the question in the following chapters.

It may be remarked in passing, for the sake of further illustration of the
subject, that this emotional accompaniment of surprise, to which we are
thus able to assign something like a fractional value, differs in two

important respects from the commonly accepted fraction of belief. In the
first place, it has what may be termed an independent existence; it is
intelligible by itself. The belief, as we endeavoured to show, needs
explanation and finds it in our consequent conduct. Not so with the
emotion; this stands upon its own footing, and may be examined in and by
itself. Hence, in the second place, it is as applicable, and as capable of any
kind of justification, in relation to the single event, as to a series of events.
In this respect, as will be remembered, it offers a complete contrast to our
state of belief about any one contingent event. May not these considerations
help to account for the general acceptance of the doctrine, that we have a
certain definite and measurable amount of belief about these events? I
cannot help thinking that what is so obviously true of the emotional portion
of the belief, has been unconsciously transferred to the other or intellectual
portion of the compound condition, to which it is not applicable, and where
it cannot find a justification.

§ 33. A further illustration may now be given of the subjective view of
Probability at present under discussion.

An appeal to common language is always of service, as the employment of
any distinct word is generally a proof that mankind have observed some
distinct properties in the things, which have caused them to be singled out
and have that name appropriated to them. There is such a class of words
assigned by popular usage to the kind of events of which Probability takes
account. If we examine them we shall find, I think, that they direct us
unmistakeably to the two-fold aspect of the question,—the objective and the
subjective, the quality in the events and the state of our minds in
considering them,—that have occupied our attention during the former
chapters.

The word ‘extraordinary’, for instance, seems to point to the observed fact,
that events are arranged in a sort of ordo or rank. No one of them might be
so exactly placed that we could have inferred its position, but when we take
a great many into account together, running our eye, as it were, along the
line, we begin to see that they really do for the most part stand in order.
Those which stand away from the line have this divergence observed, and
are called extraordinary, the rest ordinary, or in the line. So too ‘irregular’

and ‘abnormal’ are doubtless used from the appearance of things, when
examined in large numbers, being that of an arrangement by rule or
measure. This only holds when there are a good many; we could not speak
of the single events being so arranged. Again the word ‘law’, in its
philosophical sense, has now become quite popularised. How the term
became introduced is not certain, but there can be little doubt that it was
somewhat in this way:—The effect of a law, in its usual application to
human conduct, is to produce regularity where it did not previously exist;
when then a regularity began to be perceived in nature, the same word was
used, whether the cause was supposed to be the same or not. In each case
there was the same generality of agreement, subject to occasional
deflection.[8]

On the other hand, observe the words ‘wonderful’, ‘unexpected’,
‘incredible’. Their connotation describes states of mind simply; they are of
course not confined to Probability, in the sense of statistical frequency, but
imply simply that the events they denote are such as from some cause we
did not expect would happen, and at which therefore, when they do happen,
we are surprised.

Now when we bear in mind that these two classes of words are in their
origin perfectly distinct;—the one denoting simply events of a certain
character; the other, though also denoting events, connoting simply states of
mind;—and yet that they are universally applied to the same events, so as to
be used as perfectly synonymous, we have in this a striking illustration of
the two sides under which Probability may be viewed, and of the universal
recognition of a close connection between them. The words are popularly
used as synonymous, and we must not press their meaning too far; but if it
were to be observed, as I am rather inclined to think it could, that the
application of the words which denote mental states is wider than that of the
others, we should have an illustration of what has been already observed,
viz. that the province of Probability is not so extensive as that over which
variation of belief might be observed. Probability only considers the case in
which this variation is brought about in a certain definite statistical way.

§ 34. It will be found in the end both interesting and important to have
devoted some attention to this subjective side of the question. In the first

place, as a mere speculative inquiry the quantity of our belief of any
proposition deserves notice. To study it at all deeply would be to trespass
into the province of Psychology, but it is so intimately connected with our
own subject that we cannot avoid all reference to it. We therefore discuss
the laws under which our expectation and surprise at isolated events
increases or diminishes, so as to account for these states of mind in any
individual instance, and, if necessary, to correct them when they vary from
their proper amount.

But there is another more important reason than this. It is quite true that
when the subjects of our discussion in any particular instance lie entirely
within the province of Probability, they may be treated without any
reference to our belief. We may or we may not employ this side of the
question according to our pleasure. If, for example, I am asked whether it is
more likely that A. B. will die this year, than that it will rain to-morrow, I
may calculate the chance (which really is at bottom the same thing as my
belief) of each, find them respectively, one-sixth and one-seventh, say, and
therefore decide that my ‘expectation’ of the former is the greater, viz. that
this is the more likely event. In this case the process is precisely the same
whether we suppose our belief to be introduced or not; our mental state is,
in fact, quite immaterial to the question. But, in other cases, it may be
different. Suppose that we are comparing two things, of which one is
wholly alien to Probability, in the sense that it is hopeless to attempt to
assign any degree of numerical frequency to it, the only ground they have in
common may be the amount of belief to which they are respectively
entitled. We cannot compare the frequency of their occurrence, for one may
occur too seldom to judge by, perhaps it may be unique. It has been already
said, that our belief of many events rests upon a very complicated and
extensive basis. My belief may be the product of many conflicting
arguments, and many analogies more or less remote; these proofs
themselves may have mostly faded from my mind, but they will leave their
effect behind them in a weak or strong conviction. At the time, therefore, I
may still be able to say, with some degree of accuracy, though a very slight
degree, what amount of belief I entertain upon the subject. Now we cannot
compare things that are heterogeneous: if, therefore, we are to decide
between this and an event determined naturally and properly by Probability,
it is impossible to appeal to chances or frequency of occurrence. The

measure of belief is the only common ground, and we must therefore
compare this quantity in each case. The test afforded will be an exceedingly
rough one, for the reasons mentioned above, but it will be better than none;
in some cases it will be found to furnish all we want.

Suppose, for example, that one letter in a million is lost in the Post Office,
and that in any given instance I wish to know which is more likely, that a
letter has been so lost, or that my servant has stolen it? If the latter
alternative could, like the former, be stated in a numerical form, the
comparison would be simple. But it cannot be reduced to this form, at least
not consciously and directly. Still, if we could feel that our belief in the
man's dishonesty was greater than one-millionth, we should then have
homogeneous things before us, and therefore comparison would be
possible.

§ 35. We are now in a position to give a tolerably accurate definition of a
phrase which we have frequently been obliged to employ, or incidentally to
suggest, and of which the reader may have looked for a definition already,
viz. the probability of an event, or what is equivalent to this, the chance of
any given event happening. I consider that these terms presuppose a series;
within the indefinitely numerous class which composes this series a smaller
class is distinguished by the presence or absence of some attribute or
attributes, as was fully illustrated and explained in a previous chapter. These
larger and smaller classes respectively are commonly spoken of as instances
of the ‘event,’ and of ‘its happening in a given particular way.’ Adopting
this phraseology, which with proper explanations is suitable enough, we
may define the probability or chance (the terms are here regarded as
synonymous) of the event happening in that particular way as the numerical
fraction which represents the proportion between the two different classes
in the long run. Thus, for example, let the probability be that of a given
infant living to be eighty years of age. The larger series will comprise all
infants, the smaller all who live to eighty. Let the proportion of the former
to the latter be 9 to 1; in other words, suppose that one infant in ten lives to
eighty. Then the chance or probability that any given infant will live to
1
eighty is the numerical fraction 10 . This assumes that the series are of
indefinite extent, and of the kind which we have described as possessing a
fixed type. If this be not the case, but the series be supposed terminable, or

regularly or irregularly fluctuating, as might be the case, for instance, in a
society where owing to sanitary or other causes the average longevity was
steadily undergoing a change, then in so far as this is the case the series
ceases to be a subject of science. What we have to do under these
circumstances, is to substitute a series of the right kind for the inappropriate
one presented by nature, choosing it, of course, with as little deflection as
possible from the observed facts. This is nothing more than has to be done,
and invariably is done, whenever natural objects are made subjects of strict
science.

§ 36. A word or two of explanation may be added about the expression
employed above, ‘the proportion in the long run.’ The run must be supposed
to be very long indeed, in fact never to stop. As we keep on taking more
terms of the series we shall find the proportion still fluctuating a little, but
its fluctuations will grow less. The proportion, in fact, will gradually
approach towards some fixed numerical value, what mathematicians term
its limit. This fractional value is the one spoken of above. In the cases in
which deductive reasoning is possible, this fraction may be obtained
without direct appeal to statistics, from reasoning about the conditions
under which the events occur, as was explained in the fourth chapter.

Here becomes apparent the full importance of the distinction so frequently
insisted on, between the actual irregular series before us and the substituted
one of calculation, and the meaning of the assertion (Ch. I. § 13), that it was
in the case of the latter only that strict scientific inferences could be made.
For how can we have a ‘limit’ in the case of those series which ultimately
exhibit irregular fluctuations? When we say, for instance, that it is an even
chance that a given person recovers from the cholera, the meaning of this
assertion is that in the long run one half of the persons attacked by that
disease do recover. But if we examined a sufficiently extensive range of
statistics, we might find that the manners and customs of society had
produced such a change in the type of the disease or its treatment, that we
were no nearer approaching towards a fixed limit than we were at first. The
conception of an ultimate limit in the ratio between the numbers of the two
classes in the series necessarily involves an absolute fixity of the type.
When therefore nature does not present us with this absolute fixity, as she
seldom or never does except in games of chance (and not demonstrably

there), our only resource is to introduce such a series, in other words, as has
so often been said, to substitute a series of the right kind.

§ 37. The above, which may be considered tolerably complete as a
definition, might equally well have been given in the last chapter. It has
been deferred however to the present place, in order to connect with it at
once a proposition involving the conceptions introduced in this chapter; viz.
the state of our own minds, in reference to the amount of belief we entertain
in contemplating any one of the events whose probability has just been
described. Reasons were given against the opinion that our belief admitted
of any exact apportionment like the numerical one just mentioned. Still, it
was shown that a reasonable explanation could be given of such an
1
expression as, ‘my belief is 10 th of certainty’, though it was an explanation
which pointed unmistakeably to a series of events, and ceased to be
intelligible, or at any rate justifiable, when it was not viewed in such a
relation to a series. In so far, then, as this explanation is adopted, we may
say that our belief is in proportion to the above fraction. This referred to the
purely intellectual part of belief which cannot be conceived to be separable,
even in thought, from the things upon which it is exercised. With this
intellectual part there are commonly associated various emotions. These we
can to a certain extent separate, and, when separated, can measure with that
degree of accuracy which is possible in the case of other emotions. They are
moreover intelligible in reference to the individual events. They will be
found to increase and diminish in accordance, to some extent, with the
fraction which represents the scarcity of the event. The emotion of surprise
does so with some degree of accuracy.

The above investigation describes, though in a very brief form, the amount
of truth which appears to me to be contained in the assertion frequently
made, that the fraction expressive of the probability represents also the
fractional part of full certainty to which our belief of the individual event
amounts. Any further analysis of the matter would seem to belong to
Psychology rather than to Probability.

1 In the ordinary signification of this term. As De Morgan uses it he makes

Formal Logic include Probability, as one of its branches, as indicated in his
title “Formal Logic, or the Calculus of Inference, necessary and probable.”
2
Formal Logic. Preface, page v.
3 An illustration of the points here insisted on has recently [1876] been

given in a quarter where few would have expected it; I allude, as many
readers will readily infer, to J. S. Mill's exceedingly interesting Essays on
Theism. It is not within our province here to criticise any of their
conclusions, but they have expressed in a very significant way the
conviction entertained by him that beliefs which are not justified by
evidence, and possibly may not be capable of justification (those for
instance of immortality and the existence of the Deity), may nevertheless
not only continue to exist in cultivated minds, but may also be profitably
encouraged there, at any rate in the shape of hopes, for certain supposed
advantages attendant on their retention, irrespective even of their truth.
4 It is necessary to take an example in which the man is forced to act, or we

should not be able to shew that he has any belief on the subject at all. He
may declare that he neither knows nor cares anything about the matter, and
that therefore there is nothing of the nature of belief to be extracted out of
his mental condition. He very likely would take this ground if we asked
him, as De Morgan does, with a slightly different reference (Formal Logic,
p. 183), whether he considers that there are volcanoes on the unseen side of
the moon larger than those on the side turned towards us; or, with Jevons
(Principles of Science, Ed. II. p. 212) whether he considers that a
Platythliptic Coefficient is positive. These do not therefore seem good
instances to illustrate the position that we always entertain a certain degree
of belief on every question which can be stated, and that utter inability to
give a reason in favour of either alternative corresponds to half belief.
5
Except indeed on the principles indicated further on in §§ 24, 25.
6 For a fuller discussion of this, see the Chapter on Causation.

7 The best example I can recall of the distinction between judging from the

subjective and the objective side, in such cases as these, occurred once in a
railway train. I met a timid old lady who was in much fear of accidents. I
endeavoured to soothe her on the usual statistical ground of the extreme
rarity of such events. She listened patiently, and then replied, “Yes, Sir, that
is all very well; but I don't see how the real danger will be a bit the less
because I don't believe in it.”
8
This would still hold of empirical laws which may be capable of being
broken: we now have very much shifted the word, to denote an ultimate law
which it is supposed cannot be broken.

CHAPTER VII.

THE RULES OF INFERENCE IN PROBABILITY.
§ 1. In the previous chapter, an investigation was made into what may be
called, from the analogy of Logic, Immediate Inferences. Given that nine
men out of ten, of any assigned age, live to forty, what could be inferred
about the prospect of life of any particular man? It was shown that,
although this step was very far from being so simple as it is frequently
supposed to be, and as the corresponding step really is in Logic, there was
nevertheless an intelligible sense in which we might speak of the amount of
our belief in any one of these ‘proportional propositions,’ as they may
succinctly be termed, and justify that amount. We must now proceed to the
consideration of inferences more properly so called, I mean inferences of
the kind analogous to those which form the staple of ordinary logical
treatises. In other words, having ascertained in what manner particular
propositions could be inferred from the general propositions which included
them, we must now examine in what cases one general proposition can be
inferred from another. By a general proposition here is meant, of course, a
general proposition of the statistical kind contemplated in Probability. The
rules of such inference being very few and simple, their consideration will
not detain us long. From the data now in our possession we are able to
deduce the rules of probability given in ordinary treatises upon the science.
It would be more correct to say that we are able to deduce some of these
rules, for, as will appear on examination, they are of two very different
kinds, resting on entirely distinct grounds. They might be divided into those
which are formal, and those which are more or less experimental. This may
be otherwise expressed by saying that, from the kind of series described in
the first chapters, some rules will follow necessarily by the mere application
of arithmetic; whilst others either depend upon peculiar hypotheses, or
demand for their establishment continually renewed appeals to experience,
and extension by the aid of the various resources of Induction. We shall
confine our attention at present principally to the former class; the latter can
only be fully understood when we have considered the connection of our
science with Induction.

§ 2. The fundamental rules of Probability strictly so called, that is the
formal rules, may be divided into two classes,—those obtained by addition

or subtraction on the one hand, corresponding to what are generally termed
the connection of exclusive or incompatible events;[1] and those obtained by
multiplication or division, on the other hand, corresponding to what are
commonly termed dependent events. We will examine these in order.

(1) We can make inferences by simple addition. If, for instance, there are
two distinct properties observable in various members of the series, which
properties do not occur in the same individual; it is plain that in any batch
the number that are of one kind or the other will be equal to the sum of
those of the two kinds separately. Thus 36.4 infants in 100 live to over sixty,
35.4 in 100 die before they are ten;[2] take a large number, say 10,000, then
there will be about 3640 who live to over sixty, and about 3540 who do not
reach ten; hence the total number who do not die within the assigned limits
will be about 2820 altogether. Of course if these proportions were
accurately assigned, the resultant sum would be equally accurate: but, as the
reader knows, in Probability this proportion is merely the limit towards
which the numbers tend in the long run, not the precise result assigned in
any particular case. Hence we can only venture to say that this is the limit
towards which we tend as the numbers become greater and greater.

This rule, in its general algebraic form, would be expressed in the language
of Probability as follows:—If the chances of two exclusive or incompatible
events be respectively m1 and 1n the chance of one or other of them happening
will be m1 + 1n or mmn
+n
. Similarly if there were more than two events of the
kind in question. On the principles adopted in this work, the rule, when thus
algebraically expressed, means precisely the same thing as when it is
expressed in the statistical form. It was shown at the conclusion of the last
chapter that to say, for example, that the chance of a given event happening
in a certain way is 16, is only another way of saying that in the long run it
does tend to happen in that way once in six times.

It is plain that a sort of corollary to this rule might be obtained, in precisely
the same way, by subtraction instead of addition. Stated generally it would
be as follows:—If the chance of one or other of two incompatible events be

1
m
and the chance of one alone be 1n, the chance of the remaining one will be
1 1 n−m
−
m n
or nm
.

1
For example, if the chance of any one dying in a year is 10 , and his chance
1
of dying of some particular disease is 100 , his chance of dying of any other
9
disease is 100.

The reader will remark here that there are two apparently different modes of
stating this rule, according as we speak of ‘one or other of two or more
events happening,’ or of ‘the same event happening in one or other of two
or more ways.’ But no confusion need arise on this ground; either way of
speaking is legitimate, the difference being merely verbal, and depending
(as was shown in the first chapter, § 8) upon whether the distinctions
between the ‘ways’ are or are not too deep and numerous to entitle the event
to be conventionally regarded as the same.

We may also here point out the justification for the common doctrine that
certainty is represented by unity, just as any given degree of probability is
represented by its appropriate fraction. If the statement that an event
happens once in m times, is equivalently expressed by saying that its chance
is m1 , it follows that to say that it happens m times in m times, or every time
without exception, is equivalent to saying that its chance is m
m
or 1. Now an
event that happens every time is of course one of whose occurrence we are
certain; hence the fraction which represents the ‘chance’ of an event which
is certain becomes unity.

It will be equally obvious that given that the chance that an event will
happen is m1 , the chance that it will not happen is 1 − m1 or mm− 1.

§ 3. (2) We can also make inferences by multiplication or division. Suppose
that two events instead of being incompatible, are connected together in the
sense that one is contingent upon the occurrence of the other. Let us be told
that a given proportion of the members of the series possess a certain
property, and a given proportion again of these possess another property,

then the proportion of the whole which possess both properties will be
found by multiplying together the two fractions which represent the above
two proportions. Of the inhabitants of London, twenty-five in a thousand,
say, will die in the course of the year; we suppose it to be known also that
one death in five is due to fever; we should then infer that one in 200 of the
inhabitants will die of fever in the course of the year. It would of course be
equally simple, by division, to make a sort of converse inference. Given the
total mortality per cent. of the population from fever, and the proportion of
fever cases to the aggregate of other cases of mortality, we might have
inferred, by dividing one fraction by the other, what was the total mortality
per cent. from all causes.

The rule as given above is variously expressed in the language of
Probability. Perhaps the simplest and best statement is that it gives us the
rule of dependent events. That is; if the chance of one event is m1 , and the
chance that if it happens another will also happen 1n, then the chance of the
1
latter is mn . In this case it is assumed that the latter is so entirely dependent
upon the former that though it does not always happen with it, it certainly
will not happen without it; the necessity of this assumption however may be
obviated by saying that what we are speaking of in the latter case is the joint
event, viz. both together if they are simultaneous events, or the latter in
consequence of the former, if they are successive.

§ 4. The above inferences are necessary, in the sense in which arithmetical
inferences are necessary, and they do not demand for their establishment
any arbitrary hypothesis. We assume in them no more than is warranted,
and in fact necessitated by the data actually given to us, and make our
inferences from these data by the help of arithmetic. In the simple examples
given above nothing is required beyond arithmetic in its most familiar form,
but it need hardly be added that in practice examples may often present
themselves which will require much profounder methods than these. It may
task all the resources of that higher and more abstract arithmetic known as
algebra to extract a solution. But as the necessity of appeal to such methods
as these does not touch the principles of this part of the subject we need not
enter upon them here.

§ 5. The formula next to be discussed stands upon a somewhat different
footing from the above in respect of its cogency and freedom from appeal to
experience, or to hypothesis. In the two former instances we considered
cases in which the data were supposed to be given under the conditions that
the properties which distinguished the different kinds of events whose
frequency was discussed, were respectively known to be disconnected and
known to be connected. Let us now suppose that no such conditions are
given to us. One man in ten, say, has black hair, and one in twelve is short-
sighted; what conclusions could we then draw as to the chance of any given
man having one only of these two attributes, or neither, or both? It is clearly
possible that the properties in question might be inconsistent with one
another, so as never to be found combined in the same person; or all the
short-sighted might have black hair; or the properties might be allotted[3] in
almost any other proportion whatever. If we are perfectly ignorant upon
these points, it would seem that no inferences whatever could be drawn
about the required chances.

Inferences however are drawn, and practically, in most cases, quite justly
drawn. An escape from the apparent indeterminateness of the problem, as
above described, is found by assuming that, not merely will one-tenth of the
whole number of men have black hair (for this was given as one of the
data), but also that one-tenth alike of those who are and who are not short-
sighted have black hair. Let us take a batch of 1200, as a sample of the
whole. Now, from the data which were originally given to us, it will easily
be seen that in every such batch there will be on the average 120 who have
black hair, and therefore 1080 who have not. And here in strict right we
ought to stop, at least until we have appealed again to experience; but we do
not stop here. From data which we assume, we go on to infer that of
the 120, 10 (i.e. one-twelfth of 120) will be short-sighted, and 110 (the
remainder) will not. Similarly we infer that of the 1080, 90 are short-
sighted, and 990 are not. On the whole, then, the 1200 are thus divided:—
black-haired short-sighted, 10; short-sighted without black hair, 90; black-
haired men who are not short-sighted, 110; men who are neither short-
sighted nor have black hair, 990.

This rule, expressed in its most general form, in the language of Probability,
would be as follows:—If the chances of a thing being p and q are

respectively m1 and 1n, then the chance of its being both p and q is mn
1
, p and
not q is nmn
−1
, q and not p is mmn
−1
, not p and not q is (m − 1)(n
mn
− 1)
, where p and q
are independent. The sum of these chances is obviously unity; as it ought to
be, since one or other of the four alternatives must necessarily exist.

§ 6. I have purposely emphasized the distinction between the inference in
this case, and that in the two preceding, to an extent which to many readers
may seem unwarranted. But it appears to me that where a science makes
use, as Probability does, of two such very distinct sources of conviction as
the necessary rules of arithmetic and the merely more or less cogent ones of
Induction, it is hardly possible to lay too much stress upon the distinction.
Few will be prepared to deny that very arbitrary assumptions have been
made by many writers on the subject, and none will deny that in the case of
what are called ‘inverse probabilities’ assumptions are sometimes made
which are at least decidedly open to question. The best course therefore is to
make a pause and stringent enquiry at the point at which the possibility of
such error and doubtfulness first exhibits itself. These remarks apply to
some of the best writers on the subject; in the case of inferior writers, or
those who appeal to Probability without having properly mastered its
principles, we may go further. It would really not be asserting too much to
say that they seem to think themselves justified in assuming that where we
know nothing about the distribution of the properties alluded to we must
assume them to be distributed as above described, and therefore apportion
our belief in the same ratio. This is called ‘assuming the events to be
independent,’ the supposition being made that the rule will certainly follow
from this independence, and that we have a right, if we know nothing to the
contrary, to assume that the events are independent.

The validity of this last claim has already been discussed in the first
chapter; it is only another of the attempts to construct à priori the series
which experience will present to us, and one for which no such strong
defence can be made as for the equality of heads and tails in the throws of a
penny. But the meaning to be assigned to the ‘independence’ of the events
in question demands a moment's consideration.

The circumstances of the problem are these. There are two different
qualities, by the presence and absence respectively of each of which,
amongst the individuals of a series, two distinct pairs of classes of these
individuals are produced. For the establishment of the rule under discussion
it was found that one supposition was both necessary and sufficient,
namely, that the division into classes caused by each of the above
distinctions should subdivide each of the classes created by the other
distinction in the same ratio in which it subdivides the whole. If the
independence be granted and so defined as to mean this, the rule of course
will stand, but, without especial attention being drawn to the point, it does
not seem that the word would naturally be so understood.

§ 7. The above, then, being the fundamental rules of inference in
probability, the question at once arises, What is their relation to the great
body of formulæ which are made use of in treatises upon the science, and in
practical applications of it? The reply would be that these formulæ, in so far
as they properly belong to the science, are nothing else in reality than
applications of the above fundamental rules. Such applications may assume
any degree of complexity, for owing to the difficulty of particular examples,
in the form in which they actually present themselves, recourse must
sometimes be made to the profoundest theorems of mathematics. Still we
ought not to regard these theorems as being anything else than convenient
and necessary abbreviations of arithmetical processes, which in practice
have become too cumbersome to be otherwise performed.

This explanation will account for some of the rules as they are ordinarily
given, but by no means for all of them. It will account for those which are
demonstrable by the certain laws of arithmetic, but not for those which in
reality rest only upon inductive generalizations. And it can hardly be
doubted that many rules of the latter description have become associated
with those of the former, so that in popular estimation they have been
blended into one system, of which all the separate rules are supposed to
possess a similar origin and equal certainty. Hints have already been
frequently given of this tendency, but the subject is one of such extreme
importance that a separate chapter (that on Induction) must be devoted to its
consideration.

§ 8. In establishing the validity of the above rules, we have taken as the
basis of our investigations, in accordance with the general scheme of this
work, the statistical frequency of the events referred to; but it was also
shown that each formula, when established, might with equal propriety be
expressed in the more familiar form of a fraction representing the ‘chance’
of the occurrence of the particular event. The question may therefore now
be raised, Can those writers who (as described in the last chapter) take as
the primary subject of the science not the degree of statistical frequency, but
the quantity of belief, with equal consistency make this the basis of their
rules, and so also regard the fraction expressive of the chance as a merely
synonymous expression? De Morgan maintains that whereas in ordinary
logic we suppose the premises to be absolutely true, the province of
Probability is to study ‘the effect which partial belief of the premises
produces with respect to the conclusion.’ It would appear therefore as if in
strictness we ought on this view to be able to determine this consequent
diminution at first hand, from introspection of the mind, that is of the
conceptions and beliefs which it entertains; instead of making any recourse
to statistics to tell us how much we ought to believe the conclusion.

Any readers who have concurred with me in the general results of the last
chapter, will naturally agree in the conclusion that nothing deserving the
name of logical science can be extracted from any results of appeal to our
consciousness as to the quantity of belief we entertain of this or that
proposition. Suppose, for example, that one person in 100 dies on the sea
passage out to India, and that one in 9 dies during a 5 years residence there.
It would commonly be said that the chance that any one, who is now going
88
out, has of living to start homewards 5 years hence, is 100 ; for his chance of
99
getting there is 100 ; and of his surviving, if he gets there, 89; hence the result
or dependent event is got by multiplying these fractions together, which
88
gives 100 . Here the real basis of the reasoning is statistical, and the processes
or results are merely translated afterwards into fractions. But can we say the
same when we look at the belief side of the question? I quite admit the
psychological fact that we have degrees of belief, more or less
corresponding to the frequency of the events to which they refer. In the
above example, for instance, we should undoubtedly admit on enquiry that
our belief in the man's return was affected by each of the risks in question,

so that we had less expectation of it than if he were subject to either risk
separately; that is, we should in some way compound the risks. But what I
cannot recognise is that we should be able to perform the process with any
approach to accuracy without appeal to the statistics, or that, even
supposing we could do so, we should have any guarantee of the correctness
of the result without similar appeal. It appears to me in fact that but little
meaning, and certainly no security, can be attained by so regarding the
process of inference. The probabilities expressed as degrees of belief, just
as those which are expressed as fractions, must, when we are put upon our
justification, first be translated into their corresponding facts of statistical
frequency of occurrence of the events, and then the inferences must be
drawn and justified there. This part of the operation, as we have already
shown, is mostly carried on by the ordinary rules of arithmetic. When we
have obtained our conclusion we may, if we please, translate it back again
into the subjective form, just as we can and do for convenience into the
fractional, but I do not see how the process of inference can be conceived as
taking place in that form, and still less how any proof of it can thus be
given. If therefore the process of inference be so expressed it must be
regarded as a symbolical process, symbolical of such an inference about
things as has been described above, and it therefore seems to me more
advisable to state and expound it in this latter form.

On Inverse Probability and the Rules required for it.
§ 9. It has been already stated that the only fundamental rules of inference
in Probability are the two described in §§ 2, 3, but there are of course
abundance of derivative rules, the nature and use of which are best obtained
from the study of any manual upon the subject. One class of these
derivative rules, however, is sufficiently distinct in respect of the questions
to which it may give rise, to deserve special examination. It involves the
distinction commonly recognised as that between Direct and Inverse
Probability. It is thus introduced by De Morgan:—

“In the preceding chapter we have calculated the chances of an event,
knowing the circumstances under which it is to happen or fail. We are now
to place ourselves in an inverted position: we know the event, and ask what

is the probability which results from the event in favour of any set of
circumstances under which the same might have happened.”[4] The
distinction might therefore be summarily described as that between finding
an effect when we are given the causes, and finding a cause when we are
given effects.

On the principles of the science involved in the definition which was
discussed and adopted in the earlier chapters of this work, the reader will
easily infer that no such distinction as this can be regarded as fundamental.
One common feature was traced in all the objects which were to be referred
to Probability, and from this feature the possible rules of inference can be
immediately derived. All other distinctions are merely those of arrangement
or management.

But although the distinction is not by any means fundamental, it is
nevertheless true that the practical treatment of such problems as those
principally occurring in Inverse Probability, does correspond to a very
serious source of ambiguity and perplexity. The arbitrary assumptions
which appear in Direct Probability are not by any means serious, but those
which invade us in a large proportion of the problems offered by Inverse
Probability are both serious and inevitable.

§ 10. This will be best seen by the examination of special examples; as any,
however simple, will serve our purpose, let us take the two following:—

(1) A ball is drawn from a bag containing nine black balls and one white:
what is the chance of its being the white ball?

(2) A ball is drawn from a bag containing ten balls, and is found to be white;
what is the chance of there having been but that one white ball in the bag?

The class of which the first example is a simple instance has been already
abundantly discussed. The interpretation of it is as follows: If balls be
continually drawn and replaced, the proportion of white ones to the whole
1
number drawn will tend towards the fraction 10 . The contemplated action is
a single one, but we view it as one of the above series; at least our opinion
is formed upon that assumption. We conclude that we are going to take one

of a series of events which may appear individually fortuitous, but in which,
in the long run, those of a given kind are one-tenth of the whole; this kind
1
(white) is then singled out by anticipation. By stating that its chance is 10 ,
we merely mean to assert this physical fact, together with such other mental
facts, emotions, inferences, &c., as may be properly associated with it.

§ 11. Have we to interpret the second example in a different way? Here also
we have a single instance, but the nature of the question would seem to
decide that the only series to which it can properly be referred is the
following:—Balls are continually drawn from different bags each
containing ten, and are always found to be white; what is ultimately the
proportion of cases in which they will be found to have been taken from
bags with only one white ball in them? Now it may be readily shown[5] that
time has nothing to do with the question; omitting therefore the
consideration of this element, we have for the two series from which our
opinions in these two examples respectively are to be formed:—(1) balls of
different colours presented to us in a given ultimate ratio; (2) bags with
different contents similarly presented. From these data respectively we have
to assign their due weight to our anticipations of (1) a white ball; (2) a bag
containing but one white ball. So stated the problems would appear to be
formally identical.

When, however, we begin the practical work of solving them we perceive a
most important distinction. In the first example there is not much that is
arbitrary; balls would under such circumstance really come out more or less
accurately in the proportion expected. Moreover, in case it should be
objected that it is difficult to prove that they will do so, it does not seem an
unfair demand to say that the balls are to be ‘well-mixed’ or ‘fairly
distributed,’ or to introduce any of the other conditions by which, under the
semblance of judging à priori, we take care to secure our prospect of a
series of the desired kind. But we cannot say the same in the case of the
second example.

§ 12. The line of proof by which it is generally attempted to solve the
second example is of this kind;—It is shown that there being one white ball
for certain in the bag, the only possible antecedents are of ten kinds, viz.
bags, each of which contains ten balls, but in which the white balls range

respectively from one to ten in number. This of course imposes limits upon
the kind of terms to be found in our series. But we want more than such
limitations, we must know the proportions in which these terms are
ultimately found to arrange themselves in the series. Now this requires an
experience about bags which may not, and indeed in a large proportion of
similar cases, cannot, be given to us. If therefore we are to solve the
question at all we must make an assumption; let us make the following;—
that each of the bags described above occurs equally often,—and see what
follows. The bags being drawn from equally often, it does not follow that
they will each yield equal numbers of white balls. On the contrary they will,
as in the last example, yield them in direct proportion to the number of such
balls which they contain. The bag with one white and nine black will yield a
white ball once in ten times; that with two white, twice; and so on. The
result of this, it will be easily seen, is that in 100 drawings there will be
obtained on the average 55 white balls and 45 black. Now with those
drawings that do not yield white balls we have, by the question, nothing to
do, for that question postulated the drawing of a white ball as an
accomplished fact. The series we want is therefore composed of those
which do yield white. Now what is the additional attribute which is found in
some members, and in some members only, of this series, and which we
mentally anticipate? Clearly it is the attribute of having been drawn from a
bag which only contained one of these white balls. Of these there is, out of
1
the 55 drawings, but one. Accordingly the required chance is 55 . That is to
say, the white ball will have been drawn from the bag containing only that
one white, once in 55 times.

§ 13. Now, with the exception of the passage in italics, the process here is
precisely the same as in the other example; it is somewhat longer only
because we are not able to appeal immediately to experience, but are forced
to try to deduce what the result will be, though the validity of this deduction
itself rests, of course, ultimately upon experience. But the above passage is
a very important one. It is scarcely necessary to point out how arbitrary it is.

For is the supposition, that the different specified kinds of bags are equally
likely, the most reasonable supposition under the circumstances in
question? One man may think it is, another may take a contrary view. In
fact in an excellent manual[6] upon the subject a totally different supposition

is made, at any rate in one example; it is taken for granted in that instance,
not that every possible number of black and white balls respectively is
equally likely, but that every possible way of getting each number is equally
likely, whence it follows that bags with an intermediate number of black
and white balls are far more likely than those with an extreme number of
either. On this supposition five black and five white being obtainable in
252 ways against the ten ways of obtaining one white and nine black, it
follows that the chance that we have drawn from a bag of the latter
description is much less than on the hypothesis first made. The chance, in
1 1
fact, becomes now 512 instead of 55 . In the one case each distinct result is
considered equally likely, in the other every distinct way of getting each
result.

§ 14. Uncertainties of this kind are peculiarly likely to arise in these inverse
probabilities, because when we are merely given an effect and told to look
out for the chance of some assigned cause, we are often given no clue as to
the relative prevalence of these causes, but are left to determine them on
general principles. Give us either their actual prevalence in statistics, or the
conditions by which such prevalence is brought about, and we know what
to do; but without the help of such data we are reduced to guessing. In the
above example, if we had been told how the bag had been originally filled,
that is by what process, or under what circumstances, we should have
known what to do. If it had been filled at random from a box containing
equal numbers of black and white balls, the supposition in Mr Whitworth's
example is the most reasonable; but in the absence of any such information
as this we are entirely in the dark, and the supposition made in § 12 is
neither more nor less trustworthy and reasonable than many others, though
it doubtless possesses the merit of superior simplicity.[7] If the reader will
recur to Ch. V. §§ 4, 5, he will find this particular difficulty fully explained.
Everybody practically admits that a certain characteristic arrangement or
distribution has to be introduced at some prior stage; and that, as soon as
this stage has been selected, there are no further theoretic difficulties to be
encountered. But when we come to decide, in examples of the class in
question, at what stage it is most reasonable to make our postulate, we are
often left without any very definite or rational guidance.

§ 15. When, however, we take what may be called, by comparison with the
above purely artificial examples, instances presented by nature, much of
this uncertainty will disappear, and then all real distinction between direct
and inverse probability will often vanish. In such cases the causes are
mostly determined by tolerably definite rules, instead of being a mere
cloud-land of capricious guesses. We may either find their relative
frequency of occurrence by reference to tables, or may be able to infer it by
examination of the circumstances under which they are brought about.
Almost any simple example would then serve to illustrate the fact that
under such circumstances the distinction between direct and inverse
probability disappears altogether, or merely resolves itself into one of time,
which, as will be more fully shown in a future chapter, is entirely foreign to
our subject.

It is not of course intended to imply that difficulties similar to those
mentioned above do not occasionally invade us here also. As already
mentioned, they are, if not inherent in the subject, at any rate almost
unavoidable in comparison with the simpler and more direct procedure of
determining what is likely to follow from assigned conditions. What is
meant is that so long as we confine ourselves within the comparatively
regular and uniform field of natural sequences and co-existences, statistics
of causes may be just as readily available as those of effects. There will not
be much more that is arbitrary in the one than in the other. But of course
this security is lost when, as will be almost immediately noticed, what may
be called metaphysical rather than natural causes are introduced into the
enquiry.

For instance, it is known that in London about 20 people die per thousand
each year. Suppose it also known that of every 100 deaths there are about 4
attributable to bronchitis. The odds therefore against any unknown person
dying of bronchitis in a given year are 1249 to 1. Exactly the same statistics
are available to solve the inverse problem:—A man is dead, what is the
chance that he died of bronchitis? Here, since the man's death is taken for
granted, we do not require to know the general average mortality. All that
we want is the proportional mortality from the disease in question as given
above. If Probability dealt only with inferences founded in this way upon
actual statistics, and these tolerably extensive, it is scarcely likely that any

distinction such as this between direct and inverse problems would ever
have been drawn.

§ 16. Considered therefore as a contribution to the theory of the subject, the
distinction between Direct and Inverse Probability must be abandoned.
When the appropriate statistics are at hand the two classes of problems
become identical in method of treatment, and when they are not we have no
more right to extract a solution in one case than in the other. The discussion
however may serve to direct renewed attention to another and far more
important distinction. It will remind us that there is one class of examples to
which the calculus of Probability is rightfully applied, because statistical
data are all we have to judge by; whereas there are other examples in regard
to which, if we will insist upon making use of these rules, we may either be
deliberately abandoning the opportunity of getting far more trustworthy
information by other means, or we may be obtaining solutions about
matters on which the human intellect has no right to any definite
quantitative opinion.

§ 17. The nearest approach to any practical justification of such judgments
that I remember to have seen is afforded by cases of which the following
example is a specimen:— “Of 10 cases treated by Lister's method, 7 did
well and 3 suffered from blood-poisoning: of 14 treated with ordinary
dressings, 9 did well and 5 had blood-poisoning; what are the odds that the
success of Lister's method was due to chance?”.[8] Or, to put it into other
words, a short experience has shown an actual superiority in one method
over the other: what are the chances that an indefinitely long experience,
under similar conditions, will confirm this superiority?

The proposer treated this as a ‘bag and balls’ problem, analogous to the
following: 10 balls from one bag gave 7 white and 3 black, 14 from another
bag gave 9 white and 5 black: what is the chance that the actual ratio of
white to black balls was greater in the former than in the latter?—this actual
ratio being of course considered a true indication of what would be the
ultimate proportions of white and black drawings. This seems to me to be
the only reasonable way of treating the problem, if it is to be considered
capable of numerical solution at all.

Of course the inevitable assumption has to be made here about the equal
prevalence of the different possible kinds of bag,—or, as the supporters of
the justice of the calculation would put it, of the obligation to assume the
equal à priori likelihood of each kind,—but I think that in this particular
example the arbitrariness of the assumption is less than usual. This is
because the problem discusses simply a balance between two extremely
similar cases, and there is a certain set-off against each other of the
objectionable assumptions on each side. Had one set of experiments only
been proposed, and had we been asked to evaluate the probability of
continued repetition of them confirming their verdict, I should have felt all
the scruples I have already mentioned. But here we have got two sets of
experiments carried on under almost exactly similar circumstances, and
there is therefore less arbitrariness in assuming that their unknown
conditions are tolerably equally prevalent.

§ 18. Examples of the description commonly introduced seem objectionable
enough, but if we wish to realize to its full extent the vagueness of some of
the problems submitted to this Inverse Probability, we have not far to seek.
In natural as in artificial examples, where statistics are unattainable the
enquiry becomes utterly hopeless, and all attempts at laying down rules for
calculation must be abandoned. Take, for instance, the question which has
given rise to some discussion,[9] whether such and such groups of stars are
or are not to be regarded as the results of an accidental distribution; or the
still wider and vaguer question, whether such and such things, or say the
world itself, have been produced by chance?

In cases of this kind the insuperable difficulty is in determining what sense
exactly is to be attached to the words ‘accidental’ and ‘random’ which enter
into the discussion. Some account was given, in the fourth chapter, of their
scientific and conventional meaning in Probability. There seem to be the
same objections to generalizing them out of such relation, as there is in
metaphysics to talking of the Infinite or the Absolute. Infinite magnitude, or
infinite power, one can to some extent comprehend, or at least one may
understand what is being talked about, but ‘the infinite’ seems to me a term
devoid of meaning. So of anything supposed to have been produced at
random: tell us the nature of the agency, the limits of its randomness and so
on, and we can venture upon the problem, but without such data we know

not what to do. The further consideration of such a problem might, I think,
without arrogance be relegated to the Chapter on Fallacies. Accordingly any
further remarks which I have to make upon the subject will be found there,
and at the conclusion of the chapter on Causation and Design.

1 It might be more accurate to speak of ‘incompatible hypotheses with

respect to any individual case’, or ‘mutually exclusive classes of events’.
2
The examples, of this kind, referring to human mortality are taken from
the Carlisle tables. These differ considerably, as is well known, from other
tables, but we have the high authority of De Morgan for regarding them as
the best representative of the average mortality of the English middle
classes at the present day.
3 I say, almost any proportion, because, as may easily be seen, arithmetic

imposes certain restrictions upon the assumptions that can be made. We
could not, for instance, suppose that all the black-haired men are short-
sighted, for in any given batch of men the former are more numerous. But
the range of these restrictions is limited, and their existence is not of
importance in the above discussion.
4 Essay on Probabilities, p. 53. I have been reminded that in his article on

Probability in the Encyclopædia Metropolitana he has stated that such rules
involve no new principle.
5
This point will be fully discussed in a future chapter, after the general
stand-point of an objective system of logic has been explained and
illustrated.
6 Whitworth's Choice and Chance, Ed. II., p. 123. See also Boole's Laws of

Thought, p. 370.
7 Opinions differ about the defence of such suppositions, as they do about

the nature of them. Some writers, admitting the above assumption to be
doubtful, call it the most impartial hypothesis. Others regard it as a sort of
mean hypothesis.
8
Educational Times; Reprint, Vol. xxxvii. p. 40. The question was proposed
by Dr. Macalister and gave rise to considerable controversy. As usual with
problems of this inverse kind hardly any two of the writers were in

agreement as to the assumptions to be made, or therefore as to the
numerical estimate of the odds.
9 See Todhunter's History, pp. 333, 4.

There is an interesting discussion upon this question by the late J. D. Forbes
in a paper in the Philosophical Magazine for Dec. 1850. It was replied to in
a subsequent number by Prof. Donkin.

CHAPTER VIII.

THE RULE OF SUCCESSION.[*]
*
A word of apology may be offered here for the introduction of a new name. The only other
alternative would have been to entitle the rule one of Induction. But such a title I cannot admit, for
reasons which will be almost immediately explained.

§ 1. In the last chapter we discussed at some length the nature of the kinds
of inference in Probability which correspond to those termed, in Logic,
immediate and mediate inferences. We ascertained what was the meaning of
saying, for example, that the chance of any given man A. B. dying in a year
is 13, when concluded from the general proposition that one man out of three
in his circumstances dies. We also discussed the nature and evidence of
rules of a more completely inferential character. But to stop at this point
would be to take a very imperfect view of the subject. If Probability is a
science of real inference about things, it must surely lead up to something
more than such merely formal conclusions; we must be able, if not by
means of it, at any rate by some means, to step beyond the limits of what
has been actually observed, and to draw conclusions about what is as yet
unobserved. This leads at once to the question, What is the connection of
Probability with Induction? This is a question into which it will be
necessary to enter now with some minuteness.

That there is a close connection between Probability and Induction, must
have been observed by almost every one who has treated of either subject; I
have not however seen any account of this connection that seemed to me to
be satisfactory. An explicit description of it should rather be sought in
treatises upon the narrower subject, Probability; but it is precisely here that
the most confusion is to be found. The province of Probability being
somewhat narrow, incursions have been constantly made from it into the
adjacent territory of Induction. In this way, amongst the arithmetical rules
discussed in the last chapter, others have been frequently introduced which
ought not in strictness to be classed with them, as they rest on an entirely
different basis.

§ 2. The origin of such confusion is easy of explanation; it arises, doubtless,
from the habit of laying undue stress upon the subjective side of Probability,

upon that which treats of the quantity of our belief upon different subjects
and the variations of which that quantity is susceptible. It has been already
urged that this variation of belief is at most but a constant accompaniment
of what is really essential to Probability, and is moreover common to other
subjects as well. By defining the science therefore from this side these other
subjects would claim admittance into it; some of these, as Induction, have
been accepted, but others have been somewhat arbitrarily rejected. Our
belief in a wider proposition gained by Induction is, prior to verification,
not so strong as that of the narrower generalization from which it is
inferred. This being observed, a so-called rule of probability has been given
by which it is supposed that this diminution of assent could in many
instances be calculated.

But time also works changes in our conviction; our belief in the happening
of almost every event, if we recur to it long afterwards, when the evidence
has faded from the mind, is less strong than it was at the time. Why are not
rules of oblivion inserted in treatises upon Probability? If a man is told how
firmly he ought to expect the tide to rise again, because it has already risen
ten times, might he not also ask for a rule which should tell him how firm
should be his belief of an event which rests upon a ten years' recollection?[1]
The infractions of a rule of this latter kind could scarcely be more numerous
and extensive, as we shall see presently, than those of the former
confessedly are. The fact is that the agencies, by which the strength of our
conviction is modified, are so indefinitely numerous that they cannot all be
assembled into one science; for purposes of definition therefore the quantity
of belief had better be omitted from consideration, or at any rate regarded as
a mere appendage, and the science, defined from the other or statistical side
of the subject, in which, as has been shown, a tolerably clear boundary-line
can be traced.

§ 3. Induction, however, from its importance does merit a separate
discussion; a single example will show its bearing upon this part of our
subject. We are considering the prospect of a given man, A. B. living
another year, and we find that nine out of ten men of his age do survive. In
forming an opinion about his surviving, however, we shall find that there
are in reality two very distinct causes which aid in determining the strength

of our conviction; distinct, but in practice so intimately connected that we
are very apt to overlook one, and attribute the effect entirely to the other.

(I.) There is that which strictly belongs to Probability; that which (as was
explained in Chap VI.) measures our belief of the individual case as
deduced from the general proposition. Granted that nine men out of ten of
the kind to which A. B. belongs do live another year, it obviously does not
follow at all necessarily that he will. We describe this state of things by
saying, that our belief of his surviving is diminished from certainty in the
9
ratio of 10 to 9, or, in other words, is measured by the fraction 10 .

(II.) But are we certain that nine men out of ten like him will live another
year? we know that they have so survived in time past, but will they
continue to do so? Since A. B. is still alive it is plain that this proposition is
to a certain extent assumed, or rather obtained by Induction. We cannot
however be as certain of the inductive inference as we are of the data from
which it was inferred. Here, therefore, is a second cause which tends to
diminish our belief; in practice these two causes always accompany each
other, but in thought they can be separated.

The two distinct causes described above are very liable to be confused
together, and the class of cases from which examples are necessarily for the
most part drawn increases this liability. The step from the statement ‘all
men have died in a certain proportion’ to the inference ‘they will continue
to die in that proportion’ is so slight a step that it is unnoticed, and the
diminution of conviction that should accompany it is unsuspected. In what
are called à priori examples the step is still slighter. We feel so certain about
the permanence of the laws of mechanics, that few people would think of
regarding it as an inference when they believe that a die will in the long run
turn up all its faces equally often, because other dice have done so in time
past.

§ 4. It has been already pointed out (in Chapter VI.) that, so far as concerns
that definition of Probability which regards it as the science which discusses
the degree and modifications of our belief, the question at issue seems to be
simply this:—Are the causes alluded to above in (II.) capable of being
reduced to one simple coherent scheme, so that any universal rules for the

modification of assent can be obtained from them? If they are, strong
grounds will have been shown for classing them with (I.), in other words,
for considering them as rules of probability. Even then they would be rules
practically of a very different kind, contingent instead of necessary (if one
may use these terms without committing oneself to any philosophical
system), but this objection might perhaps be overruled by the greater
simplicity secured by classing them together. This view is, with various
modifications, generally adopted by writers on Probability, or at least, as I
understand the matter, implied by their methods of definition and treatment.
Or, on the other hand, must these causes be regarded as a vast system, one
might almost say a chaos, of perfectly distinct agencies; which may indeed
be classified and arranged to some extent, but from which we can never
hope to obtain any rules of perfect generality which shall not be subject to
constant exception? If so, but one course is left; to exclude them all alike
from Probability. In other words, we must assume the general proposition,
viz. that which has been described throughout as our starting-point, to be
given to us; it may be obtained by any of the numerous rules furnished by
Induction, or it may be inferred deductively, or given by our own
observation; its value may be diminished by its depending upon the
testimony of witnesses, or its being recalled by our own memory. Its real
value may be influenced by these causes or any combinations of them; but
all these are preliminary questions with which we have nothing directly to
do. We assume our statistical proposition to be true, neglecting the
diminution of its value by the process of attainment; we take it up first at
this point and then apply our rules to it. We receive it in fact, if one may use
the expression, ready-made, and ask no questions about the process or
completeness of its manufacture.

§ 5. It is not to be supposed, of course, that any writers have seriously
attempted to reduce to one system of calculation all the causes mentioned
above, and to embrace in one formula the diminution of certainty to which
the inclusion of them subjects us. But on the other hand, they have been
unwilling to restrain themselves from all appeal to them. From an early
period in the study of the science attempts have been made to proceed, by
the Calculus of Probability, from the observed cases to adjacent and similar
cases. In practice, as has been already said, it is not possible to avoid some
extension of this kind. But it should be observed, that in these instances the

divergence from the strict ground of experience is not in reality recognized,
at least not as a part of our logical procedure. We have, it is true, wandered
somewhat beyond it, and so obtained a wider proposition than our data
strictly necessitated, and therefore one of less certainty. Still we assume the
conclusion given by induction to be equally certain with the data, or rather
omit all notice of the divergence from consideration. It is assumed that the
unexamined instances will resemble the examined, an assumption for which
abundant warrant may exist; the theory of the calculation rests upon the
supposition that there will be no difference between them, and the practical
error is insignificant simply because this difference is small.

§ 6. But the rule we are now about to discuss, and which may be called the
Rule of Succession, is of a very different kind. It not only recognizes the
fact that we are leaving the ground of past experience, but takes the
consequences of this divergence as the express subject of its calculation. It
professes to give a general rule for the measure of expectation that we
should have of the reappearance of a phenomenon that has been already
observed any number of times. This rule is generally stated somewhat as
follows: “To find the chance of the recurrence of an event already observed,
divide the number of times the event has been observed, increased by one,
by the same number increased by two.”

§ 7. It will be instructive to point out the origin of this rule; if only to
remind the reader of the necessity of keeping mathematical formulæ to their
proper province, and to show what astonishing conclusions are apt to be
accepted on the supposed warrant of mathematics. Revert then to the
example of Inverse Probability on p. 182. We saw that under certain
assumptions, it would follow that when a single white ball had been drawn
from a bag known to contain 10 balls which were white or black, the chance
could be determined that there was only one white ball in it. Having done
this we readily calculate ‘directly’ the chance that this white ball will be
drawn next time. Similarly we can reckon the chances of there being two,
three, &c. up to ten white balls in it, and determine on each of these
suppositions the chance of a white ball being drawn next time. Adding these
together we have the answer to the question:—a white ball has been drawn
once from a bag known to contain ten balls, white or black; what is the
chance of a second time drawing a white ball?

So far only arithmetic is required. For the next step we need higher
mathematics, and by its aid we solve this problem:—A white ball has been
drawn m times from a bag which contains any number, we know not what,
of balls each of which is white or black, find the chance of the next drawing
also yielding a white ball. The answer is

m + 1.
m+2

Thus far mathematics. Then comes in the physical assumption that the
universe may be likened to such a bag as the above, in the sense that the
above rule may be applied to solve this question:—an event has been
observed to happen m times in a certain way, find the chance that it will
happen in that way next time. Laplace, for instance, has pointed out that at
the date of the writing of his Essai Philosophique, the odds in favour of the
sun's rising again (on the old assumption as to the age of the world) were
1,826,214 to 1. De Morgan says that a man who standing on the bank of a
river has seen ten ships pass by with flags should judge it to be 11 to 1 that
the next ship will also carry a flag.

§ 8. It is hard to take such a rule as this seriously, for there does not seem to
be even that moderate confirmation of it which we shall find to hold good
in the case of the application of abstract formulæ to the estimation of the
evidence of witnesses. If however its validity is to be discussed there appear
to be two very distinct lines of enquiry along which we may be led.

(1) In the first place we may take it for what it professes to be, and for what
it is commonly understood to be, viz. a rule which assigns the measure of
expectation we ought to entertain of the recurrence of the event under the
circumstances in question. Of course, on the view adopted in this work, we
insist on enquiring whether it is really true that on the average events do
thus repeat their performance in accordance with this law. Thus tested, no
one surely would attempt to defend such a formula. So far from past
occurrence being a ground for belief in future recurrence, there are (as will
be more fully pointed out in the Chapter on Fallacies) plenty of cases in
which the direct contrary holds good. Then again a rule of this kind is
subject to the very serious perplexity to be explained in our next chapter,

arising out of the necessary arbitrariness of such inverse reference. That is,
when an event has happened but a few times, we have no certain guide; and
when it has happened but once,[2] we have no guide whatever, as to the
class of cases to which it is to be referred. In the example above, about the
flags, why did we stop short at this notion simply, instead of specifying the
size, shape, &c. of the flags?

De Morgan, it must be remembered, only accepts this rule in a qualified
sense. He regards it as furnishing a minimum value for the amount of our
expectation. He terms it “the rule of probability of a pure induction,” and
says of it, “The probabilities shown by the above rules are merely minima
which may be augmented by other sources of knowledge.” That is, he
recognizes only those instances in which our belief in the Uniformity of
Nature and in the existence of special laws of causation comes in to
supplement that which arises from the mere frequency of past occurrence.
This however does not meet those cases in which past occurrence is a
positive ground of disbelief in future recurrence.

§ 9. (2) There is however another and very different view which might be
taken of such a rule. It is one, an obscure recognition of which has very
likely had much to do with the acceptance which the rule has received.

What we might suppose ourselves to be thus expressing is,—not the
measure of rational expectation which might be held by minds sufficiently
advanced to be able to classify and to draw conscious inferences, but,—the
law according to which the primitive elements of belief were started and
developed. Of course such an interpretation as this would be equivalent to
quitting the province of Logic altogether and crossing over into that of
Psychology; but it would be a perfectly valid line of enquiry. We should be
attempting nothing more than a development of the researches of Fechner
and his followers in psychophysical measurement. Only then we ought, like
them, not to start with any analogy of a ballot box and its contents, but to
base our enquiry on careful determination of the actual mental phenomena
experienced. We know how the law has been determined in accordance with
which the intensity of the feeling of light varies with that of its objective
source. We see how it is possible to measure the growth of memory
according to the number of repetitions of a sentence or a succession of mere

syllables. In this latter case, for instance, we just try experiments, and
determine how much better a man can remember any utterances after eight
hearings than after seven.[3]

Now this case furnishes a very close parallel to our supposed attempt to
measure the increase of intensity of belief after repeated recurrence. That is,
if it were possible to experiment in this order of mental phenomena, we
ought simply to repeat a phenomenon a certain number of times and then
ascertain by actual introspection or by some simple test, how fast the belief
was increasing. Thus viewed the problem seems to me a hopeless one. The
difficulties are serious enough, when we are trying to measure our simple
sensations, of laying aside the effects of past training, and of attempting, as
it were, to leave the mind open and passive to mere reception of stimuli.
But if we were to attempt in this way to measure our belief these difficulties
would become quite insuperable. We can no more divest ourselves of past
training here than we can of intelligence or thought. I do not see how any
one could possibly avoid classing the observed recurrences with others
which he had experienced, and of being thus guided by special analogies
and inductions instead of trusting solely to De Morgan's ‘pure induction’.
The same considerations tend to rebut another form of defence for the rule
in question. It is urged, for instance, that we may at least resort to it in those
cases in which we are in entire ignorance as to the number and nature of the
antecedents. This is a position to which I can hardly conceive it possible
that we should ever be reduced. However remote or exceptional may be the
phenomenon selected we may yet bring it into relation with some accepted
generalizations and thus draw our conclusions from these rather than from
purely à priori considerations.

§ 10. Since then past acquisitions cannot be laid aside or allowed for, the
only remaining resource would be to experiment upon the infant mind. One
would not like to pronounce that any line of enquiry is impossible; but the
difficulties would certainly be enormous. And interesting as the facts would
be, supposing that we had succeeded in securing them, they would not be of
the slightest importance in Logic. However the question were settled:—
whether, for instance, we proved that the sentiment or emotion of belief
grew up slowly and gradually from a sort of zero point under the impress of
repetition of experience; or whether we proved that a single occurrence

produced complete belief in the repetition of the event, so that experience
gradually untaught us and weakened our convictions;—in no case would
the mature mind gain any aid as to what it ought to believe.

I cannot but think that some such view as this must occasionally underlie
the acceptance which this rule has received. For instance, Laplace, though
unhesitatingly adopting it as a real, that is, objective rule of inference, has
gone into so much physiological and psychological matter towards the end
of his discussion (Essai philosophique) as to suggest that what he had in
view was the natural history of belief rather than its subsequent
justification.

Again, the curious doctrine adopted by Jevons, that the principles of
Induction rest entirely upon the theory of Probability,—a very different
doctrine from that which is conveyed by saying that all knowledge of facts
is probable only, i.e. not necessary,—seems unintelligible except on some
such interpretation. We shall have more to say on this subject in our next
chapter. It will be enough here to remark that in our present reflective and
rational stage we find that every inference in Probability involves some
appeal to, or support from, Induction, but that it is impossible to base either
upon the other. However far back we try to push our way, and however
disposed we might be to account for our ultimate beliefs by Association, it
seems to me that so long as we consider ourselves to be dealing with rules
of inference we must still distinguish between Induction and Probability.

1 John Craig, in his often named work, Theologiæ Christianæ Principia

Mathematica (Lond. 1699) attempted something in this direction when he
proposed to solve such problems as:—Quando evanescet probabilitas
cujusvis Historiæ, cujus subjectum est transiens, vivâ tantum voce
transmissæ, determinare.
2
When m = 1 the fraction becomes 23; i.e. the odds are 2 to 1 in favour of
recurrence. And there are writers who accept this result. For instance,
Jevons (Principles of Science p. 258) says “Thus on the first occasion on

which a person sees a shark, and notices that it is accompanied by a little
pilot fish, the odds are 2 to 1 that the next shark will be so accompanied.”
To say nothing of the fact that recognizing and naming the fish implies that
they have often been seen before, how many of the observed characteristics
of that single ‘event’ are to be considered essential? Must the pilot precede;
and at the same distance? Must we consider the latitude, the ocean, the
season, the species of shark, as matter also of repetition on the next
occasion? and so on. I cannot see how the Inductive problem can be even
intelligibly stated, for quantitative purposes, on the first occurrence of any
event.
3 See in Mind (x. 454) Mr Jacob's account of the researches of Herr
Ebbinghaus as described in his work Ueber das Gedächtniss.

CHAPTER IX.

INDUCTION AND ITS CONNECTION WITH
PROBABILITY.
§ 1. We were occupied, during the last chapter, with the examination of a
rule, the object of which was to enable us to make inferences about
instances as yet unexamined. It was professedly, therefore, a rule of an
inductive character. But, in the form in which it is commonly expressed, it
was found to fail utterly. It is reasonable therefore to enquire at this point
whether Probability is entirely a formal or deductive science, or whether, on
the other hand, we are able, by means of it, to make valid inferences about
instances as yet unexamined. This question has been already in part
answered by implication in the course of the last two chapters. It is
proposed in the present chapter to devote a fuller investigation to this
subject, and to describe, as minutely as limits will allow, the nature of the
connection between Probability and Induction. We shall find it advisable for
clearness of conception to commence our enquiry at a somewhat early
stage. We will travel over the ground, however, as rapidly as possible, until
we approach the boundary of what can properly be termed Probability.

§ 2. Let us then conceive some one setting to work to investigate nature,
under its broadest aspect, with the view of systematizing the facts of
experience that are known, and thence (in case he should find that this is
possible) discovering others which are at present unknown. He observes a
multitude of phenomena, physical and mental, contemporary and
successive. He enquires what connections are there between them? what
rules can be found, so that some of these things being observed I can infer
others from them? We suppose him, let it be observed, deliberately
resolving to investigate the things themselves, and not to be turned aside by
any prior enquiry as to there being laws under which the mind is compelled
to judge of the things. This may arise either from a disbelief in the existence
of any independent and necessary mental laws, and a consequent conviction
that the mind is perfectly competent to observe and believe anything that
experience offers, and should believe nothing else, or simply from a
preference for investigations of the latter kind. In other words, we suppose

him to reject Formal Logic, and to apply himself to a study of objective
existences.

It must not for a moment be supposed that we are here doing more than
conceiving a fictitious case for the purpose of more vividly setting before
the reader the nature of the inductive process, the assumptions it has to
make, and the character of the materials to which it is applied. It is not
psychologically possible that any one should come to the study of nature
with all his mental faculties in full perfection, but void of all materials of
knowledge, and free from any bias as to the uniformities which might be
found to prevail around him. In practice, of course, the form and the matter
—the laws of belief or association, and the objects to which they are
applied—act and react upon one another, and neither can exist in any but a
low degree without presupposing the existence of the other. But the
supposition is perfectly legitimate for the purpose of calling attention to the
requirements of such a system of Logic, and is indeed nothing more than
what has to be done at almost every step in psychological enquiry.[1]

§ 3. His task at first might be conceived to be a slow and tedious one. It
would consist of a gradual accumulation of individual instances, as marked
out from one another by various points of distinction, and connected with
one another by points of resemblance. These would have to be respectively
distinguished and associated in the mind, and the consequent results would
then be summed up in general propositions, from which inferences could
afterwards be drawn. These inferences could, of course, contain no new
facts, they would only be repetitions of what he or others had previously
observed. All that we should have so far done would have been to make our
classifications of things and then to appeal to them again. We should
therefore be keeping well within the province of ordinary logic, the
processes of which (whatever their ultimate explanation) may of course
always be expressed, in accordance with Aristotle's Dictum, as ways of
determining whether or not we can show that one given class is included
wholly or partly within another, or excluded from it, as the case may be.

§ 4. But a very short course of observation would suggest the possibility of
a wide extension of his information. Experience itself would soon detect
that events were connected together in a regular way; he would ascertain

that there are ‘laws of nature.’ Coming with no à priori necessity of
believing in them, he would soon find that as a matter of fact they do exist,
though he could not feel any certainty as to the extent of their prevalence.
The discovery of this arrangement in nature would at once alter the plan of
his proceedings, and set the tone to the whole range of his methods of
investigation. His main work now would be to find out by what means he
could best discover these laws of nature.

An illustration may assist. Suppose I were engaged in breaking up a vast
piece of rock, say slate, into small pieces. I should begin by wearily
working through it inch by inch. But I should soon find the process
completely changed owing to the existence of cleavage. By this
arrangement of things a very few blows would do the work—not, as I might
possibly have at first supposed, to the extent of a few inches—but right
through the whole mass. In other words, by the process itself of cutting, as
shown in experience, and by nothing else, a constitution would be detected
in the things that would make that process vastly more easy and extensive.
Such a discovery would of course change our tactics. Our principal object
would thenceforth be to ascertain the extent and direction of this cleavage.

Something resembling this is found in Induction. The discovery of laws of
nature enables the mind to dart with its inferences from a few facts
completely through a whole class of objects, and thus to acquire results the
successive individual attainment of which would have involved long and
wearisome investigation, and would indeed in multitudes of instances have
been out of the question. We have no demonstrative proof that this state of
things is universal; but having found it prevail extensively, we go on with
the resolution at least to try for it everywhere else, and we are not
disappointed. From propositions obtained in this way, or rather from the
original facts on which these propositions rest, we can make new
inferences, not indeed with absolute certainty, but with a degree of
conviction that is of the utmost practical use. We have gained the great step
of being able to make trustworthy generalizations. We conclude, for
instance, not merely that John and Henry die, but that all men die.

§ 5. The above brief investigation contains, it is hoped, a tolerably correct
outline of the nature of the Inductive inference, as it presents itself in

Material or Scientific Logic. It involves the distinction drawn by Mill, and
with which the reader of his System of Logic will be familiar, between an
inference drawn according to a formula and one drawn from a formula. We
do in reality make our inference from the data afforded by experience
directly to the conclusion; it is a mere arrangement of convenience to do so
by passing through the generalization. But it is one of such extreme
convenience, and one so necessarily forced upon us when we are appealing
to our own past experience or to that of others for the grounds of our
conclusion, that practically we find it the best plan to divide the process of
inference into two parts. The first part is concerned with establishing the
generalization; the second (which contains the rules of ordinary logic)
determines what conclusions can be drawn from this generalization.

§ 6. We may now see our way to ascertaining the province of Probability
and its relation to kindred sciences. Inductive Logic gives rules for
discovering such generalizations as those spoken of above, and for testing
their correctness. If they are expressed in universal propositions it is the part
of ordinary logic to determine what inferences can be made from and by
them; if, on the other hand, they are expressed in proportional propositions,
that is, propositions of the kind described in our first chapter, they are
handed over to Probability. We find, for example, that three infants out of
ten die in their first four years. It belongs to Induction to say whether we are
justified in generalizing our observation into the assertion, All infants die in
that proportion. When such a proposition is obtained, whatever may be the
value to be assigned to it, we recognize in it a series of a familiar kind, and
it is at once claimed by Probability.

In this latter case the division into two parts, the inductive and the
ratiocinative, seems decidedly more than one of convenience; it is indeed
imperatively necessary for clearness of thought and cogency of treatment. It
is true that in almost every example that can be selected we shall find both
of the above elements existing together and combining to determine the
degree of our conviction, but when we come to examine them closely it
appears to me that the grounds of their cogency, the kind of conviction they
produce, and consequently the rules which they give rise to, are so entirely
distinct that they cannot possibly be harmonized into a single consistent
system.

The opinion therefore according to which certain Inductive formulæ are
regarded as composing a portion of Probability, and which finds utterance
in the Rule of Succession criticised in our last chapter, cannot, I think, be
maintained. It would be more correct to say, as stated above, that Induction
is quite distinct from Probability, yet co-operates in almost all its inferences.
By Induction we determine, for example, whether, and how far, we can
safely generalize the proposition that four men in ten live to be fifty-six;
supposing such a proposition to be safely generalized, we hand it over to
Probability to say what sort of inferences can be deduced from it.

§ 7. So much then for the opinion which tends to regard pure Induction as a
subdivision of Probability. By the majority of philosophical and logical
writers a widely different view has of course been entertained. They are
mostly disposed to distinguish these sciences very sharply from, not to say
to contrast them with, one another; the one being accepted as philosophical
or logical, and the other rejected as mathematical. This may without offence
be termed the popular prejudice against Probability.

A somewhat different view, however, must be noticed here, which, by a sort
of reaction against the latter, seems even to go beyond the former; and
which occasionally finds expression in the statement that all inductive
reasoning of every kind is merely a matter of Probability. Two examples of
this may be given.

Beginning with the older authority, there is an often quoted saying by Butler
at the commencement of his Analogy, that ‘probability is the very guide of
life’; a saying which seems frequently to be understood to signify that the
rules or principles of Probability are thus all-prevalent when we are drawing
conclusions in practical life. Judging by the drift of the context, indeed, this
seems a fair interpretation of his meaning, in so far of course as there could
be said to be any such thing as a science of Probability in those days. Prof.
Jevons, in his Principles of Science (p. 197), has expressed a somewhat
similar view, of course in a way more consistent with the principles of
modern science, physical and mathematical. He says, “I am convinced that
it is impossible to expound the methods of induction in a sound manner,
without resting them on the theory of Probability. Perfect knowledge alone
can give certainty, and in nature perfect knowledge would be infinite

knowledge, which is clearly beyond our capacities. We have, therefore, to
content ourselves with partial knowledge,—knowledge mingled with
ignorance, producing doubt.”[2]

§ 8. There are two senses in which this disposition to merge the two
sciences into one may be understood. Using the word Probability in its
vague popular signification, nothing more may be intended than to call
attention to the fact, that in every case alike our conclusions are nothing
more than ‘probable,’ that is, that they are not, and cannot be, absolutely
certain. This must be fully admitted, for of course no one acquainted with
the complexity of physical and other evidence would seriously maintain
that absolute ideal certainty can be attained in any branch of applied logic.
Hypothetical certainty, in abstract science, may be possible, but not absolute
certainty in the domain of the concrete. This has been already noticed in a
former chapter, where, however, it was pointed out that whatever
justification may exist, on the subjective view of logic, for regarding this
common prevalence of absence of certainty as warranting us in fusing the
sciences into one, no such justification is admitted when we take the
objective view.

§ 9. What may be meant, however, is that the grounds of this absence of
certainty are always of the same general character. This argument, if
admitted, would have real force, and must therefore be briefly noticed. We
have seen abundantly that when we say of a conclusion within the strict
province of Probability, that it is not certain, all that we mean is that in
some proportion of cases only will such conclusion be right, in the other
cases it will be wrong. Now when we say, in reference to any inductive
conclusion, that we feel uncertain about its absolute cogency, are we
conscious of the same interpretation? It seems to me that we are not. It is
indeed quite possible that on ultimate analysis it might be proved that
experience of failure in the past employment of our methods of
investigation was the main cause of our present want of perfect confidence
in them. But this, as we have repeatedly insisted, does not belong to the
province of logical, but to that of Psychological enquiry. It is surely not the
case that we are, as a rule, consciously guided by such occasional or
repeated instances of past failure. In so far as they are at all influential, they
seem to do their work by infusing a vague want of confidence which cannot

be referred to any statistical grounds for its justification, at least not in a
quantitative way. Part of our want of confidence is derived sympathetically
from those who have investigated the matter more nearly at first hand. Here
again, analysis might detect that a given proportion of past failures lay at
the root of the distrust, but it does not show at the surface. Moreover, one
reason why we cannot feel perfectly certain about our inductions is, that the
memory has to be appealed to for some of our data; and will any one assert
that the only reason why we do not place absolute reliance on our memory
of events long past is that we have been deceived in that way before?

In any other sense, therefore, than as a needful protest against attaching too
great demonstrative force to the conclusions of Inductive Logic, it seems
decidedly misleading to speak of its reasonings as resting upon Probability.

§ 10. We may now see clearly the reasons for the limits within which
causation[3] is necessarily required, but beyond which it is not needed. To
be able to generalize a formula so as to extend it from the observed to the
unobserved, it is clearly essential that there should be a certain permanence
in the order of nature; this permanence is one form of what is implied in the
term causation. If the circumstances under which men live and die
remaining the same, we did not feel warranted in inferring that four men out
of ten would continue to live to fifty, because in the case of those whom we
had observed this proportion had hitherto done so, it is clear that we should
be admitting that the same antecedents need not be followed by the same
consequents. This uniformity being what the Law of Causation asserts, the
truth of the law is clearly necessary to enable us to obtain our
generalizations: in other words, it is necessary for the Inductive part of the
process. But it seems to be equally clear that causation is not necessary for
that part of the process which belongs to Probability. Provided only that the
truth of our generalizations is secured to us, in the way just mentioned, what
does it matter to us whether or not the individual members are subject to
causation? For it is not in reality about these individuals that we make
inferences. As this last point has been already fully treated in Chapter VI.,
any further allusion to it need not be made here.

§ 11. The above description, or rather indication, of the process of obtaining
these generalizations must suffice for the present. Let us now turn and

consider the means by which we are practically to make use of them when
they are obtained. The point which we had reached in the course of the
investigations entered into in the sixth and seventh chapters was this:—
Given a series of a certain kind, we could draw inferences about the
members which composed it; inferences, that is, of a peculiar kind, the
value and meaning of which were fully discussed in their proper place.

We must now shift our point of view a little; instead of starting, as in the
former chapters, with a determinate series supposed to be given to us, let us
assume that the individual only is given, and that the work is imposed upon
us of finding out the appropriate series. How are we to set about the task? In
the former case our data were of this kind:—Eight out of ten men, aged
fifty, will live eleven years more, and we ascertained in what sense, and
with what certainty, we could infer that, say, John Smith, aged fifty, would
live to sixty-one.

§ 12. Let us then suppose, instead, that John Smith presents himself, how
should we in this case set about obtaining a series for him? In other words,
how should we collect the appropriate statistics? It should be borne in mind
that when we are attempting to make real inferences about things as yet
unknown, it is in this form that the problem will practically present itself.

At first sight the answer to this question may seem to be obtained by a very
simple process, viz. by counting how many men of the age of John Smith,
respectively do and do not live for eleven years. In reality however the
process is far from being so simple as it appears. For it must be remembered
that each individual thing has not one distinct and appropriate class or
group, to which, and to which alone, it properly belongs. We may indeed be
practically in the habit of considering it under such a single aspect, and it
may therefore seem to us more familiar when it occupies a place in one
series rather than in another; but such a practice is merely customary on our
part, not obligatory. It is obvious that every individual thing or event has an
indefinite number of properties or attributes observable in it, and might
therefore be considered as belonging to an indefinite number of different
classes of things. By belonging to any one class it of course becomes at the
same time a member of all the higher classes, the genera, of which that
class was a species. But, moreover, by virtue of each accidental attribute

which it possesses, it becomes a member of a class intersecting, so to say,
some of the other classes. John Smith is a consumptive man say, and a
native of a northern climate. Being a man he is of course included in the
class of vertebrates, also in that of animals, as well as in any higher such
classes that there may be. The property of being consumptive refers him to
another class, narrower than any of the above; whilst that of being born in a
northern climate refers him to a new and distinct class, not conterminous
with any of the rest, for there are things born in the north which are not
men.

§ 13. When therefore John Smith presents himself to our notice without, so
to say, any particular label attached to him informing us under which of his
various aspects he is to be viewed, the process of thus referring him to a
class becomes to a great extent arbitrary. If he had been indicated to us by a
general name, that, of course, would have been some clue; for the name
having a determinate connotation would specify at any rate a fixed group of
attributes within which our selection was to be confined. But names and
attributes being connected together, we are here supposed to be just as much
in ignorance what name he is to be called by, as what group out of all his
innumerable attributes is to be taken account of; for to tell us one of these
things would be precisely the same in effect as to tell us the other. In saying
that it is thus arbitrary under which class he is placed, we mean, of course,
that there are no logical grounds of decision; the selection must be
determined by some extraneous considerations. Mere inspection of the
individual would simply show us that he could equally be referred to an
indefinite number of classes, but would in itself give no inducement to
prefer, for our special purpose, one of these classes to another.

This variety of classes to which the individual may be referred owing to his
possession of a multiplicity of attributes, has an important bearing on the
process of inference which was indicated in the earlier sections of this
chapter, and which we must now examine in more special reference to our
particular subject.

§ 14. It will serve to bring out more clearly the nature of some of those
peculiarities of the step which we are now about to take in the case of
Probability, if we first examine the form which the corresponding step

assumes in the case of ordinary Logic. Suppose then that we wished to
ascertain whether a certain John Smith, a man of thirty, who is amongst
other things a resident in India, and distinctly affected with cancer, will
continue to survive there for twenty years longer. The terms in which the
man is thus introduced to us refer him to different classes in the way already
indicated. Corresponding to these classes there will be a number of
propositions which have been obtained by previous observations and
inductions, and which we may therefore assume to be available and ready at
hand when we want to make use of them. Let us conceive them to be such
as these following:—Some men live to fifty; some Indian residents live to
fifty; no man suffering thus from cancer lives for five years. From the first
and second of these premises nothing whatever can be inferred, for they are
both[4] particular propositions, and therefore lead to no conclusion in this
case. The third answers our enquiry decisively.

To the logical reader it will hardly be necessary to point out that the process
here under consideration is that of finding middle terms which shall serve to
connect the subject and predicate of our conclusion. This subject and
predicate in the case in question, are the individual before us and his death
within the stated period. Regarded by themselves there is nothing in
common between them, and therefore no link by which they may be
connected or disconnected with each other. The various classes above
referred to are a set of such middle terms, and the propositions belonging to
them are a corresponding set of major premises. By the help of any one of
them we are enabled, under suitable circumstances, to connect together the
subject and predicate of the conclusion, that is, to infer whether the man
will or will not live twenty years.

§ 15. Now in the performance of such a logical process there are two
considerations to which the reader's attention must for a moment be
directed. They are simple enough in this case, but will need careful
explanation in the corresponding case in Probability. In the first place, it is
clear that whenever we can make any inference at all, we can do so with
absolute certainty. Logic, within its own domain, knows nothing of
hesitation or doubt. If the middle term is appropriate it serves to connect the
extremes in such a way as to preclude all uncertainty about the conclusion;
if it is not, there is so far an end of the matter: no conclusion can be drawn,

and we are therefore left where we were. Assuming our premises to be
correct, we either know our conclusion for certain, or we know nothing
whatever about it. In the second place, it should be noticed that none of the
possible alternatives in the shape of such major premises as those given
above can ever contradict any of the others, or be at all inconsistent with
them. Regarded as isolated propositions, there is of course nothing to secure
such harmony; they have very different predicates, and may seem quite out
of each other's reach for either support or opposition. But by means of the
other premise they are in each case brought into relation with one another,
and the general interests of truth and consistency prevent them therefore
from contradicting one another. As isolated propositions it might have been
the case that all men live to fifty, and that no Indian residents do so, but
having recognised that some men are residents in India, we see at once that
these premises are inconsistent, and therefore that one or other of them must
be rejected. In all applied logic this necessity of avoiding self-contradiction
is so obvious and imperious that no one would think it necessary to lay
down the formal postulate that all such possible major premises are to be
mutually consistent. To suppose that this postulate is not complied with,
would be in effect to make two or more contradictory assumptions about
matters of fact.

§ 16. But now observe the difference when we attempt to take the
corresponding step in Probability. For ordinary propositions, universal or
particular, substitute statistical propositions of what we have been in the
habit of calling the ‘proportional’ kind. In other words, instead of asking
whether the man will live for twenty years, let us ask whether he will live
for one year? We shall be unable to find any universal propositions which
will cover the case, but we may without difficulty obtain an abundance of
appropriate proportional ones. They will be of the following description:—
Of men aged 30, 98 in 100 live another year; of residents in India a smaller
proportion survive, let us for example say 90 in 100; of men suffering from
cancer a smaller proportion still, let us say 20 in 100.

Now in both of the respects to which attention has just been drawn,
propositions of this kind offer a marked contrast with those last considered.
In the first place, they do not, like ordinary propositions, either assert
unequivocally yes or no, or else refuse to open their lips; but they give

instead a sort of qualified or hesitating answer concerning the individuals
included in them. This is of course nothing more than the familiar
characteristic of what may be called ‘probability propositions.’ But it leads
up to, and indeed renders possible, the second and more important point;
viz. that these various answers, though they cannot directly and formally
contradict each other (this their nature as proportional propositions, will not
as a rule permit), may yet, in a way which will now have to be pointed out,
be found to be more or less in conflict with each other.

Hence it follows that in the attempt to draw a conclusion from premises of
the kind in question, we may be placed in a position of some perplexity; but
it is a perplexity which may present itself in two forms, a mild and an
aggravated form. We will notice them in turn.

§ 17. The mild form occurs when the different classes to which the
individual case may be appropriately referred are successively included one
within another; for here our sets of statistics, though leading to different
results, will not often be found to be very seriously at variance with one
another. All that comes of it is that as we ascend in the scale by appealing to
higher and higher genera, the statistics grow continually less appropriate to
the particular case in point, and such information therefore as they afford
becomes gradually less explicit and accurate.

The question that we originally wanted to determine, be it remembered, is
whether John Smith will die within one year. But all knowledge of this fact
being unattainable, owing to the absence of suitable inductions, we felt
justified (with the explanation, and under the restrictions mentioned in
Chap VI.), in substituting, as the only available equivalent for such
individual knowledge, the answer to the following statistical enquiry, What
proportion of men in his circumstances die?

§ 18. But then at once there begins to arise some doubt and ambiguity as to
what exactly is to be understood by his circumstances. We may know very
well what these circumstances are in themselves, and yet be in perplexity as
to how many of them we ought to take into account when endeavouring to
estimate his fate. We might conceivably, for a beginning, choose to confine
our attention to those properties only which he has in common with all
animals. If so, and statistics on the subject were attainable, they would

presumably be of some such character as this, Ninety-nine animals out of a
hundred die within a year. Unusual as such a reference would be, we
should, logically speaking, be doing nothing more than taking a wider class
than the one we were accustomed to. Similarly we might, if we pleased,
take our stand at the class of vertebrates, or at that of mammalia, if
zoologists were able to give us the requisite information. Of course we
reject these wide classes and prefer a narrower one. If asked why we reject
them, the natural answer is that they are so general, and resemble the
particular case before us in so few points, that we should be exceedingly
likely to go astray in trusting to them. Though accuracy cannot be insured,
we may at least avoid any needless exaggeration of the relative number and
magnitude of our errors.

§ 19. The above answer is quite valid; but whilst cautioning us against
appealing to too wide a class, it seems to suggest that we cannot go wrong
in the opposite direction, that is in taking too narrow a class. And yet we do
avoid any such extremes. John Smith is not only an Englishman; he may
also be a native of such a part of England, be living in such a Presidency,
and so on. An indefinite number of such additional characteristics might be
brought out into notice, many of which at any rate have some bearing upon
the question of vitality. Why do we reject any consideration of these
narrower classes? We do reject them, but it is for what may be termed a
practical rather than a theoretical reason. As was explained in the first
chapters, it is essential that our series should contain a considerable number
of terms if they are to be of any service to us. Now many of the attributes of
any individual are so rare that to take them into account would be at
variance with the fundamental assumption of our science, viz. that we are
properly concerned only with the averages of large numbers. The more
special and minute our statistics the better, provided only that we can get
enough of them, and so make up the requisite large number of instances.
This is, however, impossible in many cases. We are therefore obliged to
neglect one attribute after another, and so to enlarge the contents of our
class; at the avowed risk of somewhat increased variety and unsuitability in
the members of it, for at each step of this kind we diverge more and more
from the sort of instances that we really want. We continue to do so, until
we no longer gain more in quantity than we lose in quality. We finally take
our stand at the point where we first obtain statistics drawn from a

sufficiently large range of observation to secure the requisite degree of
stability and uniformity.

§ 20. In such an example as the one just mentioned, where one of the
successive classes—man—is a well-defined natural kind or species, there is
such a complete break in each direction at this point, that every one is
prompted to take his stand here. On the one hand, no enquirer would ever
think of introducing any reference to the higher classes with fewer
attributes, such as animal or organized being: and on the other hand, the
inferior classes, created by our taking notice of his employment or place of
residence, &c., do not as a rule differ sufficiently in their characteristics
from the class man to make it worth our while to attend to them.

Now and then indeed these characteristics do rise into importance, and
whenever this is the case we concentrate our attention upon the class to
which they correspond, that is, the class which is marked off by their
presence. Thus, for instance, the quality of consumptiveness separates any
one off so widely from the majority of his fellow-men in all questions
pertaining to mortality, that statistics about the lives of consumptive men
differ materially from those which refer to men in general. And we see the
result; if a consumptive man can effect an insurance at all, he must do it for
a much higher premium, calculated upon his special circumstances. In other
words, the attribute is sufficiently important to mark off a fresh class or
series. So with insurance against accident. It is not indeed attempted to
make a special rate of insurance for the members of each separate trade, but
the differences of risk to which they are liable oblige us to take such facts to
some degree into account. Hence, trades are roughly divided into two or
three classes, such as the ordinary, the hazardous, and the extra-hazardous,
each having to pay its own rate of premium.

§ 21. Where one or other of the classes thus corresponds to natural kinds, or
involves distinctions of co-ordinate importance with those of natural kinds,
the process is not difficult; there is almost always some one of these classes
which is so universally recognised to be the appropriate one, that most
persons are quite unaware of there being any necessity for a process of
selection. Except in the cases where a man has a sickly constitution, or
follows a dangerous employment, we seldom have occasion to collect

statistics for him from any class but that of men in general of his age in the
country.

When, however, these successive classes are not ready marked out for us by
nature, and thence arranged in easily distinguishable groups, the process is
more obviously arbitrary. Suppose we were considering the chance of a
man's house being burnt down, with what collection of attributes should we
rest content in this instance? Should we include all kinds of buildings, or
only dwelling-houses, or confine ourselves to those where there is much
wood, or those which have stoves? All these attributes, and a multitude of
others may be present, and, if so, they are all circumstances which help to
modify our judgment. We must be guided here by the statistics which we
happen to be able to obtain in sufficient numbers. Here again, rough
distinctions of this kind are practically drawn in Insurance Offices, by
dividing risks into ordinary, hazardous, and extra-hazardous. We examine
our case, refer it to one or other of these classes, and then form our
judgment upon its prospects by the statistics appropriate to its class.

§ 22. So much for what may be called the mild form in which the ambiguity
occurs; but there is an aggravated form in which it may show itself, and
which at first sight seems to place us in far greater perplexity.

Suppose that the different classes mentioned above are not included
successively one within the other. We may then be quite at a loss which of
the statistical tables to employ. Let us assume, for example, that nine out of
ten Englishmen are injured by residence in Madeira, but that nine out of ten
consumptive persons are benefited by such a residence. These statistics,
though fanciful, are conceivable and perfectly compatible. John Smith is a
consumptive Englishman; are we to recommend a visit to Madeira in his
case or not? In other words, what inferences are we to draw about the
probability of his death? Both of the statistical tables apply to his case, but
they would lead us to directly contradictory conclusions. This does not
mean, of course, contradictory precisely in the logical sense of that word,
for one of these propositions does not assert that an event must happen and
the other deny that it must; but contradictory in the sense that one would
cause us in some considerable degree to believe what the other would cause
us in some considerable degree to disbelieve. This refers, of course, to the

individual events; the statistics are by supposition in no degree
contradictory. Without further data, therefore, we can come to no decision.

§ 23. Practically, of course, if we were forced to a decision with only these
data before us, we should make our choice by the consideration that the
state of a man's lungs has probably more to do with his health than the place
of his birth has; that is, we should conclude that the duration of life of
consumptive Englishmen corresponds much more closely with that of
consumptive persons in general than with that of their healthy countrymen.
But this is, of course, to import empirical considerations into the question.
The data, as they are given to us, and if we confine ourselves to them, leave
us in absolute uncertainty upon the point. It may be that the consumptive
Englishmen almost all die when transported into the other climate; it may
be that they almost all recover. If they die, this is in obvious accordance
with the first set of statistics; it will be found in accordance with the second
set through the fact of the foreign consumptives profiting by the change of
climate in more than what might be termed their due proportion. A similar
explanation will apply to the other alternative, viz. to the supposition that
the consumptive Englishmen mostly recover. The problem is, therefore, left
absolutely indeterminate, for we cannot here appeal to any general rule so
simple and so obviously applicable as that which, in a former case,
recommended us always to prefer the more special statistics, when
sufficiently extensive, to those which are wider and more general. We have
no means here of knowing whether one set is more special than the other.

And in this no difficulty can be found, so long as we confine ourselves to a
just view of the subject. Let me again recall to the reader's mind what our
present position is; we have substituted for knowledge of the individual
(finding that unattainable) a knowledge of what occurs in the average of
similar cases. This step had to be taken the moment the problem was
handed over to Probability. But the conception of similarity in the cases
introduces us to a perplexity; we manage indeed to evade it in many
instances, but here it is inevitably forced upon our notice. There are here
two aspects of this similarity, and they introduce us to two distinct averages.
Two assertions are made as to what happens in the long run, and both of
these assertions, by supposition, are verified. Of their truth there need be no
doubt, for both were supposed to be obtained from experience.

§ 24. It may perhaps be supposed that such an example as this is a reductio
ad absurdum of the principle upon which Life and other Insurances are
founded. But a moment's consideration will show that this is quite a
mistake, and that the principle of insurance is just as applicable to examples
of this kind as to any other. An office need find no difficulty in the case
supposed. They might (for a reason to be mentioned presently, they
probably would not) insure the individual without inconsistency at a rate
determined by either average. They might say to him, “You are an
Englishman. Out of the multitude of English who come to us nine in ten die
if they go to Madeira. We will insure you at a rate assigned by these
statistics, knowing that in the long run all will come right so far as we are
concerned. You are also consumptive, it is true, and we do not know what
proportion of the English are consumptive, nor what proportion of English
consumptives die in Madeira. But this does not really matter for our
purpose. The formula, nine in ten die, is in reality calculated by taking into
account these unknown proportions; for, though we do not know them in
themselves, statistics tell us all that we care to know about their results. In
other words, whatever unknown elements may exist, must, in regard to all
the effects which they can produce, have been already taken into account,
so that our ignorance about them cannot in the least degree invalidate such
conclusions as we are able to draw. And this is sufficient for our purpose.”
But precisely the same language might be held to him if he presented
himself as a consumptive man; that is to say, the office could safely carry
on its proceedings upon either alternative.

This would, of course, be a very imperfect state for the matter to be left in.
The only rational plan would be to isolate the case of consumptive
Englishmen, so as to make a separate calculation for their circumstances.
This calculation would then at once supersede all other tables so far as they
were concerned; for though, in the end, it could not arrogate to itself any
superiority over the others, it would in the mean time be marked by fewer
and slighter aberrations from the truth.

§ 25. The real reason why the Insurance office could not long work on the
above terms is of a very different kind from that which some readers might
contemplate, and belongs to a class of considerations which have been
much neglected in the attempts to construct sciences of the different

branches of human conduct. It is nothing else than that annoying
contingency to which prophets since the time of Jonah have been subject, of
uttering suicidal prophecies; of publishing conclusions which are perfectly
certain when every condition and cause but one have been taken into
account, that one being the effect of the prophecy itself upon those to whom
it refers.

In our example above, the office (in so far as the particular cases in Madeira
are concerned) would get on very well until the consumptive Englishmen in
question found out what much better terms they could make by announcing
themselves as consumptives, and paying the premium appropriate to that
class, instead of announcing themselves as Englishmen. But if they did this
they would of course be disturbing the statistics. The tables were based
upon the assumption that a certain fixed proportion (it does not matter what
proportion) of the English lives would continue to be consumptive lives,
which, under the supposed circumstances, would probably soon cease to be
true. When it is said that nine Englishmen out of ten die in Madeira, it is
meant that of those who come to the office, as the phrase is, at random, or
in their fair proportions, nine-tenths die. The consumptives are supposed to
go there just like red-haired men, or poets, or any other special class. Or
they might go in any proportions greater or less than those of other classes,
so long as they adhered to the same proportion throughout. The tables are
then calculated on the continuance of this state of things; the practical
contradiction is in supposing such a state of things to continue after the
people had once had a look at the tables. If we merely make the assumption
that the publication of these tables made no such alteration in the conduct of
those to whom it referred, no hitch of this kind need occur.

§ 26. The assumptions here made, as has been said, are not in any way
contradictory, but they need some explanation. It will readily be seen that,
taken together, they are inconsistent with the supposition that each of these
classes is homogeneous, that is, that the statistical proportions which hold
of the whole of either of them will also hold of any portion of them which
we may take. There are certain individuals (viz. the consumptive
Englishmen) who belong to each class, and of course the two different sets
of statistics cannot both be true of them taken by themselves. They might
coincide in their characteristics with either class, but not with both;

probably in most practical cases they will coincide with neither, but be of a
somewhat intermediate character. Now when it is said of any such
heterogeneous body that, say, nine-tenths die, what is meant (or rather
implied) is that the class might be broken up into smaller subdivisions of a
more homogeneous character, in some of which, of course, more than nine-
tenths die, whilst in others less, the differences depending upon their
character, constitution, profession, &c.; the number of such divisions and
the amount of their divergence from one another being perhaps very
considerable.

Now when we speak of either class as a whole and say that nine-tenths die,
the most natural and soundest meaning is that that would be the proportion
if all without exception went abroad, or (what comes to the same thing) if
each of these various subdivisions was represented in fair proportion to its
numbers. Or it might only be meant that they go in some other proportion,
depending upon their tastes, pursuits, and so on. But whatever meaning be
adopted one condition is necessary, viz. that the proportion of each class
that went at the time the statistics were drawn up must be adhered to
throughout. When the class is homogeneous this is not needed, but when it
is heterogeneous the statistics would be interfered with unless this condition
were secured.

We are here supposed to have two sets of statistics, one for the English and
one for the consumptives, so that the consumptive English are in a sense
counted twice over. If their mortality is of an intermediate amount,
therefore, they serve to keep down the mortality of one class and to keep up
that of the other. If the statistics are supposed to be exhaustive, by referring
to the whole of each class, it follows that actually the same individuals must
be counted each time; but if representatives only of each class are taken, the
same individuals need not be inserted in each set of tables.

§ 27. When therefore they come to insure (our remarks are still confined to
our supposed Madeira case), we have some English consumptives counted
as English, and paying the high rate; and others counted as consumptives
and paying the low rate. Logically indeed we may suppose them all entered
in each class, and paying therefore each rate. What we have said above is
that any individual may be conceived to present himself for either of these

classes. Conceive that some one else pays his premium for him, so that it is
a matter of indifference to him personally at which rate he insures, and there
is nothing to prevent some of the class (or for that matter all) going to one
class, and others (or all again) going to the other class.

So long therefore as we make the logically possible though practically
absurd supposition that some men will continue to pay a higher rate than
they need, there is nothing to prevent the English consumptives (some or
all) from insuring in each category and paying its appropriate premium. As
soon as they gave any thought to the matter, of course they would, in the
case supposed, all prefer to insure as consumptives. But their doing this
would disturb each set of statistics. The English mortality in Madeira would
instantly become heavier, so far as the Insurance company was concerned,
by the loss of all their best lives; whilst the consumptive statistics (unless
all the English consumptives had already been taken for insurance) would
be in the same way deteriorated.[5] A slight readjustment therefore of each
scale of insurance would then be needed; this is the disturbance mentioned
just above. It must be clearly understood, however, that it is not our original
statistics which have proved to be inconsistent, but simply that there were
practical obstacles to carrying out a system of insurance upon them.

§ 28. Examples subject to the difficulty now under consideration will
doubtless seem perplexing to the student unacquainted with the subject.
They are difficult to reconcile with any other view of the science than that
insisted on throughout this Essay, viz. that we are only concerned with
averages. It will perhaps be urged that there are two different values of the
man's life in these cases, and that they cannot both be true. Why not? The
‘value’ of his life is simply the number of years to which men in his
circumstances do, on the average, attain; we have the man set before us
under two different circumstances; what wonder, therefore, that these
should offer different averages? In such an objection it is forgotten that we
have had to substitute for the unattainable result about the individual, the
really attainable result about a set of men as much like him as possible. The
difficulty and apparent contradiction only arise when people will try to find
some justification for their belief in the individual case. What can we
possibly conclude, it may be asked, about this particular man John Smith's
prospects when we are thus offered two different values for his life?

Nothing whatever, it must be replied; nor could we in reality draw a
conclusion, be it remembered, in the former case, when we were practically
confined to one set of statistics. There also we had what we called the
‘value’ of his life, and since we only knew of one such value, we came to
regard it as in some sense appropriate to him as an individual. Here, on the
other hand, we have two values, belonging to different series, and as these
values are really different it may be complained that they are discordant, but
such a complaint can only be made when we do what we have no right to
do, viz. assign a value to the individual which shall admit of individual
justification.

§ 29. Is it then perfectly arbitrary what series or class of instances we select
by which to judge? By no means; it has been stated repeatedly that in
choosing a series, we must seek for one the members of which shall
resemble our individual in as many of his attributes as possible, subject only
to the restriction that it must be a sufficiently extensive series. What is
meant is, that in the above case, where we have two series, we cannot fairly
call them contradictory; the only valid charge is one of incompleteness or
insufficiency for their purpose, a charge which applies in exactly the same
sense, be it remembered, to all statistics which comprise genera
unnecessarily wider than the species with which we are concerned. The
only difference between the two different classes of cases is, that in the one
instance we are on a path which we know will lead at the last, through
many errors, towards the truth (in the sense in which truth can be attained
here), and we took it for want of a better. In the other instance we have two
such paths, perfectly different paths, either of which however will lead us
towards the truth as before. Contradiction can only seem to arise when it is
attempted to justify each separate step on our paths, as well as their ultimate
tendency.

Still it cannot be denied that these objections are a serious drawback to the
completeness and validity of any anticipations which are merely founded
upon statistical frequency, at any rate in an early stage of experience, when
but few statistics have been collected. Such knowledge as Probability can
give is not in any individual case of a high order, being subject to the
characteristic infirmity of repeated error; but even when measured by its
own standard it commences at a very low stage of proficiency. The errors

are then relatively very numerous and large compared with what they may
ultimately be reduced to.

§ 30. Here as elsewhere there is a continuous process of specialization
going on. The needs of a gradually widening experience are perpetually
calling upon us to subdivide classes which are found to be too
heterogeneous. Sometimes the only complaint that has to be made is that
the class to which we are obliged to refer is found to be somewhat too broad
to suit our purpose, and that it might be subdivided with convenience. This
is the case, as has been shown above, when an Insurance office finds that its
increasing business makes it possible and desirable to separate off the men
who follow some particular trades from the rest of their fellow-countrymen.
Similarly in every other department in which statistics are made use of. This
increased demand for specificness leads, in fact, as naturally in this
direction, as does the progress of civilization to the subdivision of trades in
any town or country. So in reference to the other kind of perplexity
mentioned above. Nothing is more common in those sciences or practical
arts, in which deduction is but little available, and where in consequence
our knowledge is for the most part of the empirical kind, than to meet with
suggestions which point more or less directly in contrary directions.
Whenever some new substance is discovered or brought into more general
use, those who have to deal with it must be familiar with such a state of
things. The medical man who has to employ a new drug may often find
himself confronted by the two distinct recommendations, that on the one
hand it should be employed for certain diseases, and that on the other hand
it should not be tried on certain constitutions. A man with such a
constitution, but suffering from such a disease, presents himself; which
recommendation is the doctor to follow? He feels at once obliged to set to
work to collect narrower and more special statistics, in order to escape from
such an ambiguity.

§ 31. In this and a multitude of analogous cases afforded by the more
practical arts it is not of course necessary that numerical data should be
quoted and appealed to; it is sufficient that the judgment is more or less
consciously determined by them. All that is necessary to make the examples
appropriate is that we should admit that in their case statistical data are our
ultimate appeal in the present state of knowledge. Of course if the empirical

laws can be resolved into their component causes we may appeal to direct
deduction, and in this case the employment of statistics, and consequently
the use of the theory of Probability, may be superseded.

In this direction therefore, as time proceeds, the advance of statistical
refinement by the incessant subdivision of classes to meet the developing
wants of man is plain enough. But if we glance backwards to a more
primitive stage, we shall soon see in what a very imperfect state the
operation commences. At this early stage, however, Probability and
Induction are so closely connected together as to be very apt to be merged
into one, or at any rate to have their functions confounded.

§ 32. Since the generalization of our statistics is found to belong to
Induction, this process of generalization may be regarded as prior to, or at
least independent of, Probability. We have, moreover, already discussed (in
Chapter VI.) the step corresponding to what are termed immediate
inferences, and (in Chapter VII.) that corresponding to syllogistic
inferences. Our present position therefore is that in which we may consider
ourselves in possession of any number of generalizations, but wish to
employ them so as to make inferences about a given individual; just as in
one department of common logic we are engaged in finding middle terms to
establish the desired conclusion. In this latter case the process is found to be
extremely simple, no accumulation of different middle terms being able to
lead to any real ambiguity or contradiction. In Probability, however, the
case is different. Here, if we attempt to draw inferences about the individual
case before us, as often is attempted—in the Rule of Succession for
example—we shall encounter the full force of this ambiguity and
contradiction. Treat the question, however, fairly, and all difficulty
disappears. Our inference really is not about the individuals as individuals,
but about series or successions of them. We wished to know whether John
Smith will die within the year; this, however, cannot be known. But John
Smith, by the possession of many attributes, belongs to many different
series. The multiplicity of middle terms, therefore, is what ought to be
expected. We can know whether a succession of men, residents in India,
consumptives, &c. die within a year. We may make our selection, therefore,
amongst these, and in the long run the belief and consequent conduct of
ourselves and other persons (as described in Chapter VI.) will become

capable of justification. With regard to choosing one of these series rather
than another, we have two opposing principles of guidance. On the one
hand, the more special the series the better; for, though not more right in the
end, we shall thus be more nearly right all along. But, on the other hand, if
we try to make the series too special, we shall generally meet the practical
objection arising from insufficient statistics.

1 Some of my readers may be familiar with a very striking digression in

Buffon's Natural History (Natural Hist. of Man, § VIII.), in which he
supposes the first man in full possession of his faculties, but with all his
experience to gain, and speculates on the gradual acquisition of his
knowledge. Whatever may be thought of his particular conclusions the
passage is very interesting and suggestive to any student of Psychology.
2
See also Dugald Stewart (Ed. by Hamilton; VII. pp. 115–119).
3 Required that is for purposes of logical inference within the limits of

Probability; it is not intended to imply any doubts as to its actual universal
prevalence, or its all-importance for scientific purposes. The subject is more
fully discussed in a future chapter.
4 As particular propositions they are both of course identical in form. The

fact that the ‘some’ in the former corresponds to a larger proportion than in
the latter, is a distinction alien to pure Logic.
5
The reason is obvious. The healthiest English lives in Madeira (viz. the
consumptive ones) have now ceased to be reckoned as English; whereas the
worst consumptive lives there (viz. the English) are now increased in
relative numbers.

CHAPTER X.

CHANCE AS OPPOSED TO CAUSATION AND
DESIGN.
§ 1. The remarks in the previous chapter will have served to clear the way
for an enquiry which probably excites more popular interest than any other
within the range of our subject, viz. the determination whether such and
such events are to be attributed to Chance on the one hand, or to Causation
or Design on the other. As the principal difficulty seems to arise from the
ambiguity with which the problem is generally conceived and stated, owing
to the extreme generality of the conceptions involved, it becomes necessary
to distinguish clearly between the several distinct issues which are apt to be
involved.

I. There is, to begin with, a very old objection, founded on the assumption
which our science is supposed to make of the existence of Chance. The
objection against chance is of course many centuries older than the Theory
of Probability; and as it seems a nearly obsolete objection at the present day
we need not pause long for its consideration. If we spelt the word with a
capital C, and maintained that it was representative of some distinct creative
or administrative agency, we should presumably be guilty of some form of
Manicheism. But the only rational meaning of the objection would appear
to be that the principles of the science compel us to assume that events
(some events, only, that is) happen without causes, and are thereby removed
from the customary control of the Deity. As repeatedly pointed out already
this is altogether a mistake. The science of Probability makes no assumption
whatever about the way in which events are brought about, whether by
causation or without it. All that we undertake to do is to establish and
explain a body of rules which are applicable to classes of cases in which we
do not or cannot make inferences about the individuals. The objection
therefore must be somewhat differently stated, and appears finally to reduce
itself to this:—that the assumptions upon which the science of Probability
rests, are not inconsistent with a disbelief in causation within certain limits;
causation being of course understood simply in the sense of regular
sequence. So stated the objection seems perfectly valid, or rather the facts
on which it is based must be admitted; though what connection there would

be between such lack of causation and absence of Divine superintendence I
quite fail to see.

As this Theological objection died away the men of physical science, and
those who sympathized with them, began to enforce the same protest; and
similar cautions are still to be found from time to time in modern treatises.
Hume, for instance, in his short essay on Probability, commences with the
remark, “though there be no such thing as chance in the world, our
ignorance of the real cause of any event has the same influence on the
understanding, &c.” De Morgan indeed goes so far as to declare that “the
foundations of the theory of Probability have ceased to exist in the mind
that has formed the conception,” “that anything ever did happen or will
happen without some particular reason why it should have been precisely
what it was and not anything else.”[1] Similar remarks might be quoted from
Laplace and others.

§ 2. In the particular form of the controversy above referred to, and which is
mostly found in the region of the natural and physical sciences, the
contention that chance and causation are irreconcileable occupies rather a
defensive position; the main fact insisted on being that, whenever in these
subjects we may happen to be ignorant of the details we have no warrant for
assuming as a consequence that the details are uncaused. But this supposed
irreconcileability is sometimes urged in a much more aggressive spirit in
reference to social enquiries. Here the attempt is often made to prove
causation in the details, from the known and admitted regularity in the
averages. A considerable amount of controversy was excited some years
ago upon this topic, in great part originated by the vigorous and outspoken
support of the necessitarian side by Buckle in his History of Civilization.

It should be remarked that in these cases the attempt is sometimes made as
it were to startle the reader into acquiescence by the singularity of the
examples chosen. Instances are selected which, though they possess no
greater logical value, are, if one may so express it, emotionally more
effective. Every reader of Buckle's History, for instance, will remember the
stress which he laid upon the observed fact, that the number of suicides in
London remains about the same, year by year; and he may remember also
the sort of panic with which the promulgation of this fact was accompanied

in many quarters. So too the way in which Laplace notices that the number
of undirected letters annually sent to the Post Office remains about the
same, and the comments of Dugald Stewart upon this particular uniformity,
seem to imply that they regarded this instance as more remarkable than
many analogous ones taken from other quarters.

That there is a certain foundation of truth in the reasonings in support of
which the above examples are advanced, cannot be denied, but their authors
appear to me very much to overrate the sort of opposition that exists
between the theory of Chances and the doctrine of Causation. As regards
first that wider conception of order or regularity which we have termed
uniformity, anything which might be called objective chance would
certainly be at variance with this in one respect. In Probability ultimate
regularity is always postulated; in tossing a die, if not merely the individual
throws were uncertain in their results, but even the average also, owing to
the nature of the die, or the number of the marks upon it, being arbitrarily
interfered with, of course no kind of science would attempt to take any
account of it.

§ 3. So much must undoubtedly be granted; but must the same admission be
made as regards the succession of the individual events? Can causation, in
the sense of invariable succession (for we are here shifting on to this
narrower ground), be denied, not indeed without suspicion of scientific
heterodoxy, but at any rate without throwing uncertainty upon the
foundations of Probability? De Morgan, as we have seen, strongly
maintains that this cannot be so. I find myself unable to agree with him
here, but this disagreement springs not so much from differences of detail,
as from those of the point of view in which we regard the science. He
always appears to incline to the opinion that the individual judgment in
probability is to admit of justification; that when we say, for instance, that
the odds in favour of some event are three to two, that we can explain and
justify our statement without any necessary reference to a series or class of
such events. It is not easy to see how this can be done in any case, but the
obstacles would doubtless be greater even than they are, if knowledge of the
individual event were not merely unattained, but, owing to the absence of
any causal connection, essentially unattainable. On the theory adopted in
this work we simply postulate ignorance of the details, but it is not regarded

as of any importance on what sort of grounds this ignorance is based. It may
be that knowledge is out of the question from the nature of the case, the
causative link, so to say, being missing. It may be that such links are known
to exist, but that either we cannot ascertain them, or should find it
troublesome to do so. It is the fact of this ignorance that makes us appeal to
the theory of Probability, the grounds of it are of no importance.

§ 4. On the view here adopted we are concerned only with averages, or with
the single event as deduced from an average and conceived to form one of a
series. We start with the assumption, grounded on experience, that there is
uniformity in this average, and, so long as this is secured to us, we can
afford to be perfectly indifferent to the fate, as regards causation, of the
individuals which compose the average. The question then assumes the
following form:—Is this assumption, of average regularity in the aggregate,
inconsistent with the admission of what may be termed causeless
irregularity in the details? It does not seem to me that it would be at all easy
to prove that this is so. As a matter of fact the two beliefs have constantly
co-existed in the same minds. This may not count for much, but it suggests
that if there be a contradiction between them it is by no means palpable and
obvious. Millions, for instance, have believed in the general uniformity of
the seasons taken one with another, who certainly did not believe in, and
would very likely have been ready distinctly to deny, the existence of
necessary sequences in the various phenomena which compose what we
call a season. So with cards and dice; almost every gambler must have
recognized that judgment and foresight are of use in the long run, but
writers on chance seem to think that gamblers need a good deal of
reasoning to convince them that each separate throw is in its nature
essentially predictable.

§ 5. In its application to moral and social subjects, what gives this
controversy its main interest is its real or supposed bearing upon the vexed
question of the freedom of the will; for in this region Causation, and
Fatalism or Necessitarianism, are regarded as one and the same thing.

Here, as in the last case, that wide and somewhat vague kind of regularity
that we have called Uniformity, must be admitted as a notorious fact.
Statistics have put it out of the power of any reasonably informed person to