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Full text of "The Logic Of Chance"

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OSMANIA UNIVERSITY LIBRARY 

Call No. / IV ^ ll Accession No. 

Author 

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This booK should be returned on or before the date last marked belo' 



THE 



LOGIC OF CHANCE, 



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 STATISTS, 



BY 

JOHN VENN, Sc.D., F.R.S., 

FELLOW AND LECTURER IN THE MORAL SCIENCES, GONVILLE AND CAIT2, 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. 

Pontoon : 
MACMILLAN AND CO. 

AND NEW YORK 
1888 



TAll Hights reserved.] 



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



PEEFACE TO FIRST EDITION. 



ANY work on Probability by a Cambridge man will be so 
likely to have its* 'scope 1 ><an$rits 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 formulae occur they will not be found to 
form any essential part of the text. 

The science of Probability occupies at present a some- 
what 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 ex- 
traordinary 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 pro- 
fundity of many of the calculations, but there seems to 



vi Preface to the First Edition. 

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 endea- 
vour 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 ex- 
periencing 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 mathe- 
ihatics 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 



Preface to the First Edition. vii 

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 acci- 
dental. 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 know- 
ledge of the principles of the science, arid even considerable 
acquaintance with the details of it. The same holds true in 
Probability; its connection with mathematics, though cer- 
'tainly 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 hr j !;!'< 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 conse : 
quences. 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 devo- 
tees. Its details of course can only be studied by those who 



viii Preface to the First Edition. 

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 ma- 
thematicians 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 funda- 
mental 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 per- 
fectly 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. 



Preface to the First Edition. ix 

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 neces- 
sity 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 

1 I am here speaking, of course, of History of the Theory of Probability 

those only who have expressly treated being, as the name denotes, mainly 

of the foundations of the science. Mr historical, such enquiries have not 

Todhunter's admirable work on the directly fallen within his province. 



x Preface to the First Edition. 

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 rein re- 
putanti, earn et magis probabilem esse et magis modestam, 
quam si facta fuisset ex parte." 

Almost the only writer who seems to me to have ex- 
pressed a just view of the nature and foundation of the rules 
of Probability is Mr Mill, in his System of Logic 1 . His 
treatment of the subject is however very brief, and a consi- 
derable 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, Proba- 
bility 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 be- 
tween it and the contiguous branches of the general science 

1 This remark, and that at the Probability. I have given a pretty 

commencement of the last paragraph, full discussion of the general prin- 

having been misunderstood, I ought ciples of this view in the tenth 

to say that the only sense in which chapter, and have there pointed out 

originality is claimed for this Essay some of the peculiarities to which it 

is in the thorough working out of the leads. 
Material view of Logic as applied to 



Preface to the First Edition. xi 

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 con- 
vinced 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 grow- 
ing 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 roust 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 sug- 
gestions 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. 



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. Be- 
sides various omissions (principally where the former treat- 
ment 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 altera- 
tions 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 un- 
changed, 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 ne- 
cessary to say this, as to a cursory reader it might seem 



xiv Preface to the Second Edition. 

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 con- 
sequently 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. xn.); the 
logical principles underlying the method of Least Squares 
(Ch. xm.); 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 promi- 
nently 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 ob- 
tained from all other external sources together. Much of this 



Preface to Second Edition. xv 

criticism has been given privately in letters, and notes on 
the proof-sheets ; but one of the most elaborate of his discus- 
sions 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. 



V. 



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 nine- 
teenth (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 altera- 
tions 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 

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

62 



xviii Preface. 

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 oc- 
cupied 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, arid 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. 



TABLE OF CONTENTS 1 . 



PART I. 

PHYSICAL FOUNDATIONS OF THE SCIENCE OF PEOBABILITY. 

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, 

C, 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 ? 

1 Chapters and sections which are nearly or entirely new are printed in italics. 



xx Contents. 

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, d'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 he distinguishable into natural kinds, 

4 6. And (2) ' agencies.' 

7. Eequisites 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. 

I 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 a priori proof. 

8. The Principle of Sufficient Reason inapplicable. 



Contents. xxi 

} 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: 
0. 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 w. 

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. 

t 1, 2. Preliminary remarks. 
3, 4. Are we accurately conscious of gradations of belief? 



xxii Contents. 

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 KULES 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. 



Contents. xxiii 

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. 

15. 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. 




xxiv Contents. 

CHAPTER X. 

CHANCE, CAUSATION AND DESIGN. 

Old Theological objection to Chance. 
Scientific version of the same. 
Statistics in reference to Free-will. 
6 8. Inconclusiveness of the common arguments here. 

9, 10. Chance, (is 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. Arbutlmott'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 XT. 

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* 



Contents. xxv 

5 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 prac- 

tically. 

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



ixvi Contents. 

CHAPTER XIY. 

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. 
1216. (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 tlte event DI Probability and in Induction. 



PART IIT. 

VARIOUS APPLICATIONS OF THE THEORY OF PROBABILITY. 
CJIH. XV XIX. 

CHAPTER XY. 

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: 



Contents. xxvii 



\ 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 credi- 
bility 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. 
1618. Meaning of the term extraordinary, and its distinction from 
miraculous 



xxviii Contents. 

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. 

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 he din- 

tin gui sited from, 

5. (3) the Maximum Ordinate average, 

6. (4) and the Median. 

7. Diagram in illustration. 

$ 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 average*. 

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 l errors'. 
23. Practical escape from these. 

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



Contents. xxix 



CHAPTER XIX. 

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

i 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 determinate & 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 oj the con- 
traction of the modulus of the average-error curve. 

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

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. 2125.] 

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

21. 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. 20 37.] 

26. (i) All errors equally probable, 

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

29, 30. Further analysis of tJie 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 
(o 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 arid 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. 
3ut 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 
v. 1 



2 On certain kinds of Groups or Series. [CHAP. I. 

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 V,,' 1 .!: ; 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 state- 
ment, and I may have obtained it by any conceivable mode 
in which truths can be obtained ; but whatever the value of 



SECT. 3.] On certain kinds of Groups or Series. 3 

the general proposition may be, that of the particulars is 
neither greater nor less. The process of inferring the par- 
ticular from the general is not accompanied by the slightest 
diminution of certainty. If one of these 'immediate infer- 
ences' 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 in- 
ferences, 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 dis- 
belief, 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 



4 On certain kinds of Groups or Series. [CHAP. i. 

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 intro- 
duced 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 com- 
bines individual irregularity with ., ... 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 propo- 
sition 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 docs not obviously present itself in con- 
nection 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 con- 
trary, 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 



SECT. 3.] On certain kinds of Groups or Series. 5 

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 un- 
certain, but the average duration of a batch of lives is be- 
coming 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 



6 On certain kinds of Groups or Series. [CHAP. I. 

of things which can be traced almost everywhere, to a 
greater or less extent, throughout the whole field of our ex- 
perience. Fires, shipwrecks, yields of harvest, births, mar- 
riages, 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 diminish- 
ing 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 irregu- 
larities 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 

1 The following statistics will give (fluctuation 1-85), and the number 

a fair idea of the wide range of ex- of pounds of manufactured tobacco 

perience over which such regularity taken for home consumption (fluctua- 

is found to exist: "As illustrations tionl*89); or out-door paupers (fluc- 

of equal amounts of fluctuation from tuation 3-45) and tonnage of British 

totally dissimilar causes, take the vessels entered in ballast (fluctuation 

deaths in the West district of London 3-43), &c." [Extracted from a paper 

in seven years (fluctuation 13-66), in the Journal of the Statistical So- 

and offences against the person (flue* ciety, by Mr Guy, March, 1858 ; the 

tuation 13*61); or deaths from apo- * fluctuation' here given is a mea- 

plexy (fluctuation 5 *54), and offences sure of the amount of irregularity, 

against property, without violence that is of departure from the average, 

(fluctuation 5'48) ; or students regis- estimated in a way which will be 

tered at the College of Surgeons described hereafter.] 



SECT. 4.] On certain kinds of Groups or Series. 7 

there should be as many as eight heads and therefore as few 
as two tails, or vice versd. 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 uni- 
versal, or, what comes to the same thing, that irregularity 
may be observed in all their qualities. All that is meant is 
that there arc 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 



8 On certain kinds of Groups or Series. [CHAP. I. 

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 de- 
scription. The objects, be it remembered, are given to us in 
nature; the order under which we view them is our own pri- 
vate 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 neces- 
sarily 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 suc- 
cession; 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 



SECT. 6.] On certain kinds of Groups or Series. 9 

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 1 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 de- 
termining 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 con- 
sider 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 sub- 
ordinated to it." As remarked above, this appears a sub- 
stantially 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, arid I there- 
fore much prefer the latter expression. 

G. 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 

1 Transactions of the Cambridge Beprinted in the collected edition of 
Philosophical Society, Vol. ix. p. 605. his writings, p. 50.. 



10 On certain kinds of Groups or Series. [CHAP. I. 

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 at- 
tempt a somewhat closer analysis. 

There are certain phrases so commonly adopted as to 
have become part of the technical vocabulary of the sub- 
ject, 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 de- 
note 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 plir;i-i-oloo\ is very convenient, and 
will often be made use of in this work, but without expla- 
nation 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 



SECT. 7.] On certain kinds of Groups or Series. 11 

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 attri- 
butes, 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 uni- 
formity. 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 suc- 
cession 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 sub- 



12 On certain kinds of Groups or Series. [CHAP. I. 

stances 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 sub- 
stances 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 Proba- 
bility 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 ia different ways 
in the different cases. When the occasional attributes how- 
ever are important, or compose the majority, this way of 
speaking becomes less appropriate ; language is somewhat 
strained by our implying that two extremely different assem- 
blages 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 mis- 
understood, it will frequently be made use of here. 

9. A series of the above-mentioned kind is, I ap- 
prehend, the ultimate basis upon which all the rules of 
Probability must be based. It is essential to a clear com- 
prehension 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 



SECT. 10.] On certain kinds of Groups or Series. 13 

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 dis- 
covered, 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 infer- 
ence ; into what these substances or attributes would them- 
selves be ultimately analysed, if taken in hand by the 
i ' ' "' or metaphysician, it is no business of ours to 
enquire here. 

104 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 consti- 
tuting 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 pro- 
bably 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 Pro- 
bability; that uniformity which is found in the long run, 
and which presents so great a contrast to the individual 



14 On certain kinds of Groups or Series. [CHAP. I. 

disorder, though durable is not everlasting. Keep on watch- 
ing 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 indi- 
vidual 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 improve- 
ment. 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 



SECT. 11.] On certain kinds of Groups or Series. 15 

length. Or, again, it is perfectly possible that future gene- 
rations 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 cha- 
racter) 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 bo 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 compari- 
son 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 ex- 
ample, or the instances of duelling or accusations of witch- 
craft, 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, un- 
dergo 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 



16 On certain kinds of Groups or Series. [CHAP. I. 

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 suffi- 
cient numbers, yet it does not follow but that in certain cases 
we may have too many of them. If they are made too ex- 
tensive, 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 a 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 af- 
forded 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 



SECT. 13.] On certain kinds of Groups or Series. 17 

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 ex- 
planatory and illustrative of the foundations of the science, 
the remark may be made here (for which subsequent justi- 
fication 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 im- 
possible to show that they were alike as that they were 
different ; we can only say generally that each sh^ws 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 1 . The 
proportion of persons, for example, who die each year at the 
age of six months is, when the numbers examined are on a 

1 "We might express it thus : a it takes a very great number to esta- 

few instances are not sufficient to blish that a change is taking place in 

display a law at all ; a considerable the law. 
number will suffice to display it; but 

v. 2 



\S On certain kind$ of Groups or Series. [CHAP. I. 



scale, utterly irregular; it becomes however regular 
when the numbers examined are on a larger scale ; but if we 
cpntinued 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 sub- 
stitution 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 substi- 
tution 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 pro- 
portion begins to prevail among the events in the long run ; 
but then 011 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 distiirbance. 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, to- 
wards which the numbers are tending, by assuming that 
these circumstances do not change ; in other words, by sub- 
stituting a series with a fixed uniformity for the actual one 
with the varying uniformity 1 . 

1 The mathematician may illustrate analogies of the 'circle of curvature' 
the nature of this substitution by the in geometry, and the 'instantaneous 



SECT. 15.] On certain kinds of Groups or Series. 19 

15. If the reader will study the following example, 
one well known to mathematicians under the name of the 
Petersburg 1 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 in- 
definite 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 suc- 
cession 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 

ellipse' in astronomy. In the cases on that supposition. 

in which these conceptions are made 1 So called from its first mathe- 

use of we have a phenomenon which matical treatment appearing in the 

is continuously varying and also Commentarii of the Petersburg Aca- 

changing its rate of variation. We demy; a variety of notices upon it 

take it at some given moment, sup- will be found in Mr Todhunter's 

pose its rate at that moment to be History of the Theory of Probability, 
fixed, and then complete its career 

22 



20 On certain kinds of Groups or Series. [CHAP. I. 

than that number of times. Here then is a series formed by 
a succession of throws. We will assume, what many per- 
sons 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 equiva- 
lent 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 in- 
ferences 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 



SECT. 16.] On certain kinds of Groups or Series. 21 

give anything approaching to 50 for such a chance. Pro- 
bably 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 for- 
gotten 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 complete- 
ness 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 2 n 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), 

2*-2 + 2"- s . 2 4. 2 n ' 4 . 2 7 + + 2 . 2 n " 3 + 2 n ~ 2 + 2"" 1 , 



22 On certain kinds of Groups or Series. [CHAP. I. 

viz. (n + 1) 2"- 2 . 

This being spread over 2 n ~' different occasions of gain his 



average gain will be \(n -f- 1). 

Now when we are referring to averages it must be re- 
membered 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. Qd.: 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 2 89 
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 1 extend- 
ing 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 2 n times he may 
have a true average receipt of (n + l) pounds, and if he 
continues long enough onwards he mil have it. 

The problem will recur for consideration in a future 
chapter. 



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 par- 
ticular kind of series, that namely, which must be considered 
to constitute the basis of the science of Probability, has re- 
ceived 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 indi- 
cation was given that the series of this kind, as they actu- 
ally 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 com- 
plicated 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 thftt 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 



24 -!/ .";/ '/: -i 1 and Formation of the Series* [CHAP. n. 

each direction, and if we continue to accumulate our mea- 
sures 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 measure- 
ments. Now suppose these heights to be marshalled in the 
order of their magnitude. What we always find is some- 
thing of the following kind; about the middle point be- 
tween 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 diminu- 
tion takes place. The word ' law ' is here used in its mathe- 
matical 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 mus- 
cular power of various limbs, the dimensions of almost every 
1 Essai de Physique Sociale, 1869. Anthropometrie, 1870. 



SECT. 3.] Arrangement and Formation of the Series. 25 

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 can- 
not be referred to any better sources than to these and other 
works by the same author 1 . 

Interesting and valuable, however, as are Quetelet's sta- 
tistical 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 can- 
not 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 in- 
dependent way. 

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

1 As regards later statistics on the Secretary (Mr C. Eoberts) that their 

same subject the reader can refer to statistics are "unique in range and 

the Keports of the Anthropometrical numbers". They embrace not merely 

Committee of the British Association military recruits like most of the 

(1879, 1880, 1881, 1883 ; especially previous tables but almost every 

this last). These reports seem to class and age, and both sexes. More- 

me to represent a great advance on over they refer not only to stature 

the results obtained by Quetelet, and but to a number of other physical 

fully to justify the claim of the characteristics. 



26 Arrangement and Formation of the Series. [CHAP. n. 

they depart from that mean, we shall find that this di- 
minution of frequency takes place according to one in- 
variable law, whatever may be the nature of these mag- 
nitudes, 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 combina- 
tions, and runs of luck, afforded by games of chance. Sup- 
pose a handful, consisting of ten coins, were tossed up a 
great many times in succession, and the results were tabu- 
lated. 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 1 . 

1 As every mathematician knows, cessive terms of the expansion of 
the relative numbers of each of these (1 + 1) 10 , viz. 1, 10, 45, 120, 210, 252, 
possible throws are given by the sue- 210, 120, 45, 10, 1. 



SECT. 4] Arrangement and Formation of the Series. 27 

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 ulti- 
mately 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 com- 
monly 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 1 

1 That is they will be more densely cessive circle, the number of shot 
aggregated. If a space the size of the marks which it contains will be sue- 
bull's-eye be examined in each sue- eessively less. The actual number 



28 A ' and Formation of the Series. [CHAP. II. 

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-agerit 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 under- 
stand 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 at- 
tached. 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 ? 

of shots which strike the bull's-eye so much less surface than any of the 
will not be the greatest, since it covers other circles. 



SECT. 5.] Arrangement and Formation of the Series. 29 

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 con- 
form with a considerable degree of accuracy to one and the 
same 1 law. At least this is made plain for the more 



1 Commonly called the exponential 
law; its equation being of the form 
y = A e~ 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 num- 
ber of values approximately equal; 
after a time it begins to slope away 
rapidly towards the axis of a; (express- 
ing 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 direc- 
tions (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 



80 Arrangement and Formation of the Series. [CHAP. II. 

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 1 . 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 formulae we draw up. 

Besides the instance of the heights of men, other classes 



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 essen- 
tially 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 re- 
cognized and was founded upon the 
assumption that extreme divergences 
were excessively rare. Hence it 
would be difficult, by mere observa- 
tion, to prove that the law of error 
in any given case was not exponen- 
tial; unless the statistics were very 
extensive, or the actual results de- 



parted considerably from the expo- 
nential form. (2) It is quite impos- 
sible by any graphic representation 
to give an adequate idea of the exces- 
sive 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. Where- 
as in the hyperbola, e.g. the rate of 
approach of the curve to its asymp- 
tote is continually decreasing, it is 
here just the reverse; this rate is 
continually increasing. Hence the 
two, viz. the curve and the axis of ar, 
appear to the eye, after a very short 
time, to merge into one another. 

1 As by Quetelet: noted, amongst 
others, by Herschel, Essays, page 
409. 



SECT. 6.] Arrangement and Formation of the Series. 31 

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 com- 
paratively 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 



32 Arrangement and Formation of the Series. [CHAP. n. 

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 suc- 
cessions 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 impos- 
sible. 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 



SECT. 7.] *\rr(iuyenteitt and Formation of the Series. 33 

to tend in the same direction, instead of, as in the other 
cases, more or less neutralizing one another's work. But 
the ,..:_" .: , 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 com- 
binations, such as those in question, has such a bewildering 
effect upon the mind that one may be sometimes apt to con- 
found the impossible with the higher degrees of the merely 
mathematically improbable. 

7. At the time the first edition of this essay was com- 
posed writers on Statistics were, I think, still for the most 
part under the influence of Quetelet, and inclined to over- 
value 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 1 that the 
" assumption which lies at the basis of the well-known law of 

1 Proc. R. Soc. Oct. 21, 1879, 

v. . 3 



34 A -/; - ' and Formation of the Series. [CHAP. II. 

'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, 1C, 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 con- 
sequences 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 sym- 
metrical 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 1 , 
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 

1 We are here considering, re- of statistics ; so that there are actual 
member, the case of a finite amount limits at each end. 



SECT. 8.] Arrangement and Formation of the Series. 35 

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 *. 

1 It must be admitted that ex- shown this asymmetry in respect of 
perience has not yet (I believe) heights. 

32 



36 Arrangement and Formation of the Series. [CHAP. n. 

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 1 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 direc- 
tion, 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. 

1 The above reasoning will proba- selves, involve somewhat of an anti- 

bly be accepted as valid at this stage cipation. They demand, and in a 

of enquiry. But in strictness, as- future chapter will receive, closer 

sumptions are made here, which how- scrutiny and criticism, 
ever justifiable they may be in them- 



. 10.] Arrangement and Formation of the Series. 37 

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 1 law mentioned in 5. 

10. It may be readily admitted from what we know 
about the production of events that something resembling 



1 A definite numerical example of 
this kind of concentration of fre- 
quency 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 + l) w . 
Now it may be shown (Quetelet, Let- 
ters, p. 263 ; Liagre, Calcul des Proba- 
bilites,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, 

-= - &c. , and connect their ver- 
1*J 

tices, the figure we obtain approxi- 
mately represents some form of the 
curve y Ae~ h **, 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 magni- 
tude without limit. Jevons has given 
(Principles of Science, p. 381) a dia- 
gram 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 co- 
efficients of a binomial of two equal 
terms as (l + l) w \ in the preceding 
paragraph are symmetrical in their 
arrangement from the first, and very 
speedily become indistinguishable in 
(graphical) outline from the final ex- 
ponential form. But if, on the other 
hand, we were to consider the suc- 
cessive terms of such a binomial as 
(l + 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 how- 
ever we suppose m to increase with- 
out limit, as in the former supposi- 
tion, the unsymmetry gradually dis- 
appears and we tend towards pre- 
cisely the same exponential form as 
if we had begun with two equal terms. 
The only difference is that the posi- 
tion 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 fail- 
ures but successes and failures in 
the ratio of 1 to 4. 



38 Arrangement and Formation of the Series. [CHAP. IT. 

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 ap- 
proaching 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 toler- 
ably 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 consider- 
ably 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 



SECT. 11.] Arrangement and Formation of the Series. 39 

draw a similar conclusion from this deductive line of argu- 
ment as from the direct appeal to statistics. The same 
general result seems to be established ; namely, that approxi- 
mately, with sufficient accuracy for all practical purposes, \ve 
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 1 . 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 esta- 
blish 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 ;, i !'.. to one invariable law. If all Maws 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 

1 'Law of Error* is the usual tioned in 4, but by a convenient 

technical term for what has been generalization it is equally applied 

elsewhere spoken of above as a Law to the other two ; so that we term 

of Divergence from a mean. It is in the amount of the divergence from 

strictness only appropriate in the the mean an * error' in every case, 

case of one, namely the third, of the however it may have been brought 

three classes of phenomena men- about. 



40 Arrangement and Formation of the Series. [CHAP. n. 

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 . V ,\* devoted a 
special chapter to the consideration of it. 

The main reason, however, against discussing this argu- 
ment 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 arrange- 
ments 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 



SECT. 13.] Arrangement and Formation of the Series. 41 

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 there- 
fore 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 pro- 
priety 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 em- 
ployed with satisfactory results affords little or no proof that 
the errors to which it is applied must be arranged i.iv^minij; 
to one fixed law. 

13. So much then for the attempt to prove the preva- 
lence, 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 typp 



42 A*- r ;:* ' and Formation of the Series. [CHAP. n. 

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 1 . 

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 deter- 

1 This however seems to be the rate works by Quetelet, viz. his 
purport, either by direct assertion Physique Sociale, and his Anthropo- 
or by implication, of two elabo- mltrt'e. 



SECT. 14.] .-1 rr<ii 'jo- <; J and Formation of the Series. 43 

mine 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 etf- 
ponential 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 d, 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 1 , in the case of Englishmen the mean is about 
1 He scarcely, however, professes to give these as an accurate measure 



44 .1 ;-i . ; / .." ' and Formation of the Series. [CHAI*. II. 

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 de- 
cisive and summary. It would be said that we are here 
mixing distinctly heterogeneous elements, and that in con- 
sequence 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 fol- 
low almost entirely the law determined by these English 
conditions, for at this point the English data are very nume- 
rous, 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, 
,-, :"' .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 hetero- 

of the mean height, nor does he afford accurate data on any large 

always give precisely the same mea- scale. The statistics given a few 

sure. Practically, none but soldiers pages further on are probably far 

being measured in any great num- more trustworthy. 
bers. the English stature did not 



SECT. 16.] Arrangement and Formation of the Series. 45 

geneous 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 some- 
where 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 indi- 
cated in the above diagram. 

The instance of the English and French heights resem- 
bles the one just given, but falls far short of it in the strin- 
gency 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 



46 A rranyement arid Formation of the Series. [CHAP. n. 

which are subject to a considerable number of local or oc- 
casional disturbing causes. Amongst Frenchmen were in- 
cluded, presumably, Bretons, Provencals, 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 impor- 
tant, 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 : . -'. * ..' *] 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 |>hvM -logical 
causes represent the constant element ; that is, given a homo- 
geneous race of people to begin with, who freely inter- 
marry, and are subject to like circumstances of climate, food, 
and occupation, the standard would remain on the whole 
constant 1 . 

In such a case the man who possessed the mean height, 
mean weight, mean strength, and so on, might then be 

1 This statement will receive some explanation and correction in the next 
chapter. 



SECT. 18.] -dnviMtt /;/*< and Formation of the Series. 47 

called, in a sort of way, a 'type*. The deviations from this 
type would then be produced by innumerable small influ- 
ences, partly physiological, partly physical and social, acting 
for the most part independently of one another, and result- 
ing 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 ex- 
hibit 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; 6871, 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 nar- 



48 Arrangement and Formation of the Series. [CHAP. 11. 

rower. 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, re- 
main 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 jijj: 1 - j.i:-- 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, in^ 
stead of being fixed, were slowly changing its position as we 
continue to fire at it; changing almost certainly to some ex- 
tent 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 at- 



SECT. 20.] -1 /Y". //"//(/' and Formation of the Series. 49 

tempts 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 correla- 
tion of mental and bodily characteristics gives high proba- 
bility 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 be- 
cause 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 1 . 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 cor- 

1 I am not speaking here of the measurement of perception sand other 
now familiar results of Psychophysics, simple states of consciousness, 
which are mainly occupied with the 

v. 4 



50 A rrari ye ment and Formation of the Series. [CHAP. II. 

respond to certain limits of excellency or deficiency. 1 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 legi- 
timate 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 con- 
ceptions 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 2 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 

1 Perhaps the best brief account with some approach to accuracy, 

of Mr Galton's method is to be found select the middlemost person in the 

in a paper in Mind (July, 1880) on row and use him as a basis of com- 

the statistics of Mental Imagery. parison with the corresponding per- 

The subject under comparison here son in any other batch. And simi 

viz. the relative power, possessed larly with those who occupy other 

by different persons, of raising clear relative positions than that of the 

visual images of objects no longer middlemost. 

present to us is one which it seems 2 I refer to the introductory and 

impossible to 'measure*, in the ordi- concluding chapters: the bulk of the 

nary sense of the term. But by book is, from the nature of the case, 

arranging all the answers in the mainly occupied with statistical and 

order in which the faculty in ques- biographical details, 
tion seems to be possessed we can, 



SECT. 22.] .1 ri'iiH'jciiienl and formation of the Series. 51 



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 7 . They would confound the mere nu- 
merical distinction, which seems in some way to imply 
double excellence, with the intrinsic superiority, which 
would mostly be represented by a very small fractional ad- 
vantage. 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 superi- 
ority 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 acci- 
dental. 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 1 , 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 

1 See Galton's Hereditary Genius, pp. 336 350, "On the comparative 
worth of different races." 

42 



52 Arrangement and Formation of the Series. [CHAP. n. 

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 ap- 
preciated 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 com- 
parison. 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 there- 
fore 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. 



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 pro- 
cess. All events are the results of numerous and com- 
plicated 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 dis- 
tinction 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 



54 Origin, or Process of Causation of the Series. [CHAP. HI. 

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 ex- 
tended 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 re- 
moved 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 indis- 
tinguishably 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 !.- -: : ,r*' j again the 
very peculiarity of the phenomena which we were under- 
taking to explain. They will be found, for instance, to 



SECT. 3.] Origin, or Process of Causation of the Series. 55 

consist of large classes of objects, throughout all the indi- 
vidual 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 com- 
pose them in particular, in the order of their strength, we 
should find just such an ..... .,,. 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 pur- 
pose 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 vegetablesjimited to a single pair, or 



56 Origin, or Process of Causation of the Series. [CHAP. III. 

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 varie- 
ties and sut-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 



SECT. 4] Origin, or Process of Causation of the Series. 57 

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 in- 
fection, and so on. 



58 Origin, or Process of Causation of the Series. [CHAP. ill. 

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 dis- 
covered in the objects at the stage when they are com- 
monly 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 
auch a character. 

6. This division, into objects and the agencies which 
affect them, is merely intended for a rough practical ar- 
rangement, sufficient to point out to the reader the imme- 
diate 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 de- 
veloped. Even if it were the case that the first parents 
of each natural kind had been specially created, instead 



SECT. 6.] Origin, or Process of Causation of the Series. 59 

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 know- 
ledge, 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 determin- 
ing 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 



60 Origin, or Process of Causation of the Series. [CHAP. in. 

made up of aggregates of small components, but they are 
still best regarded as being by comparison simple and dis- 
tinct, for their component parts act mostly in groups col- 
lectively, 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 per- 
manent, 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 combi- 
nation of these causes then occurring with a regular de- 
gree 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 in- 
stance, suppose a man throwing a quoit at a mark. Here 
our fixed causes are his strength, the weight of the quoit, 

1 As stated above, this is really that which we are called upon to ex- 
little more than a re- statement, a plain in the concrete details pre- 
stage further back, of the existence sented to us in experience. 
of the same kind of uniformity as 



SECT. 8.] Origin, or Process of Causation of the Series. 61 

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 1 . 

8. The necessity of the conditions just hinted at 
will best be seen by a reference to cases in which any of 

1 "It would seem in fact that in having been till then masked and 

coarse and rude observations the overshadowed by the graver errors 

errors proceed from a very few prin- which had been now approximately 

cipal causes, and in consequence our removed There were errors of 

hypothesis [as to the Exponential graduation, and many others in the 

Law of Error] will probably repre- contraction of instruments ; other 

sent the facts only imperfectly, and errors of their adjustments ; errors 

the frequency of the errors will only "(technically so called) of observation; 

approximate roughly and vaguely to errors from the changes of tempera- 

the law which follows from it. But ture, of weather, from slight irregular 

when astronomers, not content with motions and vibrations ; in short, the 

the degree of accuracy they had thousand minute disturbing influ- 

reached, prosecuted their researches ences with which modern astrono- 

into the remaining sources of error, mers are familiar." (Extracted from 

they found that not three or four, a paper by Mr Crofton in the Vol. of 

but a great number of minor sources the Philosophical Transaction* for 

of error of nearly co-ordinate import- 1870, p. 177.) 
ance began to reveal themselves, 



62 Origin, or Process of Causation of the Series. [CHAP. ill. 

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 de- 
pendent 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 ab- 
solutely insure us. 

9. If it be replied that this want of statistical regu- 
larity only arises from the fact of our having confined our- 
selves to too limited a time, and that we should find 



SECT. 10.] Origin^ or Process of Causation of the Series. 63 

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 con- 
clusive 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 pro- 
gress in society, either for the better or the worse, the argu- 
ment 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 in- 
fluences, but on the contrary show a steady tendency to in- 
crease their power, the statistics will, with consequent steadi- 
ness and permanence, take the impress of such a change. 

10. Briefly then, if we were asked where the dis- 
tinctive 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 pro- 
perties of natural kinds, both in the ultimate and in the de- 
rivative 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 



64 Origin, or Process of Causation of the Series. [CHAP. in. 

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 im- 
portance 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 



SECT. ll.J Origin, or Process of Causation of the Series. 65 

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 uni- 
formity 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 as- 
sertion 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 pro- 
blems 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 observa- 
tions 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 
v. 5 



66 Origin, or Process of Causation of the Series. [CHAP. ni. 

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 ner- 
vous 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 



SECT. 14.] Origin, or Process of Causation of the Series. 67 

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 in- 
fluencing, 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 con- 
ditions, such as those of health, the casualties of employ- 
ment, &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 imme- 

52 



68 Origin, or Process of Carnation of the Series. [CHAP. HI. 

diate 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 im- 
perfect 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 im- 
perfect 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 for- 
jnula; but unless he takes some natural series as a copy, he 
will never be able to construct one mentally which shall per- 
manently imitate the uri^iimK 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 



SECT. 15.] Origin, or Process of Causation of the Series. 69 

effect which these disturbances are able to produce is so in- 
appreciably 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 observa- 
tions, 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 impro- 
bably 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 uni- 
formity, 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 indepen- 
dently. 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 



70 Origin, or Process of Causation of the Series. [CHAP. in. 

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, how- 
ever, of certain statistical or sociological enquiries will de- 
mand 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 ob- 
served. 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 cer- 
tainly 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 



SECT. 17.] Origin, or Process of Causation of the Series. 71 

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 ^ 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 ii-vmin^ to 
the same general law, i.e. the now familiar law in ques- 
tion, 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 Gtitttlrftny], 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 character- 
istic 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 circum- 
stances, 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 other- 
wise continually increasing dispersion. 

17. The facts were first tested by careful experiment. 



72 Origin, or Process of Causation of the Series. [CHAP. m. 

At the date of Mr Galton's original paper on the subject 1 , 
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 charac- 
teristics 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 in- 
dependent. 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 arid 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 ' re version. 5 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. to- 
wards 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 

1 Typical Laws of Heredity; read 1877. See also Journal of the An- 
before the Boyal Institution, Feb. 9, throp. Inst. Nov. 1885. 



SECT. 18.] Origin, or Process of Causation of the Series. 73 

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 in- 
troduced. 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 diver- 
gence of fortunes of the supposed gamblers. 



CHAPTER IV. 

ON THE MODES OF ESTABLISHING AND DETERMINING THE 
EXISTENCE AND NUMERICAL PROPORTIONS OF THE CHA- 
RACTERISTIC 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 pro- 
cess 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 concep- 
tions, 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 

1 This latter enquiry belongs to logical part of this volume, and is 
what may be termed the more purely entered on in the course of Chapter vi. 



SECT. 2.] Modes o/WrV-' .'. </ the Groups or Series. 75 

mean to employ the term in its widest signification; we 
mean experience supplemented by all the aids which in- 
ductive 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 proce- 
dure 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 fcefore. 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 cal- 
culate 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, espe- 
cially the former one, nobody would ever think of appealing 
to statistics. This would be a tedious process to adopt when, 



76 Modes of establishing the Groups or Series. [CHAP. iv. 

as here, the mechanical and other conditions upon which the 
production of the events depend are comparatively few, de- 
terminate, 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, d priori determination is often easy, whilst d poste- 
riori appeal to experience would be not merely tedious but 
utterly impracticable. This, combined with the frequent 
simplicity and attractiveness of such examples when deduct- 
ively 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 ap- 
parently 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. 



SECT. 3.] Modes of establishing the Groups or Series. 77 

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 1 
HH. t HT. y TH., TT. So much is certain. We are more- 
over 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 indefi- 
nite 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 men- 
tioned 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 ex- 
pectation 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 

1 For the use of those not ac- two successive throws of the penny 

quainted with the common notation give head ; HT that the first of 

employed in this subject, it may be them gives head, and the second 

remarked that H H is simply an tail ; and so on with the remaining 

abbreviated way of saying that the symbols. 



78 Modes of establishing the Groups or Series. [CHAP. iv. 

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 chap- 
ter ; 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 me-; in ing, 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 ex- 
perience, 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 d 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 in- 
troduced. In the first place it must be an ideal coin, with 
its sides equal and fair. This restriction is perfectly intelli- 
gible ; the study of solid geometry enables us to idealize a 
penny into a circular or cylindrical lamina. But this condition 



SECT. 5.] Modes of establishing the Groups or Series. 79 

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 ideal- 
ized? 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 per- 
sisted 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 what- 
ever 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 par- 
ticular 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 



80 Modes of establishing the Groups or Series. [CHAP. iv. 

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 at- 
tached 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 facili- 
tating 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 precau- 
tion ; 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, 



SECT. 7.] Modes o/V-s-///7,/,\/,//,>/ the Groups or Series. 81 

therefore, if we scrutinize our language, we shall find that 
any supposed d 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 , does it follow necessarily that the chance of 
HH (with two pence) is |? 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 



82 Modes of establishing the Groups or Series. [CHAP. IV. 

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 1 . 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 there- 
fore 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 

1 I am endeavouring to treat this tween the terms of the series ? The 

rule of Sufficient Reason in a way succession seems then reduced to a 

that shall be legitimate in the opinion dull uniformity, a mere iteration of 

of those who accept it, but there the same thing many times; the 

seem very great doubts whether a series we contemplated has disap- 

contradiction is not involved when peared. If the sides are not abso- 

we attempt to extract results from lutely alike, what becomes of the 

it. If the sides are absolutely alike, applicability of the rule? 
how can there be any difference be- 



SECT. 9.] Modes of establishing the Groups or Series. 83 

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 a priori, 
(as might have been concluded,) can it be obtained by ex- 
perience ? 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 1 . 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 con- 
siderable 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 repe- 
titions 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 

1 Formal Logic, p. 185. Principles of Science, p. 208. 

62 



84 Modes of establishing the Groups or Series. [CHAP, iv, 

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 proba- 
bilities a 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 up- 
permost, arid 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 



SECT. 11.] Modes of establishing the Groups or Series. 85 

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 reason- 
ing, it would seem to point with some likelihood to the con- 
clusion 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 pro- 
duced by the act of throwing might introduce infinite in- 
dividual 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 dif- 
ficulty, 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 nar- 
rowed, for we have reduced it to a series of ' : , 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 involun- 
tarily 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 com- 
monly received result does in reality depend upon the ful- 
filment 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 



86 Modes of establishing the Groups or Series. [CHAP. iv. 

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 substi- 
tuted or idealized series which generally has to be introduced 
as the basis of our calculation. The danger to be guarded 
against is that of iv^smlin^ 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 con- 
sciousness, and too freely pushing it to unwarranted lengths. 

Or on the other hand, we may consider that we are treat- 
ing 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 any- 
thing more than an nl^rlnviinil truth, depends of course upon 
there being a close correspondence between these two views 



SECT. 13.] Modes of establishing the Groups or Series. 87 

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 formulae. 
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 apply- 
ing 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 arith- 
metical doctrine of Permutations and Combinations 1 . 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 considera- 
tions of number or magnitude, and we terminate with them, 

1 The close connection between title of Mr Whitworth's treatise, 
these subjects is well indicated in the Choice and Chance. 



88 Modes of ulnlJiiiliif'i the Groups or Series. [CHAP. iv. 

having only arithmetical calculations to connect them to- 
gether. 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. Permu- 
tations 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 our- 
selves 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 ob- 
jective 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 



SECT. 14] Modes of establishing the Groups or Series, 89 

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 in- 
troduced 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 sug- 
gest 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 V 

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 

1 Esaai Philosophique. Ed. 1825, p. 74. 



90 Modes of establishing the Groups or Semes. [CHAP. iv. 

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 re- 
spective 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 nume- 
rical statement of the proportions in which they occur. 

15. It will very likely be replied that by the proba- 
bility above mentioned is meant, not the mere numerical 
proportion between the births, but some fact in our consti- 
tution 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 consti- 
tution 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 ob- 
served numbers might be supposed capable of being deter- 
mined 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 1 . But without going into particulars, one may 

1 An opinion prevailed rather at Quetelet amongst others) that the 
one time (quoted and supported by relative ages of the parents had 



SECT. 16.] Modes of establishing the Groups or Series. 91 

assert with some confidence that these conditions cannot well 
be altogether independent of the health, circumstances, man- 
ners 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 ad- 
duced 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 

something to do with the sex of the to 100 does not seem by any means 

offspring. If this were so, it would universal in all countries or at all 

quite bear out the above remarks. times. For various statistical tables 

As a matter of fact, it should be on the subject see Quetelet, Physique 

observed, that the proportion of 106 Sociale, Vol. i. 166, 173, 238. 



92 Modes of establishing the Groups or Series. [CHAP. iv. 

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 re- 
mote 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 be- 
forehand ; 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 develope into the sequence which 
exhibits the uniformity. Finally, this very questionable ob- 
jective probability is assumed to exist, with the same faculty 
of development, in all the cases in which uniformity is ob- 
served, 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 



SECT. 18.] Modes of < i "/*/ ';/ tf/ie Groups or Series. 93 

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 uni- 
formity 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 pro- 
duced the observed regularity, a few only, which should con- 
stitute 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 suf- 
ficient 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 accu- 
mulate 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 ixrmlnalh or sud- 
denly 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 



94 Modes of MlnlJluHwj the Groups or Series. [CHAP. iv. 

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 pro- 
duction 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 ap- 
proximation to truth. On the other hand, in all social 
applications of Probability, the unchangeable causes can only 
be regarded as really unchangeable under many qualifica- 
tions. 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 cull<-ctiii<r too many of them. 

19. The results of the last two chapters may be 
summed up as follows: We have extended the conception 



SECT. 19.] Modes of establishing the Groups or Series. 95 

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 statis- 
tical 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 ex- 
cessive deductive or d priori treatment of the science has 
been encouraged. 



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 oc- 
casion 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 n^n ^;ii<- 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 per- 
formance, and it has reference rather to the single action 
than to a group or series of actions. Thus, when a man 

1 According to Prof. Skeat (Ely- nected with the Teutonic word rand 

mological Dictionary) the earliest (brim), and implies the furious and 

known meaning is that of furious irregular action of a river full to the 

action, as in a charge of cavalry. brim. 
The etymology, he considers, is con- 



SECT. 2.] Randomness and its scientific treatment 97 

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 repeti- 
tion. 

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 where- 
abouts at any instant a drop will fall, but we know that if 
we put out a sheet of paper it will gradually become uni- 
formly 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 as- 
sumption of random arrangement must postulate the oc- 
currence of this particular state of things at some stage or 
other. But there is often considerable difficulty, leading oc- 
casionally to some arbitrariness, in deciding the particular 
stage at which it ought to be introduced. 

(1) Thus, in many of the problems discussed by mathe- 
maticians, 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, 
v. 7 



98 Randomness and its scientific treatment. [CHAP. v. 

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 1 . 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 mathe- 
matician thinks of nothing but this final result, and pays no 
heed to the process by which it may be brought about. Ac- 
cordingly the problem is again reduced to one of mensura- 
tion, 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 

1 See the problem paper of Jan. 18, 1854, in the Cambridge Mathema- 
tical Tripos. 



SECT. 4.] Randomness and its scientific treatment. 99 

the horizontal being equally represented in the long run. 
We have therefore to contemplate a surface of uniform dis- 
tribution, 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 1 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 in- 
differently, 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 un- 
certainty as to what stage should be selected for that of uni- 
form 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 

1 As, according to Mr H. Godfray, proposed in an examination. See 
the majority of the candidates did the Educational Times (Beprint, Vol. 
assume when the problem was once vn. p. 99.) 

72 



100 Randomness and its scientific treatment. [CHAP. v. 

and inquire what would generally happen in practice, sup- 
posing 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 in- 
creased 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 inde- 
pendent 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 state- 
ment 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 discus- 
sion 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 



SECT. 5.] Randomness and its scientific treatment 101 

entire accord amongst the various writers as to the assump- 
tions practically to be made in any particular case, and there- 
fore as to the conclusion to be drawn, combined with a very 
considerable amount of difference as to the axioms and defi- 
nitions to be employed. Thus Mr M. W. Crofton, with the 
substantial agreement of Mr Woolhouse, laid it down un- 
hesitatingly that "at random " has "a very clear and definite 
meaning; one which cannot be better conveyed than by Mr 
Wilson's definition, ' .'* to no law'; and in this sense 
alone I mean to use it." According to any scientific inter- 
pretation 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" (vn. 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 
& case of regularity or law. 

It may be remarked that wherever any serious practical 
consequences turn upon duly securing the desired random- 
ness, 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 



102 Randomness and its scientific treatment. [CHAP. v. 

individual occurrence. We might, in our statement of a 
problem, as easily postulate 'a number thought o/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 dis- 
tribution 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 



.SECT. 7.] Randomness and its scientific treatment. 103 

inquire for the chance of its passing through without touch- 
ing 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 ve- 
locity below which the stick may pass through without con- 
tact, 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 in- 
stance ; 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 ft, 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 in- 
finity of possible length. 



104 Randomness and its scientific treatment. [CHAP. V. 

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 con- 
tinual 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 tendecj 
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 com- 
binations 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 



SECT* 8.] Randomness and its scientific treatment. 105 

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 J." 

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 out- 
side 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 under- 
stand 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 



106 Randomness and its scientific treatment. [CHAP. V, 

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 indefi- 
nite 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 in- 
tersected 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 othei 
essential characteristic, viz. that of ultimately uniform dis- 
tribution, will be missing. There will be a special form oi 
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 



SECT. 11.] Randomness and its scientific treatment 107 

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 sup- 
posed 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 ques- 
tion 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 ran- 
dom, 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 ma- 
terials 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- 



108 Randomness and its scientific treatment. [CHAP. v. 

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 ap- 
parent 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 re- 
solvable nebulae, as they present themselves to us, are as ob- 
vious an infraction of such an arrangement as would be the 
occurrence here and there of patches of ground in a rain-fall 
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 
t^^\\ P ri : - ' may be apparent only, arising from the fact that 



SECT. 12.] Randomness and its scientific treatment. 109 

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 ar- 
rangement 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 under- 
stood, 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 



110 Randomness and its scientific treatment. [CHAP. v. 

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 
oards 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 
iar 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 'ran- 
dom' 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 pro- 
duce directly the semblance of randomness 1 , 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 

1 Vide p. 68. 



SECT. 13.] Randomness and its scientific treatment. Ill 

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 arrange- 
ments 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 TT 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 developes 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 con- 
template 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. 



112 Randomness and its scientific treatment. [CHAP. V* 

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 re- 
solution 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 suc- 
cessions of digits as those under consideration. But the 
following remarks may serve to shift the onus of unlikeli- 
hood by suggesting that the preponderance of analogy is- 
rather in favour of the existence. 

Take the well-known constant IT 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 TT to several hundred 
figures, that by the application of continued higher magni- 
fying 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 



SECT. 15.] Randomness and its scientific treatment 113 

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 incom- 
mensurable 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 ex- 
tremities 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 TT, that incommen- 
surability 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 



114 Randomness and its scientific treatment. [CHAP. v. 

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 sug- 
gested 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 instanta- 
neously varying dictates as to its direction of motion. Nor can 
we conceive or picture such a line in its ultimate or ideal con- 
dition. 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* in- 
dicated by these lines and the bisections of the contained 
right angles \ For suggesting the random selection amongst 

1 It would of course be more com- digits ; but this is much more trou- 
plete to take ten alternatives of direc- blesome in practice than to confine 
tion, and thus to omit none of the ourselves to eight. 



SECT. 16.] Randomness and its scientific treatment. 115 

these directions let them be numbered from to 7, and 
let us say that a line measured due * north* shall be de- 
signated 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 to 7 as they present themselves in the calculated 
value of TT. 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 pro- 
posed, 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 ran- 
domness. 

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 com- 
pletely 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 diffi- 
culty ; 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 

82 



116 Randomness and its scientific treatment [CHAP, v, 

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 dis- 
continuous, 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 con- 
ceivable 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 \ 
There is in fact no ' limit ' here, intelligible to the under- 
standing 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 our- 
selves continually approaching, and which therefore we are 
apt to conceive ourselves as ultimately attaining The usual 
assumption therefore which underlies the Newtonian in- 
finitesimal 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 

1 Any more than we picture the shape of an equiangular spiral at the 
centre. 



SECT. 17.] Randomness and its scientific treatment 117 

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 gra- 
phical 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 ad- 
missible 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 ' there- 
fore 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 out- 
come of such a number of tosses is that ultimately the 
proportional approximation to the a 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 dis- 
tance 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 trust- 
worthy, I feel no doubt as to the generally 'random* 



118 



Randomness and its scientific treatment. [CHAP. V, 



character of the rows of figures displayed by the incommen- 
surable 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 to 7 in the value of TT. 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. 



Start 



Finish 




CHAPTER VI 1 . 

THE SUBJECTIVE SIDE OF PROBABILITY. 
MEASUREMENT OF BELIEF. 

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 rounds 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 indivi- 
duals of them will sometimes transgress their licensed irregu- 
larity. Hence they have to be pruned a little into shape, as 

1 Originally written in somewhat deavour to express myself with less 
of a spirit of protest against what emphasis, and I have made altera- 
seemed to me the prevalent dis- tions in that direction. The reader 
position to follow De Morgan in who wishes to see a view not sub- 
taking too subjective a view of the stantially very different from mine, 
science. In reading it through now but expressed with a somewhat oppo- 
I cannot find any single sentence to site emphasis, can refer to Mr F. Y. 
which I could take distinct objection, Edgeworth's article on " The Philoso- 
though I must admit that if I were phy of Chance " (Mind, Vol. ix. ) 
writing it entirely afresh I should en- 



120 Measurement of Belief. [CHAP. vi. 

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 vari- 
ance with the assertion previously made, that Probability is a 
science of inference about real things; it is only by a substi- 
tution of the above kind that we are enabled to reason about 
the things. In nature nearly all phenomena present them- 
selves 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 de- 
flection from accuracy in the formal process of inference. 

It may be remarked also that the adoption of this imagi- 
nary 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 



SECT. 3.] Measurement of Belief 121 

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 pro- 
minently in the case of games of chance we find the alter- 
ations, 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 con- 
stantly come before us a few at a time, or even as indivi- 
duals, and we have to form some opinion about them in this 
state. An insurance office, for instance, deals with num- 
bers 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 proceed- 
ings? for to him this question becomes an individual one. 
And even the office itself receives its cases singly, and would 



122 Measurement of Belief [CHAP. vi. 

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, cer- 
tainty 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, in- 
cluding 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 



SECT. 5.] Measurement of Belief. 

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 con- 
sequences of supposing the former to be absolutely true 2 ". 
In other words, there is a science Formal Logic which in- 
vestigates 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 ac- 
cording 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 re- 
mark 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, 

1 In the ordinary signification of indicated in his title "Formal Logic, 

this term. As De Morgan uses it or the Calculus of Inference, neces- 

he makes Formal Logic include Pro- sary and probable." 

bability, as one of its branches, as 2 Formal Logic. Preface, page v. 



124 Measurement of Belief. [CHAP. vi. 

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 conclu- 
sion 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 pre- 
mises 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 practi- 
cally 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 



SECT, 7.] Measurement of Belief. 125 

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 unim passioned 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 con- 
sideration 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 



126 Measurement of Belief. [CHAP. vi. 

feelings are, of course, exceptional, but we should neverthe- 
less 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 refer- 
ence 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 gene- 
rally 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 emo- 
tional 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 ob- 
jections 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 intro- 
duced into oral or written controversies, we must not con- 
clude 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 dis- 



SECT. 9.] Measurement of Belief. 127 

torted 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 conscious- 
ness 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 ordi- 
nary 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 examina- 
tion 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 mo- 
ment, 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 al- 
most 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 



128 Measurement of Belief . [CHAP. vi. 

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 con- 
viction, 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 super- 
stitious motives, from belief in omens, dreams, and so on. 
But apart from these, and supposing any one to come forti- 
fied 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 appor- 
tioning 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 de- 
preciate the risk in proportion to the contingent advantage. 
They would very likely, when argued with, attach disparag- 
ing epithets to this state of feeling, by calling it an un- 
accountable 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 educa- 
tion and thought. If, however, educated persons have sue- 



SECT. 10.] Measurement of Belief . 129 

ceeded 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 ac- 
cordance 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 in- 
tensity which subsequent experience will justify ? Our 
instincts arc implanted in us for good purposes, send are in- 
tended 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 sub- 
duing 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 
v. 9 



130 Measurement of Belief. [CHAR vi 

a strong argument against the expediency of commencing 
the study of the science from the subjective side, or even o 
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 in- 
duces 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 righi 
value, how would it be possible to get through the practical 
work of life ? Men would be stopping indoors because ii 
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 1 . What is most to our present purpose, it 

1 An illustration of the points been given in a quarter where few 
here insisted on has recently [1876] would have expected it ; I allude, as 



SECT. 11.] Measurement of Belief. 131 

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 

many readers will readily infer, to stance of immortality and the exist- 

J. S. Mill's exceedingly interesting ence of the Deity), may nevertheless 

Essays on Theism. It is not within not only continue to exist in culti* 

our province here to criticise any of vated minds, but may also be pro- 

their conclusions, but they have ex- fitably encouraged there, at any rate 

pressed in a very significant way the in the shape of hopes, for certain 

conviction entertained by him that supposed advantages attendant on 

beliefs which are not justified by their retention, irrespective even of 

evidence, and possibly may not be their truth, 
capable of justification (those for in- 

92 



132 Measurement of Belief. [CHAP. vi. 

express understanding that we do not guarantee its accuracy. 
Our belief in some single events, for example, might be cor- 
rect, and yet that in a compound of several (if derived merely 
from our instinctive laws of belief) very possibly might noi> 
be correct, but might lead us into practical mistakes if we 
determined to act upon it. Even if the two were in accord- 
ance, 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 neces- 
sary 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 sub- 
jective and objective necessities. The two exist in perfect 
harmony side by side, the one being the accurate counter- 
part of the other. So precise is the correspondence between 
them, that few persons would notice, until study of meta- 
physics 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 indiffer- 
ently, 'the thing must be so/ or, 'we cannot conceive its being 



SECT. 13.] Measurement of Belief. 133 

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 conse- 
quence by inveterate association of the objective uniformity, 
or else that this so-called necessity (say in the Law of Con- 
tradiction) 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 possible by a 
momentary lapse of attention, that is of consciousness. 

13. But though this perfect harmony between sub- 
jective 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 riot 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 in- 
stincts 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 



134 Measurement of Belief. [CHAP, vi, 

it was exercised, or would suppose that the former had any- 
thing 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 inad- 
missible in the case of probability. However we might 
contrive to 'think' aright without appeal to facts, we can- 
not believe aright without incessantly checking our pro- 
ceedings by such appeals. Whatever then may be the 
claims of Formal Logic to rank as a separate science, ii> 
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, alterna- 
tives 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 feelinga 
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 par- 
ticulars 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 not 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 



SECT. 15.] Measurement of Belief. 135 

^ypothetically ; 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 ob- 
jective 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 Con- 
ceptualist. 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 sepa- 
rable 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 fnlln\\iii^ example. A man 1 finds himself on the 

1 It is necessary to take an ex- fore there is nothing of the nature 

ample in which the man is forced to of belief to be extracted out of his 

act, or we should not be able to shew mental condition. He very likely 

that he has any belief on the subject would take this ground if we asked 

at all. He may declare that he him, as De Morgan does, with a 

neither knows nor cares anything slightly different reference (Formal 

about the matter, and that there- Logic, p. 183), whether he considers 



136 Measurement of Belief. [CHAP. vi. 

sands of the Wash or Morecambe Bay, in a dense mist, when 
>he spring-tide is coming in ; and knows therefore that to 
be once caught by the tide would be fatal. He hears a 
diurch-bell at a distance, but has no means of knowing 
whether it is on the same side of the water with himself or 
an 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 indis- 
tinguishably 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 

that there are volcanoes on the unseen seem good instances to illustrate the 

side of the moon larger than those position that we always entertain a 

on the side turned towards us; or, certain degree of belief on every 

with Jevons (Principles of Science, question which can be stated, and 

Ed. 11. p. 212) whether he considers that utter inability to give a reason 

that a Platythliptic Coefficient is in favour of either alternative cor- 

positive. These do not therefore responds to half belief. 



SECT. 16.] Measurement of Belief. 137 

been contrived that there shall be no appreciable difference 
between the alternatives, by which to d.ecide in favour of 
one or other, and we accordingly feel no confidence in the 
particular result. Having to decide, however, we decide ac- 
cording 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 re- 
peated 1 , 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 deposi- 
taries of an important state secret, they may decide to part. 
In other words, we have to select here between the two alter- 
natives of the certainty of a single loss, and the even chance 

1 Except indeed on the principles indicated further on in 24, 25. 



138 Measurement of Belief. [CHAP. vi. 

of a double loss ; alternatives which the common mathe- 
matical 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 else- 
where, 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 presump- 
tuous 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 determi- 
nation (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 de- 
serving of examination, seems a mere appendage of the objec- 
tive, 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 enter- 
tain 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 



SECT. 17.] Measurement of Belief. 139 

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 proposi- 
tions which are used in common conversation, let us take but 
one simple example from amongst those which furnish nume- 
rical 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 popula- 
tion 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 propo- 
sitions 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? 



140 Measurement of Belief. [CHAP. vi. 

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 refer- 
ence 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 incident- 
ally 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 ascer- 
taining 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 



SECT. 19.] Measurement of Belief. 141 

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 infer- 
ences 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 singu- 
larly 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 men- 
tioned 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 v 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 con- 
tingent event. Let us take another example, slightly differ- 
ing from the last. A man is to receive 1 if a die gives six, 



142 Measurement of Belief. [CHAP. vi. 

to pay Is. if it gives any other number. It will generally be 
admitted that he ought to give 2s. tid. for the chance, and 
that if he does so he will be paying a fair sum. This ex- 
ample only differs from the last in the fact that instead of 
simple belief in a proposition, we have taken what mathema- 
ticians call 'the value of the expectation'. In other words, 
we have brought into greater prominence, not merely the be- 
lief, 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 em- 
ployment 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 confes- 
sion of such ignorance. 

21. So far then we do not seem to have made the 
slightest approximation to a solution of the particular 



SECT. 21.] Measurement of Belief. 143 

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 yefc 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 dis- 
belief 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 re- 
sult, 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 proposi- 
tion 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 



144 Measurement of Belief. [CHAP. VI. 

readiness to act upon the proposition believed is an insepar- 
able 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 con- 
template 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 per- 
fectly 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 transla- 
tion, as we may call it, into practice, of our belief that head 
and tail occur equally often. 



SECT. 23.] Measurement of Belief. 145 

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 dif- 
ference may seem slight ; but mark the effect when our con- 
duct is translated back again into the subjective condition 
upon which it depends, viz. into our belief. It is in conse- 
quence of such a translation, as it appears to me, that the 
notion has been acquired that we have an accurately deter- 
minable 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 1 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 to- 
wards establishing a theory. But the reader will bear in 
mind that almost all our conduct tends towards the same 
v. 10 



146 Measurement of Belief. [CHAP. vi. 

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 associa- 
tion, 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 



SECT. 24.] Measurement of Belief. 147 

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 can- 
not be viewed as being one of a series. Experience intro- 
duces 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 there- 
fore 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 

102 



148 Measurement of Belief. [CHAP. VL 

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 r 
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 there- 
fore his belief, measured by the result in each individual 
instance, will riot be justified, but the reverse, as shewn 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 accord- 
ance 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 ex- 
istence, 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 con- 
duct 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 



SECT. 25.] Measurement of Belief. 149 

have been giving what is to be regarded as a justification of 
his belief from the point of view of the individual agent him- 
self. If we suppose the existence of an enlarged fellow-feel- 
ing, 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 form that my belief in 
my so doing is represented by the fraction -^^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 contin- 
gencies 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 
<lo 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, 1 should feel a certain propriety in demanding 



150 Measurement of Belief. [CHAP. VL 

7000 in case the office declared that it would only pay in 
the event of my dying on a Sunday. /, 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 sum- 
marised 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 cer- 
tainly their only justification, are to be sought in the series 
of corresponding events to which they belong ; in regard to 
which it may be shewn 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 under- 
stood to denote a certain amount of belief which should be a, 
fraction of certainty. Probably on the principles of associa- 
tion, 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 da 
when it is in company with the rest of the series. 

27. We may now see, more clearly than we could 



SECT. 27.] Measurement of Belief. 151 

before, why it is that we are free from any necessity of as- 
suming 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 for- 
tuitously? 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' expecta- 
tion 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 know- 
ledge 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 1 . 

1 For a fuller discussion of this, see the Chapter on Causation. 



162 Measurement of Belief. [CHAP. vi. 

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 
rf vividness of his conviction, of which no one else can judge, 
ao 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 satisfac- 
tory 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 



SECT. 29.] Measurement of Belief. 153 

)ften the case, to import other sentiments besides pure belief, 
md to find a justification for them also in the results of our 
icience, 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 
jontemplated here: "In applying the doctrine of chances to 
/hat subject in connexion with which it was invented games 
)f chance, the principles of what has been happily termed 
moral arithmetic ' must not be forgotten. Not only would 
t be difficult for a gamester to find an antagonist on terms, 
is to fortune and needs, precisely equal, but also it is im- 
>ossible that with such an equality the advantage of a 
:orisiderable gain should balance the harm of a serious loss. 
If two men,' says Buffon, 'were to determine to play for 
heir whole property, what would be the effect of this agree- 
nent? The one would only double his fortune, and the 
ther reduce his to naught. What proportion is there be- 
ween the loss and the gain? The same that there is between 
11 and nothing. The gain of the one is but a moderate 
um, the loss of the other is numerically infinite, and 
aorally so great that the labour of his whole life may not 
ierhaps suffice to restore his property.' " 

As moral advice this is all very true and good. But if it 
ie regarded as a contribution to the science of the subject it 
3 quite inappropriate, and seems calculated to cause con- 
ision. The doctrine of chances pronounces upon certain 
inds of events in respect of number and magnitude ; it has 
bsolutely nothing to do with any particular person's feelings 
bout these relations. We might as well append a corollary 
D the rules of arithmetic, to point out that although it is 
ery true that twice two are four it does not follow that four 
orses will give twice as much pleasure to the owner as two 
ill. If two men play on equal terms their chances are 



154 Measurement of Belief. [CHAP, vi, 

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 ques- 
tion, 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 out- 
weigh 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 produc- 
tion 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 applica- 
tions 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 



SECT. 30.] Measurement of Belief. 155 

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 



156 Measurement of Belief. [CHAP. vi. 

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 ex- 
cepted, which, from the apparent uniformity and consist- 
ency 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 intel- 
lectual, 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 en- 
tirely merge in one result as to produce what might be 
called an indistinguishable compound. Every kind of infer- 
ence, 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 



SECT. 32.] Measurement of Belief. 157 

his death -bed. The only resource is to check and correct 
his belief by appealing to past and current experience 1 . This 
was advanced as an objection to the theory on which proba- 
bility is regarded as concerned primarily with laws of belief. 
But on the view taken in this Essay in which we are sup- 
posed 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 con- 

1 The best example I can recall usual statistical ground of the ex- 
of the distinction between judging treme rarity of such events. She 
from the subjective and the objec- listened patiently, and then replied, 
tive side, in such cases as these, "Yes, Sir, that is all very well; but 
occurred once in a railway train. I don't see how the real danger will 
I met a timid old lady who was be a bit the less because I don't be- 
in much fear of accidents. I en- lieve in it." 
deavoured to soothe her on the 



158 Measurement of Belief. [CHAP. vi. 

templation 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 oc- 
casionally 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 accompani- 
ment of surprise, to which we are thus able to assign some- 
thing 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 con- 
duct. 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 con- 
tingent 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 uncon- 
sciously 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 discus- 
sion. 

An appeal to common language is always of service, as 



SECT. 33.] Measurement of Belief. 159 

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 uninistakeably to the 
two-fold aspect of the question, the objective and the sub- 
jective, 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 ex- 
traordinary, the rest ordinary, or in the line. So too irre- 
gular ' and ' abnormal ' are doubtless used from the appear- 
ance 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 
wa y : 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 per- 
ceived in nature, the same word was used, whether the cause 
was supposed to be the same or not. In each case there 



160 Measurement of Belief. [CHAP. VL 

was the same generality of agreement, subject to occasional 
deflection 1 . 

On the other hand, observe the words ' wonderful ', ' un- 
expected ', ' incredible '. Their connotation describes states 
of mind simply ; they are of course not confined to Proba- 
bility, 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 de- 
noting 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 recog- 
nition 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 

1 This would still hold of empiri- shifted the word, to denote an ulti- 
cal laws which may be capable of be- mate law which it is supposed cannot 
ing broken : we now have very much be broken. 



SECT. 34.] Measurement of Belief. 161 

side of the question. In the first place, as a mere specu- 
lative inquiry the quantity of our belief of any proposition 
deserves notice. To study it at all deeply would be to tres- 
pass 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 com- 
paring two things, of which one is wholly alien to Proba- 
bility, 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 
v. 11 



162 Measurement of Belief. [CHAP. vi. 

said, that our belief of many events rests upon a very com- 
plicated 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 rnind, 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 enter- 
tain 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 accu- 
rate 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 



SECT. 35.] Measurement of Belief. 163 

terms presuppose a series ; within the indefinitely numerous 
class which composes this series a smaller class is distin- 
guished by the presence or absence of some attribute or 
attributes, as was fully illustrated and explained in a pre- 
vious 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 
phra-c"l<i^v, 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 repre- 
sents the proportion between the two different classes in the 
long rim. 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 eighty is the numerical fraction -fa. 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 termi- 
nable, 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 in- 
variably is done, whenever natural objects are made subjects 
of strict science. 

112 



164 Measurement of Belief. [CHAP. vi. 

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 con- 
ditions under which the events occur, as was explained in 
the fourth chapter. 

Here becomes apparent the full importance of the dis- 
tinction 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 



SECT. 37.] Measurement of Belief. 165 

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 ad- 
mitted of any exact apportionment like the numerical one 
just mentioned. Still, it was shown that a reasonable expla- 
nation could be given of such an expression as, * my belief is 
T ^th of certainty', though it was an explanation which pointed 
unmistakeably to a series of events, and ceased to be intel- 
ligible, or at any rate justifiable, when it was not viewed in 
such a relation to a series. In so far, then, as this expla- 
nation is adopted, we may say that our belief is in pro- 
portion 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 com- 
monly 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 



166 Measurement of Belief. [CHAP. vi. 

form, the amount of truth which appears to me to be con- 
tained 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. 



CHAPTER VII. 

THE 1WLES 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 infer- 
ence being very few and simple, their consideration will not 
detain us long. From the data now in our possession we are 



168 The Rules of Inference in Probability. [CHAP. vn. 

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 hypo- 
theses, or demand for their establishment continually re- 
newed appeals to experience, and extension by the aid of the 
various resources of Induction. We shall confine our atten- 
tion at present principally to the former class ; the latter can 
only be fully understood when we have considered the con- 
nection 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 

1 It might be more accurate to or 'mutually exclusive classes of 
speak of 'incompatible hypotheses events', 
with respect to any individual case', 



SECT. 2.] The Rules of Inference in Probability. 169 

in 100 live to over sixty, 35.4 in 100 die before they are 
ten 1 ; 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 ex- 
pressed in the \\\ .: .ro of Probability as follows: If the 
chances of two exclusive or incompatible events be re- 
spectively and - the chance of one or other of them 
r J m n 

happening will be h - or . Similarly if there were 

rr n m n mn J 

more than two events of the kind in question. On the prin- 
ciples 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 con- 
clusion of the last chapter that to say, for example, that the 

chance of a given event happening in a certain way is ^ , 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 

1 The examples, of this kind, re- high authority of De Morgan for re- 
ferring to human mortality are taken garding them as the best representa- 
from the Carlisle tables. These tive of the average mortality of the 
differ considerably, as is well known, English middle classes at the present 
from other tables, but we have the day. 



170 The Rules of Inference in P ; "' . [CHAP. vn. 

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 and 

r m 

the chance of one alone be -, the chance of the remaining 

. n , 1 1 n m 

one will be or . 

m n nm 

For example, if the chance of any one dying in a year is 
^ > and his chance of dying of some particular disease is ^yr-r , 

9 
his chance of dying of any other disease is . 

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 conven- 
tionally regarded as the same. 

We may also here point out the justification for the com- 
mon doctrine that certainty is represented by unity, just as 
any given degree of probability is represented by its appro- 
priate fraction. If the statement that an event happens once 
in m times, is equivalently expressed by saying that its chance 

is , it follows that to say that it happens m times in m 
m J 

times, or every time without exception, is equivalent to 

Yfl 

saying that its chance is or 1. Now an event that happens 
every time is of course one of whose occurrence we are 



SECT, 3.] The Rules of Inference in Probability. 171 

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 , the chance that it will not happen 

. - 1 m-l 
is 1 or - 

m m 

3. (2) We can also make inferences by multiplication 
or division. Suppose that the 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 multi- 
plying 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 popula- 
tion 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 , and the chance that 

m 

if it happens another will also happen is , then the chance 



172 The Rules of Inference in Probability. [CHAR vn. 

of the latter is . In this case it is assumed that the latter 
mn 

is so entirely dependent upon the former that though it does 
not always happen with it, it certainly will not happen with- 
out 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 them- 
selves 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 hypo- 
thesis. 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. 



SECT, 5.] The Rules of Inference in Probability. 173 

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 1 in almost any other propor- 
tion 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 

1 I say, almost any proportion, be- men are short-sighted, for in any 

cause, as may easily be seen, arith- given batch of men the former are 

metic imposes certain restrictions more numerous. But the range of 

upon the assumptions that can be these restrictions is limited, and 

made. We could not, for instance, their existence is not of importance 

suppose that all the black-haired in the above discussion. 



174 The Rules of Inference in Probability. [CHAP. VII. 

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 and - , then the 

.1 n 1 
chance of its being both p and q is , p and not q is , 

r J mn r * mn 

, ^ . m 1 , , , . (m 1) (n 1) 

q and not p is - , not p and not q is -, 

* mn mn 

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 be- 
tween the inference in this case, and that in the two preced- 
ing, to an extent which to many readers may seem unwar- 
ranted. 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 proba- 
bilities ' 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 sub- 
ject ; in the case of inferior writers, or those who appeal to 



SECT. 7.] The Rules of Inference in Probability. 175 

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 dis- 
tributed 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 d '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 respec- 
tively of each of which, amongst the individuals of a series, 
two distinct pairs of classes of these individuals are pro- 
duced. 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 



176 The Rules of Inference in Probability. [CHAP. VIL 

inference in probability, the question at once arises, What is 
their relation to the great body of formulae which are made 
use of in treatises upon the science, and in practical applica- 
tions of it ? The reply would be that these formulae, 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 some- 
times be made to the profoundest theorems of mathematics. 
Still we ought not to regard these theorems as being any- 
thing 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 cer- 
tain 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 popu- 
lar 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 fre- 
quently 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 in \r>iig!i lion-, 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 ex- 
pressed in the more familiar form of a fraction representing 



SECT. 8.] The Rules of Inference in TVA/iW/iVj/ 177 

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 pro- 
duces 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 conclu- 
sion 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 pro- 
position. Suppose, for example, that one person in 100 dies 
on the sea passage out to India, and that one in 9 dies dur- 
ing a 5 years residence there. It would commonly be said 
that the chance that any one, who is now going out, has of 
living to start homewards 5 years hence, is $& ; for his chance 
of getting there is y 9 ^; and of his surviving, if he gets 
there, f ; hence the result or dependent event is got by 
multiplying these fractions together, which gives ^j. 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 
v. 12 



178 The Rules of Inference in Probability. [CHAP. vii. 

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 ex- 
ample, 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 statis- 
tics, or that, even supposing we could do so, we should 
have any guarantee of the correctness of the result with- 
out similar appeal. It appears to me in fact that but little 
in< -n hint; and certainly no security, can be attained by so 
regarding the process of inference. The probabilities ex- 
pressed 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 in- 
ferences 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 in- 
ference about things as has been described above, and it 
therefore seems to me more advisable to state and expound 
it in this latter form. 



SECT. 9.] Inverse Probability. 179 



On Inverse Probability and the Rules required for it. 

9. It has been already stated that the only funda- 
mental 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 recog- 
nised 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 circum- 
stances under which the same might have happened 1 /' 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 

1 Essay on Probabilities, p. 53. I padia Metropolitana he has stated 
have been reminded that in his arti- that such rules involve no new prin- 
cle on Probability in the Encyclo- cipie. 

122 



180 Inverse Probability. [CHAP. vil. 

inference can be immediately derived. All other distinctions 
are merely those of arrangement or management. 

But although the distinction is not by any means fun- 
damental, 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 prob- 
lems 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 re- 
placed, the proportion of white ones to the whole number 
drawn will tend towards the fraction -fa. 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 (white) is tjien singled out by anticipa- 
tion. By stating that its chance is fa we merely mean to 
assert this physical fact, together with such other mental 



SECT. 11.] Inverse Probability. 181 

facts, emotions, inferences, &c., as may be properly associ- 
ated 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 contain- 
ing 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 1 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 simi- 
larly 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 accu- 
rately 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 A priori, we take care to secure our prospect of a 

1 This point will be fully discussed of logic has been explained and illus- 
in a future chapter, after the general trated. 
stand- point of an objective system 



182 Inverse Probability. [CHAP. vn. 

series of the desired kind. I^ut we cannot say the same in 
the case of the second example. 

12. The line of proof by which it is generally at- 
tempted 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 ulti- 
mately found to arrange themselves in the series. Now this 
requires an experience about bags which may not, and in- 
deed 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 



SECT. 13.] Inverse Probability. 183 

having been drawn from a bag which only contained one of 
these white balls. Of these there is, out of the 55 drawings, 
but one. Accordingly the required chance is ^. That is to 
say, the white ball will have been drawn from the bag con- 
taining 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 exam- 
ple ; 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. Tt 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 excel- 
lent manual 1 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 ex- 
treme 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 fol- 
lows 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 fact, becomes now ^ instead 
of -fa. In the one case each distinct result is considered 

1 Whitworth's Choice and Chance, Ed. n., p. 123. See also Boole's 
Laws of Thought, p. 370. 



184 Inverse Probability. [CHAP. vn. 

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 rela- 
tive 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 infor- 
mation as this we are entirely in the dark, and the supposi- 
tion made in 12 is neither more nor less trustworthy and 
reasonable than many others, though it doubtless possesses 
the merit of superior simplicity 1 . 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 intro- 
duced 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 



1 Opinions differ about the defence doubtful, call it tbe most impartial 

of such suppositions, as they do about hypothesis. Others regard it as a 

the nature of them. Some writers, sort of mean hypothesis, 
admitting the above assumption to be 



SECT. 15.] Inverse Probability. 185 

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 refer- 
ence 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 coexistences, 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 na- 
tural causes are introduced into the enquiry. 

For instance, it is known that in London about 20 people 



186 Inverse P- '.; y ..'V\: [CHAP. vn. 

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. Ail 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 toler- 
ably 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 Pro- 
bability must be abandoned. When the appropriate statis- 
tics are at hand the two classes of problems become identical 
in method of treatment, and when they are nob we have no 
more right to extract a solution in one case than in the other. 
The discussion however may serve to direct renewed atten- 
tion 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 



SECT. 17.] Inverse PnAabllity. 187 

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 ?*". 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 expe- 
rience, 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 arid 
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 a priori 
likelihood of each kind, but I tfcink 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 assump- 

1 Educational Times ; Reprint, Vol. kind hardly any two of the writers 

xxxvii. p. 40. The question was were in agreement as to the assump- 

proposed by Dr Macalister and gave tions to be made, or therefore as to 

rise to considerable controversy. As the numerical estimate of the odds, 
usual with problems of this inverse 



188 Inverse P/'v/M/Wv. [CHAP. vn. 

tions 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 submit- 
ted to this Inverse Probability, we have not far to seek. In 
natural as in artificial examples, where statistics are unattain- 
able 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 1 , whether such and such groups of stars are or are 
not to be regarded as the results of an accidental distribu- 
tion ; or the still wider and vaguer question, whether such and 
such things, or say the world itself, have been produced by 
ohance ? 

In cases of this kind the insuperable difficulty is in deter- 
mining 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 inii r r ii!:ii<lr. or infinite 

1 See Todhunter's History, pp. Forbes in a paper in the Philosophi- 

333, 4. cat Magazine for Deo. 1850. It was 

There is an interesting discussion replied to in a subsequent number by 

upon this question by the late J. D. Prof. Donkin. 



SECT. 18.] Inverse Probability. 189 

power, one can to some extent comprehend, or at least one 
may understand what is being talked about, but 'the infi- 
nite ' 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. 



CHAPTER VIII 

THE RULE OF SUCCESSION 1 . 

1. IN the last chapter we discussed at some length the 
nature of the kinds of inference in Probability which corre- 
spond to those termed, in Logic, immediate and mediate infer- 
ences. We ascertained what was the meaning of saying, for 
example, that the chance of any given man A. B. dying 
in a year is , 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 ques- 
tion, 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 

1 A word of apology may be offered one of Induction. But such a title I 

here for the introduction of a new cannot admit, for reasons which will 

name. The only other alternative be almost immediately explained, 
would have been to entitle the rule 



SECT. 2.] The Rule of Succession. 191 

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 ad- 
jacent 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 



2. The origin of such confusion is easy of explana- 
tion ; 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 sus- 
ceptible. 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 verifi- 
cation, 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 



192 The Rule of Succession. [CHAP. vm. 

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 pros- 
pect 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; 

1 John Craig, in his often named problems as : Quando evanescet pro- 
work, Theoloffice Christiana Prin- babilitas cujusvis Historise, cujus sub- 
cipia Mathematica (Lond. 1699) at- jectum eat transiens, viva tantum 
tempted something in this direction voce transmissae, determinare. 
when he proposed to solve such 



SECT. 3.] The Rule of Succession. 193 

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 ratio of 10 to 9, or, in 
other words, is measured by the fraction -f$. 

(II) But are we certain that nine men out of ten like 
him will live another year ? we know that they have so sur- 
vived 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 con- 
tinue 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 d 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.) 
v. 13 



194 The Rule of Succession. [CHAP. viti. 

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 con- 
sidering them as rules of probability. Even then they 
would be rules practically of a very different kind, contin- 
gent instead of necessary (if one may use these terms with- 
out committing oneself to any philosophical system), but this 
objection might perhaps be overruled by the greater simpli- 
city secured by classing them together. This view is, with 
various modifications, generally adopted by writers on Pro- 
bability, 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 depend- 
ing 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 



SECT. 6.] The Rule of Succession. 195 

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 pro- 
cess 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 calcula- 
tion 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 induc- 
tion 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 

132 



196 The Rule of Succession. [CHAP. VIIL 

leaving the ground of past experience, but takes the conse- 
quences of this divergence as the express subject of its calcu- 
lation. 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 formulae 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 



SECT. 8.] The Rule of Succession. 197 

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 
ra+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 formulae to the estimation of the 
evidence of witnesses. If however its validity is to be dis- 
cussed 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 pro- 
fesses 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 



198 The Rule of Succession. [CHAP. viu. 

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 *, 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, &a 
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 

1 When w=l the fraction becomes seen before, how many of the ob- 

J; i.e. the odds are 2 to 1 in favour served characteristics of that single 

of recurrence. And there are writers ' event ' are to be considered essential ? 

who accept this result. For instance, Must the pilot precede ; and at the 

Jevons (Principles of Science p. 258) same distance? Must we consider 

says "Thus on the first occasion on the latitude, the ocean, the season, 

which a person sees a shark, and the species of shark, as matter also 

notices that it is accompanied by a of repetition on the next occasion? 

little pilot fish, the odds are 2 to 1 and so on. I cannot see how the 

that the next shark will be so accom- Inductive problem can be even in- 

panied." To say nothing of the fact telligibly stated, for quantitative 

that recognizing and naming the fish purposes, on the first occurrence of 

implies that they have often been any event. 



SECT. 9.] The Rule of Succession. 199 

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 psycho-physical 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 1 . 

1 See in Mind (x. 454) Mr Jacob's Ebbinghaus as described in his work 
account of the researches of Herr Ueber das Gedachtniss. 



200 The Rule of Succession. [CHAP. vm. 

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 our- 
selves 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 
1 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 d priori considera- 
tions. 

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 



SECT, 10.] The Rule of Succession. 201 

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 im- 
portance 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 con- 
victions ; 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 re- 
ceived. For instance, Laplace, though unhesitatingly adopt- 
ing 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 im- 
possible to base either upon the other. However far back 
we try to push our way, and however disposed we might be 



202 The Rule of Succession. [CHAP, vni. 

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. 



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 cha- 
racter. 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 chap- 
ters. 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 Pro- 
bability and Induction. We shall find it advisable for clear- 
ness 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) discover- 



204 , Induction. [CHAP. ix. 

ing 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 investi- 
gations 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 legi- 
timate for the purpose of calling attention to the require- 
ments of such a system of Logic, and is indeed nothing more 



SECT. 4.] Induction. 205 

than what has to be done at almost every step in psycho- 
logical enquiry 1 . 

3. His task at first might be conceived to be a slow 
and tedious one. It would consist of a gradual accumula- 
tion 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 pro- 
cesses 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 sug- 
gest the possibility of a wide extension of his infonnation. 
Experience itself would soon detect that events were con- 
nected together in a regular way; he would ascertain that 
there are 'laws of nature/ Coming with no a priori neces- 
sity of believing in them, he would soon find that as a matter 

1 Some of my readers may be fa- his experience to gain, and speculates 
miliar with a very striking digression on the gradual acquisition of his 
in Buflon's Natural History (Natural knowledge. Whatever may be thought 
Hist, of Man, vni.), in which he of his particular conclusions the pas- 
supposes the first man in full pos- sage is very interesting and sugges- 
session of his faculties, but with all tive to any student of Psychology. 



206 Induction. [CHAP. IX. 

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 pro- 
ceedings, 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 pos- 
sibly 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 exten- 
sive. 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 indivi- 
dual 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 



SECT. 6.] Induction. 207 

certainty, but with a degree of conviction that is of the 
utmost practical use. We have gained the great step of 
beii$gr&ble to make trustworthy generalizations. We con- 
clude; 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 ordi- 
nary logic) determines what conclusions can be drawn from 
this generalization. 

6. We may now see our way to ascertaining the pro- 
vince of Probability and its relation to kindred sciences. 
Inductive Logic gives rules for discovering such generaliza- 
tions as those spoken of above, and for testing their correct- 
ness. 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 



208 Induction. [CHAP. ix. 

out of ten die in their first four years. It belongs to Induc- 
tion to say whether we are justified in generalizing our obser- 
vation 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 induc- 
tive 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 en- 
tirely distinct that they cannot possibly be harmonized into 
a single consistent system. 

The opinion therefore according to which certain In- 
ductive formulae are regarded as composing a portion of 
Probability, and which finds utterance in the Rule of Suc- 
cession 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 



SECT. 7.] Induction. 209 

view has of course been entertained. They are mostly dis- 
posed 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 Ana- 
logy, 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 driffc of the context, indeed, this seems a fair inter- 
pretation 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 con- 
vinced that it is impossible to expound the methods of in- 
duction 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 1 ." 

1 See also Dugald Stewart (Ed. by Hamilton; vn. pp. 115119). 
v. 14 



210 Induction. [CHAP. ix. 

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 justifica- 
tion 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 



SECT. 10.] Induction. 211 

them. But this, as we have repeatedly insisted, does not 
belong to the province of logical, but to that of Psycholo- 
gical 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 confi- 
dence 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 con- 
clusions 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 1 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 unob- 
served, 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 

1 Required that is for purposes of versal prevalence, or its all-import- 
logical inference within the limits of ance for scientific purposes. The 
Probability; it is not intended to im- subject is more fully discussed in a 
ply any doubts as to its actual uni- future chapter. 

142 



212 Induction. [CHAP. IX. 

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 con- 
sequents. 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 tieed 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 ob- 
tained. 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 find- 
ing out the appropriate series. How are we to set about the 



SECT. 12.] Induction. 213 

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 ob- 
taining 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 re- 
membered 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 attri- 
bute which it possesses, it becomes a member of a class 
intersecting, so to say, some of the other classes. John Smith 



214 Induction. [CHAP. ix. 

is a consumptive man say, and a native of a northern climate. 
Being a man he is of course included in the class of ver- 
tebrates, also in that of animals, as well as in any higher 
such classes that there may be. The property of being con- 
sumptive 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 be- 
comes 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 ex- 
traneous 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 attri- 



SECT. 14] Induction. 215 

butes, 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 1 
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 

1 As particular propositions they corresponds to a larger proportion 
are both of course identical in form. than in the latter, is a distinction 
The fact that the ' some ' in the former alien to pure Logic. 



216 Induction. [CHAP. ix. 

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 dis- 
connected with each other. The various classes above 
referred to are a set of such middle terms, and the proposi- 
tions 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 correspond- 
ing 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 conclu- 
sion 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 har- 
mony; they have very different predicates, and may seem 
quite out of each other's reach for either support or opposi- 
tion. But by means of the other premise they are in each 
case brought into relation with one another, and the general 



SECT. 16.] Induction. 217 

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 con- 
tradictory assumptions about matters of fact. 

1 6. 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 pro- 
portional 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 con- 
trast 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 



218 Induction. [CHAP. ix. 

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 pro- 
portional 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 aggra- 
vated 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 sta- 
tistics 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 unattain- 
able, owing to the absence of suitable inductions, we felt 
justified (with the explanation, and under the restrictions 
mentioned in Chap, vr.), 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 ? 



SECT. 19.] Induction. 219 

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 circum- 
stances 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 *: -V- .:. 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 need- 
less exaggeration of the relative number and magnitude of 
our errors. 

19. The above answer is quite valid ; but whilst cau- 
tioning 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 



220 Induction. [CHAP. ix. 

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 unsu.it- 
ability 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 



SECT. 21.] Induction. 221 

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 con- 
sumptive 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 scries. So with in- 
surance 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 im- 
portance 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 foi 
him from any class but that of men in general of his age in 
the country. 



222 Induction. [CHAP. IX. 

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 arbi- 
trary. 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 sta- 
tistics 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 sta- 



SECT. 23.] Induction. 223 

tistical 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 suppo- 
sition in no degree contradictory. Without further data, 
therefore, we can come to no decision. 

23. Practically, of course, if we were forced to a deci- 
sion 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 our^ 
selves 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 



224 Induction. [CHAP. ix. 

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 in- 
stances, 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 



SECT. 25.] Induction. 225 

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 un- 
known 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 cal- 
culation 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 slightei 
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 dif- 
ferent branches of human conduct. It is nothing else thai 
v. 15 



226 Induction. [CHAP. ix. 

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 parti- 
cular 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 ap- 
propriate 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 circum- 
statices, 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 assump- 
tion that the publication of these tables made no such altera- 
tion 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 



SECT. 26.] Induction. 227 

not in any way contradictory, but they need some explana- 
tion. 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, consti- 
tution, 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 mean- 
ing 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 homo- 
geneous this is not needed, but when it is heterogeneous the 

152 



228 Induction. [CHAP. ix. 

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 pay- 
ing 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 no- 
thing to prevent the English consumptives (some or all) from 
insuring in each category and paying its appropriate pre- 
mium. 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 



SECT. 28.] Induction. 229 

of statistics. The English mortality in Madeira would in- 
stantly become heavier, so far as the Insurance company was 
concerned, by the loss of all their best lives ; whilst the con- 
sumptive statistics (unless all the English consumptives had 
already been taken for insurance) would be in the same way 
deteriorated 1 . 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, how- 
ever, 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 recon- 
cile 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 con- 

1 The reason is obvious. The whereas the worst consumptive lives 

healthiest English lives in Madeira there (viz. the English) are now in- 

(viz. the consumptive ones) have now creased in relative numbers, 
ceased to be reckoned as English; 



230 Induction. [CHAP. ix. 

elude, 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 indi- 
vidual 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 differ- 
ent 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 



SECT. 30.] Induction. 231 

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 fre- 
quency, 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 re- 
peated error ; but even when measured by its own stand- 
ard 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 widen- 
ing 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 speci- 
ficness 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 



232 Induction. [CHAP. IX. 

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 him- 
self 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 con- 
stitutions. A man with such a constitution, but suffering 
from such a disease, presents himself; which recommenda- 
tion 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 neces- 
sary that numerical data should be quoted and appealed 
to ; it is sufficient that the judgment is more or less con- 
sciously 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 employ- 
ment 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 com- 
mences. At this early stage, however, Probability and In- 
duction are so closely connected together as to be very apt to 



SECT. 32.] Induction. 233 

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 vn.) that corresponding to syllogistic infer- 
ences. Our present position therefore is that in which we 
may consider ourselves in possession of any number of gene- 
ralizations, 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 ambi- 
guity 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 multi- 
plicity of middle terms, therefore, is what ought to be 
expected. We can know whether a succession of men, resi- 
dents 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 



234 Induction. [CHAP. ix. 

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 insuffi- 
cient statistics. 



CHAPTEE 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 Pro- 
bability ; 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 adminis- 
trative agency, we should presumably be guilty of some form 
of Manicheism. But the only rational meaning of the ob- 



236 Chance, Causation, and Design. [CHAP. x. 

jection 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 Pro- 
bability 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 some- 
what differently stated, and appears finally to reduce itself 
to this ; that the assumptions upon which the science of 
Probability rests, are riot 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 con- 
nection 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 ProlwliUti/, com- 
mences 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 



SECT. 2.] Chance, Causation, and Design. 237 

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 refer- 
ence 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 con- 
troversy 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 acqui- 
escence 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 

1 Essay on Probabilities, p. 114. 



238 Chance, Causation, and Design. [CHAP. x. 

this instance as more remarkable than many analogous ones 
taken from other quarters. 

That there is a certain foundation of truth in the reason- 
ings 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 indivi- 
dual 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 



SECT. 4.] Chance, Causation, and Design. 239 

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 sug- 
gests 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 



240 Chance, Causation, and Design. [CHAP. X. 

of necessary sequences in the various phenomena which com- 
pose 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 es- 
sentially 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 feel any 
hesitation upon this point. Some idea has already been 
gained, in the earlier chapters, of the nature and amount 
of the evidence which might be furnished of this fact, and 
any quantity more might be supplied from the works of 
professed writers upon the subject. If, therefore, Free-will be 
so interpreted as to imply such essential irregularity as defies 
prediction both in the average, and also in the single case, 
then the negation of free-will follows, not as a remote logical 
consequence, but as an obvious inference from indisputable 
facts of experience. 

Few persons, however, would go so far as to interpret it 
in this sense. All that troubles them is the fear that some- 
how this general regularity may be found to carry with it 
causation, certainly in the sense of regular invariable se- 
quence, and probably also with the further association of 
compulsion. Rejecting the latter association as utterly 
unphilosophical, I cannot even see that the former conse- 



SECT. 6.] Chance, Causation, and Design. 241 

quence can be admitted as really proved, though it doubtless 
gains some confirmation from this source. 

6. The nature of the argument against free-will, 
drawn from statistics, at least in the form in which it is very 
commonly expressed, seems to me exceedingly defective. 
The antecedents and consequents, in the case of our voli- 
tions, must clearly be supposed to be very nearly immediately 
in succession, if anything approaching to causation is to be 
established : whereas in statistical enquiries the data are 
often widely separate, if indeed they do not apply merely 
to single groups of actions or results. For instance, in the 
case of the misdirected letters, what it is attempted to prove 
is that each writer was so much the * victim of circumstances ' 
(to use a common but misleading expression) that he could 
not have done otherwise than he did under his circumstances. 
But really no accumulation of figures to prove that the 
number of such letters remains the same year by year, can 
have much bearing upon this doctrine, even though they 
were accompanied by corresponding figures which should 
connect the forgetfulness thus indicated with some other 
characteristics in the writers. So with the number of 
suicides. If 250 people do, or lately did, annually put an 
end to themselves in London, the fact, as it thus stands by 
itself, may be one of importance to the philanthropist and 
statesman, but it needs bringing into much closer relation 
with psychological elements if it is to convince us that the 
actions of men are always instances of inflexible order. In 
fact, instead of having secured our A and B here in closest 
intimacy of succession to one another, to employ the sym- 
bolic notation commonly used in works on Inductive Logic 
to illustrate the causal connection, we find them sepa- 
rated by a considerable interval ; often indeed we merely 
have an A or a B by itself. 

v. 16 



242 Chance, Causation, and Design. [CHAP. x. 

7, Again, another deficiency in such reasoning seems 
to be the laying undue weight upon the mere regularity or 
persistency of the statistics. These may lead to very im- 
portant results, but they are not exactly what is wanted 
for the purpose of proving anything against the freedom 
of the will ; it is not indeed easy to see what connection 
this has with such facts as that the annual number of thefts 
or of suicides remains at pretty nearly the same figure. 
Statistical uniformity seems to me to establish nothing else, 
at least directly, in the case of human actions, than it does 
in that of physical characteristics. Take but one instance, 
that of the misdirected letters. We were already aware 
that the height, weight, chest measurement, and so on, of 
a large number of persons preserved a tolerably regular 
average amidst innumerable deflections, and we were pre- 
pared by analogy to anticipate the same regularity in their 
mental characteristics. All that we gain, by counting the 
numbers of letters which are posted without addresses, is 
a certain amount of direct evidence that this is the case. 
Just as observations of the former kind had already shown 
that statistics of the strength and stature of the human 
body grouped themselves about a mean, so do those of the 
latter that a similar state of things prevails in respect of the 
readiness and general trustworthiness of the memory. The 
evidence is not so direct and conclusive in the latter case, 
for the memory is not singled out and subjected to measure- 
ment by itself, but is taken in combination with innumerable 
other influencing circumstances. Still there can be little 
doubt that the statistics tell on the whole in this direction, 
and that by duly varying and extending them they may 
obtain considerable probative force. 

The fact is that Probability has nothing more to do with 
Natural Theology, either in its favour or against it, than the 



SECT. 8.] Chance, Causation, and Design. 243 

general principles of Logic or Induction have. It is simply a 
body of rules for drawing inferences about classes of events 
which are distinguished by a certain quality. The believer 
in a Deity will, by the study of nature, be led to form an 
opinion about His works, and so to a certain extent about 
His attributes. But it is surely unreasonable to propose 
that he should abandon his belief because the sequence of 
events, not, observe, their general tendency towards happi- 
ness or misery, good or evil, is brought about in a way 
different from what he had expected ; whether it be by dis- 
playing order where he had expected irregularity, or by 
involving the machinery of secondary causes where he had 
expected immediate agency. 

8. It is both amusing and instructive to consider 
what very different feelings might have been excited in our 
minds by this co-existence of, what may be called, ignorance 
of individuals and knowledge of a^ir^r-, . if they had pre- 
sented themselves to our observation in a reverse order. 
Being utterly unable to make assured predictions about a 
single life, or the conduct of individuals, people are some- 
times startled, and occasionally even dismayed, at the un- 
expected discovery that such predictions can be confidently 
made when we are speaking of large numbers. And so 
some are prompted to exclaim, This is denying Providence ! 
it is utter Fatalism ! But let us assume, for a moment, that 
our familiarity with the subject had been experienced, in the 
first instance, in reference to the aggregates instead of the 
individual lives. It is difficult, perhaps, to carry out such a 
supposition completely ; though we may readily conceive 
something approaching to it in the case of an ignorant clerk 
in a Life Assurance Office, who had never thought of life, ex- 
cept as having such a * value ' at such an age, and who had 
hardly estimated it except in the form of averages. Might 



244 Chance, Carnation, and Design. [CHAP. x. 

we not suppose him, in some moment of reflectiveness, being 
astonished and dismayed at the sudden realization of the 
utter uncertainty in which the single life is involved ? And 
might not his exclamation in turn be, Why this is denying 
Providence ! It is utter chaos and chance ! A belief in a 
Creator and Administrator of the world is not confined to 
any particular assumption about the nature of the immediate 
sequence of events, but those who have been accustomed 
hitherto to regard the events under one of the aspects above 
referred to, will often for a time feel at a loss how to connect 
them with the other. 

9. So far we have been touching on a very general 
question ; viz. the relation of the fundamental postulates of 
Probability to the conception of Order or Uniformity in the 
world, physical or moral. The difficulties which thence arise 
are mainly theological, metaphysical or i-\ 1 --j: V What 
we must now consider are problems of a more detailed or 
logical character. They are prominently these two ; (1) the 
distinction between chance arrangement and causal arrange- 
ment in physical phenomena; and (2) the distinction be- 
tween chance arrangement and designed arrangement where 
we are supposed to be contemplating rational agency as 
acting on one side at least. 

II. The first of these questions raises the antithesis- 
between chance and causation, not as a general characteristic 
pervading all phenomena, but in reference to some specified 
occurrence : Is this a case of chance or not ? The most 
strenuous supporters of the universal prevalence of causation 
and order admit that the question is a relevant one, and 
they must therefore be supposed to have some rule for 
testing the answers to it. 

Suppose, for instance, a man is seized with a fit in a 
house where he has gone to dine, and dies there ; and some 



SECT. 10.] Chance, Causation, and Design. 245 

one remarks that that was the very house in which he was 
born. We begin to wonder if this was an odd coincidence 
and nothing more. But if our informant goes on to. tell us 
that the house was an old family one, and was occupied by 
the brother of the deceased, we should feel at once that 
these facts put the matter in a rather different light. Or 
again, as Cournot suggests, if we hear that two brothers 
have been killed in battle on the same day, it makes a great 
difference in our estimation of the case whether they were 
killed fighting in the same engagement or whether one fell 
in the north of France and the other in the south. The 
latter we should at once class with mere coincidences, whereas 
the former might admit of explanation. 

10. The problem, as thus conceived, seems to be one 
rather of Inductive Logic than of Probability, because there 
is not the slightest attempt to calculate chances. But it 
deserves some notice here. Of course no accurate thinker 
who was under the sway of modern physical notions would 
for a moment doubt that each of the two elements in question 
had its own ' cause ' behind it, from which (assuming perfect 
knowledge) it might have been confidently inferred. No 
more would he doubt, I apprehend, that if we could take a 
sufficiently minute and comprehensive view, and penetrate 
sufficiently far back into the past, we should reach a stage at 
which (again assuming perfect knowledge) the coexistence of 
the two events could equally have been foreseen. The 
employment of the word casual therefore does not imply any 
rejection of a cause ; but it does nevertheless correspond to a 
distinction of some practical importance. We call a coinci- 
dence casual, I apprehend, when we mean to imply that no 
knowledge of one of the two elements, which we can suppose 
to be practically attainable, would enable us to expect the 
other. We know of no generalization which covers them 



246 Chance, Causation, and Design. [CHAP. X. 

both, except of course such as are taken for granted to be 
inoperative. In such an application it seems that the word 
1 casual ' is not used in antithesis to ' causal ' or to 'designed', 
but rather to that broader conception of order or regularity 
to which I should apply the term Uniformity. The casual 
coincidence is one which we cannot bring under any special 
generalization; certain, probable, or even plausible. 

A slightly different way of expressing this distinction is 
to regard these * mere coincidences ' as being simply cases in 
point of independent events, in the sense in which in- 
dependence was described in a former chapter. We saw 
that any two events, A and B, were so described when each 
happens with precisely the same relative statistical frequency 
whether the other happens or not. This state of things 
seems to hold good of the successions of heads and tails in 
tossing coins, as in that of male and female births in a 
town, or that of the digits in many mathematical tables. 
Thus we suppose that when men are picked up in the street 
and taken into a house to die, there will not be in the long 
run any preferential selection for or against the house in 
which they were born. And all that we necessarily mean to 
claim when we deny of such an occurrence, in any particular 
case, that it is a mere coincidence, is that that particular 
case must be taken out of the common list and transferred 
to one in which there is some such preferential selection. 

11. III. The next problem is a somewhat more in- 
tricate one, and will therefore require rather careful sub- 
division. It involves the antithesis between Chance and 
Design. That is, we are not now (as in the preceding case) 
considering objects in their physical aspect alone, and taking 
account only of the relative frequency of their coexistence or 
sequence ; but we are considering the agency by which they 
are produced, and we are enquiring whether that agency 



SECT. 11.] Ghance y Causation, and Design. 247 

trusted to what we call chance, or whether it employed what 
we call design. 

The reader must clearly understand that we are not now 
discussing the mere question of fact whether a certain 
assigned arrangement is what we call a chance one. This, 
as was fully pointed out in the fourth chapter, can be settled 
by mere inspection, provided the materials are extensive 
enough. What we are now proposing to do is to carry on 
the enquiry from the point at which we then had to leave it 
off, by solving the question, Given a certain arrangement, is 
it more likely that this was produced by design, or by some 
of the methods commonly called chance methods ? The dis- 
tinction will be obvious if we revert to the succession of 
figures which constitute the ratio TT. As I have said, this 
arrangement, regarded as a mere succession of digits, appears 
to fulfil perfectly the characteristics of a chance arrange- 
ment. If we were to omit the first four or five digits, 
which are familiar to most of us, we might safely defy any 
one to whom it was shown to say that it was not got at by 
simply drawing figures from a bag. He might look at it for 
his whole life without detecting that it was anything but the 
result of such a chance selection. And rightly so, because 
regarded as a mere arrangement it is a chance one : it fulfils 
all the requirements of such an arrangement 1 . The question 

1 Doubts have been expressed means. If such a question were 

about the truly random character asked in relation to any unusual 

of the digits in this case (v. De divergence from the a priori chance 

Morgan, Budget of Paradoxes, p. in a case of throwing dice, say, we 

291), and Jevons has gone so far as should probably substitute for it the 

to ask (Principles of Science, p. 529), following, as being more appropriate 

4 * Why should the value of TT, when to our science : Assign the degree 

expressed to a great number of of improbability of the event in 

figures, contain the digit 7 much less question; i.e. its statistical rarity, 

frequently than any other digit!" And we should then proceed to 

I do not quite understand what this judge, in the way indicated in the 



248 Chance, Causation, and Design. [CHAP. x. 

we are now proceeding to discuss is this : Given any such 
arrangement how are we to determine the process by which 
it was arrived at ? 

We are supposed to have some event before us which 
might have been produced in either of two alternative 
ways, i.e. by chance or by some kind of deliberate design; 
and we are asked to determine the odds in favour of one 
or other of these alternatives. It is therefore a problem in 
Inverse Probability and is liable to all the difficulties to 
which problems of this class are apt to be exposed. 

text, whether this improbability gave in the chapter on Randomness. We 



rise to any grounds of suspicion. 

The calculation is simple. The 
actual number of 7's, in the 708 
digits, is 53 : whilst the fair average 
would be 71. The question is, What 
ia the chance of such a departure 
from the average in 708 turns? By 
the usual methods of calculation 
(v. Galloway on Probability) the 
chances against an excess or defect 
of 18 are about 44 : 1, in respect of 
any specified digit. But of course 
what we want to decide are the 
chances against some one of the ten 
showing this divergence. This I 
estimate as being approximately 
determined by the fraction (|) 10 
viz. *8. This represents odds of only 
about 4 : 1 against such an occur- 
rence, which is nothing remarkable. 
As a matter of fact several digits in 
the two other magnitudes which 
Mr Shanks had calculated to the 
same length, viz. Tan- 1 ! and 
Tan- 1 ^, show the same diver- 
gencies (v. Proc. Roy. Soc. xxi. 319). 

I may call attention here to a 
point which should have been noticed 



must be cautious when we decide 
upon the random character by mere 
inspection. It is very instructive 
here to compare the digits in TT with 
those within the 'period' of a circu- 
lating decimal of very long period. 
That of l-=-7699, which yields the 
full period of 7698 figures, was cal- 
culated some years ago by two Cam- 
bridge graduates (Mr Lunn and Mr 
Suffield), and privately printed. If we 
confine our examination to a portion 
of the succession the random cha- 
racter seems plausible ; i.e. the digits, 
and their various combinations, come 
out in nearly, but not exactly, equal 
numbers. So if we take batches of 
10; the averages hover nicely about 
45. But if we took the whole period 
which ' circulates,' we should find 
these characteristics overdone, and 
the random character would dis- 
appear. That is, instead of a merely 
ultimate approximation to equality 
we should have (as far as this is 
possible) an absolute attainment 
of it. 



SECT. 12.] Chance, Causation, and Design. 249 

12. For the theoretic solution of such a question we 
require the two following data : 

(1) The relative frequency of the two classes of agencies, 
viz. that which is to act in a chance way and that which 
is to act designedly. 

(2) The probability that each of these agencies, if it 
were the really operative one, would produce the event in 
question. 

The latter of these data can generally be secured without 
any difficulty. The determination of the various contin- 
gencies on the chance hypothesis ought not, if the example 
were a suitable one, to offer any other than arithmetical 
difficulties. And as regards the design alternative, it is 
generally taken for granted that if this had been operative 
it would certainly have produced the result aimed at. For 
instance, if ten pence are found on a table, all with head 
uppermost, and it be asked whether chance or design had 
been at work here ; we feel no difficulty up to a certain 
point. Had the pence been tossed we should have got ten 
heads only once in 1024 throws; but had they been placed 
designedly the result would have been achieved with cer- 
tainty. 

But the other postulate, viz. that of the relative preva- 
lence of these two classes of agencies, opens up a far more 
serious class of difficulties. Cases can be found no doubt, 
though they are not very frequent, in which this question 
can be answered approximately, and then there is no further 
trouble. For instance, if in a school class-list I were to see 
the four names Brown, Jones, Robinson, Smith, standing 
in this order, it might occur to me to enquire whether this 
arrangement were alphabetical or one of merit. In our 
enlarged sense of the terms this is equivalent to chance 
and design as the alternatives ; for, since the initial letter of 



250 Chance, Causation, and Design. [CHAP. x. 

a boy's name has no known connection with his attainments, 
the successive arrangement of these letters on any other 
than the alphabetical plan will display the random features, 
just as we found to be the case with the digits of an incom- 
mensurable magnitude. The odds are 23 to 1 against 4 
names coming undesignedly in alphabetical order ; they are 
equivalent to certainty in favour of their doing so if this 
order had been designed. As regards the relative frequency 
of the two kinds of orders in school examinations I do not 
know that statistics are at hand, though they could easily 
be procured if necessary, but it is pretty certain that the 
majority adopt the order of merit. Put for hypothesis the 
proportion as high as 9 to 1, and it would still be found more 
likely than not that in the case in question the order was 
really an alphabetical one. 

13. But in the vast majority of cases we have no 
such statistics at hand, and then we find ourselves exposed 
to very serious ambiguities. These may be divided into 
two distinct classes, the nature of which will best be seen 
by the discussion of examples. 

In the first place we are especially liable to the draw- 
back already described in a former chapter as rendering 
mere statistics so untrustworthy, which consists in the fact 
that the proportions are so apt to be disturbed almost from 
moment to moment by the possession of fresh hints or infor- 
mation. We saw for instance why it was that statistics of 
mortality were so very unserviceable in the midst of a 
disease or in the crisis of a battle. Suppose now that on 
coming into a room I see on the table ten coins lying face 
uppermost, and am asked what was the likelihood that the 
arrangement was brought about by design. Everything 
turns upon special knowledge of the circumstances of the 
case. Who had been in the room ? Were they children, 



SECT. 15.] Chance, Causation, and Design. 251 

or coin-collectors, or persons who might have been supposed 
to have indulged in tossing for sport or for gambling pur- 
poses ? Were the coins new or old ones ? a distinction of 
this kind would be very pertinent when we were considering 
the existence of any motive for arranging them the same 
way uppermost. And so on ; we feel that our statistics are 
at the mercy of any momentary fragment of information. 

14. But there is another consideration besides this. 
Not only should we be thus influenced by what may be 
called external circumstances of a general kind, such as the 
character and position of the agents, we should also be in- 
fluenced by what we supposed to be the conventional 1 estimate 
with which this or that particular chance arrangement was 
then regarded. Thus from time to time as new games of 
cards become popular new combinations acquire significance ; 
and therefore when the question of design takes the form of 
possible cheating a knowledge of the current estimate of 
such combinations becomes exceedingly important. 

15. The full significance of these difficulties will best 
be apprehended by the discussion of a case which is not 
fictitious or invented for the purpose, but which has actually 
given rise to serious dispute. Some years ago Prof. Piazzi 
Smyth published a work 2 upon the great pyramid of Ghizeh, 
the general object of which was to show that that building 
contained, in its magnitude, proportions and contents, a 
number of almost imperishable natural standards of length, 
volume, &c. Amongst other things it was determined that 

1 Of course this conventional esti- or seven heads and three tails, &c. 
mate is nothing different in kind Its distinction only consists in its 
from that which may attach to any almost universal acceptance as re- 
order or succession. Ten heads in markable. 

succession is intrinsically or ob-. 2 Our Inheritance in the Great 

jectively indistinguishable in cha- Pyramid, Ed. in. 1877. 
racter from alternate heads and tails, 



252 Chance, Causation, and Design. [CHAP. x. 

the value of TT was accurately (the degree of accuracy is not, 
I think, assigned) indicated by the ratio of the sides to the 
height. The contention was that this result could not be 
accidental but must have been designed. 

As regards the estimation of the value of the chance 
hypothesis the calculation is not quite so clear as in the 
case of dice or cards. We cannot indeed suppose that, 
for a given length of base, any height can be equally possible. 
We must limit ourselves to a certain range here ; for if too 
high the building would be insecure, and if too low it would 
be ridiculous. Again, we must decide to how close an 
approximation the measurements are made. If they are 
guaranteed to the hundredth of an inch the coincidence 
would be of a quite different order from one where the 
guarantee extended only to an inch. Suppose that this 
has been decided, and that we have ascertained that out 
of 10,000 possible heights for a pyramid of given base just 
that one has been selected which would most nearly yield 
the ratio of the radius to the circumference of a circle. 

The 7> i-.,iiii'i lp , r consideration would be the relative fre- 
quency of the ' design ' alternative, what is called its d 
priori probability, that is, the relative frequency with 
which such builders can be supposed to have aimed at that 
ratio ; with the obvious implied assumption that if they did 
aim at it they would certainly secure it. Considering our 
extreme ignorance of the attainments of the builders it is 
obvious that no attempt at numerical appreciation is here 
possible. If indeed the ' design ' was interpreted to mean 
conscious resolve to produce that ratio, instead of mere re- 
solve to employ some method which happened to produce 
it, few persons would feel much hesitation. Not only do 
we feel tolerably certain that the builders did not know the 
value of TT, except in the rude way in which all artificers 



SECT. 15.] Chance, Causation, and Design. 253 

must know it ; but we can see no rational motive, if they 
did know it, which should induce them to perpetuate it 
in their building. If, however, to adopt an ingenious sug- 
gestion 1 , we suppose that the builder may have proceeded 
in the following fashion, the matter assumes a different 
aspect. Suppose that having decided on the height of his 
pyramid he drew a circle with that as radius : that, laying 
down a cord along the line of this circle, he drew this cord 
out into a square, which square marked the base of the 
building. Hardly any simpler means could be devised in 
a comparatively rude age ; and it is obvious that the cir- 
cumference of the base, being equal to the length of the 
cord, would bear exactly the admitted ratio to the height. 
In other words, the exact attainment of a geometric value 
does not imply a knowledge of that ratio, but merely of some 
method which involves and displays it. A teredo can bore, 
as well as any of us, a hole which displays the geometric pro- 
perties of a circle, but we do not credit it with correspond- 
ing knowledge. 

As before said, all numerical appreciation of the like- 
lihood of the design alternative is out of the question. But, 
if the precision is equal to what Mr Smyth claimed, I sup- 
pose that most persons (with the above suggestion before 
them) will think it somewhat more likely that the coinci- 
dence was not a chance one. 

1 Made in Nature (Jan. 24, 1878) lengths of the four straight sides 
by Mr J. G. Jackson. It must be of the actual and practical square 
remarked that Mr Smyth's alter- base." As regards the alternatives 
native statement of his case leads up of chance and design, here, it must 
to that explanation : "The vertical be remembered in justice to Mr 
height of the great pyramid is the Smyth's argument that the anti- 
radius of a theoretical circle the thesis he admits to chance is not 
length of whose curved circumference human, but divine design, 
is exactly equal to the sum of the 



254 Chance, Carnation, and Design. [CHAP. x. 

16. There still remains a serious, and highly interesting 
speculative consideration. In the above argument we took 
it for granted, in calculating the chance alternative, that 
only one of the 10,000 possible values was favourable ; that 
is, we took it for granted that the ratio 77- was the only one 
whose claims, so to say, were before the court. But it is 
clear that if we had obtained just double this ratio the result 
would have been of similar significance, for it would have 
been simply the ratio of the circumference to the diameter. 
In fact, Mr Smyth's selected ratio, the height to twice the 
breadth of the base as compared with the diameter to the 
circumference, is obviously only one of a plurality of ratios. 
Again ; if the measured results had shown that the ratio of 
the height to one side of the base was 1 : V2 (i.e. that of a 
side to a diagonal of a square) or 1 : V3 (i.e. that of a side 
to a diagonal of a cube) would not such results equally show 
evidence of design ? Proceeding in this way, we might 
suggest one known mathematical ratio after another until 
most of the 10,000 supposed possible values had been taken 
into account. We might then argue thus: since almost 
every possible height of the pyramid would correspond to 
some mathematical ratio, a builder, ignorant of them all 
alike, would be not at all unlikely to stumble upon one or 
other of them : why then attribute design to him in one case 
rather than another ? 

17. The answer to this objection has been already 
hinted at. Everything turns upon the conventional estimate 
of one result as compared with another. Revert, for sim- 
plicity, to the coins. Ten heads is just as likely as alternate 
heads and tails, or five heads followed by five tails ; or, in 
fact, as any one of the remaining 1023 possible cases. But 
universal convention has picked out a run of ten as being 
remarkable. Here, of course, the convention seems a very 



SECT. 17.] Chance, Causation, and Design. 255 

natural and indeed inevitable one, but in other cases it is 
wholly arbitrary. For instance, in cards, " queen of spades 
and knave of diamonds " is exactly as uncommon as any 
other such pair : moreover, till bezique was introduced it 
offered presumably no superior interest over any other 
specified pair. But during the time when that game was 
very popular this combination was brought into the category 
of coincidences in which interest was felt ; and, given dis- 
honesty amongst the players, its chance of being designed 
stood at once on a much better footing 1 . 

Returning then to the pyramid, we see that in balancing 
the claims of chance and design we must, in fairness to the 
latter, reckon to its account several other values as well as 
that of TT, e.g. \/2 and V3, and a few more such simple and 
familiar ratios, as well as some of their multiples. But 
though the number of such values which might be reckoned, 
on the ground that they are actually known to us, is infinite, 
yet the number that ought to be reckoned, on the ground 
that they could have been familiar to the builders of a 
pyramid, are very few. The order of probability for or against 
the existence of design will not therefore be seriously altered 
here by such considerations 2 . 

1 See Cournot, Essai sur les fonde- more court cards than chance could 
mente de nos connaissances. Vol. i. be expected to assign him ; and that 
p. 71. in consequence his average gains for 

2 It deserves notice that con- several years in succession were un- 
siderations of this kind have found usually large. The counsel for the 
their way into the Law Courts defence urged that still larger gains 
though of course without any at- had been secured by other players 
tempt at numerical valuation. Thus, without suspicion of unfairness, 
in the celebrated De Bos trial, in (I cannot find that it was explained 
so far as the evidence was indirect, over how large an area of experience 
one main ground of suspicion seems these instances had been sought ; 
to have been that Lord De Bos, nor how far the magnitude of the 
when dealing at whist, obtained far stakes, as distinguished from the 



256 Chance, Causation, and Design. [CHAP. x. 

18, Up to this point it will be observed that what we 
have been balancing against each other are two forms of 
agency, of human agency, that is, one acting through 
chance, and the other by direct design. In this case we 
know where we are, for we can thoroughly understand agency 
of this kind. The problem is indeed but seldom numerically 
soluble, and in most cases not soluble at all, but it is at any 
rate capable of being clearly stated. We know the kind of 
answer to be expected and the reasons which would serve to 
determine it, if they were attainable. 

The next stage in the enquiry would be that of balancing 
ordinary human chance agency against, I will not call it 
direct spiritualist agency, for that would be narrowing the 
hypothesis unnecessarily, but against all other possible 
causes. Some of the investigations of the Society for 
Psychical Research will furnish an admirable illustration of 
what is intended by this statement. There is a full dis- 
cussion of these applications in a recent essay by Mr F. Y. 
Edge worth 1 ; but as his account of the matter is connected 
with other calculations and diagrams I can only quote it in 
part. But I am in substantial agreement with him. 

"It is recorded that 1833 guesses were made by a 'per- 
cipient ' as to the suit of cards which the ' agent ' had fixed 
upon. The number of successful guesses was 510, consider- 
ably above 458, the number which, as being the quarter of 
1833, would, on the supposition of pure chance, be more 
likely than any other number. Now, by the Law of Error, 
we are able approximately to determine the probability of 

number of successes, accounted for Feb. 11, 1837.) 

that of the actual gains), and that l Metretike. At the end of this 

large allowance must be made for volume will be found a useful list 

skill where the actual gains were of a number of other publications 

computed. (See the Times' report, by the same author on allied topics. 



SECT. 19.] Chance, Causation, and Design. 257 

such an excess occurring by chance. It is equal to the ex- 
tremity of the tail of a probability-curve such as [those we 
have already had occasion to examine].... The proportion 
of this extremity of the tail to the whole body is *003 to 1. 
That fraction, then, is the probability of a chance shot 
striking that extremity of the tail ; the probability that, if 
the guessing were governed by pure chance, a number of 
successful guesses equal or greater than 510 would occur": 
odds, that is, of about 332 to 1 against such occurrence. 

19. Mr Edge worth holds, as strongly as I do, that for pur- 
poses of calculation, in any strict sense of the word, we ought 
to have some determination of the data on the non-chance side 
of the hypothesis. We ought to know its relative frequency 
of occurrence, and the relative frequency with which it 
attains its aims. I am also in agreement with him that 
"what that other cause may be, whether some trick, or 
unconscious illusion, or ;7 / 7 '-" _" ~ence of the sort which 
is vindicated by the investigators it is for common-sense 
and ordinary Logic to consider." 

I am in agreement therefore with those who think that 
though we cannot form a quantitative opinion we can in 
certain cases form a tolerably decisive one, Of course if we 
allow the last word to the supporters of the chance hypo- 
thesis we can never reach proof, for it will always be open to 
them to revise and re-fix the antecedent probability of the 
counter hypothesis. What we may fairly require is that 
those who deny the chance explanation should assign some 
sort of minimum value to the probability of occurrence on 
the other supposition, and we can then try to surmount this 
by increasing the rarity of the actually produced phenomenon 
on the chance hypothesis. If, for instance, they declare that 
in their estimation the odds against any other than the chance 
agency being at work are greater than 332 to 1, we must 



?58 Chance, Causation, and Design. [CHAP. x. 

try to secure a yet uncommoner occurrence than that in 
question. If the supporters of thought-transference have 
the courage of their convictions, as they most assuredly 
have, they would not shrink from accepting this test. I 
am inclined to think that even at present, on such evidence 
as that above, the probability that the results were got at by 
ordinary guessing is very small. 

20. The problems discussed in the preceding sections 
are at least intelligible even if they are not always resol- 
vable. But before finishing this chapter we must take notice 
of some speculations upon this part of the subject which do not 
seem to keep quite within the limits of what is intelligible. 
Take for instance the question discussed by Arbutlmott (in 
a paper in the Phil. Transactions, Vol. xxvii.) under the 
title " An Argument for Divine Providence, taken from the 
constant Regularity observed in the birth of both sexes." Had 
his argument been of the ordinary teleological kind ; that 
is, had he simply maintained that the existent ratio of ap- 
proximate equality, with a six per cent, surplusage of males, 
was a beneficent one, there would have been nothing here to 
object against. But what he contemplated was just such a 
balance of alternate hypotheses between chance and design 
as we are here- considering. His conclusion in his own words 
is, " it is art, not chance, that governs." 

It is difficult to render such an argument precise without 
rendering it simply ridiculous. Strictly understood it can 
surely bear only one of two interpretations. On the one 
hand we may be personifying Chance: regarding it as an 
agent which must be reckoned with as being quite capable 
of having produced man, or at any rate having arranged the 
proportion of the sexes. And then the decision must be 
drawn, as between this agent and the Creator, which of the 
two produced the existent arrangement. If so, and Chance 



SECT. 21.] Chance, Causation, and Design. 25& 

be defined as any agent which produces a chance or random 
arrangement, I am afraid there can be little doubt that it 
was this agent that was at work in the case in question. 
The arrangement of male and female births presents, so far 
as we can see, one of the most perfect examples of chance : 
there is ultimate uniformity uiu-r^in^ out of individual 
irregularity : all the ' runs ' or successions of each alternative 
are duly represented : the fact of, say, five sons having been 
already born in a family does not seem to have any certain 
effect in diminishing the likelihood of the next being a son, 
and so on. Such a nearly perfect instance of ' independent 
events ' is comparatively very rare in physical phenomena- . 
It is all that we can claim from a chance arrangement 1 . The 
only other interpretation I can see is to suggest that there 
was but one agent who might, like any one of us, have either 
tossed up or designed, and we have to ascertain which course 
he probably adopted in the case in question. Here too, if 
we are to judge of his mode of action by the tests we should 
apply to any work of our own, it would certainly look very 
much as if he had adopted some scheme of tossing. 

21. The simple fact is that any rational attempt to 

1 That is, if we look simply to a preponderance, agencies are at 
statistical results, as Arbuthnott did, once set in motion which tend to 
and as we should do if we were ex- redress the balance. This is a modi- 
amining the tosses of a penny. If fication and improvement of the older 
the remarkable theory of Dr Diising theory, that the relative age of the 
(Die Regulierung des Geschlechts- parents has something to do with 
verhtiltnisses... Jena, 1834) be con- the sex of the offspring, 
firmed, the matter would assume Quetelet (Letters, p, 61) has at- 
a somewhat different aspect. He at- tempted to prove a proposition about 
tempts to show, both on physio- the succession of male and female 
logical grounds, and by analysis of births by certain experiments sup- 
statistics referring to men and posed to be tried upon an urn with 
animals, that there is a decidedly black and white balls in it. But 
compensatory process at work. That this is going too far. (See the note 
is, if for any cause either sex attains at the end of this chapter.) 

172 



260 Chance, Causation, and Design. [CHAP. x. 

decide between chance and design as agencies must be con- 
fined to the case of finite intelligences. One of the im- 
portant determining elements here, as we have seen, is the 
state of knowledge of the agent, and the conventional esti- 
mate entertained about this or that particular arrangement ; 
and these can be appreciated only when we are dealing with 
beings like ourselves. 

For instance, to return to that much debated question 
about the arrangement of the stars, there can hardly be any 
doubt that what Mitchell, who started the discussion, had 
in view was the decision between Chance and Design. He 
says (Trans. Roy. Soc. 1767) "The argument I intend to 
make use of... is of that kind which infers either design or 
some general law from a general analogy and from the great- 
ness of the odds against things having been in the present 
situation if it was not owing to some such cause." And he 
concludes that had the stars " been scattered by mere chance 
as it might happen " there would be " odds of near 500,000 
to 1 that no six stars out of that number [1500], scattered at 
random in the whole heavens, would be within so small a 
distance from each other as the Pleiades are." Under any 
such interpretation the controversy seems to me to be idle. 
I do not for a moment dispute that there is some force in 
the ordinary teleological . i ii > iriiiu<-iii which seeks to trace signs 
of goodness and wisdom in the general tendency of things. 
But what do we possibly understand about the nature of 
creation, or the designs of the Creator, which should enable 
us to decide about the likelihood of his putting the stars in 
one shape rather than in another, or which should allow any 
significance to "mere chance" as contrasted with his sup- 
posed all-pervading agency ? 

22, Reduced to intelligible terms the two following 
questions seem to me to emerge from the controversy : 



SECT. 23,] .Chance, Carnation, and Design. 261 

(I) The stars being distributed through space, some of 
them would of course be nearly in a straight line behind 
others when looked at from our planet. Supposing that 
they were tolerably uniformly distributed, we could calculate 
about how many of them would thus be seen in apparent 
close proximity to one another. The question is then put, 
Are there more of them near to each other, two and two, 
than such calculation would account for ? The answer is that 
there are many more. So far as I can see the only direct 
inference that can be drawn from this is that they are not 
uniformly distributed, but have a tendency to go in pairs. 
This, however, is a perfectly sound and reasonable applica- 
tion of the theory. Any further conclusions, such as that 
these pairs of stars will form systems, as it were, to them- 
selves, revolving about one another, and for all practical pur- 
poses unaffected by the rest of the sidereal system, are of 
course derived from astronomical considerations 1 . Probability 
confines itself to the simple answer that the distribution is 
not uniform ; it cannot pretend to say whether, and by what 
physical process, these binary systems of stars have been 
' caused' 2 . 

23 (II). The second question is this, Does the distri- 
bution of the stars, after allowing for the case of the binary 

1 It is precisely analogous to the speak of the probability of a "physt- 
conclusion that the flowers of the cal connection" between these double 
daisies (as distinguished from the stars. The phrase seems mislead- 
plants, v. p. 109) are not distributed ing, for on the usual hypothesis of 
at random, but have a tendency to universal gravitation all stars are 
KO in groups of two or more. Mere physically connected, by gravitation, 
observation shows this: and then, It is therefore better, as above, to 
from our knowledge of the growth of make it simply a question of relative 
plants we may infer that these little proximity, and to leave it to astro- 
groups spring from the same root. nomy to infer what follows from un- 

3 In this discussion, writers often usual proximity. 



262 .Change, Causation, and Design. [CHAP. x. 

stars just mentioned, resemble that which would be pro- 
duced by human agency sprinkling things f at random > ? 
(We are speaking, of course, of their distribution as it ap- 
pears to us, on the visible heavens, for this is nearly all that 
we can observe ; but if they extend beyond the telescopic 
range in every direction, this would lead to practically much 
the same discussion as if we considered their actual arrange- 
ment in space.) We have fully discussed, in a former chap- 
ter, the meaning of ' randomness.' Applying it to the case 
before us, the question becomes this, Is the distribution 
tolerably uniform on the whole, but with innumerable indi- 
vidual deflections ? That is, when we compare large areas, 
are the ratios of the number of stars in each equal area 
approximately equal, whilst, as we compare smaller and 
smaller areas, do the relative numbers become more and 
more irregular? With certain exceptions, such as that of 
the Milky Way and other nebular clusters, this seems to be 
pretty much the case, at any rate as regards the bulk of the 
stars 1 . 

1 Professor Forbes in the paper that the stars are decidedly more 

in the Philosophical Magazine al- thickly aggregated in the Milky Way 

ready referred to (Ch. vii. 18) gave than elsewhere. So far as this is to 

several diagrams to show what were be relied on the argument is the 

the actual arrangements of a random same as in the case of the double 

distribution. He scattered peas over stars ; it tends to prove that the 

a chess-board, and then counted the proximity of the stars in the Milky 

number which rested on each square. Way is not merely apparent, but 

His figures seem to show that the actual. (2) He has ascertained that 

general appearance of the stars is there are two large areas, in the 

much the same as that produced by North and South hemispheres, in 

such a plan of scattering. which the stars are much more 

Some recent investigations by thickly aggregated than elsewhere. 

Mr E. A. Proctor seem to show, Here, it seems to me, Probability 

however, that there are at least two proves nothing : we are simply deny- 

exceptions to this tolerably uniform ing that the distribution is uniform, 

distribution. (1) He has ascertained What may follow in the way of in- 



SECT. 23.] Chance, Causation, and Design. 263 

All further questions: the decision, for instance, for or 
against any form of the Nebular Hypothesis : or, admitting 
this, the decision whether such and such parts of the visible 
heavens have sprung from the same nebula, must be left to 
Astronomy to adjudicate. 



NOTE ON THE PROPORTIONS OF THE SEXES, 

The following remarks were rather too long for convenient insertion on 
p. 259, and are therefore appended here. 

The 'random' character of male and female births has generally been 
rested almost entirely on statistics of place and time. But what is more 
wanted, surely, is the proportion displayed when we compare a number of 
families. This seems so obvious that I cannot but suppose that the investi- 
gation must have been already made somewhere, though I have not found 
any trace of it in the most likely quarters. Thus Prof. Lexis (Massener- 
scheinungen) when supporting his view that the proportion between the 
sexes at birth is almost the only instance known to him, in natural pheno- 
mena, of true normal dispersion about a mean, rests his conclusions on the 
ordinary statistics of the registers of different countries. 

It certainly needs proof that the same characteristics will hold good 
when the family is taken as the unit, especially as some theories (e.g. that 
of Sadler) would imply that 'runs' of boys or girls would be proportionally 
commoner than pure chance would assign. Lexis has shown that this is 
most markedly the case with twins: i.e., to use an obviously intelligible 
notation, (M for male, F for female), that M.M. and F.F. are very much 
commoner in proportion than M.F. 

I have collected statistics including over. 13,000 male and female births, 
arranged in families of four and upwards. They were taken from the 
pedigrees in the Herald's Visitations, and therefore represent as a rule a 
somewhat select class, viz. the families of the eldest sons of English country 
gentlemen in the sixteenth century. They are not sufficiently extensive yet 
for publication, but I give a summary of the results to indicate their tendency 
so far. The upper line of figures in each case gives the obsemed results : i.e. 

ferences as to the physical process of says on Astronomy, p. 297. Also a 

causation by which the stars have series of Essays in The Universe and 

been disposed is a question for the the coming Transits. 
Astronomer. See Mr Proctor's 



264 Chance, Causation, and Design. [CHAP. x. 

in the case of a family of four, the numbers which had four male, three male 
and one female, two male and two female, and so on. The lower line gives 
the calculated results; i.e. the corresponding numbers which would have 
been obtained had batches of M.s and F.s been drawn from a bag in which 
they were mixed in the ratio assigned by the total observed numbers for 
those families. 



512 families of 4 ; 
yielding 
1188 M. : 860 F. 

512 families of 5 ; 
yielding 
1402 M. : 1158 F. 

512 families of 6; 
yielding 
1612 M. : 1460 F. 


m 4 m 3 / m 2 / 2 m/ 3 / 4 
81 + 148 + 161 + 98 + 24 (observed.) 
57 + 168 + 184 + 88+15 (calculated.) 

m 5 w 4 / wi 3 / 2 m 2 / 3 w/ 4 / 5 
60 + 82 + 161 + 143 + 61 + 15 (obs.) 
25 + 103 + 172 + 143 + 59 + 10 (calc.) 

w 8 w 5 / m 4 / 2 w 8 / 3 m 2 / 4 mf 5 / 6 
30 + 48 + 115 + 146 + 126 + 40 + 7 (obs.) 
10 + 56 + 133 + 159 + 108 + 41 + 5 (calo.) 



The numbers for the larger families are as yet too small to be worth 
giving, but they show the same tendency. It will be seen that in every case 
the observed central values are less than the calculated ; and that the 
observed extreme values are much greater than the calculated. The results 
seem to suggest (so far) that a family cannot be likened to a chance drawing 
Of the requisite number from one bag. A better analogy would be to suppose 
two bags, one with M.s in excess and the other with F.s in less excess, and 
that some persons draw from one and some from the other. But fuller 
statistics are needed. 

It will be observed that the total excess of male births is large. This 
may arise from undue omission of females ; but I have carefully confined 
myself to the two or three last generations, in each pedigree, for greater 
security. 



CHAPTER XI. 

ON CERTAIN CONSEQUENCES OF THE OBJECTIVE TREAT- 
MENT OF A SCIENCE OF INFERENCE *. 

1. STUDENTS of Logic are familiar with that broad 
distinction between the two methods of treatment to which 
the names of Material and Conceptualist may be applied. 
The distinction was one which had been gradually growing 
up under other names before it was emphasized, and treated 
as a distinction within the field of Logic proper, by the pub- 
lication of Mill's well known work. No one, for instance, 
can read Whe well's treatises on Induction, or Herschel's 
Discourse, without seeing that they are treating of much the 
same subject matter, and regarding it in much the same 
way, as that which Mill discussed under the name of Logic, 
though they were not disposed to give it that name. That 
is, these writers throughout took it for granted that what 
they had to do was to systematise the facts of nature in 
their objective form, and under their widest possible treat- 
ment, and to expound the principal modes of inference and 
the principal practical aids in the investigation of these 

1 In the previous edition a large detailed discussion of the Law of 

part of this chapter was devoted to Causation, as I hope before very long 

the general consideration of the dis- to express my opinions on these sub* 

tinction between a Material and a jects more fully, and more appro- 

Conceptualist view of Logic. I have priately, in a treatise on the general 

omitted most of this here, as also a principles of Inductive Logic, 
large part of a chapter devoted to the 



266 Objective Treatment of Logic. [CHAP. xi. 

modes of inference, which reason could suggest and which 
experience could justify. What Mill did was to bring these 
methods into close relation with such portions of the old 
scholastic Logic as he felt able to retain, to work them out 
into much fuller detail, to systematize them by giving them 
a certain philosophical and psychological foundation, and 
to entitle the result Logic. 

The practical treatment of a science will seldom corre- 
spond closely to the ideal which its supporters propose to 
themselves, and still seldomer to that which its antagonists 
insist upon demanding from the supporters. If we were to 
take our account of the distinction between the two views of 
Logic expounded respectively by Hamilton and by Mill, 
from Mill and Hamilton respectively, we should certainly 
not find it easy to bring them under one common definition. 
By such a test, the material Logic would be regarded as 
nothing more than a somewhat arbitrary selection from the 
domain of Physical Science in general, and the conccptualist 
Logic nothing more than a somewhat arbitrary selection from 
the domain of Psychology. The former would omit all con- 
sideration of the laws of thought and the latter all considera- 
tion of the truth or falsehood of our conclusions. 

Of course, in practice, such extremes as these are soon 
seen to be avoidable, and in spite of all controversial exagge- 
rations the expounders of the opposite views do contrive to 
retain a large area of speculation in common. I do not pro- 
pose here to examine in detail the restrictions by which this 
accommodation is brought about, or the very real and im- 
portant distinctions of method, aim, tests, and limits which 
in spite of all approach to agreement are still found to subsist. 
To attempt this would be to open up rather too wide an 
enquiry to be suitable in a treatise on one subdivision only 
of the general science of Inference. 



SECT. 3.] Objective Treatment of Logic. 267 

2. One subdivision of this enquiry is however really 
forced upon our notice. It does become important to con- 
sider the restrictions to which the ultra-material account of 
the province of Logic has to be subjected, because we shall 
thus have our attention drawn to an aspect of the subject 
which, slight and fleeting as it is within the region of Induc- 
tion becomes very prominent and comparatively permanent 
in that of Probability. According to this ultra-material view, 
Inductive Logic would generally be considered to have no- 
thing to do with anything but objective facts : its duty is to 
start from facts and to confine itself to such methods as will 
yield nothing but facts. What is doubtful it either estab- 
lishes or it lets alone for the present, what is unattainable 
it rejects, and in this way it proceeds to build up by slow 
accretion a vast fabric of certain knowledge. 

But of course all this is supposed to be done by human 
minds, and therefore if we enquire whether notions or con- 
cepts. call them what we will, have no place in such a 
scheme it must necessarily be admitted that they have some 
place. The facts which form our starting point must be 
grasped by an intelligent being before inference can be built 
upon them; and the 'facts' which form the conclusion have 
often, at any rate for some time, no place anywhere else than 
in the mind of man. But no one can read Mill's treatise, for 
instance, without noticing how slight is his reference to this 
aspect of the question. He remarks, in almost contemptuous, 
indifference, that the man who digs must of course have a 
notion of the ground he digs and of the spade he puts into 
it, but he evidently considers that these ' notions' need not 
much more occupy the attention of the speculative logician, 
in so far as his mere inferences are concerned, than they 
occupy that of the husbandman. 

3. It must be admitted that there is some warrant 



268 Objective Treatment of Logic. [CHAP. XL 

for this omission of all reference to the subjective side of 
inference so long as we are dealing with Inductive Logic. 
The inductive discoverer is of course in a very different 
position. If he is worthy of the name his mind at every 
moment will be teeming with notions which he would be as 
far as any one from calling facts : he is busy making them 
such to the best of his power. But the logician who follows 
in his steps, and whose business it is to explain and justify 
what his leader has discovered, is rather apt to overlook this 
mental or uncertain stage. What he mostly deals in are the 
'complete inductions' and 'well-grounded generalizations' 
and so forth, or the exploded errors which contradict them: 
the prisoners and the corpses respectively, which the real 
discoverer leaves on the field behind him whilst he presses 
on to complete his victory. The whole method of science, 
expository as contrasted with militant, is to emphasize the 
distinction between fact and non-fact, and to treat of little 
else but these two. In other words a treatise on Inductive 
Logic can be written without any occasion being found to 
define what is meant by a notion or concept, or even to employ 
such terms. 

4. And yet, when we come to look more closely, signs 
may be detected even within the field of Inductive Logic, 
of an occasional breaking down of the sharp distinction in 
question ; we may meet now and then with entities (to use 
the widest term attainable) in reference to which it would 
be hard to say that they are either facts or conceptions. 
For instance, Inductive Logic has often occasion to make use 
of Hypotheses : to which of the above two classes are these 
to be referred ? They do not seem in strictness to belong to 
either ; nor are they, as will presently be pointed out, by any 
means a solitary instance of the kind. 

It is true that within the province of Inductive Logic 



SECT. 4.] Objective Treatment of Logic. 269 

these hypotheses do not give much trouble on this score. 
However vague may be the form in which they first present 
themselves to the philosopher's mind, they have not much 
business to come before us in our capacity of logicians until 
they are well on their way, so to say, towards becoming 
facts: until they are beginning to harden into that firm 
tangible shape in which they will eventually appear. We 
generally have some such recommendations given to us as 
that our hypotheses shall be well-grounded and reasonable. 
This seems only another way of telling us that however 
freely the philosopher may make his guesses in the privacy 
of his own study, he had better not bring them out into 
public until they can with fair propriety be termed facts, 
even though the name be given with some qualification, as 
by terming them 'probable facts/ The reason, therefore, 
why we do not take much account of this intermediate state 
in the hypothesis, when we are dealing with the inductive 
processes, is that here at any rate it plays only a temporary 
part ; its appearance in that guise is but very fugitive. If 
the hypothesis be a sound one, it will soon take its place as 
an admitted fact ; if not, it will soon be rejected altogether* 
Its state as a hypothesis is not a normal one, and therefore 
we have not much occasion to scrutinize its characteristics. 
In so saying, it must of course be % understood that we are 
speaking as inductive logicians ; the philosopher in his work- 
shop ought, as already remarked, to be familiar enough with 
the hypothesis in every stage of its existence from its origin ; 
but the logician's duty is different, dealing as he does with 
proof rather than with the processes of original investigation 
and discovery. 

We might indeed even go further, and say that in many 
cases the hypothesis does not present itself to the reader, 
that is to the recipient of the knowledge, until it has ceased 



270 Objective Treatment of Logic. [CHAP. xi. 

to deserve that name at all It may be first suggested to 
him along with the proof which establishes it, he not having 
had occasion to think of it before. It thus comes at a single 
step out of the obscurity of the unknown into the full pos- 
session of its rights as a fact, skipping practically the inter- 
mediate or hypothetical stage altogether. The original in- 
vestigator himself may have long pondered over it, and kept 
it present to his mind, in this its dubious stage, but finally 
have given it to the world with that amount of evidence 
which raises it at once in the minds of others to the level of 
commonly accepted facts. 

Still this doubtful stage exists in every hypothesis, 
though for logical purposes, and to most minds, it exists 
in a very fugitive way only. When attention has been 
directed to it, it may be also detected elsewhere in Logic. 
Take the case, for instance, of the reference of names. 
Mill gives the examples of the sun, and a battle, as dis- 
tinguished from the ideas of them which we, or children, 
may entertain. Here the distinction is plain and obvious 
enough. But if, on the other hand, we take the case of 
things whose existence is doubtful or disputed, the difficulty 
above mentioned begins to show itself. The case of merely 
extinct things, or such as have not yet come into existence, 
offers indeed no trouble, since of course actually present 
existence is not necessary to constitute a fact. The usual 
distinction may even be retained also in the case of mythical 
existences. Centaur and Griffin have as universally recog- 
nised a significance amongst the poets, painters, and heralds 
as lion and leopard have. Hence we may claim, even here, 
that our conceptions shall be * truthful/ 'consistent with 
fact/ and so on, by which we mean that they are to be in 
accordance with universal convention upon such subjects. 
Necessary and universal accordance is sometimes claimed 



SECT. 6.] Objective Treatment of Logic. 271 

to be all that is meant by c objective/ and since universal 
accordance is attainable in the case of the notoriously fic- 
titious, our fundamental distinction between fact and con- 
ception, and our determination that our terms shall refer to 
what is objective rather than to what is subjective, may with 
some degree of strain be still conceived to be tenable even here. 

5. But when we come to the case of disputed phe- 
nomena the difficulty re-emerges. A supposed planet or 
new mineral, a doubtful fact in history, a disputed theological 
doctrine, are but a few examples out of many that might be 
offered. What some persons strenuously assert, others as 
strenuously deny, and whatever hope there may be of speedy 
agreement in the case of physical phenomena, experience 
shows that there is not much prospect of this in the case of 
those which are moral and historical, to say nothing of theo- 
logical. So long as those who are in agreement confine their 
intercourse to themselves, their ' facts ' are accepted as such, 
but as soon as they come to communicate with others all 
distinction between fact and conception is lost at once, the 
< facts ' of one party being mere groundless * conceptions ' to 
their opponents. There is therefore, I think, in these cases 
a real difficulty in carrying out distinctly and consistently 
the account which the Materialist logician offers as to the 
reference of names. It need hardly be pointed out that 
what thus applies to names or terms applies equally to 
propositions in which particular or general statements are 
made involving names. 

6. But when we step into Probability, and treat this 
from the same material or Phenomenal point of view, we can 
no longer neglect the question which is thus presented to us. 
The difficulty cannot here be rejected, as referring to what is 
merely temporary or occasional. The intermediate condition 
between conjecture and fact, so far from being temporary 



272 Objective Treatment of Logic. [CHAP. XL 

or occasional only, is here normal.. It is just the condition 
which is specially characteristic of Probability. Hence it 
follows that however decidedly we may reject the Con- 
ceptualist theory we cannot altogether reject the use of 
Conceptualist language. If we can prove that a given man 
will die next year, or attain sufficiently near to proof to 
leave us practically certain on the point, we may speak of 
his death as a (future) fact. But if we merely contemplate 
his death as probable? This is the sort of inference, or 
substitute for inference, with which Probability is specially 
concerned. We may, if we so please, speak of ' probable 
facts/ but if we examine the meaning of the words we 
may find them not merely obscure, but self-contradictory. 
Doubtless there are facts here, in the fullest sense of the 
term, namely the statistics upon which our opinion is ulti- 
mately based, for these are known and admitted by all who 
have looked into the matter. The same language may also 
be applied to that extension of these statistics by induction 
which is involved in the assertion that similar statistics 
will be found to prevail elsewhere, for these also may right- 
fully claim universal acceptance. But these statements, as 
was abundantly shown in the earlier chapters, stand on a 
very different footing from a statement concerning the in- 
dividual event ; the establishment and discussion of the 
former belong by rights to Induction, and only the latter 
to Probability. 

7. It is true that for want of appropriate terms to 
express such things we are often induced, indeed compelled, 
to apply the same name of * facts' to such individual contin- 
gencies. We should not, for instance, hesitate to speak of 
the fact of the man dying being probable, possible, unlikely, 
or whatever it might be. But I cannot help regarding such 
expressions as a ^ strictly incorrect usage arising out of a 



SECT. 8.] Objective Treatment of Logic. 273 

deficiency of appropriate technical terms. It is doubtless 
certain that one or other of the two alternatives must happen, 
but this alternative certainty is not the subject of our con- 
templation ; what we have before us is the single alternative, 
which is notoriously uncertain. It is this, and this only, 
which is at present under notice, and whose occurrence has 
to be estimated. We have surely no right to dignify this 
with the name of a fact, under any qualifications, when the 
opposite alternative has claims, not perhaps actually equal to, 
but at any rate not much inferior to its own. Such Inn^ua^t 1 . 
as already remarked, may be quite right in Inductive logic, 
where we are only concerned with conjectures of such a high 
degree of likelihood that their non-occurrence need not be 
taken into practical account, and which are moreover regard- 
ed as merely temporary. But in Probability the conjecture 
may have any degree of likelihood about it ; it may be just 
as likely as the other alternative, nay it may be much less 
likely. In these latter cases, for instance, if the chances are 
very much against the man's death, it is surely an abuse of 
liiiijiiiiij:^ to speak of the 'fact' of his dying, even though we 
qualify it by declaring it to be highly improbable. The 
subject-matter essential to Probability being the uncertain, 
we can never with propriety employ upon it language which 
in its original and correct application is only appropriate to 
what is actually or approximately certain. 

8. It should be remembered also that this state of 
things, thus characteristic of Probability, is permanent there. 
So long as they remain under the treatment of that science 
our conjectures, or whatever we like to call them, never 
develop into facts. I calculate, for instance, the chance that 
% die will give ace, or that a man will live beyond a certain 
ige. Such an approximation to knowledge as is thus ac- 
quired is as much as we can ever afterwards hope to get, 
v. 18 



274 Objective Treatment of Logic. [CHAP. xi. 

unless we resort to other methods of enquiry. We do not, as 
in Induction, feel ourselves on the brink of some experi- 
mental or other proof which at any moment may raise it 
into certainty. It is nothing but a conjecture of a certain 
degree of strength, and such it will ever remain, so long as 
Probability is left to deal with it. If anything more is ever 
to be made out of it we must appeal to direct experience, or 
to some kind of inductive proof. As we have so often said, 
individual facts can never be determined here, but merely 
ultimate tendencies and averages of many events. I may, 
indeed, by a second appeal to Probability improve the 
character of my conjecture, through being able to refer it to 
a narrower and better class of statistics; but its essential 
nature remains throughout what it was. 

It appears to me therefore that the account of the Material- 
ist view of logic indicated at the commencement of this 
chapter, though substantially sound, needs some slight recon- 
sideration and restatement. It answers admirably so far as 
ordinary Induction is concerned, but needs some revision 
if it is to be equally applicable to that wider view of 
the nature and processes of acquiring knowledge wherein 
the science of logic is considered to involve Probability 
$lso as well as Induction. 

$, Briefly then it is this. We regard the scientific 
thinker, whether he be the original investigator who dis- 
<jovers, or the logician who analyses and describes the proofs 
that may be offered, as surrounded by a world of objective 
phenomena extending indefinitely both ways in time, and in 
ev$ry direction in space. Most of them are, and always will 
in, unknown. If we speak of them as facts we mean 
they are potential objects of human knowledge, tb$t 
under appropriate circumstances men could come to deter- 
minate ajjd final agreement about them. The scientific or 



SECT. 9.] Objective Treatment of Logic. 275 

material logician has to superintend the process of converting 
as much as possible of these unknown phenomena into what 
are known, of ._ ;... ^.\'\: .,: them, as we have said above, about 
the nucleus of certain data which experience and observation 
had to start with. In so doing his principal resources are 
the Methods of Induction, of which something has been 
said in a former chapter ; another resource is found in the 
Theory of Probability, and another in Deduction. 

Now, however such language may be objected to as 
savouring of Conceptualism, I can see no better compendious 
way of describing these processes than by saying that we are 
engaged in getting at conceptions of these external pheno- 
mena, and as far as possible converting these conceptions 
into facts. What is the natural history of 'facts' if we trace 
them back to their origin ? They first come into being as 
mere guesses or conjectures, as contemplated possibilities 
whose correspondence with reality is either altogether dis- 
believed or regarded as entirely doubtful. In this stage, of 
course, their contrast with facts is sharp enough. How they 
arise it does not belong to Logic but to Psychology to say. 
Logic indeed has little or nothing to do with them whilst 
they are in this form. Everyone is busy all his life in enter- 
taining such guesses upon various subjects, the superiority of 
the philosopher over the common man being mainly found in 
the quality of his guesses, and in the skill and persistence 
with which he sifts and examines them. In the next stage 
they mostly go by the name of theories or hypotheses, when 
they are comprehensive in their scope, or are in any way on 
a scale of grandeur and importance : when however they are 
of a trivial kind, or refer to details, we really have no distinc- 
tive or appropriate name for them, and must be content 
therefore to call them 'conceptions/ Through this stage 
they flit with great rapidity in Inductive Logic ; often the 

182 



276 Objective Treatment of Logic. [CHAP. xi. 

logician keeps them back until their evidence is so strong 
that they come before the world at once in the full dignity 
of facts. Hence, as already remarked, this stage of their 
career is not much dwelt upon in Logic. But the whole 
business of Probability is to discuss and estimate them at 
this point. Consequently, so far as this science is concerned, 
the explanation of the Material logician as to the reference 
of names and propositions has to be modified. 

10. The best way therefore of describing our position 
in Probability is as follows : We are entertaining a concep- 
tion of some event, past, present, or future. From the nature 
of the case this conception is all that can be actually enter- 
tained by the mind. In its present condition it would be 
incorrect to call it a fact, though we would willingly, if we 
could, convert it into such by making certain of it one way 
or the other. But so long as our conclusions are to be 
effected by considerations of Probability only, we cannot do 
this. The utmost we can do is to estimate or evaluate it. 
The whole function of Probability is to give rules for so 
doing. By means of reference to statistics or by direct 
deduction, as the case may be, we are enabled to say how 
much this conception is to be believed, that is in what pro- 
portion out of the total number of cases we shall be right 
in so doing. Our position, therefore, in these cases seems 
distinctly that of entertaining a conception, and the process 
of inference is that of ascertaining to what extent we are 
justified in adding this conception to the already received 
body of truth and fact. 

So long, then, as we are confined to Probability these 
conceptions remain such. But if we turn to Induction we 
see that they are meant to go a step further. Their final 
stage is not reached until they have ripened into facts, and 
so taken their place amongst uncontested truths. This is 



SECT, 11.] Objective Treatment of Logic. 277 

their final destination in Logic, and our task is not accom- 
plished until they have reached it. 

11. Such language as this in which we speak of our 
position in Probability as being that of entertaining a con- 
ception, and being occupied in determining what degree of 
belief is to be assigned to it, may savour of Conceptualism, 
but is in spirit perfectly different from it. Our ultimate 
reference is always to facts. We start from them as our data, 
and reach them again eventually in our results whenever it 
is possible. In Probability, of course, we cannot do this in 
the individual result, but even then (as shown in Ch. vi.) we 
always justify our conclusions by appeal to facts, viz. to what 
happens in the long run. 

The discussion which has been thus given to this part of 
the subject may seem somewhat tedious, but it was so ob- 
viously forced upon us when considering the distinction 
between the two main views of Logic, that it was impossible 
to pass it over without fear of misapprehension and confu- 
sion. Moreover, as will be seen in the course of the next 
chapter, several important conclusions could not have been 
properly explained and justified without first taking pains 
to make this part of our ground perfectly plain and satis- 
factory. 



CHAPTER XII. 

CONSEQUENCES OF THE FOREGOING DISTINCTIONS. 

1. WE are now in a position to explain and justify some 
important conclusions which, if not direct consequences of 
the distinctions laid down in the last chapter, will at any 
rate be more readily appreciated and accepted after that 
exposition. 

In the first place, it will be seen that in Probability time 
has nothing to do with the question ; in other words, it does 
not matter whether the event, whose probability we are dis- 
cussing, be past, present, or future. The problem before us, 
in its simplest form, is this : Statistics (extended by Induc- 
tion, and practically often gained by Deduction) inform us 
that a certain event has happened, does happen, or will 
happen, in a certain way in a certain proportion of cases. 
We form a conception of that event, and regard it as possible ; 
but we want to do more ; we want to know how much we 
ought to expect it (under the explanations given in a for- 
mer chapter about quantity of belief). There is therefore a 
sort of relative futurity about the event, inasmuch as our 
knowledge of the fact, and therefore our justification or 
otherwise of the correctness of our surmise, almost neces- 
sarily comes after the surmise was formed ; but the futurity 
is only relative. The evidence by which the question is to 
be settled may not be forthcoming yet, or we may have it by 



SECT. 2.] Probability before and after the event. 279 

us but only consult it afterwards. It is from the fact of the 
futurity being, as above described, only relative, that I have 
preferred to speak of the conception of the event rather than 
of the anticipation of it. The latter term, which in some re- 
spects would have seemed more intelligible and appropriate, 
is open to the objection, that it does rather, in popular esti- 
mation, convey the notion of an absolute as opposed to a 
relative futurity. 

2. For example ; a die is thrown. Once in six times 
it gives ace ; if therefore we assume, without examination, 
that the throw is ace, we shall be right once in six times. 
In so doing we may, according to the usual plan, go forwards 
in time ; that is, form our opinion about the throw before- 
hand, when no one can tell what it will be. Or we might go 
backwards; that is, form an opinion about dice that had 
been cast on some occasion in time past, and then correct 
our opinion by the testimony of some one who had been a 
witness of the throws. In either case the mental operation 
is precisely the same ; an opinion formed merely on statisti- 
cal grounds is afterwards corrected by specific evidence. The 
opinion may have been formed upon a past, present, or future 
event ; the evidence which corrects it afterwards may be our 
own eyesight, or the testimony of others, or any kind of in- 
ference; by the evidence is merely meant such subsequent 
examination of the case as is assumed to set the matter at 
rest. It is quite possible, of course, that this specific evi- 
dence should never be f : '. ; the conception in that 
case remains as a conception, and never obtains that degree 
of conviction which qualifies it to be regarded as a 'fact/ 
This is clearly the case with all past throws of dice the re- 
sults of which do not happen to have been recorded. 

In discussing games of chance there are obvious advan- 
tages in confining ourselves to what is really, as well as 



280 Probability before and after the event [CHAP. xii. 

relatively, future, for in that case direct information concern- 
ing the contemplated result being impossible, all persons are 
on precisely the same footing of comparative ignorance, and 
must form their opinion entirely from the known or inferred 
frequency of occurrence of the event in question. On the 
other hand, if the event be passed, there is almost always evi- 
dence of some kind and of some value, however slight, to 
inform us what the event really was ; if this evidence is not 
actually at hand, we can generally, by waiting a little, obtain 
something that shall be at least of some use to us in forming 
our opinion. Practically therefore we generally confine our- 
selves, in anticipations of this kind, to what is really future, 
and so in popular estimation futurity becomes indissolubly 
associated with probability. 

3. There is however an error closely connected with 
the above view of the subject, or at least an inaccuracy of 
expression which is constantly liable to lead to error, which 
has found wide acceptance, and has been sanctioned by 
writers of the greatest authority. For instance, both Butler, 
in his Analogy, and Mill, have drawn attention, under one 
form of expression or another, to the distinction between im- 
probability before the event and improbability after the 
event, which they consider to be perfectly different things. 
That this phraseology indicates a distinction of importance 
cannot be denied, but it seems to me that the language in 
which it is often expressed requires to be amended. 

Butler's remarks on this subject occur in his Analogy, in 
the chapter on miracles. Admitting that there is a strong 
presumption against miracles (his equivalent for the ordi- 
nary expression, an 'improbability before the event*) he 
strives to obtain assent for them by showing that other 
events, which also have a strong presumption against them, 
are received on what is in reality very slight evidence. He 



SECT. 4.] Probability before and after the event. 281 

says, " There is a very strong presumption against common 
speculative truths, and against the most ordinary facts, be- 
fore the proof of them ; which yet is overcome by almost any 
proof. There is a presumption of millions to one against the 
story of Caesar, or of any other man. For, suppose a number 
Qf common facts so and so circumstanced, of which one had 
no kind of proof, should happen to come into one's thoughts, 
svery one would without any possible doubt conclude them 
bo be false. And the like may be said of a single common 
fact." 

4. These remarks have been a good deal criticized, 
md they certainly seem to me misleading and obscure in 
;heir reference. If one may judge by the context, and by 
mother passage in which the same argument is afterwards 
Deferred to 1 , it would certainly appear that Butler drew no 
listinction between miraculous accounts, and other accounts 
kvhich, to use any of the various expressions in common use, 
ire unlikely or improbable or have a presumption against 
;hem ; and concluded that since some of the latter were in- 
stantly accepted upon somewhat mediocre testimony, it was 
il together irrational to reject the former when similarly or 
setter supported 2 . This subject will come again under our 
lotice, and demand fuller discussion, in the chapter on the 
Credibility of extraordinary stories. It will suffice here to 

1 " Is it not self-evident that inter- to common natural events ; or to 
lal improbabilities of all kinds weak- events which, though uncommon, 
m external proof? Doubtless, but are similar to what we daily expe- 
o what practical purpose can this be rience ; but to the extraordinary 
Jleged here, when it has been proved phenomena of nature. And then 
Before, that real internal improbabi- the comparison will be between the 
ities, which rise even to moral cer- presumption against miracles, and 
ainty, are overcome by the most the presumption against such un- 
rdinary testimony." Part II. ch. in. common appearances, suppose as 

2 "Miracles must not be compared comets," Part II. ch. n. 



282 Probability before and after the event. [CHAP. xn. 

remark that, however satisfactory such a view of the matter 
might be to some theologians, no antagonist of miracles 
would for a moment accept it. He would naturally object 
that, instead of the miraculous element being (as Butler 
considers) "a small additional presumption" against the 
narrative, it involved the events in a totally distinct class of 
incredibility ; that it multiplied, rather than merely added 
to, the difficulties and objections in the way of accepting 
the account. 

Mill's remarks (Logic, Bk. HI. ch. xxv. 4) are of a dif- 
ferent character. Discussing the grounds of disbelief he 
speaks of people making the mistake of " overlooking the 
distinction between (what may be called) improbability 
before the fact, and improbability after it, two different 
properties, the latter of which is always a ground of dis- 
belief, the former not always/' He instances the throwing 
of a die. It is improbable beforehand that it should turn 
up ace, and yet afterwards, " there is no reason for disbe- 
lieving it if any credible witness asserts it." So again, " the 
chances are greatly against A. B.'s dying, yet if any one tells 
us that he died yesterday we believe it." 

5. That there is some difficulty about such problems 
as these must be admitted. The fact that so many people 
find them a source of perplexity, and that such various 
explanations are offered to solve the perplexity, are a suf- 
ficient proof of this 1 . The considerations of the last chapter, 

1 For instance, Sir J. F. Stephen babilities and chances. The proba- 

explains it by drawing a distinction bility of an event is its capability of 

between chances and probabilities, being proved. Its chance is the 

which he says that Butler has con- numerical proportion between the 

fused together ; " the objection that number of possible cases supposed 

very ordinary proof will overcome a to be equally favourable favourable 

presumption of millions to one is to its occurrence; and the number 

based upon a confusion between pro- of possible cases unfavourable to its 



SECT. 6.] Prdwlility before and after the event. 283 

however, over-technical and even scholastic as some of the 
language in which it was expressed may have seemed to the 
reader, will I hope guide us to a more satisfactory way of 
regarding the matter. 

When we speak of an improbable event, it must be 
remembered that, objectively considered, an event can only 
be more or less rare ; the extreme degree of rarity being of 
course that in which the event does not occur at all. Now, 
as was shown in the last chapter, our position, when forming 
judgments of the time in question, is that of entertaining 
a conception or conjecture (call it what we will), and as- 
signing a certain weight of trustworthiness to it. The real 
distinction, therefore, between the two classes of examples 
respectively, which are adduced both by Butler and by Mill, 
consists in the way in which those conceptions are obtained ; 
they being obtained in one case by the process of guessing, 
and in the other by that of giving heed to the reports of 
witnesses. 

6. Take Butler's instance first. In the 'presumption 
before the proof we have represented to us a man thinking 
of the story of Coesar, that is, making a guess about certain 
historical events without any definite grounds for it, and 
then speculating as to what value is to be attached to the 
probability of its truth. Such a guess is of course, as he 
says, concluded to be false. But what does he understand 
by the 'presumption after the proof'? That a story not 
adopted at random, but actually suggested and supported by 
witnesses, should be true. The latter might be accepted, 
whilst, the former would undoubtedly be rejected; but all 
that this proves, or rather illustrates, is that the testimony 

occurrence " (General view of the Cri- 1851), employs the terms improba- 
minal Law of England, p. 255). bility and incredibility to mark the 
Donkin, again (Phil. Magazine, June, same distinction. 



284 Probability before and after the event. [CHAP. xii. 

of almost any witness is in most cases vastly better than 
a mere guess 1 . We may in both cases alike speak of 'the 
event 1 if we will; in fact, as was admitted in the last 
chapter, common language will not readily lend itself to any 
other way of speaking. But it should be clearly under- 
stood that, phrase it how we will, what is really present to 
the man's mind, and what is to have its probable value 
assigned to it, is the conception of an event, in the sense in 
which that expression has already been explained. And 
surely no two conceptions can have a much more important 
distinction put between them than that which is involved in 
supposing one to rest on a mere guess, and the other on the 
report of a witness. Precisely the same remarks apply to 
the example given by Mill. Before A. B.'s death our 
opinion upon the subject was nothing but a guess of our 
own founded upon life statistics; after his death it was 
founded upon the evidence of some one who presumably 
had tolerable opportunities of knowing what the facts really 
were. 

7. That the distinction before us has no essential con- 
nection whatever with time is indeed obvious on a moment's 
consideration. Conceive for a moment that some one had 
opportunities of knowing whether A. B. would die or not. 
If he told us that A. B. would die to-morrow, we should in 
that case be just as ready to believe him as when he tells us 
that A. B. has died. If we continued to feel any doubt 
about the statement (supposing always that we had full 

1 In the extreme case of the wit* again in Chapter xvn. It may be 
ness himself merely guessing, or remarked that there are several sub- 
being as untrustworthy as if he tleties here which cannot be ade- 
merely guessed, the two stories will quately noticed without some previ- 
of course stand on precisely the same ous investigation into the question 
footing. This case will be noticed of the credibility of witnesses. 



SECT, 8.J rrobabitity before and after the event 285 

confidence about his veracity in matters into which he had 
duly enquired), it would be because we thought that in his 
case, as in ours, it was equivalent to a guess, and nothing 
more. So with the event when past, the fact of its being 
past makes no difference whatever; until the credible wit- 
ness informs us of what he knows to have occurred, we 
should doubt it if it happened to come into our minds, just 
as much as if it were future. 

The distinction, therefore, between probability before the 
event and probability after the event seems to resolve itself 
simply into this ; before the event we often have no better 
means of information than to appeal to statistics in some 
form or other, and so to guess amongst the various possible 
alternatives ; after the event the guess may most commonly 
be improved or superseded by appeal to specific evidence, 
in the shape of testimony or observation. Hence, naturally, 
our estimate in the latter case is commonly of much more 
value. But if these characteristics were anyhow inverted ; 
if, that is, we were to confine ourselves to guessing about the 
past, and if we could find any additional evidence about the 
future, the respective values of the different estimates would 
also be inverted. The difference between these values has 
no necessary connection with time, but depends entirely 
upon the different grounds upon which our conception or 
conjecture about the event in question rests. 

8. The following imaginary example will serve to 
bring out the point indicated above. Conceive a people with 
very short memories, and who preserved no kind of record to 
perpetuate their hold upon the events which happened 
amongst them 1 . The whole region of the past would then be 

1 According to Dante, something cardinals and others whom he there 
resembling this prevailed amongst meets are able to give information 
the occupants of the Inferno. The about many events which were yet 



286 Probability before and after the event. [CHAP. XII. 

to them what much of the future is to us ; viz. a region of 
guesses and conjectures, one in reference to which they 
could only judge upon general considerations of probability, 
rather than by direct and specific evidence. But conceive 
also that they had amongst them a race of prophets who 
could succeed in foretelling the future with as near an 
approach to accuracy and trustworthiness as our various 
histories, and biographies, and recollections, can attain in 
respect to the past. The present and usual functions of 
direct evidence or testimony, and of probability, would then 
be simply inverted ; and so in consequence would the pre- 
sent accidental characteristics of improbability before and 
after the event. It would then be the latter which would 
by comparison be regarded as * not always a ground of dis- 
belief/ whereas in the case of the former we should then 
have it maintained that it always was so. 

9. The origin of the mistake just discussed is worth 
enquiring into. I take it to be as follows. It is often the 
case, as above remarked, when we are speculating about 
a future event, and almost always the case when that future 
event is taken from a game of chance, that all persons are in 
precisely the same condition of ignorance in respect to it. 
The limit of available information is confined to statistics, 
and amounts to the knowledge that the unknown event 
must assume some one of various alternative forms. The 
conjecture, therefore, of any one man about it is as valuable 
as that of any other. But in regard to the past the case is 
very different. Here we are not in the habit of relying 
upon statistical information. Hence the conjectures of dif- 
ferent men are of extremely different values ; in the case of 
many they amount to what we call positive knowledge. 

to happen upon earth, but they had actually had happened, 
to oik it for many events which 



SECT. 10.] Probability before and after the event. 287 

This puts a broad distinction, in popular estimation, between 
what may be called the objective certainty of the past and of 
the future, a distinction, however, which from the -funrimg- 
point of a science of inference ought to have no existence. 

In consequence of this, when we apply to the past and 
the future respectively the somewhat ambiguous expression 
4 the chance of the event/ it commonly comes to bear very 
different significations. Applied to the future it bears its 
proper meaning, namely, the value to be assigned to a con- 
jecture upon statistical grounds. It does so, because in this 
case hardly any one has more to judge by than such con- 
jectures. But applied to the past it shifts its meaning, 
owing to the fact that whereas some men have conjectures 
only, others have positive knowledge. By the chance of the 
event is now often meant, not the value to be assigned to a 
conjecture founded on statistics, but to such a conjecture 
derived from and enforced by any body else's conjecture, that 
is by his knowledge and his testimony. 

10. There is a class of cases in apparent opposition to 
some of the statements in this chapter, but which will be 
found, when examined closely, decidedly to confirm them. 
I am walking, say, in a remote part of the country, and sud- 
denly meet with a friend. At this I am naturally surprised. 
Yet if the view be correct that we cannot properly speak 
about events in themselves being probable or improbable, 
but only say this of our conjectures about them, how do we 
explain this? We had formed no conjecture beforehand, 
for we were not thinking about anything of the kind, but 
yet few would fail to feel surprise at such an incident. 

The reply might fairly be made that we had formed 
such anticipations tacitly. On any such occasion every 
one unconsciously divides things into those which are known 
to him and those which are not. During a considerable 



288 Probability before and after the event. [CHAP. xji. 

previous period a countless number of persons had. met us, 
and all fallen into the list of the unknown to us. There 
was nothing to remind us of having formed the anticipa- 
tion or distinction at all, until it was suddenly called out 
into vivid consciousness by the exceptional event. The 
words which we should instinctively use in our surprise seem 
to show this: 'Who would have thought of seeing you 
here ?' viz. Who would have given any weight to the latent 
thought if it had been called out into consciousness before- 
hand ? We put our words into the past tense, showing that 
we have had the distinction lurking in our minds all the 
time. We always have a multitude of such ready-made 
classes of events in our minds, and when a thing happens to 
fall into one of those classes which are very small we cannot 
help noticing the fact. 

Or suppose I am one of a regiment into which a shot 
flies, and it strikes me, and me only. At this I am sur- 
prised, and why? Our common language will guide us to the 
reason. ' How strange that it should just have hit me of all 
men !' We are thinking of the very natural two-fold division 
of mankind into, ourselves, and everybody else ; our surprise 
is again, as it were, retrospective, and in reference to this 
division. No anticipation was distinctly formed, because 
we did not think beforehand of the event, but the event, 
when it has happened, is at once assigned to its appropriate 
class. 

11. This view is confirmed by the following considera- 
tions. Tell the story to a friend, and he will be a little 
surprised, but less so than we were, his division in this 
particular case being, his friends (of whom we are but one), 
and the rest of mankind. It is not a necessary division, but 
it is the one which will be most likely suggested to him. 

Tell it again to a perfect stranger, and his division being 



SECT. 12.] Probability before and after the event. 28D 

different (viz. we falling into the majority) we shall fail to 
make him perceive that there is anything at all remarkable 
in the event. 

It is not of course attempted in these remarks to justify 
our surprise in every case in which it exists. Different 
persons might be differently affected in the cases supposed, 
and the examples are therefore given mainly for illustration. 
Still on principles already discussed (Ch. vi. 32) we might 
expect to find something like a general justification of the 
amount of surprise. 

12. The answer commonly given in these cases is 
confined to attempting to show that the surprise should not 
arise, rather than to explaining how it does arise. It takes 
the following form, ' You have no right to be surprised, for 
nothing remarkable has really occurred. If this particular 
thing had not happened something equally improbable 
must. If the shot had not hit you or your friend, it must 
have hit some one else who was A priori as unlikely to be 
hit/ 

For one thing this answer does not explain the fact 
that almost every one is surprised in such cases, and sur- 
prised somewhat in the different proportions mentioned 
above. Moreover it has the inherent unsatisfactoriness 
of admitting that something improbable has really hap- 
pened, but getting over the difficulty by saying that all the 
other alternatives were equally improbable. A natural in- 
ference from this is that there is a class of things, in them- 
selves really improbable, which can yet be established upon 
very slight evidence. Butler accepted this inference, and 
worked it out to the strange conclusion given above. Mill 
attempts to avoid it by the consideration of the very differ- 
ent values to be assigned to improbability before and after 
the event. Some further discussion of this point will be 

V. 19 



290 The Relativity of Probability. [CHAP, xil, 

found in the chapter on Fallacies, and in that on the Credi- 
bility of Extraordinary Stories. 

13. In connection with the subject at present under 
discussion we will now take notice of a distinction which we 
shall often find insisted on in works on Probability, but to 
which apparently needless importance has been attached. 
It is frequently said that probability is relative, in the sense 
that it has a different value to different persons according 
to their respective information upon the subject in ques- 
tion. For example, two persons, A and B, are going to draw 
a ball from a bag containing 4 balls : A knows that the 
balls are black and white, but does not know more ; 
B knows that three are black and one white. It would 
be said that the probability of a white ball to A is ^, and 

toJH 

When however we regard the subject from the material 
standing point, there really does not seem to me much more 
in this than the principle, equally true in every other science, 
that our inferences will vary according to the data we as- 
sume. We might on logical grounds with almost equal 
propriety speak of the area of a field or the height of a 
mouxitain being relative, and therefore having one value to 
one person and another to another. The real meaning of the 
example cited above is this : A supposes that he is choosing 
white at random out of a series which in the long run would 
give white and black equally often ; B supposes that he 
is choosing white out of a series wtich in the long run would 
give three black to one white. By the application, there- 
fore, of a precisely similar rule they draw different conclu- 
sions ; but so they would under the same circumstances in 
any other science. If two men are measuring the height of 
a mountain, and one supposes his base to be 1000 feet, 
whilst the other takes it to be 1001, they would of course 



SECT. 14.] The Relativity of Probability. 291 

form different opinions about the height. The science of 
mensuration is not supposed to have anything to do with 
the truth of the data, but assumes them to have been cor- 
rectly taken ; why should not this be equally the case with 
Probability, making of course due allowance for the peculiar 
character of the data with which it is concerned ? 

14. This view of the relativeness of probability is 
connected, as it appears to me, with the subjective view of 
the science, and is indeed characteristic of it. It seems a 
fair illustration of the weak side of that view, that it should 
lead us to lay any stress on such an expression. As was 
fully explained in the last chapter, in proportion as we work 
out the Conceptualist principle we are led away from the 
fundamental question of the material logic, viz. Is our belief 
actually correct, or not ? and, if the former, to what extent 
and degree is it correct ? We are directed rather to ask, 
What belief does any one as a matter of fact hold ? And, 
since the belief thus entertained naturally varies according 
to the circumstances and other sources of information of the 
person in question, its relativeness comes to be admitted as 
inevitable, or at least it is not to be wondered at if such 
should be the case. 

On our view of Probability, therefore, its ' relativeness ' in 
any given case is a misleading expression, and it will be 
found much preferable to speak of the effect produced by 
variations in the nature and amount of the data which we 
have before us. Now it must be admitted that there are 
frequently cases in our science in which such variations are 
peculiarly likely to be found. For instance, I am expecting 
a friend who is a passenger in an ocean steamer. There are 
a hundred passengers on board, and the crew also numbers 
a hundred. I read in the papers that one person was lost by 
falling overboard ; my anticipation that it was my friend who 



292 The Relativity of Pliability. [CHAP. xn. 

was lost is but small, of course. On turning to another 
paper, I see that the man who was lost was a passenger, not 
one of the crew; my slight anxiety is at once doubled. But 
another account adds that it was an Englishman, and on 
that line at that season the English passengers are known 
to be few ; I at once begin to entertain decided fears. And 
so on, every trifling bit of information instantly affecting my 
expectations. 

15. Now since it is peculiarly characteristic of Proba- 
bility, as distinguished from Induction, to be thus at the 
mercy, so to say, of every little fact that may be floating 
about when we are in the act of forming our opinion, what 
can be the harm (it may be urged) of expressing this state 
of things by terming our state of expectation relative ? 

There seem to me to be two objections. In the first place, 
as just mentioned, we are induced to reject such an expres- 
sion on grounds of consistency. It is inconsistent with the 
general spirit and treatment of the subject hitherto adopted, 
and tends to divorce Probability from Inductive logic instead 
of regarding them as cognate sciences. We are aiming at 
truth, as far as that goal can be reached by our road, and 
therefore we dislike to regard our conclusions as relative in 
any other sense than that in which truth itself may be said 
to be relative. 

In the second place, this condition of unstable assent, 
this constant liability to have our judgment affected, to any 
degree and at any moment, by the accession of new know- 
ledge, though doubtless characteristic of Probability, does 
not seem to me characteristic of it in its sounder and more 
legitimate applications. It seems rather appropriate to a 
precipitate judgment formed in accordance with the rules, 
than a strict example of their natural employment. Such 
precipitate judgments may occur in the case of ordinary de- 



SECT. 16.] The Relativity of Probability. 293 

ductive conclusions. In the practical exigencies of life we 
are constantly in the habit of forming a hasty opinion with 
nearly full confidence, at any rate temporarily, upon the 
strength of evidence which we must well know at the time 
cannot be final. We wait a short time, and something else 
turns up which induces us to alter our opinion, perhaps to 
reverse it. Here our conclusions may have been perfectly 
sound under the given circumstances, that is, they may be 
such as every one else would have drawn who was bound to 
make up his mind upon the data before us, and they are 
unquestionably ' relative' judgments in the sense now under 
discussion. And yet, I think, every one would shrink from 
so terming them who wished systematically to carry out the 
view that Logic was to be regarded as an organon of truth. 

16. In the examples of Probability which we have 
hitherto employed, we have for the most part assumed that 
there was a certain body of statistics set before us on which 
our conclusion was to rest. It was assumed, on the one 
hand, that no direct specific evidence could be got, so that 
the judgment was really to be one of Probability, and to rest 
on these statistics ; in other words, that nothing better than 
them was available for us. But it was equally assumed, on 
the other hand, that these statistics were open to the obser- 
vation of every one, so that we need not have to put up with 
anything inferior to them in forming our opinion. In other 
words, we have been assuming that here, as in the case of 
most other sciences, those who have to draw a conclusion 
start from the same footing of opportunity and information. 
This, for instance, clearly is or ought to be the case when 
\ve are concerned with games of chance ; ignorance or mis- 
apprehension of the common data is never contemplated 
there. So with the statistics of life, or other insurance : so 
long as our judgment is to be accurate (after its fashion) or 



294 The Relativity of JV-W/iT/'j/. [CHAP. xii. 

justifiable, the common tables of mortality are all that any 
one has to go by. 

17. It is true that in the case of a man's prospect of 
death we should each qualify our judgment by what we 
knew or reasonably supposed as to his health, habits, pro- 
fession, and so on, and should thus arrive at varying esti- 
mates. But no one could justify his own estimate without 
appealing explicitly or implicitly to the statistical grounds on 
which he had relied, and if these were not previously avail- 
able to other persons, he must now set them before their 
notice. In other words, the judgments we entertain, here as 
elsewhere, are only relative so long as we rest them on 
grounds peculiar to ourselves. The process of justification, 
which I consider to be essential to logic, has a tendency to 
correct such individualities of judgment, and to set all ob- 
servers on the same basis as regards their data. 

It is better therefore to regard the conclusions of Proba- 
bility as being absolute and objective, in the same sense as, 
though doubtless in a far less degree than, they are in Induc- 
tion. Fully admitting that our conclusions will in many 
cases vary exceedingly from time to time by fresh accessions 
of knowledge, it is preferable to regard such fluctuations of 
assent as partaking of the nature of precipitate judgments, 
founded on special statistics, instead of depending only on 
those which are common to all observers. In calling such 
judgments precipitate it is not implied that there is any 
blame in entertaining them, but simply that, for one reason 
or another, we have been induced to form them without 
waiting for the possession of the full amount of evidence, 
statistical or otherwise, which might ultimately be looked 
for. This explanation will suit the facts equally well, and is 
more consistent with the general philosophical position main- 
tained in this work* 



CHAPTER XIII. 

ON THE CONCEPTION AND TREATMENT OF MODALITY. 

1. THE reader who knows anything of the scholastic 
Logic will have perceived before now that we have been 
touching in a variety of places upon that most thorny and 
repulsive of districts in the logical territory ; modality. It 
will be advisable, however, to put together, somewhat more 
definitely, what has to be said upon the subject. I propose, 
therefore, to devote this chapter to a brief account of the 
principal varieties of treatment which the modals have re- 
ceived at the hands of professed logicians. 

It must be remarked at the outset that the sense in 
which modality and modal propositions have been at various 
times understood, is by no means fixed and invariably the 
same. This diversity of view has arisen partly from cor- 
responding differences in the view taken of the province and 
nature of logic, and partly from differences in the philo- 
sophical and scientific opinions entertained as to the con- 
stitution and order of nature. In later times, moreover, 
another very powerful agent in bringing about a change in 
the treatment of the subject must be recognized in the 
gradual and steady growth of the theory of Probability, as 
worked out by the mathematicians from their own point of 
view. 

2. In spite, however, of these differences of treatment, 
there has always been some community of subject-matter in 
the discussions upon this topic. There has almost always 



296 Modality. [CHAP. xm. 

been some reference to quantity of belief; enough perhaps 
to justify De Morgan's 1 remark, that Probability was "the 
unknown God whom the schoolmen ignorantly worshipped 
when they so dealt with this species of enunciation, that it 
was said to be beyond human determination whether they 
most tortured the modals, or the modals them." But this 
reference to quantity of belief has sometimes been direct and 
immediate, sometimes indirect and arising out of the nature 
of the subject-matter of the proposition. The fact is, that 
that distinction between the purely subjective and purely ob- 
jective views of logic, which I have endeavoured to bring out 
into prominence in the eleventh chapter, was not by any 
means clearly recognized in early times, nor indeed before 
the time of Kant, and the view to be taken of modality 
naturally shared in the consequent confusion. This will, I 
hope, be made clear in the course of the following chapter, 
which is intended to give a brief sketch of the principal 
different ways in which the modality of propositions has 
been treated in logic. As it is not proposed to give any- 
thing like a regular history of the subject, there will be no 
necessity to adhere to any strict sequence of time, or to 
discuss the opinions of any writers, except those who may be 
taken as representative of tolerably distinct views. The 
outcome of such investigation will be, I hope, to convince 
the reader (if, indeed, he had not come to that conviction 
before), that the logicians, after having had a long and fair 
trial, have failed to make anything satisfactory out of this 
subject of the modals by their methods of enquiry and treat- 
ment ; and that it ought, therefore, to be banished entirely 
from that science, and relegated to Probability. 

3. From the earliest study of the syllogistic process 
it was seen that, complete as that process is within its own 

1 Formal Logic, p. 232. 



SECT. 4.] Modality. 297 

domain, the domain, at any rate under its simplest treatment, 
is a very limited one. Propositions of the pure form, All 
(or some) A is (or is not) .B, are found in practice to form 
but a small portion even of our categorical statements* We 
are perpetually meeting with others which express the re- 
lation of B to A with various degrees of necessity or pro- 
bability; e.g. A must be B, A may be jB; or the effect of 
such facts upon our judgment, e.g. I am perfectly certain 
that A is 5, I think that A may be B ; with many others of 
a more or less similar type. The question at once arises, 
How are such propositions to be treated ? It does not seem 
to have occurred to the old logicians, as to some of their suc- 
cessors in modern times, simply to reject all consideration of 
this topic. Their faith in the truth and completeness of 
their system of inference was far too firm for them to sup- 
pose it possible that forms of proposition universally recog- 
nized as significant in popular speech, and forms of inference 
universally recognized there as valid, were to be omitted 
because they were inconvenient or complicated. 

4. One very simple plan suggests itself, and has in- 
deed been repeatedly advocated, viz. just to transfer all that 
is characteristic of such propositions into that convenient re- 
ceptacle for what is troublesome elsewhere, the predicate 1 . 
Has not another so-called modality been thus got rid of 2 ? 

1 This appears to be the purport draws apparently no such distinction 

of some statements in a very con- as that between the true and false 

fused passage in Whately's Logic modality referred to in the next note. 

<Bk. II., ch. rv. 1). "A modal What is really surprising is that 

proposition may be stated as a pure even Hamilton puts the two (the true 

one by attaching the mode to one of and the false modality) upon the 

the terms, and the proposition will same footing. "In regard to these 

in all respects fall under the forego- [the former] the case is precisely the 

ing rules;...* It is probable that all same; the mode is merely a part of 

knowledge is useful;' probably use- the predicate." Logic, i. 257. 

ful' is here the predicate." He * I allude of course to such ex- 



298 Modality. [CHAP. xin. 

and has it not been attempted by the same device to abolish 
the distinctive characteristic of negative propositions, viz. by 
shifting the negative particle into the predicate ? It must 
be admitted that, up to a certain point, something may be 
done in this way. Given the reasoning, l Those who take 
arsenic will probably die ; A has taken it, therefore he will 
probably die;' it is easy to convert this into an ordinary 
syllogism of the pure type, by simply wording the major, 
' Those who take arsenic are people-who-will-probably-die/ 
when the conclusion follows in the same form, 'A is one 
who-will-probably-die. J But this device will only carry us 
a very little way. Suppose that the minor premise also is 
of the same modal description, e.g. 'A has probably taken 
arsenic,' and it will be seen that we cannot relegate the 
modality here also to the predicate without being brought to 
a stop by finding that there are four terms in the syllogism. 

But even if there were not this particular objection, it 
does not appear that anything is to be gained in the way of 
intelligibility or method by such a device as the above. For 

amples as ' A killed B unjustly, ' in ' material modality ' and the genuine 

which the killing of B by A was kind * formal modality '. The former 

sometimes said to be asserted not included all the cases in which the 

simply but with a modification. modification belonged by right either 

(Hamilton's Logic, i. 256.) It is to the predicate or to the subject; 

obvious that the modification in the latter was reserved for the cases 

such cases is by rights merely a in which the modification affected 

part of the predicate, there being no the real conjunction of the predicate 

formal distinction between *A is the with the subject. (Keckermann, 

killer of B y and 'A is the unjust Sy sterna Logica, Lib. n. ch. 3.) It 

killer of B. 3 Indeed some logicians was, I believe, a common scholastic 

who were too conservative to reject distinction. 

the generic name of modality in this For some account of the dispute 

application adopted the common ex- as to whether the negative particle 

pedient of introducing a specific dis- was to be considered to belong to 

tinction which did away with its the copula or to the predicate, see 

meaning, terming the spurious kind Hamilton's Logic, i. 253. 



SECT. 5.] Modality. 29& 

what is meant by a modal predicate, by the predicate 
' probably mortal/ for instance, in the proposition 'All poison- 
ings by arsenic are probably mortal' ? If the analogy with 
ordinary pure propositions is to hold good, it must be a 
predicate referring to the whole of the subject, for the sub- 
ject is distributed. But then we are at once launched into 
the difficulties discussed in a former chapter (Ch. VI. 19 
25), when we attempt to justify or verify the application of 
the predicate. We have to enquire (at least on the view 
adopted in this work) whether the application of the pre- 
dicate ' probably mortal ' to the whole of the subject, really 
means at bottom anything else than that the predicate 
' mortal ' is to be applied to a portion (more than half) of the 
members denoted by the subject. When the transference of 
the modality to the predicate raises such intricate questions 
as to the sense in which the predicate is to be interpreted, 
there is surely nothing gained by the step. 

5. A second, and more summary way of shelving all 
difficulties of the subject, so far at least as logic, or the 
writers upon logic, are concerned, is found by simply denying 
that modality has any connection whatever with logic. This 
is the course adopted by many modern writers, for instance, 
by Hamilton and Mansel, in reference to whom one cannot 
help remarking that an unduly large, portion of their logical 
writings seems occupied with telling us what does not belong 
to logic. They justify their rejection on the ground that 
the mode belongs to the matter, and must be determined by 
a consideration of the matter, and therefore is extralogical. 
To a certain extent I agree with their grounds of rejection, 
for (as explained in Chapter vi.) it is not easy to see how 
the degree of modality of any proposition, whether premise 
or conclusion, can be justified without appeal to the matter. 
But then questions of justification, in any adequate sense of 



300 Modality. [CHAP. xiu. 

the term, belong to a range of considerations somewhat alien 
to Hamilton's and Hansel's way of regarding the science. 
The complete justification of our inferences is a matter 
which involves their truth or falsehood, a point with which 
these writers do not much concern themselves, being only 
occupied with the consistency of our reasonings, not with 
their conformity with fact. Were I speaking as a Hamil- 
tonian I should say that modality is formal rather than 
material, for though we cannot justify the degree of our 
belief of a proposition without appeal to the matter, we can 
to a moderate degree of accuracy estimate it without any 
such appeal; and this would seem to be quite enough to 
warrant its being regarded as formal. 

It must be admitted that Hamilton's account of the 
matter when he is recommending the rejection of the modals, 
is not by any means clear and consistent. He not only fails, 
as already remarked, to distinguish between the formal and 
the material (in other words, the true and the false) mo- 
dality ; but when treating of the former he fails to distinguish 
between the extremely diverse aspects of modality when 
viewed from the Aristotelian and the Kantian stand-points. 
Of the amount and significance of this difference we shall 
speak presently, but it may be just pointed out here that 
Hamilton begins (Vol. I. p. 257) by rejecting the modals on 
the ground that the distinctions between the necessary, the 
contingent, the possible, and the impossible, must be wholly 
rested on an appeal to the matter of the propositions, in 
which he is, I think, quite correct. But then a little further 
on (p. 260), in explaining ' the meaning of three terms which 
are used in relation to pure and modal propositions/ he gives 
the widely different Kantian, or threefold division into the 
apodeictic, the assertory, and the problematic. He does not 
take the precaution of pointing out to his hearers the very 



SECT. 6.J Modality. 301 

different general views of logic from which these two accounts 
of modality spring 1 . 

6. There is one kind of modal -ilj-ji-m which it 
would seem unreasonable to reject on the ground of its not 
being formal, and which we may notice in passing. The 
premise ' Any A is probably B' is equivalent to ' Most A are 
B. 1 Now it is obvious that from two such premises as ' Most 
A are J?/ ' Most A are C,' we can deduce the consequence, 
'Some G are B.' Since this holds good whatever may be 
the nature of A, J5, and (7, it is, according to ordinary usage 
of the term, a formal syllogism. Mansel, however, refuses to 
admit that any such syllogisms belong to formal logic. His 
reasons are given in a rather elaborate review 2 and criticism 
of some of the logical works of De Morgan, to whom the 
introduction of 'numerically definite syllogisms' is mainly 
due. Mansel does not take the particular example given 
above, as he is discussing a somewhat more comprehensive 
algebraic form. He examines it in a special numerical 
example 3 : 18 out of 21 Fs are X\ 15 out of 21 Fs are Z\ 
the conclusion that 12 Zs are X is rejected from formal logic 
on the ground that the arithmetical judgment involved is 
synthetical, not analytical, and rests upon an intuition of 
quantity. We cannot enter upon any examination of these 

1 He has also given a short 2 Letters, Lectures and Reviews, 

discussion of the subject elsewhere p. 61. Elsewhere in the review (p. 

(Discussicms, Ed. n. p. 702), in which 45) he gives what appears to me a 

a somewhat different view is taken. somewhat different decision. 
The modes are indeed here admitted 3 It must be remembered that 

into logic, but only in so far as they this is not one of the proportional 

fall by subdivision under the relation propositions with which we have 

of genus and species, which is of been concerned in previous chapters : 

course tantamount to their entire it is meant that there are exactly 21 

rejection ; for they then differ in no 3fs, of which just 18 are X, not that 

essential way from any other exam- on the average 18 out of 21 may be 

pies of that relation. 00 regarded. 



302 Modality. [CHAP. xm. 

reasons here; but it may merely be remarked that his 
criticism demands the acceptance of the Kantian doctrines 
us to the nature of arithmetical judgments, and that it would 
be better to base the rejection not on the ground that the 
syllogism is not formal, but on the ground that it is not 
analytical. 

7. There is another and practical way of getting rid 
of the perplexities of modal reasoning which must be noticed 
here. It is the resource of ordinary reasoners rather than the 
decision of professed logicians 1 , and, like the first method of 
evasion already pointed out in this chapter, is of very partial 
application. It consists in treating the premises, during the 
process of reasoning, as if they were pure, and then re- 
introducing the modality into the conclusion, as a sort of 
qualification of its full certainty. When each of the pre- 
mises is nearly certain, or when from any cause we are not 
concerned with the extent of their departure from full cer- 
tainty, this rough expedient will answer well enough. It is, 
I apprehend, the process which passes through the minds of 
most persons in such cases, in so far as they reason consciously. 
They would, presumably, in such an example as that pre- 
viously given ( 4), proceed as if the premises that * those 
who take arsenic will die/ and that 'the man in question 
has taken it/ were quite true, instead of being only probably 
true, and they would consequently draw the conclusion that 
4 he would die/ But bearing in mind that the premises are 
not certain, they would remember that the conclusion was 
only to be held with a qualified assent. This they would 

1 I consider however, as I have this, though they looked at the mat- 
said further on (p. 320), that the treat- ter from a different point of view, 
ment in the older logics of Probable and expressed it in very different 
syllogisms, and Dialectic syllogisms, 
-came to somewhat the same thing as 



SECT. 8.] Modality. 303 

express quite correctly, if the mere nature and not the 
degree of that assent is taken into account, by saying that 
'he is likely to die/ In this case the modality is rejected 
temporarily from the premises to be reintroduced into the 
conclusion. 

It is obvious that such a process as this is of a very 
rough and imperfect kind. It does, in fact, omit from accu- 
rate consideration just the one point now under discussion. 
It takes no account of the varying shades of expression by 
which the degree of departure from perfect conviction is 
indicated, which is of course the very thing with which 
modality is intended to occupy itself. At best, therefore, it 
could only claim to be an extremely rude way of deciding 
questions, the accurate and scientific methods of treating 
which are demanded of us. 

8. In any employment of applied logic we have of 
course to go through such a process as that just mentioned. 
Outside of pure mathematics it can hardly ever be the case 
that the premises from which we reason are held with abso- 
lute conviction. Hence there must be a lapse from absolute 
conviction in the conclusion. But we reason on the hypo- 
thesis that the premises are true, and any trifling defection 
from certainty, of which we may be conscious, is mentally 
reserved as a qualification to the conclusion. But such con- 
siderations as these belong rather to ordinary applied logic ; 
they amount to nothing more than a caution or hint- to be 
borne in mind when the rules of the syllogism, or of in- 
duction, are applied in practice. When, however, we are 
treating of modality, the extent of the defection from full 
certainty is supposed to be sufficiently great for our language 
to indicate and appreciate it. What we then want is of 
course a scientific discussion of the principles in accordance 
with which this departure is to be measured and expressed, 



304 Modality. [CHAP. xui. 

both in our premises and in our conclusion. Such a plan 
therefore for treating modality, as the one under discussion, 
is just as much a banishment of it from the field of real 
logical enquiry, as if we had determined avowedly to reject it 
from consideration. 

9. Before proceeding to a discussion of the various 
ways in which modality may be treated by those who admit 
it into logic, something must be said to clear up a possi- 
ble source of confusion in this part of the subject. In the 
cases with which we have hitherto been mostly concerned, 
in the earlier chapters of this work, the characteristic of 
modality (for in this chapter we may with propriety use this 
logical term) has generally been found in singular and par- 
ticular propositions. It presented itself when we had to 
judge of individual cases from a knowledge of the average, 
and was an expression of the fact that the proposition re- 
lating to these individuals referred to a portion only of the 
whole class from which the average was taken. Given that 
of men of fifty-five, three out of five will die in the course of 
twenty years, we have had to do with propositions of the 
vague form, ' It is probable that AB (of that age) will die/ 
or of the more precise form, ' It is three to two that AB will 
die,' within the specified time. Here the modal proposition 
naturally presents itself in the form of a singular or par- 
ticular proposition. 

10. But when we turn to ordinary logic we may find 
wdversal propositions spoken of as modal. This must mostly 
be the case with those which are termed necessary or im- 
possible, but it may also be the case with the probable. We 
may meet with the form 'All X is probably F.' Adopting 
the same explanation here as has been throughout adopted in 
analogous cases, we must say that what is meant by the 
modality of such a proposition is the proportional number of 



SECT. 11.] Modality. 305 

times in which the universal proposition would be correctly 
made. And in this there is, so far, no difficulty. The only 
difference is that whereas the justification of the former, viz. 
the particular or individual kind of modal, was obtainable 
within the limits of the universal proposition which included 
it, the justification of the modality of a universal proposition 
has to be sought in a group or succession of other propo- 
sitions. The proposition has to be referred to some group of 
similar ones and we have to consider the proportion of cases 
in which it will be true. But this distinction is not at all 
fundamental. 

It is quite true that universal propositions from their 
nature are much less likely than individual ones to be justi- 
fied, in practice, by such appeal. But, as has been already 
frequently pointed out, we are not concerned with the way 
in which our propositions are practically obtained, nor with 
the way in which men might find it most natural to test 
them ; but with that ultimate justification to which we ap- 
peal in the last resort, and which has been abundantly shown 
to be of a statistical character. When, therefore, we say that 
' it is probable that all X is F,' what we mean is, that in 
more than half the cases we come across we should be right 
in so judging, and in less than half the cases we should be 
wrong. 

11. It is at this step that the possible ambiguity is 
encountered. When we talk of the chance that All X is F, 
we contemplate or imply the complementary chance that it is 
not so. Now this latter alternative is not free from am- 
biguity. It might happen, for instance, in the cases of failure, 
that no X is F, or it might happen that some X, only, is not 
F; for both of these suppositions contradict the original pro- 
position, and are therefore instances of its failure. In prac- 
tice, no doubt, we should have various recognized rules and 
v. 20 



Modality. [CHAP. xm. 

inductions to fall back upon in order to decide between these 
alternatives, though, of course, the appeal to them would be 
in strictness extralogical. But the mere existence of such an 
ambiguity, and the fact that it can only be cleared up by 
appeal to the subject-matter, are in themselves no real 
difficulty in the application of the conception of modality to 
universal propositions as well as to individual ones. 

12. Having noticed some of the ways in which the 
introduction of modality into logic has been evaded or Te- 
jected, we must now enter into a brief account of its treat- 
ment by those who have more or less deliberately admitted 
its claims to acceptance. 

The first enquiry will be, What opinions have been held 
as to the nature of modality ? that is, Is it primarily an af- 
fection of the matter of the proposition, and, if not, what is it 
exactly ? In reference to this enquiry it appears to me, as 
already remarked, that amongst the earlier logicians no such 
clear and consistent distinction between the subjective and 
objective views of logic as is now commonly maintained, can 
be detected 1 . The result of this appears in their treatment 
of modality. This always had some reference to the sub- 
jective side of the proposition, viz. in this case to the nature 
or quantity of the belief with which it was entertained ; but 
it is equally clear that this characteristic was not estimated 
at first hand, so to say, and in itself, but rather from a con- 
sideration of the matter determining what it should be. The 
commonly accepted scholastic or Aristotelian division, for 
instance, is into the necessary, the contingent, the possible, 
and the impossible. This is clearly a division according to 

1 The distinction is however by sity,says, " certitudo ad cognitionem 

no means entirely neglected. Thus spectat: necessitas vero est in re" 

Smiglecius, when discussing the mo- (Disputationes ; Disp. xm. , Quasst. 

dal affections of certainty and neces- xn.). 



SECT. 13.] Modality. 307 

the matter almost entirely, for on the purely mental side the 
necessary and the impossible would be just the same ; one 
implying full conviction of the truth of a proposition, and the 
other of that of its contradictory. So too, on the same side, it 
would not be easy to distinguish between the contingent and 
the possible. On the view in question, therefore, the modality 
of a proposition was determined by a reference to the nature 
of the subject-matter. In some propositions the nature of 
the subject-matter decided that the predicate was necessarily 
joined to the subject ; in others that it was impossible that 
they should be joined ; and so on. * 

13. The artificial character of such a four- fold division 
will be too obvious to modern minds for it to be necessary to 
criticize it. A very slight study of nature and consequent 
appreciation of inductive evidence suffice to convince us that 
those uniformities upon which all connections of phenomena, 
whether called necessary or contingent, depend, demand ex- 
tremely profound and extensive enquiry ; that they admit of 
no such simple division into clearly marked groups ; and 
that, therefore, the pure logician had better not meddle with 
them 1 . 

The following extract from Grote's Aristotle (Vol. I. p. 
192) will serve to show the origin of this four-fold division, 
its conformity with the science of the day, and consequently 
its utter want of conformity with that of our own time : 
"The distinction of Problematical and Necessary Propositions 
corresponds, in the mind of Aristotle, to that capital and 
characteristic doctrine of his Ontology and Physics, already 
touched on in this chapter. He thought, as we have seen, 
that in the vast circumferential region of the Kosmos, from 

1 It may be remarked that Whate- gent matter, without any apparent 
ly (Logic, Bk. ir. oh. n. 2) speaks suspicion that they belong entirely 
of necessary, impossible and contra- to an obsolete point of view. 

202 



308 Modality. [CHAP. 

the outer sidereal sphere down to the lunar sphere, celestial 
substance was a necessary existence and energy, sempiternal 
and uniform in its rotations and influence ; and that through 
its beneficent influence, pervading the concavity between the 
lunar sphere and the terrestrial centre (which included the 
four elements with their compounds) there prevailed a regu- 
larizing tendency called Nature ; modified, however, and 
partly counteracted by independent and irregular forces called 
Spontaneity and Chance, essentially unknowable and unpre- 
dictable. The irregular sequences thus named by Aristotle 
were the objective correlate of the Problematical Proposition 
in Logic. In these sublunary sequences, as to future time, 
way or may not, was all that could be attained, even by the 
highest knowledge ; certainty, either of affirmation or nega- 
tion, was out of the question. On the other hand, the neces- 
sary and uniform energies of the celestial substance, formed 
the objective correlate of the Necessary Proposition in Logic ;. 
this substance was not merely an existence, but an existence 
necessary and unchangeable... he considers the Problematical 
Proposition in Logic to be not purely subjective, as an 
expression of the speaker's ignorance, but something more, 
namely, to correlate with an objective essentially unknowable 
to all." 

14. Even after this philosophy began to pass away, 
the divisions of modality originally founded upon it might 
have proved, as De Morgan has remarked 1 , of considerable 
service in mediaeval times. As he says, people were much 
more frequently required to decide in one way or the other 
upon a single testimony, without there being a sufficiency of 
specific knowledge to test the statements made. The old 
logician " did not know but that any day of the week might 
bring from Cathay or Tartary an account of men who ran on 

1 Formal Logic, p. 233, 



SECT.. 15.] Modality. 309 

four wheels of flesh and blood, or grew planted in the ground, 
like Polydorus in the ^Eneid, as well evidenced as a great 
many nearly as marvellous stories." Hence, in default of 
better inductions, it might have been convenient to make 
rough classifications of the facts which were and which were 
not to be accepted on testimony (the necessary, the impos- 
sible, &c.), and to employ these provisional inductions (which 
is all we should now regard them) as testing the stories 
which reached him. Propositions belonging to the class of 
the impossible might be regarded as having an antecedent 
presumption against them so great as to prevail over almost 
any testimony worth taking account of, and so on. 

15. But this old four-fold division of modals con- 
tinued to be accepted and perpetuated by the logicians long 
after all philosophical justification for it had passed away. 
So far as I have been able to ascertain, scarcely any logician 
of repute or popularity before Kant, was bold enough to 
make any important change in the way of regarding them 1 . 
Even the Port-Royal Logic, founded as it is on Cartesianism, 
repeats the traditional statements, though with extreme 
brevity. This adherence to the old forms led, it need not be 
remarked, to considerable inconsistency and confusion in 
many cases. These forms were founded, as we have seen, on 
an objective view of the province of logic, and this view was 

1 The subject was sometimes al- tant are referred to in one passage 

together omitted, as by Wolf. He (Philosophia Rationalis sive Logica, 

says a good deal however about pro- 593). 

bable propositions and syllogisms, Lambert stands quite apart. In 

and, like Leibnitz before him, looked this respect, as in most others where 

forward to a "logica probabilium" mathematical conceptions and sym- 

as something new and desirable. I bols are involved, his logical attitude 

imagine that he had been influenced is thoroughly unconventional. See, 

by the writers on Chances, as of the for instance, his chapter 'Von dem 

few who had already treated that Wahrscheinlichen', in his Neues 

subject nearly all the most impor- Organon. 



310 Modality. [CHAP. XIIL 

by no means rigidly carried out in many cases. In fact it 
was beginning to be abandoned, to an extent and in direc- 
tions which we have not opportunity here to discuss, before 
the influence of Kant was felt. Many, for instance, added to 
the list of the four, by including the true and the false; 
occasionally also the probable, the supposed, and the certain 
were added. This seems to show some tendency towards 
abandoning the objective for the subjective view, or at least 
indicates a hesitation between them. 

16. With Kant's view of modality almost every one is. 
familiar. He divides judgments, under this head, into the 
apodeictic, the assertory, and the problematic. We shall have 
to say something about the number and mutual relations of 
these divisions presently ; we are now only concerned with 
the general view which they carry out. In this respect it will 
be obvious at once what a complete change of position has 
been reached. The * necessary' and the 'impossible* de- 
manded an appeal to the matter of a proposition in order to 
recognize them ; the ' apodeictic ' and the ' assertory ', on the 
other hand, may be true of almost any matter, for they 
demand nothing but an appeal to our consciousness in order 
to distinguish between them. Moreover, the distinction 
between the assertory and the problematic is so entirely 
subjective and personal, that it may vary not only between 
one person and another, but in the case of the same person 
at different times. What one man knows to be true, another 
may happen to be in doubt about. The apodeictic judgment 
is one which we not only accept, but which we find ourselves 
unable to reverse in thought; the assertory is simply ac- 
cepted; the problematic is one about which we feel in 
doubt. 

This way of looking at the matter is the necessary out- 
come of the conceptualist or Kantian view of logic. It has 



SECT., 17.] Modality. 3H 

been followed by many /'.- not only by those who tnay 
be called followers of Kant, but by almost all who have felt 
his influence. Ueberweg, for instance, who is altogether at 
issue with Kant on some fundamental points, adopts it. 

17. The next question to be discussed is, How many 
subdivisions of modality are to be recognized ? The Aristo- 
telian or scholastic logicians, as we have seen, adopted a four- 
fold division. The exact relations of some of these to each 
other, especially the possible and the contingent, is an ex- 
tremely obscure point, and one about which the commenta- 
tors are by no means agreed. As, however, it seems tolerably 
clear that it was not consciously intended by the use of these 
four terms to exhibit a graduated scale of intensity of con- 
viction, their correspondence with the province of modern 
probability is but slight, and the discussion of them, there- 
fore, becomes rather a matter of special or antiquarian in- 
terest. De Morgan, indeed (Formal Logic, p. 232), says that 
the schoolmen understood by : 'Yj ' more likely than 
not, and by possible less likely than not. I do not know 
on what authority this statement rests, but it credits them 
with a much nearer approach to the modern views of proba- 
bility than one would have expected, and decidedly nearer than 
that of most of their successors 1 . The general conclusion at 
which I have arrived, after a reasonable amount of investiga- 
tion, is that there were two prevalent views on the subject. 
Some (e.g. Burgersdyck, Bk. I. ch. 32) admitted that there 
were at bottom only two kinds of modality ; the contingent 
and the possible being equipollent, as also the necessary and 
the impossible, provided the one asserts and the other 
denies. This is the view to which those would naturally 
be led who looked mainly to the nature of the subject-matter. 

1 I cannot find the slightest authority for the statement in the 
elaborate history of Logic by PrantL, 



812 Modality. [CHAP. xm. 

On the other hand, those who looked mainly at the form of 
expression, would be led by the analogy of the four forms of 
proposition, and the necessity that each of them should stand 
in definite opposition to each other, to insist upon a distinc- 
tion between the four modals 1 . They, therefore, endeavoured 
to introduce a distinction by maintaining (e.g. Crackanthorpe, 
Bk. III. ch. 11) that the contingent is that which now is but 
may not be, and the possible that which now is not but may 
be. A few appear to have made the distinction corre- 
spondent to that between the physically and the logically 
possible. 

18. When we get to the Kantian division we have 
reached much clearer ground. The meaning of each of these 
terms is quite explicit, and it is also beyond doubt that they 
have a more definite tendency in the direction of assigning a 
graduated scale of conviction. So long as they are rega,rded 
from a metaphysical rather than a logical standing point, 
there is much to be said in their favour. If we use intro- 
spection merely, confining ourselves to a study of the judg- 
ments themselves, to the exclusion of the grounds on which 
they rest, there certainly does seem a clear and well-marked 
distinction between judgments which we cannot even con- 
ceive to be reversed in thought ; those which we could 
reverse, but which we accept as true; and those which we 
merely entertain as possible. 

Regarded, however, as a logical division, Kant's arrange- 
ment seems to me of very little service. For such logical 
purposes indeed, as we are now concerned with, it really 
seems to resolve itself into a two-fold division. The dis- 
tinction between the apodeictic and the assertory will be 

1 "Hi qnatnor modi magnam titateet qualitate varietate" (WaUiti's 
oeneeri sclent analogiam habere cum Instit. Logic. Bk. n. ch. 8). 
quadruplici proposition am in quan- 



SECT. 19.] Modality. 313 

admitted, I presume, even by those who accept the meta- 
physical or psychological theory upon which it rests, to be a 
difference which concerns, not the quantity of belief with 
which the judgments are entertained, but rather the violence 
which would have to be done to the mind by the attempt to 
upset them. Each is fully believed, but the one can, and 
the other cannot, be controverted. The belief with which an 
assertory judgment is entertained is full belief, else it would 
not differ from the problematic ; and therefore in regard to 
the quantity of belief, as distinguished from the quality or 
character of it, there is no difference between it and the apo- 
deictic. It is as though, to offer an illustration, the index 
had been already moved to the top of the scale in the as- 
sertory judgment, and all that was done to convert this into 
an apodeictic one, was to clamp it there. The only logical 
difference which then remains is that between problematic 
and assertory, the former comprehending all the judgments 
as to the truth of which we have any degree of doubt, and 
the latter those of which we have no doubt. The whole 
range of the former, therefore, with which Probability is 
appropriately occupied, is thrown undivided into a single 
compartment. We can hardly speak of a ' division * where 
one class includes everything up to the boundary line, and 
the other is confined to that boundary line. Practically, 
therefore, on this view, modality, as the mathematical stu- 
dent of Probability would expect to find it, as completely 
disappears as if it were intended to reject it. 

19. By less consistent and systematic thinkers, and 
by those in whom ingenuity was an over prominent feature, 
a variety of other arrangements have been accepted or pro- 
posed There is, of course, spine justification for such attempts 
in the laudable desire to bring our logical forms into better 
harmony with ordinary thought and language. In practice, 



314 Modality. [CHAP. xm. 

as was pointed out in an earlier chapter, every one recog- 
nizes a great variety of modal forms, such as * likely/ 'very 
likely/ ' almost certainly/ and so on almost without limit in 
each direction. It was doubtless supposed that, by neglect- 
ing to make use of technical equivalents for some of these 
forms, we should lose our logical control over certain possible 
kinds of inference, and so far fall short even of the precision 
of ordinary thought. 

With regard to such additional forms, it appears to me 
that all those which have been introduced by writers who 
were uninfluenced by the Theory of Probability, have done 
little else than create additional confusion, as such writers do 
not attempt to marshal their terms in order, or to ascertain 
their mutual relations. Omitting, of course, forms obviously 
of material modality, we have already mentioned the true 
and the false; the probable, the supposed, and the certain. 
These subdivisions seem to have reached their climax at a 
very early stage in Occam (Prantl, in. 380), who held that a 
proposition might be modally affected by being ' vera, scita, 
falsa, ignota, scripta, prolata, concept a, credit^-, opiriata, du- 
bitata.' 

20. Since the growth of the science of Probability, 
logicians have had better opportunities of knowing what 
they had to aim at ; and, though it cannot be said that their 
attempts have been really successful, these are at any rate a 
decided improvement upon those of their predecessors. Dr 
Thomson 1 , for instance, gives a nine-fold division. He says 
that, arranging the degrees of modality in an ascending 
scale, we find that a judgment may be either possible, 
doubtful, probable, morally certain for the thinker himself, 
morally certain for a class or school, morally certain for all, 
physically certain with a limit, physically certain without 

1 Laws of Thought, 118. 



.SECT. 21.] Modality. 315 

limitation, and mathematically certain. Many other divi- 
sions might doubtless be mentioned, but, as every mathema- 
tician will recognize, the attempt to secure any general 
agreement in such a matter of arrangement is quite hopeless. 
It is here that the beneficial influence of the mathematical 
theory of Probability is to be gratefully acknowledged. As- 
soon as this came to be studied it must have been perceived 
that in attempting to mark off clearly from one another 
certain gradations of belief, we should be seeking for breaches 
in a continuous magnitude. In the advance from a slight 
presumption to a strong presumption, and from that to moral 
certainty, we are making a gradual ascent, in the course of 
which there are no natural halting-places. The proof of this 
continuity need not be entered upon here, for the materials 
for it will have been gathered from almost every chapter of 
this work. The reader need merely be reminded that the 
grounds of our belief, in all cases which admit of number and 
measurement, are clearly seen to be of this description ; and 
that therefore unless the belief itself is to be divorced from 
the grounds on which it rests, what thus holds as to their 
characteristics must hold also as to its own. 

It follows, therefore, that modality in the old sense of the 
word, wherein an attempt was made to obtain certain natural 
divisions in the scale of conviction, must be finally abandoned. 
All that it endeavoured to do can now be done incomparably 
better by the theory of Probability, with its numerical scale 
which admits of indefinite subdivision. None of the old sys- 
tems of division can be regarded as a really natural one ; 
those which admit but few divisions being found to leave the 
whole range of the probable in one unbroken class, and those 
which adopt many divisions lapsing into unavoidable vague- 
ness and uncertainty. 

21. Corresponding to the distinction between pure 



Modality. [CHAP. xin. 

-and modal propositions, but even more complicated and 
unsatisfactory in its treatment, was that between pure and 
modal syllogisms. The thing discussed in the case of the 
latter was, of course, the effect produced upon the conclusion 
in respect of modality, by the modal affection of one or both 
premises. It is only when we reach such considerations as 
these that we are at all getting on to the ground appropriate 
to Probability; but it is obvious that very little could be 
done with such rude materials, and the inherent clumsiness 
and complication of the whole modal system come out very 
clearly here. It was in reference probably to this complica- 
tion that some of the bitter sayings 1 of the schoolmen and 
others which have been recorded, were uttered. 

Aristotle has given an intricate investigation of this sub- 
ject, and his followers naturally were led along a similar 
track. It would be quite foreign to my purpose in the slight 
sketch in this chapter to attempt to give any account of 
these enquiries, even were I competent to do so ; for, as has 
been pointed out, the connection between the Aristotelian 
modals and the modern view of the nature of Probability, 
though real, is exceedingly slight. It need only be remarked 
that what was complicated enough with four modals to be 
taken account of, grows intricate beyond all endurance when 
such as the 'probable' and the 'true' and the 'false' have 
also to be assigned a place in the list. The following exam- 
ples 8 will show the kind of discussions with which the logi- 

1 "Haud scio magis ne doctri- 2 Smiglecii Disputationes, Ingol- 

nam modalium scholastic! exercue- stadt, 1618. 

rint, quam ea illos vexarit. Certe See also Prantl's Geschichte der 

usque adeo sudatum hie fuit, ut die- Logik (under Occam and Buridan) 

terio locus sit datus ; De modalibus for accounts of the excessive com- 

non gustdbit asinus. " Keckermann, plication which the subtlety of those 

JSyst. Log. Bk. n. ch. 3. learned schoolmen evolved out of 

such suitable materials. 



EOT. 23.] Modality, 317 

tans exercised themselves. 'Whether, with one premise 
ertain, and the other probable, a certain conclusion may be 
aferred': ' Whether, from the impossible, the necessary can 
e inferred'; 'Whether, with one premise necessary and the 
ther de inesse, the conclusion is necessary', and so on > 
ndlessly. 

22. On the Kantian view of modality the discussion 
f such kinds of syllogisms becomes at once decidedly more 
imple (for here but three modes are recognized), and alsa 
omewhat more closely connected with strict Probability, (for 
he modes are more nearly of the nature of gradations of 
onviction). But, on the other hand, there is less justification 
ar their introduction, as logicians might really be expected 
o know that what they are aiming to effect by their clumsy 
ontrivances is the very thing which Probability can carry 
ut to the highest desired degree of accuracy. The former 
nethods are as coarse and inaccurate, compared with the 
atter, as were the roughest measurements of Babylonian 
dght-watchers compared with the refined calculations of the 
aodern astronomer. It is indeed only some of the general 
Adherents of the Kantian Logic who enter upon any such 
sonsiderations as these ; some, such as Hamilton and Mansel, 
entirely reject them, as we have seen. By those who do 
Teat of the subject, such conclusions as the following are laid 
lown ; that when both premises are apodeictic the conclusion 
vill be the same ; so when both are assertory or problematic, 
[f one is apodeictic and the other assertory, the latter, or 
weaker/ is all that is to be admitted for the conclusion; 
ind so on. The English reader will find some account of 
ihese rules in Ueberweg's Logic 1 . 

23. But jiHi'Mi^h those modals, regarded as instru- 
nents of accurate thought, have been thus superseded by the 
1 Translation by T. M. Lindsay, p. 439, 



318 Modality. [CHAP. xm. 

precise arithmetical expressions of Probability, the question 
still remains whether what may be termed our popular modal 
expressions could not be improved and adapted to more 
accurate use. It is true that the attempt to separate them 
from one another by any fundamental distinctions is futile, 
for the magnitude of which they take cognizance is, as we 
have remarked, continuous ; but considering the enormous 
importance of accurate terminology, and of recognizing 
numerical distinctions wherever possible, it would be a real 
advance if any agreement could be arrived at with regard to 
the use of modal expressions. We have already noticed (ch. 
II. 16) some suggestions by Mr Gal ton as to the possibility 
of a natural system of classification, resting upon the regu- 
larity with which most kinds of magnitudes tend to group 
themselves about a mean. It might be proposed, for in- 
stance, that we should agree to apply the term 'good' 
to the first quarter, measuring from the best downwards; 
''indifferent' to the middle half, and 'bad' to the last quarter. 
There seems no reason why a similarly improved terminology 
should not some day be introduced into the ordinary modal 
language of common life. It might be agreed, for instance, 
that ' very improbable ' should as far as possible be confined 
to those events which had odds of (say) more than 99 to 1 
against them ; and so on, with other similar expressions. 
There would, no doubt, be difficulties in the way, for in all 
applications of classification we have to surmount the two- 
fold obstacles which lie in the way, firstly (to use Kant's 
expression) of the faculty of making rules, and secondly of 
that of subsumption under rules. That is to say, even if we 
had agreed upon our classes, there would still be much doubt 
and dispute, in the ^ase of things which did not readily lend 
themselves to be cotmted or measured, as to whether the 
odds were more .or less than the assigned quantity. 



SECT. 24.] Modality. 319 

It is true that when we know the odds for or against an 
event, we can always state them explicitly without the neces- 
sity of first agreeing as to the usage of terms which shall 
imply them. But there would often be circumlocution and 
pedantry in so doing, and as long as modal terms are in 
practical use it would seem that there could be no harm, and 
might be great good, in arriving at some agreement as to the 
degree of probability which they should be generally under- 
stood to indicate. Bentham, as is well known, in despair of 
ever obtaining anything accurate out of the language of com- 
mon life on this subject, was in favour of a direct appeal to 
the numerical standard. He proposed the employment, in 
judicial trials, of an instrument, graduated from to 10, on 
which scale the witness was to be asked to indicate the de- 
gree of his belief of the facts to which he testified : similarly 
the judge might express the force with which he held his 
conclusion. The use of such a numerical scale, however, was 
to be optional only, not compulsory, as Bentham admitted 
that many persons might feel at a loss thus to measure the 
degree of their belief. (Rationale of Judicial Evidence, 
Bk. i., Ch. vi.) 

24. Throughout this chapter we have regarded the 
modals as the nearest counterpart to modern Probability 
which was afforded by the old systems of logic. The reason 
for so regarding them is, that they represented some slight 
attempt, rude as it was, to recognize and measure certain 
gradations ip the degree of our conviction, and to examine 
the bearing of such considerations upon our logical inferences. 

But although it is amongst the modals that the germs of 
the methods of Probability are thus to be sought ; the true 
subject-matter of our science, that is, the classes of objects 
with which it is most appropriately concerned, are rather 
represented by another part of the scholastic logic. This 



320 



Modality. 



[CHAP. xni. 



was the branch commonly called Dialectic, in the old sense 
of that term. Dialectic, according to Aristotle, seems to- 
have been a sort of sister art to Rhetoric. It was concerned 
with syllogisms differing in no way from demonstrative syl- 
logisms, except that their premises were probable instead of 
certain. Premises of this kind he termed topics, and the 
syllogisms which dealt with them enthymemes. They were 
said to start from < signs and likelihoods ' rather than from 
axioms 1 . 

25. The terms in which such reasonings are com- 
monly described sound very much like those applicable to 
Probability, as we now understand it. When we hear of 
likelihood, and of probable syllogisms, our first impression 
might be that the inferences involved would be of a similar 
character 2 . This, however, would be erroneous. In the 



1 "The ef/c6s and ffweiov them- 
selves are propositions ; the former 
stating a general probability, the 
latter a fact, which is known to be 
an indication, more or less certain, 
of the truth of some further state- 
ment, whether of a single fact, or 
of a general belief. The former is a 
general proposition, nearly, though 
not quite, universal ; as, ' most men 
who envy hate'; the latter is a 
singular proposition, which however 
is not regarded as a sign, except 
relatively to some other proposition, 
which it is supposed may be inferred 
from it." (Hansel's Aldrich ; Appen- 
dix F, where an account will be 
found of the Aristotelian enthy- 
meme, and dialectic syllogism. Also, 
of course, in Grote's Aristotle, Topics, 
and elsewhere.) 

2 "Nam in hoc etiam differt de- 



monstratio, seu demonstrativa argu- 
mentatio, a probabili, quia in ilia, 
tarn conclusio quam prasmissee neces- 
sarise sunt ; in probabili autem argu- 
mentatione sicut conclusio ut proba- 
bilis infertur ita prsemissae ut pro- 
babiles afferuntur" (Crackanthorpe, 
Bk. v., Ch. 1) ; almost the words with 
which De Morgan distinguishes be- 
tween logic and probability in a 
passage already cited (see Oh. vi. 3). 
Perhaps it was a development of 
some such view as this that Leibnitz 
looked forward to. " J'ai dit plus d'une 
fois qu'il faudrait une nouvelle espece 
de Logique, qui traiteroit des degre's 
de Probability puisqu'Aristote dans. 
ses Topiques n'a rien moins fait que 
cela" (Nouveaux essais, Lib. rv. ch. 
xvi). It is possible, indeed, that he 
may have had in his mind more what 
we now understand by the mathema- 



SECT. 26.] Modality. 321 

first place the province of this Dialectic was much too wide, 
for it covered in addition the whole field of what we should 
now term Scientific or Material Induction. The distinctive 
characteristic of the dialectic premises was their want of 
certainty, and of such uncertain premises Probability (as I 
have frequently insisted) takes account of one class only, 
Induction concerning itself with another class. Again, not the 
slightest attempt was made to enter upon the enquiry, How 
uncertain are the premises ? It is only when this is at- 
tempted that we can be considered to enter upon the field of 
Probability, and it is because, after a rude fashion, the modals 
attempted to grapple with this problem, that we have regarded 
them as in any way occupied with our special subject-matter. 
26. Amongst the older logics with which I have made 
any acquaintance, that of Crackanthorpe gives the fullest dis- 
cussion upon this subject. He divides his treatment of the 
syllogism into two parts, occupied respectively with the ' de- 
monstrative' and the * probable' s-\llogiMn. To the latter a 
whole book is devoted. In this the nature and consequences 
of thirteen different 'loci' 1 are investigated, though it is not 
very clear in what sense they can every one of them be re- 
garded as being ' probable.* 

tical theory of Probability, but in the the term. Crackanthorpe says of 
infancy of a science it is of course them, " sed duci a loco probabiliter 
hard to say whether any particular arguendi, hoc vere proprium est Ar- 
subject is definitely contemplated or gumentationis probabilis ; et in hoc 
not. Leibnitz (as Todhunter has a Demonstratione differt, quia De- 
shown in his history) took the great- monstrator utitur solummodo qua- 
est interest in such chance problems tuor Locis eisque necessariis.... Pr- 
as had yet been discussed. ter hos autem, ex quibus quoque 
1 By loci were understood certain probabiliter arguere licet, sunt znulto 
general classes of premises. They plures Loci arguendi probabiliter; 
stood, in fact, to the major premise ut a Genere, a Specie, ab Adjuncto, 
in somewhat the same relation that ab Oppositis, et similia" (Logica, 
the Category or Predicament did to Lib. v., oh. n.). 

v. 21 



322 Modality. [CHAP. xin. 

It is doubtless true, that if the old logicians had been 
in possession of such premises as modern Probability is con- 
cerned with, and had adhered to their own way of treating 
them, they would have had to place them amongst such loci, 
and thus to make the consideration of them a part of their 
Dialectic. But inasmuch as there does not seem to have 
been the slightest attempt on their part to do more here 
than recognize the fact of the premises being probable ; that 
is, since it was not attempted to measure their probability 
and that of the conclusion, I cannot but regard this part of 
Logic as having only the very slightest relation to Proba- 
bility as now conceived. It seems to me little more than 
one of the ways (described at the commencement of this 
chapter) by which the problem of Modality is not indeed re- 
jected, but practically evaded. 

27. As Logic is not the only science which is directly 
and prominently occupied with questions about belief and 
evidence, so the difficulties which have arisen there have 
been by no means unknown elsewhere. In respect of the 
modals, this seems to have been manifestly the case in Juris- 
prudence. Some remarks, therefore, may be conveniently 
made here upon this application of the subject, though of 
course with the brevity suitable on the part of a layman who 
has to touch upon professional topics. 

Recall for a moment what are the essentials of modality. 
These I understand to be the attempt to mark off from one 
another, without any resort to numerical notation, varying 
degrees of conviction or belief, and to determine the conse- 
quent effect of premises, thus affected, upon our conclusions. 
Moreover, as we cannot construct or retain a scale of any 
kind without employing -a standard from and by which to 
measure it, the attainment and recognition of a standard of 
certainty, or^of one of the other degrees of conviction, is 



81XJT. 28.] Modality. 323 

almost inseparably involved in the same enquiry. In this 
sense of the term, modal difficulties have certainly shown 
themselves in the department of Law. There have been 
similar attempts here, encountered by similar difficulties, to 
come to some definite agreement as to a scale of arrange- 
ment of the degrees of our assent. It is of course much 
more practicable to secure such agreement in the case of a 
special science, confined more or less to the experts, than in 
subjects into which all classes of outsiders have almost equal 
right of entry. The range of application under the former 
circumstances is narrower, and the professional experts have 
acquired habits and traditions by which the standards may 
be retained in considerable integrity. It does not appear, 
however, according to all accounts, as if any very striking 
success had been attained in this direction by the lawyers. 

28. The difficulty in its scientific, or strictly jurispru- 
dential shape, seems to have shown itself principally in the 
^attempt to arrange legal evidence into classes in respect of 
the degree of its cogency. This, I understand, was the case 
in the Roman law, and in some of the continental systems of 
jurisprudence which took their rise from the Roman law. 
"The direct evidence of so many witnesses was plena pro- 
batio. Then came minus plena probatio, then semiplend 
major and semiplend minor; and by adding together a 
certain number of half-proofs for instance, by the pro- 
duction of a tradesman's account-books, plus his supple- 
mentary oath full proof might be made out. It was on 
this principle that torture was employed to obtain a con- 
fession. The confession was evidence suppletory to the cir- 
cumstances which were held to justify its employment 1 ." 

According to Bentham 2 , the corresponding scale in the 

1 Stephen's General View of the a Rationale of Judicial Evidence ; 
Criminal Law of England, p. 241. Bk. i. ch. vi. 

212 



324 Modality. [CHAP. xin. 

English school was: Positive proof, Violent presumption. 
Probable presumption, Light or Rash presumption. Though 
admitted by Blackstone and others, I understand that these 
divisions are not at all generally accepted at the present 
day. 

29. In the above we are reminded rather of modal 
syllogisms. The principal practical form in which the diffi- 
culty underlying the simple modal propositions presents- 
itself, is in the attempt to obtain some criterion of judicial 
certainty. By ' certainty ' here we mean, of course, not what 
the metaphysicians term apodeictic 1 , for that can seldom or 
never be secured in practical affairs, but such a degree of 
conviction, short of this, as every reasonable person will feel 
to be sufficient for all his wants. Here again, one would 
think, the quest must appear, to accurate thinkers, an utterly 
hopeless one ; an effort to discover natural breaks in a con- 
tinuous magnitude. There cannot indeed be the least doubt 
that, amongst limited classes of keen and practised intellects,. 
a standard of certainty, as of everything else, might be re- 
tained and handed down with considerable accuracy : this is. 
possible in matters of taste and opinion where personal pecu- 
liarities of judgment are far more liable to cause disagreement 
and confusion. But then such a consensus is almost entirely 
an affair of tact and custom ; whereas what is wanted in the 
case in question is some criterion to which the comparatively 
uninitiated may be able to appeal. The standard, therefore,, 
must not merely be retained by recollection, but be generally 
recognizable by its characteristics. If such a criterion could 

1 Though this is claimed by some eingestanden. Denn wenn auoh alle 

Kantian logicians ; Nie darf an Zeugnisse und die ubrigen Anzeigen 

einem angeblichen Yerbrecher die wider ihn war en, so bleibt doch das- 

gesetzliche Strafe vollzogeu werden, Gegentheil immer moglich" (Krug, 

bevor er nicht selbst das Verbrechen Denklehre, 131). 



SECT. 30.] Modality. 325 

be secured, its importance could hardly be overrated. But 
so far as one may judge from the speeches of counsel, the 
charges of judges, and the verdicts of juries, nothing really 
deserving the name is ever attained. 

30. The nearest approach, perhaps, to a recognized 
standard is to be found in the frequent assurance that juries 
are not bound to convict only in case they have no doubt of 
the guilt of the accused ; for the absolute exclusion of all 
doubt, the utter impossibility of suggesting any counter 
hypothesis which this assumes, is unattainable in human 
affairs. But, it is frequently said, they are to convict if 
they have no 'reasonable doubt/ no such doubt, that is, 
as would be ' a hindrance to acting in the important affairs 
of life/ As a caution against seeking after unattainable 
certainty, such advice may be very useful ; but it need 
hardly be remarked that the certainty upon which we act in 
the important affairs of life is no fixed standard, but varies 
exceedingly according to the nature of those affairs. The 
greater the reward at stake, the greater the risk we are 
prepared to run, and conversely. Hardly any degree of cer- 
tainty can exist, upon the security of which we should not 
be prepared to act under appropriate circumstances 1 . 

Some writers indeed altogether deny that any standard, 
in the common sense of the word, either is, or ought to be, 
aimed at in legal proceedings. For instance, Sir J. F. 

1 As Mr C. J. Monro puts it : not do this would very often prevent 
" Suppose that a man is suspected of a Chancery judge from appointing 
murdering his daughter. Evidence the man guardian to a ward of the 
which would not convict him before court ; evidence which would not 
an ordinary jury might make a grand affect the judge's mind might make 
jury find a true bill ; evidence which a father think twice on his death- 
would not do this might make a bed before he appointed the man 
coroner's jury bring in a verdict guardian to his daughter." 
against him; evidence which would 



326 Modality. . [CHAP. XIIL 

Stephen, in his work on English Criminal Law 1 , after 
noticing and rejecting such standards as that last indi- 
cated, comes to the conclusion that the only standard recog- 
nized by our law is that which induces juries to convict : 
"What is judicial proof? That which being permitted by 
law to be given in evidence, induces twelve men, chosen 
according to the Jury Act, to say that, having heard it, their 
minds are satisfied of the truth of the proposition which it> 
affirms. They may be prejudiced, they may be timid, they 
may be rash, they may be ignorant; but the oath, the 
number, and the property qualification, are intended, as far 
as possible, to neutralize these disadvantages, and answer 
precisely to the conditions imposed upon standards of value 
or length." (p. 263.) 

To admit this is much about the same thing as to abandon 
such a standard as unattainable. Evidence which induces 
a jury to convict may doubtless be a standard to me and 
others of what we ought to consider * reasonably certain/ 
provided of course that the various juries are tolerably uni- 
form in their conclusions. But it clearly cannot be proposed 
as a standard to the juries themselves ; if their decisions are 
to be consistent and uniform, they want some external indi- 
cation to guide them. When a man is asking, How certain 
ought I to feel? to give such an answer as the above is, 
surely, merely telling him that he is to be as certain as 

1 The portions of this work which what should satisfy them. He conf- 

treat of the nature of proof in gene- pares the legislative standard of cer- 

ral, and of judioial proof in particu- tainty with that of value ; this latter 

lar, are well worth reading by every is declared to be a certain weight of 

logical student. It appears to me, gold, irrespective of the rarity or 

however, that the author goes much commonness of that metal. So with 

too far in the direction of regarding certainty ; if people grow more credu- 

proof as subjective, that is as what lous the intrinsic value of the stand* 

does satisfy people, rather than as ard will vary. 



SECT. 31.] Modality. 327 

he is. If, indeed, juries composed a close profession, they 
might, as was said above, retain a traditional standard. But 
being, as they are, a selection from the ordinary lay public, 
their own decisions in the past can hardly be held up to 
them as a direction what they are to do in future. 

31. It would appear therefore that we may fairly say 
that the English law, at any rate, definitely rejects the main 
assumption upon which the logical doctrine of modality and 
its legal counterpart are based : the assumption, namely, that 
different grades of conviction can be marked off from one 
another with sufficient accuracy for us to be able to refer 
individual cases to their corresponding classes. And that 
with regard to the collateral question of fixing a standard of 
certainty, it will go no further than pronouncing, or im- 
plying, that we are to be content with nothing short of, but 
need not go beyond, ' reasonable certainty.' 

This is a statement of the standard, with which the 
logician and scientific man can easily quarrel; and they 
may with much reason maintain that it has not the slightest 
claim to accuracy, even if it had one to strict intelligibility. 
If a man wishes to know whether his present degree of cer- 
tainty is reasonable, whither is he to appeal? He can 
scarcely compare his mental state with that which is ex- 
perienced in 'the important affairs of life/ for these, as 
already remarked, would indicate no fixed value. At the 
same time, one cannot suppose that such an expression is 
destitute of all signification. People would not continue to 
use language, especially in matters of paramount importance 
and interest, without meaning something by it. We are 
driven therefore to conclude that ' reasonable certainty ' does 
in a rude sort of way represent a traditional standard to 
which it is attempted to adhere. As already remarked, this 
is perfectly practicable in the case of any class of professional 



328 Modality. [CHAP. xm. 

men, and therefore not altogether impossible in the case of 
those who are often and closely brought into connection with 
such a class. Though it is hard to believe that any such 
expressions, when used for purposes of ordinary life, attain 
at all near enough to any conventional standard to be worth 
discussion; yet in the special case of a jury, acting under 
the direct influence of a judge, it seems quite possible that 
their deliberate assertion that they are 'fully convinced' 
may reach somewhat more nearly to a tolerably fixed standard 
than ordinary outsiders would at first think likely. 

32. Are there then any means by which we could 
ascertain what this standard is ; in other words, by which we 
could determine what is the real worth, in respect of accu- 
racy, of this ' reasonable certainty ' which the juries are sup- 
posed to secure ? In the absence of authoritative declara- 
tions upon the subject, the student of Logic and Probability 
would naturally resort to two means, with a' momentary 
notice of which we will conclude this enquiry. 

The first of these would aim at determining the standard 
of judicial certainty indirectly, by simply determining the 
statistical frequency with which the decisions (say) of a jury 
were found to be correct. This may seem to be a hopeless 
task ; and so indeed it is, but not so much on any theoretic 
insufficiency of the determining elements as on account of 
the numerous arbitrary assumptions which attach to most 
of the problems which deal with the probability of testimony 
and judgments. It is not necessary for this purpose that we 
should have an infallible superior court which revised the 
decisions of the one under consideration 1 ; it is sufficient if a 

1 The question will be more fully to the simplest possible elements by 

discussed in a future chapter, but a supposing only two judges or courts, 

few words may be inserted here by of the same average correctness of 

way of indication. Reduce the case decision. Let this be indicated by 



SECT. 32.] Modality. 329 

large number of ordinary representative cases are submitted 
to a court consisting even of exactly similar materials to the 
one whose decisions we wish to test. Provided always that we 
make the monstrous assumption that the judgments of men 
about matters which deeply affect them are 'independent' 
in the sense in which the tosses of pence are independent, 
then the statistics of mere agreement and disagreement will 
serve our purpose. We might be able to say, for instance, 
that a jury of a given number, deciding by a given majority, 
were right nine times out of ten in their verdict. Conclu- 
sions of this kind, in reference to the French courts, are 
what Poisson has attempted at the end of his great work on 
the Probability of Judgments ; though I do not suppose that 
he attached much numerical accuracy to his results. 

A scarcely more hopeful means would be found by a 
reference to certain cases of legal ' presumptions.' A ' con- 
clusive presumption ' is defined as follows : " Conclusive, or 
as they are elsewhere termed imperative or absolute pre- 
sumptions of law, are rules determining the quantity of evi- 
dence requisite for the support of any particular averment 
which is not permitted to be overcome by any proof that the 
fact is otherwise 1 ." A large number of such presumptions 
will be found described in the text-books, but they seem to 
refer to matters far too vague, for the most part, to admit of 
any reduction to statistical frequency of occurrence. It is 
indeed maintained by some authorities that any assignment 
of degree of Probability is not their present object, but that 
they are simply meant to exclude the troublesome delays 

x. Then the chance of their agree- ment is confirmed by the second, we 

ing is # 2 + (l x) 2 , for they agree if have the means of determining x. 

both are right or both wrong. If * Taylor on Evidence: the latter 

the statistical frequency of this agree- part of the extract does not seem 

ment is known, -that is, the fre- very clear, 
quency with which the first judg- 



330 Modality. [CHAP. xm. 

that would ensue if everything were considered open to- 
doubt and question. Moreover, even if they did assign a, 
degree of certainty this would rather be an indication of 
what legislators or judges thought reasonable than of what 
was so considered by the juries themselves. 

There are indeed presumptions as to the time after which 
a man, if not heard of, is supposed to be dead (capable of 
disproof, of course, by his reappearance). If this time varied 
with the age of the man in question, we should at once have 
some such standard as we desire, for a reference to the Life 
tables would fix his probable duration of life, and so deter- 
mine indirectly the measure of probability which satisfied 
the law. But this is not the case ; the period chosen is- 
entirely irrespective of age. The nearest case in point (and 
that does not amount to much) which I have been able to 
ascertain is that of the age after which it has been pre- 
sumed that a woman was incapable of bearing children* 
This was the age of 53. A certain approach to a statistical 
assignment of the chances in this case is to be found in 
Quetelet's Physique Sociale (Vol. I. p. 184, note). According 
to the authorities which he there quotes it would seem that 
in about one birth in 5500 the mother was of the age of 50 
or upwards. This does not quite assign the degree of what 
may be called the & priori chance against the occurrence of a. 
birth at that age, because the fact of having commenced a. 
family at an early age represents some diminution of the 
probability of continuing it into later life. But it serves to 
give some indication of what may be called the odds against 
such an event. 

It need not be remarked that any such clues as these to 
the measure of judicial certainty are far too slight to be of 
any real value. They only deserve passing notice as a pos- 
sible logical solution of the problem in question, or rather as- 



SECT. 32.] Modality. 331 

an indication of the mode in which, in theory, such a solution 
would have to be sought, were the English law, on these 
subjects, a perfectly consistent scheme of scientific evidence. 
This is the mode in which one would, under those circum- 
stances, attempt to extract from its proceedings an admission 
of the exact measure of that standard of certainty which it 
adopted, but which it declined openly to enunciate. 



CHAPTER XIV. 

FALLACIES. 

1. IN works on Logic a chapter is generally devoted 
to the discussion of Fallacies, that is, to the description 
and classification of the different ways in which the rules 
of Logic may be transgressed. The analogy of Probability 
to Logic is sufficiently close to make it advisable to adopt 
the same plan here. In describing his own opinions an 
author is, of course, perpetually obliged to describe and 
criticise those of others which he considers erroneous. But 
some of the most widely spread errors find no supporters 
worth mentioning, and exist only in vague popular misap- 
prehension. It will be found the best arrangement, there- 
fore, at the risk of occasional repetition, to collect a few of 
the errors that occur most frequently, and as far as possible 
to trace them to their sources; but it will hardly be worth 
the trouble to attempt any regular system of arrangement 
and classification. We shall mainly confine ourselves, in 
accordance with the special province of this work, to problems 
which involve questions of logical interest, or to those 
which refer to the application of Probability to moral and 
social science. We shall avoid the discussion of isolated 
problems in games of chance and skill except when some 
error of principle seems to be involved in them. 



SECT. 2.] Fallacies. 33 

2, (I.) One of the most fertile sources of error and 
confusion upon the subject has been already several times- 
alluded to, and in part discussed in a previous chapter. 
This consists in choosing the class to which to refer an 
event, and therefore judging of the rarity of the event and 
the consequent improbability of foretelling it, after it has 
happened, and then transferring the impressions we expe- 
rience to a supposed contemplation of the event beforehand. 
The process in itself is perfectly legitimate (however unne- 
cessary it may be), since time does not in strictness enter 
at all into questions of Probability. No error therefore 
need arise in this way, if we were careful as to the class 
which we thus selected; but such carefulness is often neg- 
lected. 

An illustration may afford help here. A man once 
pointed to a small target chalked upon a door, the target 
having a bullet hole through the centre of it, and surprised 
some spectators by declaring that he had fired that shot 
from an old fowling-piece at a distance of a hundred yards. 
His statement was true enough, but he suppressed a rather 
important fact. The shot had really been aimed in a general 
way at the barn-door, and had hit it; the target was after- 
wards chalked round the spot where the bullet struck. A 
deception analogous to this is, I tbink, often practised uncon- 
sciously in other matters. We judge of events on a similar 
principle, feeling and expressing surprise in an equally un- 
reasonable way, and deciding as to their occurrence on 
grounds which are really merely a subsequent adjunct of our 
own. Butler's remarks about 'the story of Caesar/ discussed 
already in the twelfth chapter, are of this character. He 
selects a series of events from history, and then imagines a 
person guessing them correctly who at the time had not the 
history before him. As I have already pointed out, it is one 



334 Fallacies. [CHAP.. XIV. 

thing to be unlikely to guess an event rightly without 
specific evidence; it is another and very different thing to 
appreciate the truth of a story which is founded partly or 
entirely upon evidence. But it is a great mistake to transfer 
to one of these ways of viewing the matter the mental im- 
pressions which properly belong to the other. It is like 
drawing the target afterwards, and then being surprised to 
iind that the shot lies in the centre of it. 

3. One aspect of this fallacy has been already dis- 
cussed, but it will serve to clear up difficulties which are 
often felt upon the subject if we reexamine the question 
under a some? what more general form. 

In the class of examples under discussion we are generally 
presented with an individual which is not indeed definitely 
referred to a class, but in regard to which we have no great 
difficulty in choosing the appropriate class. Now suppose 
we were contemplating such an event as the throwing of 
sixes with a pair of dice four times running. Such a throw 
would be termed a very unlikely event, as the odds against 
its happening would be 36 x 36 x 36 x 36 - 1 to 1 or 1679615 
to 1. The meaning of these phrases, as has been abundantly 
pointed out, is simply that the event in question occurs very 
rarely; that, stated with numerical accuracy, it occurs once in 
1679616 times. 

4. But now let us make the assumption that the 
throw has actually occurred; let us put ourselves into the 
position of contemplating sixes four times running when it is 
known or reported that this throw has happened. The same 
phrase, namely that the event is a very unlikely one, will 
often be used in relation to it, but we shall find that this 
phrase may be employed to indicate, on one occasion or 
another, extremely different meanings. 

(1) There is, firstly, the most correct meaning. The 



SECT. 6.] Fallacies. 335 

ovent, it is true, has happened, and we know what it is, and 
therefore, we have not really any occasion to resort to the 
rules of Probability; but we can nevertheless conceive our- 
selves as being in the position of a person who does not 
know, and who has only Probability to appeal to. By calling 
the chances 1679615 to 1 against the throw we then mean 
to imply the fact, that inasmuch as such a throw occurs only 
once in 1679616 times, our guess, were we to guess, would 
be correct only once in the same number of times; provided, 
that is, that it is a fair guess, based simply on these statis- 
tical grounds. 

5. (2) But there is a second and very different con- 
ception sometimes introduced, especially when the event in 
question is supposed to be known, not as above by the evi- 
dence of our experience, but by the report of a witness. We 
may then mean by the 'chances against the event' (as was 
pointed out in Chapter xii.) not the proportional number of 
times we should be right in guessing the event, but the 
proportional number of times the witness will be right in 
reporting it. The bases of our inference are here shifted 
on to new ground. In the former case the statistics were 
the throws and their respective frequency, now they are the 
witnesses' statements and their respective truthfulness. 

6. (3) But there is yet another meaning sometimes 
intended to be conveyed when persons talk of the chances 
against such an event as the throw in question. They may 
mean not, Here is an event, how often should I have 
guessed it? nor, Here is a report, how often will it be 
correct ? but something different from either, namely, Here 
is an event, how often will it be found to be produced by 
some one particular kind of cause ? 

When, for example, a man hears of dice giving the same 
throw several times running, and speaks of this as very 



336 Fallacies. [CHAP. xiv. 

extraordinary, we shall often find that he is not merely 
thinking of the improbability of his guess being right, or of 
the report being true, but, that along with this, he is intro- 
ducing the question of the throw having been produced by 
fair dice. There is, of course, no reason whatever why such 
a question as this should not also be referred to Probability, 
provided always that we could find the appropriate statistics- 
by which to judge. These statistics would be composed, not- 
of throws of the particular dice, nor of reports of the parti- 
cular witness, but of the occasions on which such a throw as- 
the one in question respectively had, and had not, been 
produced fairly. The objection to entering upon this view 
of the question would be that no such statistics are obtain- 
able, and that if they were, we should prefer to form our 
opinion (on principles to be described in Chapter xvi.) from 
the special circumstances of the case rather than from an 
appeal to the average. 

7. The reader will easily be able to supply examples, 
in illustration of the distinctions just given ; we will briefly 
examine but one. I hide a banknote in a certain book in a- 
large library, and leave the room. A person tells me that, 
after I went out, a stranger came in, walked straight up to 
that particular book, and took it away with him. Many 
people on hearing this account would reply, How extremely 
improbable! On analysing the phrase, I think we shall 
find that certainly two, and possibly all three, of the above 
meanings might be involved in this exclamation. (1) What 
may be meant is this, Assuming that the report is true,, 
and the stranger innocent, a rare event has occurred. Many 
books might have been thus taken without that particular 
one being selected. I should not therefore have expected 
the event, and when it has happened I am surprised. Now 
a man has a perfect right to be surprised, but he has no 



SECT. 8.] Fallacies. 337 

logical right (so long as we confine ourselves to this view) to 
make his surprise a ground for disbelieving the event. To 
do this is to fall into the fallacy described at the commence- 
ment of this chapter. The fact of my not having been likely 
to have guessed a thing beforehand is no reason in itself for 
doubting it when I am informed of it. (2) Or I may stop 
short of the events reported, and apply the rules of Proba- 
bility to the report itself. If so, what I mean is that such a 
story as this now before me is of a kind very generally false, 
and that I cannot therefore attach much credit to it now. 
(3) Or I may accept the truth of the report, but doubt the 
fact of the stranger having taken the book at random. If 
so, what I mean is, that of men who take books in the way 
described, only a small proportion will be found to have 
taken them really at random ; the majority will do so because 
they had by some means ascertained, or come to suspect, 
what there was inside the book. 

Each of the above three meanings is a possible and a 
legitimate meaning. The only requisite is that we should be 
careful to ascertain which of them is present to the mind, so 
as to select the appropriate statistics. The first makes in 
itself the most legitimate use of Probability ; the drawback 
being that at the time in question the functions of Pro- 
bability are superseded by the event being otherwise known. 
The second or third, therefore, is the more likely meaning to 
be present to the mind, for in these cases Probability, if it 
could be practically made use of, would, at the time in 
question, be a means of drawing really important inferences. 
The drawbacks are the difficulty of finding such statistics, 
and the extreme disturbing influence upon these statistics of 
the circumstances of the special case. 

8, (II.) Closely connected with the tendency just 
mentioned is that which prompts us to confound a true 
v. 22 



338 Fallacies. [CHAP. xiv. 

chance selection with one which is more or less picked. 
When we are dealing with familiar objects in a concrete 
way, especially when the greater rarity corresponds to su- 
periority of quality, almost every one has learnt to recognize 
the distinction. No one, for instance, on observing a fine 
body of troops in a foreign town, but would be prompted to 
ask whether they came from an average regiment or from 
one that was picked. When however the distinction refers 
to unfamiliar objects, and especially when only comparative 
rarity seems to be involved, the fallacy may assume a rather 
subtle and misleading form, and seems to deserve special 
notice by the consideration of a few examples. 

Sometimes the result is not so much an actual fallacy as 
a slight misreckoning of the order of probability of the event 
under consideration. For instance, in the Pyramid question, 
we saw that it made some difference whether we considered 
that TT alone was to be taken into account or whether we 
put this constant into a class with a small number of other 
similar ones. In deciding, however, whether or not there is 
anything remarkable in the actual falling short of the 
representation of the number 7 in the evaluation of TT 
(v. p. 248) the whole question turns upon considerations of 
this kind. The only enquiry raised is whether there is any- 
thing remarkable in this departure from the mean, and the 
answer depends upon whether we suppose that we are re- 
ferring to a predetermined digit, or to whatever digit of the 
ten happens to be most above or below the average. Or, 
take the case raised by Cournot (Exposition de la TMorie 
des Chances, 102, 114), that a certain deviation from the 
mean in the case of Departmental returns of the proportion 
between male and female births is significant and indicative 
of a difference in kind, provided that we select at random a 
single French Department ; but that the same deviation may 



SECT. 9.] Fallacies. 339 

be accidental if it is the maximum of the respective returns 
for several Departments 1 . The answer may be given one 
way or the other according as we bear this consideration in 
mind. 

9. We are peculiarly liable to be misled in this way 
when we are endeavouring to determine the cause of some 
phenomenon, by mere statistics, in entire ignorance as to the 
direction in which the cause should be expected. In such 
cases an ingenious person who chooses to look about over a 
large field can never fail to hit upon an explanation which is 
plausible in the sense that it fits in with the hitherto ob- 
served facts. With a tithe of the trouble which Mr Piazzi 
Smyth expended upon the measurement of the great pyramid, 
I think I would undertake to find plausible intimations of 
several of the important constants and standards which he 
discovered there, in the dimensions of the desk at which I am 
writing. The oddest instance of this sort of conclusion is 
perhaps to be found in the researches of a writer who has 
discovered 2 that there is a connection of a striking kind 
between the respective successes of the Oxford and the 
Cambridge boat in the annual race, and the greater and less 
frequency of sun-spots. 

Of course our usual practical resource in such cases is to 
make appeal to our previous knowledge of the subject in 
question, which enables us to reject as absurd a great number 
-of hypotheses which can nevertheless make a fair show when 
they are allowed to rest upon a limited amount of adroitly 
selected instances. But it must be remembered that if any 
theory chooses to appeal to statistics, to statistics it must be 
.suffered to go for judgment. Even the boat race theory 

1 Discussed by Mr F.Y.Edgeworth, 2 Journal of the Statistical Soc. 
in the Phil. Mag. for April, 1887. (Vol. XLII. p. 328) Dare one suspect 

a joke? 

222 



340 Fallacies. [CHAP. xiv. 

could be established (if sound) on this ground alone. That 
is, if it really could be shown that experience in the long run 
confirmed the preponderance of successes on one side or the 
other , -, : V ., to the relative frequency of the sun-spots, 
we should have to accept the fact that the two classes of 
events were not really independent. One of the two, which- 
ever it may be, must be suspected of causing or influencing 
the other; or both must be caused or influenced by some 
common circumstances. 

10. (III.) The fallacy described at the commencement 
of this chapter arose from determining to judge of an ob- 
served or reported event by the rules of Probability, but 
employing a wrong set of statistics in the process of judging. 
Another fallacy, closely connected with this, arises from the 
practice of taking some only of the characteristics of such an 
event, and arbitrarily confining to these the appeal to Pro- 
bability. Suppose I toss up twelve pence, and find that eleven 
of them give heads. Many persons on witnessing such an occur- 
rence would experience a feeling which they would express by 
the remark, How near that was to getting all heads ! And if 
any thing very important were staked on the throw they 
would be much excited at the occurrence. But in what 
sense were we near to twelve ? There is a not uncommon 
error, I apprehend, which consists in unconsciously regarding 
the eleven heads as a thing which is already somehow 
secured, so that one might as it were keep them, and then 
take our chance for securing the remaining one. The eleven 
are mentally set aside, looked upon as certain (for they have 
already happened), and we then introduce the notion of 
chance merely for the twelfth. But this twelfth, having 
also happened, has no better claim to such a distinction than 
any of the others. If we will introduce the notion of chance 
in the case of the one that gave tail we must do the same in 



SECT. 11.] Fallacies. 341 

the case of all the others as well. In other words, if the 
tosser be dissatisfied at the appearance of the one tail, and 
wish to cancel it and try his luck again, he must toss up the 
whole lot of pence again fairly together. In this case, of 
course, so far from his having a better prospect for the next 
throw he may think himself in very good luck if he makes 
again as good a throw as the one he rejected. What he is 
doing is confounding this case with that in which the throws 
are really successive. If eleven heads have been tossed up in 
turn, we are of course within an even chance of getting a 
twelfth ; but the circumstances are quite different in the 
instance proposed. 

11. In the above example the error is transparent. 
But in forming a judgment upon matters of greater com- 
plexity than dice and pence, especially in the case of what are 
called ' narrow escapes/ a mistake of an analogous kind is, I 
apprehend, far from uncommon. A person, for example, who 
has just experienced a narrow escape will often be filled 
with surprise and anxiety amounting almost to terror. The 
event being past, these feelings are, at the time, in strictness 
inappropriate. If, as is quite possible, they are merely in- 
stinctive, or the result of association, they do not fall within 
the province of any kind of Logic. If, however, as seems 
more likely, they partially arise from a supposed transference 
of ourselves into that point of past time at which the event 
was just about to happen, and the production by imagina- 
tion of the feelings we should then expect to experience, 
this process partakes of the nature of an inference, and can 
be right or wrong. In other words, the alarm may be pro- 
portionate or disproportionate to the amount of danger that 
might fairly have been reckoned upon in such a hypothetical 
anticipation. If the supposed transfer were completely 
carried out, there would be no fallacy ; but it is often very 



342 Fallacies. [CHAP. xiv. 

incompletely done, some of the component parts of the event 
being supposed to be determined or 'arranged* (to use a 
sporting phrase) in the form in which we now know that 
they actually have happened, and only the remaining ones 
being fairly contemplated as future chances. 

A man, for example, is out with a friend, whose rifle goes 
off by accident, and the bullet passes through his hat. He 
trembles with anxiety at thinking what might have hap- 
pened, and perhaps remarks, 'How very near I was to being 
killed!' Now we may safely assume that he means some- 
thing more than that a shot passed very close to him. He 
has some vague idea that, as he would probably say, 'his 
chance of being killed then was very great/ His surprise 
and terror may be in great part physical and instinctive, 
arising simply from the knowledge that the shot had passed 
very near him. But his mental state may be analysed, and 
we shall then most likely find, at bottom, a fallacy of the 
kind described above. To speak or think of chance in con- 
nection with the incident, is to refer the particular incident 
to a class of incidents of a similar character, and then to con- 
sider the comparative frequency with which the contem- 
plated result ensues. Now the series which we may suppose 
to be most naturally selected in this case is one composed of 
shooting excursions with his friend; up to this point the 
proceedings are assumed to be designed, beyond it only, 
in the subsequent event, was there accident. Once in a 
thousand times perhaps on such occasions the gun will go 
off accidentally; one in a thousand only of those discharges 
will be directed near his friend's head. If we will make the 
accident a matter of Probability, we ought by rights in this 
way (to adopt the language of the first example), to 'toss up 
again* fairly. But we do not do this; we seem to assume for 
certain that the shot goes within an inch of our heads, de- 



SECT. 12.] Fallacies. 343 

tach that from the notion of chance at all, and then begin to 
introduce this notion again for possible deflections from that 
saving inch. 

12. (IV.) We will now notice a fallacy connected with 
the subjects of betting and gambling. Many or most of the 
popular misapprehensions on this subject imply such utter 
ignorance and confusion as to the foundations of the science 
that it would be needless to discuss them here. The follow- 
ing however is of a far more plausible kind, and has been a 
source of perplexity to persons of considerable acuteness. 

The case, put into the simplest form, is as follows 1 . 
Suppose that a person A is playing against B, B being 
either another individual or a group of individuals, say a 
gambling bank. They begin by tossing for a shilling, and 
A maintains that he is in possession of a device which will 
insure his winning. If he does win on the first occasion he 
has clearly gained his point so far. If he loses, he stakes 
next time two shillings instead of one. The result of course 
is that if he wins on the second occasion he replaces his 
former loss, and is left with one shilling profit as well. So 
he goes on, doubling his stake after every loss, with the 
obvious result that on the first occasion of success he makes 
good all his previous losses, and is left with a shilling over. 
But such an occasion must come sooner or later, by the 
assumptions of chance on which the game is founded. Hence 
it follows that he can insure, sooner or later, being left a 
final winner. Moreover he may win to any amount; firstly 
from the obvious consideration that he might make his 
initial stake as large as he pleased, a hundred pounds, for 

1 It appears to have been long Edinburgh, for 1823) which discusses 

known to gamblers under the name certain points connected with it, but 

of the Martingale. There is a paper scarcely touches on the subject of the 

by Babbage (Trans, of Royal Soc. of sections which follow. 



344 Fallacies. [CHAP, xiv. 

instance, instead of a shilling; and secondly, because what 
he has done once he may do again. He may put his shilling 
by, and have a second spell of play, long or short as the case 
may be, with the same termination to it. Accordingly by 
mere persistency he may accumulate any sum of money he 
pleases, in apparent defiance of all that is meant by luck. 

13. I have classed this opinion among fallacies, as the 
present is the most convenient opportunity of discussing it, 
though in strictness it should rather be termed a paradox, 
since the conclusion is perfectly sound. The only fallacy 
consists in regarding such a way of obtaining the result as 
mysterious. On the contrary, there is nothing more easy 
than to insure ultimate success under the given conditions. 
The point is worth enquiry, from the principles it involves, 
and because the answers commonly given do not quite meet 
the difficulty. It is sometimes urged, for instance, that no 
bank would or does allow the speculator to choose at will the 
amount of his stake, but puts a limit to the amount for 
which it will consent to play. This is quite true, but is of 
course no answer to the hypothetical enquiry before us, which 
assumes that such a state of things is allowed. Again, it has 
been urged that the possibility in question turns entirely 
upon the fact that credit must be supposed to be given, for 
otherwise the fortune of the player may not hold out until 
his turn of luck arrives: that, in fact, sooner or later, if he 
goes on long enough, his fortune will not hold out long 
enough, and all his gains will be swept away. It is quite 
true that credit is a condition of success, but it is in no sense 
the cause. We may suppose both parties to agree at the 
outset that there shall be no payments until the game be 
ended, A having the right to decide when it shall be con- 
sidered to be ended. It still remains true that whereas in 
ordinary -j imli:ii..:. i.e. with fixed or haphazard stakes, A 



SECT, 14.] Fallacies. 345 

could not ensure winning eventually to any extent, he can 
do so if he adopt such a scheme as the one in question. And 
this is the state of things which seems to call for explanation. 

14. What causes perplexity here is the supposed fact 
that in some mysterious way certainty has been conjured out 
of uncertainty; that in a game where the detailed events are 
utterly inscrutable, and where the average, by supposition, 
shows no preference for either side, one party is nevertheless 
succeeding somehow in steadily drawing the luck his own 
way. It looks as if it were a parallel case with that of a 
man who should succeed by some device in permanently 
securing more than half of the tosses with a penny which 
was nevertheless to be regarded as a perfectly fair one. 

This is quite a mistake. The real fact is that A does 
not expose his gains to chance at all; all that he so exposes 
is the number of times he has to wait until he gains. Put 
such a case as this. I offer to give a man any sum of money 
he chooses to mention provided he will at once give it back 
again to me with one pound more. It does not need much 
acuteness to see that it is a matter of indifference to me 
whether he chooses to mention one pound, or ten, or a 
hundred. Now suppose that instead of leaving it to his 
choice which of these sums is to be selected each time, the 
two parties agree to leave it to chance. Let them, for 
instance, draw a number out of a bag each time, and let that 
be the sum which A gives to B under the prescribed condi- 
tions. The case is not altered. A still gains his pound each 
time, for the introduction of the element of chance has not 
in any way touched this. All that it does is to make this 
pound the result of an uncertain subtraction, sometimes 10 
minus 9, sometimes 50 minus 49, and so on. It is these 
numbers only, not their difference, which he submits to luck, 
and this is of no consequence whatever. 



346 Fallacies. [CHAP. xiv. 

To suggest to any individual or company that they 
should consent to go on playing upon such terms as these 
would be too barefaced a proposal. And yet the case in 
question is identical in principle, and almost identical in 
form, with this. To offer to give a man any sum he likes to 
name provided he gives you back again that same sum plus 
one, and to offer him any number of terms he pleases of the 
series 1, 2, 4, 8, 16, &c., provided you have the next term of 
the set, are equivalent. The only difference is that in the 
latter case the result is attained with somewhat more of 
arithmetical parade. Similarly equivalent are the processes 
in case we prefer to leave it to chance, instead of to choice, 
to decide what sum or what number of terms shall be fixed 
upon. This latter is what is really done in the case in 
question. A man who consents to go on doubling his stake 
every time he wins, is leaving nothing else to chance than 
the determination of the particular number of terms of such 
a geometrical series which shall be allowed to pass before he 
stops. 

15. It may be added that there is no special virtue in 
the particular series in question, viz. that in accordance with 
which the stake is doubled each time. All that is needed is 
that the last term of the series should more than balance all 
the preceding ones. Any other series which increased faster 
than this geometrical one, would answer the purpose as well 
or better. Nor is it necessary, again, that the game should 
be an even or ' fair* one. Chance, be it remembered, affects 
nothing here but the number of terms to which the series 
attains on each occasion, its final result being always arith- 
metically fixed. When a penny is tossed up it is only on 
one of every two occasions that the series runs to more than 
two terms, and so his fixed gains come in pretty regularly. 
But unless he was playing for a limited time only, it would 



SECT. 16.] Fallacies. 347 

not affect him if the series ran to two hundred terms; it 
would merely take him somewhat longer to win his stakes. 
A man might safely, for instance, continue to lay an even 
bet that he would get the single prize in a lottery of a thou- 
sand tickets, provided he thus doubled, or more than doubled, 
his stake each time, and unlimited credit was given. 

16. So regarded, the problem is simple enough, but 
there are two points in it to which attention may conveni- 
ently be directed. 

In the first place, it serves very pointedly to remind us of 
the distinction between a series of events (in this case the 
tosses of the penny) which really are subjects of chance, and 
our conduct founded upon these events, which may or may 
not be so subject 1 . It is quite possible that this latter may 
be so contrived as to be in many respects a matter of abso- 
lute certainty, a consideration, I presume, familiar enough 
to professional betting men. Why is the ordinary way of 
betting on the throws of a penny fair to both parties? Be- 
cause a 'fair' series is 'fairly' treated. The heads and tails 
occur at random, but on an average equally often, and the 
stakes are either fixed or also arranged at random. If a man 
backs heads every time for the same amount, he will of 
course in the long run neither win nor lose. Neither will he 
if he varies the stake every time, provided he does not vary 
it in such a way as to make its amount dependent on the 
fact of his having won or lost the time before. But he 
may, if he pleases, and the other party consents, so arrange 
his stakes (as in the case in question) that Chance, if one 
might so express it, does not get a fair chance. Here the 
human elements of choice and design have been so brought 
to bear upon a series of events which, regarded by them- 

1 Attention will be further directed to this distinction in the chapter on 
Insurance and Gambling. 



348 Fallacies. [CHAP.XIV. 

selves, exhibit nothing but the physical characteristics of 
chance, that the latter elements disappear, and we get a 
result which is arithmetically certain. Other analogous 
instances might be suggested, but the one before us has the 
merit of most ingeniously disguising the actual process. 

17. The meaning of the remark just made will be 
better seen by a comparison with the following case. It has 
been attempted 1 to explain the preponderance of male births 
over female by assuming that the chances of the two are 
equal, but that the general desire to have a male heir tends 
to induce many unions to persist until the occurrence of this 
event, and no longer. It is supposed that in this way there 
would be a slight preponderance of families which consisted 
of one son only, or of two sons and one daughter, and so forth. 

This is quite fallacious (as had been noticed by Laplace, 
in his Essai)\ and there could not be a better instance 
chosen than this to show just what we can do and what we 
cannot do in the way of altering the luck in a real chance- 
succession of events. To suppose that the number of actual 
births could be influenced in the way in question is exactly 
the same thing as to suppose that a number of gamblers 
could increase the ratio of heads to tails, to something over 
one-half, by each handing the coin to his neighbour as soon 
as he had thrown a head : that they have only to leave off as 
soon as head has appeared; an absurdity which we need not 
pause to explain at this stage. The essential point about 
the 'Martingale' is that, whereas the occurrence of the 
events on which the stakes are laid is unaffected, the stakes 
themselves can be so adjusted as to make the luck swing one 
way. 

1 As by Prevost in the BibliothZ- parently accepted, by Quetelet (Phy- 
que Universelle de Gentve, Oct. 1829. sique Social* i. 171), 
The explanation is noted, and ap- 



SECT. 19.] Fallacies. 349 

18. In the second place, this example brings before us 
what has had to be so often mentioned already, namely, that 
the series of Probability are in strictness supposed to be 
interminable. If therefore we allow either party to call 
upon us to stop, especially at a point which just happens to 
suit him, we may get results decidedly opposed to the 
integrity of the theory. In the case before us it is a neces- 
sary stipulation for A that he may be allowed to leave off 
when he wishes, that is at one of the points at which the 
throw is in his favour. Without this stipulation he may be 
left a loser to any amount. 

Introduce the supposition that one party may arbitrarily 
call for a stoppage when it suits him and refuse to permit it 
sooner, and almost any system of what would be otherwise fair 
play may be converted into a very one-sided arrangement. 
Indeed, in the case in question, A need not adopt this device 
of doubling the stakes every time he loses. He may play 
with a fixed stake, and nevertheless insure that one party 
shall win any assigned sum, assuming that the game is even 
and that he is permitted to play on credit. 

19. (V.) A common mistake is to assume that a very 
unlikely thing will not happen at all. It is a mistake which, 
when thus stated in words, is too obvious to be committed, 
for the meaning of an unlikely thing is one that happens at 
rare intervals; if it were not assumed that the event would 
happen sometimes it would not be called unlikely, but im- 
possible. This is an error which could scarcely occur except 
in vague popular misapprehension, and is so abundantly re- 
futed in works on Probability, that it need only be touched 
upon briefly here. It follows of course, from our definition 
of Probability, that to speak of a very rare combination of 
events as one that is 'sure never to happen/ is to use lan- 
guage incorrectly. Such a phrase may pass current as a 



J50 Fallacies. [CHAP, xiv, 

loose popular exaggeration, but in strictness it involves a 
contradiction. The truth about such rare events cannot be 
better described than in the following quotation from De 
Morgan 1 : 

"It is said that no person ever does arrive at such ex- 
tremely improbable cases as the one just cited [drawing the 
same ball five times running out of a bag containing twenty 
balls]. That a given individual should never throw an ace 
twelve times running on a single die, is by far the most 
likely; indeed, so remote are the chances of such an event 
in any twelve trials (more than 2,000,000,000 to 1 against it) 
that it is unlikely the experience of any given country, in 
any given century, should furnish it. But let us stop for a 
moment, and ask ourselves to what this argument applies. 
A. person who rarely touches dice will hardly believe that 
doublets sometimes occur three times running; one who han- 
dles them frequently knows that such is sometimes the fact. 
Every very practised user of those implements has seen still 
rarer sequences. Now suppose that a society of persons had 
thrown the dice so often as to secure a run of six aces ob- 
served and recorded, the preceding argument would still be 
used against twelve. And if another society had practised 
long enough to see twelve aces following each other, they 
might still employ the same method of doubting as to a run 
of twenty-four; and so on, ad inftnitum. The power of ima- 
gining cases which contain long combinations so much ex- 
ceeds that of exhibiting and arranging them, that it is easy 
to assign a telegraph which should make a separate signal 
for every grain of sand in a globe as large as the visible uni- 
verse, upon the hypothesis of the most space-penetrating 
astronomer. The fallacy of the preceding objection lies in 
supposing events in number beyond our experience, composed 
1 Essay on Probabilities p. 126. 



SECT. 20.] Fallacies. 351 

entirely of sequences such as fall within our experience. It 
makes the past necessarily contain the whole, as to the qua- 
lity of its components; and judges by samples. Now the 
least cautious buyer of grain requires to examine a handful 
before he judges of a bushel, and a bushel before he judges 
of a load. But relatively to such enormous numbers of com- 
binations as are frequently proposed, our experience does not 
deserve the title of a handful as compared with a bushel, or 
even of a single grain." 

20. The origin of this inveterate mistake is not diffi- 
cult to be accounted for. It arises, no doubt, from the exi- 
gencies of our practical life. No man can bear in mind 
every contingency to which he may be exposed. If therefore 
we are ever to do anything at all in the world, a large num- 
ber of the rarer contingencies must be left entirely out of 
account. And the necessity of this oblivion is strengthened 
by the shortness of our life. Mathematically speaking, it 
would be said to be certain that any one who lives long 
enough will be bitten by a mad dog, for the event is not an 
impossible, but only an improbable one, and must therefore 
come to pass in time. But this and an indefinite number 
of other disagreeable contingencies have on most occasions 
to be entirely ignored in practice, and thence they come 
almost necessarily to drop equally out of our thought and 
expectation. And when the event is one in itself of no im- 
portance, like a rare throw of the dice, a great effort of 
imagination may be required, on the part of persons not ac- 
customed to abstract mathematical calculation, to enable 
them to realize the throw as being even possible. 

Attempts have sometimes been made to estimate what 
extremity of unlikelihood ought to be considered as equi- 
valent to this practical zero point of belief. In so far as 
such attempts are carried out by logicians, or by those who 



352 Fallacies. [CHAP. XIV. 

are unwilling to resort to mathematical valuation of chances, 
they must be regarded as merely a special form of the modal 
difficulties discussed in the last chapter, and need not there- 
fore be reconsidered here ; but a word or two may be added 
concerning the views of some who have looked at the matter 
from the mathematician's point of view. 

The principal of these is perhaps Buffon. He has ar- 
rived at the estimate (Arithm&ique Morale vra.) that this 
practical zero is equivalent to a chance of y^,^. The 
grounds for selecting this fraction are found in the f$ct that, 
according to the tables of mortality accessible to him, it 
represents the chance of a man of 56 dying in the course of 
the next day. But since no man under common circum- 
stances takes the chance into the slightest consideration, it 
follows that it is practically estimated as having no value. 

It is obvious that this result is almost entirely arbitrary, 
and in fact his reasons cannot be regarded as anything more 
than a slender justification from experience for adopting a 
conveniently simple fraction ; a justification however which 
would apparently have been equally available in the case of 
any other fractions lying within wide limits of the one 
selected 1 . 

21. There is one particular form of this error, which, 
from the importance occasionally attached to it, deserves 
perhaps more special examination. As stated above, there 
can be no doubt that, however unlikely an event may be, if 
we (loosely speaking) vary the circumstances sufficiently, or 

1 This theoretical or absolute neg- tion will be found described in Mr 
leot of what is very rare must not be Merriman's Least Squares (p. 166). 
confused with the practical neglect But this rests on the understanding 
sometimes recommended by astro- that a smaller balance of error would 
nomical and other observers. A thus result in the long run. The 
criterion, known as Chauvenet's, for very rare event is deliberately re- 
indicating the limits of such rejec- jected, not overlooked. 



SECT. 21.] Fallacies. 

if, in other words, we keep on trying long enough, we shall 
meet with such an event at last. If we toss up a pair of 
dice a few times we shall get doublets ; if we try longer with 
three we shall get triplets, and so on. However unusual the 
event may be, even were it sixes a thousand times running, 
it will come some time or other if we have only patience and 
vitality enough. Now apply this result to the letters of the 
alphabet. Suppose that one letter at a time is drawn from 
a bag which contains them all, and is then replaced. If the 
letters were written down one after another as they occurred, 
it would commonly be expected that they would be found to 
make mere nonsense, and would never arrange themselves 
into the words of any language known to men. No more 
they would in general, but it is a commonly accepted result 
of the theory, and one which we may assume the reader to 
be ready to admit without further discussion, that, if the 
process were continued long enough, words making sense 
would appear; nay more, that any book we chose to men- 
tion, Milton's Paradise Lost or the plays of Shakespeare, 
for example, would be produced in this way at last. It 
would take more days than we have space in this volume to 
represent in figures, to make tolerably certain of obtaining 
the former of these works by thus drawing letters out of a 
bag, but the desired result would be obtained at length 1 . 

1 The process of calculation may combinations is favourable, if we 

be readily indicated. There are, say, reject variations of spelling. Hence 

about 350,000 letters in the work in unity divided by this number would 

question. Since any of the 26 letters represent the chance of getting the 

of the alphabet may be drawn each desired result by successive random 

time, the possible number of com- selection of the required number of 

binations would be 26 M0 ' 000 ; a num- 350,000 letters, 

ber which, as may easily be inferred If this chance is thought too small, 

from a table of logarithms, would and any one asks how often the 

demand for its expression nearly above random selection must be re- 

500,000 figures. Only one of these peated in order to give him odds of 2 

v. 23 



354 Fallacies. [CHAP. xiv. 

Now many people have not unnaturally thought it de- 
rogatory to genius to suggest that its productions could 
have also been obtained by chance, whilst others have gone 
on to argue, If this be the case, might not the world itself in 
this manner have been produced by chance ? 

22. We will begin with the comparatively simple, de- 
terminate, and intelligible problem of the possible production 
of the works of a great human genius by chance. With 
regard to this possibility, it may be a consolation to some 
timid minds to be reminded that the power of producing the 
works of a Shakespeare, in time, is not confined to consum- 
mate genius and to mere chance. There is a third alterna- 
tive, viz. that of purely mechanical procedure. Any one, 
down almost to an idiot, might do it, if he took sufficient 
time about the task. For suppose that the required number 
of letters were procured and arranged, not by chance, but 
designedly, and according to rules suggested by the theory 
of permutations : the letters of the alphabet and the number 
of them to be employed being finite, every order in which 
they could occur would come in its due turn, and therefore 
every thing which can be expressed in language would be 
arrived at some time or other. 

There is really nothing that need shock any one in such 
a result. Its possibility arises from the following cause. 

to 1 in favour of success, this also shown algebraically to be equivalent 
can be easily shown. If the chance to odds of about 2 to 1. That is, 

. . 1 when we have drawn the requisite 

of an event on each occasion is -, ,. ., , /,. 

n quantity of letters a number of times 

the chance of getting it once at least equal to the inconceivably great 

. ., /n 1\* . number above represented, it is still 

in n trull is 1-^ J ! for we only 2 to 1 that we shall have 86- 

shall do this unless we fail n times cured what we want: and then we 

running. When (as in the case in have to recognize it. 
question) n is very large, this may be 



SECT. 23.] Fallacies. 355 

The number of letters, and therefore of words, at our disposal 
is limited ; whatever therefore we may desire to express 
in language necessarily becomes subject to corresponding 
limitation. The possible variations of thought are literally 
infinite, so are those of spoken language (by intonation of 
the voice, &c.) ; but when we come to words there is a limit- 
ation, the nature of which is distinctly conceivable by the 
mind, though the restriction is one that in practice will 
never be appreciable, owing to the fact that the number of 
combinations which may be produced is so enormous as to 
surpass all power of the imagination to realize 1 . The answer 
therefore is plain, and it is one that will apply to many other 
cases as well, that to put a finite limit upon the number of 
ways in which a thing can be done, is to determine that any 
one who is able and willing to try long enough shall succeed 
in doing it. If a great genius condescends to perform it 
under these circumstances, he must submit to the possibility 
of having his claims rivalled or disputed by the chance-man 
and idiot. If Shakespeare were limited to the use of eight 
or nine assigned words, the time within which the latter 
agents might claim equality with him would not be very 
great. As it is, having had the range of the English lan- 
guage at his disposal, his reputation is not in danger of being 
assailed by any such methods. 

23. The case of the possible production of the world 
by chance leads us into an altogether different region of dis- 
cussion. We are not here dealing with figures the nature 
and use of which are within the fair powers of the under- 
standing, however the imagination may break down in at- 
tempting to realize the smallest fraction of their full signi- 

1 The longest life which could utter insignificance in the face of 
reasonably be attributed to any Ian- such periods of time as are being 
guage would of course dwindle into here arithmetically contemplated. 

232 



356 Fallacies. [CHAP. xiv. 

ficance. The understanding itself is wandering out of its 
proper province, for the conditions of the problem cannot be 
assigned. When we draw letters out of a bag we know very 
well what we are doing ; but what is really meant by pro- 
ducing a world by chance ? By analogy of the former case, 
we may assume that some kind of agent is presupposed ; 
perhaps therefore the following supposition is less absurd 
than any other. Imagine some being, not a Creator but a 
sort of Demiurgus, who has had a quantity of materials put 
into his hands, and he assigns them their collocations and 
their laws of action, blindly and at haphazard : what are the 
odds that such a world as we actually experience should have 
been brought about in this way ? 

If it were worth while seriously to set about answering 
such a question, and if some one would furnish us with the 
number of the letters of such an alphabet, and the length of 
the work to be written with them, we could proceed to indi- 
cate the result. But so much as this may surely be affirmed 
about it ; that, far from merely finding the length of this 
small volume insufficient for containing the figures in which 
the adverse odds would be given, all the paper which the 
world has hitherto produced would be used up before we had 
got far on our way in writing them down. 

24. The most seductive form in which the difficulty 
about the occurrence of very rare events generally presents 
itself is probably this. 'You admit (some persons will be 
disposed to say) that such an event may sometimes happen ; 
nay, that it does sometimes happen in the infinite course of 
time. How then am I to know that this occasion is not one 
of these possible occurrences ?' To this, one answer only can 
be given, the same which must always be given where 
statistics and probability are concerned, ' The present may 
be such an occasion, but it is inconceivably unlikely that it 



SECT. 25.] Fallacies. 357 

should be one. Amongst countless billions of times in which 
you, and such as you, urge this, one person only will be 
justified ; and it is not likely that you are that one, or that 
this is that occasion/ 

25. There is another form of this practical inability to 
distinguish between one high number and another in the 
estimation of chances, which deserves passing notice from its 
importance in arguments about heredity. People will often 
urge an objection to the doctrine that qualities, mental and 
bodily, are transmitted from the parents to the offspring, on 
the ground that there are a multitude of instances to the 
contrary, in fact a great majority of such instances. To 
raise this objection implies an utter want of appreciation of 
the very great odds which possibly may exist, and which the 
argument in support of heredity implies do exist against any 
given* person being distinguished for intellectual or other 
eminence. This is doubtless partly a matter of definition, 
depending upon the degree of rarity which we consider to be 
implied by eminence ; but taking any reasonable sense of the 
term, we shall readily see that a very great proportion of 
failures may still leave an enormous preponderance of evi- 
dence in favour of the heredity doctrine. Take, for instance, 
that degree of eminence which is implied by being one of 
four thousand. This is a considerable distinction, though, 
since there are about two thousand such persons to be found 
amongst the total adult male population of Great Britain, it 
is far from implying any conspicuous genius. Now suppose 
that in examining the cases of a large number of the chil- 
dren of such persons, we had found that 199 out of 200 of 
them failed to reach the same distinction. Many persons 
would conclude that this was pretty conclusive evidence 
against any hereditary transmission. To be able to adduce 
only one favourable, as against 199 hostile instances, would 



358 Fallacies. [CHAP. xiv. 

to them represent the entire break-down of any such theory. 
The error, of course, is obvious enough, and one which, with 
the figures thus before him, hardly any one could fail to 
avoid. But if one may judge from common conversation 
and other such sources of information, it is found in practice 
exceedingly difficult adequately to retain the conviction that 
even though only one in 200 instances were favourable, 
this would represent odds of about 20 to 1 in favour of the 
theory. If hereditary transmission did not prevail, only one 
in 4000 sons would thus rival their fathers; but we find 
actually, let us say (we are of course taking imaginary pro- 
portions here), that one in 200 does. Hence, if the statistics 
are large enough to be satisfactory, there has been some 
influence at work which has improved the chances of mere 
coincidence in the ratio of 20 to 1. We are in fact so little 
able to realise the meaning of very large numbers, that is, 
to retain the ratios in the mind, where large numbers are 
concerned, that unless we repeatedly check ourselves by 
arithmetical considerations we are too apt to treat and esti- 
mate all beyond certain limits as equally vast and vague. 

26. (VI.) In discussing the nature of the connexion 
between Probability and Induction, we examined the claims 
of a rule commonly given for inferring the probability that 
an event which had been repeatedly observed would recur 
again. I endeavoured to show that all attempts to obtain 
and prove such a rule were necessarily futile ; if these reasons 
were conclusive the employment of such a rule must of 
course be regarded as fallacious. A few examples may con- 
veniently be added here, tending to show how instead of 
there being merely a single rule of succession we might better 
divide the possible forms into three classes. 

(1) In some cases when a thing has been observed to 
happen several times it becomes in consequence more likely 



SECT. 27.] Fallacies. 

that the thing should happen again. This agrees with the 
ordinary form of the rule, and is probably the case of most 
frequent occurrence. The necessary vagueness of expression 
when we talk of the ' happening of a thing ' makes it quite 
impossible to tolerate the rule in this general form, but if we 
specialize it a little we shall find it assume a more familiar 
shape. If, for example, we have observed two or more pro- 
perties to be frequently associated together in a succession of 
individuals, we shall conclude with some force that they will 
be found to be so connected in future. The strength of our 
conviction however will depend not merely on the number 
of observed coincidences, but on far more complicated con- 
siderations ; for a discussion of which the reader must be 
referred to regular treatises on Inductive evidence. Or again, 
if we have observed one of two events succeed the other 
several times, the occurrence of the former will excite in 
most cases some degree of expectation of the latter. As 
before, however, the degree of our expectation is not to be 
assigned by any simple formula ; it will depend in part upon 
the supposed intimacy with which the events are connected. 
To attempt to lay down definite rules upon the subject 
would lead to a discussion upon laws of causation, and the 
circumstances under which their existence may be inferred, 
and therefore any further consideration of the matter must 
be abandoned here. 

27. (2) Or, secondly, the past recurrence may in it- 
self give no valid grounds for inference about the future; 
this is the case which most properly belongs to Probability 1 . 

1 We are here assuming of course assuming that e.g. the die is known 

that the ultimate limit to which our to be a fair one; if this is not known 

average tends is known, either from hut a possible bias has to be inferred 

knowledge of the causes or from pre- from its observed performances, the 

vious extensive experience. We are case falls under the former head. 



360 Fallacies. [CHAP. xiv. 

That it does so belong will be easily seen if we bear in mind 
the fundamental conception of the science. We are there 
introduced to a series, for purposes of inference an indefi- 
nitely extended series, of terms, about the details of which, 
information, except on certain points, is not given ; our know- 
ledge being confined to the statistical fact, that, say, one in 
ten of them has some attribute which we will call X. Sup- 
pose now that five of these terms in succession have been X, 
what hint does this give about the sixth being also an XI 
Clearly none at all ; this past fact tells us nothing ; the for- 
mula for our inference is still precisely what it was before, 
that one in ten being X it is one to nine that the next term 
is X. And however many terms in succession had been of 
one kind, precisely the same formula would still be given. 

28. The way in which events will justify the answer 
given by this formula is often misunderstood. For the 
benefit therefore of those unacquainted with some of the 
conceptions familiar to mathematicians, a few words of ex- 
planation may be added. Suppose then that we have had 
X twelve times in succession. This is clearly an anomalous 
state of things. To suppose anything like this continuing 
to occur would be obviously in opposition to the statistics, 
which assert that in the long run only one in ten is X. 
But how is this anomaly got over? In other words, how 
do we obviate the conclusion that X's must occur more 
frequently than once in ten times, after such a long succes- 
sion of them as we have now had ? Many people seem to 
believe that there must be a diminution of JTs afterwards 
to counterbalance their past preponderance. This however 
would be quite a mistake; the proportion in which they 
occur in future must remain the same throughout ; it cannot 
be altered if we are to adhere to our statistical formula. 
The fact is that the rectification of the exceptional disturb- 



SECT. 28.] Fallacies. 361 

ance in the proportion will be brought about simply by the 
continual influx of fresh terms in the series. These will in 
the long run neutralize the disturbance, not by any special 
adaptation, as it were, for the purpose, but by the mere 
weight of their overwhelming numbers. At every stage 
therefore, in the succession, whatever might have been the 
number and nature of the preceding terms, it will still be 
true to say that one in ten of the terms will be an X. 

If we had to do only with a finite number of terms, 
however large that number might be, such a disturbance as 
we have spoken of would, it is true, need a special alteration 
in the subsequent proportions to neutralize its effects. But 
when we have to do with an infinite number of terms, this 
is not the case ; the ' limit ' of the series, which is what we 
then have to deal with, is unaffected by these temporary 
disturbances. In the continued progress of the series we 
shall find, as a matter of fact, more and more of such dis- 
turbances, and these of a more and more exceptional character. 
But whatever the point we may occupy at any time, if we 
look forward or backward into the indefinite extension of the 
series, we shall still see that the ultimate limit to the pro- 
portion in which its terms are arranged remains the same ; 
and it is with this limit, as above mentioned, that we are 
concerned in the strict rules of Probability. 

The most familiar example, perhaps, of this kind is that 
of tossing up a penny. Suppose we have had four heads in 
succession; people 1 have tolerably realized by now that 'head 
the fifth time ' is still an even chance, as ' head * was each 

1 Except indeed the gamblers. Ac- it will not be repeated at the next 

cording to a gambling acquaintance coup: this is the groundwork of all 

whom Houdin, the conjurer, describes theories of probabilities and is term- 

himself as having met at Spa, "the ed the maturity of chances" (Card- 

oftener a particular combination has sharping expoted, p. 85). 
ocourred the more certain it is that 



362 Fallacies. [CHAP. xiv. 

time before, and will be ever after. The preceding para- 
graph explains how it is that these occasional disturbances 
in the average become neutralized in the long run. 

29. (3) There are other cases which, though rare, 
are by no means unknown, in which such an inference as 
that obtained from the Rule of Succession would be the di- 
rect reverse of the truth. The oftener a thing happens, it 
may be, the more unlikely it is to happen again. This is the 
case whenever we are drawing things from a limited source 
(as balls from a bag without replacing them), or whenever 
the act of repetition itself tends to prevent the succession 
(as in giving false alarms). 

I am quite ready to admit that we believe the results de- 
scribed in the last two classes on the strength of some such 
general Inductive rule, or rather principle, as that involved 
in the first. But it would be a great error to confound this 
with an admission of the validity of the rule in each special 
instance. We are speaking about the application of the rule 
to individual cases, or classes of cases ; this is quite a dis- 
tinct, thing, as was pointed out in a previous chapter, from 
giving the grounds on which we rest the rule itself. If a 
man were to lay it down as a universal rule, that the testi- 
mony of all persons was to be believed, and we adduced an 
instance of a man having lied, it would not be considered 
that he saved his rule by showing that we believed that it 
was a lie on the word of other persons. But it is perfectly 
consistent to give as a merely general, but not universal, 
rule, that the testimony of men is credible ; then to separate 
off a second class of men whose word is not to be trusted, 
and finally, if any one wants to know our ground for the 
second rule, to rest it upon the first. If we were speaking 
of necessary laws, such a conflict as this would be as 
hopeless as the old ' Cretan ' puzzle in logic ; but in in- 



SECT. 31.] Fallacies. 363 

stances of Inductive and Analogical extension it is perfectly 
harmless. 

30. A familiar example will serve to bring out the 
three different possible conclusions mentioned above. We 
have observed it rain on ten successive days. A and B con- 
clude respectively for and against rain on the eleventh day ; 
C maintains that the past rain affords no data whatever for 
an opinion. Which is right ? We really cannot determine 
a priori. An appeal must be made to direct observation, or 
means must be found for deciding on independent grounds 
to which class we are to refer the instance. If, for example, 
it were known that every country produces its own rain, we 
should choose the third rule, for it would be a case of drawing 
from a limited supply. If again we had reasons to believe 
that the rain for our country might be produced anywhere 
on the globe, we should probably conclude that the past 
rainfall threw no light whatever on the prospect of a con- 
tinuance of wet weather, and therefore take the second. 
Or if, finally, we knew that rain came in long spells or sea- 
sons, as in the tropics, then the occurrence of ten wet days 
in succession would make us believe that we had entered on 
one of these seasons, and that therefore the next day would 
probably resemble the preceding ten. 

Since then all these forms of such an Inductive rule are 
possible, and we have often no d, priori grounds for preferring 
one to another, it would seem to be unreasonable to attempt 
bo establish any universal formula of anticipation. All that 
we can do is to ascertain what are the circumstances under 
which one or other of these rules is, as a matter of fact, 
found to be applicable, and to make use of it under those 
3ircumstances. 

31. (VII.) In the cases discussed in (V.) the almost 
infinitely small chances with which we were concerned were 



364 Fallacies. [CHAP. xiv. 

rightly neglected from all practical consideration, however 
proper it might be, on speculative grounds, to keep our minds 
open to their actual existence. But it has often occurred to 
me that there is a common error in neglecting to take them into 
account when they may, though individually small, make up 
for their minuteness by their number. As the mathematician 
would express it, they may occasionally be capable of being 
integrated into a finite or even considerable magnitude. 

For instance, we may be confronted with a difficulty out 
of which there appears to be only one appreciably possible 
mode of escape. The attempt is made to force us into 
accepting this, however great the odds apparently are against 
it, on the ground that improbable as it may seem, it is at 
any rate vastly more probable than any of the others. I 
can quite admit that, on practical grounds, we may often find 
it reasonable to adopt this course ; for we can only act on 
one supposition, and we naturally and rightly choose, out 
of a quantity of improbabilities, the least improbable. But 
when we are not forced to act, no such decisive preference is 
demanded of us. It is then perfectly reasonable to refuse 
assent to the proposed explanation ; even to say distinctly 
that we do not believe it, and at the same time to decline, 
at present, to accept any other explanation. We remain, in 
tact, in a state of suspense of judgment, a state perfectly 
right and reasonable so long as no action demanding a spe- 
cific choice is forced upon us. One alternative may be 
decidedly probable as compared with any other individually, 
but decidedly improbable as compared with all others col- 
lectively. This in itself is intelligible enough ; what people 
often fail to see is that there is no necessary contradiction 
between saying and feeling this, and yet being prepared 
vigorously to act, when action is forced upon us, as though 
this alternative were really the true one. 



SECT. 33.] Fallacies. 865 

32. To take a specific instance, this way of regarding 
the matter has often occurred to me in disputes upon ' Spi- 
ritualist ' manifestations. Assent is urged upon us because, 
it is said, no other possible solution can be suggested. It 
may be quite true that apparently overwhelming difficulties 
may lie as against each separate alternative solution ; but is 
it always sufficiently realized how numerous such solutions 
may be ? No matter that each individually may be almost 
incredible : they ought all to be massed together and thrown 
into the scale against the proffered solution, when the only 
question asked is, Are we to accept this solution ? There is 
no unfairness in such a course. We are perfectly ready to 
adopt the same plan against any other individual alterna- 
tive, whenever any person takes to claiming this as the 
solution of the difficulty. We are looking at the matter 
from a purely logical point of view, and are quite willing, so 
far, to place every solution, spiritualist or otherwise, upon the 
same footing. The partisans of every alternative are in 
somewhat the same position as the members of a deliberative 
assembly, in which no one will support the motion of any 
other member. Every one can aid effectively in rejecting 
every other motion, but no one can succeed in passing his 
own. Pressure of urgent necessity may possibly force them 
out of this state of practical inaction, by, so to say, breaking 
through the opposition at some point of least resistance; but 
unless aided by some such pressure they are left in a state of 
hopeless dead-lock. 

33. Assuming that the spiritualistic solution admits 
of, and is to receive, scientific treatment, this, it seems to 
me, is the conclusion to which one might sometimes be led 
in the face of the evidence offered. We might have to say to 
every individual explanation, It is incredible, I cannot accept 
it ; and unless circumstances should (which it is hardly pos- 



866 Fallacies. [CHAP. xiv. 

sible that they should) force us to a hasty decision, a deci- 
sion, remember, which need indicate no preference of the 
judgment beyond what is just sufficient to turn the scale in its 
favour as against any other single alternative, we leave the 
matter thus in abeyance. It will very likely be urged that 
one of the explanations (assuming that all the possible ones 
had been included) must be true ; this we readily admit. 
It will probably also be urged that (on the often-quoted 
principle of Butler) we ought forthwith to accept the one 
which, as compared with the others, is the most plausible, 
whatever its absolute worth may be. This seems distinctly 
an error. To say that such and such an explanation is the 
one we should accept, if circumstances compelled us to anti- 
cipate our decision, is quite compatible with its present 
rejection. The only rational position surely is that of 
admitting that the truth is somewhere amongst the various 
alternatives, but confessing plainly that we have no such 
preference for one over another as to permit our saying any- 
thing else than that we disbelieve each one of them. 

34. (VIII.) The very common fallacy of 'judging by 
the event/ as it is generally termed, deserves passing notice 
here, as it clearly belongs to Probability rather than to Logic ; 
though its nature is so obvious to those who have grasped the 
general principles of our science, that a very few words of 
remark will suffice. In one sense every proposition must 
consent to be judged by the event, since this is merely, in 
other words, submitting it to the test of experience. But 
there is the widest difference between the test appropriate 
to a universal proposition and that appropriate to a merely 
proportional or statistical one. The former is subverted by a 
single exception; the latter not merely admits exceptions, 
but implies them. Nothing, however, is more common than 
to blame advice (in others) because it has happened to turn 



SECT. 35.] Fallacies. 367 

out unfortunately, or to claim credit for it (in oneself) because 
it has happened to succeed. Of course if the conclusion was 
avowedly one of a probable kind we must be prepared with 
complacency to accept a hostile event, or even a succession of 
them ; it is not until the succession shows a disposition to 
continue over long that suspicion and doubt should arise, and 
then only by a comparison of the degree of the assigned 
probability, and the magnitude of the departure from it 
which experience exhibits. For any single failure the reply 
must be, ' the advice was sound ' (supposing, that is, that it 
was to be justified in the long run), 'and I shall offer it again 
under the same circumstances.' 

35. The distinction drawn in the above instance 
deserves careful consideration ; for owing to the wide differ- 
ence between the kind of propositions dealt with in Proba- 
bility and in ordinary Logic, and the consequent difference in 
the nature of the proof offered, it is quite possible for argu- 
ments of the same general appearance to be valid in the 
former and fallacious in the latter, and conversely. 

For instance, take the well-known fallacy which consists 
in simply converting a universal affirmative, i.e. in passing 
from All A is B to All B is A. When, as in common Logic, 
the conclusion is to be as certain as the premise, there is not 
a word to be said for such a step. But if we look at the 
process with the more indulgent eye of Induction or Proba- 
bility we see that a very fair case may sometimes be made 
out for it. The mere fact that ' Some B is A ' raises a cer- 
tain presumption that any particular B taken at random will 
be an A. There is some reason, at any rate, for the belief, 
though in the absence of statistics as to the relative fre- 
quency of A and B we are unable to assign a value to this 
belief. I suspect that there may be many cases in which a 
man has inferred that some particular B is an A on the 



368 Fallacies. [CHAP. xiv. 

ground that All A is 5, who might justly plead in his behalf 
that he never meant it to be a necessary, but only a pro- 
bable inference. The same remarks will of course apply 
also to the logical fallacy of Undistributed Middle. 

Now for a ease of the opposite kind, i.e. one in which 
Probability fails us, whereas the circumstances seem closely 
analogous to those in which ordinary inference would be 
able to make a stand. Suppose that I know that one letter 
in a million is lost when in charge of the post. I write to a 
friend and get no answer. Have I any reason to suppose 
that the fault lies with him ? Here is an event (viz. the 
loss of the letter) which has certainly happened ; and we sup- 
pose that, of the only two causes to which it can be assigned, 
the ' value/ i.e. statistical frequency, of one is accurately 
assigned, does it not seem natural to suppose that something 
can be inferred as to the likelihood that the other cause had 
been operative ? To say that nothing can be known about 
its adequacy under these circumstances looks at first sight 
like asserting that an equation in which there is only one 
unknown term is theoretically insoluble. 

As examples of this kind have been amply discussed in 
the chapter upon Inverse rules of Probability I need do no 
more here than remind the reader that no conclusion what- 
ever can be drawn as to the likelihood that the fiault lay 
with my friend rather than with the Post Office. Unless we 
either know, or make some assumption about, the frequency 
with which he neglects to answer the letters he receives, the 
problem remains insoluble. 

The reason why the apparent analogy, indicated above, 
to an equation with only one unknown quantity, fails to hold 
good, is that for the purposes of Probability there are really 
two unknown quantities. What we deal with are propor- 
tional or statistical propositions. Now we are only told that 



SECT. 35.] Fallacies. 369 

in the instance in question the letter was lost, not that they 
were found to be lost in such and such a proportion of cases. 
Had this latter information been given to us we should really 
have had but one unknown quantity to determine, viz. the 
relative frequency with which my correspondent neglects to 
answer his letters, and we could then have determined this 
with the greatest ease. 



v. 24 



CHAPTER XV. 

INSURANCE AND GAMBLING. 

1. IP the reader will recall to mind the fundamental 
postulate of the Science of Probability, established and ex- 
plained in the first few chapters, and so abundantly illus- 
trated since, he will readily recognize that the two opposite 
characteristics of individual irregularity and average regu- 
larity will naturally be differently estimated by different 
minds. To some persons the elements of uncertainty may 
be so painful, either in themselves or in their consequences, 
that they are anxious to adopt some means of diminishing 
them. To others the ultimate regularity of life, at any rate 
within certain departments, its monotony as they consider it, 
may be so wearisome that they equally wish to effect some 
alteration and improvement in its characteristics. We shall 
discuss briefly these mental tendencies, and the most simple 
and obvious modes of satisfying them. 

To some persons, as we have said, the world is all too full 
of change and irregularity and consequent uncertainty. Civi- 
lization has done much to dimmish these characteristics in 
certain directions, but it has unquestionably ._: ' them 
in other directions, and it might not be very easy to say with 
certainty in which of these respects its operation has been, at 
present, on the whole most effective. The diminution of 
irregularity is exemplified, amongst other things, in the case 
of the staple products which supply our necessary food and 



SECT. 2.] Insurance and finmldiny. 371 

clothing. With respect to them, famine and scarcity are by 
comparison almost unknown now, at any rate in tolerably 
civilized communities. As a consequence of this, and of the 
vast improvements in the means of transporting goods and 
conveying intelligence, the fluctuations in the price of such 
articles are much less than they once were. In other direc- 
tions, however, the reverse has been the case. Fashion, for 
instance, now induces so many people in every large com- 
munity simultaneously to desire the same thing, that great 
fluctuations in value may ensue. Moreover a whole group of 
causes (to enter upon any discussion of which would be to 
trench upon the ground of Political Economy) combine to 
produce great and frequent variations in matters concerning 
credit and the currency, which formerly had no existence. 
Bankruptcy, for instance, is from the nature of the case, 
almost wholly a creation of modern times. We will not at- 
tempt to strike any balance between these opposite results 
of modern civilization, beyond remarking that in matters of 
prime importance the actual uncertainties have been pro- 
bably on the whole diminished, whereas in those which affect 
the pocket rather than the life, they have been rather in- 
creased. It might also be argued with some plausibility 
that in cases where the actual uncertainties have not become 
greater, they have for all practical purposes done so, by their 
consequences frequently becoming more serious, or by our 
estimate of these consequences becoming higher. 

2. However the above question, as to the ultimate 
balance of gain or loss, should be decided, there can be no 
doubt that many persons find the present amount of uncer- 
tainty in some of the affairs of life greater than suits their 
taste. How are they to diminish it ? Something of course 
may be done, as regards the individual cases, by prudence 
and foresight. Our houses may be built with a view not to 

242 



572 Insurance and Gambling. , [CHAP. xv.. 

take fire so readily, or precautions may be taken that there 
shall be fire-engines at hand. In the warding off of death 
from disease and accident, something may be done by every 
one who chooses to live prudently. Precautions of the above 
kind, however, do not introduce any questions of Probability. 
These latter considerations only come in when we begin to 
invoke the regularity of the average to save us from the 
irregularities of the details. We cannot, it is true, remove 
the uncertainty in itself, but we can so act that the conse- 
quences of that uncertainty shall be less to us, or to those in 
whom we are interested. Take the case of Life Insurance. 
A professional man who has nothing but the income he earns 
to depend upon, knows that the whole of that income may 
vanish in a moment by his death. This is a state of things 
which he cannot prevent; and if he were the only one in 
such a position, or were unable or unwilling to combine with 
his fellow-men, there would be nothing more to be done in 
the matter except to live within his income as much as pos- 
sible, and so leave a margin of savings. 

3. There is however an easy mode of escape for him. 
All that he has to do is to agree with a number of others, 
who are in the same position as himself, to make up, so to 
say, a common purse. They may resolve that those of their 
number who live to work beyond the average length of life 
shall contribute to support the families of those who die 
earlier. If a few only concurred in such a resolution they 
would not gain very much, for they would still be removed 
by but a slight step from that uncertainty which they are 
seeking to escape. What is essential is that a considerable 
number should thus combine so as to get the benefit of that 
comparative regularity which the average, as is well known, 
almost always tends to exhibit. 

4. The above simple considerations really contain the 



SECT. 4.] Insurance and Gambling. 373 

essence of all insurance. Such points as the fact that the 
agreement for indemnity extends only to a certain definite 
sum of money; and that instead of calling for an occasional 
general contribution at the time of the death of each member 
they substitute a fixed annual premium, out of the proceeds 
of which the payment is to be made, are merely accidents of 
convenience and arrangement. Insurance is simply equiva- 
lent to a mutual contract amongst those who dread the con- 
sequences of the uncertainty of their life or employment, 
that they will employ the aggregate regularity to neutralize 
as far as possible the individual irregularity. They know 
that for every one who gains by such a contract another will 
lose as much; or if one gains a great deal many must have 
lost a little. They know also that hardly any of their num- 
ber can expect to find the arrangement a 'fair' one, in the 
sense that they just get back again what they have paid in 
premiums, after deducting the necessary expenses of man- 
agement; but they deliberately prefer this state of things. 
They consist of a body of persons who think it decidedly 
better to leave behind them a comparatively fixed fortune, 
rather than one which is extremely uncertain in amount; 
although they are perfectly aware that, owing to the un- 
avoidable expenses of managing the affairs of such a society, 
the comparatively fixed sum, so to be left, will be a trifle less 
than the average fortunes which would have been left had 
no such system of insurance been adopted. 

As this is not a regular treatise upon Insurance no more 
need be said upon the exact nature of such societies, beyond 
pointing out that they are of various different kinds. Some- 
times they really are what we have compared them with, 
viz. mutual agreements amongst a group of persons to make 
up each other's losses to a certain extent. Into this category 
fall the Mutual Insurance Societies, Benefit Societies, Trades 



874 Insurance and Gambling. [CHAP. xv. 

Unions (in respect of some of their functions), together with 
innumerable other societies which go by various names. 
Sometimes they are companies worked by proprietors or 
shareholders for a profit, like any other industrial enter- 
prise. This is the case, I believe, with the majority of the 
ordinary Life Insurance Societies. Sometimes, again, it is 
the State which undertakes the management, as in the case 
of our Post Office Insurance business. 

5. It is clear that there is no necessary limit to the 
range of application of this principle 1 . It is quite conceiv- 
able that the majority of the inhabitants of some nation 
might be so enamoured of security that they should devise 
a grand insurance society to cover almost every concern in 
life. They could not indeed abolish uncertainty, for the 
conditions of life are very far from permitting this, but they 
could without much difficulty get rid of the worst of the 
consequences of it. They might determine to insure not 
merely their lives, houses, ships, and other things in re- 
spect of which sudden and total loss is possible, but also 
to insure their business ; in the sense of avoiding not only 

1 The question of the advisability moderate loss and contingent great 

of inoculation against the small-pox, loss. In the seventeenth century it 

which gave rise to much discussion seems to have been an occasional 

amongst the writers on Probability practice, before a journey into the 

during the last century, is a case in Mediterranean, to insure against 

point of the same principles applied capture by Moorish pirates, with a 

to a very different kind of instance, view to secure having the ransom 

The loss against which the insurance paid. (See, for an account of some 

was directed was death by small-pox, extraordinary developments of the 

the premium paid was the illness insurance principle, Walford's In- 

and other inconvenience, and the surance Guide and Handbook. It is 

very small risk of death, from the not written in a very scientific spi- 

iaoculation. The disputes which rit, but it contains much infonna- 

thence arose amongst writers on the tion on all matters connected with 

subject involved the same difficulties insurance.) 
aa to the balance between certain 



SECT. 5.] Insurance and 1 375 

bankruptcy, but even casual bad years, on the same principle 
of commutation. Unfamiliar as such an aim may appear 
when introduced in this language, it is nevertheless one 
which under a name of suspicious import to the conservative 
classes has had a good deal of attention directed to it. It is 
really scarcely anything else than Communism, which might 
indeed be defined as a universal and compulsory 1 insurance 
society which is to take account of all departments of busi- 
ness, and, in some at least of its forms, to invade the province 
of social and domestic life as well. 

Although nothing so comprehensive as this is likely to 
be practically carried out on any very large scale, it deserves 
notice that the principle itself is steadily spreading in every 
direction in matters of detail. It is, for instance, the great 
complaint against Trades Unions that they too often seek to 
secure these results in respect of the equalization of the 
workmen's wages, thus insuring to some degree against in- 
competence, as they rightly and wisely do against illness and 
loss of work. Again, there is the Tradesman's Mutual Pro- 
tection Society, which insures against the occasional loss 
entailed by the necessity of having to conduct prosecutions 
at law. There are societies in many towns for the prose- 
cution of petty thefts, with the object of escaping the same 
uncertain and perhaps serious loss. Amongst instances of 
insurance for the people rathdr than by them, there is of 
course the giant example of the English Poor Law, in 
which the resemblance to an initial Communistic system 
becomes very marked. The poor are insured against loss 

1 All that is meant by the above forcing it, the matter would of course 

comparison is that the ideal aimed assume a very different aspect, 

at by Communism is similar to that Similarly with the action of Trades 

of Insurance. If we look at the Unionism referred to in the next 

processes by which it would be paragraph, 
carried out, and the means for en- 



376 Insurance and T v. ... [CHAP. xv. 

of Work arising not only from illness and old age, but from 
any cause except wilful idleness. They do not, it is true, 
pay the whole premium, but since they mostly bear some 
portion of the burden of municipal and county taxation 
they must certainly be considered as paying a part of 
the premium. In some branches also of the public and 
private services the system is adopted of deducting a per- 
centage from the wage or salary, for the purpose of a 
semi-compulsory insurance against death, illness or super- 
annuation. 

6. Closely connected with Insurance, as an application 
of Probability, though of course by contrast, stands Gambling. 
Though we cannot, in strictness, term either of these prac- 
tices the converse of the other, it seems nevertheless correct 
to say that they spring from opposite mental tendencies. 
Some persons, as has been said, find life too monotonous for 
their taste, or rather the region of what can be predicted 
with certainty is too large and predominant in their estima- 
tion. They can easily adopt two courses for securing the 
changes they desire. They may, for one thing, aggravate 
and intensify the results of events which are comparatively 
incapable of prevision, these events not being in themselves 
of sufficient importance to excite any strong emotions. The 
most obvious way of doing this is by betting upon them. 
Or again, they may invent games or other pursuits, the 
individual contingencies of which are entirely removed from 
all possible human prevision, and then make heavy money 
consequences depend upon these contingencies. This is 
gambling proper, carried on mostly by means of cards and 
dice and the roulette. 

The gambling spirit, as we have said, seeks for the ex- 
citement of uncertainty and variety. When therefore people 
make a long continued practice of playing, especially if the 



SECT. 7.] Insurance and Gambling. 377 

stakes for which they play are moderate in comparison with 
their fortune, this uncertainty from the nature of the case 
begins to diminish. The thoroughly practised gambler, if 
he possesses more than usual skill (in games where skill 
counts for something), must be regarded as a man following 
a profession, though a profession for the most part of a risky 
and exciting kind, to say nothing of its ignoble and often 
dishonest character. If, on the other hand, his skill is below 
the average, or the game is one in which skill does not tell 
and the odds are slightly in favour of his antagonist, as in 
the gaming tables, one light in which he can be regarded 
is that of a man who is following a favourite amusement ; 
if this amusement involves a constant annual outlay on his 
part, that is nothing more than what has to be said of most 
other amusements. 

7. We cannot, of course, give such a rational expla- 
nation as the above in every case. There are plenty of 
novices, and plenty of fanatics, who go on steadily losing in 
the full conviction that they will eventually come out win- 
ners. But it is hard to believe that such ignorance, or such 
intellectual twist, can really be so widely prevalent as would 
be requisite to constitute them the rule rather than the ex- 
ception. There must surely be some very general impulse 
which is gratified by such resources, and it is not easy to see 
what else this can be than a love of that variety and con- 
sequent excitement which can only be found in perfection 
where exact prevision is impossible. 

It is of course very difficult to make any generalization 
here as to the comparative prevalence of various motives 
amongst mankind; but when one considers what is the 
difference which most quiet ordinary whist players feel 
between a game for ' love ' and one in which there is a 
small stake, one cannot but assign a high value to the 



378 Insurance and Gambling. [CHAP, xv, 

influence of a wish to emphasize the excitement of loss 
and gain. 

I would not for a moment underrate the practical dangers 
which are found to attend the practice of gambling. It 
is remarked that the gambler, if he continues to play for a 
long time, is under an almost irresistible impulse to increase 
his stakes, and so re-introduce the element of uncertainty. 
It is in fact this tendency to be thus led on, which makes 
the principal danger and mischief of the practice. Risk and 
uncertainty are still such normal characteristics of even civi- 
lized life, that the mere extension of such tendencies into 
new fields does not in itself offer any very alarming pros- 
pect. It is only to be deprecated in so far as there is a 
danger, which experience shows to be no trifling one, that 
the fascination found in the pursuit should lead men into 
following it up into excessive lengths 1 . 

8. The above general treatment of Gambling and 
Insurance seems to me the only rational and sound prin- 
ciple of division ; namely, that on which the different prac- 
tices which, under various names, are known as gambling 
or insurance, are arranged in accordance with the spirit of 
which they are the outcome, and therefore of the results 
which they are designed to secure. If we were to attempt 

1 One of the best discussions that sum-total of what they distribute in 
I have recently seen on these sub- prizes is less than that of what they 
jects, by a writer at once thoroughly receive in payments. The difference, 
competent and well informed, is in in respect of information deliberate- 
Mr Proctor's Chance and Luck. It ly withheld and false reports wilfully 
appears to me however that he runs spread, between most of the lotteries 
into an extreme in his denunciation that have been supported, and the 
not of the folly but of the dishonesty bubble companies which justly de- 
of all gambling. Surely also it is a serve the name of swindles, ought to 
strained use of language to speak of prevent the same name being applied 
all lotteries as 'unfair* and even to both. 
'swindling 1 on the ground that the 



SECT. 8.] Insurance and I 1 !'/ ';! 379 

to judge and arrange them according to the names which 
they currently bear, we should find ourselves led to no 
kind of systematic division whatever; the fact being that 
since they all alike involve, as their essential characteristic, 
payments and receipts, one or both of which are necessarily 
uncertain in their date or amount, the names may often be 
interchanged. 

For instance, a lottery and an ordinary insurance society 
against accident, if we merely look to the processes per- 
formed in them, are to all intents and purposes identical. 
In each alike there is a small payment which is certain in 
amount, and a great receipt which is uncertain in amount. 
A great many persons pay the small premium, whereas a 
few only of their number obtain a prize, the rest getting 
no return whatever for their outlay. In each case alike, 
also, the aggregate receipts and losses are intended to 
balance each other, after allowing for the profits of those 
who carry on the undertaking. But of course when we 
take into account the occasions upon which the insurers 
get their prizes, we see that there is all the difference in the 
world between receiving them at haphazard, as in a lottery, 
and receiving them as a partial set-off to a broken limb or 
injured constitution, as in the insurance society. 

Again, the language of betting may be easily made to 
cover almost every kind of insurance. Indeed De Morgan 
has described life insurance as a bet which the individual 
makes with the company, that he will not live beyond a 
certain age. If he dies young, he is pecuniarily a gainer, if 
he dies late he is a loser 1 . Here, too, though the expression 
1 "A fire insurance is a simple transaction in which the results of 
bet between the office and the party, a future event are to bring gain or 
and a life insurance is a collection loss." Penny Cyclopedia, under the 
x>f wagers. There is something of head of Wager. 
the principle of a wager in every 



380 Insurance and Gambling. [CHAP. xv. 

is technically quite correct (since any such deliberate risk of 
money, upon an unproductive venture, may fall under the 
definition of a bet), there is the broadest distinction between 
betting with no other view whatever than that of risking 
money, and betting with the view of diminishing risk and 
loss as much as possible. In fact, if the language of sporting 
life is to be introduced into the matter, we ought, I presume, 
to speak of the insurer as ' hedging ' against his death. 

9. Again, in Tontines we have a system of what is 
often called Insurance, and in certain points rightly so, 
but which is to all intents and purposes simply and abso- 
lutely a gambling transaction. They have been entirely 
abandoned, I believe, for some time, but were once rather 
popular, especially in France. On this plan the State, or 
whatever society manages the business, does not gain any- 
thing until the last member of the Tontine is dead. As the 
number of the survivors diminishes, the same sum-total of 
annuities still continues to be paid amongst them, as long as 
any are left alive, so that each receives a gradually increasing 
sum. Hence those who die early, instead of receiving the 
most, as on the ordinary plan, receive the least ; for at the 
death of each member the annuity ceases absolutely, so far 
as he and his relations are concerned. The whole affair 
therefore is to all intents and purposes a gigantic system 
of betting, to see which can live the longest; the State 
being the common stake-holder, and receiving a heavy com- 
mission for its superintendence, this commission being natur- 
ally its sole motive for encouraging such a transaction. It is 
recorded of one of the French Tontines 1 that a widow of 97 
was left, as the last survivor, to receive an annuity of 73,500 
livres during the rest of the life which she could manage to 
drag on after that age ; she having originally subscribed a 
1 Encyclopedic Methodique, under the bead of Tontines, 



SECT. 10.] Insurance and Gambling. 381 

single sum of 300 livres only. It is obvious that such a system 
as this, though it may sometimes go by the name of insur- 
ance, is utterly opposed to the spirit of true insurance, 
since it tends to aggravate existing inequalities of fortune 
instead of to mitigate them. The insurer here bets that 
he will die old ; in ordinary insurance he bets that he will 
die young. 

Again, to take one final instance, common opinion often 
regards the bank or company which keeps a rouge et nvir 
table, and the individuals who risk their money at it, as being 
both alike engaged in gambling. So they may be, tech- 
nically, but for all practical purposes such a bank is as sure 
and safe a business as that of any ordinary insurance society, 
and probably far steadier in its receipts than the majority of 
ordinary trades in a manufacturing or commercial city. The 
bank goes in for many and small transactions, in proportion 
to its capital ; their customers, very often, in proportion to 
their incomes go in for very heavy transactions. That the 
former comes out a gainer year after year depends, of course, 
upon the fact that the tables are notoriously slightly in their 
favour. But the steadiness of these gains when compared 
with the unsteadiness of the individual losses depends simply 
upon, in fact, is merely an illustration of, the one great 
permanent contrast which lies at the basis of all reasoning 
in Probability. 

10. We have so far regarded Insurance and Gambling 
as being each the product of a natural impulse, and as having 
each, if we look merely to experience, a great mass of human 
judgment in its favour. The popular moral judgment, however, 
which applauds the one and condemns the other rests in great 
part upon an assumption, which has doubtless much truth 
in it, but which is often interpreted with an absoluteness which 
leads to error in each direction ; the duty of insurance being 



382 Insurance and Gambling. [CHAP. xv. 

too peremptorily urged upon every one, and the practice of 
gambling too universally regarded as involving a sacrifice of 
real self-interest, as being in fact little better than a per- 
sistent blunder. The assumption in question seems to be 
extracted from the acknowledged ad vantages of insurance, 
and then invoked to condemn the practice of gambling. But 
in so doing the fact does not seem to be sufficiently recog- 
nized that the latter practice, if we merely look to the ex- 
tent and antiquity of the tacit vote of mankind in its favour, 
might surely claim to carry the day. 

It is of course obvious that in all cases with which we 
are concerned, the ,.,., i wealth is unaltered; money 
being merely transferred from one person to another. The 
loss of one is precisely equivalent to the gain of another. 
At least this is the approximation to the truth with which we 
find it convenient to start 1 . Now if the happiness which is 
yielded by wealth were always in direct proportion to its 
amount, it is not easy to see why insurance should be advo- 
cated or gambling condemned. In the case of the latter this 
is obvious enough. I have lost 50, say, but others (one or 
more as the case may be) have gained it, and the increase 
of their happiness would exactly balance the diminution of 
mine. In the case of Insurance there is a slight complica- 
tion, arising from the fact that the falling in of the policy 
does not happen at random (otherwise, as already pointed 

1 Of coarse, if we introduce con- otherwise be the case. Again, in 

Biderations of Political Economy, the case of gambling, a large loss of 

corrections wiU have to be made. capital by any one will almost neces- 

For one thing, every Insurance Office sarily involve an actual destruction 

is, aa De Morgan repeatedly insists, of wealth; to say nothing of the 

a Saving* Bank as well as an Insur- fact that, practically, gambling often 

ance Office. The Office invests the causes a constant transfer of wealth 

premiums, and can therefore afford from productive to unproductive pur- 

to pay a larger sum than would poses. 



SECT. 11.] Insurance and Gambling. 385 

out, it would be simply a lottery), but is made contingent 
upon some kind of loss, which it is intended as far as possible 
to balance. I insure myself on a railway journey, break my 
leg in an accident, and, having paid threepence for my 
ticket, receive say 200 compensation from the insurance 
company. The same remarks, however, apply here; the 
happiness I acquire by this 200 would only just balance the 
aggregate loss of the 16,000 who have paid their threepences 
and received no return for them, were happiness always 
directly proportional to wealth. 

11. The practice of Insurance does not, I think, give 
rise to many questions of theoretic interest, and need not 
therefore detain us longer. The fact is that it has hardly 
yet been applied sufficiently long and widely, or to matters 
which admit of sufficiently accurate statistical treatment, 
except in one department. This, of course, is Life Insurance ; 
but the subject is one which requires constant attention to 
details of statistics, and is (rightly) mainly carried out in 
strict accordance with routine. As an illustration of this 
we need merely refer to the works of De Morgan, a profes- 
sional actuary as well as a writer on the theory of Probability, 
who has found but little opportunity to aid his speculative 
treatment of Probability by examples drawn from this class 
of considerations. 

With Gambling it is otherwise* Not only have a variety 
of interesting single problems been discussed (of which the 
Petersburg problem is the best known) but several specula- 
tive questions of considerable importance have been raised. 
One of these concerns the disadvantages of the practice of 
gambling. There have been a number of writers who, not 
content with dwelling upon the obvious moral and indirect 
mischief which results, in the shape of over-excitement, 
consequent greed, withdrawal from the steady business 



384 Insurance and Gambling. [CHAP. xv. 

habits which alone insure prosperity in the long run, diver- 
sion of wealth into dishonest hands, &c., have endeavoured 
to demonstrate the necessary loss caused by the practice. 

12. These attempts may be divided into two classes. 
There are (1) those which appeal to merely numerical con- 
siderations, and (2) those which introduce what is called the 
' moral ' as distinguished from the mathematical value of a 
future contingency. 

(1) For instance, an ingenious attempt has been made 
by Mr Whitworth to prove that gambling is necessarily 
disadvantageous on purely mathematical grounds. 

When two persons play against each other one of the two 
must be ruined sooner or later, even though the game be a 
fair one, supposing that they go on playing long enough; the 
one with the smaller income having of course the worst 
chance of being the lucky survivor. If one of them has a 
finite, and the other an infinite income, it must clearly be 
the former who will be the ultimate sufferer if they go on 
long enough. It is then maintained that this is in fact every 
individual gambler's position, "no one is restricted to gamb- 
ling with one single opponent; the speculator deals with the 
public at large, with a world whose resources are practically 
unlimited. There is a prospect that his operations may 
terminate to his own disadvantage, through his having no- 
thing more to stake ; but there is no prospect that it will 
terminate to his advantage through the exhaustion of the 
resources of the world. Every one who gambles is carrying 
on an unequal warfare: he is ranged with a restricted capital 
against an adversary whose means are infinite 1 ." 

In the above argument it is surely overlooked that the 
adversaries against whom he plays are not one body with a 
common purse, like the bank in a gambling establishment. 
1 Choice and Chance, Ed. n. p. 308. 



SECT. 13.] Insurance and Gambling. 385 

Each of these adversaries is in exactly the same position as 
he himself is, and a precisely similar proof might be em- 
ployed to show that each of them must be eventually ruined 
which is of course a reduction to absurdity. Gambling can 
only transfer money from one player to another, and there- 
fore none of it can be actually lost. 

13. What really becomes of the money, when they 
play to extremity, is not difficult to see. First suppose a 
limited number of players. If they go on long enough, the 
money will at last all find its way into the pocket of some 
one of their number. If their fortunes were originally equal, 
each stands the same chance of being the lucky survivor; 
in which case we cannot assert, on any numerical grounds, 
that the prospect of the play is disadvantageous to any one 
of them. If their fortunes were unequal, the one who had 
the largest sum to begin with can be shown to have the 
best chance, according to some assignable law, of being left 
the final winner; in which case it must be just as advan- 
tageous for him, as it was disadvantageous for his less wealthy 
competitors. 

When, instead of a limited number of players, we sup- 
pose an unlimited number, each as he is ruined retiring 
from the table and letting another come in, the results are 
more complicated, but their general tendency can be readily 
distinguished. If we supposed , that no one retired except 
when he was ruined, we should have a state of things in 
which all the old players were growing gradually richer. In 
this case the prospect before the new comers would steadily 
grow worse and worse, for their chance of winning against 
such rich opponents would be exceedingly small. But as 
this is an unreasonable supposition, we ought rather to 
assume that not only do the ruined victims retire, but also 
that those who have gained fortunes of a certain amount 
v. 25 



386 Insurance and Gwnibling. [CHAP. xv. 

retire also, so that the aggregate and average wealth of the 
gambling body remains pretty steady. What chance any 
given player has of being ruined, and how long he may 
expect to hold out before being ruined, will depend of course 
upon the initial incomes of the players, the rules of the 
game, the stakes for which they play, and other considera- 
tions. But it is clear that for all that is lost by one, a 
precisely equal sum must be gained by others, and that 
therefore any particular gambler can only be cautioned be- 
forehand that his conduct is not to be recommended, by 
appealing to some siich suppositions as those already men- 
tioned in a former section. 

14. As an additional justification of this view the 
reader may observe that the state of things in the last 
example is one which, expressed in somewhat different 
language and with a slight alteration of circumstances, is 
being incessantly carried on upon a gigantic scale upon 
every side of us. Call it the competition of merchants and 
traders in a commercial country, and the general results are 
familiar enough. It is true that in so far as skill comes into 
the question, they are not properly gamblers; but in so far 
as chance and risk do, they may be fairly so termed, and in 
many branches of business this must necessarily be the case 
to a very considerable extent. Whenever business is carried 
on in a reckless way, the comparison is on general grounds 
fair enough. In each case alike we find some retiring ruined, 
and some making their fortunes; and in each case alike 
also the chances, cceteris paribus, lie with those who have 
the largest fortunes. Every one is, in a sense, struggling 
against the collective commercial world, but since each of his 
competitors is doing the same, we clearly could not caution 
any of them (except indeed the poorer ones) that their efforts 
must finally end in disadvantage. 



SECT. 16.] Insurance and Gambling. 387 

15. If we wish to see this result displayed in its 
most decisive form we may find a good analogy in a very 
different class of events, viz. in the fate of surnames. We 
are all gamblers in this respect, and the game is carried 
out to the last farthing with a rigour unknown at New- 
market or Monte Carlo. In its complete treatment the 
subject is a very intricate one 1 , but a simple example will 
serve to display the general tendency. Suppose a colony 
comprising 1000 couples of different surnames, and suppose 
that each of these has four children who grow up to marry. 
Approximately, one in 16 of these families will consist of 
girls only; and therefore, under ordinary conventions, about 
62 of the names will have disappeared for ever after the 
next generation. Four again out of 16 will have but one 
boy, each of whom will of course be in the same position 
as his father, viz. the sole representative of his name. 
Accordingly in the next generation one in 16 of these names 
will again drop out, and so the process continues. The 
number which disappears in each successive generation be- 
comes smaller, as the stability of the survivors becomes 
greater owing to their larger numbers. But there is no 
check to the process. 

16. The analogy here is a very close one, the names 
which thus disappear corresponding to the gamblers who 
retire ruined and those which increase in number corre- 
sponding to the lucky winners. The ultimate goal in each 
case alike, of course an exceedingly remote one, is the 
exclusive survival of one at the expense of all the others. 
That one surname does thus drop out after another must 
have struck every one who has made any enquiry into family 

1 It was, I believe, first treated as iv. 1875, where a complete mathe- 
& serious problem by Mr Galton. matical solution is indicated by Mr 
(See the Journal Anthrop. Imt. Vol. H. W. Watson.) 

252 



888 Insurance and Gambling. [CHAP. xv. 

genealogy, and various fanciful accounts have been given 
by those unfamiliar with the theory of probability. What 
is often apt to be overlooked is the extreme slightness of 
what may be termed the "turn of the tables" in favour 
of the survival at each generation. In the above numerical 
example we have made an extravagantly favourable sup- 
position, by assuming that the population doubles at every 
generation. In an old and thickly populated country where 
the numbers increase very slowly, we should be much nearer 
the mark in assuming that the average effective family, 
that is, the average number of children who live to marry, 
was only two. In this case every family which was repre- 
sented at any time by but a single male would have but 
three chances in four of surviving extinction, and of course 
the process of thinning out would be a more rapid one. 

17. The most interesting class of attempts to prove 
the disadvantages of gambling appeal to what is tech- 
nically called * moral expectation' as distinguished from 
* mathematical expectation/ The latter may be defined 
simply as the average money value of the venture in 
question; that is, it is the product of the amount to be 
gained (or lost) and the chance of gaining (or losing) it. 
For instance, if I bet four to one in sovereigns against the 
occurrence of ace with a single die there would be, on the 
average of many throws, a loss of four pounds against a gain 
of five pounds on each set of six occurrences; i.e. there 
would be an average gain of three shillings and fourpence 
on each throw. This is called the true or mathematical 
expectation. The so-called 'moral expectation', on the other 
hand, is the subjective value of this mathematical expec- 
tation. That is, instead of reckoning a money fortune in 
the ordinary way, as what it is, the attempt is made to 
reckon it at what it is felt to be. The elements of compu- 



SECT. 18.] Insurance and Gambling. 389 

tation therefore become, not pounds and ^hillings but sums 
of pleasure enjoyed actually or in prospect. Accordingly 
when reckoning the present value of a future gain, we must 
now multiply, not the objective but the subjective value, 
by the chance we have of securing that gain. 

With regard to the exact relation of this moral fortune 
to the physical various more or less arbitrary assumptions 
have been made. One writer (Buffon) considers that the 
moral value of any given sum varies inversely with the 
total wealth of the person who gains it. Another (D. Ber- 
noulli) starting from a different assumption, which we shall 
presently have to notice more particularly, makes the moral 
value of a fortune vary as the logarithm of its actual amount 1 . 
A third (Cramer) makes it vary with the square root of the 
amount. 

18. Historically, these proposals have sprung from 
the wish to reconcile the conclusions of the Petersburg 
problem with the dictates of practical common sense; for, 
by substituting the moral for the physical estimate the 
total value of the expectation could be reduced to a finite 
sum. On this ground therefore such proposals have no great 
interest, for, as we have seen, there is no serious difficulty in 
the problem when rightly understood. 

These same proposals however have been employed in 

1 Bernoulli himself does not seem every psychologist. It is what is 

to have based his conclusions upon commonly called Fechner's Law, 

actual experience. But it is a note- which he has established by aid of 

worthy fact that the assumption with an enormous amount of careful ex- 

which he starts, viz. that the sub- periment in the case of a number of 

jective value of any small increment our simple sensations. But I do not 

(dx) is inversely proportional to the believe that he has made any claim 

sum then possessed (x), and which that such a law holds good in the 

leads at once to the logarithmic law far more intricate dependence of 

above mentioned, is identical with happiness upon wealth* 
one which is now familiar enough to 



390 Insurance and Gambling. [CHAP. xv. 

order to prove that gambling is necessarily disadvantageous, 
and this to both parties. Take, for instance, Bernoulli's sup- 
position. It can be readily shown that if two persons each 
with a sum of 50 to start with choose to risk, say, 10 upon 
an even wager there will be a loss of happiness as a result ; 
for the pleasure gained by the possessor of 60 will not 
be equal to that which is lost by the man who leaves off 
with 40*. 

19. This is the form of argument commonly adopted ; 
but, as it stands, it does not seem conclusive. It may surely 
be replied that all which is thus proved is that inequality is 
bad, on the ground that two fortunes of 50 are better than 
one of 60 and one of 40. Conceive for instance that the 
original fortunes had been 60 and 40 respectively, the 
event may result in an increase of happiness ; for this will 
certainly be the case if the richer man loses and the fortunes 
are thus equalized. This is quite true; and we are therefore 
obliged to show, what can be very easily shown, that if 
the other alternative had taken place and the two fortunes 
had been made still more unequal (viz. 65 and 35 respec- 
tively) the happiness thus lost would more than balance 
what would have been gained by the equalization. And 
since these two suppositions are equally likely there will be a 
loss in the long run. 

The consideration just adduced seems however to show 

1 The formula expressive of this bet. Then the balance, as regards 

. , . . x . happiness, must be drawn between 

moral happiness is c log -; where x rr 






stands for the physical fortune poa- 

sessed at the time, and a for that or log a; 2 and log(ar+ y)(x-y) t 

small value of it at which happiness or ar 3 and x^-y 1 , 

is supposed to disappear: c being an the former of which is necessarily 

arbitrary constant. Let two persons, the greater. 

whose fortune ia x, risk y on an even 



SECT. 20.] Insurance and Gambling. 391 

that the common way of stating the conclusion is rather 
misleading; and that, on the assumption in question as 
to the law of dependence of happiness on wealth, it really is 
the case that the effective element in rendering gambling 
disadvantageous is its tendency to the increase of the in- 
equality in the distribution of wealth. 

20. This raises two questions, one of some speculative 
interest in connection with our subject, and the other of 
supreme importance in the conduct of life. The first is this : 
quite apart from any particular assumption which we make 
about moral fortunes or laws of variation of happiness, is 
it the fact that gambling tends to increase the existing 
inequalities of wealth ? Theoretically there is no doubt that 
this is so. Take the simplest case and suppose two people 
tossing for a pound. If their fortunes were equal to begin 
with there must be resultant inequality. If they were 
unequal there is an even chance of the inequality being 
increased or diminished; but since the increase is propor- 
tionally greater than the decrease, the final result remains of 
the same kind as when the fortunes were equal 1 . Taking a 
more general view the same conclusion underlies all our 
reasoning as to the averages of large numbers, viz. that the 
resultant divergencies increase absolutely (however they 
diminish relatively) as the numbers become greater. And of 
course we refer to these absolute divergencies when we 
are talking of the distribution of wealth, 

1 This may be seen more clearly of the average; for there are only 
as follows. Suppose two pair of two alternatives and these will be 
gamblers, each pair consisting of equally frequent in the long run. 
men possessing 50 and 30 re* It is obvious that we have had two 
speotively. Now if we suppose the fortunes of 50 and two of 30 con- 
richer man to win in one case and verted into one of 20, two of 40, 
the poorer in the other these two and one of 60. And this is clearly 
results will be a fair representation an increase of inequality. 



392 Insurance and Gambling. [CHAP. xv 

21. This is the theoretic conclusion. How far the 
actual practice of gambling introduces counteracting agencies 
must be left to the determination of those who are com- 
petent to pronounce. So far as outsiders are authorised 
to judge from what they read in the newspapers and other 
public sources of information, it would appear that these 
counteracting agencies are very considerable, and that in 
consequence it is a rather insecure argument to advance 
against gambling. Many a large fortune has notoriously 
been squandered on the race-course or in gambling saloons, 
and most certainly a large portion, if not the major part, 
has gone to swell the incomes of many who were by compari- 
son poor. But the solution of this question must clearly be 
left to those who have better opportunities of knowing 
the facts than is to be expected on the. part of writers on 
Probability. 

22. The general conclusion to be drawn is that those 
who invoked this principle of moral fortune as an argument 
against gambling were really raising a much more intricate 
and far-reaching problem than they were aware of. What 
they were at work upon was the question, What is the 
distribution of wealth which tends to secure the maximum 
of happiness ? Is this best secured by equality or inequality? 
Had they, really followed out the doctrine on which their 
denunciation of gambling was founded they ought to have 
adopted the Socialist's ideal as being distinctly that which 
tends to increase happiness. And they ought to have 
brought under the same disapprobation which they ex- 
pressed against gambling all those tendencies of modern 
civilized life which work in the same direction. For in- 
stance ; keen competition, speculative operations, extended 
facilities of credit, mechanical inventions, enlargement of 
business .operations into vast firms: all these, and other 



SECT. 22.] Insurance and Gambling. 393 

similar tendencies too numerous to mention here, have had 
some influence in the way of adding to existing inequalities. 
They are, or have been, in consequence denounced by 
socialists : are we honestly to bring them to this test in 
order to ascertain whether or not they are to be condemned ? 
The reader who wishes to see what sort of problems this 
assumption of 'moral fortune 1 ought to introduce may be 
recommended to read Mr F. Y. Edgeworth's Mathematical 
Psychics, the only work with which I am acquainted which 
treats of these questions. 



CHAPTER XVI. 

THE APPLICATION OF PROBABILITY TO TESTIMONY. 

1. ON the principles which have been adopted in this 
work, it becomes questionable whether several classes of 
problems which may seem to have acquired a prescriptive 
right to admission, will not have to be excluded from the 
science of Probability. The most important, perhaps, of 
these refer to what is commonly called the credibility of 
testimony, estimated either at first hand and directly, or as 
influencing a juryman, and so reaching us through his 
sagacity and trustworthiness. Almost every treatise upon 
the science contains a discussion of the principles according 
to which credit is to be attached to combinations of the 
reports of witnesses of various degrees of trustworthiness, or 
the verdicts of juries consisting of larger or smaller numbers. 
A great modern mathematician, Poisson, has written an 
elaborate treatise expressly upon this subject ; whilst a con- 
siderable portion of the works of Laplace, De Morgan, and 
others, is devoted to an examination of similar enquiries. It 
would be presumptuous to differ from such authorities as 
these, except upon the strongest grounds ; but I confess that 
the extraordinary ingenuity and mathematical ability which 
have been devoted to these problems, considered as questions 
in Probability, fails to convince me that they ought to have 
been so considered. The following are the principal grounds 
for this opinion. 



SECT, 3.] Testimony. 395 

2. It will be remembered that in the course of the 
chapter on Induction we entered into a detailed investiga- 
tion of the process demanded of us when, instead of the 
appropriate propositions from which the inference was to be 
made being set before us, the individual presented himself, 
and the task was imposed upon us of selecting the requisite 
groups or series to which to refer him. In other words, in- 
stead of calculating the chance of an event from determinate 
conditions of frequency of its occurrence (these being either 
obtained by direct experience, or deductively inferred) we 
have to select the conditions of frequency out of a plurality 
of more or less suitable ones. When the problem is pre- 
sented to us at such a stage as this, we may of course assume 
that the preliminary process of obtaining the statistics 
which are extended into the proportional propositions has 
been already performed ; we may suppose therefore that we 
are already in possession of a quantity of such propositions, 
our principal remaining doubt being as to which of them 
we should then employ. This selection was shown to be 
to a certain extent arbitrary ; for, owing to the fact of the 
individual possessing a large number of different properties, 
he became in consequence a member of different series or 
groups, which might present different averages. We must 
now examine, somewhat more fully than we did before, 
the practical conditions under which any difficulty arising 
from this source ceases to be of importance. 

3. One condition of this kind is very simple and ob- 
vious. It is that the different statistics with which we are 
presented should not in reality offer materially different 
results. If, for instance, we were enquiring into the pro- 
bability of a man aged forty dying within the year, we might 
if we pleased take into account the fact of his having red 
hair, or his having been born in a certain county or town. 



396 Testimony. [CHAP. xvi. 

Each of these circumstances would serve to specialize the 
individual, and therefore to restrict the limits of the statistics 
which were applicable to his case. But the consideration of 
such qualities as these would either leave the average pre- 
cisely as it was, or produce such an unimportant alteration 
in it as no one would think of taking into account. Though 
we could hardly say with certainty of any conceivable 
characteristic that it has absolutely no bearing on the 
result, we may still feel very confident that the bearing of 
such characteristics as these is utterly insignificant. Of 
course in the extreme case of the things most perfectly 
suited to the Calculus of Probability, viz. games of pure 
chance, these subsidiary characteristics are quite irrelevant. 
Any further particulars about the characteristics of the cards 
in a really fair pack, beyond those which are familiar to all 
the players, would convey no information whatever about the 
result. 

Or again ; although the different sets of statistics may 
not as above give almost identical results, yet they may do 
what practically comes to very much the same thing, that 
is, arrange themselves into a small number of groups, all of 
the statistics in any one group practically coinciding in their 
results. If for example a consumptive man desired to insure 
his life, there would be a marked difference in the statistics 
according as we took his peculiar state of health into account 
or not. We should here have two sets of statistics, so clearly 
marked off from one another that they might almost rank 
with the distinctions of natural kinds, and which would in 
consequence offer decidedly different results. If we were 
to specialize still further, by taking into account insignificant 
qualities like those mentioned in the last paragraph, we 
might indeed get more limited sets of statistics applicable 
to persons still more closely resembling the individual in 



SECT. 4.] , Testimony. 397 

question, but these would not differ sufficiently in their 
results to make it worth our while to do so. In other words, 
the different propositions which are applicable to the case in 
point arrange themselves into a limited number of groups, 
which, and which only, need be taken into account ; whence 
the range of choice amongst them is very much diminished 
in practice. 

4, The reasons for the conditions above described are 
not difficult to detect. Where these conditions exist the 
process of selecting a series or class to which to refer any 
individual is ver}' simple, and the selection is, for the ' par- 
ticular purposes of inference, final. In any case of insurance, 
for example, the question we have to decide is of the very 
simple kind ; Is A.B. a man of a certain age ? If so one in 
fifty in his circumstances will die in the course of the year. 
If any further questions have to be decided they would be of 
the following description. Is A.B. a healthy man ? Does he 
follow a dangerous trade ? But here too the classes in 
question are but few, and the limits by which they are 
bounded are tolerably precise ; so that the reference of an 
individual to on& or other of them is easy. And when we 
have once chosen our class we remain untroubled by any 
further considerations ; for since no other statistics are sup- 
posed to offer a materially different average, we have no 
occasion to take account of any other properties than those 
already noticed. 

The case of games of chance, already referred to, offers 
of course an instance of these conditions in an almost ideal 
state of perfection ; the same circumstances which fit them 
so eminently for the purposes of fair gambling, fitting them 
equally to become examples in Probability. When a die is 
to be thrown, all persons alike stand on precisely the same 
footing of knowledge and of ignorance about the result ; the 



398 Testimony. [CHAP, xvi. 

only data to which any one could appeal being that each 
face turns up on an average once in six times. 

5. Let us now examine how far the above conditions 
are fulfilled in the case of problems which discuss what is 
called the credibility of testimony. The following would be 
a fair specimen of one of the elementary enquiries out of 
which these problems are composed; Here is a statement 
made by a witness who lies once in ten times, what am I to 
conclude about its truth? Objections might fairly be raised 
against the possibility of thus assigning a man his place 
upon a graduated scale of mendacity. This however we will 
pass over, and will assume that the witness goes about the 
world bearing stamped somehow on his face the appropriate 
class to which he belongs, and consequently, the degree of 
credit to which he has a claim on such general grounds. 
But there are other and stronger reasons against the ad- 
missibility of this class of problems. 

6. That which has been described in the previous 
sections as the 'individual' which had to be assigned to an 
appropriate class or series of statistics is, of course, in this 
case, a statement In the particular instance in question this 
individual statement is already assigned to a class, that 
namely of statements made by a witness of a given degree of 
veracity; but it is clearly optional with us whether or not we 
choose to confine our attention to this class in forming our 
judgment; at least it would be optional whenever we were 
practically called on to form an opinion. But in the case of 
this statement, as in that of the mortality of the man whose 
insurance we were discussing, there are a multitude of other 
properties observable, besides the one which is supposed to 
mark the given class. Just as in the latter there were 
(besides his age), the place of his birth, the nature of his 
occupation, and so on; so in the former there are (besides its 



SECT, 7.] .Testimony. 399 

being a statement by a certain kind of witness), the fact of its 
being uttered at a certain time and place and under certain 
circumstances. At the time the statement is made all these 
qualities or attributes of the statement are present to us, and 
we clearly have a right to take into account as many of them 
as we please. Now the question at present before us seems to 
be simply this; Are the considerations, which we might thus 
introduce, as immaterial to the result in the case of the truth 
of a statement of a witness, as the -,-rr-, -h'-h-lin:; considera- 
tions are in the case of the insurance of a life? There can 
surely be no hesitation in the reply to such a question. 
Under ordinary circumstances we soon know all that we can 
know about the conditions which determine us in judging of 
the prospect of a man's death, and we therefore rest content 
with general statistics of mortality; but no one who heard a 
witness speak would think of simply appealing to his figure 
of veracity, even supposing that this had been authoritatively 
communicated to us. The circumstances under which the 
statement is made instead of being insignificant, are of over- 
whelming importance. The appearance of the witness, the 
tone of his voice, the fact of his having objects to gain, 
together with a countless multitude of other circumstances 
which would gradually come to light as we reflect upon the 
matter, would make any sensible man discard the assigned 
average from his consideration. He would, in fact, no more 
think of judging in this way than he would of ;IJIH jilinir to 
the Carlisle or Northampton tables of mortality to determine 
the probable length of life of a soldier who was already in 
the midst of a battle. 

7. It cannot be replied that under these circumstances 
we still refer the witness to a class, and judge of his veracity 
by an average of a more limited kind; that we infer, for ex- 
ample, that of men who look and act like him under such 



400 .Testimony. [CHAP, xvi. 

circumstances, a much larger proportion, say nine-tenths, 
are found to lie. There is no appeal to a class in this way at 
all, there is no immediate reference to statistics of any kind 
whatever ; at least none which we are conscious of using at 
the time, or to which we should think of resorting for justifi- 
cation afterwards. The decision seems to depend upon the 
quickness of the observer's senses and of his apprehension 
generally. 

Statistics about the veracity of witnesses seem in fact to 
be permanently as inappropriate as all other statistics occa- 
sionally may be. We may know accurately the percentage 
of recoveries after amputation of the leg; but what surgeon 
would think of forming his judgment solely by such tables 
when he had a case before him? We need not deny, of 
course, that the opinion he might form about the patient's 
prospects of recovery might ultimately rest upon the propor- 
tions of deaths and recoveries he might have previously wit- 
nessed. But if this were the case, these data are lying, as 
one may say, obscurely in the background. He does not 
appeal to them directly and immediately in forming his 
judgment. There has been a far more important interme- 
diate process of apprehension and estimation of what is 
essential to the case and what is not. Sharp senses, memory, 
judgment, and practical sagacity have had to be called into 
play, and there is not therefore the same direct conscious 
and sole appeal to statistics that there was before. The 
surgeon may have in his mind two or three instances in 
which the operation performed was equally severe, but in 
which the patient's constitution was different; the latter 
element therefore has to be properly allowed for. There may 
be other instances in which the constitution was similar, but 
the operation more severe ; and so on. Hence, although the 
ultimate appeal may be to the statistics, it is not so directly ; 



SECT. 8.] Testimony. 401 

their value has to be estimated through the somewhat hazy 
medium of our judgment and memory, which places them 
under a very different aspect. 

8. Any one who knows anything of the game of whist 
may supply an apposite example of the distinction here 
insisted on, by recalling to mind the alteration in the nature 
of our inferences as the game progresses. At the commence- 
ment of the game our sole appeal is rightfully made to the 
theory of Probability. All the rules upon which each player 
acts, and therefore upon which he infers that the others will 
act, rest upon the observed frequency (or rather upon the 
frequency which calculation assures us will be observed) with 
which such and such combinations of cards are found to 
occur. Why are we told, if we have more than four trumps, 
to lead them out at once? Because we are convinced, on 
pure groupds of probability, capable of being stated in the 
strictest statistical form, that in a majority of instances we 
shall draw our opponent's trumps, and therefore be left with 
the command. Similarly with every other rule which is 
recognized in the early part of the play. 

But as the play progresses all this is changed, and 
towards its conclusion there is but little relunce upon any 
rules which either we or others could base upon statistical 
frequency of occurrence, observed or inferred. A multitude 
of other considerations have come in; we begin to be in- 
fluenced partly by our knowledge of the character and 
practice of our partner and opponents; partly by a rapid 
combination of a multitude of judgments, founded upon 
our observation of the actual course of play, the grounds 
of which we could hardly realize or describe at the time 
and which may have been forgotten . since. That is, the 
particular combination of cards, now before us, does not 
. readily fall into any well-marked class to which alone it can 
v. 26 



402 Testimony. [CHAP. xvi. 

reasonably be referred by every one who has the facts before 
him. 

9. A criticism somewhat resembling the above has 
been given by Mill (Logic, Bk. in. Chap, xviii. 3) upon 
the applicability of the theory of Probability to the credi- 
bility of witnesses. But he has added other reasons which 
do not appear to me to be equally valid ; he says " common 
sense would dictate that it is impossible to strike a general 
average of the veracity, and other qualifications for true 
testimony, of mankind or any class of them ; and if it were 
possible, such an average would be no guide, the credibility of 
almost every witness being either below or above the average." 
The latter objection would however apply with equal force 
to estimating the length of a man's life from tables of mor- 
tality ; for the credibility of different witnesses can scarcely 
have a wider range of variation than the length of different 
lives. If statistics of credibility could be obtained, and 
could be conveniently appealed to when they were obtained, 
they might furnish us in the long run with as accurate 
inferences as any other statistics of the same general de- 
scription. These statistics would however in practice natu- 
rally and rightly be neglected, because there can hardly fail 
to be circumstances in each individual statement which would 
more appropriately refer it to some new class depending on 
different statistics, and affording a far better chance of our 
being right in that particular case. In most instances of 
the kind in question, indeed, such a change is thus produced 
in the mode of formation of our opinion, that, as already 
pointed out, the mental operation ceases to be in any proper 
sense founded on appeal to statistics 1 . 

1 It may be remarked also that principles of Probability in the 
there is another reason which tends majority of the cases where testi- 
to dissuade us from appealing to mony has to be estimated. It often, 



SECT. 11.] Testimony. 403 

10. The Chance problems which are concerned with tes- 
timony are not altogether confined to such instances as those 
hitherto referred to. Though we must, as it appears to me, 
reject all attempts to estimate the credibility of any par- 
ticular witness, or to refer him to any assigned class in 
respect of his trustworthiness, and consequently abandon as 
unsuitable any of the numerous problems which start from 
such data as 'a witness who is wrong once in ten times/ 
yet it does not follow that testimony may not to a slight 
extent be treated by our science in a somewhat different 
manner. We may be quite unable to estimate, except in 
the roughest possible way, the veracity of any particular 
witness, and yet it n ay be possible to form some kind of 
opinion upon the veraodty of certain classes of witnesses ; 
to say, for instance, that v Europeans are superior in this way 
to Orientals. So we might attempt to explain why, and to 
what extent, an opinion in which the judgments of ten per- 
sons, say jurors, concur, is superior to one in which five only 
concur. S--in :l::i.-jf may also be done towards laying down 
the principles in accordance with which we are to decide 
whether, and why, extraordinary stories deserve less credence 
than ordinary ones, even if we cannot arrive at any precise 
and definite decision upon the point. This last question is 
further discussed in the course of the next chapter. 

11. The change of view in accordance with which it 
follows that questions of the kind just mentioned need not 
be entirely rejected from scientific consideration, presents it- 

perhaps usually happens, that we on the average, but truth in each 

are not absolutely forced to come individual instance, so that we had 

to a decision ; at least so far as the rather not form an opinion at all 

.acquitting of an accused person may than form one of which we can only 

be considered as avoiding a decision. say in its justification that it will 

It may be of much greater import- tend to lead us right in the long 

ance to us to attain not merely truth run. 

262 



404 Testimony. [CHAP. xvi. 

self in other directions also. It has, for instance, been already 
pointed out that the individual characteristics of any sick 
man's disease would be quite sufficiently important in most 
cases to prevent any surgeon from judging about his recovery 
by a genuine and direct appeal to statistics, however such 
considerations might indirectly operate upon his judgment. 
But if an opinion had to be formed about a considerable 
number of gases, say in a large hospital, statistics might 
again come prominently into play, and be rightly recognized 
as the principal source of appeal. We should feel able to 
compare one hospital, or one method of treatment, with 
another, The ground of the difference is obvious, It arises 
from the fact that the characteristics of the individuals, 
which made us so ready to desert the average when we had 
to judge of them separately, do not produce the same dis- 
turbance when we have to judge about a group of cases. 
The averages then become the most secure and available 
ground on which to form an opinion, and therefore Pro- 
bability again becomes applicable. 

But although some resort to Probability may be ad- 
mitted in such cases as these, it nevertheless does not 
appear to me that they can ever be regarded as particularly 
appropriate examples to illustrate the methods and resources 
of the theory. Indeed it is scarcely possible to resist the 
conviction that the refinements of mathematical calculation 
have here been pushed to lengths utterly unjustifiable, when 
we bear in mind the impossibility of obtaining any corre- 
sponding degree of accuracy and precision in the data from 
which we have to start. To cite but one instance. It would 
be hard to find a case in which love of consistency has pre- 
vailed over common sense to such an extent as in the ad- 
mission of the conclusion that it is unimportant what are 
the numbers for and against a particular statement, provided 



SECT. 11.] Testimony. 405 

the actual majority is the same. That is, the unanimous 
judgment of a jury of eight is to count for the same as a 
majority of ten to two in a jury of twelve. And yet this 
conclusion is admitted by Poisson. The assumptions under 
which it follows will be indicated in the course of the next 
chapter. 

Again, perfect independence amongst the witnesses or 
jurors is an almost necessary postulate. But where can this 
be secured? To say nothing of direct collusion, human 
beings are in almost all instances greatly under the influence 
of sympathy in forming their opinions. This influence, under 
the various names of political bias, class prejudice, local 
feeling, and so on, always exists to a sufficient degree to 
induce a cautious person to make many of those individual 
corrections which we saw to be necessary when we were 
estimating the trustworthiness, in any given case, of a single 
witness ; that is, they are sufficient to destroy much, if not 
all, of the confidence with which we resort to statistics and 
averages in forming our judgment. Since then this Essay is 
mainly devoted to explaining and establishing the general 
principles of the science of Probability, we may very fairly 
be excused from any further treatment of this subject, beyond 
the brief discussions which are given in the next chapter. 



CHAPTER XVII. 

ON THE CREDIBILITY OF EXTRAORDINARY STORIES. 

1. IT is now time to recur for fuller investigation to an 
enquiry which has been already briefly touched upon more 
than once; that is, the validity of testimony to establish, 
as it is frequently expressed, an otherwise improbable story. 
It will be remembered that in a previous chapter (the 
twelfth) we devoted some examination to an assertion by 
Butler, which seemed to be to some extent countenanced 
by Mill, that a great improbability before the proof might 
become but a very small improbability after the proof. In 
opposition to this it was pointed out that the different 
estimates which we undoubtedly formed of the credibility 
of the examples adduced, had nothing to do with the 
fact of the event being past or future, but arose from a 
very different cause; that the conception of the event 
which we entertain at the moment (which is all that is then 
and there actually present to us, and as to the correctness 
of which as a representation of facts we have to make up our 
minds) coines before us in two very different ways. In one 
instance it was a mere guess of our own which we knew 
from statistics would be right in a certain proportion of 
cases ; in the other instance it was the assertion of a witness, 
and therefore the appeal was not now primarily to statistics of 
the event, but to the trustworthiness of the witness. The con- 



SECT. 3.] Credibility of Extraordinary Storite. 407 

ception, or ' event ' if we will so term it, had in fact passed 
out of the category of guesses (on statistical grounds), into 
that of assertions (most likely resting on some specific evi- 
dence), and would therefore be naturally regarded in a very 
different light. 

2. But it may seem as if this principle would lead us 
to somewhat startling conclusions. For, by transferring the 
appeal from the frequency with which the event occurs to 
the trustworthiness of the witness who makes the assertion, 
is it not implied that the probability or improbability of an 
assertion depends solely upon the veracity of the witness ? 
If so, ought not any 'Story whatever to be believed when 
it is asserted by a truthful person ? 

In order to settle this question we must look a little 
more closely into the circumstances under which such 
testimony is commonly presented to us. As it is of course 
necessary, for clearness of exposition, to take a numerical 
example, let us suppose that a given statement is made by 
a witness who, on the whole and in the long run, is right 
in what he says nine times out of ten 1 . Here then is an 
average given to us, an average veracity that is, which 
includes all the particular statements which the witness has 
made or will make. 

3. Now it has been abundantly shown in a former 
chapter (Ch. ix. 1432) that the mere fact of a par- 

1 Reasons were given in the last another; such a rough practical 

chapter against the propriety of ap- distinction will be quite sufficient 

plying the rules of Probability with for the purposes of this chapter, 

any strictness to such examples as For convenience, and to illustrate 

these. But although all approach the theory, the examples are beat 

to numerical accuracy is unattain- stated in a numerical form, but it 

able, we do undoubtedly recognize is not intended thereby to imply 

in ordinary life a distinction between that any such accuracy is really 

the credibility of one witness and attainable in practice. 



408 Credibility of Extraordinary Stories. [CHAP. xVli. 

ticular average having been assigned, is no reason for our 
being forced invariably to adhere to it, even in those cases 
in which our most natural and appropriate ground of judg- 
ment is found in an appeal to statistics and averages. 
The general average may constantly have to be corrected 
in order to meet more accurately the circumstances of par- 
ticular cases. In statistics of mortality, for instance, instead 
of resorting to the wider tables furnished by people in 
general of a given age, we often prefer the narrower tables 
furnished by men of a particular profession, abode, or mode 
of life. The reader may however be conveniently reminded 
here that in so doing we must not suppose that we are able, 
by any such device, in any special or peculiar way to secure 
truth. The general average, if persistently adhered to 
throughout a sufficiently wide and varied experience, would 
in the long run tend to give us the truth ; all the advantage 
which the more special averages can secure for us is to give 
us the same tendency to the truth with fewer and slighter 
aberrations. 

4. Returning then to our witness, we know that if 
we have a very great many statements from him upon all 
possible subjects, we may feel convinced that in nine out of 
ten of these he will tell us the truth, and that in the tenth 
case he will go wrong. This is nothing more than a matter 
of definition or consistency. But cannot we do better than 
thus rely upon his general average ? Cannot we, in almost 
any given case, specialize it by attending to various cha- 
racteristic circumstances in the nature of the statement 
which he makes; just as we specialize his prospects of 
mortality by attending to circumstances in his constitution 
or mode of life ? 

Undoubtedly we may do this ; and in any of the practical 
contingencies of life, supposing that we were at all guided 



SECT. 5.] Credibility of Extraordinary Stories. 409 

by considerations of this nature, we should act very foolishly 
if we did not adopt some such plan. Two methods of thus 
correcting the average may be suggested : one of them being 
that which practical sagacity would be most likely to employ, 
the other that which is almost universally adopted by writers 
on Probability. The former attempts to make the correction 
by the following considerations : instead of relying upon the 
witness' general average, we assign to it a sort of conjectural 
correction to meet the case before us, founded on our expe- 
rience or observation ; that is, we appeal to experience to 
establish that stories of such and such a kind are more or 
less likely to be true, as the case may be, than stories in 
general. The other proceeds upon a different and some- 
what more methodical plan. It is here endeavoured to 
show, by an analysis of the nature and number of the 
sources of error in the cases in question, that such and such 
kinds of stories must be more or less likely to be correctly 
reported, and this in certain numerical proportions. 

5. Before proceeding to a discussion of these methods 
a distinction must be pointed out to which writers upon the 
subject have not always attended, or at any rate to which 
they have not generally sufficiently directed their readers' 
attention 1 . There are, broadly speaking, two different ways 
in which we may suppose testimony to be given. It may, in 
the first place, take the form of a reply to an alternative 
question, a question, that is, framed to be answered by yes 
or no. Here, of course, the possible answers are mutually 
contradictory, so that if one of them is not correct the other 
must be so : Has A happened, yes or no ? The common 
mode of illustrating this kind of testimony numerically is by 

1 I must plead guilty to this make the treatment of this part of 
charge myself, in the first edition the subject obscure and imperfect, 
of this work. The result was to and in some respects erroneous. 



410 Credibility of Extraordinary Stories. [CHAP. xvil. 

supposing a lottery with a prize and blanks, or a bag of balls 
of two colours only, the witness knowing that there are only 
two, or at any rate being confined to naming one or other of 
them. If they are black and white, and he errs when black 
is drawn, he must say ' white/ The reason for the promi- 
nence assigned to examples of this class is, probably, that 
they correspond to the very important case of verdicts of 
juries ; juries being supposed to have nothing else to do than 
to say ' guilty* or ' not guilty/ 

On the other hand, the testimony may take the form of a 
more original statement or piece of information. Instead of 
saying, Did A happen ? we may ask, What happened ? Here 
if the witness speaks truth he must be supposed, as before, 
to have but one way of doing so ; for the occurrence of some 
specific event was of course contemplated. But if he errs he 
has many ways of going wrong, possibly an infinite number. 
Ordinarily however his possible false statements are assumed 
to be limited in number, as must generally be more or less 
the result in practice. This case is represented numerically 
by supposing the balls in the bag not to be of two colours 
only, but to be all distinct from each other; say by their 
being all numbered successively. It may of course be ob- 
jected that a large number of the statements that are made 
in the world are not in any way answers to questions, either 
of the alternative or of the open kind. For instance, a man 
simply asserts that he has drawn the seven of spades from a 
pack of cards ; and we do not know perhaps whether he had 
been asked ' Has that card been drawn or ' What card has 
been drawn? 1 or indeed whether he had been asked anything 
at all. Still more might this be so in the case of any ordi- 
nary historical statement. 

This objection is quite to the point, and must be recog- 
nized as constituting an additional difficulty. All that we 



SECT. 6.] Credibility of Extraordinary Stories. 411 

can do is to endeavour, as best we may, to ascertain, from 
the circumstances of the case, what number of alternatives 
the witness may be supposed to have had before him. When 
he simply testifies to some matter well known to be in dis- 
pute, and does not go much into detail, we may fairly con- 
sider that there were practically only the two alternatives 
before him of saying ' yes ' or ' no. 1 When, on the other hand, 
he tells a story of a more original kind, or (what comes to 
much the same thing) goes into details, we must regard him 
as having a wide comparative range of alternatives before 
him. 

These two classes of examples, viz. that of the black and 
white balls, in which only one form of error is possible, and 
the numbered balls, in which there may be many forms of 
error, are the only two which we need notice. In practice it 
would seem that they may gradually merge into each other, 
according to the varying ways in which we choose to frame 
our question. Besides asking, Did you see A strike B ? and, 
What did you see ? we may introduce any number of inter- 
mediate leading questions, as, What did A do ? What did 
he do to JB? and so on. In this way we may gradually narrow 
the possible openings to wrong statement, and so approach 
to the direct alternative question. But it is clear that all 
these cases may be represented numerically by a supposed 
diminution in the number of the balls which are thus distin- 
guished from each other. 

6. Of the two plans mentioned in 4 we will begin 
with the latter, as it is the only methodical and scientific one 
which has been proposed. Suppose that there is a bag with 
1000 balls, only one of which is white, the rest being all 
black. A ball is drawn at random, and our witness whose 
veracity is T e ^ reports that the white ball was drawn. Take 
a great many of his statements upon this particular subject, 



412 Credibility of Extraordinary Stories. [CHAP. XVII. 

say 10,000 ; that is, suppose that 10,000 balls having been 
successively drawn out of this bag, or bags of exactly the 
same kind, he makes his report in each case. His 10,000 
statements being taken as a fair sample of his general ave- 
rage, we shall find, by supposition, that 9 out of every 10 of 
them are true and the remaining one false. What will be 
the nature of these false statements ? Under the circum- 
stances in question, he having only one way of going wrong, 
the answer is easy. In the 10,000 drawings the white ball 
would come out 10 times, and therefore be rightly asserted 
9 times, whilst on the one of these occasions on which he 
goes wrong he has nothing to say but ' black/ So with the 
9990 occasions on which black is drawn; he is right and 
says black on 8991 of them, and is wrong and therefore says 
white on 999 of them. On the whole, therefore, we conclude 
that out of every 1008 times on which he says that white is 
drawn he is wrong 999 times and right only 9 times. That 
is, his special veracity, as we may term it, for cases of this 
description, has been reduced from -fa to T?J 9 ^. As it would 
commonly be expressed, the latter fraction represents the 
chance that this particular statement of his is true 1 . 

1 The generalized algebraical form be presently made, the reader will 

of this result is as follows. Let p notice that on making either of these 

be the a priori probability of an expressions =p t we obtain in each 

event, and x be the credibility of the case x=b. That is, a witness whose 

witness. Then, if he asserts that veracity = leaves the a priori prob- 

the event happened, the probability ability of an event (of this kind) un- 

that it really did happen is affected. 

If, on the other hand, we make 

ex P re8sions e( l ual to x an< * 



pa-Mi- - 

l-ac respectively, we obtain in each 

whilst if he asserts that it did not case ^ = 4. That is, when an event 

happen the probability that it did ( O f t hi 8 kind) is as likely to happen 

happen is p (1 - a?) ^ as not, the ordinary veracity of the 

jp (1 - x) + (1 -p) x ' witness in respect of it remains un- 

In illustration of some remarks to affected. 



SECT. 8.] Credibility of Extraordinary Stories. 413 

7. We will now take the case in which the witness 
has many ways of going wrong, instead of merely one. Sup- 
pose that the balls were all numbered, from 1 to 1,000, and 
the witness knows this fact. A ball is drawn, and he tells 
me that it was numbered 25, what are the odds that he is 
right? Proceeding as before, in 10,000 drawings this ball 
would be obtained 10 times, and correctly named 9 times. 
But on the 9990 occasions on which it was not drawn there 
would be a difference, for the witness has now many open- 
ings for error before him. It is, however, generally considered 
reasonable to assume that his errors will all take the form of 
announcing wrong numbers; and that, there being no apparent 
reason why he should choose one number rather than another, 
he will be likely to announce all the wrong ones equally 
often. Hence his 999 errors, instead of all leading him now 
back again to one spot, will be uniformly spread over as 
many distinct ways of going wrong. On one only of these 
occasions, therefore, will he mention 25 as having been 
drawn. It follows therefore that out of every 10 times that 
he names 25 he is right 9 times ; so that in this case his 
average or general truthfulness applies equally well to the 
special case in point. 

8. With regard to the truth of these conclusions, it 
must of course be admitted that if we grant the validity of 
the assumptions about the limits within which the blunder- 
ing or mendacity of the witness are confined, and the com- 
plete impartiality with which his answers are disposed within 
those limits, the reasoning is perfectly sound. But are not 
these assumptions extremely arbitrary, that is, are not our 
lotteries and bags of balls rendered perfectly precise in many 
respects in which, in ordinary life, the conditions supposed to 
correspond to them are so vague and uncertain that no such 
method of reasoning becomes practically available ? Suppose 



414 Credibility of Extraordinary Stories. [CHAP. XVII. 

that a person whom I have long known, and of whose mea- 
sure of veracity and judgment I may be supposed therefore 
to have acquired some knowledge, informs me that there is 
something to my advantage if I choose to go to certain 
trouble or expense in order to secure it. As regards the 
general veracity of the witness, then, there is no difficulty; 
we suppose that this is determined for us. But as regards 
his story, difficulty arid vagueness emerge at every point. 
What is the number of balls in the bag here ? What in fact 
are the nature and contents of the bag out of which we sup- 
pose the drawing to have been made? It does not seem 
that the materials for any rational judgment exist here. 
But if we are to get at any such amended figure of veracity 
as those attained in the above example, these questions must 
necessarily be answered with some degree of accuracy ; for 
the main point of the method consists in determining how 
often the event must be considered not to happen, and thence 
inferring how often the witness will be led wrongly to assert 
that it has happened. 

It is not of course denied that considerations of the kind 
in question have some influence upon our decision, but only 
that this influence could under any ordinary circumstances 
be submitted to numerical determination. We are doubt- 
less liable to have information given to us that we have 
come in for some kind of fortune, for instance, when no 
such good luck has really befallen us ; and this not once 
only but repeatedly. But who can give the faintest inti- 
mation of the nature and number of the occasions on which, 
a blank being thus really drawn, a prize will nevertheless 
be falsely announced? It appears to me therefore that 
numerical results of any practical value can seldom, if ever, 
l>e looked for from this method of procedure. 

9. Our conclusion in the case of the lottery, or, what 



SECT. 9.] Credibility of Extraordinary Stories. 415 

comes to the same thing, in the case of the bag with black 
and white balls, has been questioned or objected to 1 on the 
ground that it is contrary to all experience to suppose that 
the testimony of a moderately good witness could be so 
enormously depreciated under such circumstances. I should 
prefer to base the objection on the ground that experience 
scarcely ever presents such circumstances as those supposed ; 
but if we postulate their existence the given conclusion seems 
correct enough. Assume that a man is merely required to 
say yes or no ; assume also a group or succession of cases in 
which no should rightly be said very much oftener than 
yes. Then, assuming almost any general truthfulness of the 
witness, we may easily suppose the rightful occasions for 
denial to be so much the more frequent that a majority of 
his affirmative answers will actually occur as false ' noes ' 
rather than as correct 'ayes.' This of course lowers the 
average value of his ' ayes,' and renders them comparatively 
untrustworthy. 

Consider the following example. I have a gardener whom 
I trust as to all ordinary matters of fact. If he were to 
tell me some morning that my dog had run away I should 
fully believe him. He tells me however that the dog has 
gone mad. Surely I should accept the statement with much 
hesitation, and on the grounds indicated above. It is not 
that he is more likely to be wrong when the dog is mad ; 
but that experience shows that there are other complaints 
(e.g. fits) which are far more common than madness, and 
that most of the assertions of madness are erroneous asser- 
tions referring to these. This seems a somewhat parallel 
case to that in which we find that most of the assertions 
that a white ball had been drawn are really false assertions 
referring to the drawing of a black ball. Practically I do 

1 Todhunter's History, p. 400. Philosophical Magazine, July, 1864. 



416 Credibility of Extraordinary Stories. [CHAP. xvn. 

not think that any one would feel a difficulty in thus ex- 
orbitantly discounting some particular assertion of a witness 
whom in most other respects he fully trusted. 

10. There is one particular case which has been re- 
garded as a difficulty in the way of this treatment of the 
problem, but which seems to me to be a decided confirma- 
tion of it ; always, be it understood, within the very narrow 
and artificial limits to which we must suppose ourselves to 
be confined. This is the case of a witness whose veracity is 
just one-half; that is, one who, when a mere yes or no is 
demanded of him, is as often wrong as right. In the case of 
any other assigned degree of veracity it is extremely difficult 
to get anything approaching to a confirmation from prac- 
tical judgment and experience. We are not accustomed to 
estimate the merits of witnesses in this way, and hardly ap- 
preciate what is meant by his numerical degree of truthful- 
ness. But as regards the man whose veracity is one-half,, we 
are (as Mr 0. J. Monro has very ingeniously suggested) only 
too well acquainted with such witnesses, though under a 
somewhat different name ; for this is really nothing else than 
the case of a person confidently answering a question about 
a subject-matter of which he knows nothing, and can there- 
fore only give a mere guess. 

Now in the case of the lottery with one prize, when the 
witness whose veracity is one-half tells us that we have 
gained the prize, we find on calculation that his testimony 
goes for absolutely nothing ; the chances that we have got 
the prize are just the same as they would be if he had never 
opened his lips, viz. y^. But clearly this is what ought 
to be the result, for the witness who knows nothing about 
the matter leaves it exactly as he found it. He is indeed, 
in strictness, Scarcely a witness at all ; for the natural func- 
tion of a witness is to examine the matter, and so to add 



SECT. 11.] Credibility of Extraordinary Stories. 417 

confirmation, more or less, according to his judgment and 
probity, but at any rate to offer an improvement upon the 
mere guesser. If, however, we will give heed to his mere 
guess we are doing just the same thing as if we were to guess 
ourselves, in which case of course the odds that we are right 
are simply measured by the frequency of occurrence of the 
events. 

We cannot quite so readily apply the same rule to the 
other case, namely to that of the numbered balls, for there 
the witness who is right every other time may really be a 
very fair, or even excellent, witness. If he has many ways 
of going wrong, and yet is right in half his statements, it is 
clear that he must have taken some degree of care, and can- 
not have merely guessed. In a case of yes or no, any one can 
be right every other time, but it is different where truth 
is single and error is manifold. To represent the case of a 
simply worthless witness when there were 1000 balls and 
the drawing of one assigned ball was in question, we should 
have to put his figure of veracity at y^. If this were 
done we should of course get a similar result. 

11. It deserves notice therefore that the figure of 
veracity, or fraction representing the general truthfulness 
of a witness, is in a way relative, not absolute ; that is, it 
depends upon, and varies with, the, general character of the 
answer which he is supposed to give. Two witnesses of 
equal intrinsic veracity and worth, one of whom confined 
himself to saying yes and no, whilst the other ventured to 
make more original assertions, would be represented by 
different fractions ; the former having set himself a much 
easier task than the latter. The real caution and truthful- 
ness of the witness are only one factor, therefore, in his 
actual figure of veracity; the other factor consists of the 
nature of his assertions, as just pointed out. The ordinary 



418 Credibility of Extraordinary Stories. [CHAP. xvu. 

plan therefore, in such problems, of assigning an average 
truthfulness to the witness, and accepting this alike in the 
case of each of the two kinds of answers, though convenient, 
seems scarcely sound. This consideration would however 
be of much more importance were not the discussions upon 
the subject mainly concerned with only one description of 
answer, namely that of the ' yes or no ' kind. 

12. So much for the methodical way of treating such 
a problem. The way in which it would be taken in hand by 
those who had made no study of Probability is very different. 
It would, I apprehend, strike them as follows. They would 
say to themselves, Here is a story related by a witness who 
tells the truth, say, nine times out of ten. But it is a story 
of a kind which experience shows to be very generally made 
untruly, say 99 times out of 100. Having then these oppo- 
site inducements to belief, they would attempt in some way 
to strike a balance between them. Nothing in the nature of 
a strict rule could be given to enable them to decide how 
they might escape out of the difficulty. Probably, in so far 
as they did not judge at haphazard, they would be guided 
by still further resort to experience, or unconscious recol- 
lections of its previous teachings, in order to settle which 
of the two opposing inductions was better entitled to carry 
the day in the particular case before them. The reader 
will readily see that any general solution of the problem, 
when thus presented, is impossible. It is simply the now 
familiar case (Chap. ix. 14 32) of an individual which 
belongs equally to two distinct, or even, in respect of their 
characteristics, opposing classes. We cannot decide offhand 
to which of the two its characteristics most naturally and 
rightly refer it. A fresh induction is needed in order to 
settle this point. 

13. Rules have indeed been suggested by various 



SECT. 14.] Credibility of Extraordinary Stvries. 419 

writers in order to extricate us from the difficulty. The con- 
troversy about miracles has probably been the most fertile 
occasion for suggestions of this kind on one side or the 
other. It is to this controversy, presumably, that the phrase 
is due, so often employed in discussions upon similar sub- 
jects, ' a contest of opposite improbabilities/ What is meant 
by such an expression is clearly this: that in forming a 
judgment upon the truth of certain assertions we may find 
that they are comprised in two very distinct classes, so that, 
according as we regarded them as belonging to one or the 
other of these distinct classes, our opinion as to their truth 
would be very different. Such an assertion belongs to one 
class, of course, by its being a statement of a particular 
witness, or kind of witness; it belongs to the other by its 
being a particular kind of story, one of what is called an 
improbable nature. Its belonging to the former class is so 
far favourable to its truth, its belonging to the latter is so far 
hostile to its truth. It seems to be assumed, in speaking of 
a contest of opposite improbabilities, -that when these different 
sources of conviction co-exist together, they would each in 
some way retain their probative force so as to produce a 
contest, ending generally in a victory to one or other of 
them. Hume, for instance, speaks of our deducting one 
probability from the other, and apportioning our belief to 
the remainder 1 . Thomson, in his Laws of Thought, speaks 
of one probability as entirely superseding the other. 

14. It does not appear to me that the slightest philoso- 
phical value can be attached to any such rules as these. 
They doubtless may, and indeed will, hold in individual 

1 "When therefore these two opinion, either on one side or the 

kinds of experience are contrary, we other, with that assurance which 

have nothing to do but subtract the arises from the remainder." (Essay 

one from the other, and embrace an on Miracles.) 



420 Credibility of Extraordinary Stories. [CHAP, xvn. 

cases, but they cannot lay claim to any generality. Even 
the notion of a contest, as any necessary ingredient in the 
case, must be laid aside. For let us refer again to the way 
in which the perplexity arises, and we shall readily see, as 
has just been remarked, that it is nothing more than a par- 
ticular exemplification of a difficulty which has already been 
recognized as incapable of solution by any general d priori 
method of treatment. All that we are supposed to have before 
us is a statement. On this occasion it is made by a witness 
who lies, say, once in ten times in the long run ; that is, who 
mostly tells the truth. But on the other hand, it is a state- 
ment which experience, derived from a variety of witnesses on 
various occasions, assures us is mostly false ; stated numerically 
it is found, let us suppose, to be false 99 times in a hundred. 
Now, as was shown in the chapter on Induction, we are 
thus brought to a complete dead lock. Our science offers no 
principles by which we can form an opinion, or attempt to 
decide the matter one way or the other; for, as we found, 
there are an indefinite number of conclusions which are all 
equally possible. For instance, all the witness* extraordinary 
assertions may be true, or they may all be false, or they may 
be divided into the true and the false in any proportion 
whatever. Having gone so far in our appeal to statistics as 
to recognize that the witness is generally right, but that his 
story is generally false, we cannot stop there. We ought to 
make still further appeal to experience, and ascertain how it 
stands with regard to his stories when they are of that 
particular nature: or rather, for this would be to make a 
needlessly narrow reference, how it stands with regard to 
stories of that kind when advanced by witnesses of his 
general character, position, sympathies, and so on 1 . 

1 Considerations of this kind have mathematical treatment of the sub- 
indeed been introduced into the ject. The common algebraical solu- 



SECT. 15.] Credibility of Extraordinary Stories. 421 

15. That extraordinary stories are in many cases, pro- 
bably in a great majority of cases, less trustworthy than 
others must be fully admitted. That is, if we were to make 
two distinct classes of such stories respectively, we should 
find that the same witness, or similar witnesses, were propor- 
tionally more often wrong when asserting the former than 
when asserting the latter. But it does not by any means 
appear to me that this must always be the case. We may 
well conceive, for instance, that with some people the mere 
fact of the story being of a very unusual character may make 
them more careful in what they state, so as actually to add 
to their veracity. If this were so we might be ready ta 
accept their extraordinary stories with even more readiness 
than their ordinary ones. 

Such a supposition as that just made does not seem to me 
by any means forced. Put such a case as this : let us sup- 
pose that two persons, one of them a man of merely ordinary 
probity and intelligence, the other a scientific naturalist, 
make a statement about some common event. We believe 

tion of the problem in 5 (to begin Here t' and t measure respectively 

with the simplest case) is of course his trustworthiness in usual and 

as follows. Let p be the antecedent unusual events. As a formal solu- 

probability of the event, and t the tion this certainly meets the objec- 

measure of the truthfulness of the tions stated above in 14 and 15. 

witness; then the chance of his state- The determination however of t' 

. , . , pt would demand, as I have remarked, 

ment being true is ,, . -. . 

pt+(l-p)(l-t) continually renewed appeal to ex- 

This supposes him to lie as much perience. In any case the practical 

when the event does not happen as methods which would be adopted, if 

when it does. But we may meet any plans of the kind indicated above 

the cases supposed in the text by were resorted to, seem to me to differ 

assuming that t' is the measure of very much from that adopted by the 

his veracity when the 'event does mathematicians, in their spirit and 

not happen, so that the above plan. 

pt 
formula becomes 



422 Credibility of Extraordinary Stories. [CHAP. xvn. 

them both. Let them now each report some extraordinary 
lusus naturce or monstrosity which they profess to have seen. 
Most persons, we may presume, would receive the statement 
of the naturalist in this latter case almost as readily as in 
the former: whereas when the same story came from the 
unscientific observer it would be received with considerable 
hesitation. Whence arises the difference? From the con- 
viction that the naturalist will be far more careful, and 
therefore to the full as accurate, in matters of this kind as 
in those of the most ordinary description, whereas with the 
other man we feel by no means the same confidence. Even 
if any one is not prepared to go this length, he will probably 
admit that the difference of credit which he would attach to 
the two kinds of story, respectively, when they came from 
the naturalist, would be much less than what it would be 
when they came from the other man. 

16. Whilst we are on this part of the subject, it must 
be pointed out that there is considerable ambiguity and 
consequent confusion about the use of the term 'an extraor- 
dinary story/ Within the province of pure Probability it 
ought to mean simply a story which asserts an unusual event. 
At least this is the view which has been adopted and main- 
tained, it is hoped consistently, throughout this work. So 
long as we adhere to this sense we know precisely what we 
mean by the term. It has a purely objective reference; it 
simply connotes a very low degree of relative statistical 
frequency, actual or prospective. Out of a great number of 
events we suppose a selection of some particular kind to be 
contemplated, which occurs relatively very seldom, and this 
is termed an unusual or extraordinary event. It follows, as 
was abundantly shown in a former chapter, that owing to 
the rarity of the event we are very little disposed to expect 
its occurrence in any given case. Our guess about it, in case 



SECT. 17.] Credibility of Extraordinary Stories. 423 

we thus anticipated it, would very seldom be justified, and 
we are therefore apt to be much surprised when it does 
occur. This, I take it, is the only legitimate sense of 'extra- 
ordinary* so far as Probability is concerned. 

But there is another and very different use of the word, 
which belongs to Induction, or rather to the science of 
evidence in general, more than to that limited portion of it 
termed Probability. In this sense the 'extraordinary/ and 
still more the 'improbable/ event is not merely one of 
extreme statistical rarity, which we could not expect to 
guess aright, but which on moderate evidence we may pretty 
readily accept; it is rather one which possesses, so to say, an 
actual evidence-resisting power. It may be something which 
affects the credibility of the witness at the fountain-head, 
which makes, that is, his statements upon such a subject 
essentially inferior to those on other subjects. This is the 
case, for instance, with anything which excites his prejudices 
or passions or superstitions. In these cases it would seem 
unreasonable to attempt to estimate the credibility of the 
witness by calculiiiin^ (as in 6) how often his errors would 
mislead us through his having been wrongly brought to an 
affirmation instead of adhering correctly to a negation. We 
should rather be disposed to put our correction on the wit- 
ness* average veracity at once. 

17. In true Probability, as has just been remarked, 
every event has its own definitely recognizable degree of 
frequency of occurrence. It may be excessively rare, rare to 
any extreme we like to postulate, but still every one who 
understands and admits the data upon which its occurrence 
depends will be able to appreciate within what range of 
experience it may be expected to present itself. We do not 
expect it in any individual case, nor within any brief range, 
but we do confidently expect it within an adequately exten-. 



424 Credibility of Extraordinary Stories. [CHAP. xvil. 

sive range. How therefore can miraculous stories be simi- 
larly taken account of, when the disputants, on one side at 
least, are not prepared to admit their actual occurrence any- 
where or at any time ? How can any arrangement of bags 
and balls, or other mechanical or numerical illustrations of 
unlikely events, be admitted as fairly illustrative of miracu- 
lous occurrences, or indeed of many of those which come 
under the designation of * very extraordinary ' or ' highly 
improbable ' ? Those who contest the occurrence of a par- 
ticular miracle, as reported by this or that narrator, do not 
admit that miracles are to be confidently expected sooner or 
later. It is not a question as to whether what must happen 
sometimes has happened some particular time, and therefore 
no illustration of the kind can be regarded as apposite. 

How unsuitable these merely rare events, however ex- 
cessive their rarity may be, are as examples of miraculous 
events, will be evident from a single consideration. No one, 
I presume, who admitted the occasional occurrence of an ex- 
ceedingly unusual combination, would be in much doubt if 
he considered that he had actually seen it himself 1 . On the 
other hand, few men of any really scientific turn would 
readily accept a miracle even if it appeared to happen under 
their very eyes, They might be -.i P> - -"\ at the time, but 

1 Laplace, for instance (Essai, ed. contains 77 figures, and is therefore 

1825, p. 149), says that if we saw utterly inappreciable by the imagi- 

100 dies (known of course to be fair nation. It must be admitted, though, 

ones) all give the same face, we that there is something hypothetical 

should be bewildered at the time, about such an example, for we could 

and need confirmation from others, not really know that the dies were 

but that, after due examination, no fair with a confidence even distantly 

one would feel obliged to postulate approaching such prodigious odds, 

hallucination in the matter. But In other words, it is difficult here to 

the chance of this occurrence is keep apart those different aspects of 

represented by a fraction whose the question discussed in Chap. xiv. 

numerator is 1, and denominator 28 83. 



SECT. 18.] Credibility of Extraordinary Stories. 425 

they would probably soon come to discredit it afterwards, 
or so explain it as to evacuate it of all that is meant by 
miraculous. 

18. It appears to me therefore, on the whole, that 
very little can be made of these problems of testimony in 
the way in which it is generally intended that they should 
be treated ; that is, in obtaining specific rules for the esti- 
mation of the testimony under any given circumstances. 
Assuming that the veracity of the witness can be measured, 
we encounter the real difficulty in the utter impossibility of 
determining the limits within which the failures of the event 
in question are to be considered to lie, and the degree of 
explicitness with which the witness is supposed to answer 
the enquiry addressed to him ; both of these being charac- 
teristics of which it is necessary to have a numerical esti- 
mate before we can consider ourselves in possession of the 
requisite data. 

Since therefore the practical resource of most persons, 
viz. that of putting a direct and immediate correction, of 
course of a somewhat conjectural nature, upon the general 
trustworthiness of the witness, by a consideration of the 
nature of the circumstances under which his statement is 
made, is essentially unscientific and irreducible to rule; it 
really seems to me that there is something to be said in 
favour of the simple plan of trusting in all cases alike to 
the witness' general veracity 1 . That is, whether his story 
is ordinary or extraordinary, we may resolve to put it on 
the same footing of credibility, provided of course that the 
event is fully recognized as one which does or may occa- 

1 In the first edition this was that (as was shown in. 7) this 

stated, as it now seems to me, in de- plan is really the best theoretical 

cidedly too unqualified a manner, one which can be adopted in certain 

It must be remembered, however, cases. 



426 Credibility of Extraordinary Stories. [CHAP. XVII. 

sionally happen. It is true that we shall thus go constantly 
astray, and may do so to a great extent, so that if there 
were any rational and precise method of specializing his 
trustworthiness, according to the nature of his story, we 
should be on much firmer ground. But at least we may 
thus know what to expect on the average. Provided we 
have a sufficient number and variety of statements from 
him, and always take them at the same constant rate or 
degree of trustworthiness, we may succeed in balancing and 
correcting our conduct in the long run so as to avoid any 
ruinous error. 

19. A few words may now be added about the combi- 
nation of testimony. No new principles are introduced here, 
though the consequent complication is naturally greater. Let 
us suppose two witnesses, the veracity of each being ^. 
Now suppose 100 statements made by the pair ; according to 
the plan of proceeding adopted before, we should have them 
both right 81 times and both wrong once, in the remaining 
18 cases one being right and the other wrong. But since 
they are both supposed to give the same account, what we 
have to compare together are the number of occasions on 
which they agree and are right, and the total number on which 
they agree whether right or wrong. The ratio of the former 
to the latter is the fraction which expresses the trustworthi- 
ness of their combination of testimony in the case in question. 

In attempting to decide this point the only difficulty is 
in determining how often they will be found to agree when 
they are both wrong, for clearly they must agree when they 
are both right. This enquiry turns of course upon the num- 
ber of ways in which they can succeed in going wrong. 
Suppose first the case of a simple yes or no (as in 6), and 
take the same example, of a bag with 1000 balls, in which 
one only is white. Proceeding as before, we should find that 



SECT. 20.] Credibility of Extraordinary Stories. 427 

out of 100,000 drawings (the number required in order to 
obtain a complete cycle of all possible occurrences, as well as 
of all possible reports about them) the two witnesses agree 
in a correct report of the appearance of white in 81, and 
agree in a wrong report of it in 999. The Probability there- 
fore of the story when so attested is yf^ ; the fact therefore 
of two such witnesses of equal veracity having concurred 
makes the report nearly 9 times as likely as when it rested 
upon the authority of only one of them 1 . 

20. When however the witnesses have many ways of 
going wrong, ibhe fact of their agreeing makes the report far 
more likely to be true. For instance, in the case of the 1000 
numbered balls, it is very unlikely that when they both mis- 
take the number they should (without collusion) happen to 
make the same misstatement. Whereas, in the last case, 
every combined misstatement necessarily led them both to 
the assertion that the event in question had happened, we 
should now find that only once in 999 x 999 times would 
they both be led to assert that some given number (say, as 
before, 25) had been drawn. The odds in favour of the 

1 It is on this principle that the right and m are wrong). And the 

remarkable conclusion mentioned on chance of its being rightly asserted 

p. 405 is based. Suppose an event as py m (1 - y) n . Therefore the chance 

whose probability is p ; and that, of that when we have an assertion 

a number of witnesses of the same before us it is a true one is 

veracity (y), m assert that it hap- py m (l-y) n 

pened, and n deny this. Generaliz- pym (i _ y) n + (1 -p) y n (l - y)** ' 
ing the arithmetical reasoning given 

above we see that the chance of the which is e< l ual to 

event being asserted varies as py m ~ n 

py m (l-y) n +(l-P)y n (l-y) m ; py- n +(i-p)(i-y) m - n * 

(viz. as the chance that the event But this last expression represents 

happens, and that m are right and n the probability of an assertion which 

are wrong; plus the chance that it is unanimously supported by m-n 

does not happen, and that n are such witnesses. 



428 Credibility of Extraordinary Stories. [CHAP, xvir, 



event in fact now become f$f$, which are enormously 
greater than when there was only one witness. 

It appears therefore that when two, and of course still 
more when many, witnesses agree in a statement in a matter 
about which they might make many and various errors, the 
combination of their favourable testimony adds enormously 
to the likelihood of the event ; provided always that there 
is no chance of collusion. And in the extreme case of the 
opportunities for error being, as they well may be, practically 
infinite in number, such combination would produce almost 
perfect certainty. But then this condition, viz. absence of 
collusion, very seldom can be secured. Practically our main 
source of error and suspicion is in the possible existence of 
some kind of collusion. Since we can seldom entirely get 
rid of this danger, and when it exists it can never be sub- 
mitted to numerical calculation, it appears to me that combi- 
nation of testimony, in regard to detailed accounts, is yet 
more unfitted for consideration in Probability than even that 
of single testimony. 

21. The impossibility of any adequate or even appro- 
priate consideration of the credibility of miraculous stories 
by the rules of Probability has been already noticed in 17. 
But, since the grounds of this impossibility are often very 
insufficiently appreciated, a few pages may conveniently be 
added here with a view to enforcing this point. If it be 
regarded as a digression, the importance of the subject and 
the persistency with which various writers have at one time 
or another attempted to treat it by the rules of our science 
must be the excuse for entering upon it. 

A necessary preliminary will be to decide upon some defi- 
nition of a miracle. It will, we may suppose, be admitted by 
most persons that in calling a miracle ' a suspension of a law 
of causation/ we are giving what, though it may not amount 



SECT. 22.] Credibility of Extraordinary Stories. 429 

to an adequate definition, is at least true as a description. 
It is true, though it may not be the whole truth. Whatever 
else the miracle may be, this is its physical aspect: this is the 
point at which it comes into contact with the subject-matter 
of science. If it were not considered that any suspension of 
causation were involved, the event would be regarded merely 
as an ordinary one to which some special significance was 
attached, that is, as a type or symbol rather than a miracle. 
It is this aspect moreover of the miracle which is now ex- 
posed to the main brunt of the attack, and in support of 
which therefore the defence has generally been carried on. 

Now it is obvious that this, like most other definitions or 
descriptions, makes some assumption as to matters of fact, 
and involves something of a theory. The assumption clearly 
is, that laws of causation prevail universally, or almost uni- 
versally, throughout nature, so that infractions of them are 
marked and exceptional. This assumption is made, but it 
does not appear that anything more than this is necessarily 
required; that is, there is nothing which need necessarily 
make us side with either of the two principal schools which 
are divided as to the nature of these laws of causation. The 
definition will serve equally well whether we understand by 
law nothing more than uniformity of antecedent and conse- 
quent, or whether we assert that there is some deeper and 
more mysterious tie between the events than mere sequence. 
The use of the term 'causation' in this minimum of signifi- 
cation is common to both schools, though the one might 
consider it inadequate; we may speak, therefore, of 'suspen- 
sions of causation* without committing ourselves to either. 

22. It should be observed that the aspect of the ques- 
tion suggested by this definition is one from which we can 
hardly escape. Attempts indeed have been sometimes made 
to avoid the necessity of any assumption as to the universal 



430 Credibility of Extraordinary Stories. [CHAP. xvu. 

prevalence of law and order in nature, by defining a miracle 
from a different point of view. A miracle may be called, for 
instance, 'an immediate exeraon of creative power/ 'a sign 
of a revelation/ or, still more vaguely, an 'extraordinary 
event/ But nothing would be gained by adopting any such 
definitions as these. However they might satisfy the theo- 
logian, the student of physical science would not rest content 
with them for a moment. He would at once assert his own 
belief, and that of other scientific men, in the existence of 
universal law, and enquire what was the connection of the 
definition with this doctrine. An answer would imperatively 
be demanded to the question, Does the miracle, as you have 
described it, imply an infraction of one of these laws, or does 
it not ? And an answer must be given, unless indeed we 
reject his assumption by denying our belief in the existence 
of this universal law, in which case of course we put our- 
selves out of the pale of argument with him. The necessity 
of having to recognize this fact is growing upon men day by 
day, with the increased study of physical science. And since 
this aspect of the question has to be met some time or other, 
it is as well to place it in the front. The difficulty, in its 
scientific form, is of course a modern one, for the doctrine out 
of which it arises is modern. But it is only one instance, out 
of many that might be mentioned, in which the growth of 
some philosophical conception has gradually affected the 
nature of the dispute, and at last shifted the position of the 
battle-ground, in some discussion with which it might not at 
first have appeared to have any connection whatever. 

23. So far our path is plain. Up to this point disciples 
of very different schools may advance together; for in laying 
down the above doctrine we have carefully abstained from 
implying or admitting that it contains the whole truth. But 
from ibis point two paths branch out before us, paths as 



SECT. 24.] Credibility of Extraordinary Stories. 431 

different from each other in their character, origin, and 
direction, as can well be conceived. As this enquiry is only 
a digression, we may confine ourselves to stating briefly what 
seem to be the characteristics of each, without attempting to 
give the arguments which might be used in their support. 

(I.) On the one hand, we may assume that this principle 
of causation is the ultimate one. By so terming it, we do 
not mean that it is one from which we consciously start in 
our investigations, as we do from the axioms of geometry, 
but rather that it is the final result towards which we find 
ourselves drawn by a study of nature. Finding that, 
throughout the scope of our enquiries, event follows event in 
never-failing uniformity, and finding moreover (some might 
add) that this experience is supported or even demanded by 
a tendency or law of our nature (it does not matter here how 
we describe it), we may come to regard this as the one 
fundamental principle on which all our enquiries should rest. 

(II.) Or, on the other hand, we may admit a class of 
principles of a very different kind. Allowing that there is 
this uniformity so far as our experience extends, we may yet 
admit what can hardly be otherwise described than by 
calling it a Superintending Providence, that is, a Scheme or 
Order, in reference to which Design may be predicated 
without using merely metaphorical language. To adopt an 
aptly chosen distinction, it is not to be understood as over- 
ruling events, but rather as underlying them. 

24. Now it is quite clear that according as we come to 
the discussion of any particular miracle or extraordinary 
story under one or other of these prepossessions, the ques- 
tion of its credibility will assume a very different aspect. It 
is sometimes overlooked that although a difference about 
facts is one of the conditions of a bond fide argument, a dif- 
ference which reaches to ultimate principles is fatal to all 



432 CrmJilility of Extraordinary Stories. [CHAP. XVII. 

argument. The possibility of present conflict is banished in 
such a case as absolutely as that of future concord. A large 
amount of popular literature on the subject of miracles 
seems to labour under this defect. Arguments are stated 
and examined for and against the credibility of miraculous 
stories without the disputants appearing to have any 
adequate conception of the chasm which separates one side 
from the other. 

25. The following illustration may serve in some 
degree to show the sort of inconsistency of which we are 
speaking. A sailor reports that in some remote coral island 
of the Pacific, on which he had landed by himself, he had 
found a number of stones on the beach disposed in the exact 
form of a cross. Now if we conceive a debate to arise about 
the truth of his story, in which it is attempted to decide the 
matter simply by considerations about the validity of testi- 
mony, without introducing the question of the existence of 
inhabitants, and the nature of their customs, we shall have 
some notion of the unsatisfactory nature of many of the 
current arguments about miracles. All illustrations of this 
subject are imperfect, but a case like this, in which a sup- 
posed trace of human agency is detected interfering with the 
orderly sequence of other and non-intelligent natural causes, 
is as much to the point as any illustration can be. The 
thing omitted here from the discussion is clearly the one im- 
portant thing. If we suppose that there is no inhabitant, we 
shall probably disbelieve the story, or consider it to be 
grossly exaggerated. If we suppose that there are inhabit- 
ants, the question is at once resolved into a different and 
somewhat more intricate one. The credibility of the witness 
is not the only element, but we should necessarily have to 
take into consideration the character of the supposed inhab- 
itants, and the object of such an action on their part. 



5CT. 27.] Credibility of Extraordinary Stories. 433 

W 

26. Considerations of this character are doubtless 
Ften introduced into the discussion, but it appears to me 
lat they are introduced to a very inadequate extent. It is 
ften urged, after Paley, ' Once believe in a God, and mira- 
les are not incredible.* Such an admission surely demands 
)me modification and extension. It should rather be stated 
aus, Believe in a God whose working may be traced through- 
ut the whole moral and physical world. It amounts, in 
ict, to this ; Admit that there may be a design which we 
sin trace somehow or other in the course of things ; admit 
tiat "we are not wholly confined to tracing the connection of 
vents, or following out their effects, but that we can form 
ome idea, feeble and imperfect though it be, of a scheme 1 . 
*aley's advice sounds too much like saying, Admit that there 
re fairies, and we can account for our cups being cracked. 
Tie admission is not to be made in so off-hand a manner. 
?o any one labouring under the difficulty we are speaking 
f, this belief in a God almost out of any constant relation to 
tature, whom we then imagine to occasionally manifest him- 
elf in a perhaps irregular manner, is altogether impossible. 
The only form under which belief in the Deity can gain en- 
rance into his mind is as the controlling Spirit of an infinite 
^nd orderly system. In fact, it appears to me, paradoxical 
the suggestion may appear, that it might even be more 
>asy for a person thoroughly imbued with the spirit of In- 
luctive science, though an atheist, to believe in a miracle 
vhich formed a part of a vast system, than for such a person, 
bs a theist, to accept an isolated miracle. 

27. It is therefore with great prudence that Hume, 
tnd others after him, have practically insisted on commencing 
vith a discussion of the credibility of the single miracle, 

1 The stress which Butler lays upon this notion of a scheme is, I think, 
me great merit of his Analogy* 

v. 28 



434 Credibility of Extraordinary Stories. [CHAP. xvu. 

treating the question as though the Christian Revelation 
could be adequately regarded as a succession of such events. 
As well might one consider the living body to be represented 
by the aggregate of the limbs which compose it. What is to 
be complained of in so many popular discussions on the sub- 
ject is the entire absence of any recognition of the different 
ground on which the attackers and defenders of miracles are 
so often really standing. Proofs and illustrations are pro- 
duced in endless number, which involving, as they almost all 
do in the mind of the disputants on one side at least, that 
very principle of causation, the absence of which in the case 
in question they are intended to establish, they fail in the 
single essential point. To attempt to induce any one to dis- 
believe in the existence of physical causation, in a given 
instance, by means of illustrations which to him seem only 
additional examples of the principle in question, is like try- 
ing to make a dam, in order to stop the flow of a river, by 
shovelling in snow. Such illustrations are plentiful in times 
of controversy, but being in reality only modified forms of 
that which they are applied to counteract, they change their 
shape at their first contact with the disbeliever's mind, and 
only help to swell the flood which they were intended to 
check. 



CHAPTER XVIII. 



ON THE NATURE AND USE OF AN AVERAGE, AND ON THE 
DIFFERENT KINDS OF AVERAGE 1 . 



1. WE have had such frequent occasion to refer to 
averages, and to the kind of uniformity which they are apt 
to display in contrast with individual objects or events, that 
it will now be convenient to discuss somewhat more minutely 
what are the different kinds of available average, and what 
exactly are the functions they perform. 



1 There is much need of some good 
account, accessible to the ordinary 
English reader, of the nature and 
properties of the principal kinds of 
Mean. The common text-books of 
Algebra suggest that there are only 
three such, viz. the arithmetical, the 
geometrical and the harmonical: 
thus including two with which the 
statistician has little or nothing to 
do, and excluding two or more with 
which he should have a great deal to 
do. The best three references I can 
give the reader are the following. 
(1) The article Moyenne in the Dic- 
tionnaire des Sciences Medicates, by 
Dr Bertillon. This is written some- 
what from the Quetelet point of 
view. (2) A paper by Feclmer in 



the Abhandlunyen d. Math. phys. 
Classe d. Ron. Slicks. Gesellschaft d. 
Wiss. 1878; pp. 176. This con- 
tains a very interesting discussion, 
especially for the statistician, of a 
number of different kinds of mean. 
His account of the median is re- 
markably full and valuable. But 
little mathematical knowledge is de- 
manded. (3) A paper by Mr F. Y. 
Edgeworth in the Camb. Phil. Trans. 
for 1885, entitled Observations and 
Statistics. This demands some ma- 
thematical knowledge. Instead of 
dealing, as such investigations gene- 
rally do, with only one Law of Error 
and with only one kind of mean, it 
covers a wide field of investigation. 

282 



436 Averages. [CHAP. xvm. 

The first vague notion of an average, as we now under- 
stand it, seems to me to involve little more than that of a 
something intermediate to a number of objects. The objects 
must of course resemble each other in certain respects, other- 
wise we should not think of classing them together; and 
they must also differ in certain respects, otherwise we should 
not distinguish between them. What the average does for 
us, under this primitive form, is to enable us conveniently to 
retain the group together as a whole. That is, it furnishes a 
sort of representative value of the quantitative aspect of the 
things in question, which will serve for certain purposes to 
take the place of any single member of the group. 

It would seem then that the first dawn of the conception 
which science reduces to accuracy under the designation of 
an average or mean, and then proceeds to subdivide into 
various distinct species of means, presents itself as per- 
forming some of the functions of a general name. For what 
is the main use of a general name ? It is to reduce a plu- 
larity of objects to unity; to group a number of things 
together by reference to some qualities which they possess 
in common. The ordinary general name rests upon a con- 
siderable variety of attributes, mostly of a qualitative 
character, whereas the average, in so far as it serves the 
same sort of purpose, rests rather upon a single quantitative 
attribute. It directs attention to a certain kind and degree 
of magnitude. When the grazier says of his sheep that ' one 
with another they will fetch about 50 shillings/ or the 
farmer buys a lot of poles which ' run to about 10 feet/ it is 
true that they are not strictly using the equivalent of either 
a general or a collective name. But they are coming very 
near to such use, in picking out a sort of type or specimen of 
the magnitude to which attention is to be directed, and in 
classing the whole group by its resemblance to this type. 



SECT. 2.] Averages. 437 

The grazier is thinking of his sheep : not in a merely general 
sense, as sheep, and therefore under that name or con- 
ception, but as sheep of a certain approximate money value. 
Some will be more, some less, but they are all near enough 
to the assigned value to be conveniently classed together as 
if by a name. Many of our rough quantitative designations 
seem to be of this kind, as when we speak of f eight-day 
clocks ' or ' twelve-stone men/ &c. ; unless of course we in- 
tend (as we sometimes do in these cases) to assign a maximum 
or minimum value. It is not indeed easy to see how else we 
could readily convey a merely general notion of the quanti- 
tative aspect of things, except by selecting a type as above, 
or by assigning certain limits within which the things are 
supposed to lie. 

2. So far there is not necessarily any idea introduced 
of comparison, of comparison, that is, of one group with 
another, by aid of such an average. As soon as we begin 
to think of this we have to be more precise in saying what 
we mean by an average. We can easily see that ,the number 
of possible kinds of average, in the sense of intermediate 
values, is very great ; is, in fact, indefinitely great. Out of 
the general conception of an intermediate value, obtained by 
some treatment of the original magnitudes, we can elicit as 
many subdivisions as we please, by various modes of treat- 
ment. There are however only three or four which for our 
purposes need be taken into account. 

(1) In the first place there is the arithmetical average 
or mean. The rule for obtaining this is very simple: add 
all the magnitudes together, and divide the sum by their 
number. This is the only kind of average with which the 
unscientific mind is thoroughly familiar. But we must not 
let this simplicity and familiarity blind us to the fact that 
there are definite reasons for the employment of this average, 



Averages. [CHAP. xvm. 

and that it is therefore appropriate only in definite circum- 
stances. The reason why it affords a safe and accurate 
intermediate value for the actual divergent values, is that 
for many of the ordinary purposes of life, such as purchase 
and sale, we come to exactly the same result, whether we 
take account of those existent divergences, or suppose all 
the objects equated to their average. What the grazier 
must be understood to mean, if he wishes to be accurate, by 
saying that the average price of his sheep is 50 shillings, is, 
that so far as that flock is concerned (and so far as he is 
concerned), it comes to exactly the same thing, whether they 
are each sold at different prices, or are all sold at the ' aver- 
age ' price. Accordingly, when he compares his sales of one 
year with those of another ; when he says that last year the 
sheep averaged 48 shillings against the 50 of this year ; the 
employment of this representative or average value is a great 
simplification, and is perfectly accurate for the purpose in 
question. 

3. (2) Now consider this case. A certain population is 
found to have doubled itself in 100 years : can we talk of an 
1 average ' increase here of 1 per cent, annually ? The cir- 
cumstances are not quite the same as in the former case, but 
the analogy is sufficiently close for our purpose. The answer 
is decidedly, No. If 100 articles of any kind are sold for 100, 
w say that the average price is 1. By this we mean that 
the total amount is the same whether the entire lot are sold 
for 100, or whether we split the lot up into individuals 
and sell each of these for 1. The average price here is a 
convenient fictitious substitute, which can be applied for 
each individual without altering the aggregate total. If 
therefore the question be, Will a supposed increase of 1 p. c. 
in each of the 100 years be equivalent to a total increase to 
double the original amount? we are proposing a closely 



SECT. 3.] . Averages. 439 

analogous question. And the answer, as just remarked, must 
be in the negative. An annual increase of 1 p. c. continued 
for 100 years will more than double the total; it will multiply 
it by about 2*7. The true annual increment required is mea- 
sured by L0 2/2 ; that is, the population may be said to have 
increased ' on the average ' 07 p. c. annually. 

We are thus directed to the second kind of average dis- 
cussed in the ordinary text-books of algebra, viz. the geome- 
trical. When only two quantities are concerned, with a single 
intermediate value between them, the geometrical mean con- 
stituting this last is best described as the mean proportional 
between the two former. Thus, since 3 : ^15 :: ^15 : 5, 
,Jl5 is the geometrical mean between 3 and 5. When a 
number of geometrical means have to be interposed between 
two quantities, they are to be so chosen that every term in 
the entire succession shall bear the same constant ratio to 
its predecessor. Thus, in the example in the last paragraph, 
99 intermediate steps were to be interposed between 1 and 2, 
with the condition that the 100 ratios thus produced were to 
be all equal. 

It would seem therefore that wherever accurate quantita- 
tive results are concerned, the selection of the appropriate 
kind of average must depend upon the answer to the ques- 
tion, What particular intermediate value may be safely 
substituted for the actual variety* of values, so far as the 
precise object in view is concerned? This is an aspect of 
the subject which will have to be more fully considered in 
the next chapter. But it may safely be laid down that for 
purposes of general comparison, where accurate numerical 
relations are not required, almost any kind of intermediate 
value will answer our purpose, provided we adhere to the 
same throughout. Thus, if we want to compare the statures 
of the inhabitants of different counties or districts in Eng- 



440 Average*. [CHAP. xviu. 

land, or of Englishmen generally with those of Frenchmen, 
or to ascertain whether the stature of some particular class 
or district is increasing or diminishing, it really does not 
seem to matter what sort of average we select provided, of 
course, that we adhere to the same throughout our investi- 
gations. A very large amount of the work performed by 
averages is of this merely comparative or non-quantitative 
description ; or, at any rate, nothing more than this is really 
required. This being so, we should naturally resort to the 
arithmetical average ; partly because, having been long in 
the field, it is universally understood and appealed to, and 
partly because it happens to be remarkably simple and easy 
to calculate. 

4. The arithmetical mean is for most ordinary pur- 
poses the simplest and best. Indeed, when we are dealing 
with a small number of somewhat artificially selected magni- 
tudes, it is the only mean which any one would think of 
employing. We should not, for instance, apply any other 
method to the results of a few dozen measurements of lengths 
or estimates of prices. 

When, however, we come to consider the results of a very 
large number of measurements of the kind which can be 
grouped together into some sort of 'probability curve ' we 
begin to find that there is more than one alternative before 
us. Begin by recurring to the familiar curve represented 
on p. 29 ; or, better still, to the initial form of it represented 
in the next chapter (p. 476). We see that there are three 
different ways in which we may describe the vertex of the 
curve. We may call it the position of the maximum ordi- 
nate ; or that of the centre of the curve ; or (as will be seen 
hereafter) the point to which the arithmetical average of 
all the different values of the variable magnitude directs us. 
These three are all distinct ways of describing a position; 



SECT. 3.] Avtragc* 441 

but when we are dealing with a symmetrical curve at all 
resembling the binomial or exponential form they all three 
coincide in giving the same result : as they obviously do in 
the case in question. 

As soon, however, as we come to consider the case of 
asymmetrical, or lop-sided curves, the indications given by 
these three methods will be as a rule quite distinct; and 
therefore the two former of these deserve brief notice as 
representing different kinds of means from the arithmetical 
or ordinary one. We shall see that there is something about 
each of them which recommends it to common sense as being 
in some way natural and appropriate. 

5. (3) The first of these selects from amongst the 
various different magnitudes that particular one which is 
most frequently represented. It has not acquired any tech- 
nical designation 1 , except in so far as it is referred to, by 
its graphical representation, as the " maximum ordinate" 
method. But I suspect that some appeal to such a mean 
or standard is really far from uncommon, and that if we 
could draw out into clearness the conceptions latent in the 
judgments of the comparatively uncultivated, we should find 
that there were various classes of cases in which this mean 
was naturally employed. Suppose, for instance, that there 
was a fishery in which the fish varied very much in size 

1 This kind of mean is called by tical average. 

Fechner and others the " dichteste This mean ought to be called the 
Werth." The most appropriate ap- 'probable' value (a name however in 
peal to it that I have seen is by Prof. possession of another) on the ground 
Lexis (Ma&senerscheinungen, p. 42) that it indicates the point of likeliest 
where he shows that it indicates occurrence; i.e. if we compare all 
clearly a sort of normal length of the indefinitely small and equal units 
human life, of about 70 years; a of variation, the one corresponding 
result which is almost entirely mask- to this will tend to be most fre- 
ed when we appeal to the arithme- quently represented. 



442 Averages. [CHAP. xvm. 

but in which the commonest size was somewhat near the 
largest or the smallest. If the men were in the habit of 
selling their fish by weight, it is probable that they would 
before long begin to acquire some kind of notion of what 
is meant by the arithmetical mean or average, and would 
perceive that this was the most appropriate test. But if the 
fish were sorted into sizes, and sold by numbers in each of 
these sizes, I suspect that this appeal to a maximum ordi- 
nate would begin to take the place of the other. That is, 
the most numerous class would come to be selected as a 
sort of type by which to compare the same fishery at one 
time and another, or one fishery with others. There is also, 
as we shall see in the next chapter, some scientific ground 
for the preference of this kind of mean in peculiar cases ; 
viz. where the quantities with which we deal are true 
' errors/ in the estimate of some magnitude, and where also 
it is of much more importance to be exactly right, or very 
nearly right, than to have merely a low average of error. 

6. (4) The remaining kind of mean is that which is 
now coming to be called the "median." It is one with 
which the writings of Mr Galton have done so much to 
familiarize statisticians, and is best described as follows. 
Conceive all the objects in question to be marshalled in the 
order of their magnitude ; or, what comes to the same thing, 
conceive them sorted into a number of equally numerous 
classes ; then the middle one of the row, or the middle one 
in the middle class, will be the median. I do not think 
that this kind of mean is at all generally recognized at 
present, but if Mr Galton's scheme of natural measurement 
by what he calls " per-centiles " should come to be gener- 
ally adopted, such a test would become an important one. 
There are some conspicuous advantages about this kind of 
mean. For one thing, in most statistical enquiries, it is 



SECT. 7.] Averages. 443 

far the simplest to calculate ; and, what is more, the process 
of determining it serves also to assign another important 
element to be presently noticed, viz. the 'probable error.' 
Then again, as Fechner notes, whereas in the arithmetical 
mean a few exceptional and extreme values will often cause 
perplexity by their comparative preponderance, in the case 
of the median (where their number only and not their ex- 
treme magnitude is taken into account) the importance of 
such disturbance is diminished. 

7. A simple illustration will serve to indicate how these 
three kinds of mean coalesce into one when we are dealing 
with symmetrical Laws of Error, but become quite distinct 
as soon as we come to consider those which are unsym- 
metrical. 



DYX 



Suppose that, in measuring a magnitude along OBDC, 
where the extreme limits are OB and OC t the law of error 
is represented by the triangle BAG: the length OD will 
be at once .the arithmetical mean, the median, and the most 
frequent length: its frequency being represented by the 
maximum ordinate AD. But now suppose, on the other 
hand, that the extreme lengths are OD and 0(7, and that 
the triangle ADC represents the law of error. The most 
frequent length will be the same as before, OD, marked by 
the maximum ordinate AD. But the mean value will now 
be OX, where DX^^DC; and the median will be OF, 

where DF=l- DC. 



444 Averages. [CHAP. XVIIL 

Another example, taken from natural phenomena, may 
be found in the heights of the barometer as taken at the 
same hour on successive days. So far as 4857 of these may 
be regarded as furnishing a sufficiently stable basis of ex- 
perience, it certainly seems that the resulting curve of fre- 
quency is asymmetrical. The mean height here was found 
to be 29*98: the median was 30*01: the most frequent 
height was 30*05. The close approximation amongst these 
is an indication that the asymmetry is slight 1 . 

8. It must be clearly understood that the average, of 
whatever kind it may be, from the mere fact of its being a 
single substitute for an actual plurality of observed values, 
must let slip a considerable amount of information. In fact 
it is only introduced for economy. It may entail no loss 
when used for some one assigned purpose, as in our ex- 
ample about the sheep; but for purposes in general it 
cannot possibly take the place of the original diversity, by 
yielding all the information which they contained. If all 
this is to be retained we must resort to some other method. 
Practically we generally do one of two things : either (1) we 
put all the figures down in statistical tables, or (2) we ap- 
peal to a diagram. This last plan is convenient when the 
data are very numerous, or when we wish to display or to 
discover the nature of the law of facility under which they 
range. 

The mere assignment of an average lets drop nearly all 
of this, confining itself to the indication of ati intermediate 

1 A diagram illustrative of this was noted. That is, 29*9 includes 
number of results was given in all values between 29*900 and 29*999. 
Nature (Sept. 1, 1887). In calculat- Thus the value most frequently en- 
ing, as above, the different means, I tered in my tables was 30:0, but on 
may remark that the original results the usual principles of interpolation 
were given to three decimal places ; this is reckoned as .80*05. 
but, in classing them, only one place 



SECT. 9.] Averages. 445 

value. It gives a "middle point" of some kind, but says 
nothing whatever as to how the original magnitudes were 
grouped about this point. For instance, whether two mag- 
nitudes had been respectively 25 and 27, or 15 and 37, they 
would yield the same arithmetical average of 26. 

9. To break off at this stage would clearly be to leave 
the problem in a very imperfect condition. We therefore 
naturally seek for some simple test which shall indicate 
how closely the separate results were grouped about their 
average, so as to recover some part of the information which 
had been let slip. 

If any one were approaching this problem entirely anew, 
that is, if he had no knowledge of the mathematical exi- 
gencies which attend the theory of " Least Squares/' I ap- 
prehend that there is but one way in which he would set 
about the business. He would say, The average which we 
have already obtained gave us a rough indication, by as- 
signing an intermediate point amongst the original magni- 
tudes. If we want to supplement this by a rough indica- 
tion as to how near together these magnitudes lie, the 
best way will be to treat their departures from the mean 
(what are technically called the "errors") in precisely the 
same way, viz. by assigning their average. Suppose there 
are 13 men whose heights vary by equal differences from 
5 feet to 6 feet, we should say that their average height 
was 66 inches, and their average departure from this average 
was 3 T ^ inches. 

Looked at from this point of view we should then pro- 
ceed to try how each of the above-named averages would 
answer the purpose. Two of them, viz. the arithmetical 
mean and the median, -will answer perfectly; and, as we 
shall immediately see, are frequently used for the purpose. 
So too we could, if we pleased, employ the geometrical 



446 Averages, [CHAP. xvm. 

mean, though such employment would be tedious, owing 
to the difficulty of calculation. The 'maximum ordinate' 
clearly would not answer, since it would generally (v. the 
diagram on p. 443) refer us back again to the average 
already obtained, and therefore give no information. 

The only point here about which any doubt could arise 
concerns what is called in algebra the sign of the errors. 
Two equal and opposite errors, added algebraically, would 
cancel each other. But when, as here, we are regarding 
the errors as substantive quantities, to be considered on 
their own account, we attend only to their real magnitude, 
and then these equal and opposite errors are to be put upon 
exactly the same footing. 

10. Of the various means already discussed, two, as 
just remarked, are in common use. One of these is fa- 
miliarly known, in astronomical and other calculations, as 
the ' Mean Error/ and is so absolutely an application of the 
same principle of the arithmetical mean to the errors, that 
has been already applied to the original magnitudes, that it 
needs no further explanation. Thus in the example in the 
last section the mean of the heights was 66 inches, the 
mean of the errors was 3^ inches. 

The other is the Median, though here it is always known 
under another name, i.e. as the 'Probable Error'; a tech- 
nical and decidedly misleading term. It is briefly defined 
as that error which we are as likely to exceed as to fall 
short of: otherwise phrased, if we were to arrange all the 
errors in the order of their magnitude, it corresponds to that 
one of them which just bisects the row. It is therefore the 
' median ' error : or, if we arrange all the magnitudes in suc- 
cessive order, and divide them into four equally numerous 
classes, what Mr Galton calls 'quartiles/ the first and 
third of the consequent divisions will mark the limits of 



SECT. 11.] Averages. 447 

the 'probable error* on each side, whilst the middle one will 
mark the ' median/ This median, as was remarked, coincides, 
in symmetrical curves, with the arithmetical mean. 

It is best to stand by accepted nomenclature, but the 
reader must understand that such an error is not in any 
strict sense ' probable. 5 It is indeed highly improbable that 
in any particular instance we should happen to get just this 
error : in fact, if we chose to be precise and to regard it as 
one exact magnitude out of an infinite number, it would be 
infinitely unlikely that we should hit upon it. Nor can it be 
said to be probable that we shall be within this limit of the 
truth, for, by definition, we are just as likely to exceed as to 
fall short. As already remarked (see note on p. 441), the 
' maximum ordinate ' would have the best right to be regarded 
as indicating the really most probable value. 

11. (5) The error of mean square. As previously 
-UL'U - ! <1. the plan which would naturally be adopted by 
any one who had no concern with the higher mathematics of 
the subject, would be to take the ' mean error ' for the pur- 
pose of the indication in view. But a very different kind of 
average is generally adopted in practice to serve as a test of 
the amount of divergence or dispersion. Suppose that we 
have the magnitudes x v %>...#; their ordinary average is 

- (x. + x 9 + . + a? ), and their ' errors ' are the differences 

n \ 1 8 n/ 

between this and x v o? 4 ,...# n . Call these errors e l9 e#...e n , then 
the arithmetical mean of these errors (irrespective of sign) is 

- (^4- e a -f ...-fe,,). The Error of Mean Square 1 , on the other 
hand, is the square root of- (e, 2 -f e* +...4- O- 

1 There is some ambiguity in the eommonly uses the expression Erroi 
phraseology in use here. Thus Airy of Mean Square' to represent, ai 



448 A vemye*. [CHAP. XVIII. 

The reasons for employing this latter kind of average in 
preference to any of the others will be indicated in the fol- 
lowing chapter. At present we are concerned only with the 
general logical nature of an average, and it is therefore 
sufficient to point out that any such intermediate value will 
answer the purpose of giving a rough and summary indica- 
tion of the degree of closeness of approximation which our 
various measures display to each other and to their common 
average. If we were to speak respectively of the ' first ' and 
the ' second average/ we might say that the former of these 
assigns a rough single substitute for the plurality of original 
values, whilst the latter gives a similar rough estimate of the 
degree of their departure from the former. 

12. So far we have only been considering the general 
nature of an average, and the principal kinds of average 
practically in use. We must now enquire more particularly 
what are the principal purposes for which averages are em- 
ployed. 

In this respect the first thing we have to do is to raise 
doubts in the readers mind on a subject on which he 
perhaps has not hitherto felt the slightest doubt. Every 
one is more or lest familiar with the practice of appealing to 
an average in order to secure accuracy, But distinctly what 
we begin by doing is to sacrifice accuracy ; for in place of 
the plurality of actual results we get a single result which 

/Se 5 , and others, of the expression * Mean 

here ' V IT' GaUowa y commonl y Error,' (widely in use in its more 
speaks of the * Mean Square of the natural signification,) as the equiva- 

. Se fl T , lent of this E. M. 8. 
Errors' to represent - . I shall The teohni<)al term , Fluotuation > 

adhere to the former usagg and re- is applied by Mr F. Y. Edgeworth to 
present it briefly by E.M.S. Still the expression 
more unfortunate (to my thinking) 
is the employment, by Mr Merriman 



SECT. 13.] Averages. 449 

very possibly does not agree with any one of them. If I find 
the temperature hi different parts of a room to be different, 
but say that the average temperature is 61, there may per- 
haps be but few parts of the room where this exact tempera- 
ture is realized. And if I say that the average stature of 
a certain small group of men is 68 inches, it is probable that 
no one of them will present precisely this height. 

The principal way in which accuracy can be thus secured 
is when what we are really aiming at is not the magnitudes 
before us but something else of which they are an indication. 
If they are themselves ' inaccurate/ we shall see presently 
that this needs some explanation, then the single average, 
which in itself agrees perhaps with none of them, may be 
much more nearly what we are actually in want of. We shall 
find it convenient to subdivide this view of the subject into 
two parts ; by considering first those cases in which quantita- 
tive considerations enter but slightly, and in which no deter- 
mination of the particular Law of Error involved is demanded, 
and secondly those in which such determination cannot be 
avoided. The latter are only noticed in passing here, as a 
separate chapter is reserved for their fuller consideration. 

13. The process, as a practical one, is familiar enough 
to almost everybody who has to work with measures of any 
kind. Suppose, for instance, that I am measuring any object 
with a brass rod which, as we know, expands and contracts 
according to the temperature. The results will vary slightly, 
being sometimes a little too great and sometimes a little too 
small. Ail these variations are physical facts, and if what 
we were concerned with was the properties of brass they 
would be the one important fact for us. But when we are 
concerned with the length of the object measured, these facts 
become superfluous and misleading. What we want to do is 
to escape their influence, and this we are enabled to effect by 
v. 29 



450 Averages. [CHAP. xvin. 

taking their (arithmetical) average, provided only they are 
as often in excess as in defect 1 . For this purpose all that is 
necessary is that equal excesses and defects should be 
equally prevalent. It is not necessary to know what is the 
law of variation, or even to be assured that it is of one par- 
ticular kind. Provided only that it is in the language of 
the diagram on p. 29, symmetrical, then the arithmetical 
average of a suitable and suitably varied number of measure- 
ments will be free from this source of disturbance. And 
what holds good of this cause of variation will hold good of 
all others which obey the same general conditions. In fact 
the equal prevalence of equal and opposite errors seems to 
be the sole and sufficient justification of the familiar process 
of taking the average in order to secure accuracy. 

14. We must now make the distinction to which at- 
tention requires so often to be drawn in these subjects 
between the cases in which there respectively is, and is not, 
some objective magnitude aimed at : a distinction which the 
common use of the same word " errors " is so apt to obscure. 
When we talked, in the case of the brass rod, of excesses 
and defects being equal, we meant exactly what we said, viz. 
that for every case in which the 'true' length (i.e. that de- 
termined by the authorized standard) is exceeded by a given 
fraction of an inch, there will be a corresponding case in 
which there is an equal defect. 

On the other hand, when there is no such fixed objective 
standard of reference, it would appear that all that we mean 
by equal excesses and defects is permanent symmetry of 
arrangement. In the case of the measuring rod we were 

1 Practically, of course, we should take this variation as a specimen of 

allow for the expansion or oontrao- one of those disturbances which may 

tion. But for purposes of logical he neutralised by resort to an ave- 

cxplanation we may conveniently rage. 



SECT. 15.] Averages. 451 

able to start with something which existed, so to say, before 
its variations ; but in many cases any starting point which 
we can find is solely determined by the average. 

Suppose, for instance, we take a great number of ob- 
servations of the height of the barometer at a certain place, 
at all times and seasons and in all weathers, we should 
generally consider that the average of all these showed the 
' true ' height for that place. What we really mean is that 
the height at any moment is determined partly (and prin- 
cipally) by the height of the column of air above it, but partly 
also by a number of other agencies such as local temperature, 
moisture, wind, &c. These are sometimes more and some- 
times less effective, but their range being tolerably con- 
stant, and their distribution through this range being 
tolerably symmetrical, the average of one large batch of 
observations will be almost exactly the same as that of any 
other. This constancy of the average is its truth. I am 
quite aware that we find it difficult not to suppose that 
there must be something more than this constancy, but we 
are probably apt to be misled by the analogy of the other 
class of cases, viz. those in which we are really aiming at 
some sort of mark. 

15. As regards the practical methods available for 
determining the various kinds of average there is very little 
to be said ; as the arithmetical rules are simple and definite, 
and involve nothing more than the inevitable drudgery 
attendant upon dealing with long rows of figures. Perhaps 
the most important contribution to this part of the subject is 
furnished by Mr Gal ton's suggestion to substitute the median 
for the mean, and thus to elicit the average with sufficient 
accuracy by the mere act of grouping a number of objects 
together. Thus he has given an ingenious suggestion for 
obtaining the average height of a number of men without 

292 



452 Averages. [CHAP. xvm. 

the trouble and risk of measuring them all. "A barbarian 
chief might often be induced to marshall his men in the 
order of their heights, or in that of the popular estimate of 
their skill in any capacity ; but it would require some ap- 
paratus and a great deal of time to measure each man 
separately, even ,! ' .. it possible to overcome the usually 
strong repugnance of uncivilized people to any such pro- 
ceeding " (Phil. Mag. Jan. 1875). That is, it being known 
from wide experience that the heights of any tolerably 
--. - set of men are apt to group themselves sym- 
metrically, the condition for the coincidence of the three 
principal kinds of mean, the middle man of a row thus 
arranged in order will represent the mean or average man, 
and him we may subject to measurement. Moreover, since 
the intermediate heights are much more thickly represented 
than the extreme ones, a moderate error in the selection of 
the central man of a long row will only entail a very small 
error in the selection of the corresponding height. 

16. We can now conveniently recur to a subject which 
has been already noticed in a former chapter, viz. the at- 
tempt which is sometimes made to establish a distinction 
between an average and a mean. It has been proposed to ; 
confine the former term to the cases in which we are dealing 
with a fictitious result of our own construction, that is, with 
a mere arithmetical deduction from the observed magni- 
tudes, and to apply the latter to cases in which there is 
supposed to be some objective magnitude peculiarly repre- 
sentative of the average. 

Recur to the three principal classes, of things appropriate 
to Probability, which were sketched out in Ch. n. 4. The 
first of these comprised the results of games of chance. Toss 
a die ten times: the total number of pips on the upper 
side may vary from ten up to sixty. Suppose it to be 



SECT. 16.] Averages. 453 

thirty. We then say that the average of this batch of 
ten is three. Take another set of ten throws, and we may 
get another average, say four. There is clearly nothing 
objective peculiarly corresponding in any way to these 
avoru#^. No doubt if we go on long enough we shall 
find that the averages tend to centre about 3*5 : we then 
call this the average, or the * probable ' number of points ; 
and this ultimate average might have been pretty con- 
stantly asserted beforehand from our knowledge of the con- 
stitution of a die. It has however no other truth or reality 
about it of the nature of a type : it is simply the limit 
towards which the averages tend. 

The next class is that occupied by the members of most 
natural groups of objects, especially as regards the charac- 
teristics of natural species. Somewhat similar remarks may 
be repeated here. There is very frequently a ' limit' towards 
which the averages of increasing numbers of individuals tend 
to approach ; and there is certainly some temptation to re- 
gard this limit as being a sort of type which all had been 
intended to resemble as closely as possible. But when we 
looked closer, we found that this view could scarcely be 
justified; all which could be safely asserted was that this 
type represented, for the time being, the most numerous 
specimens, or those which under existing conditions could 
most easily be produced. 

The remaining class stands on a somewhat different 
ground. When we make a succession of more or less suc- 
cessful attempts of any kind, we get a corresponding series 
of deviations from the mark at which we aimed. These we 
may treat arithmetically, and obtain their averages, just as 
in the former cases. These averages are fictiojis, that is to 
say, they are artificial deductions of our own which need 
not necessarily have anything objective corresponding to 



454 ~\ mages. [CHAP. xvin. 

them. In fact, if they be averages of a few only they 
most probably will not have anything thus corresponding 
to them. Anything answering to a type can only be sought 
in the ' limit ' towards which they ultimately tend, for this 
limit coincides with the fixed point or object aimed at. 

17. Fully admitting the great value and interest of 
.Quetelet's work in this direction, he was certainly the first 
to direct public attention to the fact that so many classes of 
natural objects display the same characteristic property, it 
nevertheless does not seem desirable to attempt to mark 
such a distinction by any special use of these technical 
terms. The objections are principally the two following. 

In the first place, a single antithesis, like this between 
an average and a mean, appears to suggest a very much 
simpler state of things than is actually found to exist in 
nature. A reference to the three classes of things just 
mentioned, and a consideration of the wide range and di- 
versity included in each of them, will serve to remind us 
not only of the very gradual and insensible advance from 
what is thus regarded as ' fictitious ' to what is claimed as 
'real;' but also of the important fact that whereas the 'real 
type ' may be of a fluctuating and evanescent character, the 
* fiction' may (as in games of chance) be apparently fixed 
for ever. Provided only that the conditions of production 
remain stable, averages of large numbers will always prac- 
tically present much the same general characteristics. The 
far more important distinction lies between the average of 
a few, with its fluctuating values and very imperfect and 
occasional attainment of its ultimate goal, and the average 
of many and its gradually close approximation to its ulti- 
mate value : i.e. to its objective point of aim if there happen 
to be such. 

Then, again, the considerations adduced in this chapter 



SECT. 18.] Averages. 455 

will show that within the field of the average itself there is 
far more variety than Quetelet seems to have recognized. 
He did not indeed quite ignore this variety, but he prac- 
tically confined himself almost entirely to those symmetrical 
arrangements in which three of the principal means coalesce 
into one. We should find it difficult to carry out his dis- 
tinction in less simple cases. For instance, when there is 
some degree of asymmetry, it is the 'maximum ordinate' 
which would have to be considered as a 'mean' to the 
exclusion of the others ; for no appeal to an arithmetical 
average would guide us to this point, which however is to 
be iv^iiiv,r<l. if any can be so regarded, as marking out the 
position of the ultimate type. 

18. We have several times pointed out that it is a 
characteristic of the things with which Probability is con- 
cerned to present, in the long run, a continually intensifying 
uniformity. And this has been frequently described as what 
happens 'on the average.' Now an objection may very 
possibly be raised against regarding an arrangement of 
things by virtue of which order thus emerges out of disorder 
as deserving any special notice, on the ground that from the 
nature of the arithmetical average it could not possibly be 
otherwise. The process by which an average is obtained, it 
may be urged, insures this tendency to equalization amongst 
the magnitudes with which it deals. For instance, let there 
be a party of ten men, of whom four are tall and four are 
short, and take the average of any five of them. Since this 
number cannot be made up of tall men only, or of short men 
only, it stands to reason that the averages cannot differ so 
much amongst themselves as the single measures can. Is 
not then the equalizing process, it may be asked, which is 
observable on increasing the range of our observations, 
one which can be shown to follow from necessary laws of 



456 Averages. [CHAP. xvm. 

arithmetic, and one therefore which might be asserted & 
priori ? 

Whatever force there may be in the above objection arises 
principally from the limitations of the example selected, in 
which the number chosen was so large a proportion of the 
total as to exclude the bare possibility of only extreme cases 
being contained within it. As much confusion is often felt 
here between what is necessary and what is matter of ex- 
perience, it will be well to look at an example somewhat 
more closely, in order to determine exactly what are the 
really necessary consequences of the averaging process. 

19. Suppose then that we take ten digits at random 
from a table (say) of logarithms. Unless in the highly un- 
likely case of our having happened upon the same digit ten 
times running, the average of the ten must be intermediate 
between the possible extremes. Every conception of an 
average of any sort not merely involves, but actually means, 
the taking of something intermediate between the extremes. 
The average therefore of the ten must lie closer to 4 '5 (the 
average of the extremes) than did some of the single digits. 

Now suppose we take 1000 such digits instead of 10. We 
can say nothing more about the larger number, with de- 
monstrative certainty, than we could before about the smaller. 
If they were unequal to begin with (i. e. if they were not all 
the same) then the average must be intermediate, but more 
than this cannot be proved arithmetically. By comparison with 
such purely arithmetical considerations there is what may be 
called a physical fact underlying our confidence in the grow- 
ing stability of the average of the larger number. It is that 
the constituent elements from which the average is deduced 
will themselves betray a growing uniformity : that the pro- 
portions in which the different digits come out will become 
more and more nearly equal as we take larger numbers of 



SECT. 20.] Averages. 457 

them. If the proportions in which the 1000 digits were 
distributed were the same as those of the 10 the averages 
would be the same. It is obvious therefore that the arith- 
metical process of obtaining an average goes a very little 
way towards securing the striking kind of uniformity which 
we find to be actually presented. 

20. There is another way in which the same thing 
may be put. It is sometimes said that whatever may have 
been the arrangement of the original elements the process of 
continual averaging will necessarily produce the peculiar 
binomial or exponential law of arrangement. This state- 
ment is perfectly true (with certain safeguards) but it is not 
in any way opposed to what has been said above. Let us 
take for consideration the example above referred to. The 
arrangement of the individual digits in the long run is the 
simplest possible. It would be represented, in a diagram, 
not by a curve but by a finite straight line, for each digit 
occurs about as often as any other, and this exhausts all the 
'arrangement' that can be detected. Now, when we con- 
sider the results of taking averages of ten such digits, we see 
at once that there is an opening for a more extensive arrange- 
ment. The totals may range from up to 100, and there- 
fore the average will have 100 values from to 9 ; and what 
we find is that the frequency of these numbers is determined 
according to the Binomial 1 or Exponential Law. The most 
frequent result is the true mean, viz. 4*5, and from this they 
diminish in each direction towards and 10, which will each 
occur but once (on the average) in 10 10 occasions. 

The explanation here is of the same kind as in the former 
case. The resultant arrangement, so far as the averages are 

1 More strictly multinomial: the efficients of the powers of x in the 
relative frequency of the different development of 
numbers being indicated by the co- (l+x+x*+ ... +x 9 ) 10 . 



458 Averages. [CHAP. xvm. 

concerned, is only ' necessary ' in the sense that it is a neces- 
sary result of certain physical assumptions or experiences. 
If all the digits tend to occur with equal frequency, and if 
they are ' independent ' (i. e. if each is associated indifferently 
with every other), then it is an arithmetical consequence 
that the averages when arranged in respect of their magni- 
tude and prevalence will display the Law of Facility above 
indicated. Experience, so far as it can be appealed to, shows 
that the true randomness of the selection of the digits, i.e. 
their equally frequent recurrence, and the impartiality of 
their combination, is very fairly secured in practice. Ac- 
- !*./; the theoretic deduction that whatever may have 
been the original Law of Facility of the individual results 
we shall always find the familiar Exponential Law asserting 
itself as the law of the averages, is fairly justified by ex- 
perience in such a case. 

The further discussion of certain corrections and refine- 
ments is reserved to the following chapter. 

21. In regard to the three kinds of average employed 
to test the amount of dispersion, i.e. the mean error, the 
probable error, and the error of mean square, two im- 
portant considerations must be borne in mind. They will 
both recur for fuller discussion and justification in the course 
of the next chapter, when we come to touch upon the Method 
of Least Squares, but their significance for logical purposes 
is so great that they ought not to be entirely passed by at 
present. 

(1) In the first place, then, it must be remarked that in 
order to know what in any case is the real value of an error 
we ought in strictness to know what is the position of the 
limit or ultimate average, for the amount of an error is 
always theoretically measured from this point. But this is 
information which we do not always possess. Recurring 



SECT. 21.] Averages. 

once more to the three principal classes of events with which 
we are concerned, we can readily see that in the case of 
games of chance we mostly do possess this knowledge. In- 
stead of appealing to experience to ascertain the limit, we 
practically deduce it by simple mechanical or arithmetical 
considerations, and then the ' error ' in any individual case or 
group of cases is obviously found by comparing the results 
thus obtained with that which theory informs us would ulti- 
mately be obtained in the long run. In the case of de- 
liberate efforts at an aim (the third class) we may or may 
not know accurately the value or position of this aim. In 
astronomical observations we do not know it, and the method 
of Least Squares is a method for helping us to ascertain it as 
well as we can ; in such experimental results as firing at a 
mark we do know it, and may thus test the nature and 
amount of our failure by direct experience. In the remain- 
ing case, namely that of what we have termed natural kinds 
or groups of things, not only do we not know the ultimate 
limit, but its existence is always at least doubtful, and in 
many cases may be confidently denied. Where it does exist, 
that is, where the type seems for all practical purposes per- 
manently fixed, we can only ascertain it by a laborious resort 
to statistics. Having done this, we may then test by it the 
results of observations on a small scale. For instance, if we 
find that the ultimate proportion of male to female births is 
about 106 to 100, we may then compare the statistics of 
some particular district or town and speak of the consequent 
'error/ viz. the departure, in that particular and special 
district, from the general average. 

What we have therefore to do in the vast majority of 
practical cases is to take the average of a finite number of 
measurements or observations, of all those, in fact, which 
we have in hand, and take this as our starting point in 



460 Averages. [CHAP. xviu. 

order to measure the errors. The errors in fact are not 
known for certain but only probably calculated. This how- 
ever is not so much of a theoretic defect as it may seem at 
first sight ; for inasmuch as we seldom have to employ these 
methods, for purposes of calculation, that is, as distinguished 
from mere illustration, except for the purpose of dis- 
covering what the ultimate average is, it would be a sort of 
petitio principii to assume that we had already secured it. 
But it is worth while considering whether it is desirable to 
employ one and the same term for 'errors' known to be 
such, and whose amount can be assigned with certainty, and 
for ' errors ' which are only probably such and whose amount 
can be only probably u--i;jin-u. In fact it has been proposed 1 
to employ the two terms ' error ' and ' residual ' respectively 
to distinguish between the magnitudes thus determined, that 
is, between the (generally unknown) actual error and the ob- 
served error. 

22. (2) The other point involves the question to what 
extent either of the first two tests (pp. 446, 7) of the close- 
ness with which the various results have grouped themselves 
about their average is trustworthy or complete. The answer 
is that they are necessarily incomplete. No single estimate 
or magnitude can possibly give us an adequate account pf a 
number of various magnitudes. The point is a very im- 
portant one ; and is not, I think, sufficiently attended to, the 
consequence being, as we shall see hereafter, that it is far 
too summarily assumed that a method which yields the 
result with the least ' error of mean square ' must necessarily 
be the best result for all purposes. It is not however by any 
means clear that a test which answers best for one purpose 
must do so for all. 

It must be clearly understood that each of these tests is 

1 By Mr Merriman, in his work on Least Squares. 



SECT. 23.] Averages. 461 

an 'average/ and that every average necessarily rejects a 
mass of varied detail by substituting for it a single result. 
We had, say, a lot of statures : so many of 60 inches, so 
many of 61, &c. We replace these by an 'average' of 68, 
and thereby drop a mass of information. A portion of this 
we then seek to recover by reconsidering the 'errors' or 
departures of these statures from their average. As before, 
however, instead of giving the full details we substitute an 
average of the errors. The only difference is that instead of 
taking the same kind of average (i.e. the arithmetical) we 
often prefer to adopt the one called the 'error of mean 
square/ 

23. A question may be raised here which is of sufficient 
importance to deserve a short consideration. When we have 
got a set of measurements before us, why is it generally 
held to be sufficient simply to assign : (1) the mean value ; 
and (2) the mean departure from this mean ? The answer 
is, of course, partly given by the fact that we are only sup- 
posed to be in want of a rough approximation : but there is 
more to be said than this. A further justification is to be 
found in the fact that we assume that we need only con- 
template the possibility of a single Law of Error, or at any 
rate that the departures from the familiar Law will be but 
trifling. In other words, if we recur to the figure on p. 29, 
we assume that there are only two unknown quantities or 
disposable constants to be assigned ; viz. first, the position of 
the centre, and, secondly, the degree of eccentricity, if one 
may so term it, of the curve. The determination of the 
mean value directly and at once assigns the former, and the 
determination of the mean error (in either of the ways re- 
ferred to already) indirectly assigns the latter by confining us 
to one alone of the possible curves indicated in the figure. 

Except for the assumption of one such Law of Error the 



462 Averages.. [CHAP. xvm. 

determination of the mean error would give but a slight 
intimation of the sort of outline of our Curve of Facility. 
We might then have found it convenient to adopt some plan 
of successive approximation, by adding a third or fourth 
' mean/ Just as we assign the mean value of the magni- 
tude, and its mean departure from this mean ; so we might 
take this mean error (however determined) as a fresh starting 
point, and assign the mean departure from it. If the point 
were worth further discussion we might easily illustrate by 
means of a diagram the sort of successive approximations 
which such indications would yield as to the ultimate form 
of the Curve of Facility or Law of Error. 



As this volume is written mainly for those who take an interest in the 
logical questions involved, rather than as an introduction to the actual 
processes of calculation, mathematical details have been throughout avoided 
as much as possible. For this reason comparatively few references have 
been made to the exponential equation of the Law of Error, or to the 
corresponding 'Probability integral,' tables of which are given in several 
handbooks on the subject. There are two points however in connection 
with these particular topics as to which difficulties are, or should be, felt by 
so many students that some notice may be taken of them here 

(1) In regard to the ordinary algebraical expression for the law of error, 

viz. y = r= tf"* 3 * 8 , it will have been observed that I have always spoken of y 
*Jir 

as being proportional to the number of errors of the particular magnitude x. 
It would hardly be correct to say, absolutely, that y represents that number, 
because of course the actual number of errors of any precise magnitude, 
where continuity of possibility is assumed, must be indefinitely small. If 
therefore we want to pass from the continuous to the discrete, by ascertaining 
the actual number of errors between two consecutive divisions of our scale, 
when, as usual in measurements, all within certain limits are referred to 
some one precise point, we must modify our formula. In accordance with 
the usual differential notation, we must say that the number of errors falling 

into one subdivision (dx) of our scale it dx -._: e " w * 9 , where dx is a (small) 

Jir 

unit of length, in which both h~ l and x must be measured. 



SECT. 23.] Averages. 463 

The difficulty felt by most students is in applying the formula to actual 
statistics, in other words in putting in the correct units. To take an actual 
numerical example, suppose that 1460 men have been measured in regard to 
their height "true to the nearest inch," and let it be known that the 
modulus here is 3-6 inches. Then dx=l (inch); h~ l =3'& inches. Now 

2 e - h * x *dx l] that is, the sum of all the consecutive possible values 
VT 
is equal to unity. When therefore we want the sum, as here, to be 1460, we 

1460 (*\* /*L^ 2 

must express the formula thus; y -. e Vse/ , or y = 228 e Va-e/ . 

*Jir x 3-6 

Here x stands for the number of inches measured from the central or mean 
height, and y stands for the number of men referred to that height in our 
statistical table. (The values of e~ <2 for successive values of t are given in 
the handbooks.) 

For illustration I give the calculated numbers by this formula for values 
of x from to 8 inches, with the actual numbers observed in the Cambridge 
measurements recently set on foot by Mr Galton. 

inches calculated observed 

#=0 2/ = 228 =231 

ar = l 2/ = 212 =218 

x = 2 2/ = 166 =170 

s = 3 2/ = lll =110 

= 4 y= 82 =66 

x-5 2/= 32 = 31 

* = 6 y= 11 = 10 

a = 7 2/= 4 =6 

x = S y= 1 =3 

Here the average height was 69 inches : dx, as stated, = 1 inch. By 
saying, 'put x = 0,' we mean, calculate the number of men who are assigned 
to 69 inches; i.e. who fall between 68'5 and 69-5. By saying, 'put a = 4,' 
we mean, calculate the number who are assigned to 65 or to 73 ; i.e. who lie 
between 64-5 and 65 '5, or between 72'5 and 73-5. The observed results, it 
will be seen, keep pretty close to the calculated : in the case of the former 
the means of equal and opposite divergences from the mean have been taken, 
the actual results not being always the same in opposite directions. 

(2) The other point concerns the interpretation of the familiar pro- 

2 f* 
bability integral, pj e~*dt. Every one who has calculated the chance 



of an event, by the help of the tables of this integral given in so many 
handbooks, knows that if we assign any numerical value to t, the 
corresponding value of the above expression assigns the chance that an 



464 Averages. [CHAP. xvm. 

error taken at random shall lie within that same limit, viz. t. Thus put 
t 1*5, and we have the result -96 ; that is, only 4 per cent, of the errors will 
exceed 'one and a half.' But when we ask, 'one and a half what? the 
answer would not always be very ready. As usual, the main difficulty of 
the beginner is not to manipulate the formulae, but to be quite clear about 
his units. 

It will be seen at once that this case differs from the preceding in that 
we cannot now choose our unit as we please. Where, as here, there is only 
one variable (t), if we were, allowed to select our own unit, the inch, foot, or 
whatever it might be, we might get quite different results. Accordingly 
some comparatively natural unit must have been chosen for us in which we 
are bound to reckon, just as in the circular measurement of an angle as 
distinguished from that by degrees. 

The answer is that the unit here is the modulus, and that to put ' = l-5 ' 
is to say, 'suppose the error half as great again as the modulus'; the 
modulus itself being an error of a certain assignable magnitude depending 
upon the nature of the measurements or observations in question. We shall 

2 f** 
see this better if we put the integral in the form -p I e~^ xZ d(hx) ; which is 

\Mo 
precisely equivalent, since the value of a definite integral is independent of 

the particular variable employed. Here kx is the same as x : - ; i.e. it is 

the ratio of x to T , or x measured in terms of - . But = is the modulus in the 
h n h 



equation (y** -Tre-*'* 8 ) for the law of error. In other words the nu- 

\ *Jir / 

merical value of an error in this formula, is the number of times, whole or 
fractional, which it contains the modulus. 



CHAPTER XIX. 

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

1. IN the last chapter we were occupied with the Average 
mainly under its qualitative rather than its quantitative 
aspect. That is, we discussed its general nature, its principal 
varieties, and the main uses to which it could be put in 
ordinary life or in reasoning processes which did not claim to 
be very exact. It is now time to enter more minutely into 
the specific question of the employment of the average in 
the way peculiarly appropriate to Probability. That is, we 
must be supposed to have a certain number of measure- 
ments, in the widest sense of that term, placed before us, 
and to be prepared to answer such questions as ; Why do we 
take their average ? With what degree of confidence ? 
Must we in all cases take the average, and, if so, one always 
of the same kind ? 

Tffe subject upon which we are thus entering is one 
which, under its most general theoretic treatment, has per- 
haps given rise to more profound investigation, to a greater 
variety of opinion, and in consequence to a more extensive 
history and literature, than any other single problem within 
the range of mathematics 1 . But, in spite of this, the main 

1 Mr Mansfield Merriman pub- necticut Acad.) a list of 408 writings 
lished in 1877 (Trans, of the Con- on the subject of Least Squares. 

v. 30 



466 Theory of the Average. [CHAP. xix. 

logical principles underlying the methods and processes in 
question are not, I apprehend, particularly difficult to grasp : 
though, owing to the extremely technical style of treatment 
adopted even in comparatively elementary discussions of the 
subject, it is far from easy for those who have but a moderate 
command of mathematical resources to disentangle these 
principles from the symbols in which they are clothed. The 
present chapter contains an attempt to remove these difficul- 
ties, so far as a general comprehension of the subject is con- 
cerned. As the treatment thus adopted involves a con- 
siderable number of subdivisions, the reader will probably 
find it convenient to refer back occasionally to the table of 
contents at the commencement of this volume. 

2. The subject, in the form in which we shall discuss 
it, will be narrowed to the consideration of the average, on 
account of the comparative simplicity and very wide preva- 
lence of this aspect of the problem. The problem is however 
very commonly referred to, even in non-mathematical treatises, 
as the Rule or Method of Least Squares; the fact being 
that, in such cases as we shall be concerned with, the Rule 
of Least Squares resolves itself into the simpler and more 
familiar process of taking the arithmetical average. A very 
simple example, one given by Herschel, will explain the 
general nature of the task under a slightly wider treatment, 
and will serve to justify the familiar designation. 

Suppose that a man had been firing for some time with a 
pistol at a small mark, say a wafer on a wall. We may take 
it for granted that the shot-marks would tend to group 
themselves about the wafer as a centre, with a density vary- 
ing in some way inversely with the distance from the centre. 
But now suppose that the wafer which marked the centre 
was removed, so that we could see nothing but the surface of 
the wall spotted with the shot-marks; and that we were 



SECT. 2.] Theory of the Average. 467 

asked to guess the position of the wafer. Had there been 
only one shot, common sense would suggest our a-Mimiitg 
(of course very precariously) that this marked the real centre. 
Had there been two, common sense would suggest our taking 
the mid-point between them. But if three or more were 
involved, common sense would be at a loss. It would feel 
that some intermediate point ought to be selected, but 
would not see its way to a more precise determination, be- 
cause its familiar reliance, the arithmetical average, does 
not seem at hand here. The rule in question tells us how to 
proceed. It directs us to select that point which will render 
the sum of the squares of all the distances of the various 
shot-marks from it the least possible \ 

This is merely by way of illustration, and to justify the 
familiar designation of the rule. The sort of cases with 
which we shall be exclusively occupied are those compara- 
tively simple ones in which only linear magnitude, or some 
quality which can be adequately represented by linear mag- 
nitude, is the object under consideration. In respect of these 
the Rule of Least Squares reduces itself to the process of 
taking the average, in the most familiar sense of that term, 
viz. the arithmetical mean ; and a single Law of Error, or its 
graphical equivalent, a Curve of Facility, will suffice accu- 
rately to indicate the comparative* frequency of the different 
amounts of the one variable magnitude involved. 

1 In other words, we are to take when we are dealing with such cases 

the ' centre of gravity " of the shot- as occur in Mensuration, where we 

marks, regarding them all as of equal have to combine or reconcile three or 

weight. This is, in reality, the ' aver- more inconsistent equations, some 

Age ' of all the marks, as the elemen- such rule as that of Least Squares 

tary geometrical construction for becomes imperative. No taking of 

obtaining the centre of gravity of a an average will get us out of the 

system of points will show ; but it ia difficulty, 
not familiarly so regarded. Of course, 

302 



468 Theory of the Average. [CHAP. xix. 

3. We may conveniently here again call attention to a 
misconception or confusion which has been already noticed 
in a former chapter. It is that of confounding the Law of 
Error with the Method of Least Squares. These are things 
of an entirely distinct kind. The former is of the nature of 
a physical fact, and its production is one which in many 
cases is entirely beyond our control. The latter, or any 
simplified application of it, such as the arithmetical average, 
is no law whatever in the physical sense. It is rather a 
precept or rule for our guidance. The Law states, in any 
given case, how the errors tend to occur in respect of their 
magnitude and frequency. The Method directs us how to 
treat these errors when any number of them are presented 
to us. No doubt there is a relation between the two, as will 
be pointed out in the course of the following pages; but 
there is nothing really to prevent us from using the same 
method for different laws of error, or different methods for 
the same law. In so doing, the question of distinct right 
and wrong would seldom be involved, but rather one of more 
or less propriety. 

4. The reader must understand, as was implied in 
the illustration about the pistol shots, that the ultimate 
problem before us is an inverse one. That is, we are sup- 
posed to have a moderate number of * errors ' before us and 
we are to undertake to say whereabouts is the centre from 
which they diverge. This resembles the determination of a 
cause from the observation of an effect. But, as mostly 
happens in inverse problems, we must commence with the 
consideration of the direct problem. In other words, so far 
as concerns the case before us, we shall have to begin by 
supposing that the ultimate object of our aim, that is, the 
true centre of our curve of frequency, is already known to 
us : in which case all that remains to be done is to sjbudy the 



SECT. 5.] Theory of the Average. 469 

consequences of taking averages of the iimgnihi<l<-< which 
constitute the errors. 

5. We shall, for the present, confine our remarks to 
what must be regarded as the typical case where con- 
siderations of Probability are concerned ; viz. that in which 
the law of arrangement or development is of the Binomial 
kind. The nature of this law was explained in Chap. II., 
where it was shown that the frequency of the respective 
numbers of occurrences was regulated in accordance with 
the magnitude of the successive terms of the expansion of 
the binomial (1 + l) n . It was also pointed out that when n 
becomes very great, that is, when the number of influencing 
circumstances is very large, and their relative individual 
influence correspondingly small, the form assumed by a 
curve drawn through the summits of ordinates representing 
these successive terms of the binomial tends towards that 
assigned by the equation 



For all practical purposes therefore we may talk in- 
differently of the Binomial or Exponential law ; if only on 
the ground that the arrangement of the actual phenomena 
on one or other of these two schemes would soon become 
indistinguishable when the numbers involved are large. 
But there is another ground than this. Even when the 
phenomena themselves represent a continuous magnitude, 
our measurements of them, which are all with which we 
can deal, are discontinuous. Suppose we had before us the 
accurate heights of a million adult men. For all practical 
purposes these would represent the variations of a con- 
tinuous magnitude, for the differences between two suc- 
cessive magnitudes, especially near the mean, would be 
inappreciably small. But our tables will probably represent 



470 Theory of the Average. [CHAP. xix. 

them only to the nearest inch. We have so many assigned 
as 69 inches; so many as 70; and so on. The tabular 
statement in fact is of much the same character as if we 
were assigning the number of ' heads' in a toss of a handful 
of pence ; that is, as if we were dealing with discontinuous 
numbers on the binomial, rather than with a continuous 
magnitude on the exponential arrangement. 

6. Confining ourselves then, for the present, to this 
general head, of the binomial or exponential law, we must 
distinguish two separate cases in respect of the knowledge 
we may possess as to the generating circumstances of the 
variable magnitudes. 

(1) There is, first, the case in which the conditions of 
the problem are determinable a priori : that is, where we are 
able to say, prior to specific experience, how frequently each 
combination will occur in the long run. In this case the 
main or ultimate object for which we are supposing that the 
average is employed, i.e. that of discovering the true mean 
value, is superseded. We are able to say what the mean 
or central value in the long run will be ; and therefore there 
is no occasion to set about determining it, with some trouble 
and uncertainty, from a small number of observations. Still 
it is necessary to discuss this case carefully, because its 
assumption is a necessary link in the reasoning in other 
cases. 

This comparatively a priori knowledge may present itself 
in two different degrees as respects its completeness. In the 
first place it may, so far as the circumstances in question 
are concerned, be absolutely complete. Consider the results 
when a handful of ten pence is repeatedly tossed up. We 
know precisely what the mean value is here, viz. equal 
division of heads and tails : we know also the chance of six 
heads and four tails, and so on. That is, if we had to plot 



SECT. 7.] Theory of the Average. 471 

out a diagram showing the relative frequency of each com- 
bination, we could do so without appealing to experience. 
We could draw the appropriate binomial curve from the 
generating conditions given in the statement of the problem. 

But now consider the results of firing at a target con- 
sisting of a long and narrow strip, of which one point is 
marked as the centre of aim 1 . Here (assuming that there 
are no causes at work to produce permanent bias) we know 
that this centre will correspond to the mean value. And we 
know also, in a general way, that the dispersion on each side 
of this will follow a binomial law. But if we attempted to 
plot out the proportions, as in the preceding case, by erecting 
ordinates which should represent each degree of frequency 
as we receded further from the mean, we should find that 
we could not do so. Fresh data must be given or inferred. 
A good marksman and a bad marksman will both distribute 
their shot according to the same general law; but the 
rapidity with which the shots thin off as we recede from the 
centre will be different in the two cases. Another ' constant ' 
is demanded before the curve of frequency could be correctly 
traced out. 

7. (2) The second division, to be next considered, 
corresponds for all logical purposes to the first. It com- 
prises the cases in which though we have no a priori know- 
ledge as to the situation about which the values will tend to 
cluster in the long run, yet we have sufficient experience at 
hand to assign it with practical certainty. Consider for 
instance the tables of human stature. These are often very 
extensive, including tens or hundreds of thousands. In such 
cases the mean or central value is de terminable with just as 

. 1 The only reason for supposing ing errors in two dimensions, would 
this exceptional shape is to secure sim- yield slightly more complicated re- 
plicity. The ordinary target, allow- suits. 



472 Theory of the Average. [CHAP. xix. 

great certainty as by any & priori rule. That is, if we took 
another hundred thousand measurements from the same 
class of population, we should feel secure that the average 
would not be altered by any magnitude which our measuring 
instruments could practically appreciate. 

8. But the mere assignment of the mean or central 
value does not here, any more than in the preceding case, 
give us all that we want to know. It might so happen that 
the mean height of two populations was the same, but that 
the law of dispersion about that mean was very different : 
so that a man who in one series was an exceptional giant or 
dwarf should, in the other, be in no wise remarkable. 

To explain the process of thus determining the actual 
magnitude of the dispersion would demand too much mathe- 
matical detail ; but some indication may be given. What 
we have to do is to determine the constant h in the equation 1 

y-j=.e~ h * x \ In technical language, what we have to do is 
VTT 

to determine the modulus of this equation. The quantity ^ 

ft 

in the above expression is called the modulus. It measures 
the degree of contraction or dispersion about the mean 
indicated by this equation. When it is large the dispersion 
is considerable; that is the magnitudes are not closely 

1 When first referred to, the general as usual, by unity. In this form of 
form of this equation was given (v. p. expression h is a quantity of the 
29). The special form here assigned, order a;- 1 ; for hx is to be a numerical 

, . , h . , ... . , . . . quantity, standing as it does as an 
in which ~.= is substituted for A, is . , "L, , ? , . ., 

J T index. Ine modulus, being the reci- 

commonly employed in Probability, procal of this, is of the same order 

because the integral of ydx, between of quantities as the errors themselves. 

+ oo and - oo , becomes equal to unity. In fact if we multiply it by -4769. . . 

That is, the sum of all the mutually we have toe so-called ' probable 

exclusive possibilities is represented, error.' 



SECT. 9.] Theory of the Average. 473 

crowded up towards the centre, when it is small they are 
thus crowded up. The smaller the modulus in the curve 
representing the thickness with which the shot-marks 
clustered about the centre of the target, the better the 
marksman. 

9. There are several ways of determining the modulus. 
In the first of the cases discussed above, where our theo- 
retical knowledge is complete, we are able to calculate it & 
priori from our knowledge of the chances. We should 
naturally adopt this plan if we were tossing up a large 
handful of pence. 

The usual a posteriori plan, when we have the measure- 
ments of the magnitudes or observations before us, is this : 
Take the mean square of the errors, and double this; the re- 
sult gives the square of the modulus. Suppose, for instance, 
that we had the five magnitudes, 4, 5, 6, 7, 8. The mean of 
these is 6: the 'errors' are respectively 2, 1, 0, 1, 2. There- 
fore the 'modulus squared' is equal to ; i.e. the modulus is 

o 

J2. Had the magnitudes been 2, 4, 6, 8, 10; representing 
the same mean (6) as before, but displaying a greater disper- 
sion about it, the modulus would have been larger, viz. */8 
instead of >/2. 

Mr Galton's method is more of a jriMhliir,! 1 nature. It 
is described in a paper on Statistics by Intercomparison 
{Phil. Mag. 1875), and elsewhere. It may be indicated as 
follows. Suppose that we were dealing with a large number 
of measurements of human stature, and conceive that all the 
persons in question were marshalled in the order of their 
height. Select the average height, as marked by the central 
man of the row. Suppose him to be 69 inches. Then raise 
(or depress) the scale from this point until it stands at such 



474 Theory of the Average. [CHAP. xix. 

a height as just to include one half of the men above (or 
below) the mean. (In practice this would be found to re- 
quire about 171 inches: that is, one quarter of any large 
group of such men will fall between 69 and 70*71 inches.) 
Divide this number by '4769 and we have the modulus. In 
the case in question it would be equal to about 3*6 inches. 

Under the assumption with which we start, viz. that the 
law of error displays itself in the familiar binomial form, or 
in some form approximating to this, the three methods indi- 
cated above will coincide in their result. Where there is 
any doubt on this head, or where we do not feel able to cal- 
culate beforehand what will be the rate of dispersion, we 
must adopt the second plan of determining the modulus. 
This is the only universally applicable mode of calculation: 
in fact that it should yield the modulus is a truth of defini- 
tion; for in determining the error of mean square we are 
really doing nothing else than determining the modulus, as 
was pointed out in the last chapter. 

10. The position then which we have now reached is 
this. Taking it for granted that the Law of Error will fall 

into the symbolic form expressed by the equation y - e~ h * x * 9 

VTT 

we have rules at hand by which h may be determined. We 
therefore, for the purposes in question, know all about the 
curve of frequency: we can trace it out on paper: given 
one value, say the central one, we can determine any 
other value at any distance from this. That is, knowing 
how many men in a million, say, are 69 inches high, we can 
determine without direct observation how many will be 67, 
68, 70, 71, and so on. 

We can now adequately discuss the principal question of 
logical interest before us; viz. why do we take averages or 
means? What is the exact nature and amount of the ad- 



SECT. 10.] Theory of the Average. 475 

vantage gained by so doing? The advanced student would 
of course prefer to work out the answers to these questions 
by appealing at once to the Law of Error in its ultimate or 
exponential form. But I feel convinced that the best 
method for those who wish to gain a clear conception of the 
logical nature of the process involved, is to begin by treating 
it as a question of combinations such as we are familiar with 
in elementary algebra; in other words to take a finite number 
of errors and to see what comes of averaging these. We can 
then proceed to work out arithmetically the results of com- 
bining two or more of the errors together so as to get a new 
series, not contenting ourselves with the general character 
merely of the new law of error, but actually calculating what 
it is in the given case. For the sake of simplicity we will 
not take a series with a very large number of terms in it, but 
it will be well to have enough of them to secure that our law 
of error shall roughly approximate in its form to the standard 
or exponential law. 

For this purpose the law of error or divergence given by 
supposing our effort to be affected by ten causes, each of 
which produces an equal error, but which error is equally 
likely to be positive and negative (or, as it might perhaps 
be expressed, 'ten equal and indifferently additive and 
subtractive causes') will suffice. This is the lowest number 
formed according to the Binomial law, which will furnish to 
the eye a fair indication of the limiting or Exponential law 1 . 
The whole number of possible cases here is 2 10 or 1024; that 
is, this is the number required to exhibit not only all the 
cases which can occur (for there are but eleven really dis- 
tinct cases), but also the relative frequency with which each 
of these cases occurs in the long run. Of this total, 252 will 

1 See, for the explanation of this, and of the graphical method of illus- 
trating it, the note on p. 29. 



476 



Theory of the Average. 



[CHAP. xix. 



be situated at the mean, representing the 'true' result, 
or that given when five of the causes of disturbance just 
neutralize the other five. Again, 210 will be at what we 
will call one unit's distance from the mean, or that given by 
six causes combining against four ; and so on ; until at the 
extreme distance of five places from the mean we get but 
one result, since in only one case out of the 1024 will all the 
causes combine together in the same direction. The set of 
1024 efforts is therefore a fair representation of the distri- 
bution of an infinite number of such efforts. A graphical 
representation of the arrangement is given here. 





u c D EGP 



11. This representing a complete set of single ob- 
servations or efforts, what will be the number and arrange- 
ment in the corresponding set of combined or reduced 
observations, say of two together ? With regard to the 
number we must bear in mind that this is not a case of the 
combinations of things which cannot be repeated; for any 
given error, say the extreme one at F, can obviously be 
repeated twice running. Such a repetition would be a piece 
of very bad luck no doubt, but being possible it must have 
its place in the set. Now the possible number of ways of 
combining 1024 things two together, where the same thing 
may be repeated twice running, is 1024 x 1024 or 1048576. 



SECT. 12.] Theory of the Average. 477 

This then is the number in a complete cycle of the results 
taken two and two together. 

12. So much for their number ; now for their arrange- 
ment or distribution. What we have to ascertain is, firstly, 
how many times each possible pair of observations will present 
itself; and, secondly, where the new results, obtained from 
the combination of each pair, are to be placed. With regard 
to the first of these enquiries ; it will be readily seen that 
on one occasion we shall have F repeated twice ; on 20 occa- 
sions we shall have F combined with E (for F coming first 
we may have it followed by any one of the 10 at E, or any 
one of these may be followed by F) ; E can be repeated in 
10 x 10, or 100 ways, and so on. 

Now for the position of each of these reduced observations, 
the relative frequency of whose component elements has thus 
been pointed out. This is easy to determine, for when we 
take two errors there is (as was seen) scarcely any other 
mode of treatment than that of selecting the mid-point be- 
tween them ; this mid-point of course becoming identical 
with each of them when the two happen to coincide. It will 
be seen therefore that F will recur once on the new arrange- 
ment, viz. by its being repeated twice on the old one. G 9 
midway between E and F, will be given 20 times. E, on 
our new arrangement, can be got at in two ways, viz. by its 
being repeated twice (which will happen 100 times), and by 
its being obtained as the mid-point between D and F (which 
will happen 90 times). Hence E will occur 190 times alto- 
gether. 

The reader who chooses to take the trouble may work out 
the frequency of all possible occurrences in this way, and if 
the object were simply to illustrate the principle in accord- 
ance with which they occur, this might be the best way of 
proceeding. But as he may soon be able to observe, and as 



478 Theory of the Average. [CHAP. xix. 

the mathematician would at once be able to prove, the new 
' law of facility of error ' can be got at more quickly deduc- 
tively, viz. by taking the successive terms of the expansion of 
(1 + 1) 20 . They are given, below the line, in the figure on p. 476. 

13. There are two apparent obstacles to any direct 
comparison between the distribution of the old set of simple 
observations, and the new set of combined or reduced ones. 
In the first place, the number of the latter is much greater. 
This, however, is readily met by reducing them both to the 
same scale, that is by making the same total number of each. 
In the second place, half of the new positions have no repre- 
sentatives amongst the old, viz. those which occur midway 
between F and E, E and J9, and so on. This can be met by 
the usual plan of interpolation, viz. by filling in such gaps 
by estimating what would have been the number at the 
missing points, on the same scale, had they been occupied. 
Draw a curve through the vertices of the ordinates at 
A y J5, C, &c., and the lengths of the ordinates at the in- 
termediate points will very fairly represent the corresponding 
frequency of the errors of those magnitudes respectively. 
When the gaps are thus filled up, and the numbers thus 
reduced to the same scale, we have a perfectly fair basis of 
comparison. (See figure on next page.) 

Similarly we might proceed to group or ' reduce ' three 
observations, or any greater number. The number of possible 
groupings naturally becomes very much larger, being (1024) 8 
when they are taken three together. As soon as we get to 
three or more observations, we have (as already pointed out) 
a variety of possible modes of treatment or reduction, of 
which that of taking the arithmetical mean is but one. 

14. The following figure is intended to illustrate the 
nature of the advantage secured by thus taking the arith- 
metical mean of several observations. 



SECT. 15.] Theory of the Average. 479 

The curve ABCD represents the arrangement of a given 
number of ' errors ' supposed to be disposed according to the 
binomial law already mentioned, when the angles have been 
smoothed off by drawing a curve through them. A' CD' 
represents the similar arrangement of the same number 
when given not as simple errors, but as averages of pairs of 
errors. A"BD", again, represents the similar arrangement 
obtained as averages of errors taken three together. They 
are drawn as carefully to scale as the small size of the figure 
permits. 




15. A glance at the above figure will explain to the 
reader, better than any verbal description, the full signi- 
ficance of the statement that the result of combining two or 
more measurements or observations together and taking the 
average of them, instead of stopping short at the single 
elements, is to make large errors comparatively more scarce. 
The advantage is of the same general description as that of 
fishing in a lake where, of the same number of fish, there 
are more big and fewer little ones than in another water : of 



480 Theory of the Average. [CHAP. xix. 

dipping in a bag where of the same number of coins there 
are more sovereigns and fewer shillings; and so on. The 
extreme importance, however, of obtaining a perfectly clear 
conception of the subject may render it desirable to work 
this out a little more fully in detail. 

For one thing, then, it must be clearly understood that 
the result of a set of ' averages ' of errors is nothing else 
than another set of 'errors/ No device can make the at- 
tainment of the true result certain, to suppose the con- 
trary would be to misconceive the very foundations of Pro- 
bability, no device even can obviate the possibility of being 
actually worse off as the result of our labour. The average 
of two, three, or any larger number of single results, may 
give a worse result, i.e. one further from the ultimate average, 
than was given by the first observation we made. We must 
simply fall back upon the justification that big deviations 
are rendered scarcer in the long run. 

Again ; it may be pointed out that though, in the above 
investigation, we have spoken only of the arithmetical average 
as commonly understood and employed, the same general 
results would be obtained by resorting to almost any sym- 
metrical and regular mode of combining our observations or 
errors. The two main features of the regularity displayed 
by the Binomial Law of facility were (1) ultimate symmetry 
about the central or true result, and (2) increasing relative 
frequency as this centre was approached. A very little con- 
sideration will show that it is no peculiar prerogative of the 
arithmetical mean to retain the former of these and to in- 
crease the latter. In saying this, however, a distinction must 
be attended to for which it will be convenient to refer to a 
figure. 

16. Suppose that 0, in the line D'OD, was the point 
aimed at by any series of measurements ; or, what comes to 



SECT. 16.] 



Theory of the Average. 



481 



the same thing for our present purpose, was the ultimate 
average of all the measurements made. What we mean by 





C JO 



a symmetrical arrangement of the values in regard to 0, is 
that for every error OB, there shall be in the long run a pre- 
cisely corresponding opposite one OH ; so that when we erect 
the ordinate BQ, indicating the frequency with which B is 
yielded, we must erect an equal one B'Q'. Accordingly the 
two halves of the curve on each side of P, viz. PQ and PQ' 
are precisely alike. 

It then readily follows that the secondary curve, viz. 
that marking the law of frequency of the averages of two or 
more simple errors, will also be symmetrical. Consider any 
three points B, C, D: to these correspond another three 
B, C', D\ It is obvious therefore that any regular and sym- 
metrical mode of dealing with all the groups, of which BCD 
is a sample, will result in symmetrical arrangement about 
the centre 0. The ordinary familiar arithmetical average is 
but one out of many such modes. One way of describing it is 
by saying that the average of J5, 0, D, is assigned by choosing 
a point such that the sum of the squares of its distances from 
v 31 



482 Theory of the Average. [CHAP. xix. 

J5, 0, D, is a minimum. But we might have selected a point 
such that the cubes, or the fourth powers, or any higher 
powers should be a minimum. These would all yield curves 
resembling in a general way the dotted line in our figure. 
Of course there would be insuperable practical objections to 
any such courses as these ; for the labour of calculation 
would be enormous, and the results so far from being better 
would be worse than those afforded by the employment of 
the ordinary average. But so far as concerns the general 
principle of dealing with discordant and erroneous results, it 
must be remembered that the familiar average is but one 
out of innumerable possible resources, all of which would 
yield the same sort of help. 

17. Once more. We saw that a resort to the average 
had the effect of 'humping up' our curve more towards the 
centre, expressive of the fact that the errors of averages are 
of a better, i.e. smaller kind. But it must be noticed that 
exactly the same characteristics will follow, as a general rule, 
from any other such mode of dealing with the individual 
errors. No strict proof of this fact can be given here, but a 
reference to one of the familiar results of taking combina- 
tions of things will show whence this tendency arises. Ex- 
treme results, as yielded by an average of any kind, can only 
be got in one way, viz. by repetitions of extremes in the 
individuals from which the averages were obtained. But 
intermediate results can be got at in two ways, viz. either by 
intermediate individuals, or by combinations of individuals 
in opposite directions. In the case of the Binomial Law 
of Error this tendency to thicken towards the centre was 
already strongly predominant in the individual values before 
we took them in hand for our average ; but owing to this 
characteristic of combinations we may lay it down (broadly 
speaking) that any sort of average applied to any sort of law 



SECT. 18.] Theory of the Average. 483 

of distribution will give a result which bears the same 
general relation to the individual values that the dotted lines 
above bear to the black line 1 . 

18. This being so, the speculative advantages of one 
method of combining, or averaging, or reducing, our observa- 
tions, over another method, irrespective, that is, of the 
practical conveniences in carrying them out, will consist 
solely in the degree of rapidity with which it tends thus to 
cluster the result about the centre. We shall have to subject 
this merit to a somewhat further analysis, but for the present 
purpose it will suffice to say that if one kind of average gave 
the higher dotted line in the figure on p. 479 and another 
gave the lower dotted line, we should say that the former 
was the better one. The mlvamagr is of the same general 
kind as that which is furnished in algebraical calculation, by 
a series which converges rapidly towards the true value as 
compared with one which converges slowly. We can do 
the work sooner or later by the aid of either; but we get 
nearer the truth by the same amount of labour, or get as 
near by a less amount of labour, on one plan than on the 
other. 

As we are here considering the case in which the indi- 
vidual observations are supposed to be grouped in accordance 

1 Broadly speaking, we may say ingly rapid, that is, when the ex- 

that the above remarks hold good of treme errors are relatively very few, 

any law of frequency of error in it still holds good. But if we were 

which there are actual limits, how- to take as our law of facility such an 

ever wide, to the possible magnitude ,. IT ,, ,. , . 

. Txi v -x A equation as y- - , (as hinted by 

of an error. If there are no limits to 1 + x* 

the possible errors, this characteristic De Morgan and noted by Mr Edge- 

of an average to heap its results up worth : Camb. Phil. Trans, vol. x. 

towards the centre will depend upon p. 184, and vol. xiv. p. 160) it does 

circumstances. When, as in the ex- not hold good. The result of aver- 

ponential curve, the approximation aging is to diminish the tendency to 

to the base, as asymptote, is exceed- cluster towards the centre. 

312 



484 Theory of the Average. [CHAP. xix, 

with the Binomial Law, it will suffice to say that in this case 
there is no doubt that the arithmetical average is not only 
the simplest and easiest to deal with, but is also the best in 
the above sense of the term. And since this Binomial Law, 
or something approximating to it, is of very wide prevalence, 
a strong prima facie case is made out for the general employ- 
ment of the familiar average. 

19. The analysis of a few pages back carried the 
results of the averaging process as far as could be con- 
veniently done by the help of mere arithmetic. To go 
further we must appeal to higher mathematics, but the 
following indication of the sort of results obtained will 
suffice for our present purpose. After all, the successive 
steps, though demanding intricate reasoning for their proof, 
are nothing more than generalizations of processes which 
could be established by simple arithmetic 1 . Briefly, what we 
do is this : 

(1) We first extend the proof from the binomial form, 
with its finite number of elements, to the limiting or ex- 
ponential form. Instead of confining ourselves to a small 
number of discrete errors, we then recognize the possibility 
of any number of errors of any magnitude whatever. 

(2) In the next place, instead of confining ourselves to 
the consideration of an average of two or three only, 
already, as we have seen, a tedious piece of arithmetic, we 
calculate the result of an average of any number, n. The 
actual result is extremely simple. If the modulus of the 
single errors is c, that of the average of n of these will be 
c + Jn. 

(3) Finally we draw similar conclusions in reference to 
the sum vr diffwence of two averages of any numbers. Sup- 

1 The reader will find the proofs in Galloway on Probability, and in 
of these and other similar formulae Airy on Errors. 



SECT. 20.] Theory of the Average. 485 

pose, for instance, that m errors were first taken and aver- 
aged, and then n similarly taken and averaged. These 
averages will be nearly, but not quite, equal. Their sum or 
difference, these, of course, are indistinguishable in the end, 
since positive and negative errors are supposed to be equal 
and opposite, will itself be an ' error', every magnitude of 
which will have a certain assignable probability or facility of 
occurrence. What we do is to assign the modulus of these 
errors. The actual result again is simple. If c had been 
the modulus of the single errors, that of the sum or dif- 
ference of the averages of m and n of them will be 



CA/ - +-. 



20. So far, the problem under investigation has been 
of a direct kind. We have supposed that the ultimate mean 
value or central position has been given to us; either & 
priori (as in many games of chance), or from more immediate 
physical considerations (as in aiming at a mark), or from ex- 
tensive statistics (as in tables of human stature). In all 
such cases therefore the main desideratum is already taken 
for granted, and it may reasonably be asked what remains to 
be done. The answers are various. For one thing we may 
want to estimate the value of an average of many when com- 
pared with an average of a few. Suppose that one man has 
collected statistics including 1000 instances, and another has 
collected 4000 similar instances. Common sense can recog- 
nize that the latter are better than the former: but it has no 
idea how much better they are. Here, as elsewhere, quanti- 
tative precision is the privilege of science. The answer we 
Deceive from this quarter is that, in the long run, the 
modulus, and with this the probable error, the mean error, 
and the error of mean square, which all vary in proportion, 



486 Theory of the Average. .[CHAP. xix. 

diminishes inversely as the square root of the number of 
measurements or observations. (This follows from the second 
of the above formulae.) Accordingly the probable error of 
the more extensive statistics here is one half that of the less 
extensive. Take another instance. Observation shows that 
"the mean height of 2,315 criminals differs from the mean 
height of 8,585 members of the general adult population by 
about two inches" (v. Edgeworth, Methods of Statistics: 
Stat. Soc. Journ. 1885). As before, common sense would feel 
little doubt that such a difference was significant, but it 
could give no numerical estimate of the significance. Ap- 
pealing to science, we see that this is an illustration of the 
third of the above formulae. What we really want to know 
is the odds against the averages of two large batches differing 
by an assigned amount: in this case by an amount equalling 
twenty-five times the modulus of the variable quantity. 
The odds against this are many billions to one. 

21. The number of direct problems which will thus 
admit of solution is very great, but we must confine ourselves 
here to the main inverse problem to which the foregoing 
discussion is a preliminary. It is this. Given a few only of 
one of these groups of measurements or observations; what 
can we do with these, in the way of determining that mean 
about which they would ultimately be found to cluster? 
Given a large number of them, they would betray the posi- 
tion of their ultimate centre with constantly increasing 
certainty: but we are now supposing that there are only a 
few of them at hand, say half a dozen, and that we have no 
power at present to add to the number. 

In other words, expressing ourselves by the aid of 
graphical illustration, which is perhaps the best method 
for the novice and for the logical student, in the direct 
problem we merely have to draw the curve of frequency from 



SECT. 22.] Theory of the Average. 487 

a knowledge of its determining elements; viz. the position 
of the centre, and the numerical value of the modulus. In 
the inverse problem, on the other hand, we have three ele- 
ments at least, to determine. For not only must we, (1), as 
before, determine whereabouts the centre may be assumed to 
lie ; and (2), as before, determine the value of the modulus 
or degree of dispersion about this centre. This does not 
complete our knowledge. Since neither of these two ele- 
ments is assigned with certainty, we want what is always 
required in the Theory of Chances, viz. some estimate of their 
probable truth. That is, after making the best assignment 
we can as to the value of these elements, we want also to 
assign numerically the ' probable error' committed in such 
assignment. Nothing more than this can be attained in Pro- 
bability, but nothing less than this should be set before us. 

22. (1) As regards the first of these questions, the 
answer is very simple. Whether the number of measure- 
ments or observations be few or many, we must make the 
assumption that their average is the point we want; that is, 
that the average of the few will coincide with the ultimate 
average. This is the best, in fact the only assumption we 
can make. We should adopt this plan, of course, in the 
extreme case of there being only one value before us, by just 
taking that one; and our confidence increases slowly with 
the number of values before us. The only difference there- 
fore here between knowledge resting upon such data, and 
knowledge resting upon complete data, lies not in the result 
obtained but in the confidence with which we entertain it. 

23. (2) As regards the second question, viz. the deter- 
mination of the modulus or degree of dispersion about the 
mean, much the same may be said. That is, we adopt the 
same rule for the determination of the E.M.S. (error of mean 
square) by which the modulus is assigned, as we should 



488 Theory of the Average. [CHAP. xix. 

adopt if we possessed full information. Or rather we are 
confined to one of the rules given on p. 473, viz. the second, 
for by supposition we have neither the a priori knowledge 
which would be able to supply the first, nor a sufficient 
number of observations to justify the third. That is, we 
reckon the errors, measured from the average, and calculate 
their mean square: twice this is equal to the square of the 
modulus of the probable curve of facility 1 . 

24. (3) The third question demands for its solution 
somewhat advanced mathematics; but the results can be 
indicated without much difficulty. A popular way of stating 
our requirement would be to say that we want to know how 
likely it is that the mean of the few, which we have thus 
accepted, shall coincide with the true mean. But this would 
be to speak loosely, for the chances are of course indefinitely 
great against such precise coincidence. What we really do 
is to assign the ' probable error'; that is, to assign a limit 
which it is as likely as not that the discrepancy between 
the inferred mean and the true mean should exceed 2 . To 
take a numerical example: suppose we had made several 

1 The formula commonly used for logical propriety we should like to 

A , _, ., a . ,, . . 2? 3 , know the probable error committed in 

the E.M.S. in this case is - and , , ,. 

n - 1 both the assignments of the preceding 

not ^. The difference is trifling, ^ section8 ' But the P rofound m - 
n thematicians who have discussed this 

unless n be small; the justification question, and who alone are compe- 

has been offered for it that since the tent to treat it, have mostly written 

sum of the squares measured from with the practical wants of Astronomy 

the true centre is a minimum (that in view; and for this purpose it is 

centre being the ultimate arithmeti- sufficient to take account of the one 

cal mean) the aum of the squares mea- great desideratum, viz. the true values 

sured from the somewhat incorrectly sought. Accordingly the only rules 

assigned centre will be somewhat commonly given refer to the probable 

larger. error of the mean. 
8 It appears to me that in strict 



SECT. 25.] Theory of the Average. 489 

measurements of a wall with a tape, and that the average of 
these was 150 feet. The scrupulous surveyor would give us 
this result, with some such correction as this added, 'pro- 
bable error 3 inches'. All that this means is that we may 
assume that the true value is 150 feet, with a confidence that 
in half the cases (of this description) in which we did so, we 
should really be within three inches of the truth. 

The expression for this probable error is a simple multiple 
of the modulus: it is the modulus multiplied by *4769... 
That it should be some function of the modulus, or E.M.S., 
seems plausible enough; for the greater the errors, in other 
words the wider the observed discrepancy amongst our 
measurements, the less must be the confidence we can feel 
in the accuracy of our determination of the mean. But, of 
course, without mathematics we should be quite unable to 
attempt any numerical assignment. 

25. The general conclusion therefore is that the de- 
termination of the curve of facility, and therefore ultimately 
of every conclusion which rests upon a knowledge of this 
curve, where only a few observations are available, is of 
just the same kind as where an infinity are available. The 
rules for obtaining it are the same, but the confidence with 
which it can be accepted is less. 

The knowledge, therefore, obtainable by an average of a 
small number of measurements of any kind, hardly differs 
except in degree from that which would be attainable by an 
indefinitely 'extensive series of them. We know the same 
sort of facts, only we are less certain about them. But, on 
the other hand, the knowledge yielded by an average even 
of a small number differs in kind from that which is yielded 
by a single measurement. Revert to our marksman, whose 
bullseye is supposed to have been afterwards removed. If 
tie had fired only a single shot, not only should we be less 



490 Theory of the Average. [CHAP. xix. 

certain of the point he had aimed at, but we should have no 
means whatever of guessing at the quality of his shooting, or 
of inferring in consequence anything about the probable 
remoteness of the next shot from that which had gone before. 
But directly we have a plurality of shots before us, we not 
merely feel more confident as to whereabouts the centre of 
aim was, but we also gain some knowledge as to how the 
future shots will cluster about the spot thus indicated. The 
quality of his shooting begins at once to be betrayed by the 
results. 

26. Thus far we have been supposing the Law of 
Facility to be of the Binomial type. There are several 
reasons for discussing this at such comparative length. For 
one thing it is the only type which, or something approxi- 
mately resembling which, is actually prevalent over a wide 
range of phenomena. Then again, in spite of its apparent 
intricacy, it is really one of the simplest to deal with ; owing 
to the fact that every curve of facility derived from it by 
taking averages simply repeats the same type again. The 
curve of the average only differs from that of the single 
elements in having a smaller modulus ; and its modulus is 
smaller in a ratio which is exceedingly easy to give. If that 

of the one is c, that of the other (derived by averaging* 

f* 

n single elements) is T ^ . 
vn 

But for understanding the theory of averages we must 
consider other cases as well. Take then one which is intrin- 
sically as simple as it possibly can be, viz. that in which all 
values within certain assigned limits are equally probable. 
This is a case familiar enough in abstract Probability, though, 
as just remarked, not so common in natural phenomena. It 
is the state of things when we act at random directly upon 



SECT. 27.] Theory of the Average. 491 

the objects of choice 1 ; as when, for instance, we choose digits 
at random out of a table of lu^nri'lnn-. 

The reader who likes to do so can without much labour 
work out the result of taking an average of two or three 
results by proceeding in exactly the same way which we 
adopted on p. 476. The 'curve of facility* with which we 
have to start in this case has become of course simply a 
finite straight line. Treating the question as one of simple 
combinations, we may divide the line into a number of equal 
parts, by equidistant points ; and then proceed to take these 
two and two together in every possible way, as we did in the 
case discussed some pages back. 

If we did so, what we should find would be this. When 
an average of two is taken, the 'curve of facility' of the 
average becomes a triangle with the initial straight line for 
base ; so that the ultimate mean or central point becomes 
the likeliest result even with this commencement of the 
averaging process. If we were to take averages of three, 
four, and so on, what we should find would be that the 
Binomial law begins to display itself here. The familiar bell 
shape of the exponential curve would be more and more 
closely approximated to, until we obtained something quite 
indistinguishable from it. 

27. The conclusion therefore is that when we are 
dealing with averages involving a considerable number it is 
not necessary, in general, to presuppose the binomial law of 
distribution in our original data. The law of arrangement of 
what we may call the derived curve, viz. that corresponding 
to the averages, will not be appreciably affected thereby. 
Accordingly we seem to be justified in bringing to beaf all 

1 i. e. as distinguished from acting ter on Randomness, may result in 
upon them indirectly. This latter giving a non-uniform distribution, 
proceeding, as explained in the chap- 



492 Theory of the Average. [CHAP. xix. 

the same apparatus of calculation as in the former case. We 
take the initial average as the probable position of the true 
centre or ultimate average : we estimate the probability that 
we are within an assignable distance of the truth in so doing 
by calculating the 'error of mean square'; and we appeal 
to this same element to determine the modulus, i.e. the 
amount of contraction or dispersion, of our derived curve of 
facility. 

The same general considerations will apply to most other 
kinds of Law of Facility. Broadly speaking, we shall come 
to the examination of certain exceptions immediately, 
whatever may have been the primitive arrangement (i.e. 
that of the single results) the arrangement of the derived 
results (i.e. that of the averages) will be more crowded up 
towards the centre. This follows from the characteristic of 
combinations already noticed, viz. that extreme values can 
only be got at by a repetition of several extremes, whereas 
intermediate values can be got at either by repetition of 
intermediates or through the counteraction of opposite ex- 
tremes. Provided the original distribution be symmetrical 
about the centre, and provided the limits of possible error be 
finite, or if infinite, that the falling off of frequency as we 
recede from the mean be very rapid, then the results of 
taking averages will be better than those of trusting to 
single results. 

28. We will now take notice of an exceptional case. 
We shall do so, not because it is one which can often 
actually occur, but because the consideration of it will force 
us to ask ourselves with some minuteness what we mean in 
the above instances by calling the results of the averages 
* better ' than those of the individual values. A diagram will 
bring home to us the point of the difficulty better than any 
verbal or symbolic description. 



SECT. 29.] Theory of the Average. 493 

The black line represents a Law of Error easily stated in 
words, and one which, as we shall subsequently see, can be 




conceived as occurring in practice. It represents a state of 
things under which up to a certain distance from 0, on each 
side, viz. to A and jB, the probability of an error diminishes 
uniformly with the distance from 0; whilst beyond these 
points, up to E and F, the probability of error remains con- 
stant. The dotted line represents the resultant Law of Error 
obtained by taking the average of the former two and two 
together. Now is the latter 'better' than the former? 
Under it, certainly, great errors are less frequent and inter- 
mediate ones more frequent ; but then on the other hand 
the small errors are less frequent : is this state of things on 
the whole an improvement or not ? This requires us to re- 
consider the whole question. 

29. In all the cases discussed in the previous sections 
the superiority of the curve of averages over that of the 
single results showed itself at every point. The big errors 
were scarcer and the small errors were commoner; it was 
only just at one intermediate point that the two were on 
terms of equality, and this point was not supposed to possess 
any particular significance or importance. Accordingly we 
had no occasion to analyse the various cases included under 
the general relation. It was enough to say that one was 
better than the other, and it was sufficient for all purposes to 



494 Theory of the Average. [CHAP. xix. 

take the ' modulus ' as the measure of this superiority. In 
fact we are quite safe in simply saying that the average of 
those average results is better than that of the individual 
ones. 

When however we proceed in what Hume calls "the 
sifting humour/' and enquire why it is sufficient thus to 
trust to the average ; we find, in addition to the considera- 
tions hitherto advanced, that some postulate was required as 
to the consequences of the errors we incur. It involved an 
estimate of what is sometimes called the * detriment ' of an 
error. It seemed to take for granted that large and small 
errors all stand upon the same general footing of being mis- 
chievous in their consequences, but that their evil effects in- 
crease in a greater ratio than that of their own magnitude. 

30. Suppose, for comparison, a case in which the im- 
portance of an error is directly proportional to its magnitude 
(of course we suppose positive and negative errors to balance 
each other in the long run): it does not appear that any 
advantage would be gained by taking averages. Something 
of this sort may be considered to prevail in cases of mere 
purchase and sale. Suppose that any one had to buy a very 
large number of yards of cloth at a constant price per yard : 
that he had to do this, say, five times a day for many days in 
succession. And conceive that the measurement of the 
cloth was roughly estimated on each separate occasion, with 
resultant errors which are as likely to be in excess as in 
defect. Would it make the slightest difference to him 
whether he paid separately for each piece; or whether 
the five estimated lengths were added together, their average 
taken, and he were charged with this average price for each 
piece ? In the latter case the errors which will be made in 
the estimation of each piece will of course be less in the long 
run than they would be in the former : will this be of any 



SECT. 31.] Theory of the Average. 495 

consequence ? The answer surely is that it will not make 
the slightest difference to either party in the bargain. In 
the long run, since the same parties are concerned, it will not 
matter whether the intermediate errors have been small or 
large. 

Of course nothing of this sort can be regarded as the 
general rule. In almost every case in which we have to 
make measurements we shall find -that large errors are much 
more mischievous than small ones, that is, mischievous in a 
greater ratio than that of their mere magnitude. Even in 
purchase and sale, where different purchasers are concerned, 
this must be so, for the pleasure of him who is overserved 
will hardly equal the pain of him who is underserved. And 
in many cases of scientific measurement large errors may be 
simply fatal, in the sense that if there were no reasonable 
prospect of avoiding them we should not care to undertake 
the measurement at all. 

31. If we were only concerned with practical con- 
siderations we might stop at this point; but if we want to 
realize the full logical import of average-taking as a means 
to this particular end, viz. of estimating some assigned 
magnitude, we must look more closely into such an ex- 
ceptional case as that which was indicated in the figure on 
p. 493. What we there assumed was a state of things in 
reference to which extremely small errors were very fre- 
quent, but that when once we got beyond a certain small 
range all other errors, within considerable limits, were equally 
likely. 

It is not difficult to imagine an example which will aptly 
illustrate the case in point: at worst it may seem a little far- 
fetched. Conceive then that some firm in England received 
a hurried order to supply a portion of a machine, say a 
steam-engine, to customers at a distant place; and that it 



496 Theory of the Average. [CHAP. xix. 



was absolutely essential that the work should be true to 
tenth of an inch for it to be of any use. But conceive also 
that two specifications had been sent, resting on different 
measurements, in one of which the length of the requisite 
piece was described as sixty and in the other sixty-one 
inches. On the assumption of any ordinary law of error, 
whether of the binomial type or not, there can be no doubt 
that the firm would make the best of a very bad job by con- 
structing a piece of 60 inches and a half: i.e. they would 
have a better chance of being within the requisite tenth of 
an inch by so doing, than by taking either of the two specifi- 
cations at random and constructing it accurately to this. 
But if the law were of the kind indicated in our diagram 1 , 
then it seems equally certain that they would be less likely 
to be within the requisite narrow margin by so doing. As a 
mere question of probability, that is, if such estimates were 
acted upon again and again, there would be fewer failures 
encountered by simply choosing one of the conflicting 
measurements at random and working exactly to this, than 
by trusting to the average of the two. 

This suggests some further reflections as to the taking of 
averages. We will turn now to another exceptional case, 
but one involving somewhat different considerations than 
those which have been just discussed. As before, it may be 
most conveniently introduced by commencing with an ex- 
ample. 

1 There is no difficulty in conceiv- in the estimate at random (within 

ing circumstances under which a certain limits), the firm having a 

law very closely resembling this knowledge of this fact but being of 

would prevail. Suppose, e.g., that course unable to assign the two to 

one of the two measurements had their authors, we should get very 

been made by a careful and skilled much such a Law of Error as is sup- 

mechanic, and the other by a man posed above. 
who to save himself trouble had put 



SECT. 33.] Theory of the Average. 497 

32. Suppose then that two scouts were sent to take 
the calibre of a gun in a hostile fort, we may conceive that 
the fort was to be occupied next day, and used against the 
enemy, and that it was important to have a supply of shot or 
shell, and that the result is that one of them reports the 
calibre to be 8 inches and the other 9. Would it be wise to 
assume that the mean of these two, viz. 8| inches, was a 
likelier value than either separately? 

The answer seems to be this. If we have reason to 
suppose that the possible calibres partake of the nature of a 
continuous magnitude, i.e. that all values, with certain 
limits, are to be considered as admissible, (an assumption 
which we always make in our ordinary inverse step from an 
observation or magnitude to the thing observed or measured) 
then we should be justified in selecting the average as the 
likelier value. But if, on the other hand, we had reason to 
suppose that whole inches are always or generally preferred, 
as is in fact the case now with heavy guns, we should do 
better to take, even at hazard, one of the two estimates set 
before us, and trust this alone instead of taking an average 
of the two. 

33. The principle upon which we act here may be 
stated thus. Just as in the direct process of calculating or 
displaying the * errors', whether in an algebraic formula or in 
a diagram, we generally assume that their possibility is 
continuous, i.e. that all intermediate values are possible; so, 
in the inverse process of determining the probable position of 
the original from the known value of two or more errors, we 
assume that that position is capable of falling at any point 
whatever between certain limits. In such an example as the 
above, where we know or suspect a discontinuity of that 
possibility of position, the value of the average may be 
entirely destroyed. 

v. 32 



498 Theory of the Average. [CHAP. xix. 

In the above example we were supposed to know that 
the calibre of the guns was likely to run in English inches or 
in some other recognized units. But if the battery were in 
China or Japan, and we knew nothing of the standards of 
length in use there, we could no longer appeal to this 
principle. It is doubtless highly probable that those calibres 
are not of the nature of continuously varying magnitudes; 
but in an entire ignorance of the standards actually adopted, 
we are to all intents and purposes in the same position as if 
they were of that continuous nature. When this is so the 
objections to trusting to the average would no longer hold 
good, and if we had only one opportunity, or a very few 
opportunities, we should do best to adhere to the customary 
practice. 

34. When however we are able to collect and compare 
a large number of measurements of various objects, this 
consideration of the probable discontinuity of the objects we 
thus measure, that is, their tendency to assume some one or 
other of a finite number of distinct magnitudes, instead of 
showing an equal readiness to adapt themselves to all inter- 
mediate values, again assumes importance. In fact, given 
a sufficient number of measurable objects, we can actually 
deduce with much probability the standard according to 
which the things in question were made. 

This is the problem which Mr Flinders Petrie has at- 
tacked with so much acuteness and industry in his work on 
Inductive Metrology, a work which, merely on the ground of 
its speculative interest, may well be commended to the 
student of Probability, The main principles on which the 
reasoning is based are these two: (1) that all artificers are 
prone to construct their works m -curding to round numbers, 
or simple fractions, of their units of measurement; and (2) 
that, aiming to secure this, they will stray from it in toler- 



SECT. 35.] Theory of the Average. 499 

able accordance with the law of error. The result of these 
two assumptions is that if we collect a very large number of 
measurements of the different parts and proportions of some 
ancient building, say an Egyptian temple, whilst no as- 
signable length is likely to be permanently unrepresented, 
yet we find a marked tendency for the measurements to 
cluster about certain determinate points in our own, or any 
other standard scale of measurement. These points mark the 
length of the standard, or of some multiple or submultiple of 
the standard, employed by the old builders. It need hardly 
be said that there are a multitude of practical considerations 
to be taken into account before this method can be expected 
to give trustworthy results, but the leading principles upon 
which it rests are comparatively simple. 

35. The case just considered is really nothing else 
than the recurrence, under a different application, of one 
which occupied our attention at a very early stage. We 
noticed (Chap. II.) the possibility of a curve of facility which 
instead of having a single vertex like that corresponding to 
the common law of error, should display two humps or 
vertices. It can readily be shown that this problem of the 
measurements of ancient buildings, is nothing more than the 
reopening of the same question, in a slightly more complex 
form, in reference to the question of the functions of an 
average. 

Take a simple example. Suppose an instance in which 
great errors, of a certain approximate magnitude, are dis- 
tinctly more likely to be committed than small ones, so that 
the curve of facility, instead of rising into one peak towards 
the centre, as in that of the familiar law of error, shows a 
depression or valley there. Imagine, in fact, two binomial 
curves, with a short interval between their centres. Now if 
we were to calculate the result of taking averages here we 



500 Theory of the Average. [CHAP. xix, 

should find that this at once tends to fill up the valley; and 
if we went on long enough, that is, if we kept on taking 
averages of sufficiently large numbers, a peak would begin to 
arise in the centre. In fact the familiar single binomial 
curve would begin to make its appearance. 

36. The question then at once suggests itself, ought we 
to do this? Shall we give the average free play to perform 
its allotted function of thus crowding things up towards the 
centre? To answer this question we must introduce a dis- 
tinction. If that peculiar double-peaked curve had been, as 
it conceivably might, a true error-curve, that is, if it had 
represented the divergences actually made in aiming at the 
real centre, the result would be just what we should want. 
It would furnish an instance of the advantages to be gained 
by taking averages even in circumstances which were origin- 
ally unfavourable. It is not difficult to suggest an appro- 
priate illustration. Suppose a man firing at a mark from 
some sheltered spot, but such that the range crossed a broad 
exposed valley up or down which a strong wind was generally 
blowing. If the shotmarks were observed we should find 
them clustering about two centres to the right and left of 
the bullseye. And if the results were plotted out in a curve 
they would yield such a double-peaked curve as we have 
described. But if the winds were equally strong and preva- 
lent in opposite directions, we should find that the averaging 
process redressed the consequent disturbance. 

If however the curve represented, as it is decidedly more 
likely to do, some outcome of natural phenomena in which 
there was, so to say, a real double aim on the part of nature, 
it would be otherwise. Take, for instance, the results of 
measuring a large number of people who belonged to two 
very heterogeneous races. The curve of facility would here 
be of the kind indicated on p. 45, and if the numbers of the 



SECT. 37.] 



Theory of the Average. 



501 



two commingled races were equal it would display a pair of 
twin peaks. Again the question arises, 'ought* we to in- 
volve the whole range within the scope of a single average ? 
The answer is that the obligation depends upon the purpose 
we have in view. If we want to compare that heterogeneous 
race, as a whole, with some other, or with itself at some 
other time, we shall do well to average without analysis. 
All statistics of population, as we have already seen (v. p. 47), 
are forced to neglect a multitude of discriminating charac- 
teristics of the kind in question. But if our object were to 
interpret the causes of this abnormal error-curve we should 
do well to break up the statistics into corresponding parts, 
and subject these to analysis separately. 

Similarly with the measurements of the ancient buildings. 
In this case if all our various ' errors ' were thrown together 
into one group of statistics we should find that the resultant 
curve of facility displayed, not two peaks only, but a suc- 
cession of them ; and these of various magnitudes, corre- 
sponding to the frequency of occurrence of each particular 
measurement. We might take an average of the whole, but 
hardly any rational purpose could be subserved in so doing ; 
whereas each separate point of maximum frequency of oc- 
currence has something significant to teach us. 

37. One other peculiar case may be noticed in con- 
clusion. Suppose a distinctly asymmetrical, or lopsided curve 
of facility, such as this : 




502 Theory of the Average. [CHAP. xix. 

Laws of error, of which this is a graphical representation', 
are, I apprehend, far from uncommon. The curve in question, 
is, in fact, but a slight exaggeration of that of barometrical 
heights as referred to in the last chapter; when it was 
explained that in such cases the mean, the median, and the 
maximum ordinate would show a mutual divergence. The 
doubt here is not, as in the preceding instances, whether or 
not a single average should be taken, but rather what kind 
of average should be selected. As before, the answer must 
depend upon the special purpose we have in view. For all 
ordinary purposes of comparison between one time or place 
and another, any average will answer, and we should there- 
fore naturally take the arithmetical, as the most familiar, or 
the median, as the simplest. 

38. Cases might however arise under which other 
kinds of average could justify themselves, with a momentary 
notice of which we may now conclude. Suppose, for instance, 
that the question involved here were one of desirability of 
climate. The ordinary mean, depending as it does so largely 
upon the number and magnitude of extreme values, might 
very reasonably be considered a less appropriate test than 
that of judging simply by the relatively most frequent value : 
in- other words, by the maximum ordinate. And various 
other points of view can be suggested in respect of which 
this particular value would be the most suitable and sig- 
nificant. 

In the foregoing case, viz. that of the weather curve, 
there was no objective or 'true' value aimed at. But a 
curve closely resembling this would be representative of 
that particular class of estimates indicated by Mr Galton, 
and for which, as he has pointed out, the geometrical mean 
becomes the only appropriate one. In this case the curve of 
facility ends abruptly at : it resembles a much foreshortened 



SECT. 39.] Theory of the Average. 503 

modification of the common exponential form. Its charac- 
teristics have been discussed in the paper by Dr Macalister 
already referred to, but any attempt to examine its properties 
here would lead us into far too intricate details. 

39. The general conclusion from all this seems quite 
in accordance with the nature and functions of an average as 
pointed out in the last chapter. Every average, it was urged, 
is but a single representative intermediate value substituted 
for a plurality of actual values. It must accordingly let slip the 
bulk of the information involved in these latter. Occasionally, 
as in most ordinary measurements, the one thing which it 
represents is obviously the thing we are in want of; and 
then the only question can be, which mean will most accord 
with the ' true ' value we are seeking. But when, as may 
happen in most of the common applications of statistics, 
there is really no ' true value ' of an objective kind behind 
the phenomena, the problem may branch out in various 
directions. We may have a variety of purposes to work out, 
and these may demand some discrimination as regards the 
average most appropriate for them. Whenever therefore we 
have any doubt whether the familiar arithmetical average is 
suitable for the purpose in hand we must first decide pre- 
cisely what that purpose is. 



INDEX. 



Accidents, 342 
Airy, G. B., 447, 484 
Anticipations, tacit, 287 
Arbuthnott, 258 
Aristotle, 205, 307 
Average, arithmetical, 437 

geometrical, 439 

median, 442 

consequences of, 482 

necessary results of, 457 

uses of, 439, 489 

Babbage, 343 

Bags and balls, 180, 411 

Belief, correctness of, 125, 131, 178 

gradations of, 139 

growth of, 199 

language of, 143 

measurement of, 119, 125, 

146 

quantity of, 133 

test of, 140, 149, 294 

undue, 129 

vagueness of, 127 
Bentham, 319, 323 
Bernoulli, 91, 117, 389 
Bertillon, 435 

Births, male and female, 90, 258, 

263 
Boat race, Oxford and Cambridge, 

339 
Boole, 183 

V. 



Buckle, 237 

Buffon, 153, 205, 352, 389 
Burgersdyck, 311 
Butler, 209, 281, 333, 366 

Carlisle Tables, 169 
Casual, meaning of, 245 
Causation, need of, 237 

proof of, 244 
Centre of gravity, 467 
Certainty, in Law, 324 

reasonable, 327 

hypothetical, 210 
Chance and Causation, 244 

Creation, 258 

Design, 256 

Genius, 353 

neglect of small, 363 

selections, 338 
Chauvenet, 352 

Classification, numerical scheme of, 

48 

Coincidences, 245 

Combinations and Permutations, 87 
Communism, 375, 392 
Conceptualism, 275 
Conflict of chances, 418 
Consumptives, insurance of, 227 
Cournot, 245, 255, 338 
Crackanthorpe, 312, 320 
Craig, J., 192 
Crofton, M. W., 61, 101, 104 

33 



506 



Dante, 285 

Deflection, causes of, 07 

from aim, 38 

De Morgan, 83, 106, 119, 122, 135, 
177, 179, 197, 236, 247, 296, 308, 
350, 379, 382, 483 

De Bos trial, 255 

Digits, random, 111, 114 

Discontinuity, 116 

Distribution, random, 106 

Diagrams, 29, 45, 118, 443, 476, 481, 
493, 501 

Dialectic, 302, 320 

Donkin, 123, 188, 283 

Duration of life, 15, 441 

Diising, 259 

Ebbinghaus, 199 

Edgeworth, F. Y., 34, 119, 256, 339, 

393, 435, 483 
Ellis, L., 9 
Epidemics, 62 
Error, law of, 29 

asymmetrical, 34, 441, 

443 

binomial, 37, 457, 469, 

480 

geometrical, 34, 502 
' heterogeneous, 45 

production of, 36 
Error, mean, 446 

probable, 446, 472, 488 

of mean square, 447, 488 
Escapes, narrow, 341 
Expectation, moral, 388 
Experience and probability, 74 
Exponential curve, 29 
Extraordinary, sense of, 159, 423 

stories, 407, 421 

Fallacies in Logic and Probability, 367 
Fatalism, 243 



Fechner, 34, 389, 435, 441 
Fluctuation, 448 

unlimited, 73 

Forbes, J. D., 188,262 
Formal Logic, 123 
Formal and Material treatment, 86 
Free will, 240 

Galloway, 248, 448, 484 

Galton, F., 33, 50, 70, 318, 442, 451, 

473, 502 . 
Gambling and Insurance, 370 

disadvantage of, 384 

final results of, 385, 391 
Godfray, H., 99 

Grote, G., 307 
Guy, 6 

Hamilton, W., 266, 297 
Happiness, human, 382 
Heads and Tails, 77 
Heredity, 50, 357 
Herschel, 80, 466 
Houdin, 361 
Hume, 236, 419, 433 
Hypotheses, 268 

Immediate inferences, 121 
Independent events, 175, 246 
Induction and Probability, 194, 201, 
208, 233, 358 

difficulty of, 213 

pure, 200 
Inequality of wealth, 382 
Inference, rules of, 167 
Inoculation, 374 
Insurance, justification of, 149 

difficulties of, 221 

life, 151 

peculiar case, 224 

theory of, 372 

varieties of, 374 



Index. 



507 



Inverse probability, 179, 196, 249 
Irregularity, absolute and relative, 6 

Jacobs, J., 199 
Jackson, J. G., 253 
Jevons, 37, 83, 136, 198, 201, 209, 
247 

Kant, 310, 317 
Keckermann, 298, 316 
Kinds, natural, 55 
Krug, 324 

Lambert, 309 
Language of Chance, 159 
Laplace, 89, 120, 197, 237, 424 
Law, absence of, 101 . 

empirical, 160 

of causation, 206 
Least squares, 41, 467 
Leibnitz, 309, 320 
Letters, lost, 162, 368 

misdirected, 67, 287, 241 
Lexis, W., 263, 441 
Limit, conception of, 18, 109, 164 

of possible fluctuation, 32 
Lines, random, 113 

Lister's method, 187 
Lotteries, 128 
Lunn, J.-B., 248* 
Likely, equally, 77, 183 

McAlister, D., 34, 187, 502 
Hansel, H. L., 299, 301, 320 
Martingale, 343 

Material and Formal Logic, 265 
Maximum ordinate, 441, 455 
Measurement of Belief, 119 

Memory, 192 

Mental qualities, measurement of, 49 
Merriman, M., 352, 448, 460, 465 
Mill, J. S., 131, 207, 266, 282, 402 



Milton, chance production of, 353 
Miracles, 428 
Michell, J., 260 
Modality, 295 

divisions of, 307 

false, 297 

formal, 298 

in Law, 319 
Modulus, 464, 472, 484 
Monro, C. J., 325, 416 

Names, reference of, 270 
Nations, comparison of, 51 
Natural Kinds, 55, 63, 71 
Necessary and impossible matter, 310 

Objects and agencies, 53 
Occam, 314 

Paley, 433 

Penny, tosses of, 144 

Petrie, F., 498 

Petersburg Problem, 19, 154 

Poisson, 405 

Prantl, 311 

Presumption, legal, 329 

Prevost, 348 

Probability, definition of, 165 

relative, 290 

integral, 463 
Probable facts, 269 

value, 441 

error, 446, 472 
Problem, Three point, 104 
Proctor, B. A., 262, 378 
Prophecies, suicidal, 226 
Providence, 89, 431 
Propositions, proportional, 2 
Psychical research, 256 
Pyramid, the great, 251 

*, digits in, 111, 247 



508 



Index. 



Quartiles, 446 

Quetelet, 23, 30, 43, 91, 259, 330, 348, 
454. 

Bandomnesa, etymology of, 96 

in firing, 98 

proof of, 107 
Bare events, 349 
Bealism, 92 

Beason, sufficient, 82 
Besiduals, 460 
Boberts, C., 25 
Bod broken at random, 98 
thrown 103 
Bules, Inductive and Deductive, 176 

of Succession, 191 

conflict of, 222 

plurality of, 217 

Series, definite proportions in, 11 

fixed and variable, 16 

ideal, 95 

peculiar, 12 
Shanks, 248 
Skeat, W. W., 96 
Smiglecius, 306, 316 
Smyth, P., 251 
Socialism, 392 L 
Spiritualism, 365 

Stars, random arrangement of, 108, 

260 
Statistics, by Intercomparison, 473 

unconscious appeal to, 400 
Statistical Journal, 6 
Stature, human, 25, 471 

French and English, 44 
Stephen, J. F., 282, 323, 326 
Stewart, D., 209, 237 
Subjective and objective terms, 160 



Succession, jteng, 860 

Bule of, 190, 362 
Suffield, G.,248 
Suicides, 67, 237 
Surnames, extinction of, 387 
Surprise, emotion of, 157 
Syllogisms, pure and modal, 316 

Taylor, 329 
Testimony, single, 411 

combined, 426 

two kinds of, 409 

worthless, 416 
Thomson, W., 153, 314, 419 
Time, influence of, 191 

in Probability, 279 
Todhunter, 415 
Tontines, 380 
Triangle, random, 103 
Tucker, A., 127 

Types, existence of, 42, 60, 453 

fixed and fluctuating, 64, 93 

Ueberweg, 311 
Uncertainty in life, 370 
Uniformity, 240 
Units of calculation, 464 

Voluntary agency, 65, 68, 85 

Walford, 374 

Wallis, J., 312 

Watson, H. W., 387 

Whately, 297, 307 

Whist, 401 

Whitworth, W. A., 87, 183, 384 

Wilson, J. M., 104 

Witnesses, independent, 405 

Wolf, 309 

Woolhouse, 101 



OAMBBIDGE : PRINTED BY 0. J. CLAY, M.A. AND SONS, AT THE UNIVEKSITY PRESS.