# The Logic of Chance, 3rd edition An Essay on the Foundations and Province of the Theory of Probability, With Especial Reference to Its Logical Bearings and Its Application to Moral and Social Science and to Statistics

THE

LOGIC OF CHANCE

AN ESSAY

ON THE FOUNDATIONS AND PROVINCE OF

THE THEORY OF PROBABILITY,

WITH ESPECIAL REFERENCE TO ITS LOGICAL BEARINGS

AND ITS APPLICATION TO

MORAL AND SOCIAL SCIENCE AND TO STATISTICS,

BY

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

FELLOW AND LECTURER IN THE MORAL SCIENCES, GONVILLE AND CAIUS COLLEGE,

CAMBRIDGE.

LATE EXAMINER IN LOGIC AND MORAL PHILOSOPHY IN THE

UNIVERSITY OF LONDON.

“So careful of the type she seems

So careless of the single life.”

*THIRD EDITION, RE-WRITTEN AND ENLARGED.*

London:

MACMILLAN AND CO.

AND NEW YORK

1888

[*All Rights reserved.*]

*First Edition printed* 1866.

*Second Edition* 1876.

*Third Edition* 1888.

## PREFACE TO FIRST EDITION.

Any work on Probability by a Cambridge man will be solikely to have its scope and its general treatment of thesubject prejudged, that it may be well to state at the outsetthat the following Essay is in no sense mathematical. Notonly, to quote a common but often delusive assurance, will‘no knowledge of mathematics beyond the simple rules ofArithmetic’ be required to understand these pages, but it isnot intended that any such knowledge should be acquired bythe process of reading them. Of the two or three occasionson which algebraical formulæ occur they will not be found toform any essential part of the text.

The science of Probability occupies at present a somewhatanomalous position. It is impossible, I think, not toobserve in it some of the marks and consequent disadvantagesof a *sectional* study. By a small body of ardent students ithas been cultivated with great assiduity, and the results theyhave obtained will always be reckoned among the most extraordinaryproducts of mathematical genius. But by thegeneral body of thinking men its principles seem to beregarded with indifference or suspicion. Such persons mayadmire the ingenuity displayed, and be struck with the profundityof many of the calculations, but there seems tovithem, if I may so express it, an *unreality* about the wholetreatment of the subject. To many persons the mention ofProbability suggests little else than the notion of a set ofrules, very ingenious and profound rules no doubt, with whichmathematicians amuse themselves by setting and solvingpuzzles.

It must be admitted that some ground has been givenfor such an opinion. The examples commonly selected bywriters on the subject, though very well adapted to illustrateits rules, are for the most part of a special and peculiarcharacter, such as those relating to dice and cards. Whenthey have searched for illustrations drawn from the practicalbusiness of life, they have very generally, but unfortunately,hit upon just the sort of instances which, as I shall endeavourto show hereafter, are among the very worst that couldbe chosen for the purpose. It is scarcely possible for anyunprejudiced person to read what has been written about thecredibility of witnesses by eminent writers, without his experiencingan invincible distrust of the principles which theyadopt. To say that the rules of evidence sometimes givenby such writers are broken in practice, would scarcely becorrect; for the rules are of such a kind as generally to defyany attempt to appeal to them in practice.

This supposed want of harmony between Probability andother branches of Philosophy is perfectly erroneous. Itarises from the belief that Probability is a branch of mathematicstrying to intrude itself on to ground which does notaltogether belong to it. I shall endeavour to show that thisbelief is unfounded. To answer correctly the sort of questionsto which the science introduces us does generally demandsome knowledge of mathematics, often a great knowledge,but the discussion of the fundamental principles on whichthe rules are based does not necessarily require any suchviiqualification. Questions might arise in other sciences, inGeology, for example, which could only be answered by theaid of arithmetical calculations. In such a case any onewould admit that the arithmetic was extraneous and accidental.However many questions of this kind there mightbe here, those persons who do not care to work out specialresults for themselves might still have an accurate knowledgeof the principles of the science, and even considerableacquaintance with the details of it. The same holds true inProbability; its connection with mathematics, though certainlyfar closer than that of most other sciences, is still ofmuch the same kind. It is principally when we wish towork out results for ourselves that mathematical knowledgeis required; without such knowledge the student may stillhave a firm grasp of the principles and even see his way tomany of the derivative results.

The opinion that Probability, instead of being a branch ofthe general science of evidence which happens to make muchuse of mathematics, *is* a portion of mathematics, erroneous asit is, has yet been very disadvantageous to the science inseveral ways. Students of Philosophy in general have thenceconceived a prejudice against Probability, which has for themost part deterred them from examining it. As soon as asubject comes to be considered ‘mathematical’ its claimsseem generally, by the mass of readers, to be either on theone hand scouted or at least courteously rejected, or on theother to be blindly accepted with all their assumed consequences.Of impartial and liberal criticism it obtains littleor nothing.

The consequences of this state of things have been, Ithink, disastrous to the students themselves of Probability.No science can safely be abandoned entirely to its own devotees.Its details of course can only be studied by those whoviiimake it their special occupation, but its general principlesare sure to be cramped if it is not exposed occasionally tothe free criticism of those whose main culture has been ofa more general character. Probability has been very muchabandoned to mathematicians, who as mathematicians havegenerally been unwilling to treat it thoroughly. They haveworked out its results, it is true, with wonderful acuteness,and the greatest ingenuity has been shown in solving variousproblems that arose, and deducing subordinate rules. Andthis was all that they could in fairness be expected to do.Any subject which has been discussed by such men asLaplace and Poisson, and on which they have exhausted alltheir powers of analysis, could not fail to be profoundlytreated, so far as it fell within their province. But from thisprovince the real principles of the science have generallybeen excluded, or so meagrely discussed that they had betterhave been omitted altogether. Treating the subject as mathematicianssuch writers have naturally taken it up at thepoint where their mathematics would best come into play,and that of course has not been at the foundations. In theworks of most writers upon the subject we should search invain for anything like a critical discussion of the fundamentalprinciples upon which its rules rest, the class ofenquiries to which it is most properly applicable, or therelation it bears to Logic and the general rules of inductiveevidence.

This want of precision as to ultimate principles is perfectlycompatible here, as it is in the departments of Moralsand Politics, with a general agreement on processes andresults. But it is, to say the least, unphilosophical, anddenotes a state of things in which positive error is alwaysliable to arise whenever the process of controversy forces usto appeal to the foundations of the science.

ixWith regard to the remarks in the last few paragraphs,prominent exceptions must be made in the case of two recentworks at least.[1]The first of these is Professor de Morgan's*Formal Logic*. He has there given an investigation into thefoundations of Probability as conceived by him, and nothingcan be more complete and precise than his statement ofprinciples, and his deductions from them. If I could at allagree with these principles there would have been no necessityfor the following essay, as I could not hope to addanything to their foundation, and should be far indeed fromrivalling his lucid statement of them. But in his schemeProbability is regarded very much from the Conceptualistpoint of view; as stated in the preface, he considers thatProbability is concerned with formal inferences in which thepremises are entertained with a conviction short of absolutecertainty. With this view I cannot agree. As I have enteredinto criticism of some points of his scheme in one of thefollowing chapters, and shall have occasion frequently to referto his work, I need say no more about it here. The otherwork to which I refer is the profound *Laws of Thought* ofthe late Professor Boole, to which somewhat similar remarksmay in part be applied. Owing however to his peculiartreatment of the subject, I have scarcely anywhere comeinto contact with any of his expressed opinions.

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

Almost the only writer who seems to me to have expresseda just view of the nature and foundation of the rulesof Probability is Mr Mill, in his *System of Logic*.[2]Histreatment of the subject is however very brief, and a considerableportion of the space which he has devoted to it isoccupied by the discussion of one or two special examples.There are moreover some errors, as it seems to me, in whathe has written, which will be referred to in some of thefollowing chapters.

The reference to the work just mentioned will serve toconvey a general idea of the view of Probability adopted inthis Essay. With what may be called the Material view ofLogic as opposed to the Formal or Conceptualist,—with thatwhich regards it as taking cognisance of laws of things andnot of the laws of our own minds in thinking about things,—Iam in entire accordance. Of the province of Logic, regardedfrom this point of view, and under its widest aspect, Probabilitymay, in my opinion, be considered to be a portion. Theprincipal objects of this Essay are to ascertain how great aportion it comprises, where we are to draw the boundary betweenit and the contiguous branches of the general sciencexiof evidence, what are the ultimate foundations upon which itsrules rest, what the nature of the evidence they are capableof affording, and to what class of subjects they may most fitlybe applied. That the science of Probability, on this view ofit, contains something more important than the results of asystem of mathematical assumptions, is obvious. I am convincedmoreover that it can and ought to be rendered bothinteresting and intelligible to ordinary readers who have anytaste for philosophy. In other words, if the large and growingbody of readers who can find pleasure in the study ofbooks like Mill's *Logic* and Whewell's *Inductive Sciences*,turn with aversion from a work on Probability, the cause inthe latter case must lie either in the view of the subject orin the manner and style of the book.

I take this opportunity of thanking several friends,amongst whom I must especially mention Mr Todhunter, ofSt John's College, and Mr H. Sidgwick, of Trinity College,for the trouble they have kindly taken in looking over theproof-sheets, whilst this work was passing through the Press.To the former in particular my thanks are due for thusadding to the obligations which I, as an old