Project Gutenberg’s A Treatise on Probability, by John Maynard Keynes This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A Treatise on Probability Author: John Maynard Keynes Release Date: May 31, 2010 [EBook #32625] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A TREATISE ON PROBABILITY *** Produced by Andrew D. Hwang, Ralph Janke, and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) transcriber’s note Minor typographical corrections and presentational changes have been made without comment. Typographical corrections and regularizations of hyphenation are documented in the L A T E Xsource file. PDF bookmarks are provided for navigation to individual sections. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the L A T E X source file for instructions. BY THE SAME AUTHOR INDIAN CURRENCY AND FINANCE 8vo. Pp. viii + 263. 1913. 7s. 6d. net. THE ECONOMIC CONSEQUENCES OF THE PEACE 8vo. Pp. vii + 279. 1919. 8s. 6d. net. A TREATISE ON PROBABILITY MACMILLAN AND CO., Limited LONDON · BOMBAY · CALCUTTA · MADRAS MELBOURNE THE MACMILLAN COMPANY NEW YORK · BOSTON · CHICAGO DALLAS · SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO A TREATISE ON PROBABILITY BY JOHN MAYNARD KEYNES FELLOW OF KING’S COLLEGE, CAMBRIDGE MACMILLAN AND CO., LIMITED ST. MARTIN’S STREET, LONDON  PREFACE The subject matter of this book was first broached in the brain of Leibniz, who, in the dissertation, written in his twenty-third year, on the mode of electing the kings of Poland, conceived of Probability as a branch of Logic. A few years before, “un problème,” in the words of Poisson, “proposé à un austère janséniste par un homme du monde, a été l’origine du calcul des probabilitiés.” In the intervening centuries the algebraical exercises, in which the Chevalier de la Méré interested Pascal, have so far predominated in the learned world over the profounder enquiries of the philosopher into those processes of human faculty which, by determining reasonable preference, guide our choice, that Probability is oftener reckoned with Mathematics than with Logic. There is much here, therefore, which is novel and, being novel, unsifted, inaccurate, or deficient. I propound my systematic conception of this subject for criticism and enlargement at the hand of others, doubtful whether I myself am likely to get much further, by waiting longer, with a work, which, beginning as a Fellowship Dissertation, and interrupted by the war, has already extended over many years. It may be perceived that I have been much influenced by W. E. Johnson, G. E. Moore, and Bertrand Russell, that is to say by Cambridge, which, with great debts to the writers of Continental Europe, yet continues in direct succession the English tradition of Locke and Berkeley and Hume, of Mill and Sidgwick, who, in spite of their divergences of doctrine, are united in a preference for what is matter of fact, and have conceived their subject as a branch rather of science than of the creative imagination, prose writers, hoping to be understood. J. M. KEYNES. King’s College, Cambridge, May 1, 1920. v CONTENTS PART I fundamental ideas CHAPTER I page The Meaning of Probability . . . . . . . . . . . . . . . . . . . . . . . . 2 CHAPTER II Probability in Relation to the Theory of Knowledge 9 CHAPTER III The Measurement of Probabilities . . . . . . . . . . . . . . . . . . 20 CHAPTER IV The Principle of Indifference. . . . . . . . . . . . . . . . . . . . . . 44 CHAPTER V Other Methods of Determining Probabilities . . . . . . . . . 71 vi CONTENTS vii CHAPTER VI page The Weight of Arguments . . . . . . . . . . . . . . . . . . . . . . . . 78 CHAPTER VII Historical Retrospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 CHAPTER VIII The Frequency Theory of Probability . . . . . . . . . . . . . 102 CHAPTER IX The Constructive Theory of Part I. Summarized . . . . 123 PART II fundamental theorems CHAPTER X Introductory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 CHAPTER XI The Theory of Groups, with special reference to Logical Consistence, Inference, and Logical Pri- ority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 CHAPTER XII The Definitions and Axioms of Inference and Proba- bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 A TREATISE ON PROBABILITY viii CHAPTER XIII page The Fundamental Theorems of Necessary Inference . 153 CHAPTER XIV The Fundamental Theorems of Probable Inference . . 161 CHAPTER XV Numerical Measurement and Approximation of Prob- abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 CHAPTER XVI Observations on the Theorems of Chapter XIV. and their Developments, including Testimony . . . . . . . 188 CHAPTER XVII Some Problems in Inverse Probability, including Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 PART III induction and analogy CHAPTER XVIII Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 CHAPTER XIX The Nature of Argument by Analogy . . . . . . . . . . . . . . 256 [...]... the conclusions which we are investigating and our new assumptions; but the old relations between the conclusions and the former assumptions still exist and are just as real as these new ones It would be as absurd to deny that an opinion was probable, when at a later stage certain objections have come to light, as to deny, when we have reached our destination, that it was ever three miles distant; and... say “b is probable” as it would be to say “b is equal,” or “b is greater than,” and as unwarranted to conclude that, because a makes b probable, therefore a and c together make b probable, as to argue that because a is less than b, therefore a and c together are less than b Thus, when in ordinary speech we name some opinion as probable without further qualification, the phrase is generally elliptical... etc., can be conceived, though not perhaps measured by us.”—“Theory of Probabilities,” Encyclopaedia Metropolitana, p 395 He is a little more guarded in his Formal Logic, pp 174, 175; but arrives at the same conclusion so far as probability is concerned pt i A TREATISE ON PROBABILITY 22 will be demonstrated at a later stage, where algebraical representation is possible Probability has become associated,... knowledge, that is to say, which is obtained in a manner strictly direct by contemplation of the objects of acquaintance and without any admixture whatever of argument and the contemplation of the logical bearing of any other knowledge on this, corresponds to certain rational belief and not to a merely probable degree of rational belief It is true that there do seem to be degrees of knowledge and rational belief,... arbitrary He may be almost certain, that is to say, that there will not be new taxes on more than one of the articles tea, sugar, and whisky; there may be an opinion abroad, reasonable or unreasonable, that the likelihood is in the order—whisky, tea, sugar; and he may, therefore be able to effect insurances for equal amounts in each at 30 per cent, 40 per cent, and 45 per cent He has thus made sure of a. .. distinction between that part of our belief which is rational and that part which is not If a man believes something for a reason which is preposterous or for no reason at all, and what he believes turns out to be true for some reason not known to him, he cannot be said to believe it rationally, although he believes it and it is in fact true On the other hand, a man may rationally believe a proposition to... a probability-relation, “secondary propositions.” 1 4 Thus knowledge of a proposition always corresponds to certainty of rational belief in it and at the same time to actual truth in the proposition itself We cannot know a proposition unless it is in fact true A probable degree of rational belief in a proposition, on the other hand, arises out of knowledge of some corresponding secondary proposition... secondary as well as primary propositions, but it is a necessary condition that all the propositions h should be known In order that we may have rational belief in p of a lower degree of probability than certainty, it is necessary that we know a set of propositions h, and also know some secondary proposition q asserting a probability-relation between p and h In the above account one possibility has been... propositions, therefore, there exists pt i A TREATISE ON PROBABILITY 6 a relation, in virtue of which, if we know the first, we can attach to the latter some degree of rational belief This relation is the subject-matter of the logic of probability A great deal of confusion and error has arisen out of a failure to take due account of this relational aspect of probability From the premisses a implies b” and... proposition A man may rationally believe a proposition to be probable when it is in fact false, if the secondary proposition on which he depends is true and certain; while a man cannot rationally believe a proposition to be probable even when it is in fact true, if the secondary proposition on which he depends is not true Thus rational belief of whatever degree can only arise out of knowledge, although . PROBABILITY MACMILLAN AND CO., Limited LONDON · BOMBAY · CALCUTTA · MADRAS MELBOURNE THE MACMILLAN COMPANY NEW YORK · BOSTON · CHICAGO DALLAS · SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO A TREATISE ON. Project Gutenberg’s A Treatise on Probability, by John Maynard Keynes This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy. typographical corrections and presentational changes have been made without comment. Typographical corrections and regularizations of hyphenation are documented in the L A T E Xsource file. PDF