Table Of ContentTH E THEORY OF
PROBABILITY
An Inquiry into the Logical and Mathematical
Foundations of the Calculus of Probability
By HANS REICHENBACH
PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF CALIFORNIA AT LOS ANGELES
ENGLISH TRANSLATION BY
ERNEST H. H U TTEN
AND
M ARIA REICHENBACH
Second Edition
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY AND LOS ANGELES • 1949
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY AND LOS ANGELES
CALIFORNIA
❖
CAMBRIDGE UNIVERSITY PRESS
LONDON, ENGLAND
COPYRIGHT, 1949, BY
THE REGENTS OF THE UNIVERSITY OF CALIFORNIA
PRINTED IN THE UNITED STATES OF AMERICA
BY MACKENZIE & HARRIS, INC., SAN FRANCISCO, CALIF.
PREFACE TO THE GERMAN EDITION
Les mathématiciens ont autant besoin
d'etre philosophes que les philosophes
d'etre mathématiciens.
G. W . Leibniz
(Lettre & Malebranche, mars, 1699)
The philosophical problems of the concept of probability have time
and again occupied the minds of philosophers and mathematicians.
Recently they have been brought to the fore with even greater
emphasis: first, because of the prominence of the concept of probability in
modern physics, where it has gradually replaced the concept of causality;
second, because of the development of the modern philosophy of nature,
which has analyzed the concept of probability for its own sake. With the
incorporation of the results of symbolic logic, the new philosophy of nature
has developed, in the meantime, from a critical investigation of the thinking
of mathematical physics to a scientific theory of knowledge. It has now
reached the stage at which it begins to replace the era of metaphysical con
structions in philosophy by the establishment of a philosophical science.
Abundant results have already been reached in the investigation of the space-
time problem, in the logical criticism of mathematics, in the analysis of the
problem of causality, and in the general criticism of scientific concepts. The
problem of probability, however, has resisted with peculiar persistence all
attempts at solution. Yet it has beconpe obvious, as never before, that this
problem, because of its intimate relation to the problem of induction, contains
the nucleus of every theory of knowledge.
The philosophical theory of the probability problem, more than any other,
is based upon mathematical analysis. Previous attempts to solve the problem
were bound to break down because the mathematical basis of the calculus of
probability had not yet been developed in the rigorous form on which philo
sophical criticism depends. Philosophical analysis was the starting point for
a new mathematical construction of the calculus of probability. Further
inquiry showed that the new mathematical construction made possible the
long-sought transition from two-valued logic to probability logic, that is,
to a logic with a continuous scale of truth values. Thus a final theory of
probability that satisfies both mathematical and logical requirements can
now be presented.
In a period in which the value of speculative philosophy is seriously ques
tioned, it will not be regarded as surprising that a philosophical theory expects
new insights from the construction and elucidation of mathematical theories.
l v ]
vi PREFACE TO THE GERMAN EDITION
I hope to demonstrate that, conversely, the mathematical theory can be
furthered by philosophical points of view. The unification of the calculus of
probability with the calculus of logic that determines the form of this expo
sition and, furthermore, the inclusion of all problems of the theory of prob
ability in a general calculus of probability, as well as the theory of the order
of probability sequences developed in this connection, seem to be results that
may be of value to the mathematician also. He may be interested likewise
in the continuous generalization of the concept of probability sequence,
leading from the familiar geometrical probabilities to continuous probability
sequences. The complete elaboration of the mathematical structure, which
is necessary for the philosophical aim, makes this book at the same time a
mathematical textbook on the calculus of probability. I hope that clarifica
tion of fundamental concepts will make the presentation particularly ade
quate for this purpose. Nothing makes the penetration into a new mathe
matical field more difficult than the improper treatment of fundamental
concepts. But the so-called mathematical difficulties vanish when the mean
ings of the concepts whose relations make up the mathematical calculus are
clarified.
I was fortunate to have had the opportunity, since 1927, of repeatedly
presenting the ideas of this book to my students at the University of Berlin.
Many details and examples were developed in class discussion, and the
congenial atmosphere greatly helped me in working out solutions.
Only a few of the many names that I should like to mention may be in
cluded. I owe valuable help to C. Hempel, A. Becker, M. Strauss, 0. Helmer,
and my old friends K. Grelling and W. Dubislav. V. Bargmann assisted in
the mathematical part and elaborated numerous details and proofs. I thank
my friend R. Carnap of Prague for reading the introduction to symbolic
logic. E. Kokott assisted in drawing the figures. C. Hempel and V. Bargmann
also aided in the correction of proofs. ^
Hans Reichenbach
Istanbul,
August, 1934
PREFACE TO THE ENGLISH EDITION
IT is now more than fifteen years since this book was written in the
German version. During this time the theory of probability presented
in it has been much discussed, and the attacks made upon it have given
me the opportunity to think over its content again and again. I am grateful
to my opponents because their criticism has compelled me to check every
item of my theory. Many details have thus been improved in this presenta
tion. In particular, the exposition of the problems of application and induction
has been rearranged and supplemented by many additions. The English
version is therefore designated as the second edition of the book.
But, on the whole, I found that I had nothing to change in my views. The
theory has stood up to the test of a critical discussion in all its fundamentals.
The generalization of the customary theory, which identifies probability se
quences with random sequences, into a comprehensive system that embraces
sequences of all types of order, including continuous probability sequences,
has turned out to be an adequate integration of the pursuit of the mathe
matician. The theory of the application of probability concepts to physical
reality and the analysis of the problem of induction have shown that an empir
icist solution of these problems can be given. It seems to me that the various
attacks launched against this part of my theory have helped very much to
make its significance clear.
The attacks came mainly from two sides, which I should like to distinguish
as the rationalists and the unshakable Humeans. Both agree so far as they
regard my justification of induction as unsatisfactory, but they do so for
different reasons. The rationalists find this justification too weak because
they wish to find a stronger one, which in some way or other bases induction
and probability on a rational belief. The unshakable Humeans also maintain
that I did not justify induction; but they do so because they believe, like
Hume, that any such justification is impossible and that Hume's position
represents the final stage in the analysis of the problem. The more I read of
these arguments, the more convinced I became that my conception is correct;
and I should like to explain in what points I see the merits of this conception.
The first achievement, I think, consists in the demonstration that the
frequency interpretation of probability can be carried through for all uses
of the term “probable”. The difficulties that others had found in interpreting
the probabilities of single events as limits of frequencies disappear when
statements about single cases are regarded, not as assertions, but as posits.
When, in the face of these results, there remain philosophers who maintain
that there is a second concept of probability, which is not reducible to fre-
[viil
viii PREFACE TO THE ENGLISH EDITION
quency notions, I must answer that there is no need for such a concept.
Whoever wishes to reserve the right of using private meanings of a non-
verifiable pattern, or of a structure useless for predictions, may do so. What
has been shown, however, is that the frequency interpretation of probability
leads to a meaning of the term that makes the usage of language conform to
human behavior. I do not think there are other grounds that a philosophical
theory could adduce for its claim to provide an adequate interpretation of
terms.
The second achievement I find in the fact that, when the frequency inter
pretation is used, all nondeductive methods of the calculus of probability
reduce to one kind of inference: the inference of induction by enumeration.
Since all inductive methods of science, including the theory of indirect evi
dence and the formation of scientific theories, are interpretable in terms of
inferences supplied by the calculus of probability, this result establishes the
thesis that all forms of inductive inference are reducible to one form, to the
inference of induction by enumeration. This thesis was implicitly contained
in some older forms of a theory of induction, in particular, in Hume’s theory.
But so far it had not been given a proof.
As the third achievement, I should like to mention the fact that the theory
of probability has been freed from all forms of a rational belief in synthetic
statements, whether they appear in the form of a synthetic a priori, or an
animal faith, or a belief in a uniformity of the world, or a principle of insuffi
cient reason. All such notions are remnants of a philosophy of rationalism,
which holds that human reason has access to knowledge of the physical world
by other means than sense observation. What has been shown is that a theory
of probability can be built up without the use of such notions. There is no
such thing as inductive self-evidence. It is unfortunate that this discovery
of David Hume has so often been forgotten, and that so many attempts
have been made to reintroduce inductive self-evidence in new forms whenever
some older forms had been proved to be untenable.
The fourth achievement of the theory I see in its successful justification
of the inference of induction by enumeration, and therefore of all forms of
inductive inference, in spite of its renunciation of arguments based on syn
thetic self-evidence. The inductive inference is regarded as permissible, not
because it appears self-evident, but because it represents an instrument of
prediction so devised that it must lead to success if success is attainable.
This conception of a justification has been attacked by various means.
Although it is now generally admitted that it would be asking too much to
require a proof that the inductive inference must lead to true conclusions,
other requirements for a justification were advanced with the intention of
ruling out my form of a justification. For instance, a justification of a method
has been defined as a proof that there exists some inductive evidence, or
PREFACE TO THE ENGLISH EDITION ix
probability, that the method will reach its aim. On the basis of such a defini
tion it is easy to prove that what I have given is not a justification of induction.
It should be clear, however, that the argument I have set forth for the use
of induction cannot be deprived of its rigor by narrowing down the meaning
of words. That it is advisable to use a certain means if it can be shown to be
the best means for a desired end, and if nothing is known about the attain
ability of this end, appears to be an argument exempt from doubt; and it
seems to me that the word “justification” has always been applied to such
situations. When Magellan sailed along the American shore with the intention
of finding a western passage, he had no inductive evidence that there was
one; but his enterprise was justified because it was a means to reach his aim
if the aim was attainable. It is in this sense of the word that a justification
of induction has been given.
What makes this justification of induction appear weak is the adherence
to rationalistic standards, the creed that we can do more for the finding of
predictions than prepare everything for the case where success is attainable.
I do not think that any such creed is compatible with the principles of em
piricism, nor that it is an indispensable backbone for action in this recalci
trant world. The search for certainty is a desire deeply rooted in human
nature; yet we must not infer that we should submit to it. I prefer a philosophy
that teaches us to walk without the crutches of any kind of faith. It is of
the essence of empiricism that it refuses to recognize any form of rationalized
belief. That such refusal need not lead to skepticism, that actions anticipating
the future can be justified without any reference to belief, has been shown
through my analysis of the problem of induction. I do not think that a justi
fication of the inductive method of prediction may be called weak if it proves
that employing this method is the best we can do for the attainment of the
aim. This proof is the result to which I came through the construction of my
theory of probability and induction some fifteen years ago. It is with the
intention of submitting this result to the judgment of the English-speaking
world that I present this English edition of my book.
Hans R eichenbach
University of California,
Los Angeles,
May, 1948
CHANGES AND ADDITIONS IN THE
ENGLISH EDITION
Throughout the book, numerous minor changes were made; but only the major changes
will be mentioned. In chapter 1 the last paragraphs of § 2 were cut off and added to 5 3,
which was shortened in other parts. The introduction to symbolic logic, given in chapter 2,
was abbreviated and adjusted to the results of a detailed exposition of symbolic logic given
in another publication (see footnote, p. 12). In chapter 3 and the following chapters the
symbolic notation was adapted to the English language; in particular, the German symbol
W was replaced by the symbol P, standing for “probability”. Some changes and additions
were made in §§ 12 and 22. A new section was inserted as § 24; furthermore, the appendix
to chapter 3, which contains exercises, was added. From §§ 1 to 23 the numbers of the
English and the German sections coincide; from $ § 25 to 46 the numbers of the English sections
exceed by one those of the German sections. In chapter 4, § 30 was supplemented by refer
ences to recent publications on the problem of randomness. Chapters 5 and 6 remained
unchanged, except for some additions in §45 and the addition of § 47. From §§ 48 to 62
the numbers of the English sections exceed by two those of the German sections. In
chapters 7 and 8 changes were made in §§ 49, 57, and 62.
Chapters 9-11 take the place of the German chapters 9-10. These chapters were greatly
changed in the order of the material presented, so that no simple rule for the comparison
of sections can be given; and many changes and additions were made in the interior of the
sections. The following sections of these chapters are new: §§ 70-74, 81-88. The appendix
of the German edition, which contains some mathematical extensions and proofs, was
omitted.
The changes made in chapters 9-11 may be summarized as follows. A distinction is made
between primitive knowledge and advanced knowledge, and the problems of induction and
of the meaning of limit statements are given different treatments for the two cases. In
advanced knowledge, which presupposes the use of previous inductions, both the inductive
inference and the meaning of limit statements can be accounted for in terms of probability
concepts. In primitive knowledge, which does not make use of previous inductions, the
meaning of limit statements can be defined by the help of a “finitization”, which elim
inates infinite sequences; the use of infinite sequences then appears as a simplification
introduced for technical reasons. The inductive inference, as far as it is applied in primitive
knowledge, is made legitimate by my justification of induction.
A further distinction is made between the object-language interpretation and the meta
language interpretation of probability. The latter is used for the construction of probability
logic and is thus also called the logical interpretation. At the time the German book was
written, little was known about the significance of this distinction. The omission of the
distinction in the German book had no bearing upon my theory of induction or on the
mathematical parts of the book; the truth tables of probability logic given in the German
edition were not affected by the omission either. But since the distinction has turned out
to be relevant from many logical aspects, it has been carried through in the new presentation.
The method of derivation in probability logic was revised and greatly elaborated.
The exposition of inductive methods in their various forms was widened, and a section
on the inductive introduction of all-statements and the probability of hypotheses was
added. Recent publications on the theory of statistical estimation were included in the
frame of the discussion.
ACKNOWLEDGMENTS
T|he significance of the distinction between object-language and
metalanguage interpretation of probability was called to my atten
tion through criticisms by R. Carnap and C. Hempel. The necessity
of a minor correction of my axiom system of the calculus of probability was
pointed out to me by J. C. C. McKinsey and S. C. Kleene (p. 56). Some
results communicated to me by N. Dalkey and H. Schott were the starting
point for the analysis of complete probability systems (§24). In the course
of oral discussions with O. Helmer, I recognized the fact that the method of
anticipative posits can be extended to advanced knowledge (§ 88). E. Nagel’s
criticism induced me to be more explicit on the subject of the probability
of hypotheses (§ 85). The University of California Press and its staff were of
great help in giving the book its. printed form. * For mathematical details
and the reading of the proofs I had the assistance of J. Freund, A. Birnbaum,
and H. Edmundson.
I wish to express my sincere gratitude to the translators of the book,
Ernest H. Hutten and my wife Maria Reichenbach. Their work has shown
that the problem of translating German into English is not insoluble if it is
attacked with the courage to deviate from sentence structure in favor of
preserving the meaning. As a result of this method, the English formulation
is often more precise than the German original. Whenever the deviations
appeared to them to demand too great a venture, I did not hesitate to make
use of an author’s right to rephrase his text. Any criticism that the transla
tion is too free should therefore not be directed against my translators, for
whose active contribution to my book I am deeply obliged.
Hans Reichenbach
University of California,
Los Angeles,
September, 1948
