Alfred Tarski - Introduction to Logic: and to the Methodology of Deductive Sciences

INTRODUCTION TO LOGIC
AND TO THE
METHODOLOGY OF
DEDUCTIVE SCIENCES

ALFRED TARSKI

T RANSLATED BY O LAF H ELMER

DOVER PUBLICATIONS, INC.
New York

TO MY WIFE

Bibliographical Note

This Dover edition, first published in 1996, is an unabridged republication of the 9th printing, 1961, of the 1946 second, revised edition of the work originally published by Oxford University Press, New York, in 1941.

Library of Congress Cataloging-in-Publication Data

Tarski, Alfred.

[O logics matematycznej i metodzie dedukeyjnej, English]

Introduction to logic and to the methodology of deductive sciences/Alfred Tarski; translated by Olaf Helmer.

p. cm.

Originally published: 2nd ed., rev. New York: Oxford University Press, 1946 (1961 printing).

Includes bibliographical references (p.) and index.

ISBN -13: 978-0-486-28462-0

ISBN -10: 0-486-28462-X

1. MathematicsPhilosophy. 2. ArithmeticFoundations.

I. Title.

QA9.T28 1995

511.8dc20

94-48412

CIP

Manufactured in the United States by Courier Corporation

28462X08

www.doverpublications.com

TABLE OF CONTENTS

F IRST P ART
ELEMENTS OF LOGIC. DEDUCTIVE METHOD

O N THE T HEORY OF C LASSES

S ECOND P ART
APPLICATIONS OF LOGIC AND METHODOLOGY IN CONSTRUCTING MATHEMATICAL THEORIES

Laws of irreflexivity for the fundamental relations; indirect proofs

PREFACE

The present book is a partially modified and extended edition of my book On Mathematical Logic and Deductive Method, which appeared first in 1936 in Polish and then in 1937 in an exact German translation (under the title: Einfhrung in die mathematische Logik und in die Methodologie der Mathematik). In its original form it was intended as a popular scientific book; its aim was to present to the educated laymanin a manner which would combine scientific exactitude with the greatest possible intelligibilitya clear idea of that powerful trend of contemporary thought which is concentrated about modern logic. This trend arose originally from the somewhat limited task of stabilizing the foundations of mathematics. In its present phase, however, it has much wider aims. For it seeks to create a unified conceptual apparatus which would supply a common basis for the whole of human knowledge. Furthermore, it tends to perfect and sharpen the deductive method, which in some sciences is regarded as the sole permitted means of establishing truths, and indeed in every domain of intellectual activity is at least an indispensable auxiliary tool for deriving conclusions from accepted assumptions.

The response accorded to the Polish and German editions, and especially some suggestions made by reviewers, gave rise to the idea of making the new edition not merely a popular scientific book, but also a textbook upon which an elementary college course in logic and the methodology of deductive sciences could be based. The experiment seemed the more desirable in view of a certain lack of suitable elementary textbooks in this domain.

In order to carry out the experiment, it was necessary to make several changes in the book.

Some very fundamental questions and notions were entirely passed over or merely touched upon in the previous editions, either because of their more technical character, or in order to avoid points of a controversial nature. As examples may be cited such topics as the difference between the usage of certain logical notions in systematic developments of logic and in the language , which is devoted to the sentential calculus, contains much new material. I have also added many new exercises to these chapters, and have increased the number of historical indications.

While in previous editions the use of special symbols was reduced to a minimum, I considered it necessary in the present edition to familiarize the reader with the elements of logical symbolism. Nevertheless, the use of this symbolism in practice remains very restricted, and is limited mostly to exercises.

In previous editions the principal domain from which examples were drawn for illustrating general and abstract considerations was high-school mathematics; for it was, and still is, my opinion that elementary mathematics, and especially algebra, because of the simplicity of its concepts and the uniformity of its methods of inference, is peculiarly appropriate for exemplifying various fundamental phenomena of a logical and methodological nature. Nevertheless, in the present edition, particularly in the newly added passages, I draw examples more frequently from other domains, especially from everyday life.

Independent of these additions, I have rewritten certain sections whose mastery by students had been found somewhat difficult.

The essential features of the book remain unchanged. The preface to the original edition, the major part of which is reprinted in the next few pages, will give the reader an idea of the general character of the book. Perhaps, however, it is desirable to point out explicitly at this place what he should not expect to find in it.

will to some extent compensate for the omission.

Secondly, apart from two rather short passages, the book gives no information about the traditional Aristotelian logic, and contains no material drawn from it. But I believe that the space here devoted to traditional logic corresponds well enough to the small role to which this logic has been reduced in modern science; and I also believe that this opinion will be shared by most contemporary logicians.

And, finally, the book is not concerned with any problems belonging to the so-called logic and methodology of empirical sciences. I must say that I am inclined to doubt whether any special logic of empirical sciences, as opposed to logic in general or the logic of deductive sciences, exists at all (at least so long as the word logic is used as in the present bookthat is to say, as the name of a discipline which analyzes the meaning of the concepts common to all the sciences, and establishes the general laws governing the concepts). But this is rather a terminological, than a factual, problem. At any rate the methodology of empirical sciences constitutes an important domain of scientific research. The knowledge of logic is of course valuable in the study of this methodology, as it is in the case of any other discipline. It must be admitted, however, that logical concepts and methods have not, up to the present, found any specific or fertile applications in this domain. And it is at least possible that this situation is not merely a consequence of the present stage of methodological researches. It arises, perhaps, from the circumstance that, for the purpose of an adequate methodological treatment, an empirical science may have to be considered, not merely as a scientific theorythat is, as a system of asserted statements arranged according to certain rules, but rather as a complex consisting partly of such statements and partly of human activities. It should be added that, in striking opposition to the high development of the empirical sciences themselves, the methodology of these sciences can hardly boast of comparably definite achievementsdespite the great efforts that have been made. Even the preliminary task of clarifying the concepts involved in this domain has not yet been carried out in a satisfactory way. Consequently, a course in the methodology of empirical sciences must have a quite different character from one in logic and must be largely confined to evaluations and criticisms of tentative gropings and unsuccessful efforts. For these and other reasons, I see little rational justification for combining the discussion of logic and the methodology of empirical sciences in the same college course.

A few remarks concerning the arrangement of the book and its use as a college text.