second edition LOGIC, SEMANTICS, METAMATHEMATICS PAPERS FROM 1923 TO 1938 BY ALFRED TARSKI TRANSLATED BY J.H. WOODGER Second edition edited and introduced by JoHN CoRcORAN HACKETT PUBLISHING COMPANY First edition published in 1956 by Oxford University Press Copyright © 1983 by Hackett Publishing Company,I nc. All rights reserved Printed in the United States of America 10 09 08 07 «45678 For further information, please address Hackett Publishing Company,I nc. P.O. Box 44937 Indianapolis, Indiana 46244-0937 Library of Congress Cataloging in Publication Data Tarski, Alfred. Logic, semantics, metamathematics. Includes bibliographical references and indexes. 1, Logic, Symbolic and mathematical—Addresses, essays, lectures. 2. Semantics (Philosophy) — Address, essays, lectures. 1. Corcoran, John. II. Title. BC135.T35 1983 160 83-10850 ISBN 0-915144-75-1 ISBN-13: 978-0-915144-75-4 (cloth) The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48-1984. TO THE MEMORY OF HIS TEACHER TADEUSZ KOTARBINSKI THE AUTHOR EDITOR’S PREFACE TO THE REVISED EDITION THE revised edition is intended to supersede the first edition in several respects. Two considerations which guided the design of the revisions were (1) the desire to increase the accuracy andc larity of the articles without altering their usefulness as historical documents and (2) the need to make Tarski’s early thought more accessible to the current generation of linguists, logicians, mathematicians, philosophers, and historians. The Editor’s Introduction and the Analytical Index are new with this edition. The Editor’s Introduction shows how Tarski’s development of much of the modern framework of logic and methodology can be traced in the articles in this volume. The Analytical Index is much more than a guide to locations of terms; it is as much an indexo f ideas as of terminology, and it lists, besides terms used by Tarski, not only current synonyms for the terminology of the articles, but also current terms for ideas used by but not named by Tarski. Included in this edition are all seventeen articles collected in the first edi- tion. All of the front material of the original has been retained unchanged. Several items have been addedt o the bibliography. Over 150 pages have been altered in various ways. The fact that Tarski had been unable to read a considerable part of the original text before publication accounts for many of the changes. Some changes in wording were neededf or sake of clarity, uniformity, or precision. Typographical errors, in prose and in mathematical formulas, have been corrected (although there were fewer of these than would have been expected given the complexity of the work). In several places, whole sentences have been rewritten in order to elimi- nate ambiguity or misleading connotations and in somep laces, where more extensive rewriting would have been called for, footnotes have been added indicating the nature of the difficulty. The bibliographical note attached to the “Wahrheitsbegriff” (article VIII) has been rewritten as hast he first paragraph oft his article’s his- torical note. Postscripts have been addedt o articles IV and V. Various new historical references have been inserted as footnotes. There was cross-referencing among the original articles which wass ig- nificantly increased in the translations and increased even more in the present edition. All the rewriting was done by Tarski andt he final form of every change has received Tarski’s explicit approval. Tarski prepared over fifty single- spaced pages of changes which have been incorporated into the present text. In addition, with the help of M. Scanlan (Oregon State University), viii EDITOR’S PREFACE H. G. Bohnert (Michigan State University) and H. Hiz (University of Pennsylvania), I prepared hundreds of suggestions to which Tarski re- spondedi n writing, by phone, or in conversation. S. Givant (Mills College) assisted Tarski. Various methods to identify changed passages were considered and found to be too awkward, too expensive, and too time consuming, espe- cially in light of the fact that changes in the articles as originally written had beeni ntroduced already in thef irst edition. For example, it is some- times difficult to determine whether phrasing, terminology, and restruc- turing of a sentence to conform to the needso f a different language could conceal subtle changes in viewpoint. For most scholarly purposes this edition can be regarded as an authoritative substitute for the first edition. However, writers involved in delicate historical questions may need to consult the original papers themselves as event he first edition may prove insufficient for such purposes. Je; IN this printing, the third of the second edition, all errata that had come to my attention have been corrected. These corrections are added to those madei n the secondp rinting. Noneo f the errata would have required Tarski’s attention. Thanks go to S. Nambiar, P. Chen, andK . Herreid (all of the Uni- versity of Buffalo) as well as to M. Schirn (University of Munich) who detected a missing fourteen-word (fifteen in the Germano riginal) passage on p. 172 in the course of comparing the English translation of the Wahrheitsbegriff with the original German text. Thet ranslation oft he pre- viously missing passage, which appearsa s a newf ootnote, is by Professor Schirn. J. C., 1997 Editorial Note to the Fourth Printing: Shortly following the 1983 first printing of the second edition, all of the papers translated here were reprinted in their original untranslated form in the 1986 four-volume work, Alfred Tarski: Collected Papers, Basel, Birkhauser, edited by S.G i- vant and R. McKenzie. The entire four volumes were reviewed in 1991 in Mathematical Reviews [MR 91h: 01101-4]. Of course, the secondaryl itera- ture on Tarski has continued to grow; especially notable is the 2004 biog- raphy, Alfred Tarski Life and Logic, by A. Feferman and S. Feferman, Cambridge University Press. Several logicians made suggestions for corrections in this printing; special thanks go to J. Gasser (Lausanne), D. Hitchcock (Hamilton), A. Koslow (New York), F Nabrasa (Sarasota), J. Sagiiillo (Santiago de Compostela), R. Santos (Lisbon), and P. Simons (Leeds). J. C., 2006 TRANSLATOR’S PREFACE Tue setting free of Poland after the First World War wasf ol- lowed byi ntensive activity in her Universities. In the depart- ments of philosophy and mathematics this took the form, in a number of places, of new and powerful investigations in the fields of mathematical logic, the foundations of mathematics, and the methodology of the sciences. Prominent in this, move- ment was the Warsaw school led by Lukasiewicz, Kotarbinski, and Lesniewski. Undert heir skilled guidance a younger genera- tion grew up and among these Alfred Tarski quickly dis- tinguished himself. Evers ince I first enjoyed the hospitality of Professor Tarski in Warsaw in 1936, it has seemed to me that the importance and scope of the Polish school of logicians were insufficiently known and appreciated in the English-speaking world. Then came the Second World War, bringing ruin once moret o Poland, killing men, destroying laboratories, and burning manuscripts and libraries. After this war it occurred to me that I should be performing a public service, as well as acknowledging in some small measure my debt to my Polish friends, if I prepared a collected edition of some of Professor Tarski’s publications in English translation. When hev isited England in 1950 to deliver the Sherman Lectures at University College, London, I men- tioned my plan to him and received his approval. This volume contains Tarski’s major contributions to logic, semantics, and metamathematics published before the Second World War; their arrangement here corresponds to the chrono- logical order in which they first appeared in print. With the exception of articles II and XI (which are too closely connected with, and too often referred to in, the other articles to be omitted), the volume does not include Tarski’s studies in the foundations of special mathematical disciplines—set theory, group theory, etc. Neither does it contain his papers of a pronouncedly mathematical character, dealing with special topics from the domain of set theory, measure theory, abstract algebra, elementary geometry, etc. Also excluded are short notes, abstracts, and preliminary reports which are closely related to some of the articles included in the volume and the contents of which are more fully presented in these articles. x TRANSLATOR’S PREFACE A longer paper from the domain of logic and methodology which has been omitted is ‘Sur la méthode déductive’ in Travaux du IX* Congrés International de Philosophie (Paris 1937); this paper is a purely expository one, and its ideas have been fully developed in Chapter VI of Tarski’s book Introduction to logic (New York, 1941). In a sense the present work is more than a volumeo f transla- tions. Naturally an attempt has been made to remove the misprints and errors which occuri n the originals. Moreover, the articles have been provided by the author with cross-references to other articles in the volume and with notes referring to later developments and recent literature. Occasionally some new remarks have been added for the purpose of clarifying certain passages in the original text. Articles II and VI contain more serious changes, Tarski having inserted in them several passages which, he hopes, will help toc larify and amplify their contents. While the work of translation was in progress, the passages which seemed to me doubtful and difficult were noted down and sent to Professor Tarski in Berkeley, California. In this way it was possible to adjust the text of the translations in many points, so as to meet the author’s wishes. However, in view of the time limit and the geographical distance between ther esi- dences of the author and the translator, it was impracticable to send the whole manuscript to Professor Tarski before it was set in print. Instead he received galley proofs, on which for obvious reasons he could not suggest too extensive changes. It also proved impossible to discuss the suggested changes in detail, and it was left to my decision which changes were actually to be carried through. Thus Professor Tarski is not responsible for the final text or the technical aspect of the book. Three articles in this volumea re joint publications of Tarski and other authors: Professor C. Kuratowski (article VII), Dr. A. Lindenbaum (article XIII), and Professor J. Lukasiewicz (article IV). We are greatly obliged to Professors Kuratowski and Lukasiewicz for their permission to include the translations of the jointly published articles in the volume, Dr. Lindenbaum fell a victim to the Gestapo during the war. The papers included in the volume originally appeared in the following periodicals and collective works: Actes du Congrés International de Philosophie Scientifique (articles XV and XVI), Comptes Rendus de la Société des Sciences et des Lettres de Varsovie TRANSLATOR’S PREFACE xi (articles III and IV), Ergebnisse eines mathematischen Kol- loquiums (articles XIII and XIV), E'rkenntnis (article X), Funda- menta Mathematicae (articles I, VI, VII, XI, XII, and XVII), Ksiega Pamiqtkowa Pierwszego Polskiego Zjazdu Matematycznego (article II), Monatshefte fiir Mathematik und Physik (articles V and IX), Przeglad Filozoficzny (articles I, X, XV, and XVI), and Travaux de la Société des Sciences et des Lettres de Varsovie (article VIII). Acknowledgements should be made to the pub- lishers and editors of these periodicals for their generosity. I am obliged to Mr. 8. W. P. Steen for kindly reading the proofs of article VIII and throughout the work I have received much help from my son, Michael Woodger. Wea re also indebted to several colleagues and students of Professor Tarski—Dr. C. C. Chang, Professor A. C. Davis, Professor J. Kalicki, Mr. R. Montague, Professor J. Myhill, Professor D. Rynin, and Mr. D. Scott—for their assistance in revising the original text of the articles and in reading galley proofs. Finally it is a pleasure to acknowledge the courtesy and help which we have received from the staff of the Clarendon Press at all stages in the production of the book. J. H.W.