LESNIEWSKI'S SYSTEMS PROTOTHETIC

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1 LESNIEWSKI'S SYSTEMS PROTOTHETIC

2 NIJHOFF INTERNATIONAL PHILOSOPHY SERIES Volume 54

3 Lesniewski' s Systems Protothetic Edited by lan T.I. SRZEDNICKI - General Editor University of Melbourne, Australia and ZBIGNIEW STACHNIAK - Volume Editor York University, Canada SPRINGER -SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress CataIoging-in-Publication Data ISBN ISBN (ebook) DOI / Printed an acid-free paper AII Rights Reserved 1998 Springer Science+Business Media Dordrecht Origina11y published by Kluwer Academic Publishers in 1998 Softcover reprint ofthe hardcover Ist edition 1998 No pari of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 TABLE OF CONTENTS EDITOR'S FOREWORD I. PETER M. SIMONS Nominalism in Poland II. V. FREDERICK RICKEY A Survey of Lesniewski's Logic III. ALFRED TAJTELBAUM-TARSKI On the Primitive Term of Logistic N. BOLESLAW SOBOCINSKI An Investigation of Protothetic V. JERZY SLUPECKI St. Lesniewski's Protothetics VI. BOLESLAW SOBOCINSKI On the Single Axioms of Protothetic VII. V. FREDERICK RICKEY Axiomatic Inscriptional Syntax. Part II: The Syntax of Protothetic VIII. AUDOENUS LE BLANC Investigations in Protothetic PROTOTHETIC BIBLIOGRAPHY AUTHOR INDEX vii

6 EDITOR'S FOREWORD This edition of papers concerning Lesniewski's logical system Protothetic completes the four volume project - The LeSniewski Collection - a collected edition of Lesniewski's papers and major contributions to LeSniewski's system of the foundations of mathematics. The three volumes published so far are: (1) LeSniewski's Systems. Ontology and Mereology Edited by J.T.J. Srzednicki, Y.F. Rickey, and J. Czelakowski. Nijhoff International Philosophy Series, 13 (1984). (2) S. LeSniewski's Lecture Notes in Logic Edited by J.T.J. Srzednicki and Z. Stachniak. Nijhoff International Philosophy Series, 24 (1988). (3) Stanislaw LeSniewski: Collected Works Edited by J.T.J. Srzednicki, S.J. Surma, and D. Barnett with an Annotated Bibliography by Y.F. Rickey. Nijhoff International Philosophy Series, 44 (1992). I. In the period between the two world wars, Stanislaw Lesniewski ( ), one of the founders and a prominent member of the Warsaw School of Logic, created a system of the foundations of mathematics comprising three deductive theories: Protothetic, Ontology, and Mereology. The point of departure for the construction of this system was his study oflogical paradoxes and, in this context, a distinction between the distributive and collective interpretations of a class. This distinction between the two interpretations was reflected in the development of two deductive theories, the theory of collective classes, which he eventually called Mereology, and the theory of distributive classes, called Ontology. Finally, in order to combine Mereology and Ontology into a logically rigorous system, he constructed Protothetic - the system of "first principles." Lesniewski's ambition was not to add one more calculus to the variety already invented, nor even to prove general metatheorems about alternative formal calculi, in the interests of "comparative logic"; it was

7 viii EDITOR'S FOREWORD instead to perfect a universally valid classical system of logic and foundations of mathematics, in which he could rigorously formulate generalizations expressible only in the metalanguages of systems poorer in means of expression, [,,. 1 and on which he could rely as a true instrument of deduction and scientific investigation (Luschei, 1962, p. 24). * This program was initiated by Lesniewski in 1914 with his studies on a general theory of sets (later to be named 'Mereology'). The first version of Mereology appeared in print in 1916 under the title Podstawy Og6lnej Teoryi Mnogosci. 1. ** In 1919, Ldniewskijoined the University of Warsaw as a professor of the philosophy of mathematics. He met a group of gifted mathematicians, Zygmunt Janiszewski, Stefan Mazurkiewicz, Waclaw Sierpinski, whose research interests, like those of Lesniewski, were focused on the foundations of mathematics. In 1920 this group, joined by Jan Lukasiewicz, founded the mathematical journal Fundamenta Mathematicae with Mazurkiewicz and Sierpinski as editors, and Lesniewski and Lukasiewicz as members of the editorial board. The name, scope, and membership of the editorial board of the journal adequately reflected the research activities of the Warsaw schools of mathematics and logic during the first decade of the journal's existence. The construction of Ontology in the period between , marked the next step in the formation of LeSniewski's system of the foundations of mathematics, although it was not until 1930 that Ontology appeared in print (cf. LeSniewski, 1930). The construction of Protothetic began in 1922 and went quickly through numerous improvements and modifications, to be concluded in By then, Lesniewski's system of the foundations of mathematics was formally ready and, to quote Jordan, it was the most thorough, original, and philosophically significant attempt to provide a logically secure foundation for the whole of mathematics (cf. Jordan, 1945). Even such a critic of the importance of LeSniewski's contribution to modem logic as Grzegorczyk admitted that Lesniewski's treatment of logic was in his times the most exact; it was simpler than Principia [Mathematical and had it been published simultaneously with the second edition of the Principia, it would have played a considerable part in the development of logic (Grzegorczyk, 1955, p. 78). The roots of Protothetic can already be found in Lesniewski's early writings between 1912 and The "deductions" in his 1916 work on the general theory of sets are based on his logical intuitions which eventually were captured in the axioms and directives of Proto the tic and Ontology. In 1921, having developed his systems of Mereology and Ontology, Lesniewski was ready to * For complete publication information for this and other works cited in Editor's Foreword, see the Protothetic Bibliography at the end of this volume. ** Cf. Stanislaw Lesniewski, Foundations of a General Theory of Sets. I. Prace Polskiego Kola Naukowego w Moskwie, Sekcya Matematyczno-Przyrodnicza, No.2, Moskwa, 1916 [in Polish].

8 EDITOR'S FOREWORD ix " ",j~\';l4 h:vii <&>'1#, P..,..lt"'1' t4~t' wrffi:ini~wli(t :.w~-dr:wltluwal~ nb h,wyjni.k~w.\~:~~ '".'".... OtT cl~:u.~ : K.~t:3.d)iili;ki~:: ~1h.'Wd~. ~n.. ~{~l~q<~ua. 'i::. ~~~~~*1l~'~~i~ ::.. ~~ib~ 1&:~ ~h:tt$11j jl!~~~lkmt", 11'4,i1l; tbj, fjoj;icwa.. l~a P. fo(>i;i,""~~> The title page of Podstawy Og61nej Teory; MnogoscL I. search for a general system of "first principles." Dissatisfied with the existing proposals (e.g., theories of deduction by J.G.P. Nicod or A.N. Whitehead and B. Russell), he embarked on constructing the calculus of propositions that would be based on a minimal set of primitive symbols, with rigorous directives of procedure, and the simplest possible axiom-system. Lesniewski carefully designed Protothetic to be general and universally valid, consistent, decidable, complete, expressively powerful, logically economic, pure, and elegant, in other words - perfect. In 1922, Alfred Tarski provided Lesniewski with just what was needed. He demonstrated (in his doctoral dissertation written under Lesniewski's supervision) that the "ordinary" theory of deduction can be based on the sign of equivalence as the sole primitive term. He proved that the logical product, and, hence, all the familiar functions of propositions,

9 x EDITOR'S FOREWORD can be defined in terms of equivalence, the universal quantifier, and variable propositional functors. Lesniewski began the development of Protothetic by constructing a weaker system 6 (which is a complete two-valued equivalential system). In 1922 he began the task of formulating the directives for Protothetic from the perspective of his theory of semantic categories. Testing and modifying various details of these directives lasted several years. But by 1922 the directives were precise enough to fonn the basis of a considerable amount of axiomatic research (Lesniewski, 1929; cf. Lesniewski, 1992, p. 422). Still in 1922, Protothetic underwent several transformations. First, 6 was strengthened to the system 61, and later to 62, which is one of the many possible mutually equivalent systems of the theory that I name protothetic (Lesniewski, 1938a; cf. McCall, 1967, p. 122). 62, in turn, was succeeded by a more elegant system 63. Finally, in 1923, Lesniewski proposed the directives for the system 65, the most mature of the different dialects of Protothetic. In the years the directives of 65 were polished and supplemented with forty-nine "terminological explanations." Concerning the formalization of Protothetic, Lesniewski remarked that Having no predilection for various 'mathematical games' that consist in writing out according to one or another conventional rule various more or less picturesque fonnulae which need not be meaningful, or even - as some of the 'mathematical gamers' might prefer - which should necessarily be meaningless, I would not have taken the trouble to systematize and to often check quite scrupulously the directives of my system, had I not imputed to its theses a certain specific and completely detennined sense, in virtue of which its axioms, definitions, and final directives (as encoded for ( 5), have for me an irresistible intuitive validity. I see no contradiction, therefore, in saying that I advocate a rather radical 'fonnalism' in the construction of my system even though I am an obdurate 'intuitionist' (Lesniewski, 1929; cf. Lesniewski, 1992, p. 487). Satisfied with the directives and general properties of 65, Lesniewski focused his attention on the problem of simplification of the single axiom of 65. His collaboration with Mordchaj Wajsberg in the years , and later with Boleslaw Sobocinski in the years , resulted in a significant simplification of the axiom. In 1945, Sobocinski found the single axiom for 65 consisting of 54 symbols (when written in the authentic Lesniewski symbolism), which still remains the shortest known single axiom for this system.* In the meantime, an implicational version of Protothetic appeared with the name 64. Its axiomatization is the result of a fruitful collaboration between Lesniewski and Tarski. A number of "computative" or "algorithmic" dialects * Lesniewski in (1929) and (1938a) discusses the details concerning the search for the shortest single axiom of Protothetic until 1937; Sobocinski (1960) describes this search until the discovery of the shortest known single axiom in 1945.

10 EDITOR'S FOREWORD xi U,q,yJ:: (i)h,pj~lh(ft}),;) H?}:H J",1(f(\hj):, U)qSl Nt!tnl(l1t), N1fr'{q}trt):,. {j}~\rt ~t (if)) ~Jiq; v, "" (,t~ {rtt{tcb}.eq),.., (t)tff~f(qi: One of the pages of Choynowski' s lecture notes taken during the course From the Foundations of Protothetic in of Protothetic were also developed between 1924 and 1935 (see Le Blanc, 1991). From the very beginning, Protothetic attracted considerable attention from Lesniewski's colleagues and students. Lesniewski discussed the directives for Protothetic in detail in his lectures at the University of Warsaw, starting with his course on the Foundations of Logistic in the academic year. Lecture notes taken by Mieczyslaw Choynowski during the course From the Foundations of Protothetic in the academic year have survived, giving a unique insight into the logical structure and successive developments of this system (cf. Lesniewski, 1988). In 1937, Lukasiewicz created a journal - Collectanea Logica - that was devoted exclusively to logic. (Sobocinski was the managing editor.) The founders hoped that the

11 xii EDITOR'S FOREWORD role and the international stature of this new journal would be analogous to Fundamenta Mathematicae. Among the eleven papers of the first volume of the journal, eight, written by Lesniewski, Sobocinski, and Slupecki, were devoted to Protothetic. Unfortunately, the first and only volume of Collectanea Logica, which was ready to appear, was destroyed during the first days of World War II. The documents which could have been used to reconstruct the volume perished during the Warsaw uprising in 1944.* The postwar history of Protothetic begins with the heroic effort of former colleagues and students of Lesniewski, especially Hii, Kotarbinski, Lejewski, Sobocinski, and Slupecki, to continue LeSniewski's work, to reconstruct and salvage his logical contributions, and to make them widely known. Tadeusz Kotarbinski was the first person to attempt to publish the collected works of Lesniewski, a task which he was unable to complete. He knew that some of the notes taken during Lesniewski's lectures circulated in Warsaw during the war and were an important source of knowledge about Lesniewski's work. These notes represented the missing link between a few papers that Lesniewski managed to publish and the wealth of results he left unpublished. Kotarbinski collected some of the notes which survived the war and gave them to Slupecki with the task of reconstructing Lesniewski's logical systems. Slupecki's articles on Protothetic (see paper V in this volume) and Ontology, which appeared in volumes I and II of the newly created Polish journal Studia Logica, proved to be among the most influential papers in the development and further studies of Lesniewski 's systems. The most important lecture notes collected by Kotarbinski were published by Kluwer Academic Publishers in 1988 (cf. LeSniewski, 1988). Boleslaw Sobocinski's involvement in the study of Protothetic goes back to his prewar investigations concerning the axiomatic foundations of this system (cf. Sobocinski, 1939; see also paper IV in this volume). After the war he continued his research on Protothetic and devoted his first postwar publications to this system (cf. the Protothetic Bibliography). His paper 'On the single axioms of protothetic' (paper VI in this volume) is regarded as one of the most important papers concerning Protothetic. Another work, 'An investigation of Proto the tic ' (paper IV in this volume), in addition to important technical contributions to Protothetic, gives a unique account of the ill-fated Collectanea Logica. At the same time, he lectured on logic attracting students and researchers to the logical ideas of LeSniewski. In 1960, Sobocinski created the Notre Dame Journal of Formal Logic making it an important forum for promoting research investigations concerning Lesniewski's system of the foundations of mathematics. * For more information on Collectanea Logica and its fate, see Sobocinski (1949) (also paper IV in this volume).

12 EDITOR'S FOREWORD xiii II. This volume is a collection of the major contributions to Protothetic. The title Lesniewski's Systems: Protothetic is not intended to suggest that all important papers devoted to Protothetic are included here, for many have been left out, either because they were similar in content to those included in the present volume or simply because they were too numerous to be included in a single volume. Also excluded are Lesniewski's own papers dealing with Protothetic, since they appear in Lesniewski (1992). The papers included in this volume are arranged by dates of publication. Those by Peter Simons on nominalism in Poland and by Frederick Rickey on the logical systems of Lesniewski, which open the volume, are the only exceptions to this order. Simons' paper discloses LeSniewski's philosophical commitments regarding his system of the foundations of mathematics, while Rickey's work surveys the most important technical contributions to Lesniewski's system. Both papers provide a philosophical and historical context for the technical contributions reported in the remaining papers of this volume. Alfred Tarski's paper 'On the primitive term of logistic' is a new English translation of his doctoral thesis that was originally published in Polish in Przeglt{d Filozoficzny under the title '0 wyrazie pierwotnym logistyki' (cf. Tarski, 1923b). The well-known translation by J. H. Woodger, which is also entitled 'On the primitive term of logistic', in the collection Tarski (1956), is a compilation of three papers: Tarski (1923a, 1923b), and (1924). In the present version, we aimed at a literal translation of the Polish original rather than a smooth one based on Tarski's other publications. We tried to retain Tarski's individual style as well as the lay-out of the Polish publication. Boleslaw Sobocinski's paper 'An investigation of Proto the tic ' was intended to appear under the title 'Z badan nad prototetyk",' in Vol. 1 of Collectanea Logica. Its English translation, preceded by the 21-page long introduction which discusses the origins and fate of Collectanea Logica as well as the contents of its first volume, was eventually published in 1949 (cf. Sobocinski, 1949). Another English translation from the Polish original by Z. Jordan appeared in 1967 in McCall's Polish Logic (see McCall, 1967). In this volume we include an edited version of Sobocinski (1949), in which the part of the introduction that is devoted to Collectanea Logica joins Jordan's translation. To retain Sobocinski's individual style only the necessary editorial alterations have been made. The Protothetic Bibliography included in this volume lists only works in which Protothetic is explicitly discussed. Papers on subjects related to Protothetic, such as equivalential calculi, calculi with variable functors, theories of definitions, quantification, syntactic categories, etc., are excluded. Many of

13 xiv EDITOR'S FOREWORD these exclusions can be found in Rickey's 'An Annotated Lesniewski Bibliography' included in Lesniewski (1992).* ACKNOWLEDGMENTS The editors of this volume are indebted to many individuals for their help during the editorial work. We wish to thank V. Frederick Rickey for many valuable suggestions concerning the contents of the volume. We thank Cynthia Bearlin, Ewa Stachniak, David Tremaine, and Michael Wharton for their generous help during the translation and edition of the material. Our gratitude is also due to Jaroslaw Malik for his many hours devoted to the preparation of the Protothetic Bibliography. Special thanks go to Maja S.M. de Keijzer of Kluwer Academic Publishers for her continuous assistance in the preparation of this volume, for her valuable support, and for her guidance through all the stages of the editorial process. Finally, we would like to thank Jan Tarski and the following institutions and publishers for granting their reprint permissions for papers included in this volume: the Institute of Philosophy and Sociology of the Polish Academy of Sciences, The Notre Dame Journal of Formal Logic, Oxford University Press, and Rodopi. We have done our utmost to contact the copyright owners of all the material reprinted in this volume. In some cases, however, locating the copyright owner proved to be an impossible task after all these years. If you are a copyright owner whom we have failed to reach, then please contact Kluwer Academic Publishers. ZBIGNIEW STACHNIAK * An updated version of 'An Annotated Lesniewski Bibliography' is available from V.F. Rickey of the Department of Mathematics and Statistics at Bowling Green State University, Ohio, U.S.A.

METAMATHEMATICS OF FUZZY LOGIC

METAMATHEMATICS OF FUZZY LOGIC TRENDS IN LOGIC Studia Logica Library VOLUME 4 Managing Editor Ryszard Wojcicki, Institute ofphilosophy and Sociology, Polish Academy ofsciences, Warsaw, Poland Editors Petr