Initiatives in Logic PDF
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INITIATIVES IN LOGIC REASON AND ARGUMENT 2 VOLUME General Editor: Jan T. J. Srzednicki Editors: E. M. Barth, Groningen. N. C. A. DaCosta, Sao Paulo. P. T. Geach, Cambridge, U.K. K. Lorenz, Saarbriicken. W. Marciszewski, Warsaw. S. J. Surma, Auckland. R. Sylvan, Canberra. R. R. Verma, Lucknow. Editorial Board: J. Bochenski, Freibourg. J. L. Cho, Oxford. J. Grattan, Guiness, Middlesex. A. Grzeqorczyk, Warsaw. R. Harre, Oxford. L. Kielkopf, Columbus, Ohio. D. Scott, Pittsburgh, Pennsylvania. K. Segerberg, Auckland. P. Suppes, Stanford, California. K. Szaniawski, Warsaw. P. Weingartner, Salzburg. R. Wojcicki, Lodz. Initiatives in Logic edited by Jan Srzednicki The University of Melbourne 1987 MARTINUS NIJHOFF PUBLISHERS A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP DORDRECHT I BOSTON I LANCASTER Distributors for the United States and Canada: Kluwer Academic Publishers, P.O. Box 358, Accord Station, Hingham, MA 02018-0358, USA for the UK and Ireland: Kluwer Academic Publishers, MTP Press Limited, Falcon House, Queen Square, Lancaster LAI IRN, UK for all other countries: Kluwer Academic Publishers Group, Distribution Center, P.O. Box 322, 3300 AH Dordrecht, The Netherlands Library of Congress Cataloging in Publication Data Injtjatjves In logjc ! edjted by Jan Srzednjckj. p. cm. -- (Reason and argument; 2) 1. LogIc, Symboljc and mathematjcal. I. Srzednjckl, Jan T. J. II. Serjes. BC135.149 1987 1S0--dc19 87-2130S CIP ISBN-13: 978-94-010-8144-3 e-ISBN-13: 978-94-009-3673-7 DOl: 10.1007/978-94-009-3673-7 Copyright © 1987 by Martinus Nijhoff Publishers, Dordrecht. Softcover reprint of the hardcover 1s t edition 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, with out the prior written permission of the publishers, Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht, The Netherlands. EDITORIAL NOTE The present volume collects together accounts of some new departures in the area of logic. Each is, or was at its time a novelty, each required inventive initiative to come to fruition, each meant an advance in logic. The samples range over a wide span of time, from the Classical Greek period to modern computer technique. Together the selection illustrates the fact that not only developing logic as such, as distinct from merely utilizing alre ady perfected logical techniques, Is an art, but that Is a typical art. By that we mean that It requires inventive Intelli gence and ingenuity, qualities that cannot be taught, only per fected and developed under guidance. This contrasts with skills that can in principle be taught to anybody of sufficient ability. A person who cannot be taught skills is dense, but a person who cannot be an artist is not dense, sometimes he could even be abler than the artistic, creative individual. Of course, logic is not the kind of art that can be practised by the less than able, ability is a prerequisite, as It is for every type of art, but that prerequisite is not enough. Like every other science, creative ingenuity is a conditio sine qua non for the work that really extends the domain of logic. To illustrate this is a modest aim of the present volume. It is of course impossible to present a history of creativity in logic at all thoroughly in a small collection of articles, yet the point is made without it, and It Is hoped that each and everyone of the contributions carries Its own direct Interest In a way that gives the whole a strong appeal. * * * The arrangement of contributions is organized as follows. The volume starts with two essays concerning the great ini tiator in logic Leibniz. Wolfgang Lenzen's essay Leibniz'.s calculus 01 strict Implication shows how Lelbnlz anticipated Lewis' modal calculi; the comparison gets possible due to the suitable axlomatlzatlons proposed by the author. A formally al ternative approach, that making use of algebraic means, is y Editorial note suggested by MacleJ luniewicz's paper Leibniz's modal calculus 01 concepts. These historical reconstructions are followed by a neW approach to conditionals, obviously related to modal logics. as adopted by Ingemund Gullv~g in his paper entitled The logic 01 conditions. The next group of papers is devoted to foundational studies. Among the pioneering thinkers In this field was Henri Poincare. His original views on the relations between logics and mathe matics, esp. set theory. are analyzed in Gerhard Heinzmann's paper Philosophical pragmatism in Poincare. The following A note on Zeno 83 by Nicholas Denyer recalls the Greek troubles with infinity which lie at the beginning of historical chain ending in set theory. Should the reader be interested in further fate of GOdel's theorem up to early 1980's, he will find it in Roman Murawski's paper Generalization and strengthenings 01 GOdel's incompleteness theorem. Some of the papers deal with less known Polish logical ini tiatives' both belonging to pure mathematical logic and to ap plication of logic to philosophy. Some of Mordchaj Wajsberg's results in mathematical logic are presented by Stanislaw Surma in the context of Lvov-Warsaw School in the essay The logical work 01 Mordchaj "ajsberg, while a specific Wajsberg result is criti cally discussed by M. N. Bezhanishvili in his Notes on "ajsberg's prool 01 separation theorem. In the Polish climate of 1930's, favourable both to logic and philosophy there appeared attempts similar to those discussed by Edward Nleznamki in his Logical analysis 01 Thomism - the Polish programme that originated in 1930' s; this covers also recent developments. In the same period Kazimierz Ajdukiewlcz Initiated a logical theory of questions; this field is the subject of Leon KoYs paper On justilication 01 questions, where he suggests a pragmatic approach. The contributions closinCi this volume have in common what may be called an alogorithmic-oriented approach. Wojciech Buszkowski's article The logic 01 types belongs to the chain of inquiries Initiated by K. AJdukiewicz' algorithm for checking syntactic connexion, and developed essentially by Lambek's re sults. Witold Marclszewski under the title System 01 computer aided reasoning lor mathematics and natural language reports on recent research in this field, especially In Poland; this dis cussion is complemented by two technical reports. by Leslaw W. Szczerba. and by Anna Zalewska, concerning the use of such a system in teaching logic and mathematics. J. S. - W. M. TABLE OF CONTENTS ltJn Srzednicki and Witold MtJrciszewski EdltoritJi Note v LLst of Contributors Ix Wolfgang Lenzen LBBNIZ'S CALCULUS OF STRICT IMPLICATION Maciej luniewicz LEIBNIZ'S MODAL CALCULUS OF COtcEPTS 36 lngemund Gullv!g ruE LOGIC OF CONDITIONS 52 Gerhard Heinzmann PHILOSOPHICAL PRACMAT lSM IN POIN:ARE 70 Nicholas Denyer A NOTE ON ZENO 53 81 Roman MurtJwski CEHERALlZATlONS AND STRe«:m-lEN.INGS OF OODEl's INCOMPLETENESS 1liEOR&1 84 Stanislaw 1. Surma THE LOGICAL WORK OF MOROCHA.J WAJSBERG 101 M. N. Bezhanishvili NOTES ON WAJSBERG's PROOF OF THE SEPARATION 1"HEQRB.1 116 viii Table of contents Edward Nieznaflskf LOOlCAL ANAL YSlS OF 11iOMlSM TIle Polish Programme that originated in 1930's 128 Leon KoJ ON JUSTIFICATION OF QUESTIONS 156 WoJcfech Buszkowski THE LOOIC OF TYPES 180 Wftold Marclszewskf SYSTEMS OF COMPlfTER-AlDED REASONING FOR MATHEMATICS AND NATURAL LANCIJACE 207 TWO REPORTS ON EDUCATIONAL APPUCATIONS OF MIZAR MSE. A SYSTEM OF COMPUTER-AIDED REASONING Anna Zalewska TIle application of MIZAR MSE in a course in logic 234 Leslaw W. Szczerba The use of MIZAR MSE in a course in foundations of geometry 240 Literature 235 Index of Nmnes 259 LIST OF CONTRIBUTORS M. N. BEZHANISHVILI Thilisl, USSR WOJCIECH BUSZKOWSKI UAM (Adam Mickiewicz University), lost. of Mathematics Poznafl, Poland NICHOLAS DENVER St. John's College, Cambridge University Cambridge, England INGEMUND GULL VAG The University of Trondheim, ColI. of Arts and Science, Philosophy Opt. Dragvoll, Norway GERHARD HEINZMANN Universitat des Saariandes, FB. Philosophie Saarbrticken, FRG MACIEJ JUNIEWICZ Uniwersytet Szczeciflskl <University of Szczecln), lost. of Mathematics Szczecin, Poland LEON KOJ UMCS (Maria Curie-Sklodowska University), lost. of Phllosophy Lublin, Poland WOLFGANG LENZEN Universitat Osnabrtick, FB. Philosophie Osnabrtick, Austria Ix List of contributors WITOLD MARClSZEWSKI UW (Warsaw University) - Bialystok Branch, Opt. of Logic Bialystok, Poland ROMAN MURAWSKI UAM (Adam Mickiewicz University), lost. of Mathematics Poznan, Poland EDWARD NIEZNANsKI ATK (Academy of catholic Theology), Philosophy Opt. Warszawa, Poland STANISI:.AW J. SURMA Wellington University Wellington, New Zealand LESI:.AW W. SZCZERBA UW (Warsaw University), lost. of Mathematics Warszawa, Poland ANNA ZALEWSKA UW (Warsaw Universlty)- Bialystok Branch, Opt. of Logic Bialystok, Poland
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