Modern Logic — A Survey: Historical, Philosophical and Mathematical Aspects of Modern Logic and its Applications PDF
Preview Modern Logic — A Survey: Historical, Philosophical and Mathematical Aspects of Modern Logic and its Applications
MODERN LOGIC - A SURVEY SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: J AAKKO HINTIKKA, Florida State University Editors: DONALD DAVIDSON, University of Chicago GABRIEL NUCHELMANS, University ofL eyden WESLEY C. SALMON, University ofA rizona VOLUME 149 MODERN LOGIC - A SURVEY Historical, Philosophical, and Mathematical Aspects ofM odem Logic and its Applications Edited by EV ANDRO AGAZZI Dept. of Philosophy, University of Genoa, Italy D. REIDEL PUBLISHING COMPANY DORDRECHT : HOLLAND / BOSTON: U.S.A. LONDON: ENGLAND Library of Congress Cataloging in Publication Data Main entry under title: Modern logic - a survey. (Synthese library; v. 149) Includes bibliographies and index. 1. Logic, Modern-Addresses, essays, lectures. I. Agazzi, Evandro. BC38.M54 160 80-22027 ISBN-13: 978-94-009-9058-6 e-ISBN-13: 978-94-009-9056-2 DOl: 978-94-009-9056-2 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland Sold and distributed in the U.S.A. and Canada by Kluwer Boston Inc., 190 Old Derby Street, Hingham, MA 02043, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland D. Reidel Publishing Company is a member of the Kluwer Group. Italian Edition by Istituto Della Enciclopedia Italiana All Rights Reserved Copyright © 1981 by D. Reidel Publishing Company, Dordrecht, Holland So/kover reprint of the hardcover I st edition 1981 No part 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 informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE vii PART 1: INTRODUCTION J. M. BOCHENSKI/ The General Sense and Character of Modem Logic 3 S. J. Su RM A / The Growth of Logic Out of the Foundational Research in Mathematics 15 PART 2: PURE LOGIC K. SCHUTTE / Proof Theory 37 A. MACINTYRE / Model Theory 45 G. KREISEL / Constructivist Approaches to Logic 67 J. BARWISE / Infmitary Logics 93 A. ROSE / Many-Valued Logics 113 N. D. BELNAP, Jr. / Modal and Relevance Logics: 1977 131 PART 3: THE INTERPLAY BETWEEN LOGIC AND MATHEMATICS E. CASARI / Logic and the Foundations of Mathematics 155 G. TAKEUTI / Logic and Set Theory 167 H. HERMES / Recursion Theory 173 A. S. TROELSTRA / The Interplay Between Logic and Mathematics: Intuit:onism 197 J. E. FENSTAD / Logic and Probability 223 G. E. REYES / Logic and Category Theory 235 PART 4: THE RELEVANCE OF LOGIC TO OTHER SCIENTIFIC DISCIPLINES E. AGAZZI / Logic and Methodology of Empirical Sciences 255 vi T ABLE OF CONTENTS J. HINTIKKA , Standard Vs. Nonstandard Logic: Higher..()rder, Modal, and First..()rder Logics 283 C. BOHM , Logic and Computers 297 G. J. MASSEY' Logic and Linguistics 311 M. L. DALLA CHIARA' Logical Foundations of Quantum Mechanics 331 L. J. COHEN' Inductive Logic 1945-1977 353 PART 5: LOGIC AND PHILOSOPHICAL TOPICS C. LEJEWSKI , Logic and Ontology 379 G. H. VON WRIGHT 'Problems and Prospects of Deontic Logic - A Survey 399 B. C. V AN FRAASSEN' Report on Tense Logic 425 R. STALNAKER' Logical Semiotic 439 CH. PERELMAN I Logic and Rhetoric 457 INDEX OF NAMES 465 INDEX OF SUBJECTS 473 PREFACE Logic has attained in our century a development incomparably greater than in any past age of its long history, and this has led to such an enrichment and proliferation of its aspects, that the problem of some kind of unified recom prehension of this discipline seems nowadays unavoidable. This splitting into several subdomains is the natural consequence of the fact that Logic has intended to adopt in our century the status of a science. This always implies that the general optics, under which a certain set of problems used to be con sidered, breaks into a lot of specialized sectors of inquiry, each of them being characterized by the introduction of specific viewpoints and of technical tools of its own. The first impression, that often accompanies the creation of one of such specialized branches in a diSCipline, is that one has succeeded in isolating the 'scientific core' of it, by restricting the somehow vague and redundant generality of its original 'philosophical' configuration. But, after a while, it appears that some of the discarded aspects are indeed important and a new specialized domain of investigation is created to explore them. By follOwing this procedure, one finally finds himself confronted with such a variety of independent fields of research, that one wonders whether the fact of labelling them under a common denomination be nothing but the contingent effect of a pure historical tradition. At this stage a need of con ceptual clarification is perceived, that is the need of understanding why we still accept to call the whole of these different fields, e.g., 'Logic', or 'Physics', or 'Biology', etc. and this question can be answered only through a work of philosophical understanding and historical reconstruction. All this is true for every science and for Logic as well, which has ac complished not only its transition from the condition of being a branch of philosophy to the fully-fledged shape of an articulate science, but has also given rise (like several 'pure' sciences) to a great lot of applications. This is why an adequate understanding of what modem logic is requires an inter disciplinary approach, and at the same time an effort of philosophical appre ciation, not in order to interpret it again as a part of Philosophy but in the sense of making it the object of a philosophical analysis and interpretation. This was the aim ofanintemationalconference on Modem Logic organized in Rome by the Institute of the Italian Encyclopedia and by the Italian vii E. Agazzi (ed.), Modern Logic - A Survey, vii-viii. Copyright ©1980 by D. Reidel Publirhing Company. viii PREFACE Society of Logic and Philosophy of Science in September 1976. The present volume contains the papers presented at that meeting, each of them being a general survey of one of the most relevant fields in which Logic is subdivided, or of one of the most important sets of problems which are connected with Logic in our days. A simple glance at the table of contents shows the width of the horizon that has been explored (thorough completeness could not be reached and was not an ambition of the organizers). The purpose of the conference was to provide a general appreciation of modem logic which could be accessible to cultivated people, without need of any specific technical competence. In other words, the cultural pOSition and role of Logic was hoped to become apparent, rather than the exact patterns of its technical machinery. This, in particular, explains the subdivision of the topics according to a line which takes into account the different kinds of intellectual 'interest' under which Logic may be approached today, and which go from its historical development, to its impact on mathematics, to its internal sub· division, to its different applications, to its philosophical commitments. Of course every contributor has conceived this task according to his own estimation and this has led to different levels of clarity, of technical com· plexity, of specialization or generality. It is left therefore to the reader to chose the papers with which to start in reading this book, according to his personal taste and background, but one can be confident that a singificant image of modem logic as a whole may be obtained from this volume, which might hardly be shaped even by the careful study of several good textbooks. EV ANDRO AGAZZI PART 1 INTRODUCTION 1. M. BOCHENSKI THE GENERAL SENSE AND CHARACTER OF MODERN LOGIC By 'Modem Logic' (abridged as 'ML') the class of studies is meant which were originated by Leibniz, developed, among others, by Boole, Peirce, Frege, Peano, Lesniewski and their followers; in other tenns the class of studies listed in Alonzo Church's Bibliography and in The Journal of Symbolic Logic. The expression 'ML' is sometimes used, it is true, in other ways, e.g. to denote studies in Hegelian dialectics. Those uses are irrelevant for the sake of the present paper which will be exclusively concerned with ML as described above. It may be only said, that no other known sort of contem porary logic can compare with the latter as far as standards of procedures and quality of results are concerned. The aim of the paper is to describe - as the title selected by the organizers of the conference indicates - the general sense and character of ML thus understood. In other tenns an attempt will be made to find the fundamen tal characteristics of ML-al studies. The method used will be comparative. We are going to ask: How does ML compare with three fields with which it is usually linked: logic, mathematics and philosophy? Is ML Logic and, if so, how does it differ from other types of logic? Is it a mathematical discipline and, if that is the case, what is the difference between it and other mathematical sciences? Is it philosophy and, this being admitted, what is its place among the other philosophical diSCiplines? The present paper will be mostly concerned with the first class of problems, the comparison between ML and the other types of logic; the other two classes of problems will be treated only marginally. As far as the main problems are concerned, the method will necessarily be historical: for, contrary to mathematics and philosophy, all other fonns of logic with which ML may be compared belong to the past. I It will be convenient to begin with a summary classification of the main parts and systems of ML. They can be classified under two headings: either 3 E. Agazzi (ed.). Modem Logic - A Survey. 3-14. Copyright e 1980 by D. Reidel Publishing Company.
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