ASPECTS OF MODERN LOGIC

ASPECTS OF MODERN LOGIC

SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIORAL SCIENCES Editors: DONALD DAVIDSON, Princeton University J AAKKO HINTIKKA, Academy of Finland and Stanford University GABRIEL NUCHELMANS, University of Leyden WESLEY C. SALMON, Indiana University

EVERT W. BETH ASPECTS OF MODERN LOGIC D. REIDEL PUBLISHING COMPANY / DORDRECHT-HOLLAND

MOD ERNE LOGICA Published by Van Gorkum & Compo N.V., Assen, 1967 with support of the Prins Bernhard Fonds Translated from the Dutch by Dr. D. H. J. de Jongh and Susan de Jongh-Kearl Library of Congress Catalog Card Number 79-135102 SBN 90 277 0173 3 ISBN-13: 978-94-010-3334-3 001: 1 0.007/978-94-01 0-3332-9 e-isbn-13:978-94010-3332-9 All Rights Reserved Copyright 1970 by D. Reidel Publishing Company, Dordrecht, Holland No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher

A NOTE ON THE EDITING This book stems from an unfinished manuscript which was found after the death of Professor Beth. The manuscript was accompanied by a detailed description of the subjects, which the author had wanted to discuss in this book, and the order as he had planned it. The first four chapters were almost ready for publication. Together they form a coherent exposition of the method of semantic and deductive tableaux designed by Professor Beth. To these four chapters we added the chapter which had been projected as the eighth, 'About the so-called thought machine'. The first part of the book, consisting of these five chapters, has thus become a coherent presentation on logic in the narrower sense. The remaining chapters had not been worked out, but the texts of three articles, the contents of which combine well with some of the unfinished chapters, and which the author very probably meant to utilize in preparing these chapters, had been joined to the manuscript by Professor Beth. These articles appear here as the Chapters VI, VII and VIII. We added to this also the two articles which appear here as Chapters IX and X. These five unchanged articles form the second part of the book. This explains why some chapters in part 2 have not been subdivided into sections, while part 1 is consistently so subdivided. We have seen fit to make some annotations to elucidate the text. These are denoted by one or more asterisks. E. M. BARTH J. J. A. MOOIJ

ACKNOWLEDGEMENTS In a conversation which I had with Drs. E. M. Barth in 1965, she brought to my attention the desirability of publishing posthumously a manuscript entitled Aspects of Modern Logic, by my deceased husband, although it was in an unfinished state. Together with Dr. J. J. A. Mooij, at my request, she prepared this book for publication. To both I express here my gratitude for the care and devotion with which they conducted this task, which, among other things, consisted of making the manuscript ready for press. I am very thankful to Drs. H. C. Doets for the assistance he gave in compiling the bibliography, the author and subject indexes, and for reading the printed proofs. For the second time I have the pleasure of expressing my indebtedness to the Prins Bernhard Fonds, which, after its financial contribution for the publication of Science, A Road to Wisdom, has now similarly subsidized Aspects of Modern Logic. Also I thank the Koninklijke Nederlandse Akademie van Wetenschappen of Amsterdam, the Uitgeversmaatschappij P. Noordhoff N.V. of Groningen, the Uitgeverij Het Spectrum N.V. of Utrecht jan twerp, the editors of Euclides and the editors of the Algemeen Nederlands Tijdschrift voor Wijsbegeerte en Psychologie for their permission to reproduce previously published studies in this book; for more detailed information I refer to 'Sources'. Finally I wish to thank Dr. D. H. J. de Jongh for his spontaneity in accepting the task of translating this book into English. Also I want to express my gratitude to Mrs. S. de Jongh-Kearl for the advice she gave her husband during the translation. c. P. C. BETH-P ASTOOR

CONTENTS A NOTE ON THE EDITING ACKNOWLEDGEMENTS v VII PART I INTRODUCTION 3 CHAPTER I. THE FUNDAMENT AL CRITERION FOR THE SOUNDNESS OF ARGUMENTS 9 1. Introduction 9 2. Fundamental criterion for the soundness of arguments 9 3. The formal character oflogic 11 4. The transition to mathematical logic 11 5. Construction of a fragment of modern logic 13 6. Natural deduction 17 7. Supplementary considerations 18 CHAPTER II. INFERENTIAL AND CLASSICAL LOGIC 21 8. Semantic and operational aspects of meaning 21 9. Purely implicationallogic 21 10. Deduction problems and deductive tableaux 22 11. Truth-value problems and semantic tableaux 25 12. Inferential and classical logic - Peirce's Law 26 13. Other aspects of meaning 28 14. Informa11ogic: G. Mannoury, A. Naess, Ch. Perelman 29 CHAPTER III. PROOF BY CONTRADICTION 30 15. Introductory remarks 30

x ASPECTS OF MODERN LOGIC 16. Conversion of closed tableaux into natural deductions 31 17. The negation 32 18. Semantic and deductive tableaux 33 19. Final considerations - completeness of classical purely implicationallogic 37 CHAPTER IV. THE PROBLEM OF LOCKE-BERKELEY 42 n~~~ ~ 21. Statement of the problem and attempts at its solution 42 22. The exposition method of Aristotle 44 23. Appeal to the method of semantic tableaux 45 24. The completeness problem - informal and heuristic deduction methods 48 CHAPTER V. ON THE SO-CALLED 'THOUGHT MACHINE' 63 25. Introduction 26. Binary arithmetic and logic 27. Specific operations of the human intellect 28. Prehistory 29. Difficulties 30. Development of modern formal logic 31. Automatization of reasoning 32. Analysis of the difficulties 33. An example 34. Heuristics or methodology 35. Methodology 36. Concluding remarks 63 63 65 67 68 69 69 70 71 72 72 73 CHAPTER VI. CHAPTER VII. THE PARADOXES PART II REASON AND INTUITION 77 86 CHAPTER VIII. FORMALIZED LANGUAGE AND COMMON USAGE 102

CONTENTS XI CHAPTER IX. CONSIDERATIONS ABOUT LOGICAL THOUGHT 118 CHAPTER X. CONSTANTS OF MATHEMATICAL THOUGHT 140 SOURCES BIBLIOGRAPHY 165 166 INDEX OF NAMES 171 INDEX OF SUBJECTS 174