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1、TECHNICAL METHODS IN PHILOSOPHY John L. Pollock UNIVERSITY OF ARIZONA Westview Press BOULDER SAN FRANCISCO LONDON Focus Series All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, re- cording, or
2、 any information storage and retrieval system, without permission in writing from the publisher. Copyright 1990 by Westview Press, Inc. Published in 1990 in the United States of America by Westview Press, Inc., 5500 Central Avenue, Boulder, Colorado 80301, and in the United Kingdom by Westview Press
3、, Inc., 13 Brunswick Centre, London WCIN IAF, England Library of Congress Cataloging-in-Publication Data Pollock, John L. Technical methods in philosophy / by John L. Pollock. p. cm.-(The Focus series) ISBN 0-8133-7871-0. ISBN 0-8133-7872-9 (pbk.). 1. Logic, Symbolic and mathematical. 2. Set theory.
4、 3. Predicate calculus. 4. First-order logic. 5. Metatheory. I. Title. 11. Series: Focus series (Westview Press) BC135.P683 1990 160-dc20 89-3821 6 CIP Printed and bound in the United States of America The paper used in this publication meets the requirements of the American (ro) National Standard f
5、or Permanence of Paper for Printed Library Materials Z39.48-1984. For Carole CONTENTS PREFACE IX CHAPTER ONE: SET THEORY 1. The Logical Framework 1 2. The Basic Concepts of Set Theory 2 2.1 Class Membership 2 2.2 Definition by Abstraction 3 2.3 Some Simple Sets 5 2.4 Subsets 6 2.5 Unions and Interse
6、ctions 8 2.6 Relative Complements 11 2.7 Power Sets 12 2.8 Generalized Union and Intersection 13 3. Relations 15 3.1 Ordered Pairs 15 3.2 Ordered n-tuples 18 3.3 The Extensions of Properties and Relations 19 3.4 Operations on Relations 21 3.5 Properties of Relations 23 3.6 Equivalence Relations 25 3
7、.7 Ordering Relations 27 4. Functions 30 4.1 Mappings Into and Onto 32 4.2 Operations on Functions 33 4.3 The Restriction of a Function 34 4.4 One-One Functions 34 4.5 Ordered n-tuples 35 4.6 Relational Structures and Isomorphisms 35 5. Recursive Definitions 37 6. Arithmetic 44 6.1 Peanos Axioms 44
8、6.2 Inductive Definitions 5 1 6.3 The Categoricity of Peanos Axioms 54 viii CONTENTS 6.4 Set-Theoretic Surrogates 56 6.5 Arithmetic 59 CHAPTER TWO: LOGIC 1. The Predicate Calculus 62 1.1 Syntax 62 1.2 Formal Semantics 66 1.3 Derivations 72 1.4 Definite Descriptions 84 1.5 First-Order Logic with Func
9、tions 87 2. First-Order Theories 87 2.1 Axiomatic Theories 87 2.2 Semantic Closure 89 2.3 Godels Theorem 96 3. Higher-Order Logic 106 SOLUTIONS TO EXERCISES Chapter One Ill Chapter Two 117 LIST OF SYMBOLS 121 INDEX 123 PREFACE The purpose of this book is to introduce the technical tools and concepts
10、 that are indispensable for advanced work in philo- sophy and to do so in a way that conveys the important concepts and techniques without becoming embroiled in unnecessary technical details. The most valuable technical tools are those provided by set theory and the predicate calculus. Knowledge of
11、the predicate calculus is indispensable if for no other reason than that it is used so widely in the formulation of philosophical theories. This is partly because it has become conventional to formulate theories in that way, but it is also because the predicate calculus provides a medium for such fo
12、rmulations that is both concise and unambiguous. Furthermore, a knowledge of the predicate calculus is required for an understanding of Godels theorem. Godels theorem is one of the intellectually most important achievements of this century. Godels theorem and related theorems concerning the predicat
13、e calculus appear to have amazing implications for epistemology and the philosophy of mathematics. No student of philosophy can be regarded as properly educated without some grasp of these theorems. An understanding of the predicate calculus requires some prior understanding of set theory. In additi
14、on, set theory is important in its own right. Set theory allows the concise and perspicuous formulation of principles that can only be formulated in a very complicated manner within nontechnical language. Given a basic grasp of set theoretic concepts, technical principles often become easy to unders
15、tand, and it is easier to see what their logical consequences are. The same principles can be formulated in nontechnical language, but those formulations tend to become convoluted and unwieldy. A familiarity with set theoretic concepts also provides a different perspective from which to view abstrac
16、t problems. The use of set theoretic concepts often reveals simple structure that would otherwise be concealed and helps one to think more clearly about the problems. x PREFACE Nontechnical philosophers are often put off by the fact that set theory is generally developed as a foundation for mathematics. One begins with obscure axioms and derives formal theorems in a rigorou
