Course of Mathematical Logic Volume I Relation and Logical Formula
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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, JAAKKO HINTIKKA,
Rockefeller University and Princeton University Academy of Finland and Stanford University
GABRIEL NUCHELMANS, WESLEY
C.
SALMON,
University of Leyden
Indiana University
ROLAND FRAÎSSE
COURSE OF MATHE·MATICAL LOGIC VOLUME 1
Relation and Logica/ Formula
rJ.M
(f[J SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
COURS DE LOGIQUE MATHEMATIQUE, TOME 1 First published by Gauthier-Villars and E. Nauwelaerts, Paris, Louvain, 1967 Second revised and improved edition, Gauthier-Villars, Paris, 1971 Translation edited by David Louvish
Library of Congress Catalog Card Number 72-95893 ISBN 978-90-277-0403-0 ISBN 978-94-010-2529-4 (eBook) DOI 10.1007/978-94-010-2529-4
Ali Rights Reserv.ed Copyright© 1973 by Springer Science+Business Media Dordrecht Originally published by D. Reide1 Publishing Company, Dordrecht, Holland in 1973 Softcover reprint of the hardcover 1st edition 1973 No part ofthis book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher
TABLE OF CONTENTS
PREFACE
IX
INTRODUCTION
XI
CHAPTER 1/WORD, FORMULA
1.1. Word, Rank, Occurrence, Concatenation, Interval, Substitution 1.2. Weight; Formula, Subformula, Occurrence of a Formula, Term Dominating Another Term 1.3. Weight of a Word 1.4. Atom; Weighted Substitution Exercises
1 2 6 8 13
CHAPTER 2/CONNECTIONS AND CONNECTIVE FORMULAS
2.1. Truth Values; Connection, Arity; Negation, Conjunction, Disjunction, Implication, Deducibility 2.2. Connective Formula; Value, Tautology, Contradiction, Deducibility 2.3. Formula in I, f\, V; Dual; Conjunctive and Disjunctive Formulas 2.4. Completeness Lemma for Tautologies 2.5. Translation of the Completeness Lemma by Means of Weighted Substitution 2.6. Active, Inactive Index or Atom; Reduction, Positive Connection, Alternating Connection 2.7. Set Closed under Formulation Exercises
17 21 24 27 31 33 36 42
VI
COURSE OF MATHEMATICAL LOGIC CHAPTER 3/RELATION, MUL TIRELATION; OPERATOR AND PREDICATE
3.1. Relation, Multirelation, Base, Arity; Concatenation; Restriction and Extension 3.2. Isomorphism, Automorphism, Embedding, Age 3.3. Deducibility of Relations 3.4. Operator, Predicate, Predicarity and Arity; Model; Selector, + and - Operators, Identifier, Connector, Rank-Changer, Dilator 3.5. Deducibility of Operators; Composite Operator; Image and Inverse Image of a Class 3.6. Active or Inactive Index; Quantifier; Composite Quantifier 3.7. Composite of Quantifier and Connector 3.8. Composite of Quantifier and Rank-Changer
60 62 65 68
Exercises
74
47 51 55
57
CHAPTER 4/LOCAL ISOMORPHISM; FREE OPERATOR AND FREE FORMULA
4.1. 4.2. 4.3. 4.4. 4.5.
Local Isomorphism; Local Automorphism Free Interpretability Free Operator Compactness Lemma Active or Inactive Predicate; the Operators Pr!/' and SJr!/'; Interpolatio
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