ESSAYS ON MATHEMATICAL AND PHILOSOPHICAL LOGIC

ESSAYS ON MATHEMATICAL AND PHILOSOPHICAL LOGIC

SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Academy of Finland and Stanford University Editors: ROBERT S. COHEN, Boston University DONALD DAVIDSON, University of Chicago GABRIEL NUCHELMANS, University of Leyden WESLEY C. SALMON, University of Arizona VOLUME 122

ESSAYS ON MATHEMATICAL AND PHILOSOPHICAL LOGIC PROCEEDINGS OF THE FOURTH SCANDINAVIAN LOGIC SYMPOSIUM AND OF THE FIRST SOVIET-FINNISH LOGIC CONFERENCE, JYVASKYLA., FINLAND, JUNE 29-JULY 6, 1976 Edited by JAAKKO HINTIKKA, ILKKA NIINILUOTO, and ESA SAARINEN D. REIDEL PUBLISHING COMPANY DORDRECHT :HOLLAND / BOSTON :U.S.A. LONDON:ENGLAND

Library of Congress Cataloging in Publication Da{a Scandinavian Logic Symposium, 4th, Jyviiskylii, Finland, 1976. Essays on mathematical and philosophical logic. (Synthese library; v. 122) Includes bibliographies and index. 1. Logic-Congresses. 1. Hintikka, Kaarlo laakko Juhani, 1929- II. Niiniluoto, Ilkka. III. Saarinen, Esa, 1953- IV. Soviet-Finnish Logic Conference, 1st, Jyviiskylii, Finland, 1976. V. Title. BC5.S28 1976 160 78-14736 ISBN-13: 978-94-009-9827-8 e-isbn-i3: 978-94-009-9825-4 DOl: 10.1007/978-94-009-9825-4 Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland Sold and distributed in the U.S.A., Canada, and Mexico by D. Reidel Publishing Company, Inc. Lincoln Building, 160 Old Derby Street, Hingham, Mass. 02043, U.S.A. All Rights Reserved Copyright 1979 by D. Reidel Publishing Company, Dordrecht, Holland and copyright holders as specified on the appropriate pages within Softcover reprint of the hardcover I st edition 1979 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

PREFACE The Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference were held in JyvaskyIa, Finland, June 29-July 6, 1976. The Conferences were organized by a committee which consisted of the editors of the present volume. The Conferences were supported financially by the Ministry of Education of Finland, by the Academy of Finland, and by the Division of Logic, Methodology, and Philosophy of Science of the International Union of History of Science. The Philosophical Society of Finland and the Jyvaskyla Summer Festival gave valuable help in various practicalities. 35 papers by authors representing 10 countries were presented at the two meetings. Of those papers 24 appear here. THE EDITORS v

TABLE OF CONTENTS PREFACE v PART 1/ PROOF THEORY GEORG KREISEL / Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory 3 DAG PRAWITZ / Proofs and the Meaning and Completeness of the Logical Constants 25 v. A. SMIRNOV / Theory of Quantification and tff-calculi 41 LARS SVENONIUS/Two Kinds of Extensions of Primitive Recursive Arithmetic 49 DIRK VAN DALEN and R. STATMAN / Equality in the Presence of Apartness 95 PART II / INFINITARY LANGUAGES VEIKKO RANTALA / Game-Theoretical Semantics and Back-and- Forth 119 MAARET KAR TTUNEN / Infinitary Languages N o o ~ and. Generalized Partial Isomorphisms 153 PART III / SET THEORY AND MODEL THEORY SOLOMON FEFERMAN/Generalizing Set-Theoretical Model Theory and an Analogue Theory on Admissible Sets 171 JUHA OIKKONEN / Hierarchies of Model Theoretic Definability- An Approach to Second Order Logics 197 JUSSI KETONEN I Open Problems in the Theory of Ultrafilters 227 PART IV / GENERALIZED QUANTIFIERS J. A. MAKOWSKY / The Reals Cannot Be Characterized Topologically with Strictly Local Properties and Countability Axioms 251 M. KR YNI CKI / On the Expressive Power of the Language Using the Henkin Quantifier 259

viii T ABLE OF CONTENTS JOUKO VAANANEN / Remarks on Free Quantifier Variables 267 PART V / RECURSION THEORY DAG NORMANN / Recursion in 3 and a Splitting Theorem 275 I. A. LAVROV / Retracts of Post's Numbering and Effectivization of Quantifiers 287 PART VI/LOGIC AND NATURAL LANGUAGE JAAKKO HINTIKKA / Quantifiers in Natural Languages: Some Logical Problems, I 295 JENS ERIK FENST AD / Models for Natural Languages 315 ESA SAARINEN / Backwards-Looking Operators in Tense Logic and in Natural Language 341 PART VII / PHILOSOPHICAL LOGIC BENGT HANSSON I Paradoxes in a Semantic Perspective 371 VIT ALI TSELISHCHEV / Hintikka's Possible Worlds and Rigid Designators 387 A. A. STARCHENKO / On the Content Analysis of Two Normative Notions 401 YU. GLADKICH / Singular Terms, Existence and Truth: Some Remarks on a First Order Logic of Existence 405 PART VIII / TRUTHLIKENESS DA VID MILLER / On Distance From the Truth as a True Distance 415 ILKKA NIINILUOTO I Truthlikeness in First-Order Languages 437 INDEX OF NAMES 459