Language, Truth and Logic in Mathematics
One can distinguish, roughly speaking, two different approaches to the philosophy of mathematics. On the one hand, some philosophers (and some mathematicians) take the nature and the results of mathematicians' activities as given, and go on to ask what ph
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JAAKKO HINTIKKA SELECTED PAPERS VOLUME 3
The titles published in this series are listed at the end of this volume
JAAKKO HINTIKKA Boston University
LANGUAGE, TRUTH AND LOGIC IN MATHEMATICS
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4923-0 DOI 10.1007/978-94-017-2045-8
ISBN 978-94-017-2045-8 (eBook)
Printed on acid-free paper
All Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 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 information storage and retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
ORIGIN OF THE ESSAYS
vii
INTRODUCTION
ix
1.
"What Is Elementary Logic? Independence-Friendly Logic as the True Core Area of Logic"
2.
(with Gabriel Sandu) "A Revolution in Logic?"
27
3.
"A Revolution in the Foundations of Mathematics?"
45
4.
"Is There Completeness in Mathematics after G6del?"
62
5.
"Hilbert Vindicated?"
84
6.
"Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics"
106
"Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics"
130
(with Gabriel Sandu) "The Skeleton in Frege's Cupboard: The Standard versus Nonstandard Distinction"
144
9.
(with Arto Mutanen) "An Alternative Concept of Computability"
174
10.
(with Gabriel Sandu) "What is the Logic of Parallel Processing?"
189
11.
"Model Minimization -
212
12.
"New Foundations for Mathematical Theories"
7.
8.
An Alternative to Circumscription"
225
ORIGIN OF THE ESSAYS
The following list indicates the first publication forums of the different essays included in the present volume (the forthcoming publication forum, if an essay ,lppears here for the first time): 1.
2. 3. 4. 5. 6.
7. 8.
9. 10. 11. 12.
"What Is Elementary Logic? Independence-Friendly Logic as the True Core Area of Logic", in K. Gavroglu et aI., editors, Physics, Philosophy and the Scientific Community, Kluwer Academic, Dordrecht, 1995, pp. 301-326. (with Gabriel Sandu) "A Revolution in Logic?", Nordic Journal for Philosophical Logic (new). "A Revolution in the Foundations of Mathematics?", Synthese vol. 111 (1997), pp. 155-170. "Is There Completeness in Mathematics after Godel?", Philosphical Topics vol. 17, no. 2 (1989), pp. 69-90. "Hilbert Vindicated?", Synthese vol. 110 (1997), pp. 15-36. "Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics", in Jaakko Hintikka, editor, From Dedekind to Giidel: Essays on the Development of the Foundations of Mathematics, Kluwer Academic, Dordrecht, 1995, pp. 21-44. "Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics", in E. Agazzi, editor, Modern Logic, D. Reidel, Dordrecht, 1981, pp. 283-296. (with Gabriel Sandu) "The Skeleton in Frege's Cupboard: The Stan
