Handbook of Philosophical Logic. Volume I: Elements of Classical Logic

503 Pages·1983·13.708 MB·English

by  Dov M. Gabbay, Franz Guenthner (eds.)

HANDBOOK OF PHILOSOPHICAL LOGIC
VOLUMEI
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
J A A K K 0 HI N T I K K A, Florida State University, Tallahassee
Editors:
DON A L D D A V I D SON, University of California, BerkeZey
GABRIEL NUCHELMANS, University of Leyden
WE S LE Y C. SAL M 0 N, University of Pittsburgh
VOLUME 164
HANDBOOK OF
PHILOSOPHICAL
LOGIC
Volume I:
Elements of Classical Logic
Edited by
D. GABBAY
Department of Mathematics and Computer Science, Bar-Ilan University, Israel
and
F. GUENTHNER
Neuphilologische Fakultaet, University of Tuebingen, West Germany
D. REIDEL PUBLISHING COMPANY
A MEMBER OF THE KLUWER ... ACADEMIC PUBLISHERS GROUP
DORDRECHT/BOSTON/LANCASTER
Library of Congress Cataloging in Publication Data
Main entry under title:
Elements of classical logic.
(Handbook of philosophicallogicj v. 1) (Synthese libraryj v. 164)
Bibliography: p.
Includes indexes.
1. Logic-Addresses, essays, lectures. I. Gabbay, Dov M.,
1945- . 11. Guenthner, Franz. Ill. Series.
BC6.H36 1983 vol. 1 (BC71] 160s (160] 83-4276
Published by D. Reidel Publishing Company,
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Sold and distributed in the U.S.A. and Canada
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In all other countries, sold and distributed
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P.O. Box 322, 3300 AH Dordrecht, Holland.
ISBN-I 3:978-94-009-7068-7 eISBN-13:978-94-009-7066-3
DOl: 10.1007/978-94-009-7066-3
4-1290-300 ts
All Rights Reserved
© 1983 by D. Reidel Publishing Company
No part of the material protected by this copyright notice may be reproduced or
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CONTENTS TO VOLUME I
ACKNOWLEDGEMENTS ri
PREFACE ~
A NOTE ON NOTATION xiii
1.1. W I L F RID HOD G E S / Elementary Predicate Logic 1
1.2. GO RAN S U ND HOLM / Systems of Deduction 133
1.3. HUG U E S LE B LA N C / Alternatives to Standard First-order
Semantics 189
1.4. JO HA N VA N B ENT HEM and KEES DO ETS / Higher-order
Logic 275
1.5. ALLEN HAZEN / Predicative Logics 331
1.6. DIRK V AN DALEN / Algorithms and Decision Problems: A
Crash Course in Recursion Theory 409
NAMEINDEX ~9
SUB JE C TIN D EX 485
TABLE OF CONTENTS TO VOLUMES 11, Ill, AND IV 495
ACKNOWLEDGEMENTS
The preparation of the Handbook of Philosophical Logic was generously
supported by the Lady Davis Fund at Bar-Ilan University, Ramat-Gan, Israel
and the Werner-Reimers-Stiftung, Bad Homburg, West Germany, which
provided us· with the chance of discussing the chapters in the Handbook
at various workshops with the contributors. It is a great pleasure to acknowl
edge the assistance of these institutions during the preparation of this col
lection. We benefitted further from the editorial and personal advice and
help from the publisher.
Most important of all, we would like to express our thanks to all the
contributors to the Handbook for their unlimited goodwill, their professional
counsel, as well as their friendly support. They have made the preparation
of this collection a stimulating and gratifying enterprise.
D. GAB BAY (Bar-Ilan University)
F. G U ENT H N E R (University of Tuebingen)
PREFACE
The aim of the first volume of the present Handbook of Philosophical Logic
is essentially two-fold: First of all, the chapters in this volume should provide
a concise overview of the main parts of classical logic. Second, these chapters
are intended to present all the relevant background material necessary for the
understanding of the contributions which are to follow in the next three
volumes. We have thought it to be of importance that the connections
between classical logic and its 'extensions' (covered in Volume 11) as well as
its most important 'alternatives' (covered in Volume Ill) be brought out
clearly from the start.
The first chapter presents a clear and detailed picture of the range of what
is generally taken to be the standard logical framework, namely, predicate
(or first-order quantificational) logic. On the one hand, this chapter surveys
both propositionai logic and first-order predicate logic and, on the other
hand, presents the main metalogical results obtained for them. Chapter 1.1
also contains a discussion of the limits of first-order logic, i.e. it presents
an answer to the question: Why has predicate logic played such a formidable
role in the formalization of mathematics and in the many areas of philo
sophical and linguistic applications?
Chapter 1.1 is prerequisite for just about all the other chapters in the
entire Handbook, while the other chapters in Volume I provide more detailed
discussions of material developed or hinted at in the first chapter.
Chapters 1.2 centers on proof-theoretic questions and outlines several
well-known frameworks for the development of a proof-theory for predicate
logic.
The semantic account for the logical systems presented in Chapter 1.1 is
essentially model-theoretic; Chapter 1.3 surveys several alternatives to the
standard model-theoretic semantics for predicate logic.
Chapter 1.4 takes a big step beyond predicate logic by introducing
languages where quantification is not only over individuals but also over
higher-order entities, such as sets, functions, sets of sets, etc. Various results
concerning second-order and higher-order logic as well as relevant material
from abstract model theory are pulled together here.
ix
D. Gabbay and F. Guenthner (eds .), Handbook of Philosophical Logic, Vol. I, ix-xi.
© 1983 by D. Reidel Publishing Company.
x PREFACE
Chapter 1.5 is essentially an introduction to ramified type theory and
its philosophical and logical significance. The reason why it is included
in the present volume is due to the fact that it covers material which over
laps greatly with the contents of all of the volumes of the Handbook. Being
most closely connected to the chapter on higher-order logic, it was decided
to make it part of Volume I.
Finally, Chapter 1.6 is a concise guide to a number of concepts from the
theory of algorithms and decidability which will turn up in many later
chapters in the discussion of the meta-mathematical properties of the logical
systems presented here.
In preparing the Handbook of Philosophical Logic, the editors have tried
to follow a number of guidelines, the most important of which are the fol
lowing. First, the field of philosophical logic being so vast and heterogeneous
with roots in general philosophy, mathematical logic and theoretical linguistics
to cite just a few, it was never our intention to aim for an exhaustive coverage
of all the work done under the term philosophical logic. Most of the domains
of investigation not covered in the present volumes were thus excluded on
purpose as they seemed to belong more to such related disciplines like the
philosophy of science or the philosophy of mathematics. Second, our pref
erence has gone from the beginning towards the form of substantial self
contained survey essays as opposed to a large number of short articles. The
reason for this is twofold: there exist already many short expository surveys
of practically all the topics covered in the Handbook, on the other hand,
relatively detailed expositions of the main results and problems in the various
fields simply cannot be accommodated in the space of a few pages. Third,
the division of the Handbook into the present four volumes is not intended
to imply a strict and theoretical division of work in philosophical logic into
exactly these subareas. On the contrary, the four volumes should rather be
regarded as a whole. Neither the place, order of appearance nor the relative
length of the various contributions should be taken as indications of the
general structure of the entire field of philosophical logic. The Handbook
was conceived to be a guide to the many areas of philosophical logic without
imposing a particular picture of either their relative merits or status within
the field. Fourth and finally, the survey articles in the Handbook should
be accessible and useful to researchers and students in a variety of fields.
It is hoped that philosophers, logicians, mathematicians, linguists, psychol
ogists, computer scientists and many others can fruitfully draw upon the
material presented in these volumes and arrive perhaps at the conclusion that
these fields are not quite as independent as they might previously have
PREFACE xi
thought. Although in some cases new results are included in the presentation,
the emphasis in every essay is on providing a general survey of the subject
in question. The level of sophistication and the required prerequisites are on
the whole rather uniform. As pointed out above, it is in addition the essential
function of Volume I to put the necessary background for later papers at the
reader's disposition.
D. M. GABBA Y
F. GUENTHNER