Table Of ContentMODERN LOGIC - A SURVEY
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
J AAKKO HINTIKKA, Florida State University
Editors:
DONALD DAVIDSON, University of Chicago
GABRIEL NUCHELMANS, University ofL eyden
WESLEY C. SALMON, University ofA rizona
VOLUME 149
MODERN LOGIC - A SURVEY
Historical, Philosophical, and Mathematical Aspects
ofM odem Logic and its Applications
Edited by
EV ANDRO AGAZZI
Dept. of Philosophy, University of Genoa, Italy
D. REIDEL PUBLISHING COMPANY
DORDRECHT : HOLLAND / BOSTON: U.S.A.
LONDON: ENGLAND
Library of Congress Cataloging in Publication Data
Main entry under title:
Modern logic - a survey.
(Synthese library; v. 149)
Includes bibliographies and index.
1. Logic, Modern-Addresses, essays, lectures. I. Agazzi, Evandro.
BC38.M54 160 80-22027
ISBN-13: 978-94-009-9058-6 e-ISBN-13: 978-94-009-9056-2
DOl: 978-94-009-9056-2
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Italian Edition by
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Copyright © 1981 by D. Reidel Publishing Company, Dordrecht, Holland
So/kover reprint of the hardcover I st edition 1981
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TABLE OF CONTENTS
PREFACE vii
PART 1: INTRODUCTION
J. M. BOCHENSKI/ The General Sense and Character of Modem Logic 3
S. J. Su RM A / The Growth of Logic Out of the Foundational Research
in Mathematics 15
PART 2: PURE LOGIC
K. SCHUTTE / Proof Theory 37
A. MACINTYRE / Model Theory 45
G. KREISEL / Constructivist Approaches to Logic 67
J. BARWISE / Infmitary Logics 93
A. ROSE / Many-Valued Logics 113
N. D. BELNAP, Jr. / Modal and Relevance Logics: 1977 131
PART 3: THE INTERPLAY BETWEEN LOGIC AND
MATHEMATICS
E. CASARI / Logic and the Foundations of Mathematics 155
G. TAKEUTI / Logic and Set Theory 167
H. HERMES / Recursion Theory 173
A. S. TROELSTRA / The Interplay Between Logic and Mathematics:
Intuit:onism 197
J. E. FENSTAD / Logic and Probability 223
G. E. REYES / Logic and Category Theory 235
PART 4: THE RELEVANCE OF LOGIC TO OTHER
SCIENTIFIC DISCIPLINES
E. AGAZZI / Logic and Methodology of Empirical Sciences 255
vi T ABLE OF CONTENTS
J. HINTIKKA , Standard Vs. Nonstandard Logic: Higher..()rder, Modal,
and First..()rder Logics 283
C. BOHM , Logic and Computers 297
G. J. MASSEY' Logic and Linguistics 311
M. L. DALLA CHIARA' Logical Foundations of Quantum Mechanics 331
L. J. COHEN' Inductive Logic 1945-1977 353
PART 5: LOGIC AND PHILOSOPHICAL TOPICS
C. LEJEWSKI , Logic and Ontology 379
G. H. VON WRIGHT 'Problems and Prospects of Deontic Logic - A
Survey 399
B. C. V AN FRAASSEN' Report on Tense Logic 425
R. STALNAKER' Logical Semiotic 439
CH. PERELMAN I Logic and Rhetoric 457
INDEX OF NAMES 465
INDEX OF SUBJECTS 473
PREFACE
Logic has attained in our century a development incomparably greater than
in any past age of its long history, and this has led to such an enrichment and
proliferation of its aspects, that the problem of some kind of unified recom
prehension of this discipline seems nowadays unavoidable. This splitting into
several subdomains is the natural consequence of the fact that Logic has
intended to adopt in our century the status of a science. This always implies
that the general optics, under which a certain set of problems used to be con
sidered, breaks into a lot of specialized sectors of inquiry, each of them being
characterized by the introduction of specific viewpoints and of technical
tools of its own. The first impression, that often accompanies the creation
of one of such specialized branches in a diSCipline, is that one has succeeded
in isolating the 'scientific core' of it, by restricting the somehow vague and
redundant generality of its original 'philosophical' configuration. But, after a
while, it appears that some of the discarded aspects are indeed important
and a new specialized domain of investigation is created to explore them. By
follOwing this procedure, one finally finds himself confronted with such a
variety of independent fields of research, that one wonders whether the fact
of labelling them under a common denomination be nothing but the
contingent effect of a pure historical tradition. At this stage a need of con
ceptual clarification is perceived, that is the need of understanding why we
still accept to call the whole of these different fields, e.g., 'Logic', or 'Physics',
or 'Biology', etc. and this question can be answered only through a work of
philosophical understanding and historical reconstruction.
All this is true for every science and for Logic as well, which has ac
complished not only its transition from the condition of being a branch of
philosophy to the fully-fledged shape of an articulate science, but has also
given rise (like several 'pure' sciences) to a great lot of applications. This is
why an adequate understanding of what modem logic is requires an inter
disciplinary approach, and at the same time an effort of philosophical appre
ciation, not in order to interpret it again as a part of Philosophy but in the
sense of making it the object of a philosophical analysis and interpretation.
This was the aim ofanintemationalconference on Modem Logic organized
in Rome by the Institute of the Italian Encyclopedia and by the Italian
vii
E. Agazzi (ed.), Modern Logic - A Survey, vii-viii.
Copyright ©1980 by D. Reidel Publirhing Company.
viii PREFACE
Society of Logic and Philosophy of Science in September 1976. The present
volume contains the papers presented at that meeting, each of them being a
general survey of one of the most relevant fields in which Logic is subdivided,
or of one of the most important sets of problems which are connected with
Logic in our days. A simple glance at the table of contents shows the width
of the horizon that has been explored (thorough completeness could not be
reached and was not an ambition of the organizers). The purpose of the
conference was to provide a general appreciation of modem logic which
could be accessible to cultivated people, without need of any specific
technical competence. In other words, the cultural pOSition and role of Logic
was hoped to become apparent, rather than the exact patterns of its technical
machinery. This, in particular, explains the subdivision of the topics
according to a line which takes into account the different kinds of intellectual
'interest' under which Logic may be approached today, and which go from
its historical development, to its impact on mathematics, to its internal sub·
division, to its different applications, to its philosophical commitments.
Of course every contributor has conceived this task according to his own
estimation and this has led to different levels of clarity, of technical com·
plexity, of specialization or generality.
It is left therefore to the reader to chose the papers with which to start
in reading this book, according to his personal taste and background, but one
can be confident that a singificant image of modem logic as a whole may
be obtained from this volume, which might hardly be shaped even by the
careful study of several good textbooks.
EV ANDRO AGAZZI
PART 1
INTRODUCTION
1. M. BOCHENSKI
THE GENERAL SENSE AND CHARACTER
OF MODERN LOGIC
By 'Modem Logic' (abridged as 'ML') the class of studies is meant which
were originated by Leibniz, developed, among others, by Boole, Peirce,
Frege, Peano, Lesniewski and their followers; in other tenns the class of
studies listed in Alonzo Church's Bibliography and in The Journal of
Symbolic Logic.
The expression 'ML' is sometimes used, it is true, in other ways, e.g.
to denote studies in Hegelian dialectics. Those uses are irrelevant for the sake
of the present paper which will be exclusively concerned with ML as
described above. It may be only said, that no other known sort of contem
porary logic can compare with the latter as far as standards of procedures and
quality of results are concerned.
The aim of the paper is to describe - as the title selected by the organizers
of the conference indicates - the general sense and character of ML thus
understood. In other tenns an attempt will be made to find the fundamen
tal characteristics of ML-al studies.
The method used will be comparative. We are going to ask: How does
ML compare with three fields with which it is usually linked: logic,
mathematics and philosophy? Is ML Logic and, if so, how does it differ from
other types of logic? Is it a mathematical discipline and, if that is the case,
what is the difference between it and other mathematical sciences? Is it
philosophy and, this being admitted, what is its place among the other
philosophical diSCiplines?
The present paper will be mostly concerned with the first class of
problems, the comparison between ML and the other types of logic; the
other two classes of problems will be treated only marginally. As far as the
main problems are concerned, the method will necessarily be historical: for,
contrary to mathematics and philosophy, all other fonns of logic with which
ML may be compared belong to the past.
I
It will be convenient to begin with a summary classification of the main
parts and systems of ML. They can be classified under two headings: either
3
E. Agazzi (ed.). Modem Logic - A Survey. 3-14.
Copyright e 1980 by D. Reidel Publishing Company.
