Table Of ContentLogic, Language and Computation
APPLIED LOGIC SERIES
VOLUMES
Managing Editor
Dov M. Gabbay, Department o/Computing, Imperial College, London, U.K.
Co-Editor
Jon Barwise, Department 0/ Philosophy, Indiana University, Bloomington, IN,
U.S.A.
Editorial Assistant
Jane Spurr, Department o/Computing, Imperial College, London, U.K.
SCOPE OF THE SERIES
Logic is applied in an increasingly wide variety of disciplines, from the traditional
subjects of philosophy and mathematics to the more recent disciplines of cognitive
science, computer science, artificial intelligence, and linguistics, leading to new
vigor in this ancient subject. Kluwer, through its Applied Logic Series, seeks to
provide a home for outstanding books and research monographs in applied logic,
and in doing so demonstrates the underlying unity and applicability of logic.
The titles published in this series are listed at the end of this volume.
Logic, Language and
Computation
edited by
SEIKIAKAMA
Computational Logic Laboratory,
Department ofl nformation Systems,
Teikyo Heisei University,
Japan
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6377-7 ISBN 978-94-011-5638-7 (eBook)
DOI 10.1007/978-94-011-5638-7
Logo design by L. Rivlin
Printed on acid-free paper
AII Rights Reserved
© 1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover 1s t edition 1997
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.
EDITORIAL PREFACE
The editors of the Applied Logic Series are happy to present to the reader
the fifth volume in the series, a collection of papers on Logic, Language
and Computation. One very striking feature of the application of logic to
language and to computation is that it requires the combination, the
integration and the use of many diverse systems and methodologies - all
in the same single application. The papers in this volume will give the
reader a glimpse into the problems of this active frontier of logic.
The Editors
CONTENTS
Preface IX
1. S. AKAMA
Recent Issues in Logic, Language and Computation 1
2. M.J. CRESSWELL
Restricted Quantification 27
3. B.H. SLATER
The Epsilon Calculus' Problematic 39
4. K. VON HEUSINGER
Definite Descriptions and Choice Functions 61
5. N. ASHER
Spatio-Temporal Structure in Text 93
6. Y. NAKAYAMA
DRT and Many-Valued Logics 131
7. S. AKAMA
On Constructive Modality 143
8. H. W ANSING
Displaying as Temporalizing: Sequent Systems for Subintuitionistic Logics 159
9. L. FARINAS DEL CERRO AND V. LUGARDON
Quantification and Dependence Logics 179
10. R. SYLVAN
Relevant Conditionals, and Relevant Application Thereof 191
Index 245
Preface
This is a collection of papers by distinguished researchers on Logic, Lin
guistics, Philosophy and Computer Science. The aim of this book is
to address a broad picture of the recent research on related areas. In
particular, the contributions focus on natural language semantics and
non-classical logics from different viewpoints.
The editor's paper surveys recent issues in Logic, Language and Com
putation to serve as an introduction to the book. The papers by Cress
well, Slater, and von Heusinger investigate natural language semantics in
the tradition of standard predicate logic. Both Slater and von Heusinger
propose to use Hilbert's f-calculus as a promising framework for formal
izing natural language discourse. Asher's and Nakayama's papers are
about Discourse Representation Theory (DRT), which is one of the suc
cessful theories for discourse semantics.
The remaining papers are concerned with several non-classical logics.
Akama's paper provides a constructivist approach to modality based on
Nelson's constructive logic with strong negation with a Kripke style se
mantics. Wansing's paper studies sequent calculi for subintuitionistic
logics which are subsystems of intuitionistic logic by dropping struc
ture rules. Farinas del Cerro and Lugardon work out a proof-theoretic
foundation for dependence logics as proposed by R. 1. Epstein. Sylvan
addresses a relevant approach to conditionals within the framework of
relevant logics and suggests possible applications in various fields.
I believe that the contributions in this book will give an interesting
overview of the recent topics in formal logic and related areas. I wish to
thank the contributors and the referees. Finally, I dedicate the book to
Prof. Akira Ikeya, who introduced me to formal logic and semantics.
Ichihara
Seiki Akama
SEIKI AKAMA
RECENT ISSUES IN LOGIC, LANGUAGE AND
COMPUTATION
1. Introduction
This paper attempts to survey recent issues in Logic, Language and
Computation, and also serves as an introduction to the present book.
Of course, my survey is not exhaustive, but it will provide background
information to the reader for understanding current topics in these ar
eas. In particular, I give an exposition of recent topics in formal logic,
formal semantics and artificial intelligence (AI). I also try to discuss
some important directions for future work in connection with several
contributions in this book.
The issues in this book clearly cover different fields, namely logic,
linguistics, philosophy and computer science (including AI). In these ar
eas, similar problems have been independently studied. However, in the
last decade, there has been a great deal of discussion and interaction
between researchers from these fields. Now we can see that there is in
fact a new interdisciplinary area that relates them. I believe that one
of the reasons for this development is the unifying role played by formal
logic: logic is used as a tool for studying problems in all these different
areas. For example, Montague semantics (an important approach to the
formal semantics of natural language) uses intensional logic, one of the
branches of modal logic, as its basis. To give another example, logic
programming is obviously the result of investigations into resolution cal
culi, whose principal aim is to develop computer-oriented proof systems.
It is thus interesting to look at these related areas from a logical point
of view. Accordingly, the present book collects papers that use logic to
focus on some important issues in this interdisciplinary area.
The rest of this introduction is organized as follows. In Section 2, I
review the development of formal logic, and in particular, non-classical
logics, in relation to some philosophical subjects. I briefly review modal,
intuitionistic and many-valued logics. Section 3 focuses on the brief his
tory of formal semantics for natural language. After sketching the basic
ideas of Montague semantics, I argue that newer theories such as dis-
S. Akama (ed.), Logic, Language and Computation, 1-26.
© 1997 Kluwer Academic Publishers.
2 SEIKI AKAMA
course representation theory, property theory and situation semantics
can be regarded as natural improvements of Montague semantics. I sur
vey the issues of reasoning about incomplete information in computer
science and AI in Section 4. In particular, I address the importance
of formalizing non-monotonic reasoning. Some interactions with for
mal logic are also discussed. In the final section, I sketch interesting
prospects in Logic, Language and Computation in connection with the
contributions in this book.
2. Classical Logic or Non-Classical Logics?
It is commonly held that the origin of modern logic can be traced back
to Frege's Begriffsschrift in which the first formal logical system was for
malized. Frege's system is now known as classical logic, which is viewed
as the "standard" logical system. More precisely, by classical logic, I
mean the two-valued propositional and predicate calculus. Frege's in
tention in Begriffsschrift seemed to be to devise the universal language
for mathematics. Later, classical logic was studied in detail by promi
nent logicians. In particular, Hilbert's metamathematical investigations,
Gentzen's proof-theoretic studies and Tarski's model theory are, I be
lieve, important achievements in the early history of logic, and since
then further significant progress has been made.
Logic has also had a great influence on philosophy of mathematics.
In fact, classical logic was considered the only formalized language for
realist mathematics. However, classical logic was challenged by Brouwer
who questioned the law of the excluded middle that is one of central
axioms of classical logic. His idea was later formalized by Heyting as
the so-called intuitionistic logic. Intuitionists claim that intuitionistic
logic is the only correct logic. One can regard intuitionistic logic as the
first non-classical logic.
Nowadays, logical systems classified as non-classical abound in the
literature. In fact, a number of new non-classical logics enable us to
apply logic to several disciplines (see Thrner (1984)) and there is no
doubt that such logics are important. However, some general remarks
on non-classical logics may be in order. As Haack (1978) pointed out,
non-classical logics include "extended" logics and "deviant" logics. No
table logics in the former category include modal logic, tense logic and
epistemic logic. On the other hand, we can view intuitionistic logic,
many-valued logic and relevance logic as deviant logics. (Of course, the
examples mentioned here are not exhaustive.) The basic distinction be-
RECENT ISSUES 3
tween these two categories is the following. Extended logics expand clas
sical logic by additional logical constructs. For example, in modal logic
modal operators are added to classical logic to express modal notions.
In contrast, deviant logics are rivals to classical logic that give up some
classical principles. For example, intuitionistic logic lacks the principle
of excluded middle to make the notion of constructive proofs essential.
In many-valued logics, we allow for many truth-values instead of two
truth-values. Relevance logics do not admit some classical inferences in
order to avoid the paradoxes of implication. In this way, non-classical
logics have been developed to deal with specific topics which cannot be
formalized within classical logic. Below I briefly sketch the main features
of three important non-classical logics, namely modal logic, intuitionistic
logic and many-valued logic.
Modal logic is a logic for formalizing the concepts of necessity and pos
sibility. Historically, modal notions were discussed by Aristotle. How
ever, the first formal approach to modality in modern logic is due to C.
I. Lewis (1918) who studied so-called strict implication. Today, modal
logic can be viewed as an extension of classical logic with modal opera
tors. Technically, only one modal operator is needed, namely either "L"
(necessarily) or "M" (possibly). If we use L as the primitive operator,
M can be introduced by the following definition:
MA =def -,L-,A.
Although the main concern of modal logic is metaphysical modality,
present-day modal logicians use modal logic for various concepts such
as knowledge, belief, time, obligation, provability, and others. In fact,
different versions of modal logics can be obtained by interpreting modal
operators in different ways. For instance, LA can be read "A is known"
in epistemic logic, in which L is epistemically interpreted. This simple
modelling of some intensional notions enables us to utilize modal logic as
a tool for studying several issues in philosophy, linguistics and computer
SClence.
One of the reasons for the great success of modal logic is that we
have an intuitive semantics now known as Kripke semantics or possible
worlds semantics originally due to Kripke (1959, 1963a, b). (A similar
insight into the semantics of modal logic can also be found in Hintikka's
(1962) work on epistemic logic.) A basic idea of Kripke semantics is to
use the notion of possible worlds in the interpretation of modal opera
tors. In fact, necessity (possibility) can be interpreted as truth in all
(some) possible worlds. In other words, truth definitions of modality
