Table Of ContentQUANTIFIERS: LOGICS, MODELS AND COMPUTATION
VOLUME ONE
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Boston University
Editors:
DIRK VAN DALEN, University of Utrecht, The Netherlands
DONALD DAVIDSON, University of California, Berkeley
THEO A.F. KUIPERS, University ofGroningen, The Netherlands
PATRICK SUPPES, Stanford University, California
JAN WOLEN-SKI, Jagiellonian University, Krakow, Poland
VOLUME 248
QUANTIFIERS:
LOGICS, MODELS AND
COMPUTATION
Volume One: Surveys
Edited by
MICHAL KRYNICKI
Department of Mathematics. University of Warsaw
MARCIN MOSTOWSKI
Department of Philosophy. University of Warsaw
and
LESLAW W. SZCZERBA
Siedlce University
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4539-3 ISBN 978-94-017-0522-6 (eBook)
DOI 10.1007/978-94-017-0522-6
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
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.
CONTENTS TO VOLUME I
PREFACE Vll
M. KRYNICKI and M. MOSTOWSKI / Quantifiers, Some Problems and
Ideas 1
P. LINDSTROM / Prologue 21
H.-D. EBBINGHAUS / On the Model Theory of Some Generalized
Quantifiers 25
L. HELLA and K. LUOSTO / Finite Generation Problem and n-ary
Quantifiers 63
1. VAANAEN / Games and Trees in Infinitary Logic: A Survey 105
H. HERRE / Theory of Linear Order in Extended Logics 139
M. KRYNICKI and M. MOSTOWSKI / Henkin Quantifiers 193
X. CAICEDO / Continuous Operations on Spaces of Structures 263
1. PLUM / Model Theory of Topological Structures 297
1. A. MAKOWSKI and Y. B. PNUELI / Computable Quantifiers and
Logics over Finite Structures 313
D. WESTERSTAHL / Quantifiers in Natural Language: A Survey
of Some Recent Work 359
INDEX OF NAMES 409
TABLE OF CONTENTS TO VOLUME II 417
Preface
This volume contains survey papers presenting results of research in se
lected, well defined areas organized around the concept quantifier. The mod
ern concept of quantifier was formed in the late 50-ties and early 50-ties.
During the following 20 years it also proved its usefulness in linguistics and
computer science. The literature on the subject is extensive, but scattered
throughout different journals. Only a few books address this subject, e.g.
J. Barwise and S. Feferman (eds) Model - Theoretic Logics (Springer 1985),
and P. Gardenfors (ed.) Genemlized Quantifiers (Reidel Pub. Co. 1987).
This volume attempts to partially fill the gap. It is the first concentrated
effort to present systematically the main research results related to this sub
ject. The majority of papers are in the style of handbook chapters. All of
them are self-contained, but at various levels of difficulty.
The introduction to the volume explains main notions and ideas and
is offered in the opening paper by Krynicki and Mostowski. It contains a
more detailed description of the volume's content. "Prologue" written by
Per Lindstrom, the author of fundamental ideas related to modern theory of
logics and quantifiers, describes the beginings of the modern concept of gen
eralized quantifier. All the other papers survey specific selected well defined
topics. These survey papers present, as comprehensively as possible, the ac
tual state of research in their respective domains. It was not our intention to
always include complete mathematical proofs of all the theorems presented.
Instead we expected the authors to focus more on general ideas rather than
technical details. However, some unpublished results have occasionally been
included. In such cases complete proofs are supplied.
We have intended to give as complete a presentation as possible of the
research area. However it was physicaly impossible to include everything.
Therefore the selection of topics covered in this volume is not complete. For
example, it does not cover some results and ideas presented in the books
mentioned above, esp. in the Parts A and B of the book edited by Barwise
and Feferman. We have omitted two important topics closely related to
viii
the main subject of the volume: (1) cardinality quantifiers (see Kaufman
The quantifier "there exists uncountably many" and some of its relatives
in Barwise and Feferman 1985), and (2) Lindstrom theorem characterizing
elementary logic, which received extensive treatment in the textbooks such
as Ebbinghaus, Flum and Thomas Mathematical Logic, Springer-Verlag 1985
and the second edition of Chang and Keisler Model Theory, Elsevier Science
Pb. B. V. 1990).
The volume can be useful for those interested in general theory and
applications of generalized quantifiers and model-theoretic logics as well as
to mathematicians, computer scientists, philosophers and linguists interested
in model theory and natural language semantics. The volume presents one
of the most active research areas in modern logic and thus it complements
traditionaly oriented expositions of logic. Individual papers can be used as
discussion material in graduaded seminars and lectures.
This volume was prepared concurrently with the volume Quantifiers:
Logics, Models and Computation II - Contributions published in
the same series and containing a collection of research papers. These volumes
are complementary. The second one presents subjects which are too new to
be summarized.
All of the papers in this volume except for, those by P. Lindstrom,
J. Viiananen, J. A. Makowsky and Y. B. Pnueli, M. Krynicki and M. Mostow
ski, are based on lectures delivered at the conference on "Quantifiers (Gen
eralizations, extensions and variants of elementary logic)" organized
by the Department of Mathematics of Siedlce College and the Section of
Logic of the Institute of Philosophy and Sociology of the Polish Academy
of Sciences and held at Chlewiska, June 10-14 , 1991. The program of the
conference, list of participants and abstracts of the talks can be found in the
Bulletin of the Section of Logic (vol. 20 No. 3/4 Polish Academy of Science,
Institute of Philosophy and Sociology, Siedlce College).
* * *
We would like to thank all the people who helped us in our work on
this volume, contributors and reviewers, participants and organizers of the
Chlewiska conference, and particularly Alicja D~bska for her careful TEXing,
Annie Kuipers from Kluwer for her limitless patience, and Jaakko Hintikka
for his friendly support.
THE EDITORS
Michal Krynicki
and Marcin Mostowski
Quantifiers, Some Problems and Ideas
1
The word quantifier comes from the latin quantitas (quantity) as con
trasted with qualitas (quality). This intuition does not mean that the no
tion of quantifier is understood as an arithmetical concept. We understand
quantifiers rather as qualifiers for distributive concepts (such as countable
nouns) unlike those applicable to nondistributive concepts.
The concept of quantifier makes the main difference between so called
traditional and modern logics. Generalized quantifiers are actually one ofthe
most important logical research area. This volume is devoted to presentation
of recent developpments in this field.
This paper is intended as an introduction to the volume, containing short
presentation of the research area of generalized quantifiers. Additionally
we discuss shortly applications and some new ideas of logical investigations
devoted to quantifiers. The final part of this paper gives short information
about survey papers presented in this volume.
Logic as the research field is treated sometimes as a sample of the most
rigorous and the most unambigous science. In general this opinion is correct.
Unfortunately some of the main logical concepts are extremely ambiguous.
Particularly this is true about the terms logic and quantifier. Usually lo
gicians have no problems with these ambiguities defining their concepts by
precise formal definitions. However in comments or informal considerations
they could be at least puzzling.
Particularly the term quantifier is used to name one of the following six
things:
1. a quantifier expression, e. g. there is ... such that;
2. a quantifier prefix, e. g. there is a: such that;
3. a grammatical construction assigning to a formula (formulae) a com
plex formula obtained by preceding it with a quantifier prefix. e. g.
assigning to a formula t.p another formula 3a:t.p;
lThis work has been supported by Polish Government Grant Projekt Badawczy
"KWANTYFIKATORY".
M. Krynicki et al. (eds.). Quantifiers: Logics, Models and Computation, Volume I, 1-19.
© 1995 Kluwer Academic Publishers.
2 M. Krynicki, M. Mostowski
4. the same as above but considered as a rule dealing with a quantifier
expression and a variable - instead of a prefix;
5. a rule assignig to any interpretation a meaning for a quantifier expres
sion, e. g. to every model M with the universe U assigning the set
3M = {X C U: X -# 0};
6. a meaning of a quantifier expression under a given interpretation, e.
g. a concrete set :1M .
In many papers of this volume all these meanings are represented by the
term quantifier. Of course any correct formal definition has to separate all
these meanings. In mathematical text all such ambiguities have to be solved
by context. However a reader has to be particularly careful trying to find
what is meant by a given use of the term quantifier.
1. Standard Definition
The term "quantifier" was introduced by C. S. Peirce as a name for universal
and existential quantifiers. The first logical theory of them was formulated
by G. Frege. However, according to modern interpretation, Aristotle's work
in formal logic was the first proposal of the formal theory of some quanti-
fi ers ( "a II ... are ...", "s ome. .. z.s .".., "s ome... z.s not ...", "n o... z.s
... "). Quantifier expressions considered by Aristotle are definable by using
universal and existential quantifiers. In general, till the second world war,
logicians worked on reducing all possible concepts to those expressible by
Fregean quantifiers. Later it has been noted that some expressions such
as "there are infinitely many" cannot be expressed by first order universal
and existential quantifiers, but are essentially weaker than the second order
Fregean quantifiers.
Consideration of several similar examples prompted Andrzej Mostowski
to formulate the first general concept of quantifier in [A. Mostowski 1957].
He proposed to explicate the general concept of the quantifier as interpreted
by a class of subsets of the universe. For instance, considering a model with
universe U, existential quantifier is interpreted as the class {X ~ U : X is
nonempty }, and the quantifier "there exist infinitely many" is interpreted as
the class {X ~ U : X is infinite }. More precisely, a quantifier is a functor Q
associating with every structure A a family Q(A) of subsets of the universe
of A. Moreover, this functor has the following property: if f is a bijection
between universes of A and 13 then for every subset X of the universe of A
we have: X E Q(A) if and only if f(X) E Q(I3). In particular, a family
Q(A) is closed under permutations of the universe of A.
Quantifiers, Some Problems and Ideas 3
The definition of the satisfaction relation is extended by the following
clause: A 1= Qx<p[a] if and only if <pA,a,x E Q(A) where <pA,a,x denotes the
set defined by <p in A under the valuation a with respect to the variable x.
As an example we can consider the following:
It can be read as there exists exactly N".
An other example of the quantifier in the above sense is the cardinality
quantifier 32:Na (there exists at least N,,). Formally:
Quantifiers 32:Na are traditionally denoted by Q".
However A. Mostowski's definition is not sufficiently general even to cover
such quantifiers as those considered by Aristotle. Further generalization pro
posed by Lindstrom in [Lindstrom 1966] covered Aristotle quantifiers as well
as others studied in the logical literature. The difference from A. Mostowski's
definition is that Lindstrom's definition allows one to interpret a quantifier as
a relation between relations on the universe. For instance, Aristotle's quan
tifier "all ... are ..." is interpreted as the relation {(X, Y) : X, Y <;;:; U and
X <;;:; Y}, and Hartig quantifier "the same number of elements is ... as ... "
is interpreted as the relation {(X, Y) : X, Y <;;:; U and card(X) = card(Y)}.
Thus the general definition would be expressed as follows.
For any natural numbers nl, ... , nk, Q is a quantifier of type (nl, ... , nd
if and only if
Q is a functor associating with every structure A a k-ary relation Q(A)
between relations on the universe of A such that if (R1, ... , Rk) E Q(A) then
Ri <;;:; lAin; for i = 1, ... , k and for every bijection f from the universe of A
to the universe of B for arbitrary relations R1, ... , Rk on IAI (Ri is ni-ary)
we have: (R1, ... ,Rk) E Q(A) if and only if ([(R1), ... , [(Rk» E Q(B), where
[(R) = {(/(ZI)' ... ,/(Zj)); (ZI, ... ,Zj) E R} for R <;;:; IAlj.
+ ... +
Syntactically a quantifier Q of type (nl, ... , nk) binds nl nk vari
ables and k formulae. The definition of the satisfaction relation is extended
by the following clause:
A 1= QXl ... Xk( '!PI, ... , vJk)[a] if and only if (vJ~,a,xl , ... , vJ~,a,Xk) E Q(A),
where for i = 1, ... , k, vJ~,a,x; denotes the relation defined by vJi in A under
the valuation a with respect to variables Xi.
