Non-Classical Logics and their Applications to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory

390 Pages·1995·8.102 MB·English

by  Ulrich Höhle, Erich Peter Klement (auth.), Ulrich Höhle, Erich Peter Klement (eds.)

NON-CLASSICAL LOGICS AND THEIR APPLICATIONS TO FUZZY SUBSETS
A HANDBOOK OF THE MATHEMATICAL FOUNDATIONS
OF FUZZY SET THEORY
THEORY AND DECISION LIBRARY
General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich)
Series A: Philosophy and Methodology of the Social Sciences
Series B: Mathematical and Statistical Methods
Series C: Game Theory, Mathematical Programming and Operations Research
Series D: System Theory, Knowledge Engineering and Problem Solving
SERIES B: MATHEMATICAL AND STATISTICAL METHODS
VOLUME 32
Editor: H. 1. Skala (Paderborn); Assistant Editor: M. Kraft (Paderborn); Editorial Board:
1. Aczel (Waterloo, Ont.), G. Bamberg (Augsburg), H. Drygas (Kassel), W. Eichhorn
(Karlsruhe), P. Fishburn (Murray Hill, N.J.), D. Fraser (Toronto), W. Janko (Vienna), P. de
Jong (Vancouver), T. Kariya (Tokyo), M. Machina (La Jolla, Calif.), A. Rapoport (Toronto),
M. Richter (Kaiserslautern), B. K. Sinha (Cattonsville, Md.), D. A. Sprott (Waterloo, Ont.),
P. Suppes (Stanford, Calif.), H. Theil (St. Augustine, Fla.), E. Trillas (Madrid), L. A. Zadeh
(Berkeley, Calif.).
Scope: The series focuses on the application of methods and ideas of logic, mathematics and
statistics to the social sciences. In particular, formal treatment of social phenomena, the
analysis of decision making, information theory and problems of inference will be central
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dissemination of recent results.
The titles published in this series are listed at the end of this volume.
NON-CLASSICAL LOGICS
AND THEIR APPLICATIONS
TO FUZZY SUBSETS
A Handbook of the
Mathematical Foundations
of Fuzzy Set Theory
edited by
ULRICH HOHLE
Fachbereich Mathematik,
Bergische Universităt, Wuppertal, Germany
and
ERICH PETER KLEMENT
Institut fUr Mathematik,
Johannes Kepler Universităt, Linz, Austria
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Non-classlcal log1cs and thelr appl1catlons to fuzzy subsets : a
handbook of the mathematlcal foundatlons of fuzzy set theory I
edlted by Ulrlch Hohle and Erlch Peter Klement.
p. cm. -- (Theory and declslon llbrary. Series B.
Mathematlcal and statlstlcal methods ; v. 32)
"Proceedlngs of the 14th Llnz Seminar on Fuzzy Set Theory ... the
second week of September 1992 at the Blldungszentrum St. Magdalena
(Llnz. Austrla)"--P. 1.
Includes blbl10graphlcal references and Index.
ISBN 978-94-010-4096-9 ISBN 978-94-011-0215-5 (eBook)
DOI 10.1007/978-94-011-0215-5
1. Nonclasslcal mathematlcal 10g1c--Congresses. 2. Fuzzy set
theory--Congresses. 1. Hohle. Ulrlch. II. Klement. E. P. (Erlch
Peter) III. Serles.
QA9.4.N64 1995
511.3--dc20 94-35630
ISBN 978-94-010-4096-9
Printed on acid-free paper
Ali Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1s t 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 inforrnation storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Authors and Editors Vll
Preface 1
Introduction 3
Part A Algebraic Foundations of Non-Classical Logics 5
I a-Complete MV-algebras,
L.P. Belluce 7
II On MV -algebras of continuous functions,
A. Di Nola and S. Sessa 23
III Free and projective Heyting and monadic Heyting algebras,
R. Grigolia 33
IV Commutative, residuated l-monoids,
u. Hahle 53
V A Proof of the completeness of the infinite-valued calculus
of Lukasiewicz with one varibale,
D. Mundici and M. Pasquetto 107
Part B Non-Classical Models and Topos-Like Categories 125
VI Presheaves over GL-monoids,
U. Hahle 127
VII Quantales: Quantal sets,
C.J. Mulvey and M. Nawaz 159
VIII Categories of fuzzy sets with values in a quantale or
projectale,
L.N. Stout 219
IX Fuzzy logic and categories of fuzzy sets,
O. Wyler 235
v
vi
Part C General Aspects of Non-Classical Logics 269
X Prolog extensions to many-valued logics,
F. Klawonn . 271
XI Epistemological aspects of many-valued logics and fuzzy
structures,
L.J. Kohout 291
XII Ultraproduct theorem and recursive properties of fuzzy
logic,
v. Novak 341
Bibliography 371
Index 385
Authors and Editors
Lawrence Peter Belluce
Department of Mathematics
University of British Columbia
121-1984 Mathematics Road
Vancouver, BC, Canada V6T 1Y4.
Antonio Di Nola
Istituto di Matematica
Facolta di Architettura
Universita di Napoli
Via Monteoliveto 3
1-80134 Napoli, Italy.
Revaz Grigolia Institute of Cybernetics
Georgian Academy of Sciences
S. Euli str.,5
380086 Tbilisi, Republic of Georgia.
Ulrich Hohle
Fachbereich 7 Mathematik
Bergische Universiat Wuppertal
GauBstraBe 20
D-42097 Wuppertal, Germany.
Frank Klawonn Institut fUr Betriebssysteme und Rechnerverband
Technische Universitat Braunschweig
Biiltenweg 74/75
D-38106 Braunschweig, Germany.
Erich Peter Klement Institut fUr Mathematik
Johannes Kepler Unversitat Linz
A-4040 Linz, Austria.
Ladislav J. Kohout
Department of Computer Science B-l73
and The Institute for Cognitive Sciences
Florida State University
Tallahassee, Florida 32306-4019, USA.
Christopher J. Mulvey
Mathematics Division
School of Mathematical and Physical Sciences
University of Sussex, Falmer
Brighton, BNl 9QH, Great Britain.
VII
viii
Daniele Mundici
Dipartimento di Scienze della Informazione
Universita di Milano
via Comelico, 39/41
1-20135 Milano, Italy.
Mohammad Nawaz
Bureau of Curriculum and Extension Centre
Quetta, Baluchistan, Pakistan.
ViIem Novak
Institute of Geonics
Academy of Sciences of Czech Republic
Studentska 1768
70800 Ostrava-Poruba, Czech Republic.
Salvatore Sessa
Istituto di Matematica
Facolta di Architetture
Universita di Napoli
Via Monteoliveto 3
1-80134 Napoli, Italy.
Lawrence Neff Stout
Department of Mathematics
Illinois Wesleyan University
Bloomington, IL 61702-2900, USA.
Oswald Wyler
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA 15213-3890, USA.
Foreword
The present volume collects the proceedings of the 14th Linz Seminar on Fuzzy
Set Theory, a research symposium dedicated exclusively to non-classical logics
and their applications to fuzzy set theory. This conference took place in the
second week of September 1992 at the Bildungszentrum St. Magdalena (Linz,
Austria). It was sponsored by the Johannes Kepler Universitiit Linz and sup
ported by the Austrian Bundesministerium fur Wissenschaft und Forschung in
Vienna and by the Linzer Hochschulfonds.
As in past years it was the aim of the organizers to bring together a small
group of leading researchers from all over the world, for this time with special
interests in the fields of intuitionistic logic, Lukasiewicz logic and fuzzy set the
ory. During this week 26 talks were presented, highlighted by three invited two
hour-talks, covering topics reaching from monoidal, lattice-theoretical struc
tures and categorical aspects of non-classical logics to epistomological problems
of fuzzy set theory.
Due to the excellent conditions provided by the Bildungszentrum St. Mag
dalena there was much opportunity for informal and private discussions, used
by most of the participants during the breaks or in the evenings. Judging from
their statements after returning home, the participants regarded this Seminar
as being most stimulating and successful.
An important result of these efforts is the present volume containing twelve
selected papers. We express our deepest appreciation to all authors contributing
to this book, who agreed to rewrite, polish, and in many cases to enlarge their
original presentation in response to the discussions and peer review during the
Seminar.
We are therefore convinced that this volume not only surveys the past, but
also presents a comp~ehensive picture of the present work and indicates reason
able directions of future research work in the field. In this sense the present
volume is not to be compared with usual conference proceedings - it can be
viewed as a first handbook of the mathematical foundations of fuzzy set theory.
The financial support by the Austrian Bundesministerium fur Wissenschaft
und Forschung in Vienna and the Linzer Hochschulfonds is very gratefully ac
knowledged, as well as the practical support by the staff of the Bildungszentrum
St. Magdalena. Finally, we acknowledge Kluwer Academic Publishers for their
cooperation in bringing this project to completion, particularly to Ms. Marie
M. Stratta for her capable assistance and patience. Our warmest thanks are
due to H.J. Skala, who strongly supported this project from the very beginning.
July 1994 The Editors
Introd uction
Non-classical logics emerged in the early twenties and thirties of this century
and go back to the famous papers of J. Lukasiewicz [1920]' E.L. Post [1921]' A.
Heyting [1930], G. Birkhoff and J.v. Neumann [1936]. From a syntactical point
of view they are always fragments of classical logic. Disregarding for a moment
quantum logic, non-classical logics can be characterized by the abandonment
of the law of the excluded middle and the maintenance of the integrality, the
exportation, importation and Duns Scotus law.
This abandonment can be expressed in various ways : Let 0 be the as
sociative and commutative logical operation appearing in the exportation and
importation law; then the requirement of the idempotency of 0 implies that
the abandonment of the law of the excluded middle is equivalent to the aban
donment of the law of double negation. Vice versa, if one insists in the law
of double negation, then the abandonment of the law of the excluded middle
forces the non-idempotency of 0. It is well known that the first case leads to
intuitionistic logic [Heyting 1930], while the second case can be understood as
a kind of integral, commutative linear logic (see also J.Y. Girard [1987]). If in
the second case we add the axiom of divisibility
(( a 1\ f3) -> (a 0 (a -> f3)))
then we obtain the so-called Wajsberg axioms and arrive at the infinite-valued
Lukasiewicz logic. Each of these logics has its own significant meaning: In
tuitionistic logic plays a fundamental role in the foundations of constructive
mathematics, Lukasiewicz logic admits antinomies and throws a special light on
the paradoxes of set theory.
Let us now turn to the semantical aspects of non-classical logics. The def
initions of notions like satisfiability or validity require a lattice-ordered monoid
M, the structure of which corresponds uniquely to the given logical axioms. In
accordance with Tarski's matrix method (J. Lukasiewicz and A. Tarski [1930]),
respectively the papers of A. Mostowski [1948] ,and H. Rasiowa and R. Sikorski
(see also [1970]) we first interpret logical operations as algebraic operations on
M and subsequently m-ary predicate symbols as M-valued maps. As an im
mediate consequence of this approach we obtain the elements of many-valued,
non-classical model theory.
Now we extend our considerations and cover also the theory of fuzzy sets,
as it was initiated by L.A. Zadeh 1965. It is not difficult to see that fuzzy
set theory is not a part of many-valued model theory, but fuzzy set theory is
very close to it. Contrary to many-valued model theory, a very special lattice,
namely the real unit interval [0,1], is taken as a basis; then once again unary
predicate symbols are interpreted as lattice-valued (in this case as [O,l]-valued)
maps, conjunction and disjunction are given by the lattice operations min and
max, and finally the negation is interpreted by the involution x -vot 1-x on [0,1].
It is a characteristic of the situation in fuzzy set theory that an explicit use of
3