Aspects of Vagueness

305 Pages·1984·12.596 MB·English

by  Jiři Bečvář (auth.), Heinz J. Skala, S. Termini, E. Trillas (eds.)

ASPECTS OF VAGUENESS
THEORY AND DECISION LIBRARY
AN INTERNATIONAL SERIES
IN THE PHILOSOPHY AND METHODOLOGY OF THE
SOCIAL AND BEHAVIORAL SCIENCES
Editors
GERALD EBERLEIN, University of Technology, Munich
WERNER LEINFELLNER, University of Nebraska
Editorial Advisory Board
K. BORCH, Norwegian School of Economics and Business Administration
M. BUNGE, McGill University
J. S. COLEMAN, University of Chicago
W. KROEBER-RIEL, University of Saarland
A. RAPOPORT, University of Toronto
F. SCHICK, Rutgers University
A. SEN, Oxford University
W. STEGMULLER, University of Munich
K. SZANIA WSKI, University of Warsaw
L. TONDL, Prague
A. TVERSKY, Stanford University
VOLUME 39
ASPECTS
OF VAGUENESS
Edited by
HEINZ J. SKALA
University of Paderborn, F.R. G.
S. TERMINI
Consiglio Nazionale delle Ricerche, Laboratorio di Cibernetica, Naples, Italy
and
E. TRILLAS
Universidad Politecnica de Barcelona, Spain
D. REIDEL PUBLISHING COMPANY
Library of Congress Cataloging in Publication Data
Aspects of vagueness
(Theory and Decision Library = v. 39)
Papers presented at the Second World Conference on Mathematics
at the Services of Man, held at the Universidad Politecnica de Las Palmas, Canary
Islands, Spain, June 23 to July 3, 1982, and sponsored by the Cabildo Insular de
Gran Canada and the Universidad.
Includes indexes.
1. Fuzzy sets-Congresses. 2. Fuzzy SYstems-Congresses. 3. Mathe
matical Models-Congresses. I. Skala, Heinz J. II. Fermini, S. (Settimo), 1945-
III. Trillas, E. (Enric) IV. Conference on mathematics at the serVices of man
(2nd: 1982: Universidad Politecnica de Las Palm as) V. Gran Canaria (Canary
Islands). Cabildo Insular, VI. Universidad Politecnica de Las Palmas.
VII. Title: Vagueness. VIII. Series.
QA248.A87 1984 508.3'2 83-26994
ISBN-13:978-94-009-6311-S e-ISBN-13 :978-94-009-6309-2
DOl: 10.1007/978-94-009-6309-2
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
190 Old Derby Street, Hingham, MA 02043, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland.
All Rights Reserved
© 1984 by D. Reidel Publishing Company, Dordrecht, Holland
and copyrightholders as specified on appropriate pages within
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
TABLE OF CONTENTS
PREFACE vii
J.BEC'VV ARV /N otes on Vagueness and Mathem'at1c s 1
S.GOTTWALD/Fuzzy Set Theory: Some Aspects of the
Early Development 13
U.HOHLE and E.P.KLEMENT/Plausibility Measures -
A General Framework for Possibility and Fuzzy
Probability Measures 31
M.KATZ/Controlled-Error Theories of Proximity and
Dominance 51
D.MILLER/Impartial Truth 75
D.MILLER/A Geometry of Logic 91
S.V.OVCHINNIKOV/Representations of Transitive
Fuzzy Relations 105
A.PULTR/Fuzziness and Fuzzy Equality 119
H.J.SKALA/Large Societies and Individual Strategy
Selection: A Case Study of Ambiguity 137
A.SOCHOR/The Alternative Set Theory and its Approach
to Cantor's Set Theory 161
S.TERMINI/Aspects of Vagueness and Some Epistemo
logical Problems Related to their Formalization 205
E.TRILLAS and L.VALVERDE/An Inquiry into Indistin
guishability Operators 231
L.A. ZADEH/A Theory of Commonsense Knowledge 257
INDEX OF NAMES 297
INDEX OF SUBJECTS 301
PREFACE
The Second World Conference on Mathematics at the Service
of Man was held at the Universidad Politecnica de Las Palmas,
Canary Islands, Spain, June 28 to July 3, 1982. The first
volume of the Proceedings of the Conference, entitled
"Functional Equations-Theory and Applications" has appeared
in the Reidel series "Mathematics and Its Applications".
The papers in this volume consist of the invited lectures
delivered at the Conference, Section 7: Non-Classical Logics
and Modelling, as well as some selected papers which offer an
introduction to the philosophy, methodology and to the lite
rature of the broad and fascinating field of vagueness,
imprecision and uncertainty. The contributed papers appeared
in the volume of photo-offset preprints distributed at the
Conference.
It is our hope that the papers present a good sample with
respect to the background, the formalism and practice of
this area of research as far as we understand it today.
As the subject "Vagueness" touches many aspects of human
thinking, the contributions have been made from a broad
spectrum ranging from philo~ophy through pure mathematics to
probability theory and mathematical economics, therefore the
careful reader should find some new insights here.
In conclusion, the editors want to thank all authors who
have contributed to this volume; the publishers of "Commenta
tiones Mathematicae Universitatis Carolinae" for permission
to reprint the paper "Fuzziness and Fuzzy Equality",
Commentationes Mathematicae Universitatis Carolinae 23 (1982),
249-267, and D. Reidel for friendly cooperation.
We would also like to thank the sponsors of the conference:
the Cabildo Insular de Gran Canaria, and the Universidad
Politecnica de Las Palmas. One of the editors was supported
by the Deutsche Forschungsgemeinschaft. Last, but not least,
we wish to express our appreciation to Mrs. J. H6xtermann,
for secretarial assistance and K. Braun, Paderborn.
The Editors
vii
H. J. Skala, S. Termini, and E. Trillas (eds.), Aspects of Vagueness, vii.
© 1984 by D. Reidel Publishing Company.
NOTES ON VAGUENESS AND MATHEMATICS
1. This paper consists of a list of notes on the issue of
vagueness, the emphasis being on its relation to "crisp"
entities. This is a classical topic, in fact any examination
of the interplay of empirical material and theoretical con
cepts has to take it into account (see,e.g., [4]). In
addition to earlier investigations, last decades have brought
substantial contributions in the direction of providing
technical mathematical and loglcal explications of aspects
of vagueness. This development has been in part initiated by
problems in the theory of systems, in computer science, in
the area of artificial intelligence, in foundations of mathe
matics, etc. The most extensive branch of the development has
been the theory of fuzzy sets, originating with [9] (for a
detailed overview, see [2] and the bibliography in [3]).
Among important contributions based on different principles
let us mention especially the alternative set theory [8].
The exposition in this paper does not lean upon a concrete
technical explication of the concept of vagueness. Also the
language used is deliberately (mostly) nontechnical; thus,
in particular, we use the word "vague" rather as a generic
term, encompassing various shades of being imprecise, indeter
minate, fuzzy, ambiguous, uncertain, etc. This corresponds
to the orientation of the paper: its aim is to recall se
lected aspects of vagueness, sometimes in less frequently
discussed connections which, nevertheless, may prove rele
vant to the formation of a general background. (The whole
picture is composed of essentially familiar items; according
ly, only a small number of bibliographical references have
been included.)
2. There is a striking contrast between our ablility in
everyday practice to successfully recognize objects, events,
properties etc., and, on the other hand, our inability to
provide complete descriptions or unambiguous characteriza
tions of them. Experience tells us that any attempt at a
complete characterization of a piece of reality eventually
reaches a point of uncertainty as to the adequacy and/or
applicability of the identification criteria suggested. In
H. J. Skala, S. Termini, and E. Trilllls (eds.), Aspects of Vagueness, 1-11.
© 1984 by D. Reidel Publishing Company.
BECVAR
2 1.
what follows we will look more closely at some of the ingre
dients of this bipolar situation.
The world in which we live is not homogeneous; it exhibits,
on all levels of magnitude and development, some relatively
stable patterns. The articulation i~ especially appealing in
the organic sphere. On the highest level known to us so far,
human organisms have gradually developed, in a continuous in
teraction with the environment, the ability to isolate, both
in the external world and in the sphere of conscious thinking,
a considerable number of more or less definite entities,
pertai,ning to all branches of human activity (objects,
relations, states, propositions, processes, concepts of
utility, causality, mathematical concepts ... ; properties
of being complex, abstract, vague also belong here) .
In these entities we are able to recognize the presence of
various degrees of vagueness. At the same time we realize
that some of them, especially the basic logical and mathe
matical constructs, appear to us as little vague as is con
ceivable at all [7], at least in the moment when they become
involved in the central zone of our mental activity.
In general, the status of being vague is influenced by the
context in which an entity appears. For example, the
process and result of an interaction between vaguely deter
mined lumps of, say, organic matter can be sometimes adequa
tely thought of in a rather crisp way. Also the human rou
tine identification and manipulation of objects proceeds so
that we seem to experience them directly as sufficiently
clear and have the impression of handling them according to
simple rules.
3. Our brain itself accomodates an ingredient which provides
an active contraposition to vagueness. It consists in that
entities once conceived in it can be treated there also in a
formal, or "combinatorial", way, largely independent of their
source, original meaning and imprecision; they - or the words
used for them - can even play the role of mere auxiliary
carriers of modified hypothetical meanings which may be in
complete, in different senses vague, or contradictory
(similar situation obtains or can be simulated in a com
puter) and may this reach the level and status which is
typical of operating with abstract objects within the frame
work of mathematics and (classical) logic. We realize that
NOTES ON VAGUENESS AND MATHEMATICS 3
also in the current use and understanding of a natural lan
guage - which, with the probable exception of its "logical"
kernel, typically reflects the complicated, non-crisp part
of our mental processes - elements of the combinatorial atti
tude are operative. (For example, when listening to a long
phrase one uses to solve the problem of understanding the
whole by making, among others, during hearing a part of the
phrase a quick mental experimentation and predictions as to
what the next parts may be, in order that the task is com
pleted without a disturbing delay. Moreover, during liste
ning only some "representative" features of the meanings of
the words and phrases are being called up, as possible
stimuli for a subsequent analysis, inquiry into the truth,
consistency, etc.)
4. We mentioned that the mathematical and logical frame seems
to us to have supreme clarity and precision. This does not
imply, however, that its basic elements are simple or "pri
me". As a matter of fact, they appear to be located in the
uppermost levels of an immense pyramid of complex automatic
and selforganizing processes, the structure, evolution and
interpretation of which are still largely unknown. (For a
parallel let us recall that our conscious interpretation of
what appears in the visual field is also preceded by a vast,
unconscious analysis and preprocessing of the patterns on
the retina.) It would not be much enlighting thus to say
that the mathematical frame is vague. We prefer just to
conclude that basing an explication of the phenomena in the
real world on a reduction to this frame is possibly a sort
of high-level translation. Accordingly, both the questions
about its directness and universal explicative capacity are
open. This proves relevant, in particular, to the attempts
at an explication of various features of vagueness, e.g. the
dichotomy discrete-continuous, etc.; an "adequate" represen
tation of phenomena in the sphere of elementary physical
particles is a related problem.
(For a computer, the underlying (technological) pyramid is
different. Hence also the question about the explicative/
modelling capacity of contemporary computers - which are
sophisticated artificial arrangements of pieces of matter,
subject to physical laws which are sufficiently well known
for the intended purpose - pOints in a somewhat different
direction. Let us note, incidentally, that computers are
not Garden-of-Eden configurations in the compl~e system
BECVAR
4 J.
of Nature; they appear within it as the result of its own
motion.)
There are still other important elements which make the
status of mathematical entities relative. One of them is of
a subjective, or, more properly speaking, psychological,
nature: A condition sine qua non for being able to operate
efficiently with abstract mathematical entities is to expe
rience ("see") them clearly - clare et distincte -, nearly
as concrete objects. The corresponding state of mind cannot
be communicated or completely described. (In this respect
it is similar to a large number of phenomena like the fusion
of two slightly different images in binocular vision,sensing
the harmonic quality of a chord as different from the notes
constituting it, etc.)
Another, similar element, but on a somewhat higher level, is
manifest if we try to use mathematics and logic as tools for
the explication of new phenomena; in particular, for an
analysis of some basic concepts of mathematics themselves.
This proves impossible unless we sufficiently clearly "un
derstand" what we are doing. Such an understanding, espe
cially if connected with the establishment of a new con
ceptual base, is usually not the result of a merely de
ductive or combinatorial manipulation of old concepts, but
is simultaneously related to new insights and intuitions -
and decisions; these then result in a collective understan
ding, specific for a group of people or an epoch. (As a rule,
in this process some old, delicate interpretative questions
are left unresolved, but the attraction and success of the
new approach prevail. Generations of students have learnt
and assimilated mathematical analysis in arithmetical form,
set theory in Cantor's spirit, probability theory in
Kolmogorov's axiomatization ... )
5. In the articulated real world many of the entities inter
acting on a certain level are aggregates of a very large
number of elements of lower levels; we may loosely call them
macroentities or macroobjects (of that level). For the re
sult of the interaction, variations of their detailed struc
ture are, within certain limits, without ·substantial effect.
(In the organic sphere this is stressed by measures taken
to count with the possibility. of errors and malfunction of
elements or parts.) Our life is adapted to the presence of
vagueness; it-both respects the resulting restrictions and