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![Contributors
● Andras Bardossy (Chapter 6): Institute of Hydraulic Engineering, Uni-
versity of Stuttgart, Pfaffenwaldring 61, D-70550 Stuttgart, Germany
[bardossy@europe.com].
● Radim Bělohlávek (Chapter 7): Department of Computer Science, Palacký
University of Olomouc, Tomkova 40, CZ-77900 Olomouc, Czech Republic
[belohlavek@inf.upol.cz].
● Istvan Bogardi (Chapter 6): Department of Civil Engineering, University of
Nebraska at Lincoln, Lincoln, Nebraska 68588, USA [ibogardi@unl.edu].
● Robert V. Demicco (Chapters 1, 3, 4, and 5): Department of Geological & Envi-
ronmental Studies and Center for Intelligent Systems, Binghamton University
(SUNY), Binghamton, New York 13902, USA [demicco@binghamton.edu].
● Lucien Duckstein (Chapter 6): Ecole Nationale du Génie Rural des Eaux et des
Forêts, 19 avenue du Maine, 75732 CEDEX 15, France [duckstein@engref.fr].
● Chongfu Huang (Chapter 8): Institute of Resources Science, Beijing Normal Uni-
versity, 19 Xinjiekouwai Street, Beijing 100875, China [hchongfu@bnu.edu.cn].
● George J. Klir (Chapters 1 and 2): Department of Systems Science & Industrial
Engineering and Center for Intelligent Systems, Binghamton University (SUNY),
Binghamton, New York 13902, USA [gklir@binghamton.edu].
● Vilem Novák (Chapter 10): Institute for Research and Applications of Fuzzy
Modeling, University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic
[Vilem.Novak@osu.cz].
● Irina Perfilieva (Chapter 9): Institute for Research and Applications of Fuzzy
Modeling, University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic
[Perfili@dynami.osu.cz].
● Rita Pongracz (Chapter 6): Department of Meteorology, Eotvos Lorand Univer-
sity, Pazmany P. setany1/A, H-1117 Budapest, Hungary [prita@caesar.elte.hu].
vii](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-13-2048.jpg)
![Glossary of Symbols
General Symbols
{x, y, . . .} Set of elements x, y, . . .
{x | p{x}} Set determined by property p
x1, x2, . . . , xn n-tuple
[xij ] Matrix
[x1, x2, . . . , xn] Vector
[a, b] Closed interval of real numbers between a and b
[a, b), (b, a] Interval of real numbers closed in a and open in b
(a, b) Open interval of real numbers
A, B, C . . . . Arbitrary sets (crisp or fuzzy)
x ∈ A Element x belongs to crisp set A
A(x) or μA(X) Membership grade of x in fuzzy set A
αA α-cut of fuzzy set A
α+A Strong α-cut of fuzzy set A
A = B Set equality
A = B Set inequality
A − B Set difference
A ⊆ B Set inclusion
A ⊂ B Proper set inclusion (A ⊆ B and A = B)
SUB(A, B) Degree of subsethood of A in B
P(X) Set of all crisp subsets of X (power set)
F(X) Set of all standard fuzzy subsets of X (fuzzy power set)
|A| Cardinality of crisp or fuzzy set A (sigma count)
hA Height of fuzzy set A
A Complement of set A
A ∩ B Set intersection
A ∪ B Set union
A × B Cartesian product of sets A and B
A2 Cartesian product A × A
f : X → Y Function from X to Y
f −1 Inverse of function f
R ◦ Q Standard composition of fuzzy relations R and Q
xv](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-21-2048.jpg)
![Chapter 1 Introduction
Robert V. Demicco and George J. Klir
Traditionally, science, engineering, and mathematics showed virtually no interest in
studying uncertainty. It was considered undesirable and the ideal was to eliminate
it. In fact, eliminating uncertainty from science was viewed as one manifestation
of progress. This attitude towards uncertainty, prevalent prior to the 20th century,
was seriously challenged by some developments in the first half of that century.
Among them were the emergence of statistical mechanics, Heisenberg’s uncertainty
principle in quantum mechanics, and Gödel’s theorems that established an inher-
ent uncertainty in formal mathematical systems. In spite of these developments, the
traditional attitude towards uncertainty changed too little and too slowly during the
first half of the century. While uncertainty became recognized as useful, or even
essential, in statistical mechanics and in some other areas (such as the actuarial pro-
fession or the design of large-scale telephone exchanges), it was for a long time
tacitly assumed that probability theory was capable of capturing the full scope of
uncertainty.
The presumed equality between uncertainty and probability was challenged only
in the second half of the 20th century. The challenge came from two important gen-
eralizations in mathematics. The first one was the generalization of classical measure
theory [Halmos, 1950] to the theory of monotone measures, which was first suggested
by Choquet [1953] in his theory of capacities. The second one was the generalization
of classical set theory to fuzzy set theory, which was introduced by Zadeh [1965]. In
the theory of monotone measures, the additivity requirement of classical measures is
replaced with a weaker requirement of monotonicity with respect to set inclusion. In
fuzzy set theory, the requirement of sharp boundaries of classical sets is abandoned.
That is, the membership of an object in a fuzzy set is not a matter of either affirma-
tion or denial, as it is in the case of any classical set, but it is in general a matter of
degree.
For historical reasons of little significance, monotone measures are often referred
to in the literature as fuzzy measures [Wang Klir, 1992]. This name is somewhat
confusing since no fuzzy sets are involved in the definition of monotone measures.
However, monotone measures can be fuzzified (i.e., defined on fuzzy sets), which
results in a more general class of monotone measures—fuzzy monotone measures
[Wang Klir, 1992, Appendix E].
1
FUZZY LOGIC IN GEOLOGY Copyright 2004, Elsevier Science (USA)
All rights of reproduction in any form reserved.
ISBN: 0-12-415146-9](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-25-2048.jpg)
![2 1 Introduction
As is well known, probability theory is based on classical measure theory which,
in turn, is based on classical set theory [Halmos, 1950]. When classical measures are
replaced with monotone measures of some type and classical sets are replaced with
fuzzy sets of some type, a framework is obtained for formalizing some new types
of uncertainty, distinct from probability. This indicates that the two generalizations
have opened a vast territory for formalizing uncertainty. At this time, only a rather
small part of this territory has been adequately explored [Klir Wierman, 1999;
Klir, 2002].
Liberating uncertainty from its narrow confines of probability theory opens new,
more expressive ways of representing scientific knowledge. As is increasingly rec-
ognized, scientific knowledge is organized, by and large, in terms of systems of
various types (or categories in the sense of mathematical theory of categories)
[Klir Rozehnal, 1996; Klir Elias, 2003]. In general, systems are viewed as
relations between states of some variables. They are constructed for various purposes
(prediction, retrodiction, prescription, diagnosis, control, etc.). In each system, its
relations are utilized, in a given purposeful way, for determining unknown states
of some variables on the basis of known states of some other variables. Systems in
which the unknown states are determined uniquely are called deterministic; all other
systems are called nondeterministic.
By definition, each nondeterministic system involves uncertainty of some type.
This uncertainty pertains to the purpose for which the system was constructed. It is
thus natural to distinguish between predictive uncertainty, retrodictive uncertainty,
diagnostic uncertainty, etc. In each nondeterministic system, the relevant uncertainty
must be properly incorporated into the description of the system in some formalized
language. To understand the full scope of uncertainty is thus essential for dealing with
nondeterministic systems.
When constructing a system for some given purpose, our ultimate goal is to obtain a
system that is as useful as possible for this purpose. This means, in turn, to construct a
system with a proper blend of the three most fundamental characteristics of systems:
credibility, complexity, and uncertainty. Ideally, we would like to obtain a system
with high credibility, low complexity, and low uncertainty. Unfortunately, these three
criteria conflict with one another. To achieve high usefulness of the system, we need
to find the right trade-off among them.
The relationship between credibility, complexity and uncertainty is quite intri-
cate and is not fully understood yet. However, it is already well established that
uncertainty has a pivotal role in any efforts to maximize the usefulness of constructed
systems.Although usually undesirable in systems when considered alone, uncertainty
becomes very valuable when considered in connection with credibility and complex-
ity of systems. Aslight increase in relevant uncertainty may often significantly reduce
complexity and, at the same time, increase credibility of the system. Uncertainty is
thus an important commodity in the knowledge business, a commodity that can be
traded for gains in the other essential characteristics of systems by which we represent](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-26-2048.jpg)
![1 Introduction 3
knowledge. Because of this important role, uncertainty is no longer viewed in science
and engineering as an unavoidable plague, but rather as an important resource that
allow us to deal effectively with problems involving very complex systems.
It is our contention that monotone measures and fuzzy sets (as well as the various
uncertainty theories opened by these two profound generalizations in mathematics)
are highly relevant to geology, and that their utility in geology should be seriously
studiedintheyearsahead. Theaimofthisbookistodemonstratethispointbyfocusing
on the role of fuzzy set theory, and especially the associated fuzzy logic, in geology.
The term “fuzzy logic” has in fact two distinct meanings. In a narrow sense, it is
viewed as a generalization of classical multivalued logics. It is concerned with the
development of syntactic aspects (based on the notion of proof ) and semantic aspects
(based on the notion of truth) of a relevant logic calculus. In order to be acceptable,
the calculus must be sound (provability implies truth) and complete (truth implies
provability). These issues have successfully been addressed for fuzzy logic in the
narrow sense by Hájek [1998].
In a broad sense, fuzzy logic is viewed as a system of concepts, principles, and
methods for dealing with modes of reasoning that are approximate rather than exact.
The two meanings are connected since the very purpose of research on fuzzy logic in
the narrow sense is to provide fuzzy logic in the broad sense with sound foundations.
In this book, we are concerned only with fuzzy logic in the broad sense, which is
surveyed in Chapter 2, and its role in geology, which is the subject of Chapters 3–10.
From the standpoint of science, as it is still predominantly understood, the ideas
of a fuzzy set and a fuzzy proposition are extremely radical. When accepted, one
has to give up classical bivalent logic, generally presumed to be the principal pillar
of science. Instead, we obtain a logic in which propositions are not required to be
either true or false, but may be true or false to different degrees. As a consequence,
some laws of bivalent logic no longer hold, such as the law of excluded middle or the
law of contradiction. At first sight, this seems to be at odds with the very purpose of
science. However, this is not the case. There are at least the following four reasons
why allowing membership degrees in sets and degrees of truth in propositions in fact
enhances scientific methodology quite considerably:
1. Fuzzy sets and fuzzy propositions possess far greater capabilities than their classi-
cal counterparts to capture irreducible measurement uncertainties in their various
manifestations. As a consequence, their use improves the bridge between mathe-
matical models and the associated physical reality considerably. It is paradoxical
that, in the face of the inevitable measurement errors, fuzzy data are always more
accurate than their crisp (i.e., nonfuzzy) counterparts. Crisp data of each vari-
able are based on a partition of the state set of the variable. The coarseness of
this partition is determined by the resolution power of the measuring instrument
employed. Measurements falling into the same block of the partition are not dis-
tinguished in crisp data, regardless of their position within the block. Thus, for](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-27-2048.jpg)
![1 Introduction 5
systems in a human-like manner, and, generally, for designing and building
human-friendly machines with high intelligence. Fuzzy sets and fuzzy propo-
sitions are also essential for studying human reasoning, decision making, and
acting that are based on perceptions rather than measurements.
It is the synergy of all these capabilities that has made fuzzy set theory and fuzzy
logic highly successful in many engineering applications over the last two decades
or so. The most visible of these applications have been in the area of control, ranging
from simple control systems in consumer products (intelligent washing machines,
vacuum cleaners, camcorders, etc.) to highly challenging control systems, such as
the one for controlling a pilotless helicopter via wireless communication of commands
expressed in natural language. Less visible but equally successful applications have
been demonstrated in the areas of database and information retrieval systems, expert
systems, decision making, pattern recognition and clustering, image processing and
computervision, manufacturing, robotics, transportation, riskandreliabilityanalyses,
and many other engineering areas. In fact, every field of engineering has already been
positively affected, in one way or another, by fuzzy set theory and fuzzy logic [Ruspini
et al., 1998].
In science, applications of fuzzy set theory and fuzzy logic have developed at
a considerably slower pace than in engineering and only in some areas of science
thus far. This is understandable if we realize how extremely radical the ideas of
fuzzy sets and fuzzy propositions actually are. Nevertheless, successful applications
have already been demonstrated in many areas of science. Examples are applications
in quantum physics [Pykacz, 1993; Cattaneo, 1993], chemistry [Rouvray, 1997],
biology [Von Sternberg Klir, 1998], geography [Gale, 1972], ecology [Libelli
Cianchi, 1996], linguistics [Rieger, 2001], economics [Billot, 1992], psychology
[Zétényi, 1988], and social sciences [Smithson, 1987]. In geology, the utility of fuzzy
set theory and fuzzy logic was recognized, by and large, only in the late 1990s, but the
number of publications dealing with applications of fuzzy logic in geology is already
substantial and is growing fast (Chapter 4). This is a clear indicator that the use of
fuzzy logic in geology has a great potential. Our motivation for publishing this book
is to help to develop this potential.
It is important to realize that fuzzy set theory and fuzzy logic are not only tools
that help us to deal with some difficult problems in science, engineering, and other
professional areas, but they also provide us with a conceptual framework for a rad-
ically new way of thinking. Sharp boundaries of classical sets and absolute truths
or falsities of classical propositions are still possible under the new thinking, when
justifiable, but they are viewed as limiting cases rather than the only possibilities.
Thinking in absolute terms is replaced with thinking in relative terms. Everything
becomes a matter of degree. This change in our thinking will undoubtedly open new,
more refined ways of looking at old issues of epistemology, ethics, law, social policy,
and other areas that affect our lives.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-29-2048.jpg)
![6 1 Introduction
The emergence of fuzzy set theory and fuzzy logic and their impact on mathematics
and logic as well as on science and science-dependent areas of human affairs possess
all distinctive features that are characteristic of a paradigm shift, as introduced in
the highly influential book by Thomas Kuhn [1962]. Since logic is fundamental to
virtually all branches of mathematics as well as science, this paradigm shift has much
broader implications than those generally recognized in the history of science and
mathematics, each of which affects only a particular area of science or mathemat-
ics. It is thus appropriate to refer to it as a “grand paradigm shift.” Various special
characteristics of this paradigm shift, which is still ongoing, are discussed by Klir
[1995, 1997, 2000]. It is generally agreed that this paradigm shift was initiated by the
publication of the seminal paper by Zadeh [1965]. However, many ideas pertaining
to fuzzy logic had appeared in the literature prior to the publication of that paper.
Unfortunately, these ideas were by and large ignored at that time [Klir, 2001].
The purpose of this book is threefold: (i) to examine how fuzzy logic has already
been applied in some areas of geology; (ii) to stimulate the development of applica-
tions of fuzzy logic in other areas of geology; and (iii) to stimulate the use of additional
tools of fuzzy logic in geology. Material covered in Chapters 2–10 was carefully
selected to accomplish this purpose. The following is a brief preview of this material.
Chapter 2 is an overview of fuzzy logic in the broad sense. It is written as a tutorial
for those readers who are not familiar with fuzzy logic. This chapter covers not only
those components of fuzzy logic that are employed in subsequent chapters, but also
some additional ones which offer new application possibilities for geology. More-
over, this chapter introduces terminology and notation that are followed consistently
throughout the whole book.
The aim of Chapter 3 is twofold: (i) to discuss reasons for using fuzzy logic in
geology; and (ii) to illustrate the use of fuzzy logic in geology by simple examples.
For geologists, some of the notions of fuzzy logic introduced in Chapter 2 are further
discussed in terms of simple geological interpretations. For researchers in fuzzy logic,
the chapter is a sort of tutorial which introduces them to some issues that are of concern
to geology.
Chapter 4 is a comprehensive overview of currently known applications of fuzzy
logic in geology. It is primarily an annotated bibliography that is grouped into the
following nine categories: (1) surface hydrology; (2) subsurface hydrology; (3)
groundwater risk assessment; (4) geotechnical engineering; (5) hydrocarbon explo-
ration; (6) seismology; (7) soil science and landscape development; (8) deposition
of sediments; and (9) miscellaneous applications. In addition, the role of fuzzy logic
within the broader area of soft computing is briefly characterized. The aim of this
chapter is to provide readers with a useful resource for further study of established
applications of fuzzy logic in geology, sometimes in the broader context of soft
computing.
Each of the remaining six chapters of this book covers in greater depth applica-
tions of fuzzy logic in some specific area of geology. The utility of fuzzy logic to](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-30-2048.jpg)
![8 1 Introduction
descriptions and interpretations of geologic phenomena. In this case, a computer pro-
gram using specialized fuzzy-set based “evaluating expressions” is taught to mimic
the linguistic geologic “rules” for both the division of Paleozoic measured sections of
limestone into a hierarchy of different cycles, and the interpretation of those cycles
in terms of ancient sea level.
References
Billot, A. [1992], Economic Theory of Fuzzy Equilibria. Springer-Verlag, New York.
Cattaneo, G. [1993], “Fuzzy quantum logic II: The logics of unsharp quantum mechanics.”
International Journal of Theoretical Physics, 32(10), 1709–1734.
Choquet, G. [1953–54], “Theory of capacities.” Annales de L’Institut Fourier, 5, 131–295.
Gale, S. [1972], “Inexactness, fuzzy sets and the foundations of behavioral geography.”
Geographical Analysis, 4, 337–349.
Hájek, P. [1998], Metamathematics of Fuzzy Logic. Kluwer, Boston, MA.
Halmos, P. R. [1950], Measure Theory. Van Nostrand, Princeton, NJ.
Klir, G. J. [1995], “From classical sets to fuzzy sets: a grand paradigm shift.” In: Wang,
P. P. (ed.), Advances in Fuzzy Theory and Technology, Vol. III, pp. 3–30. Duke University,
Durham, NC.
Klir, G. J. [1997], “From classical mathematics to fuzzy mathematics: emergence of a new
paradigm for theoretical science.” In: Rouvray, D. H. (ed.), Fuzzy Logic in Chemistry,
pp. 31–63. Academic Press, San Diego, CA.
Klir, G. J. [2000], Fuzzy Sets: An Overview of Fundamentals, Applications, and Personal
Views. Beijing Normal University Press, Beijing, China.
Klir, G. J. [2001], “Foundations of fuzzy set theory and fuzzy logic: A historical overview.”
International Journal of General Systems, 30(2), 91–132.
Klir, G. J. [2002], “Uncertainty-based information.” In: Melo-Pinto and H.-N. Teodorescu
(eds.), Systemic Organisation of Information in Fuzzy Systems, pp. 21–52. IOS Press,
Amsterdam.
Klir, G. J., Elias, D. [2003], Architecture of Systems Problem Solving (2nd edition).
Kluwer/Plenum, New York.
Klir, G. J., Rozehnal, I. [1996], “Epistemological categories of systems: an overview.”
International Journal of General Systems, 24(1–2), 207–224.
Klir, G. J., Wierman, M. J. [1999], Uncertainty-Based Information: Elements of Gener-
alized Information Theory (2nd edition). Physica-Verlag/Springer-Verlag, Heidelberg and
New York.
Kuhn, T. S. [1962], The Structure of Scientific Revolutions. University of Chicago Press,
Chicago, IL.
Libelli, S. M., Cianchi, P. [1996], “Fuzzy ecological models.” In: Pedrycz, W. (ed.), Fuzzy
Modelling Paradigms and Practice, pp.141–164. Kluwer, Boston, MA.
Pykacz, J. [1993], “Fuzzy quantum logic I.” International Journal of Theoretical Physics,
32(10), 1691–1707.
Rieger, B. B. [2001], “Computing granular word meanings: A fuzzy linguistic approach in
computational semiotics.” In: Wang, P. P. (ed.), Computing with Words, pp. 147–208.
John Wiley, New York.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-32-2048.jpg)
![References 9
Rouvray, D. H. (ed.) [1997], Fuzzy Logic in Chemistry. Academic Press, San Diego, CA.
Ruspini, E. H., Bonissone, P. P., Pedrycz, W. (eds.) [1988], Handbook of Fuzzy Computation.
Institute of Physics Publishing, Bristol (UK) and Philadelphia, PA.
Smithson, M. [1987], Fuzzy Set Analysis for Behavioral and Social Sciences. Springer-Verlag,
New York.
Von Sternberg, R., Klir, G. J. [1998], “Generative archetypes and taxa: A fuzzy set
formalization.” Biology Forum, 91, 403–424.
Wang, Z., Klir, G. J. [1992], Fuzzy Measure Theory. Plenum Press, New York.
Zadeh, L. A. [1965], “Fuzzy sets.” Information and Control, 8(3), 338–353.
Zétényi, T. (ed.) [1988], Fuzzy Sets in Psychology. North-Holland, Amsterdam and New York.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-33-2048.jpg)
![Chapter 2 Fuzzy Logic: A Specialized Tutorial
George J. Klir
2.1 Introduction 11
2.2 Basic Concepts of Fuzzy Sets 14
2.3 Operations on Fuzzy Sets 19
2.3.1 Modifiers 19
2.3.2 Complements 21
2.3.3 Intersections and unions 22
2.3.4 Averaging operations 25
2.3.5 Arithmetic operations 28
2.4 Fuzzy Relations 31
2.4.1 Projections, cylindric extensions, and cylindric closures 32
2.4.2 Inverses, compositions, and joins 33
2.4.3 Fuzzy relation equations 34
2.4.4 Fuzzy relations on a single set 36
2.5 Fuzzy Logic 38
2.5.1 Basic types of propositional forms 41
2.5.2 Approximate reasoning 44
2.6 Possibility Theory 46
2.7 Fuzzy Systems 49
2.8 Constructing Fuzzy Sets and Operations 53
2.9 Nonstandard Fuzzy Sets 55
2.10 Principal Sources for Further Study 57
References 59
2.1 Introduction
The term “fuzzy logic,” as currently used in the literature, has two distinct meanings.
In the narrow sense, it is viewed as a generalization of the various many-valued log-
ics that have been investigated in the area of mathematical logic since the beginning
of the 20th century. An excellent historical overview of the emergence and devel-
opment of many-valued logics was prepared by Rescher [1969]; the various issues
11
FUZZY LOGIC IN GEOLOGY Copyright 2004, Elsevier Science (USA)
All rights of reproduction in any form reserved.
ISBN: 0-12-415146-9](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-35-2048.jpg)
![12 2 Fuzzy Logic: A Specialized Tutorial
involved in generalizing many-valued logics into fuzzy logic are thoroughly covered
in monographs by Hájek [1998] and Novák et al. [1999].
In the alternative, broad sense, fuzzy logic is viewed as a system of concepts,
principles, and methods for dealing with modes of reasoning that are approximate
rather than exact [Novák Perfilieva, 2000]. In this book, we are interested in fuzzy
logic only in this broad sense. In this sense, fuzzy logic is based upon fuzzy set theory.
It utilizes the apparatus of fuzzy set theory for formulating various forms of sound
approximate reasoning in natural language. It is thus essential to begin our tutorial
with an overview of basic concepts of fuzzy set theory.
Fuzzy set theory, introduced by Zadeh [1965], is an outgrowth of classical set
theory. Contrary to the classical concept of a set, or crisp set, the boundary of a
fuzzy set is not precise. That is, the change from nonmembership to membership
in a fuzzy set may be gradual rather than abrupt. This gradual change is expressed
by a membership function, which completely and uniquely characterizes a particular
fuzzy set.
Every geologist is familiar with the terms clay, silt, and gravel, terms used to
describe the “size” of sedimentary particles (Figure 2.1a). These terms stand for crisp
sets as they are most commonly used, insofar as a grain can only belong to one size
grade at a time. Thus, in the traditional “pigeon hole” view of grain sizes, a spherical
grain with a diameter of 1.999 mm would be sand whereas a grain 2.001 mm in diam-
eter would be gravel. An alternative representation of the crisp set “sand” would be to
assign a value of 1 to grain diameters that are members of the set “sand” (the domain
interval (0.0625–2] mm) and a 0 to grain diameters that are not sand. In contrast,
Figure 2.1 Comparison of crisp-set (a) versus fuzzy-set (b) representation of the geologic
variable “grain size.”](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-36-2048.jpg)
![2.1 Introduction 13
one possible representation of the sedimentary size terms clay, silt, sand, and gravel
with fuzzy sets is shown in Figure 2.1b. In a fuzzy set representation the range of
membership in a given set (e.g., “sand”) is not limited to 0 or 1 but can take on any
value between and including [0, 1]. Our hypothetical 1.999 and 2.001 mm diame-
ter grains are simultaneously members of both sets, sand and gravel, to a degree of
about 0.5. The simple trapezoids represent the membership functions.
Two distinct notations are most commonly employed in the literature to denote
membership functions. In one of them, the membership function of a fuzzy set A is
denoted by μA(x) and usually has the form
μA: X → [0, 1], (2.1)
where X denotes the universal set under consideration and A is a label of the fuzzy
set defined by this function. The universal set is always assumed to be a crisp set.
For each x ∈ X, the value μA(x) expresses the degree (or grade) of membership of
element x of X in fuzzy set A.
In the second notation, the symbol A of a fuzzy set is also used to denote the
membership function of A. However, no ambiguity results from this double use of
the same symbol since each fuzzy set is completely and uniquely defined by one
particular membership function. That is, A(x) in the second notation has the same
meaning as μA in the first notation; (2.1) is thus written in the second notation as
A: X → [0, 1]. (2.2)
In this book, the second notation is adopted. It is simpler and, by and large,
more popular in current literature on fuzzy set theory. Classical (crisp) sets may
be viewed from the standpoint of fuzzy set theory as special fuzzy sets, in which
A(x) is either 0 or 1 for each x ∈ X. Hence, we use the same notation for fuzzy sets
and crisp sets.
Fuzzy sets whose membership functions have the form (2.2), which are called
standard fuzzy sets, do not capture the full variety of fuzzy sets. Since standard fuzzy
sets are currently predominant in the literature, this tutorial is largely devoted to
them. However, basic properties of several nonstandard types of fuzzy sets, whose
importance in some applications has lately been recognized, are introduced in
Section 2.9.
Additional examples of membership functions are shown in Figure 2.2. These func-
tions may be considered as candidates for representing the meaning of the linguistic
expression “around 3” in the context of a given application. The width of each of these
functions is, of course, strongly dependent on the application context. In general, a
membership function that is supposed to capture the intended meaning of a linguistic
expression in the context of a particular application must be somehow constructed.
This issue is discussed in Section 2.8.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-37-2048.jpg)
![2.2 Basic Concepts of Fuzzy Sets 15
universal set X, we have
SUB(A, B) =
x∈X
A(x) −
x∈X
max[0, A(x) − B(x)]
x∈X
A(x)
. (2.3)
The negative term in the numerator describes the sum of the degrees to which the
subset inequality A(x) ≤ B(x) is violated, the positive term describes the largest
possible violation of the inequality, the difference in the numerator describes the sum
of the degrees to which the inequality is not violated, and the term in the denominator
is a normalizing factor to obtain the range
0 ≤ SUB(A, B) ≤ 1.
When sets A and B are defined on a bounded subset of real numbers (i.e., X is a
closed interval of real numbers), the three terms in (2.3) are replaced with integrals
over X.
For any fuzzy set A defined on a finite universal set X, its scalar cardinality, |A|,
is defined by the formula
|A| =
x∈X
A(x).
Scalar cardinality is sometimes referred to in the literature as a sigma count.
Among the most important concepts of standard fuzzy sets are the concepts of an
α-cut and a strong α-cut. Given a fuzzy set A defined on X and a particular number
α in the unit interval [0, 1], the α-cut of A, denoted by αA, is a crisp set that consists
of all elements of X whose membership degrees in A are greater than or equal to α.
This can formally be written as
α
A = {x|A(x) ≥ α}.
The strong α-cut, α+A, has a similar meaning, but the condition “greater than or equal
to” is replaced with the stronger condition “greater than.” Formally,
α+
A = {x|A(x) α}.
The set 0+A is called the support of A and the set 1A is called the core of A. When
the core A is not empty, A is called normal; otherwise, it is called subnormal. The
largest value of A is called the height of A and it is denoted by hA. The set of distinct
values A(x) for all x ∈ X is called the level set of A and is denoted by ΛA.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-39-2048.jpg)
![16 2 Fuzzy Logic: A Specialized Tutorial
Figure 2.3 Illustration of some basic characteristics of fuzzy sets.
All the introduced concepts are illustrated in Figure 2.3. We can see that
α1A ⊆ α2A and α1+
A ⊆ α2+
A2
when α1 ≥ α2. This implies that the set of all distinct α-cuts (as well as strong α-cuts)
is always a nested family of crisp sets. When α is increased, the new α-cut (strong
α-cut) is always a subset of the previous one. Clearly, 0A = X and 1+A = ∅.
It is well established [Klir Yuan, 1995] that each fuzzy set is uniquely represented
by the associated family of its α-cuts via the formula
A(x) = sup {α · α
A(x)|α ∈ [0, 1]}, (2.4)
or by the associated family of its strong α-cuts via the formula
A(x) = sup {α · α+
A(x)|α ∈ [0, 1]}, (2.5)
where sup denotes the supremum of the respective set and αA (or α+A) denotes for
each α ∈ [0, 1] the special membership function (characteristic function) of the α-cut
(or strong α-cut, respectively).
The significance of the α-cut (or strong α-cut) representation of fuzzy sets is that
it connects fuzzy sets with crisp sets. While each crisp set is a collection of objects
that are conceived as a whole, each fuzzy set is a collection of nested crisp sets that
are also conceived as a whole. Fuzzy sets are thus wholes of a higher category.
The α-cut representation of fuzzy sets allows us to extend the various properties
of crisp sets, established in classical set theory, into their fuzzy counterparts. This
is accomplished by requiring that the classical property be satisfied by all α-cuts of
the fuzzy set concerned. Any property that is extended in this way from classical](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-40-2048.jpg)
 = B(y), (2.7)
for all x ∈ X, where y = f (x). Clearly,
F−1
[F(A)] ⊇ A](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-41-2048.jpg)
![18 2 Fuzzy Logic: A Specialized Tutorial
Figure 2.4 Illustration of the extension principle.
for all A ∈ F(X), where the equality is obtained when f is a one-to-one function.
The use of the extension principle is illustrated in Figure 2.4, where it is shown
how fuzzy set A is mapped to fuzzy set B via function F that is consistent with the
given function f . That is, B = F(A). For example, since
b = f (a1) = f (a2) = f (a3),
we have
B(b) = max[A(a1), A(a2), A(a3)]
by Equation (2.6). Conversely,
F−1
(B)(a1) = F−1
(B)(a2) = F−1
(B)(a3) = B(b)
by (2.7).
The introduced extension principle, by which functions are fuzzified, is basically
described by Equations (2.6) and (2.7). These equations are direct generalizations
of similar equations describing the extension principle of classical set theory. In the
latter, symbols A and B denote characteristic functions of crisp sets.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-42-2048.jpg)
![2.3 Operations on Fuzzy Sets 19
2.3 Operations on Fuzzy Sets
Operations on fuzzy sets possess a considerably greater variety than those on classical
sets. In fact, most operations on fuzzy sets do not have any counterparts in classical set
theory. The following five types of operations on fuzzy sets are currently recognized:
(a) modifiers;
(b) complements;
(c) intersections;
(d) unions;
(e) averaging operations.
Modifiers and complements operate on one fuzzy set. Intersections and unions oper-
ate on two fuzzy sets, but their application can be extended to any number of fuzzy
sets via their property of associativity. The averaging operations, which are not asso-
ciative, operate, in general, on n fuzzy sets (n ≥ 2). In addition to these five types
of operations, special fuzzy sets referred to as fuzzy intervals are also subject to
arithmetic operations.
As can be seen from this overall characterization of operations on fuzzy sets, this
subject is very extensive. It is also a subject that has been investigated by many
researchers, and that is now quite well developed. Due to the enormous scope of
the subject, we are able to present in this section only a very brief characterization
of each of the introduced types of operations, but we provide the reader with ample
references for further study.
2.3.1 Modifiers
Modifiers are unary operations whose primary purpose is to modify fuzzy sets to
account for linguistic hedges, such as very, fairly, extremely, moderately, etc., in
representing expressions of natural language. Each modifier, m, is an increasing
(and usually continuous) one-to-one function of the form
m: [0, 1] → [0, 1],
which assigns to each membership grade A(x) of a given fuzzy set A a modified
grade m(A(x)). The modified grades for all x ∈ X define a new, modified fuzzy set.
Denoting conveniently this modified set by MA, we have
m(A(x)) = MA(x)
Observe that function m is totally independent of elements x to which values A(x)
are assigned; it depends only on the values themselves. In describing its formal
properties, we may thus ignore x and assume that the argument of m is an arbitrary
number a in the unit interval [0, 1].](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-43-2048.jpg)
![20 2 Fuzzy Logic: A Specialized Tutorial
In general, a modifier increases or decreases values of the membership functions to
which it is applied, but preserves the order. That is, if a ≤ b then m(a) ≤ m(b) for all
a, b ∈ [0, 1]or, recognizingthemeaningofa andb, ifA(x) ≤ A(y)forsomex, y ∈ X,
then MA(x) ≤ MA(y). Sometimes, it is also required that m(0) = 0 and m(1) = 1.
Modifiers are basically of three types, depending on which values of the
membership functions they increase or decrease:
(i) modifiers that increase all values;
(ii) modifiers that decrease all values;
(iii) modifiers that increase some values and decrease other values.
To illustrate these types of modifiers, let us consider the fuzzy set A in Figure 2.2.
For each x ∈ R, A is clearly defined by the formula
A(x) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
(x − 1)/2 when x ∈ [1,3]
(5 − x)/2 when x ∈ [3,5]
0 otherwise
Assume that this fuzzy set represents, in a given application context, the linguistic
concept “close to 3.” To modify A for representing the concept “very close to 3,”
we need to reduce in some way the values of A. This can be done by choosing an
appropriate modifier from the class of functions
mλ(a) = aλ
, (2.8)
where a is the value of A to which mλ is applied and λ is a parameter whose value
determines how strongly mλ modifies A. For each value of λ, which must be in this
case greater than 1, we obtain a particular modifier. When applying the modifier to
A, we obtain a new membership function, mλ[A(x)], a composite of functions A and
m, which for each x ∈ R is defined by the formula
mλ[A(x)] =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
[(x − 1)/2]λ when x ∈ [1,3]
[(5 − x)/2]λ when x ∈ [3,5]
0 otherwise
This modified membership function has a shape exemplified by the function labeled
as C in Figure 2.2. Its width is determined by the value λ of the chosen modifier: the
larger the value, the narrower the function. The proper value of λ must be determined
in the context of each particular application.
Assume now that we want to modify the same set A for representing the concept
“fairly close to 3.” In this case, we need to increase the values of A. This can be done
with modifiers of the form (2.8), provided that λ ∈ (0, 1). Applying these modifiers to](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-44-2048.jpg)
![2.3 Operations on Fuzzy Sets 21
A results in a new membership function whose shape is exemplified by the function
labeled as F in Figure 2.2. The smaller the value of λ, the wider is the modified
membership function.
It should be mentioned at this point that (2.8) is given here solely as an example of
a possible class of modifiers of fuzzy sets. As is well known, these modifiers do not
always properly capture the meaning of linguistic hedges in natural language. Amore
comprehensive treatment of linguistic hedges is presented in Chapter 10; see also
Novák [1989].
2.3.2 Complements
Similarly to modifiers, complements of fuzzy sets may be defined via appropri-
ate unary operations on [0, 1]. While modifiers preserve the order of membership
degrees, complements reverse the order. In particular, each fuzzy complement, c,
must satisfy at least the following two requirements:
(c1) c(0) = 1 and c(1) = 0;
(c2) for all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b).
Requirement (c1) guarantees that all fuzzy complements collapse to the unique clas-
sical complement for crisp sets. Requirement (c2) guarantees that increases in the
degree of membership in A do not result in increases in the degree of membership
in the complement of A. This is essential since any increase in the degree of mem-
bership of an object in a fuzzy set cannot simultaneously increase the degree of
nonmembership of the same object in the same fuzzy set.
When used as a fuzzy complement, function c is always applied to membership
degrees A(x) of some fuzzy set A. It depends only on the values A(x) and not on the
objects x to which the values are assigned. For the purpose of characterizing fuzzy
complements, we may thus ignore these objects and observe only how function c
depends on numbers in [0, 1]. This is the reason why no reference is made to specific
degrees A(x) in the requirements (c1) and (c2). However, when function c defines a
complement of a particular fuzzy set A, we must keep track of the relevant objects x
to make the connection between A(x) and c[A(x)].
Although requirements (c1) and (c2) are sufficient to characterize the largest class
of acceptable fuzzy complements, two additional requirements are imposed on fuzzy
complements by most applications of fuzzy set theory:
(c3) c is a continuous function;
(c4) c(c(a)) = a for all a ∈ [0, 1].
Requirement (c3) guarantees that infinitesimal changes in the argument do not result
in discontinuous changes in the function. Requirement (c4) guarantees that fuzzy sets](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-45-2048.jpg)
![22 2 Fuzzy Logic: A Specialized Tutorial
are not changed by double complementation. Fuzzy complements that satisfy (c4)
are called involutive.
A practical class of fuzzy complements that satisfy requirements (c1)–(c4) is
defined for each a ∈ [0, 1] by the formula
cλ(a) = (1 − aλ
)1/λ
, (2.9)
where λ ∈ (0, ∞); it is called the Yager class of fuzzy complements. One particular
fuzzy complement is obtained for each value of the parameter λ. The complement
obtained for λ = 1, which is called a standard fuzzy complement, is the most com-
mon complement in applications of fuzzy set theory. Clearly, the standard fuzzy
complement of a fuzzy set A, usually denoted by
A, is defined for each x ∈ X by the
equation
A(x) = 1 − A(x).
Other parameter-based formulas for describing classes of fuzzy complements have
been proposed in the literature. In fact, some procedures have been developed by
which new classes of fuzzy complements can be generated [Klir Yuan, 1995].
However, this theoretical topic is beyond the scope of this tutorial.
To determine the most fitting complement in the context of each particular appli-
cation is a problem of knowledge acquisition, somewhat similar to the problem of
constructing membership functions. Given a class of fuzzy complements, such as the
Yager class, the constructing problem reduces to the problem of determining the right
value of the relevant parameter.
2.3.3 Intersections and unions
Intersections and unions of fuzzy sets, denoted by i and u respectively, are general-
izations of the classical operations of intersections and unions of crisp sets. They may
be defined via appropriate functions that map each pair of real numbers from [0, 1]
(representing degrees A(x) and B(x) of given fuzzy sets A and B for some x ∈ X)
into a single number in [0, 1] (representing membership degree (A ∩ B)(x) of the
intersection of A and B or membership degree of the union of A and B for the given
x). Hence,
(A ∩ B)(x) = i[A(x), B(x)]
and
(A ∪ B)(x) = u[A(x), B(x)]
for all x ∈ X. To discuss properties of functions i and u, which do not depend on x,
we may view i and u as functions from [0, 1] × [0, 1] to [0, 1].](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-46-2048.jpg)
![2.3 Operations on Fuzzy Sets 23
Contrary to their classical counterparts, fuzzy intersections and unions are not
unique. This is a natural consequence of the well-established fact that the linguistic
expressions “x is a member of A and B” and “x is a member of A or B” have different
meanings when applied by human beings to different vague concepts in different
contexts. To be able to capture the different meanings, we need to characterize the
classes of fuzzy intersections and fuzzy unions as broadly as possible.
It has been established that operations known in the literature as triangular
norms or t-norms and triangular conorms or t-conorms, which have been exten-
sively studied in mathematics, possess exactly those properties that are requisite, on
intuitive grounds, for fuzzy intersections and fuzzy unions, respectively. The class
of t-norms/fuzzy intersections is characterized by four requirements; the class of
t-conorms/fuzzy unions is also characterized by four requirements, three of which are
identical with the requirements for t-norms. In the following list, the requirements for
t-norms/fuzzy intersections i are paired with their counterparts for t-conorms/fuzzy
unions u, and must be satisfied for all a, b, d ∈ [0, 1]:
(i1) i(a, 1) = a (boundary requirement for i);
(u1) u(a, 0) = a (boundary requirement for u);
(i2) b ≤ d implies i(a, b) ≤ i(a, d)
(monotonicity);
(u2) b ≤ d implies u(a, b) ≤ u(a, d)
(i3) i(a, b) = i(b, a)
(commutativity);
(u3) u(a, b) = u(b, a)
(i4) i(a, i(b, d)) = i(i(a, b), d)
(associativity).
(u4) u(a, u(b, d)) = u(u(a, b), d)
It is easy to see that the first three requirements for i ensure that fuzzy intersections
collapse to the classical set intersection when applied to crisp sets: i(0, 1) = 0 and
i(1, 1) = 1 follow directly from the boundary requirement; i(1, 0) = 0 and i(0, 0) =
0 follow then from commutativity and monotonicity, respectively. Similarly, the
first three requirements for u ensure that fuzzy unions collapse to the classical set
union when applied to crisp sets. Commutativity requirements ensure that fuzzy
intersections and unions are symmetric operations, indifferent to the order in which
sets to be combined are considered; together with monotonicity requirements, they
guarantee that fuzzy intersections and unions do not decrease when any of their
arguments are increased, and do not increase when any arguments are decreased.
Associativity requirements allow us to extend fuzzy intersections and unions to more
than two sets, in perfect analogy with their classical counterparts.
The following are examples of some common fuzzy intersections and fuzzy unions
with their usual names (each defined for all a, b ∈ [0, 1]).
Standard fuzzy intersection: i(a, b) = min(a, b)
Algebraic product: i(a, b) = ab](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-47-2048.jpg)
![24 2 Fuzzy Logic: A Specialized Tutorial
Bounded difference: i(a, b) = max(0, a + b − 1)
Drastic intersection: imin(a, b) =
⎧
⎨
⎩
a when b = 1
b when a = 1
0 otherwise
Standard fuzzy union: u(a, b) = max(a, b)
Algebraic sum: u(a, b) = a + b − ab
Bounded sum: u(a, b) = min(1, a + b)
Drastic union: umax(a, b) =
⎧
⎨
⎩
a when b = 0
b when a = 0
1 otherwise
It is easy to verify that the inequalities
imin(a, b) ≤ i(a, b) ≤ min(a, b)
max(a, b) ≤ u(a, b) ≤ umax(a, b)
are satisfied for all a, b ∈ [0, 1] by any fuzzy intersection i and any fuzzy union u,
respectively. These inequalities specify, in effect, the full ranges of fuzzy intersections
and fuzzy unions.
Examples of classes of fuzzy intersections, iw, and fuzzy union, uw, that cover the
full ranges of these operations are defined for all a, b ∈ [0, 1] by the formulas
iw(a, b) = 1 − min{1, [(1 − a)w + (1 − b)w]1/w}
uw(a, b) = min[1, (aw + bw)1/w]
, (2.10)
where w is a parameter whose range is (0, ∞). One particular fuzzy intersection
and one particular fuzzy union are obtained for each value of the parameter. These
operations are often referred to in the literature as the Yager classes of intersections
and unions. Although it is not obvious from the formulas, it is relatively easy to prove
that the standard fuzzy operations are obtained in the limit for w → ∞.
Since Yager intersections increase as the value of w increases, they become less
restrictive or weaker with increasing w. The drastic intersection is the strongest and
the standard intersection is the weakest. For Yager unions, this pattern is inverted;
they become more restrictive or stronger with increasing w. The standard union is
the strongest, the drastic union the weakest.
It should be mentioned that various other classes of fuzzy intersections and
unions have been examined in the literature. Moreover, special procedures are now
available by which new classes of fuzzy intersection and unions can be generated
[Klir Yuan, 1995].
Among the great variety of fuzzy intersections and unions, the standard operations
possess certain properties that give them special significance. First, we recognize
that they are located at opposite ends of the respected ranges of these operations.](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-48-2048.jpg)
![2.3 Operations on Fuzzy Sets 25
While the standard intersection is the weakest one among all fuzzy intersections,
the standard union is the strongest one among all fuzzy unions. Second, the standard
operationsaretheonlycutworthyoperationsamongallfuzzyintersectionsandunions.
Third, they are also the only operations among fuzzy intersections and unions that
are idempotent. This means that is(a, a) = us(a, a) = a for all a ∈ [0, 1]. Non-
standard fuzzy intersections are only subidempotent, while nonstandard fuzzy unions
are superidempotent; this means that
i(a, a) a and u(a, a) a
for all a ∈ (0, 1). In addition, when using the standard fuzzy operations, errors of the
operands do not compound. This is a desirable property from the computational point
of view, which other fuzzy operations do not possess.
Whatever combination of fuzzy counterparts of the three classical set-theoretic
operations (complement, intersection, union) we choose, some properties of the clas-
sical operations (properties of the underlying Boolean algebra) are inevitably violated.
This is a consequence of imprecise boundaries of fuzzy sets. The standard fuzzy oper-
ations violate only the law of excluded middle and the law of contradiction. Some
other combinations preserve these laws, but violate distributivity and idempotence
[Klir Yuan, 1995].
2.3.4 Averaging operations
Fuzzy intersections (t-norms) and fuzzy unions (t-conorms) are special types of oper-
ations for aggregating fuzzy sets: given two or more fuzzy sets, they produce a single
fuzzy set, an aggregate of the given sets. While they do not cover all aggregating oper-
ations, they cover all aggregating operations that are associative. Because of the lack
of associativity, the remaining aggregating operations must be defined as functions
of n arguments for each n ≥ 2. These remaining aggregation operations are called
averaging operations. As the name suggests, they average in various ways member-
ship functions of two or more fuzzy sets defined on the same universal set. They
do not have any counterparts in classical set theory. Indeed, an average of several
characteristic functions of classical sets is not, in general, a characteristic function!
However, classical sets can be averaged if they are viewed as special fuzzy sets.
For each n ≥ 2, an averaging operation, h, aggregates n fuzzy sets defined on the
same universal set X, say sets A1, A2, . . . , An. Denoting conveniently the aggregate
fuzzy set by H(A1, A2, . . . , An), we have
H(A1, A2, . . . , An)(x) = h[A1(x), A2(x), . . . , An(x)]
for all x ∈ X. Since properties of various averaging operations h do not depend on
x, but only on the membership degrees A1(x), A2(x), . . . , An(x) ∈ [0, 1], we may
view these operations as functions from [0, 1]n to [0, 1].](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-49-2048.jpg)
![26 2 Fuzzy Logic: A Specialized Tutorial
The following two requirements are requisite for any averaging operation h with
n arguments (n ≥ 2):
(h1) for all a ∈ [0, 1], h(a, a, . . . , a) = a (idempotency);
(h2) for any pair of n-tuples of real numbers in [0,1], a1, a2, . . . , an and
b1, b2, . . . , bn, if ai ≤ bi for all i ∈ Nn, then
h(a1, a2, . . . , an) ≤ h(b1, b2, . . . , bn) (monotonicity).
Requirement (h1) expresses our intuition that an average of equal numbers must result
in the same number. Requirement (h2) guarantees that the average does not decrease
when any of the arguments increase.
In addition to these essential and easily understood requirements, averaging
operations on fuzzy sets are usually expected to satisfy two additional requirements:
(h3) h is a continuous function;
(h4) h is a symmetric function in all its arguments, which means that
h(a1, a2, . . . , an) = h(ap(1), ap(2), . . . , ap(n))
for any permutation p on Nn. Requirement (h3) guarantees that small changes in any
of the arguments do not result in discontinuous changes in the average. Requirement
(h4) captures the usual assumption that the aggregated fuzzy sets are equally impor-
tant. If this assumption is not warranted in some application contexts, the symmetry
requirement must be dropped.
It is significant that any averaging operation h that satisfies the two basic require-
ments (h1) and (h2) produces numbers that for each n-tuple a1, a2, . . . , an ∈ [0, 1]n
lie within the interval defined by the inequalities
min(a1, a2, . . . , an) ≤ h(a1, a2, . . . , an) ≤ max(a1, a2, . . . , an).
To see this, let
a∗ = min(a1, a2, . . . , an) and a∗
= max(a1, a2, . . . , an).
If h satisfies requirements (h1) and (h2), then a∗ = h(a∗, a∗, . . . , a∗) ≤ h(a1,
a2, . . . , an) ≤ h(a∗
1, a∗
2, . . . , a∗
n) = a∗. Conversely, if h produces numbers within the
interval bounded by the min and max operations, then it must also satisfy requirement
(h1) of idempotency; indeed,
a = min(a, a, . . . , a) ≤ h(a, a, . . . , a) ≤ max(a, a, . . . , a) = a
for all a ∈ [0, 1]. That is, averaging operations cover the whole range between the
standard fuzzy intersection and the standard fuzzy union. The standard operations](https://image.slidesharecdn.com/71683-250226063937-6b294fc7/75/Fuzzy-Logic-in-Geology-1st-Edition-Robert-V-Demicco-50-2048.jpg)
![grievance, to feel as if the whole world were a football, made for
him to play with,—
My lord, said he, drawing himself up into an attitude, I am sure
I need not, at this time of day, enter into any discussion with your
lordship on the vast importance of the rights and privileges of
Englishmen—of the original establishment of the trial by jury in this
country. It would be worse than idle to occupy your valuable time
and that of this court, by dilating upon the merits of our constitution
—the chiefest of which has, I may say—been always—and I will say
—wisely, considerately, and prudently held to be that peculiar mode
of administering justice between man and man. But, my lord, if in
civil cases the deliberation and decision of a jury are considered
adequate safeguards to the rights and property of the people, the
law, still more careful of their lives and liberties, has interposed in
criminal cases another and a higher tribunal, in the nature of a
grand jury. [Hereabouts the judge, having bowed his head
graciously, omitted to raise it again, having dropped into a sound
slumber.]
That tribunal of mediation in the first instance, is full of
importance; and whatever subsequent proceedings may be taken in
a case, I do say, for myself and my fellows, that the decision upon
ex-parte evidence requires more circumspection, more care, and
more consideration than a verdict delivered after a case had been
argued, and after witnesses have been heard on both sides.
If, my lord, your lordship concedes this point, I will merely say,
generally, that when the mind is occupied by any important object,
more especially in matters of jurisprudence, it is absolutely
necessary that nothing, if possible, should occur to irritate or
exacerbate the feelings—all should be calm, and at rest.
Several people turned their eyes towards his lordship, and some
smiled.
No incidental annoyance should be permitted to interpose itself;
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Fuzzy Logic in Geology 1st Edition Robert V. Demicco
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- 5. Fuzzy Logic inGeology 1st Edition Robert V. Demicco Digital Instant Download Author(s): Robert V. Demicco, George J. Klir ISBN(s): 9780124151468, 0124151469 Edition: 1 File Details: PDF, 7.64 MB Year: 2003 Language: english
- 7. Fuzzy Logic inGeology
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- 9. Fuzzy Logic inGeology Edited by Robert V. Demicco and George J. Klir CENTER FOR INTELLIGENT SYSTEMS BINGHAMTON UNIVERSITY (SUNY) BINGHAMTON, NEW YORK, USA Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
- 10. This book isprinted on acid-free paper. Copyright © 2004, Elsevier Science (USA) All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com.uk. You may also complete your request on-line via the Elsevier Science homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Academic Press An imprint of Elsevier Science 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press 84 Theobald’s Road, London WC1X 8RR, UK http://www.academicpress.com Library of Congress Cataloging-in-Publication Data ISBN 0-12-415146-9 PRINTED IN THE UNITED STATES OF AMERICA 03 04 05 06 07 08 9 8 7 6 5 4 3 2 1
- 11. Contents Contributors vii Foreword byLotfi A. Zadeh ix Preface xiii Glossary of Symbols xv Chapter 1 Introduction 1 Chapter 2 Fuzzy Logic: A Specialized Tutorial 11 Chapter 3 Fuzzy Logic and Earth Science: An Overview 63 Chapter 4 Fuzzy Logic in Geological Sciences: A Literature Review 103 Chapter 5 Applications of Fuzzy Logic to Stratigraphic Modeling 121 Chapter 6 Fuzzy Logic in Hydrology and Water Resources 153 Chapter 7 Formal Concept Analysis in Geology 191 Chapter 8 Fuzzy Logic and Earthquake Research 239 Chapter 9 Fuzzy Transform: Application to the Reef Growth Problem 275 Chapter 10 Ancient Sea Level Estimation 301 Acknowledgments 337 Index 339 v
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- 13. Contributors ● Andras Bardossy(Chapter 6): Institute of Hydraulic Engineering, Uni- versity of Stuttgart, Pfaffenwaldring 61, D-70550 Stuttgart, Germany [bardossy@europe.com]. ● Radim Bělohlávek (Chapter 7): Department of Computer Science, Palacký University of Olomouc, Tomkova 40, CZ-77900 Olomouc, Czech Republic [belohlavek@inf.upol.cz]. ● Istvan Bogardi (Chapter 6): Department of Civil Engineering, University of Nebraska at Lincoln, Lincoln, Nebraska 68588, USA [ibogardi@unl.edu]. ● Robert V. Demicco (Chapters 1, 3, 4, and 5): Department of Geological & Envi- ronmental Studies and Center for Intelligent Systems, Binghamton University (SUNY), Binghamton, New York 13902, USA [demicco@binghamton.edu]. ● Lucien Duckstein (Chapter 6): Ecole Nationale du Génie Rural des Eaux et des Forêts, 19 avenue du Maine, 75732 CEDEX 15, France [duckstein@engref.fr]. ● Chongfu Huang (Chapter 8): Institute of Resources Science, Beijing Normal Uni- versity, 19 Xinjiekouwai Street, Beijing 100875, China [hchongfu@bnu.edu.cn]. ● George J. Klir (Chapters 1 and 2): Department of Systems Science & Industrial Engineering and Center for Intelligent Systems, Binghamton University (SUNY), Binghamton, New York 13902, USA [gklir@binghamton.edu]. ● Vilem Novák (Chapter 10): Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic [Vilem.Novak@osu.cz]. ● Irina Perfilieva (Chapter 9): Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic [Perfili@dynami.osu.cz]. ● Rita Pongracz (Chapter 6): Department of Meteorology, Eotvos Lorand Univer- sity, Pazmany P. setany1/A, H-1117 Budapest, Hungary [prita@caesar.elte.hu]. vii
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- 15. Foreword In October 1999,at the invitation of my eminent friend, Professor George Klir, I visited the Binghamton campus of the State University of New York. In the course of my visit, I became aware of the fact that Professor Klir, a leading contributor to fuzzy logic and theories of uncertainty, was collaborating with Professor Robert Demicco, a leading contributor to geology and an expert on sedimentology, on an NSF-supported research project involving an exploration of possible applications of fuzzy logic to geology. What could be more obvious than suggesting to Professors Klir and Demicco to edit a book entitled “Fuzzy Logic in Geology.” No such book was in existence at the time. I was delighted when Professors Klir and Demicco accepted my suggestion. And, needless to say, I am gratified that the book has become a reality. But, what is really important is that Professors Klir and Demicco, the contributors and the publisher, Academic Press, have produced a book that is superlative in all respects. As the editors state in the preface, Fuzzy Logic in Geology is intended to serve three principal purposes: (1) to examine what has been done in this field; (2) to explore new directions; and (3) to expand the use of fuzzy logic in geology and related fields through exposition of new tools. To say that Fuzzy Logic in Geology achieves its aims with distinction is an under- statement. The excellence of organization, the wealth of new material, the profusion of applications, and the high expository skill of contributors, including Professors Klir and Demicco, combine to make the book an invaluable reference and an important source of new ideas. There is no doubt that Fuzzy Logic in Geology will be viewed as a landmark in its field. In the preface, Professors Klir and Demicco note that applications of fuzzy logic in science are far less visible than in engineering and, especially, in the realm of consumer products. Is there an explanation? In science, there is a deep-seated tradition of striving for the ultimate in rigor and precision.Although fuzzy logic is a mathematically based theory, as is seen in Chapter 2, there is a misperception, reflecting the connotation of its label, that fuzzy logic is imprecise and not well-founded. In fact, fuzzy logic may be viewed as an attempt to deal precisely with imprecision, just as probability theory may be viewed as an attempt to deal precisely with uncertainty. ix
- 16. x Foreword A relatedpoint is that in many of its applications, a concept which plays a key role is that of a linguistic variable, that is, a variable where values are words rather than numbers. Words are less precise than numbers. That is why the use of linguistic variables in fuzzy logic drew critical comments from some of the leading members of the scientific establishment. As an illustration, when I gave my first lecture on linguistic variables in 1972, Professor Rudolf Kalman, a brilliant scientist/engineer, had this to say: I would like to comment briefly on Professor Zadeh’s presentation. His proposals could be severely, ferociously, even brutally criticized from a technical point of view. This would be out of place here. But a blunt question remains: Is Professor Zadeh presenting important ideas or is he indulging in wishful thinking? No doubt Professor Zadeh’s enthusiasm for fuzziness has been reinforced by the prevailing climate in the US—one of unprecedented permissiveness. ‘Fuzzification’ is a kind of scientific permissiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation. In a similar vein, a colleague of mine at UCB and a friend, Professor William Kahan, wrote: Fuzzy theory is wrong, wrong, and pernicious. I cannot think of any problem that could not be solved better by ordinary logic. . . . What Zadeh is saying is the same sort of things as, ‘Technology got us into this mess and now it can’t get us out’. Well, technology did not get us into this mess. Greed and weakness and ambivalence got us into this mess. What we need is more logical thinking, not less. The danger of fuzzy theory is that it will encourage the sort of imprecise thinking that has brought us so much trouble. What Professors Kalman, Kahan, and other prominent members of the scientific establishment did not realize is that mathematically based use of words enhances the ability of scientific theories to deal with real-world problems. In particular, in both science and engineering, the use of words makes it possible to exploit the tolerance for imprecision to achieve tractability, robustness, simplicity and low cost of solution. The use of linguistic variable is the basis for the calculus of fuzzy if-then rules—a calculus which plays a key role in many of the applications of fuzzy logic—including its applications in geology. During the past few years, the use of words in fuzzy logic has evolved into method- ology labeled computing with words and perceptions (CWP)—a methodology which casts a new light on fuzzy logic and may lead to a radical enlargement of the role of natural languages in science and engineering. Computing with words and perceptions is inspired by the remarkable human capa- bility to perform a wide variety of physical and mental tasks, e.g., driving a car in city traffic or playing tennis, without any measurements and any computations. In performing such tasks, humans employ perceptions—perceptions of distance, speed, direction, intent, likelihood, and other attributes of physical and mental objects.
- 17. Foreword xi There isan enormous literature on perceptions, spanning psychology, philosophy, linguistics, and other fields. But what has not been in existence is a theory in which perceptions can be operated on as objects of computation. Fuzzy logic provides a basis for such a theory—a theory which is referred to as the computational theory of perceptions (CTP). Inthecomputationaltheoryofperceptions, perceptionsaredealtwithnotaspatterns of brain activity, but through their descriptions in a natural language. In this sense, a natural language may be viewed as a system for describing perceptions. Thus, if classical, bivalent logic is viewed as the logic of measurements, then fuzzy logic may be viewed as the logic of perceptions. Although the methodology of computing with words and perceptions is not treated explicitly in the book, the basic ideas which underlie it are in evidence throughout. Furthermore, Fuzzy Logic in Geology ventures beyond well-established techniques and presents authoritative expositions of methods which lie on the frontiers of fuzzy logic. In this respect, particularly worthy of note are the chapters on for- mal concept analysis (R. Bělohlávek), F-transformation (I. Perfilieva), and linguistic theory (V. Novák). In sum, Fuzzy Logic in Geology is a true role model. It is a high quality work which opens the door to application of new methods and new viewpoints to a variety of basic problems in geology, geophysics, and related fields. It is well-organized and reader-friendly. The editors, the contributors, and the publisher deserve our thanks and accolades. Lotfi A. Zadeh May 13, 2003 Berkeley, CA
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- 19. Preface This book hasthree purposes. Its first purpose is to demonstrate that fuzzy logic opens a radically new way to represent geological knowledge and to deal with geological problems, and that this new approach has been surprisingly successful in many areas of geology. This book’s second purpose is to help geologists understand the main facets of fuzzy logic and the role of these facets in geology. The final purpose of this book is to make researchers in fuzzy logic aware of the emerging opportunities for the application of their expertise in geology. This book is a chimera in that it is oriented not only at theoreticians, practitioners, and teachers of geology, but also at members of the fuzzy-set community. For geol- ogists, thebookcontainsaspecializedtutorialonfuzzylogic(Chapter2), abasicintro- duction to the application of fuzzy logic to model geological situations (Chapter 3), an overview of currently known applications of fuzzy logic in geology (Chapter 4), and six additional chapters with more extensive examples of applications of fuzzy logic to problems in a broad range of geological disciplines. For fuzzy logicians, the book is an overview of areas of geology in which fuzzy logic is already well established or is promising. Thus, our overall aim in preparing this book is to provide a useful link between the two communities and further stimulate interdisciplinary research. The book is a product of a close cooperation between the editors and the several contributing authors. The authors were commissioned to write chapters on specific topics. Great care has been taken to assure that the mathematical terminology and notation are uniform throughout the book. Moreover, care was also taken to assure that the structure of individual chapters and the style of referencing were consistent throughout. Furthermore, authors were requested to focus on clarity of presentation, adding summaries of technical content where appropriate. All these features make the book attractive and appropriate as a text for graduate courses and seminars. The book is written, by and large, in a narrative style, with the exception of a few sections in Chapters 7 and 9. These chapters are dependent on fairly complex mathematical preliminaries. It is far more efficient to introduce these preliminaries in a more formal style, typical of mathematical literature, using numbered definitions, lemmas, theorems, and examples. Although this formal presentation in Chapters 7 and 9 is essential for understanding operational details of the described methods, it is not necessary for a conceptual understanding of the methods and their geological applications. In fact, these chapters are structured conceptually. With this structure, xiii
- 20. xiv Preface the readermay still get the gist of the chapter without studying the details of the formal presentation. The idea of preparing a book on fuzzy logic in geology was suggested to the editors by Lotfi Zadeh, the founder of fuzzy logic, during his visit to Binghamton University in October 1999. Our opinion then, and now, is that it was a good idea. While fuzzy logic is now well established as an important tool in engineering, its applications in science are far less developed. Nevertheless, the utility of fuzzy logic in various areas of science has been increasingly recognized since at least the mid 1990s. A good example is in chemistry, where the role of fuzzy logic is examined in the excellent book Fuzzy Logic in Chemistry, edited by Dennis H. Rouvray and published by Academic Press in 1997. It thus seemed natural to propose this book, which examines the role of fuzzy logic in geology, to Academic Press, with an eye toward obtaining a synergistic effect. We hope that this book will not only serve its purpose well, but that it will stimulate publication of other books exploring the role of fuzzy logic in other areas of natural sciences such as biology and physics as well as in the social sciences such as geography and economics. Robert V. Demicco and George J. Klir Binghamton, New York December 2002
- 21. Glossary of Symbols GeneralSymbols {x, y, . . .} Set of elements x, y, . . . {x | p{x}} Set determined by property p x1, x2, . . . , xn n-tuple [xij ] Matrix [x1, x2, . . . , xn] Vector [a, b] Closed interval of real numbers between a and b [a, b), (b, a] Interval of real numbers closed in a and open in b (a, b) Open interval of real numbers A, B, C . . . . Arbitrary sets (crisp or fuzzy) x ∈ A Element x belongs to crisp set A A(x) or μA(X) Membership grade of x in fuzzy set A αA α-cut of fuzzy set A α+A Strong α-cut of fuzzy set A A = B Set equality A = B Set inequality A − B Set difference A ⊆ B Set inclusion A ⊂ B Proper set inclusion (A ⊆ B and A = B) SUB(A, B) Degree of subsethood of A in B P(X) Set of all crisp subsets of X (power set) F(X) Set of all standard fuzzy subsets of X (fuzzy power set) |A| Cardinality of crisp or fuzzy set A (sigma count) hA Height of fuzzy set A A Complement of set A A ∩ B Set intersection A ∪ B Set union A × B Cartesian product of sets A and B A2 Cartesian product A × A f : X → Y Function from X to Y f −1 Inverse of function f R ◦ Q Standard composition of fuzzy relations R and Q xv
- 22. xvi Glossary ofSymbols R ∗ Q Join of fuzzy relations R and Q R−1 Inverse of a binary fuzzy relation Less than ≤ Less than or equal to (also used for a partial ordering) x | y x given y x ⇒ y x implies y x ⇔ y x if and only if y Summation Product max(a1, a2, . . . , an) Maximum of (a1, a2, . . . , an) min(a1, a2, . . . , an) Minimum of (a1, a2, . . . , an) N Set of positive integers (natural numbers) Nn Set {1, 2, . . . , n} R Set of all real numbers Special Symbols B(X, Y, I) The set of all fuzzy concepts in a given context X, Y, I c Fuzzy complement d(A) Defuzzified value of fuzzy set A E Similarity relation (fuzzy equivalence) h Averaging operation hp Generalized means i Fuzzy intersection or t-norm imin Drastic fuzzy intersection iw Fuzzy intersection of Yager class J Fuzzy implication operator L Set of truth degrees L Complete residuated lattice LX The set of all fuzzy sets in X with truth values in L m Fuzzy modifier NecE Necessity measure corresponding to PosE pA Fuzzy propositional form and truth assignment p Fuzzy probability qualifier PosE Possibility measure associated with a proposition “ν is E” S(Q, R) Solution set of fuzzy relation equation R ◦ Q = R T Fuzzy truth qualifier X, Y Variables X, Z, I Fuzzy context u Fuzzy union or t-conorm umax Drastic fuzzy union
- 23. Glossary of Symbolsxvii uw Fuzzy union of Yager class W Set of possible worlds X Universal set (universe of discourse) Ø Empty set ⊗ Operation on L corresponding to conjunction (t-norm) → Operation on L corresponding to implication ∧ Classical operation of conjunction or minimum operation ∨ Classical operation of disjunction or maximum operation
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- 25. Chapter 1 Introduction RobertV. Demicco and George J. Klir Traditionally, science, engineering, and mathematics showed virtually no interest in studying uncertainty. It was considered undesirable and the ideal was to eliminate it. In fact, eliminating uncertainty from science was viewed as one manifestation of progress. This attitude towards uncertainty, prevalent prior to the 20th century, was seriously challenged by some developments in the first half of that century. Among them were the emergence of statistical mechanics, Heisenberg’s uncertainty principle in quantum mechanics, and Gödel’s theorems that established an inher- ent uncertainty in formal mathematical systems. In spite of these developments, the traditional attitude towards uncertainty changed too little and too slowly during the first half of the century. While uncertainty became recognized as useful, or even essential, in statistical mechanics and in some other areas (such as the actuarial pro- fession or the design of large-scale telephone exchanges), it was for a long time tacitly assumed that probability theory was capable of capturing the full scope of uncertainty. The presumed equality between uncertainty and probability was challenged only in the second half of the 20th century. The challenge came from two important gen- eralizations in mathematics. The first one was the generalization of classical measure theory [Halmos, 1950] to the theory of monotone measures, which was first suggested by Choquet [1953] in his theory of capacities. The second one was the generalization of classical set theory to fuzzy set theory, which was introduced by Zadeh [1965]. In the theory of monotone measures, the additivity requirement of classical measures is replaced with a weaker requirement of monotonicity with respect to set inclusion. In fuzzy set theory, the requirement of sharp boundaries of classical sets is abandoned. That is, the membership of an object in a fuzzy set is not a matter of either affirma- tion or denial, as it is in the case of any classical set, but it is in general a matter of degree. For historical reasons of little significance, monotone measures are often referred to in the literature as fuzzy measures [Wang Klir, 1992]. This name is somewhat confusing since no fuzzy sets are involved in the definition of monotone measures. However, monotone measures can be fuzzified (i.e., defined on fuzzy sets), which results in a more general class of monotone measures—fuzzy monotone measures [Wang Klir, 1992, Appendix E]. 1 FUZZY LOGIC IN GEOLOGY Copyright 2004, Elsevier Science (USA) All rights of reproduction in any form reserved. ISBN: 0-12-415146-9
- 26. 2 1 Introduction Asis well known, probability theory is based on classical measure theory which, in turn, is based on classical set theory [Halmos, 1950]. When classical measures are replaced with monotone measures of some type and classical sets are replaced with fuzzy sets of some type, a framework is obtained for formalizing some new types of uncertainty, distinct from probability. This indicates that the two generalizations have opened a vast territory for formalizing uncertainty. At this time, only a rather small part of this territory has been adequately explored [Klir Wierman, 1999; Klir, 2002]. Liberating uncertainty from its narrow confines of probability theory opens new, more expressive ways of representing scientific knowledge. As is increasingly rec- ognized, scientific knowledge is organized, by and large, in terms of systems of various types (or categories in the sense of mathematical theory of categories) [Klir Rozehnal, 1996; Klir Elias, 2003]. In general, systems are viewed as relations between states of some variables. They are constructed for various purposes (prediction, retrodiction, prescription, diagnosis, control, etc.). In each system, its relations are utilized, in a given purposeful way, for determining unknown states of some variables on the basis of known states of some other variables. Systems in which the unknown states are determined uniquely are called deterministic; all other systems are called nondeterministic. By definition, each nondeterministic system involves uncertainty of some type. This uncertainty pertains to the purpose for which the system was constructed. It is thus natural to distinguish between predictive uncertainty, retrodictive uncertainty, diagnostic uncertainty, etc. In each nondeterministic system, the relevant uncertainty must be properly incorporated into the description of the system in some formalized language. To understand the full scope of uncertainty is thus essential for dealing with nondeterministic systems. When constructing a system for some given purpose, our ultimate goal is to obtain a system that is as useful as possible for this purpose. This means, in turn, to construct a system with a proper blend of the three most fundamental characteristics of systems: credibility, complexity, and uncertainty. Ideally, we would like to obtain a system with high credibility, low complexity, and low uncertainty. Unfortunately, these three criteria conflict with one another. To achieve high usefulness of the system, we need to find the right trade-off among them. The relationship between credibility, complexity and uncertainty is quite intri- cate and is not fully understood yet. However, it is already well established that uncertainty has a pivotal role in any efforts to maximize the usefulness of constructed systems.Although usually undesirable in systems when considered alone, uncertainty becomes very valuable when considered in connection with credibility and complex- ity of systems. Aslight increase in relevant uncertainty may often significantly reduce complexity and, at the same time, increase credibility of the system. Uncertainty is thus an important commodity in the knowledge business, a commodity that can be traded for gains in the other essential characteristics of systems by which we represent
- 27. 1 Introduction 3 knowledge.Because of this important role, uncertainty is no longer viewed in science and engineering as an unavoidable plague, but rather as an important resource that allow us to deal effectively with problems involving very complex systems. It is our contention that monotone measures and fuzzy sets (as well as the various uncertainty theories opened by these two profound generalizations in mathematics) are highly relevant to geology, and that their utility in geology should be seriously studiedintheyearsahead. Theaimofthisbookistodemonstratethispointbyfocusing on the role of fuzzy set theory, and especially the associated fuzzy logic, in geology. The term “fuzzy logic” has in fact two distinct meanings. In a narrow sense, it is viewed as a generalization of classical multivalued logics. It is concerned with the development of syntactic aspects (based on the notion of proof ) and semantic aspects (based on the notion of truth) of a relevant logic calculus. In order to be acceptable, the calculus must be sound (provability implies truth) and complete (truth implies provability). These issues have successfully been addressed for fuzzy logic in the narrow sense by Hájek [1998]. In a broad sense, fuzzy logic is viewed as a system of concepts, principles, and methods for dealing with modes of reasoning that are approximate rather than exact. The two meanings are connected since the very purpose of research on fuzzy logic in the narrow sense is to provide fuzzy logic in the broad sense with sound foundations. In this book, we are concerned only with fuzzy logic in the broad sense, which is surveyed in Chapter 2, and its role in geology, which is the subject of Chapters 3–10. From the standpoint of science, as it is still predominantly understood, the ideas of a fuzzy set and a fuzzy proposition are extremely radical. When accepted, one has to give up classical bivalent logic, generally presumed to be the principal pillar of science. Instead, we obtain a logic in which propositions are not required to be either true or false, but may be true or false to different degrees. As a consequence, some laws of bivalent logic no longer hold, such as the law of excluded middle or the law of contradiction. At first sight, this seems to be at odds with the very purpose of science. However, this is not the case. There are at least the following four reasons why allowing membership degrees in sets and degrees of truth in propositions in fact enhances scientific methodology quite considerably: 1. Fuzzy sets and fuzzy propositions possess far greater capabilities than their classi- cal counterparts to capture irreducible measurement uncertainties in their various manifestations. As a consequence, their use improves the bridge between mathe- matical models and the associated physical reality considerably. It is paradoxical that, in the face of the inevitable measurement errors, fuzzy data are always more accurate than their crisp (i.e., nonfuzzy) counterparts. Crisp data of each vari- able are based on a partition of the state set of the variable. The coarseness of this partition is determined by the resolution power of the measuring instrument employed. Measurements falling into the same block of the partition are not dis- tinguished in crisp data, regardless of their position within the block. Thus, for
- 28. 4 1 Introduction example,a measurement that is at the mid-point of the block is not distinguished from those at the borders with adjacent blocks. While the former is uncertainty free, provided that the block is sufficiently large relative to the resolution power of the measuring instrument employed, the latter involves considerable uncertainty due to the inevitability of measurement errors. This fundamental distinction is not captured at all in crisp data. On the contrary, fuzzy data can capture this and other measurement distinctions of this kind in terms of distinct member- ship degrees. Fuzzy data are thus more accurate than crisp data in this sense. Membership degrees that accompany fuzzy data express indirectly pertinent mea- surement uncertainties. When fuzzy data are processed, the membership degrees are processed as well. This implies that any results obtained by this processing are again more accurate (in the empirical sense) than their counterparts obtained by processing the less accurate crisp data. 2. An important feature of fuzzy logic in the broad sense is its capability to capture the vagueness of linguistic terms in statements that are expressed in natural lan- guages. Vagueness of a symbol (a linguistic term) in a given language results from the existence of objects for which it is intrinsically impossible to decide whether the symbol does or does not apply to them according to linguistic habits of some speech community using the language. That is, vagueness is a kind of uncertainty that does not result from information deficiency, but rather from imprecise mean- ings of linguistic terms, which are particularly abundant in natural languages. Classical set theory and classical bivalent logic are not capable of expressing the imprecision in meanings of vague terms. Hence, propositions in natural language that contain vague terms were traditionally viewed as unscientific. However, this view is extremely restrictive. As has increasingly been recognized in many areas of science, including especially geology, natural language is often the only way in which meaningful knowledge can be expressed. 3. Fuzzy sets and fuzzy propositions are powerful tools for managing complexity and controlling computational cost. This is primarily due to granulation of systems variables, which is a fuzzy counterpart of the classical quantization of variables. In quantization, states of a given variable are grouped into subsets (quanta) that are pairwise disjoint. In granulation, they are grouped into suitable fuzzy subsets (granules). The aim of both quantization and granulation is to make precision compatible with a given task. The advantage of granulation is that, contrary to quantization, it allows us to express gradual transitions from each granule to its neighbors. In quantization, the transition from one quantum to another is always abrupt and, hence, rather superficial. Granulation is thus a better way than quantization to adjust precision of systems as needed. 4. The apparatus of fuzzy set theory and fuzzy logic enhances our capabilities of modeling human common-sense reasoning, decision-making, and other aspects of human cognition. These capabilities are essential for acquiring knowledge from human experts, for representating and manipulating knowledge in expert
- 29. 1 Introduction 5 systemsin a human-like manner, and, generally, for designing and building human-friendly machines with high intelligence. Fuzzy sets and fuzzy propo- sitions are also essential for studying human reasoning, decision making, and acting that are based on perceptions rather than measurements. It is the synergy of all these capabilities that has made fuzzy set theory and fuzzy logic highly successful in many engineering applications over the last two decades or so. The most visible of these applications have been in the area of control, ranging from simple control systems in consumer products (intelligent washing machines, vacuum cleaners, camcorders, etc.) to highly challenging control systems, such as the one for controlling a pilotless helicopter via wireless communication of commands expressed in natural language. Less visible but equally successful applications have been demonstrated in the areas of database and information retrieval systems, expert systems, decision making, pattern recognition and clustering, image processing and computervision, manufacturing, robotics, transportation, riskandreliabilityanalyses, and many other engineering areas. In fact, every field of engineering has already been positively affected, in one way or another, by fuzzy set theory and fuzzy logic [Ruspini et al., 1998]. In science, applications of fuzzy set theory and fuzzy logic have developed at a considerably slower pace than in engineering and only in some areas of science thus far. This is understandable if we realize how extremely radical the ideas of fuzzy sets and fuzzy propositions actually are. Nevertheless, successful applications have already been demonstrated in many areas of science. Examples are applications in quantum physics [Pykacz, 1993; Cattaneo, 1993], chemistry [Rouvray, 1997], biology [Von Sternberg Klir, 1998], geography [Gale, 1972], ecology [Libelli Cianchi, 1996], linguistics [Rieger, 2001], economics [Billot, 1992], psychology [Zétényi, 1988], and social sciences [Smithson, 1987]. In geology, the utility of fuzzy set theory and fuzzy logic was recognized, by and large, only in the late 1990s, but the number of publications dealing with applications of fuzzy logic in geology is already substantial and is growing fast (Chapter 4). This is a clear indicator that the use of fuzzy logic in geology has a great potential. Our motivation for publishing this book is to help to develop this potential. It is important to realize that fuzzy set theory and fuzzy logic are not only tools that help us to deal with some difficult problems in science, engineering, and other professional areas, but they also provide us with a conceptual framework for a rad- ically new way of thinking. Sharp boundaries of classical sets and absolute truths or falsities of classical propositions are still possible under the new thinking, when justifiable, but they are viewed as limiting cases rather than the only possibilities. Thinking in absolute terms is replaced with thinking in relative terms. Everything becomes a matter of degree. This change in our thinking will undoubtedly open new, more refined ways of looking at old issues of epistemology, ethics, law, social policy, and other areas that affect our lives.
- 30. 6 1 Introduction Theemergence of fuzzy set theory and fuzzy logic and their impact on mathematics and logic as well as on science and science-dependent areas of human affairs possess all distinctive features that are characteristic of a paradigm shift, as introduced in the highly influential book by Thomas Kuhn [1962]. Since logic is fundamental to virtually all branches of mathematics as well as science, this paradigm shift has much broader implications than those generally recognized in the history of science and mathematics, each of which affects only a particular area of science or mathemat- ics. It is thus appropriate to refer to it as a “grand paradigm shift.” Various special characteristics of this paradigm shift, which is still ongoing, are discussed by Klir [1995, 1997, 2000]. It is generally agreed that this paradigm shift was initiated by the publication of the seminal paper by Zadeh [1965]. However, many ideas pertaining to fuzzy logic had appeared in the literature prior to the publication of that paper. Unfortunately, these ideas were by and large ignored at that time [Klir, 2001]. The purpose of this book is threefold: (i) to examine how fuzzy logic has already been applied in some areas of geology; (ii) to stimulate the development of applica- tions of fuzzy logic in other areas of geology; and (iii) to stimulate the use of additional tools of fuzzy logic in geology. Material covered in Chapters 2–10 was carefully selected to accomplish this purpose. The following is a brief preview of this material. Chapter 2 is an overview of fuzzy logic in the broad sense. It is written as a tutorial for those readers who are not familiar with fuzzy logic. This chapter covers not only those components of fuzzy logic that are employed in subsequent chapters, but also some additional ones which offer new application possibilities for geology. More- over, this chapter introduces terminology and notation that are followed consistently throughout the whole book. The aim of Chapter 3 is twofold: (i) to discuss reasons for using fuzzy logic in geology; and (ii) to illustrate the use of fuzzy logic in geology by simple examples. For geologists, some of the notions of fuzzy logic introduced in Chapter 2 are further discussed in terms of simple geological interpretations. For researchers in fuzzy logic, the chapter is a sort of tutorial which introduces them to some issues that are of concern to geology. Chapter 4 is a comprehensive overview of currently known applications of fuzzy logic in geology. It is primarily an annotated bibliography that is grouped into the following nine categories: (1) surface hydrology; (2) subsurface hydrology; (3) groundwater risk assessment; (4) geotechnical engineering; (5) hydrocarbon explo- ration; (6) seismology; (7) soil science and landscape development; (8) deposition of sediments; and (9) miscellaneous applications. In addition, the role of fuzzy logic within the broader area of soft computing is briefly characterized. The aim of this chapter is to provide readers with a useful resource for further study of established applications of fuzzy logic in geology, sometimes in the broader context of soft computing. Each of the remaining six chapters of this book covers in greater depth applica- tions of fuzzy logic in some specific area of geology. The utility of fuzzy logic to
- 31. 1 Introduction 7 stratigraphicmodeling is demonstrated in Chapter 5 via several case studies. The chapter describes two-dimensional and three-dimensional stratigraphic simulations that use fuzzy logic to model sediment production, sediment erosion, sediment trans- port, and sediment deposition. It is shown that fuzzy logic offers a robust, easily adaptable, and computationally efficient alternative to the traditional numerical solu- tion of complex, coupled differential equations commonly used to model sediment dispersal in stratigraphic models. Chapter 6 examines the utility of fuzzy logic in hydrology and water resources. These are areas of geology where applications of fuzzy logic are well established. After the various applications of fuzzy logic in these areas are surveyed, one major area of hydrology is chosen to describe the use of fuzzy logic in detail: the area of hydro-climatic modeling of hydrological extremes (i.e., droughts and intensive precipitation). Results over four regions (Arizona, Nebraska, Germany, and Hungary) and under three different climates (semiarid, dry, and wet continental) suggest that the use of fuzzy logic is successful in predicting statistical properties of monthly precipitation and drought index from the joint forcing of macrocirculation patterns and ENSO information. The purpose of Chapter 7 is to present formal concept analysis of fuzzy data and to explore its prospective applications in geology and paleontology. Formal concept analysis is concerned with analyzing data in terms of objects and their attributes. It is capable of answering questions such as: (i) What are the natural concepts that are hidden in the object-attribute data (e.g., important classes of organisms, minerals, or fossils)?; or(ii)Whatarethedependenciesthatareimplicitintheobject-attributedata? Fuzzified formal concept analysis, which is a relatively new methodological tool, is described in detail in the chapter and is illustrated by an example from paleontology. Chapter 8 is a comprehensive overview of the role of fuzzy logic in seismology and some closely related areas. Basic terminology of seismology is introduced to help readers who are not familiar with this area of geology. The focus in the chapter is on applications of fuzzy logic and other areas of fuzzy mathematics to earthquake prediction, assessment of earthquake intensity, assessment of earthquake damage, and study of the relationship between isoseismal area and earthquake magnitude. Thelasttwochaptersofthebookexploresomenewideasemergingfromfuzzylogic that can be applied to a broad range of geological problems. These chapters require some mathematical sophistication, but they are self-contained in the sense that the reader is provided with the relevant preliminaries and specific examples of applica- tions. Chapter 9 describes a new numerical technique—fuzzy transformation—that allowscomplexfunctionstobeapproximatedtoahighorder. Moreover, usefulmanip- ulations (such as numerical integration) are, in a number of cases, easier for the transformed expressions than for the originals. This technique is then applied to a solution of an ordinary differential equation used to model long-term reef growth under a variable sea level regime. Chapter 10 provides an example of how fuzzy logic can mathematically formalize what heretofore were primarily only linguistic
- 32. 8 1 Introduction descriptionsand interpretations of geologic phenomena. In this case, a computer pro- gram using specialized fuzzy-set based “evaluating expressions” is taught to mimic the linguistic geologic “rules” for both the division of Paleozoic measured sections of limestone into a hierarchy of different cycles, and the interpretation of those cycles in terms of ancient sea level. References Billot, A. [1992], Economic Theory of Fuzzy Equilibria. Springer-Verlag, New York. Cattaneo, G. [1993], “Fuzzy quantum logic II: The logics of unsharp quantum mechanics.” International Journal of Theoretical Physics, 32(10), 1709–1734. Choquet, G. [1953–54], “Theory of capacities.” Annales de L’Institut Fourier, 5, 131–295. Gale, S. [1972], “Inexactness, fuzzy sets and the foundations of behavioral geography.” Geographical Analysis, 4, 337–349. Hájek, P. [1998], Metamathematics of Fuzzy Logic. Kluwer, Boston, MA. Halmos, P. R. [1950], Measure Theory. Van Nostrand, Princeton, NJ. Klir, G. J. [1995], “From classical sets to fuzzy sets: a grand paradigm shift.” In: Wang, P. P. (ed.), Advances in Fuzzy Theory and Technology, Vol. III, pp. 3–30. Duke University, Durham, NC. Klir, G. J. [1997], “From classical mathematics to fuzzy mathematics: emergence of a new paradigm for theoretical science.” In: Rouvray, D. H. (ed.), Fuzzy Logic in Chemistry, pp. 31–63. Academic Press, San Diego, CA. Klir, G. J. [2000], Fuzzy Sets: An Overview of Fundamentals, Applications, and Personal Views. Beijing Normal University Press, Beijing, China. Klir, G. J. [2001], “Foundations of fuzzy set theory and fuzzy logic: A historical overview.” International Journal of General Systems, 30(2), 91–132. Klir, G. J. [2002], “Uncertainty-based information.” In: Melo-Pinto and H.-N. Teodorescu (eds.), Systemic Organisation of Information in Fuzzy Systems, pp. 21–52. IOS Press, Amsterdam. Klir, G. J., Elias, D. [2003], Architecture of Systems Problem Solving (2nd edition). Kluwer/Plenum, New York. Klir, G. J., Rozehnal, I. [1996], “Epistemological categories of systems: an overview.” International Journal of General Systems, 24(1–2), 207–224. Klir, G. J., Wierman, M. J. [1999], Uncertainty-Based Information: Elements of Gener- alized Information Theory (2nd edition). Physica-Verlag/Springer-Verlag, Heidelberg and New York. Kuhn, T. S. [1962], The Structure of Scientific Revolutions. University of Chicago Press, Chicago, IL. Libelli, S. M., Cianchi, P. [1996], “Fuzzy ecological models.” In: Pedrycz, W. (ed.), Fuzzy Modelling Paradigms and Practice, pp.141–164. Kluwer, Boston, MA. Pykacz, J. [1993], “Fuzzy quantum logic I.” International Journal of Theoretical Physics, 32(10), 1691–1707. Rieger, B. B. [2001], “Computing granular word meanings: A fuzzy linguistic approach in computational semiotics.” In: Wang, P. P. (ed.), Computing with Words, pp. 147–208. John Wiley, New York.
- 33. References 9 Rouvray, D.H. (ed.) [1997], Fuzzy Logic in Chemistry. Academic Press, San Diego, CA. Ruspini, E. H., Bonissone, P. P., Pedrycz, W. (eds.) [1988], Handbook of Fuzzy Computation. Institute of Physics Publishing, Bristol (UK) and Philadelphia, PA. Smithson, M. [1987], Fuzzy Set Analysis for Behavioral and Social Sciences. Springer-Verlag, New York. Von Sternberg, R., Klir, G. J. [1998], “Generative archetypes and taxa: A fuzzy set formalization.” Biology Forum, 91, 403–424. Wang, Z., Klir, G. J. [1992], Fuzzy Measure Theory. Plenum Press, New York. Zadeh, L. A. [1965], “Fuzzy sets.” Information and Control, 8(3), 338–353. Zétényi, T. (ed.) [1988], Fuzzy Sets in Psychology. North-Holland, Amsterdam and New York.
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- 35. Chapter 2 FuzzyLogic: A Specialized Tutorial George J. Klir 2.1 Introduction 11 2.2 Basic Concepts of Fuzzy Sets 14 2.3 Operations on Fuzzy Sets 19 2.3.1 Modifiers 19 2.3.2 Complements 21 2.3.3 Intersections and unions 22 2.3.4 Averaging operations 25 2.3.5 Arithmetic operations 28 2.4 Fuzzy Relations 31 2.4.1 Projections, cylindric extensions, and cylindric closures 32 2.4.2 Inverses, compositions, and joins 33 2.4.3 Fuzzy relation equations 34 2.4.4 Fuzzy relations on a single set 36 2.5 Fuzzy Logic 38 2.5.1 Basic types of propositional forms 41 2.5.2 Approximate reasoning 44 2.6 Possibility Theory 46 2.7 Fuzzy Systems 49 2.8 Constructing Fuzzy Sets and Operations 53 2.9 Nonstandard Fuzzy Sets 55 2.10 Principal Sources for Further Study 57 References 59 2.1 Introduction The term “fuzzy logic,” as currently used in the literature, has two distinct meanings. In the narrow sense, it is viewed as a generalization of the various many-valued log- ics that have been investigated in the area of mathematical logic since the beginning of the 20th century. An excellent historical overview of the emergence and devel- opment of many-valued logics was prepared by Rescher [1969]; the various issues 11 FUZZY LOGIC IN GEOLOGY Copyright 2004, Elsevier Science (USA) All rights of reproduction in any form reserved. ISBN: 0-12-415146-9
- 36. 12 2 FuzzyLogic: A Specialized Tutorial involved in generalizing many-valued logics into fuzzy logic are thoroughly covered in monographs by Hájek [1998] and Novák et al. [1999]. In the alternative, broad sense, fuzzy logic is viewed as a system of concepts, principles, and methods for dealing with modes of reasoning that are approximate rather than exact [Novák Perfilieva, 2000]. In this book, we are interested in fuzzy logic only in this broad sense. In this sense, fuzzy logic is based upon fuzzy set theory. It utilizes the apparatus of fuzzy set theory for formulating various forms of sound approximate reasoning in natural language. It is thus essential to begin our tutorial with an overview of basic concepts of fuzzy set theory. Fuzzy set theory, introduced by Zadeh [1965], is an outgrowth of classical set theory. Contrary to the classical concept of a set, or crisp set, the boundary of a fuzzy set is not precise. That is, the change from nonmembership to membership in a fuzzy set may be gradual rather than abrupt. This gradual change is expressed by a membership function, which completely and uniquely characterizes a particular fuzzy set. Every geologist is familiar with the terms clay, silt, and gravel, terms used to describe the “size” of sedimentary particles (Figure 2.1a). These terms stand for crisp sets as they are most commonly used, insofar as a grain can only belong to one size grade at a time. Thus, in the traditional “pigeon hole” view of grain sizes, a spherical grain with a diameter of 1.999 mm would be sand whereas a grain 2.001 mm in diam- eter would be gravel. An alternative representation of the crisp set “sand” would be to assign a value of 1 to grain diameters that are members of the set “sand” (the domain interval (0.0625–2] mm) and a 0 to grain diameters that are not sand. In contrast, Figure 2.1 Comparison of crisp-set (a) versus fuzzy-set (b) representation of the geologic variable “grain size.”
- 37. 2.1 Introduction 13 onepossible representation of the sedimentary size terms clay, silt, sand, and gravel with fuzzy sets is shown in Figure 2.1b. In a fuzzy set representation the range of membership in a given set (e.g., “sand”) is not limited to 0 or 1 but can take on any value between and including [0, 1]. Our hypothetical 1.999 and 2.001 mm diame- ter grains are simultaneously members of both sets, sand and gravel, to a degree of about 0.5. The simple trapezoids represent the membership functions. Two distinct notations are most commonly employed in the literature to denote membership functions. In one of them, the membership function of a fuzzy set A is denoted by μA(x) and usually has the form μA: X → [0, 1], (2.1) where X denotes the universal set under consideration and A is a label of the fuzzy set defined by this function. The universal set is always assumed to be a crisp set. For each x ∈ X, the value μA(x) expresses the degree (or grade) of membership of element x of X in fuzzy set A. In the second notation, the symbol A of a fuzzy set is also used to denote the membership function of A. However, no ambiguity results from this double use of the same symbol since each fuzzy set is completely and uniquely defined by one particular membership function. That is, A(x) in the second notation has the same meaning as μA in the first notation; (2.1) is thus written in the second notation as A: X → [0, 1]. (2.2) In this book, the second notation is adopted. It is simpler and, by and large, more popular in current literature on fuzzy set theory. Classical (crisp) sets may be viewed from the standpoint of fuzzy set theory as special fuzzy sets, in which A(x) is either 0 or 1 for each x ∈ X. Hence, we use the same notation for fuzzy sets and crisp sets. Fuzzy sets whose membership functions have the form (2.2), which are called standard fuzzy sets, do not capture the full variety of fuzzy sets. Since standard fuzzy sets are currently predominant in the literature, this tutorial is largely devoted to them. However, basic properties of several nonstandard types of fuzzy sets, whose importance in some applications has lately been recognized, are introduced in Section 2.9. Additional examples of membership functions are shown in Figure 2.2. These func- tions may be considered as candidates for representing the meaning of the linguistic expression “around 3” in the context of a given application. The width of each of these functions is, of course, strongly dependent on the application context. In general, a membership function that is supposed to capture the intended meaning of a linguistic expression in the context of a particular application must be somehow constructed. This issue is discussed in Section 2.8.
- 38. 14 2 FuzzyLogic: A Specialized Tutorial Figure 2.2 Possible shapes of membership functions whose purpose is to capture the meaning of the linguistic expression “around 3” in the context of a given application. 2.2 Basic Concepts of Fuzzy Sets Given two fuzzy sets A, B defined on the same universal set X, A is said to be a subset of B if and only if A(x) ≤ B(x) for all x ∈ X. The usual notation, A ⊆ B, is used to signify the subsethood relation. The set of all fuzzy subsets of X is called the fuzzy power set of X and is denoted by F(X). Observe that this set is crisp, even though its members are fuzzy sets. Moreover, this set is always infinite, even if X is finite. It is also useful to define a degree of subsethood, SUB(A, B), of A in B. When the sets are defined on a finite
- 39. 2.2 Basic Conceptsof Fuzzy Sets 15 universal set X, we have SUB(A, B) = x∈X A(x) − x∈X max[0, A(x) − B(x)] x∈X A(x) . (2.3) The negative term in the numerator describes the sum of the degrees to which the subset inequality A(x) ≤ B(x) is violated, the positive term describes the largest possible violation of the inequality, the difference in the numerator describes the sum of the degrees to which the inequality is not violated, and the term in the denominator is a normalizing factor to obtain the range 0 ≤ SUB(A, B) ≤ 1. When sets A and B are defined on a bounded subset of real numbers (i.e., X is a closed interval of real numbers), the three terms in (2.3) are replaced with integrals over X. For any fuzzy set A defined on a finite universal set X, its scalar cardinality, |A|, is defined by the formula |A| = x∈X A(x). Scalar cardinality is sometimes referred to in the literature as a sigma count. Among the most important concepts of standard fuzzy sets are the concepts of an α-cut and a strong α-cut. Given a fuzzy set A defined on X and a particular number α in the unit interval [0, 1], the α-cut of A, denoted by αA, is a crisp set that consists of all elements of X whose membership degrees in A are greater than or equal to α. This can formally be written as α A = {x|A(x) ≥ α}. The strong α-cut, α+A, has a similar meaning, but the condition “greater than or equal to” is replaced with the stronger condition “greater than.” Formally, α+ A = {x|A(x) α}. The set 0+A is called the support of A and the set 1A is called the core of A. When the core A is not empty, A is called normal; otherwise, it is called subnormal. The largest value of A is called the height of A and it is denoted by hA. The set of distinct values A(x) for all x ∈ X is called the level set of A and is denoted by ΛA.
- 40. 16 2 FuzzyLogic: A Specialized Tutorial Figure 2.3 Illustration of some basic characteristics of fuzzy sets. All the introduced concepts are illustrated in Figure 2.3. We can see that α1A ⊆ α2A and α1+ A ⊆ α2+ A2 when α1 ≥ α2. This implies that the set of all distinct α-cuts (as well as strong α-cuts) is always a nested family of crisp sets. When α is increased, the new α-cut (strong α-cut) is always a subset of the previous one. Clearly, 0A = X and 1+A = ∅. It is well established [Klir Yuan, 1995] that each fuzzy set is uniquely represented by the associated family of its α-cuts via the formula A(x) = sup {α · α A(x)|α ∈ [0, 1]}, (2.4) or by the associated family of its strong α-cuts via the formula A(x) = sup {α · α+ A(x)|α ∈ [0, 1]}, (2.5) where sup denotes the supremum of the respective set and αA (or α+A) denotes for each α ∈ [0, 1] the special membership function (characteristic function) of the α-cut (or strong α-cut, respectively). The significance of the α-cut (or strong α-cut) representation of fuzzy sets is that it connects fuzzy sets with crisp sets. While each crisp set is a collection of objects that are conceived as a whole, each fuzzy set is a collection of nested crisp sets that are also conceived as a whole. Fuzzy sets are thus wholes of a higher category. The α-cut representation of fuzzy sets allows us to extend the various properties of crisp sets, established in classical set theory, into their fuzzy counterparts. This is accomplished by requiring that the classical property be satisfied by all α-cuts of the fuzzy set concerned. Any property that is extended in this way from classical
- 41. 2.2 Basic Conceptsof Fuzzy Sets 17 set theory into the domain of fuzzy set theory is called a cutworthy property. For example, when convexity of fuzzy sets is defined by the requirement that all α-cuts of a fuzzy convex set be convex in the classical sense, this conception of fuzzy convexityiscutworthy. Otherimportantexamplesaretheconceptsofafuzzypartition, fuzzy equivalence, fuzzy compatibility, and various kinds of fuzzy orderings that are cutworthy (Section 2.4). It is important to realize that many (perhaps most) properties of fuzzy sets, perfectly meaningful and useful, are not cutworthy. These properties cannot be derived from classical set theory. Another way of connecting classical set theory and fuzzy set theory is to fuzzify functions. Given a function f : X → Y, where X and Y are crisp sets, we say that the function is fuzzified when it is extended to act on fuzzy sets defined on X and Y. That is, the fuzzified function maps, in general, fuzzy sets defined on X to fuzzy sets defined on Y. Formally, the fuzzified function, F, has the form F: F(X) → F(Y), where F(X) and F(Y) denote the fuzzy power set (the set of all fuzzy subsets) of X and Y, respectively. To qualify as a fuzzified version of f , function F must conform to f within the extended domain F(X) and F(Y). This is guaranteed when a principle is employed that is called an extension principle. According to this principle, B = F(A) is determined for any given fuzzy set A ∈ F(X) via the formula B(y) = max x|y=f (x) A(x) (2.6) for all y ∈ Y. Clearly, when the maximum in (2.6) does not exist, it is replaced with the supremum. The inverse function F−1 : F(Y) → F(X), of F is defined, according to the extension principle, for any given B ∈ F(Y), by the formula [F−1 (B)](x) = B(y), (2.7) for all x ∈ X, where y = f (x). Clearly, F−1 [F(A)] ⊇ A
- 42. 18 2 FuzzyLogic: A Specialized Tutorial Figure 2.4 Illustration of the extension principle. for all A ∈ F(X), where the equality is obtained when f is a one-to-one function. The use of the extension principle is illustrated in Figure 2.4, where it is shown how fuzzy set A is mapped to fuzzy set B via function F that is consistent with the given function f . That is, B = F(A). For example, since b = f (a1) = f (a2) = f (a3), we have B(b) = max[A(a1), A(a2), A(a3)] by Equation (2.6). Conversely, F−1 (B)(a1) = F−1 (B)(a2) = F−1 (B)(a3) = B(b) by (2.7). The introduced extension principle, by which functions are fuzzified, is basically described by Equations (2.6) and (2.7). These equations are direct generalizations of similar equations describing the extension principle of classical set theory. In the latter, symbols A and B denote characteristic functions of crisp sets.
- 43. 2.3 Operations onFuzzy Sets 19 2.3 Operations on Fuzzy Sets Operations on fuzzy sets possess a considerably greater variety than those on classical sets. In fact, most operations on fuzzy sets do not have any counterparts in classical set theory. The following five types of operations on fuzzy sets are currently recognized: (a) modifiers; (b) complements; (c) intersections; (d) unions; (e) averaging operations. Modifiers and complements operate on one fuzzy set. Intersections and unions oper- ate on two fuzzy sets, but their application can be extended to any number of fuzzy sets via their property of associativity. The averaging operations, which are not asso- ciative, operate, in general, on n fuzzy sets (n ≥ 2). In addition to these five types of operations, special fuzzy sets referred to as fuzzy intervals are also subject to arithmetic operations. As can be seen from this overall characterization of operations on fuzzy sets, this subject is very extensive. It is also a subject that has been investigated by many researchers, and that is now quite well developed. Due to the enormous scope of the subject, we are able to present in this section only a very brief characterization of each of the introduced types of operations, but we provide the reader with ample references for further study. 2.3.1 Modifiers Modifiers are unary operations whose primary purpose is to modify fuzzy sets to account for linguistic hedges, such as very, fairly, extremely, moderately, etc., in representing expressions of natural language. Each modifier, m, is an increasing (and usually continuous) one-to-one function of the form m: [0, 1] → [0, 1], which assigns to each membership grade A(x) of a given fuzzy set A a modified grade m(A(x)). The modified grades for all x ∈ X define a new, modified fuzzy set. Denoting conveniently this modified set by MA, we have m(A(x)) = MA(x) Observe that function m is totally independent of elements x to which values A(x) are assigned; it depends only on the values themselves. In describing its formal properties, we may thus ignore x and assume that the argument of m is an arbitrary number a in the unit interval [0, 1].
- 44. 20 2 FuzzyLogic: A Specialized Tutorial In general, a modifier increases or decreases values of the membership functions to which it is applied, but preserves the order. That is, if a ≤ b then m(a) ≤ m(b) for all a, b ∈ [0, 1]or, recognizingthemeaningofa andb, ifA(x) ≤ A(y)forsomex, y ∈ X, then MA(x) ≤ MA(y). Sometimes, it is also required that m(0) = 0 and m(1) = 1. Modifiers are basically of three types, depending on which values of the membership functions they increase or decrease: (i) modifiers that increase all values; (ii) modifiers that decrease all values; (iii) modifiers that increase some values and decrease other values. To illustrate these types of modifiers, let us consider the fuzzy set A in Figure 2.2. For each x ∈ R, A is clearly defined by the formula A(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (x − 1)/2 when x ∈ [1,3] (5 − x)/2 when x ∈ [3,5] 0 otherwise Assume that this fuzzy set represents, in a given application context, the linguistic concept “close to 3.” To modify A for representing the concept “very close to 3,” we need to reduce in some way the values of A. This can be done by choosing an appropriate modifier from the class of functions mλ(a) = aλ , (2.8) where a is the value of A to which mλ is applied and λ is a parameter whose value determines how strongly mλ modifies A. For each value of λ, which must be in this case greater than 1, we obtain a particular modifier. When applying the modifier to A, we obtain a new membership function, mλ[A(x)], a composite of functions A and m, which for each x ∈ R is defined by the formula mλ[A(x)] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [(x − 1)/2]λ when x ∈ [1,3] [(5 − x)/2]λ when x ∈ [3,5] 0 otherwise This modified membership function has a shape exemplified by the function labeled as C in Figure 2.2. Its width is determined by the value λ of the chosen modifier: the larger the value, the narrower the function. The proper value of λ must be determined in the context of each particular application. Assume now that we want to modify the same set A for representing the concept “fairly close to 3.” In this case, we need to increase the values of A. This can be done with modifiers of the form (2.8), provided that λ ∈ (0, 1). Applying these modifiers to
- 45. 2.3 Operations onFuzzy Sets 21 A results in a new membership function whose shape is exemplified by the function labeled as F in Figure 2.2. The smaller the value of λ, the wider is the modified membership function. It should be mentioned at this point that (2.8) is given here solely as an example of a possible class of modifiers of fuzzy sets. As is well known, these modifiers do not always properly capture the meaning of linguistic hedges in natural language. Amore comprehensive treatment of linguistic hedges is presented in Chapter 10; see also Novák [1989]. 2.3.2 Complements Similarly to modifiers, complements of fuzzy sets may be defined via appropri- ate unary operations on [0, 1]. While modifiers preserve the order of membership degrees, complements reverse the order. In particular, each fuzzy complement, c, must satisfy at least the following two requirements: (c1) c(0) = 1 and c(1) = 0; (c2) for all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b). Requirement (c1) guarantees that all fuzzy complements collapse to the unique clas- sical complement for crisp sets. Requirement (c2) guarantees that increases in the degree of membership in A do not result in increases in the degree of membership in the complement of A. This is essential since any increase in the degree of mem- bership of an object in a fuzzy set cannot simultaneously increase the degree of nonmembership of the same object in the same fuzzy set. When used as a fuzzy complement, function c is always applied to membership degrees A(x) of some fuzzy set A. It depends only on the values A(x) and not on the objects x to which the values are assigned. For the purpose of characterizing fuzzy complements, we may thus ignore these objects and observe only how function c depends on numbers in [0, 1]. This is the reason why no reference is made to specific degrees A(x) in the requirements (c1) and (c2). However, when function c defines a complement of a particular fuzzy set A, we must keep track of the relevant objects x to make the connection between A(x) and c[A(x)]. Although requirements (c1) and (c2) are sufficient to characterize the largest class of acceptable fuzzy complements, two additional requirements are imposed on fuzzy complements by most applications of fuzzy set theory: (c3) c is a continuous function; (c4) c(c(a)) = a for all a ∈ [0, 1]. Requirement (c3) guarantees that infinitesimal changes in the argument do not result in discontinuous changes in the function. Requirement (c4) guarantees that fuzzy sets
- 46. 22 2 FuzzyLogic: A Specialized Tutorial are not changed by double complementation. Fuzzy complements that satisfy (c4) are called involutive. A practical class of fuzzy complements that satisfy requirements (c1)–(c4) is defined for each a ∈ [0, 1] by the formula cλ(a) = (1 − aλ )1/λ , (2.9) where λ ∈ (0, ∞); it is called the Yager class of fuzzy complements. One particular fuzzy complement is obtained for each value of the parameter λ. The complement obtained for λ = 1, which is called a standard fuzzy complement, is the most com- mon complement in applications of fuzzy set theory. Clearly, the standard fuzzy complement of a fuzzy set A, usually denoted by A, is defined for each x ∈ X by the equation A(x) = 1 − A(x). Other parameter-based formulas for describing classes of fuzzy complements have been proposed in the literature. In fact, some procedures have been developed by which new classes of fuzzy complements can be generated [Klir Yuan, 1995]. However, this theoretical topic is beyond the scope of this tutorial. To determine the most fitting complement in the context of each particular appli- cation is a problem of knowledge acquisition, somewhat similar to the problem of constructing membership functions. Given a class of fuzzy complements, such as the Yager class, the constructing problem reduces to the problem of determining the right value of the relevant parameter. 2.3.3 Intersections and unions Intersections and unions of fuzzy sets, denoted by i and u respectively, are general- izations of the classical operations of intersections and unions of crisp sets. They may be defined via appropriate functions that map each pair of real numbers from [0, 1] (representing degrees A(x) and B(x) of given fuzzy sets A and B for some x ∈ X) into a single number in [0, 1] (representing membership degree (A ∩ B)(x) of the intersection of A and B or membership degree of the union of A and B for the given x). Hence, (A ∩ B)(x) = i[A(x), B(x)] and (A ∪ B)(x) = u[A(x), B(x)] for all x ∈ X. To discuss properties of functions i and u, which do not depend on x, we may view i and u as functions from [0, 1] × [0, 1] to [0, 1].
- 47. 2.3 Operations onFuzzy Sets 23 Contrary to their classical counterparts, fuzzy intersections and unions are not unique. This is a natural consequence of the well-established fact that the linguistic expressions “x is a member of A and B” and “x is a member of A or B” have different meanings when applied by human beings to different vague concepts in different contexts. To be able to capture the different meanings, we need to characterize the classes of fuzzy intersections and fuzzy unions as broadly as possible. It has been established that operations known in the literature as triangular norms or t-norms and triangular conorms or t-conorms, which have been exten- sively studied in mathematics, possess exactly those properties that are requisite, on intuitive grounds, for fuzzy intersections and fuzzy unions, respectively. The class of t-norms/fuzzy intersections is characterized by four requirements; the class of t-conorms/fuzzy unions is also characterized by four requirements, three of which are identical with the requirements for t-norms. In the following list, the requirements for t-norms/fuzzy intersections i are paired with their counterparts for t-conorms/fuzzy unions u, and must be satisfied for all a, b, d ∈ [0, 1]: (i1) i(a, 1) = a (boundary requirement for i); (u1) u(a, 0) = a (boundary requirement for u); (i2) b ≤ d implies i(a, b) ≤ i(a, d) (monotonicity); (u2) b ≤ d implies u(a, b) ≤ u(a, d) (i3) i(a, b) = i(b, a) (commutativity); (u3) u(a, b) = u(b, a) (i4) i(a, i(b, d)) = i(i(a, b), d) (associativity). (u4) u(a, u(b, d)) = u(u(a, b), d) It is easy to see that the first three requirements for i ensure that fuzzy intersections collapse to the classical set intersection when applied to crisp sets: i(0, 1) = 0 and i(1, 1) = 1 follow directly from the boundary requirement; i(1, 0) = 0 and i(0, 0) = 0 follow then from commutativity and monotonicity, respectively. Similarly, the first three requirements for u ensure that fuzzy unions collapse to the classical set union when applied to crisp sets. Commutativity requirements ensure that fuzzy intersections and unions are symmetric operations, indifferent to the order in which sets to be combined are considered; together with monotonicity requirements, they guarantee that fuzzy intersections and unions do not decrease when any of their arguments are increased, and do not increase when any arguments are decreased. Associativity requirements allow us to extend fuzzy intersections and unions to more than two sets, in perfect analogy with their classical counterparts. The following are examples of some common fuzzy intersections and fuzzy unions with their usual names (each defined for all a, b ∈ [0, 1]). Standard fuzzy intersection: i(a, b) = min(a, b) Algebraic product: i(a, b) = ab
- 48. 24 2 FuzzyLogic: A Specialized Tutorial Bounded difference: i(a, b) = max(0, a + b − 1) Drastic intersection: imin(a, b) = ⎧ ⎨ ⎩ a when b = 1 b when a = 1 0 otherwise Standard fuzzy union: u(a, b) = max(a, b) Algebraic sum: u(a, b) = a + b − ab Bounded sum: u(a, b) = min(1, a + b) Drastic union: umax(a, b) = ⎧ ⎨ ⎩ a when b = 0 b when a = 0 1 otherwise It is easy to verify that the inequalities imin(a, b) ≤ i(a, b) ≤ min(a, b) max(a, b) ≤ u(a, b) ≤ umax(a, b) are satisfied for all a, b ∈ [0, 1] by any fuzzy intersection i and any fuzzy union u, respectively. These inequalities specify, in effect, the full ranges of fuzzy intersections and fuzzy unions. Examples of classes of fuzzy intersections, iw, and fuzzy union, uw, that cover the full ranges of these operations are defined for all a, b ∈ [0, 1] by the formulas iw(a, b) = 1 − min{1, [(1 − a)w + (1 − b)w]1/w} uw(a, b) = min[1, (aw + bw)1/w] , (2.10) where w is a parameter whose range is (0, ∞). One particular fuzzy intersection and one particular fuzzy union are obtained for each value of the parameter. These operations are often referred to in the literature as the Yager classes of intersections and unions. Although it is not obvious from the formulas, it is relatively easy to prove that the standard fuzzy operations are obtained in the limit for w → ∞. Since Yager intersections increase as the value of w increases, they become less restrictive or weaker with increasing w. The drastic intersection is the strongest and the standard intersection is the weakest. For Yager unions, this pattern is inverted; they become more restrictive or stronger with increasing w. The standard union is the strongest, the drastic union the weakest. It should be mentioned that various other classes of fuzzy intersections and unions have been examined in the literature. Moreover, special procedures are now available by which new classes of fuzzy intersection and unions can be generated [Klir Yuan, 1995]. Among the great variety of fuzzy intersections and unions, the standard operations possess certain properties that give them special significance. First, we recognize that they are located at opposite ends of the respected ranges of these operations.
- 49. 2.3 Operations onFuzzy Sets 25 While the standard intersection is the weakest one among all fuzzy intersections, the standard union is the strongest one among all fuzzy unions. Second, the standard operationsaretheonlycutworthyoperationsamongallfuzzyintersectionsandunions. Third, they are also the only operations among fuzzy intersections and unions that are idempotent. This means that is(a, a) = us(a, a) = a for all a ∈ [0, 1]. Non- standard fuzzy intersections are only subidempotent, while nonstandard fuzzy unions are superidempotent; this means that i(a, a) a and u(a, a) a for all a ∈ (0, 1). In addition, when using the standard fuzzy operations, errors of the operands do not compound. This is a desirable property from the computational point of view, which other fuzzy operations do not possess. Whatever combination of fuzzy counterparts of the three classical set-theoretic operations (complement, intersection, union) we choose, some properties of the clas- sical operations (properties of the underlying Boolean algebra) are inevitably violated. This is a consequence of imprecise boundaries of fuzzy sets. The standard fuzzy oper- ations violate only the law of excluded middle and the law of contradiction. Some other combinations preserve these laws, but violate distributivity and idempotence [Klir Yuan, 1995]. 2.3.4 Averaging operations Fuzzy intersections (t-norms) and fuzzy unions (t-conorms) are special types of oper- ations for aggregating fuzzy sets: given two or more fuzzy sets, they produce a single fuzzy set, an aggregate of the given sets. While they do not cover all aggregating oper- ations, they cover all aggregating operations that are associative. Because of the lack of associativity, the remaining aggregating operations must be defined as functions of n arguments for each n ≥ 2. These remaining aggregation operations are called averaging operations. As the name suggests, they average in various ways member- ship functions of two or more fuzzy sets defined on the same universal set. They do not have any counterparts in classical set theory. Indeed, an average of several characteristic functions of classical sets is not, in general, a characteristic function! However, classical sets can be averaged if they are viewed as special fuzzy sets. For each n ≥ 2, an averaging operation, h, aggregates n fuzzy sets defined on the same universal set X, say sets A1, A2, . . . , An. Denoting conveniently the aggregate fuzzy set by H(A1, A2, . . . , An), we have H(A1, A2, . . . , An)(x) = h[A1(x), A2(x), . . . , An(x)] for all x ∈ X. Since properties of various averaging operations h do not depend on x, but only on the membership degrees A1(x), A2(x), . . . , An(x) ∈ [0, 1], we may view these operations as functions from [0, 1]n to [0, 1].
- 50. 26 2 FuzzyLogic: A Specialized Tutorial The following two requirements are requisite for any averaging operation h with n arguments (n ≥ 2): (h1) for all a ∈ [0, 1], h(a, a, . . . , a) = a (idempotency); (h2) for any pair of n-tuples of real numbers in [0,1], a1, a2, . . . , an and b1, b2, . . . , bn, if ai ≤ bi for all i ∈ Nn, then h(a1, a2, . . . , an) ≤ h(b1, b2, . . . , bn) (monotonicity). Requirement (h1) expresses our intuition that an average of equal numbers must result in the same number. Requirement (h2) guarantees that the average does not decrease when any of the arguments increase. In addition to these essential and easily understood requirements, averaging operations on fuzzy sets are usually expected to satisfy two additional requirements: (h3) h is a continuous function; (h4) h is a symmetric function in all its arguments, which means that h(a1, a2, . . . , an) = h(ap(1), ap(2), . . . , ap(n)) for any permutation p on Nn. Requirement (h3) guarantees that small changes in any of the arguments do not result in discontinuous changes in the average. Requirement (h4) captures the usual assumption that the aggregated fuzzy sets are equally impor- tant. If this assumption is not warranted in some application contexts, the symmetry requirement must be dropped. It is significant that any averaging operation h that satisfies the two basic require- ments (h1) and (h2) produces numbers that for each n-tuple a1, a2, . . . , an ∈ [0, 1]n lie within the interval defined by the inequalities min(a1, a2, . . . , an) ≤ h(a1, a2, . . . , an) ≤ max(a1, a2, . . . , an). To see this, let a∗ = min(a1, a2, . . . , an) and a∗ = max(a1, a2, . . . , an). If h satisfies requirements (h1) and (h2), then a∗ = h(a∗, a∗, . . . , a∗) ≤ h(a1, a2, . . . , an) ≤ h(a∗ 1, a∗ 2, . . . , a∗ n) = a∗. Conversely, if h produces numbers within the interval bounded by the min and max operations, then it must also satisfy requirement (h1) of idempotency; indeed, a = min(a, a, . . . , a) ≤ h(a, a, . . . , a) ≤ max(a, a, . . . , a) = a for all a ∈ [0, 1]. That is, averaging operations cover the whole range between the standard fuzzy intersection and the standard fuzzy union. The standard operations
- 51. Another Random ScribdDocument with Unrelated Content
- 52. Maraschino—till at lastthe clock striking two, reminded her it was time to go to bed. 'Ah,' said I, 'that is extremely just and proper. But, alas! I am like my melancholy little friend who was very gentil, but whose hair came a leetle through the top of his hat,—I have no bed to go to.' 'It's very provoking,' said the landlady, 'so tired as you are, too.' 'It is, indeed,' replied I—seeing a proposition of some sort or other on the tip of her tongue. 'Now,' said she, looking remarkably serious, 'can I trust you—will you promise me, if I give you a bed, to do as I bid you, Mr. Daly?' 'Your commands,' said I, 'shall be obeyed to the letter—only let me rest myself quietly and comfortably—it is all I ask—for never was poor devil so tired in his life as I.' 'Take a drop more punch, Mr. Daly,' said my landlady, 'it will make you sleep the sounder.' 'No fear of that,' said I; 'but what do you propose?' 'Why,' said mine hostess, 'I have one bed unoccupied.' 'Why didn't you say so before?' cried I. 'I'll tell you why,' said my fair friend; 'it's a double-bedded room, and the other bed is occupied by a——' '——snoring farmer, from Farnham,' said I; 'or perhaps a tight- skinned sailor, walking his way to town from Portsmouth.' 'No,' said she, looking very pathetic—and very pretty by the way —'by a lady.' 'A lady,' said I, 'oh, charming thought!——' 'There it is,' interrupted the lady, 'that's just what I expected, you are all fire and tow—alight in a moment—now I shall not say another word, and you must sleep, if you will sleep here, in the arm-chair by the fire.'
- 53. 'No,' said I,'no—don't be angry—I didn't know—I thought——' 'Yes, Mr. Daly, that's what you are always thinking, I believe,' said mine hostess, 'but that won't do—the lady who occupies the other bed in the double-bedded room is a sad invalid; she has been stopping here for some time, and the only rest she gets is by dint of laudanum, which the doctor gives her in large doses, and she sleeps soundly during the night, which makes up for the sufferings she endures by day. If you choose to behave well—and, tired as you are, I don't like to turn you out or leave you here—you shall have the other bed. You must go gently into the room, and when you are in bed I will come and take away your candle; and as I sleep in the next room, if you don't remain perfectly quiet I shall insist upon your getting up and coming down again here into the bar.' 'Agreed,' said I, 'I only ask for a bed—all I want is rest—I am scarcely able to walk or stand, therefore I agree to your condition; let me finish my punch, and marshal me the way I should go.' After looking at me suspiciously and hesitatingly for a minute or two, my dear landlady agreed to trust me; and accordingly having seen that my bed was properly prepared she returned, and, lighting a candle, preceded me upstairs, and opening the door of the room put her finger to her lips to enforce silence, whispering me, that when I was in bed I should knock against the wainscot which separated her room from that in which I was to repose, and that she would come and fetch my candle. I promised to obey all her injunctions. The curtains of the other bed were closely drawn—I never felt so awkward in my life—but I had promised; yet one peep before the light vanished—no—perhaps the lady would wake and scream, and I should be forthwith ejected. I resolved to keep my faith, at all events till mine hostess was herself asleep, and then see—as far as utter darkness would permit—how the affair would terminate. Accordingly, I hurried off my clothes—washed my face, hands, and mouth as gently and quietly as possible, and having concluded
- 54. my brief preparationsfor depositing myself on my much longed-for couch, gave the concerted signal, and scarcely was well in my place before my dear landlady entered the room on tip-toe, and, coming up close to the bedside and having whispered 'Now, remember your promise,' took the glimmering light away, and left me in the dark with my fair and slumbering companion. There was something very strange in my position; I was tired to death, but somehow I could not sleep. I lay and listened to hear whether my fair incognita would sneeze—or cough—or cry 'hem'—or play off any little coquettish trick which, under the circumstances, I thought probable enough. I durst not move, for I knew I was watched; however, I sat up in the bed and began to wonder. Is it, thought I, an old woman or a young woman?—an invalid is interesting, and, bless her, she must be uncommonly genteel, for she does not snore in the least—a few minutes served to convince me that my landlady did—and I rather rejoiced in the sound of her slumbers, since I thought I might perhaps succeed in attracting the attention of my sleeping partner; and the fact that a gentleman of my very respectable pretensions was so whimsically associated with her—knowing mine hostess's archness—induced me to attribute her readiness to quarter me upon the slumbering beauty, to a foreknowledge on her part that my introduction would not be considered altogether an intrusion. After I had satisfied myself that my landlady was really safe, I had recourse to some slight coughs, which do occasionally infest one; but no, my signals were not answered: the dose of laudanum had been particularly strong that night. At last I thought I heard a slight movement. I began to listen till I heard the beating of my own heart, and felt a sort of drumming palpitation in my ears. I held my breath: pshaw, thought I, this woman has been cheating me, the other bed is tenantless,—a trick to try me,—and for what a stupid dolt she will set me down if I don't convince her that I had at least curiosity enough in my composition to ascertain what was in it.
- 55. My feelings firedat the thought. Up I got,—groped my way across the room,—the white dimity drapery being just visible amidst the 'palpable obscure.' I reached the bed,—I paused,—I heard nothing;—I partly opened the curtains at the side, and said in a soft, very soft voice, 'Hem!' No answer. 'Ma'am,—ma'am,' still silent;—'are you there?' said I;—and, placing my hand on the pillow, found she was. Dear, unconscious creature, there she lay, comfortably cuddled up in the clothes, and sleeping, or seeming to sleep, soundly. I was, I admit, on the point of proceeding to awaken her, in order to announce my presence, when, in stepping towards the head of the bed, my foot came in contact with a chair which stood on its right- hand side, which was overthrown with a crash that, in an instant, roused—not my dear opium-drinker—but my lynx-like landlady. I heard her jump out of bed. I jumped into mine, but, in less than two minutes, there she was, like Margaret's 'grimly ghost,' standing before me, loading me with reproaches, and ordering me, in the most peremptory terms, to take the candle, descend the stairs, and dress myself in the parlour behind the bar, and wait until she came down to eject me from the house; seeing that she could have no kind of confidence in a gentleman who had so much confidence in himself. Vain were my pantomimic supplications: she would listen to nothing but immediate abdication; and I could not well be angry with her, for she had put faith in me, and perhaps run the risk of losing a valuable customer by indulging me with the luxuries of ease and rest which, under no other circumstances, she could have afforded me. I implicitly obeyed her commands; and, as soon as she had retired to dress herself, collected my wearing apparel, and slunk downstairs to prepare for my departure, which seemed inevitable. As I passed along the passages, I heard multifarious snorings in all directions, which convinced me of the truth of my landlady's assertions as to the influx of company, and made me repent more sorely than before, that I could not for once in my life act with discretion and decorum.
- 56. I had scarcelyfinished dressing myself when my landlady made her appearance in the parlour. 'I really am surprised, sir,' said she, 'at your conduct. I thought, as a gentleman, you might have been trusted, considering the circumstances under which I ventured to put you into that room.' 'Really,' said I, 'I thought you were playing me a trick, and I could not bear your having the laugh against me, and so I certainly did venture just to ascertain——' 'Ascertain!' cried the landlady; 'that's just the very thing you ought upon no consideration whatever to have done. Did I not tell you the lady was an invalid? Oh! Mr. Daly, Mr. Daly! I believe you are the d——' '——evil be, ma'am,' said I, interrupting her, 'to him who evil thinks. I meant no harm, and——' 'You might have ruined me, sir,' said mine hostess. 'Might I?' said I; 'when?' 'This very night, sir,' said she; 'this very hour. Why, what would have been thought of me and my house, if it had been known that I had allowed you to sleep in that room? Nobody would have believed that I did it out of pure regard for your comfort, tired and knocked up as you were, and because I had not a hole or corner besides into which you could have poked yourself: however, it will be a lesson for me another time; and now, Mr. Daly, if you will take my advice,—the lads are getting up in the yard,—you will let me order out a chaise and pair, and go on to Guildford, where, I have no doubt, they have plenty of beds, and where you may get some comfortable rest; and as the brother of the lady in No. 3 is sleeping here to-night, something unpleasant to all parties might happen in the morning, and you would do me a very great favour if you would go.' I felt considerably inclined, for many reasons, to accede to what appeared the very reasonable desire of mine hostess: first of all, I might do her a mischief by staying; in the second place, the lady
- 57. might complain toher brother; in the third place, the White Hart at Guildford was a remarkably good inn; and a well-made bed, and a well-warmed bed-room, would be extremely comfortable by comparison with the chilly atmosphere and the chair-slumber of the parlour behind the bar at Ripley. To Guildford I must eventually proceed,—and why not now? So, with the best possible grace, I told mine hostess that I was at her command, and begged of her to dispose of me as she thought fit. I paid her liberally for the horses, the repast, and the portion of my night's rest which I ought to have had; and when I stepped into the 'yellow and two,' I shook hands with her, and she gave me a look as much as to say, again and again, 'Daly, Daly! you are not to be trusted.' Well, sir, away I went, glasses rattling, and wind whistling (a short stage, you know); and, before four, we reached the White Hart. I had forestalled my Guildford sleep in the chaise; however, we soon made them hear at the inn, and in less than three quarters of an hour I was again rolled up in the sheets, having before I went to bed written a note to my servant at Wrigglesworth, which I desired might be sent off early in the morning, directing him, after leaving word with Sir Marmaduke's man that I was alive, if not merry, to come to me with my clothes and other requisites for dressing by ten o'clock; and certainly, I must say, I never did enjoy my rest and quietness so entirely and completely as upon that particular occasion. Instead of ten o'clock—having desired that I might not be disturbed—I did not awake until past noon, and then regretted that my balmy comfort had been broken in upon. From my servant, when I saw him, I learned that my friends at Wrigglesworth had really expressed great anxiety on my account, which did not displease me,—I rather like to create an effect,—but I did not hear that my dear Lady Wrigglesworth had either absented herself from dinner or disappeared for the evening in consequence of my absence, which I confess mortified my vanity a little. I dressed, and having ensconced myself in the drawing-room of the
- 58. White Hart, thewalls of which apartment were most constitutionally decorated with loyal and orthodox prints, and which immediately faces the Gothic House, I delighted myself by watching the movements of two uncommonly pretty girls in the said antiquated edifice, who appeared to be in full possession, in the absence, as I surmised, of some greater, and probably graver, personages. After breakfast I strolled out. I like Guildford: it is a nice, clean, handsome, healthy town; the hill in the street I admit to be a nuisance; the alternation between climbing up and sliding down is tiresome, and even dangerous. These little objections did not affect me—nothing affects me when I am on the hunt for subjects—so away I went—smack bang into a Quaker's shop to buy myself a pair of gloves—and there—there I saw what I had never before seen— two Quaker children playing about the place, thee'ing and thou'ing each other with perfect French familiarity. Now, do you know, continued Daly, it is quite worthy of remark,—that nobody—always, I presume, excepting Quakers themselves—has ever seen a Quaker baby in arms, a Quaker lady enceinte, or a Quaker gentleman with a wooden leg—eh? I like these statistical speculations. So, having bought my gloves, I returned to 'mine inn,' about one, intending forthwith to proceed to Wrigglesworth. Just as I reached the door of the White Hart, and just as my man was bringing out my horses, my eye was attracted by a funeral procession, consisting merely of a hearse, one mourning coach, and a private carriage, which had halted before the door; two persons who had occupied the coach having entered the house while fresh horses were put to the three vehicles. A natural and not very blameable curiosity prompted me to ask a jolly, merry-looking undertaker, whose funeral it was, whither they were going, and whence they had come? 'Why, sir,' said the man, 'what you see here isn't the regular job as I hopes to turn it out at Chichester next Tuesday, which is the day fixed for the interment of the corpse. Short notice, you see, sir; could not do everything in a minute, sir.'
- 59. 'What is thename of the——?' I hesitatingly asked. 'Miss Barmingfield, sir,' said the man, 'is the name of the corpse. Poor young lady, it was as well as you and me three days ago, and was a coming down to Chichester to spend a month with its mother; when, just in a minute, it was taken ill at Ripley, and out it went for all the world like the snuff of a candle.' 'At Ripley!' said I; 'did she live at Ripley?' 'No, sir, she didn't,' said the undertaker; 'you'll excuse me—she died there.' 'But she must have lived there first, I presume,' said I, rather angrily; for a joker hates to be joked upon. 'A very short time indeed,' said the jolly undertaker. 'She arrived at the Talbot the day before yesterday, about twelve o'clock, in high health, and by six at night, as I said afore, she was a corpse.' 'At the Talbot!' said I. 'And are you bringing the body from the Talbot now?' 'Yes, sir,' said the man; 'on our way to Chichester. We could not move her, poor dear young lady, afore, because I couldn't get things ready till this morning.' 'Pray,' said I, with a degree of agitation which evidently astonished my companion in the crape, 'where—in what part of the Talbot at Ripley did the young lady die?' 'In Number 3; that 'ere double-bedded room right over the gateway,' said the man. 'We only packed her up this morning.' My dear Gurney, you may easily imagine what my feelings were. Only conceive the idea of having been turned into a double-bedded room in the dark with a dead woman! It was lucky that the horses were pronounced ready, and that Major Barmingfield, whose residence at Ripley mine hostess had so truly announced, made his appearance just at the moment that the undertaker had enlightened me on the subject. I felt a mingled sensation of horror at the event,
- 60. of joy atmy escape from the place where it occurred, and of repentance for my misconduct towards my landlady, who had so good-naturedly strained a point for my accommodation, which nearly overset me; and I have not a notion what I should have done, had it not been that the coldness of the weather afforded me an excuse for drinking off a glass of brandy, and the lateness of the hour forced me to mount my nag and begin my canter to Wrigglesworth forthwith. A VISIT TO THE OLD BAILEY. As I entered the Court, a case of some importance had terminated, and the judge just concluded his summing up, when the clerk of the arraigns put the customary question to the jury, How say ye, gentlemen—is the prisoner at the bar guilty or not guilty? Upon which the jurymen laid their heads together, and I heard something in a whisper from their foreman, who immediately pronounced the agreeable verdict, Not guilty. The prisoner bowed gracefully—he was a pickpocket—and retired. The prompt decision of the jury convinced me that it must have been a clear case; and I rejoiced at the departure of the now exonerated sufferer. That's a reg'lar rascal, said the sheriff to me in a whisper; never was such a case heard on, to be sure—seventeen watches, thirty- two pocket handkerchiefs, four pair of spectacles, and five snuff- boxes, all found upon his person! Yet, said I, the evidence could not have been very strong against him—the jury acquitted him after a minute's consultation. Evidence, Mr. Gurney! said the sheriff, how little do you know of the Old Bailey!—why, if these London juries were to wait to consider evidence, we never should get through the business—the way we do here is to make a zig-zag of it.
- 61. I did notexactly comprehend the term as it was now applied, although Daly had often used it in my society with reference to a pin and a card universally employed at the interesting game of rouge et noir; and I therefore made no scruple of expressing my ignorance. Don't you understand, sir? said the sheriff—why, the next prisoner will be found guilty—the last was acquitted—the one after the next will be acquitted too—it comes alternate like—save half, convict half—that's what we call a zig-zag; and taking the haggregate, it comes to the same pint, and I think justice is done as fair here as in any court in Christendom. This explanation rendered the next prisoner who made his appearance an object of considerable interest to me. He was a little dirty boy, who stood charged with having stolen a pound of bacon and a peg-top from a boy somewhat his junior. The young prosecutor produced a witness, who, as far as appearances went, might, without any great injustice, have taken the place of the prisoner, and who gave his evidence with considerable fluency and flippancy. His manner attracted the notice of one of the leading barristers of the court, Mr. Flappertrap, who, in cross-examining him, inquired whether he knew the nature of an oath. Yes, I does, said the boy. Explain it, said Flappertrap. You may be d——d, replied the lad; that's a hoath, arn't it? What does he say? said the judge—who, as I about this period discovered, was as deaf as a post. He says, 'You may be d——d,' my lord, said Flappertrap, who appeared particularly glad of an opportunity to borrow a phrase, which he might use for the occasion. What does he mean by that? said the judge. That is the way, my lord, he exhibits his knowledge of the nature of an oath. Pah! pah! said the judge—Boy, d'ye hear me?
- 62. Yes, said theboy, I hears. Have you ever been to school? Yes, said the boy, in St. Giles's parish for three years. Do you know your catechism? The boy muttered something which was not audible to the court generally, and was utterly lost upon the judge personally. What does he say? said his lordship. Speak up, sir, said Mr. Flappertrap. The boy muttered again, looking down and seeming embarrassed. Speak louder, sir, said another barrister, whose name I did not know, but who was remarkable for a most unequivocal obliquity of vision—speak to his lordship—look at him—look as I do, sir. I can't, said the boy, you squints! A laugh followed this bit of naïvete, which greatly abashed the counsellor, and somewhat puzzled the judge. What does he say? said his lordship. He says he knows his catechism, my lord. Oh—does not know his catechism—why then, what— Does know, my lord, whispered the lord mayor, who was in the chair. Oh—ah—does know—I know—here, boy, said his lordship, you know your catechism, do you? Yes, replied he, sullenly. We'll see, then—what is your name? said his lordship. My name, said the intelligent lad—what, in the catechism? Yes, what is your name? M. or N. as the case may be, said the boy.
- 63. Go down, godown, said the judge, angrily, and down he went. Gentlemen of the jury, said his lordship, this case will require very little of your attention—the only evidence against the prisoner at the bar which goes to fasten the crime upon him, is that which has been offered by the last witness, who evidently is ignorant of the nature and obligation of an oath. With respect to the pig's toes which the prisoner stands charged with stealing—— A peg-top, my lord! said Flappertrap, standing up, turning round, and speaking over the bench into the judge's ears. Peg-top, said his lordship—oh—ah—I see—very bad pen—it looks in my notes like pig's toes. Well—peg-top—of the peg-top which it is alleged he took from the prosecutor, there has not been one syllable mentioned by the prosecutor himself; nor do I see that the charge of taking the bacon is by any means proved. There is no point for me to direct your attention to, and you will say whether the prisoner at the bar is guilty or not; and a very trumpery case it is altogether, that I must admit. His lordship ceased, and the jury again laid their heads together; again the foreman gave the little hem of conscious readiness for decision; again did the clerk of the arraigns ask the important question, How say ye, gentlemen, is the prisoner at the bar guilty or not guilty? Guilty, said the foreman to the clerk of the arraigns; and I told you so, said the sheriff to me. The next case was a short one. The prisoner a woman, the evidence clear and straightforward; but no great interest was excited, because it was known that the case, for the trial of which in point of fact the learned judge had, for particular reasons, given his attendance, and which accounted for his lordship's presence at the close of the session, was very speedily to come on. This extraordinary combination of circumstances afforded me the most favourable opportunity of seeing all the sights of this half awful, half amusing scene, even to the discharge of the grand jury, who had been specially kept together for the purpose of finding or ignoring
- 64. the bill preferredagainst the eminent culprit, who was evidently the great attraction of the day—having found which, they had but three more to decide upon. It was in the middle of the defence of the female prisoner, now coram nobis, and just as she was making a beautiful but useless appeal to the gentlemen of the jury, that a bustle in the court announced some coming event. I am, said the weeping prisoner, an orphan—I lost my mother while I was yet a child—my father married again, and I was driven from what had been before a happy home—I have only to pray—— Bang went a door—the scuffle of feet were heard—down went some benches—Make way—make way! cried some of the officers. Stand back, sir, stand back—the gentlemen of the grand jury are coming into court. To what the moaning prisoner at the bar might have limited her supplications, I never had an opportunity of ascertaining, for the noise I have mentioned was succeeded by the appearance of eighteen or nineteen men, dressed up in something like the shabbiest dominoes I had seen at Lady Wolverhampton's masquerade, trimmed with very dirty fur—the leader, or foreman, carrying in his hand three bits of parchment. As these gentlemen advanced to a space reserved for them in the centre of the court, the judge kept exchanging bows with them until they had all reached their destination—the foreman then delivered to the clerk of arraigns the three bits of parchment, who, putting his glasses on his nose, read—James Hickson, larceny—not found.—John Hogg, felony —true bill.—Mary Ann Hodges, felony—not found. The clerk then informed his lordship, partly by words, and partly by signs, the result of the deliberations of the grand jury, and the fact that there were no more bills to set before them. Having thus far proceeded, that officer inquired if the gentlemen of the grand jury had any presentment to make; whereupon the foreman, one of the largest and dirtiest-looking persons imaginable, but whose countenance was indicative of love of power and command, and who appeared, at the moment he prepared himself to unburthen his great soul of a
- 65. grievance, to feelas if the whole world were a football, made for him to play with,— My lord, said he, drawing himself up into an attitude, I am sure I need not, at this time of day, enter into any discussion with your lordship on the vast importance of the rights and privileges of Englishmen—of the original establishment of the trial by jury in this country. It would be worse than idle to occupy your valuable time and that of this court, by dilating upon the merits of our constitution —the chiefest of which has, I may say—been always—and I will say —wisely, considerately, and prudently held to be that peculiar mode of administering justice between man and man. But, my lord, if in civil cases the deliberation and decision of a jury are considered adequate safeguards to the rights and property of the people, the law, still more careful of their lives and liberties, has interposed in criminal cases another and a higher tribunal, in the nature of a grand jury. [Hereabouts the judge, having bowed his head graciously, omitted to raise it again, having dropped into a sound slumber.] That tribunal of mediation in the first instance, is full of importance; and whatever subsequent proceedings may be taken in a case, I do say, for myself and my fellows, that the decision upon ex-parte evidence requires more circumspection, more care, and more consideration than a verdict delivered after a case had been argued, and after witnesses have been heard on both sides. If, my lord, your lordship concedes this point, I will merely say, generally, that when the mind is occupied by any important object, more especially in matters of jurisprudence, it is absolutely necessary that nothing, if possible, should occur to irritate or exacerbate the feelings—all should be calm, and at rest. Several people turned their eyes towards his lordship, and some smiled. No incidental annoyance should be permitted to interpose itself; nothing which could divert the judge from the point to which his
- 66. intellectual faculties oughtto be directed, and to which, my lord, under suitable circumstances, they would as they should naturally converge. But, my lord, we are finite beings—creatures of habit— subject to all the weaknesses of our nature, and liable to be acted upon by impulses almost unaccountable to ourselves. For myself and my fellows, I may, perhaps, hope for a favourable interpretation of our intentions, and a lenient judgment of our conduct. We have, my lord, struggled hard to do our duty, and I hope we have done it serviceably and effectually—conscientiously and faithfully, I am sure we have. But, my lord, we do think it necessary to call your lordship's most serious attention to a fact which is embodied in the presentment I hold in my hand. It is one which occurs to us to be of paramount importance, as far as the tempering of justice with mercy is involved: we have suffered grievously from the existence of the evil to which we point; and although at this time of the year its effects are of course not so heavily felt as in the winter season, we have considered it a duty we owe to this court, to our fellow- countrymen, and, we may say, to every man intimately or remotely connected with the administration of criminal justice, spread as they may be over the whole surface of the globe, to state that the chimney in the grand jury-room smokes so much and so continually, that it is impossible to endure its effects calmly or patiently; and we therefore think it right to bring the matter thus formally before your lordship, and to desire that measures may be taken to abate a nuisance which, by its effects, is calculated to thwart, impede, and even distort the course of justice, and produce evils, the magnitude of which it is scarcely possible to imagine, and certainly not to express. A buzz of approbation from the gentlemen of the grand jury, who had been undergoing the process of smoke-drying for several days, created a stir in the court, in the midst of which the learned judge awoke; and the lord mayor having whispered into his lordship's wig, his lordship bowed, and the clerk took the parchment. Mr. Foreman, and gentlemen of the grand jury, said his lordship, I am happy to say that your labours for the present are concluded;
- 67. there are nomore bills for your consideration. Your presentment shall be attended to, and I have to acknowledge your great zeal and attention, and to give you thanks for your services:—gentlemen, you are now discharged. The bows, and scufflings, and cries of Make way there for the gentlemen of the grand jury, who are coming out of court, were resumed, and the orator and his peers retired, leaving the poor girl at the bar, wondering what had happened, and what could be the reason that the worshipful community with the cat-skin tippets should have interposed themselves in the middle of her pathetic defence, in order to discuss the irritating characteristic of a smoky chimney. I admit that the pompous oratory of the foreman, the mons parturiens—a splendid exhibition, and the ridiculus mus, which eventually presented itself, were to me treats of no common order, and I regretted that Daly was not with me to participate with me in devouring the grave absurdities which we should have had before us. The trial of the girl was concluded, and I had no doubt as to her fate, now that I became acquainted with the principle,—she was acquitted, and never shall I forget the effect which this result of her trial produced upon her manners and features. The moment my friend Zig-zag had pronounced the words, Not guilty, the pathetic expression which had characterised her countenance turned into the most humorous, and having winked her eye at the learned judge, who, poor man, had summed up decidedly against her, she proceeded to place her two hands extended in a right line from the tip of her nose, in the direction of his lordship's seat, after the fashion of what is called taking a double sight, and then, making a noise which, if not indescribable by imitation, is certainly irreducible to writing, something between that which a hackney-coachman utters to encourage his tired horses, and that which a duck makes when it sees either a ditch or a drake in dry weather, she turned herself suddenly round with the least graceful pirouette I ever saw,
- 68. leaving one ofthe hands which she had previously elevated for observation the last part of her person visible. A short case of pot-stealing followed—the prisoner was found guilty in ten minutes; and then came the case. It was a curious and intricate one, and I felt quite assured, when I saw the prisoner, a genteel-looking young man, take his place under the inverted mirror, contrived with an almost diabolical ingenuity, so as to refract and reflect the light upon his face from the huge window at his back; I said to myself, having got both hardened and hungry during my short probation in court, We shall not dine at six to-day. It might, perhaps, injure the feelings of the individual himself, or, if he is dead, those of his friends and relations, to detail the particular case, the more especially as nothing could be clearer than that the crime laid to his charge was amply and satisfactorily—to everybody except himself—proved and substantiated. Just as the last witness for the defence was under cross- examination, I saw one of the lord mayor's servants put his powdered head in a little hole, and whisper something to the ordinary of Newgate, a remarkably pious-looking man, in full canonicals, with a bag-wig, which, to use Foote's phraseology, speaking of Dr. Simony (by whom, as of course everybody knows, he meant the unfortunate Dr. Dodd), looked as white as a curd, and as close as a cauliflower. It struck me that either the pretty wanton who had just been acquitted desired some serious conversation with the clergyman, or that the last convicted pot-stealer felt some qualms of conscience, and had sent for spiritual assistance; but no, —my friend Mr. Sheriff Bucklesbury relieved my mind from any such apprehensions, by inviting me to a whisper, with an expression of countenance which convinced me that it was nothing of so serious a character which had suddenly summoned the reverend divine from the court. Good news! said the sheriff; land is in sight. What? said I, not exactly catching the idea.
- 69. Dinner is notfar distant, said the sheriff, the ordinary has been sent for to dress the salad. Well, thought I, that ever a man so dressed, and so addressed, as the reverend divine opposite, should quit the seat of justice tempered with mercy, to mix oil and vinegar in a salad-bowl, does seem strange. It was evident to me, from the manner in which my friend spoke of the chaplain's secular vocations, that his respect for the table was infinitely greater than that which he entertained for the cloth, and never from that day have I seen painted over suburban inns, an ordinary on Sundays at two o'clock, without thinking of the reverend functionary so styled in the Old Bailey, and the probable duties he would be called upon to perform. The evidence having terminated, and the clock pointing to fifteen minutes after six, his lordship began summing up. I have already mentioned that his lordship was deaf, and the strange blunders which I noticed in his early charges will perhaps serve to inform the reader of these papers, whoever he may be, that his lordship's handwriting was utterly unintelligible, even to himself; indeed, so completely illegible were his notes, that the only resource his lordship had, if ever they were called for upon motions for new trials (which perhaps I need not here add, was in his lordship's case by no means an unfrequent occurrence), was to send them to be printed— printers being proverbially the best decypherers in the world. His lordship's charge—barring the inevitable blunders and hesitations, rendered absolutely necessary by this almost hopeless illegibility—was exceedingly minute and elaborated. He recapitulated the evidence of the three first witnesses verbatim, and continued thus of the fourth:— Now, gentlemen of the jury, here is Amos Hardy—Handy—no, not Handy—Harding—Amos Harding tells you, that on Tuesday—no, not Tuesday—I see—Friday the 14th—that is, the 24th—he was going along Liverpool—no—Liquorpond Street—near Gug's Island—Guy's— no—Gray's Inn Lane—yes—going along Liquorpond Street, Gray's Inn Lane—at about eight o'clock in the morning—and saw the fire
- 70. break out ofMr. Stephenson's windows. This, gentlemen of the jury, is a very remarkable fact—and in connection with some other circumstances to which we shall presently come, is quite worthy of your particular attention—you perceive that he swears to eight o'clock in the morning. Evening, my lord, said Mr. Flappertrap, standing up and whispering his lordship audibly. Evening is it? said his lordship—ay, so it is—evening—no matter —he swears to the time at which he saw the fire break out—and hence will naturally arise in your minds a chain of circumstances which it will be my duty to endeavour to unravel. In the first place —— Hereabouts one of the servants of the court put his head in at one of the doors at the back of the bench, and whispered the lord mayor much after the same manner in which Mr. Flappertrap had just before whispered the judge. His lordship immediately pulled out his watch—then looked at the clock—and then wrote a few words upon a slip of paper, and laid that slip of paper upon his lordship's notes. The judge took up the memorandum, and tore it in pieces—as I thought indignantly. You know what that means? said my friend, the sheriff. No, said I. Dinner's waiting, replied my friend—an announcement which startled me, as it seemed impossible but that it would be kept waiting for some time. This little scene, however, was followed by the arrival of the recorder, who, after bowing to the lord mayor, took his seat on the bench. I told you so, said the sheriff; Mr. Recorder is come to try the remaining cases—— A cry of Silence—pray, silence, indicated that Mr. Sheriff Bucklesbury and I were speaking somewhat too loudly. The circumstances to which I allude, continued his lordship, after he had torn up the note, are in fact so clearly detailed in the
- 71. evidence you haveheard, that to men of intelligence and experience, like those I am now addressing, any attempt at explanation on my part would be superfluous. The case appears a very clear one—you have to decide upon the value of the evidence, and return your verdict accordingly, giving the prisoner the benefit of any doubts you may entertain on the question. Never was I more surprised than at finding the promised explanations and comparisons of fact and testimony so suddenly cut short, after the manner of the story of the Bear and Fiddle, and I could not help, while the clerk of the arraigns was putting his accustomed question to the jury, noticing the circumstance to my worshipful friend. To be sure, said the sheriff, don't you see—the time is up—he smells the marrow puddings. The jury, emulating the expedition of the judge, in one minute, according to the zig-zag system, acquitted the prisoner; whereupon, his lordship rising to depart, addressed that individual in words to this effect:— Prisoner at the bar, you have been tried by an able, patient, and conscientious jury of your countrymen, who, convinced like myself of the enormity of your crime, and of the wicked intentions by which you were actuated in its commission, have returned the only verdict which they could justly and honestly return—they have well discharged their duty. And although it is not my province in this place to pronounce the awful sentence of the law upon you, I shall take care—— Here Mr. Flappertrap whispered his lordship that the jury had acquitted the prisoner. By-and-by, sir, said his lordship, angry at being interrupted—I shall take care, young man, that an example shall be made in your person of the—— The lord mayor here ventured to suggest that the young man was found not guilty.
- 72. Very well, mylord—presently, presently, said his lordship —even-handedness of justice; and that an enormous offender of your class may not be suffered to escape the just vengeance of the laws which he has outraged. Here Mr. Flappertrap whipped a bit of paper over the desk of the bench into the very place which the announcement of dinner had so recently occupied. His lordship looked at it, and exclaimed, unconsciously—Oh! ah!—umph! and then continued—It is true that upon the present occasion the mercy and forbearance of the jury have been exercised in a signal manner; and I trust their benevolence and indulgence will not be thrown away upon you. I maintain my own opinion still—yet they have decided, and I have only to receive that decision—you are discharged, sir, and may go about your business; but I can tell you this, young man, you have had a very narrow escape indeed. There was not a man in court who did not tacitly admit the truth and justice of at least the concluding passage of his lordship's address to the acquitted prisoner; nor was that individual himself the least astonished of his lordship's auditors. The incident, however, was worthy of its place in the day's proceedings, as producing a climax to the judicial operations of the learned lord, and leaving upon the minds of all his majesty's liege subjects then and there present, a conviction, that however classical it maybe to picture Justice blind, it is not, as a matter of convenience and utility, at all desirable that she should also be deaf. THE TOOTHPICK-MAKERS' COMPANY. The day was extremely fine; the windows of the rooms opening to the water, the house smelling of fried fish and mud, and the little boys with naked legs screaming, please to make a scramble, we having attained this enviable position in the building which looked like a race-stand, by treading a labyrinth of the dirtiest alleys and
- 73. stable-yards that everpauper or pony inhabited. It was, however, a joyous scene; and Hull, who was good enough to be my Mentor on the occasion, pooh-poohed the waiters into allowing us to look at the dinner-room, all laid out for the company; more than a hundred were expected, partitions had been pulled down, holes cut out here, and props poked in there, to afford the required accommodation; in short, everything gave token of a goodly day. Hull, who was at home everywhere, and everywhere popular, appeared, as soon as he arrived, to supersede everybody else. My dear friend, said he, I happen to know these people—the Toothpick Makers are one of the most ancient corporations of the city. My dear sir, the Mercers were incorporated in the 17th of Richard the Second—I have a tract that will prove it—1393 they were embodied—I know the clerk of the company at this day—so do you. No, I do not, said I. Pooh, pooh, said Hull, don't tell me—Jemmy Hobbs—everybody knows Jemmy Hobbs—married Miss Ball of Blackheath—'Splendid fellow, Jemmy. Well! these Mercers are a fine company, so are the Grocers,—St. Anthony is their patron. My dear sir, I am forced to know all these things. Then there are the Drapers, and the Fishmongers—pooh, pooh—Doctors, and Proctors, and Princes of the Blood, are all fishmongers—Walworth was a fishmonger—eh—my dear friend, you should see their paintings—splendid things— Spiridiona Roma—fish in all seasons. Then there are the Goldsmiths and the Skinners, and the Merchant Tailors—Linen Armourers—eh— queer fellows, some of them; but I do assure you— (this was said in a whisper,) you will see some men here to-day worth seeing. I suppose, said I, the Toothpick Makers' Company was founded by Curius Dentatus—whence comes the French cure-dent. Pooh, pooh, said Hull, no such thing—much older than Curius Dentatus—I happen to know—founded in the reign of Edward the Fifth, my dear friend.
- 74. About this periodthe company began to arrive thicker and faster, and certainly I had never seen any one of them before, which gave, at least, an air of novelty to the scene. Generally speaking, they ran fat, and wore white waistcoats, such as that to which I had likened the bow window of 77, St. James's Street: they looked all very hot, and puffed a good deal;—however, they kept coming and coming, until the drawing-room, as a sort of thing like a bad conservatory, well placed to the south-west, was called, was so full that I began to be as hot as my companions. Six o'clock arrived, but no dinner; the master of the house (who, from wearing a similar sort of uniform waistcoat, I took to be a Toothpick Maker,) came in and spoke to some of the fattest persons of the community, evidently intimating that the banquet was ready—nevertheless no move was made, because it appeared that Mr. Hicks had not arrived. You had better, said one of the more important persons in the room, let men be placed ready to see when Mr. Hicks arrives at the end of the lane by the stables. Yes, sir, was the answer; and from that time I heard nothing but Hicks and Mr. Hicks talked of, until I was driven by extreme curiosity to inquire of my omniscient friend Hull, who Mr. Hicks was. Hicks! exclaimed Hull—why, my dear friend, you know Mr. Hicks —the great Mr. Hicks—everybody knows Hicks. I for one, said I, do not— and it turned out that at the moment I was not likely to be enlightened, for, just as Hull was about to give me an account of this important personage, a hubbub and bustle near the door, which speedily pervaded the whole assembly, proclaimed his arrival. In a moment the buzz of conversation ceased, a sort of circle was made round Mr. Hicks, and several of the most distinguished members of the community hurried up to take their places near him. Hull dragged me towards this sanctum, this magic ring, and, with a look of the greatest importance, assured me, that it was right that I should immediately be presented to Mr. Hicks. The presentation accordingly took place, and no sooner was it over, than one of the grandees came up to me,
- 75. and, in aconfidential whisper, informed me that my place at dinner was on the left of Mr. Hicks, as being a friend of the master. I concluded that the arrangement was attributable to Hull, who, I found, was to be my neighbour on the left, and, although I could have dispensed with the honour of so close an approximation to the hero of the day, I rejoiced mightily that I was placed so near my friend Hull, who would be as useful to me upon such an occasion as is a catalogue of the pictures at an exhibition anywhere else. In a very short time dinner was announced, and Mr. Hicks, having the master on his right hand, led the way to the large room upstairs, round the whole of which the table ran, exhibiting, as I entered the apartment, a lengthened line of tin covers, looking like a collection of cuirasses, glittering on the board;—the heat was tremendous, and the air redolent with fried flounders. A few minutes sufficed to arrange us, grace was said by the chaplain, and we fell to. As in all similar cases, the exercise of eating and drinking superseded conversation or remark, and I, who did but little in that way myself, and having therefore an opportunity of seeing the modus operandi at my leisure, became suddenly enlightened as to the extent to which such pleasures may be carried. Of each and every dish did each and every man partake, from turtle to white-bait, both inclusive; by comparison with the individuals now before and around me, my friend Bucklesbury, whom I had a week before considered a prodigy in the way of feeding, sank into insignificance; to the elaborated course of fish succeeded a host of fowls, cutlets, hashes, stews, and other things of that nature, accompanied by sundry haunches of venison, and succeeded again by ducks innumerable, and peas immeasurable. The destruction of all these articles was, however, effected with ease in less than an hour and a half, during which the attentions paid to Mr. Hicks were most marked and gratifying: if the sun shone in upon the tip of his nose, the waiters were ordered to pull down the blinds before him; if the gentlest breeze wantoned about the back of his neck, the master of the house was called to shut the window behind him; for him the chairman culled the choicest bits; to him the landlord tendered his
- 76. most particular wines:every eye was fixed on his actions, every ear seemed open to his words; he had, however, as yet spoken little, but had eaten the more. All sublunary pleasures must have an end, so had this dinner; and a call of silence, and the thumping of the president's hammer upon the table, announced that some professional gentlemen were about to sing Non nobis, Domine. They began—we all standing up—I with the sun full in my eyes, setting over London in all its glory. The voices harmonised beautifully; but fine and melodious as they were, I felt that the canon, or whatever it is called, very much resembled a fire which, smouldering and smouldering in the low notes, kept perpetually bursting out in a fresh place, when one fancied it out. As far as the religious feeling of the thing goes, it was misplaced; and as for its duration, it seemed to be more like three graces than one. This over, the wine began to pass, and beards to wag; Hicks grew condescending, and the day began to mend; the King's health was given—song, God save the King—chorus by the company, all standing—The Queen—The Prince of Wales—then the Duke of York and the Army—the Duke of Clarence and the Navy—the Memory of St. Ursula, the mother of all Toothpick Makers, with an appropriate glee, received with loud cheers. The Master then rose and begged to propose a toast. No sooner had he uttered these words, than the whole room rang with applause, the wine-glasses danced hornpipes upon the table to the music of the forks and spoons, and the noise was tremendous. I see, continued the worthy president, that you anticipate my intentions; gentlemen, there could be no doubt upon your minds what the toast would be (more cheering). I will not occupy your time, nor hinder you from the gratification of your feelings upon this topic by dilating upon the merits of the illustrious individual whose health I am about to propose; whether we regard him in public life, guiding by his zeal and energy the community which he fosters and protects by his influence, or view him in private society, the ornament of the circle of which he is the centre, our gratitude and
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