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Intelligent Systems Reference Library 239
Seiki Akama
Yotaro Nakayama
Tetsuya Murai
Epistemic
Situation Calculus
Based on Granular
Computing
A New Approach to Common-Sense
Reasoning
Intelligent Systems Reference Library
Volume 239
Series Editors
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
Lakhmi C. Jain, KES International, Shoreham-by-Sea, UK
The aim of this series is to publish a Reference Library, including novel advances
and developments in all aspects of Intelligent Systems in an easily accessible and
well structured form. The series includes reference works, handbooks, compendia,
textbooks, well-structured monographs, dictionaries, and encyclopedias. It contains
well integrated knowledge and current information in the field of Intelligent Systems.
The series covers the theory, applications, and design methods of Intelligent Systems.
Virtually all disciplines such as engineering, computer science, avionics, business,
e-commerce, environment, healthcare, physics and life science are included. The list
of topics spans all the areas of modern intelligent systems such as: Ambient intelli-
gence, Computational intelligence, Social intelligence, Computational neuroscience,
Artificial life, Virtual society, Cognitive systems, DNA and immunity-based systems,
e-Learning and teaching, Human-centred computing and Machine ethics, Intelligent
control, Intelligent data analysis, Knowledge-based paradigms, Knowledge manage-
ment, Intelligent agents, Intelligent decision making, Intelligent network security,
Interactive entertainment, Learning paradigms, Recommender systems, Robotics
and Mechatronics including human-machine teaming, Self-organizing and adap-
tive systems, Soft computing including Neural systems, Fuzzy systems, Evolu-
tionary computing and the Fusion of these paradigms, Perception and Vision, Web
intelligence and Multimedia.
Indexed by SCOPUS, DBLP, zbMATH, SCImago.
All books published in the series are submitted for consideration in Web of Science.
Seiki Akama · Yotaro Nakayama · Tetsuya Murai
Epistemic Situation Calculus

Based on Granular
Computing
A New Approach to Common-Sense
Reasoning
Seiki Akama Yotaro Nakayama
Kawasaki, Kanagawa, Japan Biprogy
Kōtō, Tokyo, Japan
Tetsuya Murai
Chitose Institute of Science and Technology
Chitose, Hokkaido, Japan
ISSN 1868-4394 ISSN 1868-4408 (electronic)

Intelligent Systems Reference Library
ISBN 978-3-031-28550-9 ISBN 978-3-031-28551-6 (eBook)
https://doi.org/10.1007/978-3-031-28551-6
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Switzerland AG 2023
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the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
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The publisher, the authors, and the editors are safe to assume that the advice and information in this book
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Foreword
Artificial Intelligence studies nowadays mostly focus on machine learning techniques

including deep learning and related research, while the fundamental aspect of the
field remains within the framework of classical as well as non-standard logics. The
latter logical framework moreover includes non-standard set-theoretical systems like
fuzzy sets and rough sets. Thus, a discussion of the latter theories is needed to proceed
with further studies of artificial intelligence in its fundamentals.
To cover all these broad topics is not easy for researchers, as they are mostly
working in a narrow area, as they are pursuing deep fundamental theoretical
properties of a logical system.
Fortunately, however, the three authors of this monograph succeeded in carrying
out this difficult task: this book encompasses various logical systems, rough sets, and
related subjects, while fundamental issues such as the results concerning soundness
and completeness are rigorously described.
The main topic in this monograph is the granular reasoning of the epistemic situ-
ation calculus. Besides the logical framework of the situation calculus, the granular
computing based on rough set theory enables the appropriate setting herein. The
predicate of possibility is introduced and used throughout. Moreover, logical deduc-
tion systems using Gentzen type calculi and semantic tableaux calculi are introduced.
These considerations finally arrive at discussing the Frame Problem, the central issue
in artificial intelligence. Readers will find the granular computing treatment of this
issue.
Readers should have a certain level of logical background: in other words, this
monograph is not an introductory textbook for logical systems. After reading an
introductory text, however, this book will be an excellent introduction and overview
of non-standard logical and set systems of current interest.
Thus, this monograph is suited to researchers of the related fields of artificial
intelligence as well as to Ph.D. course students who are motivated to find appropriate
subjects of studies.
v
vi Foreword
Overall, it should be emphasized again that the contents herein are unique and
strongly stimulating future studies of the related areas of research, and the results of
the authors should deeply be appreciated in view of their wide perspective and deep
insight.
Tsukuba, Japan Sadaaki Miyamoto

September 2022
Preface
Artificial Intelligence (AI) is the research area of science and engineering for intelli-
gent machines, especially intelligent computer programs. One of the basic objectives
is to study computer-based advanced knowledge representation and reasoning for
developing knowledge-based systems.
It is very important to deal with common-sense reasoning in knowledge-based
systems. If we employ a logic-based framework, classical logic is not suited for
the purpose of describing common-sense reasoning. It is well known that there are
several difficulties with logic-based approaches, e.g., the so-called Frame Problem.
This book explores a new approach to common-sense reasoning using epistemic
situation calculus which integrates the ideas of situation calculus and epistemic logic.
We try to formalize common-sense reasoning in the context of granular computing
based on rough set theory.
The structure of the book is as follows. Chapter 1 gives the motivations and the
organization of the present book. We first discuss the importance of the formalization
of common-sense reasoning in knowledge-based systems. Second, we sketch our
approach using epistemic situation calculus which integrates the ideas of situation
calculus and epistemic logic. We try to formalize common-sense reasoning in the
context of granular computing based on rough set theory. We also briefly show the
organization of the book.
Chapter 2 introduces the foundations for rough set theory [1]. We outline Pawlak’s
motivating idea and give a technical exposition. Besides variable precision, rough set
model is also presented with some applications based on rough set theory. In addition,
we describe some non-classical logics. They are closely related to the foundations
of rough set theory. We also provide the basics of modal and many-valued logic.
In Chap. 3, we present the consequence relation and Gentzen type sequent calculus
for many-valued logics, and also describe the partial semantics interpreted with rough
sets.
In Chap. 4, we present another deduction system based on tableaux calculus for
four-valued logic and introduce the analytic tableaux as a basis for an automated
deductive system.
vii
viii Preface
In Chap. 5, we apply granular reasoning to the epistemic situation calculus ES

of Lakemeyer and Levesque by interpreting actions as modalities and granules of
possible worlds as states. The zoom reasoning proposed by Murai et al. is regarded as
an epistemic action and is incorporated into the ES as an abstraction and refinement
action by the granularity of the situation.
In Chap. 6, we discuss some aspects of the Frame Problem in the context of
granular reasoning. We describe the essential concern behind the Frame Problem
and discuss our point of view on the Frame Problem.
In Chap. 7, we give our conclusions about the book. First, we summarize the
results, second, we show further technical questions which should be solved in our
future work.
The book proposes a novel approach to common-sense reasoning. We assume
that the reader has mastered the material ordinarily covered in AI and mathematical
logic. We believe that the book is suitable for those, like experts and students, who
wish to get involved in the field.
We are grateful to Prof. L. Jain, Prof. Y. Kudo, Prof. S. Miyamoto, and the referees
for their constructive comments.
Kawasaki, Japan Seiki Akama

Kōtō, Japan Yotaro Nakayama
Chitose, Japan Tetsuya Murai
September 2022
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Rough Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Pawlak’s Rough Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Variable Precision Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Decision Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Decision Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Non-Classical Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Many-Valued Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 Bilattice Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.4 Relevance Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Basic Situation Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 Foundational Axioms for Situations . . . . . . . . . . . . . . . . . . . . 48
2.4.2 Domain Axioms and Basic Theories of Actions . . . . . . . . . . 49
2.4.3 The Frame Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.4 Frame Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Deduction Systems Based on Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Rough Sets and Decision Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Decision Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Decision Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Belnap’s Four-Valued Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Rough Sets and Partial Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
ix
x Contents
3.5 Consequence Relation and Sequent Calculus . . . . . . . . . . . . . . . . . . . 68

3.6 Extension of Many-Valued Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Soundness and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.8 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Tableaux Calculi for Many-Valued Logics . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.1 Rough Set and Decision Logic . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.3 Belnap’s Four-Valued Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.4 Analytic Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3 Tableaux Calculi for Many-Valued Logics . . . . . . . . . . . . . . . . . . . . . . 99
4.3.1 Relationship with Four-Valued Semantics . . . . . . . . . . . . . . . 99
4.3.2 Many-Valued Tableaux Calculi . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Soundness and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Granular Reasoning for the Epistemic Situation Calculus . . . . . . . . . . 111
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Epistemic Situation Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2.1 Background of Lakemeyer & Levesque’s Logic ES . . . . . . . 113
5.2.2 Semantics of ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.3 Basic Action Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3 Rough Set and Decision Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1 Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.2 Decision Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4 Zooming Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.1 Kripke Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.2 Granularized Possible World and Zooming Reasoning . . . . . 121
5.5 Consequence Relation for Partial Semantics . . . . . . . . . . . . . . . . . . . . 123
5.5.1 Belnap’s Four-Valued Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.2 Four-Valued Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.5.3 Semantic Relation with Four-Valued Logic . . . . . . . . . . . . . . 125
5.5.4 Consequence Relation and Sequent Calculus . . . . . . . . . . . . . 127
5.6 Zooming Reasoning as Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6.1 Zooming Reasoning in ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6.2 Action Theory for Zooming Reasoning . . . . . . . . . . . . . . . . . . 130
5.6.3 Semantics for Zooming Reasoning in ES . . . . . . . . . . . . . . . . 132
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Contents xi
6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1 The Frame Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.1 What is the Frame Problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.2 The Frame Problem in Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1.3 The Epistemological Frame Problem . . . . . . . . . . . . . . . . . . . . 143
6.2 Re-Examination of the Frame Problem in the Context
of Granular Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.1 Solution to the Frame Problem with
the Situation Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2.2 Dennett’s Robot Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.1 Summary and Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2 Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Chapter 1
Introduction
Abstract Chapter 1 gives the motivations and the organization of the present book.
We first discuss the importance of the formalization of common-sense reasoning in
knowledge-based systems. Second, we sketch our approach using epistemic situation
calculus which integrates the ideas of situation calculus and epistemic logic. We try
to formalize common-sense reasoning in the context of granular computing based
on rough set theory. We also briefly show the organization of the book.
1.1 Motivations
Artificial Intelligence (AI) is the research area of science and engineering for intelli-
gent machines, especially intelligent computer programs. One of the basic objectives
is to study computer-based advanced knowledge representation and reasoning for
developing knowledge-based systems.
It is very important to deal with common-sense reasoning in knowledge-based
systems. If we employ a logic-based framework, classical logic is not suited for
the purpose of describing common-sense reasoning. It is well known that there are
several difficulties with logic-based approaches. e.g., the so-called Frame Problem.
This book explores a new approach to common-sense reasoning using epistemic
situation calculus which integrates the ideas of situation calculus and epistemic logic.
We try to formalize common-sense reasoning in the context of granular computing
based on rough set theory. Thus, the proposed approach is based on non-classical
logic rather than classical logic as a tool for reasoning and knowledge representation.
We generalize epistemic situation calculus (ES) of Lakemeyer and Levesque [2]
which integrates ideas of situation calculus and epistemic logic for formalizing
common-sense reasoning in the context of granular computing based on rough set
theory.
Why do we need such a framework? For incomplete and inconsistent information
in knowledge representation and reasoning, we assume that the concept of partiality
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 1
S. Akama et al., Epistemic Situation Calculus Based on Granular Computing,
Intelligent Systems Reference Library 239,
https://doi.org/10.1007/978-3-031-28551-6_1
2 1 Introduction
and overcompleteness play an essential role. The partiality and overcompleteness of

information are causes of incomplete or inconsistent epistemic states on the knowl-
edge and lead to uncertain and ambiguous interpretations of the world.
Reasoning for partial and inconsistent information inherits ambiguity, and uncer-
tainty induces incompleteness and inconsistency. Incompleteness and inconsistency
are in fact caused by the lack of information and the excessiveness of information.
In classical logic, sentences are either true or false, and it is assumed that at
any time, every sentence has exactly one of these two truth-values regardless of
the available information about it. In knowledge representation, information is not
complete. Therefore, we need non-classical logic as a model for reasoning with
inconsistent and incomplete information.
We will treat here non-classical logic, also sometimes referred to as partial logic
that is needed in order to formulate and understand the proposed framework of
knowledge representation and reasoning. The idea of our knowledge representation
and reasoning concern two main aspects of AI technology: cognitive adequacy and
computational feasibility.
The goal is thus to develop an appropriate methodology for knowledge repre-
sentation and reasoning systems. The point of departure is the kind of knowledge
representation achieved by the epistemic situation calculus and reasoning system
based on non-classical logic.
Non-classical logics can be classified into two classes. One class is a rival to clas-
sical logic, including intuitionistic logic and many-valued logic. The logics in this
class deny some principles of classical logic. For example, many-valued logic denies
the two-valuedness of classical logic, allowing several truth-values. Kleene’s three-
valued logic does not adopt the law of excluded middle A ∨ ¬A as an axiom. Con-
sequently, many-valued logic can deal with the features of information like degree
and vagueness.
The second class is an extension of classical logic. The logics in this class thus
expand classical logic with some logical types of machinery. One of the most notable
logics in the class is modal logic, which extends classical logic with modal operators
to formalize the notions of necessity and possibility.
A lack of information and contradictory information is an epistemic state which
is captured similar to ambiguity or uncertainty, but the gap and glut should be treated
as distinguished. In the glut of information, the undecidability of the truth results
from the lack of information, so the contradiction is also the problem of partiality of
information.
Our position is that reasoning as a cognitive activity is based on the knowledge
available to the cognitive agent through some form of representation. The goal of
knowledge representation (KR) in computer science is to improve the existing KR
technology and theory.
In the real world, an agent makes judgments and draws a conclusion even if he
does not have complete information about a problem, and even if the information at
hand contains inconsistency. This kind of practical reasoning is distinct from standard
deductive and probabilistic reasoning, dealing with exact predicates without truth-
value gaps excluding the possibility of incoherent information.
1.1 Motivations 3
Thus, the aim is to establish a formal system that: (1) captures a practically rel-
evant body of cognitive facilities employed by humans which can be reasonably
implemented, and (2) extends the knowledge representation and reasoning capabil-
ities of humans by capitalizing on the technical strength of the system including
formal techniques to master gluts (situations where we are inclined to accept con-
tradicting statements) and gaps (situations where we want to draw a conclusion but
are uncertain because of lack of information) in a principled way.
In this book, we study how these fundamental theories–non-classical logics and
their deduction systems–can work for the reasoning in the situation calculus that
includes frame axioms used in the Frame Problem.
J. McCarthy and P. Hayes [3] first presented the Frame Problem. Moreover, Den-
nett [1] redefined it as the generalized Frame Problem. It is well known that the
Frame Problem is the challenge of the representation of the effects of action in logic
having to represent them in tractable fashion.
However, to many philosophers, the AI researchers’ Frame Problem is suggestive
of wider epistemological issues [1], and the Frame Problem gives epistemological
issues to philosophers.
The problem is the possibility to limit the scope of the reasoning required to
derive the consequences of an action. Moreover, more generally, the problem is that
to account for an agent able to make decisions on the basis only of what is relevant
to an ongoing situation without having explicitly to consider all that is not relevant.
The Frame Problem is a question of how to process only the necessary information
from the information making up the world and this is raised as the first serious problem
in the world’s description.
The second problem is the amount of processing of inference and is said as a
problem due to processing with partiality. This can be seen as a problem of difficulty
in identifying problems related to the subject of actions posed in the problem of
Dennett’s robot example. The fundamental problem on the descriptiveness of the
Frame Problem is expected to be a problem of correspondence to partiality,
Our approach does not aim to give a perfect solution for the Frame Problem, but
we provide the interpretation based on the partiality to the problem of description of
the information, which is considered as the cause of it.
To consider the partiality of the information, we give another interpretation to
the Frame Problem. The fundamental difficulty is here due to the impossibility of
description for the world and situation. The Frame Problem was assumed to be based
on a complete description of the required condition for actions.
On the other hand, granular reasoning is a reasoning framework based on the
analytics method of information considering the partiality and approximation, and
the degree of validity.
In this book, we study the knowledge representation and reasoning for an agent
in the world in which partiality plays a key concept. In other words, we have to
cope with knowledge representation with incomplete and inconsistent information
adequately.
4 1 Introduction
Therefore we adopt many-valued logics and modal logics, which are non-classical
logics, and rough set theory, which can serve as the theoretical basis for data analysis
and the semantic basis for our non-classical logics.
When we focus on the knowledge and the recognition of an agent, we need
to handle partiality, ambiguity, vagueness, incompleteness and inconsistency. This
phenomenon cannot be captured by classical bivalence logic and we need plausible
and defeasible system as underlying theory.
Partiality is the primary concern in the research of artificial intelligence, and it
is still considered as a significant problem in this field. To handle the partiality for
knowledge representation, there are various approaches.
In this research, as a tool, we focus on the rough set theory, granular computing,
many-valued logics, and modal logics. In fact, these theories and logics are closely
related.
1.2 Organization
The contents of the book are essentially based on those presented in Nakayama [4].
This book consists of two parts: The first is about the deduction system with many-
valued logics and the second is about the granular reasoning in the epistemic situation
calculus.
The deduction system treats non-classical logic as a basis for reasoning with partial
information, and granular reasoning is embedded in the ES for describing common-
sense reasoning including non-monotonic reasoning with partial semantics.
The structure of the book is as follows. In Chap. 2, we introduce the foundations
for rough set theory [5]. We outline Pawlak’s motivating idea and give a technical
exposition. Besides variable precision rough set model is also presented with some
applications based on rough set theory. In addition, we describe some non-classical
logics. They are closely related to the foundations of rough set theory. We also provide
the basics of modal and many-valued logic.
In Chap. 3, we present the consequence relation and Gentzen type sequent calculus
for many-valued logics, and also describe the partial semantics interpreted with rough
sets.
In Chap. 4, we present another deduction system based on tableaux calculus for
four-valued logic and introduce the analytic tableau as a basis for an automated
deductive system.
In Chap. 5, we apply granular reasoning to the epistemic situation calculus ES
of Lakemeyer and Levesque by interpreting actions as modalities and granules of
possible worlds as states. The zoom reasoning proposed by Murai et al. is regarded as
an epistemic action and is incorporated into the ES as an abstraction and refinement
action by the granularity of the situation.
In Chap. 6, we discuss some aspects of the Frame Problem in the context of
granular reasoning. We describe the essential concern behind the Frame Problem
and discuss our point of view for the Frame Problem.
References 5
In Chap. 7, we give our conclusions of the book. First, we summarize the results,
Second, we show further technical questions which should be solved in our future
work.
References
1. Dennett, D.: Cognitive wheels: The frame problem of AI. In: Hookway, C. (ed.) Minds, Machines
and Evolution, pp. 129–151. Cambridge University Press, Cambridge (1984)
2. Lakemeyer, G., Levesque, H.: A semantic characterization of a useful fragment of the situation
calculus with knowledge. Artif. Intell. 175, 142-164 (2011)
3. McCarthy, J., Hayes, P.: Some philosophical problems from the standpoint of artificial intelli-
gence. Mach. Intell. 4, 463-502 (1969)
4. Nakayama, Y.: A study on the epistemic situation calculus in the framework of granular rea-
soning. Ph.D. Thesis, Chitose Institute of Science and Technology (CIST), Chitose, Hokkaido,
Japan (2020)
5. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341-356 (1982)
Chapter 2
Preliminaries
Abstract Chapter 2 introduces the foundations for rough set theory. We outline
Pawlak’s motivating idea and give a technical exposition. Besides variable precision
rough set model is also presented with some applications based on rough set theory.
In addition, we describe some non-classical logics. They are closely related to the
foundations of rough set theory. We also provide the basics of modal and many-valued
logic.
2.1 Rough Set Theory
This section describes the foundations for rough set theory. We outline Pawlak’s
motivating idea and give a technical exposition. Basics of Pawlak’s rough set theory
and variable precision rough set model are presented with some related topics.
Rough set theory is interesting theoretically as well as practically, and a quick
survey on the subject, including overview, history and applications, is helpful to the
readers.
2.1.1 Pawlak’s Rough Set Theory
Rough set theory, proposed by Pawlak [25], provides a theoretical basis of sets based
on approximation concepts. A rough set can be seen as an approximation of a set.
These approximations can be described by two operators on subsets of the universe.
Rough set theory is, in particular, helpful in extracting knowledge from data tables
and it has been successfully applied to the areas such as data analysis, decision
making, machine learning and other various applications
We begin with an exposition of Pawlak’s approach to rough set theory based on
Pawlak [26]. His motivation is to provide a theory of knowledge and classification
by introducing a new concept of set, i.e., rough set.
By object, we mean anything we can think of, for example, real things, states,
abstract concepts, etc. We can assume that knowledge is based on the ability to
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 7
S. Akama et al., Epistemic Situation Calculus Based on Granular Computing,
Intelligent Systems Reference Library 239,
https://doi.org/10.1007/978-3-031-28551-6_2
8 2 Preliminaries
classify objects. Thus, knowledge is necessarily connected with the variety of clas-
sification patterns related to specific parts of the real or abstract world, called the
universe of discourse (or the universe).
Now, we turn to a formal presentation. We assume the usual notation for set theory.
Let U be non-empty finite set of objects called the universe of discourse. Any subset
X ⊆ U of the universe is called a concept or a category in U . Any family of concepts
in U is called knowledge about U . Note that the empty set ∅ is also a concept.
We mainly deal with concepts which form a partition (classification) of a certain
universe U , i.e. in families C = {X 1 , X 2 , . . . , X n } such that X i ⊆ U , X i = ∅, X i ∩
X j = ∅ for i = j, i, j = 1, . . . , n and X i = U . A family of classifications over
U is called a knowledge base over U .
Classifications can be specified by using equivalence relations. If R is an equiv-
alence relation over U , then U/R means the family of all equivalence classes of R
(or classification of U ) referred to as categories or concepts of R. [x] R denotes a
category in R containing an element x ∈ U .
A knowledge base is defined as a relational system, K = (U, R), where U = ∅
is a finite set called the universe, and R is a family of equivalence relations over U .
I N D(K ) means the family of all equivalence relations defined in K , i.e., I N D(K ) =
{I N D(P)|∅ = P ⊆ R}. Thus, I N D(K ) is the minimal set of equivalence relations,
containing all elementary relations of K , and closed under set-theoretical intersection
of equivalence relations.
If P ⊆ R and P = ∅, then ∩P denotes the intersection of all equivalence relations
belonging to P, denoted I N D(P), called an indiscernibility relation of P. It is also
an equivalence relation, and satisfies:

[x] I N D(P) = [x] R .
R∈P
Thus, the family of all equivalence classes of the equivalence relation I N D(P),
i.e., U/I N D(P) denotes knowledge associated with the family of equivalence rela-
tions P. For simplicity, we will write U/P instead of U/I N D(P).
P is also called P-basic knowledge. Equivalence classes of I N D(P) are called
basic categories (concepts) of knowledge P. In particular, if Q ∈ R, then Q is called
a Q-elementary knowledge (about U in K ) and equivalence classes of Q are referred
to as Q-elementary concepts (categories) of knowledge R.
Now, we describe the fundamentals of rough sets. Let X ⊆ U and R be an equiv-
alence relation. We say that X is R-definable if X is the union of some R-basic
categories; otherwise X is R-undefinable.
The R-definable sets are those subsets of the universe which can be exactly defined
in the knowledge base K , whereas the R-undefinable sets cannot be defined in K . The
R-definable sets are called R-exact sets, and R-undefinable sets are called R-inexact
or R-rough.
Set X ⊆ U is called exact in K if there exists an equivalence relation R ∈
I N D(K ) such that X is R-exact, and X is said to be rough in K if X is R-rough for
any R ∈ I N D(K ).
2.1 Rough Set Theory 9
Observe that rough sets can also be defined approximately by using two exact sets,
referred as a lower and an upper approximation of the set. Now, rough set theory is
outlined.
Definition 2.1 A knowledge base K is a pair K = (U, R), where U is a universe of

objects, and R is a set of equivalence relations on the objects in U .
Definition 2.2 Let R ∈ R be an equivalence relation of the knowledge base K =

(U, R) and X any subset of U . Then, the lower and upper approximations of X for
R are defined
as follows:
R X = {Y∈ U/R | Y ⊆ X },

R X = {Y∈ U/R | Y ∩ X = 0}.
R X is called the R-lower approximation and R X the R-upper approximation of X,

respectively. They will be simply called the lower-approximation and the upper-
approximation if the context is clear.
It is also possible to define the lower approximation and the upper approximation
in the following two equivalent forms:
R X = {x ∈ U | [x]R ⊆ X },
R X = {x ∈ U | [x]R ∩ X= ∅}.
or
x ∈ R X iff [x] R ⊆ X ,
x ∈ R X iff [x] R ∩ X = ∅.
The above three are interpreted as follows. The set R X is the set of all elements
of U which can be certainly classified as elements of X in the knowledge R. The set
R X is the set of elements of U which can be possibly classified as elements of X in
R.
We define R-positive region (P O S R (X )), R-negative region (N E G R (X )), and
R-borderline region (B N R (X )) of X as follows:
Definition 2.3 If K = (U, R), R ∈ R, and X ⊆ U , then the R-positive, R-negative,

and R-boundary regions of X with respect to R are defined respectively as follows:
P O S R (X ) = R X,
N E G R (X ) = U − R X,
B N R (X ) = R X − R X.
The positive region P O S R (X ) (or the lower approximation) of X is the collection

of those objects which can be classified with full certainty as members of the set X ,
using knowledge R.
The negative region N E G R (X ) is the collection of objects with which it can be
determined without any ambiguity, employing knowledge R, that they do not belong
to the set X , that is, they belong to the complement of X .
The borderline region B N R (X ) is the set of elements which cannot be classified
either to X or to −X in R. It is the undecidable area of the universe, i.e., none of the
10 2 Preliminaries
objects belonging to the boundary can be classified with certainty into X or −X as

far as R is concerned.
Now, we list basic formal results. Their proofs may be found in Pawlak [26].
Proposition 2.1 is obvious.
Proposition 2.1 The following hold:

1. X is R-definable iff R X = R X
2. X is rough with respect to R X = R X
Proposition 2.2 shows the basic properties of approximations:
Proposition 2.2 The R-lower and R-upper approximations satisfy the following
properties:
1. RX ⊆ X ⊆ RX
2. R∅ = R∅ = ∅, RU = RU = U
3. R(X ∪ Y ) = R X ∪ RY
4. R(X ∩ Y ) = R X ∩ RY
5. X ⊆ Y implies R X ⊆ RY
6. X ⊆ Y implies R X ⊆ RY
7. R(X ∪ Y ) ⊆ R X ∪ RY
8. R(X ∩ Y ) ⊆ R X ∩ RY
9. R(−X ) = −R X
10. R(−X ) = −R X
11. RRX = RRX = RX
12. RRX = RRX = RX
The concept of approximations of sets can also be applied to that of membership

relation. In rough set theory, since the definition of a set is associated with knowledge
about the set, a membership relation must be related to the knowledge.
Then, we can define two membership relations ∈ R and ∈ R . x∈ R reads “x surely
belongs to X ” and x∈ R reads “x possibly belongs to X ”. ∈ R and ∈ R are called the
R-lower membership and R-upper membership, respectively.
Proposition 2.3 states the basic properties of membership relations:
Proposition 2.3 The R-lower and R-upper membership relations satisfy the follow-
ing properties:
1. x∈ R X implies x ∈ X implies x∈ R
2. X ⊆ Y implies (x∈ R X implies x∈ R Y and x∈ R X implies x∈ R Y )
3. x∈ R (X ∪ Y ) iff x∈ R X or x∈ R Y
4. x∈ R (X ∩ Y ) iff x∈ R X and x∈ R Y
5. x∈ R X or x∈ R Y implies x∈ R (X ∪ Y )
6. x∈ R (X ∩ Y ) implies x∈ R X and x∈ R Y
7. x∈ R (−X ) iff non x∈ R X
8. x∈ R (−X ) iff non x∈ R X
Approximate (rough) equality is the concept of equality in rough set theory. Three
kinds of approximate equality can be introduced. Let K = (U, R) be a knowledge
base, X, Y ⊆ U and R ∈ I N D(K ).
1. Sets X and Y are bottom R-equal (X ∼ R Y ) if R X = RY

2. Sets X and Y are top R-equal (X R Y ) if R X = RY
3. Sets X and Y are R-equal (X ≈ R Y ) if (X ∼ R Y ) and (X R Y)
These equalities are equivalence relations for any indiscernibility relation R. They
are interpreted as follows: (X ∼ R Y ) means that positive example of the sets X and
Y are the same, (X R Y ) means that negative example of the sets X and Y are the
same, and (X ≈ R Y ) means that both positive and negative examples of X and Y are
the same.
These equalities satisfy the following proposition (we omit subscript R for sim-
plicity):
Proposition 2.4 For any equivalence relation, we have the following properties:
1. (1) X ∼ Y iff X ∩ X ∼ Y and X ∩ Y ∼ Y
2. X Y iff X ∪ Y X and X ∪ Y Y
3. If X X and Y Y , then X ∪ Y X ∪Y
4. If X ∼ X and Y ∼ Y , then X ∩ Y ∼ X ∩ Y
5. If X X Y , then X ∪ −Y U
6. If X ∼ Y , then X ∩ −Y ∼ ∅
7. If X ⊆ Y and Y ∅, then X ∅
8. If X ⊆ Y and Y U , then X U
9. X Y iff −X ∼ − Y
10. If X ∼ ∅ or Y ∼ ∅, then X ∩ Y ∼ ∅
11. If X U or Y U , then X ∪ Y U .
The following Proposition 2.5 shows that lower and upper approximations of sets
can be expressed by rough equalities:
Proposition 2.5 For any equivalence relation R:

1. R X is the intersection of all Y ⊆ U such that X ∼ R Y
2. R X is the union of all Y ⊆ U such that X R Y .
Similarly, we can define rough inclusion of sets. It is possible to define three kinds
of rough inclusions.
Let X = (U, R) be a knowledge base, X, Y ⊆ U , and R ∈ I N D(K ). Then, we
have:
⊂
1. Set X is bottom R-included in Y (X ∼ R Y ) iff R X ⊆ RY
∼
2. Set X is top R-included in Y (X ⊂ R Y ) iff R X ⊆ RY
∼
⊂ ⊂ ∼
3. Set X is R-included in Y (X ∼ R Y ) iff (X ∼ R Y ) and (X ⊂ R Y )
12 2 Preliminaries
Proposition 2.6 Rough inclusion satisfies the following:

∼
⊂ ∼ ⊂
(1) If X ⊆ Y , then X ∼ Y, X ⊂ Y and X ∼ Y
⊂ ⊂ −−
(2) If X ∼ Y and Y ∼ X , then X ∼ Y
∼ ∼
(3) If X ⊂ Y and Y ⊂ X , then X Y
∼ ∼
⊂ ⊂
(4) If X ∼ Y and Y ∼ X , then X ≈ Y
∼
(5) If X ⊂ Y iff X ∪ Y Y
⊂ −−
(6) If X ∼ Y iff X ∩ Y ∼ Y
−− −− ⊂
(7) If X ⊆ Y, X ∼ X and Y ∼ Y , then X ∼ Y
∼
(8) If X ⊆ Y, X X and Y Y , then X ⊂ Y
∼
⊂
(9) If X ⊆ Y, X ≈ X and Y ≈ Y , then X ∼ Y
∼ ∼ ∼
(10) If X ⊂ X and Y ⊂ Y , then X ∪ Y ⊂ X ∪ Y
⊂ ⊂ ⊂
(11) X ∼ X and Y ∼ Y , then X ∩ Y ∼ X ∩ Y
⊂ ∼
(12) X ∩ Y ∼ Y ⊂ X ∪ Y
⊂ −− ⊂
(13) If X ∼ Y and X ∼ Z , then Z ∼ Y
∼ ∼
(14) If X ⊂ Y and X Z , then Z ⊂ Y
∼ ∼
⊂ ⊂
(15) If X ∼ Y and X ≈ Z , then Z ∼ Y
−−
The above properties are not valid if we replace ∼ by (or conversely). If R is
an equivalence relation, then all three inclusions reduce to ordinary inclusion.
∼
⊂ ∼ ⊂
Note that ∼ R , ⊂ R and ∼ R are quasi ordering relations. They are called the lower,
upper and rough inclusion relation, respectively. Observe that rough inclusion of sets
does not imply the inclusion of sets.
The following proposition shows the properties of rough inclusion:
Proposition 2.7 Rough inclusion satisfies the following:

∼
⊂ ∼ ⊂
(1) If X ⊆ Y , then X ∼ Y, X ⊂ Y and X ∼ Y
⊂ ⊂ −−
(2) If X ∼ Y and Y ∼ X , then X ∼ Y
∼ ∼
(3) If X ⊂ Y and Y ⊂ X , then X Y
∼ ∼
⊂ ⊂
(4) If X ∼ Y and Y ∼ X , then X ≈ Y
∼
(5) If X ⊂ Y iff X ∪ Y Y
⊂ −−
(6) If X ∼ Y iff X ∩ Y ∼ Y
−− −− ⊂
(7) If X ⊆ Y, X ∼ X and Y ∼ Y , then X ∼ Y
∼
(8) If X ⊆ Y, X X and Y Y , then X ⊂ Y
∼
⊂
(9) If X ⊆ Y, X ≈ X and Y ≈ Y , then X ∼ Y
∼ ∼ ∼
(10) If X ⊂ X and Y ⊂ Y , then X ∪ Y ⊂ X ∪ Y
⊂ ⊂ ⊂
(11) X ∼ X and Y ∼ Y , then X ∩ Y ∼ X ∩ Y
⊂ ∼
(12) X ∩ Y ∼ Y ⊂ X ∪ Y
⊂ −− ⊂
(13) If X ∼ Y and X ∼ Z , then Z ∼ Y
∼ ∼
(14) If X ⊂ Y and X Z , then Z ⊂ Y
∼ ∼
⊂ ⊂
(15) If X ∼ Y and X ≈ Z , then Z ∼ Y
−−
The above properties are not valid if we replace ∼ by (or conversely). If R is
an equivalence relation, then all three inclusions reduce to ordinary inclusion.
2.1.2 Variable Precision Rough Set
The variable precision rough set model (VPRS model) proposed by Ziarko [34] is
an extension of Pawlak’s rough set theory, which provides a theoretical basis to treat
probabilistic or inconsistent information in the framework of rough sets.
Ziarko generalized Pawlak’s original rough set model in Ziarko [34], which is
called the variable precision rough set model (VPRS model) to overcome the inability
to model uncertain information, and is directly derived from the original model
without any additional assumptions.
In addition, VPRS model is an extension of Pawlak’s rough set theory, which
provides a theoretical basis to treat probabilistic or inconsistent information in the
framework of rough sets.
As the limitations of Pawlak’s rough set model, Ziarko discussed two points.
First, it cannot provide a classification with a controlled degree of uncertainty. Sec-
ond, some level of uncertainty in the classification process gives a deeper or better
understanding of data analysis.
VPRS model generalizes the standard set inclusion relation, capable of allowing
for some degree of misclassification in the largely correct classification.
Let X and Y be non-empty subsets of a finite universe U . X is included in Y ,
denoted Y ⊇ X , if for all e ∈ X implies e ∈ Y . Here, we introduce the measure
c(X, Y ) of the relative degree of misclassification of the set X with respect to set Y
defined as:
14 2 Preliminaries
⎧
⎨ |X ∩ Y |
1− , ifX = ∅,
c(X, Y ) =def |X | ,
⎩ 0, otherwise.
where |X | represents the cardinality of the set X .

VPRS is based on the majority inclusion relation. Let X , Y ⊆U be any subsets of
U . The majority inclusion relation is defined by the following measure c(X, Y ) of
the relative degree of misclassification of X with respect to Y .
The quantity c(X, Y ) will be referred to as the relative classification error. The
actual number of misclassification is given by the product c(X, Y ) ∗ car d(X ) which
is referred to as an absolute classification error.
We can define the inclusion relationship between X and Y without explicitly using
a general quantifier:
X ⊆ Y iff c(X, Y ) = 0.
The majority requirement implies that more than 50% of X elements should
be in common with Y . The specified majority requirement imposes an additional
requirement. The number of elements of X in common with Y should be above 50%
and not below a certain limit, e.g. 85%.
β
Formally, the majority inclusion relation ⊆ with a fixed precision β ∈ [0, 0.5) is
defined using the relative degree of misclassification as follows:
β
X ⊆ Y iff c(X, Y ) ≤ β,
where precision β provides the limit of permissible misclassification.

The above definition covers the whole family of β-majority relation. However,
the majority inclusion relation does not have the transitivity relation.
The following two propositions indicate some useful properties of the majority
inclusion relation:
β β
Proposition 2.8 If A ∩ B = ∅ and B ⊇ X , then it is not true that A ⊇ X .
β2 β2
Proposition 2.9 If β1 < β2 , then Y ⊇ X implies Y ⊇ X .
For the VPRS-model, we define the approximation space as a pair A = (U, R),
where U is a non-empty finite universe and R is the equivalence relation on U . The
equivalence relation R, referred to as an indiscernibility relation, corresponds to a
partitioning of the universe U into a collection of equivalence classes or elementary
set R = {E 1 , E 2 , . . . , E n }.
Using a majority inclusion relation instead of the inclusion relation, we can obtain
generalized notions of β-lower approximation (or β-positive region P O S Rβ (X )) of
the set U ⊇ X :
β
R β X = {E ∈ R ∗ : X ⊇ E} or, equivalently,
R β X = {E ∈ R ∗ : c(E, X ) ≤ β}
The β-upper approximation of the set U ⊇ X can also be defined as follows:

R β X = {E ∈ R ∗ : c(E, X ) < 1 − β}
The β-boundary region of a set is given by

B N Rβ X = {E ∈ R ∗ : β < c(E, X ) < 1 − β}.
The β-negative region of X is defined as a complement of the β-upper approximation:

N E G Rβ X = {E ∈ R ∗ : c(E, X ) ≥ 1 − β}.
The lower approximation of the set X can be interpreted as the collection of all those
elements of U which can be classified into X with the classification error not greater
than β.
The β-negative region of X is the collection of all those elements of U which can
be classified into the complement of X , with the classification error not greater than
β. The latter interpretation follows from Proposition 2.10:
Proposition 2.10 For every X ⊆ Y , the following relationship is satisfied:

P O S Rβ (−X ) = N E G Rβ X .
The β-boundary region of X consists of all those elements of U which cannot be

classified either into X or into −X with the classification error not greater than β.
Notice here that the law of excluded middle, i.e., p ∨ ¬ p, where ¬ p is the negation
of p, holds in general for imprecisely specified sets.
Finally, the β-upper approximation R β X of X includes all those elements of U
which cannot be classified into −X with the error not greater than β. If β = 0 then
the original rough set model is a special case of VPRS-model, as the following
proposition states:
Proposition 2.11 Let X be an arbitrary subset of the universe U :

1. R 0 X = R X , where R X is a lower approximation defined as R X = {E ∈
R ∗ : X ⊇ E}
R 0 X = R X , where R X is an upper approximation defined as R X =
2.
{E ∈ R ∗ : E ∩ X = ∅}
3. B N R0 X = B N R X , where B N R X is the set X boundary region defined as
B NR X = R X − R X
4. N E G R0 X = N E G R X , where N E G R X is the set X negative region
defined as N E G R X = U − R X
Proposition 2.12 If 0 ≤ β < 0.5 then the properties listed in Proposition 2.10 and
the following are also satisfied:
16 2 Preliminaries
Rβ X ⊇ R X ,
R X ⊇ Rβ X ,
B N R X ⊇ B N Rβ X ,
N E G Rβ X ⊇ N E G R X .
Intuitively, with the decrease of the classification error β the size of the positive
and negative regions of X will shrink, whereas the size of the boundary region will
grow.
With the reduction of β fewer elementary sets will satisfy the criterion for inclusion
in β-positive or β-negative regions. Thus, the size of the boundary will increase. The
reverse process can be done with the increase of β.
Proposition 2.13 With the β approaching the limit 0.5, i.e., β → 0.5, we obtain the
following:

R β X → R 0.5 X = {E ∈ R ∗ : c(E, X ) < 0.5},

R β X → R 0.5 X = {E ∈ R ∗ : c(E, X ) ≤ 0.5},
B N Rβ X → B N R0.5 X = {E∈ R ∗ : c(E, X ) = 0.5},
N E G Rβ X → N E G R0.5 X = {E ∈ R ∗ : c(E, X ) > 0.5}.
The set B N R0.5 X is called an absolute boundary of X because it is included in
every other boundary region of X .
The following Proposition 2.14 summarizes the primary relationships between
set X discernibility regions computed on 0.5 accuracy level and higher levels.
Proposition 2.14 For boundary regions of X, the following hold:

B R N0.5 X = B N Rβ X ,
β

R0.5X = R β X ,
β

β
R0.5X = Rβ X ,
β
N E G R0.5X = N E G Rβ X .
The absolute boundary is very “narrow”, consisting only of those sets which have
50/50 aplite of elements among set X interior and its exterior. All other elementary
sets are classified either into positive region R 0.5 X or the negative region N E G R0.5 .
We turn to the measure of approximation. To express the degree with which a set
X can be approximately characterized by means of elementary sets of the approx-
imation space A = (U, R), we will generalize the accuracy measure introduced in
Pawlak [25].
The β-accuracy for 0 ≤ β < 0.5 is defined as
α(R, β, X ) = car d(R β X )/car d(R β X ).
The β-accuracy represents the imprecision of the approximate characterization

of the set X relative to assumed classification error β.
Note that with the increase of β the cardinality of the β-upper approximation will
tend downward and the size of the β-lower approximation will tend upward which
leads to the conclusion that is consistent with intuition that relative accuracy may
increase at the expense of a higher classification error.
The notion of discernibility of set boundaries is relative. If a large classification
error is allowed then the set X can be highly discernible within assumed classification
limits. When smaller values of the classification tolerance are assumed it may become
more difficult to discern positive and negative regions of the set to meet the narrow
tolerance limits.
The set X is said to be β-discernible if its β-boundary region is empty or, equiv-
alently, if
R β X = R β X.
For the β-discernible sets the relative accuracy α(R, β, X ) is equal to unity. The
discernible status of a set changes depending on the value of β. In general, the
following properties hold:
Proposition 2.15 If X is discernible on the classification error level 0 ≤ β < 0.5,

then X is also discernible at any level β1 > β.
Proposition 2.16 If R 0.5 X = R 0.5 X , then X is not discernible on every classifica-

tion error level 0 ≤ β < 0.5.
Proposition 2.16 emphasizes that a set with a non-empty absolute boundary can
never be discerned. In general, one can easily demonstrate the following:
Proposition 2.17 If X is not discernible on the classification error level 0 ≤ β <

0.5, then X is also not discernible at any level β1 < β.
Any set X which is not discernible for every β is called indiscernible or abso-
lutely rough. The set X is absolutely rough iff B N R0.5 X = ∅. Any set which is not
absolutely rough will be referred to as relatively rough or weakly discernible.
For each relatively rough set X , there exists such a classification error level β that
X is discernible on this level.
Let N D I S(R, X ) = {0 ≤ β < 0.5 : B N Rβ (X ) = ∅}. Then, N D I S(R, X ) is a
range of all those β values for which X is indiscernible.
The least value of classification error β which makes X discernible will be referred
to as discernibility threshold. The value of the threshold is equal to the least upper
bound ζ (R, X ) of N D I S(X ), i.e.,
ζ (R, X ) = sup N D I S(R, X ).
Proposition 2.18 states a simple property which can be used to find the discerni-
bility threshold of a weakly discernible set X :
Proposition 2.18 ζ (R, X ) = max(m1, m2), where

18 2 Preliminaries
m 1 = 1 − min{c(E, X ) : E ∈ R ∗ and 0.5 < c(E, X )},

m2 = max{c(E, X ) : E ∈ R ∗ and c(E, X ) < 0.5}.
The discernibility threshold of the set X equals a minimal classification error β

which can be allowed to make this set β-discernible. We give some fundamental
properties of β-approximations.
Proposition 2.19 For every 0 ≤ β < 0.5, the following hold:
(1a) X ⊇ Rβ X
β
(1b) Rβ X ⊇ Rβ X
(2) R β ∅ = R β ∅ = ∅; R β U = R β U = U
(3) R β (X ∪ Y ) ⊇ R β X ∪ R β Y
(4) R β X ∩ R β Y ⊇ R β (X ∩ Y )
(5) R β (X ∪ Y ) ⊇ R β ∪ R β Y
(6) R β X ∩ R β Y ⊇ R β (X ∩ Y )
(7) R β (−X ) = −R β (X )
(8) R β (−X ) = −R β (X )
We finish the outline of variable precision rough set model, which can be regarded
as a direct generalization of the original rough set model. Consult Ziarko [34] for
more details. As we will be discussed later, it plays an important role in our approach
to rough set based reasoning.
Shen and Wang [32] proposed the VPRS model over two universes using inclusion
degree. They introduced the concepts of the reverse lower and upper approximation
operators and studied their properties. They introduced the approximation operators
with two parameters as a generalization of the VPRS-model over two universes.
2.1.3 Decision Logic
Pawlak developed decision logic (DL) for reasoning about knowledge. His main
goal is reasoning about knowledge concerning reality. Knowledge is represented as
a value-attribute table, called knowledge representation system.
There are several advantages to represent knowledge in tabular form. The data
table can be interpreted differently, namely it can be formalized as a logical system.
The idea leads to decision logic.
We review the foundations of rough set-based decision logic [25, 26]. Let S =
(U, A) be a knowledge representation system. The language of DL consists of atomic
formulas, which are attribute-value pairs, combined with logical connectives to form
compound formulas. The alphabet of the language consists of:
1. A: the set of attribute constants

2. V = Va : the set of attribute constants a ∈ A
3. Set {∼, ∨, ∧, →, ≡} of propositional connectives, called negation, dis-

junction, conjunction, implication and equivalence, respectively.
The set of formulas in DL-language is the least set satisfying the following con-
ditions:
1. Expressions of the form (a, v), or in short av , called atomic formulas, are
formulas of DL-language for any a ∈ A and v ∈ Va .
2. If φ and ψ are formulas of DL-language, then so are ∼, φ, (φ ∨ ψ), (φ ∧
ψ), (φ → ψ) and (φ ≡ ψ).
Formulas are used as descriptions of objects of the universe. In particular, atomic

formula of the form (a, v) is interpreted as a description of all objects having value
v for attribute a.
The semantics for DL is given by a model. For DL, the model is KR-system
S = (U, A), which describes the meaning of symbols of predicates (a, v) in U ,
and if we properly interpret formulas in the model, then each formula becomes a
meaningful sentence, expressing properties of some objects.
An object x ∈ U satisfies a formula φ in S = (U, A), denoted x |= S φ or in shorts
x |= φ, iff the following conditions are satisfied:
1. x |= (a, v) iff a(x) = v

2. x |=∼ φ iff x |= φ
3. x |= φ ∨ ψ iff x |= φ or x |= ψ
4. x |= φ ∧ ψ iff x |= φ and x |= ψ
The following are clear from the above truth definition:
5. x |= φ → ψ iff x |=∼ φ ∨ ψ
6. x |= φ ≡ ψ iff x |= φ → ψ and x |= ψ → φ
If φ is a formula, then the set | φ | S defined as follows:

| φ |s = {x ∈ U | x |= S φ}
will be called the meaning of the formula φ in S.
Proposition 2.20 The meaning of arbitrary formulas satisfies the following:
| (a, v) | S = {x ∈ U | a(x) = v}
|∼ φ | S = − | φ | S
| φ ∨ ψ | S =| φ | S ∪ | ψ | S
| φ ∧ ψ | S =| φ | S ∩ | ψ | S
| φ → ψ |S = − | φ |S ∪ | ψ |S
| φ ≡ ψ | S = (| φ | S ∩ | ψ | S ) ∪ (− | φ | S ∩− | ψ | S )
20 2 Preliminaries
Thus, the meaning of the formula φ is the set of all objects having the property
expressed by the formula φ, or the meaning of the formula φ is the description in the
KR-language of the set objects | φ |.
A formula φ is said to be true in a KR-system S, denoted |= S φ, iff | φ | S = U ,
i.e., the formula is satisfied by all objects of the universe in the system S. Formulas
φ and ψ are equivalent in S iff | φ | S =| ψ | S .
Proposition 2.21 The following are the simple properties of the meaning of a for-
mula.
|= S φ iff | φ | = U
|= S ∼ φ iff | φ | = ∅
φ → ψ iff | ψ | ⊆ | ψ |
φ ≡ ψ iff | ψ | = | ψ |
The meaning of the formula depends on the knowledge we have about the universe,
i.e., on the knowledge representation system. In particular, a formula may be true in
one knowledge representation system, but false in another one.
However, there are formulas which are true independent of the actual values of
attributes appearing them. But, they depend only on their formal structure.
Note that in order to find the meaning of such a formula, one need not be
acquainted with the knowledge contained in any specific knowledge representation
system because their meaning is determined by its formal structure only.
Hence, if we ask whether a certain fact is true in light of our actual knowledge,
it is sufficient to use this knowledge in an appropriate way. For formulas which are
true (or not) in every possible knowledge representation system, we do not need in
any particular knowledge, but only suitable logical tools.
To deal with deduction in DL, we need suitable axioms and inference rules. Here,
axioms will correspond closely to axioms of classical propositional logic, but some
specific axioms for the specific properties of knowledge representation systems are
also needed. The only inference rule will be modus ponens.
We will use the following abbreviations:
φ∧ ∼ φ =def 0
φ∨ ∼ φ =def 1
Obviously, |= 1 and |=∼ 0. Thus, 0 and 1 can be assumed to denote falsity and
truth, respectively.
Formula of the form:
(a1 , v1 ) ∧ (a2 , v2 ) ∧ ... ∧ (an , vn )
where vi ∈ Va , P = {a1 , a2 , ..., an } and P ⊆ A is called a P-basic formula or in short
P-formula. Atomic formulas is called A-basic formula or in short basic formula.
Let P ⊆ A, φ be a P-formula and x ∈ U . If x |= φ then φ is called the P-
description of x in S. The set of all A-basic formulas satisfiable in the knowledge
representation system S = (U, A) is called the basic knowledge in S.

We write (P), or in short (P), to denote the disjunction of all P-formulas
S
satisfied in S. If P = A then (A) is called the characteristic formula of S.
The knowledge representation system can be represented by a data table. And its
columns are labelled by attributes and its rows are labelled by objects. Thus, each
row in the table is represented by a certain A-basic formula, and the whole table is
represented by the set of all such formulas. In DL, instead of tables, we can use
sentences to represent knowledge.
There are specific axioms of DL:
1. (a, v) ∧ (a, u) ≡ 0 for any a ∈ A, u, v ∈ V and v = u

2. (a, v) ≡ 1 for every a ∈ A
v∈Va
3. ∼ (a, v) ≡ a∈Va ,u=v (a, u) for every a ∈ A
The axiom (1) states that each object can have exactly one value of each attribute.
The axiom (2) assumes that each attribute must take one of the values of its domain
for every object in the system.
The axiom (3) allows us to eliminate negation in such a way that instead of saying
that an object does not possess a given property we can say that it has one of the
remaining properties.
Proposition 2.22 The following holds for DL:

|= S (P) ≡ 1 for any P ⊆ A.
S
Proposition 2.22 means that the knowledge contained in the knowledge represen-
tation system is the whole knowledge available at the present stage. and corresponds
to the so-called closed world assumption (CWA).
We say that a formula φ is derivable from a set of formulas Ω, denoted Ω φ, iff
it is derivable from axioms and formulas of Ω by finite application of modus ponens.
Formula φ is a theorem of DL, denoted φ, if it is derivable from the axioms only.
A set of formulas Ω is consistent iff the formula φ∧ ∼ φ is not derivable from Ω.
Note that the set of theorems of DL is identical with the set of theorems of
classical propositional logic with specific axioms (1)–(3), in which negation can be
eliminated.
Formulas in the K R-language can be represented in a special form called normal
form, which is similar to that in classical propositional logic.
Let P ⊆ A be a subset of attributes and let φ be a formula in K R-language. We
say that φ is in a P-normal form in S, in short in P-normal form, iff either φ is 0 or
φ is 1, or φ is a disjunction of non-empty P-basic formulas in S. (The formula φ is
non-empty if | φ |= ∅).
A-normal form will be referred to as normal form. The following is an important
property in the DL-language.
22 2 Preliminaries
Table 2.1 K R-system 1 U A B C

1 1 0 2
2 2 0 3
3 1 1 1
4 1 1 1
5 2 1 3
6 1 0 3
Proposition 2.23 Let φ be a formula in DL-languageand let P contain all attributes

occurring in φ. Moreover, (1)–(3) and the formula (A). Then, there is a formula
S
ψ in the P-normal form such that φ ≡ ψ.
Here is the example from Pawlak [26]. Consider the following K R-system.
The following a1 b0 c2 , a2 b0 c3 , a1 b1 c1 , a2 b1 c3 , a1 b0 c3 are all basic formulas (basic
knowledge) in the K R-system. For simplicity, we will omit the symbol of conjunction
∧ in basic formulas.
The characteristic formula of the system is:
a2 b0 c2 ∨ a2 b0 c3 ∨ a1 b1 c1 ∨ a2 b1 c3 ∨ a1 b0 c3
Here, we give the following meanings of some formulas in the system:
| a1 ∨ b0 c2 | = {1, 2, 4, 6}
|∼ (a2 b1 ) | = {1, 2, 3, 4, 6}
| b0 → c2 | = {1, 3, 4, 5}
| a2 ≡ b0 | = {2, 3, 4}
Below are given normal forms of formulas considered in the above example for
K R-system 1:
a1 ∨ b0 c2 = a1 b0 c2 ∨ a1 b1 c1 ∨ a1 b0 c3
∼ (a2 b1 ) = a1 b0 c2 ∨ a2 b0 c3 ∨ a1 b1 c1 ∨ a1 b0 c3
b0 → c2 = a1 b0 c2 ∨ a1 b1 c1 ∨ a2 b1 c3
a2 ≡ b0 = a2 b0 c1 ∨ a2 b0 c2 ∨ a2 b0 c3 ∨ a1 b1 c1 ∨ a1 b1 c2 ∨ a1 b1 c3
Examples of formulas in {a, b}-normal form are:
∼ (a2 b1 ) = a1 b0 ∨ a2 b0 ∨ a1 b1 ∨ a1 b0
a1 ≡ b0 = a2 b0 ∨ a1 b1
The following is an example of a formula in {b, c}-normal form:

b0 → c2 = b0 c2 ∨ b1 c1 ∨ b1 c3
Thus, in order to compute the normal form of a formula, we have to transform by

using propositional logic and the specific axioms for a given K R-system.
Any implication φ → ψ is called a decision rule in the K R-language. φ and ψ
are referred to as the predecessor and successor of φ → ψ, respectively.
If a decision rule φ → ψ is true in S, we say that the decision rule is consistent
in S; otherwise the decision rule is inconsistent in S.
If φ → ψ is a decision rule and φ and ψ are P-basic and Q-basic formulas
respectively, then the decision rule φ → ψ is called a P Q-basic decision rule (in
short P Q-rule).
A P Q-rule φ → ψ is admissible in S if φ ∧ ψ is satisfiable in S.
Proposition 2.24 A P Q-rule is true (consistent) in S iff all {P, Q}-basic formulas
which occur in the {P, Q}-normal form of the predecessor of the rule, also occur in
{P, Q}-normal form of the successor of the rule; otherwise the rule is false (incon-
sistent).
The rule b0 → c2 is false in the above table for K R-system 1, since the {b, c}-
normal form of b0 is b0 c2 ∨ b0 c3 , {b, c}-normal form of c2 is b0 c2 , and the formula
b0 c3 does not occur in the successor of the rule.
On the other hand, the rule a2 → c3 is true in the table, because the {a, c}-normal
form of a2 is a2 c3 , whereas the {a, c}-normal form of c3 is a2 c3 ∨ a1 c3 .
Any finite set of decision rules in a DL-language is referred to as a decision
algorithm in the DL-language. If all decision rules in a basic decision algorithm are
P Q-decision rules, then the algorithm is said to be P Q-decision algorithm, or in
short P Q-algorithm, and will be denoted by (P, Q).
A P Q-algorithm is admissible in S, if the algorithm is the set of all P Q-rules
admissible in S.
A P Q-algorithm is complete in S, iff for every x ∈ U there exists a P Q-decision
rule φ → ψ in the algorithm such that x |= φ ∧ ψ in S; otherwise the algorithm is
incomplete in S.
A P Q-algorithm is consistent in S iff all its decision rules are consistent (true) in
S; otherwise the algorithm is inconsistent.
Sometimes consistency (inconsistency) may be interpreted as determinism (inde-
terminism).
Given a K R-system, any two arbitrary, non-empty subset of attributes P, Q in
the system determines uniquely a P Q-decision algorithm.
Consider the following K R-system from Pawlak [26].
Assume that P = {a, b, c} and Q = {d, c} are condition and decision attributes,
respectively. Set P and Q uniquely associate the following P Q-decision algorithm
with the table.
a1 b0 c2 → d1 e1
a2 b1 c0 → d1 e0
a2 b2 c2 → d0 e2
a1 b2 c2 → d1 e1
24 2 Preliminaries
Table 2.2 K R-system 2 U A B C D E

1 1 0 2 1 1
2 2 1 0 1 0
3 2 1 2 0 2
4 1 2 2 1 1
5 1 2 0 0 2
a1 b2 c0 → d0 e2
If assume that R = {a, b} and T = {c, d} are condition and decision attributes,
respectively, then the RT -algorithm determined by Table 2.2 is the following:
a1 b0 → c2 d1
a2 b1 → c0 d1
a2 b1 → c2 d0
a1 b2 → c2 d1
a1 b2 → c0 d0
Of course, both algorithms are admissible and complete.
In order to check whether or not a decision algorithm is consistent, we have to
check whether all its decision rules are true. The following proposition gives a much
simpler method to solve this problem.
Proposition 2.25 A P Q-decision rule φ → ψ in a P Q-decision algorithm is con-

sistent (true) in S iff for any P Q-decision rule φ → ψ in P Q-decision algorithm,
φ = φ implies ψ = ψ .
In Proposition 2.25, order of terms is important, since we require equality of

expressions. Note also that in order to check whether or not a decision rule φ → ψ
is true we have to show that the predecessor of the rule (the formula φ) discerns the
decision class ψ from the remaining decision classes of the decision algorithm in
question. Thus, the concept of truth is somehow replaced by the concept of indis-
cernibility.
Consider the K R-system 2 again. With P = {a, b, c} and Q as condition and
decision attributes. Let us check whether the P Q-algorithm:
a1 b0 c2 → d1 e1
a2 b1 c0 → d1 e0
a2 b2 c2 → d0 e2
a1 b2 c2 → d1 e1
a1 b2 c0 → d0 e2
is consistent or not. Because the predecessors of all decision rules in the algorithm
are different (i.e., all decision rules are discernible by predecessors of all decision
Table 2.3 K R-system 2

U A B C D E
1 1 0 2 1 1
4 1 2 2 1 1
2 2 1 0 1 0
3 2 1 2 0 2
5 1 2 0 0 2
rules in the algorithm), all decision rules in the algorithm are consistent (true) and
consequently the algorithm is consistent.
This can also be seen directly from Table 2.3.
The RT -algorithm, where R = {a, b} and T {c, d}
a1 b0 → c2 d1
a2 b1 → c0 d1
a2 b1 → c2 d0
a1 b2 → c2 d1
a1 b2 → c0 d0
is inconsistent because the rules
a2 b1 → c0 d1
a2 b1 → c2 d0
have the same predecessors and different successors, i.e., we are unable to discern
c0 d1 and c2 d0 by means of condition a2 b1 . Thus, both rules are inconsistent (false)
in the K R-system. Similarly, the rules
a1 b2 → c2 d1
a1 b2 → c0 d0
are also inconsistent (false).

We turn to dependency of attributes. Formally, the dependency is defined as below.
Let K = (U, R) be a knowledge base and P, Q ⊆ R.
(1) Knowledge Q depends on knowledge P iff I N D(P) ⊆ I N D(Q).

(2) Knowledge P and Q are equivalent, denoted P ≡ Q, if P ⇒ Q and Q ⇒ P.
(3) Knowledge P and Q are independent, denoted P ≡ Q, iff neither P ⇒ Q
nor Q ⇒ P hold.
Obviously, P ≡ Q iff I N D(P) ≡ I N D(Q).

The dependency can be interpreted in different ways as Proposition 2.26 indicates:
26 2 Preliminaries
Proposition 2.26 The following conditions are equivalent:

(1) P ⇒ Q
(2) I N D(P ∪ Q) = I N S(P)
(3) P O SP (Q) = U
(4) PX for all X ∈ U/Q
where PX denotes I N D(P)/ X .
By Proposition 2.27, we can see the following: if Q depends on P then knowledge

Q is superfluous within the knowledge base in the sense that the knowledge P ∪ Q
and P provide the same characterization of objects.
Proposition 2.27 If P is a reduct of Q, then P ⇒ Q − P and I N D(P) = I N D(Q).
Proposition 2.28 The following hold.

(1) If P is dependent, then there exists a subset Q ⊂ P such that Q is a reduct
of P.
(2) If P ⊆ Q and P is independent, then all basic relations in P are pairwise
independent.
(3) If P ⊆ Q and P is independent, then every subset R of P is independent.

(1) If P ⇒ Q and P ⊃ P, then P ⇒ Q.
(2) If P ⇒ Q and Q ⊂ Q, then P ⇒ Q .
(3) P ⇒ Q and Q ⇒ R imply P ⇒ R.
(4) P ⇒ R and Q ⇒ R imply P ∪ Q ⇒ R.
(5) P ⇒ R ∪ Q imply P ⇒ R and P ∪ Q ⇒ R.
(6) P ⇒ Q and Q ∪ R ⇒ T imply P ∪ R ⇒ T
(7) P ⇒ Q and R ⇒ T imply P ∪ R ⇒ Q ∪ T.
The derivation (dependency) can be partial, which means that only part of knowl-
edge Q is derivable from knowledge P. We can define the partial derivability using
the notion of the positive region of knowledge.
Let K = (U, R) be the knowledge base and P, Q ⊂ R. Knowledge Q depends in
a degree k (0 ≤ k ≤ 1) from knowledge P, in symbol P ⇒k Q, iff
car d(P O SP (Q))
k = γP (Q) =
car d(U )
where car d denotes cardinality of the set.
If k = 1, we say that Q totally depends from P; if 0 < k < 1, we say that Q roughly
(partially) depends from P, and if k = 1 we say that Q is totally independent from
P, If P ⇒1 Q, we shall also write P ⇒ Q.
The above ideas can also be interpreted as an ability to classify objects. More
precisely, if k = 1, then all elements of the universe can be classified to elementary
categories of U/Q by using knowledge P.
Thus, the coefficient γP (Q) can be understood as a degree of dependency between,
Q and P. In other words, if we restrict the set of objects in the knowledge base to
the set P O SP (Q), we would obtain the knowledge base in which P ⇒ Q is a total
dependency.
The measure k of dependency P ⇒k Q does not capture how this partial depen-
dency is actually distributed among classes of U/Q. For example, some decision
classes can be fully characterized by P, whereas others may be characterised only
partially.
We will also need a coefficient γ (X ) = car d(PX )/car d(X ) where X ∈ U/Q
which shows how many elements of each class of U/Q can be classified by employing
knowledge P.
Thus, the two numbers γ (Q) and γ (X ), x ∈ U/Q give us full information about
“classification power” of the knowledge P with respect to the classification U/Q.
(1) If R ⇒k P and Q ⇒l P, then R ∪ Q ⇒ P, for some m ≥ max(k, l).

(2) If R ∪ P ⇒k Q then R ⇒l Q and P ⇒m Q, for some l, m, ≤ k.
(3) If R ⇒k Q and R ⇒l P, then R ⇒m Q ∪ P, for some m ≤ max(k, l).
(4) If R ⇒k Q ∪ P, then R ⇒l Q and R ⇒m P, for some l, m ≥ k.
(5) If R ⇒k P and P ⇒l Q, then R ⇒m Q, for some m ≥ l + k − 1.
Here, we return to the decision algorithm for dependency. We say that the set of
attributes Q depends totally, (or in short depends) on the set of attributes P in S,
if there exists a consistent P Q-algorithm in S. If Q depends on P in S, we write
P ⇒ S S, or in short P ⇒ Q.
We can also define partial dependency of attributes. We say that the set of attributes
Q depends partially on the set of attributes P in S, if there exists an inconsistent
P Q-algorithm in S.
The degree of dependency between attributes can be defined. Let (P, Q) be a P Q-
algorithm in S. By a positive region of the algorithm (P, Q), denoted P O S(P, Q),
we mean the set of all consistent (true) P Q-rules in the algorithm.
The positive region of the decision algorithm (P, Q) is the consistent part (possi-
bly empty) of the inconsistent algorithm. Obviously, a P Q-algorithm is inconsistent
iff P O S(P, Q) = (P, Q) or what is the same car d(P O S(P, Q)) = car d(P, Q).
With every P Q-decision algorithm, we can associate a number
k = car d(P O S(P, Q))/car d(P, Q), called the degree of consistency, of the algo-
rithm, or in short the degree of the algorithm, and we say that the P Q-algorithm has
the degree (of consistency) k.
Obviously, 0 ≤ k ≤ 1. If a P Q-algorithm has degree k, we can say that the set of
attributes Q depend in degree k on the set of attributes P, denoted P ⇒k Q.
28 2 Preliminaries
Naturally, the algorithm is consistent iff k = 1; otherwise, i.e., if k = 1, the algo-

rithm. All these concepts are the same as in those discussed above. Note that in
the consistent algorithm all decisions are uniquely determined by conditions in the
decision algorithm.
In other words, this means that all decisions in a consistent algorithm are dis-
cernible by means of conditions available in the decision algorithm.
Decision logic provides a simple means for reasoning about knowledge only by
using propositional logic, and is suitable to some applications. Note here that the
so-called decision table can serve as a K R-system.
However, the usability of decision logic seems to be restrictive. In other words,
it is far from a general system for reasoning in general. In this book, we will lay
general frameworks for reasoning based on rough set theory.
2.2 Decision Tables
Decision tables can be seen as a special, important class of knowledge represen-

tation systems, and can be used for applications. Let K = (U, A) be a knowledge
representation system (KR-system) and C, D ⊂ A be two subsets of attributes, called
condition and decision attributes, respectively.
2.2.1 Definitions
KR-system with distinguished condition and decision attributes is called a decision

table, denoted T = (U, A, C, D) or in short DC. Equivalence classes of the relations
I N D(C) and I N D(D) are called condition and decision classes, respectively.
With every x ∈ U , we associate a function dx : A → V , such that dx (a) = a(x)
for every a ∈ C ∪ D; the function dx is called a decision rule (in T ), and x is referred
as a label of the decision rule dx .
Note that elements of the set U in a decision table do not represent in general any
real objects, but are simple identifiers of decision rules.
If dx is a decision rule, then the restriction of dx to C, denoted dx | C, and the
restriction of dx to D, denoted dx | D are called conditions and decisions (actions)
of dx , respectively.
The decision rule dx is consistent (in T ) if for every y = x, dx | C = d y | C
implies dx | D = d y | D; otherwise the decision rule is inconsistent.
A decision table is consistent if all its decision rules are consistent; otherwise
the decision table is inconsistent. Consistency (inconsistency) sometimes may be
interpreted as determinism (non-determinism).
Proposition 2.31 A decision table T = (U, A, C, D) is consistent iff C ⇒ D.

2.2 Decision Tables 29
Table 2.4 Decision Table 2.1 U A B C D E

1 1 0 2 2 0
2 0 1 1 1 2
3 2 0 0 1 1
4 1 1 0 2 2
5 1 0 2 0 1
6 2 2 0 1 1
7 2 1 1 1 2
8 0 1 1 0 1
From Proposition 2.31, it follows that the practical method of checking consis-
tency of a decision table is by simply computing the degree of dependency between
condition and decision attributes. If the degree of dependency equals to 1, then we
conclude that the table is consistent; otherwise it is inconsistent.
Proposition 2.32 Each decision table T = (U, A, C, D) can be uniquely decom-

posed into two decision tables T1 = (U, A, C, D) and T2 = (U, A, C, D) such
that C ⇒1 D in T1 and C ⇒0 D in T2 such that U1 = P O SC (D) and U2 =
B NC (X ).
X ∈U/I N D(D)
Proposition 2.32 states that we can decompose the table into two subtables; one
totally inconsistent with dependency coefficient equal to 0, and the second entirely
consistent with the dependency equal to 1. This decomposition however is possible
only if the degree of dependency is greater than 0 and different from 1.
Consider Table 2.4 from Pawlak [26].
Assume that a, b and c are condition attributes, and d and e are decision attributes.
In this table, for instance, the decision rule 1 is inconsistent, whereas the decision
rule 3 is consistent. By Proposition 2.32, we can decompose Decision Table 2.1 into
the following two tables:
Table 2.2 is consistent, whereas Table 2.3 is totally inconsistent, which means
all decision rules in Table 2.2 are consistent, and in Table 2.3 all decision rules are
inconsistent.
Simplification of decision tables is very important in many applications, e.g.
software engineering. An example of simplification is the reduction of condition
attributes in a decision table.
In the reduced decision table, the same decisions can be based on a smaller number
of conditions. This kind of simplification eliminates the need for checking unneces-
sary conditions.
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We also find philosophers now-a-days calmly discussing a
question which most people considered settled very long ago,
namely, whether blue and yellow together really make green.
It is of no use for the artist to lift up his eyes with astonishment at
any one being so insane as to question so generally admitted a
statement. In vain does he point to his pictures, in which his greens
have been actually so produced. The strict photologist at once puts
him down, by informing him that he knows little or nothing of the real
state of the case: his (the artist’s) colours are negative, or hues of
more or less complete darkness; whereas in nature, the colour
question is to be decided by positive colours, or hues in which all the
light used is of one kind. The meaning of this will be best understood
by an example: When a ray of white light falls on a green leaf, part of
the ray is absorbed and part reflected, and the object is therefore
only seen with the part that is reflected. That which is absorbed
consists of some of each of the colour rays, and the resulting
reflected light is nothing more than a mixture of what remains after
this partial absorption. The green we see consists of the original
white light deprived of a portion of its rays. It is not a pure and
absolute green, but only a residual group of coloured rays, and thus
in so far the green colour is negative, or consists of rays not
absorbed. It is therefore partial darkness, and not absolute light. If,
however, on the other hand, a ray of white light is passed through a
transparent medium (e. g. some chemical salt) which has the
property of entirely absorbing all but one or more of the colour rays,
and no part of the remainder, then all the light that passes through
this medium is of the one colour, or a mixture of the several colours
that pass: and if such light is thrown on a white ground, the reflected
colour will be positive, and not negative, and is far purer as well as
brighter than the colour obtained in the other way. It has been found
by actual experiment, that when positive blue, thus obtained, is
thrown on positive yellow, the resulting reflected colour bears no
resemblance to green. Sir John Herschel considers, that whether
green is a primitive colour—in other words, whether we really have
three or four primitive colours—remains yet an open question.
It was necessary to explain these matters about colour before
directly referring to the subject of this paper, namely, blindness to
certain colour rays. It should also be clearly understood that the
persons subject to this peculiar condition of vision have not
necessarily any mechanical or optical defect in the eye as an optical
instrument, which may be strong or weak, long-sighted or short-
sighted, quite independently of it. Colour blindness does not in any
way interfere with the ordinary requirements of vision, nor is there
the smallest reason to imagine that it can get worse by neglect, or
admit of any improvement by education or treatment.
Assuming that persons of ordinary vision see three simple colours,
red, yellow, and blue, and that all the rest of the colours are mixtures
of these with each other and with white light, let us try to picture to
ourselves what must be the visual condition of a person who is
unable to recognize certain rays; and as it appears that there is but
one kind of colour-blindness known, we will assume that the person
is unable to recognize those rays of white light which consist of pure
red and nothing else. In other words, let us investigate the
sensations of a person blind so far only as pure red is concerned.
All visible objects either reflect the same kind of light as that which
falls on them, absorbing part and reflecting the rest, or else they
absorb more of some colour rays than others, and reflect only a
negative tint, made up of a mixture of all the colour-rays not
absorbed. To a colour-blind person, the mixed light, as it proceeds
from the sun, is probably white, as seen by those having perfect
vision; for, as we have explained already, positive blue and yellow
(the colour rays when red is excluded) do not make green, and the
absence of the red ray is likely to produce only a slight darkening
effect. So far, then, there is no difference. But how must it be with
regard to colour.
Bearing in mind what has been said above, it is evident that in
withdrawing the red rays from the spectrum, we affect all the colours.
The orange is no longer red and yellow, but darkened yellow; the
yellow is purer, the green is quite distinct, the blue purer, and the
indigo and violet no longer red and blue, but blue mingled with more
or less of darkness, the violet being the darkest, as containing least
blue in proportion to red, while the red part itself, though not seen as
a colour, is not absolutely black, inasmuch as its part of the spectrum
is faintly coloured with the few mixed rays of blue and yellow and
white that escape from their proper place. The red then ought to be
seen as a gray neutral tint, the orange a dingy yellow, the indigo a
dirty indigo, and the violet a sickly, disagreeable tint of pale blue,
darkened considerably with black and gray.
Next let us take the case of an intelligent person affected with
colour blindness, but who is not yet aware of the fact. He has been
taught from childhood that certain shades, some darker and some
brighter, but all of neutral tint, and not really presenting to him colour
at all, are to be called by various names—scarlet, crimson, pale red,
dark red, bright red, dark green, dark purple, brown, and others. With
all these he can only associate an idea of gray; nor can he possibly
know that any one else sees more than he does. Having been taught
the names they are called by, he remembers the names, with more
or less accuracy, and thus passes muster. There is a real difference
of tint, because each of these colours consists of more or less blue,
yellow, and white, mixed with the red; and our friend is enabled to
recognize and name them, more or less correctly, according to his
acuteness of perception and accuracy of memory.
If we desire to experiment on such a person, we must ask no
names whatever, but simply place before him a number of similar
objects differently coloured. Taking, for example, skeins of coloured
wools, let us select a complete series of shades of tint, from red,
through yellow, green, and blue, to violet, and request him to arrange
them as well as he is able, placing the darkest shades first, and
putting those tints together that are most like each other. It is curious
then to watch the progress of the arrangement. In a case lately tried
by the writer of this article, the colour-blind person first threw aside at
once a particular shade of pale green as undoubted white, and then
several dark blues, dark reds, dark greens, and browns, were put
together as black. The yellows and pure blues were placed correctly,
as far as name was concerned, by arranging several shades in order
of brightness—but the order was very different from that which
another person would have selected. The greens were grouped,
some with yellows, and some with blues.
The colours in this experiment were all negative and impure, but
we may also obtain something like the same result with positive
colour, transmitted by the aid of polarized light through plates of
mica. In a case of this kind described by Sir J. Herschel, the only
colours seen were blue and yellow, while pale pinks and greens
were regarded as cloudy white, fine pink as very pale blue, and
crimson as blue; white red, ruddy pink, and brick red were all
yellows, and fine pink blue, with much yellow. Dark shades of red,
blue, or brown, were considered as merely dark, no colour being
recognized.
The account of Dr. Dalton’s own peculiarity of vision by himself,
offers considerable interest. He says, speaking of flowers: “With
respect to colours that were white, yellow, or green, I readily
assented to the appropriate term; blue, purple, pink, and crimson
appeared rather less distinguishable, being, according to my idea, all
referable to blue. I have often seriously asked a person whether a
flower was blue or pink, but was generally considered to be in jest.”
He goes on further to say, as the result of his experience: “1st. In the
solar spectrum three colours appear, yellow, blue, and purple. The
two former make a contrast; the two latter seem to differ more in
degree than in kind. 2nd. Pink appears by daylight to be sky-blue a
little faded; by candlelight it assumes an orange or yellowish
appearance, which forms a strong contrast to blue. 3rd. Crimson
appears muddy blue by day, and crimson woollen yarn is much the
same as dark blue. 4th. Red and scarlet have a more vivid and
flaming appearance by candlelight than by daylight” (owing probably
to the quantity of yellow light thrown upon them).
As anecdotes concerning this curious defect of colour vision, we
may quote also the following: “All crimsons appeared to me (Dr.
Dalton) to be chiefly of dark blue, but many of them have a strong
tinge of dark brown. I have seen specimens of crimson claret and
mud which were very nearly alike. Crimson has a grave appearance,
being the reverse of every showy or splendid colour.” Again: “The
colour of a florid complexion appears to me that of a dull, opaque,
blackish blue upon a white ground. Dilute black ink upon white paper
gives a colour much resembling that of a florid complexion. It has no
resemblance to the colour of blood.” We have a detailed account of
the case of a young Swiss, who did not perceive any great difference
between the colour of the leaf and that of the ripe fruit of the cherry,
and who confounded the colour of a sea-green paper with the scarlet
of a riband placed close to it. The flower of the rose seemed to him
greenish blue, and the ash gray colour of quick-lime light green. On
a very careful comparison of polarized light by the same individual,
the blue, white, and yellow were seen correctly, but the purple, lilac,
and brown were confounded with red and blue. There was in this
case a remarkable difference noticed according to the nature and
quantity of light employed; and as the lad seemed a remarkably
favourable example of the defect, the following curious experiment
was tried. A human head was painted, and shown to the colour-blind
person, the hair and eyebrows being white, the flesh brownish, the
lips and cheeks green. When asked what he thought of this head?
the reply was, that it appeared natural, but that the hair was covered
with a nearly white cap, and the carnation of the cheeks was that of
a person heated by a long walk.
There is an interesting account in the Philosophical Transactions
for 1859 (p. 325), which well illustrates the ideas entertained by
persons in this condition with regard to their own state. The author,
Mr. W. Pole, a well-known civil engineer, thus describes his case:—“I
was about eight years old when the mistaking of a piece of red cloth
for a green leaf betrayed the existence of some peculiarity in my
ideas of colour; and as I grew older, continued errors of a similar kind
led my friends to suspect that my eyesight was defective; but I
myself could not comprehend this, insisting that I saw colours clearly
enough, and only mistook their names.
“I was articled to a civil engineer, and had to go through many
years’ practice in making drawings of the kind connected with this
profession. These are frequently coloured, and I recollect often being
obliged to ask in copying a drawing what colours I ought to use; but
these difficulties left no permanent impression, and up to a mature
age I had no suspicion that my vision was different from that of other
people. I frequently made mistakes, and noticed many
circumstances in regard to colours, which temporarily perplexed me.
I recollect, in particular, having wondered why the beautiful rose light
of sunset on the Alps, which threw my friends into raptures, seemed
all a delusion to me. I still, however, adhered to my first opinion, that
I was only at fault in regard to the names of colours, and not as to
the ideas of them; and this opinion was strengthened by observing
that the persons who were attempting to point out my mistakes, often
disputed among themselves as to what certain hues of colour ought
to be called.” Mr. Pole adds that he was nearly thirty years of age
when a glaring blunder obliged him to investigate his case closely,
and led to the conclusion that he was really colour-blind.
All colour-blind persons do not seem to make exactly the same
mistakes, or see colours in the same way; and there are, no doubt,
many minor defects in appreciating, remembering, or comparing
colours which are sufficiently common, and which may be
superadded to the true defect—that of the optic nerve being
insensible to the stimulus of pure red light. It has been asserted by
Dr. Wilson, the author of an elaborate work on the subject, that as
large a proportion as one person in every eighteen is colour-blind in
some marked degree, and that one in every fifty-five confounds red
with green. Certainly the number is large, for every inquiry brings out
several cases; but, as Sir John Herschel remarks, were the average
anything like this, it seems inconceivable that the existence of the
defect should not be one of vulgar notoriety, or that it should strike
almost all uneducated persons, when told of it, as something
approaching to absurdity. He also remarks, that if one soldier out of
every fifty-five was unable to distinguish a scarlet coat from green
grass, the result would involve grave inconveniences that must have
attracted notice. Perhaps the fact that a difference of tint is
recognized, although the eye of the colour-blind person does not
appreciate any difference of colour, when red, green, and other
colours are compared together, and that every one is educated to
call certain things by certain names, whether he understands the
true meaning of the name or not, may help to explain both the
slowness of the defective sight to discover its own peculiarity, and
the unwillingness of the person of ordinary vision to admit that his
neighbour really does not see as red what he agrees to call red.
There is, however, another consideration that this curious subject
leads to. It is known that out of every 10,000 rays issuing from the
sun, and penetrating space at the calculated rate of 200,000 miles in
each second of time, about one-fifth part is altogether lost and
absorbed in passing through the atmosphere, and never reaches the
outer envelope of the human eye. It is also known that of the rays
that proceed from the sun, some produce light, some heat, and
some a peculiar kind of chemical action to which the marvels of
photography are due. Of these only the light rays are appreciated
specially by the eye, although the others are certainly quite as
important in preserving life and carrying on the business of the world.
Who can tell whether, in addition to the rays of coloured light that
together form a beam of white light, four-fifths of which only pass
through the atmosphere, there may not have emanated from the sun
other rays altogether absorbed and lost? or whether in entering the
human eye, or being received on the retina at the back of the eye, or
made sensitive by the optic nerve, there may not have been losses
and absorptions sufficient to shut out from us, who enjoy what we
call perfect vision, some other sources of information. How, in a
word, do we who see clearly only three or four colours, and their
various combinations, together with their combined white light—how
do we know that to beings otherwise organized, the heat, or
chemical rays, or others we are not aware of, may not give distinct
optical impressions? We may meet one person whose sense of
hearing is sufficiently acute to enable him to hear plainly the shrill
night-cry of the bat, often totally inaudible, while his friend and daily
companion cannot perhaps distinguish the noise of the grasshopper,
or the croaking of frogs, and yet neither of these differs sufficiently
from the generality of mankind to attract attention, and both may
pass through life without finding out their differences in organization,
or knowing that the sense of hearing of either is peculiar. So
undoubtedly it is with light. There may be some endowed with visual
powers extraordinarily acute, seeing clearly what is generally
altogether invisible; and this may have reference to light generally, or
to any of the various parts of which a complete sunbeam is
composed. Such persons may habitually see what few others ever
see, and yet be altogether unaware of their powers, as the rest of the
world would be of their own deficiency.
The case of the colour-blind person is the converse. He sees, it is
true, no green in the fields, or on the trees, no shade of pink mantling
in the countenance, no brilliant scarlet in the geranium flower, but still
he talks of these things as if he saw them, and he believes he does
see them, until by a long process of investigation he finds out that
the idea he receives from them is very different from that received by
his fellows. He often, however, lives on for years, and many have
certainly lived out their lives without guessing at their deficiency.
These results of physical defects of certain kinds remaining totally
unknown, either to the subject of them or his friends, even when all
are educated and intelligent, are certainly very curious; but it will
readily be seen that they are inevitable in the present development
of our faculties. In almost everything, whether moral or intellectual,
we measure our fellows by our own standard. He whose faculties are
powerful, and whose intellect is clear, looks over the cloud that
hovers over lower natures, and wonders why they, too, will not see
truth and right as he sees them. Those, on the other hand, who dwell
below among the mists of error and the trammels of prejudice, will
not believe that their neighbour, intellectually loftier, sees clearly over
the fog and malaria of their daily atmosphere.
In taking leave of the question of colour blindness, it should be
mentioned that hitherto no case has been recorded in which this
defect extends to any other ray than the red.
There seems no reason for this, and possibly, if they were looked
for, cases might be found in which the insensibility of the optic nerve
had reference to the blue instead of the red ray—the least instead of
the most refrangible part of the beam of light. It would also be well
worth the trial if those who have any reason to suppose that they
enjoy a superiority of vision would determine by actual experiment
the extent of their unusual powers, and learn whether they refer to
an optical appreciation of the chemical or heat rays, or show any
modification of the solar spectrum by enlargement or otherwise.
Lastly, it would be well, when children show an unusual difficulty in
describing colours, to try by some such experiments as those here
related whether any defect of colour blindness exists or not. It would
clearly be undesirable that such children as have this defect should
waste time in learning accomplishments or professions which they
must always be unable to practise. They, their parents and teachers,
may thus be saved some of that disappointment which is always
experienced when presumed tastes and talents are cultivated or
forced contrary to the natural powers of the individual. It must clearly
be hopeless to endeavour to obtain good taste in colours, when most
of the colours themselves are not seen at all, or are so recognized
as to present appearances altogether different from those seen by
the rest of the world.
Spring.
Here, where the tall plantation firs
Slope to the river, down the hill,
Strange impulses—like vernal stirs—
Have made me wander at their will.
I see, with half-attentive eyes,

The buds and flowers that mark the Spring,
And Nature’s myriad prophecies
Of what the Summer suns will bring.
For every sense I find delight—

The new-wed cushat’s murmurous tones,
Young blossoms bursting into light,
And the rich odour of the cones.
The larch, with tassels purple-pink,

Whispers like distant falling brooks;
And sun-forgotten dewdrops wink
Amid the grass, in shady nooks.
The breeze, that hangs round every bush,

Steals sweetness from the tender shoots,
With, here and there, a perfumed gush
From violets among the roots.
See—where behind the ivied rock

Grow drifts of white anemonies,
As if the Spring—in Winter’s mock—
Were mimicking his snows with these.
The single bloom yon furzes bear

Gleams like the fiery planet Mars:—
The creamy primroses appear
In galaxies of vernal stars;—
And, grouped in Pleiad clusters round,

Lent-lilies blow—some six or seven;—
With blossom-constellations crown’d,
This quiet nook resembles Heaven.
Thomas Hood.
Inside Canton.
The mere notion that I was in possession of a room inside Canton
—with freedom to wander through every quarter of that hitherto
mysterious city, of which former travellers had only conveyed a
notion from glances taken from the White Cloud Mountain, revealing
nothing but an expanse of tiles and trees, with a pagoda-top or two,
and a few mandarin flag-poles—was sufficient to banish anything
like sleep. And apart from this constant wondering at perpetually
finding myself where I was—the sharp “tung” of the mosquitoes
before settling down for their gory banquet, the calls of the French
and English bugles answering each other from the five-storied
pagoda to the joss-house barracks, the terribly breathless
atmosphere, and the grim, gigantic Chinese gods, who sat in the
moonlight like pantomime ogres round my chamber, were quite
enough to have kept one awake, and would have done so even if a
genius had descended to read a paper on Art, which they might have
discussed with him afterwards.
At last the quickly-rising tropical sun fired a ray like a shell into my
eyes through a broken pane in the mother-of-pearl window of my
joss-haunted room. This drove me out of bed, or, rather, off my
matting, as quickly as though a real shrapnell had hissed its intention
of immediately exploding beneath me. For this fearful sun of a
Canton summer falls in red-hot death upon the European whose
brain it can reach. Our soldiers were struck down before it in the
White Cloud expedition as though a crane had dropped a woolsack
on their heads.
We have all of us, at some time or another, said, “I never felt so
hot in my life!” This has been less with relation to actual caloric than
to a sudden flush of awkwardness attendant upon having asked
people after their dead relations, or uncomfortable family affairs; or in
expectation of some accidental and unintentional revelation of a
circumstance in our own lives, of which we were not remarkably
proud. Or, more especially, on being introduced by a gushing man to
an enemy you had long since cut, with the assurance that you ought
both to know each other. But I find this morning that I feel hotter still.
The wind blows against me as from the door of a glasshouse; and
the sun comes straight down like a red-hot nail, even through my
double umbrella (which I am careful to put up before I venture out on
the terrace), and my light but thick pith hat. At such times your claret
is self-mulled, and butter becomes thick oil. You cannot find a cool
place on your hard-stuffed pillow. The sun apparently twists its rays
—sends them round corners, and through venetians, and under
porticos; the light being so vivid that its mere reflection banishes
shade. The swinging punkah—which A-wa, whose picture you have
seen on cheap grocers’ tea-papers, pulls night and day, awake and
asleep, as though he were a slightly vitalized lever-escapement—
this flounced and flirting terror of all bilious people gets up a delusive
breeze, and when it stops the heat comes rushing back with double
force. Everything you wear clings to you; or, if flannel, fetches out the
“prickly heat” until you are beside yourself. In every draught, one
side is chilled whilst the other is burned, as happens at the fireplace
of an old country house, where one side is roasted, whilst on the
other you are nearly blown up the chimney. And when you are
actually out and about, you appear to live and move in the focus of
one large burning-glass. It is a dead thick heat, that you fancy might
be cut into blocks, and stored in Arctic ships for gradual distribution.
The kindness of General Straubenzee had consigned me to a
Buddhist temple for my residence. It was the last costly work of Yeh,
on Magazine Hill, and was barely finished when we took the city. An
elaborate bell, yet unhung, stood sentinel at my door. I afterwards
watched its departure to be taken to England, by Captain Maguire, in
the Sanspareil, and it may now be seen in the Crystal Palace.
Magazine Hill is to Canton what Montmartre is to Paris, and is
covered with joss-houses, now all used as barracks for our men. It is
to the extreme north of the city, which it commands, as well as the
country outside, and is the only high ground within the walls, which
here come close to it. Gazing from this on the open country, one is
reminded of the view from the walls of our own Chester, near the jail,
looking over the Roodee towards the Welsh mountains. To continue
the comparison with places which may be familiar to my readers, the
look-out towards the south, comprising the entire city, is marvellously
like the eye-stretch over Lyons from the Fourvières, when the air is
too hazy to see the Alps. There is, however, one localized object—a
tall pagoda, rising high above the expanse of red roofs. One
involuntary thought of Kew Gardens brings one back, for the
moment, to home; and as this pagoda is not considered safe to
ascend—on the authority of Major Luard, who gallantly tried it—and
as it promises at some future time, if not taken down, to form a
gigantic accident (as all columns and pagodas must do one of these
days) the likeness is more perfect.
I found a sturdy little unshod pony waiting for me at the foot of the
hill, with a tidy little pigtailed boy to guide him. The pony was for sale
for seven dollars—it sounded cheap, but the expense of keep was
the great question. My little friend made a speech:—“Chin-chin! my
talkee A No. 1 Inglis, all a plopper (proper).” But I found his
vocabulary of even the scanty “Canton English” very limited. I made
out, however, that he was going to London to learn “all sort pigeon;”
and he was very much delighted at pointing out to me some
signboards over a few little shops, edging a pond, and reading:
—“Best Wash from Hong Kong,” “A No. 1, Washsoap,” &c. And
when we passed two culprits, tied together by their pigtails, and lying
full-length upon the ground, guarded by an Irishman in front of a
baraque, inscribed “Paddy-goose” (a favourite sobriquet at the dram-
shops), he roared with laughter, and said:—“Soger hab catchee two
piecey pilat, too muchee drunkee—wanchee chokee-pigeon: no
loast duck.” This interpreted expressing, with the Chinese
substitution of the l for the r, that two pirates had been captured by
the police in an extreme state of intoxication, and that they would go
to prison, where roast duck would be a novelty.
After passing over a desert of brick rubbish—the remains of
houses destroyed because they formed ambuscades from which the
lurking braves captured or shot at stragglers on the walls, I was fairly
inside Canton. Here the streets are all so exactly alike, that in
endeavouring to give a notion of one, I may describe all. The
majority appeared to vary from seven to ten feet in breadth—the
crowded Cranbourn Passage, which runs from St. Martin’s Lane to
Castle Street could be soon transformed into one, by a handful of
theatrical mechanics. The houses are two or three stories high, and
their signboards, in gaudy paint or gilding, either hang in front of
them, or are set up in stone sockets, and all at right angles to the
houses, so that, as the China character is written perpendicularly,
they can be read going up or down the street. The manner in which
they intrude on the thoroughfare braves all notices of Commissioners
and Boards. The streets are all paved with granite in large flags, and
this has acquired a peculiarly polished appearance from the absence
of all wheel and quadrupedal traffic, and the constant shuffling along
of the soft soles or naked feet of the natives. For the Cantonese do
not appear to understand the use of wheels, or beasts of burden;
everything is carried on bamboo poles by the intensely hard-working
coolie population. Where they can do it, the streets are shaded with
matting.
And now it was that all my childish associations connected with
China were on the point of realization. For in the “pigeon” of Lord
Elgin and Sir Michael Seymour—who must shake hands, and
understand how much and how honestly both are respected by all of
us—in the China Mail information that Patna opium is at 770 dollars,
Malwa dull, and for Turkey no demand; and that Bank bills are 4s.
9d.; Sycee silver, 5½ per cent. premium, and Shanghai green-tea
quotations are unchanged—in a whirl of treaties, and Peiho forts,
and conferences totally misunderstood on either side, from the
dismal ignorance of the practical Chinese language amongst our
professed Chinese students (who could translate the great
metaphysical work of Fo, but would be sadly bothered to decide a
simple police “row”);—in all this, there is nothing in common with our
old China. But here these associations crowded on us. Men ran
along with slung tea-packages, as they did on the gaily-varnished
canisters of the “Canton T Company,” in the High Street of my
boyhood. Women with their bismuthed faces peered from windows,
as they did on the fans and plates from which I formed my earliest
notions of what was then called “the Celestial Empire.” And then
came another memory, clinging to that delightful time when a belief
in the reality of everything was our principal mental characteristic,
extending even to “Bogey” in the cellar, and the dustman who threw
sand in the eyes of sleepy little boys on the staircase, and the black
dog in the passage; nay, even to that celebrated silver spade with
which the doctor dug up our little baby brother or sister from out of
the parsley-bed—when story-books had that astonishing hold on me
that, out of our town, I perfectly established the field along which
Christian ran with his fingers in his ears when his neighbours tried to
call him back. (And if ever there was a case for the parochial
authorities of a man deserting his wife and children, Christian’s was
one.) In this happy time I had associations with China, and they now
come back from one of the most charming of the attractive stories in
the Arabian Nights Entertainments. I was now looking—practically,
with my own eyes—on a Chinese town, and a group of idle boys
playing. A grave stranger of a foreign and travelled aspect was
watching them. I should not have been at all surprised if he had
recognized, in one of the urchins, the son of his dead brother—had
clothed him at a ready-made tailor’s, and then introduced him, by
lifting up a stone with a ring in it, to those wonderful nursery grounds
of Hunt and Roskill, and Phillips, and Garrard, where the dew was all
diamonds, and the wall-fruit all stones. And was it not likely that, in
this very street, the stranger might have subsequently passed when
anxious to exchange his new moderator lamps for any old argands,
or solars, or camphines that might be dust-collecting about the
house? Here again was an open space of ground, on which that
palace might have stood, which went away one night in such a hurry.
And strange to say, there was a palace here, and it did disappear
one early January morning. It belonged to that old miscreant Yeh,
and its sudden absence was owing rather to the sponging of
practical guns than the rubbing of wonderful lamps. And although I
heard nothing, both here and at Hong Kong, but of Hall of the
Calcutta, and Mr. Oliphant; Telesio’s pale ale “chop” (or boat store);
John Dent’s French cook’s chow-chow; the arrival of the Fei-man
steamer; Colonel Stevenson’s bamboo balcony on the hill: the 59th;
Sir John Bowring and Mr. Chisholm Anstey: and innumerable
“shaves:” yet my thoughts ran upon Confucius and pagodas,
nodding mandarins, chop-sticks, and the feast of lanterns, and
above all, on Aladdin.
I was to join Mr. Parkes at the yamun of the Allied Commissioners,
and go with him to pay a visit to Peh-kwei, the Governor of Canton.
This yamun had been the palace of the Tartar general, but was now
filled with English and French officials, soldiers, marines,
compradors, coolies, and Chinese rabble, attending the police
cases. We here formed a small procession, and our revolvers came
into show; for Mr. Parkes was the most unpopular man in the city
with the Cantonese. They called him “the red-bristled barbarian,” and
had let fly various jingals at him, at different times, in the streets. But
he had the courage of the——anybody you please; and the more
they annoyed him, the more he would ride them down, and bang
them back into their ambuscades. We were all on ponies or in chairs,
with the exception of our guards; and we rode so fast along the
narrow streets, and through the bustling crowds of passengers, and
almost over the wares displayed out of doors, that a fire-engine
going through the Lowther Arcade in a hurry could not have created
greater confusion. On entering the first court of Peh-kwei’s yamun,
we were saluted with guns, and standards were hoisted on the
mandarin poles. These courts are large paved areas, with a very
broad flag-path up the middle, and fine trees at the sides; they are
divided from each other by vast wooden buildings, like barns, with
Chinese roofs, and stone lions guarding them. The patient ingenuity
of the makers is shown in these animals; they have a large ball in
their mouths, which you can turn round behind the teeth, but cannot
take out; it has evidently been cut from the solid. We rode through
the centre of these barns, up the stairs, to a higher court beyond, but
our attendants filed off round the sides; and then we dismounted,
and were introduced to Peh-kwei. I had often seen him wagging his
head, and tongue, and hands, in old china-shops; but now he stood
upright, in a long, white silk peignoir: and then he and Mr. Parkes
began bowing to one another in such continuity, that they looked
wound up, and minutes elapsed before either of them would take a
seat. Then tea was brought in, and for a little time the talk was
exactly like the twaddle that passes at a morning call in England
between people who don’t care a straw about each other, never
have, and are never likely to. But Mr. Parkes began to pull some
Chinese documents from his pocket; and as I had been introduced
as “a mandarin on his travels,” Peh-kwei made a very lucky
suggestion that I should see his grounds.
This was just what I wanted—liberty to invade what would have
been deemed a privacy even by the Cantonese; but the acres of
unkept, overgrown wilderness, with its rotting pavilions, tumble-down
temples, dried-up lakes, crumbling rockwork, and broken seats and
tables, formed the spring of all the impressions I afterwards received
in and about Canton. Nothing so dreary—not even Vauxhall on a wet
Christmas Day—ever could be imagined. It was not the breakdown
of acute organic lesion, but the decay of long, long-continued
atrophy: and I formed a theory at the moment, which the appearance
of every other yamun, or temple, strengthened, that the Chinese had
for ages so jealously shut up their vaunted city, not from any terror of
the barbarians becoming acquainted with their secrets of trade,
government, or manufacture, but from a positive idea of shame that
any one should see the mouldering neglected “lions” of their
southern capital. True to the estimated value of their curios,
everything was in a state of “crackle.” Combine all you can call to
mind of dreary places—Miss Linwood’s old room in Leicester
Square, and the present aspect of the Square itself; the gaunt,
cheerless show-rooms of palaces generally: the “Moated Grange”
and “Haunted House;” the old pavilion on Monkey Island, and indeed
“pavilions” generally, from that in Hans Place to any damp ceiling-
stained summer-house, dedicated to friendship or nature, that you
know of—mix them together, and extract their essence, and then you
will not have the least idea of the general rot and ruin that is
spreading, like an ulcer, throughout Canton.
William Hogarth:
PAINTER, ENGRAVER, AND PHILOSOPHER.
Essays on the Man, the Work, and the Time.
III.—A long Ladder, and Hard to Climb.

When a cathedral chapter have received their congé d’élire—so
runs the popular and perfectly erroneous tradition—and have made
choice of a Bishop, the pastor elect simpers, blushes, and says that
really he is much obliged, but that he would rather not accept the
proffered dignity. “Nolo episcopari,” he urges in graceful deprecation.
Nobody in or out of the chapter believes in his reluctance, and
nobody now-a-days believes in the harmless legend. Thus, too,
when the Commons elect a Speaker, a tradition with little more
foundation assumes that the right honourable gentleman approaches
the foot of the Throne, hints in the most delicate manner that he, the
chosen of the Commons, is a blockhead and an impostor, declares
that he shall make but an indifferent Speaker, and seeks to be
relieved from his onerous charge. At that same moment, perhaps,
Messrs. Adams and Ede are embroidering Mr. Speaker’s gold robe;
and experienced tonsors near Lincoln’s Inn are finishing the last row
of curls on the ambrosial horse-hair which to-morrow will be a wig.
When you ask a young lady to take a little more Mayonnaise de
homard, or entreat her to oblige the company with “Entends tu les
gondoles?”—that charming Venetian barcarole—does she not
ordinarily, and up to a certain degree of pressure, refuse—say that
she would rather not, or that she has a cold? Whose health is
proposed and drunk amid repeated cheers, but he rises, and
assures the assembled guests that he is about the last person in the
world who should have been toasted; that he never felt so
embarrassed in his life—he leads at the common law bar, and on
breaches of promise is immense—and that he wants words to, &c.
&c.? At the bar mess he is known as “Talking Smith,” and at school
his comrades used to call him “Captain Jaw.” My friends, we do not
place any faith in these denials; and forthwith clap the mitre on the
Prelate’s head, bow to the Speaker, help the young lady to arrange
the music stool, and intone nine times nine with one cheer more.
It is strange—it is vexatious; but I cannot persuade the ladies and
gentlemen who peruse these papers to believe that I am not writing
the Life of William Hogarth, and that these are merely discursive
Essays on the Man, the Work, and the Time. People persist in
thinking that it is with him who is now writing a case of nolo
episcopari. Indeed it is no such thing. I should dearly wish to write
myself Biographer. “Fain would I climb, but that I fear to fall.” I told
you in the outset that this Endeavour was no Life. I disclaimed any
possession of exclusive information. I claimed a liberal benefit-of-
clergy as to names and dates. I have had no access to muniment
rooms. I have explored the contents of no charter chests. I have
disentombed no dusty records, and rescued no parish registers from
the degrading fate of serving to singe a goose. I am timorous, and
seek not to be heard as one speaking with authority. I am
anonymous, and risk no fame. But the north country won’t believe
me, and the south and the midland shake their heads incredulously
when I say this is not Hogarth’s Life, but only so much gossip about
him and his pictures and times. I say so again; and if the public won’t
be enlightened—si vult decipi—all I can add is, Decipiatur.
Now as to the exact date of the expiration of Hogarth’s
apprenticeship—when was it? I have but an impression. I cannot
speak from any certain knowledge, and assume, therefore, that the
expiry was circa 1720. Ireland opines that it was in 1718, William
having then attained his twenty-first year. The registers of the
Goldsmiths’ Company might be more explicit, or, better still, Mr.
Scott, the chamberlain of London, might enlighten us all, to a month,
and to a day. For of old the chamberlain was the official Nemesis to
the ofttimes unruly ’prentices of London. The idle, or rebellious, or
truant novice, was arraigned before this dread functionary. He had
power to relegate the offender to the carcere duro of Bridewell, there

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and overcompleteness play an essential role. The partiality and overcompleteness of

2.1 Rough Set Theory

2.1.1 Pawlak’s Rough Set Theory

Definition 2.1 A knowledge base K is a pair K = (U, R), where U is a universe of

Definition 2.2 Let R ∈ R be an equivalence relation of the knowledge base K =

R X is called the R-lower approximation and R X the R-upper approximation of X,

Definition 2.3 If K = (U, R), R ∈ R, and X ⊆ U , then the R-positive, R-negative,

The positive region P O S R (X ) (or the lower approximation) of X is the collection

objects belonging to the boundary can be classified with certainty into X or −X as

Proposition 2.1 The following hold:

Proposition 2.2 shows the basic properties of approximations:

The concept of approximations of sets can also be applied to that of membership

1. Sets X and Y are bottom R-equal (X ∼ R Y ) if R X = RY

Proposition 2.5 For any equivalence relation R:

Proposition 2.6 Rough inclusion satisfies the following:

Proposition 2.7 Rough inclusion satisfies the following:

2.1.2 Variable Precision Rough Set

where |X | represents the cardinality of the set X .

where precision β provides the limit of permissible misclassification.

Proposition 2.10 For every X ⊆ Y , the following relationship is satisfied:

The β-boundary region of X consists of all those elements of U which cannot be

Proposition 2.11 Let X be an arbitrary subset of the universe U :

α(R, β, X ) = car d(R β X )/car d(R β X ).

The β-accuracy represents the imprecision of the approximate characterization

Proposition 2.15 If X is discernible on the classification error level 0 ≤ β < 0.5,

Proposition 2.16 If R 0.5 X = R 0.5 X , then X is not discernible on every classifica-

Proposition 2.17 If X is not discernible on the classification error level 0 ≤ β <

ζ (R, X ) = sup N D I S(R, X ).

Proposition 2.18 ζ (R, X ) = max(m1, m2), where

m 1 = 1 − min{c(E, X ) : E ∈ R ∗ and 0.5 < c(E, X )},

The discernibility threshold of the set X equals a minimal classification error β

2.1.3 Decision Logic

1. A: the set of attribute constants

3. Set {∼, ∨, ∧, →, ≡} of propositional connectives, called negation, dis-

Formulas are used as descriptions of objects of the universe. In particular, atomic

1. x |= (a, v) iff a(x) = v

The following are clear from the above truth definition:

If φ is a formula, then the set | φ | S defined as follows:

Proposition 2.20 The meaning of arbitrary formulas satisfies the following:

1. (a, v) ∧ (a, u) ≡ 0 for any a ∈ A, u, v ∈ V and v = u

Proposition 2.22 The following holds for DL:

Proposition 2.23 Let φ be a formula in DL-languageand let P contain all attributes