MATHEMATICAL LOGIC FOR COMPUTER SCIENCE

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MATHEMATICAL LOGIC

FOR

COMPUTER SCIENCE

Second Edition

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Series in Computer Science Vol. 47

MATHEMATICAL LOGIC

FOR

COMPUTER SCIENCE

Second Edition

World Scientific

Singapore • New Jersey'London • Hong Kong

Lu Zhongwan

Chinese Academy of Sciences

Beijing

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MATHEMATICAL LOGIC FOR COMPUTER SCIENCE (Second Edition)

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PREFACE

Mathematical logic studies logical problems with mathematical meth-ods, principally logical problems in mathematics. It is a branch of mathe-matics. There are two kinds of mathematical research, proof and computa-tion, which are essentially related to each other. Hence mathematical logic is essentially related to computer science, and many branches of mathe-matical logic have applications in it. This book describes those aspects of mathematical logic which are closely related to each other, including clas-sical and non-clasclas-sical logics. Roughly, non-clasclas-sical logics can be divided into two groups, those that rival classical logic and those which extend it. This first group includes, for instance, constructive logic and multi-valued logics. The second includes modal and temporal logics, etc. Of non-classical logics, this book chooses to describe constructive and modal logics.

Materials adopted in this book are intended to attend to both the pe-culiarities of logical systems and the requirements of computer science, but those concerning the applications of mathematical logic in computer science are not involved. Topics are discussed concisely with the essentials empha-sized and the minor details excluded. For various logics, their background, language, semantics, formal deduction, soundness and completeness are the main topics introduced. Formal deduction is treated in the form of natural deduction which resembles ordinary mathematical reasoning.

This book consists of an introduction, nine chapters, and an appen-dix. In the Introduction, the nature of mathematical logic is explained. In Chapter 1 of prerequisites, the basic concepts of set theory, including the fundamental theorems of countable sets, are reviewed; and inductive defi-nitions and proofs are explained in detail, since many concepts in mathe-matical logic are defined inductively. Besides these prerequisites, this book is self-contained.

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VI

Chapters 2-5 describe classical logics. Classical propositional logic may be regarded as part of classical first-order logic; but these logics are de-scribed separately in Chapters 2 and 3 because classical propositional logic has its own characteristics. Classical logic is the basis of this book; its soundness and completeness are studied in Chapter 5. Especially, the com-pleteness problem of classical propositional logic and the different cases of classical first-order logic with and without equality are treated separately, in order to show the distinction of these cases in the treatment of complete-ness. Chapter 4 introduces the axiomatic deduction system, and proves the equivalence between it and the natural deduction system.

Chapter 6 studies, on the basis of soundness and completeness, Com-pactness Theorem, Lowenheim-Skolem Theorem, and Herbrand Theorem, which is the basis of one approach of automatic theorem proving in artificial intelligence.

Chapters 7-9 describe constructive and modal logics, and discuss the relationship between classical logic and these non-classical logics.

In Appendix, a simple form of formal proof in natural deduction system is introduced.

The first edition of this book was printed in 1989. The revisions in this edition are essentially concerned with rewriting proofs and expanding the explanations in the remarks. New terms and notations are adopted instead of original ones; for instance, "propositional logic" and "first-order logic" are renamed as "classical propositional logic" and "classical first-order logic", and "interpretation" and "assignment" are combined into one term "valuation". Furthermore, Sec. 6.4 of Chapter 6 is eliminated.

I would like to offer my deepest thanks to many people. Professor Hu Shihua taught me mathamatical logic selflessly. In the writing of this book, Professor Wang Shiqiang, Professor Tang Zhisong, Professor Xu Kongshi, Professor Yang Dongping, and the late Professor Wu Yunzeng provided much criticism and advice. Mr. Zhang Li helped in making suggestions and preparing the revisions. The Graduate School of University of Science and Technology of China (in Beijing) and Tsinghua University provided me with the opportunity to teach the materials of this book. The discussions with the students during my years of teaching in the universities have been very helpful in the revision of this book.

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VII

I would also like to thank the staff of World Scientific Publishing

Com-pany, first Professor K. K. Phua, and then Mr. S. J. Han, Ms. G. K. Tan,

Ms. Jennifer Gan, Ms. H. M. Ho, and Ms. S. H. Gan, for their friendly

and efficient help in the production of this book.

Finally I would like to express gratitude to my wife Ding Yi for her

patient typing and encouragement during the long writing period.

Lu Zhongwan

Institute of Software, Chinese Academy of Sciences

Garduate School of University of Science and Technology of China (in Beijing)

October 1996

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3. Classical First-Order Logic 69

3.1 Proposition functions and quantifiers 70

3.2 First-order language 74

3.3 Semantics 83

3.4 Logical consequence 93

3.6 Prenex normal form 106

ix

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Contents XI

9.2 Semantics 206

9.4 Soundness 211

9.5 Completeness 212

9.6 Equality 217

Appendix (a simple form of formal proof in natural deduction) 221

Bibliography 227

List of Symbols 229

Index

233

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MATHEMATICAL LOGIC FOR COMPUTER SCIENCE