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Mathematical logic studies logical problems with mathematical methods, principally logical problems in mathematics. It is a branch of mathematics. There are two kinds of mathematical research, proof and computation, which are essentially related to each other. Hence mathematical logic is essentially related to computer science, and many branches of mathematical logic have applications in it. This book describes those aspects of mathematical logic which are closely related to each other, including classical and non-classical logics. Roughly, non-classical logics can be divided into two groups, those tha t 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 peculiarities 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 emphasized 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 appendix. 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 definitions and proofs are explained in detail, since many concepts in mathematical logic are defined inductively. Besides these prerequisites, this book is self-contained. Chapters 2-5 describe classical logics. Classical propositional logic may be regarded as part of classical first-order logic but these logics are described 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 completeness 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 completeness. 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, Compactness 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