Proof Theory and Algebra in Logic

1. Front Matter

   Pages i-viii

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2. Proof Theory

   1. Front Matter

      Pages 1-2

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   2. Sequent Systems

      • Hiroakira Ono

      Pages 3-22

   3. Cut Elimination for Sequent Systems

      • Hiroakira Ono

      Pages 23-33

   4. Proof-Theoretic Analysis of Logical Properties

      • Hiroakira Ono

      Pages 35-46

   5. Modal and Substructural Logics

      • Hiroakira Ono

      Pages 47-60

   6. Deducibility and Axiomatic Extensions

      • Hiroakira Ono

      Pages 61-73

3. Algebra in Logic

   1. Front Matter

      Pages 75-76

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   2. From Algebra to Logic

      • Hiroakira Ono

      Pages 77-95

   3. Basics of Algebraic Logic

      • Hiroakira Ono

      Pages 97-111

   4. Logics and Varieties

      • Hiroakira Ono

      Pages 113-128

   5. Residuated Structures

      • Hiroakira Ono

      Pages 129-138

   6. Modal Algebras

      • Hiroakira Ono

      Pages 139-149

4. Back Matter

   Pages 151-160

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This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate or graduate level courses.
The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed inthe second part.  Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic.

• Proof theory introduction
• Algebraic semantics
• Nonclassical logics
• Sequent systems
• Cut elimination
• Algebraic logic
• Universal algebra
• Deduction theorems
• Boolean algebras
• Algebra Logic
• syntactic and semantic logic
• Algebraic method logic
• Algebraic method syntactic logic
• Algebraic method semantic logic
• Modal Logic Introduction
• many-valued logic introduction
• superintuitionistic logic introduction
• substructural logic introduction
• algebraic semantics introduction
• nonclassical logic introduction

“I cannot think of a better textbook as the backbone of an introductory course in proof theory, given how clear and self-contained the material is. … ‘Proof Theory and Algebra in Logic’ is an inspiring book that sows the seeds for further studies, and it is the perfect bridge towards more advanced research topics.” (Gianluca Caterina, MAA Reviews, June 21, 2021)

• Japan Advanced Institute of Science and Technology, Nomi, Japan

  Hiroakira Ono

Hiroakira Ono is an emeritus professor of Japan Advanced Institute for Science and Technology (JAIST). He moved to JAIST in 1993 after working at Hiroshima University for twenty years. Later he became a distinguished professor until his retirement from JAIST in 2012. His main interest is study of nonclassical logics, in particular substructural logics and intermediate logics, from both syntactic and semantical point of view. In addition to his co-authored book on substructural logics in 2007, he published six textbooks in Japanese, including those on logic and algebra, one of which gained a reputation as one of most popular textbooks on logic in Japan since its publication in 1994.

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