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Proof Theory and Algebra in Logic

  • Textbook
  • © 2019


  • A textbook for short introductory courses on nonclassical logic at the undergraduate or graduate level
  • Offers a concise introduction to two major techniques in the study of nonclassical logic: proof theory and algebraic methods, and highlights a combination of proof theory with algebraic methods
  • Provides concrete examples showing how these techniques are applied in nonclassical logic
  • Demonstrates the complementary features of proof theory and algebraic methods by describing both their differences and similarities, as well as their connections

Part of the book series: Short Textbooks in Logic (STXLO)

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About this book

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.

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Table of contents (10 chapters)

  1. Proof Theory

  2. Algebra in Logic


“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)

Authors and Affiliations

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

    Hiroakira Ono

About the author

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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