Hiroakira Ono’s Proof Theory and Algebra in Logic, the first title of the new Springer series Short Textbook in Logic, stands out as a minimalist gem amongst modern introductory books in logic.

As implied by the title, the book is structured in two self-contained parts that are can be read and understood as standalone texts. 

Part I consists of about seventy concise pages in which the author lays out a streamlined, yet rigorous and engaging, introduction to proof theory. Using Gentzen’s sequent systems, in two short chapters Professor Ono provides the reader with the background needed for a thorough understanding of the cut elimination theorem and the subformula property. Relevant results such as the decidability of intuitionistic logic, Craig’s interpolation property and Givenko’s theorem are outlined in chapter three, whereas standard sequent systems for normal modal logic and the role of structural rules are thoroughly presented in chapter four. Part I ends with a discussion on local deduction theorems and a more abstract framework for modal and substructural logics. 

Part II has more of a survey flavor. A presentation of the basic notions of algebraic logic, mostly outlined within the concrete context of Heyting algebras, paves the way to lift the reader to a more abstract understanding of a universal algebraic approach to logic, followed by a substantial list of relevant applications and examples. 

Professor Ono, who is currently an emeritus professor at the Japan Advanced Institute for Science and Technology (JAIST) since he retired in 2013, throughout his long and fruitful career has brought major contributions to the field of logic, focusing on the study of substructural logics and working towards a unifying approach to non-classical logics within an algebraic framework. His deep knowledge of the subject translates into a very intelligible, reader-friendly work: throughout the text, words are carefully used, symbols and notation are kept to a minimal level of difficulty and complex ideas are smoothly conveyed in a very direct style, which is never too terse or dry.

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. Since exercises are not included, instructors may find it useful to complement the textbook with a companion set of lecture notes, problems sets, and external references to help the students navigating the material, especially at the undergraduate level. Ultimately, “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 is Professor of Mathematics at Endicott College in Beverly, MA. His current research focuses on logic, philosophy of science and applied category theory.