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Proof theory and algebra in logic
Ono H., Springer International Publishing, New York, NY, 2019. 168 pp. Type: Book
Date Reviewed: Mar 3 2021

If you are interested in mathematical logic and its relationship to algebraic structures, this brief introductory book gives an excellent overview for you. The author of the book, Hiroakira Ono, has been a very active researcher in this area for decades; he gives a very good overview of the theme. Starting with traditional two-valued propositional logic, various nonclassical ones are introduced, for instance, kinds of intuitionistic logic, (normal) many-valued logic, modal logic, and substructural logic.

The two main parts of the book cover syntactic and semantic aspects, respectively: on the one hand, proof theory of formal inference systems; on the other hand, abstract algebraic models.

Part 1 presents important notions and features (decidability, completeness, deducibility, for example), in parallel with the introduction of different logic systems. Inference rules are grouped and their roles are shown in connection with properties of the systems. Cut rule elimination and handling the excluded middle is treated throughout.

Part 2 gives the algebraic counterparts of the logic systems treated in the first part. It not only covers the relationships among different algebraic structures, but also refers to the pursuant logic system discussed in Part 1. It introduces partial and total orders, lattices, chains, Boolean algebras, semigroups, monoids, relationships, and operations (subalgebras, homomorphism, and direct products). By introducing representations, the author shows the algebraic completeness of classical logic. Different algebras and their properties are then discussed: Heyting algebras, the Lindenbaum-Tarski algebra, locally finite algebras, correspondence among varieties of intuitionistic logic and Heyting algebras, residuated structures, two kinds of many-valued chains, and modal algebras.

Each chapter introduces its theme via an informal introduction. Within the exposition of the theme, the reader is involved in some lemmas: the proof is left to him/her as an exercise. For the sake of beginners, and for ease of reading at some (indicated) points, the notation is simplified. A rich list of references is given at the end, and an index of special terms indicates the defining occurrence of each one.

Reviewer:  K. Balogh Review #: CR147204 (2105-0100)
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Proof Theory (F.4.1 ... )
 
 
Algebraic Approaches To Semantics (F.3.2 ... )
 
 
Temporal Logic (F.4.1 ... )
 
 
Theory (K.2 ... )
 
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