Proof Theory and Algebra in Logic

Description

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 in the 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. Front Matter ....Pages i-viii Front Matter ....Pages 1-2 Sequent Systems (Hiroakira Ono)....Pages 3-22 Cut Elimination for Sequent Systems (Hiroakira Ono)....Pages 23-33 Proof-Theoretic Analysis of Logical Properties (Hiroakira Ono)....Pages 35-46 Modal and Substructural Logics (Hiroakira Ono)....Pages 47-60 Deducibility and Axiomatic Extensions (Hiroakira Ono)....Pages 61-73 Front Matter ....Pages 75-76 From Algebra to Logic (Hiroakira Ono)....Pages 77-95 Basics of Algebraic Logic (Hiroakira Ono)....Pages 97-111 Logics and Varieties (Hiroakira Ono)....Pages 113-128 Residuated Structures (Hiroakira Ono)....Pages 129-138 Modal Algebras (Hiroakira Ono)....Pages 139-149 Back Matter ....Pages 151-160