Books by Norihiro Kamide

The present book is the first monograph ever with a central focus on the proof theory of paraconsistent logics in the vicinity of the four-valued, constructive paraconsistent logic N4 by David Nelson. The volume brings together a number of papers the authors have written separately or jointly on various systems of inconsistency-tolerant logic. The material covers the structural proof theory of ¿ N4, ¿ its fragments, including first-degree entailment logic, ¿ related logics, such as trilattice logics, connexive systems, systems of symmetric and dual paraconsistent logic, and variations of bi-intuitionistic logic, ¿ paraconsistent temporal logics, ¿ substructural subsystems of N4, such as paraconsistent intuitionistic linear logics, paraconsistent logics based on involutive quantales, and paraconsistent Lambek logics. Although the proof-theory of N4 and N4-related logics is the central theme of the present monograph, models and model-theoretic semantics also play an important role in the presentation. The relational, Kripke-style models that are dealt with provide a motivating and intuitively appealing insight into the logics with respect to which they are shown to be sound and complete. Nevertheless, the emphasis is on Gentzen-style proof systems -in particular sequent calculi of a standard and less standard kind- for paraconsistent logics, and cut-elimination and its consequences are a central topic throughout. A unifying element of the presentation is the repeated application of embedding theorems in order to transfer results from other logics such as intuitionistic logic to the paraconsistent case.

This book introduces some extensions of resource- aware logics, and shows some fundamental (e.g., cut- elimination and completeness) theorems for these extensions. By resource-aware logics, we mean substructural logics without the structural rule of contraction in Gentzen-type sequent calculi. Typical examples of these resource-aware logics are linear logics, which are known to be useful for describing concurrent systems, and non-commutative logics, which are known to be useful for formalizing grammatical reasoning. These logics are known to be appropriate for representing resource-sensitive reasoning. In this book, we study (1) some temporal extensions (i.e., extensions by adding some temporal operators) of linear and non-commutative logics, (2) some generalizations of these temporal linear and non-commutative logics, (3) some paraconsistent and involutive extensions (i.e., extensions by adding a paraconsistent negation connective or an involution operator) of linear and non-commutative logics.