Constructive negations and paraconsistency
type: Item
instance of: scholarly article
title: Constructive negations and paraconsistency
zbMATH Open document ID: 1161.03014
DOI: 10.1007/978-1-4020-6867-6
author: S. P. Odintsov
published in: Trends in Logic -- Studia Logica Library
publication date: 2008-07-10T00:00:00Z
review text: This is the first book-length algebraic study of constructive paraconsistent logics. It contains a comprehensive algebraic investigation of the lattices of extensions of both (i) Johansson's minimal propositional logic \textbf{Lj} and (ii) David Nelson's paraconsistent constructive propositional logic with strong negation \textbf{N4}, as well as its extension by a falsum constant, the system \textbf{N4}\(^{\bot}\). These logics are of interest for quite a few reasons, one being that they are paraconsistent analogues of the better known intuitionistic propositional logic \textbf{Li} and Nelson's intuitionistic logic with strong negation \textbf{N3}, respectively. Whereas the relational (Kripke-style) semantics for \textbf{N3} is three-valued, the relational semantics for the paraconsistent logic \textbf{N4} is four-valued, hence the notation. The monograph under review has a very clear structure. Chapter one is a brief introduction to the subject matter of the book, to the history of constructive paraconsistent logics, and to the presented results. The rest of the material is arranged into two parts, Chapters 2--7 and Chapters 8--10. Moreover, there are some concluding remarks in a rather short Chapter 11, a comprehensive and very useful bibliography, and an index of key terms and some notation. Each part of the book is devoted to a particular treatment of negation in constructive logic. Part I deals with the purely implication treatment of negation as (leading to) absurdity, and Part II is devoted to strong negation in the sense of definite falsity (or denial). The first part of the book offers an in-depth study of the lattice of extensions of \textbf{Lj}. This lattice is shown to be the disjoint union of the class of intermediate logics, the class of all properly paraconsistent extensions of \textbf{Lj}, and the class of extension of \textbf{Lj} containing the scheme \(\neg p\). The notions of intuitionistic and negative counterparts of a paraconsistent logic are d…
reviewed by: Heinrich Wansing
Mathematics Subject Classification ID: 03B53
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03B60
zbMATH DE Number: 5299202
zbMATH Keywords: paraconsistency
zbMATH Keywords: negation
zbMATH Keywords: strong negation
zbMATH Keywords: algebraic semantics
zbMATH Keywords: twist-structures
zbMATH Keywords: constructive logic
zbMATH Keywords: Johansson's minimal logic
zbMATH Keywords: Nelson's logics
zbMATH Keywords: absurdity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4020-6867-6
OpenAlex ID: W87187940
description: scientific article; zbMATH DE number 5299202