Citations of:
The Class of Extensions of Nelson's Paraconsistent Logic
Studia Logica 80 (23):291320 (2005)
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In this study, we introduce Gentzentype sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzentype sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cutelimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding (...) 

The variety of N4? lattices provides an algebraic semantics for the logic N4?, a version of Nelson 's logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4?lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski. 

Various four and threevalued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's fourvalued logic of firstdegree entailment. Threevalued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to (...) 

It was proved by Odintsov and Pearce that the logic is a deductive base for paraconsistent answer set semantics of logic programs with two kinds of negation. Here we describe the lattice of logics extending, characterise these logics via classes of models, and prove that none of the proper extensions of is a deductive base for PAS. 

Gentzentype sequent calculi GBD+, GBDe, GBD1, and GBD2 are respectively introduced for De and Omori’s axiomatic extensions BD+, BDe, BD1, and BD2 of Belnap–Dunn logic by adding classical negation. These calculi are constructed based on a small modification of the original characteristic axiom scheme for negated implication. Theorems for syntactically and semantically embedding these calculi into a Gentzentype sequent calculus LK for classical logic are proved. The cutelimination, decidability, and completeness theorems for these calculi are obtained using these embedding theorems. (...) 

In this introduction to the special issue “40 years of FDE”, we offer an overview of the field and put the papers included in the special issue into perspective. More specifically, we first present various semantics and proof systems for FDE, and then survey some expansions of FDE by adding various operators starting with constants. We then turn to unary and binary connectives, which are classified in a systematic manner. Firstorder FDE is also briefly revisited, and we conclude by listing (...) 

A classical paraconsistent logic, which is regarded as a modified extension of firstdegree entailment logic, is introduced as a Gentzentype sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzentype sequent calculus LK for classical logic and vice versa are proved. The cutelimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained for an intuitionistic (...) 

Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of socalled explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions (...) 

The variety of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}lattices provides an algebraic semantics for the logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}, a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson (...) 

We investigate the notion of classical negation from a nonclassical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent fourvalued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of fourvalued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a fourvalued modal logic of Béziau and (...) 

The paper is devoted to the contributions of Helena Rasiowa to the theory of nonclassical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation. 

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent fourvalued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, (...) 

