On extensions of intermediate logics by strong negation
type: Item
instance of: scholarly article
title: On extensions of intermediate logics by strong negation
zbMATH Open document ID: 0929.03035
DOI: 10.1023/A:1004222213212
author: Marcus Kracht
published in: Journal of Philosophical Logic
publication date: 2000-01-12T00:00:00Z
review text: The structure \({\mathcal E} {\mathbf N}\) of extensions of Nelson logic \(N\) (intuitionistic logic with strong negation) and its correlations with the structure \({\mathcal E} {\mathbf {Int}}\) of intermediate logics is studied. The mapping from \(\Lambda\) to \(n( \Lambda) = N + \Lambda \) is an embedding of the complete lattice \({\mathcal E} {\mathbf {Int}}\) into \({\mathcal E} {\mathbf N}\), preserving and reflecting finite model property, tabularity, completeness, frame-compactness, interpolation and decidability; moreover, \(n( \Lambda)\) is a conservative extension of \(\Lambda\). The family of inference rules is described which are admissible exactly in all logics of the form \(n ( \Lambda)\). Finally, the structure of extensions of \(n( {\mathbf {LC}})\) is described; all these extensions are finitely axiomatizable, decidable and have the fmp. The key instrument is the construction of Nelson algebras (algebras for the logic \(N\)) of a special kind -- so-called twist algebras, corresponding to Heyting algebras, and representing every Nelson algebra as a subalgebra of a twist algebra. Reviewer's remark. Some passages in the paper are slightly inaccurate. E.g., the condition \(\forall x \forall y [( x \leftrightarrow y) = ( \tilde x \leftrightarrow \tilde y) = 1 \Rightarrow ( x = y) ]\) does not follow from the given definition of Nelson algebra. The proof of preserving frame-compactness contains a minor gap; namely, the formula \(\gamma _{v( \phi)} \wedge \widehat { \phi }\) should be used instead of \(\gamma _{v( \phi)} \to \widehat { \phi }\) . And the description of extensions of \(n( {\mathbf {LC} })\) needs the following additional lemma: Every subdirectly irreducible Nelson algebra is embeddable into a twist algebra of a subdirectly irreducible Heyting algebra.
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06D20
zbMATH DE Number: 1156517
zbMATH Keywords: intermediate logics
zbMATH Keywords: strong negation
zbMATH Keywords: Nelson algebras
zbMATH Keywords: extensions of Nelson logic
zbMATH Keywords: twist algebras
reviewed by: Dmitrij Skvortsov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1156517