On extensions of intermediate logics by strong negation
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.
scientific article; zbMATH DE number 1156517
scientific article