Constructive logic with strong negation as a substructural logic. (English) Zbl 1205.03040

Busaniche, Manuela; Cignoli, Roberto

J. Log. Comput. 20, No. 4, 761-793 (2010).

Spinks and Veroff have shown that constructive logic with strong negation (CLSN) can be considered as a substructural logic.
This result paves the way for the application of algebraic techniques developed for the study of substructural logics to CLSN.
The authors prove that nilpotent minimum logic is the extension of CLSN by a prelinearity equation. This generalizes the well-known result by Monteiro and Vakarelov that three-valued Łukasiewicz logic is an extension of CLSN. A Glivenko-like theorem relating CLSN and three-valued Łukasiewicz logic is proved.

Reviewer: Dana Piciu (Craiova)

MSC:

03B60	Other nonclassical logic
03B47	Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G25	Other algebras related to logic

Keywords:

constructive logic; strong negation; nilpotent minimum logic; Nelson algebras; residuated lattices; Heyting algebras