Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Fragments of quasi-Nelson: residuation

Fragments of quasi-Nelson: residuation Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Non-Classical Logics Taylor & Francis

Fragments of quasi-Nelson: residuation

Journal of Applied Non-Classical Logics , Volume 33 (1): 68 – Jan 2, 2023
68 pages

Loading next page...
 
/lp/taylor-francis/fragments-of-quasi-nelson-residuation-Xi0tD5p0C1

References (24)

Publisher
Taylor & Francis
Copyright
© 2023 Informa UK Limited, trading as Taylor & Francis Group
ISSN
1958-5780
eISSN
1166-3081
DOI
10.1080/11663081.2023.2203312
Publisher site
See Article on Publisher Site

Abstract

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts.

Journal

Journal of Applied Non-Classical LogicsTaylor & Francis

Published: Jan 2, 2023

Keywords: Nelson's constructive logic with strong negation; non-involutive; twist-structures; pocrims; subreducts

There are no references for this article.