, Volume 80, Issue 2, pp 393430
First online:
Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation
 Dimiter VakarelovAffiliated withDepartment of Mathematical Logic with Laboratory for Applied Logic Faculty of Mathematics and Computer Science, Sofia University Email author
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Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized Nlattices. We show that generalized Nlattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the cominimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3valued logic of Lukasiewicz, two 12valued logics and one 48valued logic. Standard questions for all these logics — decidability, Kripkestyle semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of selfdual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.
Keywords
Nelson negation subminimal logic counterexample semantics manyvalued logics Title
 Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation
 Journal

Studia Logica
Volume 80, Issue 23 , pp 393430
 Cover Date
 200508
 DOI
 10.1007/s1122500584765
 Print ISSN
 00393215
 Online ISSN
 15728730
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Nelson negation
 subminimal logic
 counterexample semantics
 manyvalued logics
 Industry Sectors
 Authors

 Dimiter Vakarelov ^{(1)}
 Author Affiliations

 1. Department of Mathematical Logic with Laboratory for Applied Logic Faculty of Mathematics and Computer Science, Sofia University, blvd James Bouchier 5, 1126, Sofia, Bulgaria