Nelson's negation on the base of weaker versions of intuitionistic negation

Studia Logica 80 (2-3):393 - 430 (2005)
Abstract
Constructive logic with <span class='Hi'>Nelson</span> 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 <span class='Hi'>Nelson</span> negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices 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 <span class='Hi'>Nelson</span> negation which is an extension of the Johansson's minimal logic with <span class='Hi'>Nelson</span> negation and its in a sense dual version — the co-minimal logic with <span class='Hi'>Nelson</span> negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.
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    Heinrich Wansing (2008). Constructive Negation, Implication, and Co-Implication. Journal of Applied Non-Classical Logics 18 (2-3):341-364.
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