Summary: Constructive logic with Nelson negation is an extension of 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 maximally weakening the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit a 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 intuitionistic logic are the minimal logic with Nelson negation, which is an extension of Johansson’s minimal logic with Nelson negation, and its, in a sense, dual version – the co-minimal logic with Nelson negation. Among the extensions that are near to classical logic are the well-known 3-valued logic of Łukasiewicz, 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 \(\mathbf{1/2}\) are also considered.