In this paper we discuss Gabbay's idea of basing nonmonotonic deduction on semantic consequence in intuitionistic logic extended by a consistency operator and Turner's suggestion of replacing the intuitionistic base system by Kleene's three-valued logic. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson's constructive logic N instead of intuitionistic logic or Kleene's system. Moreover, in N a more general notion of consistency can be defined and nonmonotonic deduction can thus be based on a logical system satisfying the Deduction Theorem.
"Semantics-based Nonmonotonic Inference." Notre Dame J. Formal Logic 36 (1) 44 - 54, Winter 1995. https://doi.org/10.1305/ndjfl/1040308828