Two kinds of formal semantics for negation can help us understand puzzles in the psychology of reasoning: the Australian Plan semantics, which generalises the account of negation found in frame semantics for intuitionistic logic, relevant and substructural logics; and negation-as-failure, found in logic programming and default logic. We provide a framework where the two kinds of negation coexist, mapping them to the celebrated distinction between System 1 and System 2 thinking. We use the framework to target the phenomenon of Matching Bias: specifically, to explain why reasoners fare differently in card selection tasks, depending on whether the target property is phrased positively or negatively.