 A note on terminology. In Dunn 1993, a negation operation validating the contraposition rule is called subminimal. The term “subminimal negation” had been used by Allen Hazen in an unpublished paper from 1992 for a richer language containing negation, conjunction, disjunction, and intuitionistic implication to denote a negation that fails to validate the intuitionistically valid De Morgan sequent \((\osim A \wedge \osim B) \vdash \osim (A \vee B)\) and the classically but not intuitionistically valid \(\osim (A \wedge B) \vdash (\osim A \vee \osim B)\). Dunn’s use of the term “subminimal” is thus different from Hazen’s. In Dunn and Zhou 2005 only a single negation as impossibility is used, the vocabulary is enriched by conjunction and disjunction, and in both, the one with negation, conjunction, and disjunction as well as the one with negation only, subminimal negations are referred to as preminimal negations. Moreover, the minimal negations from Dunn 1993, 1996 are called quasi-minimal in Dunn and Zhou 2005 because they lack negative ex contradictione, a property of negation in what is usually called (Johansson’s) minimal logic, see Kolmogorov 1925 and Johansson 1937. In Vakarelov 2005, the term “subminimal negation” is used to denote a negation weaker than Kolmogorov and Johansson’s minimal negation. Vakarelov introduces his subminimal negation in combination with David Nelson’s strong negation (see below) in a language that contains a truth constant \(\top\), which is definable in minimal logic. Vakarelov’s subminimal negation, \(\neg\), validates the contraposition rule but fails to validate \(\neg \neg \top\). Colacito, de Jongh, and Vargas 2017 define a system called “subminimal logic”, see also Niki (forthcoming). Their subminimal negation is added to the language of positive intuitionistic logic and thus, in particular, contains intuitionistic implication. The basic subminimal logic is obtained from positive intuitionistic propositional logic by adding the congruence axiom scheme \((A \leftrightarrow B ) \rightarrow (\neg A \leftrightarrow \neg B)\). We shall come back to subminimal negation in subminimal logic in  Section 2.3.