 Humberstone (2000) calls a logic contra-classical just in case not every formula provable in the logic is provable in classical logic (and, moreover, considers a more demanding notion of a contra-classical logic by requiring that there is no way of translating its connectives in such a way that one obtains a subsystem of classical logic). There are several different kinds of contra-classical logics, such as, for example, Abelian logics containing the axiom schema ((A → B) → B) → A, connexive logics, and logics of logical bilattices. The negation, truth order conjunction, weak implication, and information order disjunction fragment of Arieli and Avron’s (1996) bilattice logic BL⊃, for example, is a standard propositional vocabulary containing a negation, a conjunction, a disjunction, and a conditional. It is a non-trivial but inconsistent logic and as such contra-classical.