A thesis concerning conditionals according to which any two conditionals $\phi \to \psi$ and $\phi \to \neg \psi$ (where $\neg$ is a negation) are considered contraries, i.e., the truth of one should entail the failure of the other. The principle is false of most deductive systems, e.g., in classical logic, both $\phi \to \psi$ and $\phi \to \neg \psi$ are true in any model in which $\phi$ is false and in logics of strict implication, both $\phi ⊰\psi$ and $\phi ⊰\neg \psi$ are true whenever $\phi$ is a contradiction. Non-classical schema such as Boethius’ thesis are intended to axiomatize this thesis, for which reason the primary deductive systems satisfying conditional non-contradiction are connexive logics.