 Many prominent systems of non-classical logic are subsystems of what is generally called ‘classical logic.’ Systems of connexive logic are contra-classical in the sense that they are neither subsystems nor extensions of classical logic. Connexive logics have a standard logical vocabulary and comprise certain non-theorems of classical logic as theses. Since classical propositional logic is Post-complete, any additional axiom in its language gives rise to the trivial system, so that any non-trivial system of connexive logic will have to leave out some theorems of classical logic. The name ‘connexive logic’ was introduced by Storrs McCall (1963, 1964) and suggests that systems of connexive logic are motivated by some ideas about coherence or connection between the premises and the conclusions of valid inferences or between the antecedent and the succedent (consequent) of valid implications. The kind of coherence in question concerns the meaning of implication and negation (see the entries on  indicative conditionals,  the logic of conditionals,  counterfactuals,  and  negation).  One basic idea is that no formula provably implies or is implied by its own negation. This conception may be expressed by requiring that for every formula A,