Connexive logic is a rather new branch of modern logic, based on the idea that no proposition ever implies its own negation. This idea goes back to Aristotle, who put forward the thesis that one and the same consequent cannot be "necessitated" by the same antecedent affirmed and denied. Many medieval logicians discovered, however, that the connexive theses fail to hold in certain cases, namely when the antecedent is impossible, or when the consequent is necessary. This book scrutinizes the theories of various medieval logicians such as Boethius, Abelard, Kilwardby, Burley, Buridan, Paul of Venice, Albert of Saxony, and the Pseudo-Scot, who endorsed a conception of conditionals as strict implications. This conception validates the "humble" version of the connexive principles, but invalidates their "hardcore" versions according to which, e.g., even the tautological disjunction 'p or not-p' is not implied by its own negation, i.e., by the self-contradictory conjunction 'p and not-p'.