 It is quite natural to obtain an FDE-based connexive logic by starting form David Nelson’s logic N4 because the latter system’s intuitionistic implication is the weakest conditional satisfying modus ponens and the deduction theorem. The conditionals studied within conditional logic in the tradition of Robert Stalnaker and David Lewis, where the conditional is usually written as ‘□→’, are much weaker than intuitionistic or relevant implication. The project of taking the basic system of conditional logic CK introduced by Brian Chellas (1975) as the point of departure for obtaining connexive conditional logics has been carried out in Wansing and Unterhuber 2019, and a similar approach is considered in Kapsner and Omori 2017. Whereas the semantics for the Lewis-Nelson models from Kapsner and Omori 2017 uses binary relations RA on a non-empty set of states, for every formula A, the Chellas-Segerberg semantics employed in Wansing and Unterhuber 2019 uses binary relations RX on a non-empty set of states, for subsets X of the set of all states. Both versions of the semantics can be equipped with sound and complete tableau calculi (although Kapsner and Omori only present the models), but the Chellas-Segerberg semantics is suitable for developing a purely structural correspondence theory in terms of properties of relations that are language-independent insofar as they are not relativized to a formula.