On connexivity in modal and conditional contexts

We define and axiomatize three new logics based on the connexive logic $\mathsf{C}$, the modal logic $\mathsf{C}\mathsf{n}\mathsf{K}$ and the conditional logics $\mathsf{C}\mathsf{n}\mathsf{C}\mathsf{K}$ and ${\mathsf{C}\mathsf{n}\mathsf{C}\mathsf{K}}_R$. These logics display strong connexivity properties and are connected to one another, since ${\mathsf{C}\mathsf{n}\mathsf{C}\mathsf{K}}_R$ is the reflexive extension of $\mathsf{C}\mathsf{n}\mathsf{C}\mathsf{K}$ and $\mathsf{C}\mathsf{n}\mathsf{K}$ is faithfully embeddable into both $\mathsf{C}\mathsf{n}\mathsf{C}\mathsf{K}$ and ${\mathsf{C}\mathsf{n}\mathsf{C}\mathsf{K}}_R$ in a multitude of natural ways. We argue that all the three logics provide (albeit in different ways) natural expansions of $\mathsf{C}$ to their respective languages that preserve and further develop several core properties of $\mathsf{C}$, especially its connexivity profile.