When Conditional Logic met Connexive Logic

Conditional logic and connexive logic are two theories whose goal is the correct formalization of the way conditionals (i.e. sentences of form if A, C) are used in natural language. However, both approaches are never combined in the literature. I present here the first formal system which allows modeling the ideas behind these two approaches in a unique framework. Furthermore, the resulting system allows explaining the different ways conditionals are negated in natural language.