An Intuitionistically Complete System of Basic Intuitionistic Conditional Logic

We introduce a basic intuitionistic conditional logic \(\textsf{IntCK}\) that we show to be complete both relative to a special type of Kripke models and relative to a standard translation into first-order intuitionistic logic. We show that \(\textsf{IntCK}\) stands in a very natural relation to other similar logics, like the basic classical conditional logic \(\textsf{CK}\) and the basic intuitionistic modal logic \(\textsf{IK}\). As for the basic intuitionistic conditional logic \(\textsf{ICK}\) proposed in Weiss (Journal of Philosophical Logic, 48, 447–469, 2019), \(\textsf{IntCK}\) extends its language with a diamond-like conditional modality \(\Diamond \hspace{-4.0pt}\rightarrow \), but its (\(\Diamond \hspace{-4.0pt}\rightarrow \))-free fragment is also a proper extension of \(\textsf{ICK}\). We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates.