An Intuitionistically Complete System of Basic Intuitionistic Conditional Logic

We introduce a basic intuitionistic conditional logic \(\textsf{IntCK}\) $\mathsf{IntCK}$ 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 \(\textsf{IntCK}\) $\mathsf{IntCK}$ stands in a very natural relation to other similar logics, like the basic classical conditional logic \(\textsf{CK}\) $\mathsf{CK}$ and the basic intuitionistic modal logic \(\textsf{IK}\) $\mathsf{IK}$ . As for the basic intuitionistic conditional logic \(\textsf{ICK}\) $\mathsf{ICK}$ proposed in Weiss (Journal of Philosophical Logic, 48, 447–469, 2019), \(\textsf{IntCK}\) $\mathsf{IntCK}$ extends its language with a diamond-like conditional modality \(\Diamond \hspace{-4.0pt}\rightarrow \) $◊\to$ , but its ( \(\Diamond \hspace{-4.0pt}\rightarrow \) $◊\to$ )-free fragment is also a proper extension of \(\textsf{ICK}\) $\mathsf{ICK}$ . 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.