A basic system of paraconsistent Nelsonian logic of conditionals

We define a Kripke semantics for a conditional logic based on the propositional logic \(\textsf{N4}\) $\mathsf{N}\mathsf{4}$ , the paraconsistent variant of Nelson’s logic of strong negation; we axiomatize the minimal system induced by this semantics. The resulting logic, which we call \(\textsf{N4CK}\) $\mathsf{N}\mathsf{4}\mathsf{CK}$ , shows strong connections both with the basic intuitionistic logic of conditionals \(\textsf{IntCK}\) $\mathsf{IntCK}$ introduced earlier in (Olkhovikov, 2023) and with the \(\textsf{N4}\) $\mathsf{N}\mathsf{4}$ -based modal logic \(\textsf{FSK}^d\) ${\mathsf{FSK}}^d$ introduced in (Odintsov and Wansing, 2004) as one of the possible counterparts to the classical modal system \(\textsf{K}\) $\mathsf{K}$ . We map these connections by looking into the embeddings which obtain between the aforementioned systems.