A basic system of paraconsistent Nelsonian logic of conditionals

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