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We define a Kripke semantics for a conditional logic based on thepropositional logic $\mathsf{N4}$, the paraconsistent variant of Nelson's logicof strong negation; we axiomatize the minimal system induced by this semantics.The resulting logic, which we call $\mathsf{N4CK}$, shows strong connectionsboth with the basic intuitionistic logic of conditionals $\mathsf{IntCK}$introduced earlier in arXiv:2306.10402 and with the $\mathsf{N4}$-based modallogic $\mathsf{FSK}^d$ introduced by S. Odintsov and H. Wansing as one of thepossible counterparts to the classical modal system $\mathsf{K}$. We map theseconnections by looking into the embeddings which obtain between theaforementioned systems.

A basic system of paraconsistent Nelsonian logic of conditionals