A basic system of paraconsistent Nelsonian logic of conditionals

We define a Kripke semantics for a conditional logic based on the propositional logic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mn>4</mml:mn></mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mn>4</mml:mn> <mml:mi>CK</mml:mi></mml:mrow> </mml:math> , shows strong connections both with the basic intuitionistic logic of conditionals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>IntCK</mml:mi></mml:math> introduced earlier in (Olkhovikov, 2023) and with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mn>4</mml:mn></mml:mrow> </mml:math> -based modal logic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>FSK</mml:mi> <mml:mi>d</mml:mi></mml:msup> </mml:math> introduced in (Odintsov and Wansing, 2004) as one of the possible counterparts to the classical modal system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> . We map these connections by looking into the embeddings which obtain between the aforementioned systems.