Paraconsistency, paracompleteness, and valuation

This paper presents a propositional system that is simultaneously paracon-sistent and paracomplete, offering a semantic framework grounded in valuation the-ory. The authors define paraconsistency as the ability of a logic to tolerate contradic-tions without collapsing into triviality, and paracompleteness as the failure to satisfy thelaw of the excluded middle. The work introduces a logical system,𝜋, with weak prim-itive negation and a definable classical negation. Through valuation semantics and themethod of valuation-tableaux, the authors demonstrate the decidability of𝜋, while alsoproving that𝜋is not characterizable by any finite-valued matrix.