Presumptive Reasoning in a Paraconsistent Setting

We explore presumptive reasoning in the paraconsistent case. Specifically, we provide semantics for non-trivial reasoning with presumptive arguments with contradictory assumptions or conclusions. We adapt the case models proposed by Verheij [25, 26] and define the paraconsistent analogues of the three types of validity defined therein: coherent, presumptively valid, and conclusive ones. To formalise the reasoning, we define case models that use BD4, an expansion of the Belnap–Dunn logic with the Baaz Delta operator. We also show how to recover presumptive reasoning in the original, classical context from our paraconsistent version of case models. Finally, we construct a two-layered logic over BD4 and biG (an expansion of Gödel logic with a coimplication -< or 4) and obtain a faithful translation of presumptive arguments into formulas.