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 BD Delta, 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 BD Delta and biG (an expansion of Godel logic with a coimplication (sic) or Delta) and obtain a faithful translation of presumptive arguments into formulas.