  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 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 the 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 and obtain a faithful translation of presumptive arguments into formulas.
