One tradition in the logical study of argumentation is to allow for arguments that combine strict and defeasible inference rules, and to derive the strict inference rules from a logic at least as strong as classical logic. An unsolved problem in this tradition is how the trivialising effect of the classical Ex Falso principle can be avoided when two arguments that use defeasible rules have contradictory conclusions. The problem is especially hard since any solution should arguably preserve current results on satisfaction of consistency and logical closure properties. One approach to solve the problem is to replace classical logic as the source for strict rules with a weaker, monotonic paraconsistent logic. This paper explores this approach in the context of the ASPIC+ framework for structured argumentation, by instantiating it with a paraconsistent consequence notion of Rescher & Manor (1970). The results are positive: satisfaction of the closure and consistency postulate is proven.