  Signed systems were introduced as a general, syntax-independent framework for
paraconsistent reasoning, that is, non-trivialised reasoning from inconsistent
information. In this paper, we show how the family of corresponding
paraconsistent consequence relations can be axiomatised by means of quantified
Boolean formulas. This approach has several benefits. First, it furnishes an
axiomatic specification of paraconsistent reasoning within the framework of
signed systems. Second, this axiomatisation allows us to identify upper bounds
for the complexity of the different signed consequence relations. We strengthen
these upper bounds by providing strict complexity results for the considered
reasoning tasks. Finally, we obtain an implementation of different forms of
paraconsistent reasoning by appeal to the existing system QUIP.
