There are many paraconsistent logics out there[1]. There are also many "trivial" logics out there[1] (which exhibit the Principle of Explosion). The former are useful, the latter not so much.
All them of them are logics. Consistency is not a requirement; it's simply a property a system may or may not have.
All them of them are logics. Consistency is not a requirement; it's simply a property a system may or may not have.
[1] http://plato.stanford.edu/entries/logic-paraconsistent/