This paper is devoted to combining the negation of Classical Logic (**CL**) and the negation of Graham Priest's **LP** in a way that is faithful to central properties of the combined logics. We give a number of desiderata for a logic **L** which combines both negations. These desiderata include the following: (a) **L** should be truth functional, (b) **L** should be strictly non-explosive for the paraconsisent negation ˜ (i.e. if *A* and ˜*A* both have a non-trivial set of consequences, then this should also be the case for the set containing both) and (c) **L** should be a conservative extension of **CL** and of **LP**. The desiderata are motivated by a particular property-theoretic perspective on paraconsistency. Next we devise the logic **CLP**. We present an axiomatization of this logic and three semantical characterizations (a non-deterministic semantics, an in nitely valued set-theoretic semantics and an in nitely valued semantics with integer numbers as values). We prove that **CLP** is the only logic satisfying all postulated desiderata. The in nitely valued semantics of **CLP** can be seen as giving rise to an interpretation in which inconsistencies and inconsistent properties come in degrees: not every sentence which involves inconsistencies is equally inconsistent.