In this talk I argue in favour of the hypothesis that combining inconsistency tolerance (there is an overlap between the extension of a property and it's negation) with the expressiveness of the classical complement operation (being able to express that something is [classically] not in the extension of the property) requires an account of paraconsistency where inconsistencies come in degrees (given a particular perspective on paraconsistency). I do this by developing a new infinitely valued logic CLP which adds a classical negation to the prototypical truth functional paraconsistent logic LP devised by Graham Priest. I will give a number of desiderata for a logic that combines the negation of classical logic and the negation of LP in a way that is faithful to central properties of the combined logics. Then I will prove that CLP is the only logic satisfying all desiderata. The infinitely 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. If time permits, I will briefly (the full discussing would require another talk) discuss an interesting non-material (maybe counterfactual) conditional which can naturally be added to CLP.