@article {verdee2013degrees, title = {Degrees of inconsistency. Carefully combining classical and paraconsistent negation.}, year = {Submitted}, abstract = {

This paper is devoted to combining the negation of Classical Logic (CL) and the negation of Graham Priest{\textquoteright}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.

}, author = {Verd{\'e}e, Peter} }