When some premises are more premise than others, the consequences are the real premises

This paper proposes two adaptive approaches to inconsistent prioritized belief bases. Both approaches rely on a selection mechanism that is not applied to the premises as they stand, but to the consequence sets of the belief levels. One is based on classical compatibility, the other on the modal logic T of Feys. For both approaches the two main strategies of inconsistency adaptive logics are formulated: the reliability strategy and the minimal abnormality strategy. All four systems are compared and found useful.