  Meyer, Routley, and Dunn (1979) call attention to one class of logics that meet conditions (a) and (b).  They are among the   paraconsistent logics,  which are logics according to which a sentence together with its negation will  not entail any arbitrary sentence. Paraconsistent logics can be used to obtain theories that resolve Russell’s paradox, and the Liar, by embracing negation inconsistency without succumbing to triviality.