  The logics being proposed here, then, are naturally paraconsistent: they do not validate the “spread” law: (\(p \amp \twiddle p) \rightarrow q\), and so can putatively handle inconsistency without collapse into triviality. (In Sociative Logics, Sylvan refers to these as pararelevant logics.) Indeed, with counterexamples to the law of non-contradiction, there would be worlds where some sentences are both true and false.