Proof theory, semantics and algebra for normative systems

This paper reports a correspondence between input/output logic and the theory of joining-system, an algebraic approach to normative system. The results have the form: every norm (a; x) is logically derivable from a set of norms G if and only if it is in the space of norms algebraically generated by G. We present three versions of correspondence: input/output logic and Boolean joining-system, intuitionistic input/output logic and Heyting joining-system, quasi input/output logic
and quasi joining-system. The algebraic approach o ers a holistic perspective on normative systems. We use isomorphism and embedding of joining-system to discuss the similarity of normative systems.