Proof Theory, Semantics and Algebra for Normative Systems

Abstract

Abstract. 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 offers a holistic perspective on normative systems. We use isomorphism and embedding of joining-system to discuss the similarity of normative systems.

New Algebraic Normative Theories for Ethical and Legal Reasoning in the LogiKEy Framework

“…The algebrization of the I/O framework shows more similarity with the theory of joining-systems [24] that is an algebraic approach for study normative-systems over Boolean algebras. We can say that norms in the I/O framework play the same role of joining in the theory of Lindahl and Odelstal [24,25]. There are important similarities between input/output logic and the theory of joining systems, such as: studying normative systems as deductive systems and representing norms as ordered pairs.…”

“…While the focus in input/output logic is deontic and factual detachment in the theory of joining systems, the central theme is intermediate concepts and representing normative systems as a network of subsystems and relations between them; for more detail, see [24]. Sun [25] built Boolean joining systems that characterize I/O logic in a sense that a norm is derivable from a set of norms if and only if it is in the set of norms algebraically generated in the Lindenbaum-Tarski algebra for propositional logic. The work of Sun [25], similar to the Bochman approach [26], has no direct connection to input/output operations.…”

“…Sun [25] built Boolean joining systems that characterize I/O logic in a sense that a norm is derivable from a set of norms if and only if it is in the set of norms algebraically generated in the Lindenbaum-Tarski algebra for propositional logic. The work of Sun [25], similar to the Bochman approach [26], has no direct connection to input/output operations. Here, we built algebraic I/O operations directly over Boolean algebras and, more generally, abstract logics.…”