Subordination Algebras as Semantic Environment of Input/Output Logic

TL;DR

This paper links input/output logic with subordination algebras, providing a new semantic foundation that enhances understanding of norms and enables modal logic techniques to analyze input/output operators.

Contribution

It introduces a formal semantics for input/output logic using subordination algebras, bridging two independent research areas and enabling modal logic analysis of output operators.

Findings

Input/output logic can be modeled on subordination algebras.

Output operators can be represented as modal operators.

This connection allows for systematic axiomatization and analysis.

Abstract

We establish a novel connection between two research areas in non-classical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical formalizations of normative reasoning in philosophical logic and AI; on the other hand, subordination algebras, investigated in the context of a research program integrating topological, algebraic, and duality-theoretic techniques in the study of the semantics of modal logic. Specifically, we propose that the basic framework of input/output logic, as well as its extensions, can be given formal semantics on (slight generalizations of) subordination algebras. The existence of this interpretation brings benefits to both research areas: on the one hand, this connection allows for a novel conceptual understanding of subordination algebras as mathematical models of the properties and behaviour of norms; on the other hand, thanks to the well developed connection between subordination algebras and modal logic, the output operators in input/output logic can be given a new formal representation as modal operators, whose properties can be explicitly axiomatised in a suitable language, and be systematically studied by means of mathematically established and powerful tools.