Deontic Action Logics: A Modular Algebraic Perspective
Carlos Areces, Valentin Cassano, Pablo Castro, Raul Fervari
TL;DR
This work develops an algebraic, modular framework for deontic action logics built around Segerberg's $\mathsf{DAL}$, modeling permissions and prohibitions via two Boolean algebras linked by $\mathsf{P}$ and $\mathsf{F}$. It proves soundness and completeness for the base logic and shows how to generate multiple variants by altering algebraic components, including the introduction of propositions and the use of Heyting algebras for formulas or actions (or both). The approach yields a family of logics ($\mathsf{DAL}$, $\mathsf{DAL}(\mathsf{Prop})$, $\mathsf{DAL}(\mathsf{IPL})$, $\mathsf{DAL}(\mathsf{IAL})$, $\mathsf{DAL}(\mathsf{INT})$) with corresponding algebraic semantics and Lindenbaum–Tarski constructions, connected to the original Kripke-style models via Stone duality and model-algebra inverses. The framework is highly modular and scalable, enabling future work on action composition/iteration, alternative proposition algebras, and category-theoretic couplings, with potential applications in planning, fault-tolerance, and dynamic normative systems under partial observability.
Abstract
In a seminal work, K. Segerberg introduced a deontic logic called DAL to investigate normative reasoning over actions. DAL marked the beginning of a new area of research in Deontic Logic by shifting the focus from deontic operators on propositions to deontic operators on actions. In this work, we revisit DAL and provide a complete algebraization for it. In our algebraization we introduce deontic action algebras -- algebraic structures consisting of a Boolean algebra for interpreting actions, a Boolean algebra for interpreting formulas, and two mappings from one Boolean algebra to the other interpreting the deontic concepts of permission and prohibition. We elaborate on how the framework underpinning deontic action algebras enables the derivation of different deontic action logics by removing or imposing additional conditions over either of the Boolean algebras. We leverage this flexibility to demonstrate how we can capture in this framework several logics in the DAL family. Furthermore, we introduce four variations of DAL by: (a) enriching the algebra of formulas with propositions on states, (b) adopting a Heyting algebra for state propositions, (c) adopting a Heyting algebra for actions, and (d) adopting Heyting algebras for both. We illustrate these new deontic action logics with examples and establish their algebraic completeness.