Deontic Action Logics: A Modular Algebraic Perspective

2025-02-19

AI Summary

K. Segerberg’s Deontic Action Logic (DAL) lacks a unified algebraic semantics, hindering its formal analysis and application in normative reasoning. Method: We introduce the first modular algebraic framework—Deontic Action Algebra—comprising a Boolean algebra of actions, a Heyting algebra of state propositions, and deontic mappings for permission and prohibition, forming a dual-algebraic structure; crucially, we employ Heyting algebras to model state propositions, thereby extending intuitionistic semantics to normative action reasoning. Contribution/Results: We systematically derive four semantically distinct variants of DAL and establish their algebraic completeness. This framework uniformly captures diverse action-oriented deontic logics, providing a rigorous, modular, and scalable algebraic foundation for compliance verification and explainable decision-making in AI systems.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Algebraization of Deontic Action Logic (DAL)
• Introduction of deontic action algebras
• Variations of DAL with Heyting algebras

Innovation

• Algebraizes deontic action logic
• Introduces deontic action algebras
• Varies DAL with Heyting algebras