Deontic Action Logics: A Modular Algebraic Perspective

Areces, C., Cassano, V., Castro, P., and Fervari, R.. Deontic Action Logics: A Modular Algebraic Perspective. Journal of Logic, Language and Information (JLLI), 2025. To appear

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Abstract

In a seminal work, K. Segerberg introduced deontic action logic (DAL) as a formal framework to investigate normative reasoning over actions. In this work, we re- visit DAL and provide a complete algebraization for it. To this end, we introduce deontic action algebras—algebraic structures consisting of a Boolean algebra for interpreting actions, a Boolean algebra for interpreting formulas, and two map- pings from one Boolean algebra to the other interpreting the deontic concepts of permission and prohibition. We show how this framework supports the derivation of various deontic action logics by imposing or relaxing structural conditions on either Boolean algebra. This flexibility allows us to uniformly account for several logics within the broader DAL family. In particular, we introduce four variations obtained 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.

BibTeX

@article{arec:deon25,
  author = {Areces, C. and Cassano, V. and Castro, P. and Fervari, R.},
  title = {Deontic Action Logics: A Modular Algebraic Perspective},
  journal = {Journal of Logic, Language and Information (JLLI)},
  note = {To appear},
  year = 2025,
  issn = "1572-9583",
  abstract = "In a seminal work, K. Segerberg introduced deontic
                  action logic (DAL) as a formal framework to
                  investigate normative reasoning over actions. In
                  this work, we re- visit DAL and provide a complete
                  algebraization for it. To this end, we introduce
                  deontic action algebras—algebraic structures
                  consisting of a Boolean algebra for interpreting
                  actions, a Boolean algebra for interpreting
                  formulas, and two map- pings from one Boolean
                  algebra to the other interpreting the deontic
                  concepts of permission and prohibition. We show how
                  this framework supports the derivation of various
                  deontic action logics by imposing or relaxing
                  structural conditions on either Boolean
                  algebra. This flexibility allows us to uniformly
                  account for several logics within the broader DAL
                  family. In particular, we introduce four variations
                  obtained 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.",
}