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Normative implications

Andrea De Domenico, Ali Farjami, Krishna Manoorkar, Alessandra Palmigiano, Mattia Panettiere, Apostolos Tzimoulis, Xiaolong Wang

TL;DR

The paper develops an algebraic framework for normative reasoning by introducing slanted Heyting and slanted co-Heyting algebras, shown to be equivalent to subordination algebras. It provides canonical extensions, residuation-like properties, and a rich set of normative axioms expressed as inequalities, enabling context-sensitive obligations through the contextual implication $\Rightarrow_{\prec}$. A central theme is the correspondence and inverse correspondence between algebraic conditions and normative constraints, extended to analytic $\mathcal{L}$-inequalities and Kracht-style formulations. The approach yields expressive modeling of deontic reasoning in diverse contexts (e.g., priorities, capacities, and sequential rules) and opens avenues for extending to permissions, alternate propositional bases, and generalized implications.

Abstract

We continue to develop a research line initiated in \cite{wollic22}, studying I/O logic from an algebraic approach based on subordination algebras. We introduce the classes of slanted (co-)Heyting algebras as equivalent presentations of distributive lattices with subordination relations. Interpreting subordination relations as the algebraic counterparts of input/output relations on formulas yields (slanted) modal operations with interesting deontic interpretations. We study the theory of slanted and co-slanted Heyting algebras, develop algorithmic correspondence and inverse correspondence, and present some deontically meaningful axiomatic extensions and examples.

Normative implications

TL;DR

The paper develops an algebraic framework for normative reasoning by introducing slanted Heyting and slanted co-Heyting algebras, shown to be equivalent to subordination algebras. It provides canonical extensions, residuation-like properties, and a rich set of normative axioms expressed as inequalities, enabling context-sensitive obligations through the contextual implication . A central theme is the correspondence and inverse correspondence between algebraic conditions and normative constraints, extended to analytic -inequalities and Kracht-style formulations. The approach yields expressive modeling of deontic reasoning in diverse contexts (e.g., priorities, capacities, and sequential rules) and opens avenues for extending to permissions, alternate propositional bases, and generalized implications.

Abstract

We continue to develop a research line initiated in \cite{wollic22}, studying I/O logic from an algebraic approach based on subordination algebras. We introduce the classes of slanted (co-)Heyting algebras as equivalent presentations of distributive lattices with subordination relations. Interpreting subordination relations as the algebraic counterparts of input/output relations on formulas yields (slanted) modal operations with interesting deontic interpretations. We study the theory of slanted and co-slanted Heyting algebras, develop algorithmic correspondence and inverse correspondence, and present some deontically meaningful axiomatic extensions and examples.

Paper Structure

This paper contains 14 sections, 8 theorems, 4 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaries
  4. Slanted Heyting algebras and subordination algebras
  5. Examples and discussion
  6. Slanted co-Heyting algebras
  7. Slanted pseudo-complements and co-complements
  8. More examples
  9. Correspondence and inverse correspondence
  10. Correspondence
  11. Inverse correspondence
  12. Modelling deontic reasoning
  13. Conclusions and future directions
  14. More examples of correspondence and inverse correspondence

Key Result

proposition 1

(cf. de2024obligations2) For any lattice $A$, all $k_1, k_2\in K(A^\delta)$, $o_1, o_2\in O(A^\delta)$, and $u_1, u_2\in A^\delta$,

Theorems & Definitions (15)

  • proposition 1
  • proposition 2
  • definition 1
  • definition 2
  • proposition 3
  • proof
  • lemma 1
  • proof
  • definition 3
  • definition 4
  • ...and 5 more