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arxiv: 2507.05519 · v8 · submitted 2025-07-07 · 💻 cs.AI · cs.LO

Modeling Deontic Modal Logic in the s(CASP) Goal-directed Predicate Answer Set Programming System

Gopal Gupta , Abhiramon Rajasekharan , Alexis R. Tudor , Elmer Salazar , Joaqu\'in Arias This is my paper

Pith reviewed 2026-05-19 05:16 UTC · model grok-4.3

classification 💻 cs.AI cs.LO
keywords deontic modal logicanswer set programmingASPnegation as failureglobal constraintsmodal operatorsparadox resolutionknowledge representation
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The pith

Deontic modal logic operators are expressed directly in answer set programming using default negation, strong negation, and global constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates a representation of deontic modal logic inside answer set programming. Operators for obligation, prohibition, and permission are captured by combining negation-as-failure with strong negation and global constraints. This encoding resolves multiple classic paradoxes of deontic logic without extra machinery. The same approach also supports conditional obligations and prohibitions in knowledge representation.

Core claim

Deontic modal operators can be elegantly and directly expressed using default negation and strong negation in ASP, with global constraints representing obligations, prohibitions, and permissions, resulting in the resolution of decades-old paradoxes.

What carries the argument

Default negation (negation-as-failure), strong negation, and global constraints in answer set programming to represent deontic modal operators.

Load-bearing premise

The semantics of default negation and strong negation together with global constraints in ASP can faithfully capture the intended meaning of deontic modal operators.

What would settle it

A concrete deontic paradox or standard axiom violation that still appears when obligations and permissions are encoded with these ASP constructs.

Figures

Figures reproduced from arXiv: 2507.05519 by Abhiramon Rajasekharan, Alexis R. Tudor, Elmer Salazar, Gopal Gupta, Joaqu\'in Arias.

Figure 1
Figure 1. Figure 1: 3-fold partition of propositions in Deontic logic compared to ASP notation 4. not p: denotes that p may be false (no evidence that p is true, i.e., non￾necessary p). 5. -p: denotes that p is unconditionally false (impossible p). As noted, the notion “unconditionally true p” maps to “necessary p”, “maybe true p” maps to “possible p”, “unconditionally false p” maps to “impossible p”, “unknown p” maps to “con… view at source ↗
Figure 2
Figure 2. Figure 2: Encoding with s(CASP) of the example from [26] modeling continuous time [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

We consider the problem of implementing deontic modal logic. We show how (deontic) modal operators can be elegantly and directly expressed using default negation (negation-as-failure) and strong negation present in answer set programming (ASP). We propose using global constraints of ASP to represent obligations, prohibitions, and permissions in deontic modal logic. We show that our proposed representation results in the various decades-old paradoxes of deontic modal logic being simply and elegantly resolved. Our method also serves as a means for modeling conditional obligations and conditional prohibitions in knowledge representation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a method for modeling deontic modal logic within the s(CASP) goal-directed predicate answer set programming system. It demonstrates how deontic operators for obligation (O), permission (P), and prohibition can be represented using default negation (negation-as-failure), strong negation, and global constraints in ASP. The central claim is that this representation resolves various long-standing paradoxes in deontic logic, such as those identified by Ross and Chisholm, in a simple and elegant manner, while also enabling the modeling of conditional obligations and prohibitions.

Significance. Should the proposed encoding faithfully capture the semantics of standard deontic logic without introducing non-monotonic artifacts that alter the intended meanings, this work could offer a valuable bridge between modal logic and practical ASP-based reasoning systems. It has potential significance for applications in AI ethics, legal reasoning, and knowledge representation where deontic concepts are crucial. The goal-directed nature of s(CASP) may provide computational benefits for querying such models.

major comments (2)
  1. [Proposed Representation and Paradox Resolution] The central claim that paradoxes are resolved by the ASP encoding (via default/strong negation and global constraints) requires explicit demonstration that key deontic properties such as O(p) → ¬O(¬p) and consistency under conditional obligations are preserved rather than being side-effects of stable-model minimality or constraint failure. Without a semantic correspondence argument or counterexample check, it is unclear whether the approach solves the paradoxes internally to deontic logic or merely avoids them through ASP mechanisms.
  2. [Modeling Conditional Obligations] The modeling of conditional obligations is described at a high level but lacks detail on how global constraints interact with default negation in s(CASP) to prevent unintended propagation or loss of intended conditional semantics; this is load-bearing for the claim that the method supports conditional obligations without introducing new inconsistencies.
minor comments (2)
  1. [Notation and Encoding] Clarify the precise ASP syntax used for representing the modal operators (e.g., how 'not' and '-' are mapped to O, P, and ~O) with at least one fully worked example that includes both a query and the resulting answer set.
  2. [Introduction] Add citations to foundational deontic logic literature on the specific paradoxes addressed (Ross, Chisholm) and to prior work on ASP encodings of modal logics to situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [Proposed Representation and Paradox Resolution] The central claim that paradoxes are resolved by the ASP encoding (via default/strong negation and global constraints) requires explicit demonstration that key deontic properties such as O(p) → ¬O(¬p) and consistency under conditional obligations are preserved rather than being side-effects of stable-model minimality or constraint failure. Without a semantic correspondence argument or counterexample check, it is unclear whether the approach solves the paradoxes internally to deontic logic or merely avoids them through ASP mechanisms.

    Authors: We agree that an explicit demonstration of the key properties would strengthen the central claim. In the revised manuscript we will add a dedicated subsection that shows O(p) → ¬O(¬p) is enforced directly by the global constraint encoding of obligation together with strong negation, and that models violating this property are eliminated. We will also supply counterexample checks for Ross’s and Chisholm’s paradoxes, demonstrating that the resolutions follow from the intended deontic mapping rather than being incidental consequences of stable-model minimality. A short semantic-correspondence argument relating the ASP stable models to the standard deontic semantics will be included. revision: yes

  2. Referee: [Modeling Conditional Obligations] The modeling of conditional obligations is described at a high level but lacks detail on how global constraints interact with default negation in s(CASP) to prevent unintended propagation or loss of intended conditional semantics; this is load-bearing for the claim that the method supports conditional obligations without introducing new inconsistencies.

    Authors: We accept that the current description of conditional obligations is too high-level. The revised version will expand this section with concrete s(CASP) program fragments and a step-by-step account of how the goal-directed solver evaluates a conditional obligation rule. We will illustrate that the global constraint is activated only when the antecedent is satisfied, that default negation prevents unintended propagation to unrelated literals, and that the resulting answer sets preserve the intended conditional semantics without introducing new inconsistencies. Additional query examples will be provided to make the interaction explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: standard ASP semantics applied to deontic modeling without self-referential reductions or fitted predictions

full rationale

The paper presents an encoding of deontic operators (obligation, permission, prohibition) via default negation, strong negation, and global constraints in s(CASP). No equations, derivations, or parameter fits are shown that reduce the claimed resolution of paradoxes (Ross, Chisholm, etc.) to the inputs by construction. The central claim rests on applying established non-monotonic ASP semantics to a new domain rather than redefining terms or importing uniqueness via self-citation chains. Any prior author work on s(CASP) is not load-bearing for the deontic encoding itself, which is presented as a direct modeling choice. The approach is self-contained as a representational proposal and does not exhibit self-definitional, fitted-input, or ansatz-smuggling patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the domain assumption that ASP negation semantics can directly encode deontic operators and that global constraints suffice to represent obligations without side effects on the logic's properties.

axioms (2)
  • domain assumption Default negation and strong negation in answer set programming can directly express deontic modal operators.
    Invoked in the abstract to map modal logic constructs to ASP features.
  • domain assumption Global constraints in ASP can represent obligations, prohibitions, and permissions.
    Stated as the proposed representation for deontic concepts.

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discussion (0)

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Reference graph

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