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Modeling Deontic Modal Logic in ASP

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Pith's one-line read The paper claims that deontic modal logic can be directly implemented in answer set programming by encoding obligations, prohibitions, and permissions as constraints, and that this dissolves its classic paradoxes.

desk verdict Nice concrete idea for deontic operators as ASP denials, but the paradox resolution rests on hand-picked preemption conditions and the paper needs a real derivation rule before that claim holds. read the letter →

arxiv 2507.05519 v10 pith:RSKFFMLH submitted 2025-07-07 cs.AI cs.LO

Gopal Gupta (The University of Texas at Dallas) , Abhiramon Rajasekharan (The University of Texas at Dallas) , Alexis R. Tudor (The University of Texas at Dallas) , Elmer Salazar (The University of Texas at Dallas) , Joaquín Arias (Universidad Rey Juan Carlos) This is my paper · ORCID
classification cs.AIcs.LO
keywords deonticmodallogicanswersetprogrammingglobalconstraintsoddloopsovernegationcontrary-to-dutyparadoxChisholm'sdefaultstrong
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the modal operators of deontic logic, obligation, prohibition, and permission, can be translated directly into answer set programming, the logic-based framework for declarative problem solving. In its encoding, an obligation that p is the denial ':- not p', a prohibition that p is ':- not -p', and a permission that p is ':- -p'; a conditional norm Q -> OBp becomes ':- not p, Q'. The authors claim that this constraint-based reading resolves long-standing deontic paradoxes such as Chisholm's contrary-to-duty paradox, Forrester's paradox, Sartre's dilemma, Ross's paradox, and the Good Samaritan paradox, because obligations are preempted when the condition that violates them is provable. If correct, the result gives a practical, executable route from normative language to reasoning programs without extra logical machinery.

What carries the argument

The load-bearing mechanism is the ASP global constraint, also called a denial: a rule of the form ':- not p, Q' that rejects every answer set in which Q is true and p is absent. Linked to it is the odd loop over negation (OLON) rule c :- not p, not c, which lets a violating condition c preempt the obligation; under the stable model semantics the rule is taken out of consideration when c is derivable by other rules, and otherwise acts as a denial forcing p. The mapping rests on treating default negation (not) as modal negation and strong negation (-) as propositional negation, so that the deontic square of opposition is reproduced by combinations of the two negations.

What would settle it

Run the paper's Chisholm encoding in a conventional answer-set solver under the standard stable-model semantics and compare the answer sets with the two worlds claimed in the paper; if the OLON rule is treated as inconsistent or yields additional worlds, the resolution depends on the goal-directed execution behavior rather than on the denial encoding alone.

Watch

Extended reading notes

Core claim

The central discovery is that the deontic operators OB, IM, and PE are not primitive modal machinery but shorthand for global constraints on the set of acceptable worlds. In answer set programming, 'it is obligatory that p' becomes the constraint that any answer set not containing p is rejected, written as the denial ':- not p'; 'impermissible that p' becomes ':- not -p'; and 'permitted that p' becomes ':- -p'. Conditional norms of the forms Q -> OBp and OB(Q -> p) both translate to the denial ':- not p, Q'. The paper further claims that when a violating condition c arises, the denial is re-encoded as an odd-loop-over-negation rule c :- not p, not c so that the obligation is dropped exactly when c is derivable, which is how the paradoxes are said to resolve. It also proves that the standard deontic logic axioms D, K, and NEC hold in this representation.

Load-bearing premise

The method presumes that for each obligation one can identify, in advance, the violating condition c that should preempt it, and in the paper these conditions are chosen by hand for each paradox rather than derived by a general procedure.

Editorial extensions

If this is right

  • Any Horn deontic formula in the supported fragment can be executed directly by an answer set programming system, making normative reasoning computable.
  • The paradoxes of contrary-to-duty reasoning disappear without introducing dyadic operators, sanctions, or preference orderings over models.
  • Conditional obligations and prohibitions become conditional constraints, so secondary obligations ('least one can do') can be represented by chained OLON rules.
  • Because constraints can be named via OLON rules, separate logic can decide when a norm applies, supporting elaboration-tolerant knowledge representation.
  • The approach distinguishes factual detachment from deontic detachment by whether the norm's antecedent is evaluated in the current or accessible world.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit is that the same constraint-as-denial translation should work for epistemic and temporal modalities, because the distinction between default and strong negation is not specific to deontic operators.
  • The paper does not give a general rule for choosing which condition c preempts an obligation; a testable extension would derive c automatically from a proof of violation, which would turn the hand-built paradox resolutions into a system.
  • Because the claimed OLON behavior is tied to the goal-directed execution system used in the paper, a natural probe is to run the same encodings on a conventional answer-set solver; if the answer sets differ, the paradox resolution depends on that system's semantics rather than on ASP itself.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a translation of a fragment of deontic modal logic into answer set programs (ASP). Obligations, impermissibilities, and permissions are represented as global constraints (denials): OBp as `:- not p`, IMp as `:- not -p`, PEp as `:- -p`, with conditional norms handled analogously. The authors further propose preempting an obligation by an OLON rule `c :- not p, not c` when a condition c violates the obligation, and they claim that this mechanism resolves Chisholm's contrary-to-duty paradox and several other classical deontic paradoxes. A theorem asserts that SDL axioms D, K, and NEC are satisfied. The paper also presents a car-borrowing example with executable s(CASP) code and justification trees.

Significance. If the central claim were fully established, the paper would offer a very simple and executable implementation of deontic reasoning in ASP, with a clean mapping between the two negations of modal logic and ASP's default and strong negation, plus a uniform treatment of contrary-to-duty obligations. The concrete encodings are a genuine strength: the Chisholm program has exactly the two admissible worlds reported, the s(CASP) examples in Section 6 are runnable, and the justification trees in Appendix C support reproducibility. However, the paradox-resolution claim is not established because the preemption condition c in Section 4.3 is chosen by hand in each example, and some appendix resolutions depart from the stated translation fragment. As written, the paper is better described as a framework for normative modeling with explicitly user-specified exceptions than as a resolution of the classical deontic paradoxes.

major comments (4)
  1. [Section 4.3 and Section 5] The preemption condition c is a free parameter with no derivation rule. The paper states that for an obligation OBp and a condition c that violates it, one should encode the OLON rule `c :- not p, not c`, but it never defines how c is obtained from the deontic theory or narrative. In each paradox, c is chosen to produce the intended answer set: c=-go in Chisholm (§5.1 line 1), c=kill in Forrester (§A.1 line 1), c=stay/join in Sartre (§A.2 lines 1-2), and c=broke in Kant (§A.2 line 1). This is load-bearing: replacing the Chisholm OLON rule with `tell :- not go, not tell` (i.e., c=tell) while keeping lines 2-4 and the fact `-go` yields no answer set, so the set of admissible worlds is not determined by the deontic formulas alone. A formal characterization of admissible preemption conditions, or a derivation procedure for c, is required before the claimed 'simple and elegant' resolution of the paradoxes can be assessed.
  2. [Appendix A.3] The Ross paradox example is not translated within the stated fragment. The premise 'mail -> post∨burn' is encoded as the two ASP rules `post :- mail` and `burn :- mail`, which assert both post and burn, not a disjunction. Since the syntax in Section 4.2 does not admit disjunction, this example does not demonstrate a resolution of Ross's paradox under the proposed encoding; it demonstrates a stronger, hand-chosen normative theory. The paper should either extend the fragment and give a translation of disjunction, or explain why the stronger encoding is the intended representation.
  3. [Appendix A.4] The Good Samaritan example does not actually resolve the paradox: the encoding `:- not rob` and `:- not help` forces rob and help in every answer set, which yields the very conclusion OBrob that the paradox calls into question. The text says 'in any world in which Smith is robbed, Jones helps him' and concludes the paradox is no longer a paradox, but this is an acceptance of the paradoxical conclusion rather than a resolution. The authors need to explain why conclusion OBrob is acceptable under the proposed semantics, or provide a different encoding that avoids it.
  4. [Section 4.2, Theorem] The theorem that SDL axioms D, K, and NEC are satisfied is stated more strongly than what is proven. The proof of D shows only that the two denials corresponding to OBp and OB¬p are jointly unsatisfiable via the built-in `:- p,-p`; this is a consistency property, not the derivation of the formula ¬OB¬p. The proof of NEC is not conclusive in the supported fragment: 'τ is a tautology' is not representable as a single atom under the Horn syntax of Section 4.2, and `not τ` is negation-as-failure over an atom, not classical negation of a formula. The theorem should be restated as a property of answer sets, with the scope of the fragment made explicit.
minor comments (4)
  1. [Section 4.3] The last paragraph says the obligation can be 'even further simplified' as `:- not p, not c`. This is indeed equivalent to the OLON rule, but stating the equivalence explicitly would avoid confusion, since the OLON form also names the exception as a head predicate.
  2. [Appendix B, Narratives 7 and 8] The explanatory text for Narrative 7 is repeated verbatim for Narrative 8, including the sentence about using classical negation `-financially_broke(jones)`, which is not present in Narrative 8. This appears to be a copy-paste error.
  3. [Figure 5] Line 24 of Figure 5 reads `same_battery_level(C, Tb, Tr). :-` with an extra period before the neck symbol; this is likely a typo.
  4. [Section 5.1] The sentence 'we will find that their are only two admissible worlds' contains a typo: 'their' should be 'there'.

Circularity Check

1 steps flagged · score 6.0 of 10

Paradox resolutions rest on hand-picked preemption conditions c, so the central 'resolution' claim reduces by construction to the chosen exceptions.

  1. fitted input called prediction [Section 4.3 (preemption rule); applied in Sections 5.1, A.1, A.2]
    "Given an obligation OBp, and a possible condition c that results in its violation, then instead of encoding the obligation as :- not p, we encode it as the OLON rule c :- not p, not c."

    Section 4.3 never derives c; it is simply 'given'. In the paradox sections c is selected so that the intended answer set exists: c=-go in Chisholm (5.1), c=kill in Forrester (A.1), c=stay in Sartre and c=broke in Kant (A.2). The paper even simplifies the rule to ':- not p, not c', i.e., OBp is enforced only when the chosen c is absent. Consequently the claimed 'resolution' of each paradox is not a consequence of the deontic-to-ASP encoding of Section 4.2; it is the effect of the hand-written exception literal. The admissible-world outcome changes with c (e.g., taking c=tell in the Chisholm program instead of c=-go leaves lines 2-4 and fact -go with no answer set), so no invariant, c-independent 'resolution' is established.

full rationale

The direct encodings (OBp -> :- not p, IMp -> :- not -p, PEp -> :- -p) and the claim that SDL axioms D, K, NEC are satisfied are definitional verifications of a proposed representation, not circular predictions. The extended car-borrowing example is a self-contained executable benchmark and does not depend on self-citation. The sole circular component is the paradox-resolution claim: Section 4.3 presupposes the violation condition c, and every paradox appendix chooses c by hand to produce the desired answer set. Since the paper gives no criterion for deriving c and no theorem that admissible worlds are invariant under the choice of c, the advertised 'simple and elegant resolution' is, for these cases, equivalent to the hand-fitted exception clauses. This is partial circularity in the central claim, not in the overall framework.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The framework's ledger: each paradox resolution depends on a per-example fitted choice of preemption condition c and abducible set, while the axioms are the norms-as-constraints thesis, the negation-mapping convention, and the standard ASP semantics. No new physical or logical entities are needed beyond the preemptible-constraint construct and the exception predicates of the car example.

free parameters (3)
  • preemption condition c per OLON obligation = -go (Chisholm), stay (Sartre), broke (Kant), fail_to_return_car / fail_to_return_by_noon / fail_to_return_ok_battery…
    Section 4.3 introduces c without a selection rule; each paradox example picks a different c so that the intended world survives, which functions as a per-example fitted choice.
  • abducible declarations = go, -go, tell, -tell in Chisholm; warning_sign, -warning_sign, dog, -dog in the dog example
    Section 5 declares which atoms are abducibles; this determines the space of possible worlds and is chosen per example.
  • obligation versus OLON split = in Chisholm only the 'go' obligation is made preemptible; lines 2 and 3 are plain denials
    Section 5.1: which norms get OLON preemption and which stay as unconditional denials is a modeling choice affecting the resulting worlds.
assumptions (5)
  • domain assumption Obligations, prohibitions, and permissions are global constraints on accessible worlds rather than truth-bearing declarative sentences.
    Section 4.1, resolving Jorgensen's dilemma; this is the philosophical core of the method, stated but not derived.
  • domain assumption Deontic formulas are Horn formulas with modal operators only at rule fronts, no nested modalities, no negated modal operators.
    Section 4.2 restricts the fragment; the paradoxes are all within it, and general deontic logic is not covered.
  • ad hoc to paper Negation next to a proposition maps to strong negation and negation next to a modal operator maps to default negation.
    Section 3 introduces this as the key translation principle; it is the paper's own convention and a premise for all encodings.
  • standard math Standard stable model semantics of ASP and the s(CASP) behavior for OLON rules.
    Section 2 relies on stable models; Section 4.3 additionally relies on the claim that an OLON rule is dropped when its head is provable elsewhere, which is not formalized.
  • ad hoc to paper Axiom D is guaranteed by the denial ':- p, -p' asserted by ASP for every predicate p.
    Sections 3 and 4.2 use this to prove D, conflating the inconsistency of contradictory constraints with the deontic axiom.
invented entities (2)
  • preemptible obligation construct (OLON rule c :- not p, not c) independent evidence
    purpose: represents an obligation that is dropped when a violation condition c holds
    The construct has well-defined ASP semantics and the paper provides runnable examples, so its behavior can be checked independently of the prose; but which c to use is chosen per example.
  • violation and exception predicates (fail_to_return_car, fail_to_return_by_noon, fail_to_return_ok_battery) independent evidence
    purpose: track whether a primary or secondary obligation was violated and cancel the relevant norm in the car-borrowing example
    These implement the preemption mechanism in a larger example; they are modeler-defined ledger entries whose consequences are verified by s(CASP) runs, but they are not derived from the deontic formulas automatically.

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Pith. "Pith review of Modeling Deontic Modal Logic in ASP." pith.science (2026). https://pith.science/paper/RSKFFMLH

@misc{pith2026250705519,
  author       = {Pith},
  title        = {Pith review of: Modeling Deontic Modal Logic in ASP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSKFFMLH}},
  note         = {Machine review of arXiv:2507.05519}
}
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.

Figures

Figures reproduced from arXiv: 2507.05519 by the authors.

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 “contingent p”… view at source ↗
Figure 1
Figure 1. Basic modal notions in Alethic modal logic and their denotation. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
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 ↗
Figures from the paper (4 more)
Figure 2
Figure 2. Figure 2: 3-fold partition of propositions in Deontic logic compared to ASP notation [PITH_FULL_IMAGE:figures/full_fig_p003_2.png]
Figure 3
Figure 3. Figure 3: Basic modal notions in Deontic modal logic and their denotation. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 4
Figure 4. Figure 4: ASP Encoding of Car-borrowing Example 7 Related Work There is significant work on resolving paradoxes of deontic logic going back decades (see [15]). To our best knowledge, none model deontic operators as constraints that the accessible worlds must satisfy [PITH_FULL_…
Figure 5
Figure 5. Figure 5: s(CASP) Encoding of Car-borrowing Example [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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    i.e.¬(join∧stay)

    Joining and staying are incompatible. i.e.¬(join∧stay). The ASP encoding is as follows. It states that staying preempts joining and vice versa./external-link-alt 1join :- not stay, not join. 2stay :- not join, not stay. 3:- stay, join. If we assign all possible combinations of...

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    I’m obligated to pay you back $10 tonight

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    encoded as: /external-link-alt 1broke :- not pay, not broke

    I can’t pay you back $10 tonight (e.g., just gambled away everything). encoded as: /external-link-alt 1broke :- not pay, not broke. 2:- broke, pay. 3broke. There is implicit knowledge that if a person is broke, they cannot pay, which is encoded as a constraint. That is, in any...

  32. [40]

    It is obligatory that the letter is mailed

  33. [41]

    If the letter is mailed, then the letter is posted or the letter is burned

  34. [42]

    This is encoded as:

    Therefore, it is obligatory that the letter is posted or the letter is burned. This is encoded as:

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    to mail a letter, we post it

    OBmail 2.mail→post∨burn. Therefore OB(post∨burn). The paradox arises due to the or-introduction rule: if the statement “to mail a letter, we post it” is true, then “to mail a letter, we post it or burn it” is also true. However, it should be noted that “to mail a letter, we po...

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    Justifications for goal-directed constraint answer set programming

    Arias, J., Carro, M., Chen, Z., and Gupta, G. Justifications for goal-directed constraint answer set programming. In ICLP-TC 2020, volume 325, 59–72. EPTCS

    2020

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.