A logic for instrumental obligation

Jialiang Yan; Qingyu He

REVIEW 4 major objections 4 minor 24 references

A logic for instrumental obligation

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Instrumental obligation is shown to be a derived notion built from causal intervention and priority

desk verdict Genuinely nice semantic idea for instrumental obligation as 'best intervention causing the goal,' but the completeness proof is not checkable as written and the permission operator is ill-formed; worth engaging with, not for the theorems as stated. read the letter →

arxiv 2505.06824 v1 pith:AGJXKY6G submitted 2025-05-11 math.LO

Jialiang Yan , Qingyu He This is my paper · ORCID

classification math.LO MSC 03B45

keywords instrumentalobligationcausaldeonticmodelsprioritystructuresinterventionformulasstructuralequationlogicNP-completenesssoundnessandcompleteness

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

The paper tries to show that instrumental obligation—the reading of 'ought' in sentences like "in order to evade arrest, Max ought to mingle with the crowd"—does not need a new primitive modal operator. It can be defined inside a causal model that has been enriched with a priority ordering over atomic facts. An action is instrumentally obligatory toward a goal exactly when the goal does not already hold, the action causally brings the goal about, and every other intervention that also brings the goal about leads to a state no better under the priority-induced ideal ordering. If the account is right, the instrumental 'ought' is the best causal route to a goal, and all reasoning about such obligations is captured by a sound and complete Hilbert system whose satisfiability problem is NP-complete.

What carries the argument

The central object is the causal deontic model: a recursive structural equation model together with a priority ordering $P$ over atoms, which induces an ideal ordering over assignments (Definition 6). The load-bearing identity is the definition of $O_i$ in Eq. (3), which unpacks instrumental obligation as a conjunction of three conditions: the goal atom is absent, the named intervention causes the goal, and every goal-causing intervention leads to a state no better in the induced ideal ordering. This identity is what reduces instrumental obligation to causal intervention plus a lexicographic comparison of satisfied atoms. The Hilbert system CIO then supplies axioms for interventions, for the strict partial order of priorities, and for a global operator naming the current exogenous assignment; completeness is shown by building a canonical model whose structural functions and priority ordering are read directly off a maximal consistent set.

What would settle it

Build the paper's John model, then add a new endogenous variable $E$ with structural equation $C = A \lor B \lor E$, $D = B$, and an added high-priority atom $H$ that only $E=1$ makes true. Keep the actual assignment at $A=B=E=0$. The semantics of Eq. (3) quantifies over all interventions, so the intervention $E=1$ counts as a competing way to achieve $C=1$, and it yields a state ranked strictly above the $A=1$ state by the priority ordering. Consequently the formula $O_i(C{=}1 : A{=}1)$ comes out false. If one judges that John is still instrumentally obligated to exercise because $E$ is not an option available to him, this is a direct counterexample to the paper's reduction of instrumental obligation to intervention plus fixed priority.

Watch

Extended reading notes

Core claim

The paper's central claim is that the valid principles of instrumental obligation are exactly those provable in the system CIO, and that the semantic content of $O_i$ is given by Eq. (3). For a goal $X=x$ and an action $Y=y$, $O_i(X=x : Y=y)$ holds in the actual assignment precisely when: $X=x$ is currently false; intervening to set $Y=y$ makes $X=x$ true; and for every alternative intervention $Z=z$ that also makes $X=x$ true, the state reached by $Z=z$ is no better than the state reached by $Y=y$, under the ideal ordering induced by the priority structure on atoms. This makes instrumental obligation a derived notion—an action is the best way to achieve the goal. The paper further claims soundness and completeness for CIO with respect to all causal deontic models (Theorems 1 and 2) and NP-completeness of satisfiability (Theorem 3), thereby locating the logic's computational boundary.

Load-bearing premise

The load-bearing premise is that a single fixed priority ordering over atomic facts fully determines which outcomes are better, and that the goal only filters which interventions are candidates, never reshaping how outcomes are compared.

Editorial extensions

If this is right

Instrumental 'ought' statements become testable causal claims: checking whether an obligation holds requires checking interventions, not consulting a separate deontic primitive.
The logic's soundness and completeness mean that any valid inference about instrumental obligation can be derived in CIO, so automated reasoning over such obligations is in principle available.
Satisfiability being NP-complete puts the logic in the same complexity class as ordinary propositional satisfiability, so there are no additional high-complexity obstacles from the causal or priority structure.
Instrumental permission is not the dual of obligation: an action can be permitted whenever it achieves the goal and is not worse than doing nothing, even if a better method exists.
Because the obligation is defined via intervention formulas, the goal's achievement is a causal consequence of the obligated action, not a conditional premise that detaches only hypothetically.

Reading between the lines

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

A natural extension is to make the priority ordering itself goal-dependent: if 'goal achieved with side effects' and 'goal missed without side effects' are incomparable rather than ranked by a fixed order, Eq. (3) would need refinement. This is a testable variant, not a claim of the paper.
Because Eq. (3) quantifies over all interventions, the semantics treats merely possible interventions on a par with available actions. Adding an availability relation over interventions would make the logic sensitive to an agent's options, a step the paper does not take.
The NP-completeness proof suggests the logic could serve as a specification language for planning problems where actions are modeled as interventions and goals as atoms; the priority structure then acts as a lexicographic objective function.
The definition handles only a single goal atom; combining goals as conjunctions is the obvious extension and is flagged as future work, but the complexity bound would likely survive because the formula size still bounds the model.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's real contribution is the semantic reduction: read instrumental obligation as "the best intervention that causes the goal," formalized via causal intervention formulas plus a priority-induced ideal ordering. That is a clean and genuinely new idea, and the weight-loss example shows it has intuitive bite. The authors also deserve credit for setting the reduction within a known framework (Halpern's causal models plus van Benthem-Grossi-Liu priority structures) and for flagging the difference between conditional and instrumental obligation at the start.

The formal part is not there yet. The stress-test note is right on target: the completeness proof in §6.1 is a sketch with a missing axiom. The canonical model construction refers to an axiom 'Gob' that never appears in the axiom list. A1 is stated only for binary values, but the semantics allow arbitrary ranges, and A2 only gives existence, not uniqueness, so the canonical model's structural functions are not well-defined by the displayed axioms. The Truth Lemma covers atoms, priority atoms, and the -u modality, but it does not handle the derived O_i operator; since O_i is introduced as an abbreviation this might be repairable by expanding definitions, but that is not done. The permission formula in Eq. (4) is genuinely ill-formed: the context u' is unbound, and the intervention variables are mixed up. These are load-bearing issues for the claimed soundness and completeness theorems.

The philosophical motivation is solid, and the approach is highly plausible: I would bet the completeness theorem can be repaired by adding the missing axiom, restricting A1, and carefully handling the derived operator. But as written, the paper does not support its main claims. The NP-completeness argument also depends on the completeness machinery, so it inherits the gaps.

Who is this for? Researchers in deontic logic, causal reasoning, or normative AI who want a causal account of instrumental 'ought.' They will get a promising framework and a clear research program, but they should not cite it for the theorems yet.

If I were an editor, I would send it to a serious referee, with explicit instructions to check the completeness proof and the permission operator. The paper needs a substantial revision, but the core idea deserves referee time.

Referee Report

4 major / 4 minor

Summary. The paper proposes a formal logic for instrumental obligation, understood as a causal notion: an action is instrumentally obligatory in order to achieve a goal if the action causes the goal and is optimal among the actions that cause it. The paper extends causal models with a priority-induced ideal ordering, defines a language LD with intervention operators, priority atoms, and a context-labeling operator, and then defines instrumental obligation O_i and permission P_i as derived operators (Eqs. (3) and (4)). The main formal claims are Theorems 1 and 2: the Hilbert calculus CIO is sound and complete with respect to causal deontic models, and Theorem 3: satisfiability for CIO is NP-complete. The paper illustrates the semantics with a weight-loss/pills/sports example and discusses philosophical motivations from Finlay's work on instrumental 'ought'.

Significance. If the results are correct, the paper provides a reductive analysis of instrumental obligation in terms of causal intervention and priority orderings, with a finitely axiomatized logic and a concrete complexity bound. A notable strength is that O_i is not taken as a primitive semantic primitive but is defined as an explicit abbreviation in the object language (Eq. (3)), which makes the reduction transparent and open to formal scrutiny. The NP-completeness theorem, if properly supported, would be a useful benchmark and would contrast with more complex deontic formalisms. The philosophical starting point, distinguishing instrumental obligation from conditional obligation, is well motivated. However, the formal core as written is not checkable: the completeness proof relies on an undefined axiom, the functionality axiom is too weak for the general language, and the permission operator is syntactically ill-formed.

major comments (4)

[Section 6.1, Definition 'Canonical model' and following paragraph] The proof that the canonical model is well-defined says: 'A1, A2 and (Gob) guarantee that each A_{\vec{U}=\vec{u}} is well-defined', but no axiom named 'Gob' appears in the calculus listed in Section 6.1. This leaves the central completeness proof (Theorem 2) unsupported at its first step. The authors must either add the missing axiom to the calculus or correct the reference.
[Section 6.1, axiom A1 and the Truth Lemma] Axiom A1 is stated only for y,y' in {0,1}, whereas Definition 1 (Language L) allows formulas X=x for arbitrary x in R. In the Truth Lemma, the step 'if [...]Y=y is not in Gamma, then by A1 and A2 there is y' != y such that [...]Y=y' is in Gamma' requires that the disjunct delivered by A2 be different from y and that no third value also be present. A1 does not deliver this for ranges with more than two values. Unless the language is restricted to binary variables or a genuinely general functionality axiom is added, the canonical structural functions f^Gamma_V are not shown to be single-valued, and the completeness theorem may fail.
[Section 5, Eq. (4)] The definition of the permission operator P_i is syntactically ill-formed. The conjunction ranges over \vec{y}', but the consequent contains an unbound variable list \vec{Z}=\vec{z} and an unbound context \vec{u}', and the two interventions in the consequent are both evaluated under \vec{u}' rather than under the actual context \vec{u}. Since instrumental permission is one of the paper's stated contributions ('The concept of instrumental permission is also taken into account in the model'), this definition must be corrected before the formal system can be evaluated.
[Section 6.2, Lemma 2 and Theorem 3] The NP upper-bound proof does not explain how the ideal-ordering comparisons in the expansion of O_i (and P_i) are checked in polynomial time. The definition of O_i in Eq. (3) includes a conjunction over all interventions \vec{Z} subset of V and \vec{z} subset of R(\vec{Z}), which is exponential in the number of variables. If the input length |phi| counts the unexpanded abbreviation, the NP bound needs a witness argument for these comparisons; if the expansion is part of the input, this must be stated explicitly. As written, the proof only indicates how structural functions and the priority ordering are guessed, and the sentence 'P can be described directly from the information of phi' is not a verification procedure.

minor comments (4)

[Section 6.1, proof of Theorem 1] The phrase 'causal-denotic' should be 'causal deontic'.
[Section 6.1, Definition 'Canonical model'] The canonical model is numbered 'Definition 1' for a second time; renumber it to avoid confusion with Definition 1 in Section 3.
[Section 5, text after Eq. (1)] Condition (ii) reads 'A'(Y)=y != A'(Y)', which is self-contradictory; it should presumably be 'A(Y)=y != A'(Y)'.
[Section 6.1, Truth Lemma] The Truth Lemma covers atoms, priority atoms, and intervention formulas, but it does not explicitly address the derived operators O_i and P_i. Since these are abbreviations rather than primitive symbols, this is not fatal, but the paper should state explicitly that formulas containing O_i and P_i are understood throughout as their expanded definitions, so that the completeness proof for the primitive language extends to the full defined language.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: instrumental obligation is introduced as an explicit stipulative abbreviation, and the metatheorems are internal to that semantics.

full rationale

The paper's central definition, Eq. (3), presents O_i(X=x : Y=y)_u as an explicit abbreviation in the object language: it conjoins (¬X=x)_u, ([Y=y]X=x)_u, and a universal comparison of all alternative interventions under the induced ideal ordering. No empirical parameter is fitted and no quantity is predicted from data; soundness/completeness (Theorems 1–2) and NP-completeness (Theorem 3) are internal metatheoretic results relative to the stipulated class of causal deontic models. The only overlapping-author citation, [24] (Xie and Yan), is used as a modeling precedent for adding a priority structure to a causal model, and the paper immediately adapts that method to P-graphs over causal atoms; it is not invoked as a uniqueness theorem, an external mathematical fact, or a forbidden alternative, so it is not load-bearing. The skeptical concerns about the formal core—the undefined axiom 'Gob' in the canonical-model construction at §6.1, the omitted O_i case in the Truth Lemma, and the binary-only statement of A1—are checkability and correctness gaps, not circular reductions: they make Theorems 1–2 unsupported as written, but they do not exhibit any conclusion that is equal to its premises by construction. The same holds for the ill-formed permission abbreviation in Eq. (4) at §5. Accordingly, no circular step is present and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim depends on standard mathematical assumptions (classical logic, finiteness, recursive equations) and on the domain assumption that instrumental obligation is best represented by comparing post-intervention states under a priority-induced ideal ordering. No empirical parameters are fitted and no new physical entities are introduced.

assumptions (5)

domain assumption The set of structural equations F is recursive (acyclic), so each setting of exogenous variables and each intervention has a unique solution.
Stated in Section 3.1 and used in Definition 2 and Definition 9 to make phi_u a nominal.
domain assumption The priority ordering over atoms is a strict partial order (asymmetric and transitive).
Axioms Asym and Trans in Section 6.1; the induced ideal ordering in Definition 6 depends on this.
domain assumption Exogenous values in the causal model are finite and fully determine the assignment through the structural equations.
Used in Definition 9 and in the O_i/P_i abbreviations; not true for cyclic or non-recursive models.
standard math The language uses classical propositional logic and standard Boolean semantics.
Axiom (P) and MP; all logical theorems are presumed.
domain assumption All variable ranges R and the domain of exogenous assignments are finite, which underlies the NP upper bound and the finite disjunctions in formulas (1)-(3).
Lemma 2 and the NP-completeness proof rely on finite signatures and finite value ranges.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A logic for instrumental obligation." pith.science (2026). https://pith.science/paper/AGJXKY6G

@misc{pith2026250506824,
  author       = {Pith},
  title        = {Pith review of: A logic for instrumental obligation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGJXKY6G}},
  note         = {Machine review of arXiv:2505.06824}
}

read the original abstract

This paper develops a logic based on causal inferences to formally capture the concept of instrumental obligation. We establish a causal deontic model that extends causal models with priority structures, allowing us to represent both the instrumental and deontic aspects of an obligation. In this framework, instrumental obligation is defined as a derived notion through intervention formulas of causal reasoning, where an action is considered obligatory if it is the best way to achieve the goal. We provide a sound and complete axiomatic system and show that the logic is NP-complete. The concept of instrumental permission is also taken into account in the model.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages

[1]

Theoria 80(2), 116–152 (2014)

van Benthem, J., Grossi, D., Liu, F.: Priority structures in deontic logic. Theoria 80(2), 116–152 (2014)

work page 2014
[2]

Journal of Logic, Language and Information4, 251–271 (1995)

Blackburn, P., Seligman, J.: Hybrid languages. Journal of Logic, Language and Information4, 251–271 (1995)

work page 1995
[3]

Journal of Philosophy60, 685–700 (1963)

Davidson, D.: Actions, reasons and causes. Journal of Philosophy60, 685–700 (1963)

work page 1963
[4]

In: Pigden, C.R

Finlay, S.: Against all reason? scepticism about the instrumental norm. In: Pigden, C.R. (ed.) Hume on Motivation and Virtue, pp. 150–173. Philosophers in Depth, Palgrave Macmillan, London (2009)

work page 2009
[5]

Philosophical studies143, 315–340 (2009)

Finlay, S.: Oughts and ends. Philosophical studies143, 315–340 (2009)

work page 2009
[6]

In: Charlow, N., Chrisman, M

Finlay, S.: ‘ought’: Out of order. In: Charlow, N., Chrisman, M. (eds.) Deontic Modality. Oxford University Press, Oxford (2016)

work page 2016
[7]

The Philosophical Re- view81(3), 305–316 (1972)

Foot, P.: Morality as a system of hypothetical imperatives. The Philosophical Re- view81(3), 305–316 (1972)

work page 1972
[8]

In: Deontic Logic and Normative Sys- tems, pp

Giordani, A.: Reason-based deontic logic. In: Deontic Logic and Normative Sys- tems, pp. 183–200. College Publications (2021)

work page 2021

Show all 24 references

[9]

Journal of Artificial Intelligence Re- search12, 317–337 (2000)

Halpern, J.Y.: Axiomatizing causal reasoning. Journal of Artificial Intelligence Re- search12, 317–337 (2000)

2000
[10]

Nous3, 373–398 (1969)

Hansson, B.: An analysis of some deontic logics. Nous3, 373–398 (1969)

1969
[11]

Philosophical Review82(4), 429–450 (1973).https://doi.org/10.2307/2183709

Hill, T.E.: The hypothetical imperative. Philosophical Review82(4), 429–450 (1973).https://doi.org/10.2307/2183709

1973 doi
[12]

Online Journal of Communication and Media Technolo- gies7(1), 126–146 (2017)

Hof, S.d.V.d., Hostut, S.: Instrumental, strategic and political conception of corpo- rate social responsibility. Online Journal of Communication and Media Technolo- gies7(1), 126–146 (2017)

2017
[13]

In: Globerson, A., Silva, R

Ibeling, D., Icard, T.: On open-universe causal reasoning. In: Globerson, A., Silva, R. (eds.) Proceedings of the Thirty-Fifth Conference on Uncertainty in Artificial Intelligence, UAI 2019, Tel Aviv, Israel, July 22-25, 2019. Proceed- ings of Machine Learning Research, vol. 1...

2019
[14]

Kratzer,A.:Modality.In:vonStechow,A.,Wunderlich,D.(eds.)Semantics:AnIn- ternational Handbook of Contemporary Research, pp. 639–650. De Gruyter (1991)

1991
[15]

Univer- sity of Amsterdam (2008)

Liu, F.: Changing for the better: Preference dynamics and agent diversity. Univer- sity of Amsterdam (2008)

2008
[16]

Liu,F.:ReasoningaboutPreferenceDynamics,SyntheseLibrary,vol.354.Springer (2011)

2011
[17]

Liu,F.:Atwo-levelperspectiveonpreference.JournalofPhilosophicalLogic40(3), 421–439 (2011)

2011
[18]

Biometrika82(4), 669–688 (1995), https://doi.org/10.1093/biomet/82.4.669

Pearl, J.: Causal diagrams for empirical research. Biometrika82(4), 669–688 (1995), https://doi.org/10.1093/biomet/82.4.669

1995 doi
[19]

In: Agotnes, T., Broersen, J., Elgesem, D

Roy, O., Anglberger, A.J.J., Gratzl, N.: The logic of obligation as weakest permis- sion. In: Agotnes, T., Broersen, J., Elgesem, D. (eds.) Deontic Logic in Computer Science, LNCS, vol. 7393, pp. 139–150. Springer, Berlin, Heidelberg (2012)

2012
[20]

Philosophical Perspectives18, 337–364 (2004)

Schroeder, M.: The scope of instrumental reason. Philosophical Perspectives18, 337–364 (2004)

2004
[21]

Australasian Journal of Philosophy 83, 357–372 (2005)

Schroeder, M.: The hypothetical imperative. Australasian Journal of Philosophy 83, 357–372 (2005)

2005
[22]

Ethics117, 649–673 (2007) A logic for instrumental obligation 17

Setiya, K.: Cognitivism about instrumental reason. Ethics117, 649–673 (2007) A logic for instrumental obligation 17

2007
[23]

In: Idowu, S.O., Capaldi, N., Zu, L., Gupta, A.D

Spitzeck, H.: Normative versus instrumental corporate responsibility. In: Idowu, S.O., Capaldi, N., Zu, L., Gupta, A.D. (eds.) Encyclopedia of Corporate Social Responsibility, pp. 1826–1833. Springer, Berlin, Heidelberg (2013)

2013
[24]

Journal of Logic and Computation34(2), 352–371 (2023)

Xie, K., Yan, J.: A logic for desire based on causal inference. Journal of Logic and Computation34(2), 352–371 (2023)

2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Theoria 80(2), 116–152 (2014)

van Benthem, J., Grossi, D., Liu, F.: Priority structures in deontic logic. Theoria 80(2), 116–152 (2014)

work page 2014

[2] [2]

Journal of Logic, Language and Information4, 251–271 (1995)

Blackburn, P., Seligman, J.: Hybrid languages. Journal of Logic, Language and Information4, 251–271 (1995)

work page 1995

[3] [3]

Journal of Philosophy60, 685–700 (1963)

Davidson, D.: Actions, reasons and causes. Journal of Philosophy60, 685–700 (1963)

work page 1963

[4] [4]

In: Pigden, C.R

Finlay, S.: Against all reason? scepticism about the instrumental norm. In: Pigden, C.R. (ed.) Hume on Motivation and Virtue, pp. 150–173. Philosophers in Depth, Palgrave Macmillan, London (2009)

work page 2009

[5] [5]

Philosophical studies143, 315–340 (2009)

Finlay, S.: Oughts and ends. Philosophical studies143, 315–340 (2009)

work page 2009

[6] [6]

In: Charlow, N., Chrisman, M

Finlay, S.: ‘ought’: Out of order. In: Charlow, N., Chrisman, M. (eds.) Deontic Modality. Oxford University Press, Oxford (2016)

work page 2016

[7] [7]

The Philosophical Re- view81(3), 305–316 (1972)

Foot, P.: Morality as a system of hypothetical imperatives. The Philosophical Re- view81(3), 305–316 (1972)

work page 1972

[8] [8]

In: Deontic Logic and Normative Sys- tems, pp

Giordani, A.: Reason-based deontic logic. In: Deontic Logic and Normative Sys- tems, pp. 183–200. College Publications (2021)

work page 2021