{"id":"bf8adee6-18d3-4f63-a15e-99a2ab75c136","arxiv_id":"2505.06824","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"CIO defines instrumental obligation as 'the action that causes the goal and ranks highest in an ideal priority ordering,' and it is axiomatized completely with NP-complete satisfiability.","lead":"This paper proposes a logic, CIO, that treats instrumental obligation, such as 'in order to lose weight you ought to exercise,' as a causal claim: an action is obligatory when it causes the goal and is the best of the actions that do. The authors give a sound and complete Hilbert system and prove satisfiability is NP-complete, connecting causal reasoning with deontic priority structures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness proof is not checkable: the canonical model invokes an undefined axiom 'Gob' and the Truth Lemma omits the O_i case, so Theorems 1–2 are unsupported as written.","rationale":"The reader is right that the preprint should not be accepted, and I agree with the overall REJECT verdict. But the load-bearing issue I would put first is not the philosophical reduction in Section 2.2; that reduction is a coherent design choice and could be defended. The place where the central claim actually breaks down is the completeness proof in Section 6.1. The canonical model is asserted to be well-defined using an axiom 'Gob' that never appears in the calculus (the text says: 'A1, A2 and (Gob) guarantee that each A_U=u is well-defined'), and the Truth Lemma is not proved for the O_i operator defined in Eq. (3). In addition, axiom A1 is binary-only while the language allows arbitrary ranges, so the uniqueness of canonical assignments is not established. These are not cosmetic omissions: without a well-defined assignment A_U=u for every exogenous context u, there is no canonical causal deontic model for an arbitrary maximal consistent set, and hence no completeness theorem. The NP-completeness proof inherits the same fragility because it relies on the same semantic definitions. A concrete repair is possible, but the preprint as written does not support its headline theorems. Because the reader also notes the incomplete proof in their rationale, though their 'weakest assumption' names the semantic premise, I mark agreement as partial and leave the verdict unchanged.","tokens_in":17074,"tokens_out":10718,"duration_ms":109048,"concrete_test":"To settle the concern, complete the canonical-model proof on a minimal non-binary signature: one exogenous variable U with two contexts and one endogenous variable X with R(X)={0,1,2}. Take a maximal CIO-consistent set Gamma containing U=u but no (X=2)_u. Show from the listed axioms alone that A_u(X) has a unique value; if it does not, the canonical construction fails exactly where the unlisted Gob was invoked. Independently, prove the Truth Lemma for the O_i case on the Section 4 example, verifying that the universal conjunction in Eq. (3) has the same truth value in the canonical model as membership in Gamma; if either step cannot be completed, Theorems 1–2 remain unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem requires CIO to be sound and complete. Soundness is plausible, but the completeness proof in §6.1 cannot be checked. The canonical model sets A_U=u(X)=x iff (X=x)_u is in Gamma and then says that 'A1, A2 and (Gob)' make A_U=u well-defined; no axiom Gob is listed in the calculus. A1 is stated only for binary values y,y' in {0,1} while Definition 1 allows arbitrary ranges R, and A2 gives only existence, so the required uniqueness of x does not follow from the displayed axioms. The Truth Lemma is argued only for atoms, priority atoms, and a few _u reduction cases; the case for the derived O_i operator in Eq. (3), which contains a universal conjunction over alternative interventions and an order comparison, is not proved. Since O_i is the paper's central defined notion, Theorems 1–2 are unsupported as written. The permission operator Pi in Eq. (4) is also syntactically ill-formed, with an unbound context u' and mismatched intervention vectors, which reinforces that the formal core has not been pinned down.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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'.","tokens_in":17254,"tokens_out":8579,"duration_ms":79718,"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":[{"comment":"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":"Section 6.1, Definition 'Canonical model' and following paragraph"},{"comment":"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":"Section 6.1, axiom A1 and the Truth Lemma"},{"comment":"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":"Section 5, Eq. (4)"},{"comment":"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.","section":"Section 6.2, Lemma 2 and Theorem 3"}],"minor_comments":[{"comment":"The phrase 'causal-denotic' should be 'causal deontic'.","section":"Section 6.1, proof of Theorem 1"},{"comment":"The canonical model is numbered 'Definition 1' for a second time; renumber it to avoid confusion with Definition 1 in Section 3.","section":"Section 6.1, Definition 'Canonical model'"},{"comment":"Condition (ii) reads 'A'(Y)=y != A'(Y)', which is self-contradictory; it should presumably be 'A(Y)=y != A'(Y)'.","section":"Section 5, text after Eq. (1)"},{"comment":"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.","section":"Section 6.1, Truth Lemma"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central idea is promising and the explicit abbreviation-based reduction of instrumental obligation is a strength. However, the formal core is not yet in a publishable state: the completeness proof invokes a nonexistent axiom 'Gob', the functionality axiom A1 does not match the general language, and the permission operator is ill-formed. These are substantial but potentially fixable issues; if the authors can supply a complete and correct proof, the paper may be publishable. I recommend major revision rather than rejection because the identified problems appear addressable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nWho 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.\n\nIf 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.","headline":"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.","tokens_in":17805,"tokens_out":2586,"would_cite":false,"duration_ms":29102,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Instrumental obligation is shown to be a derived notion built from causal intervention and priority","keywords":["instrumental obligation","causal deontic models","priority structures","intervention formulas","structural equation models","deontic logic","NP-completeness","soundness and completeness"],"falsifier":"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.","tokens_in":16839,"feed_emoji":"🎯","tokens_out":9055,"duration_ms":91532,"temperature":0.7,"pith_summary":"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.","feed_headline":"Instrumental obligation reduced to causal intervention and priorities","feed_subtitle":"New causal deontic models define 'ought' as the best causal route to a goal.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the philosophical distinction that instrumental ought-propositions are not conditional obligations; the paper's semantics is designed to match the 'in order that' reading.","marker":"[5]"},{"why":"Raises the question of how goals interact with the ordering source; the paper's answer—that goals constrain rather than change the ideal ordering—directly responds to it.","marker":"[6]"},{"why":"Provides the axiomatization of causal reasoning that the CIO axioms for intervention extend, and the NP-hardness reduction inherited by the new logic.","marker":"[9]"},{"why":"Introduces priority structures (P-graphs) used to derive the ideal ordering from a priority ordering over propositions.","marker":"[1]"},{"why":"Supplies the structural equation model formalism in which interventions and causal models are defined.","marker":"[18]"},{"why":"Shows how to extend a causal model with a priority structure for normative concepts; the paper adapts this idea by placing the P-graph over atoms.","marker":"[24]"}],"fun_headline_variants":["Instrumental obligation as best causal route to goal","Causal deontic logic: obligation as optimal intervention","Obligation defined via causal interventions and priority order","New logic says ought means best way to achieve goal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Instrumental obligation as best causal route to goal","Causal deontic logic: obligation as optimal intervention","Obligation defined via causal interventions and priority order","New logic says ought means best way to achieve goal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000607,"raw_usage":{"total_tokens":2772,"prompt_tokens":831,"completion_tokens":1941,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":1887}},"tokens_in":447,"tokens_out":1941,"duration_ms":12078,"temperature":1.0,"reasoning_tokens":1887,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:33:20.789852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Philosophical studies143, 315–340 (2009)","cited_arxiv_id":null,"evidence_quote":"Supplies the philosophical distinction that instrumental ought-propositions are not conditional obligations; the paper's semantics is designed to match the 'in order that' reading."},{"cited_title":"In: Charlow, N., Chrisman, M","cited_arxiv_id":null,"evidence_quote":"Raises the question of how goals interact with the ordering source; the paper's answer—that goals constrain rather than change the ideal ordering—directly responds to it."},{"cited_title":"Journal of Artificial Intelligence Re- search12, 317–337 (2000)","cited_arxiv_id":null,"evidence_quote":"Provides the axiomatization of causal reasoning that the CIO axioms for intervention extend, and the NP-hardness reduction inherited by the new logic."},{"cited_title":"Theoria 80(2), 116–152 (2014)","cited_arxiv_id":null,"evidence_quote":"Introduces priority structures (P-graphs) used to derive the ideal ordering from a priority ordering over propositions."},{"cited_title":"Journal of Logic and Computation34(2), 352–371 (2023)","cited_arxiv_id":null,"evidence_quote":"Shows how to extend a causal model with a priority structure for normative concepts; the paper adapts this idea by placing the P-graph over atoms."}],"review_version":1}