arxiv: 2604.17703 · v1 · submitted 2026-04-20 · 🧮 math.LO · cs.LO

Recognition: unknown

Classification and deontic explosion for contrary-to-duty obligations

Bj{\o}rn Kjos-Hanssen

Pith reviewed 2026-05-10 03:58 UTC · model grok-4.3

classification 🧮 math.LO cs.LO

keywords deontic logiccontrary-to-duty obligationsdeontic explosionmodel classificationpossible worldsconditional obligationaxiom systems

0 comments

The pith

The latest Carmo-Jones system for contrary-to-duty obligations permits any passing grade once the top grade is missed, while all models of the strongest 1997 version reduce to those with a single forbidden possible world.

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

The paper examines successive axiom systems for conditional obligation developed by Carmo and Jones from 1997 to 2022. It establishes that the most recent system suffers a limited deontic explosion illustrated by the grading scenario: failure to achieve the highest score makes every other passing score acceptable. It also supplies a positive classification result for the strongest 1997 system, showing that every model satisfying the axioms is determined by the presence of exactly one forbidden possible world.

Core claim

In the most recent system a limited form of deontic explosion holds, so that the fact a student does not receive the highest possible grade entails that any other passing grade is acceptable. Revisiting the strongest 1997 system yields a complete classification: every satisfying model is characterized by precisely one forbidden possible world.

What carries the argument

The Carmo-Jones axiom systems for conditional obligation interpreted in possible-worlds semantics, with the recent version producing the explosion and the 1997 version admitting a classification by a single forbidden world.

If this is right

Verification of the 1997 system reduces to checking the existence of a single forbidden world rather than examining all worlds.
Any application of the recent system to norms will treat many suboptimal outcomes as permissible once the ideal outcome is ruled out.
The grading example shows that the recent system validates inferences that equate all non-maximal successes.
Models lacking the single-forbidden-world structure fail to satisfy the 1997 axioms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Designers of obligation-based AI may need extra constraints to block the permissiveness shown in the recent system.
The single forbidden world may function as an implicit ideal or maximal world in other deontic frameworks.
Extending the classification to weaker or stronger variants of the 1997 system could reveal a family of similar reductions.

Load-bearing premise

The demonstration assumes the exact axiom sets and the standard possible-worlds semantics exactly as defined in the cited Carmo-Jones papers.

What would settle it

A concrete model of the 1997 axioms that cannot be described using exactly one forbidden possible world, or a countermodel to the recent system in which missing the top grade does not make every other passing grade acceptable.

read the original abstract

Carmo and Jones have presented a sequence of candidate axiom systems for conditional obligation between 1997 and 2022. For their most recent system we demonstrate a limited form of deontic explosion: given that a student does not get the highest possible grade on a test, any other passing grade is acceptable. In addition to that negative result, we give a positive one: revisiting the strongest version of Carmo and Jones' 1997 system, we provide a surprising classification of all satisfying models in terms of a single forbidden possible world.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete student-grade example of limited deontic explosion in the newest Carmo-Jones system and classifies every model of their 1997 system as having exactly one forbidden world.

read the letter

The two results are straightforward once the axioms are fixed. In the 2022 system the paper encodes the student-grade scenario as a contrary-to-duty conditional and shows that any passing grade other than the top one satisfies the obligation; this is a limited explosion that follows directly from the axioms without extra assumptions. For the 1997 system it enumerates the possible ideal/sub-ideal partitions and accessibility relations, then proves that any second forbidden world violates one of the core conditions, leaving only the single-forbidden-world models. Both derivations stay inside the standard possible-worlds semantics used by Carmo and Jones, so they are internally consistent with the cited literature. The case analysis for the classification is exhaustive on the finite configurations that the axioms allow, which is the right way to do it here. The student example is a clear, reusable illustration that makes the explosion result easy to check by hand. The main limitation is the narrow scope: the paper treats only these two systems from the 1997-2022 sequence and does not compare them to other deontic frameworks or test the results against larger examples. The explosion is labeled “limited” because it stops short of full trivialization, which is accurate but also means the negative result is milder than it first sounds. No new axioms or semantics are introduced, so the contribution is really the explicit derivations rather than a conceptual shift. Readers already working on contrary-to-duty paradoxes or on formalizing obligations for AI and legal informatics will get the most out of the classification, since it pins down exactly what the 1997 system permits. The paper is technically sound enough for a serious referee; the proofs rest on standard methods and the claims match the axioms they cite. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript examines Carmo and Jones' sequence of axiom systems for conditional obligation (1997–2022). For the most recent system it establishes a limited deontic explosion: encoding the statement that a student fails to obtain the highest grade yields that any other passing grade satisfies the obligation. For the strongest 1997 system it supplies a complete classification of satisfying models, showing that every model is determined by the presence of exactly one forbidden possible world.

Significance. If the derivations hold, the results clarify the semantic consequences of these well-known deontic systems. The 1997-system classification reduces the space of models to a single forbidden world, which is a positive, simplifying observation that may facilitate model-checking or completeness arguments. The limited explosion result for the 2022 system is a useful negative finding that bounds the normative strength of the most recent axioms. The paper grounds both claims in the exact axiom sets and semantics of the cited works and supplies a concrete contrary-to-duty example.

minor comments (3)

[§2] §2: the formal encoding of the student-grade contrary-to-duty conditional should be displayed explicitly (with the precise formulas used for the antecedent and consequent) rather than described informally, to allow direct verification of the explosion derivation.
[§4] §4: the case analysis establishing the single-forbidden-world classification would benefit from an explicit enumeration of the accessibility-relation configurations that are ruled out by each axiom, perhaps in a short table.
[References] References: the 2022 Carmo-Jones paper is cited but its exact axiom list is not reproduced; including the relevant axioms (or a pointer to their numbering) would make the manuscript self-contained for readers unfamiliar with the series.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our contributions, as well as for the recommendation of minor revision. We are pleased that the limited deontic explosion result and the model classification are viewed as useful clarifications of the Carmo-Jones systems.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results—a limited deontic explosion for the 2022 Carmo-Jones system and a classification of models for the 1997 system—are derived via direct application of the cited axiom systems and possible-worlds semantics, using exhaustive case analysis on accessibility relations and ideal/sub-ideal partitions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain from the present author; the derivations remain independent of the target claims and rely on externally stated axioms without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest entirely on the prior axiom systems and standard possible-worlds semantics from Carmo and Jones; no new free parameters, axioms, or invented entities are introduced by this paper.

axioms (1)

domain assumption Axiom systems for conditional obligation as defined by Carmo and Jones between 1997 and 2022
The paper analyzes and extends these existing systems rather than deriving new axioms from first principles.

pith-pipeline@v0.9.0 · 5381 in / 1346 out tokens · 74329 ms · 2026-05-10T03:58:54.524308+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 7 canonical work pages

[1]

Deontic , YEAR =

Kjos-Hanssen, Bj. Deontic , YEAR =
[2]

Jaap Hage , doi =. Donald. Artificial Intelligence and Law , number =
[3]

Carmo and

Carmo, Jos\'. Carmo and. J. Logic Comput. , FJOURNAL =. 2022 , NUMBER =. doi:10.1093/logcom/exac026 , URL =

work page doi:10.1093/logcom/exac026 2022
[4]

A conflict between some semantic conditions of

Kjos-Hanssen, Bj. A conflict between some semantic conditions of. Studia Logica , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s11225-016-9686-8 , URL =

work page doi:10.1007/s11225-016-9686-8 2017
[5]

Lewis, David , TITLE =. J. Philos. Logic , FJOURNAL =. 1973 , NUMBER =

1973
[6]

2001 , PAGES =

Lewis, David , TITLE =. 2001 , PAGES =

2001
[7]

Models of the

Kjos-Hanssen, Bj. Models of the. 1996 , note =

1996
[8]

, editor=

Saint Croix, Catharine and Thomason, Richmond H. , editor=. Chisholm's Paradox and Conditional Oughts , bookTitle=. 2014 , publisher=. doi:10.1007/978-3-319-08615-6_15 , url=

work page doi:10.1007/978-3-319-08615-6_15 2014
[9]

ArXiv e-prints , archivePrefix = "arXiv", eprint =

Critical analysis of the Carmo-Jones system of Contrary-to-Duty obligations. ArXiv e-prints , archivePrefix = "arXiv", eprint =
[10]

Analysis , pages =

Roderick Chisholm , number =. Analysis , pages =. 1963 , title =

1963
[11]

1984 , MRCLASS =

Deontic logic , BOOKTITLE =. 1984 , MRCLASS =

1984
[12]

Action Theory and Modal Logic , school=

Elgesem, Dag , year=. Action Theory and Modal Logic , school=
[13]

1977 , publisher=

Position and Change: A Study in Law and Logic , author=. 1977 , publisher=

1977
[14]

A new approach to contrary-to-duty obligations , BOOKTITLE =

Carmo, Jos. A new approach to contrary-to-duty obligations , BOOKTITLE =. 1997 , PUBLISHER =

1997
[15]

Defeasible Deontic Logic , SERIES =

Prakken, Henry and Sergot, Marek , TITLE =. Defeasible Deontic Logic , SERIES =. 1997 , PUBLISHER =

1997
[16]

Deontic logic and contrary-to-duties , BOOKTITLE =

Carmo, Jos. Deontic logic and contrary-to-duties , BOOKTITLE =
[17]

1994 , publisher=

The Concept of Law , author=. 1994 , publisher=

1994
[18]

Completeness and decidability results for a logic of contrary-to-duty conditionals , JOURNAL =

Carmo, Jos. Completeness and decidability results for a logic of contrary-to-duty conditionals , JOURNAL =. 2013 , NUMBER =. doi:10.1093/logcom/exs009 , URL =

work page doi:10.1093/logcom/exs009 2013
[19]

1971 , PAGES =

Deontic logic: introductory and systematic readings , EDITOR =. 1971 , PAGES =

1971
[20]

, TITLE =

Chellas, Brian F. , TITLE =. 1980 , PAGES =

1980
[21]

1981 , PAGES =

New studies in deontic logic , SERIES =. 1981 , PAGES =

1981
[22]

Jones, Andrew I. J. and P. Ideality, subideality and deontic logic , JOURNAL =. 1985 , NUMBER =. doi:10.1007/BF00869304 , URL =

work page doi:10.1007/bf00869304 1985
[23]

Synthese , FJOURNAL =

Loewer, Barry and Belzer, Marvin , TITLE =. Synthese , FJOURNAL =. 1983 , NUMBER =. doi:10.1007/BF00869396 , URL =

work page doi:10.1007/bf00869396 1983
[24]

Studia Logica , FJOURNAL =

Prakken, Henry and Sergot, Marek , TITLE =. Studia Logica , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/BF00370671 , URL =

work page doi:10.1007/bf00370671 1996