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Classification and deontic explosion for contrary-to-duty obligations

This paper critiques Carmo and Jones' recent axiom system for conditional obligations by demonstrating a limited form of deontic explosion, while simultaneously offering a positive classification of all satisfying models for their strongest 1997 system in terms of a single forbidden possible world.

Original authors: Bjørn Kjos-Hanssen

Published 2026-04-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bjørn Kjos-Hanssen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Trying to Write the Rules of "Should"

Imagine you are trying to write a rulebook for a very strict, logical robot that decides what people ought to do. This isn't about what people want to do, but what they should do.

In philosophy and logic, this is called Deontic Logic. The paper looks at a specific set of rules (axioms) created by researchers named Carmo and Jones over the last 25 years. These rules try to handle Contrary-to-Duty (CTD) obligations.

What is a CTD obligation?
It's a rule that kicks in when you mess up.

Normal Rule: "You ought to visit your sick grandmother."
CTD Rule: "If you don't visit her, you ought to at least call her."

The paper asks: Do these rules work perfectly together, or do they break the robot's brain?

Part 1: The "Deontic Explosion" (The Brain Glitch)

The author, Bjørn Kjos-Hanssen, found a weird glitch in the most recent version of these rules. He calls it a "limited form of deontic explosion."

The Analogy: The Grading System
Imagine a student, James, taking a test. The grades are A (Best), B, C, D, and F (Worst).

The Ideal: If James gets an A or a B, he should get an A. (Makes sense).
The Backup: If James gets a C or a D, he should get a C. (Also makes sense; if you can't get an A, settle for a C).
The Glitch: The logic rules say that if you combine these ideas, something crazy happens.
- The rules imply that if James doesn't get an A (the best), then getting a C becomes mandatory.
- Even worse, if James gets a B, the logic says he should have gotten a C? No, that's not quite it.
- The Real Glitch: The logic concludes that if the "perfect" outcome (A) is impossible, then any "okay" outcome (like a C) becomes the only acceptable thing.

The "Bee Gees" Moment
The author compares this to a heartbroken person.

Scenario: You want to marry your true love (Alice).
Reality: You can't marry Alice.
The Glitchy Logic: Because you can't have Alice, the rules say you must settle for your second choice (Beatrice) or even just stay single. But the logic gets twisted: it suggests that if you can't have the best, then any other option that isn't the worst becomes "obligatory."

It's like a GPS that says: "You missed the highway. Therefore, you are now legally required to drive on the sidewalk." It's a paradox where the rules force you into a mediocre situation just because the perfect one is gone.

Part 2: The Silver Lining (The Mathematical Treasure)

You might think, "Well, if the rules are broken, why bother?"

The author says: Wait! Even though the rules have a glitch, they are mathematically beautiful.

He looked at the strongest version of the rules (from 1997) and realized that all possible models (all the ways the robot could think) fall into a very simple pattern.

The Analogy: The "Forbidden World"
Imagine a universe of possible worlds (like different versions of reality).
The author discovered that for the rules to work perfectly, the universe can only have one specific "Bad World."

The Rule: "Avoid the Bad World."
The Result: If you are in a situation where you can avoid the Bad World, you must do so. If you are already in the Bad World, the rules break down (which makes sense).

It's like a game of "Red Light, Green Light" where there is only one specific spot on the floor that is "Red Light."

If you are anywhere else, you are free to move.
If you step on that one spot, you are in trouble.
The entire complex system of "what you should do" boils down to: "Don't step on the red spot."

This is a "Classification Theorem." In math, this is like saying, "All the different shapes you can make with these blocks are actually just variations of a single cube." It turns a messy, complicated problem into a simple, clean solution.

Part 3: The Takeaway

1. The Problem:
When we try to write logical rules for "what we should do when things go wrong," we often run into paradoxes. The rules can force us to accept mediocre outcomes just because the perfect one is unavailable. It's like a strict teacher who says, "Since you didn't get 100%, you must get a C, or you're failing."

2. The Discovery:
Despite the glitch, the underlying math is surprisingly simple. The "perfect" system of rules basically just says: "There is one bad thing to avoid. If you can avoid it, you must."

3. The Future:
The author used computer software (like a digital proof-checker) to verify all these claims. He suggests that while the current rules have flaws, understanding their simple mathematical structure helps us build better, more human-like logic systems in the future.

Summary in One Sentence

The paper shows that while trying to logically program "what we should do when we fail" leads to some weird paradoxes, the underlying math of the best system is actually quite simple: it just boils down to avoiding one single "forbidden" outcome.

1. Problem Statement

The paper addresses the foundational challenges in deontic logic, specifically regarding Contrary-to-Duty (CTD) obligations. CTD obligations arise when a primary duty is violated, triggering a secondary duty (e.g., "If you go to the neighbor's, you should call; if you don't go, you should not call").

The author focuses on the sequence of axiom systems proposed by Carmo and Jones between 1997 and 2022. While these systems aim to formalize conditional obligations ($ob(X|Y)$, meaning $X$ is obligatory in context $Y$ ), the paper investigates whether these axioms lead to logical anomalies (paradoxes) or possess a coherent mathematical structure. The central tension is between the intuitive desire to model CTD scenarios and the risk of deontic explosion (where a violation of a primary duty leads to the obligation of any arbitrary or suboptimal outcome).

2. Methodology

The author employs a rigorous axiomatic and model-theoretic approach:

Axiomatic Analysis: The paper aggregates and analyzes a specific set of axioms (labeled 5(a) through 5(g)) proposed by Carmo and Jones. These axioms govern a function $ob: P(W) \to P(P(W))$ , where $W$ is a finite set of possible worlds.
Counter-Example Construction: The author constructs a specific scenario involving exam grades (A, B, C, D, F) to test the logical consistency of the axioms. By applying axioms 5(b), 5(e), and 5(f) sequentially, the author derives a "weak form of deontic explosion."
Model Classification: The paper shifts from negative results to positive structural analysis. It defines "least models" (minimal sets of obligations satisfying the axioms) and attempts to classify all models satisfying the full system (Axioms 5(a)–5(e)).
Formal Verification: All mathematical claims are verified using the Lean proof assistant, and initial anomaly detection utilized scripts from the Maple computer algebra system.

3. Key Contributions

A. Demonstration of Deontic Explosion

The paper identifies a significant flaw in the Carmo and Jones systems (specifically the interaction of axioms 5(b), 5(e), and 5(f)).

The Scenario: Consider a student who ought to get an A (given A or B) and ought to get a C (given C or D).
The Derivation:
1. By Axiom 5(b), if the grade is C or D, it ought to be A or C.
2. By Axiom 5(f), if the grade is A, B, C, or D, it ought to be A or C.
3. By Axiom 5(e) (monotonicity in sub-contexts), since "A or C" is obligatory in the context {A, B, C, D}, it remains obligatory in the sub-context {B, C, D} (where A is impossible).
4. By Axiom 5(b) again, in the context {B, C, D}, if "A or C" is obligatory and A is impossible, then C becomes obligatory.
The Anomaly: The system concludes that if a student cannot get an A, they are obligated to get a C, even if a B is available. This is termed a "weak deontic explosion," where the unavailability of the best outcome forces the obligation of a mediocre outcome, rendering the logic counter-intuitive (e.g., "If I can't have you, I don't want nobody").

B. Classification of Models (The Positive Result)

Despite the anomaly, the author demonstrates that the strongest version of the Carmo and Jones system (Axioms 5(a)–5(e) with the strong intersection closure 5(c)) admits a complete mathematical classification.

Theorem 6: For any finite set of worlds $W$ $W$ , a function $ob$ satisfies the full system if and only if it falls into one of three categories:
1. Trivial: No obligations exist ( $ob(X) = \emptyset$ ).
2. Universal: Everything containing the context is obligatory ( $ob(X) = \{Y \mid \emptyset \neq X \subseteq Y\}$ ).
3. The "Forbidden World" Model: There exists exactly one "bad" world $a$ (the forbidden world). An obligation $Y$ holds in context $X$ if and only if $Y$ intersects $X$ and $Y$ contains all of $X$ except possibly the forbidden world $a$ .
- Significance: This implies that the only non-trivial way to satisfy these axioms is to define a single "worst-case scenario" that must be avoided. All other obligations are derived from the necessity of avoiding this single world.

C. Characterization of Least Models

The paper provides precise characterizations for the "least models" (minimal sets of obligations) for subsystems:

Theorem 4: The least model for Axiom 5(b) plus basic CTD assumptions is characterized by the structure canon2,II.
Theorem 5: The least model for Axioms 5(b), 5(d), and 5(f) is characterized by canon2.
These results offer a formal basis for understanding how different subsets of axioms handle conflicting obligations.

4. Results

Negative Result: The Carmo and Jones systems (specifically the 2022 iteration) suffer from a logical anomaly where the unavailability of a primary ideal outcome forces the obligation of a suboptimal outcome, effectively collapsing the distinction between "acceptable" and "obligatory" in specific contexts.
Positive Result: The full system (5(a)–5(e)) is mathematically rigid. It does not allow for complex, multi-faceted moral landscapes; it essentially reduces to a logic of avoiding a single forbidden world.
Independence: The axioms 5(a) through 5(e) are proven to be independent; no single axiom is a logical consequence of the others.

5. Significance

Theoretical Impact: The paper challenges the viability of the Carmo and Jones axioms as a comprehensive solution to CTD problems. It suggests that the "Standard System" may be too restrictive (leading to the "forbidden world" classification) or too permissive (leading to deontic explosion).
Methodological Contribution: It bridges the gap between philosophical deontic logic and formal mathematics. By using proof assistants (Lean) and computer algebra (Maple), the author ensures the validity of the paradox and the classification theorems.
Future Directions: The paper suggests that future research should focus on subsystems (e.g., replacing the strong intersection axiom 5(c) with the weak version) to see if they allow for richer, more nuanced classifications that avoid both the explosion and the "single forbidden world" rigidity.

In summary, the paper argues that while the Carmo and Jones systems fail to capture the full nuance of human moral reasoning due to deontic explosion, they possess a surprising and elegant mathematical structure that classifies all valid models as variations of a "single forbidden world" logic.