💻 computer science

The Varieties of Ought-Implies-Can and Deontic STIT Logic

This paper presents a modular framework and sound, complete sequent calculi for deontic STIT logic that formalizes, compares, and analyzes the logical relationships among ten distinct interpretations of the Ought-implies-Can principle.

Original authors: Kees van Berkel, Tim S. Lyon

Published 2026-04-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kees van Berkel, Tim S. Lyon

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

Imagine you are the captain of a ship. You have a map (your duties) and a compass (your choices). The ancient philosophical rule "Ought-implies-Can" (OiC) is like a safety check on your map. It says: "You can only be ordered to sail to a place if your ship is actually capable of getting there." You can't be blamed for not flying to the moon if you don't have a rocket.

However, philosophers have been arguing for centuries about exactly how strict this rule should be. Does "can" mean "physically possible"? Does it mean "I have the muscle to do it"? Or does it mean "I have the freedom to choose not to do it"?

This paper by Kees van Berkel and Tim Lyon is like a universal translator and a giant logic puzzle designed to sort out these ten different arguments. They take the messy, abstract debates of philosophy and translate them into a precise, mathematical language called STIT Logic (Seeing To It That).

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Can" is Too Vague

The authors start by saying that the phrase "Ought implies Can" is like a Swiss Army knife with ten different blades. Some people use the "screwdriver" blade (logical possibility), while others use the "bottle opener" blade (physical ability).

They identified 10 distinct versions of this rule, ranging from very weak to very strong:

The "Dreamer" (Weak): "You ought to do X" implies "X isn't a logical contradiction." (e.g., You can't be ordered to make a square circle).
The "Realist" (Medium): "You ought to do X" implies "X is actually possible right now." (e.g., You can't be ordered to open a window if you are tied to a chair).
The "Agent" (Strong): "You ought to do X" implies "You have the specific ability to choose X."
The "Rebel" (Very Strong): "You ought to do X" implies "You also have the ability to refuse to do X." (If you can't say no, you aren't really free to say yes).

2. The Solution: A Modular Lego Kit

The authors realized that old logic systems were like a pre-built house; you couldn't easily change the walls to fit different "Can" rules. So, they built a modular Lego kit (called a "Deontic STIT Framework").

The Base: A standard logic system that handles agents making choices.
The Bricks: They created specific "rules" (like D2, D3, D4, D5) that act like Lego bricks.
- If you snap on the "Possibility Brick," your logic system now only allows duties that are possible.
- If you snap on the "Control Brick," your system now requires that the agent has total control over the outcome.

This allows them to build 16 different logic systems, each representing a different philosophical stance on what it means to "be able to" do something.

3. The Tool: The "Proof-Search" Machine

To test these systems, they built a Labelled Sequent Calculus. Think of this as a high-tech detective's checklist.

How it works: You write down a rule (e.g., "If you ought to do X, then you can do X"). The machine tries to prove it.
The Magic: If the machine gets stuck and can't prove it, it doesn't just say "Error." It automatically builds a Counter-Model.
- Analogy: Imagine you claim, "I can fly." The machine tries to prove you can. It fails. Instead of just saying "No," it builds a tiny, perfect simulation of a world where you can't fly (maybe you have no wings, or gravity is too strong). This proves your claim is false in that specific scenario.

4. The Discovery: The "Endorsement Ladder"

By running their checklist on all 10 versions of "Ought-implies-Can," they discovered a hierarchy (a ladder).

The Endorsement Principle: This is their biggest finding. It's like a "Ripple Effect" in logic.
- If you adopt the Strongest version (e.g., "I must have total control to be obligated"), you are automatically committed to accepting all the Weaker versions (e.g., "It must be logically possible," "It must be physically possible").
- However, if you only accept the Weakest version, you are not committed to the Stronger ones.

They mapped out exactly which rules force you to accept others. It's like saying: "If you agree that a driver must have a license to drive, you automatically agree that the driver must be alive. But if you agree the driver must be alive, you don't necessarily agree they need a license."

5. Why This Matters

Before this paper, philosophers were shouting past each other, using the same words ("Ought," "Can") but meaning different things.

For Philosophers: This paper provides a "Rosetta Stone." It shows exactly how different interpretations of moral responsibility relate to each other.
For AI and Law: As we build AI agents that make decisions (like self-driving cars or legal bots), we need to know exactly what "duty" means. If an AI is told "You ought to save the pedestrian," does it matter if the AI could have chosen not to? This paper gives the engineers the precise mathematical tools to define those boundaries.

Summary

In short, the authors took a fuzzy philosophical debate, built a precise mathematical toolbox to dissect it, and produced a family tree of moral rules. They showed us that "Ought implies Can" isn't just one rule; it's a spectrum, and choosing one point on that spectrum logically forces you to accept the points below it.

1. Problem Statement

The paper addresses a gap in the formalization of Deontic STIT Logic (Seeing To It That), a prominent framework for analyzing multi-agent choice and agency. While traditional deontic STIT logic incorporates a version of the Ought-implies-Can (OiC) principle (the metaethical postulate that "what an agent ought to do, the agent can do"), it relies on a specific, utilitarian-based interpretation.

However, philosophical literature contains a wide variety of OiC interpretations ranging from weak (logical possibility) to strong (full control). Existing frameworks struggle to accommodate this diversity because:

Traditional STIT relies on a utilitarian semantics (maximizing utility) which makes it difficult to isolate specific modal properties of "can."
There is no unified formal taxonomy establishing the logical relationships (subsumption, independence, equivalence) between these ten distinct philosophical readings of OiC.

The authors aim to create a modular framework that can formally capture, compare, and analyze these ten different OiC principles within a single logical system.

2. Methodology

The authors employ a modular relational semantics approach combined with labelled sequent calculi.

A. Semantic Framework (Relational Semantics)

Instead of the traditional branching-time semantics with utility functions, the authors adopt a relational semantics (based on [14]) that separates the structural properties of the logic from specific utility constraints.

Language ( $L_n$ ): A multi-agent modal language including:
- $[i]\phi$ : Agent $i$ sees to it that $\phi$ .
- $\Box\phi$ : $\phi$ is settled true (historical necessity).
- $\Diamond\phi$ : $\phi$ is actualizable (dual of $\Box$ ).
- $\otimes_i\phi$ : It ought to be the case for agent $i$ that $\phi$ .
Frames ( $DS_n$ ): Defined as tuples $\langle W, R_\Box, \{R_{[i]}\}, \{R_{\otimes_i}\} \rangle$ $⟨ W, R_{□}, {R_{[i]}}, {R_{\otimes_{i}}}⟩$ .
- $R_\Box$ partitions worlds into "moments."
- $R_{[i]}$ partitions moments into agent choices (equivalence relations).
- Independence of Agents (C3): Ensures simultaneous choices are consistent.
- Ideal World Constraints (D1-D5): The authors introduce four specific frame properties to modulate the behavior of the obligation operator ( $\otimes_i$ $\otimes_{i}$ ):
  - D2 (Seriality): Obligations are consistent (ideal worlds exist).
  - D3 (Moment-Constraint): Ideal worlds are confined to the current moment (realizable).
  - D4 (Choice-Constraint): Obligations are about specific choices (agent sees to it).
  - D5 (Ideal Choice): There exists an ideal choice for every agent.
Deliberative Operators: To handle "deliberative" readings of OiC (where obligations must be violable), the authors define derived operators:
- $\otimes^d_i \phi \equiv \otimes_i \phi \land \Diamond \neg \phi$ (Weak deliberative: obligation can be violated).
- $\otimes^c_i \phi \equiv \otimes_i \phi \land \Diamond [i]\neg \phi$ (Strong deliberative: agent can choose to violate).

B. Proof-Theoretic Approach (Labelled Sequent Calculi)

The authors construct sound and complete labelled sequent calculi ( $G3DS_nX$ ) for every combination of the frame properties $X \subseteq \{D2, D3, D4, D5\}$ .

Structure: Rules are derived directly from the semantic clauses and frame properties.
Features:
- Eigenvariables: Used in rules like $(\Box)$ and $([i])$ to handle universal quantification over accessible worlds.
- Path Conditions: Rules use $\Box$ -paths and $[i]$ -paths (equivalence relation simulations) to handle transitivity and symmetry without explicit structural rules.
- Geometric Axioms: Property D5 is handled via a system of rules $(D5_i)$ , a technique for handling geometric axioms in labelled calculi.

3. Key Contributions

A. Taxonomy of Ten OiC Principles

The paper formally defines and categorizes ten distinct interpretations of OiC found in philosophy, mapping them to specific logical formulas:

OiLP (Logical Possibility): $\otimes_i \phi \to \neg \otimes_i \neg \phi$ (Consistency).
OiAP (Actually Possible): $\otimes_i \phi \to \Diamond \phi$ .
OiA (Ability): $\otimes_i \phi \to \Diamond [i]\phi$ .
OiV (Violability): $\otimes^d_i \phi \to \Diamond \neg \phi$ .
OiR (Refrainability): $\otimes^d_i \phi \to \Diamond [i]\neg [i]\phi$ .
OiO (Opportunity): $\otimes^d_i \phi \to (\Diamond \phi \land \Diamond \neg \phi)$ .
OiA+O (Ability & Opportunity): $\otimes^d_i \phi \to (\Diamond [i]\phi \land \Diamond \phi \land \Diamond \neg \phi)$ .
OiCtrl (Control): $\otimes^c_i \phi \to (\Diamond [i]\phi \land \Diamond [i]\neg \phi)$ .
OiNC (Normatively Can): $\otimes_i \phi \to \otimes_i \Diamond \phi$ .
OiNA (Normatively Able): $\otimes_i \phi \to \otimes_i \Diamond [i]\phi$ .

B. Logical Taxonomy and the "Endorsement Principle"

Using the calculi, the authors construct a lattice of logics (Fig. 4) based on the strength of the underlying frame properties.

Subsumption: They determine which logics entail others. For example, the logic with properties $\{D3, D5\}$ is sufficient to derive the traditional STIT version of OiC (OiA).
The Endorsement Principle: This is a novel conceptual contribution. It states that endorsing a stronger OiC principle (e.g., Control) logically commits an agent to endorsing all weaker principles entailed by the logic (e.g., Logical Possibility, Ability). Conversely, rejecting a weaker principle forces the rejection of all stronger ones.
Independence: The authors prove the mutual independence of certain principles (e.g., OiNA is independent of OiA) by constructing counter-models from failed proof searches.

C. Formal Verification

Soundness and Completeness: The paper proves that the sequent calculi $G3DS_nX$ are sound and complete with respect to the relational semantics $DS_nX$ .
Counter-Model Extraction: The calculi allow for the constructive extraction of counter-models when a proof search fails, providing a mechanical way to demonstrate that one principle does not entail another.

4. Results

Modular Framework: The authors successfully decoupled the utilitarian constraints of traditional STIT, allowing for a fine-grained analysis of OiC.
Hierarchy of Logics: A complete lattice of 16 deontic STIT logics was generated.
- The weakest system ( $DS_n$ ) validates only basic consistency.
- The strongest system ( $DS_n\{D2, D3, D4\}$ ) corresponds to the traditional "dominance ought" logic of Horty.
Resolution of Philosophical Debates:
- The paper clarifies that the traditional STIT logic (Horty) corresponds to OiA (Ability) but requires specific frame properties ( $D3, D5$ ) to be valid.
- It shows that OiCtrl (Control) requires the strongest constraints ( $D3, D5$ ) plus the deliberative operator.
- It demonstrates that OiNA (Normatively Able) is strictly independent of the standard ability-based principles, challenging the assumption that normative ability follows from physical ability.

5. Significance

Unification: This work provides the first comprehensive formal taxonomy of the diverse philosophical interpretations of "Ought-implies-Can."
Proof-Theoretic Rigor: By using labelled sequent calculi, the authors move beyond axiomatic systems, offering a computational method to determine logical relationships and generate counter-examples.
Philosophical Clarity: The "Endorsement Principle" offers a clear logical structure for ethical debates. It suggests that one cannot consistently hold a strong view of agency (e.g., "I am only obliged if I have total control") without implicitly accepting weaker views (e.g., "I am only obliged if it is logically possible").
Future Applications: The framework lays the groundwork for automated normative reasoning, decision algorithms, and further investigation into the intersection of epistemic, temporal, and deontic logic.

In summary, the paper transforms the philosophical debate on "Ought-implies-Can" into a precise logical landscape, demonstrating how different interpretations of agency and possibility relate to one another through a modular, proof-theoretic framework.