arxiv: 2604.27917 · v1 · submitted 2026-04-30 · 💻 cs.LO · math.LO

Recognition: unknown

A Logic of Inability

Shanxia Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:33 UTC · model grok-4.3

classification 💻 cs.LO math.LO

keywords coalition logicinability operatormulti-agent systemsmodal logicconservative extensionagencyimpossibilitynegation

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The pith

Coalition Logic gains an explicit inability operator by defining it as the negation of ability, creating a conservative extension with its own modal laws.

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

The paper adds an operator for what coalitions cannot achieve to Coalition Logic, treating it simply as the logical opposite of what they can achieve. This produces a sound, complete, and conservative extension, so all prior theorems about ability remain unchanged while new facts about limits become provable. The operator displays anti-monotonicity in coalition size, contravariance in goal strength, asymmetric behavior under conjunction and disjunction, failure of superadditivity, non-equivalence to opponent ability, and a link between grand-coalition inability and overall systemic impossibility. These properties matter in safety-critical multi-agent systems where designers must specify not only permitted actions but also outcomes that agents are provably unable to bring about.

Core claim

Defining the inability operator as the negation of the coalition ability operator yields a conservative extension of Coalition Logic. The extension is sound and complete with respect to the standard semantics, and the new operator obeys anti-monotonicity with respect to coalition inclusion, contravariance with respect to goal strength, asymmetric interaction with conjunction and disjunction, failure of superadditivity, non-equivalence to the ability of opposing coalitions, and the equivalence of grand-coalition inability with systemic impossibility.

What carries the argument

The inability operator, defined as the negation of the standard coalition ability operator and interpreted over the same neighbourhood models.

If this is right

Larger coalitions are less prone to inability: if a small group cannot achieve a goal, larger groups containing it may still be unable, but the reverse need not hold.
Stronger goals are less prone to inability: inability to achieve a weak goal implies inability to achieve any stronger goal.
Inability distributes asymmetrically over conjunction and disjunction, unlike standard ability.
Inability lacks superadditivity: the inability of two disjoint coalitions to achieve separate goals does not imply their union cannot achieve both.
A coalition's inability to achieve a goal is not equivalent to the opposing coalition's ability to prevent it.
If the grand coalition is unable to achieve a goal, the goal is systemically impossible in the model.

Where Pith is reading between the lines

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

Specifications in multi-agent safety can now directly encode negative capabilities without indirect workarounds that only rule out ability.
The same definitional move could be applied to other ability logics such as ATL to obtain corresponding inability operators without new model classes.
Reasoning about constraints and impossibility in distributed systems becomes first-class rather than derived from absence of ability.
Verification tools could check both positive and negative agency statements in one uniform language.

Load-bearing premise

That inability is adequately captured simply by negating ability without any additional semantic primitives or dedicated axioms.

What would settle it

A concrete neighbourhood model in which a coalition cannot force a certain outcome yet the derived theorems about inability produce a contradiction with the original ability facts or violate one of the listed modal properties.

read the original abstract

Coalition Logic is primarily concerned with what coalitions can achieve, whereas what coalitions cannot achieve -- their \emph{inability} -- has received comparatively little explicit attention. This asymmetry matters in artificial intelligence and safety-critical multi-agent systems, where one often needs to specify not merely what agents are instructed or disposed not to do, but what they are \emph{unable} to bring about. We develop a conservative extension of Coalition Logic with an explicit inability operator, interpreted as the negation of coalition ability. This operator is introduced as a conservative and formally tractable starting point for studying inability as a modal concept in its own right. We prove soundness, completeness, and conservativity over standard Coalition Logic, and analyse the resulting modal behaviour: anti-monotonicity with respect to coalition inclusion, contravariance with respect to goal strength, asymmetric interaction with conjunction and disjunction, failure of superadditivity, non-equivalence with opponent ability, and the connection between grand-coalition inability and systemic impossibility. Making this definable operator explicit reveals a systematic modal structure governing the limits of agency, and supports reasoning about constraints, negative capabilities, and impossibility in multi-agent systems.

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

This paper defines inability as the negation of ability in Coalition Logic, proves the expected soundness and completeness results, and catalogs modal properties that follow directly from that definition.

read the letter

The main thing to know is that this paper defines an inability operator in Coalition Logic simply as the negation of the ability operator. It then claims to prove soundness, completeness, and conservativity, and analyzes properties such as anti-monotonicity with coalition size, contravariance on goals, and failure of superadditivity. What works here is the focus on a gap in the literature. Coalition Logic has been about what groups can achieve, but inability gets less direct treatment, and spelling out its modal behavior could help with safety specs in AI where you need to express hard constraints. The paper lists several interactions that differ from ability, like how it handles conjunction and disjunction asymmetrically, and how grand coalition inability relates to overall impossibility. That catalog is the useful part. The limitation is that the extension is conservative by design, so the technical results don't require new semantics or independent axioms. The properties derive straight from negating the existing ability semantics. This makes the work solid but modest; it's more about making something implicit explicit and exploring the consequences than introducing a genuinely new concept or proof technique. If the full paper has only the abstract-level claims without substantial examples or comparisons, the contribution stays on the lighter side. This paper suits logicians in the agency and multi-agent logic area, or AI folks who want formal tools for impossibility. Someone looking for a clean treatment of negative capabilities in coalitions would get something out of it. It should go to peer review because the claims are checkable and the application motivation is reasonable, though reviewers will likely note how definitional the core move is. Recommendation: Send it for review with a note to assess if the modal analysis adds enough beyond the definition to justify publication in a logic journal.

Referee Report

0 major / 3 minor

Summary. The paper develops a conservative extension of Coalition Logic by introducing an explicit inability operator interpreted directly as the negation of the coalition ability operator. It claims to prove soundness, completeness, and conservativity with respect to the standard Coalition Logic semantics, and analyzes the resulting modal properties: anti-monotonicity with respect to coalition inclusion, contravariance with respect to goal strength, asymmetric interaction with conjunction and disjunction, failure of superadditivity, non-equivalence with opponent ability, and the connection between grand-coalition inability and systemic impossibility. The extension is motivated by the need to reason explicitly about constraints and negative capabilities in multi-agent systems and AI safety.

Significance. If the central claims hold, the work provides a simple, formally tractable way to make inability explicit within an established logic without introducing new semantic primitives or axioms. This supports direct reasoning about limitations in multi-agent settings, which is relevant for specification and verification tasks. The modal analysis derives non-trivial properties (such as failure of superadditivity and the distinction from opponent ability) as consequences of the base semantics and negation, offering insight into the structure of negative agency. The conservative nature is a strength, as it ensures compatibility with existing Coalition Logic results and tools.

minor comments (3)

The abstract lists six modal properties; the main text should include explicit cross-references (e.g., to lemmas or propositions) showing where each is derived from the semantics of negation and the Coalition Logic axioms.
A short illustrative example in the introduction or semantics section, showing a concrete multi-agent scenario where the inability operator is used to express a constraint that cannot be captured as concisely without it, would improve readability.
Notation for the inability operator (e.g., whether it is written as a new primitive or explicitly as ¬[C]φ) should be fixed consistently from the first use onward.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of the contribution. The recommendation to accept is appreciated, as is the recognition that the explicit inability operator provides a simple, conservative way to reason about negative agency without new semantic primitives. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper explicitly defines the inability operator as the negation of the coalition ability operator and proves that the resulting extension is conservative over standard Coalition Logic. Soundness, completeness, and the listed modal properties (anti-monotonicity, contravariance, failure of superadditivity, etc.) are direct logical consequences of this definition together with the semantics of the base logic; they do not reduce any claimed result to a fitted parameter, self-citation chain, or hidden ansatz. No load-bearing step equates a derived theorem to its own input by construction, and the derivation remains self-contained against the external benchmark of Coalition Logic.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established semantics and axioms of Coalition Logic and introduces the inability operator purely as its negation; no free parameters, new entities, or ad-hoc axioms beyond the standard framework are indicated.

axioms (2)

standard math Standard Coalition Logic axioms and semantics
The extension is defined as conservative over standard Coalition Logic.
domain assumption Inability operator defined as negation of ability operator
Explicitly interpreted as the negation of coalition ability as a conservative starting point.

pith-pipeline@v0.9.0 · 5492 in / 1271 out tokens · 65909 ms · 2026-05-07T05:33:11.147118+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Beyond Ability: The Four-Fold Spectrum of Power and the Logic of Full Inability
cs.LO 2026-05 unverdicted novelty 7.0

Coalition Logic is extended by defining Full Inability (FI) as a distinct modality alongside Full Control, Positive Determination, and Adverse Determination, with algebraic structure, Klein four-group symmetry, and a ...

Reference graph

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