arxiv: 2605.04452 · v1 · submitted 2026-05-06 · 💻 cs.LO · math.LO

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Beyond Ability: The Four-Fold Spectrum of Power and the Logic of Full Inability

Shanxia Wang

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Pith reviewed 2026-05-08 17:11 UTC · model grok-4.3

classification 💻 cs.LO math.LO

keywords coalition logicfull inabilityeffectivity functionsklein four-groupdefinitional extensionstrategic powerplayable modelsorder convexity

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The pith

Coalition power splits exhaustively into four categories—full control, positive determination, adverse determination, and full inability—where the last means enforcing neither a claim nor its negation.

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

Coalition Logic has treated a coalition's inability to force a proposition simply as the negation of its ability. The paper shows this overlooks a distinct case in which the coalition can force neither the proposition nor its negation. Adding this full inability case to the existing notions produces four mutually exclusive and exhaustive categories of strategic status. These categories are closed under the operations of propositional negation and coalition complementation, forming a Klein four-group. The paper then defines a conservative extension CLFI of Coalition Logic that treats full inability as a primitive modality while preserving the original logic's expressive power and PSPACE complexity.

Core claim

The four categories FC, PD, AD, and FI partition a coalition's strategic status exhaustively and exclusively. Under α-duality generated by propositional negation and coalition complementation they form a Klein four-group. In playable models the four power regions are order-convex in the powerset lattice. CLFI is a sound, complete, conservative definitional extension of Coalition Logic that preserves PSPACE-completeness and supplies direct proof-theoretic access to symmetric inability, strategic dependence, propositional dummyhood, and containment verification.

What carries the argument

The full inability operator FI(C, φ), defined as the case in which coalition C enforces neither φ nor ¬φ, which completes the four-way partition and supplies the missing element for the Klein-group symmetry under α-duality.

If this is right

Every coalition's power relative to a given proposition belongs to exactly one of the four categories.
The categories are symmetric under the duality operations and therefore satisfy the algebraic identities of a Klein four-group.
In playable models each category corresponds to an interval, so inability properties can be verified by checking interval membership.
The logic CLFI adds full inability as a primitive while remaining equally expressive and PSPACE-complete.
Properties such as strategic dependence and propositional dummyhood receive direct syntactic representations.

Where Pith is reading between the lines

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

The interval-convexity result may support more efficient model-checking procedures that avoid enumerating all subsets.
The same four-way partition and duality could be examined in neighbouring formalisms such as alternating-time temporal logic.
Direct axiomatization of full inability may simplify proofs about coalitions that are strategically irrelevant for a given claim.
The conservative extension property ensures that existing Coalition Logic theorems remain valid inside the new system.

Load-bearing premise

The underlying effectivity functions must satisfy the standard Coalition Logic properties and the models must be playable.

What would settle it

A playable model in which the effectivity sets for some coalition and proposition place its status outside the four categories or in which the power regions fail to form convex intervals in the powerset lattice.

Figures

Figures reproduced from arXiv: 2605.04452 by Shanxia Wang.

**Figure 1.** Figure 1: gives a schematic Venn-style visualization of this decomposition. P(W) Ew(C) E∗ w(C) Rw FC(C) R Full Control w PD(C) Positive Determination Rw AD(C) Adverse Determination Rw FI(C) Full Inability view at source ↗

**Figure 2.** Figure 2: Strategy-cell geometric interpretation. Full Inability manifests as a universal view at source ↗

read the original abstract

Coalition Logic studies what coalitions can enforce. Recent work treats inability as simple non-ability: $\neg\Eff{C}\varphi$. This conflates two distinct configurations -- a coalition unable to force $\varphi$ may still force $\neg\varphi$, retaining adversarial control rather than genuine inability. We introduce \textbf{Full Inability} ($\FI$): the symmetric condition in which a coalition can enforce neither a proposition nor its negation. Combining coalitional effectivity with propositional negation yields a four-fold spectrum: \textbf{Full Control} ($\FC$), \textbf{Positive Determination} ($\PD$), \textbf{Adverse Determination} ($\AD$), and \textbf{Full Inability} ($\FI$). These categories partition a coalition's strategic status exhaustively and exclusively. We establish their algebraic and order-theoretic structure. Under $\alpha$-duality, propositional negation and coalition complementation generate a Klein four-group symmetry. In playable models, the four power regions are order-convex in the powerset lattice, yielding interval-stable verification of inability. We axiomatize $\CLFI$, a definitional extension treating Full Inability as a primitive modality. Via elimination translation, we prove soundness, completeness, and conservativity over Coalition Logic. The extension preserves expressive power and complexity ($\PSPACE$-complete), but provides direct proof-theoretic access to symmetric inability, strategic dependence, propositional dummyhood, and containment verification.

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 adds Full Inability as a primitive to Coalition Logic and shows the resulting four categories form a Klein four-group with order-convexity in playable models.

read the letter

The central move is to stop treating inability as plain negation of ability. Instead the paper defines Full Inability directly as the case where a coalition can force neither phi nor its negation. This produces four exhaustive labels: Full Control, Positive Determination, Adverse Determination, and Full Inability. The partition follows immediately from the two boolean conditions on effectivity functions, so it holds by construction in any effectivity model. Under alpha-duality the two involutions generate the Klein four-group symmetry, and the paper notes that the regions are order-convex inside playable models. The logic CLFI is then introduced as a definitional extension with an elimination translation that inherits soundness, completeness, and PSPACE-completeness from the base Coalition Logic. That is the actual new material: the primitive modality, the explicit four-fold spectrum, and the algebraic and order-theoretic claims. The translation approach is standard and keeps the extension conservative, which is a clean way to add expressive power without raising complexity. The limitation is that order-convexity is stated only for playable models, which already obey the usual monotonicity and consistency conditions; outside those models the convexity claim does not apply. The abstract asserts the translation works but does not display the detailed model definitions or the faithfulness proof, so a referee would need to verify there are no edge cases in the translation. This is aimed at researchers already working inside coalition logic and multi-agent modal logics who want a more direct language for symmetric inability and strategic dependence. It is a precise refinement rather than a foundational shift, so it will be useful to that narrow group. The work is coherent on its own terms and uses standard techniques, so it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces Full Inability (FI) as a primitive modality in Coalition Logic to distinguish genuine inability from cases where a coalition can still enforce the negation of a proposition. It defines four categories—Full Control (FC), Positive Determination (PD), Adverse Determination (AD), and Full Inability (FI)—that exhaustively and exclusively partition a coalition's strategic status based on effectivity functions. The paper demonstrates that these categories form a Klein four-group under α-duality and are order-convex in playable models. It axiomatizes CLFI as a definitional extension of Coalition Logic and uses an elimination translation to establish soundness, completeness, conservativity, and preservation of PSPACE-completeness.

Significance. This work provides a more granular analysis of coalitional power by formalizing symmetric inability, which has implications for strategic dependence and verification in multi-agent systems. The algebraic symmetry and order-theoretic properties add depth to the theory. A key strength is the definitional extension approach with elimination translation, which ensures the new logic inherits key properties from Coalition Logic without increasing complexity, facilitating adoption in existing frameworks.

minor comments (2)

The abstract is information-dense; consider breaking the description of the four-fold spectrum and the technical results (soundness via translation) into separate sentences for improved readability.
When discussing order-convexity in playable models, explicitly recall or cite the standard Coalition Logic axioms (monotonicity, superadditivity) that underpin the powerset interval behavior, even if they are assumed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, including the recognition of the four-fold spectrum, the Klein four-group structure under α-duality, the order-convexity in playable models, and the definitional extension approach that preserves soundness, completeness, conservativity, and PSPACE-completeness. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; all claims follow from explicit definitions and standard properties of Coalition Logic

full rationale

The four categories are introduced by direct combination of Eff_C(φ) with propositional negation and coalition complementation, so their exhaustive partition is definitional rather than derived. The Klein four-group structure is obtained from the two involutions (negation and complement) under α-duality, which commute and square to identity by elementary group theory on the labels. Order-convexity holds inside playable models by the monotonicity, superadditivity and consistency axioms already present in the base Coalition Logic. CLFI is presented as a conservative definitional extension equipped with an elimination translation; soundness, completeness and PSPACE preservation are therefore inherited once the translation is shown faithful, without any reduction of new results to fitted parameters or self-referential premises. No load-bearing step collapses to a self-citation chain or to an ansatz smuggled from prior work by the same author.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework introduces one new modality and relies on standard assumptions from Coalition Logic; no numerical free parameters are present as this is a logical axiomatization rather than a fitted model.

axioms (2)

domain assumption Standard Coalition Logic effectivity function properties (e.g., monotonicity, coalition monotonicity)
Invoked throughout to define the spectrum and prove partition and duality properties.
domain assumption Playable models for order-convexity of power regions
Required for the claim that the four regions are order-convex in the powerset lattice.

invented entities (1)

Full Inability modality (FI) no independent evidence
purpose: To primitively capture the symmetric case where a coalition enforces neither φ nor ¬φ
New primitive introduced to enable direct proof-theoretic access; no independent evidence provided beyond the definitional extension.

pith-pipeline@v0.9.0 · 9329 in / 1390 out tokens · 53854 ms · 2026-05-08T17:11:12.286467+00:00 · methodology

discussion (0)

Reference graph

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