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

TL;DR
This paper introduces a four-fold spectrum of coalition power, including full inability, and develops a logical framework to analyze these strategic states with algebraic, order-theoretic, and proof-theoretic tools.
Contribution
It defines and formalizes a symmetric four-region classification of coalition power, extending Coalition Logic with a primitive for Full Inability and analyzing its properties.
Findings
The four power regions form an order-convex structure in the powerset lattice.
Axiomatization of the logic CLFI with soundness and completeness results.
The extension maintains the same expressive power and complexity as standard Coalition Logic.
Abstract
Coalition Logic studies what coalitions can enforce. Recent work treats inability as simple non-ability: . This conflates two distinct configurations -- a coalition unable to force may still force , retaining adversarial control rather than genuine inability. We introduce \textbf{Full Inability} (): 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} (), \textbf{Positive Determination} (), \textbf{Adverse Determination} (), and \textbf{Full Inability} (). These categories partition a coalition's strategic status exhaustively and exclusively. We establish their algebraic and order-theoretic structure. Under -duality, propositional negation and coalition…
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