Beyond Ability: The Four-Fold Spectrum of Power and the Logic of Full Inability
For logicians and AI researchers working on multi-agent systems, this work refines the logical treatment of coalitional power by distinguishing genuine inability from adversarial control, though it is an incremental extension of existing Coalition Logic.
The paper identifies a flaw in Coalition Logic where inability is conflated with non-ability, and introduces Full Inability (FI) as a symmetric condition where a coalition can force neither a proposition nor its negation. It axiomatizes an extension CLFI, proving soundness, completeness, and PSPACE-completeness, providing direct access to symmetric inability and strategic dependence.
Coalition Logic studies what coalitions can enforce. Recent work treats inability as simple non-ability: $\neg\Eff{C}φ$. This conflates two distinct configurations -- a coalition unable to force $φ$ may still force $\negφ$, 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 $α$-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.