A Logic of Inability

TL;DR

This paper extends Coalition Logic with an explicit inability operator to better reason about what coalitions cannot achieve, providing a formal modal framework for understanding systemic impossibilities in multi-agent systems.

Contribution

It introduces a conservative, tractable extension of Coalition Logic with an inability operator, analyzing its modal properties and implications for reasoning about limitations.

Findings

Proves soundness, completeness, and conservativity of the extension.

Analyzes modal properties like anti-monotonicity and contravariance.

Connects inability with systemic impossibility and constraints.

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.