Decomposing Inconsistencies: Marginal Contributions and Pooling Techniques

Abstract

Inconsistency measures quantify the degree of conflict within a set of propositions. They can be broadly categorized into global measures, which assess the overall inconsistency of a set, and local measures, which evaluate the contribution of single formulas to the overall inconsistency. This paper investigates the relationship between these two classes of measures through the lens of marginal contributions and pooling mechanisms. We propose a systematic framework for deriving local inconsistency measures from global ones by employing notions of marginal contributions inspired by cooperative game theory, including Shapley and Banzhaf values. Conversely, we explore methods for constructing global inconsistency measures by aggregating local contributions using various pooling techniques. A key research question arises: which combinations of marginal contribution notions (maC) and pooling mechanisms (P) are compatible? Compatibility is defined such that, given a global measure I, applying (P) to the marginal contributions derived from I yields the same result as directly applying I, and vice versa. We analyze this compatibility condition and identify specific pairs of methods, (maC) and (P), that satisfy it across various inconsistency frameworks. Our findings provide a deeper understanding of the interplay between global and local inconsistency measures, providing a foundation for designing principled and interpretable inconsistency evaluation methods in logic-based systems.