A general framework for measuring inconsistency through minimal inconsistent sets
Hunter and Konieczny explored the relationships between measures of inconsistency for a belief base and the minimal inconsistent
subsets of that belief base in several of their papers. In particular, an inconsistency value termed MIV
, defined from minimal inconsistent subsets, can be considered as a Shapley Inconsistency Value. Moreover, it can be axiomatized
completely in terms of five simple axioms. MinInc, one of the five axioms, states that each minimal inconsistent set has the
same amount of conflict. However, it conflicts with the intuition illustrated by the lottery paradox, which states that as
the size of a minimal inconsistent belief base increases, the degree of inconsistency of that belief base becomes smaller.
To address this, we present two kinds of revised inconsistency measures for a belief base from its minimal inconsistent subsets.
Each of these measures considers the size of each minimal inconsistent subset as well as the number of minimal inconsistent
subsets of a belief base. More specifically, we first present a vectorial measure to capture the inconsistency for a belief
base, which is more discriminative than MIV
. Then we present a family of weighted inconsistency measures based on the vectorial inconsistency measure, which allow us
to capture the inconsistency for a belief base in terms of a single numerical value as usual. We also show that each of the
two kinds of revised inconsistency measures can be considered as a particular Shapley Inconsistency Value, and can be axiomatically
characterized by the corresponding revised axioms presented in this paper.