This study examines the notion of inconsistency in pairwise comparisons for providing an axiomatization for it. It also proposes two inconsistency indicators for pairwise comparisons. The primary motivation for the inconsistency reduction is expressed by a computer industry concept “garbage in, garbage out”. The quality of the output depends on the quality of the input.

A complete proof of convergence of a certain class of reduction algorithms for distance-based inconsistency (defined in 1993) for pairwise comparisons is presented in this paper. Using pairwise comparisons is a powerful method for synthesizing measurements and subjective assessments. From the mathematical point of view, the pairwise comparisons method generates a matrix (say A) of ratio values (a_ij) of the ith entity compared with the jth entity according to a given criterion. Entities/criteria can be both quantitative or qualitative allowing this method to deal with complex decisions. However, subjective assessments often involve inconsistency, which is usually undesirable. The assessment can be refined via analysis of inconsistency, leading to reduction of the latter. The proposed method of localizing the inconsistency may conceivably be of relevance for nonclassical logics (e.g., paraconsistent logic) and for uncertainty reasoning since it accommodates inconsistency by treating inconsistent data as still useful information.