Inconsistency is a common problem in knowledge, and so there is a need to analyse it. Inconsistency measures assess its severity, but there is a more basic question: "where is the inconsistency?". Typically, not all subsets of a knowledgebase are causing the inconsistency, and minimal inconsistent sets have been the standard way to localise the germane ones, even though there are shortcomings in some scenarios. Recently, ⋆-conflicts were proposed as a more suitable definition to localise inconsistency when considering a method to repair it. But in general there is no way to tell what is a sensible definition to capture the germane conflicts. This work provides a set of desirable properties to assess definitions for germane conflicts. Also, a new conflict definition, based on substitution, is presented and evaluated via the proposed properties, and the related computational complexity is analysed.

The aim of measuring inconsistency is to obtain an evaluation of the imperfections in a set of formulas, and this evaluation may then be used to help decide on some course of action (such as rejecting some of the formulas, resolving the inconsistency, seeking better sources of information, etc). A number of proposals have been made to define measures of inconsistency. Each has its rationale. But to date, it is not clear how to delineate the space of options for measures, nor is it clear how we can classify measures systematically. To address these problems, we introduce a general framework for comparing syntactic measures of inconsistency. It is based on the notion of an inconsistency graph for each knowledgebase (a bipartite graph with a set of vertices representing formulas in the knowledgebase, a set of vertices representing minimal inconsistent subsets of the knowledgebase, and edges representing that a formula belongs to a minimal inconsistent subset). We then show that various measures can be computed using the inconsistency graph. Then we introduce abstractions of the inconsistency graph and use them to construct a hierarchy of syntactic inconsistency measures. Furthermore, we extend the inconsistency graph concept with a labeling that extends the hierarchy to include some other types of inconsistency measures.

A number of proposals have been made to define inconsistency measures. Each has its rationale. But to date, it is not clear how to delineate the space of options for measures, nor is it clear how we can classify measures systematically. In this paper, we introduce a general framework for comparing syntactic inconsistency measures. It uses the construction of an inconsistency graph for each knowledgebase. We then introduce abstractions of the inconsistency graph and use the hierarchy of the abstractions to classify a range of inconsistency measures.

We study algorithms for fragments of first order logic ex- tended with counting quantifiers, which are known to be highly complex in general. We propose a fragment over unary predicates that is NP-complete and for which there is a nor- mal form where Counting Quantification sentences have a single Unary predicate, thus call it the CQU fragment. We provide an algebraic formulation of the CQU satisfiability problem in terms of Integer Linear Programming based on which two algorithms are proposed, a direct reduction to SAT instances and an Integer Linear Programming version extended with a column generation mechanism. The latter is shown to lead to a viable implementation and experiments shows this algorithm presents a phase transition behavior.