We introduce a novel framework for encoding inconsistency into relational tuples and tackling query answering for union of con-junctive queries (UCQs) with respect to a set of denial constraints (DCs). We define a notion of inconsistent tuple with respect to a set of DCs and define four measures of inconsistency degree of an answer tuple of a query. Two of these measures revolve around the minimal number of inconsistent tuples necessary to compute the answer tuples of a UCQ, whereas the other two rely on the maximum number of inconsistent tuples under set-and bag-semantics, respectively. In order to compute these measures of inconsistency degree, we leverage two models of provenance semiring, namely why-provenance and provenance polynomials, which can be computed in polynomial time in the size of the relational instances for UCQs. Hence, these measures of inconsistency degree are also computable in polynomial time in data complexity. We also investigate top-k and bounded query answering by ranking the answer tuples by their inconsistency degrees. We explore both a full materialized approach and a semi-materialized approach for the computation of top-k and bounded query results.