This paper concerns the paraconsistent logic LPQ$^{\supset,\mathsf{F}}$ and an application of it in the area of relational database theory. The notions of a relational database, a query applicable to a relational database, and a consistent answer to a query with respect to a possibly inconsistent relational database are considered from the perspective of this logic. This perspective enables among other things the definition of a consistent answer to a query with respect to a possibly inconsistent database without resort to database repairs. In a previous paper, LPQ$^{\supset,\mathsf{F}}$ is presented with a sequent-style natural deduction proof system. In this paper, a sequent calculus proof system is presented because it is common to use a sequent calculus proof system as the basis of proof search procedures and such procedures may form the core of algorithms for computing consistent answers to queries.