  This paper is concerned with the first-order paraconsistent logic
LPQ$^{\supset,\mathsf{F}}$. A sequent-style natural deduction proof system for
this logic is presented and, for this proof system, both a model-theoretic
justification and a logical justification by means of an embedding into
first-order classical logic is given. For no logic that is essentially the same
as LPQ$^{\supset,\mathsf{F}}$, a natural deduction proof system is currently
available in the literature. The given embedding provides both a
classical-logic explanation of this logic and a logical justification of its
proof system. The major properties of LPQ$^{\supset,\mathsf{F}}$ are also
treated.
