An Annotated Logic Defined by a Matrix

Of special interest in abstract algebraic logic currently is the problem of extending the general theory of algebraization to logical systems that fail to be structural. The annotated logics PL were introduced in the late eighties as a logical framework to deal with deductive databases that contain inconsistent, conflicting or contradictory information. Like many paraconsistent logics they are non-structural, and for this reason they do not have an algebraic semantics in the usual sense. In this paper a structural and algebraizable annotated logic PM(L) is constructed that simulates the deductive process of PL in a natural way. Unlike PL, PM(L) is semantically defined by a single matrix M(L). The matrix is simpler than other matrix semantics of similar kind that have appeared in the literature. If M(L) is finite, a sound and complete axiomatization of PM(L) is given and this is used to prove that the equivalent algebraic semantics of PM(L) is the quasivariety generated by the Leibniz reduction of the underlying M(L) of M(L), and to find an equational axiomatization for this quasivariety.