  Abstract algebraic logic is a theory that provides general tools for the
algebraic study of arbitrary propositional logics. According to this theory,
every logic L is associated with a matrix semantics Mod*(L). This paper is a
contribution to the systematic study of the so-called "truth sets" of the
matrices in Mod*(L). In particular, we show that the fact that the truth sets
of Mod*(L) can be defined by means of equations with universally quantified
parameters is captured by an order-theoretic property of the Leibniz operator
restricted to deductive filters of L. This result was previously known for
equational definability without parameters. Similarly, it was known that the
truth sets of Mod*(L) are implicitly definable if and only if the Leibniz
operator is injective on deductive filters of L over every algebra. However, it
was an open problem whether the injectivity of the Leibniz operator transfers
from the theories of L to its deductive filters over arbitrary algebras. We
show that this is the case for logics expressed in a countable language, and
that it need not be true in general. Finally we consider an intermediate
condition on the truth sets in Mod*(L) that corresponds to the order-reflection
of the Leibniz operator.
