  The four-valued semantics of Belnap--Dunn logic, consisting of the truth
values True, False, Neither, and Both, gives rise to several non-classical
logics depending on which feature of propositions we wish to preserve: truth,
non-falsity, or exact truth (truth and non-falsity). Interpreting equality of
truth values in this semantics as material equivalence of propositions, we can
moreover see the equational consequence relation of this four-element algebra
as a logic of material equivalence. In this paper we axiomatize all
combinations of these four-valued logics, for example the logic of truth and
exact truth or the logic of truth and material equivalence. These combined
systems are consequence relations which allow us to express implications
involving more than one of these features of propositions.
