  This paper focuses on order-preserving logics defined from varieties of
distributive lattices with negation, and in particular on the problem of
whether these can be axiomatized by means of finite Hilbert calculi. On the
side of negative results, we provide a syntactic condition on the equational
presentation of a variety that entails failure of finite axiomatizability for
the corresponding logic. An application of this result is that the logic of all
distributive lattices with negation is not finitely axiomatizable; likewise, we
establish that the order-preserving logic of the variety of all Ockham algebras
is also not finitely axiomatizable. On the positive side, we show that an
arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite
number of equations if and only if the corresponding order-preserving logic is
axiomatized by a finite Hilbert calculus. This equivalence also holds for every
subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a
new proof that the implication-free fragment of intuitionistic logic is
finitely axiomatizable, as well as a new Hilbert calculus for it. Our proofs
are constructive in that they allow us to effectively convert an equational
presentation of a variety of algebras into a Hilbert calculus for the
corresponding order-preserving logic, and vice versa. We also consider the
assertional logics associated to the above-mentioned varieties, showing in
particular that the assertional logics of finitely axiomatizable subvarieties
of semi-De Morgan algebras are finitely axiomatizable as well.
