Multi-type Display Calculus for Semi De Morgan Logic

Semi De Morgan algebras form a variety of normal distributive lattice expansions [7] introduced by H.P. Sankappanavar [16] as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. A fully selfextensional logic SDM naturally arises from semi De Morgan algebras, which has been studied from a duality-theoretic perspective [13], from the perspective of canonical extensions [15], and from a proof-theoretic perspective [14]. Related to the proof theoretic perspective, the G3-style sequent calculus introduced in [14] is shown to be cut-free. However, the proof of cut elimination is quite involved, due to the fact that, along with the standard introduction rules for conjunction and disjunction, this calculus includes also introduction rules under the scope of structural connectives. These difficulties can be explained by the fact that the axiomatization of SDM is not analytic inductive in the sense of [10, Definition 55], due to the presence of the following axioms