  We introduce a proper display calculus for (non-distributive) Lattice Logic
which is sound, complete, conservative, and enjoys cut-elimination and
sub-formula property. Properness (i.e. closure under uniform substitution of
all parametric parts in rules) is the main interest and added value of the
present proposal, and allows for the smoothest Belnap-style proof of
cut-elimination. Our proposal builds on an algebraic and order-theoretic
analysis of the semantic environment of lattice logic, and applies the
guidelines of the multi-type methodology in the design of display calculi.
