  We revisit the duality between Kripke and algebraic semantics of
intuitionistic and intuitionistic modal logic. We find that there is a certain
mismatch between the two semantics, which means that not all algebraic models
can be embedded into a Kripke model. This leads to an alternative proposal for
a relational semantics, the stable semantics. Instead of an arbitrary partial
order, the stable semantics requires a distributive lattice of worlds. We
constructively show that the stable semantics is exactly as complete as the
algebraic semantics. Categorifying these results leads to a 2-duality between
two-dimensional stable semantics and categories of product-preserving
presheaves, i.e. models of algebraic theories in the style of Lawvere.
