  We investigate a recently-devised polyhedral semantics for intermediate
logics, in which formulas are interpreted in n-dimensional polyhedra. An
intermediate logic is polyhedrally complete if it is complete with respect to
some class of polyhedra. The first main result of this paper is a necessary and
sufficient condition for the polyhedral-completeness of a logic. This
condition, which we call the Nerve Criterion, is expressed in terms of
Alexandrov's notion of the nerve of a poset. It affords a purely combinatorial
characterisation of polyhedrally-complete logics.
  Using the Nerve Criterion we show, easily, that there are continuum many
intermediate logics that are not polyhedrally-complete but which have the
finite model property. We also provide, at considerable combinatorial labour, a
countably infinite class of logics axiomatised by the Jankov-Fine formulas of
'starlike trees' all of which are polyhedrally-complete. The polyhedral
completeness theorem for these 'starlike logics' is the second main result of
this paper.
