The nerve criterion and polyhedral completeness of intermediate logics
Sam Adam-Day, Nick Bezhanishvili, David Gabelaia, Vincenzo Marra

Abstract:
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. We provide 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 the so-called ‘nerve’ of a poset, a construction which we employ from polyhedral geometry.
The criterion allows for the investigation of the polyhedral completeness phe- nomenon using purely combinatorial methods. Utilising it, we show that there are continuum many intermediate logics that are not polyhedrally-complete. We also provide a countably infinite class of logics axiomatised by the Jankov-Fine formulas of ‘starlike trees’, which includes Scott’s Logic, all of which are polyhedrally-complete.