Polyhedral completeness of intermediate logics: the Nerve Criterion

TL;DR

This paper introduces the Nerve Criterion, a combinatorial condition based on polyhedral semantics, to characterize when intermediate logics are complete with respect to polyhedra, expanding understanding of their structural properties.

Contribution

It provides a necessary and sufficient combinatorial criterion for polyhedral completeness of intermediate logics, and applies this to identify classes of such logics.

Findings

Many intermediate logics with the finite model property are not polyhedrally-complete.

A countable class of 'starlike' logics are shown to be polyhedrally-complete.

The Nerve Criterion offers a purely combinatorial way to determine polyhedral completeness.

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. 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.