Logic on the n-cube

TL;DR

This paper introduces a logical framework based on the faces of an n-cube, extending known algebraic structures and resulting in a compact, coNP-complete logic that handles inconsistency and nonmonotonicity.

Contribution

It extends the Rota-Metropolis operations to the face lattice of the n-cube, establishing a term-equivalence with Post algebras of order 3 and analyzing the resulting logic.

Findings

Structures are term-equivalent to Post algebras of order 3

The inclusion order matches the De Luca-Termini sharpening order

The logic is compact and coNP-complete, tolerating inconsistency

Abstract

We endow the partially ordered set of nonempty faces of the n-cube with a distinguished 0-dimensional face and three operations that naturally extend the Rota-Metropolis partial operations. While the structures thus obtained turn out to be term-equivalent to Post algebras of order 3, the inclusion order between faces coincides with the De Luca-Termini sharpening order, and yields a compact coNP-complete logic that tolerates a modicum of inconsistency and nonmonotonicity.