The lattice of super-Belnap logics

TL;DR

This paper explores the structure of the lattice of super-Belnap logics, revealing its complex substructure, completeness properties, and connections to graph theory, resulting in a continuum of such logics and algebraic implications.

Contribution

It provides a detailed structural analysis of super-Belnap logics, introduces the antiaxiomatic operator, and establishes a novel link to graph theory, expanding understanding of these logics.

Findings

The lattice of super-Belnap logics can be split into subintervals with specific properties.

There is a continuum of finitary super-Belnap logics.

Connections to graph theory allow construction of non-finitary super-Belnap logics.

Abstract

We study the lattice of extensions of four-valued Belnap--Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap--Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap--Dunn logic is iso\-morphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety.