Logics of upsets of De Morgan lattices

TL;DR

This paper investigates logics based on De Morgan lattices, introduces the concept of n-filters, and provides finite axiomatizations for various logics, including Shramko's logic of anything but falsehood.

Contribution

It develops a lattice-theoretic framework for analyzing and axiomatizing logics derived from De Morgan lattices, extending previous three-valued logics.

Findings

Finite Hilbert-style axiomatization for logics from prime upsets.

Finite Gentzen-style axiomatization for logics from filters.

Axiomatization of Shramko's logic of anything but falsehood.

Abstract

We study logics determined by matrices consisting of a De~Morgan lattice with an upward closed set of designated values, such as the logic of non-falsity preservation in a given finite Boolean algebra and Shramko's logic of non-falsity preservation in the four-element subdirectly irreducible De Morgan lattice. The key tool in the study of these logics is the lattice-theoretic notion of an $n$-filter. We study the logics of all (complete, consistent, and classical) $n$-filters on De Morgan lattices, which are non-adjunctive generalizations of the four-valued logic of Belnap and Dunn (of the three-valued logics of Priest and Kleene, and of classical logic). We then show how to find a finite Hilbert-style axiomatization of any logic determined by a finite family of prime upsets of finite De Morgan lattices and a finite Gentzen-style axiomatization of any logic determined by a finite family of filters on finite De Morgan lattices. As an application, we axiomatize Shramko's logic of anything but falsehood.