On Modal Companions of Logics with Strong Negation

TL;DR

This paper explores the modal companions of logics with strong negation, constructing algebraic representations and identifying which N4 extensions have modal companions, revealing a rich and complex landscape.

Contribution

It introduces a novel algebraic representation for N4-lattices and characterizes the N4 extensions that possess modal companions.

Findings

All N3- extensions have modal companions.

A continuum of N4- extensions lack modal companions.

Constructed a representation similar to Heyting algebras for N4-lattices.

Abstract

BS4 is a natural Belnapian conservative extension of Lewis modal system S4 via strong negation. In [24] it was proved that the translation TB that naturally generalises the Godel-Tarski translation T embeds faithfully Nelsons logic N4 into BS4. So it is natural to define a modal companion of a logic extending N4 as an extension of BS4. In this paper we construct a representation of an N4-lattice similar to the representation of a Heyting algebra as an open elements algebra for a suitable topoboolean algebra. Using this algebraic result we construct a wide class of N4- extensions, elements of which have modal companions. In particular, all N3- extensions have modal companions. Also we prove that there are a continuum of N4- extensions that have no modal companions.