On Modal Companions of Logics with Strong Negation
Dmitry M. Anishchenko
TL;DR
This work addresses the existence of modal companions for extensions of Nelson's constructive logic with strong negation by developing an algebraic framework based on twist-structures over topoboolean algebras and Belnapian modalities. It introduces a translation-driven pathway, via $T$ and $T_{\mathbf{B}}$, to embed $N4^{\bot}$- and $N3^{\bot}$-extensions into Belnapian modal logics such as $BS4$, and constructs Lindenbaum twist-structures to analyze modal-companion existence. The key contributions are (i) a general representation enabling a wide class of $N4^{\bot}$-extensions (including all $N3^{\bot}$-extensions) to have modal companions, and (ii) a counterpoint showing a continuum of $N4^{\bot}$-extensions without modal companions, illustrating limits of the companion phenomenon. The results link four-valued Nelson logics with Belnapian modal logics through an explicit algebraic apparatus, offering practical criteria and constructions for when modal companions exist and how to obtain them through $\tau_{\mathbf{B}} L$.
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.