Expanding Belnap: dualities for a new class of default bilattices

TL;DR

This paper introduces a new class of prioritised default bilattices extending Belnap's original structure, providing novel duality representations that enhance the mathematical understanding of these algebraic models.

Contribution

It develops a new family of bilattices, $ extbf{J}_n$, and establishes both single-sorted and multi-sorted topological dualities for their generated quasivarieties and varieties.

Findings

Single-sorted duality for quasivariety is complex

Multi-sorted duality for variety is simpler

Provides a new algebraic framework for default bilattices

Abstract

Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled \emph{How a computer should think}. Belnap argued that instead of using a logic with two values, for `true' ($t$) and `false' ($f$), a computer should use a logic with two further values, for `contradiction' ($\top$) and `no information' ($\bot$). The resulting structure is equipped with two lattice orders, a \emph{knowledge order} and a \emph{truth order}, and hence is called a \emph{bilattice}. Prioritised default bilattices include not only values for `true' ($t_0$), `false' ($f_0$), `contradiction' and `no information', but also indexed families of default values, $t_1, \dots, t_n$ and $f_1, \dots, f_n$, for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: $\mathbf J_n$, for $n \in \omega$. The bilattice $\mathbf J_0$ is precisely Belnap's seminal example. We address mathematical rather than logical aspects of our prioritised default bilattices. We obtain a single-sorted topological representation for the bilattices in the quasivariety $\mathcal J_n$ generated by $\mathbf J_n$, and separately a multi-sorted topological representation for the bilattices in the variety $\mathcal V_n$ generated by $\mathbf J_n$. Our results provide an interesting example where the multi-sorted duality for the variety has a simpler structure than the single-sorted duality for the quasivariety.