Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled How a computer should think. Belnap argued that instead of using a logic with two values, for ‘true’ (\(\varvec{t}\)) and ‘false’ (\(\varvec{f}\)), 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 knowledge order and a truth order, and hence is called a bilattice.

Prioritised default bilattices include not only values for ‘true’ (\(\varvec{t}_0\)), ‘false’ (\(\varvec{f}_0\)), ‘contradiction’ and ‘no information’, but also indexed families of default values, \(\varvec{t}_1, \dots , \varvec{t}_n\) and \(\varvec{f}_1, \dots , \varvec{f}_n\), 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 obtain a multi-sorted duality for the variety

双语提供了一种代数工具，可用来同时建模知识和真理。它们是由Belnap在1977年的一篇名为“计算机应该如何思考”的论文中介绍的。Belnap认为，对于“ true”（\（\ varvec {t} \））和“ false”（\（\ varvec {f} \）），不要使用具有两个值的逻辑，计算机应使用带有另外两个值分别是'contradiction'（\（\ top \））和'no information'（\（\ bot \））。生成的结构具有两个晶格顺序，一个知识顺序和一个真相顺序，因此被称为bilattice。

优先的默认能力不仅包括'true'（\（\ varvec {t} _0 \）），'false'（\（\ varvec {f} _0 \）），'contradiction'和'no information'的值，而且缺省值也索引家庭，\（\ varvec {吨} _1，\点，\ varvec {吨} _n \）和\（\ varvec {F} _1，\点，\ varvec {F} _n \） ，用于知识和真理程度的同步建模。

我们专注于一个新的优先级默认功能家族：\（\ mathbf {J} _n \），表示\（n \ in \ omega \）。bilattice \（\ mathbf {J} _0 \）正是Belnap的开创性例子。我们获得了该品种的多分类对偶