In recent research, some of the present authors introduced the concept of an n-dimensional Boolean algebra and its corresponding propositional logic nCL, generalising the Boolean propositional calculus to n>= 2 perfectly symmetric truth values. This paper presents a sound and complete sequent calculus for nCL, named nLK. We provide two proofs of completeness: one syntactic and one semantic. The former implies as a corollary that nLK enjoys the cut admissibility property. The latter relies on the generalisation to the n-ary case of the classical proof based on the Lindenbaum algebra of formulas and Boolean ultrafilters.

翻译：在近期研究中，部分作者引入了n维布尔代数及其相应命题逻辑nCL的概念，将布尔命题演算推广至n≥2个完全对称的真值。本文提出了nCL的可靠且完备的相继式演算系统，命名为nLK。我们提供了两种完备性证明：一种基于语法，另一种基于语义。前者推论出nLK具有切割可消去性质，后者则依赖于经典证明中基于林登鲍姆公式代数和布尔超滤子方法在n元情形下的推广。