We present $\cal L$, an extension of Parigot's $\lambda\mu$-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction that extends $\lambda\mu$'s reduction system with two new reduction rules, and show that the system satisfies subject reduction. Using Aczel's generalisation of Tait and Martin-L\"of's notion of parallel reduction, we show that this extended reduction is confluent. Although the notion of type assignment has its limitations with respect to representation of proofs in natural deduction with implication and negation, we will show that all propositions that can be shown in there have a witness in $\cal L$. Using Girard's approach of reducibility candidates, we show that all typeable terms are strongly normalisable, and conclude the paper by showing that type assignment for $\cal L$ enjoys the principal typing property.

翻译：我们提出$\cal L$，这是Parigot的$\lambda\mu$-演算的扩展，通过将否定作为类型构造子引入，并附带表示否定引入和消去的语法结构。我们将定义一种归约概念，通过两条新的归约规则扩展$\lambda\mu$的归约系统，并证明该系统满足主题归约。利用Aczel对Tait和Martin-Löf的平行归约概念的推广，我们证明了该扩展归约系统具有合流性。尽管类型指派概念在表示蕴含和否定的自然演绎证明方面存在局限性，但我们将证明该系统中所有可证明的命题在$\cal L$中都有见证项。利用Girard的可归约性候选项方法，我们证明了所有可类型化项都是强规范化的，并通过证明$\cal L$的类型指派满足主型性质来结束本文。