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.

中文翻译：

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向 Lambda Mu 添加否定

我们提出了 $\cal L$，它是 Parigot 的 $\lambda\mu$ 演算的扩展，通过添加否定作为类型构造函数，以及表示否定引入和消除的句法构造。我们将定义一个归约概念，它用两个新的归约规则扩展 $\lambda\mu$ 的归约系统，并证明该系统满足主题归约。使用 Aczel 对 Tait 的推广和 Martin-L"of 的并行归约概念，我们证明了这种扩展归约是融合的。尽管类型分配的概念在用蕴涵和否定的自然演绎中的证明表示方面有其局限性，但我们将表明所有可以在那里显示的命题在 $\cal L$ 中都有一个见证。使用 Girard 的可约性候选方法，