We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and G\"{o}del's system $\mathbf{T}$, by defining simultaneously the strong negation $A^{\mathbf{N}}$ of a formula $A$ and the strong negation $P^{\mathbf{N}}$ of a predicate $P$ in TCF. As a special case of the latter, we get the strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of the double negation elimination for strong negation in TCF, and we study the so-called tight formulas of TCF i.e., formulas implied from the weak negation of their strong negation. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics.

翻译：我们在可计算泛函理论TCF（Plotkin的PCF与Gödel系统$\mathbf{T}$的共同扩展）中引入强否定，通过同步定义TCF中公式$A$的强否定$A^{\mathbf{N}}$与谓词$P$的强否定$P^{\mathbf{N}}$实现。作为后者的特例，我们得到了TCF中归纳谓词与余归纳谓词的强否定。我们证明了TCF中强否定的适当版本的"假言推理"（Ex falso quodlibet）与双重否定消去规则，并研究了TCF中所谓的紧致公式（即由其强否定的弱否定蕴含的公式）。通过多种案例研究与示例，我们展示了TCF中强否定定义的自然性，并论证了TCF作为Bishop式构造性数学主要部分形式系统的适用性。