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 strong negation $A^{\mathbf{N}}$ of a formula $A$ and strong negation $P^{\mathbf{N}}$ of a predicate $P$ in TCF. As a special case of the latter, we get strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of double negation elimination for strong negation in TCF. We introduce the so-called tight formulas of TCF i.e., formulas implied by the weak negation of their strong negation, and the relative tight formulas. 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：我们通过在可计算泛函理论TCF中同时定义公式$A$的强否定$A^{\mathbf{N}}$与谓词$P$的强否定$P^{\mathbf{N}}$，将强否定纳入该理论体系。TCF是Plotkin的PCF与G\"{o}del系统$\mathbf{T}$的共同扩展。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中强否定对应的Ex falso quodlibet原理与双重否定消去原理的适当形式。我们引入了TCF中的紧致公式（即被其强否定的弱否定所蕴含的公式）以及相对紧致公式。通过多个案例研究与实例展示，我们揭示了TCF中强否定定义的自然性，并论证了TCF可作为Bishop风格构造性数学大部分内容的形​​式化系统。