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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 from 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.
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