  We consider the complexity (in terms of the arithmetical hierarchy) of the
various quantifier levels of the diagram of a computably presented metric
structure. As the truth value of a sentence of continuous logic may be any real
in $[0,1]$, we introduce two kinds of diagrams at each level: the closed
diagram, which encapsulates weak inequalities of the form $\phi^\mathcal{M}
\leq r$, and the open diagram, which encapsulates strict inequalities of the
form $\phi^\mathcal{M} < r$. We show that the closed and open $\Sigma_N$
diagrams are $\Pi^0_{N+1}$ and $\Sigma_N$ respectively, and that the closed and
open $\Pi_N$ diagrams are $\Pi^0_N$ and $\Sigma^0_{N + 1}$ respectively. We
then introduce effective infinitary formulas of continuous logic and extend our
results to the hyperarithmetical hierarchy. Finally, we demonstrate that our
results are optimal.
