In this paper, we consider pure infiniteness of generalized Cuntz–Krieger algebras associated to labeled spaces
$(E,{\mathcal{L}},{\mathcal{E}})$
. It is shown that a
$C^{\ast }$
-algebra
$C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$
is purely infinite in the sense that every non-zero hereditary subalgebra contains an infinite projection (we call this property (IH)) if
$(E,{\mathcal{L}},{\mathcal{E}})$
is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs,
$C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})=C^{\ast }(p_{A},s_{a})$
is purely infinite in the sense of Kirchberg and Rørdam if and only if every generating projection
$p_{A}$
,
$A\in {\mathcal{E}}$
, is properly infinite, and also if and only if every quotient of
$C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$
has property (IH).