  We introduce the notion of weakly extendible cardinals and show that these
cardinals are characterized in terms of weak compactness of second order logic.
The consistency strength and largeness of weakly extendible cardinals are
located strictly between that of strongly unfoldable (i.e. shrewd) cardinals,
and strongly uplifting cardinals. Weak compactness of many other logics can be
connected to certain variants of the notion of weakly extendible cardinals. We
also show that, under V=L, a cardinal $\kappa$ is the weak compactness number
of ${\cal L}^{\aleph_0,II}_{stat,\kappa,\omega}$ if and only if it is the weak
compactness number of ${\cal L}^{II}_{\kappa,\omega}$. The latter condition is
equivalent to the condition that $\kappa$ is weakly extendible by the
characterization mentioned above (this equivalence holds without the assumption
of V=L).
