Weakly extendible cardinals and compactness of extended logics

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).