  In this paper we analyze the connection between some properties of partially
strongly compact cardinals: the completion of filters of certain size and
instances of the compactness of $\mathcal{L}_{\kappa,\kappa}$. Using this
equivalence we show that if any $\kappa$-complete filter on $\lambda$ can be
extended to a $\kappa$-complete ultrafilter and $\lambda^{<\kappa} = \lambda$
then $\square(\mu)$ fails for all regular $\mu\in[\kappa,2^\lambda]$. As an
application, we improve the lower bound for the consistency strength of
$\kappa$-compactness, a case which was explicitly considered by Mitchell.
