  It is well known that the completeness theorem for
$\mathrm{L}_{\omega_1\omega}$ fails with respect to Tarski semantics. Mansfield
showed that it holds for $\mathrm{L}_{\infty\infty}$ if one replaces Tarski
semantics with boolean valued semantics. We use forcing to improve his result
in order to obtain a stronger form of boolean completeness (but only for
$\mathrm{L}_{\infty\omega}$). Leveraging on our completeness result, we
establish the Craig interpolation property and a strong version of the omitting
types theorem for $\mathrm{L}_{\infty\omega}$ with respect to boolean valued
semantics. We also show that a weak version of these results holds for
$\mathrm{L}_{\infty\infty}$ (if one leverages instead on Mansfield's
completeness theorem). Furthermore we bring to light (or in some cases just
revive) several connections between the infinitary logic
$\mathrm{L}_{\infty\omega}$ and the forcing method in set theory.
