One application I know is Scott's construction of forcing extensions of models of higher order theories of the Real numbers. Scott quickly after the invention of forcing, used a forcing argument to show that the Continuum Hypothesis is independent of an axiomatization of Second Order Analysis.

He interpreted first order variables (usually real numbers) as Random Variables over a complete Boolean Algebra with the ccc, second order variables (usually functions from the reals to the reals) as functions from RV to RV, satisfying a certain technical condition and then defined the Boolean value of each statement, which is an element of the Boolean algebra. He then showed that the axioms have Boolean Value $\mathbb{1}$ and that the inference rules respect the Boolean value (i.e. do not decrease the Boolean value) and finally exhibits a Boolean algebra in which the statement $CH$ has Boolean value $\ne \mathbb{1}$. See here for more details: http://link.springer.com/content/pdf/10.1007%2FBF01705520.pdf

All in all this line of argumentation preshadows the way (set theoretic) forcing is presented in (for example) Jech's book, which explains forcing as forcing with Boolean valued models of the universe $V$.