The criterion of the soundness and the comleteness of the classical logic with respect to the $V$-realizability

Let $L$ be an extension of the language of arithmetic, $V$ a class of number-theoretical functions. A notion of the $V$-realizability for predicate formulas is defined in such a way that predicate variables are substituted by formulas of the language $L$. It is proved that the classical logic is sound and complete with respect to the semantics of the $V$-realizability if $V$ contains all $L$-definable functions.