 Solution

A Boolean valuation is an arbitrary function $v$ assigning one of the two truth values to each propositional variable. It extends uniquely to all propositional formulas by following the construction of the propositional formula:For logical implication, $\tilde{v}(A\to B)$ is false exactly when $\tilde{v}(A)$ is true and $\tilde{v}(B)$ is false; otherwise it is true. Here the operations on the right are Boolean operations on truth values. Structural recursion gives existence, and structural induction gives uniqueness: once the immediate subformulas have their values, the outer connective fixes the resulting value.