= 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>:
$$
\widetilde v(p)=v(p),\quad
\widetilde v(\neg A)=\neg\widetilde v(A),\quad
\widetilde v(A\wedge B)=\widetilde v(A)\wedge\widetilde v(B),\quad
\widetilde v(A\vee B)=\widetilde v(A)\vee\widetilde v(B).
$$
For <logical implication>, $\widetilde v(A\to B)$ is false exactly when $\widetilde v(A)$ is true and $\widetilde v(B)$ 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.
Back to article page
