I am reading Derek Goldrei's "Propositional and Predicate Calculus".
Let $P$ be the set of all propositional variables and $S=\{ \neg, \land, \lor, \rightarrow, \leftrightarrow \}$ be the set of connectives. Let $Form (P,S)$ be the set of all propositional formulas that can be constructed from $P$ and $S$. This is what I want to show:
For each variable assignment function $v: P \to \{ T,F \}$, I want to show that there is a unique function $\overline v : Form (P,S) \to \{ T,F \}$ such that $\overline v (p)= v(p)$ for each $p\in P$.
On the side note, the author writes that the proof is done by showing that the existence of a function $\overline v$ can be shown by the Recursion principle and I understand the proof of the existence that author provides. He first defines the function $\overline v$ for all formulas having no connectives and then assumes that the function has been defined for all formulas having at most $n$ connectives and naturally defines $\overline v$ for all formulas having all formulas having exactly $n$ connectives in the way that $\overline v$ respects the connectives.
Here's the recursion principle that I know:
Let $h: A \to A$ be a function and let $a_0\in A$. Then there is a unique function $f:\omega \to A$ such that $f(0)=a_0$ and $f(n^{+})=h(f(n))$ for each $n\in\omega$.
How do I see the existence of $\overline v$ as a consequence of the recursion principle as the author says? I tried several things but nothing seemed to work. Hints will be appreciated.
