Structural Induction: Metalogic Proof I need to show that by structural induction that the truth value $V$ of a WFF depends only on the truth values of its sentence letters. In other words, I need to show show that for any WFF $\theta$ and any PL-interpretations $I$ and $I'$, if $I(\alpha) = I'(\alpha)$ for every sentence letter $\alpha$ in $\theta$, then $V_I (\theta) = V_ I' (\theta).$
I'm not sure where to begin because I'm not sure how to translate the former assertion into English.
Does anyone have any tips about how to proceed with the structural inductive proof? 
 A: Hint
Induction on the complexity of the formula $\theta$, i.e. on the number of occurrences of connectives. 
1) Base step: $0$ occurrence of connectives; thus $\theta$ is made of a single sentence letter $\alpha$. Obvious.
2) Induction step: consider two significant cases: $¬$ and $∨$ (others binary conncetives: $\land, \to$, are similar). 
2a) $θ$ is $¬ψ$, where we assume that the induction hypotheses holds for $ψ$. Obvious.
2b) $θ$ is $ψ∨χ$, where we assume that the induction hypotheses holds for $ψ$ and $χ$. 
By induction hypothesis, for any interpretations $I$ and $I'$, we have that if $I(α)=I′(α)$ for every sentence letter $α$ in $\psi$ and $\chi$, then $V_I(\psi)=V_{I'}(\psi)$, and the same for $\chi$.
We ahve several sub-cases; consider the case $V_I(\psi)=$ t and $V_I(\chi)=$ f. Then $V_I(\psi \lor \chi)=$ t. But $V_{I'}(\psi)=$ t and $V_{I'}(\chi)=$ f and thus also: $V_{I'}(\psi \lor \chi)=$ t.
Conclusion: $V_I(\theta)=V_I(\psi \lor \chi)=V_{I'}(\psi \lor \chi)=V_{I'}(\theta)$.
And so on...
